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def modelfit(alg, dtrain, predictors, performCV=True, printFeatureImportance=True, cv_folds=5):
import numpy as np
import pandas as pd
from sklearn.model_selection import cross_val_score
from sklearn import metrics
import matplotlib.pylab as plt
from matplotlib.pylab import rcParams
rcParams['figure.figsize'] = 12, 4
# Fit the algorithm on the data
alg.fit(dtrain[predictors], dtrain['Survived'])
# Predict training set:
dtrain_predictions = alg.predict(dtrain[predictors])
dtrain_predprob = alg.predict_proba(dtrain[predictors])[:, 1]
if performCV:
cv_score = cross_val_score(alg, dtrain[predictors], dtrain['Survived'], cv=cv_folds,
scoring='roc_auc')
# Print model report:
print("\nModel Report")
print("Accuracy : %.4g" % metrics.accuracy_score(dtrain['Survived'].values, dtrain_predictions))
print("AUC Score (Train): %f" % metrics.roc_auc_score(dtrain['Survived'], dtrain_predprob))
if performCV:
print("CV Score : Mean - %.7g | Std - %.7g | Min - %.7g | Max - %.7g" % (np.mean(cv_score), np.std(cv_score), np.min(cv_score), np.max(cv_score)))
# Print Feature Importance:
if printFeatureImportance:
plt.figure(figsize=(12, 8))
feat_imp = pd.Series(alg.feature_importances_, predictors).sort_values(ascending=False)
feat_imp.plot(kind='bar', title='Feature Importance')
plt.ylabel('Feature Importance Score')
plt.xticks(rotation=30)
plt.show() | |
# emacs: -*- mode: python; py-indent-offset: 4; indent-tabs-mode: nil -*-
# vi: set ft=python sts=4 ts=4 sw=4 et:
### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ##
#
# See COPYING file distributed along with the PyMVPA package for the
# copyright and license terms.
#
### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ### ##
"""Penalized logistic regression classifier."""
__docformat__ = 'restructuredtext'
import numpy as np
from mvpa2.misc.exceptions import ConvergenceError
from mvpa2.base.learner import FailedToTrainError
from mvpa2.clfs.base import Classifier, accepts_dataset_as_samples
if __debug__:
from mvpa2.base import debug
class PLR(Classifier):
"""Penalized logistic regression `Classifier`.
"""
__tags__ = [ 'plr', 'binary', 'linear', 'has_sensitivity' ]
def __init__(self, lm=1, criterion=1, reduced=0.0, maxiter=20, **kwargs):
"""
Initialize a penalized logistic regression analysis
Parameters
----------
lm : int
the penalty term lambda.
criterion : int
the criterion applied to judge convergence.
reduced : float
if not 0, the rank of the data is reduced before
performing the calculations. In that case, reduce is taken
as the fraction of the first singular value, at which a
dimension is not considered significant anymore. A
reasonable criterion is reduced=0.01
maxiter : int
maximum number of iterations. If no convergence occurs
after this number of iterations, an exception is raised.
"""
# init base class first
Classifier.__init__(self, **kwargs)
self.__lm = lm
self.__criterion = criterion
self.__reduced = reduced
self.__maxiter = maxiter
def __repr__(self):
"""String summary over the object
"""
return """PLR(lm=%f, criterion=%d, reduced=%s, maxiter=%d, enable_ca=%s)""" % \
(self.__lm, self.__criterion, self.__reduced, self.__maxiter,
str(self.ca.enabled))
def _train(self, data):
"""Train the classifier using `data` (`Dataset`).
"""
# Set up the environment for fitting the data
X = data.samples.T
d = self._attrmap.to_numeric(data.sa[self.get_space()].value)
if set(d) != set([0, 1]):
raise ValueError, \
"Regressors for logistic regression should be [0,1]. Got %s" \
%(set(d),)
if self.__reduced != 0 :
# Data have reduced rank
from scipy.linalg import svd
# Compensate for reduced rank:
# Select only the n largest eigenvectors
U, S, V = svd(X.T)
if S[0] == 0:
raise FailedToTrainError(
"Data provided to PLR seems to be degenerate -- "
"0-th singular value is 0")
S /= S[0]
V = np.matrix(V[:, :np.max(np.where(S > self.__reduced)) + 1])
# Map Data to the subspace spanned by the eigenvectors
X = (X.T * V).T
nfeatures, npatterns = X.shape
# Weighting vector
w = np.matrix(np.zeros( (nfeatures + 1, 1), 'd'))
# Error for convergence criterion
dw = np.matrix(np.ones( (nfeatures + 1, 1), 'd'))
# Patterns of interest in the columns
X = np.matrix( \
np.concatenate((X, np.ones((1, npatterns), 'd')), 0) \
)
p = np.matrix(np.zeros((1, npatterns), 'd'))
# Matrix implementation of penalty term
Lambda = self.__lm * np.identity(nfeatures + 1, 'd')
Lambda[nfeatures, nfeatures] = 0
# Gradient
g = np.matrix(np.zeros((nfeatures + 1, 1), 'd'))
# Fisher information matrix
H = np.matrix(np.identity(nfeatures + 1, 'd'))
# Optimize
k = 0
while np.sum(np.ravel(dw.A ** 2)) > self.__criterion:
p[:, :] = self.__f(w.T * X)
g[:, :] = X * (d - p).T - Lambda * w
H[:, :] = X * np.diag(p.A1 * (1 - p.A1)) * X.T + Lambda
dw[:, :] = H.I * g
w += dw
k += 1
if k > self.__maxiter:
raise ConvergenceError, \
"More than %d Iterations without convergence" % \
(self.__maxiter)
if __debug__:
debug("PLR", \
"PLR converged after %d steps. Error: %g" % \
(k, np.sum(np.ravel(dw.A ** 2))))
if self.__reduced:
# We have computed in rank reduced space ->
# Project to original space
self.w = V * w[:-1]
self.bias = w[-1]
else:
self.w = w[:-1]
self.bias = w[-1]
def __f(self, y):
"""This is the logistic function f, that is used for determination of
the vector w"""
return 1. / (1 + np.exp(-y))
@accepts_dataset_as_samples
def _predict(self, data):
"""
Predict the class labels for the provided data
Returns a list of class labels
"""
# make sure the data are in matrix form
data = np.matrix(np.asarray(data))
# get the values and then predictions
values = np.ravel(self.__f(self.bias + data * self.w))
predictions = (values > 0.5).astype(int)
# save the state if desired, relying on State._setitem_ to
# decide if we will actually save the values
self.ca.predictions = predictions
self.ca.estimates = values
return predictions
def get_sensitivity_analyzer(self, **kwargs):
"""Returns a sensitivity analyzer for PLR."""
return PLRWeights(self, **kwargs)
from mvpa2.base.state import ConditionalAttribute
from mvpa2.base.types import asobjarray
from mvpa2.measures.base import Sensitivity
from mvpa2.datasets.base import Dataset
class PLRWeights(Sensitivity):
"""`Sensitivity` reporting linear weights of PLR"""
_LEGAL_CLFS = [ PLR ]
def _call(self, dataset=None):
"""Extract weights from PLR classifier.
PLR always has weights available, so nothing has to be computed here.
"""
clf = self.clf
attrmap = clf._attrmap
if attrmap:
# labels (values of the corresponding space) which were used
# for mapping Here we rely on the fact that they are sorted
# originally (just an arange())
labels_num = attrmap.values()
labels = attrmap.to_literal(asobjarray([tuple(sorted(labels_num))]),
recurse=True)
else:
labels = [(0, 1)] # we just had our good old numeric ones
ds = Dataset(clf.w.T, sa={clf.get_space(): labels,
'biases' : [clf.bias]})
return ds | |
import ngl
import numpy as np
from hdff import *
import hdtopology as hdt
n = 1000
d = 3
sample = np.random.uniform(-1.0,1.0,n*d).astype('f')
sample = sample.reshape(n, d)
###### test function ######
def ackley(domain, d=3):
# print "domain", domain.shape
Sum = np.zeros(domain.shape[0], dtype=float)
for i in range(d-1):
theta1 = 6*domain[:,i] - 3
theta2 = 6*domain[:,i+1] - 3
# sum -= exp(-0.2) * sqrt(pow(theta1, 2) + pow(theta2, 2)) + 3 * (cos(2 * theta1) + sin(2 * theta2));
# print(theta1.shape, Sum)
Sum = Sum - np.exp(-0.2) * np.sqrt(np.power(theta1, 2) + np.power(theta2, 2)) + 3 * (np.cos(2 * theta1) + np.sin(2 * theta2));
Sum = np.squeeze(np.array(Sum.T))
# print(Sum.shape)
return Sum
f = ackley(sample)
method = "RelaxedGabriel"
max_neighbors = 500
beta = 1.0
### provide recarray for data input ###
data = np.concatenate((sample, np.matrix(f).T), axis=1).astype('f')
names = ['X1', 'X2', 'X3', 'f']
types = ['<f4']*(d+1)
data = data.view(dtype=list(zip(names,types)) ).view(np.recarray)
print(data.dtype)
### provide array of unint32 for the edges
edges = ngl.getSymmetricNeighborGraph(method, sample, max_neighbors,beta)
print(edges, type(edges), edges.dtype)
### compute topology
eg = hdt.ExtremumGraphExt()
flag_array = np.array([0],dtype=np.uint8)
eg.initialize(data, flag_array, edges, True ,10, 1)
mc = DataBlockHandle()
mc.idString("TDA");
eg.save(mc)
dataset = DatasetHandle()
dataset.add(mc)
group = DataCollectionHandle("summaryTopologyTest.hdff")
group.add(dataset)
group.write() | |
# This file contains a short example of the evaluation process
# including training and testing.
# Author: Stefan Kahl, 2018, Chemnitz University of Technology
import os
import numpy as np
import config as cfg
from model import lasagne_net as birdnet
from model import lasagne_io as io
from utils import stats
from utils import log
import train
import test
###################### EVALUATION #######################
def evaluate():
# Clear stats
stats.clearStats(True)
# Parse Dataset
cfg.CLASSES, TRAIN, VAL = train.parseDataset()
# Build Model
NET = birdnet.build_model()
# Train and return best net
best_net = train.train(NET, TRAIN, VAL)
# Load trained net
SNAPSHOT = io.loadModel(best_net)
# Test snapshot
MLRAP, TIME_PER_EPOCH = test.test(SNAPSHOT)
result = np.array([[MLRAP]], dtype='float32')
return result
if __name__ == '__main__':
cfg.LOG_MODE = 'all'
r = evaluate()
log.export() | |
#!/usr/bin/env python
# coding: utf-8
# # CE-40717: Machine Learning
# ## HW5-Support Vector Machine
# ### Please fill this part
#
#
# 1. Full Name: AmirPourmand
# 2. Student Number: 99210259
#
#
# *You are just allowded to change those parts that start with "TO DO". Please do not change other parts.*
#
# *It is highly recommended to read each codeline carefully and try to understand what it exactly does. Best of luck and have fun!*
# In[2]:
# You are not allowed to import other packages.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.svm import SVC
import cvxopt
# #### About the Data:
# Heart diseases, also known as [Cardiovascular diseases (CVDs)](https://en.wikipedia.org/wiki/Cardiovascular_disease), are the first cause of death worldwide, taking an estimated 17.9 million lives each year which is about 32% of all deaths all over the world.
#
# In the present HomeWork, we are going to implement Support Vector Machines (SVM) algorithm that determines which patient is in danger and which is not.
#
# For this perpose, `Heart_Disease_Dataset.csv` file can be used that is attached to the HomeWork folder. Use `Dataset_Description.pdf` for more detail.
#
# In[3]:
df = pd.read_csv("./Heart_Disease_Dataset.csv")
df
# ### Pre-Processing - (15pts)
# #### Exploratory Data Analysis (EDA):
# In statistics, exploratory data analysis is an approach to analyze datasets to summarize their main characteristics, often using statistical graphics and other data visualization methods.
#
# This is a general approach that should be applied when you encounter a dataset.
# In[4]:
###############################################################################
## TODO: Find the shape of the dataset. ##
###############################################################################
shape = df.shape
print("shape of dataset is: " , shape)
###############################################################################
## TODO: Check if there is missing entries in the dataset columnwise. ##
###############################################################################
missings = df.info()
print("this dataset doesn't have any missing value as shown above.")
###############################################################################
## TODO: Check whether the dataset is balanced or not. ##
## If the difference between 2 classes was less than 100 for our dataset, ##
## it is called "ballanced". ##
###############################################################################
values=df.target.value_counts()
print("ballanced: ",np.abs(values[0]-values[1])<100)
###############################################################################
## TODO: plot the age distirbution and gender distrbution for both normal ##
## and heart diseses patients.(4 plots) ##
###############################################################################
print("--------------------- Plots --------------------------")
plt.figure(figsize=(10,10))
plt.subplot(2,2,1)
plt.hist(df[df.target==0].age,bins=100)
plt.title('Age distribution for Normal People')
plt.subplot(2,2,2)
plt.hist(df[df.target==0].sex)
plt.title('Gender distribution for Normal People')
plt.subplot(2,2,3)
plt.hist(df[df.target==1].age,bins=100)
plt.title('Age distribution for Ill People')
plt.subplot(2,2,4)
plt.hist(df[df.target==1].sex)
plt.title('Gender distribution for Ill People')
# #### Question 1: What do you conclude from the plots?
# #### Answer:
# #### I simply get the idea that ill people's age follows (roughly) from normal distribution. Also, the percentage of men who have heart attack is far higher than woman which means men should be more careful :)
# #### Outlier Detection & Removal:
# We will filter ouliers using Z-test.
# 
# Z-test formula:
# \begin{equation*}
# Z = \bigg|\frac {x - mu} {std}\bigg|
# \end{equation*}
# In[5]:
################################################################################
## TODO: Suppose that, based on our prior knowledge, we know some columns have##
## outliers. Calculate z-score for each featuer and determine the outliers ##
## with threshold=3, then eliminate them. Target dataframe has(1173,12)shape. ##
################################################################################
columns = ["age","resting bp s","cholesterol","max heart rate"]
threshold = 3
count = df.shape[0]
for col in columns:
z_test_age = (df[col]-df[col].mean())/df[col].std()
df=df[np.abs(z_test_age)<threshold]
df
################################################################################
# END OF YOUR CODE #
################################################################################
# #### Feature Engineering:
# Sometimes the collected data are raw; they are either incompatible with your model or hinders its performance. That’s when feature engineering comes to rescue. It encompasses preprocessing techniques to compile a dataset by extracting features from raw data.
#
# In[6]:
################################################################################
## TODO: Normalize numerical features to be between 0 and 1 ##
## Note that just numerical fetures should be normalized. type of features is ##
## determined in dataset description file. ##
################################################################################
def min_max_scaler(feature):
return (feature-feature.min())/(feature.max()-feature.min())
numberical_columns = ['age','resting bp s','cholesterol','max heart rate','oldpeak']
for col in numberical_columns:
df[col] = min_max_scaler(df[col])
df
################################################################################
# END OF YOUR CODE #
################################################################################
# ### SVM - (25pts)
# #### spliting data
# In[7]:
# The original dataset labels is 0 and 1 and in the following code we change it to -1 and 1.
df.target.replace(0 , -1 , inplace = True)
# Turn pandas dataframe to numpy array type
df = df.to_numpy()
# Splitting data into train and test part. 70% for train and 30% for test
train = df[:int(len(df) * 0.7)]
test = df[int(len(df) * 0.7):]
# Getting features
X_train = train[: , :-1]
y_train = train[: , -1]
# Getting labels
X_test = test[: , :-1]
y_test = test[: , -1]
# shapes should be:
# Train: (821, 11) (821,)
# Test: (352, 11) (352,)
print("Train: ", X_train.shape ,y_train.shape)
print("Test: " ,X_test.shape ,y_test.shape)
# #### SVM Using sklearn:
# Use the standard libarary SVM classifier (SVC) on the training data, and then test the classifier on the test data. You will need to call SVM with 3 kernels: (1) Linear, (2) Polynomial and (3) RBF. You can change C to achive better results. For "RBF" find "gamma" witch takes 90% accuracy, at least. For polynomial kernel you are allowed to change "degree" to find best results.
#
# For each kernel, reportting the followings is required:
# Accuracy, Precision, Recall, F1score.
# In[8]:
def classification_report(y_true, y_pred):
#################################################################################
## TODO: Define a function that returns the followings: ##
## Accuracy, Precision, Recall, F1score. ##
#################################################################################
True_Positive = np.sum((y_true==1)&(y_pred==1))
False_Positive = np.sum((y_true==-1)&(y_pred==1))
False_Negative = np.sum((y_true==1)&(y_pred==-1))
True_Negative = np.sum((y_true==-1)&(y_pred==-1))
Accuracy = np.mean(np.equal(y_true,y_pred))
Precision = True_Positive/(True_Positive+False_Positive)
Recall = True_Positive/(True_Positive+False_Negative)
F1score = 2*Precision*Recall/(Precision+Recall)
#################################################################################
# END OF YOUR CODE #
#################################################################################
return f'{Accuracy:.3f}', f'{Precision:.3f}', f'{Recall:.3f}', f'{F1score:.3f}'
# In[9]:
#########################################################################################
## TODO: Use svm of sklearn package (imported above) with 3 kernels. ##
## You should define model, fit using X_train, predict using X_test. ##
## your predictions known as y_pred. ##
## then use classification_report function to evaluate model. ##
#########################################################################################
linearClf=svm.SVC(kernel='linear',C=10)
linearClf.fit(X_train,y_train)
y_pred=linearClf.predict(X_test)
# linear kernel
print("results of sklearn svm linear kernel:", classification_report(y_test, y_pred))
polyClf=svm.SVC(kernel='poly',C=1,degree=2)
polyClf.fit(X_train,y_train)
y_pred=polyClf.predict(X_test)
# polynomial kernel
print("results of sklearn svm polynomial kernel:", classification_report(y_test, y_pred))
RBFClf=svm.SVC(kernel='rbf',C=50,gamma=20)
RBFClf.fit(X_train,y_train)
y_pred=RBFClf.predict(X_test)
# rbf kernel
print("results of sklearn svm RBF kernel:", classification_report(y_test, y_pred))
#########################################################################################
# END OF YOUR CODE #
#########################################################################################
# #### SVM:
# Now that you know how the standard library SVM works on the dataset, attempt to implement your own version of SVM. Implement SVM using Quadratic Programming(QP) approach. Remember that SVM objective fuction with QP is:
#
# \begin{equation*}
# min_{\alpha}\quad\frac{1}{2}\alpha^T\,Q\,\alpha-1^T\,\alpha\\
# s.t.\qquad y^T\,\alpha=0,\,\alpha\ge0
# \end{equation*}
#
# where:
# \begin{equation*}
# Q_{i,j}=y_i\,y_j\,\langle x_i\,,\,x_j\rangle
# \end{equation*}
#
# and:
# \begin{equation*}
# \text{if}\;(\alpha_n>0)\;\text{then}\;x_n\;\text{is a support vector}
# \end{equation*}
#
# For this perpose, complete the following code. You are allowed to use "cvxopt" package. It's an optimization package for Quadratic Programming. Below is the user's guide for the QP from CVXOPT:
#
# [Quadratic Programming](https://cvxopt.org/userguide/coneprog.html#quadratic-programming)
# In[17]:
# Hide cvxopt output
cvxopt.solvers.options["show_progress"] = False
#####################################################################################
## TODO: Use the information from the lecture slides to formulate the SVM ##
## kernels. These kernel functions will be called in the SVM class. ##
#####################################################################################
def linear_kernel(x,z,gamma=None,poly=None):
return np.dot(x,z)
def polynomial_kernel(x,z,gamma=None,polynomial=1):
return (1 + np.dot(x,z)) ** polynomial
def rbf_kernel(x,z,gamma,poly=None):
return np.exp(-gamma*np.linalg.norm(x-z)**2)
#####################################################################################
# END OF YOUR CODE #
#####################################################################################
class MySVM(object):
def __init__(self, kernel=linear_kernel, C=None,gamma=None,degree=None):
self.kernel = kernel
self.C = C
if self.C is not None: self.C = float(self.C)
self.gamma = gamma
self.degree = degree
def fit(self, X, y):
n_samples, n_features = X.shape
#####################################################################################
## TODO: Compute Gram matrix "K" for the given kernel. ##
#####################################################################################
K = np.zeros(shape=(n_samples,n_samples))
for i in np.arange(n_samples):
for j in np.arange(n_samples):
K[i,j] = self.kernel(X[i],X[j],self.gamma,self.degree)
#####################################################################################
# END OF YOUR CODE #
#####################################################################################
#####################################################################################
## TODO: Setup SVM objective function in QP form (Notation from attached link). ##
## Guidance: G and h have defferent definition if C is used or not. ##
#####################################################################################
P = cvxopt.matrix(y[:,None]*y[None,:]* K)
q = cvxopt.matrix(-1*np.ones(n_samples))
A = cvxopt.matrix(y, (1, n_samples))
b = cvxopt.matrix(0.0)
if self.C is None:
G = cvxopt.matrix(-1*np.identity(n_samples) )
h = cvxopt.matrix(np.zeros(n_samples))
else:
G = cvxopt.matrix(np.vstack((-1*np.identity(n_samples), np.identity(n_samples))))
h = cvxopt.matrix(np.hstack((np.zeros(n_samples), np.ones(n_samples) * self.C)))
#####################################################################################
# END OF YOUR CODE #
#####################################################################################
# solve QP problem
solution = cvxopt.solvers.qp(P, q, G, h, A, b)
# Lagrange multipliers
alpha = np.ravel(solution['x'])
# Support vectors have non zero lagrange multipliers
sv = alpha > 1e-5
#this will actually give the indices of the support vectors
ind = np.arange(len(alpha))[sv]
# get alphas of support vector , Xs and ys too.
self.alpha = alpha[sv]
self.sv = X[sv]
self.sv_y = y[sv]
#####################################################################################
## TODO: Compute the Intercept b and Weight vector w. ##
#####################################################################################
# Intercept
self.b = 0
diff = 0
for n in range(len(self.alpha)):
if self.C is not None and self.alphan[n] > self.C - 1e-5:
continue
diff =diff+ 1
self.b =self.b+ self.sv_y[n]
self.b =self.b- np.sum(self.alpha * self.sv_y * K[ind[n],sv])
if diff>0:
self.b=self.b/diff
else:
self.b = 0
# Weight vector
if self.kernel == linear_kernel:
self.w = np.zeros(n_features)
for n in range(len(self.alpha)):
self.w = self.w + self.alpha[n] * self.sv_y[n] * self.sv[n]
else:
self.w = None #Guidance: for non-linear case this should be None. (do not change)
#####################################################################################
# END OF YOUR CODE #
#####################################################################################
def predict(self, X):
if self.w is not None:
return np.dot(X, self.w) + self.b
else:
#####################################################################################
## TODO: For non-linear case, implement the kernel trick to predict the label. ##
#####################################################################################
y_predict = np.zeros(len(X))
for i in range(len(X)):
s = 0
for alpha, sv_y, sv in zip(self.alpha, self.sv_y, self.sv):
s += alpha * sv_y * self.kernel(X[i], sv,self.gamma,self.degree)
y_predict[i] = s
return y_predict + self.b
#####################################################################################
# END OF YOUR CODE #
#####################################################################################
# In[13]:
###################################################################################
## TODO: define 3 model same as previous part (SVM Using sklearn) and evaluate ##
## them. Note that for comaparing your result with that part for each kernel use ##
## same parameters in both parts. ##
###################################################################################
# linear kernel
linearSVM = MySVM(linear_kernel,C=10)
linearSVM.fit(X_train,y_train)
y_pred=np.sign(linearSVM.predict(X_test))
print("results of MySVM linear kernel:", classification_report(y_test , y_pred))
# polynomial kernel
#best degree for polynomial is 1!
polySVM=MySVM(polynomial_kernel,C=1,degree=2)
polySVM.fit(X_train,y_train)
y_pred=np.sign(polySVM.predict(X_test))
print("results of sklearn svm polynomial kernel:", classification_report(y_test, y_pred))
Rbf_SVM=MySVM(rbf_kernel,C=50,gamma=20)
Rbf_SVM.fit(X_train,y_train)
y_pred=np.sign(Rbf_SVM.predict(X_test))
# rbf kernel
print("results of sklearn svm RBF kernel:", classification_report(y_test, y_pred))
# #### Question 2: Report best results.
#
#
#
# 1. Best kernel:
# 2. Best Accuracy:
#
#
#
# #### Question 2: Report best results.
#
#
#
# 1. Best kernel: RBF Kernel
# 2. Best Accuracy: 93.18%
# ### Bonus Score - (5pts)
#
# In this step you can check other kernel functions or change parameters or any idea to get better result in compare with last section's results.
# In[14]:
df = pd.read_csv("./Heart_Disease_Dataset.csv")
columns = ["age","resting bp s","cholesterol","max heart rate"]
threshold = 3
count = df.shape[0]
for col in columns:
z_test_age = (df[col]-df[col].mean())/df[col].std()
df=df[np.abs(z_test_age)<threshold]
##here is my idea###
### we should scale all features. Not just numerical ones.
for col in df.columns:
df[col] = min_max_scaler(df[col])
df.target.replace(0 , -1 , inplace = True)
df = df.to_numpy()
train = df[:int(len(df) * 0.7)]
test = df[int(len(df) * 0.7):]
X_train = train[: , :-1]
y_train = train[: , -1]
X_test = test[: , :-1]
y_test = test[: , -1]
Rbf_SVM=MySVM(rbf_kernel,C=50,gamma=20)
Rbf_SVM.fit(X_train,y_train)
y_pred=np.sign(Rbf_SVM.predict(X_test))
print("results of sklearn svm RBF kernel:", classification_report(y_test, y_pred))
# #### My idea is simple: we should scale all features not just numerical ones.
# #### you can see the overall performance of F1_score which is the best one we have is increased 1% from 0.92 to 0.93 percent and accuracy went from 93.2 to 93.8
# In[ ]:
# In[ ]:
# In[ ]:
# In[ ]:
# In[ ]:
# In[ ]:
# In[ ]:
# In[ ]: | |
# -*- coding: utf-8 -*-
"""
Created on Fri Oct 9 13:48:37 2020
@author: bensr
"""
import sys
from neuron import h, gui
import numpy as np
import matplotlib.pyplot as plt
h.load_file("runModel.hoc")
param_list = np.loadtxt('./params/params.csv')
pc = h.ParallelContext()
h.dt = 0.1
ntimesteps = 3168
tstop = ntimesteps*h.dt
pc.set_maxstep(10)
#root_name = h.secname(sec=h.cell.soma[0])
class Cell:
def __init__(self, gid):
self.hoc_cell = h.cADpyr232_L5_TTPC1_0fb1ca4724()
self.gid = gid
pc.set_gid2node(gid, pc.id())
curr_p = param_list[gid]*gid
self.update_params(curr_p)
self.ic = h.IClamp(self.soma[0](.5))
self.ic.delay = 100
self.ic.dur = 100
self.ic.amp = 0.5
self.v = h.Vector().record(self.soma[0](.5)._ref_v,sec=self.soma[0])
def __getattr__(self, name):
# we don't have it, see if the hoc_cell has it?
return getattr(self.hoc_cell, name)
def update_params(self,p):
for curr_sec in self.hoc_cell.axonal:
curr_sec.nseg=1
curr_sec.gNaTa_tbar_NaTa_t = p[0]
curr_sec.gK_Tstbar_K_Tst = p[2]
curr_sec.gSKv3_1bar_SKv3_1 = p[5]
for curr_sec in self.hoc_cell.somatic:
curr_sec.nseg=1
curr_sec.gNaTs2_tbar_NaTs2_t = p[1]
for curr_sec in self.hoc_cell.apical:
curr_sec.nseg=1
curr_sec.gIhbar_Ih = p[3]
curr_sec.gImbar_Im = p[4]
ncell = len(param_list)
ncell = 10
gids = range(pc.id(), ncell) # round robin
cells = [Cell(gid) for gid in gids]
# make sure to enable cache efficiency
h.cvode.cache_efficient(True)
#print(pc.dt())
def prun():
h.finitialize(-65)
h.tstop = tstop
pc.psolve(tstop)
prun()
allvs = []
for curr_cell in cells:
curr_vs = curr_cell.v.to_python()
allvs.append(curr_vs)
plt.plot(curr_vs) | |
import numpy as np
import torch
def compute_border_indices(J, i0, i1):
"""
Computes border indices at all scales which correspond to the original
signal boundaries after padding.
At the finest resolution,
original_signal = padded_signal[..., i0:i1].
This function finds the integers i0, i1 for all temporal subsamplings
by 2**J, being conservative on the indices.
Parameters
----------
J : int
maximal subsampling by 2**J
i0 : int
start index of the original signal at the finest resolution
i1 : int
end index (excluded) of the original signal at the finest resolution
Returns
-------
ind_start, ind_end: dictionaries with keys in [0, ..., J] such that the
original signal is in padded_signal[ind_start[j]:ind_end[j]]
after subsampling by 2**j
"""
ind_start = {0: i0}
ind_end = {0: i1}
for j in range(1, J + 1):
ind_start[j] = (ind_start[j - 1] // 2) + (ind_start[j - 1] % 2)
ind_end[j] = (ind_end[j - 1] // 2) + (ind_end[j - 1] % 2)
return ind_start, ind_end
def cast_psi(Psi, _type):
"""
Casts the filters contained in Psi to the required type, by following
the dictionary structure.
Parameters
----------
Psi : dictionary
dictionary of dictionary of filters, should be psi1_f or psi2_f
_type : torch type
required type to cast the filters to. Should be a torch.FloatTensor
Returns
-------
Nothing - function modifies the input
"""
for filt in Psi:
for k in filt.keys():
if torch.is_tensor(filt[k]):
filt[k] = filt[k].type(_type).contiguous().requires_grad_(False)
def cast_phi(Phi, _type):
"""
Casts the filters contained in Phi to the required type, by following
the dictionary structure.
Parameters
----------
Psi : dictionary
dictionary of filters, should be phi_f
_type : torch type
required type to cast the filters to. Should be a torch.FloatTensor
Returns
-------
Nothing - function modifies the input
"""
for k in Phi.keys():
if torch.is_tensor(Phi[k]):
Phi[k] = Phi[k].type(_type).contiguous().requires_grad_(False)
def compute_padding(J_pad, T):
"""
Computes the padding to be added on the left and on the right
of the signal.
It should hold that 2**J_pad >= T
Parameters
----------
J_pad : int
2**J_pad is the support of the padded signal
T : int
original signal support size
Returns
-------
pad_left: amount to pad on the left ("beginning" of the support)
pad_right: amount to pad on the right ("end" of the support)
"""
T_pad = 2**J_pad
if T_pad < T:
raise ValueError('Padding support should be larger than the original' +
'signal size!')
to_add = 2**J_pad - T
pad_left = to_add // 2
pad_right = to_add - pad_left
if max(pad_left, pad_right) >= T:
raise ValueError('Too large padding value, will lead to NaN errors')
return pad_left, pad_right | |
'''
Copyright (c) 2020, Martel Lab, Sunnybrook Research Institute
Description: This code will convert an SQL .csv file that has reports
seperated into indidual text lines into a .csv of individual reports.
Input: saved .csv file of sql database needed to be converted.
Output: a .csv file of the input file with Lines collected into organized
reports.
'''
import argparse
import pandas as pd
import sys
from tqdm import tqdm
import numpy as np
parser = argparse.ArgumentParser()
parser.add_argument("--input_sql", type=str,
help="File location of SQL data.")
parser.add_argument("--save_name", type=str,
help="File location you wish to csv file.")
parser.add_argument("--session_col", type=str, default="AccNum",
help="Label for the column that holds information on exam "
"session number ")
parser.add_argument("--report_col", type=str, default="ReportTxt",
help="Label for the column that contains report text.")
opt = parser.parse_args()
print('-'*80)
print(opt)
print('-'*80)
def eliminate_sequence_number(sql_data,
accession_label='AccNum',
textlabel='ReportTxt'):
"""
This function takes a csv input that has a SequenceNumber column that
give the sequence information of report text. This will then append all
string data in the proper sequence to merge multiple lines of the same
session into one row of information in a dataframe.
:param sql_data: csv file containing sql output where the report data is
split into lines characterizeed by the sequence number. Requires a
SequenceNumber column and an ExamDate column.
:param accession_label: str, name of the column that contains session
number
:param textlabel: str, name of the column that contains report text
:return: Dataframe of all rows of sql_data file compressed into sessions
"""
sql_data = sql_data[[textlabel, accession_label, 'SequenceNumber',
'ExamDate']].replace(np.nan, '', regex=True).astype({textlabel: str})
acc_nums = list(set(sql_data[accession_label]))
output_data = []
acc_nums_iterator = tqdm(acc_nums, desc="SQLtoDataframe Progress: ",
position=0, leave=True)
for num in acc_nums_iterator:
temp = sql_data.loc[sql_data[accession_label] == num]
temp = temp.drop_duplicates()
temp = temp.sort_values(['SequenceNumber'])
save_value = list(temp.values[0])
col_oi = list(temp.columns).index(textlabel)
for row in temp.itertuples():
if row.SequenceNumber == 0:
continue
else:
if row[col_oi+1] == '(null)':
save_value[col_oi] += '\n'
else:
save_value[col_oi] += '\n' + row[col_oi+1]
output_data.append(save_value)
sys.stdout.write(
'\rFinished analyzing ' + str(len(output_data)/len(acc_nums))
)
sys.stdout.flush()
op_cols = list(sql_data.columns)
rt_df = pd.DataFrame(output_data,
columns=op_cols)
rt_df = rt_df.drop('SequenceNumber', 'columns')
return rt_df
sql_reports = pd.read_csv(opt.input_sql)
reports_processed = eliminate_sequence_number(sql_reports,
accession_label=opt.session_col,
textlabel=opt.report_col)
reports_processed.to_csv(opt.save_name)
print('End of SQLtoDataframe Script') | |
'''
Comparing single layer MLP with deep MLP (using TensorFlow)
'''
import tensorflow as tf
import numpy as np
import pickle
import timeit
start = timeit.default_timer()
# Create model
# Add more hidden layers to create deeper networks
# Remember to connect the final hidden layer to the out_layer
def create_multilayer_perceptron():
# Network Parameters
n_hidden_1 = 256 # 1st layer number of features
#n_hidden_2 = 256 # 2nd layer number of features
#n_hidden_3 = 256
#n_hidden_4 = 256
#n_hidden_5 = 256
#n_hidden_6 = 256
#n_hidden_7 = 256
n_input = 784 # data input
n_classes = 2
# Store layers weight & bias
weights = {
'h1': tf.Variable(tf.random_normal([n_input, n_hidden_1])),
#'h2': tf.Variable(tf.random_normal([n_hidden_1, n_hidden_2])),
#'h3': tf.Variable(tf.random_normal([n_hidden_2, n_hidden_3])),
#'h4': tf.Variable(tf.random_normal([n_hidden_3, n_hidden_4])),
#'h5': tf.Variable(tf.random_normal([n_hidden_4, n_hidden_5])),
#'h6': tf.Variable(tf.random_normal([n_hidden_5, n_hidden_6])),
#'h7': tf.Variable(tf.random_normal([n_hidden_6, n_hidden_7])),
'out': tf.Variable(tf.random_normal([n_hidden_1, n_classes]))
}
biases = {
'b1': tf.Variable(tf.random_normal([n_hidden_1])),
#'b2': tf.Variable(tf.random_normal([n_hidden_2])),
#'b3': tf.Variable(tf.random_normal([n_hidden_3])),
#'b4': tf.Variable(tf.random_normal([n_hidden_4])),
#'b5': tf.Variable(tf.random_normal([n_hidden_5])),
#'b6': tf.Variable(tf.random_normal([n_hidden_6])),
#'b7': tf.Variable(tf.random_normal([n_hidden_7])),
'out': tf.Variable(tf.random_normal([n_classes]))
}
# tf Graph input
x = tf.placeholder("float", [None, n_input])
y = tf.placeholder("float", [None, n_classes])
# Hidden layer with RELU activation
layer_1 = tf.add(tf.matmul(x, weights['h1']), biases['b1'])
layer_1 = tf.nn.relu(layer_1)
# Hidden layer with RELU activation
#layer_2 = tf.add(tf.matmul(layer_1, weights['h2']), biases['b2'])
#layer_2 = tf.nn.relu(layer_2)
# Hidden layer with RELU activation
#layer_3 = tf.add(tf.matmul(layer_2, weights['h3']), biases['b3'])
#layer_3 = tf.nn.relu(layer_3)
# Hidden layer with RELU activation
#layer_4 = tf.add(tf.matmul(layer_3, weights['h4']), biases['b4'])
#layer_4 = tf.nn.relu(layer_4)
# Hidden layer with RELU activation
#layer_5 = tf.add(tf.matmul(layer_4, weights['h5']), biases['b5'])
#layer_5 = tf.nn.relu(layer_5)
# Hidden layer with RELU activation
#layer_6 = tf.add(tf.matmul(layer_5, weights['h6']), biases['b6'])
#layer_6 = tf.nn.relu(layer_6)
# Hidden layer with RELU activation
#layer_7 = tf.add(tf.matmul(layer_6, weights['h7']), biases['b7'])
#layer_7 = tf.nn.relu(layer_7)
# Output layer with linear activation
out_layer = tf.matmul(layer_1, weights['out']) + biases['out']
return out_layer,x,y
# Do not change this
def preprocess():
mat = loadmat('mnist_all.mat') # loads the MAT object as a Dictionary
# Pick a reasonable size for validation data
# ------------Initialize preprocess arrays----------------------#
train_preprocess = np.zeros(shape=(50000, 784))
validation_preprocess = np.zeros(shape=(10000, 784))
test_preprocess = np.zeros(shape=(10000, 784))
train_label_preprocess = np.zeros(shape=(50000,))
validation_label_preprocess = np.zeros(shape=(10000,))
test_label_preprocess = np.zeros(shape=(10000,))
# ------------Initialize flag variables----------------------#
train_len = 0
validation_len = 0
test_len = 0
train_label_len = 0
validation_label_len = 0
# ------------Start to split the data set into 6 arrays-----------#
for key in mat:
# -----------when the set is training set--------------------#
if "train" in key:
label = key[-1] # record the corresponding label
tup = mat.get(key)
sap = range(tup.shape[0])
tup_perm = np.random.permutation(sap)
tup_len = len(tup) # get the length of current training set
tag_len = tup_len - 1000 # defines the number of examples which will be added into the training set
# ---------------------adding data to training set-------------------------#
train_preprocess[train_len:train_len + tag_len] = tup[tup_perm[1000:], :]
train_len += tag_len
train_label_preprocess[train_label_len:train_label_len + tag_len] = label
train_label_len += tag_len
# ---------------------adding data to validation set-------------------------#
validation_preprocess[validation_len:validation_len + 1000] = tup[tup_perm[0:1000], :]
validation_len += 1000
validation_label_preprocess[validation_label_len:validation_label_len + 1000] = label
validation_label_len += 1000
# ---------------------adding data to test set-------------------------#
elif "test" in key:
label = key[-1]
tup = mat.get(key)
sap = range(tup.shape[0])
tup_perm = np.random.permutation(sap)
tup_len = len(tup)
test_label_preprocess[test_len:test_len + tup_len] = label
test_preprocess[test_len:test_len + tup_len] = tup[tup_perm]
test_len += tup_len
# ---------------------Shuffle,double and normalize-------------------------#
train_size = range(train_preprocess.shape[0])
train_perm = np.random.permutation(train_size)
train_data = train_preprocess[train_perm]
train_data = np.double(train_data)
train_data = train_data / 255.0
train_label = train_label_preprocess[train_perm]
validation_size = range(validation_preprocess.shape[0])
vali_perm = np.random.permutation(validation_size)
validation_data = validation_preprocess[vali_perm]
validation_data = np.double(validation_data)
validation_data = validation_data / 255.0
validation_label = validation_label_preprocess[vali_perm]
test_size = range(test_preprocess.shape[0])
test_perm = np.random.permutation(test_size)
test_data = test_preprocess[test_perm]
test_data = np.double(test_data)
test_data = test_data / 255.0
test_label = test_label_preprocess[test_perm]
# Feature selection
# Your code here.
#train_data1 = np.delete(train_data,np.where(np.amax(train_data,axis=0)==0),1)
#test_data1 = np.delete(test_data,np.where(np.amax(train_data,axis=0)==0),1)
#validation_data1 = np.delete(validation_data,np.where(np.amax(train_data,axis=0)==0),1)
print('preprocess done')
return train_data, train_label, validation_data, validation_label, test_data, test_label
# Parameters
learning_rate = 0.0001
training_epochs = 100
batch_size = 100
# Construct model
pred,x,y = create_multilayer_perceptron()
# Define loss and optimizer
cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=pred, labels=y))
optimizer = tf.train.AdamOptimizer(learning_rate=learning_rate).minimize(cost)
timer =1
# Initializing the variables
init = tf.global_variables_initializer()
# load data
train_features, train_labels, valid_features, valid_labels, test_features, test_labels = preprocess()
# Launch the graph
with tf.Session() as sess:
sess.run(init)
# Training cycle
for epoch in range(training_epochs):
print(timer)
timer = timer +1
avg_cost = 0.
total_batch = int(train_features.shape[0] / batch_size)
# Loop over all batches
for i in range(total_batch):
batch_x, batch_y = train_features[i * batch_size: (i + 1) * batch_size], train_labels[i * batch_size: (i + 1) * batch_size]
# Run optimization op (backprop) and cost op (to get loss value)
_, c = sess.run([optimizer, cost], feed_dict={x: batch_x, y: batch_y})
# Compute average loss
avg_cost += c / total_batch
print("Optimization Finished!")
correct_prediction = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
accuracy = tf.reduce_mean(tf.cast(correct_prediction, "float"))
print("Accuracy:", accuracy.eval({x: test_features, y: test_labels}))
stop = timeit.default_timer()
print('\n Time Taken: ' + str(stop - start)) | |
# -*- coding: utf-8 -*-
"""
This is the original file that the client sent me.
When I (Omar Trejo) solved this problem, I chose
not to reuse this code, and start from scratch.
"""
import wave
import numpy as np
import matplotlib.pyplot as plt
#import utility
import scipy
import scipy.signal
import numba
import scipy.io.wavfile as wavfile
import scikits.audiolab as audiolab
f = audiolab.Sndfile("/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001MS.wav", 'r')
#f = wave.open("/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001MS.wav",'r')
data = np.array(f.read_frames(f.nframes), dtype=np.float64)
f.close()
rate = f.samplerate;
print("rate: ", rate)
wavfile.write("/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SROO1MSout.wav", rate, data)
with wave.open('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001MSout.wav') as w:
channels = w.getnchannels()
fs = w.getframerate()
assert w.getsampwidth() == 2
data = w.readframes(w.getnframes())
print("Initial Sampling Frequency: ", fs)
print("Channels: ", channels)
print("Length: ", len(data))
mic_audio = data
#sig = np.frombuffer(data, dtype='<i2').reshape(-1, channels)
#normalized = utility.pcm2float(sig, np.float64)
#mic_audio = normalized
plt.figure()
plt.plot(mic_audio)
plt.title("Original Signals")
#print("Length Normalized: ", (len(mic_audio)))
@numba.jit
def L(mic_audio):
secs = len(mic_audio)/44100.0 # Number of seconds in mic_audio
return (secs)
(secs) = L(mic_audio)
(secs) = L(mic_audio)
print ("Secs in mic_audio: ", secs)
p2 = np.floor(np.log2(len(mic_audio)))
nextpow2 = np.power(2, p2-1)
print("Next power of 2: ", nextpow2)
mic_audio2 = mic_audio[0:nextpow2]
secs2 = len(mic_audio2)/44100.0
@numba.jit
def M(secs2):
samps = secs2*1024.0 # Number of samples to downsample
#samps = secs*256.0 # Number of samples to downsample
return (samps)
(samps) = M(secs2)
print ("# of samples to downsample: ", samps)
@numba.jit
def N(mic_audio2, samps):
Y = scipy.signal.resample(mic_audio2, samps)
return (Y)
(Y) = N(mic_audio2, samps)
fs2 = samps/secs2
print("fs2: ", fs2) # Sampling frequency of downsampled signal (Y)
print("Length Y: ", len(Y)) # Number of samples in Y
## ----- PLOT data ----- ##
plt.figure()
plt.plot(mic_audio)
plt.hold()
plt.plot(Y, color = 'm', linestyle=':')
plt.title("Original vs Downsampled Signals")
plt.figure()
plt.plot(mic_audio[0:2000])
plt.hold()
plt.plot(Y[0:2000])
plt.title("Original vs Downsampled Signals")
#--------------------------------------------------------------------------
#----------------------------------------------------------------------------
#----------------------------------------------------------------------------
#@numba.jit
#def samp(mic_audio):
# secs = len(mic_audio)/44100.0
# samps = secs*1024.0
# Y = scipy.signal.resample(mic_audio, samps)
# return (samps, secs, Y)
#
#(samps, secs, Y) = samp(mic_audio)
#
#secs = len(mic_audio)/44100.0 # Number of seconds in signal X
#samps = secs*1024 # Number of samples to downsample
#Y = scipy.signal.resample(mic_audio, samps)
#def secs(X):
# return len(mic_audio)/44100
#
#@numba.jit
#def samps(secs):
# return secs*1024
#
#@numba.jit
#def Y(samps):
# return scipy.signal.resample(mic_audio, samps)
#
#
#plt.figure()
#mic_audioslice = mic_audio[0:2646000]
#plt.plot(mic_audioslice)
#
#plt.figure()
#mic_audioslice2 = mic_audio[0:1323000]
#plt.plot(mic_audioslice2)
#
#plt.figure()
#mic_audioslice3 = mic_audio[0:44100]
#plt.plot(mic_audioslice3)
#
#plt.figure()
#mic_audioslice4 = mic_audio[0:22050]
#plt.plot(mic_audioslice4)
#with wave.open('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav') as w2:
# channels2 = w2.getnchannels()
# wave.Wave_write(w2)
#
# fs_mic = w2.setframerate(1024)
# fs2 = w2.getframerate()
# assert w2.getsampwidth() == 2
# data2 = w2.readframes(w2.getnframes())
#
# print("New Sampling Frequency: ", fs2)
# #print("Channels: ", channels2)
# print(len(data2))
#
#
#sig2 = np.frombuffer(data2, dtype='<i2').reshape(-1, channels2)
#
#normalized2 = utility.pcm2float(sig2, np.float32)
#normchunk2 = normalized2[0:10240]
#mic_audio2 = normchunk2
##plt.plot(normchunk)
#plt.plot(mic_audio2)
########################## Drafts #####################################
#import wave
#import audioop
#import scipy
#from scipy import
##
#from path import Path
#d = Path('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01')
#for f in d.files('*.wav'):
#scipy.io.wavfile.read('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav')
#import wave
#XY = wave.open('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav')
#wave.Wave_read.getnchannels(XY)
#wave.Wave_read.getframerate(XY)
#wave.Wave_read.getparams(XY)
#-----------------------------------
#import numpy as np
#import wave
#import struct
#import sys
#
#def wav_to_floats(wave_file):
# w = wave.open(wave_file)
# astr = w.readframes(w.getnframes())
# a = struct.unpack("%ih" % (w.getnframes()* w.getnchannels()), astr)
# a = [float(val)/pow(2,15) for val in a]
# return a
#
#signal = wav_to_floats(sys.argv[1])
#print ("read "+str(len(signal))+" frames")
#print ("in the range "+str(min(signal))+" to "+str(min(signal)))
#--------------------------------------------
#import wavefile
# returns the contents of the wav file as a double precision float array
#def wav_to_floats(filename = 'SR001XY.wav'):
# path = '/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/'
# w = wavefile.load(path+filename)
#
#
# return w[1][0]
#
#signal = wav_to_floats(sys.argv[1])
#print ("read "+str(len(signal))+" frames")
#print ("in the range "+str(min(signal))+" to "+str(min(signal)))
#----------------this works-------------
#import numpy as np
#from scipy.io import wavfile ## This works!! but data is in integer format
#fs, data = np.array(wavfile.read('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav'))
#----------------------------------------
#import utility
#from scipy.io import wavfile
#from utility import pcm2float
#
#fs, data = wavfile.read('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav')
#normalized = pcm2float(data,'float32')
#
#print ("sampling rate = {} Hz, length = {} samples, channels = {}".format(fs, *data.shape))
#print (data)
#plot(data)
#----------------------------------------------
#/////////////// From https://github.com/mgeier/python-audio/blob/master/audio-files/audio-files-with-wave.ipynb /////////
#import matplotlib.pyplot as plt
#import numpy as np
#import wave
#
#
#with wave.open('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav') as w:
# framerate = w.getframerate()
# frames = w.getnframes()
# channels = w.getnchannels()
# width = w.getsampwidth()
# print("sampling rate:", framerate, "Hz, length:", frames, "samples,",
# "channels:", channels, "sample width:", width, "bytes")
#
# data = w.readframes(frames)
#
#print (len(data))
#print(type(data))
#print (data)
#
#sig = np.frombuffer(data, dtype='B')
#print ("signal B: ", sig)
#
#sig = np.frombuffer(data, dtype='<i2').reshape(-1, channels)
#print ("signal i2: ", sig)
#plt.plot(sig)
#
#print ("sig base is data: ", sig.base.base is data)
#print ("Flags", sig.flags)
#
#import utility
#normalized = utility.pcm2float(sig, 'float32')
#plt.plot(normalized)
#
#print ("normalized flags: ", normalized.flags)
#
#with wave.open('/Users/<username_obfuscated>/LocalDura/AUDIO/tests/audio_test_2/mic/4CH/FOLDER01/SR001XY.wav') as w:
# framerate = w.getframerate()
# frames = w.getnframes()
# channels = w.getnchannels()
# width = w.getsampwidth()
# print("sampling rate = {framerate} Hz, length = {frames} samples, channels = {channels}, sample width = {width} bytes".format(**locals()))
#
# data = w.readframes(frames)
#
#
# #assert width == 3
# temp = bytearray()
#
#for i in range(0, len(data), 3):
# temp.append(0)
# temp.extend(data[i:i+3])
#
## Using += instead of .extend() may be faster
## (see https://youtu.be/z9Hmys8ojno?t=35m50s).
## But starting with an empty bytearray and
## extending it on each iteration might be slow, anyway.
## See further below for how to reserve all necessary memory in the beginning.
#
# four_bytes = np.frombuffer(temp, dtype='B').reshape(-1, 4)
# four_bytes
#
#
# sig = np.frombuffer(temp, dtype='<i4').reshape(-1, channels)
# sig
#
#
#normalized = utility.pcm2float(sig, 'float32')
#plt.plot(normalized)
#//////////////////////////////////////////////////////////////////////////
# frames = audioop.reverse(frames, params.sampwidth)
#
# with wave.open(d/'SR0001XY.wav', 'wb') as XY:
# XY.setparams(params)
# XY.writeparams(frames)
# | |
from collections import namedtuple
import numpy as np
import talib
from jesse.helpers import get_candle_source
from jesse.helpers import slice_candles
MAMA = namedtuple('MAMA', ['mama', 'fama'])
def mama(candles: np.ndarray, fastlimit: float = 0.5, slowlimit: float = 0.05, source_type: str = "close",
sequential: bool = False) -> MAMA:
"""
MAMA - MESA Adaptive Moving Average
:param candles: np.ndarray
:param fastlimit: float - default: 0.5
:param slowlimit: float - default: 0.05
:param source_type: str - default: "close"
:param sequential: bool - default=False
:return: MAMA(mama, fama)
"""
candles = slice_candles(candles, sequential)
source = get_candle_source(candles, source_type=source_type)
mama, fama = talib.MAMA(source, fastlimit=fastlimit, slowlimit=slowlimit)
if sequential:
return MAMA(mama, fama)
else:
return MAMA(mama[-1], fama[-1]) | |
def add_ellipse(ax, scores, group, comp1 = 0, comp2 = 1, palette=None, alpha=0.95, **kwargs):
"""Add ellipses to a PCA ordination plot based on categorical variables. The indexes of scores and group must match.
Parameters
----------
ax : matplotlib axes
The axis of the plot
scores : :py:class:`pandas.DataFrame`
A pandas dataframe containing Scores from PCA
group : :py:class:`pandas.Series`
A pandas series containing group labels
comp1 : int
Index of the component to put on the X-axis, by default 0 or the first component
comp2 : int
Index of the component to put on the Y-axis, by default 1 or the second component
palette : list
Palette for colors
alpha : float
Alpha-value for the confidence interval, 0.95 by default
**kwargs:
Keyword arguments are passed on to matplotlib.patches.Ellipse
"""
import scipy.stats as st
import numpy as np
from matplotlib import cm
from matplotlib.patches import Ellipse
if (palette == None):
palette = cm.get_cmap("tab20").colors
try:
bb = (scores.index == group.index).all()
except:
print("indexes of scores and group must match!")
return
if (bb == False):
print("indexes of scores and group must match!")
return
_ellipse_kwargs = {"linewidth": 2}
_ellipse_kwargs.update(**kwargs)
for i, u in enumerate(group.unique()):
c1h = scores.loc[group[group == u].index, scores.columns[comp1]]
c2h = scores.loc[group[group == u].index, scores.columns[comp2]]
w = st.t.interval(alpha=alpha, df=len(c1h)-1, loc=np.mean(c1h), scale=st.sem(c1h))[1] - st.t.interval(alpha=alpha, df=len(c1h)-1, loc=np.mean(c1h), scale=st.sem(c1h))[0]
h = st.t.interval(alpha=alpha, df=len(c2h)-1, loc=np.mean(c2h), scale=st.sem(c2h))[1] - st.t.interval(alpha=alpha, df=len(c2h)-1, loc=np.mean(c2h), scale=st.sem(c2h))[0]
elps = Ellipse((c1h.mean(), c2h.mean()), w, h, edgecolor=palette[i],facecolor='none', **_ellipse_kwargs)
ax.add_artist(elps) | |
import pytest
import os,shutil,sys
import numpy as np
from mpi4py import MPI
from pypospack.pyposmat.data import PyposmatConfigurationFile
from pypospack.pyposmat.engines import PyposmatIterativeSampler
pyposmat_data_dir = 'data'
config_fn = os.path.join(pyposmat_data_dir,'pyposmat.config.in')
def test__initialize_data_directory__with_relative_path_arg():
test_data_dir_path = 'test_dir'
pyposmat_app = PyposmatIterativeSampler(configuration_filename = config_fn)
pyposmat_app.read_configuration_file()
is_created,s = pyposmat_app.initialize_data_directory(data_directory=test_data_dir_path)
assert s == os.path.join(os.getcwd(),test_data_dir_path)
assert pyposmat_app.data_directory == os.path.join(os.getcwd(),test_data_dir_path)
shutil.rmtree(test_data_dir_path)
def test__initialize_data_directory__with_absolute_path_arg():
test_data_dir_path = os.path.join(os.getcwd(),'test_dir')
pyposmat_app = PyposmatIterativeSampler(configuration_filename=config_fn)
pyposmat_app.read_configuration_file()
is_created,s = pyposmat_app.initialize_data_directory(data_directory=test_data_dir_path)
assert s == os.path.join(os.getcwd(),test_data_dir_path)
assert pyposmat_app.data_directory == test_data_dir_path
shutil.rmtree(test_data_dir_path) | |
"""Objects that hold timeseries objects defined over multiple locations."""
import datetime
from dataclasses import dataclass, field, fields, replace
from typing import Dict, Optional, Tuple, Union
import numpy as np
import pandas as pd
from loguru import logger
from .._typing import ArrayLike, PathLike
from ..numerical_libs import sync_numerical_libs, xp
from .adm_mapping import AdminLevel, AdminLevelMapping
# TODO now that this holds the adm_mapping, the adm_ids column can probably be replaced...
@dataclass(frozen=True)
class SpatialStratifiedTimeseries:
"""Class representing a generic timeseries that is defined over admin regions."""
adm_level: int
adm_ids: ArrayLike
dates: ArrayLike # TODO cupy cant handle this...
adm_mapping: AdminLevelMapping # = field(init=False)
def __post_init__(self):
valid_shape = self.dates.shape + self.adm_ids.shape
for f in fields(self):
if "data_field" in f.metadata:
field_shape = getattr(self, f.name).shape
if field_shape != valid_shape:
logger.error("Invalid timeseries shape {}; expected {}.", field_shape, valid_shape)
raise ValueError
def __repr__(self) -> str:
names = [f.name for f in fields(self) if f.name not in ["adm_ids", "dates"]]
return (
f"{names} for {self.adm_ids.shape[0]} adm{self.adm_level} regions from {self.start_date} to {self.end_date}"
)
@property
def start_date(self) -> datetime.date:
"""The first date of valid data."""
return self.dates[0]
@property
def end_date(self) -> datetime.date:
"""The last date of valid data."""
return self.dates[-1]
@property
def n_days(self) -> int:
"""Total number of days of data."""
return len(self.dates)
@property
def n_loc(self) -> int:
"""Total number of locations for which we have timeseries at the base admin level."""
return len(self.adm_ids)
@staticmethod
@sync_numerical_libs
def _generic_from_csv(
filename: PathLike,
n_days: Optional[int] = None,
valid_date_range=(None, None),
force_enddate: Optional[datetime.date] = None,
force_enddate_dow: Optional[int] = None,
adm_col: str = "adm2",
date_col: str = "date",
column_names: Dict[str, str] = None,
):
"""Return a dict containing args to a subclass's constructor"""
df = pd.read_csv(
filename,
index_col=[adm_col, date_col],
engine="c",
parse_dates=["date"],
).sort_index()
dates = df.index.unique(level=date_col).values
dates = dates.astype("datetime64[s]").astype(datetime.date)
date_mask = _mask_date_range(dates, n_days, valid_date_range, force_enddate, force_enddate_dow)
ret = {
"dates": dates[date_mask],
"adm_ids": xp.array(df.index.unique(level=adm_col).values),
}
for fcolumn, out_name in column_names.items():
var_full_hist = xp.array(df[fcolumn].unstack().fillna(0.0).values).T
ret[out_name] = var_full_hist[date_mask]
return ret
def to_dict(self, level=None):
"""Return the timeseries as a dict containing the data it's indices."""
# get data to requested adm level
obj = self.sum_adm_level(level) if level is not None else self
ret = {f.name: getattr(obj, f.name) for f in fields(obj) if f.name not in ("adm_level", "adm_mapping")}
# reshape index columns and get the right name for the adm id column
ret_shp = ret["dates"].shape + ret["adm_ids"].shape
ret["date"] = np.broadcast_to(ret.pop("dates")[..., None], ret_shp)
adm_col_name = f"adm{obj.adm_level}"
ret[adm_col_name] = np.broadcast_to(ret.pop("adm_ids")[None, ...], ret_shp)
# Flatten arrays
ret = {k: np.ravel(xp.to_cpu(v)) for k, v in ret.items()}
return ret
def to_dataframe(self, level=None):
"""Return the timeseries as a pandas.DataFrame."""
data_dict = self.to_dict(level)
df = pd.DataFrame(data_dict)
adm_col = df.columns[df.columns.str.match("adm[0-9]")].item()
return df.set_index([adm_col, "date"]).sort_index()
def to_csv(self, filename, level=None):
"""Save the timeseries data to a csv."""
# TODO log
df = self.to_dataframe(level)
df.to_csv(filename, index=True)
def replace(self, **changes):
"""Replace data columns."""
return replace(self, **changes)
def sum_adm_level(self, level: Union[int, str]):
"""Sum the values to a different admin level then they are defined on."""
if type(level) == str:
# Check string begins with 'adm'
if level[:3] == "adm":
level = int(level.split("adm")[-1])
else:
logger.error("String admin aggregation level must begin with adm")
raise ValueError
# TODO masking, weighting?
if level > self.adm_level:
logger.error("Requested sum to finer adm level than the data.")
raise ValueError
if level == self.adm_level:
return self
# out_id_map = self.adm_mapping.levels[level].idx
out_id_map = self.adm_mapping.levels[level].idx[self.adm_mapping.levels[self.adm_level].level_idx]
new_ids = self.adm_mapping.levels[level].ids
new_data = {"adm_level": level, "adm_ids": new_ids, "dates": self.dates, "adm_mapping": self.adm_mapping}
for f in fields(self):
if "summable" in f.metadata:
orig_ts = getattr(self, f.name)
new_ts = xp.zeros(new_ids.shape + self.dates.shape, dtype=orig_ts.dtype)
xp.scatter_add(new_ts, out_id_map, orig_ts.T)
new_data[f.name] = new_ts.T
return self.__class__(**new_data)
def validate_isfinite(self):
"""Check that there are no invalid values in the timeseries (nan, inf, etc)."""
for f in fields(self):
if "data_field" in f.metadata:
col_name = f.name
data = getattr(self, f.name)
finite = xp.isfinite(data)
if xp.any(~finite):
locs = xp.argwhere(~finite)
logger.error(
f"Nonfinite values found in column {col_name} of {self.__class__.__qualname__} at {locs}!",
)
raise RuntimeError
def _mask_date_range(
dates: np.ndarray,
n_days: Optional[int] = None,
valid_date_range: Tuple[Optional[datetime.date], Optional[datetime.date]] = (None, None),
force_enddate: Optional[datetime.date] = None,
force_enddate_dow: Optional[int] = None,
):
valid_date_range = list(valid_date_range)
if valid_date_range[0] is None:
valid_date_range[0] = dates[0]
if valid_date_range[1] is None:
valid_date_range[1] = dates[-1]
# Set the end of the date range to the last valid date that is the requested day of the week
if force_enddate_dow is not None:
end_date_dow = valid_date_range[1].weekday()
days_after_forced_dow = (end_date_dow - force_enddate_dow + 7) % 7
valid_date_range[1] = dates[-(days_after_forced_dow + 1)]
if force_enddate is not None:
if force_enddate_dow is not None:
if force_enddate.weekday() != force_enddate_dow:
logger.error("Start date not consistant with required day of week")
raise RuntimeError
valid_date_range[1] = force_enddate
# only grab the requested amount of history
if n_days is not None:
valid_date_range[0] = valid_date_range[-1] - datetime.timedelta(days=n_days - 1)
# Mask out dates not in request range
date_mask = (dates >= valid_date_range[0]) & (dates <= valid_date_range[1])
return date_mask
@dataclass(frozen=True, repr=False)
class CSSEData(SpatialStratifiedTimeseries):
cumulative_cases: ArrayLike = field(metadata={"data_field": True, "summable": True})
cumulative_deaths: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_cases: ArrayLike = field(default=None, metadata={"data_field": True, "summable": True})
incident_deaths: ArrayLike = field(default=None, metadata={"data_field": True, "summable": True})
def __post_init__(self):
if self.incident_cases is None:
object.__setattr__(self, "incident_cases", xp.gradient(self.cumulative_cases, axis=0, edge_order=2))
if self.incident_deaths is None:
object.__setattr__(self, "incident_deaths", xp.gradient(self.cumulative_deaths, axis=0, edge_order=2))
super().__post_init__()
@staticmethod
def from_csv(
file: PathLike,
n_days: Optional[int] = None,
valid_date_range=(None, None),
force_enddate: Optional[datetime.date] = None,
force_enddate_dow: Optional[int] = None,
adm_mapping: Optional[AdminLevelMapping] = None,
):
logger.info("Reading historical CSSE data from {}", file)
adm_level = "adm2"
var_dict = SpatialStratifiedTimeseries._generic_from_csv(
file,
n_days,
valid_date_range,
force_enddate,
force_enddate_dow,
adm_level,
column_names={"cumulative_reported_cases": "cumulative_cases", "cumulative_deaths": "cumulative_deaths"},
)
var_dict["adm_mapping"] = adm_mapping
return CSSEData(2, **var_dict)
@dataclass(frozen=True, repr=False)
class HHSData(SpatialStratifiedTimeseries):
current_hospitalizations: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_hospitalizations: ArrayLike = field(metadata={"data_field": True, "summable": True})
# TODO we probably need to store a AdminLevelMapping in each timeseries b/c the hhs adm_ids dont line up with the csse ones after we aggregate them to adm1...
@staticmethod
def from_csv(
file: PathLike,
n_days: Optional[int] = None,
valid_date_range=(None, None),
force_enddate: Optional[datetime.date] = None,
force_enddate_dow: Optional[int] = None,
adm_mapping: Optional[AdminLevelMapping] = None,
):
logger.info("Reading historical HHS hospitalization data from {}", file)
adm_level = "adm1"
var_dict = SpatialStratifiedTimeseries._generic_from_csv(
file,
n_days,
valid_date_range,
force_enddate,
force_enddate_dow,
adm_col=adm_level,
column_names={
"incident_hospitalizations": "incident_hospitalizations",
"current_hospitalizations": "current_hospitalizations",
},
)
var_dict["adm_mapping"] = adm_mapping
return HHSData(1, **var_dict)
@dataclass(frozen=True, repr=False)
class BuckyFittedData(SpatialStratifiedTimeseries):
cumulative_cases: ArrayLike = field(metadata={"data_field": True, "summable": True})
cumulative_deaths: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_cases: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_deaths: ArrayLike = field(metadata={"data_field": True, "summable": True})
@dataclass(frozen=True, repr=False)
class BuckyFittedCaseData(SpatialStratifiedTimeseries):
cumulative_cases: ArrayLike = field(metadata={"data_field": True, "summable": True})
cumulative_deaths: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_cases: ArrayLike = field(metadata={"data_field": True, "summable": True})
incident_deaths: ArrayLike = field(metadata={"data_field": True, "summable": True})
@staticmethod
def from_csv(
file: PathLike,
n_days: Optional[int] = None,
valid_date_range=(None, None),
force_enddate: Optional[datetime.date] = None,
force_enddate_dow: Optional[int] = None,
adm_mapping: Optional[AdminLevelMapping] = None,
):
logger.info("Reading historical CSSE data from {}", file)
adm_level = "adm2"
var_dict = SpatialStratifiedTimeseries._generic_from_csv(
file,
n_days,
valid_date_range,
force_enddate,
force_enddate_dow,
adm_level,
column_names={
"cumulative_cases": "cumulative_cases",
"cumulative_deaths": "cumulative_deaths",
"incident_cases": "incident_cases",
"incident_deaths": "incident_deaths",
},
)
var_dict["adm_mapping"] = adm_mapping
return BuckyFittedCaseData(2, **var_dict) | |
import numpy as np
from PIL import Image
from random import randint, choice
# im = Image.open('./data/bank-2.png')
# a = np.asarray(im)
# print(a.shape)
colors = [
[222, 193, 158],
[206, 181, 146],
[195, 174, 137],
[185, 168, 131],
[199, 179, 139],
[212, 193, 135],
[212, 208, 159],
[185, 170, 123],
[211, 196, 139],
[212, 196, 148],
[181, 172, 143],
[196, 182, 146],
[234, 229, 196],
[223, 217, 184],
[214, 198, 153],
[219, 198, 146],
[206, 184, 136],
[218, 204, 160],
[206, 184, 136],
[216, 196, 153],
]
array = np.zeros((360, 640, 4), dtype=np.uint8)
array[:, :, 0:3] = 5
print(array.shape)
print(array)
array[:, :, 3] = 255
step = 2
for row in range(0, 720, step):
curr_col = 0
while curr_col < 1280:
width = randint(4, 8)
color = choice(colors) + [255]
array[row: row + step, curr_col: curr_col + width, :] = color
curr_col += width
print(array.shape)
print(array)
im = Image.fromarray(array)
im.save('./data/background.png') | |
"""
SCRIPT TO CONVERT WRITE CHARMM RTF AND PRM FILES
FROM BOSS ZMATRIX
Created on Mon Feb 15 15:40:05 2016
@author: Leela S. Dodda leela.dodda@yale.edu
@author: William L. Jorgensen Lab
Usage: python OPM_Routines.py -z phenol.z -r PHN
REQUIREMENTS:
BOSS (need to set BOSSdir in bashrc and cshrc)
Preferably Anaconda python with following modules
pandas
argparse
numpy
"""
from LigParGen.BOSSReader import bossPdbAtom2Element,bossElement2Mass,ucomb
import pickle
import pandas as pd
import numpy as np
def retDihedImp(df):
odihed = []
if np.sum([df['V' + str(pot)] for pot in range(1, 5)]) != 0.0:
for pot in range(1, 5):
if (df['V' + str(pot)] != 0.0):
odihed.append('%s %4.5f %d %4.5f \n' % (df['NAME'].replace(
"-", " "), df['V' + str(pot)], pot, 180.00 * abs(pot % 2 - 1)))
else:
pot = 2
odihed.append('%s %4.5f %d %4.5f \n' % (df['NAME'].replace(
"-", " "), df['V' + str(pot)], pot, 180.00 * abs(pot % 2 - 1)))
return (odihed)
def retDihed(df):
odihed = []
for pot in range(1, 5):
odihed.append('%s %4.5f %d %4.5f \n' % (df['NAME'].replace(
"-", " "), df['V' + str(pot)], pot, 180.00 * abs(pot % 2 - 1)))
return (odihed)
def Boss2CharmmRTF(num2typ2symb, Qs, resid, bnd_df, imps):
charges = [float(Qs[i][1]) for i in range(len(Qs))]
rtf = open(resid + '.rtf', 'w+')
rtf.write('! LigParGen generated RFT file for NAMD/CHARMM \n')
Mass = ['MASS %d %s %3.4f %s \n' % ((i + 1), num2typ2symb[i][2], bossElement2Mass(
bossPdbAtom2Element(num2typ2symb[i][0])), bossPdbAtom2Element(num2typ2symb[i][0])) for i in range(len(Qs))]
for i in range(len(Mass)):
rtf.write('%s' % Mass[i])
rtf.write('AUTO ANGLES DIHE \n')
rtf.write('RESI %5s %3.3f \n' % (resid, sum(charges)))
for i in range(len(Qs)):
rtf.write('ATOM %s %s %s \n' % (num2typ2symb[i][0], bossPdbAtom2Element(
num2typ2symb[i][0]) + num2typ2symb[i][1][-3:], Qs[i][1]))
for (x, y) in zip(bnd_df.cl1, bnd_df.cl2):
rtf.write('BOND %s %s \n' % (num2typ2symb[x][0], num2typ2symb[y][0]))
for i in imps:
rtf.write('IMPR %s \n' % (i.replace("-", " ")))
rtf.write('PATCH FIRST NONE LAST NONE \n')
rtf.write('END \n')
rtf.close()
return None
def Boss2CharmmPRM(resid, num2typ2symb, Qs, bnd_df, ang_df, tor_df):
#### COLLECTING NONBONDING PART #######
prm = open(resid + '.prm', 'w+')
prm.write('! LigParGen generated PRM file for NAMD/CHARMM \n')
# bnd_df = bnd_df.drop_duplicates(['TIJ'])
prm.write('\nBOND \n')
for i in bnd_df.index:
prm.write('%s %s %4.3f %4.3f \n' % (num2typ2symb[bnd_df.cl1[i]][
2], num2typ2symb[bnd_df.cl2[i]][2], bnd_df.KIJ[i], bnd_df.RIJ[i]))
# ang_df = ang_df.drop_duplicates(['TY', 'R', 'K'])
prm.write('\nANGLE \n')
for i in ang_df.index:
prm.write('%s %s %s %4.3f %4.3f \n' % (num2typ2symb[ang_df.cl1[i]][2], num2typ2symb[
ang_df.cl2[i]][2], num2typ2symb[ang_df.cl3[i]][2], ang_df.K[i], ang_df.R[i]))
prm.write('\nDIHEDRAL \n')
if len(tor_df.index) > 0:
tor_df = tor_df.drop_duplicates(['NAME', 'TY'])
pro_df = tor_df[tor_df.TY == 'Proper']
for i in list(pro_df.index):
ndf = pro_df.ix[i]
pro_out = retDihed(ndf.to_dict())
for i in range(4):
prm.write('%s' % pro_out[i])
prm.write(
'X X X X 0.00000 1 0.000000 ! WILD CARD FOR MISSING TORSION PARAMETERS\n')
prm.write('\nIMPROPER \n')
imp_df = tor_df[tor_df.TY == 'Improper']
for i in list(imp_df.index):
ndf = tor_df.ix[i]
imp_out = retDihedImp(ndf.to_dict())
for i in range(len(imp_out)):
prm.write('%s' % imp_out[i])
prm.write(
'X X X X 0.00000 1 0.000000 ! WILD CARD FOR MISSING IMPROPER PARAMETERS \n')
prm.write(
'\nNONBONDED nbxmod 5 atom cdiel switch vatom vdistance vswitch - \ncutnb 14.0 ctofnb 12.0 ctonnb 11.5 eps 1.0 e14fac 0.5 geom\n')
Qlines = ['%s 0.00 %3.6f %3.6f 0.00 %3.6f %3.6f \n' %
(num2typ2symb[i][2], float(Qs[i][3]) * -1.00, float(Qs[i][2]) * 0.561231, float(Qs[i][3]) * -0.50,
float(Qs[i][2]) * 0.561231) for i in range(len(Qs))]
# uniqQs = list(set(Qlines))
for i in range(len(Qlines)):
prm.write('%s' % Qlines[i])
prm.close()
return None
def Boss2CharmmTorsion(bnd_df, num2opls, st_no, molecule_data, num2typ2symb):
# print num2opls
dhd = []
for line in molecule_data.MolData['TORSIONS']:
dt = [float(l) for l in line]
dhd.append(dt)
dhd = np.array(dhd)
dhd = dhd # kcal to kj conversion
dhd = dhd / 2.0 # Komm = Vopls/2
dhd_df = pd.DataFrame(dhd, columns=['V1', 'V2', 'V3', 'V4'])
ats = []
for line in molecule_data.MolData['ATOMS'][3:]:
dt = [line.split()[0], line.split()[4],
line.split()[6], line.split()[8]]
dt = [int(d) for d in dt]
ats.append(dt)
for line in molecule_data.MolData['ADD_DIHED']:
dt = [int(l) for l in line]
ats.append(dt)
assert len(ats) == len(
dhd), 'Number of Dihedral angles in Zmatrix and Out file dont match'
ats = np.array(ats) - st_no
for i in range(len(ats)):
for j in range(len(ats[0])):
if ats[i][j] < 0:
ats[i][j] = 0
at_df = pd.DataFrame(ats, columns=['I', 'J', 'K', 'L'])
final_df = pd.concat([dhd_df, at_df], axis=1, join_axes=[at_df.index])
bndlist = list(bnd_df.UR) + (list(bnd_df.UR))
final_df['TY'] = ['Proper' if ucomb(list([final_df.I[n], final_df.J[n], final_df.K[
n], final_df.L[n]]), bndlist) == 3 else 'Improper' for n in range(len(final_df.I))]
# final_df['SumV'] = np.abs(
# final_df.V1) + np.abs(final_df.V2) + np.abs(final_df.V3) + np.abs(final_df.V4)
# final_df = final_df[final_df['SumV'] != 0.00]
final_df['TI'] = [num2typ2symb[j][2] for j in final_df.I]
final_df['TJ'] = [num2typ2symb[j][2] for j in final_df.J]
final_df['TK'] = [num2typ2symb[j][2] for j in final_df.K]
final_df['TL'] = [num2typ2symb[j][2] for j in final_df.L]
final_df['SYMB'] = ['-'.join([num2typ2symb[final_df.I[i]][0], num2typ2symb[final_df.J[i]][
0], num2typ2symb[final_df.K[i]][0], num2typ2symb[final_df.L[i]][0]]) for i in final_df.index]
if len(final_df.index) > 0:
final_df['NAME'] = final_df.TI + '-' + final_df.TJ + \
'-' + final_df.TK + '-' + final_df.TL
return final_df
def boss2CharmmBond(molecule_data, st_no):
bdat = molecule_data.MolData['BONDS']
bdat['cl1'] = [x - st_no if not x - st_no < 0 else 0 for x in bdat['cl1']]
bdat['cl2'] = [x - st_no if not x - st_no < 0 else 0 for x in bdat['cl2']]
bnd_df = pd.DataFrame(bdat)
bnd_df['UF'] = ((bnd_df.cl1 + bnd_df.cl2) *
(bnd_df.cl1 + bnd_df.cl2 + 1) * 0.5) + bnd_df.cl2
bnd_df['UR'] = ((bnd_df.cl1 + bnd_df.cl2) *
(bnd_df.cl1 + bnd_df.cl2 + 1) * 0.5) + bnd_df.cl1
# bnd_df.to_csv('bos_bonds.csv', index=False)
hb_df = bnd_df.drop(['cl1', 'cl2', 'UF', 'UR'], 1)
hb_df = hb_df.drop_duplicates()
return bnd_df
def boss2CharmmAngle(anglefile, num2opls, st_no):
adat = anglefile
adat['cl1'] = [x - st_no if not x - st_no < 0 else 0 for x in adat['cl1']]
adat['cl2'] = [x - st_no if not x - st_no < 0 else 0 for x in adat['cl2']]
adat['cl3'] = [x - st_no if not x - st_no < 0 else 0 for x in adat['cl3']]
ang_df = pd.DataFrame(adat)
ang_df = ang_df[ang_df.K > 0]
# ang_df.to_csv('bos_angles.csv', index=False)
ang_df['TY'] = np.array([num2opls[i] + '-' + num2opls[j] + '-' + num2opls[k]
for i, j, k in zip(ang_df.cl1, ang_df.cl2, ang_df.cl3)])
return ang_df
def bossData(molecule_data):
ats_file = molecule_data.MolData['ATOMS']
types = []
for i in enumerate(ats_file):
types.append([i[1].split()[1], 'opls_' + i[1].split()[2]])
st_no = 3
Qs = molecule_data.MolData['Q_LJ']
assert len(Qs) == len(types), 'Please check the at_info and Q_LJ_dat files'
num2opls = {}
for i in range(0, len(types)):
num2opls[i] = Qs[i][0]
num2typ2symb = {i: types[i] for i in range(len(Qs))}
for i in range(len(Qs)):
num2typ2symb[i].append(bossPdbAtom2Element(
num2typ2symb[i][0]) + num2typ2symb[i][1][-3:])
num2typ2symb[i].append(bossPdbAtom2Element(num2typ2symb[i][0]))
num2typ2symb[i].append(bossElement2Mass(num2typ2symb[i][3]))
num2typ2symb[i].append(Qs[i][0])
return (types, Qs, num2opls, st_no, num2typ2symb)
def Boss2Charmm(resid, molecule_data):
types, Qs, num2opls, st_no, num2typ2symb = bossData(molecule_data)
bnd_df = boss2CharmmBond(molecule_data, st_no)
ang_df = boss2CharmmAngle(molecule_data.MolData['ANGLES'], num2opls, st_no)
tor_df = Boss2CharmmTorsion(bnd_df, num2opls, st_no,
molecule_data, num2typ2symb)
Boss2CharmmRTF(num2typ2symb, Qs, resid, bnd_df, list(
tor_df[tor_df.TY == 'Improper']['SYMB']))
Boss2CharmmPRM(resid, num2typ2symb, Qs, bnd_df, ang_df, tor_df)
return None
def mainBOSS2CHARMM(resid, clu=False):
mol = pickle.load(open(resid + ".p", "rb"))
Boss2Charmm(resid, mol)
return None | |
"""
@Time : 2021/9/21 2:08
@File : bciciv2.py
@Software: PyCharm
@Desc :
"""
import os
import warnings
from typing import List
import numpy as np
import scipy.io as sio
import torch
import torch.nn as nn
from tqdm.std import tqdm
from torch.utils.data import Dataset
from .utils import minmax_scale, standard_scale
class BCICIV2aDataset(Dataset):
def __init__(self, data_path: str, seq_len: int, subject_list: List = None, modal: str = 'eeg',
return_idx: bool = False,
transform: nn.Module = None, verbose: bool = True, standardize: str = 'none'):
assert isinstance(subject_list, list)
self.data_path = data_path
self.transform = transform
self.subject_list = subject_list
self.modal = modal
self.return_idx = return_idx
event_types = [769, 770, 771, 772, 783]
assert modal in ['eeg', 'pps', 'all']
self.data = []
self.labels = []
for i, patient in enumerate(tqdm(subject_list, desc='::: LOADING BCICIV2 DATA ::::')):
print(f'[INFO] Processing patient {patient}...')
data = np.load(os.path.join(data_path, patient))
raw = data['s']
types = data['etyp']
pos = data['epos'].flatten()
durations = data['edur'].flatten()
event_idx = np.array([i for i in range(len(types)) if types[i] in event_types])
event_pos = pos[event_idx]
event_durations = durations[event_idx]
assert (event_durations == event_durations[0]).all()
raw = raw.T
raw_segments = [raw[:, p: p + event_durations[idx]] for idx, p in enumerate(event_pos)]
raw_segments = np.stack(raw_segments, axis=0)
if modal == 'eeg':
recordings = raw_segments[:, :22]
elif modal == 'pps':
recordings = raw_segments[:, 22:]
elif modal == 'all':
recordings = raw_segments
else:
raise ValueError
# TODO: add standardization
label = sio.loadmat(os.path.join(data_path, '2a_label', patient.split('.')[0] + '.mat'))
annotations = label['classlabel'].flatten() - 1
recordings = recordings[:(recordings.shape[0] // seq_len) * seq_len].reshape(-1, seq_len,
*recordings.shape[1:])
annotations = annotations[:(annotations.shape[0] // seq_len) * seq_len].reshape(-1, seq_len)
assert recordings.shape[:2] == annotations.shape[:2], f'{patient}: {recordings.shape} - {annotations.shape}'
self.data.append(recordings)
self.labels.append(annotations)
self.data = np.concatenate(self.data, axis=0)
self.labels = np.concatenate(self.labels)
self.idx = np.arange(self.data.shape[0] * self.data.shape[1]).reshape(-1, self.data.shape[1])
self.full_shape = self.data[0].shape
def __getitem__(self, item):
x = self.data[item]
y = self.labels[item]
x = x.astype(np.float32)
y = y.astype(np.long)
if self.transform is not None:
x = self.transform(x)
x = torch.from_numpy(x)
y = torch.from_numpy(y)
if self.return_idx:
return x, y, torch.from_numpy(self.idx[item].astype(np.long))
else:
return x, y
def __len__(self):
return len(self.data) | |
""" Utilities for aiding in testing.
Not tests of utilities... That could be confusing."""
import os.path
import numpy as np
import pandas as pd
from chardet.universaldetector import UniversalDetector
import pysd
def runner(model_file):
directory = os.path.dirname(model_file)
# load model
if model_file.endswith('.mdl'):
model = pysd.read_vensim(model_file)
elif model_file.endswith(".xmile"):
model = pysd.read_xmile(model_file)
else:
raise AttributeError('Modelfile should be *.mdl or *.xmile')
# load canonical output
try:
encoding = detect_encoding(directory + '/output.csv')
canon = pd.read_csv(directory + '/output.csv', encoding=encoding, index_col='Time')
except IOError:
try:
encoding = detect_encoding(directory + '/output.tab')
canon = pd.read_table(directory + '/output.tab', encoding=encoding, index_col='Time')
except IOError:
raise IOError('Canonical output file not found')
# run model
output = model.run(return_columns=canon.columns)
return output, canon
def assert_frames_close(actual, expected, **kwargs):
"""
Compare DataFrame items by column and
raise AssertionError if any column is not equal.
Ordering of columns is unimportant, items are compared only by label.
NaN and infinite values are supported.
Parameters
----------
actual: pandas.DataFrame
expected: pandas.DataFrame
kwargs:
Examples
--------
>>> assert_frames_close(pd.DataFrame(100, index=range(5), columns=range(3)),
... pd.DataFrame(100, index=range(5), columns=range(3)))
>>> assert_frames_close(pd.DataFrame(100, index=range(5), columns=range(3)),
... pd.DataFrame(110, index=range(5), columns=range(3)),
... rtol=.2)
>>> assert_frames_close(pd.DataFrame(100, index=range(5), columns=range(3)),
... pd.DataFrame(150, index=range(5), columns=range(3)),
... rtol=.2) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
AssertionError:
...
References
----------
Derived from: http://nbviewer.jupyter.org/gist/jiffyclub/ac2e7506428d5e1d587b
"""
assert (isinstance(actual, pd.DataFrame) and
isinstance(expected, pd.DataFrame)), \
'Inputs must both be pandas DataFrames.'
assert set(expected.columns) == set(actual.columns), \
'test set columns must be equal to those in actual/observed set.'
assert np.all(np.equal(expected.index.values, actual.index.values)), \
'test set and actual set must share a common index' \
'instead found' + expected.index.values + 'vs' + actual.index.values
for col in expected.columns:
try:
assert_allclose(expected[col].values,
actual[col].values,
**kwargs)
except AssertionError as e:
assertion_details = 'Expected values: ' + np.array2string(expected[col].values, precision=2, separator=', ') + \
'\nActual values: ' + np.array2string(actual[col].values, precision=2, separator=',', suppress_small=True)
raise AssertionError('Column: ' + str(col) + ' is not close.\n' + assertion_details)
def assert_allclose(x, y, rtol=1.e-5, atol=1.e-5):
assert np.all(np.less_equal(abs(x - y), atol + rtol * abs(y)))
def detect_encoding(file):
detector = UniversalDetector()
for line in open(file, 'rb').readlines():
detector.feed(line)
if detector.done: break
detector.close()
return detector.result['encoding'] | |
import pandas as pd
import numpy as np
from typing import Dict, List
from collections import Counter
import collections
import csv
import itertools
import os.path
import random
# for reproducable results, remove when re-running for different results
# random.seed(42)
np.set_printoptions(suppress=True)
class Suitor(object):
'''
Class for the suitor.
'''
def __init__(self, id, prefList, capacity):
self.prefList = prefList
self.capacity = capacity # capacity of the Suitor, a constraint
self.held = set()
self.proposals = 0
self.id = id
def preference(self):
# number of proposals is also the index of the next available option as it increments
# returns the next most prefered suited (size 1)
return self.prefList[self.proposals]
def __repr__(self):
return repr(self.id)
class Suited(object):
'''
Class for the suited.
'''
def __init__(self, id, prefList, capacity):
self.prefList = prefList
self.capacity = capacity # the capacity of the Suited, a constraint
self.held = set()
self.id = id
def reject(self):
'''
Trim the self.held set down to its capacity, returning the list of rejected suitors
'''
if len(self.held) < self.capacity:
# do not reject if held < capacity
return set()
else:
# sort the list of held suitors, by suited's preference
# this is to keep the suited's best preferred suitors (at each iteration)
sortedSuitors = sorted(
list(self.held), key=lambda suitor: self.prefList.index(suitor.id))
# held is the number of suitors up to the suited's capacity
self.held = set(sortedSuitors[:self.capacity])
# reject all of the other suitors not held and return the rejected suitors
return set(sortedSuitors[self.capacity:])
def __repr__(self):
return repr(self.id)
def stableMarriage(suitors: List, suiteds: List):
'''
Implement the stableMarriage algorithm, a.k.a. Deferred Acceptance
A stable marriage is a polygamous (many to many) marriage between suitors and suiteds.
One suited can match to many suitors, and vice versa.
Return the matches from the suiteds' perspective.
e.g. Each SS room: 14 unique suitors, 50 unique suiteds
suitors: List of Suitors
suiteds: List of Suiteds
'''
unassigned = suitors
# randomizes to ensure fairness of order
random.shuffle(suitors)
counter = 0
round_num = 1
for suitor in suitors:
if len(suitor.prefList) == 0:
# print("\n\nSuitor", suitor.id, "has submitted no rankings and is removed from the matching process.")
unassigned.remove(suitor)
for suited in suiteds:
if len(suited.prefList) == 0:
# print("\n\nSuited", suited.id, "has submitted no rankings and is removed from the matching process.")
suiteds.remove(suited)
# all_removed = []
# while loop until there are no more unassigned suitors
while unassigned:
print(f'Unassigned remaining: {len(unassigned)}')
print(f'Round: {round_num}')
round_num += 1
for suitor in unassigned:
counter += 1
# print("\n\n-------------Matching number", counter, "for suitor", suitor.id,"-------------\n\n")
# print(f'This suitor {suitor} has proposed {suitor.proposals} times so far, and the suitor wants to be next matched with suited {suitor.preference()}')
# Before adding to held:
# 1) suitor.held < suitor.capacity
# 2) suitor.proposals < len(prefList) to (strictly less to prevent index errors)
if (len(suitor.held) < suitor.capacity) and (suitor.proposals < len(suitor.prefList)):
if suitor.id in suiteds[suitor.preference()].prefList:
suiteds[suitor.preference()].held.add(suitor)
suitor.held.add(suiteds[suitor.preference()])
# print("\nSuitor", suitor.id, "is paired with", suitor.preference())
# else:
# print("\nSuitor", suitor.id,"is NOT in the preference list of suited", suitor.preference(),"and is queued to be matched with another suited")
# else:
# the suitor cannot be paired with anymore suited
# print(f"This suitor {suitor} has hit its capacity or has proposed to all of its preferred suiteds.")
for suitor in unassigned:
# incrementing all unassigned suitors' number of proposals
# they will try their next preferred
suitor.proposals += 1
for suited in suiteds:
# reject existing suitor(s) if a more preferred suitor comes along and fills the capacity
removed_suitors = suited.reject()
for removed_suitor in removed_suitors:
# remove suited from suitors held since they were rejected,
# allowing them to be potentially paired with another suited
# print(f'suited removed: {suited}')
removed_suitor.held.remove(suited)
# remove from unassigned if:
# 1) suitor has proposed to all of its preferred suited or
# 2) if suitor is at capacity
removed = [suitor for suitor in unassigned if (suitor.proposals >= len(
suitor.prefList) or (len(suitor.held) >= suitor.capacity))]
# update unassigned to smaller set
unassigned = set(unassigned)-set(removed)
# if len(removed) > 0:
# print("\nThe following suitors are removed in this round", removed, f'Count: {len(removed)}')
# all_removed.extend(removed)
# print("\nList of suitors available to be matched after this round is", unassigned, f'Count: {len(unassigned)}')
# randomize unassigned at each iteration
unassigned = list(unassigned)
random.shuffle(unassigned)
# ordered from least to biggest # of suitor id; only for readability
return dict([(suited, list(sorted(suited.held, key=lambda x: x.id))) for suited in suiteds])
def setup(suitorFile: str, suitedFile: str):
'''
Load files and return a ([Suitors], [Suiteds])
Order matters. Be careful of the file order you input!
suitorFile: csv of suitor file, with capacity and ranking
suitedFile: csv of suited file, with capacity and ranking
'''
# note to self, potentially refactor with column headings in csv,
# so we can just pull the column names directly
with open(suitorFile) as csvfile:
reader = csv.reader(csvfile, delimiter=',')
suitors = list(reader)
for i in range(len(suitors)):
while '' in suitors[i]:
suitors[i].remove('')
for i in range(len(suitors)):
suitors[i] = list(map(lambda x: int(float(x)), suitors[i]))
with open(suitedFile) as csvfile:
reader = csv.reader(csvfile, delimiter=',')
suiteds = list(reader)
for i in range(len(suiteds)):
while '' in suiteds[i]:
suiteds[i].remove('')
for i in range(len(suiteds)):
suiteds[i] = [int(float(suited)) for suited in suiteds[i]]
list_of_suitors = []
list_of_suiteds = []
# for each suitor, append the suitor's ID, preference and capacity
for i in range(len(suitors)):
list_of_suitors.append(
Suitor(suitors[i][0], suitors[i][2:], suitors[i][1]))
# capacity is position 1
print("Suitor", suitors[i][0], "meets", suitors[i][1], "suiteds")
# For each suited, append the suited's ID, preference and capacity
for j in range(len(suiteds)):
list_of_suiteds.append(
Suited(suiteds[j][0], suiteds[j][2:], suiteds[j][1]))
# capacity is position 1
print("Suited", suiteds[j][0], "accepts", suiteds[j][1], "suitors")
return list_of_suitors, list_of_suiteds
def rank(suitor, list_of_suited, mode, suitors, suiteds):
'''
Utility function for report function.
Returns a list of ranks for where_the_suitor_ranked_its_matched_suited
or where_the_matched_suited_ranked_this_suitor for suitor_rank and suited_rank
respectively.
'''
rank_list = []
for suited in list_of_suited:
if mode == 'suitor_rank':
# rank of where the suitor ranked its matched suited
rank_list.append(suitors[suitor].prefList.index(suited)+1)
elif mode == 'suited_rank':
# rank of where the suited ranked its matched suitor
rank_list.append(suiteds[suited].prefList.index(suitor)+1)
else:
raise ValueError(
'Invalid mode. Please choose "suitor_rank" or "suited_rank"')
return rank_list
def report(suitors, suiteds, matchFile):
'''
Create a report of matches for each suited with explanation of results.
'''
matches = pd.read_csv(matchFile)
for suited in range(len(suiteds)):
print("Generating Report for Suited", suited)
suited_preferences = suiteds[suited].prefList
result = pd.DataFrame(
np.empty((len(suited_preferences), 5)), dtype=int)
# suited_matched_to
# can be multiple suiteds in the many to many case
# make it a list to show commas, np array uses spaces by default
suited_matched_to = pd.Series(suited_preferences).apply(
lambda suitor: repr(np.where(matches.values == suitor)[1]))
# drop suited with no matches
suited_matched_to_ = suited_matched_to[suited_matched_to.str.len() > 0]
# where_the_suitor_ranked_you
where_the_suitor_ranked_you = pd.Series(suited_preferences).apply(
lambda suitor: (suitors[suitor].prefList.index(suited))+1)
# where_the_suitor_ranked_its_matched_suited
df = pd.concat([pd.Series(suited_preferences), suited_matched_to_],
axis=1).dropna().rename(columns={0: 'suitor', 1: 'suited'})
# this is a list of rankings instead of a single ranking in the many to many case
where_the_suitor_ranked_its_matched_suited = df.apply(lambda my_df: rank(
my_df.suitor, my_df.suited, 'suitor_rank', suitors, suiteds), axis=1)
# where_the_matched_suited_ranked_this_suitor
# this is a list of rankings instead of a single ranking in the many to many case
where_the_matched_suited_ranked_this_suitor = df.apply(lambda my_df: rank(
my_df.suitor, my_df.suited, 'suited_rank', suitors, suiteds), axis=1)
result = pd.DataFrame({'Suitor rankings by you': suited_preferences,
'Suited matched to': suited_matched_to_,
'Where the suitor ranked you': where_the_suitor_ranked_you,
'Where the suitor ranked its matched suited': where_the_suitor_ranked_its_matched_suited,
'Where the matched suited ranked this suitor': where_the_matched_suited_ranked_this_suitor})
result['Where the suitor ranked you'].fillna(
"Suitor did not rank you", inplace=True)
result.fillna(
"Suited rejected suitor - not in top pref list", inplace=True)
result.to_csv(
f'./ss_match_sim/report_for_suited_{suited}.csv')
def pipeline(suitorFile, suitedFile, run_report=True, matchFile=None):
'''
Run the pipeline to create a Matching and
return the matching csv and associated reports
'''
suitors, suiteds = setup(suitorFile, suitedFile)
if run_report:
# report in the perspective for the suited
report(suitors, suiteds, matchFile)
else:
matches = stableMarriage(suitors.copy(), suiteds.copy())
# use Series since different row lengths (each suited takes in a variable number of suitors) as if blank, default is NaN
df = pd.DataFrame({k: pd.Series([x.id for x in v])
for k, v in matches.items()})
df.columns = [f'Suited {suited}' for suited in suiteds]
print("\n-------------------------Matching is Complete-------------------------\n")
# most suited have 5 suitors
print(pd.DataFrame({k: [v] for k, v in Counter([len(v) for k, v in matches.items()]).items()}).T.reset_index(
).sort_values(by='index').rename(columns={'index': 'matched_meetings_for_suiteds', 0: 'count'}))
print('Verifying stability...')
print(verifyStable(suitors, suiteds, matches))
return df
def verifyStable(suitors, suiteds, marriage):
'''
verifyStable: [Suitor], [Suited], {Suited -> [Suitor]} -> bool
Check that the assignment of suitors to suited is a stable marriage.
'''
def precedes(L, item1, item2): return L.index(item1) < L.index(item2)
# find the partner(s) of suitor
def partner(suitor): return filter(
lambda s: suitor in marriage[s], suiteds)
def suitorPrefers(suitor, suited):
'''
Return True if the suitor prefers the suited over its existing matches.
Otherwise, returns False.
'''
return any(map(lambda x: precedes(suitor.prefList, suited.id, x.id), list(partner(suitor))))
def suitedPrefers(suited, suitor):
'''
Return True if the suited prefers the suitor over its existing matches
'''
# map this function to each of the suited's prefList
# check if there is any suitor that the suited prefers over than its existing paired suitor
# if there is any True in the iterator return True
# e.g. Suited 0 is married to [25, 33, 104, 237, 281]
# (Suitor = 93, Suited = 0)
# check if Suitor 93 is preferred over Suitor 25 (True!)
# check if Suitor 93 is preferred over Suitor 3 (False) ...
# -> [True, False, False, False, False]
# > True
return any(map(lambda x: precedes(suited.prefList, suitor.id, x.id), marriage[suited]))
# iterate over the cartesian products of each suitor, suited set
counter = 0
for (suitor, suited) in itertools.product(suitors, suiteds):
counter += 1
if counter % (len(suiteds)*len(suitors)/10) == 0:
print(f'{counter/(len(suiteds)*len(suitors))*100}% checks complete')
# stability condition
if (suitor not in marriage[suited]) and suitorPrefers(suitor, suited) and suitedPrefers(suited, suitor):
print(
f'Suitor {suitor} and Suited {suited} is not a stable pairing')
print(f'suitorPrefers: {suitorPrefers(suitor, suited)}')
print(f'suitedPrefers: {suitedPrefers(suited, suitor)}')
return False
return True
if __name__ == "__main__":
ventureFile = './ss_match_sim/venture_ranking_sim_room_1.csv'
mentorFile = './ss_match_sim/mentor_ranking_sim_room_1.csv'
# matchFile = './ss_match_sim/ss_matches_room_0.csv'
suitedFile = ventureFile
suitorFile = mentorFile
# run pipeline
result_df = pipeline(suitorFile, suitedFile,
run_report=False, matchFile=None)
result_df.to_csv("./ss_match_sim/ss_matches_room_0.csv", index=False)
# run report
# pipeline(suitorFile, suitedFile, run_report=True, matchFile=matchFile)
# friends toy example
# A = './matching/DA1920/ss_match_sim/friends_toy_ex_suited_1_to_1.csv'
# B = './matching/DA1920/ss_match_sim/friends_toy_ex_suitors_1_to_1.csv'
# suitedFile = A
# suitorFile = B
# result_df = pipeline(suitorFile, suitedFile, run_report=False)
# result_df.to_csv("./matching/DA1920/ss_match_sim/test_marriage4.csv", index=False)
# code to check average ranking of mentor13
# np.mean(np.where(pd.read_csv(ventureFile, header=None).iloc[:,2:]==13)[1]+1) | |
import tensorflow as tf
import logging
import numpy as np
from . import feature_generation
from . import feature_utilities
import six
def input_fn(
metadata,
filenames,
num_features,
max_time_delta,
window_size,
min_timeslice_size,
objectives,
parallelism=4,
num_slices_per_id=4):
random_state = np.random.RandomState()
def xform(id_, movement_features):
def _xform(id_, features):
id_ = metadata.id_map_int2bytes[id_]
return feature_utilities.extract_n_random_fixed_times(
random_state, features, num_slices_per_id, max_time_delta,
window_size, id_, min_timeslice_size)
features, timestamps, time_ranges, id_ = tf.py_func(
_xform,
[id_, movement_features],
[tf.float32, tf.int32, tf.int32, tf.string])
return (features, timestamps, time_ranges, id_)
def add_labels(features, timestamps, time_bounds, id_):
def _add_labels(id_, timestamps):
labels = [o.create_label(id_, timestamps) for o in objectives]
return labels
labels = tf.py_func(
_add_labels,
[id_, timestamps],
[tf.float32] * len(objectives))
return ((features, timestamps, time_bounds, id_), tuple(labels))
def set_shapes(all_features, labels):
feature_generation.set_feature_shapes(all_features, num_features, window_size)
for i, obj in enumerate(objectives):
t = labels[i]
t.set_shape(obj.output_shape)
return all_features, labels
def lbls_as_dict(features, labels):
d = {obj.name : labels[i] for (i, obj) in enumerate(objectives)}
return features, d
def features_as_dict(features, labels):
features, timestamps, time_bounds, id_ = features
d = {'features' : features, 'timestamps' : timestamps, 'time_ranges' : time_bounds, 'id' : id_}
return d, labels
raw_data = feature_generation.read_input_fn_infinite(
filenames,
num_features,
num_parallel_reads=parallelism,
random_state=random_state)
return (raw_data
.map(xform, num_parallel_calls=parallelism)
.flat_map(feature_generation.flatten_features)
.map(add_labels, num_parallel_calls=parallelism)
.map(set_shapes, num_parallel_calls=parallelism)
.map(lbls_as_dict, num_parallel_calls=parallelism)
.map(features_as_dict, num_parallel_calls=parallelism)
)
def predict_input_fn(paths,
num_features,
time_ranges,
window_size,
min_timeslice_size,
parallelism=4):
random_state = np.random.RandomState()
def xform(id_, movement_features):
def _xform(id_, features):
return feature_utilities.np_array_extract_slices_for_time_ranges(
random_state, features, id_, time_ranges,
window_size, min_timeslice_size)
raw_features = tf.cast(movement_features, tf.float32)
features, timestamps, time_ranges_tensor, id_ = tf.py_func(
_xform,
[id_, raw_features],
[tf.float32, tf.int32, tf.int32, tf.string])
return (features, timestamps, time_ranges_tensor, id_)
def set_shapes(features, timestamps, time_bounds, id_):
all_features = features, timestamps, time_bounds, id_
feature_generation.set_feature_shapes(all_features, num_features, window_size)
return all_features
def features_as_dict(features, timestamps, time_bounds, id_):
d = {'features' : features, 'timestamps' : timestamps, 'time_ranges' : time_bounds, 'id' : id_}
return d
raw_data = feature_generation.read_input_fn_one_shot(paths, num_features, num_parallel_reads=parallelism)
return (raw_data
.map(xform, num_parallel_calls=parallelism)
.flat_map(feature_generation.flatten_features)
.map(set_shapes)
.map(features_as_dict)
) | |
# This file implements a flask server for inference. You can modify the file to align with your own inference logic.
from __future__ import print_function
import io
import json
import os
import pickle
import signal
import sys
import traceback
import flask
from flask import request
import tensorflow_hub as hub
import boto3
import base64
import tensorflow as tf
import tensorflow_hub as hub
from PIL import Image
import numpy as np
from io import BytesIO
import json
import pandas as pd
prefix = "/opt/ml"
model_path = os.path.join(prefix, "model")
def load_img(image_object):
"""
Load image from Amazon S3 bucket and processed it.
"""
max_dim = 512
img = tf.keras.preprocessing.image.img_to_array(image_object)
img = tf.convert_to_tensor(img, dtype=tf.float32)
shape = tf.cast(tf.shape(img)[:-1], tf.float32)
long_dim = max(shape)
scale = max_dim / long_dim
new_shape = tf.cast(shape * scale, tf.int32)
img = tf.image.resize(img, new_shape)
img = img[tf.newaxis, :] / 255
return img
def read_image_from_s3(bucket_name, key):
"""S3 to PIL Image"""
s3 = boto3.resource('s3')
bucket = s3.Bucket(bucket_name)
object = bucket.Object(key)
response = object.get()
return Image.open(response['Body'])
def tensor_to_image(tensor):
"""
Transform tensor to image.
"""
tensor = tensor * 255
tensor = np.array(tensor, dtype=np.uint8)
if np.ndim(tensor) > 3:
assert tensor.shape[0] == 1
tensor = tensor[0]
return Image.fromarray(tensor)
class TensorflowService(object):
model = None # Where we keep the model when it's loaded
@classmethod
def get_model(cls):
"""Get the model object for this instance, loading it if it's not already loaded."""
if cls.model == None:
# Load ML Model from TensorflowHub
cls.model = hub.load(model_path)
print("Tensorflow hub model loaded!")
return cls.model
@classmethod
def predict(cls, content_image, style_image):
"""For the input, do the predictions and return them.
Args:
input (a pandas dataframe): The data on which to do the predictions. There will be
one prediction per row in the dataframe"""
clf = cls.get_model()
return clf(tf.constant(content_image), tf.constant(style_image))[0]
# The flask app for serving predictions
app = flask.Flask(__name__)
@app.route("/ping", methods=["GET"])
def ping():
"""Determine if the container is working and healthy.
In this sample container, we declare
it healthy if we can load the model successfully."""
health = TensorflowService.get_model() is not None # You can insert a health check here
status = 200 if health else 404
return flask.Response(response="\n", status=status, mimetype="application/json")
@app.route("/invocations", methods=["POST"])
def inference():
"""Performed an inference on incoming data.
In this sample server, we take data as application/json,
print it out to confirm that the server received it.
"""
content_type = flask.request.content_type
if flask.request.content_type != "application/json":
msg = "I just take json, and I am fed with {}".format(content_type)
else:
msg = "I am fed with json. Therefore, I am happy"
data = flask.request.data.decode("utf-8")
data = io.StringIO(data)
data = json.loads(data.read())
account_id = boto3.client("sts").get_caller_identity()["Account"]
region = boto3.Session().region_name
bucket_name = f"photo-to-sketch-{account_id}"
dict_style = {"1":"style/1.jpeg","2":"style/2.jpeg","3":"style/3.jpeg","4":"style/4.jpeg"}
effect_type = dict_style[data["effectType"]]
#Style image
style_image_object = read_image_from_s3(bucket_name, effect_type)
style_image = load_img(style_image_object)
print("Style image loaded!")
# Content image
input_image = data['image']
content_image_object = Image.open(BytesIO(base64.b64decode(input_image)))
content_image = load_img(content_image_object)
print("Content image loaded!")
stylized_image = TensorflowService.predict(content_image,style_image)
stylized_image = tensor_to_image(stylized_image)
#Encode the response to base64
buffered = BytesIO()
stylized_image.save(buffered, format="JPEG")
img_str = base64.b64encode(buffered.getvalue())
print("Stylized image generated!")
return flask.Response(
response=json.dumps({"image": img_str.decode("utf-8")}),
status=200,
mimetype="text/plain",
) | |
#!/usr/bin/env python
# coding: utf-8
# # Lexical normalization pipeline
#
# author - AR Dirkson
# date - 15-7-2019
#
# Python 3 script
#
# This pipeline takes raw text data and performs:
# - Removes URLs, email addresses
# - Tokenization with NLTK
# - Removes non_English posts (conservatively) using langid module with top 10 languages and threshold of 100
# - British English to American English
# - Normalization of contractions
# - Normalization of generic abbreviations and slang
# - Normalization of domain-specific (patient forum) abbreviations
# - Spelling correction
# In[1]:
import pickle
import numpy as np
import random
import pandas as pd
from collections import Counter, defaultdict, OrderedDict
from nltk import pos_tag, word_tokenize
import re
import seaborn as sns
import matplotlib.pyplot as plt
import editdistance
import kenlm
from sklearn.metrics import recall_score, precision_score, f1_score, fbeta_score
from nltk.tokenize.treebank import TreebankWordDetokenizer
from gensim.models import KeyedVectors
import langid
# In[2]:
class Normalizer ():
def __init__(self):
pass
#to use this function the files need to be sorted in the same folder as the script under /obj_lex/
def load_obj(self, name):
with open('obj_lex\\' + name + '.pkl', 'rb') as f:
return pickle.load(f, encoding='latin1')
def load_files(self):
self.abbr_dict = self.load_obj ('abbreviations_dict')
self.aspell_dict = self.load_obj ('aspell_dict_lower')
self.short_expanse_dict = self.load_obj ('short_expansions_dict')
self.cList = self.load_obj ('contractionslistone')
self.cList2 = self.load_obj ('contractionslisttwo')
self.drugnames = self.load_obj ('fdadrugslist')
def change_tup_to_list(self, tup):
thelist = list(tup)
return thelist
def change_list_to_tup(self,thelist):
tup = tuple(thelist)
return tup
#---------Remove URls, email addresses and personal pronouns ------------------
def replace_urls(self,list_of_msgs):
list_of_msgs2 = []
for msg in list_of_msgs:
nw_msg = re.sub(
r'\b' + r'((\(<{0,1}https|\(<{0,1}http|\[<{0,1}https|\[<{0,1}http|<{0,1}https|<{0,1}http)(:|;| |: )\/\/|www.)[\w\.\/#\?\=\+\;\,\&\%_\n-]+(\.[a-z]{2,4}\]{0,1}\){0,1}|\.html\]{0,1}\){0,1}|\/[\w\.\?\=#\+\;\,\&\%_-]+|[\w\/\.\?\=#\+\;\,\&\%_-]+|[0-9]+#m[0-9]+)+(\n|\b|\s|\/|\]|\)|>)',
'-URL-', msg)
list_of_msgs2.append(nw_msg)
return list_of_msgs2
def replace_email(self,list_of_msgs):
list_of_msgs2 = []
for msg in list_of_msgs:
nw_msg = re.sub (r"([a-zA-Z0-9_.+-]+@[a-zA-Z0-9-]+\.[a-zA-Z0-9-.]+[. ])", ' ', msg) #remove email
nw_msg2 = re.sub (r"(@[a-zA-Z0-9]+[. ])", ' ', nw_msg) #remove usernames
# nw_msg3 = re.sub(r"(@ [a-zA-Z0-9]+[. ])", ' ', nw_msg2) #remove usernames
list_of_msgs2.append(nw_msg2)
return list_of_msgs2
def remove_empty (self,list_of_msgs):
empty = []
check_msgs3 =[]
for a, i in enumerate (list_of_msgs):
if len(i) == 0:
print('empty')
else:
check_msgs3.append(i)
return check_msgs3
def remove_registered_icon (self, msg):
nw_msg = re.sub ('\u00AE', '', msg)
nw_msg2 = re.sub ('\u00E9', 'e', nw_msg)
return nw_msg2
def escape_char (self, msg):
msg1 = msg.replace('\x08', '')
msg2 = msg1.replace ('\x8d', '')
msg3 = msg2.replace('ðŸ', '')
return msg3
def anonymize (self, posts):
posts2 = self.replace_urls (posts)
posts3 = self.replace_email (posts2)
posts4 = self.remove_empty(posts3)
posts5 = [self.remove_registered_icon(p) for p in posts4]
posts6 = [self.escape_char(p) for p in posts5]
return posts6
#---------Convert to lowercase ----------------------------------------------------
def lowercase (self, post):
post1 = []
for word in post:
word1 = word.lower()
post1.append (word1)
return post1
#---------Remove non_English posts -------------------------------------------------
def language_identify_basic (self, posts):
nw = []
tally = 0
list_removed = []
for post in posts:
out = langid.classify (post)
out2 = list(out)
if out2[0]=='en':
nw.append(post)
else:
tally += 1
list_removed.append(tuple ([post, out2[0], out2[1]]))
return nw, tally, list_removed
def language_identify_thres (self, msgs, lang_list, thres):
nw = []
tally = 0
list_removed = []
for post in msgs:
langid.set_languages(lang_list)
out = langid.classify (post)
out2 = list(out)
if out2[0]=='en':
nw.append(post)
elif out2[1] > thres:
nw.append(post)
else:
tally += 1
list_removed.append(tuple ([post, out2[0], out2[1]]))
return nw, tally, list_removed
def remove_non_english(self, posts):
# d = TreebankWordDetokenizer()
# posts2 = [d.detokenize(m) for m in posts]
posts_temp, tally, list_removed = self.language_identify_basic(posts)
lang = []
for itm in list_removed:
lang.append(itm[1])
c = Counter(lang)
lang_list = ['en']
for itm in c.most_common(10):
z = list(itm)
lang_list.append(z[0])
print("Most common 10 languages in the data are:" + str(lang_list))
posts3, tally_nw, list_removed_nw = self.language_identify_thres(posts, lang_list, thres = -100)
return posts3
## --- Contraction expansions ------------------------------##
def prepareContractions(self):
self.c_re = re.compile('(%s)' % '|'.join(self.cList.keys()))
self.c_re2 = re.compile('(%s)' % '|'.join(self.cList2.keys()))
def remove_apos (self, sent):
sent2 = re.sub ("'",'', sent)
return sent2
# except TypeError:
# pass
def expandContractions (self, text):
def replace(match):
return self.cList[match.group(0)]
return self.c_re.sub(replace, text)
#needs to happen after tokenization
def expandContractions_second (self, text):
text2 = []
for w in text:
if w.lower() in self.cList2:
v = word_tokenize(self.cList2[w.lower()])
for i in v:
text2.append(i)
else:
text2.append(w)
return text2
###--- 1-2 letter expansions -------------------------------##
def load_ngrammodel(self):
path = 'obj_lex\\tetragram_model.binary'
self.model = kenlm.Model(path)
def get_parameters_ngram_model (self, word, sent):
i = sent.index(word)
if ((i-2) >= 0) and (len(sent)>(i+2)):
out = sent[(i-2):(i+3)]
bos = False
eos = False
elif ((i-2) < 0) and (len(sent)> (i+2)) : #problem with beginning
bos = True
eos = False
out = sent[0:(i+3)]
elif ((i-2) >= 0) and (len(sent) <= (i+2)): #problem with end
bos = False
eos = True
out = sent[(i-2):]
else: #problem with both
out = sent
eos = True
bos = True
d = TreebankWordDetokenizer()
out2 = d.detokenize(out)
return bos, eos, out2
def get_prob(self, word, token, out, bos, eos): #token is candidate
out_nw = out.replace(word, token)
p = self.model.score(out_nw, bos = bos, eos = eos)
return p
def short_abbr_expansion(self, sent):
sent2 = []
for word in sent:
if len(word) > 2:
sent2.append(word)
else:
if word in self.short_expanse_dict .keys():
cand = self.short_expanse_dict [word]
final_p = -100
bos, eos, out = self.get_parameters_ngram_model(word,sent)
for i in cand:
p = self.get_prob(word, i, out, bos, eos)
if p > final_p:
final_p = p
correct = i
sent2.append(correct)
else:
sent2.append(word)
return sent2
#---------Lexical normalization pipeline (Sarker, 2017) -------------------------------
def loadItems(self):
'''
This is the primary load function.. calls other loader functions as required..
'''
global english_to_american
global noslang_dict
global IGNORE_LIST_TRAIN
global IGNORE_LIST
english_to_american = {}
lexnorm_oovs = []
IGNORE_LIST_TRAIN = []
IGNORE_LIST = []
english_to_american = self.loadEnglishToAmericanDict()
noslang_dict = self.loadDictionaryData()
for key, value in noslang_dict.items ():
value2 = value.lower ()
value3 = word_tokenize (value2)
noslang_dict[key] = value3
return None
def loadEnglishToAmericanDict(self):
etoa = {}
english = open('obj_lex/englishspellings.txt')
american = open('obj_lex/americanspellings.txt')
for line in english:
etoa[line.strip()] = american.readline().strip()
return etoa
def loadDictionaryData(self):
'''
this function loads the various dictionaries which can be used for mapping from oov to iv
'''
n_dict = {}
infile = open('obj_lex/noslang_mod.txt')
for line in infile:
items = line.split(' - ')
if len(items[0]) > 0 and len(items) > 1:
n_dict[items[0].strip()] = items[1].strip()
return n_dict
def preprocessText(self, tokens, IGNORE_LIST, ignore_username=False, ignore_hashtag=False, ignore_repeated_chars=True, eng_to_am=True, ignore_urls=False):
'''
Note the reason it ignores hashtags, @ etc. is because there is a preprocessing technique that is
designed to remove them
'''
normalized_tokens =[]
#print tokens
text_string = ''
# NOTE: if nesting if/else statements, be careful about execution sequence...
# tokens2 = [t[0] for t in tokens]
for t in tokens:
t_lower = t.strip().lower()
# if the token is not in the IGNORE_LIST, do various transformations (e.g., ignore usernames and hashtags, english to american conversion
# and others..
if t_lower not in IGNORE_LIST:
# ignore usernames '@'
if re.match('@', t) and ignore_username:
IGNORE_LIST.append(t_lower)
text_string += t_lower + ' '
#ignore hashtags
elif re.match('#', t_lower) and ignore_hashtag:
IGNORE_LIST.append(t_lower)
text_string += t_lower + ' '
#convert english spelling to american spelling
elif t.strip().lower() in english_to_american.keys() and eng_to_am:
text_string += english_to_american[t.strip().lower()] + ' '
#URLS
elif re.search('http[s]?://(?:[a-zA-Z]|[0-9]|[$-_@.&+]|[!*\(\),]|(?:%[0-9a-fA-F][0-9a-fA-F]))+', t_lower) and ignore_urls:
IGNORE_LIST.append(t_lower)
text_string += t_lower + ' '
elif not ignore_repeated_chars and not re.search(r'[^a-zA-Z]', t_lower):
# if t_lower only contains alphabetic characters
t_lower = re.sub(r'([a-z])\1+', r'\1\1', t_lower)
text_string += t_lower + ' '
# print t_lower
# if none of the conditions match, just add the token without any changes..
else:
text_string += t + ' '
else: # i.e., if the token is in the ignorelist..
text_string += t_lower + ' '
normalized_tokens = text_string.split()
return normalized_tokens, IGNORE_LIST
def dictionaryBasedNormalization(self, tokens, I_LIST, M_LIST):
tokens2 =[]
for t in (tokens):
t_lower = t.strip().lower()
if t_lower in noslang_dict.keys() and len(t_lower)>2:
nt = noslang_dict[t_lower]
[tokens2.append(m) for m in nt]
if not t_lower in M_LIST:
M_LIST.append(t_lower)
if not nt in M_LIST:
M_LIST.append(nt)
else:
tokens2.append (t)
return tokens2, I_LIST, M_LIST
#----Using the Sarker normalization functions ----------------------------
#Step 1 is the English normalization and step 2 is the abbreviation normalization
def normalize_step1(self, tokens, oovoutfile=None):
global IGNORE_LIST
global il
MOD_LIST = []
# Step 1: preprocess the text
normalized_tokens, il = self.preprocessText(tokens, IGNORE_LIST)
return normalized_tokens
def normalize_step2(self, normalized_tokens, oovoutfile=None):
global IGNORE_LIST
global il
MOD_LIST = []
ml = MOD_LIST
normalized_tokens, il, ml = self.dictionaryBasedNormalization(normalized_tokens, il, ml)
return normalized_tokens
def sarker_normalize (self,list_of_msgs):
self.loadItems()
msgs_normalized = [self.normalize_step1(m) for m in list_of_msgs]
msgs_normalized2 = [self.normalize_step2(m) for m in msgs_normalized]
return msgs_normalized2
#-------Domain specific abreviation expansion ----------------------------
# The list of abbreviations is input as a dictionary with tokenized output
def domain_specific_abbr (self, tokens, abbr):
post2 = []
for t in tokens:
if t.lower() in abbr.keys():
nt = abbr[t.lower()]
[post2.append(m) for m in nt]
else:
post2.append(t)
return post2
def expand_abbr (self, data, abbr):
data2 = []
for post in data:
post2 = self.domain_specific_abbr (tokens = post, abbr= abbr)
data2.append(post2)
return data2
#-------Spelling correction -------------------------------------------------
def flev_rel (self, cand, token):
abs_edit_dist = editdistance.eval(cand, token)
rel_edit_dist = abs_edit_dist / len(token)
return rel_edit_dist
def modelsim (self,cand,token, model):
try:
similarity = model.similarity(cand, token)
except KeyError:
similarity = 0
return similarity
def run_low_emb (self, word, voc, model, w1 =0.4, w2= 0.6):
replacement = [' ',100]
for token in voc:
sim1 = self.flev_rel(word, token) #lower is better
sim2 = self.modelsim (word, token, model)
sim = w1 * sim1 + w2 * (1-sim2)
if sim < replacement[1]:
replacement[1] = sim
replacement[0] = token
return replacement
def wrong_concatenation(self, token, token_freq):
best_plausibility = 0
best_split = 0
t = token_freq[token]
limit = 9*t
NUMBER = re.compile('[0-9]+') # Only letters and dashes
if '-' in token:
return token
else:
for i in range(3, len(token)):
left, right = token[:i], token[i:]
if len(right) < 3:
continue
elif NUMBER.fullmatch(left) and right in token_freq:
best_split= (left,right)
elif NUMBER.fullmatch(right) and left in token_freq:
best_split = (left, right)
else:
if left not in token_freq or right not in token_freq:
continue
if token_freq[left] < limit or token_freq[right] < limit:
# print('too low')
continue
plausibility = min(token_freq[left], token_freq[right])
if plausibility > best_plausibility:
best_plausibility = plausibility
best_split = (left, right)
if best_split != 0:
return list(best_split)
else:
return token
def spelling_correction (self, post, min_rel_freq = 9, max_flev_rel = 0.76):
post2 = []
cnt = 0
tagged_post = pos_tag(post)
tags = [t[1] for t in tagged_post]
for a, token in enumerate (post):
token2 = token.lower()
if (tags[a] == 'NNP') or (tags[a] == 'NNPS'):
post2.append(token)
else:
if self.TRUE_WORD.fullmatch(token2) and (token2 != '-url-') and (token2 != '-') and (token2 != '--'):
# if token2 in self.spelling_corrections:
# correct = self.spelling_corrections[token2]
# if len(correct) >1:
# [post2.append(i) for i in correct]
# else:
# post2.append(correct)
# cnt +=1
# self.replaced.append(token2)
# self.replaced_with.append(correct)
if token2 in self.aspell_dict:
post2.append(token)
elif token2 in self.drugnames:
post2.append(token)
else:
freq_word = self.token_freq[token2]
limit = freq_word * min_rel_freq
subset = [t[0] for t in self.token_freq_ordered2 if t[1]>= limit]
candidate = self.run_low_emb(token2, subset, self.model2)
if candidate[1] > max_flev_rel:
x = self.wrong_concatenation(token2, self.token_freq)
if x != token2:
[post2.append(i) for i in x]
cnt +=1
self.replaced.append(token2)
self.replaced_with.append( " ".join(x))
self.spelling_corrections [token2] = x
else:
post2.append(token)
else:
post2.append(candidate[0])
cnt +=1
self.replaced.append(token2)
self.replaced_with.append(candidate[0])
self.spelling_corrections [token2] = candidate[0]
else:
post2.append(token)
self.total_cnt.append (cnt)
return post2
def initialize_files_for_spelling(self):
total_cnt = []
replaced = []
replaced_with = []
spelling_corrections= {}
return total_cnt, replaced, replaced_with, spelling_corrections
def change_tup_to_list (self, tup):
thelist = list(tup)
return thelist
def load_model (self):
filename = 'obj_lex//Health_2.5mreviews.s200.w10.n5.v15.cbow.bin'
self.model2 = KeyedVectors.load_word2vec_format(filename, binary=True)
def create_token_freq (self, data):
flat_data = [item for sublist in data for item in sublist]
flat_data2 = [i.lower() for i in flat_data]
flat_data3 = []
for token2 in flat_data2:
if self.TRUE_WORD.fullmatch(token2) and (token2 != '-url-') and (token2 != '-') and (token2 != '--'):
flat_data3.append(token2)
self.token_freq = Counter(flat_data3)
token_freq_ordered = self.token_freq.most_common ()
self.token_freq_ordered2 = [self.change_tup_to_list(m) for m in token_freq_ordered]
def correct_spelling_mistakes(self, data, different_token_freq = False):
self.load_model()
self.load_files ()
self.total_cnt, self.replaced, self.replaced_with, self.spelling_corrections = self.initialize_files_for_spelling()
self.TRUE_WORD = re.compile('[-a-z]+') # Only letters and dashes
if different_token_freq == False:
self.create_token_freq(data)
else:
self.token_freq = self.load_obj('token_freq')
token_freq_ordered = self.token_freq.most_common ()
self.token_freq_ordered2 = [self.change_tup_to_list(m) for m in token_freq_ordered]
out = []
for num, m in enumerate(data):
if num%1000 == 0:
print(num)
out.append(self.spelling_correction (m))
return out, self.total_cnt, self.replaced, self.replaced_with, self.spelling_corrections
#--------Overall normalization function--------------------------------------
def normalize (self, posts, anonymize = True, remove_foreign = False):
self.load_files ()
posts0 = [str(m) for m in posts]
if anonymize == True:
posts1 = self.anonymize(posts0)
print(posts1[0])
else:
posts1 = posts0
if remove_foreign == True:
posts1b = self.remove_non_english(posts1)
else:
posts1b = posts1
posts2 = [i.replace ('’', "'") for i in posts1b]
self.prepareContractions()
posts3 = [self.expandContractions(m) for m in posts2]
posts4 = [self.remove_apos(m) for m in posts3]
posts5 = [word_tokenize(m) for m in posts4]
print('done with tokenizing')
print(posts5[0])
posts6 = [self.expandContractions_second(m) for m in posts5]
print(posts6[0])
self.load_ngrammodel()
posts8 = [self.sarker_normalize(posts6)]
posts8b = posts8[0]
posts9 = [self.short_abbr_expansion(m) for m in posts8b]
posts10 = [self.expand_abbr(posts9, self.abbr_dict)]
print(posts10[0][0])
return posts10[0]
# In[4]:
#example of usage
test = ['my bff is 4ever', 'the colour of the moon is grey']
#normalize but not correct spelling mistakes
test2 = Normalizer().normalize(test)
print(test2)
#correct spelling mistakes - input must be tokenized
test3, total_cnt, replaced, replaced_with, spelling_corrections = Normalizer().correct_spelling_mistakes(test2)
print(test3)
# In[ ]:
print('The total number of spelling mistakes found is: ' + str(sum(total_cnt)))
# In[ ]: | |
import csv
import numpy as np
import pickle
import random
import tensorflow as tf
import os
import sys
from data_generator import DataGenerator
from sl import SupervisedLearning
from tensorflow.python.platform import flags
from constants import *
from tensorflow.python.client import timeline
from data_processing import load_data, build_dictionary, loadEmbedding_rand, sentences_to_padded_index_sequences, load_dictionary, get_rev_dict
from utils import get_exp_string, restore_most
from tqdm import tqdm
from config import params
from train_test_helper import test_on_dataset, train_hex, test_hex, biased_test, train_supervised, test_supervised
def main():
train_datasets = params["train_datasets"].split(',')
test_datasets = params["test_datasets"].split(',')
train_paths = [DATASET_PATHS[d] for d in train_datasets]
test_paths = [DATASET_PATHS[d] for d in test_datasets]
tokenized_train_datasets = [load_data(p) for p in train_paths]
tokenized_test_datasets = [load_data(p) for p in test_paths]
if params["load_dict"] is False:
word_indices = build_dictionary(tokenized_train_datasets, params["dict_path"], params)
else:
word_indices = load_dictionary(params["dict_path"], params)
rev_dict = get_rev_dict
pre_trained_emb = loadEmbedding_rand(params["pretrain_embedding_path"], word_indices)
params["embeddings"] =pre_trained_emb
sentences_to_padded_index_sequences(word_indices, tokenized_train_datasets, params)
sentences_to_padded_index_sequences(word_indices, tokenized_test_datasets, params)
data_generator = DataGenerator(params["batch_size"], task="nli", config=params)
data_generator.set_datasets(tokenized_train_datasets, tokenized_test_datasets, train_datasets, test_datasets)
if params["task"] == 'nli':
prem_ph, hypo_ph, y_nli_ph = data_generator.get_place_holders(params["seq_length"])
input_tensors = {'input':(prem_ph, hypo_ph), 'label':y_nli_ph}
else:
input_tensors = None
sess = tf.InteractiveSession()
model = SupervisedLearning(params)
with sess.as_default():
model.construct_model(params, input_tensors=input_tensors, prefix = "mtl_")
saver = loader = tf.train.Saver(tf.get_collection(tf.GraphKeys.TRAINABLE_VARIABLES), max_to_keep=5)
exp_string = get_exp_string(params)
tf.global_variables_initializer().run()
#load model
resume_itr = 0
model_file = None
if params["resume"] or not params["train"]:
assert(params['part_load_path']=='')
model_file = tf.train.latest_checkpoint(params["logdir"] + '/' + exp_string)
if params["test_iter"] > 0:
model_file = model_file[:model_file.rindex('model')] + 'model' + str(params["test_iter"])
elif params["test_iter"] == -4:
# select the most recent checkpoint
model_file = model_file
elif params["test_iter"] == -2:
model_file = model_file[:model_file.rindex('model')] + 'model' + "_BEST"
elif params["test_iter"] == -3:
model_file = params["logdir"] + '/' + 'pretrain_model_BEST'
if model_file:
ind1 = model_file.rindex('model')
resume_itr = model_file[ind1+5:]
if resume_itr != '_BEST':
resume_itr = int(resume_itr)
else:
resume_itr = 0
print("Restoring model weights from " + model_file)
saver.restore(sess, model_file)
if params['part_load_path']!='':
restore_most(params['part_load_path'], sess)
if params["train"]:
if params['neg_reg'] == 'hex':
train_hex(params, model, saver, sess, exp_string, data_generator, resume_itr)
else:
train_supervised(params,model, saver, sess, exp_string, data_generator, resume_itr)
else:
if params['bias_test'] == True:
biased_test(params, model, saver, sess, exp_string, data_generator, resume_itr, params['major_class'])
else:
if params['neg_reg'] == 'hex':
test_hex(params, model, saver, sess, exp_string, data_generator, resume_itr)
else:
test_supervised(params,model, saver, sess, exp_string, data_generator, resume_itr)
if __name__ == "__main__":
main() | |
import warnings
from collections import Counter
from typing import Any, Dict, List, Optional, Tuple, Union
import numpy as np
import torch
import torch.nn as nn
import torchvision
from sentence_transformers import SentenceTransformer
from sklearn.feature_extraction.text import CountVectorizer, TfidfVectorizer
from torchvision import transforms
from torchvision.datasets.folder import pil_loader
from tqdm.auto import tqdm
from transformers import AutoTokenizer, AutoModel
from .basedataset import BaseDataset
def check_weak_labels(dataset: Union[BaseDataset, np.ndarray]) -> np.ndarray:
if isinstance(dataset, BaseDataset):
assert dataset.weak_labels is not None, f'Input dataset has no weak labels!'
L = np.array(dataset.weak_labels)
else:
L = dataset
return L
def split_labeled_unlabeled(dataset: BaseDataset, cut_tied: Optional[bool] = False) -> Tuple[BaseDataset, BaseDataset]:
weak_labels = np.array(dataset.weak_labels)
labeled_idx = []
for i, l in enumerate(weak_labels):
counter = Counter()
weaklabels_cnt = 0
for l_ in l:
if l_ >= 0:
counter[l_] += 1
weaklabels_cnt += 1
lst = counter.most_common() # .values()
if np.max(l) > -1:
if not cut_tied or (len(lst) > 1 and lst[0][-1] - lst[1][-1] > 0): # or weaklabels_cnt<=1:
labeled_idx.append(i) # = []
if len(labeled_idx) == len(weak_labels):
warnings.warn('No unlabeled data found! Use full dataset us unlabeled dataset')
return dataset, dataset
else:
return dataset.create_split(labeled_idx)
def split_conf_unconf_by_percentile(dataset: BaseDataset, y: Optional[np.ndarray] = None, percentile: float = 0.2,
return_y=True, return_thres=False):
assert percentile > 0 and percentile < 1, f'percentile should be in (0, 1), now it\'s {percentile}!'
assert len(y) == len(dataset)
path = dataset.path
split = dataset.split
conf_dataset = type(dataset)(path=path)
conf_dataset.split = split
unconf_dataset = type(dataset)(path=path)
unconf_dataset.split = split
ids = np.array(dataset.ids)
labels = np.array(dataset.labels)
examples = np.array(dataset.examples)
weak_labels = np.array(dataset.weak_labels)
features = dataset.features
max_probs = np.max(y, axis=1)
n = int(len(max_probs) * percentile)
sort_idx = np.argsort(-max_probs)
thres = max_probs[sort_idx[n]]
split_point = n
while split_point < len(max_probs):
if max_probs[sort_idx[split_point]] < thres:
break
else:
split_point += 1
conf_idx = sort_idx[:split_point]
unconf_idx = sort_idx[split_point:]
unconf_dataset.ids = ids[unconf_idx]
unconf_dataset.labels = labels[unconf_idx]
unconf_dataset.weak_labels = weak_labels[unconf_idx]
unconf_dataset.examples = examples[unconf_idx]
if features is not None:
unconf_dataset.features = features[unconf_idx]
conf_dataset.ids = ids[conf_idx]
conf_dataset.labels = labels[conf_idx]
conf_dataset.weak_labels = weak_labels[conf_idx]
conf_dataset.examples = examples[conf_idx]
if features is not None:
conf_dataset.features = features[conf_idx]
if return_thres and return_y:
return (conf_dataset, y[conf_idx]), (unconf_dataset, y[unconf_idx]), (max_probs.max(), max_probs.min(), thres)
elif return_thres:
return conf_dataset, unconf_dataset, (max_probs.max(), max_probs.min(), thres)
elif return_y:
return (conf_dataset, y[conf_idx]), (unconf_dataset, y[unconf_idx])
else:
return conf_dataset, unconf_dataset
def split_conf_unconf(dataset: BaseDataset, y: Optional[np.ndarray] = None, mode: Optional[str] = 'thres',
theta: float = 0.2, return_y=True, return_thres=False):
assert theta > 0 and theta < 1, f'theta should be in (0, 1), now it\'s {theta}!'
assert len(y) == len(dataset)
path = dataset.path
split = dataset.split
conf_dataset = type(dataset)(path=path)
conf_dataset.split = split
unconf_dataset = type(dataset)(path=path)
unconf_dataset.split = split
ids = np.array(dataset.ids)
labels = np.array(dataset.labels)
examples = np.array(dataset.examples)
weak_labels = np.array(dataset.weak_labels)
features = dataset.features
max_probs = np.max(y, axis=1)
if mode == 'thres':
thres = theta
conf_idx = np.where(max_probs >= thres)[0]
unconf_idx = np.where(max_probs < thres)[0]
elif mode == 'percentile':
n = int(len(max_probs) * theta)
split_point = n
sort_idx = np.argsort(-max_probs)
while split_point < len(max_probs):
thres = max_probs[sort_idx[split_point]]
if max_probs[sort_idx[split_point - 1]] == thres:
split_point += 1
else:
break
conf_idx = sort_idx[:split_point]
unconf_idx = sort_idx[split_point:]
else:
raise NotImplementedError
unconf_dataset.ids = ids[unconf_idx]
unconf_dataset.labels = labels[unconf_idx]
unconf_dataset.weak_labels = weak_labels[unconf_idx]
unconf_dataset.examples = examples[unconf_idx]
if features is not None:
unconf_dataset.features = features[unconf_idx]
conf_dataset.ids = ids[conf_idx]
conf_dataset.labels = labels[conf_idx]
conf_dataset.weak_labels = weak_labels[conf_idx]
conf_dataset.examples = examples[conf_idx]
if features is not None:
conf_dataset.features = features[conf_idx]
if return_thres and return_y:
return (conf_dataset, y[conf_idx]), (unconf_dataset, y[unconf_idx]), (max_probs.max(), max_probs.min(), thres)
elif return_thres:
return conf_dataset, unconf_dataset, (max_probs.max(), max_probs.min(), thres)
elif return_y:
return (conf_dataset, y[conf_idx]), (unconf_dataset, y[unconf_idx])
else:
return conf_dataset, unconf_dataset
#### feature extraction for TextDataset
def bag_of_words_extractor(data: List[Dict], **kwargs: Any):
corpus = list(map(lambda x: x['text'], data))
count_vect = CountVectorizer(**kwargs)
corpus_counts = count_vect.fit_transform(corpus).toarray().astype('float32')
def extractor(data: List[Dict]):
corpus = list(map(lambda x: x['text'], data))
return count_vect.transform(corpus).toarray().astype('float32')
return corpus_counts, extractor
def tf_idf_extractor(data: List[Dict], **kwargs: Any):
corpus = list(map(lambda x: x['text'], data))
tfidf_transformer = TfidfVectorizer(**kwargs)
tfidf = tfidf_transformer.fit_transform(corpus).toarray().astype('float32')
def extractor(data: List[Dict]):
corpus = list(map(lambda x: x['text'], data))
return tfidf_transformer.transform(corpus).toarray().astype('float32')
return tfidf, extractor
def sentence_transformer_extractor(data: List[Dict], model_name: Optional[str] = 'paraphrase-distilroberta-base-v1', **kwargs: Any):
corpus = list(map(lambda x: x['text'], data))
model = SentenceTransformer(model_name, **kwargs)
embeddings = model.encode(corpus)
return embeddings, model.encode
def bert_text_extractor(data: List[Dict], device: torch.device = None, model_name: Optional[str] = 'bert-base-cased',
feature: Optional[str] = 'cls', **kwargs: Any):
"""
Extract text features (semantic representation) using pretrain models either by retriving embedding of [CLS] token
or by averaging all tokens of given sentences.
:param data: Data in json format.
:param device: Torch device to be used.
:param model_name: transformer (Huggingface) model name to be used.
:param feature: Two options are: "cls" for [CLS], "avr" for average of all tokens.
:param kwargs: misc arguments for the pretrained model.
:return: text feature as np array of size (corpus_size, output_dim)
"""
# assert feature == 'cls' or feature == 'avr', "Please choose from cls and avr as text feature."
@torch.no_grad()
def extractor(data: List[Dict]):
corpus = list(map(lambda x: x['text'], data))
model = AutoModel.from_pretrained(model_name, **kwargs).to(device)
tokenizer = AutoTokenizer.from_pretrained(model_name) # e.g. 'bert-base-cased'
text_features = []
for sentence in tqdm(corpus):
inputs = tokenizer(sentence, return_tensors="pt", truncation=True, return_attention_mask=False, return_token_type_ids=False)
inputs = inputs['input_ids'].to(device)
if feature == 'cls':
output = model(inputs).pooler_output.cpu().squeeze().numpy() # [len(sentence), dim_out]
text_features.append(output)
elif feature == 'avr':
output = model(inputs).last_hidden_state.cpu().squeeze().numpy() # [len(sentence), dim_out]
text_features.append(np.average(output, axis=0))
else:
raise NotImplementedError
return np.array(text_features)
embeddings = extractor(data)
return embeddings, extractor
#### feature extraction for RelationDataset
def bert_relation_extractor(data: List[Dict], device: torch.device = None,
model_name: Optional[str] = 'bert-base-cased',
feature: Optional[str] = 'cat', **kwargs: Any):
"""
R-BERT feature extractor for relation extraction (github.com/monologg/R-BERT).
:param data: Data in json format.
:param device: Torch device to be used.
:param model_name: transformer (Huggingface) model name to be used.
:param feature: "cat" for concatenated [cls;ent1;ent2]. "avr" for averaged cls, ent1, ent2.
:param kwargs: Misc arguments for the pretrained model.
:return: Text feature as np array of size (corpus_size, output_dim * 3 = [cls;ent1;ent2]) for "cat",
or (corpus_size, output_dim) for "avr",
"""
# assert feature == 'cat' or feature == 'avr', "Please choose from cat and avr as text feature."
@torch.no_grad()
def extractor(data: List[Dict]):
assert (('span1' in data[0]) and ('span2' in data[0])), "Passed data missing span index."
assert (('entity1' in data[0]) and ('entity2' in data[0])), "Passed data missing entity name."
corpus = list(map(lambda x: x['text'], data))
span1_list, span2_list = list(map(lambda x: x['span1'], data)), list(map(lambda x: x['span2'], data)) # char level
ent1_list, ent2_list = list(map(lambda x: x['entity1'], data)), list(map(lambda x: x['entity2'], data))
model = AutoModel.from_pretrained(model_name, **kwargs).to(device)
tokenizer = AutoTokenizer.from_pretrained(model_name) # e.g. 'bert-base-cased'
text_features = []
for i, sentence in tqdm(enumerate(corpus)):
span1s, span1n = span1_list[i] # ent 1 start, ent 1 end
span2s, span2n = span2_list[i] # ent 2 start, ent 2 end
# Assumption - e1s < e2s, e1 and e2 no overlap.
e1_tkns = tokenizer.tokenize(ent1_list[i])
e2_tkns = tokenizer.tokenize(ent2_list[i])
e1_first = span1s < span2s
if e1_first:
left_text = sentence[:span1s]
between_text = sentence[span1n:span2s]
right_text = sentence[span2n:]
else:
left_text = sentence[:span2s]
between_text = sentence[span2n:span1s]
right_text = sentence[span1n:]
left_tkns = tokenizer.tokenize(left_text)
between_tkns = tokenizer.tokenize(between_text)
right_tkns = tokenizer.tokenize(right_text)
if e1_first:
tokens = ["[CLS]"] + left_tkns + ["$"] + e1_tkns + ["$"] + between_tkns + ["#"] + e2_tkns + [
"#"] + right_tkns + ["[SEP]"]
e1s = len(left_tkns) + 1 # inclusive
e1n = e1s + len(e1_tkns) + 2 # exclusive
e2s = e1n + len(between_tkns)
e2n = e2s + len(e2_tkns) + 2
end = e2n
else:
tokens = ["[CLS]"] + left_tkns + ["#"] + e2_tkns + ["#"] + between_tkns + ["$"] + e1_tkns + [
"$"] + right_tkns + ["[SEP]"]
e2s = len(left_tkns) + 1 # inclusive
e2n = e2s + len(e2_tkns) + 2 # exclusive
e1s = e2n + len(between_tkns)
e1n = e1s + len(e1_tkns) + 2
end = e1n
if len(tokens) > 512:
if end >= 512:
len_truncated = len(between_tkns) + len(e1_tkns) + len(e2_tkns) + 6
if len_truncated > 512:
diff = len_truncated - 512
len_between = len(between_tkns)
between_tkns = between_tkns[:(len_between - diff) // 2] + between_tkns[(len_between - diff) // 2 + diff:]
if e1_first:
truncated = ["[CLS]"] + ["$"] + e1_tkns + ["$"] + between_tkns + ["#"] + e2_tkns + ["#"] + [
"[SEP]"]
e1s = 1 # inclusive
e1n = e1s + len(e1_tkns) + 2 # exclusive
e2s = e1n + len(between_tkns)
e2n = e2s + len(e2_tkns) + 2
else:
truncated = ["[CLS]"] + ["#"] + e2_tkns + ["#"] + between_tkns + ["$"] + e1_tkns + ["$"] + [
"[SEP]"]
e2s = 1 # inclusive
e2n = e2s + len(e2_tkns) + 2 # exclusive
e1s = e2n + len(between_tkns)
e1n = e1s + len(e1_tkns) + 2
tokens = truncated
assert len(tokens) <= 512
else:
tokens = tokens[:512]
assert e1_tkns == tokens[e1s + 1:e1n - 1]
assert e2_tkns == tokens[e2s + 1:e2n - 1]
assert len(tokens) <= 512
inputs = torch.tensor(tokenizer.convert_tokens_to_ids(tokens), device=device).unsqueeze(0)
output = model(inputs).last_hidden_state.cpu().squeeze().numpy() # [len(sentence), dim_out]
cls_emb = output[0, :]
ent1_emb, ent2_emb = np.average(output[e1s:e1n, :], axis=0), np.average(output[e2s:e2n, :], axis=0)
if feature == "cat":
text_features.append(np.concatenate([cls_emb, ent1_emb, ent2_emb]))
elif feature == "avr":
text_features.append(np.average(np.concatenate([cls_emb, ent1_emb, ent2_emb]), axis=0))
else:
raise NotImplementedError
return np.array(text_features)
embeddings = extractor(data)
return embeddings, extractor
#### feature extraction for ImageDataset
def image_feature_extractor(data: List[Dict], device: torch.device = None, model_name: Optional[str] = 'resnet18', **kwargs: Any):
data_transform = transforms.Compose([
transforms.Resize(224),
transforms.CenterCrop(224),
transforms.ToTensor(),
transforms.Normalize([0.485, 0.456, 0.406], [0.229, 0.224, 0.225])
])
batch_size = 128
@torch.no_grad()
def extractor(data: List[Dict]):
image_paths = list(map(lambda x: x['image_path'], data))
pretrained_model = getattr(torchvision.models, model_name)(pretrained=True)
model = nn.Sequential(*list(pretrained_model.children())[:-1])
model.to(device)
model.eval()
image_features = []
imgs = []
for ip in tqdm(image_paths):
imgs.append(data_transform(pil_loader(ip)))
if len(imgs) == batch_size:
img_input = torch.stack(imgs).to(device)
output = model(img_input)
image_features.append(output.squeeze().cpu().numpy())
imgs = []
if len(imgs) > 0:
img_input = torch.stack(imgs).to(device)
output = model(img_input)
image_features.append(output.squeeze().cpu().numpy())
return np.vstack(image_features)
embeddings = extractor(data)
return embeddings, extractor
#### Loading Glove Embedding for Sequence Dataset
def get_glove_embedding(embedding_file_path=None, PAD='PAD', UNK='UNK'):
f = open(embedding_file_path, 'r', encoding='utf-8').readlines()
word_dict = {}
embedding = []
for i, line in enumerate(f):
split_line = line.split()
word = split_line[0]
embedding.append(np.array([float(val) for val in split_line[1:]]))
word_dict[word] = i
embedding = np.array(embedding)
word_dict[PAD] = len(word_dict)
word_dict[UNK] = len(word_dict)
dict_len, embed_size = embedding.shape
scale = np.sqrt(3.0 / embed_size)
spec_word = np.random.uniform(-scale, scale, [2, embed_size])
embedding = np.concatenate([embedding, spec_word], axis=0).astype(np.float)
return word_dict, embedding | |
import logging
import os
import numpy as np
import torch
import torchvision
from PIL import Image
from torch.utils.data import SubsetRandomSampler, Subset, Dataset
from torchvision.transforms import transforms
from sklearn.model_selection import StratifiedShuffleSplit
from theconf import Config as C
from archive import autoaug_policy, autoaug_paper_cifar10, fa_reduced_cifar10
from augmentations import *
from common import get_logger
from samplers.stratified_sampler import StratifiedSampler
logger = get_logger('Unsupervised Data Augmentation')
logger.setLevel(logging.INFO)
def train_val_split(labels, n_labeled_per_class, mode = None):
labels = np.array(labels)
train_labeled_idxs = []
train_unlabeled_idxs = []
val_idxs = []
# Set some strategies
print(f'Current sampling strategy is {mode}')
if mode == 'PureRandom':
idxs = np.arange(labels.shape[0])
np.random.shuffle(idxs)
train_labeled_idxs = idxs[:n_labeled_per_class*10]
train_unlabeled_idxs = idxs[n_labeled_per_class*10:-5000]
val_idxs = idxs[-5000:]
else:
for i in range(10):
idxs = np.where(labels == i)[0]
np.random.shuffle(idxs)
train_labeled_idxs.extend(idxs[:n_labeled_per_class])
train_unlabeled_idxs.extend(idxs[n_labeled_per_class:-500])
val_idxs.extend(idxs[-500:])
rec = np.zeros(10)
for i in range(10):
rec[i] = np.sum(labels[train_labeled_idxs] == i)
print(f'We sampled {rec} instances for each class')
np.random.shuffle(train_labeled_idxs)
np.random.shuffle(train_unlabeled_idxs)
np.random.shuffle(val_idxs)
return train_labeled_idxs, train_unlabeled_idxs, val_idxs
def get_dataloaders(dataset, batch, batch_unsup, dataroot, mode = None, n_labeled = 250):
if 'cifar' in dataset:
transform_train = transforms.Compose([
transforms.RandomCrop(32, padding=4),
transforms.RandomHorizontalFlip(),
transforms.ToTensor(),
transforms.Normalize((0.4914, 0.4822, 0.4465), (0.2023, 0.1994, 0.2010)),
])
transform_valid = transforms.Compose([
transforms.RandomCrop(32, padding=4),
transforms.RandomHorizontalFlip(),
transforms.ToTensor(),
transforms.Normalize((0.4914, 0.4822, 0.4465), (0.2023, 0.1994, 0.2010)),
])
transform_test = transforms.Compose([
transforms.ToTensor(),
transforms.Normalize((0.4914, 0.4822, 0.4465), (0.2023, 0.1994, 0.2010)),
])
else:
raise ValueError('dataset=%s' % dataset)
autoaug = transforms.Compose([])
if isinstance(C.get()['aug'], list):
logger.debug('augmentation provided.')
autoaug.transforms.insert(0, Augmentation(C.get()['aug']))
else:
logger.debug('augmentation: %s' % C.get()['aug'])
if C.get()['aug'] == 'fa_reduced_cifar10':
autoaug.transforms.insert(0, Augmentation(fa_reduced_cifar10()))
elif C.get()['aug'] == 'autoaug_cifar10':
autoaug.transforms.insert(0, Augmentation(autoaug_paper_cifar10()))
elif C.get()['aug'] == 'autoaug_extend':
autoaug.transforms.insert(0, Augmentation(autoaug_policy()))
elif C.get()['aug'] == 'default':
pass
else:
raise ValueError('not found augmentations. %s' % C.get()['aug'])
transform_train.transforms.insert(0, autoaug)
if C.get()['cutout'] > 0:
transform_train.transforms.append(CutoutDefault(C.get()['cutout']))
if dataset in ['cifar10', 'cifar100']:
if dataset == 'cifar10':
total_trainset = torchvision.datasets.CIFAR10(root=dataroot, train=True, download=True, transform=transform_train)
unsup_trainset = torchvision.datasets.CIFAR10(root=dataroot, train=True, download=True, transform=None)
testset = torchvision.datasets.CIFAR10(root=dataroot, train=False, download=True, transform=transform_test)
elif dataset == 'cifar100':
total_trainset = torchvision.datasets.CIFAR100(root=dataroot, train=True, download=True, transform=transform_train)
unsup_trainset = torchvision.datasets.CIFAR100(root=dataroot, train=True, download=True, transform=None)
testset = torchvision.datasets.CIFAR100(root=dataroot, train=False, download=True, transform=transform_test)
else:
raise ValueError
# n_labeled = 250
train_idx, valid_idx, _ = train_val_split(total_trainset.targets, int(n_labeled/10), mode = mode) # only for cifar10
# sss = StratifiedShuffleSplit(n_splits=1, test_size=49750, random_state=0) # 250 trainset
# sss = sss.split(list(range(len(total_trainset))), total_trainset.targets)
# train_idx, valid_idx = next(sss)
train_labels = [total_trainset.targets[idx] for idx in train_idx]
trainset = Subset(total_trainset, train_idx) # for supervised
trainset.train_labels = train_labels
otherset = Subset(unsup_trainset, valid_idx) # for unsupervised
# otherset = unsup_trainset
otherset = UnsupervisedDataset(otherset, transform_valid, autoaug, cutout=C.get()['cutout'])
else:
raise ValueError('invalid dataset name=%s' % dataset)
trainloader = torch.utils.data.DataLoader(
trainset, batch_size=batch, shuffle=False, num_workers=8, pin_memory=True,
sampler=StratifiedSampler(trainset.train_labels), drop_last=True)
unsuploader = torch.utils.data.DataLoader(
otherset, batch_size=batch_unsup, shuffle=True, num_workers=8, pin_memory=True,
sampler=None, drop_last=True)
testloader = torch.utils.data.DataLoader(
testset, batch_size=batch, shuffle=False, num_workers=32, pin_memory=True,
drop_last=False
)
return trainloader, unsuploader, testloader
class CutoutDefault(object):
"""
Reference : https://github.com/quark0/darts/blob/master/cnn/utils.py
"""
def __init__(self, length):
self.length = length
def __call__(self, img):
if self.length <= 0:
return img
h, w = img.size(1), img.size(2)
mask = np.ones((h, w), np.float32)
y = np.random.randint(h)
x = np.random.randint(w)
y1 = np.clip(y - self.length // 2, 0, h)
y2 = np.clip(y + self.length // 2, 0, h)
x1 = np.clip(x - self.length // 2, 0, w)
x2 = np.clip(x + self.length // 2, 0, w)
mask[y1: y2, x1: x2] = 0.
mask = torch.from_numpy(mask)
mask = mask.expand_as(img)
img *= mask
return img
class Augmentation(object):
def __init__(self, policies):
self.policies = policies
def __call__(self, img):
for _ in range(1):
policy = random.choice(self.policies)
for name, pr, level in policy:
if random.random() > pr:
continue
img = apply_augment(img, name, level)
return img
class UnsupervisedDataset(Dataset):
def __init__(self, dataset, transform_default, transform_aug, cutout=0):
self.dataset = dataset
self.transform_default = transform_default
self.transform_aug = transform_aug
self.transform_cutout = CutoutDefault(cutout) # issue 4 : https://github.com/ildoonet/unsupervised-data-augmentation/issues/4
def __getitem__(self, index):
img, _ = self.dataset[index]
img1 = self.transform_default(img)
img2 = self.transform_default(self.transform_aug(img))
img2 = self.transform_cutout(img2)
return img1, img2
def __len__(self):
return len(self.dataset) | |
#!/usr/bin/env
"""
@author: briantaylor
@author: giladgoldfarb
@author: jeanluc-auge
Transmission setup example:
reads from network json (default = examples/edfa/edfa_example_network.json)
propagates a 96 channels comb
"""
from gnpy.core.utils import load_json
from convert import convert_file
from gnpy.core.equipment import *
from argparse import ArgumentParser
from sys import exit
from pathlib import Path
from logging import getLogger, basicConfig, INFO, ERROR, DEBUG
from matplotlib.pyplot import show, axis, figure, title
from networkx import (draw_networkx_nodes, draw_networkx_edges,
draw_networkx_labels, dijkstra_path)
from gnpy.core import network_from_json, build_network
from gnpy.core.elements import Transceiver, Fiber, Edfa
from gnpy.core.info import SpectralInformation, Channel, Power
#from gnpy.core.algorithms import closed_paths
logger = getLogger(__package__ or __file__)
eqpt_library = {}
EQPT_LIBRARY_FILENAME = Path(__file__).parent / 'eqpt_config.json'
def format_si(spectral_infos):
return '\n'.join([
f'#{idx} Carrier(frequency={c.frequency},\n power=Power(signal={c.power.signal}, nli={c.power.nli}, ase={c.power.ase}))'
for idx, si in sorted(set(spectral_infos))
for c in set(si.carriers)
])
logger = getLogger('gnpy.core')
def plot_network_graph(network, path, source, sink):
nodelist = [n for n in network.nodes() if isinstance(n, (Transceiver, Fiber))]
pathnodes = [n for n in path if isinstance(n, (Transceiver, Fiber))]
edgelist = [(u, v) for u, v in zip(pathnodes, pathnodes[1:])]
node_color = ['#ff0000' if n is source or n is sink else
'#900000' if n in path else '#ffdfdf'
for n in nodelist]
edge_color = ['#ff9090' if u in path and v in path else '#ababab'
for u, v in edgelist]
labels = {n: n.location.city if isinstance(n, Transceiver) else ''
for n in pathnodes}
fig = figure()
pos = {n: (n.lng, n.lat) for n in nodelist}
kwargs = {'figure': fig, 'pos': pos}
plot = draw_networkx_nodes(network, nodelist=nodelist, node_color=node_color, **kwargs)
draw_networkx_edges(network, edgelist=edgelist, edge_color=edge_color, **kwargs)
draw_networkx_labels(network, labels=labels, font_size=14, **kwargs)
title(f'Propagating from {source.loc.city} to {sink.loc.city}')
axis('off')
show()
def main(args):
input_filename = str(args.filename)
split_filename = input_filename.split(".")
json_filename = split_filename[0]+'.json'
try:
assert split_filename[1] in ('json','xls','csv','xlsm')
except AssertionError as e:
print(f'invalid file extension .{split_filename[1]}')
raise e
if split_filename[1] != 'json':
print(f'parse excel input to {json_filename}')
convert_file(input_filename)
json_data = load_json(json_filename)
read_eqpt_library(EQPT_LIBRARY_FILENAME)
network = network_from_json(json_data)
build_network(network)
spacing = 0.05 #THz
si = SpectralInformation() # !! SI units W, Hz
si = si.update(carriers=tuple(Channel(f, (191.3+spacing*f)*1e12,
32e9, 0.15, Power(1e-3, 0, 0)) for f in range(1,97)))
trx = [n for n in network.nodes() if isinstance(n, Transceiver)]
if args.list>=1:
print(*[el.uid for el in trx], sep='\n')
else:
try:
source = next(el for el in trx if el.uid == args.source)
except StopIteration as e:
source = trx[0]
print(f'invalid souce node specified: {args.source!r}, replaced with {source}')
try:
sink = next(el for el in trx if el.uid == args.sink)
except StopIteration as e:
sink = trx[1]
print(f'invalid destination node specified: {args.sink!r}, replaced with {sink}')
path = dijkstra_path(network, source, sink)
print(f'There are {len(path)} network elements between {source} and {sink}')
for el in path:
si = el(si)
print(el)
#plot_network_graph(network, path, source, sink)
parser = ArgumentParser()
parser.add_argument('filename', nargs='?', type=Path,
default= Path(__file__).parent / 'edfa_example_network.json')
parser.add_argument('source', type=str, nargs='?', default="", help='source node')
parser.add_argument('sink', type=str, nargs='?', default="", help='sink node')
parser.add_argument('-v', '--verbose', action='count')
parser.add_argument('-l', '--list', action='count', default=0, help='list all network nodes')
if __name__ == '__main__':
args = parser.parse_args()
level = {1: INFO, 2: DEBUG}.get(args.verbose, ERROR)
logger.setLevel(level)
basicConfig()
exit(main(args)) | |
2104.03847 | \section{Introduction}
We derive a stable reformulation of the convex semidefinite
programming, \textbf{SDP}, model for the
key rate calculation for quantum key distribution, \textbf{QKD}, problems.
We use this to derive efficient,
accurate, algorithms for the problem, in particular, for finding
provable lower bounds for the problem. The reformulation is based on
a novel facial reduction, \textbf{FR}, approach. We exploit the Kronecker structure
and do \textbf{FR}\, first for the linear constraints to guarantee a
positive definite, strictly feasible solution. Second we exploit the
properties of the completely positive maps and do \textbf{FR}\,
on the nonlinear, quantum relative entropy objective function,
to guarantee positive definiteness of its arguments.
This allows for a Gauss-Newton type interior-point approach that avoids
the need for perturbations to obtain positive definiteness,
a technique currently used in the literature. The result is high
accuracy solutions with provable lower (and upper) bounds for the convex
minimization problem. We note that the convex minimization problem
is designed to provide a \emph{lower bound} for the key rate.
\index{\textbf{QKD}, quantum key distribution}
\textdef{Quantum key distribution, \textbf{QKD}}~\cite{Scarani2009, Xu2020}, is the art of distributing
a secret key between two honest parties, traditionally known as Alice and
Bob. A secret key rate\footnote{the number
of bits of secret key obtained per exchange of quantum signal}
calculation is at the core of a security proof for
any \textbf{QKD}\, protocol. It has been formulated as a convex optimization problem
for both asymptotic~\cite{coles2016numerical,winick2017reliable} and
finite-size regimes~\cite{george2020finite}. We note that finite-size
key rate problems involve a variation of the asymptotic key rate
formulation. In this paper we consider the specific problem setup for the asymptotic key rate calculation~\cite{winick2017reliable}:
\index{$\Hn$, set of $n$-by-$n$ Hermitian matrices}
\index{set of $n$-by-$n$ Hermitian matrices, $\Hn$}
\begin{equation}\label{eq:keyrateopt}
\begin{array}{rcllll}
&& \min_\rho &
D({\mathcal G} (\rho)\|{\mathcal Z} ({\mathcal G} (\rho))) \\
&&\text{s.t.} & \Gamma(\rho) = \gamma,
\\ &&& \rho \succeq 0.
\end{array}
\end{equation}
Here: \index{$m$, number of linear constraints}
the objective function $D(\delta\|\sigma)=$\textdef{$f(\delta,\sigma)=
\tr \left( \delta (\log \delta - \log \sigma)\right)$}
is the quantum relative entropy;
$\Gamma: \mathbb{H}^{n} \rightarrow \mathbb{R}^{m}$ is a linear map
defined by $\Gamma(\rho) = \left(\tr (\Gamma_i \rho)\right)$;
$\Hn$ is the linear space of Hermitian matrices over the reals;
and $\gamma \in {\mathbb{R}^m\,}$. In this problem, $\{\Gamma_i\}$ is a set of
Hermitian matrices corresponding to physical observables. The
data pairs $\Gamma_i, \gamma_i$ are known observation statistics that
include the $\trace (\rho) =1$ constraint. The maps
${\mathcal G} $ and ${\mathcal Z} $ are linear, completely positive maps that are specified
according to the description of a \textbf{QKD}\, protocol. In general, ${\mathcal G} $ is
trace-non-increasing, while ${\mathcal Z} $ is trace-preserving and its Kraus
operators are a resolution of identity. The maps are usually represented via
the so-called operator-sum (or Kraus operator) representation.
(More details on these representation are given below as
needed; see also~\Cref{def:operGandZ}.)
Without loss of generality we can assume that the
feasible set, a spectrahedron, is nonempty. This is because
our problem is related to a physical scenario, and
we can trivially set the
key rate to be zero when the feasible set is empty.
Note that the Hermitian (positive semidefinite, density) matrix
$\rho$ is the only variable in the above~\cref{eq:keyrateopt}
optimization problem. Motivated by the fact that the mappings ${\mathcal G} ,
{\mathcal Z} \circ {\mathcal G} $ are positive semidefinite preserving but possibly
not positive definite preserving, we rewrite
\cref{eq:keyrateopt} as follows:\footnote{This allows us to regularize
below using facial reduction, \textbf{FR}.}
\begin{equation}
\label{eq:keyrateoptequivsigdel} \quad
\begin{array}{rcllll}
&& \min_{\rho,\sigma,\delta} &
\trace(\delta(\log \delta -\log \sigma)) \\
&&\text{s.t.} & \Gamma(\rho) = \gamma
\\ &&& \sigma = {\mathcal Z} (\delta)
\\ &&& \delta = {\mathcal G} (\rho)
\\ &&& \rho, \sigma, \delta \succeq 0.
\end{array}
\end{equation}
The asymptotic key rate is obtained by getting a reliable lower bound of
this problem and then removing the cost of error correction.
The latter is determined experimentally or directly estimated from the
observation statistics.
In principle, any (device-dependent) \textbf{QKD}\, protocol
can be analyzed in this framework given in~\cref{eq:keyrateoptequivsigdel}.
This includes measurement-device-independent, and both
discrete-variable and continuous-variable protocols.
This is typically done
after introducing suitable tools to reduce dimension, e.g.,~the squashing
models~\cite{zhang2021security}, or the dimension reduction
method~\cite{upadhyaya2021dimension}. In reality, the success of
this security proof method is often limited by computational resources,
as well as the efficiency and accuracy of underlying algorithms.
The work in~\cite{winick2017reliable} provides a reliable framework to compute
the key rate using a two-step routine. In the first step, one tries to
efficiently find a near optimal, feasible point, of the optimization
problem~\cref{eq:keyrateopt}. In the second step, one then obtains a
reliable lower bound from this
feasible point by a linearization and duality argument. In terms of numerical
computation, the bottleneck of this approach
for large-size \textbf{QKD}\, problems comes from the
first step, as it involves semidefinite optimization with a nonlinear
objective function. In particular, the work in~\cite{winick2017reliable} proposes an algorithm based on the Frank-Wolfe
method to solve the first step. However hthis can converge slowly in practice.
We note that Faybusovich and Zhou~\cite{Faybusovich2020longstep}
also attempted to provide a more efficient algorithm based on the
long-step path-following interior-point method for \textbf{QKD}\, key rate
calculation problem. However, their discussions were restricted to real
symmetric matrices, while for \textbf{QKD}\, key rate calculation, it is important
to handle Hermitian matrices. Although it might be possible to extend
the algorithm in~\cite{Faybusovich2020longstep} to deal with Hermitian
matrices, currently the extension was not done and thus, we cannot
directly compare our algorithms with theirs.
In addition, the problem formulation used in~\cite{Faybusovich2020longstep}
does not guarantee positive definiteness of the matrices involved in
the objective function. Therefore, they perturb the solution by adding a
small identity matrix. This perturbation is not required in our new method
in this paper due to the regularization using facial reduction, \textbf{FR}.
Due to the structure of the linear mapping ${\mathcal G} $, the matrix $\delta$
is often singular in~\cref{eq:keyrateoptequivsigdel}. Therefore
strict feasibility fails in~\cref{eq:keyrateoptequivsigdel}.
This indicates that the objective function, the
\textdef{relative entropy function} is evaluated on
singular matrices in both~\cref{eq:keyrateopt,eq:keyrateoptequivsigdel},
creating theoretical
and numerical difficulties. In fact, the domain of the problem
that guarantees finiteness for the objective function, requires
restrictions on the ranges of the linear mappings.
By moving back and forth between equivalent formulations of the types
in~\cref{eq:keyrateopt,eq:keyrateoptequivsigdel}, we derive a
regularized model that simplifies type~\cref{eq:keyrateopt}, and where positive
definiteness is preserved. In particular, the regularization allows for an
efficient interior point method even though the objective
function is not differentiable on the boundary of the
semidefinite cone. This allows for efficient algorithmic developments.
In addition, this enables us to accurately solve previously
intractable problems.
\subsection{Outline and Main Results}
In~\Cref{sect:prel} we present the preliminary notations and convex
analysis tools that we need.
In particular, we include details about the linear maps
and adjoints and basic \emph{facial reduction, \textbf{FR}}, needed for our
algorithms; see~\Cref{sect:lintrsadj,sect:confacered}.
The details and need for facial reduction, \textbf{FR}, is discussed
in~\Cref{sect:FR}. This is
due to the loss of strict feasiblity for the linear constraints
for some classes of
instances, as well as the loss of rank in the linear map ${\mathcal G} $ in
the nonlinear objective function. The \textbf{FR}\, guarantees that the objective function
$f$ is well defined for all positive definite density matrices.
Therefore, the domain of $f$ is not implicitly defined by conditions
that guarantee finiteness in~\cref{eq:keyrateopt}. Equivalently, we have
that strict feasibility holds in~\cref{eq:keyrateoptequivsigdel}.
A partial \textbf{FR}\, is based on singularity sometimes encountered
from the \emph{reduced density operator constraint},~\Cref{sect:partialFR}.
A second type of \textbf{FR}\, is done on the completely positive mappings
of the objective function,~\Cref{sec:FRontheobj}. Both of these are based on
spectral decompositions and rotations and are therefore very accurate.
The result is a much simplified problem~\cref{eq:finalredinrho} where
strict feasibility holds and the objective function arguments preserve
positive definiteness.
In addition, we discuss the differentiability,
both first and second order, in \Cref{thm:gradf}.
In~\Cref{sect:intpt} we begin with the optimality conditions and
a projected Gauss-Newton, {\bf GN}, interior point method.
This uses the modified objective function
that is well defined for positive definite density matrices, $\rho\succ 0$.
We use the \emph{stable} {\bf GN\,} search direction for the primal-dual
interior-point, \textbf{p-d i-p}, method. This avoids unstable backsolve steps for
the search direction. We also use a sparse preserving
nullspace representation for the primal feasibility in ~\Cref{sect:sparsenullrep}.
This provides for exact primal feasibility steps during the algorithm.
Optimal diagonal precondition for the linear system is presented
in~\Cref{sect:precond}.
Our upper and lower bounding techniques are give
in~\Cref{sect:dualbnds}.
In particular, we provide novel theoretical based lower bounding
techniques for the \textbf{FR}\, and original problem
in~\Cref{cor:lowerbound,cor:lowerboundpert}, respectively.\footnote{This
appears to be a novel contribution for general nonlinear convex \textbf{SDP}\,
optimization.}
Applications to the security analysis of some selected \textbf{QKD}\, protocols
are given in~\Cref{sect:tests}. This includes comparisons
with other codes as well as solutions of problems that could not be
solved previously. We include the lower bounds and
illustrate its strength by including the relative
gaps between lower and upper bounds;
and we compare with the abalytical optimal values when it is possible to do so.
We provide concluding remarks in~\Cref{sect:concl}.
Technical proofs, further references and results, appear in
\Cref{app:pfstech,app:compasp}. The details for six protocol examples
used in our tests are given in~\Cref{sect:testexamples}.
\section{Preliminaries}
\label{sect:prel}
We now present the notations and convex analysis background.
The asymptotic key rate $R^{\infty}$ is given by the Devetak-Winter
formula~\cite{Devetak2005} that can be written in the following form \cite{winick2017reliable}:
\begin{equation}\label{eq:dwkeyrate}
R^{\infty} = \min_\rho D({\mathcal G} (\rho)\|{\mathcal Z} ({\mathcal G} (\rho))) - p_{\text{pass}}\delta_{\text{EC}},
\end{equation}
where the first term is the quantum relative entropy function
from~\cref{eq:keyrateopt},
\textdef{$p_{\text{pass}}$} is the probability that a given signal is
used for the key generation rounds, and \textdef{$\delta_{\text{EC}}$} is the cost of error correction per round. The last two parameters are directly determined by observed data. Thus, the essential part of the quantum key distribution rate computation is to solve the following nonlinear convex semidefinite program as in~\cref{eq:keyrateopt}:
\begin{equation}
\label{eq:absQKD}
\qquad \min \{f(\rho) : \Gamma(\rho) =\gamma, \ \rho\succeq 0 \},
\end{equation}
where the objective function $f$ is the quantum relative entropy
function as shown in \cref{eq:dwkeyrate}, and the constraint set is a spectrahedron, i.e.,~the
intersection of an affine manifold and the positive semidefinite cone.
The affine manifold is defined using the linear map
for the linear equality constraints in~\cref{eq:absQKD}:
\[
\Gamma(\rho) = \left(\tr (\Gamma_i \rho)\right), i=1,\ldots, m, \quad
\Gamma : \Hn \to {\mathbb{R}^m\,}.
\]
These are divided into two sets: the observational and reduced density operator
constraint sets, i.e.,~$S_R\cap S_O$
The set of \textdef{state} $\rho$ satisfying the \textdef{observational constraints} is given by
\index{$S_O$, observational constraints}
\index{constraint sets}
\index{constraint sets!observational, $S_O$}
\index{constraint sets!reduced density, $S_R$}
\index{$\rho$, state}
\begin{equation}
\label{eq:constrobser}
S_O = \left\{\rho \succeq 0 \,:\, \langle P^A_s\otimes P^B_t,\rho\rangle =
p_{st},\, \forall st\right\},
\end{equation}
where we let \textdef{$n_A,n_B$} be the sizes
$P^A_s\in \mathbb{H}^{n_A},P^B_t\in \mathbb{H}^{n_B}$, respectively; and
we denote the \textdef{Kronecker product, $\otimes$}.
We set \textdef{$n=n_An_B$ which is the size of $\rho$}.
\index{size of $\rho$, $n=n_An_B$}
\index{$\otimes$, Kronecker product}
The set of state $\rho$ satisfying the constraints with respect to the
\emph{reduced density operator, $\rho_A$}, is
\index{reduced density operator constraint, $S_R$}
\index{$S_R$, reduced density operator constraint}
\begin{equation} \label{eq:constrreduced}
\begin{array}{rrl}
S_R &=& \left\{\rho\succeq 0 \,:\, \tr_{B}(\rho) = \rho_{A} \right\} \\
&=&\left\{\rho\succeq 0 \,:\, \langle \Theta_j\otimes \mathbbm{1}_B,\rho\rangle = \theta_j,\, \forall j = 1,\ldots,m_R\right\},
\end{array}
\end{equation}
where $\theta_j = \langle \Theta_j,\rho_A\rangle$ and $\{\Theta_j\}$
forms an orthonormal basis for the real vector space of Hermitian matrices
on system A. This implicitly defines the linear map and
constraint in \textdef{$\tr_B(\rho) = \rho_A$}.
Here we denote the identity matrix \textdef{$\mathbbm{1}_B \in \mathbb{H}^{n_B}$}.
Here, we may assume that $\Gamma_1 = I$ and $\gamma_1 =1$ to guarantee that
we restrict our variables to \textdef{density matrices}, i.e.,~semidefinite
and unit trace. (See~\cite[Theorem 2.5]{MR1796805}.)
We now continue with the terminology and preliminary background for the paper.
\subsection{Notations}
\index{$\mathbb{S}^n$, set of real symmetric $n$-by-$n$ matrices}
\index{set of real symmetric $n$-by-$n$ matrices, $\mathbb{S}^n$}
\index{$\mathbb{R}(X)$, real part of $X$}
\index{real part of $X$, $\mathbb{R}(X)$}
\index{$\imag(X)$, imaginary part of $X$}
\index{imaginary part of $X$, $\imag(X)$}
\index{$\mathbb{S}_+^n$, positive semidefinite cone of $n$-by-$n$ real symmetric matrices}
\index{positive semidefinite cone of $n$-by-$n$ real symmetric matrices, $\mathbb{S}_+^n$}
\index{$\mathbb{S}_{++}^n$, positive definite cone of $n$-by-$n$ real symmetric matrices}
\index{positive definite cone of $n$-by-$n$ real symmetric matrices, $\mathbb{S}_{++}^n$}
\index{$\Hnp$, positive semidefinite cone of $n$-by-$n$ Hermitian matrices}
\index{positive semidefinite cone of $n$-by-$n$ Hermitian matrices, $\Hnp$}
\index{$\Hnpp$, positive definite cone of $n$-by-$n$ Hermitian matrices}
\index{positive definite cone of $n$-by-$n$ Hermitian matrices, $\Hnpp$}
\index{$X\succeq 0$}
\index{$X\succ 0$}
We use $\mathbb{C}^{n\times n}$ to denote the space of $n$-by-$n$ complex matrices, and
$\Hn$ to denote the \emph{subset} of $n$-by-$n$ Hermitian matrices; we use
$\mathbb{H}$ when the dimension is clear. We use $\mathbb{S}^n,\mathbb{S}$ for the subspaces of $\Hn$
of real symmetric matrices. Given a matrix $X\in \mathbb{C}^{n\times n}$, we use $\mathbb{R}(X)$ and $\imag(X)$ to denote the real and the imaginary parts of $X$, respectively.
We use $\Hnp,\mathbb{S}_+^n$ ($\Hnpp, \mathbb{S}_{++}^n$, resp) to denote the positive
semidefinite cone (the positive definite cone, resp); and again
we leave out the dimension when it is clear.
We use the partial order notations $X\succeq 0,
X\succ 0$ for semidefinite and definite, respectively.
We let $\mathbb{R}^n$ denote the usual vector space of real $n$-coordinates;
$\mathcal{P}_C(X)$ denotes the projection of $X$ onto the
closed convex set $C$.
For a matrix $X$, we use $\range(X)$ and $\nul(X)$ to denote the
\textdef{range} and the \textdef{nullspace} of $X$, respectively.
We let \textdef{$\BlkDiag$}$(A_1,A_2,\ldots,A_k)$ denote the block diagonal matrix with diagonal blocks $A_i$.
\index{$\mathbb{R}^n$, vector space of real $n$-coordinates}
\index{vector space of real $n$-coordinates, $\mathbb{R}^n$}
\index{$\mathcal{P}_C(X)$, projection of $X$ onto $C$}
\index{$\range(X)$, range of $X$}
\index{$\nul(X)$, nullspace of $X$}
\index{$\BlkDiag(A_1,A_2)$, block diagonal matrix with diagonal blocks $A_1,A_2$}
\subsection{Real Inner Product Space $\mathbb{C}^{n\times n}$}
In general, $\Hn$ is not a subspace of $\mathbb{C}^{n\times n}$ unless we treat both as
vector spaces over $\mathbb{R}$. To do this we define a
\textdef{real inner product in $\mathbb{C}^{n\times n}$} that takes the standard inner
products of the real and imaginary parts:
\begin{equation}
\label{def:complex_innerprod}
\begin{array}{rcl}
\langle Y,X\rangle
&=&
\langle\Re(Y), \Re(X)\rangle+\langle\imag(Y), \imag(X)\rangle
\\&=&
\Tr \left(\Re(Y)^T \Re(X)\right)+\Tr \left(\imag(Y)^T \imag(X)\right)
\\&=&
\Re(\Tr \left(Y^\dagger X\right)).
\end{array}
\end{equation}
We note that
\[
\Re(\<Y,X\> ) = \<\Re(Y), \Re(X) \> + \< \imag(Y), \imag(X) \>,\quad
\imag(\<Y,X\> ) = - \<\Re(Y), \imag(X) \> + \< \imag(Y), \Re(X) \> .
\]
Over the reals, $\dim (\Hn) = n^2, \dim (\mathbb{C}^{n\times n}) = 2n^2$. The induced
norm is the Frobenius norm $\|X\|^2_F = \<X,X\>=\trace \left(X^\dagger X\right)$, where
we denote the \textdef{conjugate transpose, $\cdot^\dagger $}.
\index{$\cdot^\dagger $, conjugate transpose}
\index{$\|X\|_F$, Frobenius norm}
\index{Frobenius norm, $\|X\|_F$}
\subsection{Linear Transformations and Adjoints}
\label{sect:lintrsadj}
Given a linear map ${\mathcal L} : \mathcal{D} \to \mathcal{R}$, we call the
unique linear map ${\mathcal L} ^\dagger : \mathcal{R} \to \mathcal{D}$ the \textdef{adjoint} of ${\mathcal L} $, if it satisfies
\[
\left\langle {\mathcal L} (X),Y\right\rangle = \left\langle X,{\mathcal L} ^\dagger (Y)\right\rangle, \, \forall X \in \mathcal{D},Y\in \mathcal{R}.
\]
Often in our study, we use vectorized computations instead of using complex matrices directly. In order to relieve the computational burden, we use isomorphic and isometric realizations of matrices by ignoring the redundant entries.
We consider $\mathbb{H}^n$ as a vector space of dimension $n^2$ over the reals. We
define $\Hvec(H)\in \mathbb{R}^{n^2}$ by stacking $\diag(H)$ followed by
$\sqrt 2$ times the strict upper triangular
parts of $\mathbb{R}(H)$ and $\imag(H)$, both columnwise:
\[
\Hvec(H) = \begin{pmatrix}
\diag(H)\cr
\sqrt 2 \mathbb{R}(upper(H))\cr
\sqrt 2 \imag(upper(H))
\end{pmatrix}\in \mathbb{R}^{n^2}, \quad \HMat = \Hvec^{-1} = \Hvec^\dagger .
\]
We note that for the real symmetric matrices $\mathbb{S}^n$, we can use the first
\textdef{triangular number, $t(n)=n(n+1)/2$} of elements in $\Hvec$,
and we denote this
by \textdef{$\svec$}$(S)\in \mathbb{R}^{t(n)}$, with adjoint \textdef{$\sMat$}.
\index{$t(n)=n(n+1)/2$, triangular number}
\index{${\mathcal L} ^\dagger $, adjoint of ${\mathcal L} $}
\index{adjoint of ${\mathcal L} $, ${\mathcal L} ^\dagger $}
We use various linear maps in a \textbf{SDP}\, framework. For given $\Gamma_i\in \Hn, i=1,\ldots,m$, define
\[
\Gamma : \Hn \to {\mathbb{R}^m\,} \text{ by }
\Gamma (H) = (\langle \Gamma_i,H\rangle)_i \in {\mathbb{R}^m\,}.
\]
The adjoint satisfies
\[
\langle \Gamma (H),y\rangle = \sum_i y_i \Tr (\Gamma_iH) =
\Tr \left( H \left(\sum_i y_i \Gamma_i\right) \right)=
\left\langle H, \Gamma^\dagger (y) \right\rangle.
\]
The matrix representation $A$ of $\Gamma$ is found from
\[
(A \Hvec(H))_i = (\Gamma (H))_i = \<\Gamma_i,H\> = \left\langle\Hvec (\Gamma_i),\Hvec (H)\right\rangle,
\]
i.e., for $g_i=\Hvec (\Gamma_i),\ \forall i$ and $h=\Hvec (H)$,
\[
\Gamma(H) \equiv A(h), \ \text{ where } A=\begin{bmatrix} g_1^T\cr \vdots\cr g_m^T \end{bmatrix}.
\]
\subsubsection{Adjoints for Matrix Multiplication}
Adjoints are essential for our interior point algorithm when using
matrix-free methods.
We define the \textdef{symmetrization linear map, $\sym$}, as
$\sym(M) = (M+M^\dagger )/2$.
The \textdef{skew-symmetrization linear map, $\sksym$}, is
$\sksym(M) = (M-M^\dagger )/2$.
\index{$\sym$, symmetrization linear map}
\index{$\sksym$, skew-symmetrization linear map}
\begin{lemma}[adjoint of ${\mathcal W} (R) := WR$]
\label{lem:WRadj}
Let $W\in \mathbb{C}^{n\times n}$ be a given square complex matrix,
and define the (left matrix
multiplication) linear map ${\mathcal W} : \mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$ by
${\mathcal W} (R)=WR$. Then the adjoint ${\mathcal W} ^\dagger : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ is defined by
\begin{equation}
\label{eq:generalWRadj}
{\mathcal W} ^\dagger (M) =
\Re(W)^T \Re(M) +\imag(W)^T \imag(M)
+ i\left( \Re(W)^T \imag(M) -\imag(W)^T \Re(M)\right).
\end{equation}
If $W\in \Hn$ and ${\mathcal W} :\Hn \to \mathbb{C}^{n\times n}$, then the adjoint ${\mathcal W} ^\dagger :\mathbb{C}^{n\times n}\to
\Hn$ is defined by
\begin{equation}
\label{eq:adjWMHerm}
{\mathcal W} ^\dagger (M) =
\sym\left[\Re(W)\Re(M) -\imag(W)\imag(M)\right] +
i \sksym\left[\imag(W) \Re(M) + \Re(W) \imag(M)\right].
\end{equation}
\end{lemma}
\begin{proof}
See \Cref{sec:proof_adjoint}.
\end{proof}
\begin{lemma}[adjoint of $\rho(S)=S\rho$]
\label{lem:Srho}
Let $\rho\in \mathbb{C}^{n\times n}$ be a given square complex matrix,
and define the (right matrix
multiplication) linear map $\rho : \mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$ by
$\rho(S)=S \rho$. Then the adjoint $\rho^\dagger : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ is defined by
\begin{equation}
\label{eq:generalSrhoadj}
\rho^\dagger (M) = {\mathcal S} \left[ \Re(M)\Re(\rho) +\imag(M) \imag(\rho)^{T} \right]
+ i \sksym \left[ -\Re(M)\imag(\rho)^{T} + \imag(M) \Re(\rho) \right] .
\end{equation}
If $\rho\in \Hn$ and $\rho:\Hn \to \mathbb{C}^{n\times n}$, then the adjoint $\rho^\dagger :\mathbb{C}^{n\times n}\to
\Hn$ is defined by
\begin{equation}
\label{eq:adjSrhoHerm}
\rho^\dagger (M) = {\mathcal S} \left[ \Re(M)\Re(\rho) -\imag(M) \imag(\rho) \right]
+ i \sksym \left[ \Re(M)\imag(\rho) + \imag(M) \Re(\rho) \right] .
\end{equation}
\end{lemma}
\begin{proof}
See \Cref{sec:proof_adjoint2}
\end{proof}
\subsection{Cones, Faces, and Facial Reduction, \textbf{FR}}
\label{sect:confacered}
The facial structure of the semidefinite cone is well understood. We
outline some of the concepts we need for facial reduction
and exposing vectors, see e.g.,~\cite{DrusWolk:16}.
We recall that a \textdef{convex cone} $K$ is defined by: $\lambda K\subseteq K,
\forall \lambda \geq 0, \, K+K\subset K$, i.e.,~it is a cone and so
contains all rays, and it is a convex set. \index{dual cone, $S^\dagger $}
For a set $S\subseteq \mathbb{H}$ we denote the \textdef{dual cone},
$S^\dagger = \{\phi \in \mathbb{H} : \langle \phi,s\rangle \geq 0, \ \forall s \in
S\}$.
\index{$S^\dagger $, dual cone}
\begin{definition}[{\textdef{face}}]
A convex cone $F$ is a face of a convex cone $K$, denoted $F\unlhd K$,
if
\[
x,y\in K, x+y\in F \implies x,y \in F.
\]
Equivalently, for a general convex set $K$ and convex subset $F\subseteq
K$, we have $F\unlhd K$, if
\[
[x,y] \subset K, z\in \relint [x,y], z\in F \implies [x,y]\subset F,
\]
where $[x,y]$ denote the line segment joining $x,y$.
\end{definition}
\index{line segment, $[x,y]$}
\index{$[x,y]$, line segment}
Faces of the positive semidefinite cone are characterized by the range or nullspace of any element in the relative interior of the faces.
\begin{lemma}
\label{lem:propfaces}
Let $F$ a convex subset of $\mathbb{H}_+^n$ with $X\in \relint F$. Let
\[
X = \begin{bmatrix} P & Q\end{bmatrix}
\begin{bmatrix} D & 0 \cr 0 & 0\end{bmatrix}
\begin{bmatrix} P & Q\end{bmatrix}^\dagger
\]
be the orthogonal spectral decomposition with $D\in \mathbb{H}_{++}^r$.
Then the following are equivalent:
\begin{enumerate}
\item
$F\unlhd \mathbb{H}_+^n$;
\item
\label{item:facesranges}
$F= \{ Y \in \mathbb{H}_+^n \,:\, \range(Y) \subset \range(X) \}
= \{ Y \in \mathbb{H}_+^n \,:\, \nul(Y) \supset \nul(X) \}$;
\item
$F= P \mathbb{H}_+^r P^\dagger $;
\item
\label{item:expvect}
$F= \mathbb{H}_+^n \cap (Q Q^\dagger )^\perp$.
\end{enumerate}
\end{lemma}
The matrix $QQ^\dagger $, in \Cref{item:expvect} of \Cref{lem:propfaces}, is called an
\textdef{exposing vector} for the face $F$.
Exposing vectors come into play throughout \Cref{sect:FR}.
\begin{definition}[minimal face]
Let $K$ be a closed convex cone and let $X\in K$. Then $\face(X)\unlhd
K$ is the \emph{minimal face}, the intersection of all faces of $K$ that
contain $X$.
\end{definition}
\index{$\face(X)$, minimal face}
\index{minimal face, $\face(X)$}
Facial reduction is a process of identifying the minimal face of $\mathbb{H}_+^n$
containing the affine subspace $\{\rho:\Gamma(\rho) = \gamma\}$.
\Cref{lem:FRfarkas} plays an important role in the heart of facial
reduction process. Essentially, either there exists a $\rho\succ 0$
that satisfies the constraints, or the alternative that there exists
a linear combination of the $\Gamma_i$ that is positive semidefinite but
has a zero expectation.
\begin{lemma}[theorem of the alternative, {\cite[Theorem 3.1.3]{DrusWolk:16}}]
\label{lem:FRfarkas}
For the feasible constraint system in \eqref{eq:absQKD}, exactly one of the following statements holds:
\begin{enumerate}
\item there exists $\rho\succ 0$ such that $\Gamma(\rho) = \gamma$;
\item there exists $y$ such that
\begin{equation}
\label{eq:little_auxsystem}
0 \ne \Gamma^\dagger (y) \succeq 0 \ , \ \<\gamma,y\> = 0.
\end{equation}
\end{enumerate}
\end{lemma}
In \Cref{lem:FRfarkas}, the matrix $\Gamma^\dagger (y)$ is an exposing vector
for the face containing the constraint set in \eqref{eq:absQKD}.
\section{Problem Formulations and Facial Reduction}
\label{sect:FR}
We now present the details on various formulations of \textbf{QKD}\,
from~\cref{eq:keyrateopt,eq:keyrateoptequivsigdel}. We show
that facial reduction allows for regularization of both the constraints
and the objective function. We include results about
\textbf{FR}\, for positive transformations and show that highly accurate \textbf{FR}\, can
be done in these cases.
\subsection{Properties of Objective Function and Mappings ${\mathcal G} ,{\mathcal Z} $}
\label{sect:objfn}
The \textdef{quantum relative entropy function}
$D : \Hnp \times \Hnp \to \R_+\cup \{+\infty\}$ is denoted by
$ D(\delta || \sigma)$, and is defined as
\begin{equation}
\label{eq:objective}
D(\delta || \sigma ) =
\left\{
\begin{array}{ll}
\trace(\delta \log \delta)-\trace (\delta \log \sigma) & \text{if } \range(\delta)\cap \nul(\sigma) = \emptyset \\
\infty & \text{otherwise.
\\
\end{array}
\right.
\end{equation}
That the quantum relative entropy $D$ is finite if $\range(\delta)\subseteq \range(\sigma)$ is shown
by extending the matrix log function to be $0$ on the nullspaces of
$\delta, \sigma$. (See~\cite[Definition 5.18]{watrous_2018}.)
It is known that $D$ is nonnegative, equal to $0$ if, and only if, $\rho
=\sigma$, and is jointly convex in both $\delta$ and
$\sigma$, see ~\cite[Section 11.3]{MR1796805}.
\begin{definition}
\label{def:operGandZ}
The linear map \textdef{${\mathcal G} : \Hn \to \Hk$} is defined as a
sum of matrix products (\textdef{Kraus representation})
\begin{equation}
\label{eq:Gmap}
\displaystyle{\mathcal G} (\rho) := \sum_{j=1}^{\ell} K_j \rho K_j^\dagger ,
\end{equation}
where $K_j \in \mathbb{C}^{k \times n}$ and $\sum_{j=1}^{\ell} K_j^\dagger K_j \preceq I$. The adjoint is ${\mathcal G} ^\dagger (\delta) := \sum_{j=1}^{\ell} K_j^\dagger \delta K_j$.
\end{definition}
Typically we have $k>n$ with $k$ being a multiple of $n$; and thus we can have ${\mathcal G} (\rho)$ rank deficient for
all $\rho\succ 0$.
\begin{definition}
\label{def:operGandG}
The self-adjoint (projection) linear map \textdef{$\mathcal{Z}:
\Hk \to \Hk$} is defined as the sum
\index{${\mathcal Z} : \Hk \to \Hk$}
\begin{equation}
\label{eq:Zmap}
\displaystyle{\mathcal Z} (\delta) := \sum_{j=1}^{N} Z_j \delta Z_j,
\end{equation}
where $Z_j = Z_j^2 = Z_j^\dagger \in \Hkp$ and $\sum_{j=1}^N Z_j = I_k$.
\end{definition}
Since $\sum_{j=1}^N Z_j = I_k$, the set $\{Z_i\}_{j=1}^N$ is a
\textdef{spectral resolution of $I$}, \Cref{prop:Zopprops} below states
some interesting properties of the operator ${\mathcal Z} $; see
also~\cite[Appendix C, (C1)]{Coles2012}.
\begin{prop}
\label{prop:Zopprops}
The linear map ${\mathcal Z} $ in \Cref{def:operGandG}
is an orthogonal projection on $\Hk$. Moreover,
\begin{equation}
\label{eq:deltaZdeltalog}
\trace (\delta) \leq 1, \delta \succ 0 \implies
\Big\{ \trace \left(\delta \log {\mathcal Z} (\delta)\right) = \trace \left({\mathcal Z} (\delta) \log {\mathcal Z} (\delta)\right)
\Big\}.
\end{equation}
\end{prop}
\begin{proof}
First we show that the matrices of ${\mathcal Z} $ satisfy
\begin{equation}
\label{eq:ZiZjzero}
Z_iZ_j=0,~\forall~i\neq j.
\end{equation}
For $i,j \in \{1,\ldots, N\}$, we have by \Cref{def:operGandG} that
\begin{equation}
\label{eq:orthogZj}
\begin{array}{rcl}
Z_i \left( \sum_{s=1}^N Z_s \right) Z_i = Z_i I_kZ_i = Z_i
& \implies &
0=\sum_{s\ne i} Z_i Z_s Z_i = \sum_{s\ne i} (Z_sZ_i)^\dagger (Z_sZ_i)
\\& \implies &
Z_jZ_i=0,\ \forall j\neq i.
\end{array}
\end{equation}
We now have ${\mathcal Z} ={\mathcal Z} ^2={\mathcal Z} ^{1/2}={\mathcal Z} ^\dagger $. Thus, ${\mathcal Z} $ is an orthogonal projection. Finally, we use the series expansion of the log function and the
properties of the $Z_j$ seen in~\cref{eq:orthogZj}
to prove~\cref{eq:deltaZdeltalog}; see \Cref{lemma:ZlogZ=logZ} for
details.
\end{proof}
Using \eqref{eq:objective}, \Cref{lemma:rangeZrelation} below shows that
the objective value of the model \eqref{eq:keyrateopt} is finite on the
feasible set. This also provides insight on the usefulness of
\textbf{FR}\, on the variable $\sigma$ done below.
\begin{lemma}
\label{lemma:rangeZrelation}
Let $X\succeq 0$. Then $\range(X) \subseteq \range({\mathcal Z} (X))$.
\end{lemma}
\begin{proof}
See \Cref{sec:proof_rangelem}.
\end{proof}
\begin{remark}
\label{rem:notposdef}
In general, the mapping ${\mathcal G} $ in~\cref{eq:Gmap}
\underline{does not preserve positive definiteness}. Therefore the
objective function $f(\rho)$, see~\cref{eq:f_in_trace} below, may need
to evaluate $\trace (\delta \log \delta)$ and $\trace (\delta \log \sigma)$
with \emph{both} $\delta = {\mathcal G} (\rho)$ and $\sigma = {\mathcal Z} {\mathcal G} (\rho)$
always singular.
Although the objective function $f$ is well-defined at singular points
$\delta, \sigma$, the gradient of $f$ at singular points $\delta,
\sigma$ is not well-defined. Our approach using \textbf{FR}\, within an
interior point method avoids these numerical
difficulties.\footnote{For objective value computations without
using the MATLAB built-in function logm, see \Cref{remark:nologm}.}
\end{remark}
\subsection{Derivatives for Quantum Relative Entropy under Positive Definite Assumptions}
We can reformulate the quantum relative entropy function defined in the key rate optimization \cref{eq:keyrateopt} as
\begin{equation}
\label{eq:f_in_trace}
\begin{array}{rcl}
f(\rho)
&=&
D({\mathcal G} (\rho)\|{\mathcal Z} ({\mathcal G} (\rho)))
\\
&=&
\Tr\left({\mathcal G} (\rho) \log {\mathcal G} (\rho)\right) -
\Tr({\mathcal G} (\rho) \log
{\mathcal Z} ({\mathcal G} (\rho)))\\
&=&
\Tr\left({\mathcal G} (\rho) \log {\mathcal G} (\rho)\right) -
\Tr({\mathcal Z} ({\mathcal G} (\rho)) \log
{\mathcal Z} ({\mathcal G} (\rho)))\\
\end{array}
\end{equation}
Here, the linear map ${\mathcal Z} $ is added to the second term in~\cref{eq:f_in_trace} above, and the equality follows from~\Cref{prop:Zopprops}.
In this section, we review the gradient (Frech\'et derivative), and the image of the Hessian, for the reformulated relative entropy function $f$ defined in~\cref{eq:f_in_trace}. We obtain the derivatives of $f$ under the
assumption that the matrix-log is acting on positive
definite matrices. This assumption is needed for differentiability.
Note that the difficulty arising from the singularity is handled by
using perturbations in~\cite{winick2017reliable,Faybusovich2020longstep}.
This emphasizes the need for the regularization below as otherwise $f$
in~\cref{eq:f_in_trace} is \emph{never} differentiable.
We avoid using perturbations in this paper by applying \textbf{FR}\, in the sections below.
We now use the chain rule and
derive the first and the second order derivatives of the composition
of a linear and entropy function.
\begin{lemma}
\label{lem:gradfromchainrule}
Let ${\mathcal H} :\mathbb{H}^n\to \mathbb{H}^k$ be a linear map that
preserves positive semidefiniteness. Assume that ${\mathcal H} (\rho) \in \mathbb{H}_{++}^k$. Define the composite function $g:\mathbb{H}^k_+ \to \mathbb{R}$ by
\[
g(\rho) = \trace \left({\mathcal H} (\rho)\log({\mathcal H} (\rho)) \right).
\]
Then the gradient of $g$ at $\rho$ is
\[
\nabla g(\rho) = {\mathcal H} ^\dagger (\log[{\mathcal H} (\rho)]) + {\mathcal H} ^\dagger (I),
\]
and the Hessian of $g$ at $\rho$ acting on $\Delta \rho$ is
\[
\nabla^2 g(\rho)(\Delta \rho) ={\mathcal H} ^\dagger \left(
\log^\prime{\mathcal H} (\rho)({\mathcal H} (\Delta \rho))
\right),
\]
where $\log^\prime$ denotes the Fr\'echet derivative.\qed
\end{lemma}
\index{$\log^\prime$, Fr\'echet derivative of $\log$}
\index{Fr\'echet derivative of $\log$, $\log^\prime$}
Under the assumption that ${\mathcal G} (\rho) \succ 0$,
we can use~\Cref{lemma:rangeZrelation} and show
that ${\mathcal Z} ({\mathcal G} (\rho)) \succ 0$.
Using \Cref{lem:gradfromchainrule} and \eqref{eq:deltaZdeltalog},
we obtain the first and the
second order derivatives of the objective function $f$ in \eqref{eq:f_in_trace}.
\begin{corollary}
\label{thm:gradf}
Suppose that $\rho\in \Hnp$ and ${\mathcal G} (\rho) \succ 0$.
Then the \textdef{gradient of $f$} at $\rho$ is
\begin{equation}
\label{eq:gradient}
\nabla f(\rho) =
{\mathcal G} ^\dagger \Big( \log[{\mathcal G} (\rho)]\Big) -
({\mathcal Z} \circ {\mathcal G} )^\dagger \Big( \log[({\mathcal Z} \circ {\mathcal G} )(\rho)] \Big).
\end{equation}
The
\textdef{Hessian at $\rho\in \Hn_{+}$ acting on the direction
$\Delta \rho \in \Hn$} is
\begin{equation}
\label{eq:hessianqkd}
\begin{array}{rcl}
\nabla^2 f(\rho)(\Delta \rho)
&=&
{\mathcal G} ^\dagger \Big(
[\log^\prime{\mathcal G} (\rho)({\mathcal G} \Delta \rho)]\Big)
-
({\mathcal Z} \circ{\mathcal G} )^\dagger \Big([\log^\prime({\mathcal Z} \circ
{\mathcal G} )(\rho)(({\mathcal Z} \circ{\mathcal G} )(\Delta \rho))\Big).
\end{array}
\end{equation}
\end{corollary}
\subsection{Reformulation via Facial Reduction (\textbf{FR})}
Using \Cref{prop:Zopprops}, we can now reformulate the objective function
in the key rate optimization problem \cref{eq:keyrateoptequivsigdel} to
obtain the following equivalent model:
\begin{equation}
\label{eq:optprob}
\begin{array}{rcllll}
&& \min_{\rho,\sigma,\delta} &
\trace(\delta\log \delta) - \trace( \sigma \log \sigma) \\
&&\text{s.t.} & \Gamma(\rho) = \gamma
\\ &&& \sigma - {\mathcal Z} (\delta) = 0
\\ &&& \delta - {\mathcal G} (\rho) = 0
\\ &&& \textdef{$\rho\in \Hnp$}, \,\textdef{$\sigma \in \Hkp$},\,\textdef{$\delta \in \Hkp$}.
\end{array}
\end{equation}
The new objective function is the key in our analysis, as it simplifies the expressions for gradient and Hessian. Next, we derive facial reduction based on the constraints in \cref{eq:optprob}.
\subsubsection{Partial \textbf{FR}\, on the Reduced Density Operator Constraint}
\label{sect:partialFR}
\index{facially reduced reduced density operator constraint}
\index{$S_R$, reduced density operator constraint}
\index{reduced density operator constraint, $S_R$}
Consider the spectrahedron $S_R$ defined by the reduced density operator
constraint in \cref{eq:constrreduced}.
We now simplify the problem via \textbf{FR}\, by using only~\cref{eq:constrreduced}
in the case that $\rho_A \in \mathbb{H}^{n_{A}}$ is singular.
We now see in~\Cref{thm:FRonObserv} that we
can do this explicitly using the spectral decomposition of $\rho_A$;
see also~\cite[Sec. II]{Ferenczi2012}).
Therefore, this step is extremely accurate.
Using the structure arising from the reduced density operator constraint,
we obtain partial \textbf{FR}\, on the constraint set in~\Cref{thm:FRonObserv}.
\begin{theorem}
\label{thm:FRonObserv}
Let $\range (P) = \range (\rho_A)\subsetneq \mathbb{H}^{n_A}, P^\dagger P=\mathbbm{1}_r$, and let $V=
P\otimes \mathbbm{1}_B$. Then
the spectrahedron $S_R$ in~\cref{eq:constrreduced} has the property that
\begin{equation}\label{eq:SO_face}
\rho \in S_R \implies \rho = VRV^\dagger , \text{ for some }
R\in \mathbb{H}^{r\cdot n_B}_{+}.
\end{equation}
\end{theorem}
\begin{proof}
Let $\begin{bmatrix} P&Q \end{bmatrix}$ be a unitary matrix such that
$\range (P) = \range (\rho_{A})$ and $\range (Q) = \nul (\rho_{A})$. Let $W = QQ^{T} \succeq 0$.
Recall that the adjoint $\Tr_B^\dagger(W) = W\otimes \mathbbm{1}_B$.
Then $\rho \in S_R$ implies that
\begin{equation} \label{eq:TRBW}
\langle W \otimes \mathbbm{1}_B, \rho \rangle = \langle W, \tr_{B}(\rho) \rangle = \langle W, \rho_{A} \rangle = 0,
\end{equation}
where \textdef{$\mathbbm{1}_B \in \mathbb{H}^{n_B}$} is the identity matrix of size
$n_{B}$, and we use~\cref{eq:constrreduced} to guarantee that
$\Tr_B(\rho)= \rho_A$.
Therefore, $W \otimes \mathbbm{1}_B \succeq 0$ is an exposing vector
for the spectrahedron $S_R$ in \cref{eq:constrreduced}. And we can write
$\rho = VRV^\dagger $ with $V=P\otimes \mathbbm{1}_B$ for any $\rho \in S_R$.
This yields an equivalent representation \cref{eq:SO_face} with a
smaller positive semidefinite constraint.\footnote{We provide a
self-contained alternate proof in~\Cref{sec:proofFRonObserv}.}
\end{proof}
We emphasize that facial reduction is not only powerful in reducing
the variable dimension, but also in reducing the number of constraints.
Indeed, if $\rho_{A}$ is not full-rank, then at least one of the
constraints in \cref{eq:constrreduced} becomes redundant and can be
discarded, see~\cite{bw3,SWW:17}.
In this case, it is equivalent to the matrix $\rho_A$ becoming smaller
in dimension.
(Our empirical observations show that many of the other observational
constraints $\Gamma_i(\rho)=\gamma_i$
also become redundant and can be discarded.)
\subsubsection{\textbf{FR}\, on the Constraints Originating from ${\mathcal G} ,{\mathcal Z} $}
\label{sec:FRontheobj}
Our motivation is that the domain of the objective function may be
restricted to the boundary of the semidefinite cone, i.e.,~the
matrices ${\mathcal G} (\rho), {\mathcal Z} ({\mathcal G} (\rho))$ are singular by the definition
of ${\mathcal G} $. We would like to guarantee
that we have a well-formulated problem with
strictly feasible points in the domain of the objective function so that the derivatives are well-defined. This guarantees basic numerical stability.
This is done by considering the constraints in the equivalent
formulation in~\eqref{eq:keyrateoptequivsigdel}.
We first note the useful equivalent form for the entropy function.
\begin{lemma}
\label{lem:VdeltaV}
Let $Y = VRV^\dagger \in \mathbb{H}_+,\, R\succ 0$ be the compact spectral decomposition of $Y$ with $V^\dagger V=I$.
Then
\[
\trace (Y \log Y)
= \trace (R\log R) .
\]
\end{lemma}
\begin{proof}
We obtain a unitary matrix $U = \begin{bmatrix}
V & P
\end{bmatrix}$ by completing the basis.
Then $Y = UDU^\dagger $, where $D = \BlkDiag(R,0)$. We conclude, with $0\cdot \log0 =0 $, that $\trace Y \log Y = \trace D \log D = \trace R \log R$.
\end{proof}
We use the following simple result to obtain the exposing vectors of the minimal face in the problem analytically.
\begin{lemma}
\label{lem:tranform}
Let ${\mathcal C} \subseteq \Hn_+$ be a given closed convex set
with nonempty interior. Let $Q_{i} \in \mathbb{H}^{k \times n},
i=1,\dots,t$, be given matrices. Define the linear map
${\mathcal A} : \Hn \rightarrow \Hk$ and matrix $V$ by
$$
{\mathcal A} (X) = \sum_i^t Q_{i}XQ_{i}^\dagger , \quad
\range(V) = \range\left( \sum_{i=1}^tQ_{i}Q_{i}^\dagger \right).
$$
Then the minimal face,
\[
\face({\mathcal A} ({\mathcal C} )) = V \mathbb{H}_{+}^{r} V^\dagger .
\]
\end{lemma}
\begin{proof}
First, note that properties of the mapping
implies that ${\mathcal A} ({\mathcal C} )\subset \Hkp$.
Nontrivial exposing vectors $0\neq W\in \mathbb{H}^n_+$ of ${\mathcal A} ({\mathcal C} )$ can be
characterized by the null space of the adjoint operator ${\mathcal A}^\dagger $:
$$
\begin{array}{rcl}
0\neq W\in \mathbb{H}^n_+, \langle W, {\mathcal A} ({\mathcal C} )\rangle = 0
&\iff &
0\neq W \succeq 0, \,
\langle W, Y \rangle = 0 , \, \forall \ Y \in {\mathcal A} ({\mathcal C} ) \quad
\\&\iff &
0\neq W \succeq 0, \,
\langle {\mathcal A}^\dagger (W), X \rangle = 0 , \, \forall \ X \in {\mathcal C} \quad
\\&\iff &
0\neq W \succeq 0, \,
W \in \nul({\mathcal A}^\dagger )
\\&\iff &
0\neq W \succeq 0, \,
Q_i^\dagger WQ_i = 0, \forall i,
\\&\iff &
0\neq \range(W) \subseteq \nul \left( \sum_i Q_iQ_i^\dagger \right),
\end{array}
$$
where the third equivalence follows from $\Int ({\mathcal C} ) \neq \emptyset$; and the
fourth equivalence follows from the properties of the sum
of mappings of a semidefinite matrix.
The choice of $V$ follows from choosing a maximal rank exposing
vector and constructing $V$ using \Cref{lem:propfaces}:
\[
\range(V) = \nul (W) = \range \left( \sum_i Q_iQ_i^\dagger \right).
\]
\end{proof}
We emphasize that the minimal face in \Cref{lem:tranform} means that
$V$ has a minimum number of columns, as without loss of
generality, we choose it to be full column
rank. In other words, this is the greatest reduction in the dimension of
the image.
The exposing vectors of ${\mathcal A} ({\mathcal C} )$ are characterized by the positive
semidefinite matrices in the null space of ${\mathcal A}^\dagger $.
This also implies the strong conclusion that
the \textdef{singularity degree} of ${\mathcal A} ({\mathcal C} )$ is one, i.e.,~\textbf{FR}\, can be
done in one step. This is an important conclusion for
stability,~\cite{DrusWolk:16,ScTuWonumeric:07}. Moreover,
we can now conclude that after \textbf{FR}\, for the initial linear equality constraints
$\Gamma(\rho) = \gamma$,
our main problem also has singularity degree one.
\begin{corollary}
Let ${\mathcal A} $ be as defined in~\Cref{lem:tranform} and
\[
{\mathcal F} := \left\{ (X,Y) \in \Hnp \times \Hkp :
\begin{bmatrix} {\mathcal A} & -I \end{bmatrix}
\begin{pmatrix} X \cr Y \end{pmatrix}
= 0\right\}.
\]
then the singularity degree of ${\mathcal F} $ is one.
\end{corollary}
\begin{proof}
According to \Cref{lem:FRfarkas}, the singularity degree of ${\mathcal F} $ is one
if $0\neq (W_{X},W_{Y}) \in (\Hnp,\Hkp)$ is an exposing vector of
the minimal face, $\face {\mathcal F} $, such that
\begin{equation}\label{eq:sing}
W_{X} = {\mathcal A} ^\dagger (-W_{Y}) \in \Hkp \text{ and } W_{Y} \in \Hkp.
\end{equation}
Let $W \in \Hkp$ be such that $\range(W) = \nul \left( \sum_i Q_iQ_i^\dagger \right)$ as in \Cref{lem:tranform}. Then $W_{X} = 0$ and $W_{Y} = W$ form an exposing vector of the minimal face for ${\mathcal F} $ and they satisfy \cref{eq:sing}.
\end{proof}
\begin{remark}
The maximum rank exposing vector $W$ can also be found by solving the following feasibility system $\min \{\ 0 \ : \ {\mathcal A}^\dagger (W) = 0, \ \tr(W) = 1, \ W \succeq 0 \}$ using the interior point method.
\end{remark}
\index{exposing vector}
\index{$V_{\rho}$}
\index{$V_{\delta}$}
\index{$V_{\sigma}$}
We describe how to apply \Cref{lem:tranform} to obtain $V_{\rho},V_{\delta},V_{\sigma}$ of the minimal face of $(\Hnp,\Hkp,\Hkp)$ containing the feasible region of \eqref{eq:optprob}. By \Cref{lem:propfaces}, we may write
$$\begin{array}{rrll}
\rho &=& V_{\rho}R_{\rho}V_{\rho}^\dagger \in \mathbb{H}_+^n , & R_\rho \in \mathbb{H}_+^{n_\rho} \\
\delta &=& V_{\delta}R_{\delta}V_{\delta}^\dagger \in \mathbb{H}_+^k , & R_\delta \in \mathbb{H}_+^{k_\delta} \\
\sigma &=& V_{\sigma}R_{\sigma}V_{\sigma}^\dagger \in \mathbb{H}_+^k, & R_\sigma \in \mathbb{H}_+^{k_\sigma}. \\
\end{array}$$
Define the linear maps
$$\begin{array}{rrrrrl}
\Gamma_V:& \mathbb{H}_{+}^{n_{\rho}} \rightarrow \mathbb{R}^{m}&\text{ by }& \Gamma_V(R_{\rho}) &=& \Gamma( V_{\rho}R_{\rho}V_{\rho}^\dagger ), \\
{\mathcal G} _V:& \mathbb{H}_{+}^{n_{\rho}} \rightarrow \Hkp &\text{ by }& {\mathcal G} _V (R_{\rho}) &=& \mathcal{G}( V_{\rho}R_{\rho}V_{\rho}^\dagger ), \\
{\mathcal Z} _V:& \mathbb{H}_{+}^{k_{\delta}} \rightarrow \Hkp &\text{ by }& {\mathcal Z} _V(R_{\delta}) &=& \mathcal{Z}( V_{\delta}R_{\delta}V_{\delta}^\dagger ).
\end{array}$$
The matrices $V_{\rho},V_{\delta},V_{\sigma}$ are obtained as follows.
\begin{enumerate}
\item We apply \textbf{FR}\, to $\{ \rho \in \Hnp : \Gamma(\rho) = \gamma\}$ to find $V_{\rho}$ for the minimal face, $\face(\Hnp,\rho)$.
\item Define \index{${\mathcal R} _{\rho}$}
\[
\mathcal{R}_{\rho}:= \{ R_\rho \in \mathbb{H}_+^{n_\rho} : \Gamma_V(R_\rho) = \gamma\}.
\]
Note that $\text{int}( \mathcal{R}_{\rho}) \neq \emptyset$.
Applying \Cref{lem:tranform} to $\{ {\mathcal G} _V (R_{\rho}) \in \Hkp : R_{\rho} \in \mathcal{R}_{\rho} \}$, the matrix $V_{\delta}$ yields the minimal face, $\face(\Hkp,\delta)$ if we choose
\begin{equation}
\label{eq:analytic_Vd}
\range ( V_\delta ) = \range \left( {\mathcal G} _V(I) \right) .
\end{equation}
\item \index{${\mathcal R} _{\delta}$}
Define $$\mathcal{R}_{\delta}:=\{ R_{\delta} \in
\mathbb{H}_+^{k_{\delta}} : V_{\delta}R_{\delta}V_{\delta}^\dagger =
{\mathcal G} _V (R_{\rho}), \ R_{\rho} \in \mathcal{R}_{\rho} \}.$$
We again note that $\text{int}( \mathcal{R}_{\delta}) \neq \emptyset$. Applying
\Cref{lem:tranform} to $\{ {\mathcal Z} _V (R_{\delta}) \in \Hkp :
R_{\delta} \in \mathcal{R}_{\delta} \}$, we find the matrix
$V_{\sigma}$ representing the minimal face $\face(\Hkp,\sigma)$.
Thus, we choose $V_\sigma$ satisfying
\begin{equation}
\label{eq:analytic_Vs}
\range( V_\sigma ) = \range \left( {\mathcal Z} _V(I) \right).
\end{equation}
\end{enumerate}
As above, this also can be seen by looking at the image of $I$ and the relative interior of the range of ${\mathcal Z} _V$.
We note, by \Cref{lemma:rangeZrelation}, that $\range(V_\sigma) \supseteq \range(V_\delta)$. Note that we have assumed the exposing vector of maximal rank for the original constraint set on $\rho$ in the first step is obtained. Without loss of generality, we can assume that the columns in $V_{\rho},V_{\delta},V_{\sigma}$ are orthonormal. This makes the subsequent computation easier.
\begin{assump}
Without loss of generality, we assume $V_M^\dagger V_M=I$ for $M=\rho,\delta,\sigma$.
\end{assump}
Define ${\mathcal V} _\delta (R_\delta) := V_\delta R_\delta V_\delta^\dagger $ and ${\mathcal V} _\sigma (R_\sigma) := V_\sigma R_\sigma V_\sigma^\dagger $. Applying \Cref{lem:VdeltaV} and substituting for $\rho,\delta,\sigma$ to \eqref{eq:optprob}, we obtain the equivalent formulation \cref{eq:optprobobjconstrthree}.
\begin{equation}
\label{eq:optprobobjconstrthree}
\begin{array}{rclll}
&\min & \Tr (R_\delta \log (R_\delta) ) - \Tr\big(
R_\sigma \log(R_\sigma)\big) \\
&\text{s.t.} & {\Gamma_{\scriptscriptstyle V}} ( R_\rho ) = \gamma
\\ && {\mathcal V} _\sigma (R_\sigma) - {\mathcal Z} _V(R_\delta) = 0
\\ &&
{\mathcal V} _\delta (R_\delta) - {\mathcal G} _V(R_\rho) = 0
\\&& R_\rho, R_\sigma, R_\delta \succeq 0.
\end{array}
\end{equation}
After facial reduction, many of the linear equality constraints in \cref{eq:optprobobjconstrthree} end up being
redundant. We may delete redundant constraints and keep a well-conditioned equality constraints.
In the next section, we show that the removal of the redundant constraints can be performed by \emph{rotating} the constraints.
\subsubsection{Reduction on the Constraints}
Recall that our primal problem after \textbf{FR}\, is given in
\cref{eq:optprobobjconstrthree}.
Moreover, by the work above we can assume that ${\Gamma_{\scriptscriptstyle V}}$ is surjective. In \Cref{thm:rotate_const} and \Cref{thm:rotate_const2} below, we show that we can simplify the last two equality constraints in \cref{eq:optprobobjconstrthree} by an appropriate rotation.
\begin{theorem} \label{thm:rotate_const}
Let $R_\rho \in \mathbb{H}_+^{n_\rho}$ and $R_\delta \in \mathbb{H}_+^{k_\delta}$. It holds that
\begin{equation}\label{eq:rotate_const}
{\mathcal V} _\delta (R_\delta) = {\mathcal G} _V(R_\rho) \quad \Longleftrightarrow \quad R_\delta = {\cG_{\scriptscriptstyle UV}} (R_\rho),
\end{equation}
where ${\cG_{\scriptscriptstyle UV}} ( \cdot) := V_\delta^\dagger {\mathcal G} _{V}(\cdot) V_{\delta}$.
\end{theorem}
\begin{proof}
Let $P$ be such that $U = \begin{bmatrix} V_\delta & P \end{bmatrix}$ is unitary. Rotating the first equality in \cref{eq:rotate_const} using the unitary matrix $U$ yields an equivalent equality $U^\dagger {\mathcal V} _\delta(R_\delta) U = U^\dagger {\mathcal G} _V(R_\rho) U$. Applying the orthogonality of $V_{\delta}$, the left-hand side above becomes
\begin{equation}
\label{eq:UdeltaU}
U ^\dagger {\mathcal V} _\delta (R_\delta) U =
\begin{bmatrix}
R_\delta & 0\cr 0 & 0
\end{bmatrix}.
\end{equation}
From facial reduction, it holds that $\range(V_\delta) = \range ({\mathcal G} _V)$ and thus $P^\dagger {\mathcal G} _V = 0$. Therefore, the right hand-side becomes
\begin{equation}
\label{eq:UGrhoU}
U^\dagger {\mathcal G} _V(R_\rho) U
= \begin{bmatrix} V_\delta^\dagger \\ P^\dagger \end{bmatrix} {\mathcal G} _V(R_\rho) \begin{bmatrix} V_\delta & P \end{bmatrix} = \begin{bmatrix}
V_{\delta}^\dagger {\mathcal G} _V(R_\rho) V_{\delta} & 0\\
0 & 0
\end{bmatrix}.
\end{equation}
\end{proof}
\begin{theorem}
\label{thm:rotate_const2}
Let $R_\sigma \in \mathbb{H}_+^{k_\sigma}$ and $R_\delta \in \mathbb{H}_+^{k_\delta}$. It holds that
\begin{equation}
{\mathcal V} _\sigma (R_\sigma ) = {\mathcal Z} _V(R_\delta) \quad \Longleftrightarrow \quad R_\sigma = {\cZ_{\scriptscriptstyle UV}} (R_\delta),
\end{equation}
where ${\cZ_{\scriptscriptstyle UV}} ( \cdot) := V_\sigma^\dagger {\mathcal Z} _{V}(\cdot) V_{\sigma}$.
\end{theorem}
\begin{proof}
Using the unitary matrix $U = \begin{bmatrix} V_\sigma & P \end{bmatrix} $ in the proof of \Cref{thm:rotate_const}, we obtain the statement.
\end{proof}
With \Cref{thm:rotate_const,thm:rotate_const2}, we reduce the number of linear constraints in \eqref{eq:optprobobjconstrthree} as below.
\begin{equation}
\label{eq:optprobobjintptFRreduced}
\begin{array}{rclll}
&\min & \Tr (R_\delta \log (R_\delta) ) - \Tr\big(
R_\sigma \log(R_\sigma)\big) \\
&\text{s.t.} & {\Gamma_{\scriptscriptstyle V}} ( R_\rho ) = \gamma
\\ && R_\sigma - {\cZ_{\scriptscriptstyle UV}}(R_\delta) = 0
\\ &&
R_\delta - {\cG_{\scriptscriptstyle UV}}(R_\rho) = 0
\\&& R_\rho \in \mathbb{H}_+^{n_\rho}, R_\sigma \in
\mathbb{H}_+^{k_\sigma}, R_\delta \in \mathbb{H}_+^{k_\delta}.
\end{array}
\end{equation}
We emphasize that the images of ${\mathcal Z} _V$ and ${\mathcal G} _V$ in \eqref{eq:optprobobjconstrthree} are both in $\mathbb{H}^k$ but the images of ${\cZ_{\scriptscriptstyle UV}}$ and ${\cG_{\scriptscriptstyle UV}}$ in \eqref{eq:optprobobjintptFRreduced} are in $\mathbb{H}^{k_\sigma}$ and $\mathbb{H}^{k_\delta}$, respectively, and $k_\delta\le k_\sigma \le k$.
\index{$k_\delta$, $k_\sigma$}
\begin{remark}
\label{lem:GUV_ZUV_preserve}
The mapping ${\cG_{\scriptscriptstyle UV}}$ satisfies
the properties for ${\mathcal G} $ in~\eqref{eq:Gmap}.
However, the properties in~\cref{eq:Zmap} do not hold
for the mapping ${\cZ_{\scriptscriptstyle UV}}$.
\end{remark}
\subsection{Final Model for (\textbf{QKD}\,) and Derivatives}
\label{sect:finalmodel}
In this section we have a main result, i.e.,~the main model that we work
on and the derivatives.
We eliminate some of variables in the model \cref{eq:optprobobjintptFRreduced} to obtain a simplified formulation.
Define $\textdef{$\widehat \cZ$} := {\cZ_{\scriptscriptstyle UV}} \circ {\cG_{\scriptscriptstyle UV}}$ and $\textdef{$\widehat \cG$} : = {\cG_{\scriptscriptstyle UV}}$. We substitute $R_\sigma = \widehat \cZ(R_\rho)$ and $R_\delta = \widehat \cG(R_\rho)$ back in the objective function in \cref{eq:optprobobjintptFRreduced}. For simplification, and by abuse of notation, we set
\[
\fbox{$\rho \leftarrow R_\rho,\, \sigma \leftarrow R_\sigma,\,
\delta \leftarrow R_\delta$.}
\]
We obtain the final model (\textbf{QKD}):
\begin{equation}
\label{eq:finalredinrho}
\begin{array}{rclll}
p^* =&\min & f(\rho) =
\Tr\big( \widehat \cG (\rho) (\log \widehat \cG (\rho) ) \big) - \Tr \Big( \widehat \cZ(\rho) \log \widehat \cZ (\rho) \Big)
\\
&\text{s.t.} & {\Gamma_{\scriptscriptstyle V}} ( \rho ) = {\gamma_{\scriptscriptstyle V}}
\\&& \textdef{$\rho \in \mathbb{H}_+^{n_\rho}$},
\end{array}
\end{equation}
where ${\gamma_{\scriptscriptstyle V}} \in \mathbb{R}^{{m_{\scriptscriptstyle V}}}$ for some ${m_{\scriptscriptstyle V}} \le m$. The final model is essentially in the same form as the original model \cref{eq:keyrateopt}, see also \Cref{prop:Zopprops}.
\index{${m_{\scriptscriptstyle V}}$, number of linear constraints after \textbf{FR}\,}
Note that the final model now has smaller number of variables compared to the original problem~\cref{eq:keyrateopt}.
Moreover, the objective function $f$, with the modified linear maps $\widehat \cG,\widehat \cZ$, is well-defined and analytic
on $\rho\in \mathbb{H}_{++}^{n_\rho}$, i.e.,~we have
\begin{equation}
\label{eq:rho_welldef}
\rho \succ 0 \implies \widehat \cG(\rho) \succ 0 \implies
\widehat \cZ(\rho) \succ 0.
\footnote{This follows from~\cite[Theorem 6.6]{con:70}, i.e.,~from
$\relint (A C) = A \relint (C)$, where $C$ is a conex set and
$A : \mathbb{E}^n \to \mathbb{E}^m$ is a linear map.}
\end{equation}
We conclude this section by presenting the derivative formulae for gradient and hessian. The simple formulae in \Cref{thm:explgradHess} are a direct application of
\Cref{lem:gradfromchainrule}. Throughout \Cref{sect:intpt} we work with these derivatives.
\begin{theorem}[derivatives of regularized objective]
\label{thm:explgradHess}
Let $\rho \succ 0$. The gradient of $f$ in~\eqref{eq:finalredinrho} is
\[
\nabla f(\rho) = \fbox{$\widehat \cG^\dagger (\log[\widehat \cG(\rho)]) +
\widehat \cG^\dagger (I)$} -
\fbox{$\widehat \cZ^\dagger (\log[\widehat \cZ(\rho)]) +
\widehat \cZ^\dagger (I)$}.
\]
The Hessian in the direction $\Delta \rho$ is then
\[
\begin{array}{rcl}
\nabla^2 f(\rho)(\Delta \rho)
&=&
\fbox{$\widehat \cG^\dagger
(\log^\prime[\widehat \cG(\rho)](\widehat \cG(\Delta \rho)) $}
-
\fbox{$\widehat \cZ^\dagger
(\log^\prime[\widehat \cZ(\rho)](\widehat \cZ(\Delta \rho)) $} .
\end{array}
\]
\end{theorem}
\begin{theorem}
\label{cor:fsubdiff}
Let $f$ be as defined in~\eqref{eq:finalredinrho} and
let $\{\rho_i\}_i \subseteq \mathbb{H}_{++}^{n_\rho}$
with $\rho_i \to \bar \rho$.
If we have the convergence $\lim_i \nabla f(\rho_i) = \phi$, then
\[
\phi \in \partial f(\bar \rho).
\]
\end{theorem}
\begin{proof}
The result follows
from the characterization of the subgradient as
containing the convex hull of all
limits of gradients, e.g.,~\cite[Theorem 25.6]{con:70}.
\end{proof}
\section{Optimality Conditions, Bounding, {\bf GN\,} Interior Point Method}
\label{sect:intpt}
In this section we apply a Gauss-Newton interior point approach to solve the model \eqref{eq:finalredinrho}.
We begin by presenting optimality conditions for the model \eqref{eq:finalredinrho} and then present the algorithm.
We finish this section with some implementation heuristics
followed by bounding strategies.
\subsection{Optimality Conditions and Duality}
We first obtain perturbed optimality conditions for~\cref{eq:finalredinrho}
with positive barrier parameters. This is most often done by
using a barrier function
and adding terms such as $\mu_\rho \log \det ( \rho )$ to the Lagrangian.
After differentiation we obtain $\mu_\rho \rho^{-1}$ that we equate with
the dual variable $Z_\rho$. After multiplying through by $\rho$ we
obtain the \textdef{perturbed complementarity equations}
e.g.,~$Z_\rho \rho -\mu_\rho I =0$.
\begin{theorem}
\label{thm:optcondfinalprob}
Let $L$ be the Lagrangian for \cref{eq:finalredinrho}, i.e.,
\[
L(\rho, y ) = f(\rho) +\langle y, {\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} \rangle,
\, y\in \mathbb{R}^{{m_{\scriptscriptstyle V}}}.
\]
The following holds for problem~\cref{eq:finalredinrho}.
\begin{enumerate}
\item
\[
\begin{array}{rcl}
p^*
&=& \max\limits_{y}
\min\limits_{\rho\succeq 0} L(\rho,y).
\end{array}
\]
\item
\label{item:strduality}
The \textdef{Lagrangian dual} of \cref{eq:finalredinrho} is
\index{dual \textbf{QKD}\,}
\[
\begin{array}{rcl}
d^*
&=& \max\limits_{Z\succeq 0,y} \left(
\min\limits_{\rho} (L(\rho,y)-\langle Z,\rho\rangle)\right),
\end{array}
\]
and strong duality holds for~\cref{eq:finalredinrho}, i.e., $d^*
=p^* $
and $d^* $ is attained for some $(y,Z) \in \mathbb{R}^{m_{\scriptscriptstyle V}}\times \mathbb{H}_+^{n_\rho}$.
\item
\label{item:optcharac}
The primal-dual pair $(\rho,(y,Z))$, with $\partial f(\rho) \neq \emptyset$,
is optimal if, and only if,
\begin{equation}
\label{eq:pdoptcond}
\begin{array}{rcll}
0 &\in& \partial f(\rho) + \Gamma_V^\dagger (y) -Z & \text{(dual feasibility)} \\
0 &=& {\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} & \text{(linear primal feasibility)} \\
0 &=& \langle \rho,Z\rangle & \text{(complementary slackness)} \\
0 &\preceq &\rho,Z& \text{(semidefiniteness primal feasibility)}.
\end{array}
\end{equation}
Moreover, $\Gamma_V^\dagger (y) \succeq 0, \, \langle y,\gamma_V\rangle < 0$, for
some $y$, implies that the primal~\cref{eq:finalredinrho} is infeasible.
\item
\label{eq:2ndorderoptcond}
Let ${{\mathcal N\,}_{\scriptscriptstyle\!\!{{\Gamma_{\scriptscriptstyle V}}}}}$ be injective with $\range ({{\mathcal N\,}_{\scriptscriptstyle\!\!{{\Gamma_{\scriptscriptstyle V}}}}})=\Null ({\Gamma_{\scriptscriptstyle V}})$.
Let $\rho,y,Z$ satisfy the optimality conditions~\cref{eq:pdoptcond}.
Then the second order optimality condition for uniqueness at $\rho$ is that
\[
\mathcal{N}^\dagger _{\scriptscriptstyle{{\Gamma_{\scriptscriptstyle V}}}}
\nabla^2 f(\rho) {{\mathcal N\,}_{\scriptscriptstyle\!\!{{\Gamma_{\scriptscriptstyle V}}}}} \succ 0.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
The dual in \Cref{item:strduality} is obtained from from standard min-max argument; See~\cite[Chapter 5]{MR2061575}.
\[
\begin{array}{rcl}
d^* = \max\limits_y \min\limits_{\rho\in \mathbb{H}_+^{n_\rho}} L(\rho,y)
&=& \max\limits_y \left\{ L(\rho,y) : Z \in
\partial f(\rho) + \Gamma_V^\dagger (y) , \, Z \in (\mathbb{H}_+^{n_\rho}-\rho)^\dagger \right\}
\\&=& \max\limits_{y,Z\succeq 0}
\min\limits_{\rho\in \mathbb{H}_+^{n_\rho}} L(\rho,y)-\langle Z,\rho\rangle.
\end{array}
\]
That strong duality holds comes from our regularization process, i.e.,
the existence of a Slater point; see~\cite[Chapter 8]{Lu:69}.
\Cref{item:optcharac} is the standard optimality conditions for convex
programming, where the dual feasibility $0\in \partial f(\rho) +
\Gamma_V^\dagger (y) -Z $ holds from \Cref{cor:fsubdiff}.
The second-order sufficient conditions in \Cref{eq:2ndorderoptcond} are standard; see~\cite[Chapter 12]{MR2001b:90002}.
\end{proof}
\subsubsection{Perturbed Optimality Conditions }
\label{sec:pertoptconds}
Many interior-point based algorithms try to solve the optimality
conditions \cref{eq:finalredinrho} by solving a sequence of perturbed
problems while driving the perturbation parameter $\mu \downarrow 0$.
The parameter $\mu$ gives a measure of the duality gap.
In this section, we present the perturbed optimality conditions for (\textbf{QKD}).
\begin{theorem}
\label{thm:optcondpd}
The barrier function for~\Cref{eq:finalredinrho} with barrier parameter
$\mu >0$ is
\[
B_\mu( \rho, y ) = f(\rho) +\langle y, {\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} \rangle
-\mu \log \det (\rho) .
\]
With $Z = \mu \rho^{-1}$ scaled to $Z\rho -\mu I =0$, we obtain the
perturbed optimality conditions for \cref{eq:finalredinrho} at $\rho, Z \succ 0$, $y$:
\begin{equation}
\label{eq:pertoptcond}
\begin{array}{lcccl}
\text{dual feasibility } \ \ \ \quad ( \nabla B_{\rho} =0)& : &
F_\mu^d & = & \nabla_\rho f(\rho) +
\Gamma_V^\dagger (y) - Z =0
\\
\text{primal feasibility } \quad (\nabla B_{y} =0 ) &:&
F_\mu^p & = & {\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} =0 \\
\text{perturbed complementary slackness }&:&
F_\mu^c & = & Z \rho - \mu I= 0 . \cr
\end{array}
\end{equation}
In fact, for each $\mu>0$ there is a unique primal-dual solution
$\rho_\mu,y_\mu,Z_\mu$
satisfying~\cref{eq:pertoptcond}. This
defines the central path as $\mu \downarrow 0$.
Moreover,
\[
(\rho_\mu,y_\mu,Z_\mu) \underset{\mu\downarrow 0}{\to}
(\rho,y,Z) \text{ satisfying } \cref{eq:pdoptcond}.
\]
\end{theorem}
\begin{proof}
The optimality condition \eqref{eq:pertoptcond} follows from the necessary and sufficient optimality conditions of convex problem
\[
\min_{\rho} \{ f(\rho) - \mu \log\det (\rho) \ : \ \Gamma_V (\rho) = {\gamma_{\scriptscriptstyle V}} \}
\]
and setting $Z = \mu \rho^{-1}$. Note that $B_\mu$ is the Lagrangian function of this convex problem.
For each $\mu>0$ there exists a unique solution to \eqref{eq:pertoptcond} due to the strict convexity of the barrier term $-\mu \log \det (\rho)$ and boundedness of the level set of the objective.
The standard log barrier argument~\cite{HillierFrederickS2008LaNP,MR2001b:90002}
and \Cref{cor:fsubdiff} together give the last claim.
\end{proof}
\Cref{thm:optcondpd} above provides an interior
point path following method, i.e.,~for each $\mu\downarrow 0$ we solve
the pertubed optimality conditions
\begin{equation}
\label{eq:optcondpert}
F_\mu (\rho,y,Z)=
\begin{bmatrix}
\nabla_\rho f(\rho) + \Gamma_V^\dagger (y) - Z \cr
{\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} \cr
Z \rho - \mu I
\end{bmatrix} = 0, \quad \rho,Z \succ 0.
\end{equation}
The question is how to do this efficiently.
The nonlinear system is overdetermined as
\[
F_\mu : \mathbb{H}^{n_\rho} \times \mathbb{R}^{{m_{\scriptscriptstyle V}}} \times \mathbb{H}^{n_\rho} \to
\mathbb{H}^{n_\rho} \times \mathbb{R}^{{m_{\scriptscriptstyle V}}}\times \mathbb{C}^{n_\rho\times n_\rho}.
\]
Therefore we cannot apply Newton's method directly because the linearization does not yield a square system.
\subsection{Gauss-Newton Search Direction}
To solve the optimality conditions~\cref{eq:optcondpert}, we consider
the equivalent nonlinear least squares problem
\[
\min_{\rho,Z\succ 0,y} g(\rho,y,Z) : =\frac 12 \|F_\mu (\rho,y,Z)\|^2
=\frac 12 \|F_\mu^d (\rho,y,Z)\|_F^2
+\frac 12 \|F_\mu^p (\rho)\|^2
+\frac 12 \|F_\mu^c (\rho,Z)\|_F^2.
\]
\index{$g(\rho,y,Z)$, nonlinear least square function}
\index{nonlinear least square function, $g(\rho,y,Z)$}
The Gauss-Newton method is a popular method for solving nonlinear least squares problems.
The \textdef{Gauss-Newton direction, ${d_{\scriptscriptstyle GN}}$}, is the least squares solution of the
linearization
\[
F_\mu^\prime(\rho,y,Z) {d_{\scriptscriptstyle GN}} = -F_\mu(\rho,y,Z),
\]
where $F_\mu^\prime$ denotes the Jacobian of $F_\mu$.
\index{$F_\mu^\prime$, Jacobian of $F_\mu$}
\index{Jacobian of $F_\mu$, $F_\mu^\prime$}
\begin{lemma}
\label{lem:GNdescent}
Under a full rank assumption of $F_\mu^\prime(\rho,y,Z)$, we get
\[
{d_{\scriptscriptstyle GN}} = -((F_\mu^\prime)(\rho,y,Z)^\dagger F_\mu^\prime(\rho,y,Z) )^{-1}
(F_\mu^\prime(\rho,y,Z))^\dagger F_\mu(\rho,y,Z).
\]
Moreover, if $\nabla g(\rho,y,Z)\neq 0$, then
${d_{\scriptscriptstyle GN}}$ is a descent direction for $g$.
\end{lemma}
\begin{proof}
The gradient of $g$ is, omitting the variables,
\[
\nabla g = (F_\mu^\prime)^\dagger (F_\mu );
\]
and the Gauss-Newton direction is the least squares solution of the
linearization $F_\mu^\prime {d_{\scriptscriptstyle GN}} = -F_\mu$, i.e.,~under a full rank
assumption, we get the solution from the normal equations as
\[
{d_{\scriptscriptstyle GN}} = -((F_\mu^\prime)^\dagger F_\mu^\prime )^{-1}
(F_\mu^\prime)^\dagger F_\mu.
\]
We see that the inner product with the gradient is indeed negative, hence a descent direction.
\end{proof}
We now give an explicit representation of the linearized system for
\eqref{eq:optcondpert}. We define the (right/left matrix multiplication)
linear maps
\[
{\cM_{\scriptscriptstyle Z}},\ {\cM_{\scriptscriptstyle \rho}} : \mathbb{H}^{n_\rho} \to \mathbb{C}^{n_\rho\times n_\rho}, \quad
\textdef{${\cM_{\scriptscriptstyle Z}}(\Delta X) = Z\Delta X$}, \,
\textdef{${\cM_{\scriptscriptstyle \rho}}(\Delta X) = \Delta X\rho$}.
\]
Then the linearization of \eqref{eq:optcondpert} is
\index{${d_{\scriptscriptstyle GN}}$, {\bf GN\,}-direction}
\begin{equation}
\label{eq:GNorig}
\begin{array}{rcl}
F_\mu^\prime {d_{\scriptscriptstyle GN}}
=
\begin{bmatrix}
\nabla^2 f(\rho) \Delta \rho + \Gamma_V^\dagger (\Delta y) -\Delta Z \cr
{\Gamma_{\scriptscriptstyle V}} (\Delta \rho) \cr
Z \Delta \rho + \Delta Z \rho
\end{bmatrix}
\vspace{.1in}
=
\begin{bmatrix}
\nabla^2 f(\rho) & \Gamma_V^\dagger & -I \cr
{\Gamma_{\scriptscriptstyle V}} && \cr
{\cM_{\scriptscriptstyle Z}} && {\cM_{\scriptscriptstyle \rho}}
\end{bmatrix}
\begin{pmatrix}
\Delta \rho\cr \Delta y \cr \Delta Z
\end{pmatrix}
\approx&
-F_\mu.
\end{array}
\end{equation}
We emphasize that the last term is in $\mathbb{C}^{n_\rho\times n_\rho}$ and the system is
overdetermined.
The adjoints of ${\cM_{\scriptscriptstyle Z}},{\cM_{\scriptscriptstyle \rho}}$ are discussed in~\Cref{sect:lintrsadj},
\Cref{lem:WRadj,lem:Srho}.
Solving the system \eqref{eq:GNorig}, we obtain the \textdef{{\bf GN\,}-direction, ${d_{\scriptscriptstyle GN}}$}$\in \mathbb{H}^{n_\rho}\times
\mathbb{R}^{{m_{\scriptscriptstyle V}}}\times \mathbb{H}^{n_\rho}$.
\subsection{Projected Gauss-Newton Directions}
The {\bf GN\,} direction in \eqref{eq:GNorig} solves a relatively large overdetermined
linear least squares system and does not explicitly exploit the zero blocks. We
now proceed to take advantage of the special structure of the linear
system.
\subsubsection{First Projected Gauss-Newton Direction}
\label{sec:1stprojGN}
Given the system \eqref{eq:GNorig}, we can make a substitution for $\Delta Z$ using the first block equation
\begin{equation}
\label{eq:deltaZ}
\Delta Z = F_\mu^d+\nabla^2 f(\rho) \Delta \rho + \Gamma_V^\dagger (\Delta y) .
\end{equation}
This leaves the two blocks of equations
\index{${d_{\scriptscriptstyle GN}}$, {\bf GN\,}-direction}
\index{{\bf GN\,}-direction, ${d_{\scriptscriptstyle GN}}$}
\[
\begin{array}{rcl}
\left(F_\mu^{pc}\right)^\prime \begin{pmatrix}
\Delta \rho \cr \Delta y
\end{pmatrix}
&=&
\begin{bmatrix}
{\Gamma_{\scriptscriptstyle V}} ( \Delta \rho ) \cr
Z \Delta \rho +
\left(\nabla^2 f(\rho) \Delta \rho + \Gamma_V^\dagger ( \Delta y ) \right) \rho
\end{bmatrix}
\vspace{.1in}
\\&=&
\begin{bmatrix}
{\Gamma_{\scriptscriptstyle V}} & \cr
{\cM_{\scriptscriptstyle Z}} +{\cM_{\scriptscriptstyle \rho}} \nabla^2f(\rho) & {\cM_{\scriptscriptstyle \rho}} \Gamma_V^\dagger
\end{bmatrix}
\begin{pmatrix}
\Delta \rho \cr \Delta y
\end{pmatrix}
\\&\approx&
-\begin{bmatrix}
F^p_\mu\cr F^c_\mu +F^d_\mu \rho
\end{bmatrix},
\end{array}
\]
where the superscript in $F_\mu^{pc}$ stands for the primal and complementary slackness constraints.
The adjoint equation follows:
\[
\left[ \left(F_\mu^{pc}\right)^\prime \right]^\dagger
\begin{pmatrix}
r_p \cr R_c
\end{pmatrix}
=
\begin{bmatrix}
\Gamma_V^\dagger & {\cM_{\scriptscriptstyle Z}}^\dagger +\nabla^2f(\rho){\cM_{\scriptscriptstyle \rho}}^\dagger \cr
0 & {\Gamma_{\scriptscriptstyle V}}{\cM_{\scriptscriptstyle \rho}}^\dagger
\end{bmatrix}
\begin{pmatrix}
r_p \cr R_c
\end{pmatrix}.
\]
In addition, we can evaluate the condition number of the system using
$ \left( \left(F_\mu^{pc}\right)^\prime \right)^\dagger \left(F_\mu^{pc}\right)^\prime $.
Note that we include the adjoints as they are needed for matrix free
methods that exploit sparsity.
\subsubsection{Second Projected Gauss-Newton Direction}
\label{sect:2ndprojGNdir}
We can further reduce the size of the linear system by making further
variable substitutions.
Recall that in \Cref{sec:1stprojGN} we solve the system with a variable in $\mathbb{H}^{n_\rho} \times \mathbb{R}^{{m_{\scriptscriptstyle V}}}$, i.e., $n_\rho^2+{m_{\scriptscriptstyle V}}$ number of unknowns.
In this section, we make an additional substitution using the second block equation in \eqref{eq:GNorig} and reduce the number of the unknowns to $n_\rho^2$.
\begin{theorem}
\label{thm:2ndprojGN}
Let $\hat{\rho}\in \mathbb{H}^{n_\rho}$ be a feasible point for ${\Gamma_{\scriptscriptstyle V}} (\cdot)= {\gamma_{\scriptscriptstyle V}}$.
Let \textdef{${\mathcal N} ^\dagger :\mathbb{R}^{n_\rho^2-{m_{\scriptscriptstyle V}}} \to \mathbb{H}^{n_\rho}$} be an
injective linear map in adjoint form
so that, again by abuse of notation and redefining the primal residual,
we have the nullspace representation:
\[
F_\mu^{p} = {\Gamma_{\scriptscriptstyle V}} (\rho) - {\gamma_{\scriptscriptstyle V}} \, \iff \, F_\mu^{p} =
{\mathcal N} ^\dagger (v)+ \hat \rho -\rho , \text{ for some }v.
\]
Then the second projected {\bf GN\,} direction,
${d_{\scriptscriptstyle GN}} = \begin{pmatrix} \Delta v\cr\Delta y \end{pmatrix}$,
is found from the least squares solution of
\begin{equation}
\label{eq:proj2GNeqn}
\begin{array}{l}
\fbox{$
\left[Z {\mathcal N} ^\dagger (\Delta v) +\nabla^2 f(\rho)
{\mathcal N} ^\dagger (\Delta v)\rho \right] + \left[ \Gamma_V^\dagger (\Delta y ) \rho \right]
=
- F_\mu^c -Z F_\mu^{p} - \left( F_\mu^d +\nabla^2 f(\rho)F_\mu^{p}
\right) \rho.$}
\end{array}
\end{equation}
\end{theorem}
\begin{proof}
Using the new primal feasibility representation, the perturbed optimality conditions in~\cref{eq:optcondpert} become:
\begin{equation}
\label{eq:modoptcondpertnullsp}
F_\mu (\rho,v,y,Z)=
\begin{bmatrix}
F_\mu^d\cr
F_\mu^{p}\cr
F_\mu^c
\end{bmatrix}
=
\begin{bmatrix}
\nabla_\rho f(\rho) + \Gamma_V^\dagger ( y ) - Z \cr
{\mathcal N} ^\dagger ( v ) + \hat \rho -\rho \cr
Z \rho - \mu I
\end{bmatrix}
= 0, \quad \rho,Z \succ 0.
\end{equation}
After linearizing the system \eqref{eq:modoptcondpertnullsp}
we use the following to find the {\bf GN\,} search direction:
\[
\begin{array}{rcl}
F_\mu^\prime {d_{\scriptscriptstyle GN}}
=
\begin{bmatrix}
\nabla^2 f(\rho) \Delta \rho + \Gamma_V^\dagger (\Delta y) -\Delta Z \cr
{\mathcal N} ^\dagger (\Delta v) -\Delta \rho\cr
Z \Delta \rho + \Delta Z \rho
\end{bmatrix}
\vspace{.1in}
\approx
-F_\mu.
\end{array}
\]
From the first block equation we have
\[
\begin{array}{rcl}
\Delta Z
&=&
F_\mu^d+\nabla^2 f(\rho) \Delta \rho + \Gamma_V^\dagger ( \Delta y )
\\&=&
F_\mu^d+\nabla^2 f(\rho) (F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
+ \Gamma_V^\dagger ( \Delta y ).
\end{array}
\]
From the second block equation, we have
\[
\Delta \rho = F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v).
\]
Substituting $\Delta Z$ and $\Delta \rho$ into $Z \Delta \rho + \Delta Z \rho $ gives
\[
\begin{array}{rclcl}
Z \Delta \rho + \Delta Z \rho
&=&
Z ( F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
+\left[F_\mu^d+\nabla^2 f(\rho)
(F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
+ \Gamma_V^\dagger (\Delta y) \right]\rho
\\&=&
\left[Z {\mathcal N} ^\dagger (\Delta v) +\nabla^2 f(\rho)
{\mathcal N} ^\dagger (\Delta v)\rho \right]
+ \left[ \Gamma_V^\dagger (\Delta y ) \rho \right]
+ Z F_\mu^{p} + \left( F_\mu^d +\nabla^2 f(\rho)F_\mu^{p} \right) \rho .
\end{array}
\]
Rearranging the terms, the third block equation becomes
\[
\begin{array}{rcl}
F_\mu^{c\prime}\begin{pmatrix}\Delta v\cr \Delta y\end{pmatrix}
& = &
\left[Z {\mathcal N} ^\dagger (\Delta v) +\nabla^2 f(\rho)
{\mathcal N} ^\dagger (\Delta v)\rho \right] + \left[ \Gamma_V^\dagger ( \Delta y ) \rho \right]
\\& = &
- F_\mu^c -Z F_\mu^{p} - \left( F_\mu^d +\nabla^2 f(\rho)F_\mu^{p} \right) \rho.
\end{array}
\]
\end{proof}
The matrix representation of \eqref{eq:proj2GNeqn} is presented in \Cref{app:matrix_repre_sys}.
It is easy to see that the adjoint satisfying
$ \langle F_\mu^{c\prime} ({d_{\scriptscriptstyle GN}} ) , R_c \rangle =
\langle {d_{\scriptscriptstyle GN}}, (F_\mu^{c\prime})^\dagger (R_c) \rangle $
now follows:
\[
(F_\mu^{c\prime})^\dagger (R_c) =
\begin{bmatrix}
{\mathcal N} \Hvec {\cM_{\scriptscriptstyle Z}}^\dagger + {\mathcal N} \nabla^2 f(\rho)\Hvec {\cM_{\scriptscriptstyle \rho}}^\dagger \cr
{\Gamma_{\scriptscriptstyle V}} {\cM_{\scriptscriptstyle \rho}}^\dagger
\end{bmatrix}(R_c) .
\]
After solving the system \eqref{eq:proj2GNeqn}, we make back substitutions to recover the original variables. In other words,
once we get $(\Delta v, \Delta y)$ from solving \eqref{eq:proj2GNeqn},
we obtain $(\Delta \rho, \Delta y, \Delta Z)$ using the original system:
\[
\Delta \rho = F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v), \, \quad
\Delta Z =
F_\mu^d+\nabla^2 f(\rho) (F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
+ \Gamma_V^\dagger (\Delta y ).
\]
\Cref{thm:exactPR} below illustrates cases where we maintain the exact primal feasibility.
\begin{theorem}
\label{thm:exactPR}
Let $\alpha$ be a steplength and consider the update
\[
\rho_+ \leftarrow \rho + \alpha \Delta \rho = \rho + F_\mu^{p}+\alpha {\mathcal N} ^\dagger (\Delta v) .
\]
\begin{enumerate}
\item
If a steplength one is taken ($\alpha=1$),
then the new primal residual is exact, i.e.,
\[
F_\mu^p={\mathcal N} ^\dagger ( v_+ ) + \hat \rho -\rho_+=0.
\]
\item
Suppose that the exact primal feasibility is achieved. Then the primal residual is $0$ throughout the iterations regardless of the steplength.
\end{enumerate}
\end{theorem}
\begin{proof}
If a steplength one is taken for updating
\[
\rho_+ \leftarrow \rho + \Delta \rho = \rho + F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v),
\]
then the new primal residual
\[
\begin{array}{rcl}
(F_\mu^p)_+
&=&
{\mathcal N} ^\dagger (v_+) + \hat \rho - \rho_+
\\&=&
{\mathcal N} ^\dagger (v+\Delta v) + \hat \rho - \rho - F_\mu^{p}-{\mathcal N} ^\dagger (\Delta v)
\\&=&
{\mathcal N} ^\dagger (v) + \hat \rho - \rho - {\mathcal N} ^\dagger (v) -\hat \rho + \rho
\\&=&
0.
\end{array}
\]
In other words, as for Newton's method, a step length of one implies that the new
residuals are zero for linear equations.
We can now change the line
search to maintain $\rho_+ = {\mathcal N} ^\dagger (v+ \alpha \Delta v) -\hat
\rho \succ 0$ and preserve exact primal feasibility.
Assume that $F_\mu^{p}=0$.
\[
\begin{array}{rcl}
\rho_+
\leftarrow
\rho + \alpha \Delta \rho
= \rho + \alpha ( F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
= \rho + \alpha {\mathcal N} ^\dagger (\Delta v)
\end{array}
\]
Now, we see that
\[
{\Gamma_{\scriptscriptstyle V}} (\rho_+) = {\Gamma_{\scriptscriptstyle V}} ( \rho + \alpha {\mathcal N} ^\dagger (\Delta v) )
= {\Gamma_{\scriptscriptstyle V}} ( \rho ) = \gamma,
\]
where the last equality follows from the exactly feasibility assumption.
\end{proof}
\subsection{Projected Gauss-Newton Primal-Dual Interior Point Algorithm}
We now present the pseudocode for the Gauss-Newton primal-dual interior point method.
\Cref{algo:GNalgo} is summarized as follow; it is a series of solving the over-determined linear system \eqref{eq:proj2GNeqn} while decreasing the perturbation parameter $\mu \downarrow 0$ and maintaining the positive definiteness of $\rho, Z$.
\index{algorithm, {\bf GN\,} interior point for \textbf{QKD}}
\begin{algorithm}
\caption{Projected Gauss-Newton Interior Point Algorithm for (\textbf{QKD})}
\begin{algorithmic}
\REQUIRE $\hat{\rho}\succ 0$, $\mu \in \mathbb{R}_{++}$, $\eta \in (0,1)$
\WHILE{stopping criteria is not met}
\STATE solve \eqref{eq:proj2GNeqn} for $(\Delta v, \Delta y)$
\STATE $\Delta \rho = F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v)$
\STATE $\Delta Z = F_\mu^d+\nabla^2 f(\rho) (F_\mu^{p}+{\mathcal N} ^\dagger (\Delta v))
+ \Gamma_V^\dagger (\Delta y) $
\STATE choose steplength $\alpha$
\STATE $(\rho,y,Z) \leftarrow (\rho,y,Z) + \alpha (\Delta \rho, \Delta y, \Delta Z)$
\STATE $\mu \leftarrow \langle \rho,Z\rangle/n_\rho;\, \mu \leftarrow \eta \mu$
\ENDWHILE
\end{algorithmic}
\label{algo:GNalgo}
\end{algorithm}
\subsection{Implementation Heuristics}
We now discuss the implementation details. This involves preprocessing
for a nullspace representation and preconditioning. The details follow.
\subsubsection{Stopping Criteria}
We terminate the algorithm when the optimality condition
\eqref{eq:optcondpert} is approximately satisfied.
Denote the residual in~\Cref{thm:2ndprojGN} by
\[
\text{RHS} = - F_\mu^c -Z F_\mu^{p} - \left( F_\mu^d +\nabla^2 f(\rho)F_\mu^{p} \right) \rho,
\]
and the denominator term by
\[
\text{denom} =
1 + \frac 12\min\left\{ \ \|\rho\|+\|Z\| \ , \
|\text{bestub}| + | \text{bestlb}| \ \right\}.
\]
Then
\begin{equation}
\label{def:stopgap}
\text{relstopgap} = \frac 1{\text{denom}}
\max \left\{ \ \text{bestub} - \text{bestlb} \ , \
\| \text{RHS} \| \ \right\}.
\end{equation}
In other words, for a pre-defined tolerance $\epsilon$, we terminate the
algorithm when the relstopgap$ < \epsilon$.
If the algorithm computes lower and upper bounds of the optimal value throughout its execution, we may terminate the algorithm when the gap between lower and upper bounds is within $\epsilon$.
Finally, a common way to terminate an algorithm is to impose restrictions on the running time, e.g., setting an upper bound on the number of iterations or the physical running time.
\subsubsection{{\bf GN\,} Direction using Sparse Nullspace Representation}
\label{sect:sparsenullrep}
We let $r = \Hvec (\rho) $, and construct
a matrix representation $H$ for the Hessian, and a matrix representation
$M$ for the linear constraints that
includes a permutation of rows and columns $rp,cp$ with inverse column
permutation $icp$, so that
\[
r=\Hvec(\rho): \quad \textdef{$r(cp) = P_{cp}r$},\,\textdef{$r =
P_{icp}r(cp)$},\quad P_{cp}P_{icp}=P_{icp}P_{cp}=I,\, P_{icp}=P_{cp}^T.
\]
We can ignore the row permutations. We have
\[
\begin{array}{rcl}
{\Gamma_{\scriptscriptstyle V}} (\rho )
&=&
({\Gamma_{\scriptscriptstyle V}} \HMat) \Hvec(\rho)
\\&=&
({\Gamma_{\scriptscriptstyle V}} \HMat) P_{icp}P_{cp}\Hvec(\rho)
\\&=&
(P_{cp}({\Gamma_{\scriptscriptstyle V}} \HMat)^T)^T P_{cp}\Hvec(\rho)
\\&=&
M r(cp)\\
& =& M P_{cp}\Hvec(\rho).
\end{array}
\]
We now get the nullspace representation:
\[
\hat r = \Hvec (\hat \rho) ;\,
{\Gamma_{\scriptscriptstyle V}} ( \hat \rho ) = M \hat r(cp) = {\gamma_{\scriptscriptstyle V}}, \, M = \begin{bmatrix} B & E
\end{bmatrix}, \,
N^\dagger = \begin{bmatrix}B^{-1} E \cr -I\end{bmatrix};
\]
\begin{equation}
\label{eq:nullspacerep}
r=\Hvec (\rho) :\quad
{\Gamma_{\scriptscriptstyle V}}( \rho) = {\gamma_{\scriptscriptstyle V}} \iff
{{\mathcal M} _{\scriptscriptstyle\!{{\Gamma_{\scriptscriptstyle V}}}}} P_{cp} r = {\gamma_{\scriptscriptstyle V}} \, \iff\, r = \hat r +
P_{icp} N^\dagger ( w ), \text{ for some } w.
\end{equation}
The permutation of rows and colums are done in order to obtain a simple,
near triangular, well conditioned $B$ so that $B^{-1}E$ can be done
simply and maintain sparsity if possible.
The permutation of the rows does not affect the problem and we can
ignore it. However the permutation of the columns cannot be ignored.
We get the following
\[
{\mathcal N} ^\dagger (v) =\HMat \left(P_{icp} N^\dagger (v)\right) , \, \Gamma_V^\dagger (\Delta y) = P_{icp}M (\Delta y) , \,
\nabla^2f(\rho){\mathcal N} ^\dagger (\Delta v) = \HMat \left(HP_{icp}N^\dagger (\Delta v) \right).
\]
By abuse of notation, the Gauss-Newton direction ${d_{\scriptscriptstyle GN}}\in \mathbb{R}^{n_\rho^2}$
can now be found from:
\begin{equation}
\label{eq:2ndprojelim}
\begin{array}{rcl}
F_\mu^{c\prime}{d_{\scriptscriptstyle GN}}
&=&
Z\HMat \left(P_{icp}N^\dagger (\Delta v)\right) +
\left(\HMat \left(\nabla^2f(\rho)(P_{icp}N^\dagger (\Delta v))\right)+\Gamma_V^\dagger (\Delta y) \right) \rho
\\&=&
\begin{bmatrix}
{\cM_{\scriptscriptstyle Z}}\left(\HMat \left(P_{icp}N^\dagger \bigcdot\right) \right) + {\cM_{\scriptscriptstyle \rho}}\left(\HMat \left(\nabla^2 f(\rho)
P_{icp}N^\dagger \bigcdot\right) \right) &
{\cM_{\scriptscriptstyle \rho}} \Gamma_V^\dagger \bigcdot
\end{bmatrix}
\begin{pmatrix}
\Delta v \cr \Delta y
\end{pmatrix}
\\&=&
-(F^c_\mu +F^d_\mu \rho + ZF^p_\mu) -
\left(\nabla^2 f(\rho)(F_\mu^{p})\right) \rho.
\end{array}
\end{equation}
\subsubsection{Preconditioning}
\label{sect:precond}
The overdetermined linear system in~\cref{eq:2ndprojelim} can be
ill-conditioned. We use diagonal preconditioning, i.e.,~we let
$d_i=\|F_\mu^{c\prime}(e_i)\|$, for unit vectors $e_i$ and then
column precondition using
\[
F_\mu^{c\prime} \leftarrow F_\mu^{c\prime} \Diag(d)^{-1}.\footnote{The
MATLAB command is:
$d_{GN} = ((F_\mu^{c\prime}/ \Diag(d))\backslash \text{RHS})./d$.}
\]
This diagonal preconditioning has been shown to be the optimal diagonal
preconditioning for the so-called $\Omega$-condition
number,~\cite{DeWo:90}. It performs exceptionally well in our tests
below.
\subsubsection{Step Lengths}
The {\bf GN\,} method is based on a linearization that suggests a steplength of
one. However, long step methods are known to be more efficient in practice for
interior point methods for linear \textbf{SDP} s. Typically steplengths are
found using backtracking to ensure primal-dual
positive definiteness of $\rho, Z$.
In our case we do not have a
linear objective and we typically experience Maratos type situations,
i.e.,~we get fast convergence for primal feasibility but slow and no
convergence for dual feasibility. However, we do have the gradient and
Hessian of the objective function and therefore can minimize the
quadratic model for the objective function in the search direction
$\Delta \rho$
\[
\min_\alpha f(\rho) + \alpha \langle \nabla f(\rho),\Delta \rho \rangle
+ \frac 12 \alpha^2 \langle \Delta \rho \nabla^2 f(\rho),\Delta \rho
\rangle, \quad \alpha^* = -\langle \nabla f(\rho),\Delta \rho\rangle/
\langle \Delta \rho , \nabla^2 f(\rho)\Delta \rho\rangle.
\]
Therefore, we begin the backtracking with this step.
Moreover, we take a steplength of one as soon as possible, and only
after this do we allow steplengths larger than one. This means
that exact primal feasibility holds for all further steps. This happens
relatively early for our numerical tests.
\subsection{Dual and Bounding}
\label{sect:dualbnds}
We first look at upper bounds\footnote{Our discussion about upper bounds here is about upper bounds for the given optimization problem, which are not necessarily key rate upper bounds of the QKD protocol under study. This is because the constraints that one feeds into the algorithm might not use all the information available to constrain Eve's attacks.} found from feasible solutions
in~\Cref{prop:iterrefine}. Then we use
the dual program to provide provable lower bounds for the \textbf{FR}\,
problem \eqref{eq:keyrateopt} thus providing lower bounds for the
original problem with the accuracy of \textbf{FR}.
\subsubsection{Upper Bounds}
A trivial upper bound is obtained as soon as we have a primal feasible
solution $\hat \rho$ by evaluating the objective function. Our algorithm is a
primal-dual \emph{infeasible} interior point approach. Therefore we
typically have approximate linear feasibility ${\Gamma_{\scriptscriptstyle V}} (\hat \rho) \approx {\gamma_{\scriptscriptstyle V}}$;
though we do maintain positive definiteness $\hat \rho\succ 0$ throughout the
iterations. Therefore, once we are close to feasibility we can project
onto the affine manifold and hopefully maintain positive definiteness,
i.e.,~we apply iterative refinement by finding the projection
\[
\min_\rho \left\{ \frac 12 \|\rho - \hat \rho\|^2 \ : \ {\Gamma_{\scriptscriptstyle V}} (\rho) = {\gamma_{\scriptscriptstyle V}} \right\}.
\]
\begin{prop}
\label{prop:iterrefine}
Let $\hat \rho \succ 0,\,F_\mu^p = {\Gamma_{\scriptscriptstyle V}} ( \hat \rho ) - {\gamma_{\scriptscriptstyle V}}$. Then
\[
\rho = \hat \rho - {\Gamma_{\scriptscriptstyle V}}^{-1} F_\mu^p
= \mathop{\rm argmin}_\rho \left\{ \frac 12 \|\rho - \hat \rho\|^2 \ : \ {\Gamma_{\scriptscriptstyle V}} (\rho) = {\gamma_{\scriptscriptstyle V}} \right\},
\]
where we denote \textdef{${\Gamma_{\scriptscriptstyle V}}^{-1}$, generalized inverse}.
If $\rho \succeq 0$, then $p^* \leq f(\rho)$.
\qed
\end{prop}
\index{generalized inverse, ${\Gamma_{\scriptscriptstyle V}}^{-1}$}
In our numerical experiments below we see
that we obtain valid upper bounds
starting in the early iterations and, as we use a Newton type method,
we maintain exact primal feasibility throughout the iterations resulting
in a zero primal residual, and no further need for the projection.
As discussed above, we take a step length of one as soon as
possible. This means that exact primal feasibility holds for the
remaining iterations and we keep improving the upper bound at each
iteration.
\subsubsection{Lower Bounds for \textbf{FR}\, Problem}
\label{sect:LBforFR}
Facial reduction for the affine constraint means that the corresponding
feasible set of the original problem lies within the minimal face
$V_\rho \mathbb{H}^{n_\rho}_+ V_\rho^\dagger$ of the semidefinite cone.
Since we maintain positive definiteness for $\rho,Z$ during the
iterations, we can obtain a lower bound using weak duality.
Note that $\rho \succ 0$ implies that the gradient
$\nabla f(\rho)$ exists.
\begin{cor}[lower bound for \textbf{FR}\,~\cref{eq:finalredinrho}]
\label{cor:lowerbound}
Consider the problem~\cref{eq:finalredinrho}.
Let $\hat \rho,\hat y$ be a primal-dual iterate with $\hat
\rho\succ 0$. Let
\[
\bar Z = \nabla f(\hat \rho) +\Gamma_V^\dagger (\hat y) .
\]
If $\bar Z\succeq 0$,
then a lower bound for problem~\cref{eq:finalredinrho} is
\[
p^* \geq f(\hat\rho) + \langle \hat y, {\Gamma_{\scriptscriptstyle V}} (\hat \rho) -{\gamma_{\scriptscriptstyle V}} \rangle -\langle
\hat \rho,\bar Z\rangle.
\]
\end{cor}
\begin{proof}
Consider the dual problem
\[
d^* = \max_{y,Z\succeq 0}
\min_{\rho\in \mathbb{H}^{n_\rho}} L(\rho,y)-\langle Z,\rho\rangle.
\]
We now have dual feasibility
\[
\bar Z\succeq 0, \,
\nabla f(\hat \rho) +\Gamma_V^\dagger (\hat y) - \bar Z = 0 \implies
\hat \rho \in \mathop{\rm argmin}_\rho L(\rho,\hat y) - \langle \bar Z,\rho\rangle.
\]
Since we have dual feasibility, weak duality
in~\Cref{thm:optcondfinalprob},~\Cref{item:strduality} as stated
in the dual problem above yields the result.
\end{proof}
\begin{remark}
We note that the lower bound in~\Cref{cor:lowerbound}
is a simplification of the approach in~\cite{winick2017reliable}, where
after a near optimal solution is found,
a dual problem of a linearized problem is solved using CVX in MATLAB. Then a strong duality
theorem is assumed to hold and is applied along with a linearization of
the objective function.
Here we do not assume strong duality, though it holds for the facially
reduced problem. And we apply weak duality
to get a theoretically guaranteed lower bound.
We emphasize that this holds within the margin of error of the \textbf{FR}.
Recall that we started with the problem in~\cref{eq:absQKD}. If we only
apply the accurate \textbf{FR}\, based on spectral decompositions, then the lower
bound from~\Cref{cor:lowerbound} is accurate and theoretically valid up
to the accuracy of the spectral decompositions.\footnote{Note that the
condition number of the spectral decomposition of Hermitian matrices is
$1$; see e.g.,~\cite{MR98m:65001}.} In fact, in our numerics, we can
obtain tiny gaps of order $10^{-13}$ when requested; and we have never
encountered a case where the lower bound is greater than the upper
bound. Thus the bound applies to our original problem as well.
Greater care must be taken if we had to apply \textbf{FR}\, to the entire
constraint $\Gamma(\rho)=\gamma$. The complexity of \textbf{SDP}\, feasibility is
still not known. Therefore, the user should be aware of the difficulties
if the full \textbf{FR}\, is done.
A corresponding result for a lower bound for the original problem is
given in~\Cref{cor:lowerboundpert}.
\end{remark}
\subsubsection{Lower Bounds for the Original Problem}
We can also obtain a lower bound for the case where \textbf{FR}\, is performed with
some error. Recall that we assume that the original
problem~\cref{eq:absQKD} is feasible. We follow the same arguments as
in~\Cref{sect:LBforFR} but apply it to the original problem. All that
changes is that we have to add a small perturbation to the optimum
$V_\rho \hat R V_\rho^\dagger$ from the \textbf{FR}\, problem in order
to ensure a positive definite $\rho$ for differentiability.
The exposing vector from \textbf{FR}\, process presents an intuitive choice for
the perturbation.
\begin{cor}
\label{cor:lowerboundpert}
Consider the original problem~\cref{eq:absQKD} and the results
from the theorem of the alternative, \Cref{lem:FRfarkas}, for fixed $y$:
\begin{equation}
\label{eq:thmalternforLB}
0 \neq W = \Gamma^\dagger (y) \succeq 0, \ \gamma^\dagger y =
\epsilon_\gamma, \quad \epsilon_\gamma \geq 0.
\end{equation}
Let the orthogonal spectral decomposition be
\[
W = \begin{bmatrix} V & N \end{bmatrix}
\begin{bmatrix} D_\delta & 0 \cr 0 & D_> \end{bmatrix}
\begin{bmatrix} V & N \end{bmatrix}^\dagger, \, D_> \in \mathbb{S}_{++}^r.
\]
Let $0\preceq \eta\approx W$ be the (approximate)
exposing vector obtained as the nearest rank $r$ positive semidefinite
matrix to W,
\[
W = N D_> N^\dagger + V D_\delta V^\dagger
= \eta + V D_\delta V^\dagger .
\]
Let $\hat R, \hat y$ be a primal-dual iterate for the \textbf{FR}\, problem,
with $\hat R\succ 0$.
Add a small perturbation matrix $\Phi \succ 0$ to guarantee that the
approximate optimal solution
\[
\hat \rho_\phi = V\hat R V^\dagger +N\Phi N^\dagger \succ 0.
\]
Without loss of generality, let $\hat y$ be a dual variable
for~\cref{eq:absQKD}, adding zeros to extend the given vector if needed. Set
\begin{equation}
\label{eq:generalZpsd}
\bar Z_\phi = \nabla f(\hat \rho_\phi )
+ \Gamma^\dagger(\hat y).
\end{equation}
If $\bar Z_\phi \succeq 0$,
then a lower bound for the original problem~\cref{eq:absQKD} is
\begin{equation}
\label{eq:generalLB}
p^* \geq f(\hat \rho_\phi) + \langle \hat y, \Gamma (\hat \rho_\phi)
-\gamma \rangle
-\langle \hat \rho_\phi,\bar Z_\phi\rangle.
\end{equation}
\end{cor}
\begin{proof}
By abuse of notation, we let $f,L$ be the objective function and
Lagrangian for ~\cref{eq:absQKD}.
Consider the dual problem
\[
d^* = \max_{y,Z\succeq 0}
\min_{\rho\in \mathbb{H}^n} \big(L(\rho,y)-\langle Z,\rho\rangle\big).
\]
We now have dual feasibility
\[
\bar Z_\phi\succeq 0, \,
\nabla f(\hat \rho_\phi) +\Gamma^\dagger (\hat y) - \bar Z_\phi = 0 \implies
\hat \rho_\phi \in \mathop{\rm argmin}_\rho \big(L(\rho,\hat y) - \langle \bar Z_\phi,\rho\rangle\big).
\]
Since we have dual feasibility, weak duality
in~\Cref{thm:optcondfinalprob},~\Cref{item:strduality} as stated
in the dual problem above yields the result.
\end{proof}
\begin{remark}
We note that~\cref{eq:generalZpsd} with $\bar Z_\phi$ is dual feasiblity
(stationarity of the Lagrangian) for an optimal $\hat \rho_\phi$.
Therefore, under continuity arguments, we expect $\bar Z_\phi\succeq 0$ to
hold as well.
In addition, for implementation we need to be able to evaluate $\nabla
f(\hat \rho_\phi)$. Therefore, we need to form the positive definite
preserving maps $\widehat \cG, \widehat \cZ$, but without performing \textbf{FR}\, on the
feasible set. That we can do this accurately using a spectral
decomposition follows from~\Cref{lem:tranform}.
\end{remark}
\section{Numerical Testing}\label{sect:tests}
We compare our algorithm to other algorithms by considering six \textbf{QKD}\, protocols including four variants of the Bennett-Brassard 1984 (BB84) protocol, twin-field \textbf{QKD}\, and discrete-modulated
continuous-variable \textbf{QKD}. In~\Cref{sect:testexamples} we include the descriptions of protocol examples
that we use to generate instances for the numerical tests.
We continue with the tests
in~\Cref{sect:compnumericstheory,sect:challengingtests,sect:comperform}.
This includes security analysis of some selected \textbf{QKD}\, protocols and
comparative performances among different algorithms.
In particular, in~\Cref{sect:compnumericstheory}, we compare the results
obtained by our algorithm with the analytical results for selected test examples where tight analytical results can be obtained.
In \Cref{sect:challengingtests}, we present results where it is quite
challenging for the previous method in~\cite{winick2017reliable} to
produce tight lower bounds. In particular, we consider the
discrete-modulated continuous-variable \textbf{QKD}\, protocol and compare results obtained in~\cite{Lin2019}. In~\Cref{sect:comperform}, we compare performances among different algorithms in terms of accuracy and running time.
\subsection{Comparison between the Algorithmic Lower Bound and the Theoretical Key Rate}
\label{sect:compnumericstheory}
We compare results from instances for which there exist tight analytical key rate expressions to demonstrate that our Gauss-Newton method can achieve high accuracy with respect to the analytical key rates. There are known analytical expressions for entanglement-based BB84, prepare-and-measure BB84 as well as measurement-device-independent BB84 variants mentioned in \Cref{sect:testexamples}. We take the measurement-device-independent BB84 as an example since it involves the largest problem size among these three examples and therefore more numerically challenging. In \Cref{fig:experiment1}, we present instances with different choices of parameters for data generation. The instances are tested with a desktop compter that runs with the operating system Ubuntu 18.04.4 LTS, MATLAB version 2019a, Intel Xeon CPU E5-2630 v3 @ 2.40GHz $\times$ 32 and 125.8 Gigabyte memory. We set the tolerance $\epsilon = 10^{-12}$ for the Gauss-Newton method.
In \Cref{fig:experiment1}, the numerical lower bounds from the Gauss-Newton method are close to the analytical results to at least 12 decimals and in many cases they agree up to 15 decimals.
\begin{figure}[ht!]
\centering
\includegraphics[height=6cm]{images/exp1test3.eps}
\caption{Comparisons of key rate for measurement-device-independent BB84 (\Cref{sect:example3}) between our Gauss-Newton method and the known analytical key rate.}
\label{fig:experiment1}
\end{figure}
As noted in \Cref{sect:example5}, analytical results are also known when the channel noise parameter $\xi$ is set to zero since in this case, one may argue the optimal eavesdropping attack is the generalized beam splitting attack. This means the feasible set contains essentially a single $\rho$ up to unitaries. Since our objective function is unitary invariant, one can analytically evaluate the key rate expression. In \Cref{fig:experiment2}, we compare the results from the Gauss-Newton method with the analytical key rate expressions for different choices of distances $L$ (See \Cref{sect:example5} for the description about instances of this protocol example). These instances were run in the same machine as in \Cref{fig:experiment1}. We set the tolerance $\epsilon = 10^{-9}$ for the Gauss-Newton method.
\subsection{Solving Numerically Challenging Instances}
\label{sect:challengingtests}
We show results where the Frank-Wolfe method without \textbf{FR}\, has difficulties in providing tight lower bounds in certain instances. In \Cref{fig:experiment2}, we plot results obtained previously in~\cite[Figure 2(b)]{Lin2019} by the Frank-Wolfe method without \textbf{FR}. In particular, results from Frank-Wolfe method have visible differences from the analytical results starting from distance $L = 60$ km. In addition the lower bounds are quite loose once the distance reaches $150$ km. In fact, there are points like the one around $180$ km where the Frank-Wolfe method cannot produce nontrivial (nonzero) lower bounds. On the other hand, the Gauss-Newton method provides much tighter lower bounds.
\begin{figure}[ht!]
\centering
\includegraphics[height=7cm]{images/exp1test5.eps}
\caption{Comparison of key rate for discrete-modulated
continuous-variable \textbf{QKD}\, (\Cref{sect:example5}) among our Gauss-Newton method, the Frank-Wolfe method and analytical key rate for the noise $\xi = 0$ case.}
\label{fig:experiment2}
\end{figure}
In \Cref{fig:experiment2test6}, we show another example to demonstrate the advantages of our method. These instances were run in the same machine as in \Cref{fig:experiment2}. For this discrete-phase-randomzied BB84 protocol with $5$ discrete global phases (see \Cref{sect:example6} for more descriptions), the previous Frank-Wolfe method was unable to find nontrivial lower bounds. This is because the previous method can only achieve an accuracy around $10^{-3}$ for this problem due to the problem size. This is insufficient to produce nontrivial lower bounds for many instances since the key rates are on the order of $10^{-3}$ or lower as shown in \Cref{fig:experiment2test6}. On the other hand, due to high accuracy of our method, we can obtain meaningful key rates. The advantage of high accuracy achieved by our method enables us to perform security analysis for protocols that involve previously numerically challenging problems. Like the discrete-phase-randomized BB84 protocol, these protocols involve more signal states, which lead to higher-dimensional problems.
\begin{figure}[ht!]
\centering
\includegraphics[height=7cm]{images/exp2test6.eps}
\caption{Key rate for discrete-phase-randomized BB84 (\Cref{sect:example6}) with the number of discrete global phases $c=5$. In this plot, the coherent state amplitude is optimized for each distance by a simple coarse-grained search over the parameter regime.}
\label{fig:experiment2test6}
\end{figure}
\subsection{Comparative Performance}
\label{sect:comperform}
In this section we examine the comparative performance among three algorithms; the Gauss-Newton method, the Frank-Wolfe method and cvxquad.
The Gauss-Newton method refers to the algorithm developed throughout this paper. The Frank-Wolfe method refers to the algorithm developed in~\cite{winick2017reliable} and cvxquad is developed in~\cite{cvxquad}.
We use \Cref{table:FINAL} to present detailed reports on some selected instances. More numerics are reported throughout \Cref{table:table1,table:table2,table:table3,table:table4,table:table5,table:table6} in \Cref{sec:additionalreport}.
The instances are tested with MATLAB version 2020a using
Dell PowerEdge R640 Two Intel Xeon Gold 6244 8-core 3.6 GHz (Cascade Lake) with 192 Gigabyte memory.
For the instances corresponds to the DMCV protocol, we used the tolerance $\epsilon = 10^{-9}$ and the tolerance $\epsilon = 10^{-12}$ was used for the remaining instances.
The maximum number of iteration was set to $80$.
\begin{table}[H]
\hspace{-0.5cm}
\tiny{
\begin{tabular}{|ccc||cc||cc||cc||cc|} \hline
\multicolumn{3}{|c||}{\textbf{Problem Data}} & \multicolumn{2}{|c||}{\textbf{Gauss-Newton}} & \multicolumn{2}{|c||}{\textbf{Frank-Wolfe with FR}} & \multicolumn{2}{|c||}{\textbf{Frank-Wolfe w/o FR}} & \multicolumn{2}{|c|}{\textbf{cvxquad with FR}} \cr\hline
\multicolumn{1}{|c|}{\textbf{protocol}} & \multicolumn{1}{|c|}{\textbf{parameter}} & \multicolumn{1}{|c||}{\textbf{size}} & \multicolumn{1}{|c|}{\textbf{gap}} & \multicolumn{1}{|c||}{\textbf{time}} & \multicolumn{1}{|c|}{\textbf{gap}} & \multicolumn{1}{|c||}{\textbf{time}} & \multicolumn{1}{|c|}{\textbf{gap}} & \multicolumn{1}{|c||}{\textbf{time}} & \multicolumn{1}{|c|}{\textbf{gap}} & \multicolumn{1}{|c|}{\textbf{time}} \cr\hline
ebBB84 & (0.50,0.05) & (4,16) &5.98e-13 & 0.63 & 1.01e-04 & 84.39 & 1.17e-04 & 94.71 & 5.46e-01 & 216.37 \cr
ebBB84 & (0.90,0.07) & (4,16) &2.33e-13 & 0.25 & 2.32e-04 & 85.09 & 2.54e-04 & 113.20 & 7.39e-01 & 647.60 \cr
pmBB84 & (0.50,0.05) & (8,32) &5.51e-13 & 0.24 & 3.13e-05 & 1.85 & 6.47e-04 & 1.47 & 5.26e-01 & 170.12 \cr
pmBB84 & (0.90,0.07) & (8,32) &1.01e-12 & 0.17 & 7.31e-05 & 1.04 & 6.25e-04 & 31.77 & 6.84e-01 & 235.89 \cr
mdiBB84 & (0.50,0.05) & (48,96) &7.86e-13 & 1.08 & 9.62e-05 & 1.54 & 5.39e-04 & 134.79 & 1.82e-01 & 588.71 \cr
mdiBB84 & (0.90,0.07) & (48,96) &2.96e-13 & 1.12 & 1.51e-04 & 101.84 & 3.48e-03 & 408.26 & 4.57e-01 & 574.31 \cr
TFQKD & (0.80,100,0.70) & (12,24) &7.67e-13 & 1.20 & 1.98e-04 & 96.08 & 1.55e-03 & 179.57 & 3.98e-03 & 990.92 \cr
TFQKD & (0.90,200,0.70) & (12,24) &3.42e-12 & 0.96 & 1.92e-05 & 2.07 & 1.65e-04 & 2.15 & 2.26e-04 & 875.44 \cr
DMCV & (10,60,0.05,0.35) & (44,176) &2.74e-09 & 510.66 & 2.44e-06 & 1015.14 & 3.36e-06 & 1709.65 & $\star\star$ & 0.86 \cr
DMCV & (11,120,0.05,0.35) & (48,192) &3.23e-09 & 720.61 & 2.60e-06 & 348.81 & 1.98e-06 & 628.25 & $\star\star$ & 1.24 \cr
dprBB84 & (1,0.08,30) & (12,48) &4.92e-13 & 0.93 & 3.79e-06 & 77.86 & 9.38e-05 & 108.50 & $\star\star$ & 119.20 \cr
dprBB84 & (2,0.14,30) & (24,96) &1.04e-12 & 10.07 & 6.19e-06 & 15.61 & 3.62e-06 & 27.79 & $\star\star$ & 105.40 \cr
dprBB84 & (3,0.10,30) & (36,144) &4.96e-13 & 61.32 & 6.48e-04 & 7.89 & 2.08e-02 & 28.46 & $\star\star$ & 614.71 \cr
dprBB84 & (4,0.12,30) & (48,192) &1.13e-12 & 272.09 & 4.41e-05 & 15.28 & 9.79e-04 & 184.42 & $\star\star$ & 3397.34 \cr
\hline
\end{tabular}
}
\caption{Numerical Report from Three Algorithms}
\label{table:FINAL}
\end{table}
In \Cref{table:FINAL} \textbf{Problem Data} refers to the data used to generate the instances.
\textbf{Gauss-Newton} refers to the Gauss-Newton method. \textbf{Frank-Wolfe} refers to the Frank-Wolfe algorithm used in~\cite{winick2017reliable} and we use `with \textbf{FR}\, (w/o \textbf{FR} , resp)' to indicate the model is solved with \textbf{FR}\, (without \textbf{FR} , resp).
The header \textbf{cvxquad with \textbf{FR}} refers to the algorithm provided by~\cite{cvxquad} with \textbf{FR}\, reformulation.
If a certain algorithm fails to give a reasonable answer within a reasonable amount of time, we give a `$\star\star$' flag in the gap followed by the time taken to obtain the error message.
The following provides details for the remaining headers in \Cref{table:FINAL}.
\begin{enumerate}
\item \textbf{protocol}: the protocol name; refer to \Cref{sect:testexamples};
\item \textbf{parameter}: the parameters used for testing; see \Cref{sect:example1} - \Cref{sect:example6} for the ordering of the parameters;
\item \textbf{size}: the size $(n,k)$ of original problem; $n,k$ are defined in \eqref{eq:Gmap};
\item \textbf{gap}:
the relative gap between the bestub and bestlb;
\begin{equation}
\label{def:relgap}
\frac
{\text{bestub - bestlb}}
{1+
\frac{|\text{bestub}| + |\text{bestlb}|}2} .
\end{equation}
\item \textbf{time}: time taken in seconds.
\end{enumerate}
We make some discussions on the formula \cref{def:relgap}.
The best upper bound from Gauss-Newton algorithm is used for all instances for `bestub' in \cref{def:relgap}.
The Gauss-Newton algorithm computes the lower bounds as presented in \Cref{cor:lowerbound}.
The Frank-Wolfe algorithm presented in~\cite{winick2017reliable} obtains the lower bound by a linearization technique near the optimal.
cvxquad presented in~\cite{cvxquad} uses the semidefinite approximations of the matrix logarithm. The lower bounds from cvxquad is often larger than the theoretical optimal values. This indicates that the lower bounds from cvxquad is not reliable.
Therefore we adopt the lower bound strategy used in~\cite{winick2017reliable} for cvxquad.
We now discuss the results in \Cref{table:FINAL}.
Comparing the two columns \textbf{gap} and \textbf{time} among
the different methods,
we see that the Gauss-Newton method outperforms other
algorithms in both the accuracy and the running time.
For example, comparing \textbf{Gauss-Newton} and
\textbf{Frank-Wolfe with FR}, the gaps and running times from \textbf{Gauss-Newton} are competitive.
There are three instances that \textbf{Gauss-Newton} took longer time. We emphasize that the gaps from \textbf{Gauss-Newton} are obtained with much higher accuracy.
We now illustrate that the reformulation strategy via \textbf{FR}\, contributes to superior algorithmic performances.
For the columns \textbf{Frank-Wolfe with FR} and
\textbf{Frank-Wolfe w/o FR} in \Cref{table:FINAL},
the \textbf{FR}\, reformulation contributes to not only giving tighter gaps but also reducing the running time significantly.
We now consider the column corresponding to
\textbf{cvxquad with FR} in \Cref{table:FINAL}.
We see that the algorithm starts to fail (marked with `$\star\star$') as the problem sizes increase.
In fact cvxquad consistently fails due to the memory shortage
when \textbf{FR}\, reformulation was \emph{not} used.
\section{Conclusion}
\label{sect:concl}
\subsection{Summary}
In this paper we have used preprocessing and novel facial reduction,
\textbf{FR}, for the \textbf{QKD}\, problem in~\cref{eq:keyrateopt}, to derive
a regularized and simplified equivalent problem~\cref{eq:finalredinrho}.
\textbf{FR}\, was applied to both the linear constraints, as well as
to the nonlinear convex objective function. These steps used spectral
decompositions, and thus they provided a very accurate equivalent facially
reduced problem. This allowed for a stable, projected Gauss-Newton,
primal-dual interior-point approach.
Our empirical evidence illustrates significant improvements in solution
time and accuracy of solutions. In particular, we solve problems to
machine accuracy and provide theoretical provable accurate lower bounds.
In fact, we obtain lower bounds within $10^{-15}$ relative
accuracy when desired.
\paragraph{Summary of the Model Reformulation}
We have reformulated the model $(\textbf{QKD})$ through the sequence
\[
\cref{eq:keyrateopt}
\xrightarrow{\text{(1)}}
\cref{eq:keyrateoptequivsigdel}
\xrightarrow{\text{(2)}}
\cref{eq:optprob}
\xrightarrow{\text{(3)}}
\cref{eq:optprobobjconstrthree}
\xrightarrow{\text{(4)}}
\cref{eq:optprobobjintptFRreduced}
\xrightarrow{\text{(5)}}
\cref{eq:finalredinrho},
\]
via (1) variable substitutions;
(2) property of ${\mathcal Z} $ from \Cref{prop:Zopprops};
(3) facial reduction on $\rho,\delta,\sigma$;
(4) rotation of the constraints;
(5) substituting the constraints back to the objective.
\subsection{Future Plans}
\label{sect:future}
There are still many improvements that can be made.
Exact primal feasibility was quickly obtained and maintained throughout
the iterations. However, accurate dual feasibility was difficult to maintain.
This is most likely due to the finite difference approximation of the Hessian.
This approximation can be improved by including a quasi-Newton approach,
as we have accurate gradient evaluations.
We maintain high accuracy results even in the cases where the Jacobian
was not full rank at the optimum. This appears to be due to the
special data structures and more theoretical analysis at the optimum can be done.
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
We thank Kun Fang and Hamza Fawzi for discussions.
H. H., J. I. and H. W. thank the support of the National Sciences and Engineering Research Council (NSERC) of Canada.
Part of this work was done at the Institute for Quantum Computing, University of Waterloo, which is supported by Innovation, Science and Economic Development Canada.
J. L. and N. L. are supported by NSERC under the Discovery Grants Program, Grant No. 341495, and also under the Collaborative Research and Development Program, Grant No. CRDP J 522308-17. Financial support for this work has been partially provided by Huawei Technologies Canada Co., Ltd.
\newpage |
1705.03942 | \section{Introduction} \label{sec1}
\addtolength{\footskip}{-0.2cm}
The fact that the minimal observable length can be useful to impose an effective cut-off in the ultraviolet domain in order to make the theory of quantum fields renormalizable was suggested very early by Heisenberg. It was Snyder, who formalized the idea for the first time in the form of an article, and showed that the noncommutative structure of space-time characterizes the minimal measurable length in a very natural way \cite{Snyder}. Since then, the notion of noncommutativity has evolved from time to time and revealed its usefulness in different contexts of modern physics \cite{Seiberg_Witten}. Some natural and desirable possibilities arise when the canonical space-time commutation relation is deformed by allowing general dependence of position and momentum \cite{Kempf_Mangano_Mann,Das_Vagenas, Gomes,Bagchi_Fring,Quesne_Tkachuk}. In such scenarios, the Heisenberg uncertainty relation necessarily modifies to a generalized version to the so-called \textit{generalized uncertainty principle} (GUP). Over last two decades, it is known that within this framework, in particular, where the space-time commutation relation involves higher powers of momenta, explicitly lead to the existence of nonzero minimal uncertainty in position coordinate, which is familiar as the \textit{minimal length} in the literature. An intimate connection between the gravitation and the existence of the fundamental length scale was proposed in \cite{Mead}. The string theory, which is the most popular approach to quantum gravity, also supports the presence of such minimal length \cite{Veneziano,Gross_Mende,Amati,Yoneya,Konishi}, since the strings are the smallest probes that exist in perturbative string theory, and so it is not possible to probe space-time below the string length scale. In loop quantum gravity, the existence of a minimum length has a very interesting consequence, as it turns the big bang into a big bounce \cite{Rovelli,Dzierzak}. Furthermore, the arguments from black hole physics suggest that any theory of quantum gravity must be equipped with a minimum length scale \cite{Maggiore,Park}, due to the fact that the energy required to probe any region of space below the Plank length is greater than the energy required to create a mini black hole in that region of space. The minimal length exists in other subjects too; such as, path integral quantum gravity \cite{Padmanabhan,Greensite}, special relativity \cite{Amelino}, doubly special relativity \cite{Magueijo,Girelli,Cortes}, etc. In short, the existence of minimal measurable length, by now, has become a universal feature in almost all approaches of quantum gravity. Moreover, some Gedankenexperiments and
thought experiments \cite{Mead,Scardigli,Ng} in the spirit of black hole physics have also supported the idea. For further informations on the subject one may follow some review articles devoted to the subject, for instance, \cite{Garay,Ng_Review,Hossenfelder_Review}.
In spite of having several serious proposals for quantizing the general relativity, unfortunately we do not have a fully consistent quantum theory of gravity yet. This has motivated the study of \textit{semi-classical quantum gravity} (SCQG), where the gravitational filed is treated as a background classical field, and the matter fields are treated quantum mechanically. Under such approximation, if $|\psi\rangle$ is the wave function of the matter field, the Einstein tensor $G_{\mu\nu}$ can be obtained in terms of the quantum mechanical energy-momentum tensor for matter fields $T^{\mu\nu}$ as $G_{\mu\nu} =8\pi G\langle \psi|T^{\mu\nu}|\psi\rangle/c^4$. However, Newtonian gravity has been observed to be the correct approximation to general relativity till the smallest length scale (0.4mm) to which general relativity has been tested \cite{Yang}. Thus, at small distances, it is expected that the semi-classical approximation can be described by a Schr\"odinger-Newton equation \cite{Ruffini,Penrose,Tod,Van,Yang_Miao}, which is the nonrelativistic limit of the Dirac equation and the Klein-Gordon equation with a classical Newtonian potential \cite{Giulini}. It has been proposed that the Schr\"odinger-Newton equation can be utilized to explore various interesting properties of gravitational systems at the given length scale \cite{Giulini_Brobardt,Manfredi,Bera}.
Some interesting consequences may follow from the combination of the above two frameworks, namely, the GUP and SCQG, and it is allowed since both of the effects occur at small scales. Technically, this can be achieved by imposing the minimal length structure into the semi-classical scheme of gravity by means of deforming the Schr\"odinger-Newton equation in accordance with the laws of GUP. The most interesting fact is that the new theory resulting from the combination of GUP and SCQG may provide an intermediate theory between a full quantum theory of gravity and the SCQG and, this is precisely the issue that we address in the present manuscript. Recently, it has been suggested that both the Schr\"odinger-Newton equation \cite{Grossardt,Gan} and the deformation of quantum mechanical structure by the GUP \cite{Pikovski} can be tested experimentally by using the opto-mechanical systems. Therefore, it becomes important to understand the effects of the combined theory, especially when there is a strong viability that the theory may be tested by using a similar type of experimental set-up in near future.
\section{GUP and minimal length}\label{sec2}
\lhead{Modified Schr\"odinger-Newton equation}
\chead{}
\rhead{}
\addtolength{\voffset}{-1.2cm}
\addtolength{\footskip}{0.2cm}
Let us commence by introducing a particular version of modified commutation relation between the position and momentum operators $X,P$ \cite{Kempf_Mangano_Mann,Das_Vagenas}
\begin{equation}\label{GUP}
[X_i, P_j] = i\hbar(\delta_{ij} + \beta\delta_{ij}P^2 + 2\beta P_i P_j), \qquad [X_i,X_j]=[P_i,P_j]=0,
\end{equation}
with $\beta$ having the dimension of inverse squared momentum and, $P^2=\sum_{j=1}^{3}P_j^2$. As obviously, in the limit $\beta\rightarrow 0$, the deformed relations \myref{GUP} reduce to the standard canonical commutation relations $[x_i,p_j]=i\hbar\delta_{ij}$. However, there exist many other similar type of deformations in the literature, which have been used to investigated many interesting phenomena in different contexts; see, for instance \cite{Bagchi_Fring,Quesne_Tkachuk,Pedram_Nozari_Taheri, Dey_Fring_Gouba,Hossenfelder_Review,Dey_Hussin}. Nevertheless, the generalized uncertainty relation or the Robertson-Schr\"odinger uncertainty relation corresponding to the deformed algebra \myref{GUP} turns out to be
\begin{eqnarray}
&& \Delta X_i\Delta P_j \geq \frac{1}{2}\Big\vert\langle[X_i,P_j]\rangle\Big\vert \\
&& \Delta X_i\Delta P_i \geq \frac{\hbar}{2}\left[1+\beta\left\{(\Delta P)^2+\langle P\rangle^2\right\}+2\beta\left\{(\Delta P_i)^2+\langle P_i\rangle^2\right\}\right],
\end{eqnarray}
from which one can compute the exact expression of the minimal observable length by using the standard minimization technique. For further information on the minimization procedure, one may refer, for instance \cite{Kempf_Mangano_Mann,Bagchi_Fring,Dey_Fring_Gouba}. A possible representation of the algebra \myref{GUP} in terms of the canonical position and momentum operators $x,p$ is given by \cite{Kempf_Mangano_Mann,Das_Vagenas}
\begin{equation}\label{Rep}
X_i = x_i, \qquad P_i = p_i(1+\beta p^2),
\end{equation}
where $p_j = -i\hbar\frac{\partial}{\partial {x}_j}$.
Certainly, the representation \myref{Rep} is not unique \cite{Kempf_Mangano_Mann},
as for instance; see \cite{Dey_Fring_Khantoul}, where the authors explore four
possible representations of the algebra \myref{GUP} with some of them being Hermitian. However, it is easy to show that \myref{Rep} satisfies \myref{GUP} up to the first order of $\beta$ (hence, we neglect higher order terms of $\beta$). Physically, the notations are understood in the sense that $p_i$ represents the momenta at low energy, while $P_i$ correspond to those at high energy.
\section{The gravitational quantum well}\label{sec3}
The gravitational quantum well (GQW) is characterized by the motion of a nonrelativistic object of mass $m$ in a gravitational field $\overrightarrow{g}=-g\overrightarrow{e}_z$ with a restriction imposed by a mirror placed at the origin $z=0$, such that the potential turns out to be
\begin{align}\label{GravPot}
\begin{aligned}
V_0(z) =\left\lbrace\begin{array}{l}
+\infty, ~~~z\leq 0, \\
mgz, ~~~z>0,
\end{array} \right.
\end{aligned}
\end{align}
with $g$ being the gravitational acceleration. The experimental set-up of the corresponding potential \myref{GravPot} has already been studied \cite{Nesvizhevsky}. Theoretically, the problem resemble a quantum mechanical particle moving in a potential well given by \myref{GravPot} subjected to a boundary condition $\psi^0(0)=0$ at $z=0$. If we consider $\psi^0(\overrightarrow{x})=\psi^0(z)\psi^0(y)$, then the solutions of the corresponding time-independent Schr\"odinger equation along the $z$ direction
\begin{equation}\label{Ham0}
H_0\psi^0(z) = E^0\psi^0(z), \qquad H_0 = \frac{p^2}{2m} +mgz,
\end{equation}
are well-known and, are given by \cite{Flugge} (Problem $40$)
\begin{eqnarray}
E_{n}^0 &=& -\frac{mg\alpha_n}{\theta}, \qquad \theta = \left(\frac{2m^2g}{\hbar^2}\right)^{1/3},\\
\psi_n^0(z) &=& N_n \text{Ai}[\theta z+\alpha_n],
\end{eqnarray}
where $\alpha_n$ is the $n$\textsuperscript{th} zero of the regular Airy function $\text{Ai}(z)$ \cite{Abramowitz_Stegun}, and $N_n=\theta^{1/2}/|\text{Ai}'(\alpha_n)|$ is the normalization factor. Along the $y$ axis, the particle is free and the corresponding wave function takes the form
\begin{equation}
\psi^0(y)=\int_{-\infty}^{\infty}g(k)e^{iky}dk,
\end{equation}
where $g(k)$ determines the shape of the wave packet in momentum space. In analogy to a classical object, a collision of a quantum particle with the impenetrable mirror along the $z$ direction will make it bounced at a critical height $h_n$
\begin{equation}\label{hight}
h_n = \frac{E_n^0}{mg} = -\frac{\alpha_n}{\theta},
\end{equation}
which is, naturally, quantized. In the GRANIT experiment \cite{Nesvizhevsky1}, the measured critical heights for the first two states are
\begin{align}
& h^{\text{exp}}_1 = 12.2\mu m \pm 1.8_{\text{syst}} \pm 0.7_{\text{stat}},\\
& h^{\text{exp}}_2 = 21.6\mu m \pm 2.2_{\text{syst}} \pm 0.7_{\text{stat}}.
\end{align}
Whereas, the theoretical values for $h_1$ and $h_2$ can be obtained from \myref{hight} as
\begin{align}
& h^{\text{th}}_1 = 13.7\mu m,\\
& h^{\text{th}}_2 = 24.0\mu m,
\end{align}
where $m=939~\text{Mev/c\textsuperscript{2}}$ and $g=9.81~\text{m/s\textsuperscript{2}}$. The variation of the heights $h_1$ and $h_2$ resulting from the theory and the experiment are, therefore, $\delta^{\text{th}}=h_2^{\text{th}}-h_1^{\text{th}}=10.3\mu m$ and $\delta^{\text{exp}}=h_2^{\text{exp}}-h_1^{\text{exp}}=9.4 \mu m \pm 5.4\mu m$, respectively. Thus, the deviation of $\delta$ between the theory and the experiment turns out to be $4.5\mu m$. The argument that we shall pursue here is that any correction in the Hamiltonian \myref{Ham0} will effectively reduce this deviation of $\delta$ between the theory and the experiment, and thus the theoretical result will become closer to the experiment. A similar argument has already been used in \cite{Bertolami_Rosa_etal,Brau,Banerjee,Buisseret}. However, we explore two types of correction here. First, we consider a braneworld model studied in \cite{Randall}, which was discovered in the course of solving the hierarchy problem of the standard model. Their theory is based on the assumption that, while all the standard model fields, gauge and matter, are confined in a (3+1) dimensional manifold, only the graviton can propagate freely in the extra dimensions which are considered to be infinite. In presence of such infinite extra dimensions, the Newtonian potential is modified to the following form
\begin{equation}\label{PotCor}
V(r)=-\frac{GMm}{r}\left(1+\frac{k_b}{r^b}\right), \quad r>>\Lambda =\sqrt[b]{|k_b|},
\end{equation}
where $\Lambda$ is the length scale at which the correction due to the infinite extra dimensions becomes dominant. If we consider the Newtonian potential $-GMm/r=V_0(r)$, we can write the above equation \myref{PotCor} as $V(r)=V_0(r)+V_b(r)$, so that $V_b(r)=-GMmk_b/r^{b+1}$ can be considered as a perturbation, which will eventually contribute to the correction over the theoretical values of $\delta$. Therefore, $\delta^{\text{th}}$ will be changed to $\delta^{\text{th}}+\Omega$, where
\begin{equation}\label{Omega}
\Omega=\frac{1}{mg}\left[\left\langle\psi_2^0(z)|V_b|\psi_2^0(z)\right\rangle -\left\langle\psi_1^0(z)|V_b|\psi_1^0(z)\right\rangle\right],
\end{equation}
and $|\Omega|\leq 4.5\mu m$. The second correction emerges from the contribution of the GUP deformed Hamiltonian. If we replace the momentum $p$ corresponding to the low energy in \myref{Ham0} by the momentum $P$ coming from the GUP deformation \myref{Rep}, we obtain
\begin{equation}\label{GUPCorr}
H=H_0+H_1, \qquad H_1=\frac{\beta}{m}p^4,
\end{equation}
where we neglect the higher order terms of $\beta$. Thus, if we denote the correction of $\delta^{\text{th}}$ coming from the GUP by $\Xi$, with
\begin{equation}
\Xi = \frac{1}{mg}\left[\left\langle\psi_2^0(z)|H_1|\psi_2^0(z)\right\rangle -\left\langle\psi_1^0(z)|H_1|\psi_1^0(z)\right\rangle\right],
\end{equation}
then
\begin{equation}
|\Omega|\leq|\Omega+\Xi|\leq 4.5\mu m.
\end{equation}
Let us now compute both of the corrections $\Omega$ and $\Xi$ in the following section.
\section{Corrections}\label{sec4}
The potential described in \myref{PotCor} corresponds to an interaction between two particles. However, we are dealing with a situation where a point particle bounces on a plane mirror placed at the surface of the Earth. Therefore, we are required to derive the effective potentials between the test particle and the Earth, as well as, between the particle and the mirror.
\subsection{Correction due to Earth-particle interaction}
Let us first derive the potential coming from the Earth-particle interaction. By considering our planet to be a spherical body with mass density $\rho_E$, the total potential acting on the particle is
\begin{align}
V_b^E(\overrightarrow{r}) = -mG\rho_E k_b\int_E\frac{d^3\overrightarrow{r}'}{|\overrightarrow{r}-\overrightarrow{r}'|^{b+1}}.
\end{align}
In spherical coordinates which becomes
\begin{align}
V_b^E(h) = -2\pi mG\rho_E k_b\int^R_0 r^2dr\int^{1}_{-1} \frac{du}{[r^2-2r(h+R)u+(h+R)^2]^{(b+1)/2}},
\end{align}
where $h$ is the altitude above Earth's surface and $R$ being the radius of the Earth. For, $b=0,1,2,3$, one obtains \cite{Buisseret}
\begin{align}\label{power-up}
\begin{aligned}
& V_0^E(h) = -\frac{GmM}{h+R}, \\
& V_1^E(h) = -\frac{3mgk_1}{4R}\left[2R - \frac{h(h+2R)}{h+R}\ln\left(\frac{h+2R}{h}\right)\right],\\
& V_2^E(h) = -\frac{3mgk_2}{2R}\left[ \ln\left(\frac{h+2R}{h}\right) - \frac{2R}{h+R}\right],\\
& V_3^E(h) = -\frac{3mgk_3}{4R}\left[\frac{2R}{h(h+2R)}- \frac{1}{h+R}\ln\left(\frac{h+2R}{h}\right)\right],
\end{aligned}
\end{align}
while for $b>3$, it becomes
\begin{equation}
V_b^E(h) = -\frac{3mgk_b}{2(b-1)(b-2)(b-3)R(h+R)}\left[\frac{(b-3)R-h}{h^{b-2}}+\frac{(b-1)R+h}{(h+2R)^{b-2}}\right].
\end{equation}
Notice that in the limit $b\rightarrow 0~(k_0=1)$, we recover the Newtonian potential in an exact form.
\subsection{Correction due to mirror-particle interaction}
Although the mass of the mirror is negligible with respect to that of the Earth, the effect originating from the mirror-particle interaction should be taken into account since the mirror is placed at a distance much closer to the particle. The mirror can be seen as a parallelepiped with density $\rho_M$, and located at $-\infty <(x,y)<\infty$ and $-L<z<0$. With respect to the size of the particle, the mirror can be considered as an infinite plane, so that the mirror behaves as a disc of infinite radius. Thus, the total interacting potential in this case becomes
\begin{align}\label{MirrorInt}
V_b^M(h) = -2\pi mG\rho_M k_b\int^0_{-L} dz\int^{R^*}_{0} \frac{rdr}{[r^2+(h-z)^2]^{(b+1)/2}},
\end{align}
where $R^*$ is taken to be very large, but finite, in order to avoid the divergent integrals. Nevertheless, the integrals \myref{MirrorInt} are computed as follows
\begin{align}\label{mirror}
\begin{aligned}
& V_1^M(h) = \frac{3mgk_1\rho}{2R}\left[L\ln(h+L)+h\ln\left(\frac{h+L}{h}\right)\right],\\
& V_2^M(h) = -\frac{3mgk_2\rho}{2R} \ln\left(\frac{h+L}{h}\right),\\
& V_{b>2}^M(h) = -\frac{3mgk_b\rho}{2R(b-1)(b-2)}\left[\frac{1}{h^{b-2}}+\frac{1}{(h+L)^{b-2}}\right].
\end{aligned}
\end{align}
\subsection{The perturbative correction from GUP deformation}
Now, let us consider the correction due to the perturbation term $H_1$ \myref{GUPCorr} arising from the GUP deformation. The correction to the energy $\Delta E_n$ at the lowest order in $\beta$ is given by
\begin{align}
\begin{aligned}
\Delta E_n^{\text{GUP}} & = \frac{\beta}{m}\left\langle\psi(\overrightarrow{x})|p^4|\psi(\overrightarrow{x})\right\rangle \\
& = \frac{\beta}{m}[4m^2\left\langle\psi_n^0(z)|p^4_z|\psi_n^0(z)\right\rangle + 2\left\langle\psi_n^0(z)|p^2_z|\psi_n^0(z)\right\rangle\left\langle\psi^0(y)|p^2_y|\psi^0(y)\right\rangle \\
& = \frac{\beta}{m}[4m^2\left\langle [E^0_n - V_0(z)]^2\right\rangle + 8m^2E_c\left\langle E^0_n-V_0(z)\right\rangle] \\
& = 4\beta m[E^0_n(E^0_n +2E_c) -2(E^0_n+E_c)\left\langle V_0(z)\right\rangle +\left\langle V_0^2(z)\right\rangle],
\end{aligned}
\end{align}
where $E_c = m\left\langle\psi^0(y)|v^2_y|\psi^0(y)\right\rangle /2$ is the kinetic energy of the particle along the horizontal direction. Note that a term proportional to $\left\langle\psi^0(y)|p^4_y|\psi^0(y)\right\rangle$ has been omitted since it only leads to a global shift of the energy spectrum. Therefore, by computing the following integrals
\begin{align}
\begin{aligned}
\left\langle V_0(z)\right\rangle & =\langle\psi_n^0(z)|V_0(z)|\psi_n^0(z)\rangle= mgN_n^2\int^{+\infty}_0 z Ai^2(\theta z+\alpha_n) = \frac{2}{3}E_n^0, \\
\left\langle V_0^2(z)\right\rangle & =\langle\psi_n^0(z)|V^2_0(z)|\psi_n^0(z)\rangle = (mg)^2N_n^2\int^{+\infty}_0 z^2 Ai^2(\theta z+\alpha_n) = \frac{8}{15}(E_n^0)^2,
\end{aligned}
\end{align}
we obtain the final expression of the correction to the energy as follows
\begin{align}
\Delta E_n^{\text{GUP}} = \frac{4}{5}\beta m(E^0_n)^2(1+\frac{10 E_c}{3 E^0_n}).
\end{align}
\section{The modified GQW spectrum}\label{sec5}
Combining the corrections coming from the SCQG, we obtain
\begin{equation}
V_b = V_b^E(z)+V^M_b(z),
\end{equation}
which when combined with the correction to the energy arising from the GUP deformation, we obtain the total energy shift
\begin{equation}
\Delta E_n = \left\langle V_b\right\rangle +\Delta E^{\text{GUP}}_n.
\end{equation}
Therefore, we have
\begin{equation}
\Omega_b + \Xi =\frac{1}{mg}\left[\left\langle\psi_2^0(z)|V_b|\psi_2^0(z)\right\rangle + \Delta E_2^{\text{GUP}} - \left\langle\psi_1^0(z)|V_b|\psi_1^0(z)\right\rangle -\Delta E_1^{\text{GUP}}\right].
\end{equation}
Let us now compute the numerical values of the above expressions, with $R = 6378 \text{km}$, $m = 939 \text{MeV}/\text{c}^2, g = 9.81 \text{m}/\text{s}^2, L = 10\text{cm}, \rho= 1$ and $E_c \simeq m\left\langle v_y\right\rangle^2/2 \simeq 0.221 \mu \text{eV}$
\begin{align}
& \Omega_1 +\Xi = k_1 \times 5.59203\times 10^{-11} + \beta\times1.62357\times 10^{-57}\text{m} < 4.5\mu \text{m}, \label{Eq54}\\
& \Omega_2 +\Xi = k_2 \times 2.38493\times 10^{-7} \text{m}^{-1} + \beta\times 1.62357\times 10^{-57}\text{m} < 4.5\mu \text{m}, \\
& \Omega_3 +\Xi = k_3 \times 0.0101239~\text{m}^{-2} + \beta\times1.62357\times 10^{-57}\text{m} < 4.5\mu \text{m}. \label{Eq56}
\end{align}
In order to obtain the bounds on the power-law parameters $\Lambda$ and $k_b$, let us consider $V_b=-mgk_b\tilde{V}_b$. Eq. \myref{PotCor} implies that $k_b = \Lambda^b$, so that $V_b=-mg\Lambda^b\tilde{V}_b$. Therefore, we can write
\begin{equation}
\Omega_b=-\Lambda^b\tilde{\Omega}_b=-\Lambda^b\left[\left\langle\psi_2^0(z)|\tilde{V}_b|\psi_2^0(z)\right\rangle - \left\langle\psi_1^0(z)|\tilde{V}_b|\psi_1^0(z)\right\rangle\right],
\end{equation}
which immediately leads to the constraint on the parameter $\Lambda$ as $|\Lambda|=\sqrt[b]{\Omega_b/\tilde{\Omega}_b}$. According to \myref{Omega}, $\Omega$ is bounded below $4.5\mu m$ and, so $\Lambda$ is bounded by the following relation
\begin{equation}
|\Lambda|=\sqrt[b]{4.5/\tilde{\Omega}_b}.
\end{equation}
\begin{table}
\centering
\begin{tabular}{c|ccc}
$b$ & 1&2&3\\
\hline\\
$|\Lambda|$(m) $<$ & $8.13\times 10^{4}$ & $4.34$ & $0.076$
\end{tabular}
\caption{Constrains on the power-law parameters}
\label{constraint}
\end{table}
Using the results from \myref{Eq54}-\myref{Eq56}, we can calculate $\tilde{\Omega}_b$ corresponding to different values of $b$. If we turn off the GUP deformation ($\beta=0$), we obtain the Table \ref{constraint}. In a similar way it is also possible to obtain the bounds for the cases when $\beta\neq 0$. On the other hand, if we turn off the power-law modification, we see that the GUP deformation parameter has the constraint
\begin{equation}\label{Eq57}
\beta <2.77167\times 10^{51},
\end{equation}
which provides a tighter upper bound than those derived in the context of gravitational fields; see, for instance \cite{Scardigli_Casadio}. However, the upper bound \myref{Eq57} is weaker in comparison to \cite{Quesne_Tkachuk,Bawaj, Scardigli_Lambiase_Vagenas}, which were obtained in different contexts in the literature. Nevertheless, what we notice is that we obtain positive contributions from both SCQG and GUP deformation, which will make the theoretical values of $\delta$ closer to the experiment.
\section{A schematic proposal for experiment}\label{sec6}
To this end, we make a proposal for an experimental quantum bouncer through an opto-mechanical set-up, which would be able to provide an understanding of the combined effect of SCQG and GUP. Opto-mechanical devices yield a promising avenue for preparing and investigating quantum states of massive objects ranging from a few picograms up to several kilograms \cite{Kippenberg}. Significant experimental progress has been achieved by using such devices in different contexts, including coherent interactions \cite{Connell}, laser cooling of nano and micro-mechanical devices into their quantum ground state \cite{Teufel}. Recently, such types of systems have been utilized for the understanding of more exciting features like the SNE in SCQG \cite{Grossardt,Gan} and GUP \cite{Pikovski}. Here, our goal is to understand the combined effect of SCQG and GUP through the given opto-mechanical system.
The underlying principle behind the experiment is that the ultra cold neutrons (UCN) move freely in the gravitational field above a mirror and make a total reflection from the surface of the mirror when the corresponding de Broglie wavelength is bigger than the interatomic distances of the matter. Thus, the set-up gives rise to a GQW, where the UCN form bound quantum states in the Earth's gravitational field. The eigenvalues are non-equidistant and, are in the range of pico-eV. These type of scenarios offer fascinating possibilities to combine the effects of Newton's gravity at short distances with the high precision resonance spectroscopy methods of quantum mechanics.
\begin{figure}[ht]
\centering \includegraphics[width=11.0cm,height=7.0cm]{Exp.png}
\caption{The schematic of the opto-mechanical setup for the proposed experiment}
\label{Fig1}
\end{figure}
The schematic diagram of the experimental set-up is given in Fig. \ref{Fig1}, which consists of an anti-vibrator granite table, a convex magnification system (mirror), an inclinometer, a ceiling scatterer and a position sensitive detector. The UCN shall be formed by passing the neutrons of the proposed wavelength between 0.75- 0.90 nm through the superfluid of \textsuperscript{4}He of a volume of about 4-5 liters. The UCN are brought to the experimental set-up by a guided system composed of a slit, a neutron collimator and an aperture. The neutrons fly in the slit and are screened in the range of 4-10 m/s. The upper wall of the slit is an efficient absorber which only lets the surviving neutrons to pass and, hence forth controlling first quantum states of the neutron flux. The mirror consists of a magnification system , which is made of a bi-metal (NiP) coated glass rod of radius between 3-3.5 nm with the roughness of about 1.5 nm. However the mirror system shall be modified to elliptical shape in order to harness the neutrons of much smaller wavelength
as well. The distribution of 100 $\mu$m in height is magnified to $\sim 2.5$ mm at the position on the detector. This gives the average magnification power of about 25 at the glancing angle at $20^0$ to make the reflection high.
The most important aspect of the position sensitive neutron detector is that the neutrons must be changed into a charged particles by a nuclear reaction. The converter materials are available as \textsuperscript{3}He, \textsuperscript{6}Li, \textsuperscript{10}B, and \textsuperscript{157}Gd, etc. A black thin CCD (charge coupled detector) could be used along with a thin layer of \textsuperscript{3}He, \textsuperscript{10}B or Ti-\textsuperscript{10}B-Ti. The ideal thickness of \textsuperscript{10}B should be about 200 nm, and must be evaporated directly on the CCD surface. The standard pixel size is 24 $\mu$m $\times$ 24 $\mu$m and the thickness of the active volume should be about 20 $\mu$m. The kinetic energy is deposited on the active area and a charge cluster is created, which typically spreads into nine pixels. The weighted center of the charge cluster is a good estimation of the incident neutron position.
At the exit of the UCN system the wavefucntion changes due to the absence of suppressing slit and as such the neutrons turn as bouncers on the mirror surface and their trajectories are measured by the Time of flight (TOF) method. They are characterized by a Gaussian distribution with a mean of 9.4 m/s and the standard deviation of 2.8 m/s which is subject to change due to various modifications of the design.
\section{Conclusions}\label{sec7}
We have studied a two-fold correction in the energy spectrum of a GQW. The first correction arises from the scheme of the SCQG itself, where we considered the SNE in a particular framework of braneworld model with infinite extra dimensions. The second correction emerges from a GUP deformation of the given SCQG framework. The combined effect of these two interesting theories provide a new bound on the energy spectrum of the GQW, which is closer to the observed values in the GRANIT experiment. Since, both of the effects of SCQG and GUP occur at small length scales, and both of the theories provide positive contributions in the correction of the energy spectrum, it raises a natural possibility that the combined theory may guide us towards a theory beyond the SCQG. Thus our proposal may yield an intermediate theory beyond the SCQG and the complete theory of quantum gravity. Moreover, we have provided a schematic experimental set-up which would help the laboratory experts to explore our theory further in the lab.
There are many natural directions that may follow up our investigation. First, it will be interesting to study the similar kind of effects of GUP in the context of other theories of gravity. There exist many other type of GUP deformation, which may be useful to study the similar theories to confirm our findings. However, the most interesting future problem lies on the understanding of the experimental realization of the combined theory of SCQG and GUP by using the opto-mechanical set-up, while it has already been used to understand each of the individual frameworks of SCQG and GUP.
\vspace{0.5cm} \noindent \textbf{\large{Acknowledgements:}} AB is supported by MHRD, Government of India and would like to thank Department of Physics and MMED at NIT Srinagar for carrying out her research pursuit. SD is supported by a CARMIN postdoctoral fellowship by IHES and IHP. The work of QZ is supported by NUS Tier 1 FRC Grant R-144-000-360-112. |
1705.03586 | \section{\label{Intro} Introduction}
A ratchet is a mechanical device that combines a pawl and a wheel such that
the former limits the rotation of the latter to only one direction.
Also, a ratchet mechanism can refer to dynamism among objects
that rectifies incoming stimulative actions into directed movement.
The mechanism of a ratchet is attributed to the nature of
a nonequilibrium (or macroscopic) system.
If the size of the ratchet is reduced to nanoscale,
the rectifying action of the ratchet becomes unreliable or probabilistic because
the influence of the surrounding molecules is comparable to the input stimuli
to the ratchet; the pawl moves erroneously and allows the wheel to rotate in the opposite
(i.e., undesired) direction.
Such a very small ratchet system is
called a Brownian ratchet (BR) or Smoluchowski--Feynman ratchet from
Smoluchowski's (and Feynman's) thought experiment \cite{Smoluchowski1912,FeynmannLecI}.
To be consistent with the Second Law of Thermodynamics,
if the temperature of the ``agents'' causing
the input stimuli to the ratchet equals the temperature of the ratchet,
there can be no net rotation of the wheel.
This contraposition implies that if net rotation does appear, the statistical property of
the input agents differs from that in thermal equilibrium, or
that the temperature of the pawl is lower than that of the input
agents \cite{FeynmannLecI,PhysRevLett.71.1477,PhysRevE.81.061104}.
The problem of how net rotation or unidirectional motion results
from unbiased stimuli in the thermal environment has been analyzed by numerous studies
with various types of ratchet model \cite{Reimann200257,RevModPhys.81.387}.
Because of its universal nature in nonequilibrium phenomena,
the concept of a ratchet mechanism has attracted a great deal of attention
from various perspectives, e.g., biological \cite{VALE199097,Wang2002,Kawaguchi20142450}
and artificial molecular motors
\cite{ANIE:ANIE200504313,RevModPhys.81.387,Kuhne14122010,C5NR08768F},
optical thermal ratchets \cite{PhysRevLett.74.1504},
dielectrophoretic ratchets \cite{PhysRevE.86.041106},
and granular ratchet systems\cite{1742-5468-2008-10-P10011,PhysRevLett.104.248001,
PhysRevLett.107.138001,PhysRevE.86.061310,PhysRevLett.110.120601,
PhysRevE.87.040101,PhysRevE.87.052209,PhysRevE.94.032910}.
In this study, we consider the rectification behavior of
two-dimensional (2D) BR models for a rotating thin rod inside
a cylinder, and its optimization for the rotational performance.
Firstly, we outline our dynamical model,
in which we suppose that
the thin rod (rotor) contacts diagonally with the cylinder (stator) at the upper and lower
edges, and rotates inside the cylinder through mutual ratchet interaction
under temporally varying fields
\cite{PhysRevE.84.061119,PhysRevE.87.022144,doi:10.7566/JPSJ.84.044004}.
Real systems that are relevant to such Brownian rotary ratchets
may be found in microscopic light-driven
rotors \cite{:/content/aip/journal/apl/78/2/10.1063/1.1339258},
the artificial molecular rotor of
caged supramolecules \cite{Kuhne14122010},
or synthetic molecular systems, e.g.,
\cite{ANIE:ANIE200603618,TCR:TCR201402007}.
As in \cite{PhysRevE.84.061119,PhysRevE.87.022144,doi:10.7566/JPSJ.84.044004},
we describe the state of rotation as a trajectory on a 2D plane.
Representing the state of the rotor tip at time $t$ as
$\boldsymbol{X}\equiv (X_t,Y_t)^{\mathrm{T}}$ (hereinafter,
$\mathrm{T}$ denotes the transpose of a vector or matrix, and
bold face represents a 2D vector),
we assume that $\boldsymbol{X}$ obeys Langevin dynamics:
\begin{equation}
\gamma\dot{\boldsymbol{X}} = -\partial_{\boldsymbol{X}}
V_{0}(\boldsymbol{X}) -\partial_{\boldsymbol{X}}
V_h(\boldsymbol{X}, t) +\boldsymbol{f}_{I}(\boldsymbol{x})
+ \boldsymbol{R}_t.
\label{LEQ}
\end{equation}
Here,
$\partial_{\boldsymbol{x}} \equiv
(\frac{\partial}{\partial x}, \frac{\partial}{\partial y})^{\mathrm{T}}$,
$\gamma$ ($=1$) denotes a viscosity coefficient,
and $\boldsymbol{R}_t$ is a random force
with properties $\langle \boldsymbol{R}_t\rangle=\boldsymbol{0}$ and
$
\langle \boldsymbol{R}_{t}\boldsymbol{R}_{t'}^{\mathrm{T}} \rangle =
2D \delta(t-t')\Hat{1}
$,
where $\Hat{1}$ and $\langle A_t\rangle$ denote a $2\times 2$ unit matrix and
the average of $A_t$ over all possible process of
$\boldsymbol{R}_t$, respectively.
Here, $\boldsymbol{R}_t$ corresponds to the thermal fluctuation, and
the noise intensity $D$ is assumed to satisfy
$D=k_{\mathrm{B}}T$ with a temperature $T$ and
the Boltzmann constant $k_{\mathrm{B}}$.
In addition, $V_0(\boldsymbol{x})$ represents the 2D ratchet potential
for the static part of the rotor--stator interaction (one can imagine
the interaction between the pawl and the wheel for this).
The function $V_h(\boldsymbol{x}, t)$ is the temporally varying part
of the interaction:
\begin{align}
V_h(\boldsymbol{x}, t)
&=
- h\boldsymbol{N}_t\cdot\boldsymbol{x}
,
\quad\boldsymbol{N}_t=(\cos\Phi_t,\sin\Phi_t)^{\mathrm{T}}
,
\label{LEQ:Poth}
\end{align}
where $h\boldsymbol{N}_t$ represents a force on the rotor.
The angle $\Phi_t$ switches successively to independent values in
$[0,2\pi)$ in a sequence of events described as a Poisson process
with a mean interval $\Omega^{-1}$.
In other words, the mean and auto-correlation function of $\boldsymbol{N}_t$ obey
\begin{equation}
\left\langle\boldsymbol{N}_t\right\rangle_{\Phi}=\boldsymbol{0},
\quad
\left\langle
\boldsymbol{N}_{t}\boldsymbol{N}_{0}^{\mathrm{T}}
\right\rangle_{\Phi}
= \frac{e^{-\Omega t}}{2}\Hat{1},
\label{cor_NN}
\end{equation}
where $\langle A_t\rangle_{\Phi}$ denotes
the average of $A_t$ over all possible process of
$\Phi_t$ (Appendix.~\ref{OnNoise}).
We can regard $h\boldsymbol{N}_t$ as an external field or
a force due to
a temporal deformation of the stator, and call this a randomly directed d.c. field (RDDF).
For simplicity, we consider the load
$\boldsymbol{f}_I(\boldsymbol{x})$ for the rotation as
\begin{equation}
\boldsymbol{f}_{I}(\boldsymbol{x})
= \frac{I}{2\pi}
\left(
\frac{y}{\lvert\boldsymbol{x}\rvert^2},
-\frac{x}{\lvert\boldsymbol{x}\rvert^2}\right)^{\mathrm{T}}
\equiv
-\frac{I}{2\pi} \partial_{\boldsymbol{x}}
\theta(\boldsymbol{x}),
\label{LEQ:PotI}
\end{equation}
where $\theta(\boldsymbol{x})\equiv\tan^{-1}\frac{y}{x}$ and
$I$ denotes the load torque.
In the absence of an external field and load ($h=I=0$) in Eq.~(\ref{LEQ}),
we have an equilibrium state that corresponds to thermal equilibrium;
there is no net circulation of $\boldsymbol{X}$ about the origin, so there is
no net rotation. As mentioned above,
net rotation requires the (agents of) external field to be
athermal \cite{PhysRevLett.71.1477,Reimann200257}.
Here, as in Eq.~(\ref{cor_NN}), the RDDF can have a sufficiently long correlation time and
be athermal.
In general, there are two basic types of 2D field:
either one in which only the field angle varies but the magnitude is constant,
or a uni-axially polarized field.
The RDDF is classed as the former type because
the force angle varies randomly without bias.
An example of the latter field type is reported in \cite{PhysRevE.84.061119};
with dynamics in a two-tooth ratchet potential under
a uni-axially polarized sinusoidal field, it is shown that a net rotation appears
with a rotational direction that depends on the polarization angle. The
ranges of angle for the clockwise and counterclockwise rotations are
asymmetric, reflecting the chirality of the ratchet potential
(cf. \cite{PhysRevE.77.061111}, which reports on
the occurrence of unidirectional rotation
with a symmetric (achiral) two-well hindered-rotation potential).
An aim of the present study is to show that a combination of the
two-tooth ratchet potential
and the RDDF (as a basic example of an athermal unbiased field)
can support net rotation in a constant direction that is
determined by only the chirality of the ratchet potential.
Such a net rotational state is also capable of producing a positive power against
the load in Eq.~(\ref{LEQ:PotI}) for a sufficiently small $I$.
Another aim is to formulate a method of optimizing the 2D ratchet potential
to maximize the efficiency of rotational output.
In previous papers by some or all of the authors
\cite{PhysRevE.84.061119,PhysRevE.87.022144,doi:10.7566/JPSJ.84.044004},
analyses of the two- and three-tooth models
with the four- and six-state approaches have been shown
\cite{PhysRevE.84.061119,PhysRevE.87.022144},
and the analytical framework for estimating energetic efficiency
\cite{doi:10.7566/JPSJ.84.044004} has been developed,
in which any optimization has been disregarded.
Here,
we define the efficiency of the rotational output.
The balance between the input power of the external field
($\boldsymbol{f}_{h}\equiv h\boldsymbol{N}_t$) and
the combined power consumed by the load
[$\boldsymbol{f}_{I}\equiv \boldsymbol{f}_{I}(\boldsymbol{X})$]
and the other resistive forces is
\begin{equation}
\overline{
\dot{\boldsymbol{X}}\cdot \boldsymbol{f}_{h}
}
=
\overline{
(-\dot{\boldsymbol{X}}\cdot\boldsymbol{f}_{I})
}
+
\gamma
\left(
\overline{\lvert\langle\Dot{\boldsymbol{X}}\rangle\rvert^2}
+
\overline{L_t'}\,\overline{\omega_t'}
\right)
+Q_T,
\label{EnergyBalanceEq}
\end{equation}
where
\begin{align}
L_t'\equiv &\;
X_t(\dot{Y}_t-\langle\dot{Y}_t\rangle ) -Y_t
(\dot{X}_t-\langle \dot{X}_t\rangle)
,
\label{def:L'}
\\
\omega_t'\equiv &\;
\frac{
X_t(\dot{Y}_t-\langle\dot{Y}_t\rangle ) -Y_t
(\dot{X}_t-\langle \dot{X}_t\rangle)
}{X_t^2+Y_t^2}
,
\label{def:omega'}
\\
Q_T
\equiv
&\;
\frac{k_{\mathrm B}T}{\gamma}(\overline{\partial_xF_x +\partial_yF_y})
+
\gamma
\overline{\left(
L_t' -
\overline{L_t'}
\right)
\left(
\omega_t'- \overline{\omega_t'}\right)}
\nonumber\\
&+
\frac{1}{\gamma}
\overline{\left(\frac{X_t\Tilde F_x+Y_t\Tilde F_y}{\sqrt{X_t^2+Y_t^2}}\right)^2},
\label{def:Qt}
\end{align}
with $\boldsymbol{F}\equiv (F_x,F_y)^{\top}\equiv
\boldsymbol{f}_{h}+\boldsymbol{f}_{I}$ and
$
\Tilde{\boldsymbol{F}}\equiv \boldsymbol{F}-\gamma
\langle \dot{\boldsymbol{X}} \rangle$.
The equality (\ref{EnergyBalanceEq}) is derived in \cite{doi:10.7566/JPSJ.84.044004}
based on
\cite{PhysRevE.75.061115,PhysRevLett.83.903,PhysRevE.68.021906,PhysRevE.70.061105}.
Here, we define the long time average of $A$ as
$
\overline{A}
\equiv \overline{A(\boldsymbol{X},\Phi_t)} \equiv \int_{0}^{T_{\mathrm{tot}}}
d t\, A(\boldsymbol{X},\Phi_t)/T_{\mathrm{tot}}$ for $T_{\mathrm{tot}} \gg \Omega^{-1}$,
and assume
$
\overline{A}
=
\left\langle\! \langle
A(\boldsymbol{X},\Phi_t)\rangle\!
\right\rangle_{\Phi}
$ (ergodic hypothesis), where
$ \left\langle\! \langle A \rangle\! \right\rangle_{\Phi}$ means
doubly averaging over all possible realization of the stochastic processes
$\{\boldsymbol{R}_t\}_{t=0}^{T_{\mathrm{tot}}}$ and $\{\Phi_t\}_{t=0}^{T_{\mathrm{tot}}}$.
The products of the dynamical variables are considered in
the Stratonovich sense \cite{Sekimoto2010}.
The left-hand side (LHS) in Eq.~(\ref{EnergyBalanceEq}) is the input power.
The first term on the right-hand side (RHS) is the power consumed by the load.
The second and third terms are the dissipation rates
associated with the mean translational and rotational motions, respectively
(these can be interpreted as the power consumed while
drawing in the surrounding molecules).
Here, $L_t'$ and $\omega_t'$ denote the angular momentum and angular velocity, respectively,
defined in coordinates fixed to the mean translational motion.
The final term $Q_T$ in Eq.~(\ref{EnergyBalanceEq})
can be regarded as an excess dissipation rate resulting from the difference
between the dissipation due to velocity fluctuations---consisting of the second (rotational component) and third (radial component) terms in
Eq.~(\ref{def:Qt})---and the input power from the thermal bath (the first term multiplied by minus one).
Using the input power and the output powers associated with the rotation
in the RHS of Eq.~(\ref{EnergyBalanceEq}),
the rectification efficiency (or generalized efficiency)
\cite{PhysRevE.75.061115,PhysRevLett.83.903,PhysRevE.68.021906,PhysRevE.70.061105}
is defined as
\begin{gather}
\eta =
\frac{
\gamma \overline{L_t'}\,\overline{\omega_t'}
-
\overline{
\dot{\boldsymbol{X}}\cdot\boldsymbol{f}_{I}
}
}{
\overline{\dot{\boldsymbol{X}}\cdot \boldsymbol{f}_h}
}.
\label{def:eta}
\end{gather}
This definition is usable even in the absence of a load ($I=0$).
There have been many studies of the rotation or transport efficiency of ratchet systems.
In one-dimensional ratchet systems in particular, proposals have been made for exact expressions
for the efficiency or for models that realize highly efficient performance, e.g.,
\cite{0295-5075-27-6-002,PhysRevE.60.2127,PhysRevE.69.021102,PhysRevE.75.061115,PhysRevE.90.032104}.
In the context of maximization of efficiency,
although there are various aspects to optimization \cite{PhysRevE.68.046125,PhysRevE.74.066602},
basic approaches may be classified into two types:
those that optimize the temporally
varying part of the ratchet potential
\cite{Tarlie03031998,PhysRevE.79.031118,PhysRevE.87.032111},
and those that optimize the static part
\cite{PhysRevE.75.061115,PhysRevE.69.021102}.
Experiments relevant to these optimization approaches can be found in
\cite{PhysRevLett.74.1504,PhysRevE.86.041106}.
However, in the present context and to the best of our knowledge,
there have been few theoretical studies on 2D ratchet models\cite{schmid2009}.
In considering the optimization of the static part of the ratchet potential,
a basic idea is to design the ratchet potential in the following form:
\begin{equation}
V_0(\boldsymbol{x}) =
\frac{1}{4}
\left[
1-\left\{
v_{0}(\boldsymbol{x})
\right\}^m
\right]^2
-
K v_{1}(\boldsymbol{x})
,
\label{LEQ:Pot0}
\end{equation}
where $m \geq 1$.
For $m\gg 1$, the curve of
$v_{0}(\boldsymbol{x})=1$
approximates a potential valley that
mimics a constraint on the rotor--stator contact and
along which the orbit of the rotational-motion concentrates.
The purpose of $v_{1}(\boldsymbol{x})$ is to create the local minima and saddles
in the valley.
The functions $v_{0}(\boldsymbol{x})$ and $v_{1}(\boldsymbol{x})$ are non-decreasing functions of
$\lvert\boldsymbol{x}\rvert$,
and the region specified by $v_{0}(\boldsymbol{x})\leq 1$ is a simply connected space.
These details are shown in Sec.~\ref{TT-Model}.
Here, an important point is that for $m\gg 1$ we can characterize a ratchet potential
with two curves specified by
$v_{0}(\boldsymbol{x})=1$ and $v_{1}(\boldsymbol{x})=E$ with a constant $E$
as shown later.
This allows us to easily design an optimized
ratchet potential that maximizes the rotational output.
In this study, we develop an optimization method by using
a 2D two-tooth ratchet potential.
Of course, our approach is applicable to more general 2D ratchet potentials
in the form of Eq.~(\ref{LEQ:Pot0}).
In Sec.~\ref{TT-Model}, for the two-tooth ratchet model, we provide
$v_{0}(\boldsymbol{x})$ and $v_{1}(\boldsymbol{x})$ and describe their details.
In Sec.~\ref{Quantities}, we define indexes with which to characterize the performance
of the ratchet model; we show analytical expressions for these,
which are obtained using the same approach as in \cite{doi:10.7566/JPSJ.84.044004}.
In Sec.~\ref{Optimization}, we formulate the optimization problem.
In Sec.~\ref{NumericalResults}, we test the results of the optimization.
In Sec.~\ref{Discuss}, we suggest a way to optimize
three-tooth ratchet models.
In. Sec.~\ref{summary}, we summarize the whole study.
\section{\label{TT-Model} Two-tooth Ratchet Model}
\begin{figure}[t]
\def7.3cm{9.5cm}
\centering
\includegraphics[width=7.3cm,keepaspectratio,clip]{fig1.pdf}
\caption{
(Color online)
Contour plot of $V_0(\boldsymbol{x})$ with a skeleton of
curves
$\mathrm{C}_{\infty}:\{\boldsymbol{x}\mid v_0(\boldsymbol{x})=1\}$,
$\mathrm{E}_{+}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E_{+}\}$ and
$\mathrm{E}^{+}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E^{+}\}$.
The parameters of $V_0(\boldsymbol{x})$ are
$(m,a, b, K, d, e, f, \lambda, \alpha, \beta) =
(2, 1.8, 1, 0.02396, 3, 8, 1, 0.27, 0.34\pi, 0.05\pi)$
(the ``$d=3$'' row of B1 in Table~\ref{TAB:B}), for which
$E_{+}=13.7888$ and $E^{+}=1.78493$.
The tangent points between $\mathrm{C}_{\infty}$ and $\mathrm{E}_{+}$
($\mathrm{E}^{+}$)
almost agree with the local minima (saddles) of $V_0(\boldsymbol{x})$,
i.e., $\boldsymbol{x}_{+}$ and $-\boldsymbol{x}_{+}$
($\boldsymbol{x}^{+}$ and $-\boldsymbol{x}^{+}$),
and so do the valleys $\mathrm{C}$ and $\mathrm{C}_{\infty}$.
$\{\boldsymbol{n}_{+}, \boldsymbol{\tau}_{+}\}$
($\{\boldsymbol{n}^{+}, \boldsymbol{\tau}^{+}\}$)
denote the eigenvectors
of $\Hat{G}_{0}(\boldsymbol{x})$ at the minima
(saddles), which also almost agree with
the common tangent and normal vectors, i.e.,
$\{\boldsymbol{n}_{v}, \boldsymbol{\tau}_{v}\}$,
between $\mathrm{C}_{\infty}$ and $\mathrm{E}_{+}$
($\mathrm{C}_{\infty}$ and $\mathrm{E}^{+}$).
}
\label{fig:show_pot}
\end{figure}
For $V_0(\boldsymbol{x})$ in Eq.~(\ref{LEQ:Pot0}),
let us consider a ratchet potential with a two-fold symmetry as shown in
Fig.~\ref{fig:show_pot}, and call it the two-tooth ratchet model.
In such a case, $v_{0}(\boldsymbol{x})$ and
$v_{1}(\boldsymbol{x})$ also have two-fold symmetry.
Here, we define them as
\begin{align}
v_{0}(\boldsymbol{x})
&=
\lvert\boldsymbol{a}\cdot \boldsymbol{x}\lvert^2
+
\lambda
\frac{\lvert\boldsymbol{e}\cdot \boldsymbol{x}\rvert^2
\lvert\boldsymbol{e}_{\perp}\cdot \boldsymbol{x}\rvert^2}
{\lvert\boldsymbol{x}\rvert^2}
,
\label{v0:core}
\\
v_{1}(\boldsymbol{x})
&= \frac{1}{2}
\lvert\boldsymbol{d}\cdot\boldsymbol{x}\rvert^2
,
\label{v1:core}
\end{align}
where
$\boldsymbol{a}$, $\boldsymbol{d}$, $\boldsymbol{e}$, and $\boldsymbol{e}_{\perp}$ are
complex vector-valued parameters:
\begin{gather*}
\boldsymbol{a}=
\begin{pmatrix}
\frac{1}{a}\\ \frac{i}{b}
\end{pmatrix}
,
\quad
\boldsymbol{e}=
\begin{pmatrix}
\cos\beta&-\sin\beta\\
\sin\beta& \cos\beta
\end{pmatrix}
\begin{pmatrix}
\frac{1}{e}\\\frac{i}{f}
\end{pmatrix}
,
\\
\boldsymbol{e}_{\perp}=
\begin{pmatrix}
0&-1\\
1& 0
\end{pmatrix}
\boldsymbol{e}
,
\quad
\boldsymbol{d}\equiv
\begin{pmatrix}
\cos\alpha &-\sin\alpha\\
\sin\alpha & \cos\alpha
\end{pmatrix}
\begin{pmatrix}
d\\
i
\end{pmatrix},
\label{ComplexVecs}
\end{gather*}
with $i^2=-1$, $a > 0$, $b > 0$, $e \geq 0$, $f \geq 0$, and $0\leq\beta<\frac{\pi}{4}$.
We assume $m\gg 1$ and $0 < K \ll 1$ in Eq.~(\ref{LEQ:Pot0}), unless stated otherwise.
Then, the curve
$\mathrm{C}_{\infty}: \{\boldsymbol{x}\mid v_0(\boldsymbol{x})=1\}$
approximately indicates the potential valley.
If $\lambda = 0$, $\mathrm{C}_{\infty}$ is an ellipse, i.e.,
$
\lvert\boldsymbol{a}\cdot \boldsymbol{x}\rvert^2
=
(\boldsymbol{a}\cdot \boldsymbol{x})
(\boldsymbol{a}^{\ast}\cdot \boldsymbol{x})
=\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1$,
otherwise, for $\lambda\ne 0$, it adds a fourth harmonic deformation, with
reference axes $(\cos\beta,\sin\beta)^{\mathrm{T}}$ and
$(-\sin\beta,\cos\beta)^{\mathrm{T}}$.
The sharpness of
the potential profile normal to $\mathrm{C}_{\infty}$
is tuned by $m$
(as shown in Sec.~\ref{subsec:Hesse},
the curvature is proportional to $m^2$ for $m\geq 1$).
Function $v_{1}(\boldsymbol{x})$ is a potential function with
an anisotropic axis $(\cos\alpha,\sin\alpha)^{\mathrm{T}}$.
The curve of
$\lvert\boldsymbol{d}\cdot\boldsymbol{x}\rvert^2 = \mathrm{constant}$
is an ellipse whose short axis is along $(\cos\alpha,\sin\alpha)^{\mathrm{T}}$
and whose eccentricity is $\sqrt{1-d^{-2}}$ ($d>1$).
If $\mathrm{C}_{\infty}$ does not have line symmetry with respect to the anisotropic axis,
the pathway along the valley has a ratchet property.
\subsection{\label{subsec:v0v1} Features of the potential function}
Let $\mathrm{O}$, $\mathrm{C}$, $\boldsymbol{x}_{\sigma}$ ($\sigma\in \{-,+\}$),
and $\boldsymbol{x}^{\mu}$ ($\mu\in \{-,+\}$) be the origin, the potential valley
of $V_{0}(\boldsymbol{x})$, the local minimum, and the saddle, respectively
(Fig.~\ref{fig:show_pot}) [
$\boldsymbol{x}_{+}$ and $\boldsymbol{x}^{+}$ are placed in $x>0$ and $y>0$,
and $\boldsymbol{x}_{-}= -\boldsymbol{x}_{+}$ and
$\boldsymbol{x}^{-}= -\boldsymbol{x}^{+}$].
The minima and saddles satisfy
$\partial_{ \boldsymbol{x}} V_{0}(\boldsymbol{x})= \boldsymbol{0}$, and
Eq.~(\ref{LEQ:Pot0}) leads to
\begin{equation}
\frac{m}{2}\left[
1-
\left\{
v_{0}(\boldsymbol{x})
\right\}^m
\right]
\left\{
v_{0}(\boldsymbol{x})
\right\}^{m-1}
\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})
+K
\partial_{\boldsymbol{x}}v_{1}(\boldsymbol{x})
=
\boldsymbol{0}
.
\label{force}
\end{equation}
Using the orthogonal vectors
\begin{equation}
\boldsymbol{\tau}_{v}
\equiv
\frac{ \partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})}
{\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert},
\quad
\boldsymbol{n}_{v}
\equiv
\begin{pmatrix}
0&-1\\ 1 &0
\end{pmatrix}
\boldsymbol{\tau}_{v}
,
\label{n_v}
\end{equation}
we decompose Eq.~(\ref{force})
in two directions as
\begin{gather}
\frac{m}{2}\left[
1-
\left\{
v_{0}(\boldsymbol{x})
\right\}^m
\right]
\left\{
v_{0}(\boldsymbol{x})
\right\}^{m-1}
=-K
\frac{ \boldsymbol{\tau}_{v}\cdot
\partial_{\boldsymbol{x}}v_{1}(\boldsymbol{x})}
{\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert}
,
\label{force_1}
\\
\boldsymbol{n}_{v}\cdot \partial_{\boldsymbol{x}}v_{1}(\boldsymbol{x})
=0
.
\label{force_2}
\end{gather}
When taking the limit $m\rightarrow \infty$
in Eqs.~(\ref{force_1}) and (\ref{force_2}),
the minima and the saddles, $\{\boldsymbol{x}_{\sigma},\boldsymbol{x}^{\mu}\}$,
satisfy
\begin{gather}
\boldsymbol{n}_{v}
\cdot
\partial_{ \boldsymbol{x}}
v_1(\boldsymbol{x})
=
0, \quad \boldsymbol{x}\in \mathrm{C}_{\infty}.
\label{3th:force_2}
\end{gather}
For a geometrical interpretation of Eq.~(\ref{3th:force_2}),
let us define $\mathrm{E}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E\}$
as a family of curves specified by the parameter $E$.
Then, Eq.~(\ref{3th:force_2}) means that with certain values of $E$, the curves
$\mathrm{E}$ and $\mathrm{C}_{\infty}$ have tangent points at
$\boldsymbol{x} \in \{\boldsymbol{x}_{\sigma},\boldsymbol{x}^{\mu}\}$
at which $\boldsymbol{n}_v$ is tangent to both curves.
As shown in Fig.~\ref{fig:show_pot}, there are two cases of tangency
depending on $E$;
let $\mathrm{E}_{+}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E_{+}\}$
[ $\mathrm{E}^{+}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E^{+}\}$]
be a curve that is tangent to
$\mathrm{C}_{\infty}$ at $\boldsymbol{x}=\boldsymbol{x}_{\sigma}$
[$\boldsymbol{x}=\boldsymbol{x}^{\mu}$]
as $E$ reaches $E_{+}$ [$E^{+}$].
Since we choose $K>0$, we have $E^{+}\leq E_{+}$. Therefore,
$\mathrm{E}_{+}$ is externally tangent to
$\mathrm{C}_{\infty}$, and
$\mathrm{E}^{+}$ is internally tangent to $\mathrm{C}_{\infty}$.
However, these describe only the local relationships between
$v_0(\boldsymbol{x})$ and $v_1(\boldsymbol{x})$ at
$\boldsymbol{x}=\boldsymbol{x}_{\sigma}$ ($E= E^{+}$)
and $\boldsymbol{x}^{\mu}$ ($E^{+}$) as they contact;
the global relationships between them remain undefined.
As global conditions in which
$\mathrm{E}_{+}$ ($\mathrm{E}^{+}$) contacts
with $\mathrm{C}_{\infty}$ only at two points
$\boldsymbol{x}=\boldsymbol{x}_{+}$
and $\boldsymbol{x}_{-}$
($\boldsymbol{x}=\boldsymbol{x}^{+}$ and $\boldsymbol{x}^{-}$),
we insist that all points
on $\mathrm{C}_{\infty}$ satisfy
\begin{equation}
E^{+} \leq
v_1(\boldsymbol{x})
\leq E_{+},
\label{ineq:E}
\end{equation}
where equal cases of the left and right sides hold
at $\boldsymbol{x}=\boldsymbol{x}^{\mu}$ and
$\boldsymbol{x}_{\sigma}$, respectively.
In this case, letting $\Delta V$ be the difference of $V_{0}(\boldsymbol{x})$ [Eq.~(\ref{LEQ:Pot0})]
between the saddle and the local minimum, we have
\begin{equation}
\Delta V = K(E_{+}-E^{+}).
\label{DV}
\end{equation}
\subsection{\label{subsec:Hesse} Hessian matrix}
The Hessian matrix
$\Hat{G}_{0}(\boldsymbol{x})\equiv
\partial_{\boldsymbol{x}}\partial_{\boldsymbol{x}}^{\mathrm{T}}
V_{0}(\boldsymbol{x})$
is diagonalized approximately for $m\gg 1$.
We denote its eigenvectors by
$\boldsymbol{n}(\boldsymbol{x})$ and $\boldsymbol{\tau}(\boldsymbol{x})$, i.e.,
\begin{align}
\Hat{G}_{0}(\boldsymbol{x})
\boldsymbol{\tau}(\boldsymbol{x})
&=\Lambda_{\tau}(\boldsymbol{x})
\boldsymbol{\tau}(\boldsymbol{x}),
\label{eigen_tau}
\\
\Hat{G}_{0}(\boldsymbol{x})
\boldsymbol{n}(\boldsymbol{x})
&=\Lambda_{n}(\boldsymbol{x})
\boldsymbol{n}(\boldsymbol{x}),
\label{eigen_n}
\end{align}
where
$\Lambda_{n}(\boldsymbol{x})$ and $\Lambda_{\tau}(\boldsymbol{x})$
are the corresponding eigenvalues, respectively;
$\boldsymbol{n}(\boldsymbol{x})$ and $\boldsymbol{\tau}(\boldsymbol{x})$
are tangent and normal to $\mathrm{C}$ at
$\boldsymbol{x}\in\{\boldsymbol{x}_{\sigma},\boldsymbol{x}^{\mu}\}$;
$\Lambda_{n}(\boldsymbol{x})$ and $\Lambda_{\tau}(\boldsymbol{x})$
are equivalent to the curvatures of $V_{0}(\boldsymbol{x})$ along the
$\boldsymbol{n}(\boldsymbol{x})$ and $\boldsymbol{\tau}(\boldsymbol{x})$ axes,
respectively.
Hereinafter, we denote these eigenvectors by
$\boldsymbol{n}(\boldsymbol{x}_{\sigma})\equiv \boldsymbol{n}_{\sigma}$,
$\boldsymbol{\tau}(\boldsymbol{x}_{\sigma})\equiv \boldsymbol{\tau}_{\sigma}$,
$\boldsymbol{n}(\boldsymbol{x}^{\mu})\equiv \boldsymbol{n}^{\mu}$, and
$\boldsymbol{\tau}(\boldsymbol{x}^{\mu})\equiv \boldsymbol{\tau}^{\mu}$.
In addition, we define
the reference direction of $\boldsymbol{n}_{\sigma}$ ($\boldsymbol{n}^{\mu}$)
as directed in the counterclockwise (clockwise)
pathway of $\mathrm{C}$, and
$\boldsymbol{\tau}_{\sigma}$ ($\boldsymbol{\tau}^{\mu}$) as directed in the right-hand side
of $\boldsymbol{n}_{\sigma}$ ($\boldsymbol{n}^{\mu}$) (see Fig.~\ref{fig:show_pot}).
From Eq.~(\ref{LEQ:Pot0}), we have
\begin{align}
\Hat G_{0}(\boldsymbol{x})
=&
-\frac{m}{2}\left[
m-1-(2m-1)
v_{0}(\boldsymbol{x})^m
\right]
v_{0}(\boldsymbol{x})^{m-2}
\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})
\partial_{\boldsymbol{x}}^{\mathrm{T}}v_{0}(\boldsymbol{x})
\nonumber
\\
&-
\frac{m}{2}\left[
1-
v_{0}(\boldsymbol{x})^m
\right]
v_{0}(\boldsymbol{x})^{m-1}
\partial_{\boldsymbol{x}}
\partial_{\boldsymbol{x}}^{\mathrm{T}}v_{0}(\boldsymbol{x})
-K
\partial_{ \boldsymbol{x}}
\partial_{ \boldsymbol{x}}^{\mathrm{T}}
v_1(\boldsymbol{x})
.
\label{Hessian}
\end{align}
Substituting $v_0(\boldsymbol{x})=1$
into the first two factors in the first term in Eq.~(\ref{Hessian}),
and $v_0(\boldsymbol{x})=1+\delta v_0$ into the second term,
we approximate $\Hat G_{0}(\boldsymbol{x})$ as
\begin{align}
\Hat G_{0}(\boldsymbol{x})
\approx &
\frac{m^2}{2}
\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert^2
\boldsymbol{\tau}_{v}
\boldsymbol{\tau}_{v}^{\mathrm T}
+
\frac{m^2}{2}\delta
v_{0}
\partial_{\boldsymbol{x}}
\partial_{\boldsymbol{x}}^{\mathrm{T}}v_{0}(\boldsymbol{x})
-K
\partial_{ \boldsymbol{x}}
\partial_{ \boldsymbol{x}}^{\mathrm{T}}
v_1(\boldsymbol{x})
,
\label{G0:pre}
\end{align}
where $\boldsymbol{\tau}_{v}$ is defined in Eq.~(\ref{n_v}), and,
from Eq.~(\ref{force_1}), $\delta v_0$ is estimated as
\begin{equation}
\delta v_{0}\approx
\frac{2K\boldsymbol{\tau}_{v}\cdot
\partial_{\boldsymbol{x}}v_{1}(\boldsymbol{x})}
{m^2 \lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert}
.
\label{delta_v0}
\end{equation}
From Eqs.~(\ref{G0:pre}) and (\ref{delta_v0}),
neglecting the nondiagonal components (which are not essential),
we obtain
\begin{gather}
\Hat G_{0}(\boldsymbol{x})
\approx
\frac{m^2}{2}
\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert^2
\boldsymbol{\tau}_{v}
\boldsymbol{\tau}_{v}^{\mathrm T}
+K g(\boldsymbol{x})
\boldsymbol{n}_{v}
\boldsymbol{n}_{v}^{\mathrm T}
,
\label{Hessian:G}
\\
g(\boldsymbol{x})\equiv
\boldsymbol{n}_{v}^{\mathrm T}
\left[
\left\{
\boldsymbol{\tau}_{v}\cdot
\partial_{ \boldsymbol{x}}
v_1(\boldsymbol{x})
\right\}
\frac{
\partial_{\boldsymbol{x}}
\partial_{\boldsymbol{x}}^{\mathrm{T}}v_{0}(\boldsymbol{x})
}{ \lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert }
-
\partial_{ \boldsymbol{x}}
\partial_{ \boldsymbol{x}}^{\mathrm{T}}
v_1(\boldsymbol{x})
\right]
\boldsymbol{n}_{v}
\label{Hessian:g}
\end{gather}
for $\boldsymbol{x} \in \{ \boldsymbol{x}_{\sigma}, \boldsymbol{x}^{\mu}\}$.
This is valid for $m \gg 1$, in which
the eigenvectors of the Hessian matrix at
$\boldsymbol{x} \in \{ \boldsymbol{x}_{\sigma}, \boldsymbol{x}^{\mu}\}$, i.e.,
$\boldsymbol{\tau}_{\sigma}$, $\boldsymbol{n}_{\sigma}$,
$\boldsymbol{\tau}^{\mu}$ and $\boldsymbol{n}^{\mu}$,
are well approximated with $\boldsymbol{\tau}_{v}$ and $\boldsymbol{n}_{v}$
in Eq.~(\ref{n_v}).
We thus have
\begin{equation}
\Lambda_{\tau}(\boldsymbol{x})\approx
\frac{m^2}{2}
\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x})\rvert^2,
\quad
\Lambda_{n}(\boldsymbol{x})\approx K g(\boldsymbol{x})
\label{eigenvals}
\end{equation}
at $\boldsymbol{x} \in \{ \boldsymbol{x}_{\sigma}, \boldsymbol{x}^{\mu}\}$
for $m\gg 1$.
\section{\label{Quantities} Performance indexes}
We characterize the rotational-motion performance
of the 2D ratchet using
the mean angular momentum (MAM)
\begin{equation}
L \equiv \overline{X_t\dot{Y}_t-Y_t\dot{X}_t},
\label{MAM}
\end{equation}
the mean angular velocity (MAV) $\omega \equiv \overline{\Dot{\Theta}_t}$, and the
efficiency
\begin{gather}
\eta =
\frac{
\gamma L\omega
+
P_I
}{P_h}
,
\label{def:eta1}
\end{gather}
where
\begin{gather}
\Theta_t
\equiv
\int_{0}^{t} d s
\left(
\frac{X_s\dot Y_s-Y_s\dot X_s
}{\lvert\boldsymbol{X}\rvert^2}
\right)
\equiv
\theta(\boldsymbol{X})
-
\theta(\boldsymbol{X}_0)
,
\label{def:theta}
\\
P_I \equiv
-
\overline{
\dot{\boldsymbol{X}}\cdot \boldsymbol{f}_{I}(\boldsymbol{X})
}
=\frac{I}{2\pi}\overline{\dot \theta(\boldsymbol{X})}
=
\frac{I \omega}{2\pi}
,
\label{def:P_I}
\\
P_h \equiv
\overline{h\boldsymbol{N}_t
\cdot \Dot{\boldsymbol{X}}(t)}
,
\label{def:P_h}
\end{gather}
i.e., the counterclockwise displacement angle about the origin,
the power consumed by the load, and the input power of the external field
(which is equivalent to the total power consumption), respectively.
We have replaced Eq.~(\ref{def:eta}) with Eq.~(\ref{def:eta1}) because
the long-time averages of
the relative angular momentum
$L_t'$ [Eq.~(\ref{def:L'})] and
the relative angular velocity $\omega_t'$ [Eq.~(\ref{def:omega'})]
agree with $L$ and $\omega$, respectively, to $o(h^2)$
(see Appendix~\ref{App:XxX}).
Hereinafter,
$O(\cdot)$ and $o(\cdot)$ denote the Landau symbols (Big- and Little-O).
In Eq.~(\ref{MAM}), the direction $L>0$ corresponds to counterclockwise
rotation.
The direction of the ratchet (chirality) is defined as
the direction in which one goes around
a circular pathway along $\mathrm{C}$ through each of the minima
from the side of steeper gradient to the more gentle one. Hence, the ratchet in Fig.~\ref{fig:show_pot}
has counterclockwise chirality.
In the following analytical and numerical simulation results,
under the RDDF,
the net rotation of the ratchet tends to be the same as the chirality.
In the numerical simulations, we examine only the case of $I=0$ and
we treat the efficiency as
\begin{gather}
\eta =
\frac{
\gamma L\omega}{P_h}.
\label{def:eta0}
\end{gather}
In this paper, we consider a ratchet system in a thermal bath
under a weak and slow external field, and we impose the following requirements:
1) the typical magnitudes of $V_{h}(\boldsymbol{x}, t)$ and $I$
(which are denoted by $O(h)$ and $O(I)$, respectively, in an energetic dimension)
are smaller than the energy barrier
$\Delta V$ [see Eq.~(\ref{DV})] to a sufficient extent, it being assumed hereinafter that $O(I)\sim O(h)$;
2) the mean switching time of the RDDF ($T_{p}\equiv \frac{2\pi}{\Omega}$)
is longer than the typical relaxation time $T_r$ of a trajectory to a sufficient extent,
i.e., $\Omega T_r \ll 1$, where $T_r^{-1}$ is related to
the curvature of $V_{0}(\boldsymbol{x})$
at the minima [or more likely is governed by the smallest eigenvalue
of $\Hat{G}_{0}(\boldsymbol{x}_{\sigma})$].
In a previous paper \cite{doi:10.7566/JPSJ.84.044004},
we proposed a framework for obtaining approximate expressions for
the performance indexes
($L$, $\omega$, and $P_h$) using a master equation
for coarse-grained states under the assumptions mentioned above.
For a self-contained description, we briefly introduce
the basic construction of the master equation and
its applications to the computation of $L$, $\omega$, and $P_h$
in Secs.~\ref{DefStates} and \ref{CGK-theory}.
In Sec.~\ref{Expressions}, we show the final expressions for $L$, $\omega$, and $P_h$
that we use in later sections.
\begin{figure}[!tb]
\def7.3cm{10.0cm}
\centering
\includegraphics[width=7.3cm,keepaspectratio,clip]{fig2.pdf}
\caption{
(Color online)
Notation for moving domain boundaries on $V(\boldsymbol{x},t)$.
With $\sigma\in \{-,+\}$ and $\mu\in \{-,+\}$,
$\Tilde{\mathrm{O}}$, $\Tilde{\boldsymbol{x}}_{\sigma}$, and $\Tilde{\boldsymbol{x}}^{\mu}$
represent the local maximum, the local minimum, and the saddle points, respectively,
of $V(\boldsymbol{x},t)$.
The 2D space is divided into four domains $\Tilde{\mathrm{D}}_{\sigma}^{\mu}$
by the ridge curves
$\Tilde{\mathrm{B}}_{\sigma}$ and $\Tilde{\mathrm{B}}^{\mu}$
of $V(\boldsymbol{x},t)$.
$\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\Tilde{\boldsymbol{x}}_{\sigma})$ and
$\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\Tilde{\boldsymbol{x}}^{\mu})$
[$\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\Tilde{\boldsymbol{x}}_{\sigma})$ and
$\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\Tilde{\boldsymbol{x}}^{\mu})$]
are the tangent (normal) vectors to
$\Tilde{\mathrm{B}}_{\sigma}$ and $\Tilde{\mathrm{B}}^{\mu}$
at the minimum and the saddle points.
$\mathrm{C}_0$ (dashed--dotted curve) denotes a closed curve
surrounding a central region
of the potential that at least includes $\mathrm{O}$, $\Tilde{\mathrm{O}}'$ and
either a cross point between $\mathrm{B}_{\sigma}$ and $\Tilde{\mathrm{B}}_{\sigma}$,
or another between $\mathrm{B}^{\mu}$ and $\Tilde{\mathrm{B}}^{\mu}$.
$\Delta\mathrm{D}_{\sigma\ast}^{\mu}$ [$\Delta\mathrm{D}_{\sigma}^{\mu \ast}$] (hatched regions)
denotes the region surrounded by $\Tilde{\mathrm{B}}_{\sigma}$ and
$\mathrm{B}^{\mu}$ [$\Tilde{\mathrm{B}}^{\mu}$ and $\mathrm{B}_{\mu}$]
but excluding the interior of $\mathrm{C}_0$.
}
\label{fig:PotShapes}
\end{figure}
\subsection{\label{DefStates} Coarse-grained states and related definitions}
As shown in Fig.~\ref{fig:PotShapes},
we denote $\mathrm{O}$, $\boldsymbol{x}_{\sigma}$, and $\boldsymbol{x}^{\mu}$
($\sigma \in \{+,-\}$ and $\mu \in \{+,-\}$) as the origin,
the local minimum, and the saddle, respectively, determined by
$\partial_{ \boldsymbol{x}} V_{0}(\boldsymbol{x})= \boldsymbol{0}$.
Hereinafter, the signs ``$+$'' and ``$-$'' are identical with
$+1$ and $-1$, thereby $\boldsymbol{x}_{\sigma} = \sigma\boldsymbol{x}_{+}$ and
$\boldsymbol{x}^{\mu} = \mu\boldsymbol{x}^{+}$, where
$\boldsymbol{x}_{+}$ and $\boldsymbol{x}^{+}$ lie
in $x>0$ and $y>0$, respectively.
Furthermore, $\mathrm{B}_{\sigma}$ ($\mathrm{B}^{\mu}$) denotes the ridge curve running from
$\mathrm{O}$ through $\boldsymbol{x}_{\sigma}$ ($\boldsymbol{x}^{\mu}$) outward;
$\mathrm{D}_{\sigma}^{\mu}$ denotes the domain
surrounded by $\mathrm{B}_{\sigma}$ and $\mathrm{B}^{\mu}$;
$\mathrm{C}$ denotes the potential valley of $V_{0}(\boldsymbol{x})$.
We extend these static ridge curves to temporally varying ridge curves
on the basis of the function
\begin{equation}
V(\boldsymbol{x}, t) \equiv V_{0}(\boldsymbol{x})
+ V_{h}(\boldsymbol{x},t) +\frac{I}{2\pi} \theta(\boldsymbol{x})
\end{equation}
with the second and third terms in
Eqs.~(\ref{LEQ:Poth}) and (\ref{LEQ:PotI});
$\Tilde{\mathrm{O}}$, $\Tilde{\boldsymbol{x}}_{\sigma}$, and
$\Tilde{\boldsymbol{x}}^{\mu}$ denote
the local maximum, the local minimum, and the saddle (Fig.~\ref{fig:PotShapes}) given by
$\partial_{ \boldsymbol{x}} V(\boldsymbol{x}, t)= \boldsymbol{0}$,
respectively, which move temporally with the external field.
Similarly, $\Tilde{\mathrm{B}}_{\sigma}$ ($\Tilde{\mathrm{B}}^{\mu}$) denotes
the ridge curves running from
$\Tilde{\mathrm{O}}$ through $\Tilde{\boldsymbol{x}}_{\sigma}$
($\Tilde{\boldsymbol{x}}^{\mu}$) outward;
$\Tilde{\mathrm{D}}_{\sigma}^{\mu}$ denotes the domain surrounded by
$\Tilde{\mathrm{B}}_{\sigma}$ and $\Tilde{\mathrm{B}}^{\mu}$;
$\Tilde{\mathrm{C}}$ denotes the potential valley of $V(\boldsymbol{x}, t)$.
Corresponding to
$\boldsymbol{\tau}(\boldsymbol{x})$ and
$\boldsymbol{n}(\boldsymbol{x})$ in
Eqs.~(\ref{eigen_tau}) and (\ref{eigen_n}),
we denote by
$\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\boldsymbol{x})$ and
$\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})$
the tangent and normal vectors at the point $\boldsymbol{x}$ on
the boundary of $\Tilde{\mathrm{D}}_{\sigma}^{\mu}$
($\boldsymbol{x}\in\Tilde{\mathrm{B}}_{\sigma}$ or
$\boldsymbol{x}\in\Tilde{\mathrm{B}}^{\mu}$), where
the reference direction of $\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})$
lies in $\Tilde{\mathrm{D}}_{\sigma}^{\mu}$, and
$\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\boldsymbol{x})$ is
oriented in the right-hand direction of
$\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})$ (Fig.~\ref{fig:PotShapes}).
The vectors
$\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\boldsymbol{x})$
and
$\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})$
are the eigenvectors of the Hessian matrix
$
\Hat{G}(\boldsymbol{x})\equiv
\partial_{\boldsymbol{x}}\partial_{\boldsymbol{x}}^{\mathrm{T}}
V(\boldsymbol{x},t)
$, i.e.,
\begin{align}
\Hat{G}(\boldsymbol{x})
\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\boldsymbol{x})
&=\Lambda_{\tau}(\boldsymbol{x})
\Tilde{\boldsymbol{\tau}}_{\sigma}^{\mu}(\boldsymbol{x}),
\label{G_eigen_tau}
\\
\Hat{G}(\boldsymbol{x})
\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})
&=\Lambda_{n}(\boldsymbol{x})
\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x}),
\label{G_eigen_n}
\end{align}
where
$\Lambda_{\tau}(\boldsymbol{x})$ and $\Lambda_{n}(\boldsymbol{x})$
are the corresponding eigenvalues.
In particular, at
$\boldsymbol{x}\in
\{\Tilde{\boldsymbol{x}}_{\sigma},\Tilde{\boldsymbol{x}}^{\mu}\}$,
$\Lambda_{\tau}(\boldsymbol{x})$ and $\Lambda_{n}(\boldsymbol{x})$
are the curvatures of $V(\boldsymbol{x},t)$ along the ridge curve and the valley, respectively;
therefore, we have
$\Lambda_{\tau}(\Tilde{\boldsymbol{x}}_{\sigma})>0$,
$\Lambda_{n}(\Tilde{\boldsymbol{x}}_{\sigma})>0$,
$\Lambda_{\tau}(\Tilde{\boldsymbol{x}}^{\mu})>0$, and
$\Lambda_{n}(\Tilde{\boldsymbol{x}}^{\mu})<0$.
\subsection{Master equation for coarse-grained states}
\label{CGK-theory}
The time evolution of
probability density function (PDF) $p(\boldsymbol{x},t)$
for $\boldsymbol{X}=\boldsymbol{x}$ obeys the Fokker--Planck equation as
\begin{gather}
\label{FPE}
\partial_{t} p(\boldsymbol{x},t) =
-
\partial_{\boldsymbol{x}}\cdot
\boldsymbol{J}(\boldsymbol{x},t)
,
\\
\label{FPE:J}
\boldsymbol{J}(\boldsymbol{x},t)
\equiv \left[
-
\partial_{\boldsymbol{x}}
V(\boldsymbol{x},t)
\right]
p(\boldsymbol{x},t)
-
D\partial_{\boldsymbol{x}}
p(\boldsymbol{x},t),
\end{gather}
where $\partial_t\equiv \frac{\partial}{\partial t}$ and
$\partial_{\boldsymbol{x}}\cdot \boldsymbol{J}(\boldsymbol{x},t)$ means the 2D divergence
of the probability current density.
In terms of $p(\boldsymbol{x},t)$, a probability for an event
$\boldsymbol{X}\in \mathrm{D}_{\sigma}^{\mu}$ is given by
\begin{equation}
P(\sigma,\mu,t)
\equiv
\int_{\boldsymbol{x}\in \mathrm{D}_{\sigma}^{\mu}}
d\boldsymbol{x}
p(\boldsymbol{x},t).
\label{Markov:Psig_mu}
\end{equation}
Using this, probabilities for events
$\boldsymbol{X}\in
\mathrm{D}_{\sigma}^{+}\cup \mathrm{D}_{\sigma}^{-}$
and
$\boldsymbol{X}\in
\mathrm{D}_{+}^{\mu}\cup \mathrm{D}_{-}^{\mu}$
are represented as
$P(\sigma,t)=\sum_{\mu} P(\sigma,\mu,t)$ and
$Q(\mu,t)=\sum_{\sigma} P(\sigma,\mu,t)$, respectively.
Furthermore, the conditional probabilities,
the relative probabilities of the event
$\boldsymbol{X}\in \mathrm{D}_{\sigma}^{\mu}$ under the conditions
$\boldsymbol{X}\in
\mathrm{D}_{\sigma}^{+}\cup \mathrm{D}_{\sigma}^{-}$
and
$\boldsymbol{X}\in
\mathrm{D}_{+}^{\mu}\cup \mathrm{D}_{-}^{\mu}$,
are defined respectively as
\begin{equation}
P(\sigma\mid\mu,t) \equiv \frac{P(\sigma,\mu,t)}{Q(\mu,t)},
\quad
Q(\mu\mid\sigma,t) \equiv \frac{P(\sigma,\mu,t)}{P(\sigma,t)}.
\label{COND_PQ}
\end{equation}
In addition to the assumptions 1) $V_{h} \ll \Delta V$ and 2) $\Omega T_{r} \ll 1$,
we assume that $D$ is so small that $D\ll \Delta V$ hereinafter.
Then, the PDF peaks sharply at
$\Tilde{\boldsymbol{x}}_{\sigma}$ [=$\boldsymbol{x}_{\sigma} + O(h)$],
otherwise almost vanishes in the other region, and
the trajectories in the transition between two states
$\boldsymbol{X}\in\Tilde{\mathrm{D}}_{+}^{\mu}$ and
$\boldsymbol{X}\in\Tilde{\mathrm{D}}_{-}^{\mu}$
concentrate to $\Tilde{\mathrm{C}}$.
From Eqs.~(\ref{FPE}) and (\ref{Markov:Psig_mu}), the time derivative of
$P(\sigma,\mu,t)$ leads to
\begin{equation}
\partial_t P(\sigma,\mu,t)=
\int_{\boldsymbol{x}\in \mathrm{D}_{\sigma}^{\mu}}
d\boldsymbol{x}
\left[
- \partial_{\boldsymbol{x}}\cdot
\boldsymbol{J}(\boldsymbol{x},t)
\right].
\end{equation}
We divide the domain of integration $\mathrm D_{\sigma}^{\mu}$ into
$\Tilde{\mathrm D}_{\sigma}^{\mu}$ and
$\Delta \mathrm D_{\sigma}^{\mu} \equiv
\mathrm D_{\sigma}^{\mu}-\Tilde{\mathrm D}_{\sigma}^{\mu}
$;
$\Delta\mathrm{D}_{\sigma}^{\mu}$ consists of two domains
$\{\boldsymbol{x}\mid\boldsymbol{x} \in \mathrm D_{\sigma}^{\mu},
\boldsymbol{x} \notin \Tilde{\mathrm D}_{\sigma}^{\mu}\}$
and
$\{\boldsymbol{x}\mid\boldsymbol{x} \in \Tilde{\mathrm D}_{\sigma}^{\mu},
\boldsymbol{x} \notin \mathrm D_{\sigma}^{\mu}\}$. Therefore,
$\Delta\mathrm{D}_{\sigma}^{\mu}$ partly possesses a ``negative domain''
for which the sign of the integral is inverted.
From the assumptions $h \ll \Delta V$ and $D\ll \Delta V$, we can regard
the PDF as actually vanishing around
$\Tilde{\mathrm O}$ and $\mathrm{O}$, or the interior of $\mathrm{C}_0$
in Fig.~\ref{fig:PotShapes}.
We can thus consider the region $\Delta\mathrm{D}_{\sigma}^{\mu}$ as a sum of
the part surrounded by $\mathrm{B}_{\sigma}$ and $\Tilde{\mathrm{B}}_{\sigma}$
excluding the interior of $\mathrm{C}_0$, and the other surrounded by
$\mathrm{B}^{\mu}$ and $\Tilde{\mathrm{B}}^{\mu}$ excluding the interior of $\mathrm{C}_0$,
as indicated by hatched regions in Fig.~\ref{fig:PotShapes}.
Hereinafter, we denote by
$\Delta\mathrm{D}_{\sigma\ast}^{\mu}$ the former region, and by
$\Delta\mathrm{D}_{\sigma}^{\mu\ast}$ the latter one.
Dividing the domain of integration $\mathrm D_{\sigma}^{\mu}$
into $\Tilde{\mathrm D}_{\sigma}^{\mu}$,
$\Delta\mathrm{D}_{\sigma\ast}^{\mu}$, and
$\Delta\mathrm{D}_{\sigma}^{\mu\ast}$, we have
\begin{align}
\partial_t P(\sigma,\mu,t)
\approx &
\int_{\boldsymbol{x}\in \Tilde{\mathrm{D}}_{\sigma}^{\mu}}
d\boldsymbol{x}
\left[
- \partial_{\boldsymbol{x}}
\cdot\boldsymbol{J}(\boldsymbol{x},t)
\right]
\nonumber
\\
&
-
\partial_t P(\sigma,\mu,t)\bigr|_{Q}
+
\partial_t P(\sigma,\mu,t)\bigr|_{P},
\label{DP:express}
\end{align}
where
$\partial_t P(\sigma,\mu,t)\bigr|_{Q} \equiv
-\int_{\Delta\mathrm{D}_{\sigma}^{\mu\ast}} d\boldsymbol{x}
\left[
- \partial_{\boldsymbol{x}}
\cdot\boldsymbol{J}(\boldsymbol{x},t)
\right]
$ and
$\partial_t P(\sigma,\mu,t)\bigr|_{P} \equiv
\int_{\Delta\mathrm{D}_{\sigma\ast}^{\mu}} d\boldsymbol{x}
\left[
- \partial_{\boldsymbol{x}}
\cdot\boldsymbol{J}(\boldsymbol{x},t)
\right]
$.
From the assumptions,
we can approximate $p(\boldsymbol{x},t)$ with the thermal equilibrium PDF
[$\propto e^{-V(\boldsymbol{x},t)/D}$]
around
the minima of $V(\boldsymbol{x},t)$, and we assume
$\boldsymbol{J}(\boldsymbol{x},t)= \boldsymbol{0}$ on
$\Tilde{\mathrm{B}}_{\sigma}$.
Applying this to the first term in Eq.~(\ref{DP:express}), we obtain
\begin{align}
\int_{\boldsymbol{x}\in \Tilde{\mathrm{D}}_{\sigma}^{\mu}}
d\boldsymbol{x}
\left[
-
\partial_{\boldsymbol{x}}
\cdot
\boldsymbol{J}(\boldsymbol{x},t)
\right]
&\approx
\int_{\boldsymbol{x}\in \Tilde{\mathrm{B}}^{\mu}}
d\boldsymbol{x}
\,\Tilde{\boldsymbol{n}}_{\sigma}^{\mu}(\boldsymbol{x})\cdot
\boldsymbol{J}(\boldsymbol{x},t)
\nonumber
\\
&\equiv
\left(
\delta_{\sigma,-\mu}-
\delta_{\sigma,\mu}
\right)
J^{\mu}(t),
\label{def:J_mu}
\end{align}
where $J^{\mu}(t)$ represents the probability current
from $\Tilde{\mathrm{D}}_{\mu}^{\mu}$ to
$\Tilde{\mathrm{D}}_{-\mu}^{\mu}$.
Terms $\partial_t P(\sigma,\mu,t)\bigr|_{Q}$ and
$\partial_t P(\sigma,\mu,t)\bigr|_{P}$ are considered as follows.
For simplicity, we show them for the case $\sigma=\mu$ as
\begin{align}
\partial_t P(\mu,\mu,t)\bigr|_{Q}
&
=
\int_{\boldsymbol{x}\in\Tilde{\mathrm{B}}^{\mu}} \,d\boldsymbol{x}\,
\Tilde{\boldsymbol{n}}_{\mu}^{\mu}(\boldsymbol{x})
\cdot
\boldsymbol{J}(\boldsymbol{x},t)
-
\int_{\boldsymbol{x}\in\mathrm{B}^{\mu}} \,d\boldsymbol{x}\,
\boldsymbol{n}^{\mu}(\boldsymbol{x})
\cdot
\boldsymbol{J}(\boldsymbol{x},t)
\nonumber
\\
&\approx
Q(\mu,t)
\int_{\boldsymbol{x}\in\mathrm{B}^{\mu}} \,d\boldsymbol{x}\,
\frac{
\boldsymbol{n}^{\mu}(\boldsymbol{x})
\cdot
\left[
\boldsymbol{J}(\Tilde{\boldsymbol{x}}(\boldsymbol{x}),t)
-
\boldsymbol{J}(\boldsymbol{x},t)
\right]}
{Q(\mu,t)}
\label{_Q_RJ}
\\
&\approx Q(\mu,t)\partial_t P(\mu\mid\mu,t)
,
\label{Q_RJ}
\\
\partial_t P(\mu,\mu,t)\bigr|_{P}
&
\approx
P(\mu,t)
\int_{\boldsymbol{x}\in\mathrm{B}_{\mu}} \,d\boldsymbol{x}\,
\frac{\boldsymbol{n}_{\mu}(\boldsymbol{x})
\cdot
\boldsymbol{J}(\boldsymbol{x},t)}
{P(\mu,t)}
\label{_P_RJ}
\\
&\approx
P(\mu,t)\partial_t Q(\mu\mid\mu,t)
,
\label{P_RJ}
\end{align}
where $\Tilde{\boldsymbol{x}}(\boldsymbol{x})$ in Eq.~(\ref{_Q_RJ}) represents a map from a point
$\boldsymbol{x}$ on $\mathrm{B}^{\mu}$ to the corresponding nearest point
on $\Tilde{\mathrm{B}}^{\mu}$.
An action of relative current density
$\boldsymbol{J}(\Tilde{\boldsymbol{x}}(\boldsymbol{x}),t)-\boldsymbol{J}(\boldsymbol{x},t)$
in the integrand in Eq.~(\ref{_Q_RJ})
[Eq.~(\ref{_P_RJ}),
in which $\boldsymbol{J}(\Tilde{\boldsymbol{x}}(\boldsymbol{x}),t)=\boldsymbol{0}$
($\Tilde{\boldsymbol{x}}(\boldsymbol{x})\in \Tilde{\mathrm{B}}_{\mu}$)]
is regarded as increasing $P(\mu\mid\mu,t)$
[decreasing $Q(\mu\mid\mu,t)$] without varying $Q(\mu,t)$ [$P(\mu,t)$].
In consequence, Eq.~(\ref{DP:express}) becomes
\begin{gather}
\partial_t
P(\sigma,\mu,t)
\approx
\left(
\delta_{\sigma,-\mu}
-\delta_{\sigma,\mu}
\right)
J^{\mu}(t)+
J_{\sigma}^{\mu}(t),
\label{DP}
\\
J_{\sigma}^{\mu}(t)
\equiv
P(\sigma,t)
\partial_t
Q(\mu\mid\sigma,t)
-
Q(\mu,t)
\partial_t
P(\sigma\mid\mu,t)
.
\label{def:J'}
\end{gather}
Based on reaction rate theory~\cite{RevModPhys.62.251} or
Langer's method~\cite{PhysRevLett.21.973},
we obtain $J^{\mu}(t)$ in Eq.~(\ref{def:J_mu}) as
\begin{gather}
J^{\mu}(t)
\approx
W(-\mu,\mu,t) P(\mu,t)- W(\mu,\mu,t)P(-\mu,t),
\label{J_mu}
\\
W(\sigma,\mu,t)
\equiv
\frac{1}{2\pi}
e^{-[
V(\boldsymbol{x}^{\mu},t) -V(\boldsymbol{x}_{-\sigma},t)]/D}
\sqrt{
\frac{\Lambda_{\tau}(\boldsymbol{x}_{-\sigma})
\Lambda_{n}(\boldsymbol{x}_{-\sigma})\lvert\Lambda_{n}(\Tilde{\boldsymbol{x}}^{\mu})\rvert}
{\Lambda_{\tau}(\Tilde{\boldsymbol{x}}^{\mu})}},
\label{W_sig_mu}
\end{gather}
where $W(-\mu,\mu,t)$ [$W(\mu,\mu,t)$] is the transition probability
from a state $\boldsymbol{X}\in \mathrm{D}_{-\mu}$ to
a state $\boldsymbol{X}\in \mathrm{D}_{\mu}^{\mu}$
[$\boldsymbol{X}\in \mathrm{D}_{\mu}$ to $\boldsymbol{X}\in \mathrm{D}_{-\mu}^{\mu}$];
$\Lambda_{\tau}(\boldsymbol{x})$ and $\Lambda_{n}(\boldsymbol{x})$ are
the eigenvalues of the Hessian matrix $\Hat{G}(\boldsymbol{x})$.
For details, see Appendix~\ref{master}.
From Eq.~(\ref{DP}), the expectation value for the time derivative of a quantity
$A(\boldsymbol{X})\equiv A$ can be approximated with
the corresponding coarse-grained variable $A(\boldsymbol{x}_{\sigma}) \equiv A_{\sigma}$
as
\begin{align}
\langle \dot{A} \rangle
&\approx
\sum_{\mu}\left(
A_{-\mu} - A_{\mu}
\right)
J^{\mu}(t)
+\sum_{\sigma,\mu}A_{\sigma}J_{\sigma}^{\mu}(t),
\label{PhysObs}
\end{align}
where $\langle A \rangle =\sum_{\sigma,\mu}A(\boldsymbol{x}_{\sigma})P(\sigma,\mu,t)$,
and $A$ is assumed to be a single-valued function of the position.
However, the MAM ($L$) and MAV ($\omega$) cannot be expressed straightforwardly
as in Eq.~(\ref{PhysObs}),
e.g., it seems that the idea regarding $\omega$ as being
$\sum_{\sigma,\mu}\theta (\boldsymbol{x}_{\sigma})\partial_t P(\sigma,\mu,t)$
fails. This may be because
the angular momentum and angular velocity are classified as
axial vectors that possess information about the rotational direction
as well as their magnitudes.
Here, apart from Eq.~(\ref{PhysObs}),
we directly relate $L$ and $\omega$ with
the currents $J^{\mu}(t)$ and $J_{\sigma}^{\mu}(t)$
on the basis of physical consideration.
For an example with $\omega$, recalling that
$J^{\mu}(t)$, $J_{\mu}^{\mu}(t)$, and $-J_{-\mu}^{\mu}(t)$
express the counterclockwise currents through
$\mathrm{B}^{\mu}$,
$
\left\{
\theta(\boldsymbol{x}_{-\mu})
-
\theta(\boldsymbol{x}_{\mu})
\right\}
J^{\mu}(t)
$,
$
\left\{
\theta(\boldsymbol{x}_{-\mu})
-
\theta(\boldsymbol{x}_{\mu})
\right\}
J_{\mu}^{\mu}(t)$,
and
$
\left\{
\theta(\boldsymbol{x}_{\mu})
-
\theta(\boldsymbol{x}_{-\mu})
\right\}
J_{-\mu}^{\mu}(t)$
approximate the phase velocities measured on the pathway from
$\theta(\boldsymbol{x}_{\mu})$ to $\theta(\boldsymbol{x}_{-\mu})$
through $\mathrm{B}^{\mu}$.
We represent $L$ and $\omega$ as a superposition of two parts
as $L= L^{(I)}+ L^{(h)}$ and $\omega = \omega^{(I)}+\omega^{(h)}$,
and express each term as
\begin{align}
L^{(I)}
&\approx
\frac{g_{L}}{2}
\sum_{\mu}
\left[
\boldsymbol{x}^{\mu}\times
(\boldsymbol{x}_{-\mu}-\boldsymbol{x}_{\mu})
\right]_{z}
\overline{
J^{\mu}(t)
},
\label{LIdef:LI}
\\
L^{(h)}
&\approx
g_{L}'
\sum_{\sigma,\mu}
(\boldsymbol{x}_{\sigma}\times \boldsymbol{x}^{\mu})_{z}
\nonumber
\\
\ &\times
\overline{
\left[
P(\sigma,t)
\partial_t Q(\mu\mid\sigma,t)
-
Q(\mu,t)
\partial_t P(\sigma\mid\mu,t)
\right]
},
\label{L_expect}
\\
\omega^{(I)} &\approx
\;g_{O}
\sum_{\mu}
\left[
\theta(\boldsymbol{x}_{-\mu})
-
\theta(\boldsymbol{x}_{\mu})
\right]
\overline{J^{\mu}(t)},
\label{Ex_omg|I}
\\
\omega^{(h)}
&\approx
\;g_{O}'
\sum_{\sigma,\mu}
\left[
\theta(\boldsymbol{x}_{-\mu})-
\theta(\boldsymbol{x}_{\mu})
\right]
\left(
\delta_{\sigma,\mu}-\delta_{\sigma,-\mu}
\right)
\nonumber
\\
\ &\times
\overline{
\left[
P(\sigma,t)\partial_t Q(\mu\mid\sigma,t)
-
Q(\mu,t) \partial_t P(\sigma\mid\mu,t)
\right]
},
\label{Ex_omg|h}
\end{align}
where $L^{(I)}$ and $L^{(h)}$, also
$\omega^{(I)}$ and $\omega^{(h)}$,
come from the two types of current,
$J^{\mu}(t)$ and $J_{\sigma}^{\mu}(t)$.
Since the coarse-grained variables for the position and velocity vectors
are not exact, we employ dimensionless parameters
$g_{L}$, $g_{L}'$, $g_{O}$, and $g_{O}'$
to adjust the approximations to the numerical results;
as shown in Sec.~\ref{NumericalResults}, their actual values are $O(1)$.
Each summand in Eq.~(\ref{LIdef:LI}) represents
the $z$-component of the angular momentum at $\boldsymbol{x}^{\mu}$
with the position $\boldsymbol{x}^{\mu}$ and the momentum
$\frac12(\boldsymbol{x}_{-\mu}-\boldsymbol{x}_{\mu})J^{\mu}(t)$,
where the latter is the mean of
$(\boldsymbol{x}_{-\mu}-\boldsymbol{x}^{\mu})J^{\mu}(t)$ and
$(\boldsymbol{x}^{\mu}-\boldsymbol{x}_{\mu})J^{\mu}(t)$.
In Eq.~(\ref{L_expect}), we regard the terms
$\boldsymbol{x}_{\mu}\times [(\boldsymbol{x}^{\mu}-\boldsymbol{x}_{\mu}) J_{\mu}^{\mu}(t)]$
($\sigma=\mu$)
and
$\boldsymbol{x}^{\mu}\times [-(\boldsymbol{x}_{-\mu}-\boldsymbol{x}^{\mu}) J_{-\mu}^{\mu}(t)]$
($\sigma=-\mu$)
as the counterclockwise angular momentum.
The interpretation of each summand in Eqs.~(\ref{Ex_omg|I})
and (\ref{Ex_omg|h}) has already been mentioned in the previous paragraph.
Note that
$\theta(\boldsymbol{x}_{-\mu})-
\theta(\boldsymbol{x}_{\mu})=\pi$.
The long time average in Eqs.~(\ref{L_expect}) and (\ref{Ex_omg|h})
reads as
\begin{align*}
&\overline{
P(\sigma,t)\partial_t Q(\mu,t\mid\sigma,t)
-
Q(\mu,t) \partial_t P(\sigma,t\mid\mu,t)
}
\\
=&
\overline{
-
P(\sigma,\mu,t) \partial_t \ln P(\sigma,t)
+
P(\sigma,\mu,t) \partial_t \ln Q(\mu,t)
}
\\
=&
\overline{
\ln \frac{P(\sigma,t)}{Q(\mu,t)}
\partial_t P(\sigma,\mu,t)
}
\end{align*}
from Eq.~(\ref{COND_PQ}) and the partial integration.
Substituting this into Eqs.~(\ref{L_expect}) and (\ref{Ex_omg|h}),
we obtain
\begin{align}
L^{(h)}
&=
g_L'
\sum_{\sigma,\mu}
(\boldsymbol{x}_{\sigma}\times \boldsymbol{x}^{\mu})_{z}
\left(
\delta_{\sigma,-\mu}
-\delta_{\sigma,\mu}
\right)
\overline{
J^{\mu}(t)
\ln \frac{P(\sigma,t)}{Q(\mu,t)}
}
\nonumber
\\
&\approx
-
g_L'
\sum_{\mu}
(\boldsymbol{x}_{\mu}\times \boldsymbol{x}^{\mu})_{z}
\overline{
\left[
\ln \frac{P(\mu,t)}{Q(\mu,t)}
+
\ln \frac{P(-\mu,t)}{Q(\mu,t)}
\right]
J^{\mu}(t)
}
,
\label{Lt}
\\
\omega^{(h)}
&\approx
-\pi g_{O}'
\sum_{\mu}
\overline{
\left[
\ln\frac{P(\mu,t)}{Q(\mu,t)}
+
\ln\frac{P(-\mu,t)}{Q(\mu,t)}
\right]
J^{\mu}(t)
}
,
\label{App:PI_h}
\end{align}
where we assume that $J_{\sigma}^{\mu}(t)$ is of higher order in $h$
than $J^{\mu}(t)$ [$\sim O(h)$] in Eqs.~(\ref{DP}) and (\ref{def:J'}).
The mean power consumption $P_h$ in Eq.~(\ref{def:P_h}) can be written as
$
\overline{h\boldsymbol{N}_t
\cdot \langle\Dot{\boldsymbol{X}}\rangle}$.
Then, $\langle\Dot{\boldsymbol{X}}\rangle$ is estimated
by applying the first term in Eq.~(\ref{PhysObs}) as
\begin{equation}
\langle\dot{\boldsymbol{X}}\rangle \approx
g_{V} \sum_{\mu} (\boldsymbol{x}_{-\mu}-\boldsymbol{x}_{\mu})
J^{\mu}(t)
\label{X_dot:app}
\end{equation}
with an adjustable parameter
$g_V$ neglecting the higher-order terms other than $O(h)$, and
we obtain
\begin{equation}
P_{h} =
-2g_V h\sum_{\mu}
\overline{
J^{\mu}(t)
\boldsymbol{N}_t} \cdot
\boldsymbol{x}_{\mu}
.
\label{P_h0}
\end{equation}
Calculations for
$L^{(I)}$, $L^{(h)}$,
$\omega^{(I)}$, $\omega^{(h)}$, and $P_h$ are shown in Appendix~\ref{LRT}.
\subsection{\label{Expressions}Expressions for $L$, $\omega$ and $P_h$}
From the details given in
Appendix~\ref{App:MAM} [Eqs.~(\ref{LI:1})--(\ref{App:omg})], we obtain
\begin{gather}
L
\approx
\frac{ g_{L} W_{0}}{2D}
\left(
\boldsymbol{x}_{+}\times\boldsymbol{x}^{+}
\right)_{z}\,
\{I_0(D)-I\}
,
\label{Lt_final}
\\
\omega
\approx
\frac{\pi g_{O} W_0 }{2D}
\left\{
I_{0}(D)
-I
\right\}
,
\label{omegaEx}
\end{gather}
where
\begin{align}
W_0
&\equiv
\frac{1}{2\pi}
e^{-[
V_{0}(\boldsymbol{x}^{+}) -V_{0}(\boldsymbol{x}_{+})]/D
}
\sqrt{
\frac{H_{\tau}H_{n}\lvert G_n\rvert}{G_{\tau}}}
,
\label{def:W0}
\\
&
\approx
\frac{K}{2\pi}\frac{
\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x}_{+})\rvert}{
\lvert\partial_{\boldsymbol{x}}v_{0}(\boldsymbol{x}^{+})\rvert
}
\sqrt{ g(\boldsymbol{x}_{+})\lvert g(\boldsymbol{x}^{+})\rvert }
e^{-\Delta V/D}
\quad (m \gg 1)
,
\label{W_sig_mug}
\\
I_{0}(D)
&\equiv
-
\frac{8 g_{L}' h^2 \Omega}{
g_{L}\sqrt{2\pi D H_{n}}
}
\frac{
\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{+}
}{\Omega + 4 W_0},
\label{TorqBalance}
\\
&\approx
-
\frac{8 g_{L}' h^2 \Omega}{
g_{L}\sqrt{2\pi K D g(\boldsymbol{x}_{+})}
}
\frac{
\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})
}{\Omega + 4 W_0}
\quad (m \gg 1)
.
\label{I0D}
\end{align}
Here, $H_{\tau}\equiv\Lambda_{\tau}(\boldsymbol{x}_{\sigma})$,
$H_{n}\equiv\Lambda_{n}(\boldsymbol{x}_{\sigma})$,
$G_{n}\equiv\Lambda_{n}(\boldsymbol{x}^{\mu})$, and
$G_{\tau}\equiv\Lambda_{\tau}(\boldsymbol{x}^{\mu})$
from Eqs.~(\ref{eigen_tau}) and (\ref{eigen_n});
$g_{L}$, $g_{L}'$, and $g_{O}$ are adjustable parameters of $O(1)$.
Equations (\ref{W_sig_mug}) and (\ref{I0D})
are obtained from Eqs.~(\ref{DV}) and (\ref{eigenvals}).
Equations (\ref{Lt_final}) and (\ref{omegaEx}) suggests that
the stimuli of the RDDF can support positive work and torque for the load
as long as $I < I_{0}(D)$ ($\gamma L$ is regarded as a viscous torque).
Thus, the quantity $\displaystyle\max_{D} {I_{0}(D)}$ indicates the maximal load for
such productive work;
it quantifies the maximal performance of the ratchet.
From Eq.~(\ref{TorqBalance}), it is found that
a higher value of $\displaystyle\max_{D} I_0(D)$ is gained if the value of
$-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{+}$ is increased.
As shown in Fig.~\ref{fig:show_pot},
the factor $-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{+}$ characterizes the
asymmetry in the ratchet shape.
Additionally, one may anticipate another way of increasing $\displaystyle\max_{D} I_0(D)$, namely by decreasing $H_n$.
However, we note that Eq.~(\ref{TorqBalance})
is not always valid for small $H_n$ either because it eventually conflicts with
the prerequisite $\Omega T_{r}\ll 1$ for small $H_n$
or, because of the time-dependent fields, the potential
with small $H_n$ possibly yields temporal minima
other than $\{\boldsymbol{x}_{\sigma}\}$.
Namely, as $H_n$ becomes vanishingly small, the influence
of the time-dependent fields becomes relatively strong, possibly
breaking the local equilibrium condition on which our theory
crucially depends (see Appendix~\ref{master}).
So, the effect of decreasing $H_n$ may be limited.
From the results in Appendix~\ref{App:Power}, we also obtain $P_h$ as
\begin{equation}
P_{h}
\approx
\frac{2 g_V h^2\left|\boldsymbol{x}_{+}\right|^2 }{D}
\frac{\Omega W_{0}}{\Omega+4W_{0}}
,
\label{Ph_final}
\end{equation}
where $g_V$ is an adjustable parameter.
\section{\label{Optimization} Optimization of ratchet potential}
\subsection{Optimization problem}
\label{sec:OPT0}
We now consider the problem of maximizing
$\omega$ and $L$ through $I_0(D)$
by optimizing $V_0(\boldsymbol{x})$
[see Eqs.~(\ref{Lt_final})--(\ref{TorqBalance})].
This also has the appreciable effect of increasing $\eta$
through the numerator $L\omega$ in Eq.~(\ref{def:eta1}), whereas
the optimization of $V_0(\boldsymbol{x})$ does not crucially affect the denominator $P_h$
according to Eq.~(\ref{Ph_final}).
As mentioned in Sec.~\ref{Quantities}, from Eq.~(\ref{TorqBalance}),
we can carry out the maximization of $I_0(D)$
by designing $V_0(\boldsymbol{x})$ so as to maximize the factor
$-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{+}(\boldsymbol{x}_{+})$,
which can be replaced with the approximation
$-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})$
for $m\gg 1$ from Eq.~(\ref{I0D}).
In addition to this, we may minimize $H_{n}$
[which corresponds to $g(\boldsymbol{x}_{+})$ in Eq.~(\ref{I0D})]
within a valid range for the local equilibrium condition around
the potential minima.
Hereinafter, we assume $m\gg 1$ even in cases in which the essential 2D ratchet characteristics are retained.
We then treat $-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})$
as the main objective function to maximize and,
if necessary, treat $g(\boldsymbol{x}_{+})$ as an optional objective function
to minimize within some limited range.
Thus, a goal of the optimization is to optimize
$v_0(\boldsymbol{x})$ or $v_1(\boldsymbol{x})$ to maximize
$-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})$.
As shown in Sec.~\ref{TT-Model}, functions
$v_0(\boldsymbol{x})$ and $v_1(\boldsymbol{x})$
set up the shape of the potential valley and
the local minima and saddles in it.
Taking these into account, we first optimize
$v_1(\boldsymbol{x})$ because it immediately affects
$-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})$
through $\boldsymbol{x}_{+}$.
Here, let $p$ be a parameter in $v_1(\boldsymbol{x})$, and
rewrite it as
$v_1(\boldsymbol{x})\equiv v_1(\boldsymbol{x};p)$
to express its dependence on $p$;
$\boldsymbol{x}_{+}$ also depends on $p$.
In Eq.~(\ref{v1:core}), $p$ corresponds to $\alpha$ or $d$.
Then, our problem is to find an optimized value of $p$ ($\equiv p_{\ast}$), i.e.,
\begin{equation}
p_{\ast} \equiv \argmax_{p}
\left\{
-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v}(\boldsymbol{x}_{+})
\right\},
\label{opt_prob}
\end{equation}
where $\boldsymbol{x}_{+}$ ($ \in \mathrm{C}_{\infty}$)
is subject to
$E_{+} = v_1(\boldsymbol{x}_{+};p)$ and
\begin{equation}
E^{+} \leq
v_1(\boldsymbol{x};p)
\leq E_{+},
\quad \forall\boldsymbol{x}\in \mathrm{C}_{\infty},
\label{G0}
\end{equation}
with $E^{+} = v_1(\boldsymbol{x}^{+};p)$
($\boldsymbol{x}^{+} \in \mathrm{C}_{\infty}$).
Because this expression is rather complicated for compact wording,
an alternative for practical computation is as follows.
Here, let us consider $v_1(\boldsymbol{x})$ with the specific form
$
v_1(\boldsymbol{x}) \equiv \boldsymbol{x}^{\mathrm T}
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
\boldsymbol{x}$,
where
\begin{align}
\Hat{O}_{\alpha}
\equiv
\begin{pmatrix}
\cos\alpha& -\sin\alpha
\\
\sin\alpha& \cos\alpha
\end{pmatrix}
,
\quad
\Hat{E}_d
\equiv
\begin{pmatrix}
d^2& 0
\\
0& 1
\end{pmatrix}
.
\label{Def:OE}
\end{align}
In the actual procedure, with $\boldsymbol{x}_{+}$ determined in
\begin{equation}
\mathrm{G}_1:\
\boldsymbol{x}_{+} = \argmax_{\boldsymbol{x} \in \mathrm{C}_{\infty}}
\left\{
-\boldsymbol{x}\cdot \boldsymbol{n}_{v}(\boldsymbol{x})
\right\},
\label{G1}
\end{equation}
we fix $(\alpha,d)$ through Eq.~(\ref{3th:force_2}) or
\begin{equation}
\mathrm{G}_2:\
\boldsymbol{n}_{v}^{\mathrm T}(\boldsymbol{x}_{+})
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
\boldsymbol{x}_{+}
=
0
.
\label{force_2b}
\end{equation}
Hereinafter, $\alpha$ and $d$ range as $0 \leq \alpha < \frac{\pi}{2}$ and $d>1$,
which makes the ratchet direction counterclockwise (see Fig.~\ref{fig:show_pot}).
Note that Eq.~(\ref{G0}) is unchanged
under $d\rightarrow \frac{1}{d}$, $\alpha\rightarrow \frac{\pi}{2}+\alpha$,
$E_{+} \rightarrow \frac{E_{+}}{d^2}$, and $E_{-} \rightarrow \frac{E_{-}}{d^2}$.
So far, either $\alpha$ or $d$ is a free parameter, but not both.
For example,
using the replacement $d\equiv\tan \delta$ and the matrix $\Hat{A}_{\alpha}$
defined as
\begin{align}
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
=
\frac{1+d^2}{2}\Hat{1}-\frac{1-d^2}{2}
\Hat{A}_{\alpha}
,
\quad \Hat{A}_{\alpha}\equiv
\begin{pmatrix}
\cos 2\alpha& \sin 2\alpha
\\
\sin 2\alpha& -\cos 2\alpha
\end{pmatrix}
,
\label{Def:ODO}
\end{align}
Eq.~(\ref{force_2b}) is read as
\begin{equation}
\mathrm{G}_2':\
\cos 2\delta = \frac{
- \boldsymbol{n}_{v}(\boldsymbol{x}_{+})\cdot\boldsymbol{x}_{+}}
{
- \boldsymbol{n}_{v}(\boldsymbol{x}_{+})^{\mathrm T}
\Hat{A}_{\alpha}
\boldsymbol{x}_{+}
}
\quad
\left(\frac{\pi}{4} <\delta < \frac{\pi}{2}\right).
\label{delta_alpha}
\end{equation}
This is useful when one chooses $\alpha$ as the free parameter, and
determines $\delta$ (also $d$) with $\alpha$.
If $d$ is given instead, $\alpha$ is determined by solving
Eq.~(\ref{force_2b}).
After determining $\boldsymbol{x}_{+}$ and $(\alpha,d)$, if the right inequality in
Eq.~(\ref{G0}) is satisfied for
$E_{+}=\boldsymbol{x}_{+}^{\mathrm T}
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
\boldsymbol{x}_{+}
$, we settle the (elliptic) curve $\mathrm{E}_{+}$
with these values.
Otherwise, if the inequality is unsatisfied,
we may search for other values
of $\boldsymbol{x}_{+}$ and $(\alpha,d)$, which may be found at the second
extreme point
$\boldsymbol{x} \in \mathrm{C}_{\infty}$ of
$-\boldsymbol{x}\cdot \boldsymbol{n}_{v}(\boldsymbol{x})$, or
may refine $v_0(\boldsymbol{x})$.
This procedure is finalized by finding
$\boldsymbol{x}^{+}$ ($\in \mathrm{C}_{\infty}$), which satisfies
$
\boldsymbol{n}_{v}^{\mathrm T}(\boldsymbol{x}^{+})
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
\boldsymbol{x}^{+}
=
0
$ and the left inequality in
Eq.~(\ref{G0}) for
$E^{+}=\boldsymbol{x}^{+\mathrm T}
\Hat{O}_{\alpha} \Hat{E}_d \Hat{O}_{\alpha}^{\mathrm T}
\boldsymbol{x}^{+}$.
The curve $\mathrm{E}^{+}$ is also settled with $\boldsymbol{x}^{+}$ and $E^{+}$.
\subsubsection{\label{sec:LamZero} Elliptic case ($\lambda=0$)}
We show analytical results for
$L$, $P_h$, and $\eta$ maximized by optimizing
$v_1(\boldsymbol{x})$, through
the parameters $\alpha$ and $d$,
with
$\mathrm{G}_1$ [Eq.~(\ref{G1})] and $\mathrm{G}_2$ [Eq.~(\ref{force_2b})]
for the elliptic $\mathrm{C}_{\infty}$ ($\lambda=0$) and $m\gg 1$.
The maximized expressions
for those in Eqs.~(\ref{Lt_final}), (\ref{W_sig_mug}), (\ref{I0D}),
(\ref{Ph_final}), and (\ref{def:eta0}) are obtained as
\begin{gather}
L
\approx
\frac{ g_{L} ab W_{0}}{2D}
\{I_0(D)-I\}
\label{lamda0:L}
,
\\
P_{h}
\approx
\frac{2 g_V h^2(a^2 + b^2 - ab)}{D}
\frac{\Omega W_{0}}{\Omega+4W_{0}}
,
\label{lamda0:Ph}
\\
\eta
\approx
\frac{
\added[]{2}
\gamma g_Og_L^{'2}
h^2 \{ab(a-b)\}^2
}{g_L g_V (a^2 + b^2 - ab) \Delta V D^2}
\frac{\Omega W_{0}}{\Omega+4W_{0}}
,
\label{lamda0:eta}
\end{gather}
where, for $a>b>0$,
\begin{gather}
W_0
\approx
\frac{\deleted[]{\cancel{2}}\Delta V}{\pi (a^2 + b^2 - ab)}
e^{-\Delta V/D}
,
\\
I_{0}(D)
\approx
\frac{4 g_{L}' h^2 \Omega}{
g_{L}\sqrt{\deleted[]{\cancel{2}}\pi \Delta V D}
}
\frac{
\sqrt{ab}(a-b)
}{\Omega + 4 W_0}
,
\label{lamda0:I0}
\\
\Delta V=
\frac{K}{2}\sqrt{ab}(a+b)(d^2-1)\sin(2\alpha)
.
\label{Zero:DE}
\end{gather}
The details of the above process are given in Appendix~\ref{App:LamZero}.
From Eq.~(\ref{omegaEx}), $\omega$ is proportional to $L$.
Corresponding to Eq.~(\ref{force_2b}) or (\ref{delta_alpha}),
$\alpha$ and $d$ ($>1$) are related as
\begin{equation}
\frac{d^2+1}{d^2-1}=
\frac{ \sqrt{ab}}{a+b}
\sin 2 \alpha
-
\frac{{a}^{2}+{b}^{2}}{a^2-b^2}
\cos 2 \alpha
.
\label{ellipse:cond2}
\end{equation}
In the elliptic case, according to Eq.~(\ref{ellipse:cond2}),
we can choose any value for $\alpha$
unless
the prerequisite $\Delta V \gg D$ in
the approximation (see Appendix A)
is violated. Furthermore, we do not need to minimize $g(\boldsymbol{x}_{+})$
(or to optimize $v_1(\boldsymbol{x})$ through $\alpha$). Note that,
in the particular case of $\alpha\rightarrow \frac{\pi}{2}$ (or $0$),
Eq.~(\ref{Zero:DE}) leads to $\Delta V \rightarrow 0$,
and $\Delta V \gg D$ is violated,
where
$\mathrm{E}_{+}$ and $\mathrm{E}^{+}$ coincide with
$\mathrm{C}_{\infty}$ [$d \rightarrow \frac{a}{b}$ (or $\frac{b}{a}$)].
\subsection{\label{sec:LamNonZero} Nonelliptic case ($\lambda\ne 0$)}
Here, as a second optimization, we consider a strategy for minimizing $E_{+}$.
In the case of $\lambda\ne 0$,
the curve $\mathrm{E}_{+}$ never coincides with $\mathrm{C}_{\infty}$
for any $(\alpha,d)$.
When minimizing $E_{+}$ with respect to $(\alpha,d)$, $E_{+} > E^{+}$ is retained, and
both $\alpha$ and $d$ acquire definitive values.
At the minimized $E_{+}$, the two curves $\mathrm{E}_{+}$ and
$\mathrm{E}^{+}$ tightly enclose $\mathrm{C}_{\infty}$.
This suggests that minimizing $E_{+}$ causes
$g(\boldsymbol{x}_{+})$ (corresponding to $H_{n}$) to decrease.
In the case of $\lambda \ne 0$,
in addition to the procedure $\mathrm{G}_1$
in Eq.~(\ref{G1}), firstly, we impose
\begin{equation}
\mathrm{G}_3:\quad
(\alpha_{\ast}, d_{\ast}) = \argmin_{ 0 \leq \alpha < \frac{\pi}{2}, d(\alpha)>1} E_{+},
\label{CondG3}
\end{equation}
where $d(\alpha)$ denotes $d$ as a function of $\alpha$ defined in
Eq.~(\ref{delta_alpha}) [or Eq.~(\ref{force_2b})]; thus,
the essential number of optimization parameters is one.
Specifically, after determining
$\boldsymbol{x}_{+}$
via $\mathrm{G}_1$,
from the set of the pairs $(\alpha, d)$ satisfied in $\mathrm{G}_2$,
$\mathrm{G}_3$ selects $\alpha_{\ast}$ and $d_{\ast}$
such that they minimize $E_{+}$
(this automates the tuning of parameters).
As mentioned above, the procedure $\mathrm{G}_3$
flattens the potential profile along the valley, and narrows
the intersection of the valley.
It is then expected that the fluctuation of the rotor trajectory
may be suppressed within the valley.
This accords with our intention to improve the rotational efficiency.
Here, we should note that
$\boldsymbol{x}_{+}$ in Eq.~(\ref{delta_alpha}) has been obtained
in the limit $m\rightarrow \infty$ and in the absence
of the external fields ($h= 0$ and $I= 0$). However,
the actual minimum point deviates from $\boldsymbol{x}_{+}$;
if determining $\boldsymbol{x}_{+}$ with
$
\partial_{ \boldsymbol{x}} V_{0}(\boldsymbol{x})
=h\boldsymbol{N}_{t}+\boldsymbol{f}_{I}(\boldsymbol{x})
$, Eqs.~(\ref{force_1}) and (\ref{force_2}) are modified.
In particular, in the case of $h\ne 0$, $I=0$, and $m \rightarrow \infty$,
Eq.~(\ref{3th:force_2}) is modified as
$
\boldsymbol{n}_{v}
\cdot
\partial_{ \boldsymbol{x}}
v_1'(\boldsymbol{x},\Phi_t)
=
0
$
($\boldsymbol{x}\in \mathrm{C}_{\infty}$) with
\begin{gather}
v_1'(\boldsymbol{x},\Phi_t)
\equiv
v_1(\boldsymbol{x})
-
\frac{h}{K} \boldsymbol{x}\cdot \boldsymbol{N}_{t}
\label{v1d}
\end{gather}
for minima and saddles.
In this case, a curve
$\mathrm{E}(\Phi_t):\{\boldsymbol{x}\mid v_1'(\boldsymbol{x},\Phi_t)=E\}$
is the same ellipse as $\mathrm{E}$ except that
the center of $\mathrm{E}(\Phi_t)$ moves around the origin.
Because of this movement,
the minimum point, at which $\mathrm{E}(\Phi_t)$ is circumscribed to $\mathrm{C}_{\infty}$,
also moves along $\mathrm{C}_{\infty}$.
There is a single circumscribed point corresponding to the global minimum
and a single inscribed point
corresponding to a saddle, which we
denote by $\boldsymbol{x}_{\ast t}$ and
$\boldsymbol{x}_t^{\ast}$, respectively.
Similarly, corresponding to Eq.~(\ref{G0}), such a minimum and a saddle satisfy
$
E^{\ast}(\Phi_t) \leq
v_1'(\boldsymbol{x},\Phi_t)
\leq E_{\ast}(\Phi_t)
$
for $\forall\boldsymbol{x}\in \mathrm{C}_{\infty}$
with
$E^{\ast}(\Phi_t) \equiv v_1'(\boldsymbol{x}_t^{\ast},\Phi_t)$ and
$E_{\ast}(\Phi_t) \equiv v_1'(\boldsymbol{x}_{\ast t},\Phi_t)$.
As the circumscribed ellipse
$\mathrm{E}_{\ast}(\Phi_t):\{\boldsymbol{x}\mid v_1'(\boldsymbol{x},\Phi_t)=E_{\ast}(\Phi_t)\}$
varies with the external field,
$\boldsymbol{x}_{\ast t}$ ($\boldsymbol{x}_t^{\ast}$) is not always
close to either $\boldsymbol{x}_{+}$ or $\boldsymbol{x}_{-}$
($\boldsymbol{x}^{+}$ or $\boldsymbol{x}^{-}$). Rather, it may
sometimes jump to another point on $\mathrm{C}_{\infty}$ away from them,
which creates a temporal minimum.
The occurrence of such events
depends on the parameters $(\alpha, d)$ or the shape of $\mathrm{C}_{\infty}$.
In the experimental observation shown in Sec.~\ref{lambda_nonzero},
the temporal minimum
is likely to arise when $\mathrm{C}_{\infty}$ (of larger $\lambda$)
is tightly enclosed by $\mathrm{E}_{+}$ and $\mathrm{E}^{+}$,
as a result of optimizing $(\alpha, d)$ in
$\mathrm{G}_3$.
It is also expected that the temporal minimum may become an obstacle
in the conversion of power to net rotational output, and
may have a negative influence on the efficiency.
Therefore, we moderate $\mathrm{G}_3$
by adding a relaxation such
that the gap between
$\mathrm{E}^{+}$ and $\mathrm{E}_{+}$ becomes wider to a sufficient extent.
Since
$d$ is minimized to $d_{\ast}$ in $\mathrm{G}_3$,
then, to relax it, we replace $d$ with
\begin{equation}
d=d_{\ast}+\epsilon\quad (\epsilon > 0),
\label{Det:d}
\end{equation}
where $\epsilon$ is a relaxation parameter.
Again applying this $d$ to
$\mathrm{G}_2$ [Eq.~(\ref{force_2b})],
we obtain a revised $\alpha$. Now,
with the ratchet potential of this $(\alpha,d)$,
we can expect that the contact point between
the ellipse $\mathrm{E}_{+}(\Phi_t)$ and $\mathrm{C}_{\infty}$ is always
close to either $\boldsymbol{x}_{+}$ or $\boldsymbol{x}_{-}$,
and that the local equilibrium can be retained.
\section{\label{NumericalResults} Numerical results}
\begin{table}[tbh]
\centering
\begin{tabular}{c|ccccl}
Label &\multicolumn{2}{c}{Key param.\,vals.} & Fig.~\ref{fig:pot_zero_lamda}
& \multicolumn{2}{c}{$\Delta V$, $K$, and/or $d$.}
\\
\hline
\multirow{3}{*}{A1} &\multirow{3}{*}{$\alpha$} & $0.03\pi$ & (a)
& \multirow{3}{*}{$(K,\Delta V, d)$} & $(0.0284, 0.1500, d_{0.4\pi})$\\
& & $0.4\pi$ & (b) & &$(0.102, 0.1498, d_{0.4\pi})$ \\
& & $0.48\pi$ & (c) & & $(0.193, 0.1501, 2.1)$ \\
\hline
\multirow{3}{*}{A2} & \multirow{3}{*}{$(a,b)$} & $(1.8,1)$& (b)&\multirow{3}{*}{$(K,\Delta V)$} & $(0.102, 0.1498)$ \\
& & $(2.7, 1.5)$ & (d)& & $(0.0454, 0.1500)$\\
& & $(3.6, 2)$ & (e) & & $(0.0255, 0.1500)$\\
\hline
\multirow{3}{*}{A3}&\multirow{3}{*}{$m$} &$1$ & (f) &\multirow{3}{*}{$\Delta V$} & $0.1896$ \\
& & $2$& (b) & & $0.1498$ \\
& & $3$& (g) & & $0.1436$ \\
\hline
\multirow{3}{*}{A4} & \multirow{3}{*}{$a$} & $1.2$ & (h) & \multirow{3}{*}{$(K, \Delta V)$} & $(0.672, 0.1501)$ \\
& & $1.8$& (b) & &$(0.102, 0.1498)$ \\
& & $2.4$& (i) & & $(0.038, 0.1495)$\\
\hline
\end{tabular}
\caption{
List of parameter families in the elliptic case ($\lambda=0$).
The families are labeled as in the first column, and
their key parameters are listed in the second and third columns.
The common parameters in each family are as follows:
A1: $(m,a,b) = (2, 1.8, 1)$,
A2: $(m,d,\alpha) = (2, d_{0.4\pi}, 0.4\pi)$,
A3: $(a,b,K,d,\alpha) = (1.8, 1.0, 0.102, d_{0.4\pi}, 0.4\pi)$,
A4: $(m,b,\alpha) = (2,1,0.4\pi)$.
In A4, $d$ is determined by Eq.~(\ref{ellipse:cond2}) for
each $(a,b,\alpha)$.
$\Delta V \approx 0.15$ is maintained by modifying $K$ (fifth and sixth columns) except for A3.
$d_{0.4\pi}\approx 1.860118$.
}
\label{TAB:A}
\end{table}
We show the numerical results of $L$ (MAM) in Eq.~(\ref{MAM}),
$\omega =
\left\langle\frac{\Theta_{T_{\mathrm{tot}}}}{T_{\mathrm{tot}}}\right\rangle_{\Phi}$ (MAV)
with $\Theta_t$ in Eq.~(\ref{def:theta}),
$P_h$ in Eq.~(\ref{def:P_h}), and
$\eta$ in Eq.~(\ref{def:eta0})
for several parameter families of $V_{0}(\boldsymbol{x})$.
We also discuss the utility of the optimization strategy described in
Secs.~\ref{sec:OPT0} and \ref{sec:LamNonZero}.
The numerical simulation of Eq.~(\ref{LEQ}) was carried out using
the second-order stochastic Runge--Kutta method
with a time increment of $0.005$ ($m=1,2$) or $0.002$ ($m=3$).
The long time average, $\overline{A(\boldsymbol{X},\Phi_t)}$, was obtained by
averaging $128$ independent trials of the time series of
$T_{\mathrm{tot}}\Omega = 2^{17}$.
Throughout this paper, the parameters of $V_h(\boldsymbol{x}, t)$
in Eq.~(\ref{LEQ:Poth}) are set to $h=0.01$ and $\Omega=0.001$;
no load is applied ($I=0$);
the fitting parameters in Eqs.~(\ref{Lt_final}), (\ref{omegaEx}),
(\ref{TorqBalance}), and (\ref{Ph_final}) are set to
$g_L=2.2$, $g_L'=1.0$, $g_O=0.82g_{L}$, and $g_V=0.75$.
\begin{figure}[!tb]
\def7.3cm{12.3cm}
\centering
\includegraphics[width=7.3cm,keepaspectratio,clip]{fig3.pdf}
\caption{
(Color online)
Contour graphs of $V_{0}(\boldsymbol{x})$
on $\{\boldsymbol{x}\mid \lvert x\rvert \leq 3.75, \lvert y\rvert\leq 2.5\}$
in
the parameter families A1, A2, A3, and A4 in Table~\ref{TAB:A}.
In (a), (b), and (c), the locations of the local minima differ (A1);
in (b), (d), and (e), the shapes of the elliptic valley have a similarity
with the ratio of diameters as $1:2:3$ (A2);
in (f), (b), and (g), $m = 1$, $2$, and $3$ (A3);
in (h), (b), and (i), the eccentricities differ (A4).
See the fourth column in Table~\ref{TAB:A} for
the correspondences.
The solid and dashed closed curves indicate
$\mathrm{C}_{\infty}$, $\mathrm{E}_{+}$ (ellipse circumscribed to $\mathrm{C}_{\infty}$), and
$\mathrm{E}^{+}$ (ellipse inscribed to $\mathrm{C}_{\infty}$), respectively.
The arrows starting at the origin and ending
at the minimum and saddle
(near the circumscribed and inscribed points) indicate $\boldsymbol{x}_{+}$ (minimum)
and $\boldsymbol{x}^{+}$ (saddle), respectively.
The arrows tangent to $\mathrm{C}_{\infty}$
at $\boldsymbol{x}_{+}$ and $\boldsymbol{x}^{+}$ indicate
$\boldsymbol{n}_{+}$ and $\boldsymbol{n}^{+}$, respectively.
}
\label{fig:pot_zero_lamda}
\end{figure}
\subsection{\label{LamZero:num} Elliptic case ($\lambda=0$)}
\begin{figure}[hbt]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig4.pdf}
\caption{
(Color online)
(a) Scaled mean angular momentum $L/h^2$,
(b) scaled mean angular velocity $\omega/\Omega$,
(c) scaled input power $P_h/(h^2\Omega)$, and (d) efficiency $\eta$
versus noise intensity $D$.
As shown in the legend box in the lower-right panel,
connected symbols ($\Box$, $\bullet$, and $\triangle$) and
(dashed, solid, and dotted) curves represent the numerical (Sim.) and approximation (Appr.)
results under the potentials of parameter family A1,
which are shown in Fig.~\ref{fig:pot_zero_lamda}(a)--(c).
}
\label{fig:ang}
\end{figure}
We show the outcome of the optimization for
the performance indexes according to the parameter families
A1--A4 in Table~\ref{TAB:A},
and test the results in Eqs.~(\ref{lamda0:L}) and (\ref{lamda0:Ph}).
The contour graphs of $V_{0}(\boldsymbol{x})$ for the parameter sets in Table~\ref{TAB:A}
are displayed in Fig.~\ref{fig:pot_zero_lamda}.
In parameter family A1, it is mainly $\alpha$ that is varied so that
the local minima are positioned near
the $x$ axis ($\alpha = 0.03\pi$) as in Fig.~\ref{fig:pot_zero_lamda}(a),
the optimized position ($\alpha = 0.4\pi$) as in (b),
and near the $y$ axis ($\alpha = 0.48\pi$) as in (c).
In the second case, the factor
$-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$ in
$I_{0}(D)$ [Eq.~(\ref{TorqBalance})] is maximized with
the optimized position $\boldsymbol{x}_{+}$ in Eq.~(\ref{G1}), and
the parameter $d$ satisfies Eq.~(\ref{ellipse:cond2})
[corresponding to Eq.~(\ref{force_2}) or Eq.~(\ref{force_2b})].
In contrast, in the first and third cases,
$\alpha$ and $d$ do not satisfy Eq.~(\ref{ellipse:cond2}).
As in Fig.~\ref{fig:pot_zero_lamda}(a) and (c),
neither $\mathrm{E}_{+}$ nor $\mathrm{E}^{+}$ are tangent to
$\mathrm{C}_{\infty}$.
Figure~\ref{fig:ang} shows the plots of
$L$, $\omega$, $P_h$, and $\eta$ for $D$ in parameter family A1.
The sets of connected symbols and the (dashed, solid, and dashed--dotted) curves
represent the results of the numerical simulations
(Sim.) and the approximations
(Appr.), i.e.,
Eqs.~(\ref{Lt_final}), (\ref{omegaEx}), (\ref{Ph_final}), and (\ref{def:eta0}),
respectively (see the legend box for the correspondences between the parameters
and the types of symbol or curve).
Each of these curves has a peak with respect to $D$ that can be
estimated from the relation $\Omega \sim W_{0}$ as the steepest point
of the factor $W_{0}/(\Omega+4W_{0})$ in Eqs.~(\ref{lamda0:L})--(\ref{lamda0:eta}).
Comparing the peaks of $L$ (also $\omega$ and $\eta$) in the series of $\alpha$,
the highest one is found at $\alpha = 0.4\pi$, where $d=d_{0.4\pi}$
[for such comparisons, we attempt to impose consistency on $\Delta V$
by modifying $K$ ($\Delta V \approx 0.15$ in Sec.~\ref{LamZero:num})].
This confirms that the optimization for
$v_1(\boldsymbol{x})$ (or $\alpha$ and $d$ in it)
via $\mathrm{G}_1$ [Eq.~(\ref{G1})]
and $\mathrm{G}_2$ [Eq.~(\ref{force_2b}) or $\mathrm{G}_2'$ in Eq.~(\ref{delta_alpha})]
works well.
\begin{figure}[tbh]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig5.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/(h^2\Omega)$, and (d) $\eta$ versus $D$
under the potentials of parameter family A2
[Fig.~\ref{fig:pot_zero_lamda}(b), (d), and (e)].
}
\label{fig:mag}
\end{figure}
In parameter family A2, the major and minor radii
of the elliptic pathway of the valley
are varied as $(a,b) = (1.8, 1)$, $(2.7, 1.5)$, and $(3.6, 2)$
while retaining the similarity. Their corresponding potential landscapes
are shown in
Fig.~\ref{fig:pot_zero_lamda}(b), (d), and (e).
With the common parameters $(m,\alpha,\lambda) = (2,0.4\pi,0)$,
we set $d$ as in Eq.~(\ref{ellipse:cond2}).
Thus, $v_1(\boldsymbol{x})$ is optimized so that
the factor $-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$
is maximized.
Figure~\ref{fig:mag} shows that the peaks of $L$, $\omega$, $P_h$, and $\eta$ increase
with the diameter of the elliptic pathway.
These are consistent with Eqs.~(\ref{lamda0:L})--(\ref{Zero:DE}).
Here, it should be noted that
as the diameter of the pathway increases,
the typical magnitude of $V_h(\boldsymbol{x}, t)$ for $\Delta V$ increases.
Then, in order to maintain the local equilibrium condition,
it is necessary to decrease $h$ and $\Omega$ with the diameter.
\begin{figure}[ht]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig6.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/(h^2\Omega)$, and (d) $\eta$ versus $D$
under the potentials of parameter family
A3 [Fig.~\ref{fig:pot_zero_lamda}(b), (f), and (g)].
}
\label{fig:m}
\end{figure}
In parameter family A3,
only $m$ is increased as $m\in \{1,2,3\}$.
The corresponding potential landscapes are shown in
Fig.~\ref{fig:pot_zero_lamda}(f), (b), and (g), respectively.
In this family,
the intersection of the valley narrows for large $m$, whereas
the diameters of the pathway are nearly equal.
In Fig.~\ref{fig:m}, we can see that
for both numerical and approximation results,
each curve of $L$, $\omega$, $P_h$, and $\eta$
is likely to approach a certain curve as $m$ increases.
The approximation result of $m=1$ deviates exceptionally from such an asymptotic
approach.
For this reason, we consider that
the influence of the external field on the thermal equilibrium condition
is relatively large at $m=1$ because of the
smaller curvature in the intersection of the valley.
\begin{figure}[tbh]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig7.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/(h^2\Omega)$, and (d) $\eta$
versus $D$ under the potentials of parameter family A4
[Fig.~\ref{fig:pot_zero_lamda}(b), (h), and (i)].
}
\label{fig:elp}
\end{figure}
In parameter family A4, the eccentricity of the elliptic pathway
is increased as $\frac{a}{b} = 1.2$ [Fig.~\ref{fig:pot_zero_lamda}(h)],
$1.8$ [(b)], and $2.4$ [(i)].
Each value of $d$ obeys Eq.~(\ref{ellipse:cond2}), in which case
$-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$ is maximized.
In Fig.~\ref{fig:elp}, we
can see that the peaks
of $L$, $\omega$, $P_h$, and $\eta$ increase with $\frac{a}{b}$.
These are consistent with Eqs.~(\ref{lamda0:L})--(\ref{Zero:DE}).
As mentioned previously, for consistency
with the local equilibrium condition at larger $\frac{a}{b}$,
it is necessary to keep
$\Omega$ and $h$ sufficiently small.
We make two remarks about the comparison of the
approximation and simulation results.
Firstly, our approximation has the adjustable parameters
$g_L$, $g_L'$, $g_O$, and $g_V$
for absorbing complexities in the coarse-grained approach,
which we have determined by eye so that the approximations
agree as much as possible with all the simulation results.
Therefore, rather than focusing on the difference in height between the two results for each individual parameter, it is reasonable to compare them
in relation to the similarities among
the plotted curves in a parameter family.
From this respect,
regarding the relationship between the peak heights in Figs.~\ref{fig:ang}--\ref{fig:elp},
the approximation is consistent with the simulation results
except for the case of $m=1$ in Fig.~\ref{fig:m}.
As mentioned above, if the local equilibrium condition holds well,
our approximation can have such a consistency.
Secondly, it can be
observed that the agreement between the two results
seems better for the lowest curves
in Figs.~\ref{fig:ang} and \ref{fig:elp}.
We consider this to be a visual effect whereby, when observing the upper and lower curves for a couple of parameter sets in a panel in these figures, the difference between the two results for the lower curve is more inconspicuous than that for the upper one.
\subsection{Weakly distorted elliptic case ($\lambda\ne 0$)}
\label{lambda_nonzero}
\begin{table}[bth]
\centering
\begin{tabular}{c|ccccl}
Label &\multicolumn{2}{c}{Key param.\,vals.} & Figs.
& \multicolumn{2}{c}{$\alpha$ and/or $d$}
\\
\hline
\multirow{3}{*}{B1} & \multirow{3}{*}{$d$} & $1.9000_{\ast}$ & Fig.~\ref{fig:TAB_B}(a) & \multirow{3}{*}{$\alpha$} & $0.4766\pi_{\ast}$ \\
& & $2$ & Fig.~\ref{fig:TAB_B}(b) & & $0.4095\pi$ \\
& & $3$ & Fig.~\ref{fig:show_pot} & & $0.3392\pi$ \\
\hline
\multirow{3}{*}{B2} & \multirow{3}{*}{$e$} &$2$& Fig.~\ref{fig:TAB_B}(c) & \multirow{3}{*}{$(\alpha,d)$} & $(0.4824\pi_{\ast}, 1.7384_{\ast})$ \\
& & $3$ & Fig.~\ref{fig:TAB_B}(d)& & $(0.4785\pi_{\ast}, 1.8270_{\ast})$\\
& & $8$ & Fig.~\ref{fig:TAB_B}(a) & & $(0.4766\pi_{\ast}, 1.9000_{\ast})$\\
\hline
\multirow{4}{*}{B3}&\multirow{4}{*}{$\lambda$} &$0.1$ & Fig.~\ref{fig:TAB_B}(e) & \multirow{4}{*}{$(\alpha, d)$} & $(0.4902\pi_{\ast}, 1.8335_{\ast})$\\
& &$0.1$ & & & $(0.4249\pi, 1.85)$\\
& & $0.27$& Fig.~\ref{fig:TAB_B}(a) & & $(0.4766\pi_{\ast},1.9000_{\ast})$ \\
& & $1.2$& Fig.~\ref{fig:TAB_B}(f) & & $(0.4297\pi_{\ast}, 2.4091_{\ast})$ \\
\hline
\multirow{5}{*}{B4} & \multirow{5}{*}{$\beta$} & $0$ & Fig.~\ref{fig:TAB_B}(g) & \multirow{5}{*}{$(\alpha, d)$}& $(0.4639\pi_{\ast},1.8784_{\ast})$ \\
& & $0$ & & & $(0.4150\pi,1.9)$ \\
& & $0.05\pi$& Fig.~\ref{fig:TAB_B}(a) & &$(0.4766\pi_{\ast},1.9000_{\ast})$ \\
& & $0.05\pi$& & &$(0.4238\pi,1.95)$ \\
& & $0.15\pi$& Fig.~\ref{fig:TAB_B}(h) & & $(0.4980\pi_{\ast}, 1.8133_{\ast})$
\\
\hline
\end{tabular}
\caption{
List of parameter families in weakly distorted elliptic case.
The first and second columns consist of labels and key parameters, respectively.
For values of $\alpha$ and $d$ in the sixth column,
those with an asterisk ``$*$'' were
determined through $\mathrm{G}_3$ [Eq.~(\ref{CondG3})]; without an asterisk,
$d$ is modified as $d=d_{\ast}+\epsilon$ with $\epsilon > 0$,
and then $\alpha$ is determined through $\mathrm{G}_2$.
The common parameters for each family are as follows:
B1:
$(m,a, b, e, f, \lambda, \beta) = (2, 1.8, 1, 8, 1, 0.27, 0.05\pi)$;
B2:
$(m,a, b, f, \lambda, \beta) = (2, 1.8, 1, 1, 0.27, 0.05\pi)$;
B3:
$(m, a, b, e, f, \beta) = (2, 1.8, 1, 8, 1, 0.05\pi)$;
B4:
$(m, a, b, e, f, \lambda) = (2, 1.8, 1, 8, 1, 0.27)$.
$\Delta V = 0.15$ is maintained by modifying $K$,
which is more precise than the elliptic case.
}
\label{TAB:B}
\end{table}
\begin{figure}[!t]
\def7.3cm{14.6cm}
\centering
\includegraphics[height=7.3cm,keepaspectratio,clip]
{fig8.pdf}
\caption{
(Color online)
Contour graphs of $V_{0}(\boldsymbol{x})$
on $\{\boldsymbol{x}\mid \lvert x\rvert \leq 2.6, \lvert y\rvert\leq 1.664\}$
in the parameter families B1, B2, B3, and B4 in Table~\ref{TAB:B}, and
the curves $\mathrm{C}_{\infty}$, $\mathrm{E}_{+}$, and $\mathrm{E}^{+}$.
In (a) and (b), $d=d_{\ast}$ and $d_{\ast}+\epsilon$ (B1);
in (c) and (d), modulation of the four-fold symmetry ($\frac{e}{f}$) differs (B2);
in (e) and (f), $\lambda$ differs (B3);
in (g) and (h), $\beta$ differs (B4).
See the fourth column in Table~\ref{TAB:B} for
the correspondences.
}
\label{fig:TAB_B}
\end{figure}
\begin{figure}[hbt]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig9.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/ (h^2\Omega)$, and (d) $\eta$
versus $D$ under the potentials of parameter family B1
[Figs.~\ref{fig:TAB_B}(a), \ref{fig:TAB_B}(b), and \ref{fig:show_pot}].
}
\label{fig:opt}
\end{figure}
\begin{figure}[tbh]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig10.pdf}
\caption{
(Color online)
(a)
$\eta$ versus $D$ for a series of $d$.
For each of $d \in \{1.90_{\ast}, 1.91, 1.93, 1.95, 1.97\}$,
retaining $\Delta V = 0.15$ and
$(m,a, b, e, f, \lambda, \beta) = (2, 1.8, 1, 8, 1, 0.27, 0.05\pi)$,
$\alpha$ is optimized via
$\mathrm{G}_2$; particularly,
in the case of $d = 1.90_{\ast}$, which being
in the parameter family B1 (the first line in Table.~\ref{TAB:B}),
$(\alpha,d)$ is optimized via $\mathrm{G}_3$.
(b)
$\displaystyle\max_{D}\eta$ as a function of $d$
for $d \geq 1.90$.
As $d$ varies, $\alpha$ is optimized simultaneously via
$\mathrm{G}_2$ with the other parameters being the same as those in (a).
For $1\leq d < 1.90$, there is no optimized value of $\alpha$, and the curve
is not drawn.
The correspondences between the parameters
and the types of symbol (numerical simulation results) and
curve (approximation results) are shown in the legend boxes.
}
\label{fig:psi.so.d}
\end{figure}
\begin{figure}[tbh]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig11.pdf}
\caption{
(Color online)
(a)
$\eta$ versus $D$ for a series of $\alpha$ from $0.35\pi$ to $0.51\pi$.
(b)
$\displaystyle\max_{D}\eta$ as a function of $\alpha$
over the range treated in (a).
While $\alpha$ varies, $d$ is fixed at $1.95$ (i.e.,
$v_1(\boldsymbol{x})$ is not optimized),
$\Delta V=0.15$ is retained, and
$(m,a, b, e, f, \lambda, \beta) = (2, 1.8, 1, 8, 1, 0.27, 0.05\pi)$.
}
\label{fig:psi.so.alp}
\end{figure}
\begin{figure}[hbt]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]
{fig12.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/ (h^2\Omega)$, and (d) $\eta$
versus $D$ under the potentials of parameter family B2
[Fig.~\ref{fig:TAB_B}(c), (d), and (a)].
}
\label{fig:ef}
\end{figure}
Outcomes of the optimization described in Sec.~\ref{sec:LamNonZero}
for $V_{0}(\boldsymbol{x})$ of nonelliptic pathway ($\mathrm{C}_{\infty}$)
are shown with the results of the performance indexes
according to the parameter families B1--B4 in Table~\ref{TAB:B}.
Firstly, let us observe the effect of the relaxation for $d$
in Eq.~(\ref{Det:d}).
In parameter family B1,
$d$ is varied as $d_{\ast}$, $2$, and $3$, i.e.,
the first one is determined by $\mathrm{G}_3$ [Eq.~(\ref{CondG3})] together with
$\alpha_{\ast}$, and the second and third ones are increased from $d_{\ast}$ in accordance with
the moderation procedure [Eq.~(\ref{Det:d})]
followed by readjustment of $\alpha$ through $\mathrm{G}_2$.
To see the curves
$\mathrm{C}_{\infty}$, $\mathrm{E}_{+}$, and $\mathrm{E}^{+}$
in Figs.~\ref{fig:TAB_B}(a), \ref{fig:TAB_B}(b), and \ref{fig:show_pot},
$\mathrm{E}_{+}$ and $\mathrm{E}^{+}$ closely contact to
$\mathrm{C}_{\infty}$ for $d=d_{\ast}$ [Fig.~\ref{fig:TAB_B}(a)]
and, as $d$ is increased, the space between $\mathrm{E}_{+}$ and $\mathrm{E}^{+}$
becomes wider [Figs.~\ref{fig:TAB_B}(b) and \ref{fig:show_pot}].
The simulation results of $L$, $\omega$, and $\eta$ in Fig.~\ref{fig:opt}
demonstrate that the curves of $d=2$ are higher than those of
$d = d_{\ast}\approx 1.900$
around the peak region.
Turning to the plot of $P_h$, the curve of $d=d_{\ast}$
has another peak around $D\approx 0.006$, while the others have
only a single peak.
A reason for this new peak in $P_h$ is, as mentioned in Sec.~\ref{sec:LamNonZero},
as follows.
In the presence of time-dependent fields,
instead of the curves $\mathrm{E}_{+}$ and $\mathrm{E}^{+}$,
which are defined for $m\rightarrow \infty$ and $h=I=0$,
we should consider the temporally moving curves $E^{\ast}(\Phi_t)$ and $E_{\ast}(\Phi_t)$
with $v_1'(\boldsymbol{x},\Phi_t)$ in Eq.~(\ref{v1d}).
The motion of the circumscribed point of $E_{\ast}(\Phi_t)$ may
temporally create another minimum at a point distant from both $\boldsymbol{x}_{+}$ and
$\boldsymbol{x}_{-}$, and then may induce a jump of state.
Such a jump motion may expend power associated with a small amount of thermal activation.
We can thus relate such a power consumption to the new peak in $P_h$.
This also suggests that the input power is not applied efficiently
to the rotation while employing $v_1(\boldsymbol{x})$ such
that $\mathrm{E}_{+}$ and $\mathrm{E}^{+}$
enclose $\mathrm{C}_{\infty}$ without sufficient room.
In contrast, when making a suitably loose gap between
$\mathrm{E}_{+}$ and $\mathrm{E}^{+}$ with
$\epsilon$ in Eq.~(\ref{Det:d}), the movement of the minimum can be restricted near
either $\boldsymbol{x}_{+}$ or $\boldsymbol{x}_{-}$, in which case the local equilibrium
is maintained.
We then expect that
incorporating the moderation brings a better efficiency.
This is consistent with the numerical results for $\eta$ in Fig.~\ref{fig:opt}.
We should also note that the presented approximation cannot predict the extra peak
of $P_h$.
This is because we have assumed that the local equilibrium always holds around the minima of $V_{0}(\boldsymbol{x})$, and have
ignored any temporally induced current due to the creation of a temporal minimum.
Thus, for the case of $V_{0}(\boldsymbol{x})$ optimized with the moderation,
we can assume a local equilibrium, and basically regard the approximation to be
consistent with the results of numerical simulation.
We give a more detailed view on the marginal behaviors of $\eta$ in
the optimization for $v_1(\boldsymbol{x})$
under the procedure $\mathrm{G}_3$ followed by the moderation Eq.~(\ref{Det:d}).
Figure~\ref{fig:psi.so.d}(a) shows the graphs of
$\eta$ versus $D$ for a series of $d$ from $1.90$
(the case of $d=1.90_{\ast}\equiv d_{\ast}$ and $\alpha=0.48\pi_{\ast}\equiv \alpha_{\ast}$ in the parameter family B1
in Table~\ref{TAB:B}) to $1.97$,
where, for each $d$, $\alpha$ is simultaneously readjusted
in accordance with $\mathrm{G}_2$, i.e.,
$\alpha = \displaystyle\argmax_{ 0 \leq \alpha < \frac{\pi}{2}}
(-\boldsymbol{x}_{+}\cdot \boldsymbol{n}_{v})$,
and $\Delta V = 0.15$ is retained by modulating $K$.
These curves indicate that the peak is higher
as $d$ is closer to $d_{\ast}$, but drops at $d=d_{\ast}$.
Figure~\ref{fig:psi.so.d}(b) shows
the dependence of the peak height on $d$
in the aforementioned settings of parameters.
The solid curve thus may approximate
$\displaystyle\max_{D,\alpha}\eta$ for $d > d_{\ast}$,
whereas it is not defined for $1 \leq d < d_{\ast}$, in which
no optimized value of $\alpha$
satisfying $\mathrm{G}_2$ exists.
One can see that the numerical results (symbols) follow the solid curve,
except for the difference in their heights.
Figure~\ref{fig:psi.so.alp} shows (a) the graphs of
$\eta$ versus $D$ as only $\alpha$ varies around $\alpha \approx 0.42\pi$
with $d=1.95$, $\Delta V = 0.15$ and
$(m,a, b, e, f, \lambda, \beta) = (2, 1.8, 1, 8, 1, 0.27, 0.05\pi)$,
and (b) $\displaystyle\max_{D}\eta$ over the range
of $\alpha$ treated in the panel (a).
Recalling that the referenced parameters $\alpha \approx 0.42\pi$ and $d=1.95$
(filled circles or solid curve)
are obtained in the moderation
procedure for the case of $\alpha_{\ast} > 0.42\pi$ and $d_{\ast}$, there is a possibility of raising
the peak of $\eta$, i.e., $\displaystyle\max_{D}\eta$, by increasing $\alpha$ from
$\alpha \approx 0.42\pi$. However, in the numerical results (symbols),
as $\alpha$ is increased, $\displaystyle\max_{D}\eta$
soon plateaus and goes down for $\alpha \geq 0.45\pi$.
For $\alpha < 0.42\pi$, the peak diminishes monotonically;
this implies that $\alpha$ moves away from the optimized point on $d =1.95$.
The solid curve for $\displaystyle\max_{D}\eta$ in Fig.~\ref{fig:psi.so.alp}(b)
has a discontinuity at $\alpha\approx 0.49\pi$,
where the original two minima of $V_0(\boldsymbol{x})$
switch to another two minima
(The number of minima of $V_0(\boldsymbol{x})$
changes as two, four, and two
for $\alpha < 0.47\pi$, $0.47\pi \leq \alpha \leq 0.49\pi$, and $0.49\pi < \alpha$,
respectively), therefore, the curve is drawn only for
the domain lower than the singular point ($\alpha\approx 0.49\pi$).
Around that point,
it is expected that the local equilibrium assumption breaks,
the rotational performance drops as mentioned above,
and also our approximation becomes inconsistent
with the original assumptions such that the potential always has two minima.
Consequently, these results reveal that
the moderation procedure works well with a
small relaxation parameter.
In parameter family B2, $e$ is increased;
with $\frac{e}{f}$ [see Eq.~(\ref{v0:core})], we can enhance the fourth-order circular harmonic distortion
of the shape of the pathway along the potential valley.
It is deformed gradually from an ellipse as $\frac{e}{f}$ differs from one.
In Fig.~\ref{fig:TAB_B}, we see the shapes
of the pathway for $e = 2$ (c), $3$ (d), and $8$ (a).
In Fig.~\ref{fig:ef}, the approximation curves indicate that
the indexes rise as $e$ increases, and the numerical results seem
to follow such a tendency, although it is not as clear.
The emergence of the peak at $D\approx 0.006$ in $P_h$ is, as mentioned above,
because of the fact that $(\alpha, d)$ is determined by $\mathrm{G}_3$ without
the moderation.
As in the figure legends, we add an asterisk ``$\ast$'' to the parameter
value(s) for which
$(\alpha, d)$ is determined in $\mathrm{G}_3$ (see Table~\ref{TAB:B}).
\begin{figure}[hbt]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig13.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/ (h^2\Omega)$, and (d) $\eta$
versus $D$ under the potentials of parameter family B3
[Fig.~\ref{fig:TAB_B}(e), (a), and (f)].
}
\label{fig:lam}
\end{figure}
In parameter family B3, $\lambda$ in Eq.~(\ref{v0:core}) is increased as
$0.1$, $0.27$, and $1.2$.
As shown in Fig.~\ref{fig:TAB_B}(e), (a), and (f)
for $\lambda = 0.1$, $0.27$, and $1.2$,
the four-fold symmetric modulation on the pathway is conspicuous with $\lambda$.
In Fig.~\ref{fig:lam},
we see that the peaks of $L$, $\omega$, $P_h$, and $\eta$ decrease
with $\lambda$, except for
the case $\lambda = 0.1$ (filled circles) in which
$(\alpha, d)$ is optimized with the modulation.
A characteristic of this decrease is that
as $\lambda$ is increased,
the factor $-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$
increases; however, the other factor
$H_n$ increases simultaneously, in which case all the performance indexes decrease.
\begin{figure}[tbh]
\def7.5cm{7.5cm}
\centering
\includegraphics[height=7.5cm,keepaspectratio,clip]{fig14.pdf}
\caption{
(Color online)
(a) $L/h^2$, (b) $\omega/\Omega$, (c) $P_h/ (h^2\Omega)$, and (d) $\eta$
versus $D$ under the potentials of parameter family B4
[Fig.~\ref{fig:TAB_B}(g), (a), and (h)].
}
\label{fig:psi}
\end{figure}
In parameter family B4, $\beta$ is varied as
$\beta = 0$, $0.05\pi$, and $0.15\pi$;
with $\beta$, the axis of the fourth-order harmonic distortion rotates.
In Fig.~\ref{fig:TAB_B}(g), (a), and (h) for $\beta = 0$, $0.05\pi$,
and $0.15\pi$, respectively, we can see such a rotation.
Figure~\ref{fig:psi} shows that
$L$, $\omega$, and $\eta$ have higher peaks
for $\beta=0.05\pi$ as
$v_1(\boldsymbol{x})$ is optimized with the moderation.
Finally, let us compare
the best result in the elliptic case ($\lambda=0$) in Sec.~\ref{LamZero:num}
with that in the parameter families B1--B4
under the same conditions of $(m,a,b)$
with respect to the peak of $\eta$.
For the former, see the case of $(m,a,b)=(2,1.8,1)$, i.e.,
the curve of $a=1.8$ in Fig.~\ref{fig:elp} (or that of $m=2$ in
Fig.~\ref{fig:m} or that of $a=1.8$ in Fig.~\ref{fig:mag}).
We can see that $\eta$ for $\beta=0.05\pi$ in Fig.~\ref{fig:psi}
has a higher peak, $\eta\approx 0.038\times 10^{-2}$, than the best one,
$\eta\approx 0.023\times 10^{-2}$,
in the elliptic case.
This result suggests that the term $\lambda$ can contribute to a better efficiency.
It also implies that the efficiency could be improved by designing
$v_0(\boldsymbol{x})$ and $v_1(\boldsymbol{x})$
more carefully.
So far,
maximizing the performance indexes under the RDDF [Eq.~(\ref{LEQ:Poth})]
has been considered by optimizing $V_0(\boldsymbol{x})$; however, the value
of $\eta$ is very small. Finally, let us discuss the reason for such small
efficiency, and a possible way of remodeling to improve it.
In the present model,
for a small $h$, the field $h\boldsymbol{N}_t$
has a role in modulating the ratchet (saw-tooth) profile along
the valley by varying the positions of
the minima and saddle (or ridge curves) of $V(\boldsymbol{x},t)$
and the slopes around the minima.
This eventually causes net rotational motion because of the circular ratchet structure
of $V_0(\boldsymbol{x})$.
However,
because the primary action of the field is
to cause a linear displacement of the minima and saddles,
not all the power of the field
is applied to the unidirectional rotational motion;
instead, a great deal of the power is scattered to other motions
(i.e., rocking motions without bias in the rotational and radial directions)
\cite{doi:10.7566/JPSJ.84.044004}.
Thus, we can conclude that
the main reason for the small efficiency
lies in the form of the field.
The problem of improving the efficiency within the non-biased fields
can therefore be recast into a problem of designing
the time-dependent part of the potential, $V_h(\boldsymbol{x},t)$,
or external fields to maximize
its power conversion efficiency.
Exploiting an idea from one-dimensional ratchet models
that incorporate a mechanism for avoiding such a rocking motion
with saw-tooth type potentials that are shifted randomly back or forth
by an appropriate distance
\cite{0295-5075-27-6-002,PhysRevE.60.2127,PhysRevE.69.021102},
we may consider a form of the field as
$\boldsymbol{f}_h(\boldsymbol{x},t) =
h q(\boldsymbol{x},t) \partial_{\boldsymbol{x}}\theta(\boldsymbol{x})$.
This represents a circular field around the origin, the
direction of which varies randomly with the spatial dependency of
$q(\boldsymbol{x},t)$.
We expect that
this can reduce the rocking motion in the radial direction, and
may also suppress such diffusive motion in the rotational direction if
we appropriately design
the spatial and temporal variations of $q(\boldsymbol{x},t)$ in accordance with
$V_0(\boldsymbol{x})$ imposing a constraint that
the spatial average of $\boldsymbol{f}_h(\boldsymbol{x},t)$
has no bias.
\section{\label{Discuss} Direction For Three-tooth Ratchet Model}
So far, we have dealt with optimizing
the two-tooth ratchet potential in
Eqs.~(\ref{LEQ:Pot0})--(\ref{v1:core}).
However, our approach could be applied to more general ratchet potentials.
Here, we show how a similar approach holds for a three-tooth ratchet potential
of the same form as $V_0(\boldsymbol{x})$ in Eq.~(\ref{LEQ:Pot0}).
It is necessary that
$v_{0}(\boldsymbol{x})$ and $v_{1}(\boldsymbol{x})$
have three-fold symmetry.
For $m\gg 1$, the curve
$\mathrm{C}_{\infty}: \{\boldsymbol{x}|v_{0}(\boldsymbol{x})=1 \}$
corresponds to a potential valley, and the region of
$v_{0}(\boldsymbol{x})<1$ must be a simply connected space.
Therefore, a simple expression is proposed as
\begin{equation}
v_{0}(\boldsymbol{x})\equiv
\lvert\boldsymbol{x}\rvert^2
+a\lvert\boldsymbol{x}\rvert^4
+b
(\boldsymbol{e}_{0} \cdot \boldsymbol{x})
(\Hat{g}_1 \boldsymbol{e}_{0} \cdot \boldsymbol{x})
(\Hat{g}_2 \boldsymbol{e}_{0} \cdot\boldsymbol{x})
,
\label{ThreeTooth:v0}
\end{equation}
where $a$ is positive, so that we have
$v_{0}(\boldsymbol{x})\rightarrow\infty$ for
$\lvert\boldsymbol{x}\rvert \rightarrow\infty$,
and $\lvert b\rvert$ is sufficiently small
for such $\mathrm{C}_{\infty}$ of a simply connected curve.
Term $\Hat{g}_{1}$ ($\Hat{g}_{2}$) represents a matrix for
a rotation of angle $+\frac{2\pi}{3}$ ($-\frac{2\pi}{3}$):
\begin{align}
\Hat g_1 \equiv
\frac12
\begin{pmatrix}
-1 & -\sqrt{3}\\
\sqrt{3} & -1
\end{pmatrix},
\quad
\Hat g_2 \equiv
\frac12
\begin{pmatrix}
-1 & \sqrt{3}\\
-\sqrt{3} & -1
\end{pmatrix}
.
\end{align}
The third term adds a third circular harmonic in $\mathrm{C}_{\infty}$;
$\boldsymbol{e}_{0}$ is a reference axis on the azimuthal angle
about the origin.
Note that as $\boldsymbol{e}_{0}$ rotates, $\mathrm{C}_{\infty}$ rotates
by the same angle about the origin.
Without loss of generality, we have $b > 0$ and $\boldsymbol{e}_{0}=(1,0)^{\mathrm{T}}$.
Similarly, $v_{1}(\boldsymbol{x})$ is given as
\begin{equation}
v_{1}(\boldsymbol{x})\equiv
\lvert\boldsymbol{x}\rvert^2
+c\lvert\boldsymbol{x}\rvert^4
+d
(\boldsymbol{e}_{1} \cdot \boldsymbol{x})
(\Hat{g}_1 \boldsymbol{e}_{1} \cdot \boldsymbol{x})
(\Hat{g}_2 \boldsymbol{e}_{1} \cdot\boldsymbol{x})
\label{eq:v1_3th}
\end{equation}
with a reference axis
$\boldsymbol{e}_{1}\equiv (\cos\alpha,\sin\alpha)^{\mathrm{T}}$ and
positive values $c$ and $d$.
\begin{figure}[t]
\def7.3cm{7.3cm}
\centering
\includegraphics[height=7.3cm,keepaspectratio,clip]{fig15.pdf}
\caption{
(Color online)
Contour plot of a three-tooth ratchet potential with skeletons
of $\mathrm{C}_{\infty}: \{v_0(\boldsymbol{x})=1\}$,
$\mathrm{E}_{+}$, and $\mathrm{E}^{+}$.
The parameters of $V_0(\boldsymbol{x})$ of
Eqs.~(\ref{LEQ:Pot0}), (\ref{ThreeTooth:v0})--(\ref{eq:v1_3th}) are
$(m,a,b,c,d,K,\alpha)=(2,1,3,1,2,1,0.52\pi)$.
$\mathrm{E}_{+}$ and $\mathrm{E}^{+}$ correspond to
the curves $\{v_1(\boldsymbol{x})=E\}$ for
$E=1.62$ (externally tangent case) and $0.67$ (internally tangent case),
respectively.
}
\label{fig:3th}
\end{figure}
Figure~\ref{fig:3th} shows a contour graph of the three-tooth ratchet potential
of Eqs.~(\ref{LEQ:Pot0}), (\ref{ThreeTooth:v0})--(\ref{eq:v1_3th}).
The curves
$\mathrm{E}_{+}$ and $\mathrm{E}^{+}$ on the graph
represent the circumscribed and inscribed curves
of $\mathrm{E}:\{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E\}$
to $\mathrm{C}_{\infty}$ with $E=1.62$ and $0.67$, respectively.
The externally (internally) tangent points correspond to
the local minima (saddles) of $V_0(\boldsymbol{x})$.
For $m \rightarrow \infty$, these minima and saddles
satisfy Eq.~(\ref{3th:force_2}).
The optimization of $L$, $\omega$, and $\eta$ can be carried out
through the maximization of a factor such as
$I_0(D)$ [Eq.~(\ref{TorqBalance})], which can be obtained
by following the procedure in Appendix~\ref{App:LamZero}.
Similarly, let us assume that
the factor $-\boldsymbol{x}_{\ast}\cdot \boldsymbol{n}(\boldsymbol{x}_{\ast})$
at a local minimum point $\boldsymbol{x}_{\ast}$ affects
the maximization of $I_0(D)$ more than it does $H_{n}$.
We then employ
the strategy to maximize
$-\boldsymbol{x}\cdot \boldsymbol{n}_{v}(\boldsymbol{x})$
using $v_{0}(\boldsymbol{x})$ and $v_{1}(\boldsymbol{x})$
with the assumption that $m\gg 1$.
In particular, letting $p$ be a target parameter in $v_1(\boldsymbol{x})$ for the optimization,
the problem is to solve Eqs.~(\ref{opt_prob}) and (\ref{G0});
the actual procedure follows
Eq.~(\ref{G1}) in Sec.~\ref{sec:OPT0} as
\begin{equation}
\boldsymbol{x}_{\ast}=
\argmax_{\boldsymbol{x} \in \mathrm{C}_{\infty}}
\left\{
-\boldsymbol{x}\cdot \boldsymbol{n}_{v}(\boldsymbol{x})
\right\},
\end{equation}
and then, with this
$\boldsymbol{x}_{\ast}$, we find such $p$ as satisfies
Eqs.~(\ref{3th:force_2}) and (\ref{G0}) [replace $\boldsymbol{x}_{\ast}$ with $\boldsymbol{x}_{+}$].
As described in Sec.~\ref{sec:LamNonZero}, if we choose
$v_1(\boldsymbol{x})$ to be a different functional form from $v_0(\boldsymbol{x})$,
we can further arrange the
values of target parameters in $v_1(\boldsymbol{x})$ to
decrease $g(\boldsymbol{x})$ within a suitable range.
For example, we can consider a $v_0(\boldsymbol{x})$ that has
the sixth circular harmonic deformation.
In such a case, as in Sec.~\ref{sec:LamNonZero},
letting $\alpha$ and $d$ in $v_1(\boldsymbol{x})$ be target parameters, we first determine
$\alpha_{\ast}$ and $d_{\ast}$ as
\begin{equation}
(\alpha_{\ast}, d_{\ast}) = \argmin_{ 0 \leq \alpha < \frac{\pi}{3}, d(\alpha)} E_{+},
\end{equation}
where $d(\alpha)$ means a function that relates $\alpha$ to $d$
through Eq.~(\ref{3th:force_2}).
Next, to prevent the creation of temporal minima,
we moderate the above minimization by replacing $d$ with
$d=d_{\ast}+\epsilon$ ($\epsilon >0$), and revise $\alpha$
to satisfy Eq.~(\ref{3th:force_2}) with this $d$.
We expect this procedure to bring about a robust local equilibrium for external fields
and to reduce the power consumption for rotation.
By observing the numerical result for $P_h$,
we can confirm whether the local equilibrium has been retained.
\section{\label{summary} Summary}
The underlying themes in this study have been to elucidate the types of ratchet model (as combinations of the 2D ratchet potential and
the unbiased randomly varying field) that produce a robust net rotation,
and to determine how to maximize the rotational output and efficiency.
In this paper, we have shown that the proposed ratchet model, consisting of
a 2D two-tooth ratchet potential and an RDDF,
generates a net rotation in the direction of the ratchet potential, i.e., the chirality.
The 2D three-tooth ratchet model also possesses
such a property \cite{PhysRevE.87.022144,doi:10.7566/JPSJ.84.044004}.
The mechanism of net rotation is not so obvious
because
the deformation along the valley in the 2D ratchet model can be composed of
various types of deformation.
The mathematical origin of the net rotation can be found in
Eq.~(\ref{Lt_2}), i.e.,
$ L,\omega \propto \overline{ \ln\left[
\frac{P(\mu,t) P(-\mu,t)}{Q(\mu,t)^2} \right] J^{\mu}(t)}$,
in which $J^{\mu}(t)$, the barrier-crossing current,
and the multiplied factor, the entropy-like measure for
the deviation of the positional distribution from the equilibrium one,
are correlated as a result of the rectification effect due to the chirality,
and the average of these products remains a bias [Eq.~(\ref{lnPQJ})].
Another explanation uses a ratchet
exposed to an external field made of superimposed
uni-axially polarized fields within the same 2D plane.
The mechanism for the net rotation of a two-tooth ratchet
under a uni-axially polarized randomly varying field can be
explained using the mechanism for the propeller rotation of a ``gee-haw whammy diddle''
or ``propeller stick'' \cite{wiki:whammydiddle} (cf. \cite{PhysRevE.84.061119}).
Employing $M$ copies of such a uni-axially polarized field,
we orient their angles of polarization to $\phi_k = \frac{2\pi k}{M}$ ($0\leq k < M$), respectively,
whereby the ratchet is exposed to the field
$\sum_k h_k(t)\boldsymbol{N}_k$ (cf. Eq.~(\ref{LEQ:Poth})), where
$\boldsymbol{N}_k=(\cos\phi_k, \sin\phi_k)$ and
$h_k(t)$ is a unbiased dichotomic noise, independent of the others and
varying between $-\frac{h}{\sqrt{M}}$ and $\frac{h}{\sqrt{M}}$
with mean frequency $\Omega$. Thus, this field mimics the RDDF.
Then, as a total of the propeller-stick-like responses to the
individual fields, we can expect this ratchet to yield a net rotation
in the direction determined by its chirality.
The optimization of the 2D ratchet potential has been considered by employing
the redesigned form of the ratchet potential in Eq.~(\ref{LEQ:Pot0}).
In the proposed potential, the parameter $m$ controls the sharpness of the valley;
thereby, for $m\gg 1$, the two curves with
$\mathrm{C}_{\infty}: \{\boldsymbol{x}\mid v_0(\boldsymbol{x})=1\}$
and $\mathrm{E}: \{\boldsymbol{x}\mid v_1(\boldsymbol{x})=E\}$
determine a skeleton of the 2D ratchet potential, and
the eigenvalues of the Hessian matrix are expressed approximately
in terms of the quantities derived from
$\mathrm{C}_{\infty}$ and $\mathrm{E}$ (Sec.~\ref{TT-Model}).
These enable us to easily design a strategy for maximizing
the performance indexes [$L$ (MAM), $\omega$ (MAV), and $\eta$ (efficiency)].
From the analytic expressions for $L$ and $\omega$
(Secs.~\ref{Expressions}),
we have specified the factor $-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$
as the main objective function to maximize,
and $H_n$ as the optional one to minimize within
the appropriate range for the local equilibrium condition.
Quantities $-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$ and $H_n$ are
relevant to the asymmetry of the potential profile along the pathway
and the curvature at the potential minimum, respectively.
Through the optimization of $v_1(\boldsymbol{x})$,
the procedure to maximize the main factor
$-\boldsymbol{x}_{+}\cdot\boldsymbol{n}_{+}$ consists of
$\mathrm{G}_1$ [Eq.~(\ref{G1})] and $\mathrm{G}_2$ [Eq.~(\ref{force_2b})]
(Sec.~\ref{sec:OPT0}), and
the one to minimize $H_n$ consists of $\mathrm{G}_3$ [Eq.~(\ref{CondG3})] and its moderation
[Eq.~(\ref{Det:d})] (Sec.~\ref{sec:LamNonZero}).
The moderation of $\mathrm{G}_3$ is required to prevent the creation of temporal minima.
We reason that such temporal minima cause extra dissipation that is observed
as another peak in $P_h$; the relaxation parameter $\epsilon$
[Eq.~(\ref{Det:d})] is determined
so that $P_h$ has no additional peak in the plot for the noise intensity $D$.
Although the proposed optimization method has
been implemented on the basis of the two-tooth ratchet model,
it is applicable to three-tooth or other similar ratchet models (Sec.~\ref{Discuss})
if $\mathrm{G}_2$ is generalized as in Eq.~(\ref{3th:force_2}).
The outcomes of the optimization have been shown
in Secs.~\ref{sec:LamZero} and \ref{NumericalResults}
for the cases of $\mathrm{C}_{\infty}$ given by elliptic or nonelliptic
curves.
The analytical expressions for the maximized $L$, $\omega$, and $\eta$
are shown in the elliptic case (Secs.~\ref{sec:LamZero} and \ref{App:LamZero}).
Consistent with the numerical simulation results in Sec.~\ref{LamZero:num},
these suggest that the peaks of
$L$, $\omega$, and $\eta$ increase as the diameter or eccentricity of
the ellipse becomes larger. A note for applying such larger values of diameter
or eccentricity is that $h$ and $\Omega$ must be sufficiently small to retain
the local equilibrium.
In the nonelliptic case (Sec.~\ref{sec:LamNonZero}),
the optimization
procedure $\mathrm{G}_3$ with the moderation is useful; compared with
no moderation, it improves the efficiency with a suitable choice of
the relaxation parameter.
In comparing the efficiency between the elliptic ($\lambda=0$) and
nonelliptic ($\lambda\ne 0$) cases under the same condition of $(m,a,b)$,
we have seen that the best result in the latter case exhibits a higher peak than the best one
in the former.
This suggests that a more sophisticated design of $v_0(\boldsymbol{x})$,
incorporating higher-order harmonic deformations,
could improve efficiency.
\begin{acknowledgments}
We used the supercomputer of the Academic Center for Computing and Media Studies (ACCMS) at Kyoto University in this research.
\end{acknowledgments} |
1303.2463 | \section{Introduction}
Dark matter remains one of the most elusive problems in astronomy.
Clues of unseen, ``dark'' matter have been found for over 80 years (e.g. \citet{Oort1932A,Zwicky1933,Karachentsev1966A}).
But only since the analysis of rotation curves of galaxies, both through radio and optical, did the problem of dark matter become clear beyond doubt.
This seems to have been first realized by \citet{Freeman1970A}, who notes that,
to fully explain the HI rotation curves of M 33 and NGC 300, there is a need for additional, unseen mass, following a very different distribution.
\citet{Rubin1970A} note a similar behavior within the optical disk of Andromeda, from emission line radial velocities, where the mass to light radio (M/L) increases with radius.
The concept of dark matter halos was convincingly demonstrated when HI-rotation curves beyond the optical disks failed to show a decline \citep{Rogstad1972A,Roberts1975A,Bosma1981A}.
The observed flatness of galaxy rotation curves out to very large radial distances, well beyond the optically observable disks, makes the presence of dark matter inevitable.
This confirmation of the presence of dark matter as a major component of galaxies is one of the early triumphs of radio interferometry \citep{Monnier2012A}.\\
The exact nature of dark matter particles remains unclear, although we can infer some of its properties.
Three key constraints lie at the basis.
The first follows from the small-scale anisotropies in the cosmic microwave background (CMB).
The CMB is a relic of the baryon-photon decoupling and as such it is tracing the baryon distribution at that time.
The Wilkinson Microwave Anisotropy Probe (WMAP) has shown that the CMB is remarkably flat,
to order order one in $10^5$, implying that the universe at that time was extremely homogeneous \citep{Bennett2012A}.
This is in strong contrast with the present universe which is, at small scales, very inhomogeneous.
The second constraint comes from the large scale structure, as measured from red-shifted galaxies \citep{Sanchez2006A} and the Lyman-alpha forest \citep{McDonald2006A}.
The rate at which structures formed in the universe, implies a matter density of $1/4 \sim \Omega_m \sim 1/3$, which is far more than can be observed in our local universe.
The third constraint comes from Big-Bang nucleosynthesis (BBNS), which explains the primordial abundance ratios of atom species in the universe \citep{Yang1984,Jose2011A}.
A part of the dark matter can be in the form of dense baryonic matter, such as rogue planets and small black holes.
BBNS however places a firm upper limit on the maximum baryonic density.
To get a consistent ratio with absorbed abundance ratios, it requires the baryon density to be far below the total matter density $\Omega_m$, with value: $\Omega_b < 0.05$.
The universe could not have formed the amount of structure it has, if we only accommodate for the amount of baryonic matter set by BBNS.
To fill the difference between the total and bayonic matter densities, we need a particle which is non-baryonic and which cannot interact with photons.
The most likely candidates are weakly interacting massive particles (WIMPs) \citep{Steigman1985A}.
If these particles were non-relativistic -- ``cold'' -- they could have been clumping together before the baryon-photon decoupling.
Once baryons decoupled from photons, they could quickly fall into the existing potential wells as generated by the dark matter, giving a kickstart to the formation of structure.
So far, the best constraint on the amount of matter is the 9-year results of WMAP, setting a limit of $\Omega_b h^2 = 0.02223 \pm 0.00033$ on the baryon density and $\Omega_d h^2 = 0.1153 \pm 0.0019$ on cold dark matter \citep{Bennett2012A}.\\
Another interpretation of the dark matter problem is that we are not missing matter, but rather that our understanding of the laws of gravity is incomplete.
One solution to this is modified Newtonian dynamics (MOND) \citep{Milgrom1983A}.
MOND proposes a modification to Newtonian gravity such that, in the limit of the weakest of forces, the actual force $g$ starts to deviate from the Newtonian force $g_N$ as $g = \sqrt{g_N a_0}$.
Here $a_0$ represents a constant acceleration of $a_0 \simeq 10^{-10}$ m s$^{-2}$.
With this simple modification, MOND is capable of reproducing a surprisingly large set of phenomena traditionally credited as evidence for dark matter \citep{Famaey2012A}.\\
Many theories exist, all with their own merits and problems.
The only way to find the true solution is by placing additional (observational) constraints.
One such constraint would be the detection of the dark matter particle itself.
A range of experiments is underway in an attempt to measure the interaction between ordinary and dark matter particles.
Potential WIMP-candidate detections have been reported by some experiments, while these have been contradicted with null detections from other experiments.
So far, the results are still under debate and more work is required \citep{Freese2012A}.\\
The importance of simulations for dark matter research was already realized early on, when \citet{Ostriker1973A} noted that spiral galaxies are intrinsically stable only when they are embedded in the potential well created by a dark matter halo.
This halo needs to be 1 to 2.5 times as heavy as the disk to maintain stability against bar formation.
With hindsight, we know that bars are far more common in galaxies, and as such the halo is not required for stability.
Modern day cosmological dark matter simulations predict that dark matter has a significant impact on the formation and evolution of galaxies.
The dark matter clumps into halos, which serve as gravitational sinks for baryonic matter to fall into.
Once inside, these baryons form the galaxies and other forms of visible structure in the universe.
The size and shape of these halos is influenced by the type of dark matter particle and its merger history.
As such, getting a grip on the shape of the halo offers a potential constraint on the dark matter model \citep{Davis1985A}.\\
The shape of dark matter halos can be classified by the shape parameter $q$, using the ratios between the vertical axis $c$ and major axis $a$, such that $q = c/a$.
This divides the potential halo shapes up into three classes: prolate ($q > 1$), oblate ($q < 1$) and spherical ($q \sim 1$).
This is, of course, only a simplified version of reality, where we can also expect tri-axial shapes and changes of shape with radius and history \citep{VeraCiro2013A}.
However, for halos with masses $\gtrsim 10^{12.3} h^{-1} M_\odot$ the halos can be adequately described by a single major-to-minor axis ratio \citep{Schneider2012A}.\\
\begin{table*}[htb]
\caption[General properties of the galaxies]{General properties of the galaxies in this survey.\\
\small Distances are based on the Tully-Fisher relationship (TF) or the tip of the Red Giant Branch (TRGB).
References: (1): This work, (2a): \citep{Tully2008A}, (2b): \citep{Radburn-Smith2011A}, (2c): \citep{Matthews2000A}, (3): \citep{Vaucouleurs1991A}, (4a): \citep{Skrutskie2003A} or (4b):\citep{Jarrett2003A} }
\label{tbl:opticalproperties}
\centering
\begin{tabular}{llllllcccc}
\textbf{Galaxy} & \textbf{RA} & \textbf{DEC} & \textbf{PA} & \textbf{Dist} && \textbf{Scale} & \textbf{Morphology} & \textbf{$H_\textrm{tot}$}\\
\textbf{ } & \textbf{ } & \textbf{ } & \textbf{($^\odot$)} & \textbf{(Mpc)} && \textbf{pc/arcsec} & & \textbf{(mag)}\\
& (1) & (1) & (1) & (2) & (2) & & (3) & (4) \\*\hline\hline
\object{IC 2531} & 9:59:55.5 & -29:37:01.9 & 15.3 & 27.2 &TF (a) & 131.7& Sc & 8.835 (b) \\*
\object{IC 5052} & 20:52:05.3 & -69:12:04.2 & 128.2 & 5.7 &TRGB (b) & 27.4 & SBd & 9.102 (b) \\*
\object{IC 5249} & 22:47:06.0 & -64:50:02.7 & 75.5 & 32.1 &TF (a) & 155.4& SBd & 12.314 (a) \\*
\object{ESO 115-G021} & 2:37:47.2 & -61:20:12.7 & 226.8 & 4.9 &TRGB (a) & 23.8 & SBdm & 14.301 (a) \\*
\object{ESO 138-G014} & 17:07:01.4 & -62:05:14.3 & 314.6 & 15.8 &TF (a) & 76.5 & SB(s)d& 10.663 (b) \\*
\object{ESO 146-G014} & 22:12:59.8 & -62:04:06.9 & 222.8 & 21.7 &TF (a) & 105.1& SBd & - \\*
\object{ESO 274-G001} & 15:14:14.6 & -46:48:21.6 & 232.7 & 3.0 &TRGB (a) & 14.6 & SAd & 8.416 (b) \\*
\object{UGC 7321} & 12:17:34.4 & +22:32:25.9 & 8.2 & 10.0 &TRGB (c) & 48.5 & Sd & 10.883 (b) \small\\*\hline\hline
\end{tabular}
\end{table*}
Direct observations of the shape of halos is tricky, as there are only few tracers which offer a clear view on the vertical potential of the halo.
The flat rotation curves of galaxies, while an excellent tracer of the radial potential of the halo, provides no information in the vertical direction.
Luckily, some tracers do exist.
The stellar streams of stripped, in-falling galaxies can be used to measure the potential the stream is traversing.
\citet{Helmi2004A} analyzed the stream of Sagittarius, as found in the 2MASS survey, around the galaxy and found that the data best fits a model with a prolate shape, with an axis ratio of 5/3.
In a similar fashion, only further out, do the satellites of the Galaxy offer such a tracer.
Globular cluster NGC 5466 was modeled by \citet{Lux2012A} and found to favor an oblate or tri-axial halo, while excluding spherical and prolate halos with a high confidence.\\
Gravitational lensing offers another measure of the vertical potential.
Strong lensing uses the Einstein lens of background sources around a single galaxy / cluster.
By modeling the lens, it is possible to create a detailed mass map of the system.
By combining this with gas and stellar kinematics, it is possible to calculate the dark matter mass distribution \citep{Treu2010A}.
For example, \citet{Barnabe2012A} applied this method to lensed galaxy SDSS J2141, to find a slightly oblate halo ($q = 0.91^{+0.15}_{−0.13}$).\\
Weak lensing lacks the clear gravitation lens seen in strong lensing.
As such it is unable to measure the halo of a single galaxy.
Instead it provides an average halo shape from a statistically large sample of sources, by modeling the alignment of the background galaxies to a large series of foreground galaxies.
This allows the sources to probe the outer edges of halos.
A recent analysis by \citet{vanUitert2012} finds, on a sample of $2.2\times10^7$ galaxies, that the halo ellipticity distribution favors oblate, with $q=0.62^{+0.25}_{-0.26}$.\\
Another way to place constraints on the halo shape is by carefully modeling local edge-on galaxies.
The thickness of the \ion{H}{I} layer in spiral galaxies depends directly on the local hydrostatic equilibrium.
The \ion{H}{I} layer flares strongly at large radii, as there is less matter gravitationally binding it to the central plane.
Because of this, flaring provides a sensitive tracer of the vertical potential as function of radius in the disk.
By combining this with the horizontal potential as gained from the rotation curve, and estimates for the gas and stellar mass distributions, one can fit the potential well created by the dark matter halo.
This method was first applied to the Galaxy by \citet{Celnik1979A}.
\citet{vanderKruit1981C} applied the method to edge-on galaxy NGC 891, finding that the halo was not as flattened as the stellar disk.
NGC 4244 was analyzed by \citet{Olling1995A}, who found a highly flattened halo of $q=0.2^{+0.3}_{-0.1}$.
\citet{Banerjee2010A} analyzed \object{UGC 7321} and found a spherical halo.
These authors assumed a constant velocity dispersion, or at max a decreasing gradient, in their work.
\citet{OBrien2010D} set out to also measure the velocity dispersion as function of radius.
Using that approach to measure the halo shape of \object{UGC 7321}, they find a spherical halo ($q = 1.0 \pm 0.1$).\\
For a more extended introduction on the history of the dark matter problem, we refer the reader to \citet{vanderKruit2010A} and \citet{Sanders2010A}.\\
In this series of papers, we will continue the work started by \citet{OBrien2010B,OBrien2010C,OBrien2010D,OBrien2010A}, which aims to use the rotation curve and flaring to derive the shape of dark matter halos in which the galaxies are embedded.
In this first paper, we present our sample of 8 nearby, edge-on galaxies and their \ion{H}{I} properties.
Since \citet{OBrien2010A}, many of these galaxies have been re-observed by various groups, and we include these observations for a significant improvement.
In paper II we present our new \ion{H}{I} fitting tool \textsc{Galactus}, which will be applied to the cubes in paper III.
In paper IV we present and model the stellar mass distributions based on optical and near-infrared imaging.
Paper V will then present the derived dark matter halo shapes.\\
This paper follows the following outline.
In section \ref{sec:SampleNObservations} the criteria which define our sample are outlined, as well as the origin of the observations.
Section \ref{sec:HIreductions} details how the data has been reduced.
The results are then presented in section \ref{sec:HIresults}, followed by a discussion in section \ref{sec:Discussion}.
For ease of comparison between figures, we have opted to present the main figures at the end of the paper.
\section{Sample \& Observations}\label{sec:SampleNObservations}
Our sample is the same as that of \citet{OBrien2010A}.
Criteria for the galaxies were that they lie in the southern sky, close to edge-on
($a/b \geq 10$), avoiding Galactic latitudes $\|b\| \leq 10^\odot$ as to avoid
optical and infrared extinction, relatively bulge-less (Hubble type Scd-Sd),
nearby to be able to resolve the flare ($d \leq 30$ Mpc) and with a minimal
integrated flux of 15 Jy km s$^{-1}$.
This led to a sample of 5 edge-on galaxies.
Several other galaxies were added for various reasons.
\object{UGC 7321} was added due to the availability of very good archival data,
from the work by \citet{Uson2003A}.
\object{ESO 274-G001}, despite its low Galactic latitude of $9.3^\circ$, was
included for its exceptional proximity.
\object{IC 2531}, a box/peanut bulge, barred galaxy, was included as a test of
halo shapes under different types of mass scales and stages of secular
evolution.
We list the global properties of the sample in table
\ref{tbl:opticalproperties}.
We also present the sample in the optical in figure \ref{fig:OpticalOverview}.\\
The observations presented here come from both archived data as well as new observations.
Almost all were observed using the Australian Telescope Compact Array (ATCA) near Narrabri, Australia.
The only exception is UGC 7321, which was based on archived observations from the
Very Large Array (VLA). \citet{OBrien2010A} also included ATCA observations for this galaxy, but we found that this
limited 4 hour observation reduced the quality of the data.
Most of the observations came from previous archived observations by various authors over a span of years, going all the way to 1993.
A significant number of these observations was already used by \citet{OBrien2010A}.
A major new set of observations came as part of the work undertaken by the The Local Volume \ion{H}{I} Survey (LVHIS) \citep{Koribalski2008}.
For \object{ESO 138-G014} and \object{ESO 274-G001} we have taken new observations using the ATCA.
The various observations and configurations are listed in table \ref{tbl:HIobservations}.
\section{Reductions}\label{sec:HIreductions}
\begin{table*}[htb]
\caption[Global properties of the \ion{H}{I} cubes]{\label{tbl:GlobalPropertiesHICubes}Global properties of the \ion{H}{I} cubes.}
\centering
\begin{tabular}{l|ccccc|ccccc}
& \multicolumn{5}{c}{\textbf{High Resolution}} & \multicolumn{5}{c}{\textbf{Low Resolution}}\\
\textbf{Galaxy} & \textbf{$\theta_\textrm{FWHM}$} & \textbf{$\Delta $v} & \textbf{$\sigma$} & \textbf{$\sigma$} & \textbf{$\sigma$} & \textbf{$\theta_\textrm{B}$} & \textbf{$\Delta $v} & \textbf{$\sigma$} & \textbf{$\sigma$} & \textbf{$\sigma$}\\*
& \textbf{(arcsec)} & \textbf{(km/s)} & \textbf{($\frac{\textrm{mJy}}{\textrm{beam}}$)} & \textbf{(K)} & \textbf{atoms/cm$^2$}& \textbf{(arcsec)} & \textbf{(km/s)} & \textbf{($\frac{\textrm{mJy}}{\textrm{beam}}$)} & \textbf{(K)} & \textbf{atoms/cm$^2$}\\*
\hline\hline
IC 2531 & 11.0 & 3.3 & 1.3 & 6.6 & $3.9 \times 10^{19}$ & 30.0 & 3.3 & 1.9 & 1.3 & $7.8 \times 10^{18}$ \\*
IC 5052 & 11.0 & 3.3 & 1.1 & 5.6 & $3.4 \times 10^{19}$ & 30.0 & 3.3 & 1.6 & 1.1 & $6.4 \times 10^{18}$ \\*
IC 5249 & 8.2 & 3.3 & 1.4 & 12.3 & $7.4 \times 10^{19}$ & 30.0 & 3.3 & 2.5 & 1.7 & $9.9 \times 10^{18}$ \\*
ESO 115-G021 & 10.6 & 3.3 & 0.9 & 4.9 & $3.0 \times 10^{19}$ & 30.0 & 3.3 & 1.4 & 0.9 & $5.4 \times 10^{18}$ \\*
ESO 138-G014 & 10.8 & 3.3 & 2.2 & 11.3 & $6.8 \times 10^{19}$ & 30.0 & 3.3 & 3.1 & 2.1 & $1.3 \times 10^{19}$ \\*
ESO 146-G014 & 9.8 & 3.3 & 1.2 & 7.4 & $4.4 \times 10^{19}$ & 30.0 & 3.3 & 1.8 & 1.2 & $7.3 \times 10^{18}$ \\*
ESO 274-G001 & 12.2 & 3.3 & 1.3 & 5.1 & $3.1 \times 10^{19}$ & 30.0 & 3.3 & 1.7 & 1.2 & $7.0 \times 10^{18}$ \\*
UGC 7321 & 13.8 & 5.2 & 0.3 & 1.1 & $9.9 \times 10^{18}$ & 30.0 & 5.2 & 0.5 & 0.4 & $3.5 \times 10^{18}$ \small\\*\hline\hline
\end{tabular}
\end{table*}
\subsection{Calibration \& Imaging}
The calibration and imaging of the data has been done in \textsc{miriad}
\citep{Sault1995A}. All subsequent analysis was done in \textsc{python}.
All archived data from the ATCA was take in the XX and YY polarizations.
The new observations also include the XY and YX polarizations.
We made use of the pre-calibrated UV data from \citet{OBrien2010A}, although we
have re-reduced parts where we believed improvements could be made.\\
The data was first imported and split into the various observations, excluding
any telescope shadowing. We then inspected all data for RFI and flagged where
necessary. The primary and secondary calibrator were then used to calculate the gains and
bandpasses. This was then applied to the main target. The continuum was then
subtracted from the UV data.\\
The imaging was done using a three pass strategy, similar to
\citet{OBrien2010A}. Since some of the data has pointing offsets for the same
galaxy, we used the \textsc{miriad} ``joint approach'' to invert the data in mosaic
mode. The data was imaged in Stokes II, which is a special case of Stokes I,
under the assumption of an unpolarized source. A robust parameter of 0.4 was used to
provide a good balance between beamsize and sidelobes.\\
In the first pass we cleaned the full cube, using the
\textsc{mossdi} task. This was then restored, which resulted in an estimate for the
major and minor axis of the beam. This was taken as the basis for the second
pass. Here we used a
square cell-size of one third of the major axis of the beamshape. The central quadrant was again
cleaned and restored, using a circular beam with the major axis beamshape from the
previous pass. The restored cube was smoothed to twice the beamsize and the RMS
noise determined. We defined a regions mask in the cube where the signal was double the
RMS. This was used in the third pass to perform a deep-clean on the data,
where we cleaned the data to a cutoff level of half the RMS noise. The final
cube was then restored. \\
We also produced low resolution cubes with a beam of $30''$ to search for any faint,
extended component. This was done by tapering the invert to $30''$. The clean
was then based on the same regions as the high resolution cube. In some cases we
found the regions to be too tight for the low resolution beam. The regions were
then broadened and both the high- and low-resolution cubes were again cleaned and restored using these wider
regions.\\
The VLA data for \object{UGC 7321} consisted of very small band-passes, such that there
was no possibility to perform a continuum subtraction in the UV. The continuum
was therefore subtracted in the image plane. The above strategy led to a beam of
15 arcsec for UGC 7321. We have applied an additional sidelobe suppression of 700 arcsec
to the beam, which led to the smaller beam as reported here.\\
\subsection{Analysis}
The noise in a mosaicked cube varies with position. It is therefore important
to measure the noise close to the galaxy, while avoiding any contamination from
the galaxy. The regions mask offered a good tool for this. We created a zeroth
order moment map from these regions, which defined the maximum extent of the
galaxy. The noise was then measured on the pixels in every channel that
were inside this moment zero map, but were outside the specific region in
that channel. We show the noise estimates in table \ref{tbl:GlobalPropertiesHICubes}.
We convert from intensity $I$ in Jy/beam to surface brightness temperature $T_B$ in Kelvin,
and to column densities $n_\ion{H}{I}$ in atoms/cm$^2$, using equations \ref{eqn:JytoK} and \ref{eqn:Ktocm}.
\begin{eqnarray}
T_B &=& 606000 \;I \: / \:\theta^2\label{eqn:JytoK}\\
n_\ion{H}{I} &=& 1.8127 \times 10^{18} \;T_B \; \Delta v \label{eqn:Ktocm}
\end{eqnarray}
The central position and position angle (PA) were derived from the high
resolution cubes. First the cube was clipped below 30K, as
the lower intensity regions were often found to be asymmetrical. A clear example
of this asymmetry can be seen in IC 5052 (see figure \ref{fig:IC5052-doublemoment0}).
We then created a moment zero image of the cube.
The center of the galaxy was found as the minimal $\chi^2$ of the
difference between the image and its $180^\odot$ rotated counterpart, as trialled
over a wide range of positions. The position angle was then found by rotating
the galaxy over a range of possible angles around this central position. The
lowest $\chi^2$ between the rotated images and its upside-down flipped
counterpart gave the position angle.
We set all position angles such that the approaching side will align with the left-hand side of the position-velocity diagram.
The results for this are accurate to within $0.1$ pixel and $0.1$ degree and are listed in table \ref{tbl:opticalproperties}.\\
Both the high and low resolution cubes were rotated, using the central position
and PA, such that the major axis of the galaxy was aligned with the horizontal.
We created zeroth moment maps for both rotated cubes, again with two times the
RMS noise clipping. These cubes are shown in \emph{c} parts of figures 1-8. The
high resolution zeroth moment map was also used to created contour levels on top
of the Digital Sky Survey (DSS) J-band image for that region. These are shown in
the \emph{a} parts of figures 1-8.\\
It was also interesting to see the maximum brightness temperature maps of the
galaxies, as these would be instrumental in detecting any potential optical
depth issues. The cube was converted from intensities $I$ in mJy/beam to
brightness temperatures $T_B$ in K, using equation \ref{eqn:JytoK}.
We created a maximum brightness temperature map by selecting the maximum
temperature along the velocity axis $v$ for each position $r,z$. These are shown in the
top plot of the \emph{b} parts of figures 1-8. The middle plot shows the maximum
temperature along the minor axis $z$ for each position $r,v$.
This creates an equivalent to a position-velocity diagram, only here it shows the maximum temperature regardless of height $z$.
The lower plot takes the maximum brightness temperature along both $v$ and $z$, and infers the corresponding optical depth,
based on a assumed spin temperature of 125 K.
We chose the values to lie close to the mean observed values for the cold neutral medium
(CNM) \citep{Dickey2009A}. As such it forms a diagnostic tool for potential self-absorption
issues. We calculated the opacity $\tau$ using equation \ref{eqn:tau}
\begin{equation}
\tau = -\log\left(1 - T_B / T_\mathrm{s}\right)\label{eqn:tau}
\end{equation}
The integrated position-velocity (PV) diagrams are shown in the \emph{d} parts
of figures 1-8. These have been created by integrating the rotated
high-resolution cubes along their minor axis and converting them to brightness
temperatures. The right-hand panels show the integrated spectra $S(v)$, which
were created by integrating the PV diagrams over the major axis. Similarly, we
show the brightness per position in the lower plots, by integrating the PV
diagrams over $v$.\\
The width of the profile at the 20\% and 50\% maximum intensity levels, $W20$
and $W50$, were measured directly from $S(v)$. These were based on the
low-resolution cubes, which were masked using the regions files. The integrated
flux $FI$ was also measured in this cube. We make no use of clipping, in
contrast to \cite{OBrien2010A}. The total \ion{H}{I} mass,
$M_\textrm{\ion{H}{I}}$, was derived using
\begin{equation}
M_\textrm{\ion{H}{I}} = 2.36 \cdot 10^5 \times D^2 \times \textrm{FI}
\end{equation}
where the adopted distance $D$ is in Mpc as shown in table \ref{tbl:opticalproperties}.
The derived values for FI, $M_\textrm{\ion{H}{I}}$, $W20$ and $W50$ are shown
in table \ref{tbl:HISpectrumProperties}.\\
\begin{table*}[htb]
\caption[\ion{H}{I} spectral properties]{Spectral properties of the galaxies}
\label{tbl:HISpectrumProperties}
\centering
\begin{tabular}{lcccccc}
\textbf{Galaxy} & \textbf{v$_\textrm{sys}$} & \textbf{v$_\textrm{max}$} & \textbf{$W_{20}$} & \textbf{$W_{50}$} & \textbf{FI} & \textbf{$M_\textrm{\ion{H}{I}}$}\\*
& \textbf{(km/s)} & \textbf{(km/s)} & \textbf{(km/s)} & \textbf{(km/s)} & \textbf{(Jy km/s)} & \textbf{(M$_\odot$)}\\*\hline\hline
\object{IC 2531} & 2463.8 & 260.5 & 484.8 & 468.3 & 41.7 & $7.3 \times 10^{9.0}$\\*
\object{IC 5052} & 596.7 & 114.4 & 197.9 & 181.4 & 117.7 & $8.9 \times 10^{8}$\\*
\object{IC 5249} & 2351.7 & 131.9 & 234.2 & 217.7 & 23.2 & $5.6 \times 10^{9}$ \\*
\object{ESO 115-G021} & 514.2 & 85.1 & 141.8 & 125.3 & 109.0 & $6.2 \times 10^{8}$ \\*
\object{ESO 138-G014} & 1492.3 & 130.9 & 237.5 & 227.6 & 48.9 & $2.9 \times 10^{9}$ \\*
\object{ESO 146-G014} & 1678.8 & 84.1 & 148.4 & 128.6 & 17.5 & $1.9 \times 10^{9}$ \\*
\object{ESO 274-G001} & 523.6 & 103.9 & 181.4 & 168.2 & 150.1 & $3.2 \times 10^{8}$ \\*
\object{UGC 7321} & 409.3 & 128.8 & 237.0 & 221.6 & 41.7 & $9.8 \times 10^{8}$ \small\\*\hline\hline
\end{tabular}
\end{table*}
The systematic velocity v$_\textrm{sys}$ was measured from the PV diagram in a
method similar to the position finding. For a wide range of possible systemic
velocities v$_\textrm{sys}$, the PV diagram was rotated $180^\circ$ around that
velocity. The rotated PV-diagram was subtracted from the original, with the
lowest $\chi^2$ giving v$_\textrm{sys}$. The maximum velocity v$_\textrm{max}$
was calculated by finding the furthest channel from v$_\textrm{sys}$ that still
contained flux above two times the RMS noise level. The results for
v$_\textrm{sys}$ and v$_\textrm{max}$ are also shown in table \ref{tbl:HISpectrumProperties}.
We have also prepared a document with all our channel-maps, which is available on request.
\section{Results}\label{sec:HIresults}
\begin{figure*}
\centering
\subfloat[][IC 2531]{%
\label{fig:IC2531-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{IC2531-overviewDSS.pdf}}}%
\subfloat[][IC 5052]{%
\label{fig:IC5052-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{IC5052-overviewDSS.pdf}}}%
\hspace{8pt}
\subfloat[][IC 5249]{%
\label{fig:IC5249-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{IC5249-overviewDSS.pdf}}}%
\subfloat[][ESO 115-G021]{%
\label{fig:ESO115-G021-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{ESO115-G021-overviewDSS.pdf}}}%
\hspace{8pt}
\subfloat[][ESO 138-G014]{%
\label{fig:ESO138-G014-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{ESO138-G014-overviewDSS.pdf}}}%
\subfloat[][ESO 146-G014]{%
\label{fig:ESO146-G014-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{ESO146-G014-overviewDSS.pdf}}}%
\hspace{8pt}
\subfloat[][ESO 274-G001]{%
\label{fig:ESO274-G001-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{ESO274-G001-overviewDSS.pdf}}}%
\subfloat[][UGC 7321]{%
\label{fig:UGC7321-optical}%
\resizebox{0.4625\textwidth}{!}{\includegraphics{UGC7321-overviewDSS.pdf}}}%
\caption{Overview of the sample in the DSS J-band.}\label{fig:OpticalOverview}
\end{figure*}
\subsection{IC 2531}
Galaxy \object{IC 2531} is the only Sc galaxy in our sample. The classification
is different from that of \citet{Bureau1999A} and \citet{OBrien2010A}, where the
galaxy has been classified as an Sb galaxy. As an Sc galaxy, it has the largest
gas mass of the sample, and has a maximum rotation velocity v$_\textrm{max}$
which is twice as high as the rest of the sample (table
\ref{tbl:HISpectrumProperties}). The \ion{H}{I} layer is remarkably flat, with
the disk seemingly having a smaller scale-height than the stellar disk (figure
\ref{fig:IC2531-DSS}).
Based on the double horns in the integrated profile, we would expect the galaxy
to be fairly symmetric (figure \ref{fig:IC2531-PV}). The PV diagram itself is,
however, fairly asymmetric. Interestingly, HIPASS also reports a more
asymmetric profile. The asymmetry is also present in the analysis of the galaxy
in \citet[figure 15]{OBrien2010C}, where the two sides of the derived surface
density can be seen to deviate strongly.
The PV-diagram shows a so-called ``figure 8''-pattern, for which
\citet{Bureau1997A} suggest that it is due to a very extended weak bar.
This bar did not show up in in the \ion{H}{$\alpha$} analysis of
\citet{Bureau1999A}, although they suffered from background problems for this
galaxy. This was not fully unexpected, as \citet{Bureau1997A} already noted that
the figure extended over a very long range. They speculated that the figure
might also be due to a warp, density ring or a spiral arm. \citet{Ann2006A} reported
that there is indeed a warp in this galaxy. We also believe it more likely that this
pattern is associated with the warp and does not represent a bar. The warp is visible in figure \ref{fig:IC2531-doublemoment0}.
The PV-diagram is also asymmetric in the rotation curve. The receding side shows
a slowly increasing rotation curve, while the approaching side show a sharp
increase followed by a flat rotation.
The maximum surface brightness temperature map (figure \ref{fig:IC2531-maxtemperature}) shows a strong central peak and some additional peaks.
There does not appear to be symmetry in the distribution of these peaks. They are most likely due to local events in the gas, such as supernova or a burst of star formation.
\subsection{IC 5052}
\object{IC 5052} is a nearby SBd galaxy. The bulge is clearly visible in figure.
\ref{fig:IC5052-DSS}. We find a previous unreported, line-of-sight warp in the
outskirts of the galaxy. This is particularly clear in the zeroth moment map of
the low resolution cube (figure \ref{fig:IC5052-doublemoment0}). Further
modeling will be required to confirm this as a true warp. For now, we note that
the channel maps are very similar to the fitted model of the line-of-sight warp
in \object{ESO 123-G13} \citep{Gentile2003A}. The warp appears to be symmetric
in pitch angle, although it is more extended towards the lower-left side.
We have overlaid the contour-maps on top of various optical and infrared images
that were available, but we can find no optical counterpart to the warp.
The PV diagram of this galaxy (figure \ref{fig:IC5052-PV}) shows signs of this
warp beyond 250 arcseconds. The main part of the profile looks like a solid-body
rotator. The high- and low velocity sides are asymmetric, which is also clear
from the integrated profile. The shape of the spectrum does agrees well with the
profile from HIPASS.
The maximum surface brightness temperature profile of figure
\ref{fig:IC5052-maxtemperature} shows a plateau around 80-100 K for most of the
inner part of the galaxy. Only the central part shows a higher temperature,
rising to a maximum of 140 K. This central over-density is also visible in the
moment 0 maps (\ref{fig:IC5052-doublemoment0}).
\citet{OBrien2010A} identify this as a star forming region.
\subsection{IC 5249}
\object{IC 5249} is the most distant galaxy in our sample, at a distance of 32
Mpc. The galaxy is only 5 arcminutes. This is still
enough to fully resolve the \ion{H}{I} disk in both width and height.
The maximum S/N of the cube has not been improved since \citet{OBrien2010A}, and is
still around 10.
The maximum temperatures exhibit a plateau between 80-100 K, over the radius
-100 and 100 arcseconds, most of the galaxy (figure
\ref{fig:IC5249-maxtemperature}). Another plateau is visible in integrated flux
along the major axis (\ref{fig:IC5249-PV}), wherein over that same region the
intensity is seen to hover around 0.06 Jy km/s / arcsec. The PV-diagram itself
is slightly asymmetric, with the higher velocity side having more mass than the
low velocity side.
\citet{vanderKruit2001A} have previously analyzed the rotation curve of the
galaxy, using \ion{H}{I} and \ion{H}{$\alpha$}. They find the galaxy to exhibit
a steep rising rotation curve, where-after it flattens out to
$v_\textrm{max} \approx 100$ km/s.
There is a faint indication of a warp, visible in the lower plot in figure \ref{fig:IC5249-doublemoment0}.
\subsection{ESO 115-G021}
\object{ESO 115-G021} is a nearby SBdm galaxy at a distance of only 4.9 Mpc. A lot of
data was available, totaling to 83.5 hours. Due to this, it is a very well
resolved galaxy. The plane of the galaxy is over $\sim80$ beams wide and the
galaxy is $\sim10$ beams in height (see figure \ref{fig:ESO115-G021-DSS}).
The derived spectrum is more symmetric than the HIPASS spectrum, which has a
larger peak at the high-velocity side. There is a gas-cloud at 02:36:59 -61:18:41
visible in multiple channels. It is shows in both the high and low resolution
cubes (figure \ref{fig:ESO115-G021-doublemoment0}) at position (180'', 300'').
In the PV diagram (figure \ref{fig:ESO115-G021-PV}) it is visible as the small
upward peak between 500 and 515 km/s. This close proximity to the galaxy in both
velocity and distance indicates the gas is associated with the galaxy. The cloud
is invisible in the DSS, I and V bands paper IV, indicating that it has
negligible stellar content. The gas cloud has a \ion{H}{I} mass of $3.75 \times
10^5$M$_\odot$.
The PV-diagram (figure \ref{fig:ESO115-G021-PV}) shows an asymmetry, with the
high velocity side being more extended than the low velocity side. This is most
likely due to the warp, which is visible in the zeroth moment maps (figure
\ref{fig:ESO115-G021-doublemoment0}). The main component of the PV-diagram shows
solid body rotation.
The maximum temperature map (figure \ref{fig:ESO115-G021-maxtemperature}) shows
that this warp has a surface brightness of 20-30 K. The more stable inner parts
of the disk show a maximum temperature plateau around 80-90 K, with the strong
exception of the region between 0'' and +50'', where there is a rise to a
maximum of 140 K. This region corresponds to a bright region of the DSS image,
and is likely due to star formation.
\subsection{ESO 138-G014}
Galaxy \object{ESO 138-G014} is the least observed galaxy in our sample.
Nevertheless, the galaxy is well resolved by the telescope. The minor axis
of the galaxy in particular is covered by roughly 6 beams.
The PV-diagram (\ref{fig:ESO138-G014-PV}) is asymmetric, with the receding side
showing an extended component. This is associated with a warp, as visible in
figure \ref{fig:ESO138-G014-doublemoment0}. No optical counterpart to this warp is
known \citep{Sanchez2003A}.
The maximum brightness temperature map resembles a plateau for most of the
galaxy, averaging between 80 and 100 K (figure
\ref{fig:ESO138-G014-maxtemperature}). The only exception is at the left side,
where a peak of 140 K is attained. This corresponds to the bright region in the
south-west region in the DSS image (figure \ref{fig:ESO138-G014-DSS}), and is
most likely a star forming region.
\subsection{ESO 146-G014}
Galaxy \object{ESO 146-G014} is the faintest galaxy in our sample.
The moment zero maps (figure \ref{fig:ESO146-G014-maxtemperature})
show that the galaxy does not appear to have a fixed plane. Instead it looks
slightly bent. This is most likely due a strong warp, which can also be seen
in the visible in the DSS J-band image (figure \ref{fig:ESO146-G014-optical}).
The PV diagram (figure \ref{fig:ESO146-G014-PV}) looks symmetrical, although the
integrated spectrum is more asymmetric than the HIPASS spectrum, with a stronger
peak at the low-velocity side. Overall the galaxy appears to exhibit solid-body
rotation.
The maximum surface brightness temperatures appear to form a plateau over major part of the galaxy, averaging around 90 K (figure \ref{fig:ESO146-G014-maxtemperature}).
On the left-hand side of the galaxy, this maximum drops relatively slowly towards the warp.
\subsection{ESO 274-G001}
\object{ESO 274-G001} is an SAd galaxy.
At only 3.02 Mpc, it is the closest galaxy in our sample.
It is also has the least \ion{H}{I} mass of the galaxies in our sample (table \ref{tbl:HISpectrumProperties}).
Due to it's proximity, it is still the largest galaxy in terms of angular size, spanning 13 arcminutes in width.
In height it is also very well resolved, covering at least 80 arcseconds.
The central hole in the zeroth moment image (figure \ref{fig:ESO274-G001-DSS}) is an artifact, due to the removal of a continuum radio source at this coordinate.
The most likely explanation is a active galactic nucleus (AGN).
It is likely at the galactic center (see table \ref{tbl:opticalproperties}) and we have therefore manually centered the galaxy on this position.
AGN are known to exert strong influence on the ISM of the galaxy (e.g. \citet{Shulevski2012A} and references therein).
It is therefore interesting to note how regular the \ion{H}{I} disk is, both in the zeroth moment as in maximum temperature.
Most of the inner disk shows a maximum surface density remarkably flat plateau between 80 and 100 K (figure \ref{fig:ESO274-G001-maxtemperature}).
The galaxy is barely visible in most optical and infrared bands, indicating only little stellar mass.
This may explain why the gas relatively undisturbed, in comparison with our other galaxies.
There is a warp visible beyond $\sim280$ arcsec on both sides of the galaxy (figure \ref{fig:ESO274-G001-doublemoment0}).
The PV-diagram is slightly asymmetric, with more gas at the low velocity side.
This is also visible in the integrated profiles (figure \ref{fig:ESO274-G001-PV}).
The shape of the spectrum matches well to HIPASS, although it is more symmetric.
\subsection{UGC 7321}
\object{UGC 7321} is a well known low surface brightness galaxy, which has already been studied using the Westerbork Synthesis Radio Telescope (WSRT) \citet{Garcia-Ruiz2002A} and the VLA \citet{Uson2003A}.
The observations by \citet{Uson2003A} form the basis of our work here.
In terms of noise levels, it is the best resolved galaxy in our sample.
It is also our only northern sky galaxy.
The zeroth moment map \ref{fig:UGC7321-doublemoment0} shows a well behaved, symmetric galaxy.
Only at the outskirts is there some indication of warping, a feature already noted by \citet{Matthews2003A}, who also notes is flaring.
Additionally, they find evidence that suggests the presence of a neutral gas halo.
We believe that this halo might be visible as the extended structure on the lower figure, between 0 and 110 arcseconds.
The PV-diagram of the galaxy is asymmetric.
The receding side looks more like a flat rotation curve than the approaching side, which continue to rise over much of the galaxy.
The maximum surface brightness map shows a clear central peak and two additional peaks around 100 arcseconds (figure \ref{fig:UGC7321-maxtemperature}).
These peaks appear to be symmetric around the center of the galaxy.
The rest of the galaxy forms a plateau, around 60 K.
We again see that the maximum temperature trails slowly off into the warp on the right-hand side.
\section{Discussion}\label{sec:Discussion}
The spatial resolution of the high resolution cubes is on average 1-2'' larger
than \citet{OBrien2010A} (table \ref{tbl:GlobalPropertiesHICubes}). This is due
to most of the new observations being in tight configurations. There are
therefore more short baselines. The weight in the UV-plane shifts to slightly
smaller scales, resulting in larger beams. The effect is small enough to be of
little importance to this work, as the vertical scale of the galaxies is still
fully resolved.\\
In the high resolution cubes, the noise is significantly lower, thanks to the
additional observations and larger beams. Where no additional observations were available, the
noise is (as expected) roughly equal. The surprising differences is in the column density
noise levels. It seems that \citet{OBrien2010A} has not integrated the column densities over channel width.
If this is taken into account, the values are again comparable.\\
Overall there is some disagreement with \citet{OBrien2010A} on the derived central positions, mostly because we include more faint extended emission in our measurements.
The biggest offset is in \object{ESO 138-G014} (see figure \ref{fig:ESO138-G014-doublemoment0}), where the non-symmetric warp is affecting the position estimate.
The position angles however, do agree well.
Similarly, the systemic velocities v$_\textrm{sys}$ and associated parameters $W20$, $W50$ and v$_\textrm{max}$ are all well determined.
Kinematic modeling will have to be performed to find the true center of each galaxy, the derived values here will only serve as first estimates in paper III.\\
We have also compared the integrated flux $FI$ with the HIPASS catalog
\citep{Doyle2005A} and the work by \citet{OBrien2010A}. In general our results
match well to HIPASS. \object{ESO 274-G001} has the largest difference, with
25 mJy km/s more detected here than in HIPASS. The other major differences are
for \object{ESO 115-G021} with a difference of 19 mJy km/s, and \object{IC
5052} with a difference of 7 mJy km/s. Both have again more flux in our
work.\\
These three galaxies are the largest in the sample. The Parkes single dish beam has a
angular resolution of 15.5 arcminutes. Most likely the Parkes beam fails
to cover the entire galaxy, missing the outskirts and is therefore getting a
lower total flux. With that variation in mind, each datacube has successfully
recovered all flux from HIPASS.\\
\citet{OBrien2010A} reports very different values for $FI$. Most striking is
\object{IC 5052}, where only 37.9 mJy km/s is reported, against 117.7 mJy km/s here. It is
unclear to us why \citet{OBrien2010A} often reports very different values. Most
likely it has to do with the masking of extended emission in their work.\\
As noted before, the distances to the galaxies are assumed further than in
\citet{OBrien2010A}. That change plus the higher values for $FI$ combine to have
a large impact on the induced total mass $M_\textrm{\ion{H}{I}}$ of the galaxy.
Most galaxies now have estimated masses anywhere between 20-300\% more.\\
All eight of our galaxies shows signs of warping.
This is consistent with the work by \citet{Garcia-Ruiz2002A}, who found that all galaxies with extended \ion{H}{I} disk have warps.
All of the warps occur at or just beyond the visible stellar disk, similar to the findings by \citet{vanderKruit2007A}.
The only possible exception to this rule could be \object{ESO 146-G14}, which seems to have the stellar disk embedded in the warp (see figure \ref{fig:ESO146-G014-DSS})
However, the overall \ion{H}{I} layer appears to be bent in this galaxy, with the gas not following a single plane.
It could therefore well be that we are dealing with a projection effect by the warp, where the true \ion{H}{I} disk is still a single plane.
Further modeling will be required before this can be confirmed.\\
\subsection{The opacity of the \ion{H}{I} layer}
The 21-cm line of \ion{H}{I} is known to be optically thick along many sight lines in our own (edge-on) Galaxy \citep{Allen2012A}.
Evidence is accumulating that flat-topped \ion{H}{I} profiles characteristic of this effect are common in M31 and M33, where hydrogen mass correction factors of order 1.3 to 1.4 have been proposed \citep{Braun2012A}.
Such self-absorption of the \ion{H}{I} emission not only biases the total hydrogen masses, it also raises questions about the use of the \ion{HI} velocity structure for studies of the gas kinematics.
Nowhere are the answers to such questions more urgent than in the study of highly-inclined galaxies, where techniques have been proposed to ``peel off'' the edges of the observed \ion{H}{I} in order to expose the detailed velocity dispersion of the ISM over large segments of the galaxy disks in both $r$ and $z$ \citep{OBrien2010B}.\\
\begin{figure}
\centering
\includegraphics[angle=270, width=0.49\textwidth]{ron3.pdf}
\caption{Overview of a sample of Galactic \ion{H}{I} clouds in a 4 X 4 degree patch of sky, as assembled by \citet[figure 3]{Allen2012A} from archival \ion{H}{I} survey data. The various clouds exhibit a remarkably uniform behavior.}\label{fig:Ron-Fig3}
\end{figure}
If a moderately inclined galaxy like M31 hides $34\%$ or its \ion{H}{I} mass \citep{Braun2012A}, what is the effect on edge-on galaxies?
In the \ion{H}{I} the dominating factor for self-absorption would be the spin temperature of the cold neutral medium, which would serve as an upper limit to the observable surface brightness temperature.
Consider now the maximum surface brightness maps for \object{ESO 274-G001} (figure \ref{fig:ESO274-G001-maxtemperature}).
There is a clear plateau around 90 Kelvin throughout almost the entire galaxy.
At the outskirts the line of sight is much shorter than near the inner parts, yet the observed peak temperature remains stable.
As a case of even shorter path-lengths, consider \citet[figure 3]{Allen2012A} (reproduced here as figure \ref{fig:Ron-Fig3}), where a 'mini-survey' has been performed of \ion{H}{I}, \ion{OH}{} and \ion{CO}{} at 30' resolution inside the Galaxy.
This gas is quite local, within about 2 kpc of the sun.
Again we see a peak temperature of 90K, only this time in individual clouds.
Why would such different path-lengths always result in the same brightness temperature?\\
The traditional view is that at these brightness temperatures, column densities are reached at which the atomic gas starts to be converted into molecular gas, thus providing a natural threshold to the observed temperatures \citep{Stecher1967A,Hollenbach1971A,Federman1979A}.
A column densities is however an artificial construct based on the position of the observer, and as such the gas cannot be expected to conform to it.
A volume density threshold would seem to be a much more physically-relevant quantity.
However, since the path-lengths between the local \ion{H}{I} clouds from \citet{Allen2012A} and the central parts of the edge-on galaxies presented here vary over roughly two orders of magnitude, the volume and the inferred density are also likely to vary in the same way.
Any volume density threshold would seem to have difficulty explaining the close similarities in the maximum brightness temperatures.\\
Opacity however can explain this naturally.
The brightness temperature of 90 Kelvin is in the same range as the expected spin temperature for the cold neutral medium \citep{Dickey1990A,Dickey2009A}.
90K is then the maximum ascertainable temperature before the gas turns optically thick.
We are thus seeing optically thick gas in \citet{Allen2012A}.
In figure \ref{fig:ESO274-G001-maxtemperature} the path-lengths have become so long compared to the sizes of the clouds, that each line is bound to hit an optically thick cloud.
Considering the radii at which these can be seen, a lot more than $30\%$ of the galaxy \ion{H}{I} mass could be hidden in and behind these clouds.\\
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{ron5.pdf}
\caption{The correlation between \ion{H}{I} and \ion{OH}{} surface brightness temperatures, reproduced from the Galactic mini-survey by \citet[figure 5]{Allen2012A}. The \ion{H}{I} and \ion{OH}{} follow a linear correlation for most of the diagram. Beyond 90K the \ion{H}{I} seems to saturate, while the \ion{OH}{} continues to rise. Is the \ion{OH}{} tracing an optically thick \ion{H}{I} layer?}
\label{fig:Ron-Fig5}
\end{figure}
Another tempting argument in favor of this explanation comes from the study of local clouds in \citet[figure 5]{Allen2012A}, reproduced here in figure \ref{fig:Ron-Fig5}.
A remarkable linear relation is shown between the molecular \ion{OH}{} (thermal 18cm emission) and \ion{H}{I} surface brightnesses of the form $T_A\left(\textrm{\ion{OH}{}}\right) \approx 1.50 \times 10^{-4} T_B\left(\textrm{\ion{H}{I}}\right)$.
The \ion{OH}{} and \ion{H}{I} follow the same velocities and distributions.
This relation holds for $T_B\left(\textrm{\ion{H}{I}}\right) \leq$ 60K.
Beyond that the \ion{H}{I} appears to saturate, leveling off completely around 90K.
Yet the \ion{OH}{} temperatures continues to rise.
Could it be tracing a now-turned-opaque \ion{H}{I} gas?\\
The lower panel in figures 4b to 10b show the inferred maximum self-absorption for a spin temperature of 125 K.
A spin temperature of a 125 K is most likely an upper-limit for the CNM, yet it already implies a maximum opacity of $\tau \sim 1$.
If true, this will have drastic implications for deriving the structure and kinematics, as traditional \ion{H}{I} modeling tools work under the assumption of an optically thin gas.
To deal with this, in our next paper we will present the new \ion{H}{I} modeling and fitting tool \textsc{GALACTUS}, which can model self-absorption.
\begin{acknowledgements}
S.P. would like to thank Renzo Sancisi, Fillipo Fraternali, Thijs van der Hulst, Marc Verheyen, Thomas Martinsson and Aleksander Schuvelski for useful debates, comments and suggestions.
The Australia Telescope is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions.
\textsc{Pyfits} and \textsc{pyraf} are products of the Space Telescope Science Institute, which is operated by AURA for NASA.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
This research has made use of NASA's Astrophysics Data System.
\end{acknowledgements}
\bibliographystyle{aa} |
1303.2214 | \section{Introduction}
The possible existence of a spin-liquid phase on the honeycomb lattice
has recently attracted considerable attention. Meng {\it et al.}\cite{meng}
investigated the Hubbard model for this system at half-filling, using large-scale
quantum Monte Carlo (QMC) calculations for clusters containing up to 648 sites.
Careful finite-size extrapolations indicated semi-metallic behavior for onsite
Coulomb interactions in the range $U\le 3.5 t$ ($t$ is the nearest neighbor
hopping) and an antiferromagnetic insulator for $U\ge 4.3t$. The intermediate
range $3.5t\le U \le 4.3t$ then corresponds to a Mott phase without long-range
order, the hallmark of a spin liquid. These findings were, however, disputed
by Sorella {\it et al.}\cite{sorella} who performed similar QMC calculations
for even larger clusters including up to 2592 sites. The new results showed a
considerably reduced spin-liquid phase, confined at most to the narrow window
$3.8t \le U\le 3.9t$.
The effect of nonlocal Coulomb correlations on the honeycomb lattice was also
studied within the cluster extension of dynamical mean field theory \cite{cdmft}
(CDMFT). Wu {\it et al.}\cite{wu} used continuous-time quantum Monte Carlo
\cite{rubtsov} (CTQMC), whereas Liebsch \cite{PRB2011} employed a
multi-orbital-multi-site extension \cite{perroni,tong} of finite-temperature exact
diagonalization \cite{ed} (ED) as impurity solver. Despite the fact that in
the ED CDMFT calculations it was possible to include only a relatively small
bath (six bath levels per six-site unit cell), the cluster self-energy components
were found to be in nearly quantitative agreement with the CTQMC CDMFT results.
(For a detailed comparison see Fig.~25 of Ref.~\onlinecite{EDR}.) In particular,
for $U\approx 5t$ both schemes revealed a Mott phase, with an excitation gap
$\Delta\approx0.6t$, in close agreement with the one found by Meng {\it et al.}
\cite{meng} With decreasing $U$, the CTQMC results at temperatures $T\ge 0.05t$
indicated the closing of the Mott gap near $U=3.8t$, \cite{wu} while the ED
results at lower temperature $T=0.005t$ revealed a weak insulating contribution
to the self-energy at the Dirac points at arbitrarily low $U$.\cite{PRB2011}
For $U\le3t$ the small gap associated with this self-energy was, however,
difficult to resolve in the spectral distributions due to the temperature
rounding of the gap edges.
Analogous ED CDMFT calculations (also for six bath levels) were carried out by
He and Lu \cite{helu} at a considerably lower effective temperature ($T=10^{-5}t$).
The excitation gap in this case was found to extend to $U\rightarrow0$.
On the basis of these results the authors concluded that the spin-liquid phase
of the honeycomb lattice at half-filling exists from $U=0$ up to the onset of
the antiferromagnetic phase near $U=4.5t$.
Closely related to these works are two calculations based on the variational
cluster approximation \cite{VCA} (VCA) by Yu {\it et al.} \cite{yu} and
Seki and Ohta.\cite{seki} In both cases, ED was used as impurity solver, with
six bath levels as in Refs.~\onlinecite{PRB2011} and \onlinecite{helu}.
Whereas Yu {\it et al.}~identified a spin-liquid phase in the range
$U\approx 3t - 4t$ and semi-metallic behavior at smaller values of $U$,
Seki and Ohta obtained a similar insulating contribution to the self-energy
at the Dirac points as in Ref.~\onlinecite{PRB2011} and concluded that the
Mott gap persists down to arbitrarily small values of $U$.
Most recently, Hassan and S\'en\'echal \cite{hassan} performed ED calculations
for the honeycomb lattice within VCA, CDMFT and the cluster dynamical impurity
approximation \cite{CDIA} (CDIA).
They argued that a bath consisting only of six levels is insufficient and leads
to the erroneous conclusion that the system is gapped for all nonzero values of
the onsite Coulomb interaction $U$.
In this context it is also important to recall the results of functional
renormalization group (FRG) calculations \cite{frg} for the honeycomb lattice
which reveal a stable semi-metallic phase below about $U\approx 3.8 t$.
In view of these contradictory results it is evident that the possible existence
and extent of the semi-metallic phase of the honeycomb lattice are difficult to
determine within present non-local many-body techniques. In particular, it is not
clear which assumptions and approximations give rise to certain consequences:
the size of the correlated cluster, the size and symmetry of the bath in ED,
the accessible temperature range, the accuracy of spectral functions at very low
energies, etc. Naturally, these uncertainties also affect the identification
of the elusive spin-liquid phase.
The purpose of this work is to shed light on some of these issues by comparing
new results derived within the dynamical cluster approximation \cite{DCA} (DCA)
with previous ones obtained within CDMFT.\cite{wu,PRB2011} As impurity solver
we use finite-temperature ED as well as CTQMC. The nearly quantitative agreement
between the ED and CTQMC self-energies, within DCA as well as CDMFT, demonstrates
that the intrinsic limitations of these impurity solvers are not the cause of the
discrepancies between the various results cited above.
Instead we show here that, in the special case of the honeycomb lattice, it is
of crucial importance to preserve the translational invariance of the system.
Obviously, any deviation from bulk symmetry opens a gap at the Dirac points.
Recall, for instance, the single-particle gaps obtained for graphene ribbons.
Thus, the semi-metallic and spin liquid phases can only be studied properly by
using many-body methods that do not violate translation symmetry.
This argument disqualifies CDMFT which is well-known to yield a self-energy
that is not translationally invariant.\cite{DCA,biroli} The self-energy components
in this scheme account for correlations within the unit cell, but not between cells.
We therefore believe that all CDMFT calculations performed until now for the
honeycomb lattice should exhibit, at low $U$ and low $T$, an excitation gap which
is an artifact caused by the lack of translation symmetry of the self-energy.
Although this gap is related to the presence of the local Coulomb interaction,
it is not a true Mott gap but merely the consequence of the intrinsic limitation
of the cluster approach. As a result, CDMFT and other schemes that do not preserve
translation invariance are not suitable for the identification of a spin-liquid
phase on the honeycomb lattice.
The comparison of the CDMFT self-energy with analogous results derived within DCA,
for ED as well as CTQMC, underlines this point. In DCA, the self-energy is by
construction translationally invariant, so that the electronic structure at low
$U$ is semi-metallic, in agreement with the predictions based on large-scale QMC
and FRG calculations.\cite{meng,sorella,frg} The spurious tail of the excitation
gap at small $U$ and low $T$ that is seen in CDMFT is absent in DCA.
As will be shown below, in the case of the honeycomb lattice, the DCA condition
that ensures translation symmetry is too rigid for the description of
correlations within the unit cell. As a result, the semi-metallic phase is still
stable near $U=5t-6t$ where CDMFT and large-scale QMC calculations already find Mott
insulating behavior. Thus, CDMFT and DCA may be viewed as complementary cluster
schemes: DCA is preferable at low $U$ since it maintains the long-range order that
is crucial for the Dirac cones, whereas CDMFT yields a more realistic description
of short-range correlations in the Mott phase when the absence of translation
symmetry plays a minor role.
We also note here that the gap tail obtained in CDMFT at small $U$ is not related
to the finite size and symmetry properties of the bath used in ED.
On the contrary, in the special case of the honeycomb lattice, a rather small
bath containing only six levels is sufficient for the description of short-range
correlations within the six-site unit cell. The reason is that, because of the
semi-metallic properties of the system, the projection of the bath Green's function
on a finite-cluster is not affected by the usual low-energy disparities that arise
in the case of correlated metals.
The outline of this paper is as follows: In Section II we discuss the
application of DCA and CDMFT to the honeycomb lattice and point out the
key difference between the self-energies obtained within these schemes.
Section III presents the main ingredients of the ED impurity solver for
both DCA and CDMFT. Section IV provides the discussion of the results obtained
within ED DCA, and the comparison with analogous CTQMC DCA results.
The summary is presented in section V. Throughout this work only paramagnetic
phases are discussed.
\section{DCA vs. CDMFT for the Honeycomb Lattice}
To describe Coulomb correlations in the honeycomb lattice we consider the
single-band Hubbard Hamiltonian
\begin{equation}
H=-t \sum_{\langle ij\rangle\sigma} ( c^+_{i\sigma} c_{j\sigma} + {\rm H.c.})
+ U \sum_i n_{i\uparrow} n_{i\downarrow} ,
\end{equation}
where $t$ is the nearest neighbor hopping term and $U$ the on-site Coulomb energy.
Throughout this paper $t=1$ defines the energy scale. The non-interacting band
dispersion is given by:
$\epsilon({\bf k})= \pm t\vert 1+e^{ik_x\sqrt{3}}+e^{i(k_x\sqrt{3}+k_y3)/2}\vert$.
The nearest neighbor spacing is assumed to be $a=1$.
We choose a six-site unit cell with positions specified as
${\bf a}_1=(0,0)$, ${\bf a}_2=(0,1)$, ${\bf a}_3=(\sqrt3/2,3/2)$,
${\bf a}_4=(\sqrt3,1)$, ${\bf a}_5=(\sqrt3,0)$, and ${\bf a}_6=(\sqrt3/2,-1/2)$.
The supercell lattice vectors are given by ${\bf A}_{1/2}=(3\sqrt3/2,\pm3/2)$.
Within CDMFT as well as DCA, the interacting lattice Green's function in
the site basis is defined as
\begin{equation}
G_{ij}(i\omega_n) = \sum_{\bf k} \left[ i\omega_n + \mu - h({\bf k})-
\Sigma(i\omega_n)\right]^{-1}_{ij} ,
\label{G}
\end{equation}
where $\omega_n=(2n+1)\pi T$ are Matsubara frequencies and $T$ is the temperature.
At half-filling, the chemical potential is $\mu =U/2$. The ${\bf k}$ sum extends
over the reduced Brillouin Zone, $h({\bf k})=-t({\bf k})$, where $t({\bf k})$ denotes
the hopping matrix for the superlattice, and $\Sigma_{ij}(i\omega_n)$ represents
the self-energy matrix in the site representation.
Within CDMFT, the elements of $t({\bf k})$ within the unit cell given by
$t_{ij}=t$ for neighboring sites. In addition, hopping between cells yields:
\begin{eqnarray}
t_{14} &=& t\, e^{-i{\bf k}\cdot {\bf A}_1} \nonumber\\
t_{25} &=& t\, e^{-i{\bf k}\cdot {\bf A}_2} \\
t_{36} &=& t\, e^{-i{\bf k}\cdot {\bf A}_3} , \nonumber
\end{eqnarray}
where ${\bf A}_3={\bf A}_2-{\bf A}_1$. The hopping matrix $t({\bf k})$ is Hermitian,
so that $t_{ji} = t^*_{ij}$. All other elements vanish.
To distinguish the hopping matrix elements within DCA, we denote them by
$\bar t_{ij}({\bf k})$. In the real-space version of DCA\cite{biroli} they are
related to those within CDMFT via a phase factor:
\begin{equation}
\bar t_{ij} = t_{ij}\, e^{-i{\bf k} \cdot {\bf a}_{ij}} ,
\label{phase}
\end{equation}
where ${\bf a}_{ij}={\bf a}_{i}-{\bf a}_{j}$. This phase relation yields the
following matrix elements:
\begin{eqnarray}
\bar t_{12}&=&\bar t_{36}=\bar t_{54}=t\,e^{-i{\bf k}\cdot{\bf a}_{12}}\nonumber\\
\bar t_{23}&=&\bar t_{41}=\bar t_{65}=t\,e^{-i{\bf k}\cdot{\bf a}_{23}} \\
\bar t_{34}&=&\bar t_{52}=\bar t_{16}=t\,e^{-i{\bf k}\cdot{\bf a}_{34}},\nonumber
\end{eqnarray}
with analogous connections among the Hermitian elements. All other matrix elements
vanish.
\begin{figure} [t!]
\begin{center}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig1a.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig1b.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Density of states $\rho(\omega)$ (solid curves) of honeycomb lattice and cluster
components $\rho_m(\omega)$ in diagonal molecular orbital basis (dashed curves)
for (a) CDMFT and (b) DCA. For clarity, these components are divided by $n_c=6$.
In CDMFT, all density components are non-symmetric and orbitals 3 and 4 are doubly
degenerate. In DCA, only $\rho_{1}$ and $\rho_{2}$ are non-symmetric, while the
degenerate components $\rho_{m=3\ldots6}$ are symmetric.
$\omega=0$ defines the Fermi energy for half-filling.
}\label{dos}\end{figure}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=5.0cm,height=6.0cm,angle=-90]{Fig2a.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig2b.ps}
\end{center}\vskip-3mm
\caption{(Color online)
(a) Segment of Brillouin Zone of honeycomb lattice (solid red lines). The reduced
Zone (dashed blue lines) is obtained by folding the Dirac points $K$ onto $\Gamma$
and the $M$ points onto $M'$.
(b) Decomposition of density of states into low-energy contribution (denoted as $K$)
corresponding to outer regions $KMK'M'$ and high-energy contribution (denoted as
$\Gamma$) corresponding to inner regions $\Gamma M'K'$ of large Brillouin Zone.\\
}\label{BZ}\end{figure}
The cluster Hamiltonian in CDMFT has the hopping matrix elements
$[\sum_{\bf k} t({\bf k})]_{ij} = t^{cl}_{ij}$ where $t^{cl}_{ij}=t=1$ for
first neighbors and $t^{cl}_{ij}=0$ otherwise. In contrast, in DCA we find
${\bar t}^{cl}_{ij}=\bar t=0.8103$ for first and third neighbors and
${\bar t}^{cl}_{ij}=0$ otherwise.
Within CDMFT as well as DCA, $G_{ij}$ is a symmetric matrix, with site-independent
diagonal components $G_{ii}$. In the case of CDMFT, there are three independent
off-diagonal elements: $G_{12}$, $G_{13}$ and $G_{14}$. Here, $G_{11}$, $G_{13}$
are imaginary and $G_{12}$, $G_{14}$ are real. Thus, the corresponding density of
states components $\rho_{11}$ and $\rho_{13}$ are even functions of energy, while
$\rho_{12}$ and $\rho_{14}$ are odd. In the case of DCA, translation symmetry is
preserved, so that one has the additional condition $\rho_{12}=\rho_{14}$ and
$G_{12}=G_{14}$ due to the equality of first- and third-neighbor hopping
interactions $\bar t$.
Because of these symmetry properties, it is useful to express the lattice Green's
function in the diagonal molecular-orbital basis whose elements $G_m(i\omega_n)$
($m=1\ldots6$) are determined by:
\begin{eqnarray}
G_{1,2} &=& (G_{11}+2G_{13}) \pm (G_{14}+2G_{12}) \nonumber \\
G_{3,4} = G_{5,6}
&=& (G_{11}- G_{13}) \pm (G_{14}-G_{12}). \label{Gk}
\end{eqnarray}
The unitary transformation $\bar T_{im}$ linking the site and molecular-orbital
bases is defined in Eq.~(6) of Ref.~\onlinecite{PRB2011}.
Evidently, in CDMFT there are two independent complex functions, $G_1=-G^*_2$ and
$G_3=-G^*_4$. In DCA, the elements $G_{m=3\ldots6}$ are degenerate and imaginary.
The onsite and intersite components of the lattice Green's function can be derived
by inverting Eq.~(\ref{Gk}):
\begin{eqnarray}
G_{11} &=& [(G_1 + G_2) + 2 (G_3 + G_4)]/6 \nonumber \\
G_{12} &=& [(G_1 - G_2) - (G_3 - G_4)]/6 \nonumber\\
G_{13} &=& [(G_1 + G_2) - (G_3 + G_4)]/6 \nonumber \\
G_{14} &=& [(G_1 - G_2) + 2 (G_3 - G_4)]/6 .
\end{eqnarray}
Figure~\ref{dos} illustrates the uncorrelated density of states components in the
diagonal molecular orbital basis, where
$\rho_m(\omega) = -\frac{1}{\pi}\,{\rm Im}\, G_m(\omega)$.
The total density of states is, of course, the same within CDMFT and DCA, but its
decomposition into molecular-orbital or intersite contributions differs for these
two schemes.
The four CDMFT densities shown in panel (a) are non-symmetric and satisfy the
relations $\rho_2(\omega)=\rho_1(-\omega)$ and $\rho_4(\omega)=\rho_3(-\omega)$.
The corresponding DCA densities are plotted in panel (b). In this case, only
$\rho_1(\omega)=\rho_2(-\omega)$ are non-symmetric, whereas
$\rho_{3}(\omega)=\rho_{4}(\omega)$ are symmetric.
Figure~\ref{BZ} (a) shows the Brillouin Zone of the honeycomb lattice together
with the three times smaller reduced Zone. Panel (b) illustrates the contributions
to the density of states stemming from the outer ${\bf k}$ regions $KMK'M'$ and
the inner regions $\Gamma M'K'$. These two contributions overlap slightly since
the point $K'$ does not lie half-way between $\Gamma$ and $M$. Thus,
the low-energy part of the density of states (denoted as $K$) extends up
$\vert \omega\vert \le 2$, while the high-energy part (denoted as $\Gamma$)
corresponds to the window $1.75\le\vert\omega\vert\le3$. The comparison with
Fig.~\ref{dos} (b) shows that the diagonal elements of the DCA density of states
correspond to the distributions indicated in Fig.~\ref{BZ} (b). Thus,
$\rho_{1,2}(\omega)$ account for the energy bands in the inner regions $\Gamma M'K'$
and $\rho_{3,4}(\omega)$ for those in the outer regions $KM K'M'$ of the original
Brillouin Zone. The momentum regions shown in Fig.~\ref{BZ} (a) therefore specify
the appropriate tiling of the Brillouin Zone within the DCA.
The self-energy matrices in CDMFT and DCA satisfy the same symmetry properties as
the lattice Green's functions so that they can be diagonalized in the same manner.
These diagonal elements will be denoted as $\Sigma_m(i\omega_n)$. In the site basis
the components $\Sigma_{11}$ and $\Sigma_{13}$ are imaginary, whereas $\Sigma_{12}$
and $\Sigma_{14}$ are real. As translation symmetry is not obeyed in CDMFT,
$\Sigma_{12}$ and $\Sigma_{14}$ differ, while in DCA they coincide.
We point out that, although the hopping matrix elements $t({\bf k})$ in CDMFT and
DCA differ only by a unitary transformation as indicated in Eq.~(\ref{phase}), the
same does not hold for the respective self-energy matrices. As discussed below,
the preservation of translation invariance in DCA and its absence in CDMFT give
rise to fundamentally different physical solutions which severely affect the phase
boundaries. Thus, the DCA and CDMFT self-energy matrices are not simply related
via a unitary transformation.
Severe differences of this kind between DCA and CDMFT do not arise in the case of
the Hubbard model for the square lattice, where the cluster Hamiltonians maintain
the same symmetry. The only difference is that the hopping interaction between
neighbors is changed from $t=1$ in CDMFT to $\bar t=1.273$ in DCA. As a result,
these cluster schemes lead to a less dramatic reorganization of spectral weight
among the cluster components than in the case of the honeycomb lattice.
\section{Exact Diagonalization}
To avoid double-counting of Coulomb interactions in the quantum impurity calculation,
the self-energy must be removed from the six-site cluster in which correlations are
treated explicitly. This removal yields the bath Green's function matrix
\begin{equation}
G_0(i\omega_n) = [G(i\omega_n)^{-1} + \Sigma(i\omega_n)]^{-1} .
\label{G0}
\end{equation}
Within the ED approach, this bath Green's function of the infinite lattice
is projected onto the corresponding function of a supercluster consisting of
$n_c=6$ correlated sites within the unit cell plus a bath consisting of $n_b$
discrete levels. Here, we choose $n_b=6$, so that the total number
of levels of the supercluster is $n_s=n_c+n_b=12$. Within the diagonal
molecular-orbital basis, this projection implies
\begin{eqnarray}
G_{0,m}(i\omega_n) &\approx& G^{cl}_{0,m}(i\omega_n) \nonumber\\
&=& \left( i\omega_n + \mu -\epsilon_m -
\sum_{k=7}^{12} \frac{\vert V_{mk}\vert^2}{i\omega_n - \epsilon_k}\right)^{-1},
\label{G0m}
\end{eqnarray}
where $\epsilon_{m=1...6}$ denotes impurity levels and $\epsilon_{k=7...12}$ bath
levels. The bath levels are defined relative to the chemical potential. We assume
that the molecular orbitals couple to independent baths so that the hybridization
matrix elements are also diagonal in this representation: $V_{mk}=\delta_{m+6,k}V_k$.
Fig.~\ref{levels}(a) illustrates the impurity and bath levels in the diagonal
molecular orbital basis. Panel (b) shows the equivalent representation when the
impurity orbitals are transformed to the original site basis. The bath remains
unchanged and the hopping terms in this basis are given by
$V_{ik} =\sum_m \bar T_{im} V_{mk}$.
This picture differs from the one in which also the bath is treated
within the site basis (see below).
\begin{figure} [t!]
\begin{center}
\includegraphics[width=5.5cm,height=6.0cm,angle=-90]{Fig3a.ps}\vskip-5mm
\includegraphics[width=5.5cm,height=6.0cm,angle=-90]{Fig3b.ps}
\end{center}\vskip-3mm
\caption{(Color online)
(a) Cluster levels in molecular orbital basis. There are six independent terms
connecting orbital levels $\epsilon_{m=1...6}$ (red dots) and bath levels
$\epsilon_{k=7...12}$ (blue dots) via hopping integrals $V_{k=7...12}$. In CDMFT
(for fixed impurity levels) one has:
$\epsilon_{1,2}=\mp 2t$, $\epsilon_{3,4}=\epsilon_{5,6}=\pm t$, and
$\epsilon_7=-\epsilon_8$, $\epsilon_9=\epsilon_{11}=-\epsilon_{10}=-\epsilon_{12}$,
$V_7=V_8$, $V_9=V_{10}=V_{11}=V_{12}$.
Thus there are four independent bath parameters.
In DCA, $ \epsilon_{1,2}=\mp 3\bar t$, $\epsilon_{3...6}=\epsilon_{9...12}=0$,
{\it i.e.}, there are only three independent fit parameters.
(b) Cluster levels in site basis $i=1...6$ (green dots) connected to molecular
orbital bath levels $\epsilon_{k=7...12}$ (blue dots) via hopping integrals $V_{ik}$.
For clarity, the hopping interactions between impurity sites are not shown.
Representations (a) and (b) are equivalent since they are connected via the unitary
transformation $\bar T$ between impurity sites $i=1...6$ and orbitals $m=1...6$.
The bath molecular orbital levels in (b) are the same as in (a). Thus, although the
cluster sites have identical levels at zero energy, the bath levels maintain the
orbital symmetry.
}\label{levels}\end{figure}
To determine the bath levels $\epsilon_k$ and hopping terms $V_{mk}$
we minimize the difference
\begin{equation}
{\rm Diff}_m = \sum_{n=0}^M W_n^N
\vert G_{0,m}(i\omega_n) - G^{cl}_{0,m}(i\omega_n)\vert^2 ,
\label{diff}
\end{equation}
where $M\approx 2^{10}$ is the total number of Matsubara points and the weight
function $W^N_n=1/\omega^N_n$ is introduced to give more weight to the low-frequency
region. We usually take $N=1$ or $N=2$. Note also that both Green's functions in the
above expression approach $1/i\omega_n$ for large $\omega_n$.
Thus the difference defined in Eq.~(\ref{diff}) automatically focuses on the low-energy
region. This is not the case when the differences of the inverse Green's functions are
minimized instead. The reason is that the hybridization functions corresponding to
$G_{0,m}$ and $G_{0,m}^{cl}$ are not normalized to the same asymptotic amplitudes.
To start the iterative procedure, we use bath parameters obtained for the uncorrelated
system, or from a converged solution for nearby Coulomb energies. The resulting
$\epsilon_k$ and $V_{mk}$ are usually very stable against variations of initial
conditions.
In the CDMFT calculations discussed in Ref.~\onlinecite{PRB2011}, not only the bath
levels $\epsilon_k$ and hopping elements $V_k$ were used as parameters in the fit of
$G_{0,m}(i\omega_n)$, but also the impurity levels $\epsilon_m$. Since the
expression Eq.~(\ref{G0m}) ensures the correct asymptotic behavior, the variation
of $\epsilon_m$ yields slightly better accuracy of the fit at the lower Matsubara
points. For each diagonal component $G_{0,m}$ three fit parameters are then available.
As there are only two independent complex functions $G_{0,m}$, the total number of
parameters to fit the bath is six. As shown in Fig.~24 of Ref.~\onlinecite{EDR}
for $U=4$ and $T=0.01$, this procedure yields a surprisingly good reproduction
of the lattice bath Green's function via the cluster Anderson Green's function,
in spite of the fact that we use only one bath level per impurity orbital.
The reason for this good fit is the semi-metallic nature of the honeycomb lattice,
giving rise to a vanishing density of states at the Fermi level. In contrast, in
ordinary correlated metals and the triangular or square lattice Hubbard models,
the density of states of the infinite lattice is finite, so that a successful fit
to a cluster Green's function usually requires at least two bath levels per orbital
and restriction to not very low temperatures (typically $T\ge0.01$).
In the DCA calculations presented below, we fix the impurity levels $\epsilon_m$
at their nominal cluster values. Thus, $\epsilon_{1,2}=\mp 3\bar t$ and
$\epsilon_{3,4}=0$. The latter value reflects the fact that the DCA density of
states components $\rho_{3,4}(\omega)$ are even functions of energy. Thus, the fit
of $G_{0,m=1,2}$ involves two parameters (the bath level $\epsilon_7=-\epsilon_8$
and the hopping element $V_7=V_8$), while $G_{0,m=3,4}$ includes only the hopping
element $V_9=V_{10}$ as fit parameter.
\begin{figure} [t!]
\begin{center}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig4.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Comparison of lattice bath Green's function $G_{0,m}(i\omega_n)$ (solid red curves)
and cluster Green's function (dashed blue curves) for $U=3$ and $T=0.01$.
As the density of states for $m=3$ is symmetric in DCA (see Fig.~1 (b)), $G_{0,3}$
is purely imaginary, while $G_{0,1}$ is complex. Thus, the latter function is
fitted with two parameters, whereas $G_{0,3}$ involves only one fit parameter.
The solid and dashed curves for $G_{0,1}$ are indistinguishable.
}\label{G0.DCA}\end{figure}
Figure \ref{G0.DCA} illustrates the quality of the fit of $G_0$ within ED DCA for
$U=3$ and $T=0.01$. The parameters used in these fits are:
$\epsilon_1=-3\bar t=-2.4309$,
$\epsilon_7=-1.85694$, $V_7=0.26270$ for $m=1$ and $\epsilon_3=\epsilon_9=0$,
$V_9=0.86701$ for $m=3$.
As pointed out above in the case of CDMFT, the excellent representation of the
lattice Green's function via the cluster Green's function using only one bath level
per impurity orbital is related to the vanishing density of states at the Fermi level.
The diagonalization of the supercluster Hamiltonian is conveniently carried out in
the site basis. At low temperatures only few excited states need to be included
in the evaluation of the cluster Green's function $G^{cl}_{ij}(i\omega_n)$. The
diagonalization can then be performed very efficiently by making use of the Arnoldi
algorithm. Details concerning this procedure are provided in
Refs.~\onlinecite{perroni,tong,EDR}.
Since the cluster Green's function obeys the same symmetry properties as the
lattice Green's function, it is diagonal in the molecular-orbital basis. These
elements will be denoted as $G^{cl}_m(i\omega_n)$. The diagonal cluster self-energy
components are then given by an expression analogous to Eq.~(\ref{G0}):
\begin{equation}
\Sigma^{cl}_{m}(i\omega_n) = 1/G^{cl}_{0,m}(i\omega_n)-1/G^{cl}_{m}(i\omega_n) .
\label{Scl}
\end{equation}
The key physical assumption in DMFT is now that this cluster self-energy provides
an accurate representation of the lattice self-energy. Thus,
\begin{equation}
\Sigma_{m}(i\omega_n) \approx \Sigma^{cl}_{m}(i\omega_n) .
\label{S}
\end{equation}
In the next iteration, these self-energy components are used as input in the
lattice Green's function Eq.~(\ref{G}). In the diagonal molecular-orbital basis
the DCA lattice Green's function is given by
\begin{equation}
G_{m}(i\omega_n)=\sum_{\bf k}\left[i\omega_n+\mu-{\bar T}^{-1}\bar h({\bf k}){\bar T}
- \Sigma(i\omega_n)\right]^{-1}_{mm} .
\label{Gm}
\end{equation}
We note here that, at real energies, the cluster quantities $G^{cl}_{m}$,
$G^{cl}_{0,m}$ and $\Sigma^{cl}_{m}$ have discrete spectra, while the corresponding
lattice spectra associated with the quantities $G_{m}$, $G_{0,m}$ and $\Sigma_{m}$
are continuous.
We close this section by pointing out that we believe the projection of the bath
Green's function within the diagonal molecular-orbital basis discussed above to be
more general and more flexible than analogous projections within the nondiagonal site
basis. As mentioned above, within CDMFT there are two independent complex functions
$G_{0,m}$ (with nonsymmetric spectral distributions) that are fitted each with
one bath level $\epsilon_k$ and one hopping term $V_k$ (assuming the impurity
level $\epsilon_m$ to be fixed). Thus, there are altogether four fit parameters.
This should be compared to only one fit parameter if the site basis is used instead.
For symmetry reasons all bath levels then are zero so that only the site independent
impurity bath hopping element remains as a single fit parameter. Introducing a
hopping interaction among bath levels as was done in Ref.~\onlinecite{helu}
increases the number of fit parameters from one to two.
Actually, since the bath can always be represented in a diagonal form, hopping
among bath levels is implicitly included in the diagonal molecular orbital picture
with four fit parameters. Analogous considerations hold for DCA.
Nevertheless, as will be shown in the next section, these slightly different
implementations of ED all yield consistent answers concerning the variation of
the excitation gap as a function of Coulomb energy.
\section{Results and Discussion}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=6.0cm,height=8.5cm,angle=-90]{Fig5.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Comparison of excitation gaps as functions of Coulomb interaction derived using
several cluster methods and impurity solvers for paramagnetic phase of honeycomb
lattice:
Meng {\it et al.}: large-scale QMC\cite{meng},
Wu {\it et al.}: CTQMC CDMFT \cite{wu},
Liebsch: ED CDMFT \cite{PRB2011},
He {\it et al.}: ED CDMFT \cite{helu},
Seki {\it et al.}: ED VCA \cite{seki}.
In contrast, both ED and CTQMC DCA yield semi-metallic behavior with $\Delta=0$
for $U\le 6$ (see text).
}\label{gap}\end{figure}
Figure~\ref{gap} shows the comparison of the excitation gaps obtained for various
cluster methods and impurity solvers. Near $U\approx5$, all calculations (except
DCA, see below) predict a Mott
phase with a gap $\Delta\approx 0.5-0.9$. At $U\le4$, the CDMFT and VCA results
that do not preserve translation symmetry exhibit a gap tail that persists
down to $U\rightarrow0$. The differences between these results are partly caused
by the different temperatures used in these studies. In particular, the gap
closing near $U=3.8$ obtained within CDMFT by Wu {\it et al.}\cite{wu} seems
to be related to the rather high temperature, $T=0.05$, employed in the CTQMC
calculation. Since the CTQMC self-energy agrees well with the ED results,
CTQMC CDMFT presumably would also yield a gap at lower $T$.
Also, the ED calculations in Ref.~\onlinecite{PRB2011} were carried out at
$T=0.005$, while those in Refs.~\onlinecite{helu,seki} essentially correspond
to the $T\rightarrow0$ limit.
In striking contrast to CDMFT, the translation invariance of DCA ensures the
existence of a semi-metallic phase at low values of $U$.
On the other hand, the condition $\Sigma_{12}=\Sigma_{14}$ cannot generally be
correct for the short-range correlations within the unit cell. Thus, at Coulomb
energies, where local Mott physics dominates and long-range translational
invariance becomes less important, DCA should be less appropriate than CDMFT.
Indeed, both ED and CTQMC DCA results suggest that the semi-metallic phase with
$\Delta=0$ extends to $U>6$, i.e., beyond the critical Coulomb energy
$U_c\approx 3.9-4.3$ of the antiferromagnetic phase.\cite{meng,sorella}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig6a.ps}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig6b.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Density of states $A_{11}(\omega)=-\frac{1}{\pi}{\rm Im}G_{11}(\omega)$
of honeycomb lattice for several Coulomb energies at. Red solid curves:
$U=6$, dashed curves: $U=3 - 5$. (a) ED DCA ($T=0.01$) (b) CTQMC DCA ($T=0.025)$.
For illustrative purpose, only the low-energy range of the ED spectra is shown.
The dotted curve denotes the bare density of states.
}\label{spectra}\end{figure}
This is illustrated in Fig.~\ref{spectra}, which shows the interacting density
of states obtained in ED and CTQMC DCA for several Coulomb energies.
The ED spectra were obtained by making use of the extrapolation routine
{\it ratint}, \cite{ratint} while the CTQMC spectra were derived via the
maximum entropy method.\cite{maxent}
For details concerning the CTQMC calculations, see Ref.~\onlinecite{wu}.
The main effect of Coulomb interactions
is seen to be the usual band narrowing and effective mass enhancement, as found
in weakly correlated systems. In contrast, the corresponding ED and CTQMC
CDMFT spectra for $U=5$ reveal a large Mott gap of about $\Delta=0.6$
(see Fig.~\ref{gap}).\cite{wu,PRB2011}
The persistence of semi-metallic behavior at large $U$ within DCA is related
to the fact that the enforcement of translation symmetry is achieved at the
expense of equating first- and third-neighbor interactions in the cluster
Hamiltonian. The self-energy in the site basis then satisfies the condition
$\Sigma_{12}=\Sigma_{14}$, whereas in CDMFT $\Sigma_{14}$ is noticeably smaller
than $\Sigma_{12}$.\cite{PRB2011,EDR}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig7a.ps}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig7b.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Green's function components $G_{1i}(i\omega_n)$ ($i=1,2,3$) of honeycomb lattice
as functions of Matsubara frequency calculated within (a) ED DCA and (b) CTQMC DCA
at $T=0.01$. Solid red curves: $U=4$; dashed curves: $U=1 - 3$.
}\label{Gij}\end{figure}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=6.5cm,height=8.5cm,angle=-90]{Fig8.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Green's function components $G_{1i}(i\omega_n)$ ($i=1,2,4$) of honeycomb lattice
as functions of Matsubara frequency calculated within ED CDMFT.
Solid red curves: $U=4$; dashed curves: $U=1 - 3$.
}\label{Gij.CDMFT}\end{figure}
The good correspondence between the DCA spectra obtained within ED and CTQMC is a
consequence of the nearly quantitative agreement of the lattice Green's functions
$G_{1i}(i\omega_n)$ which are shown in Fig.~\ref{Gij}. As pointed out in the
preceding section, for symmetry reasons $G_{11}$ and $G_{13}$ are imaginary,
while $G_{12}=G_{14}$ are real. Both impurity solvers yield
Im\,$G_{11}(i\omega_n)\rightarrow0$
in the limit $\omega_n\rightarrow 0$, implying that the local density of states,
$\rho(\omega)= -\frac{1}{\pi}\,{\rm Im}\,G_{11}(\omega)$ vanishes at $\omega=0$.
Also, both schemes indicate that with increasing values of $U$ the initial
slope of Im\,$G_{11}$ and Im\,$G_{13}$ increases. Thus, the Dirac cones become
steeper and spectral weight is shifted towards the Fermi level.
The results obtained within DCA differ in two qualitative aspects from those
derived previously in CDMFT. As shown in Fig.~\ref{Gij.CDMFT}, the components
$G_{12}$ and $G_{14}$ in CDMFT do not coincide.
Moreover, the initial slopes of Im\,$G_{11}$ and Im\,$G_{13}$ become smaller
with increasing $U$ rather than larger as within DCA. In Ref.~\onlinecite{PRB2011}
it was demonstrated that for $U\ge 4$ a Mott gap opens in the density of states,
in approximate agreement with the large-scale QMC calculations by Meng {\it et al.}
\cite{meng} At smaller values of $U$, a tiny gap or pseudogap was also found (see
below), which is however difficult to resolve within ED at finite $T$. As the opening
of a gap in the density of states implies a reduction of
$\vert{\rm Im}\,G_{11}(i\omega_n)\vert$
at small values of $\omega_n$, the results shown in Figs.~\ref{Gij} and \ref{Gij.CDMFT}
underline the fundamental difference between DCA and CDMFT for the honeycomb lattice:
Whereas DCA yields a weakly correlated semi-metal, CDMFT gives rise to insulating
behavior even at small $U$.
To illustrate the effect of Coulomb correlations in more detail, we show in
Fig.~\ref{Sij.ED} the self-energy components in the site basis for several values
of $U$. The corresponding results obtained within CTQMC DCA are
depicted in Fig.~\ref{Sij.QMC}. There is good overall correspondence between
these two impurity solvers, except for slightly different magnitudes of the
off-diagonal components. We note, however, that Re\,$\Sigma_{12}$ and Im\,$\Sigma_{13}$
are approximately one and two orders of magnitude smaller than Im\,$\Sigma_{11}$,
respectively. As can be seen in Fig.~\ref{Gij}, these differences have only a minor
effect on the variation of the Green's function components with increasing Coulomb
energy.
\begin{figure} [t!]
\begin{center}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig9a.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig9b.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig9c.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Self-energy components $\Sigma_{1i}(i\omega_n)$ ($i=1,2,3$) of honeycomb lattice
as functions of Matsubara frequency calculated within ED DCA for $U=1 - 4$
at $T=0.01$.
}\label{Sij.ED}\end{figure}
\begin{figure} [t!]
\begin{center}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig10a.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig10b.ps}
\includegraphics[width=5.0cm,height=8.5cm,angle=-90]{Fig10c.ps}
\end{center}\vskip-3mm
\caption{(Color online)
Self-energy components $\Sigma_{1i}(i\omega_n)$ ($i=1,2,3$) of honeycomb lattice
as functions of Matsubara frequency calculated within CTQMC DCA for $U=1 - 4$ at
$T=0.01$.
}\label{Sij.QMC}\end{figure}
The crucial question in the case of the honeycomb lattice is how Coulomb correlations
influence the energy bands in the vicinity of the Dirac points. The self-energy
at these points can be shown to have the simple form:
\cite{PRB2011}
\begin{equation}
\Sigma(K,i\omega_n) \approx i\omega_na + \frac{b^2}{i\omega_n(1-a)},
\hskip3mm \omega_n\rightarrow0,
\label{SK}
\end{equation}
where the coefficients are given by
\begin{eqnarray}
a &=& {\rm Im}\,[\Sigma_{11}(i\omega_n)-\Sigma_{13}(i\omega_n)]/\omega_n \\
b &=& {\rm Re}\,[\Sigma_{12}(i\omega_n)-\Sigma_{14}(i\omega_n)]
\end{eqnarray}
in the limit $\omega_n\rightarrow0$.
Thus, $\Sigma(K,i\omega_n)$ is imaginary as expected for particle-hole symmetry
near the Dirac points. Moreover, this self-energy consists of metallic
($\sim i\omega_n$) and insulating ($\sim 1/i\omega_n$) contributions, where the
latter term is a direct consequence of the fact that $\Sigma_{12}\ne\Sigma_{14}$.
The presence of this term implies
Re\,$\Sigma(K,\omega) \approx b^2/[\omega(1-a)]$ at real $\omega$. In the
low-temperature limit, this expression yields an excitation gap of magnitude
$\Delta \approx 2 \sqrt{\vert c \vert}$, where $c= b^2/(1-a)$.
A similar insulating contribution to the self-energy was recently found in
Ref.~\onlinecite{seki}. Presumably, this insulating term is also present in the
ED calculations reported in Refs.~\onlinecite{helu} and \onlinecite{yu}.
At finite $T$, the gap
is smoothened out so that it becomes difficult to determine its boundaries.
In contrast, as discussed in Section II, DCA preserves the bulk symmetry,
so that $\Sigma_{12}=\Sigma_{14}$ and $\Delta=0$.
Thus, the DCA self-energy at the Dirac points is purely metallic, where the
increasing magnitude of the coefficient $a$ implies increasing quasi-particle
broadening and shift of spectral weight towards the Fermi level as $U$ increases.
From the initial slope of Im\,$\Sigma_{11}$ at $U=4$ we obtain an effective mass
enhancement of about $m^*/m\approx 1.3$.
The above discussion demonstrates that the presence or absence of the insulating
contribution to $\Sigma(K,i\omega_n)$ is not caused by the impurity solver used
in the CDMFT or DCA calculations. In fact, the good agreement between ED and CTQMC,
for both CDMFT and DCA, suggests that in the case of the honeycomb lattice one bath
level per impurity orbital is sufficient for an accurate fit of the bath Green's
function. The reason is that, because of the semi-metallic nature of the honeycomb
lattice, the projection of the bath Green's function of the infinite lattice onto
a finite-cluster Anderson Green's function is not plagued by the
low-energy-low-temperature discrepancies that usually occur in the case of correlated
metals. In these systems at least two bath levels per impurity orbital are
typically required and very low temperatures must be avoided.\cite{EDR}
\section{Summary}
The role of Coulomb correlations in Hubbard model for the honeycomb lattice has
been studied within finite-temperature exact diagonalization and continuous-time
quantum Monte Carlo combined with the dynamical cluster approximation.
The unique feature of DCA is that it preserves the translation invariance so
that the system at small values of $U$ is semi-metallic. In contrast, CDMFT
violates translation symmetry which implies the opening of an excitation gap
at arbitrarily small $U$, regardless of the impurity solver.
This gap is therefore an artifact caused by the lack of long-range crystal
symmetry and does not correspond to a true Mott gap. At larger values of $U$,
however, many-body interactions are dominated by short-range correlations and
translation symmetry seizes to be important. DCA then becomes less accurate since
it overemphasizes semi-metallic behavior. Thus, for $U \approx 5$, CDMFT is
preferable and reveals a Mott gap in qualitative agreement with large-scale QMC
calculations.
In the case of the honeycomb lattice, DCA and CDMFT may therefore be viewed
as complementary cluster approaches. As DCA preserves translation symmetry,
it is more appropriate in the semi-metallic phase at small $U$ where long-range
order is a prerequisite for the description of the weakly correlated Dirac cones.
The condition $\Sigma_{12}=\Sigma_{14}$ which guaranties this symmetry, however,
is unrealistic at larger $U$, when short-range correlations within the six-site
unit cell begin to dominate. Thus, in the region of the Mott phase, CDMFT is more
suitable. As a result of these inherent limitations of both cluster schemes, the
critical Coulomb interaction defining the precise boundary between these phases
is at present difficult to determine within either CDMFT or DCA.
We emphasize that this difficulty is not related to the finite size or symmetry
of the bath used in ED. On the contrary, within CDMFT as well as DCA, the ED
self-energies agree well with the corresponding CTQMC results.
It is interesting to inquire why the remarkable difference between CDMFT and DCA
for the honeycomb lattice discussed in this paper does not also manifest itself
in other systems, such as the Hubbard models for square and triangular lattices.
In these cases, long-range order is mainly responsible for the logarithmic
divergence of the van Hove singularities of the density of states.
Thus, any lack of perfect translation symmetry would give rise to a rounding
of this peak, an effect that would be difficult to distinguish from broadening
induced by finite temperature and quasi-particle damping. In contrast, any
rounding of Dirac cones induces the opening of a gap. In this regard,
the Dirac cones of the honeycomb lattice correspond to a rather peculiar special
situation that does not arise in most cases which have been studied previously
within CDMFT and DCA at finite temperatures.
\bigskip
{\bf Acknowledgements}\ \
The ED DCA calculations were carried out on the J\"ulich Juropa machine.
W. W. acknowledges support from the Natural Sciences and Engineering Research
Council of Canada. A. L. likes to thank Profs. Lu and Seki for sending their
data shown in Fig.~\ref{gap}. |
1303.2372 | \section{Introduction}
Non-singular bouncing cosmology \cite{Novello:2008ra} has gained significant interest in recent studies of the early universe. The main reason for such a research direction is that the most popular paradigm of the early universe, namely inflation, still suffers from the ``Big-Bang singularity'' problem, which however can be naturally avoided in non-singular bouncing or cyclic cosmologies. Additionally, these paradigms can also solve the horizon, flatness and monopole problems, and make compatible observational predictions such as nearly scale-invariant power spectrum and moderate non-Gaussianities \cite{Cai:2007zv,Cai:2008ed,Qiu:2010ch}. Therefore, they are recently considered as good alternatives to inflation.
In order to realize a successful bounce several requirements must be
fulfilled. First of all, the basic condition is to have the Hubble
parameter change its sign from negative to positive at the bounce, which
implies that during the bouncing phase the Null Energy Condition (NEC) must
be violated, with the total EoS of the universe going below $-1$
\cite{Cai:2007qw,Cai:2009zp}. The NEC violation in the context of General
Relativity is nontrivial \cite{Nojiri:2013ru}, usually leading to ghost
degree(s) of freedom \cite{Carroll:2003st,Cline:2003gs}, which would demand
either ghost-elimination mechanisms or an extended analysis to a modified
gravity context \cite{Capozziello:2011et,Biswas:2005qr}.
Apart from the above basic condition, in order for a bounce to be a
successful alternative to inflation it should also solve the other Big-Bang
problems, and moreover it should produce a nearly scale-invariant power
spectrum as required by observations \cite{Bennett:2012fp}. These
impose more stringent constraints on the bounce evolution, especially in
the contracting phase. For instance, the horizon problem can be solved if
the quantum fluctuations in the far past lie deep inside the horizon, while
they should exit the horizon in the contracting phase in order to generate
perturbations compatible with observations, provided that inflation is
absent in bouncing scenario. This requires a total EoS satisfying $w>-1/3$
in the contracting phase \cite{Piao:2004jg,Qiu:2012ia}. However, the
scale-invariance of the perturbations is even harder to be achieved. In
particular, as it was initially shown in \cite{Finelli:2001sr}, if
the perturbations generated in the contracting phase are purely adiabatic,
the EoS of the contracting universe should satisfy $w\approx0$ in order to
produce the desired spectrum.
However, although bounce models with total EoS $w\approx0$ before the
bounce, namely the ``matter bounce'', could lead to nearly scale-invariant
power spectrum, they generally suffer from the ``anisotropy problem''
\cite{Kunze:1999xp} in the contracting phase. In
particular, in 4D General Relativity a tiny amount of anisotropic
fluctuation from the simple isotropic Friedmann-Robertson-Walker (FRW)
geometry in the contracting phase, would increase as $a^{-6}$, where $a$ is
the scale factor. Thus it would finally dominate over
matter-like background, leading to a Big-Crunch singularity with complete
anisotropy instead of a bounce, unless one impose a strong fine-tuning of
the model parameters and the initial amount of anisotropy in order to
obtain a bounce before the domination of the anisotropic term. In that
sense the ``matter bounce" scenario is not stable against cosmological
anisotropy (for its similar problem in the presence of radiation see
\cite{Karouby:2010wt}). For this reason, we must construct scenarios with
total EoS larger than $1$ in order to prevent the dominance of anisotropy.
However, as we mentioned above, a different EoS may not be able to provide
the scale-invariant power spectrum, if we insist on applying the simple
adiabatic mechanism of generating primordial perturbations\footnote{We
would like to mention that here by ``simple adiabatic mechanism'' we mean
that the perturbations generated are purely adiabatic, and the EoS remains
constant. However, the term ``adiabatic mechanism'' which was first
proposed in \cite{Khoury:2009my} in ``Ekpyrotic'' scenarios
\cite{Khoury:2001wf}, refers to the mechanism that generates adiabatic
perturbations via varying EoS.}. Therefore, we should resort to alternative
mechanisms, such as adiabatic, entropy and conformal ones
\cite{Khoury:2009my,Finelli:2002we,Lehners:2007ac,
Buchbinder:2007ad,Lehners:2007wc, Hinterbichler:2011qk}
\footnote{Note that the stability of isotropic solutions in anisotropic
perturbations has been studied in \cite{Aref'eva:2009vf}.}.
In the present work we investigate the bounce realization in the framework
of recently proposed generalized Galileon cosmology
\cite{Nicolis:2008in,Deffayet:2009mn} (see also
\cite{Silva:2009km,DeFelice:2011bh} for various
developments). Due to the delicate design of the Lagrangian form such
a theory, which contains higher-order derivatives, can keep its equation of
motion up to second-order and thus is free of ghosts (this was pioneered by
the work by Horndeski \cite{Horndeski:1974wa}), but it can
indeed provide extra degree(s) of freedom in order to violate NEC.
Recently, in \cite{Qiu:2011cy}, the first ghost-free bounce model
based on Galileon cosmology was constructed by one of the present
authors and collaborators (see also \cite{Easson:2011zy}), and hence in
this article we will consummate this class of models by addressing the
problems mentioned above. Note that alternative scenarios addressing the
anisotropy problem in Galileon bouncing cosmologies have been presented
in \cite{Cai:2012va,Cai:2013vm}, of which before contracting with $w>1$, the
universe
can be dominated by cold matter \cite{Lin:2010pf}, where scale-invariant
perturbations could be generated.
First of all, by introducing an Ekpyrotic-like
negative potential we can easily obtain a very large EoS in the
contracting phase, thus the anisotropy problem will be eliminated. However,
as mentioned above, a large EoS forbids the Galileon field to generate the
desired form of perturbations, thus as a next step we additionally
introduce the curvaton field which is suitably coupled to the Galileon
field, such that the nearly scale-invariant spectrum can be produced.
Finally, we perform a complete analysis of the behavior around the bounce
point of the full Galileon-curvaton system, making use of the ``inverse"
reconstruction procedure \cite{Cai:2009in}, showing that with a proper
choice of the Lagrangian functional forms a non-singular bounce
can be reconstructed, which can connect smoothly to the matter-domination
era and moreover alleviate the Big-Rip singularity which
appears in \cite{Qiu:2011cy}.
The plan of the work is the following: In section \ref{anisotropyproblem}
we briefly review the anisotropy problem. In section \ref{Galileonbounce}
we present the bouncing background evolution before,
during and after the bounce, and we show that the
perturbations are stable and free of ghosts. In section
\ref{curvatonmechanism} we analyze the curvaton mechanism that produces
nearly scale-invariant perturbations. In section \ref{Reconstructing} we
perform a semi-analytical procedure in order to reconstruct an exact
bouncing solution that is not followed by a Big-Rip. Finally, in section
\ref{Conclusions} we summarize and we discuss the obtained results.
Throughout the manuscript we use the $(-,+,+,+)$ metric signature,
and units in which $M_{Pl}=1/\sqrt{8\pi G}=1$.
\section{The anisotropy problem}
\label{anisotropyproblem}
The anisotropy problem is a notorious problem that generally exists in
bouncing models with $w<1$ in contracting phase
\cite{Kunze:1999xp}. In General Relativity, if we
allow the existence of a non-zero anisotropy at the beginning of the
contraction it will evolve scaling as $a^{-6}(t)$. In order to demonstrate
this more transparently, without loss of generality we consider as an
example the simple anisotropic Bianchi-IX metric \cite{Misner:1974qy}:
\be
ds^2=-dt^2+a^2(t)\sum_{i=1}^3e^{2\beta_i(t)}d{x^i}^2~,
\ee
with $\beta_1(t)+\beta_2(t)+\beta_3(t)=0$. The Friedmann Equation writes as:
\be\label{friedmannani}
3H^2=\rho_u+\frac{1}{2}\left(\sum_{i=1}^3\dot\beta_i^2\right)~,
\ee
where $H=\dot a/a$ the Hubble parameter and with $\rho_u$ incorporating
all the fluids in the universe. The $\beta_i$'s satisfy the equations
\be
\ddot\beta_i+3H\dot\beta_i=0~,
\ee
which provide the solutions $\beta_i\propto a^{-3}(t)$. Since the
second part in the right hand side of equation (\ref{friedmannani}) can be
considered as an effective anisotropy term $\chi^2$, we conclude that
\be
\chi^2\equiv\frac{1}{2}\left(\sum_{i=1}^3\dot\beta_i^2\right)\propto
a^{-6}(t)~,
\ee
and thus the anisotropy term corresponds to an effective energy density
with EoS $w=1$. Although in an expanding universe this term is always
sub-dominant and thus isotropization can be achieved, in a contracting
case, as long as it is initially non-zero (even arbitrarily small), the
anisotropy will grow fast and become dominant over all species with EoS
less than 1, leading finally to a collapsing anisotropic universe. For this
reason, in order to avoid the domination of a possible anisotropic
fluctuation, one has to realize a contracting background that evolves even
faster, which requires an EoS larger than unity in the contracting phase
\footnote{For some Grand Unification Theories where anisotropic stresses
and collisionless particles are taken into account, there may still be
anisotropy problems, see \cite{Barrow:2010rx} for more details. However,
it is not the case that we're currently considering. We thank John Barrow
for pointing it out to us.}.
\section{The Galileon bounce}
\label{Galileonbounce}
In the previous section we briefly showed that in order to realize a bounce
we need an effective EoS $w>1$ in the contracting phase, in order to avoid
the domination of an anisotropic fluctuation.
In this section we formulate the bounce realization in generalized Galileon cosmology.
In the generalized Galileon scenario, where the coefficients of the various
action-terms are considered as functions of the scalar field, the
corresponding action can be written as \cite{Deffayet:2009mn}:
\begin{equation}
{\cal L}=\sum_{i=2}^{5}{\cal L}_{i}\,,\label{Ltotal}
\end{equation}
where
\begin{eqnarray}
&&{\cal L}_{2} = K(\phi,X)\nonumber\\
&&{\cal L}_{3} = -G_{3}(\phi,X)\Box\phi\nonumber\\
&&{\cal L}_{4} = G_{4}(\phi,X)\,
R+G_{4,X}\,[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)\,(\nabla^{\mu}
\nabla^{\nu}\phi)]\nonumber\\
&&{\cal L}_{5} = G_{5}(\phi,X)\,
G_{\mu\nu}\,(\nabla^{\mu}\nabla^{\nu}\phi)\nonumber\\
&&\ \ \ \ \ \ \ -\frac{1}{6}\,
G_{5,X}\left[(\Box\phi)^{3}
-3(\Box\phi)\,(\nabla_{\mu}\nabla_{\nu}\phi)\,
(\nabla^{\mu}\nabla^{\nu}\phi)\right.
\nonumber\\
&&\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2(\nabla^{\mu}\nabla_{\alpha}\phi)\,(\nabla^
{\alpha}\nabla_{\beta}\phi)\,(\nabla^{\beta}\nabla_{\mu}\phi)\right]\,
.\label {
eachlag5}
\end{eqnarray}
In this action the functions $K$ and $G_{i}$ ($i=3,4,5$) depend on the
scalar field $\phi$ and its kinetic energy
$X\equiv-\frac{1}{2}\nabla_\mu\phi\nabla^\mu\phi$, while $R$ is the Ricci
scalar and $G_{\mu\nu}$ is the Einstein tensor. Moreover, $G_{i,X}$ and
$G_{i,\phi}$ ($i=3,4,5$) denote the partial derivatives of $G_{i}$ with
respect to $X$ and $\phi$, ($G_{i,X}\equiv\partial G_{i}/\partial X$ and
$G_{i,\phi}\equiv\partial
G_{i}/\partial\phi$), and the box operator is constructed from covariant derivatives: $\Box\phi\equiv g^{\mu\nu} \nabla_\mu \nabla_\nu\phi$. In the following we focus on the case
\begin{eqnarray}
\label{model}
&&K(\phi,X)=X-V(\phi),~G_3(\phi,X)=g(\phi)X,\nonumber\\
&&G_4(\phi,X)=
\frac{1}{2}, ~G_5(\phi,X)=0.
\end{eqnarray}
Therefore, the action that we are going to use reads:
\bea
\label{action}
&&{\cal
S}=\int
d^4x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}
\nabla_\mu\phi\nabla^\mu\phi-V(\phi)\right.\ \ \ \ \ \ \ \ \ \ \
\
\nonumber\\
&&\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ +\frac{g}{2}
\nabla_\mu\phi\nabla^\mu\phi
\Box\phi \right].
\eea
We now proceed to a detailed investigation of the above scenario. Firstly,
in the following subsection we provide approximate analytical solutions at
the far past before the bounce, around the bouncing regime, and after the
bounce, at the background level. Then in the next subsection, we analyze
the perturbation behavior.
\subsection{Background evolution: analytical results}
\label{galileonbackground}
In the following we impose a flat Friedmann-Robertson-Walker (FRW) background metric of the form $ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)d{\bf{x}}^{2}$, where $t$ is the cosmic time, $x^i$ are the comoving spatial coordinates, $N(t)$ is the lapse function, and $a(t)$ is the scale factor. Varying the action (\ref{action}) with
respect to $N(t)$ and $a(t)$ respectively, and setting $N=1$, we obtain the Friedmann equations
\be
\label{friedmann}
H^2=\frac{1}{3}\rho~,~~~\dot{H}=-\frac{1}{2}(\rho+P)~.
\ee
Additionally, we have defined the effective energy density and pressure as:
\bea
\label{rho}
\rho &=& \frac{1}{2}\dot\phi^2+V(\phi)+3gH\dot\phi^3~,\\
\label{pressure} P &=&
\frac{1}{2}\dot\phi^2-V(\phi)-g\dot\phi^2\ddot\phi~,
\eea
and thus the total EoS of the universe is just
\bea
\label{wtot}
w\equiv\frac{P}{\rho}=\frac{\dot\phi^2-2V(\phi)-2g\dot\phi^2\ddot\phi}{\dot\phi^2+2V(\phi)+6gH\dot\phi^3}~.
\eea
Finally, variation of (\ref{action}) with respect to the Galileon field provides its evolution equation:
\be
\label{eom}
\mathcal{D}\ddot{\phi}+\Gamma\dot{\phi}+V_{\phi}=0~,
\ee
where
\bea
\label{mathcalD}
\mathcal{D}&=&1+6gH\dot{\phi}+\frac{3}{2}g^{2}\dot{\phi}^{4}~, \\
\Gamma&=&\frac{3}{2}\left(1+3gH\dot{\phi}\right)\left(2H-g\dot{\phi}^{3}
\right)~.
\eea
In the following we will suitably choose the potential $V(\phi)$ in
order to obtain a very large positive equation of state in relation
(\ref{wtot}), namely $w>1$, so that our model will not suffer from the
anisotropy problem in the contracting phase. Meanwhile, when the Galileon
term $G\Box\phi$ becomes more and more important, the EoS becomes negative
and eventually triggers the bounce.
As a specific example, we choose the potential to be
\be
V(\phi)=-V_0e^{c\phi},
\label{potential}
\ee
with $V_0,c>0$, namely a negative exponential potential, which is the usual
one in Ekpyrotic scenarios \cite{Khoury:2001wf}. Within this choice, when
the nonlinear kinetic term is relatively small we obtain $\rho<P$ and thus
$w>1$. One could ask whether the negative potential would lead to a
negative energy density, however the more negative the potential is, the
steeper it is, and thus it gives rise to larger kinetic term, which can
compensate the potential negativity (this will be verified later on).
Let us proceed to a qualitative investigation of the dynamics of such
scenario. For simplicity we consider that $\phi$ increases from negative to
positive during the evolution, therefore at the beginning when $\phi$
starts at a large negative value the potential (\ref{potential}) is in its
``slow-varying'' region, and the field moves slowly. In this region the
nonlinear term $G\Box\phi$ will have a small contribution to the action.
Similarly to the Ekpyrotic models, the universe will contract with a very
large positive EoS, but as time passes $\phi$ moves towards positive
values and its velocity increases, the effect of the $G\Box\phi$ term is
enhanced, and it can trigger the bounce. However, after the bounce and due
to the large slope of the potential, the field kinetic energy could
increase to unacceptably large values which could spoil the validity of the
effective theory. Therefore, we should need some mechanisms to slow the
field motion down, and this will be demonstrated in detail in the next
sections.
In the following we proceed to the quantitative investigation of the
scenario, extracting approximate analytical solutions for the background
evolution in the far past before the bounce, around the bouncing point,
and in the far future after the bounce.
\subsubsection{Solution far before the bounce}
Far before the bounce, as we assumed, $\phi$ begins with a large negative value and moves slowly, while the nonlinear term of $\dot\phi$ is negligible. The energy density and pressure reduces to
\be
\rho\simeq\frac{1}{2}\dot\phi^2+V(\phi)~,~P\simeq\frac{1}{2}\dot\phi^2-V(\phi)~,
\ee
similarly to a single scalar field with an Ekpyrotic potential. Thus, we can choose the model parameters and the initial conditions of $\phi$ in order to acquire scaling solutions, namely to obtain a scale factor evolving as
\be
\label{scalea}
a(t)\sim (t_\ast-t)^p\sim|\eta_\ast-\eta|^\frac{p}{1-p}
~,~\ \ p\equiv\frac{2}{3(1+w)},~
\ee
where the subscript `$\ast$' denotes the time where the non-linear term in
equations (\ref{rho})-(\ref{eom}) becomes important. Note that since we are
considering the region where $t<t_\ast$, we have $t_\ast-t>0$.
For completeness we have also expressed the above solution using the
conformal time $\eta$, related to the cosmic time $t$ through
$dt=ad\eta$. Similarly,
the field $\phi(t)$ scales as
\be
\label{ansatz}
\phi(t)\simeq-\frac{2}{c}\ln(t_\ast-t)~,~p=\frac{2}{c^2}~.
\ee
Moreover, taking the derivatives of the above expressions we find
\bea
\label{dotphi0}
&&\dot\phi(t)\simeq\frac{2}{c(t_\ast-t)},\\
&&H(t)\simeq\frac{p}{
t-t_\ast } ~,
\label{Hsol0}
\eea
and therefore the energy density from expression (\ref{rho}) becomes
\bea
\label{energybg}
\rho(t)_{t<t_\ast}=3H^2\propto (t-t_\ast)^{-2}
\eea
Finally, using relations (\ref{scalea}) and (\ref{ansatz}), for the total
EoS using (\ref{wtot}) we obtain
\bea
\label{wtot1}
w\simeq\frac{1}{3}c^2-1=const.
\eea
\subsubsection{Solution around the bounce}
Around the bouncing point $t_B$ the nonlinear term becomes important. From the Friedmann equation (\ref{friedmann}) we can express $H$ as:
\be
\label{solH} H=\frac{1}{2}g\dot\phi^3\pm\frac{1}{6}\sqrt{
9g^2\dot\phi^6+6(\dot\phi^2-2V_0e^{c\phi})}~,
\ee
in which the nonlinear term becomes important and the last terms in
(\ref{rho}) and (\ref{pressure}) can no longer be neglected. Without loss
of generality we can keep the minus-sign branch in order to obtain a
positive $\dot\phi(t)$.\footnote{Note that in general the Hubble parameter
could transit from the minus-sign branch to the plus-sign branch either
before (for $\dot\phi_{initial}<0$) or after the bounce (for
$\dot\phi_{initial}>0$). Thus, above we assume that if such transition
exist it will take place after the bounce. However, as we will shortly see
below, at late times and under the curvaton backreaction,
expression (\ref{solH}) is not exactly valid any more. Hence, we do not
examine in detail the relation between the two branches.} Therefore, when
the universe goes from the contracting ($H<0$) to the expanding phase
($H>0$), one has $\dot\phi>\sqrt{2V_0}e^{c\phi/2}$ before and
$\dot\phi<\sqrt{2V_0}e^{c\phi/2}$ after the bounce. It is therefore natural
to consider the solution at the bounce region to be
\be\label{phibounce}
\dot\phi=\sqrt{2V_0}(\alpha t+\beta)e^{\frac{c\phi}{2}}~,
\ee
with $\alpha$ and $\beta$ being two parameters satisfying the conditions
\be
\alpha<0~,~\alpha t_B+\beta=1~.
\ee
Note that substitution of (\ref{phibounce}) into the $\phi$-equation of
motion (\ref{eom}), provides the necessary value for
the coefficient
$\alpha$ in order to have self-consistency.
Integrating equation (\ref{phibounce}) leads to
\be
e^{-\frac{c\phi_0}{2}}-e^{-\frac{c\phi}{2}}=\sqrt{2V_0}\Big[\frac{\alpha}{2
}(t_0^2-t^2)+\beta(t_0-t)\Big]~,
\ee
with $t_0$ a boundary value for $t$, and $\phi_0$ the corresponding value of $\phi$. The above formula will be simplified if we set $t_0$ in the far future when $\phi_0$ becomes very large, and thus $e^{-\frac{c\phi_0}{2}}\approx 0$. In this case the expression for $\phi(t)$ becomes
\be
\phi=-\frac{2}{c}\ln\Big\{-\sqrt{2V_0}(t_0-t)\Big[\frac{\alpha}{2}
(t-t_0)+\alpha t_0+\beta\Big]\Big\}~,
\label{solfull}
\ee
which leads to
\bea
\label{solfull1}
\dot\phi&=& \frac{4(\alpha t+\beta)}{ c(t_0-t)[\alpha(t-t_0)+2(\alpha
t_0+\beta)]} ~,\\
\ddot\phi&=&\frac{2}{c}\left\{\frac{1}{(t_0-t)^2}+\frac{\alpha^2}{[
\alpha(t+t_0)+2\beta]^2}\right\}~.
\label{solfull2}
\eea
It would be useful if we could approximate the above expressions around the bouncing point $t_B$, namely $t\rightarrow t_B=(1-\beta)/\alpha$. In this case, and neglecting the constant terms in order to extract the pure scaling behavior, we obtain
\bea
&&\phi\simeq
\frac{4 \alpha(t-t_B)}{c\left[(\beta +\alpha
t_0)^2-1\right]},\nonumber\\
&&\dot\phi\simeq\frac{4\alpha^2\left[(\beta+\alpha
t_0)^2+1\right](t-t_B)}{c\left[ (\beta+\alpha
t_0)^2-1\right]^2},
\nonumber\\
&&\ddot\phi\simeq\frac{8\alpha^3\left[3(\beta+\alpha
t_0)^2+1\right](t-t_B)}{c (\beta+\alpha t_0-1)^3(\beta+\alpha t_0+1)^3},
\label{phiapprox0}
\eea
that is $\phi$ exhibits a linear behavior around $t_B$. Following the
same way, one could also insert these expressions into equation
(\ref{solH}) and (\ref{wtot}) to straightforwardly obtain the approximate
solutions for $H(t)$ and $w(t)$, respectively.
\subsubsection{Solution after the bounce}
Far after the bounce, when $t$ approaches $t_0$, the solutions are still
given by (\ref{solfull})-(\ref{solfull2}). Thus, approximating
them at $t\rightarrow t_0$, and neglecting the constant terms in order to
extract the pure scaling behavior, we acquire
\bea
\label{phiapprox}
\phi&\simeq&-\frac{2}{c}\ln(t_0-t)~,
\\
\dot\phi&\simeq&
\frac{2}{c}\frac{1}{t_0-t}~,
\\
\ddot\phi& \simeq&\frac{2}{c}\frac{1}{(t_0-t)^2}~.
\eea
In this case, inserting these expressions into (\ref{solH}), we can extract a
simple approximate expression for $H(t)$ too, namely
\be
\label{scaleH}
H\simeq \frac{1}{(t_0-t)^3}~,
\ee
which leads to a Big-Rip singularity when $t$ approaches $t_0$.
This can be easily explained since when the nonlinear terms become very
important the last terms in (\ref{rho}) and (\ref{pressure}) become
dominant. Thus, when the energy density is dominated by the $3gH\dot\phi^3$
term, namely $3H^2=\rho\sim 3gH\dot\phi^3$, we straightforwardly find that
the Hubble parameter $H$ will be proportional to $\dot\phi^3$, and
(\ref{scaleH}) is verified. Finally, the total EoS can also be obtained
by inserting these expressions into (\ref{wtot}).
\subsubsection{Numerical verification}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.31]{phiscaling.eps}
\caption{{\it{The evolution of the Galileon field $\phi$ with respect to
$t$. We choose
the initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$ and
$\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$, $g=1$,
and $V_0=1/12$, respectively.}}} \label{phi}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.31]{dotphiscaling.eps}
\caption{{\it{The evolution of the speed of the Galileon field $\dot\phi$
with respect
to $t$. We choose the initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$
and $\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$,
$g=1$, and $V_0=1/12$, respectively.}}} \label{dotphi}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.31]{Hscaling.eps}
\caption{{\it{The evolution of the Hubble parameter $H$ with respect to
$t$. We
choose the initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$ and
$\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$, $g=1$,
and $V_0=1/12$, respectively.}}} \label{hubble}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.31]{wscaling.eps}
\caption{{\it{The evolution of the EoS $w$ with respect to $t$. We choose
the
initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$ and
$\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$, $g=1$,
and $V_0=1/12$, respectively.}}} \label{eos}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.31]{rhoscaling.eps}
\caption{{\it{The evolution of the total energy
density $\rho$ of the Galileon field, with respect to $t$. We choose
the
initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$ and
$\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$, $g=1$,
and $V_0=1/12$, respectively.}}} \label{rhoplot}
\end{center}
\end{figure}
We close the background investigation by performing an exact numerical
elaboration in order to verify the above approximate expressions in the
various regimes. In particular, we numerically solve equations
(\ref{friedmann}) and (\ref{eom}), imposing (\ref{ansatz}) as our
initial conditions. In Figures \ref{phi}, \ref{dotphi}, \ref{hubble} and
\ref{eos}, we respectively present $\phi(t)$, $\dot\phi(t)$, the Hubble
parameter $H(t)$ as well as the EoS $w(t)$. As we observe, both $\phi(t)$
and $\dot\phi(t)$ are monotonically increasing. Moreover, before the bounce
the universe contracts with a scaling solution with EoS $w$ being constant
and larger than unity (in this specific example $w=3$), and thus the
scenario is free from the anisotropy problem discussed in section
\ref{anisotropyproblem}.
As time passes the nonlinear term becomes important and triggers the
bounce, which forces $H(t)$ to change from negative to positive. Note that
the numerical results confirm that $\dot\phi(t)$ has a positive value
during the bouncing period, and thus it justifies our choice of the minus
sign in (\ref{solH}). After the bounce, the nonlinear kinetic term makes
the scalar-field energy density increasing, leading the universe to a
Big-Rip. One can also see that the total EoS $w(t)$ indeed indicates the
bounce followed by the Big-Rip, verifying the analytical results that has
been obtained in preceding paragraphs.
Finally, in Fig. \ref{rhoplot}, we present the evolution of the total energy
density $\rho$ of the Galileon field, calculated through (\ref{rho}). As
we observe $\rho$ is always positive, despite the use of a negative
potential, due to the increase of the kinetic energy, and it becomes zero
only at the bounce point as expected. Therefore, in the present work we do
not need mechanisms that could transit the universe to positive potential
energy \cite{Felder:2002jk}, however it would be desirable to consider a
mechanism that could smooth the increase of the Galileon kinetic
energy, by either considering a bound in the potential, or couple
$\phi$ to other matter fields such is radiation, which could lead to
energy transfer
away from it (a procedure that could lead to the universe preheating too).
These mechanisms lie beyond the scope of the present work and are left for a
future investigation.
\subsection{Perturbations}
In the previous subsection we analyzed the background evolution of the
Galileon bounce. Thus, we can now proceed to the investigation of the
perturbations, focusing on their stabilities. It proves convenient to
foliate the FRW metric in an Arnowitt-Deser-Misner (ADM) form
\cite{Arnowitt:1962hi}:
\be
ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)~,
\ee
where $N$ is the lapse function, $N^i$ is the shift vector, and $h_{ij}$ is the induced 3-metric. One can then perturb these functions as:
\be
N=1+A~,~N_i=\partial_i\psi~,~h_{ij}=a^2(t)e^{2\zeta}\delta_{ij}~,
\ee
where $A$, $\psi$ and $\zeta$ are the scalar metric perturbations. As usual it is useful to define the (gauge-invariant) comoving curvature perturbation through
\be
\mathcal{R}\equiv\zeta+\frac{H}{\dot\phi}\delta\phi~,
\label{curvpert}
\ee
and hence in the uniform $\phi$ gauge we acquire $\delta\phi=0$ and
$\zeta=\mathcal{R}$.
Under the above perturbation scheme, the action (\ref{action}) perturbed up
to second order becomes:
\be
\label{pertaction}
\mathcal{S}^{(2)}=\int d\eta d^3xa^2\frac{Q_{\cal R}}{c_{s}^2}\Bigl[{\cal
R}^{\prime 2}-c_{s}^2(\partial{\cal R})^2\Bigr]~,
\ee
where $\eta$ is the conformal time. In the above expression we have
introduced the sound-speed squared $c_s^2$, and the quantity $Q_{\cal R}$
related to instabilities, which in our specific scenario read as
\bea
c_s^2&=&Q_{\cal
R}^{-1}[1+2g(\ddot\phi+2H\dot\phi)-\frac{1}{2}g^2\dot\phi^4]~,
\\
Q_{\cal
R}&=&1+6gH\dot\phi+\frac{3}{2}g^2\dot\phi^4~.
\eea
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.3]{cs2scaling.eps}
\caption{{\it{The evolution of the squared speed of sound $c_s^2$ with
respect
to $t$. We choose the initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$
and $\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$,
$g=1$, and $V_0=1/12$, respectively.}}}
\label{cs2}
\end{center}
\end{figure}
Note that according to the definition of $\mathcal{D}$ in (\ref{mathcalD})
we obtain $2Q_{\cal R}=\mathcal{D}$.
The perturbative action (\ref{pertaction}) could in principle lead to ghosts
and gradient instabilities, which would be catastrophic since it is this
action that can be written as a canonical form and then be quantized. From
its form one can see that the avoidance of ghosts requires the factor in
front of the kinetic term of the perturbation variable ${\cal R}$ to be
positive, namely $Q_R/c_s^2>0$, while the absence of gradient
instabilities requires $c_s^2\geq0$ \cite{DeFelice:2011bh}, which means the
ratio of the factors of spatial and time derivatives must be positive.
In order to show that this is the case for
the exact behavior too, in Figures \ref{cs2} and \ref{QR} we respectively
depict $c_s^2$ and $Q_{\cal R}$, arising from the numerical elaboration
of the full system.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.3]{QRscaling.eps}
\caption{{\it{The evolution of the instability-related quantity $Q_{\cal
R}$ with respect to $t$. We choose the
initial conditions to be $\phi_i=-\sqrt{3}(\ln20)/3$ and
$\dot\phi_i=\sqrt{3}/60$, and the parameters to be $c=2\sqrt{3}$, $g=1$,
and $V_0=1/12$, respectively.}}}
\label{QR}
\end{center}
\end{figure}
From the above analysis we can see that our model is stable under both
ghost and gradient instabilities. However, it generally generates a (deep)
blue tilted power spectrum, which cannot be consistent with
observations. Note that in \cite{Qiu:2011cy} this was shown for
$w=1/3$, therefore in our present case where $w>1$ the blue tilt of the
power spectrum is even stronger. This implies that the anisotropy-free
requirement $w>1$ and the scale-invariant perturbation generation cannot
be obtained simultaneously in the current case, as was already mentioned
in the Introduction.
For this reason, we should introduce an additional mechanism in order to be
able to generate the nearly invariant perturbation spectrum, without
spoiling the solution to the anisotropy problem. This can be performed by
the curvaton mechanism, as we will present in the next section.
\section{The curvaton mechanism and the Scale-invariant spectrum}
\label{curvatonmechanism}
In the previous section we showed that the Galileon scenario under an
Ekpyrotic-like potential can exhibit a bouncing solution naturally. In the
contracting phase the corresponding total EoS of the universe satisfies
$w>1$ in order for the evolution to be free of the anisotropy problem
discussed in section \ref{anisotropyproblem}. However, under this
requirement the corresponding perturbations, although free of ghost and
gradient instabilities, cannot give rise to a nearly scale-invariant power
spectrum, as it is required by observational data
\cite{Bennett:2012fp}.
In \cite{Qiu:2011cy} it was shown that this problem can be solved by
introducing a curvaton field $\sigma$ coupled to the Galileon $\phi$. As it
is usual for curvaton fields \cite{Lyth:2001nq}, $\sigma$ does not affect
the bouncing background behavior, but it can lead to a power spectrum in
agreement with observations. In particular, through a specific
Galileon-curvaton coupling, the curvaton field lies effectively in a
``fake'' de-Sitter expansion or matter-like contraction, and thus it can
generate a nearly scale-invariant power spectrum. This is the idea behind
the ``conformal'' mechanism \cite{Hinterbichler:2011qk} and in this section
we investigate the necessary form of the coupling functions.
Let us consider the curvaton action as
\be
\label{action_curvaton}
{\cal S}_{\sigma}=\frac{1}{2}\int d^4x
\sqrt{-g}[-{\cal F}(\phi)(\partial\sigma)^2-2{\cal G}(\phi)W(\sigma)]~,
\ee
allowing for the most general form of coupling between $\sigma$ and $\phi$. The functions ${\cal F(\phi)}$ and ${\cal G(\phi)}$ depend on $\phi$, while $W(\sigma)$ is the potential for $\sigma$. Variation of action (\ref{action_curvaton}) with respect to $\sigma$ gives its background evolution equation as:
\be
\label{eomsigma}
\ddot\sigma_0+\frac{(a^3{\cal F})^\cdot}{a^3{\cal F}}\dot\sigma_0+\frac{{\cal G}}{{\cal F}}W_{\sigma_0}=0~,
\ee
where $\sigma_0$ is the background value of $\sigma$ and $W_{\sigma_0}$
corresponds to $\partial W/\partial\sigma|_{\sigma_0}$. In the following,
and up to the end of this section, we omit the subscript ``0'', denoting
the background by a simple $\sigma$, unless explicitly mentioned.
Additionally, the energy density and pressure of $\sigma$ can be
respectively written as:
\be
\label{rhosigma}
\rho_\sigma=\frac{1}{2}{\cal F}\dot\sigma^2+{\cal
G}W~,~P_\sigma=\frac{1}{2}{\cal F}\dot\sigma^2-{\cal G}W~.
\ee
We now proceed to the examination of the perturbations generated from the
curvaton field. Perturbing it by $\delta\sigma$ and defining $u\equiv
a\sqrt{{\cal F}}\delta\sigma$, we can extract the corresponding
perturbation equation as a second-order differential equation:
\be
u''+\left(k^2+a^2\frac{{\cal G}}{{\cal
F}}W_{\sigma\sigma}-\frac{z''}{z}\right)u=0~,
\ee
where the prime denotes derivative with respect to the conformal time $\eta$, $W_{\sigma\sigma}\equiv\partial^2W(\sigma)/(\partial\sigma)^2$ is the second derivative of the potential with respect to $\sigma$, and we have defined $z\equiv a\sqrt{\cal F}$. As usual, the power spectrum generated by $\delta\sigma$ is defined as:
\be
\label{sigmaspectrum}
{\cal P}_{\delta\sigma}=\frac{k^3}{2\pi^2}\Big|\frac{u}{z}\Big|^2~.
\ee
Therefore, we deduce that the condition for obtaining a scale-invariant power spectrum of $\delta\sigma$ is:
\be
\label{curvcon}
\frac{a^2{\cal G}W_{\sigma\sigma}}{\cal
F}-\frac{z''}{z}\simeq-\frac{2}{|\eta_\ast-\eta|^2}~.
\ee
There are several ways to satisfy the condition (\ref{curvcon}). The
simplest one is to set $W(\sigma)=0$, in which the first term in the above
condition disappears, and thus we need just to suitably choose ${\cal F}$
in order to obtain $z''/z\simeq2|\eta_\ast-\eta|^{-2}$. Alternatively we
can incorporate the
effects of both the kinetic and the potential terms of $\sigma$. In the following subsections we consider these cases separately.
\subsection{$W(\sigma)=0$}
\label{Wzero}
Under $W(\sigma)=0$, the condition (\ref{curvcon}) of obtaining scale-invariant power spectrum becomes
\be
\label{z}
z\propto |\eta_\ast-\eta|^2\,\,\,{\text{or}}\,\,\,|\eta_\ast-\eta|^{-1}~.
\ee
For the case of $z\propto |\eta_\ast-\eta|^2$ we obtain
\be
\delta\sigma=\frac{u}{z}\sim k^\frac{3}{2}~,~~~k^{-\frac{3}{2}}|\eta_\ast-\eta|^{-3}~,
\ee
the latter of which dominates over the former. Therefore, using expression
(\ref{sigmaspectrum}) we can obtain the power spectrum as
\be
{\cal P}_{\delta\sigma}\sim k^0 |\eta_\ast-\eta|^{-6}~.
\ee
As we observe, the spectrum is indeed scale-invariant but it has an increasing amplitude.
On the other hand, for the case of $z\propto|\eta_\ast-\eta|^{-1}$ we acquire
\be
\delta\sigma=\frac{u}{z}\sim k^{-\frac{3}{2}}~,~~~k^\frac{3}{2}|\eta_\ast-\eta|^{3}~,
\ee
the former of which dominates over the latter. Therefore, using
(\ref{sigmaspectrum}) we can obtain the power spectrum as
\be
{\cal P}_{\delta\sigma}\sim k^0.
\ee
In this case the spectrum is scale-invariant and moreover it is conserved on super-horizon scales.
The absence of $W(\sigma)$ leads to an absence of ${\cal G}(\phi)$ too.
Thus, we only need to suitably determine the form of ${\cal F}(\phi)$
according to the condition (\ref{z}). Note that far before the bounce we
have already assumed the scale-factor ansatz (\ref{scalea}), and therefore
(\ref{z}) requires just
\be
\label{scalef}
{\cal F}\propto
|\eta_\ast-\eta|^\frac{2(2-3p)}{1-p}\,\,\, {\text{or}}
\,\,\,|\eta_\ast-\eta|^\frac{2}{p-1}~.
\ee
Furthermore, from the ansatz solution (\ref{ansatz}) for $\phi$ and the above expressions we deduce that the suitable choice of the form of ${\cal F}$ in terms of $\phi$ might be
\be
\label{f}
{\cal F}\propto e^{2(3-c^2)\phi/c}\,\,\, {\text{or}}\,\,\,e^{c\phi}~,
\ee
where we have made use of the relation $p=2/c^2$ (note that since in contracting phase $w\gg 1$, we have $0<p\ll 1$).
We close this subsection by examining the backreaction of $\sigma$ field on
the background evolution, since although the curvaton is necessary for the
correct perturbation generation we would not desire it to spoil the
background bouncing behavior itself. In the contraction region where
$t<t_\ast$, the background energy density of the system scales as
$\rho_{t<t_\ast}\sim (t_\ast-t)^{-2}$, as it was found in
(\ref{energybg}). On the other hand, the evolution equation of the curvaton
field (\ref{eomsigma}) gives $\dot\sigma\sim a^{-3}{\cal F}^{-1}$, and thus
its energy density in (\ref{rhosigma}) becomes $\rho_\sigma\simeq{\cal
F}\dot\sigma^2/2 \sim a^{-6}{\cal F}^{-1}$. Since we know the
time-dependence of the scale factor from (\ref{scalea}) and the
time-dependence of ${\cal F}$ from (\ref{scalef}), we straightforwardly
deduce that for the solution branch where $z\propto|\eta_\ast-\eta|^2$ we
obtain $\rho_\sigma\sim(t_\ast-t)^{-4}$,
while for the solution branch where $z\propto|\eta_\ast-\eta|^{-1}$ we acquire $\rho_\sigma\sim(t_\ast-t)^{2(1-3p)}\sim (t_\ast-t)^2$ (in the last step we used that $p\ll 1$). Therefore, our analysis indicates that in the solution branch where $z\propto|\eta_\ast-\eta|^{2}$ (with ${\cal F}\propto e^{2(3-c^2)\phi/c})$, one has to suitably tune the initial conditions in order for the curvaton not to destroy the background bouncing behavior. However, in the second solution branch where $z\propto|\eta_\ast-\eta|^{-1}$ (with ${\cal F}\propto e^{c\phi})$, the energy density of the curvaton field grows slower than that of the background, and thus the background bouncing evolution is not altered by the backreaction of the curvaton.
\subsection{$W(\sigma)\neq0$}
\label{Wnonzero}
We now examine the case where the curvaton potential is non-zero, in order to investigate its effect on the perturbation generation. Without loss of generality and for simplicity we assume that ${\cal F}$ is approximately a constant (we set ${\cal F}=1$), although extension to general ${\cal F}$ is straightforward.
In order to see what condition (\ref{curvcon}) gives in this case, we
recall that at the early stage of the bouncing phase, where the
perturbation $\delta\sigma$ is generated, the scale factor evolves
according to (\ref{scalea}), and therefore since $z\equiv a\sqrt{{\cal
F}}$ we obtain
\be
\label{zWnonzero}
\frac{z''}{z}=\frac{a''}{a}\simeq\frac{p}{1-p}\left(\frac{p}{1-p}
-1\right)\frac{1}{\left|\eta-\eta_{\ast}\right|^{2}}~.
\ee
Furthermore, introducing $a_\ast=a(\eta_\ast)$ we can write
\be
\label{a2Wnonzero}
a^{2}\frac{{\cal G}}{{\cal F}}W_{,\sigma\sigma}=a_\ast^2{\cal
G}W_{,\sigma\sigma}\left|\eta_{\ast}-\eta\right|^{\frac{2p}{1-p}}~.
\ee
Inserting equations (\ref{zWnonzero}) and (\ref{a2Wnonzero}) into
(\ref{curvcon}) we deduce that the condition for obtaining a
scale-invariant power spectrum of $\delta\sigma$ becomes
\be
\label{condWnonzero}
a_\ast^2{\cal
G}W_{\sigma\sigma}=\frac{3p-2}{(1-p)^2}\left|\eta_{\ast}-\eta\right|^{
-\frac{2}{1-p}}~.
\ee
As a specific example we consider the well-studied case of a quadratic
potential, namely $W(\sigma)=m_\sigma^2\sigma^2/2$, in which case
$W_{\sigma\sigma}=const$. Hence, condition (\ref{condWnonzero}), using
also the $\phi$-evolution from (\ref{ansatz}), gives
\be
\label{GWnonzero}
{\cal G}\propto e^{c\phi}~.
\ee
Therefore, we extract that the field perturbation $\delta\sigma$ scales as:
\be
\delta\sigma=\frac{u}{z}\sim
k^{-\frac{3}{2}}|\eta_\ast-\eta|^{\frac{1}{p-1}}~,~~~k^\frac{3}{2}
|\eta_\ast-\eta|^{\frac{3p-2}{p-1}}~.
\ee
We mention that since we assume ${\cal F}=1$ (that is $z=a$) the ``fake'' effect is absent, and thus the dominating mode of the perturbations is always the growing mode.
We close this subsection by examining the backreaction of $\sigma$ field on the background evolution, since we would not want the curvaton to spoil the background bouncing behavior itself. The $\sigma$-evolution equation (\ref{eomsigma}) becomes
\be
\label{sigmaeqWnonzero}
\ddot\sigma+3H\dot\sigma+m_\sigma^2{\cal G}\sigma=0~.
\ee
Since according to relations (\ref{GWnonzero}) and (\ref{ansatz}) we have
${\cal G}\sim e^{c\phi}\sim(t_\ast-t)^{-2}$, we can write ${\cal G}={\cal
G}_0(t_\ast-t)^{-2}$ with ${\cal G}_0$ being an arbitrary constant.
Therefore, equation (\ref{sigmaeqWnonzero}) accepts the solution
\be
\label{solsigma}
\sigma\sim(t_\ast-t)^{\frac{1}{2}[1-3p\pm\sqrt{(1-3p)^2-4m_\sigma^2{\cal
G}_0}]}~.
\ee
Finally, substituting it into (\ref{rhosigma}) for the curvaton energy
density we obtain
\be
\label{curvenergWnonzero}
\rho_\sigma\sim(t_\ast-t)^{[1-3p\pm\sqrt{(1-3p)^2-4m_\sigma^2{\cal G}_0}]-2}~.
\ee
Comparing the background energy-density evolution (\ref{energybg}) with the curvaton energy-density evolution (\ref{curvenergWnonzero}) we deduce that the requirement for the latter to grow slower than the former is to have $1-3p\pm\sqrt{(1-3p)^2-4m_\sigma^2{\cal G}_0}>0$. However, since in order to solve the anisotropy problem we focus on $w>1$ (or equivalently $p<1/3$), then provided ${\cal G}_0>0$ the above requirement is always satisfied. Therefore, in the scenario at hand the energy density of the curvaton field will never dominate over the background evolution, that is the background bouncing behavior will not
be destroyed by its backreaction.
We close this section with a comment on the preservation of the scale
invariance across and after the bounce, which is in general a crucial
question in bouncing scenarios, and on the matching conditions we impose. In
the above analysis we required
$\delta\sigma$ and $\delta\sigma^\prime$ to be continuous across the bounce,
which is consistent with Deruelle-Mukhanov matching conditions
\cite{Deruelle:1995kd}. Considering the expanding phase, since it usually contains two
modes, namely the constant and the growing/decaying one, the perturbation
spectrum may or may not get altered
depending on whether mode-mixing is realized or not, or depending on
whether the varying modes in contracting/expanding phase are
growing/decaying, which is determined by the background.
For instance, in the above
case where ${\cal F}(\phi)\sim e^{c\phi}$ and $W(\sigma)=0$, the contracting
modes are decaying, thus by using the aforementioned matching conditions
scale-invariance will be maintained if the expanding mode is growing, which
requires the background EoS (or the effective EoS, if there is a ``faking'')
to be no larger than 1, and therefore we may easily preserve the
scale-invariance in the expanding phase by slightly constraining the
background evolution. A more detailed discussion on these will be taken
on in a following-up paper. Similar results can be found in
\cite{Cai:2012va}.
\section{Reconstructing the exact solution around the bounce}
\label{Reconstructing}
In the previous sections we constructed the Galileon bounce scenario free of
the anisotropy problem, in which we added the curvaton field in order to
obtain a scale-invariant power spectrum of primordial perturbations generated
in the contracting phase. Additionally, we showed that under soft
requirements on the choice of ${\cal F}(\phi)$ or ${\cal G}(\phi)$, the
backreaction of the curvaton field will not alter the background bouncing
evolution.
However, after the bounce the effect of the curvaton field can be
significant, and in particular it can regularize the universe evolution in
order not to result to a Big-Rip. In order to examine what classes of
coupling functions can provide this overall behavior, in this section we
semi-analytically reconstruct them following the ``inverse'' procedure
\cite{Cai:2009in}, in which we impose as input the desired bouncing scale
factor, reconstructing suitably the various function in order to correspond
to a consistent and exact solution of the full system of equations.
The complete action of the Galileon-curvaton system, consisted of both (\ref{action}) and (\ref{action_curvaton}), can be written as:
\bea
\label{fullaction}
&&{\cal
S}_{total}=\int
d^4x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}
\nabla_\mu\phi\nabla^\mu\phi-V(\phi)\right.\ \ \ \ \ \ \
\nonumber\\
&&\left.\ \ \ \ +\frac{g}{2}
\nabla_\mu\phi\nabla^\mu\phi
\Box\phi-{\cal F}(\phi)(\partial\sigma)^2-2{\cal
G}(\phi)W(\sigma)\right].
\eea
Note that although matter and radiation could be included
straightforwardly, in the above action we have neglected them in order to
examine the pure effects of the Galileon-curvaton evolution. Thus, the
cosmological equations in the FRW metric are the first Friedmann equation:
\be
\label{FR1}
3H^2=\frac{\dot{\phi}^2}{2}+V(\phi)+3gH\dot{\phi}^3+\frac{1}{2}{\cal
F}(\phi)\dot{\sigma}^2+{\cal G}(\phi)W(\sigma)~,
\ee
and the evolution equations for the two fields, namely
\be
\label{sigmaevol}
\ddot{\sigma}+\dot{\sigma}\left[3H+\frac{\partial{\cal
F}(\phi)}{\partial\phi}\frac{\dot{\phi}}{{\cal F}(\phi)}\right]+\frac{{\cal
G}(\phi)}{{\cal F}(\phi)}\frac{\partial W(\sigma)}{\partial\sigma}=0~
\ee
and
\bea
\label{phievol}
&&\ddot{\phi}
\left[1+6gH\dot{\phi}+\frac{3}{2}g^2\dot{\phi}^4\right]-\frac{1}{2}\frac{
\partial{\cal F}(\phi)}{\partial\phi}\dot{\sigma}^2+\frac{\partial{\cal
G}(\phi)}{\partial\phi}W(\sigma)\nonumber\\
&&+\frac{3}{2}\dot{\phi}\left\{2H+g\dot{\phi}\left[6H^2-\dot{\phi}^2-{\cal
F}(\phi)\dot{\sigma}^2-3gH\dot{\phi}^3
\right]\right\}\nonumber\\&&+\frac{\partial V(\phi)}{\partial\phi}=0~,
\eea
respectively.
One could solve the above equations fully numerically, imposing specific
ansantzes and initial conditions, however doing so he does not have
control of what functions-ansantzes, parameter choices and initial
conditions, lead to bouncing solutions. That is why the aforementioned,
semi-analytical, ``inverse'' procedure, where the scale factor is imposed
{\it a priori}, is better and more appropriate for the analysis of this
section, allowing for a systematic control on the conditions of the bounce
realization. We mention that since there are more unknown functions than
equations, one can in general always reconstruct the desired evolution.
Let us impose a desired bouncing scale factor $a(t)$ as an input, by which
$H(t)$ is also known. Furthermore, we consider $V(\phi)$ as usual, and we
impose $\phi(t)$ at will too. Thus, only three free functions remain,
namely ${\cal F}(\phi)$, ${\cal G}(\phi)$ and $W(\sigma)$ which must be
derived by the equations, along with the solution for $\sigma(t)$. Since
there are three independent cosmological equations, namely equations
(\ref{FR1}), (\ref{sigmaevol}) and (\ref{phievol}), we must also impose by
hand one more of the above four functions. We prefer to set $W(\sigma)=0$,
since this was one (simpler) case that was approximately analyzed in
subsection \ref{Wzero} (however one could easily consider other $W(\sigma)$
forms too). Such a choice simplifies things since the function ${\cal
G}(\phi)$ also disappears from the equations, and therefore the
cosmological equations (\ref{FR1})-(\ref{phievol}) are considered as
differential equations for $\sigma(t)$ and ${\cal F}(t)$. Thus, after
obtaining the solution, and since we know $\phi(t)$, we can reconstruct
${\cal F}(\phi)$.
Equation (\ref{FR1}) can be algebraically solved in order to obtain
$\dot{\sigma}^2$ as
\bea \dot{\sigma}^2(t)&=&\frac{1}{{\cal
F}(t)}\left[6H(t)^2-2V(\phi(t))-\dot\phi^2(t)\right.\nonumber\\
\label{auxil4}
&&\left.\ \ \ \ \ \ \ \ -6gH(t)\dot\phi^3(t)\right]~.
\eea
Substituting this into (\ref{sigmaevol}) gives a simple first
order differential equation for ${\cal F}(t)$ of the form
\be\label{auxil5}
h(t,{\cal F}(t),\dot{\cal F}(t))=0~,
\ee
which can be easily solved. Thus, from the solution of ${\cal F}(t)$ and the known $\phi(t)$ we can reconstruct ${\cal F}(\phi)$.
We mention here that the above procedure holds for every input functions,
with the only requirement being the obtained $\dot{\sigma}^2$ from relation
(\ref{auxil4}) to be positive, otherwise there is no solution that can
correspond to the input functions. With the above semi-analytical procedure
one has full control on how to choose the model parameters and the initial
conditions in order to get a positive $\dot{\sigma}^2$ in
(\ref{auxil4}). On the other hand, if one tries to solve fully numerically
the three equations (\ref{FR1})-(\ref{phievol}) simultaneously, it is very
hard to determine the model parameters and the initial conditions in order
to get a consistent bouncing solution.
In order to apply explicitly the above reconstructing procedure, without loss of generality we choose a bouncing scale factor of the form
\be
\label{at}
a(t)=a_B\left(1+\frac{3}{2}\omega t^{2}\right) ^{1/3}~,
\ee
where $a_B$ is the scale factor at the bouncing point and $\omega$ is a
positive parameter which describes how fast the bounce takes place. The
above ansatz presents the bouncing behavior, where $t$ varies in the
bounce region, that is between the times $t_{B-}$ and $t_{B+}$, with
$t=t_B=0$ the bouncing point, however one could use at will any other
bouncing ansatz. Straightforwardly we find
\be
H(t)=\frac{\omega t}{(1+3\omega t^{2}/2)}~,
\ee
and thus the universe is free of a Big-Rip after the bounce.
\begin{figure}[ht]
\includegraphics[scale=0.32]{tphiF.eps}
\caption{(Colored online) {\it{The solution for the coupling function
${\cal F}(t)$ and the imposed Galileon field $\phi(t)$, under the imposed
bouncing ansatz (\ref{at}). We choose the parameters as $g=1$,
$V_0=1/12$, $c=2\sqrt{3}$, $\omega=1$, $\phi_I=0.1$, $t_I=-100$, and
$t_{B\pm}=\pm1$.}}}
\label{phiFtt}
\end{figure}
\begin{figure}[ht]
\includegraphics[scale=0.32]{Fphi.eps}
\caption{(Colored online) {\it{The reconstructed coupling function
${\cal F}(\phi)$ under the imposed
bouncing ansatz (\ref{at}), using Fig. \ref{phiFtt}. We choose
the parameters as $g=1$, $V_0=1/12$, $c=2\sqrt{3}$, $\omega=1$,
$\phi_I=0.1$, $t_I=-100$, and $t_{B\pm}=\pm1$.}}}
\label{Fphi}
\end{figure}
For the field $\phi$ and the potential $V(\phi)$, enlightened by the
analysis of subsection \ref{galileonbackground} and without loss of
generality, respectively we assume
\be
\label{phisample}
\phi(t)=\phi_I\ln(t-t_I)~
\ee
and
\be
V(\phi)=-V_0e^{c\phi}~,
\ee
while as we mentioned we set $W(\sigma)=0$.
We follow the procedure described above, and for the model parameters we
choose $g=1$, $V_0=1/12$, $c=2\sqrt{3}$, $\omega=1$, $\phi_I=0.1$,
$t_I=-100$, and $t_{B\pm}=\pm1$. In Fig. \ref{phiFtt} we depict the
solution for ${\cal F}(t)$ and also the known $\phi(t)$ from
(\ref{phisample}), and in Fig. \ref{Fphi} we present the corresponding
reconstructed ${\cal F}(\phi)$. Finally, for completeness in Fig.
\ref{sigma} we show the solution for $\sigma(t)$.
\begin{figure}[ht]
\includegraphics[scale=0.32]{sigmat.eps}
\caption{(Colored online) {\it{The solution for the curvaton field
$\sigma(t)$, under the imposed
bouncing ansatz (\ref{at}). We choose the
parameters as $g=1$, $V_0=1/12$, $c=2\sqrt{3}$, $\omega=1$, $\phi_I=0.1$,
$t_I=-100$, and $t_{B\pm}=\pm1$.}}}
\label{sigma}
\end{figure}
We close this section by mentioning that in principle one could think of other reconstructing procedures, for instance setting ${\cal F}(\phi)$ and reconstruct ${\cal G}(\phi)$ and $W(\sigma)$, or even setting $\phi(t)$ and $\sigma(t)$ and reconstruct ${\cal F}(\phi)$, ${\cal G}(\phi)$ and $W(\sigma)$. So there can actually be many possibilities to realize the non-singular bounce.
\section{Conclusions}
\label{Conclusions}
Bounce cosmology is an interesting paradigm since it alleviates the
Big-Bang singularity problem. Additionally, it can solve the
Big-Bang problems, and nearly scale-invariant primordial
perturbations can be incorporated too. These features make bouncing
cosmologies successful alternatives to inflation. However, there are many
detailed issues that should be carefully addressed during the establishment
of bouncing cosmology, and in the present work we tried to confront some of
them.
First of all, the NEC violation, which is required for the bounce
realization, may bring ghost degrees of freedom. In the above analysis we
were based on the Galileon scenario, which is a higher-derivative
construction free of ghosts, and thus we obtained a bouncing evolution free
of ghost and gradient instabilities.
However, there is a second problem that may disturb the bounce
construction, namely that in the contracting phase even a tiny anisotropic
fluctuation from the totally isotropic FRW geometry will be radically
enhanced and destroy completely the FRW evolution. The solution of this
``anisotropy problem'' requires the total EoS of the universe to lie in
the regime $w>1$ in the contracting phase. Thus, starting from
\cite{Qiu:2011cy} where the anisotropy problem was present, in this work we
were able to solve it and obtain $w>1$ by considering an Ekpyrotic-like
potential with negative value. This is one of the main contributions of the
present article.
The above solution of the anisotropy problem through a large EoS has an
undesired effect, namely it spoils the generation of a nearly
scale-invariant power spectrum, that a scalar with smaller EoS can bring
through adiabatic perturbation. Therefore, in order to still be able to
produce a power spectrum in agreement with observations, we additionally
introduced in the scenario a second, curvaton field, coupled to the
Galileon one, which can indeed generate the desired perturbations in an
isocurvature way. In our analysis we presented this mechanism in general,
and we analyzed explicitly two specific examples where nearly
scale-invariant perturbations are generated. Finally, we examined the
conditions under which the curvaton field does not cause a significant
backreaction on the background bouncing behavior caused by the Galileon
field.
Furthermore, the curvaton field, apart from the generation of the desired
perturbations, has another important role, namely after the bounce it can
regularize the background evolution in order to avoid a Big-Rip
singularity, which is caused by the Galileon field itself. In particular,
although the curvaton backreaction is not significant at the background
level before the bounce, during and after the bounce it becomes important
and changes the background evolution. In order to see this effect
we performed a semi-analytically ``inverse'' analysis, reconstructing
suitably the desired bouncing evolution of the scale factor, which is free
of a Big-Rip without any fine-tuning. We mention here that since
the region where the scale-invariant spectrum is generated lies in
the contracting phase, while the region where the Big-Bang is avoided is
around and after the bounce point, the conditions on the functions
that generate scale-invariance perturbations should hold in the
contracting phase while those for the Big-Bang avoidance should hold around
and after the
bounce. Therefore, one can always match the required function form of the
contracting regime with the required form of the bounce regime, to obtain
both perturbation scale invariance and Big-Bang avoidance, although not
always analytically.
We close this work by mentioning that there could be other
possibilities to avoid the Big-Rip singularity. For instance, an
alternative evolution after the bounce would be to assume that the Galileon
and curvaton fields decay to standard model particles
\cite{Langlois:2013dh}. In this case the decaying Galileon energy density
cannot trigger the Big Rip anymore, and additionally it can produce the
matter content of the universe. Such a detailed analysis of the post-bounce
evolution, and its relation to the subsequent thermal history of the
universe, lies beyond the scope of the present work and it is left for a
future investigation.
{\bf Note added:} After completing our manuscript, we came to know that
studies of bounce cosmology aiming to the same issue has been done in
\cite{Osipov:2013ssa}, in which similar results are obtained for different
(conformal) Galileon models but with the same (Ekpyrotic-like) potential.
\begin{acknowledgments}
T.Q. thanks Robert Brandenberger for his useful comments. The work of T.Q.
is funded in part by the National Science Council of R.O.C. under Grant No.
NSC99-2112-M-033-005-MY3 and No. NSC99-2811-M-033-008 and by the National
Center for Theoretical Sciences. X.G. was supported by ANR (Agence Nationale
de la Recherche) grant ``STR-COSMO" ANR-09-BLAN-0157-01. The research of
E.N.S. is implemented within the framework of the Action ``Supporting
Postdoctoral Researchers'' of the Operational Program ``Education and
Lifelong Learning'' (Actions Beneficiary: General Secretariat for Research
and Technology), and is co-financed by the European Social Fund (ESF) and
the Greek State.
\end{acknowledgments} |
0912.2950 | \section*{\textmd{\normalsize COLO-HEP-549}}
\begin{center}
\textbf{\Large Classical and Quantum SUSY Breaking Effects in IIB
Local Models}
\par\end{center}{\Large \par}
\begin{center}
{\large S. P. de Alwis$^{\dagger}$}
\par\end{center}
\begin{center}
Physics Department, University of Colorado, \\
Boulder, CO 80309 USA
\par\end{center}
\begin{center}
\vspace{0.3cm}
\par\end{center}
\begin{center}
\textbf{Abstract}
\par\end{center}
\begin{center}
\vspace{0.3cm}
\par\end{center}
We discuss the calculation of soft supersymmetry breaking terms in
type IIB string theoretic models in the Large Volume Scenario (LVS).
The suppression of FCNC gives a lower bound on the size of the compactification
volume. This leads to soft terms which are strongly suppressed relative
to the gravitino mass so that the dominant contribution to the gaugino
masses comes from the Weyl anomaly. The other soft terms are essentially
generated by the renormalization group running from the string scale
to the TeV scale.
\vfill
$^{\dagger}$ dealwiss@colorado.edu
\eject
\section{Introduction}
Theories of supersymmetry (SUSY) breaking and transmission from a
hidden to a visible sector has been the subject of much discussion
over the last two to three decades. Much of this discussion has had
little to do with string theory and often it has been conducted purely
within a global SUSY framework. However a theory of supersymmetry
breaking must necessarily be embedded within a (${\cal N}=1$) supergravity
(SUGRA) which is derived from string theory. The following is a summary
of the arguments leading to this assertion.
\begin{enumerate}
\item \textcolor{black}{Adding a set of explicit soft SUSY breaking terms
to a global theory (like the Minimally Supersymmetric Standard Model
(MSSM)) leads to far too much arbitrariness - it does not give us
a theory.}
\item Spontaneous SUSY breaking in Global SUSY leads to a cosmological constant
(CC) at the SUSY breaking scale which cannot be fine-tuned to zero.
\item \textcolor{black}{A theory of SUSY breaking is therefore necessarily
a SUGRA with a scalar potential which has a minimum that breaks SUSY
spontaneously.}
\item A SUGRA needs to be embedded in string theory in order to have a quantum
mechanically consistent and complete theory.
\end{enumerate}
\textcolor{black}{The main problem in relating string theory to phenomenology
is that the starting point of the theory is in ten dimensions. While
there are only five weakly coupled string theories (which are related
to each other through various dualities) the number of four dimensional
`compactifications' is extremely large. At the time of the second
string revolution of the mid eighties, it was hoped that in spite
of the existence of many compactifications, the number of theories
with stabilized moduli (the fields governing the size and shape of
the compact manifold) is small if not just one. However it was realized
through the work of many authors (for a review see \citep{Grana:2005jc,Douglas:2006es})
culminating in that of \citep{Giddings:2001yu} (GKP) and \citep{Kachru:2003aw}
(KKLT) that the number of such four dimensional models is extremely
large. Thus at the current stage any discussion of the phenomenological
consequences of string theory must proceed by first imposing a set
of experimental inputs (in addition to requiring a compactification
to four dimensional ${\cal N}=1$ supergravity). These are: }
\begin{itemize}
\item \textcolor{black}{CC is tiny $\sim O((10^{-3}eV)^{4})$ }
\item \textcolor{black}{No light scalars with gravitational strength coupling}
\item \textcolor{black}{SUSY partner masses $\gtrsim O(100GeV)$ }
\item \textcolor{black}{Lightest Higgs $>114GeV$}
\item \textcolor{black}{Flavor changing neutral currents (FCNC) suppressed}
\item \textcolor{black}{No large CP violating phases}
\end{itemize}
\textcolor{black}{The first of these is achieved by ensuring that
there is a sufficiently large number of flux configurations such that
there would be many solutions that realize this value. For typical
Calabi-Yau compactifications of IIB string theory this certainly is
the case. The second is achieved by a combination of fluxes and non-perturbative
(NP) effects. The remaining four constraints are dependent on the
particular mechanism of supersymmetry breaking and transmission. }
In this paper we will discuss a theory of supersymmetry breaking that
emerges from the so-called Large Volume Scenario (LVS) of type IIB
string compactifications \citep{Balasubramanian:2005zx}. In addition
to fluxes which stabilize the dilaton and complex structure moduli,
as in the original KKLT model \citep{Kachru:2003aw} non-perturbative
(NP) effects may be used to stabilize the Kaehler moduli. Although
there appears to have been some controversy about the latter in the
literature the issue seems to have been settled - at least when the
cycles in question are not wrapped by branes that support a gauge
theory with chiral fermions. (For a recent comprehensive discussion
and for references to earlier work see \textcolor{black}{\citep{Berglund:2005dm}}).
However for a four-cycle which is wrapped by D7 branes carrying a
chiral gauge theory (such as the MSSM) the situation appears to be
different. In this case it has been argued in \citep{Blumenhagen:2007sm}
that the chirality (of the MSSM) precludes the stabilization of the
four cycle which they wrap by NP effects. The argument depends on
the observation that in a D-brane construction of chiral theory there
would be an anomalous $U(1)$ gauge group, which in effect requires
the presence of charged matter field factors in the NP superpotential
contribution that depends on the relevant four cycle volume. It has
been argued in \citep{Blumenhagen:2007sm} (see also \citep{Conlon:2008wa,Blumenhagen:2009gk})
that such matter fields must have zero vacuum values, so that effectively
this contribution would be absent. This means also that this cycle
would shrink below the string scale (this follows from examining the
associated D-term potential). Effectively the situation becomes similar
to that of having a D3 brane at a singularity (for more discussion
see below). It is not clear to the author that this argument has been
rigorously established, however it appears that requiring a reasonable
phenomenology (in particular that MSSM fields should not acquire vaccum
values at the scales at which the moduli are stabilized) seems to
justify such a scenario. In any case we will take the attitude that
such an outcome yields an interesting set up whose phenomenology is
worth investigating.
After discussing the basic physical inputs and reviewing the pertinent
results of \citep{Conlon:2008wa} and \citep{Blumenhagen:2009gk},
we go on to discuss the classical soft terms that arise from this
class of theories (the details of the calculations are given in Appendix
A). In particular we will find that the classical soft mass squared
is positive definite. This and the classical CC are both highly suppressed
by a power of the (large) volume. Furthermore we show that the suppression
of FCNC effects imply that there is a lower bound on the volume. If
we ignore Weyl anomaly effects then comparison of the classical FCNC
effects with the flavor diagonal classical masses, leads to a large
volume ${\cal V}\gtrsim10^{12}$ in Planck units.
However Weyl anomaly effects (usually called AMSB) changes the phenomenology
of this class of theories. As shown in Appendix B the Weyl anomaly
gives an additional set of terms (calculated by Kaplunovsky and Louis
(KL) \citep{Kaplunovsky:1994fg}) to the gauge coupling function,
leading to a contribution to the gaugino mass that is much larger
than the classical one. This in turn drives the scalar masses by the
mechanism of gaugino mediation \citep{Kaplan:1999ac,Chacko:1999mi}.
However the lower bound on the CYO volume implies a tension between
having TeV scale soft masses (to address the hierarchy problem) and
making the sGoldstino heavy enough to avoid the cosmological modulus
problem.
\section{Generalities}
We follow the notation and discussion of \citep{Conlon:2008wa} and
\citep{Blumenhagen:2009gk}. We also set $M_{P}\equiv(8\pi G_{N})^{-1/2}=2.4\times10^{18}GeV=1$.
The superpotential, Kaehler potential and gauge kinetic function for
the theory under discussion are,
\begin{eqnarray}
W & = & \hat{W}(\Phi)+\mu(\Phi)H_{1}H_{2}+\frac{1}{6}Y_{\alpha\beta\gamma}(\Phi)C^{\alpha}C^{\beta}C^{\gamma}+\ldots,\label{eq:W}\\
K & = & \hat{K}(\Phi,\bar{\Phi})+\tilde{K}_{\alpha\bar{\beta}}(\Phi,\bar{\Phi})C^{\alpha}C^{\bar{\beta}}+[Z(\Phi,\bar{\Phi})H_{1}H_{2}+h.c.]+\ldots\label{eq:K}\\
f_{a} & = & f_{a}(\Phi).\label{eq:f_a}\end{eqnarray}
Here $\Phi=\{\Phi^{A}\}$ and $C^{\alpha}$ are chiral superfields
(including the two Higgs doublets $H_{1,2}$) that correspond to the
moduli and MSSM/GUT fields respectively. Also\begin{eqnarray}
\hat{K} & = & -2\ln\left({\cal V}+\frac{\xi}{2}\left(\frac{(S+\bar{S})}{2}\right)\right)-\ln\left(i\int\Omega\wedge\bar{\Omega}(U,\bar{U})\right)-\ln(S+\bar{S}),\label{eq:hatK}\\
\hat{W} & = & \int G_{3}\wedge\Omega+\sum_{i}A_{i}e^{-a_{i}T^{i}}.\label{eq:hatW}\end{eqnarray}
Here ${\cal V}$ is the volume (in Einstein frame) of the internal
manifold and the $\xi$= -($\chi\zeta(3)/2(2\pi)^{3}$) term is a
correction term that is higher order in the $\alpha'$ expansion.
For typical Calabi-Yau manifolds $\xi\sim O(1)$. $S$ is the axio-dilaton,
$U=\{U^{m}\}$ represents the set of ($m=1,\ldots,h_{21}$) complex
structure moduli and $T^{i}$ ($i=1,\ldots,h_{11}$) are the (complexified)
Kaehler moduli. The type of Calabi-Yau manifolds that we consider
are of the `Swiss cheese' type. In the simplest such manifold consistent
with our requirements the volume may be written as%
\footnote{In order to simplify the notation we have rescaled the moduli by replacing
those used in \citep{Blumenhagen:2009gk} by $\tau_{i}\rightarrow\eta_{i}^{-1}\tau_{i}$
. Correspondingly we have also replaced $a_{i}\rightarrow\eta_{i}a_{i}$.%
}\begin{equation}
{\cal V}=\tau_{b}^{3/2}-\tau_{s}^{3/2}-\tau_{a}^{3/2}.\label{eq:Swisscheese}\end{equation}
In the above the tau's are Kaehler moduli which control the volume
of the four cycles with $\tau_{b}$ effectively determining the overall
size of the CY. While in explicit calculations in the the rest of
the paper, we will use \eqref{eq:Swisscheese} for the sake of simplicity
it should be clear from the discussion that the results would hold
even in a more general CY manifold of this type.
It turns out that in order to realize the Large Volume Scenario (LVS)
with the MSSM located either on a $D3$ brane at a singularity or
a seven-brane wrapping a four cycle on a CY orientifold, the total
number of 2/4 cycles $h_{11}\ge3$. At least one of these cycles ($\tau_{s}$
in the above) must either be wrapped by a Euclidean D3 instanton or
by a stack of seven branes with a condensing gauge group that will
generate non-perturbative effects. As pointed out in \citep{Blumenhagen:2007sm}
the MSSM cannot be located on this cycle - hence the need for (at
least) one more cycle ($\tau_{a}$). It has been argued in \citep{Conlon:2008wa,Blumenhagen:2009gk}
that this cycle shrinks to zero unless stabilized at the string scale
by non-perturbative string effects. In any case the F/D term associated
with the corresponding Kaehler modulus is zero. Let us briefly review
this argument.
The potential for the moduli is (assuming that the minimum would be
at large ${\cal {\cal {\cal V}}}$ and expanding in it) \begin{eqnarray}
V & = & V_{F}+V_{D}.\label{eq:pot1}\\
V_{F} & = & \frac{4}{3}g_{s}(a|A|)^{2}\frac{\sqrt{\tau_{s}}e^{-2a\tau_{s}}}{{\cal V}}-2g_{s}a|AW_{0}|\frac{\tau_{s}e^{-a\tau_{s}}}{{\cal V}}+\frac{3}{8}\frac{\xi|W_{0}|^{2}}{g_{s}^{1/2}{\cal V}^{3}}+\ldots,\label{eq:VF}\\
V_{D} & = & \frac{f}{2}D^{2},\, D=f^{-1}k^{i}K_{i}.\label{eq:VD}\end{eqnarray}
In the above we've used the absence of a non-perturbative superpotential
for the modulus of the MSSM cycle and the fact that the effective
axionic partner of $\tau_{s}$ is stabilized at an odd multiple of
$\pi$ (giving the sign flip of the second term of $V_{F}$). The
phases of $A$ and $W_{0}$ (the flux superpotential) can then be
set to zero without loss of generality \citep{Balasubramanian:2005zx}.
The D-term comes from the anomalous $U(1)$ (with Killing vector field
$k$) living on the MSSM cycle under which the standard model fields
and the modulus $\tau_{a}$ are charged and we have set the matter
fields which are charged under this $U(1)$ to zero following the
arguments of \citep{Blumenhagen:2007sm} \citep{Conlon:2008wa,Blumenhagen:2009gk}.
The $U(1)$ gauge coupling function is linear in $\tau_{a}$ and $S$.
Also one has $K_{a}\sim\tau_{a}^{\alpha}/{\cal V},\,\alpha>0$, and
$D\propto1/{\cal V}$. The F-term potential is minimized at\begin{eqnarray}
e^{-a\tau_{s}} & \simeq & \frac{3}{4}\frac{W_{0}}{aA{\cal V}}\sqrt{\tau_{s}}\left(1-\frac{3}{4a\tau_{s}}\right),\label{eq:sol1}\\
\tau_{s}^{3/2} & \simeq & \frac{\hat{\xi}}{2}(1+\frac{1}{2a\tau_{s}}),\label{eq:sol2}\end{eqnarray}
where we've written $\hat{\xi}=(\frac{S+\bar{S}}{2})^{3/2}\xi$. Note
that extremizing with respect to $\tau_{s}$ gives us an exponentially
large volume and the three displayed terms in $V_{F}$ are all of
order ${\cal V}^{-3}$. This would mean that that at the classical
(negative) minimum found in \citep{Balasubramanian:2005zx}, the contribution
to the F-term potential from the dilaton and complex-structure moduli%
\footnote{At this point we ignore uplifting issues.%
} are zero. Also $V_{D}=0$ since it is positive definite and of order
$1/{\cal V}^{2}$. This would mean that, at least based on classical
considerations, $\tau_{a}\rightarrow0$, so that the standard model
cycle shrinks to zero or at least shrinks below the string scale.
Of course one might need to include all $\alpha'$ corrections in
such a case, but even so the important point here is that both the
D term and hence also the F-term of the MSSM cycle modulus $\tau_{a}$
become negligible at this classical minimum \citep{Blumenhagen:2009gk}.
We also note for future use that \eqref{eq:sol1} implies that\begin{equation}
a\tau_{s}=|\ln m_{3/2}|+O(1),\label{eq:ataus}\end{equation}
and for $m_{3/2}\sim10-100TeV$ this is a number of $O(10)$.
The minimum found in \citep{Balasubramanian:2005zx} is at a negative
value of the (classical) cosmological constant\begin{equation}
V_{0}=-\frac{3\hat{\xi}}{16a\tau_{s}}\frac{m_{3/2}^{2}}{{\cal V}}.\label{eq:V0}\end{equation}
Note that here and in the rest of the paper we will be using the formula
for the gravitino mass\begin{equation}
m_{3/2}^{2}=e^{K}|W|^{2}\sim\frac{|W|^{2}}{{\cal V}^{2}}.\label{eq:gravitinomass}\end{equation}
\section{Classical soft terms}
Let us first compute the classical soft mass term in this model. The
general expression for the squared soft mass in SUGRA is \citep{Kaplunovsky:1993rd}\citep{Brignole:1997dp}\begin{eqnarray}
m_{\alpha\bar{\beta}}^{2} & = & V_{class}|_{0}\tilde{K}_{\alpha\bar{\beta}}+m_{3/2}^{2}\tilde{K}_{\alpha\bar{\beta}}-F^{A}F^{\bar{B}}R_{A\bar{B}\alpha\bar{\beta}}\label{eq:softmass1}\\
& = & V_{class}|_{0}\tilde{K}_{\alpha\bar{\beta}}+m_{3/2}^{2}\tilde{K}_{\alpha\bar{\beta}}-F^{b}F^{\bar{b}}R_{b\bar{b}\alpha\bar{\beta}}\nonumber \\
& & -2ReF^{b}F^{\bar{s}}R_{b\bar{s}\alpha\bar{\beta}}-F^{s}F^{\bar{s}}R_{s\bar{s}\alpha\bar{\beta}}.\label{eq:softmass2}\end{eqnarray}
In the second equality we've used the fact that in the classical vacuum
before uplifting the only source of SUSY breaking are the F-terms
of $T^{b}$ and $T^{s}$. Also we will set $\tau_{a}\rightarrow0$
(and hence also set the corresponding F-term to zero) following the
arguments of \citep{Conlon:2008wa,Blumenhagen:2009gk}. To calculate
\eqref{eq:softmass2} explicitly we need these F-terms as well as
the matter metric $K_{\alpha\bar{\beta}}$. We find (see Appendix
A for details)
\begin{eqnarray}
F^{b} & = & -\tau^{b}\left(2+\frac{3}{8}\frac{\hat{\xi}}{a\tau^{s}}\frac{1}{{\cal V}}+O\left(\frac{1}{(a\tau^{s})^{2}{\cal V}}\right)\right)m_{3/2},\label{eq:Fb}\\
F^{s} & = & -\frac{3}{2}\frac{\tau^{s}}{a\tau^{s}}m_{3/2}(1+O({\cal V}^{-1})).\label{eq:Fs}\end{eqnarray}
For the MSSM on D3 branes the matter metric can be calculated (see
Appendix A) from the formulae for the Kaehler coordinates given in
\citep{Grana:2003ek} (assuming that the formulae given in that reference
remain valid for D3 branes at a singularity). Since we expect D7 branes
on a collapsed cycle to act like D3 branes, these formulae should
be valid in that case too. We have \begin{equation}
K_{\alpha\bar{\beta}}=\frac{c}{{\cal V}+\hat{\xi}/2}(\sqrt{\tau^{b}}\omega_{\alpha\bar{\beta}}^{b}-\sqrt{\tau^{s}}\omega_{\alpha\bar{\beta}}^{s}),\label{eq:mattermetric}\end{equation}
where $\omega^{b(s)}$ is the harmonic $(1,1)$ form associated with
the big(small) modulus evaluated at the position of the D3 brane or
the collapsed cycle wrapped by the D7 brane, and $c$ is an $O(1)$
constant.
Let us pause here for a moment to discuss the validity of (\ref{eq:mattermetric}),
beyond the context of smooth CY orientifolds within which it was derived
(see Appendix A) following the formula for the embedding of $ $D3
branes given in \citep{Grana:2003ek}. The basic argument in Appendix
A depended essentially on the field redefinition that is necessary
to obtain the correct holomorphic coordinates on moduli space, given
in the above reference. This is schematically of the form (see equation
(3.13) of \citep{Grana:2003ek})\begin{equation}
T^{i}+T^{\bar{i}}=2\tau^{i}+2\mu l^{2}i\omega_{\alpha\bar{\beta}}^{i}C^{\alpha}C^{\bar{\beta}}+(CUC+\bar{C}\bar{U}\bar{C}){\rm term}+\ldots\label{eq:Ttau0}\end{equation}
This formula was obtained by comparing the effective action of IIB
string theory with D3 branes located at some (generic smooth) point
on the CY orientifold with the standard supergravity formula for the
kinetic terms of the effective action which are written in terms of
a Kaehler potential. Now unfortunately a similar derivation has not
been given when the branes are located at a singularity. In so far
as one still expects a supergravity description of the low energy
physics, the question is to what extent a formula such as \eqref{eq:Ttau0}
remains valid. For us the essential feature of this formula is the
dependence on the matter fields to quadratic order. In particular
for the above calculation of the dependence of the metric on the Kaehler
moduli, what is relevant is the second term. So our assumption here
is that this structure, i.e. the proportionality to the $1,1$ form
$\omega^{i}$ (of the cycles which are stabilized) is preserved beyond
the original calculation at a generic smooth point. It is hard to
imagine that (apart from the possible modification of the coefficient)
that this structure can be qualitatively modified when the D3 brane
location is at some singularity. Indeed what we are using for our
calculations are just very generic features of the formula for the
metric \eqref{eq:mattermetric}. The numerical value of the normalized
diagonal mass term (see equation \eqref{eq:softmassclassical} will
certainly not change since it depends only on the part of the curvature
on moduli space that is proportional to the matter metric $K_{\alpha\bar{\beta}}$
and is independent on details of the dependence on the two $\omega$'s
. Furthermore the relative numerical coefficients in \eqref{eq:mattermetric}
(provided they are not changed by more than $O(1)$ numbers) is not
going to affect the qualitative features pertaining to the FCNC issue
discussed below either. In fact what is relevant for the latter is
the different dependence on the two moduli that is a feature of the
two terms in the metric, and this is unlikely to change for the metric
extracted from D3 branes at a singularity or D7 branes wrapping a
collapsing cycle.
The dependence on $\omega^{s}$ tends to give FCNC effects and we'll
postpone that discussion to the next section. Dropping that term the
metric is of the form $K_{\alpha\bar{\beta}}=f(\tau^{b},\tau^{s})\omega_{\alpha\bar{\beta}}^{b}$
and the Riemann tensor can be calculated from the formula $R_{i\bar{j}\alpha\bar{\beta}}=(\partial_{i}\partial_{\bar{j}}\ln f)K_{\alpha\bar{\beta}}$.
This gives (after using (\ref{eq:sol2}) and neglecting $O(1/(a\tau^{s})^{2}{\cal V}$)
terms %
\footnote{Typically $\hat{\xi}$ is a number of $O(1)$ and $a\tau^{s}$ is
a number around $30$ so these corrections can be safely ignored.
For instance even if $\hat{\xi}\sim5$ the ratio is expected to be
of around 15-20\% and that is the order of corrections that we would
expect to these formulae. This is certainly does not affect the qualitative
features of the calculations of the classical soft masses.%
})\begin{eqnarray}
R_{b\bar{b}\alpha\bar{\beta}} & = & \frac{1}{4(\tau^{b})^{2}}(1+\frac{15}{16}\frac{\hat{\xi}}{a\tau^{s}{\cal V}})K_{\alpha\bar{\beta}},\label{eq:Rbb}\\
R_{b\bar{s}\alpha\bar{\beta}} & = & -\frac{9}{16}\frac{(\tau^{s})^{1/2}}{(\tau^{b})^{5/2}}K_{\alpha\bar{\beta}},\label{eq:Rbs}\\
R_{s\bar{s}\alpha\bar{\beta}} & = & \frac{3}{16}\frac{(\tau^{s})^{-1/2}}{(\tau^{b})^{3/2}}K_{\alpha\bar{\beta}}.\label{eq:Rss}\end{eqnarray}
Using the above and equations (\ref{eq:Fb})\eqref{eq:Fs} in \eqref{eq:softmass2}
we find%
\footnote{To leading order in the matter fields $K_{\alpha\bar{\beta}}$ is
the same as $\tilde{K}_{\alpha\bar{\beta}}$ so we will not distinguish
between them in the rest of the paper.%
} to leading order in $1/a\tau_{s}{\cal V}$ (see Appendix A for details),\begin{equation}
m_{\alpha\bar{\beta}}^{2}=V_{0}K_{\alpha\bar{\beta}}+\frac{3}{8}\frac{\hat{\xi}}{a\tau^{s}}\frac{m_{3/2}^{2}}{{\cal V}}K_{\alpha\bar{\beta}}=+\frac{3}{16}\frac{\hat{\xi}}{a\tau^{s}}\frac{m_{3/2}^{2}}{{\cal V}}K_{\alpha\bar{\beta}}.\label{eq:softmassclassical}\end{equation}
Note that the second term differs in sign from the result quoted in
\citep{Blumenhagen:2009gk}. The reason is that we have here included
the contribution of the $F^{s}$ terms (and also the sub leading corrections
to $R_{b\bar{b}\alpha\bar{\beta}}$). Thus the classical squared masses
(even after including the contribution of the negative CC) is actually
positive.
\subsection{Uplift issues}
So far we have worked with the LVS minimum, which breaks supersymmetry,
but has a negative cosmological constant. It was argued in \citep{Balasubramanian:2005zx}
that at this minimum one expects the positive definite contributions
to the potential coming from the dilaton and the complex-structure
moduli to be actually zero since they scaled as ${\cal V}^{-2}$ whereas
the negative minimum scaled as ${\cal V}^{-3}$ . However this minimum
needs to be uplifted and one way that could happen is if the dilaton
($S$) and the complex structure moduli ($U^{m}$) acquired F-terms.
Generically one would expect at the new uplifted minimum (given that
$V_{0}\sim-m_{3/2}^{2}/({\cal V}\ln m_{3/2})$)
\begin{eqnarray}
F^{S}\bar{F}^{\bar{S}}K_{S\bar{S}} & \lesssim & \frac{m_{3/2}^{2}}{\ln m_{3/2}{\cal V}},\label{eq:FS}\\
F^{m}\bar{F}^{\bar{n}}K_{m\bar{n}} & \lesssim & \frac{m_{3/2}^{2}}{\ln m_{3/2}{\cal V}}.\label{eq:FU}\end{eqnarray}
Since much of the rest of the discussion focuses on the large modulus
we will often replace $T^{b}\rightarrow T$ and $\tau^{b}\rightarrow\tau$.
We have used above the argument that the seven brane when wrapping
a shrinking 4-cycle should behave like a D3 brane. For the same reason
we expect the leading classical contribution to the gauge coupling
function for both the D3 and D7 cases to be \begin{equation}
f_{a}=S+\kappa T^{a},\label{eq:fS}\end{equation}
(with $\kappa=0$ in the D3 case). Since as argued in \citep{Conlon:2008wa,Blumenhagen:2009gk}
the F-term of the modulus $T^{a}$ corresponding to the shrinking
cycle is vanishingly small, we see that without the uplift contribution
from the dilaton, there would be no (classical) contribution to the
gaugino mass in both the D3 and the D7 brane cases.
After uplift (assuming that $F^{S},F^{U}$take generic values consistent
with \eqref{eq:FS}\eqref{eq:FU}) we have the following expressions
for the gaugino mass $M$, the scalar mass $m$, the $A$ term, the
effective $\mu$ term %
\footnote{Assuming that the supersymmetric one is zero as is the case for the
D3 branes \citep{Grana:2003ek}.%
} and $B$ terms (see for example \citep{Kaplunovsky:1993rd} for definitions
and general formulae and Appendix A):\begin{eqnarray}
M_{a} & = & \frac{F^{i}\partial_{i}f_{a}}{2f_{a}}=\frac{F^{S}}{2S}\lesssim O\left(\frac{m_{3/2}}{\sqrt{\ln m_{3/2}{\cal V}}}\right),\label{eq:Mclassical}\\
m_{\alpha\bar{\beta}}^{2} & = & (m_{3/2}^{2}K_{\alpha\bar{\beta}}-F^{i}F^{\bar{j}}R_{i\bar{j}\alpha\bar{\beta}})=(O\left(\frac{m_{3/2}^{2}}{\ln m_{3/2}{\cal V}}\right))K_{\alpha\bar{\beta}}+\ldots,\label{eq:mclassical}\\
A_{\alpha\beta\gamma} & = & e^{K/2}F^{i}D_{i}y_{\alpha\beta\gamma}\lesssim O(\frac{m_{3/2}}{{\cal V}})y_{\alpha\beta\gamma},\label{eq:Aclassical}\\
\mu & \sim & B\mu/\mu\lesssim O\left(\sqrt{h_{21}}\frac{m_{3/2}}{\sqrt{\ln m_{3/2}{\cal V}}}\right).\label{eq:Bclassical}\end{eqnarray}
Note that in the last equation $h_{21}$ is the number of complex
structure moduli.
\subsection{Classical FCNC effects\label{sub:Classical-FCNC-effects}}
In computing the soft mass using \eqref{eq:mattermetric} we ignored
the second term inside the parenthesis. Let us now compute its contribution
(for details see Appendix A). To do so we must evaluate the Riemann
tensor. The leading contribution comes from the sectional curvature
\begin{eqnarray}
R_{T\bar{T}\alpha\bar{\beta}} & = & \partial_{T}\partial_{\bar{T}}K_{\alpha\bar{\beta}}-K^{\gamma\bar{\delta}}\partial_{T}K_{\alpha\bar{\delta}}\partial_{\bar{T}}K_{\gamma\bar{\beta}}+O(C)\nonumber \\
& = & \frac{1}{3}K_{T\bar{T}}c[\frac{\omega_{b}}{\tau_{b}}-\frac{7}{4}\frac{\omega_{s}}{\tau_{b}}\sqrt{\frac{\tau_{s}}{\tau_{b}}}]_{\alpha\bar{\beta}}+O(C).\label{eq:R}\end{eqnarray}
The problem is that this is not proportional to $K_{\alpha\bar{\beta}}$
- see \eqref{eq:mattermetric}. The $\omega^{s}$ dependence in \eqref{eq:R}
gives (from the third term on the RHS of \eqref{eq:softmass2}) an
additional term to the expression in \eqref{eq:softmassclassical},
i.e. \begin{eqnarray}
m_{\alpha\bar{\beta}}^{2} & = & m_{3/2}^{2}K_{\alpha\bar{\beta}}-F^{T}F^{\bar{T}}R_{T\bar{T}\alpha\bar{\beta}}+\ldots\nonumber \\
& = & \frac{3}{16}\hat{\xi}\frac{m_{3/2}^{2}}{\ln m_{3/2}{\cal V}}K_{\alpha\bar{\beta}}+m_{3/2}^{2}\frac{3}{4}\sqrt{\frac{\tau_{s}}{\tau_{b}}}K'_{\alpha\bar{\beta}},\label{eq:softFCNC}\\
& \equiv & (m^{2}\delta_{\alpha}^{\gamma}+\Delta m_{\alpha}^{2\gamma})K_{\gamma\bar{\beta}}.\end{eqnarray}
Here $K'_{\alpha\bar{\beta}}\equiv c\omega_{\alpha\bar{\beta}}^{s}/\tau_{b}$
and is not proportional to $K_{\alpha\bar{\beta}}$ so $\Delta m_{\alpha}^{2\gamma}=m_{3/2}^{2}\frac{3}{4}\sqrt{\frac{\tau_{s}}{\tau_{b}}}(K'K^{-1})_{\alpha}^{\gamma}$
is not a diagonal matrix. Also $m^{2}=\frac{3}{16}\hat{\xi}\frac{m_{3/2}^{2}}{\ln m_{3/2}{\cal V}}$
. If the two harmonic one-one forms $\omega^{b}$ and $\omega^{s}$
evaluated at the position of the D3 branes (or the collapsed cycle
wrapped by the D7 branes) are of the same order of magnitude then
the dominant contribution is a flavor violating one.
Clearly this would be a phenomenological disaster since at least for
the first two generations the flavor violating non-diagonal contributions
to the squared soft masses should be suppressed relative to the flavor
conserving ones. The relevant bound may be expressed by the following
relation (see for example \citep{Luty:2005sn}) \begin{equation}
\frac{\Delta m^{2}}{m^{2}}\lesssim10^{-3}\frac{m}{500GeV}.\label{eq:phenoconstraint}\end{equation}
So if we want soft masses $m\lesssim1TeV$ in order to address the
hierarchy problem the FCNC effect must be suppressed by at least a
factor $10^{-3}$. Then the following alternatives may be pursued.
\begin{itemize}
\item To be consistent with \eqref{eq:phenoconstraint} we need $\omega^{s}\lesssim10^{-3}\frac{1}{\ln m_{3/2}\tau^{b}}\omega^{b}$
at the MSSM point. We can get this as follows. The small cycle may
be regarded as a blow up (by a $P^{2}$) of a singularity. Then (at
least in a non-compact Calabi-Yau) it has been shown in \citep{Lutken:1987ny}
that the corresponding harmonic 1,1 form falls off as $R^{-6}$ where
$R$ is the distance to the singularity%
\footnote{We thank Joe Conlon for suggesting this and bringing reference \citep{Lutken:1987ny}
to our attention.%
}. It is quite plausible that this behavior applies to the small cycle
of a compact Calabi-Yau. Then indeed the desired suppression can be
obtained if the distance $R$ is identified with the location of the
$D3$ brane (or the MSSM collapsed cycle) and is of the order of ${\cal V}^{1/6}\sim\tau^{1/4}$.
The FCNC suppression is then obtained provided ${\cal V}\gtrsim10^{12}$.
This would imply a string scale that is well below the Planck scale
since $M_{string}\sim M_{P}/\sqrt{{\cal V}}\lesssim10^{12}GeV$$ $.
Any hope of getting a GUT scenario within LVS is then eliminated,
but we would still have a viable intermediate scale phenomenology.
This of course is consistent with the usual LVS scenario as discussed
in \citep{Conlon:2006wz} and references therein. Nevertheless it
should be stressed that this conclusion holds only if we ignore quantum
- in particular Weyl anomaly and gaugino mediation - contributions
to scalar masses (see next section).
\item The alternative within LVS is to have a very heavy soft mass scale
$\gtrsim10^{3}TeV$. In this case the SUSY solution to the hierarchy
problem is much more fine-tuned than in the previous one. Nevertheless
it is interesting to note that the constraint on the volume is now
much weaker and reads ${\cal V}\gtrsim10^{3}$. Note that in this
case one might hope that the gaugino masses are still at the TeV scale
(this could have been the case according to \eqref{eq:Mclassical}
if the dilaton $ $F-term is not responsible for the uplift and the
gaugino mass arises from loop corrections to the gauge coupling function).
This scenario is essentially that of split supersymmetry. However
again the Weyl anomaly and gaugino mediation effects will modify this.
\item The third possibility is to consider compactifications with just one
Kaehler modulus. In this case an LVS solution is not possible. But
one could by including the $\alpha'$ corrections and race track terms
find an intermediate volume (${\cal V}\sim10^{3-4}$) solution. Of
course in this case $W_{0}$ the flux superpotential would have to
be fine tuned to extremely low values in order to get TeV scale soft
masses. This scenario has been discussed in \citep{deAlwis:2008kt}.
\end{itemize}
In the following we will pursue only the first of these alternatives.
\section{Quantum Effects}
\subsection{String loop effects}
String loop contributions to the classical contributions considered
in the previous section can be estimated. This can be done either
from an effective field theory calculation as in \citep{Choi:1997de,deAlwis:2008kt}
or from arguments based on calculations in toy models in string theory
\citep{Blumenhagen:2009gk}. They agree if certain cancellations take
place. This is essential if the original LVS minimum is not to be
destabilized. In this case these do not give a significant correction
to the classical soft terms discussed above.
\subsection{Weyl Anomaly and Gaugino masses}
Significant corrections to the soft terms can arise from Weyl anomaly
contributions \citep{Kaplunovsky:1994fg,Randall:1998uk,Giudice:1998xp,Bagger:1999rd,Dine:2007me,deAlwis:2008aq}.
These are independent of the size of the compactification once the
value of the gravitino mass is chosen. In particular the gaugino masses
are given by the expressions (see Appendix B for a discussion)\begin{equation}
M_{a}=-b_{a}\left(\frac{\alpha_{a}}{4\pi}\right)m_{3/2}.\label{eq:AMSBgaugino}\end{equation}
Here $a=1,2,3$ index the three standard model gauge groups - respectively
$U(1),\, SU(2),\, SU(3)$, with couplings $\alpha_{a}=g_{a}^{2}/4\pi$.
These expressions when evaluated at the UV scale (assumed to be at
or close to the unification scale so that $\alpha_{a}\sim\alpha_{GUT}\sim1/25$)
give \begin{equation}
M_{1}=\frac{33}{5}\frac{\alpha_{GUT}}{4\pi}m_{3/2},\, M_{2}=\frac{\alpha_{GUT}}{4\pi}m_{3/2},\, M_{3}=-3\frac{\alpha_{GUT}}{4\pi}m_{3/2}.\label{eq:gauginomass}\end{equation}
These should be treated as initial values for the RG evolution down
to the MSSM scale - and numerically have values $O(10^{-2}-10^{-3})m_{3/2}$.
The important point here is that these values are larger than the
classical contribution which gave gaugino masses $\lesssim O(m_{3/2}/\ln m_{3/2}\sqrt{{\cal V}})$
(see \eqref{eq:Mclassical}), unless ${\cal V}\lesssim10^{2}$ which
is far too small a volume for an LVS scenario because of the FCNC
problem.
\subsection{Gaugino Mediation}
Now using the above values as boundary conditions for the RG evolution
down to the MSSM scale scalar masses are generated by the gaugino
mediation mechanism \citep{Kaplan:1999ac,Chacko:1999mi}. As shown
in Appendix B we get,
\begin{equation}
m_{1}^{2}\sim m_{2}^{2}\sim10^{-6}m_{3/2}^{2},\, m_{3}^{2}\sim10^{-4}m_{3/2}^{2}.\label{eq:AMSBscalar2}\end{equation}
This is to be compared to the (diagonal) classical contribution \eqref{eq:softFCNC}
$m^{2}\sim m_{3/2}^{2}1/(\ln m_{3/2}{\cal V})$ which would dominate
over (\ref{eq:AMSBscalar2}) only if ${\cal V}\lesssim10^{3}$. But
in that case the flavor non-diagonal contribution (see \eqref{eq:softFCNC})
would be far too large (even assuming the suppression by a factor
${\cal V}$ of the 1,1 form $\omega^{s}$ that we argued for in the
discussion after \eqref{eq:softFCNC}).
Thus we require that \eqref{eq:AMSBscalar2} dominates the FCNC contribution
in \eqref{eq:softFCNC} which is $\Delta m^{2}\sim K'K^{-1}m_{3/2}^{2}/\sqrt{\tau^{b}}\sim m_{3/2}^{2}/(\tau^{b})^{2}$
(see \eqref{eq:softFCNC}\eqref{eq:phenoconstraint}) , by at least
a factor of $10^{3}$. i.e. we need to have \begin{equation}
\frac{\Delta m^{2}}{m_{3}^{2}}\sim\frac{10^{4}}{(\tau^{b})^{2}}\lesssim10^{-3}.\label{eq:FCNC2}\end{equation}
This gives \[
{\cal V}\sim(\tau^{b})^{3/2}\gtrsim10^{5}.\]
This would yield an effective string scale of $M_{string}\lesssim1/\sqrt{{\cal V}}\sim10^{-2.5}M_{P}\sim10^{15.5}GeV$
which may just accommodate a GUT scenario.
Be that as it may, to have a SUSY solution to the hierarchy problem
in a GUT scenario clearly needs fine tuning of the flux superpotential.
From \eqref{eq:AMSBscalar2} we see that to get $TeV$ scale squark
masses $m_{3}\sim1TeV$, the gravitino mass must be $m_{3/2}\sim\frac{|W|}{{\cal V}}\sim10^{2}TeV$.
For $W\sim O(1)$ this gives ${\cal V}\sim10^{13}$ (well above our
lower limit) and a string scale $M_{string}\sim M_{P}/\sqrt{{\cal V}}\sim10^{12}$
which is certainly well below the GUT scale. To get a scale close
to the GUT scale would need a highly fine-tuned $W\sim10^{-8}$.
\section{Conclusions\label{sec:Conclusions}}
In this paper we have discussed type IIB compactifications on ``Swiss
Cheese'' type manifolds with the MSSM either on a D3 brane(s) at
a singularity or on a stack of D7 branes which wrap a 4-cycle. The
conflict between chirality and the generation of a non-perturbative
superpotential leads to the conclusion that the latter (MSSM) cycle
actually collapses below the string scale (or to a singularity) and
its modulus does not contribute to SUSY breaking, so that effectively
one can ignore it. The following results were derived in this set
up.
\begin{enumerate}
\item The (classical) diagonal soft mass is given by $m_{soft}=\sqrt{\frac{3\hat{\xi}}{16\ln m_{3/2}{\cal V}}}m_{3/2}$.
However there is also a flavor violating contribution to the mass
matrix which needs to be suppressed. Assuming that the 1,1 form associated
with the small cycle falls off with distance, as in the non-compact
case studied in \citep{Lutken:1987ny}, we get a lower bound ${\cal V}\gtrsim10^{12}$
in order that there is sufficient suppression of FCNC effects relative
to the classical soft mass.
\item However the Weyl anomaly contribution to the gaugino mass actually
dominates the classical contribution. Furthermore this generates,
through the mechanism of gaugino mediation, a quantum contribution
at the MSSM scale to the scalar soft masses, that is also larger than
the classical contribution. Hence the scenario of item 1. above gets
modified. These LVS models appear to give a string theoretic construction
of a sequestered situation as envisaged in \citep{Randall:1998uk}%
\footnote{To truly establish sequestering one would of course need to demonstrate
that the couplings of the bulk and brane fields are suppressed at
the quantum level as well. This would require a calculation of string
loop effects beyond that considered in the literature. For some conjectures
regarding this see \citep{Conlon:2008wa} and references therein.
It should also be noted that these assumptions are in agreement with
the effective field theory estimates given in \citep{deAlwis:2008kt}.
Essentially the point is that the quantum corrections are also expected
to be suppressed by large volume factors, so that it seems unlikely
that these corrections could significantly change the classical results.
However it would be nice if this could be supported by a detailed
calculation.%
}. The suppression of FCNC effects now only gives the weaker constraint
${\cal V}\gtrsim10^{5}$ (see discussion after \eqref{eq:FCNC2}).
\item The gravitino mass in this scenario is $m_{3/2}\sim10^{2}TeV$ if
we wish to have TeV scale SUSY breaking MSSM soft terms. The gravitino
gives no cosmological problems but the sGoldstino (light modulus)
mass would be $m_{mod}\sim m_{3/2}/\sqrt{{\cal V}}<1TeV$ so this
scenario appears to suffer from the cosmological modulus problem.
Again the lower bound on ${\cal V}$ is compatible with this estimate
of the gravitino mass only if $W_{0}$ is highly fine-tuned to values
around $10^{-7}$. If $W_{0}\sim O(1)$ then ${\cal V}\sim10^{13}$
, we have an intermediate string scale, no possibility of Grand Unification,
and a very light modulus $\sim100MeV$! Furthermore there would be
a serious $\mu$ problem since as we see from \eqref{eq:Bclassical}
$\mu$ is highly suppressed for large volumes.
\item The resolution of the cosmological modulus problem necessitates raising
the gravitino mass to $m_{3/2}\sim10^{3}TeV$. However the Weyl anomaly
generated gaugino mass is now $\sim10TeV$ with gaugino mediated soft
masses which are at least a few TeV. We may take the volume ${\cal V}$
close to its minimum possible value $10^{3}$, consistent with the
now somewhat less stringent FCNC constraint \eqref{eq:phenoconstraint}
(since the squark mass is higher). Then we have a somewhat less fine-tuned
$W_{0}\sim10^{-6}$ and a light modulus $\sim10TeV$. But of course
addressing the hierarchy problem would require somewhat more fine-tuning.
If the cosmological modulus problem is taken seriously then this should
be considered the preferred solution within this class of models.
\end{enumerate}
The LVS compactification of type IIB string theory thus gives us an
appealing class of models of supersymmetry breaking and transmission.
It is consistent with all theoretical constraints and satisfies phenomenological
constraints (and in the case of 4 above cosmological ones as well)
. It is predictive and is expected to be ultra-violet complete. As
argued in the introduction a bottom up approach ignores the necessary
embedding of the low energy (or intermediate scale) hidden sector
dynamics in the larger framework of a string theory. The essential
point here is that one cannot ignore the dynamics of the string theory
moduli and focus on some additional sector (as is usually done in
GMSB) since in SUGRA, both open string fields and closed string moduli
are coupled together in a highly non-linear fashion. This class of
models, with LVS compactifications, generically break supersymmetry
and provide, within a string theory context, a very compelling and
phenomenologically viable scenario in which to discuss MSSM SUSY breaking.
The detailed phenomenology of these models will be discussed elsewhere
\citep{Baer:2009tp}.
\section{Acknowledgements}
I'm very grateful to Fernando Quevedo for collaboration in the early
stages of this work and for extensive discussions on LVS models. Special
thanks are also due to Joe Conlon for discussions and for drawing
my attention to \citep{Lutken:1987ny}. I also wish to acknowledge
discussions with Cliff Burgess, Oliver DeWolfe and Matthew Headrick
. This research is supported in part by the United States Department
of Energy under grant DE-FG02-91-ER-40672.
\section*{Appendix A}
We give here some details of the calculation of the F-terms of the
two Kaehler moduli $\tau^{b},\tau^{s}$. Useful formulae for calculating
the metrics inverse metrics and Riemann tensors for Kaehler potentials
of the form $K=-n\ln Y$are given in Appendix A of \citep{Covi:2008ea}.
In our case $n=2$ and $Y={\cal V}+\frac{\hat{\xi}}{2}$ where ${\cal V}=(\tau^{b})^{3/2}-(\tau^{s})^{3/2}$
where $\tau^{i}=\frac{1}{2}(T^{i}+\bar{T}^{\bar{i}})$ with $T^{i}$
being the holomorphic Kaehler moduli. As usual $K_{i}\equiv\partial_{T^{i}}K,$
etc. We find \begin{equation}
K^{i\bar{j}}K_{\bar{j}}\sim-2\tau^{i}-\frac{3}{2}\hat{\xi}\frac{\tau^{i}}{{\cal V}},\label{eq:KijbarKjbar}\end{equation}
and \begin{eqnarray}
K^{s\bar{s}} & = & -2({\cal V}+\frac{\hat{\xi}}{2})(-\frac{4}{3}(\tau^{s})^{1/2}+4(\tau^{s})^{2}+O({\cal V}^{-1})),\label{eq:Kssbar}\\
K^{b\bar{s}} & = & 4\tau^{b}\tau^{s}(1+O({\cal V}^{-1})).\label{eq:Kbsbar}\end{eqnarray}
Also note that the stabilization of the axion corresponding to the
small modulus results in a sign flip (i.e. the axion takes a value
that is an odd multiple of $\pi$ after choosing without loss of generality
the phases of $W_{0},A$ to be zero) so that effectively the superpotential
is $W=W_{0}+Ae^{-aT^{s}}=W_{0}-Ae^{-a\tau^{s}}$ \citep{Balasubramanian:2005zx}.
Then we find (note that $F^{i}\equiv e^{K/2}K^{i\bar{j}}(\partial_{\bar{j}}\bar{W}+K_{\bar{j}}\bar{W})$)
using the solution \eqref{eq:sol1}\eqref{eq:sol2}, \begin{equation}
F^{b}=-\tau^{b}(2+\frac{3}{2}.\frac{\hat{\xi}}{4a\tau^{s}}\frac{1}{{\cal V}})m_{3/2},\label{eq:Fbapp}\end{equation}
\begin{equation}
F^{s}=-\frac{3\tau^{s}}{2a\tau^{s}}m_{3/2}(1+O({\cal V}^{-1})).\label{eq:Fsapp}\end{equation}
($F^{b}$ was also calculated in \citep{Blumenhagen:2009gk}).
Here we compute the matter metric for the matter located on a D3 brane
which sits at a point in the internal Calabi-Yau space. We expect
that a similar formula will be valid for matter on a stack of D7 branes
wrapping a collapsing four cycle.
The holomorphic Kaehler moduli $T^{i}$ are related to the moduli
$\tau^{i}$ (in terms of which the volume is given by \eqref{eq:Swisscheese})
by (see equation (3.13) of \citep{Grana:2003ek})\begin{equation}
T^{i}+T^{\bar{i}}=2\tau^{i}+2\mu l^{2}i\omega_{\alpha\bar{\beta}}^{i}C^{\alpha}C^{\bar{\beta}}+(CUC+\bar{C}\bar{U}\bar{C}){\rm term}+\ldots\label{eq:Ttau}\end{equation}
to linear order in the complex structure moduli $U$. Here $\omega^{i}$
are the harmonic 1,1 forms on the CY orientifold evaluated at the
position of the D3 brane, $C^{\alpha}$ are the matter fields on the
D3 brane, $l$ is the axionic partner of the dilaton ($S\equiv e^{-\phi}-il$),
and $\mu$ is the tension of the D3 brane. Writing ${\cal V}_{i}\equiv\partial{\cal V}/\partial\tau^{i}$,\[
K_{\alpha}\equiv\frac{\partial K}{\partial C^{\alpha}}|_{T,U,S}=-2\frac{{\cal V}_{i}}{Y}\frac{\partial\tau^{i}}{\partial C^{\alpha}}|_{T,U,S}\]
Differentiating \eqref{eq:Ttau} with respect to $C$ keeping the
moduli $T,U,S$ fixed we have \begin{eqnarray*}
0 & = & 2\frac{\partial\tau^{i}}{\partial C^{\alpha}}+2\mu(i\omega_{\alpha\bar{\beta}}^{i})C^{\bar{\beta}}+O(UC)\\
0 & = & 2\frac{\partial^{2}\tau^{i}}{\partial C^{\alpha}\partial C^{\bar{\beta}}}+2\mu(i\omega_{\alpha\bar{\beta}}^{i})\end{eqnarray*}
Hence we have\begin{equation}
K_{\alpha\bar{\beta}}=2\mu i\omega_{\alpha\bar{\beta}}^{i}\frac{{\cal V}_{i}}{Y}+O(C^{2})=\frac{3\mu}{Y}(i\omega_{\alpha\bar{\beta}}^{b}\sqrt{\tau^{b}}-i\omega_{\alpha\bar{\beta}}^{s}\sqrt{\tau^{s}})+O(C^{2}),\label{eq:mattmetapp}\end{equation}
where in the last step we specialized to the Swiss cheese CY manifold
\eqref{eq:Swisscheese} (with $\tau^{a}\rightarrow0$). Note also
for future reference that \begin{equation}
Z=K_{H_{1}H_{2}}\sim\frac{{\cal V}_{i}}{Y}\frac{\partial^{2}\tau^{i}}{\partial H_{1}\partial H_{2}}.\label{eq:Z}\end{equation}
To compute the soft masses (and other soft terms) we need the sectional
curvatures.
First let us ignore the second term in parenthesis in \eqref{eq:mattmetapp}.
In this case the matter metric is conformal to the 1,1 harmonic form
of the large modulus $K_{\alpha\bar{\beta}}=f(\tau^{b},\tau^{s})i\omega_{\alpha\bar{\beta}}^{b},\, f\equiv3\mu\sqrt{\tau^{b}}/Y$
and the relevant components of the Riemann tensor are easily computed
using the formula $R_{i\bar{j}\alpha\bar{\beta}}=\partial_{i}\partial_{\bar{j}}\ln fK_{\alpha\bar{\beta}}$.
This gives \begin{eqnarray}
R_{b\bar{b}\alpha\bar{\beta}} & = & \frac{1}{4(\tau^{b})^{2}}(1+\frac{15}{16}\frac{\hat{\xi}}{a\tau^{s}{\cal V}})K_{\alpha\bar{\beta}},\label{eq:Rbbapp}\\
R_{b\bar{s}\alpha\bar{\beta}} & = & -\frac{9}{16}\frac{(\tau^{s})^{1/2}}{(\tau^{b})^{5/2}}K_{\alpha\bar{\beta}},\label{eq:Rbsapp}\\
R_{s\bar{s}\alpha\bar{\beta}} & = & \frac{3}{16}\frac{(\tau^{s})^{-1/2}}{(\tau^{b})^{3/2}}K_{\alpha\bar{\beta}}.\label{eq:Rssapp}\end{eqnarray}
For future use we will recalculate $R_{b\bar{b}\alpha\bar{\beta}}$
keeping both terms in the parenthesis in \eqref{eq:mattmetapp} (but
ignoring the $\hat{\xi}$ dependence for simplicity), and using the
general formula for the Riemann tensor in Kaehler geometry\begin{eqnarray}
R_{b\bar{b}\alpha\bar{\beta}} & = & \partial_{b}\partial_{\bar{b}}K_{\alpha\bar{\beta}}-K^{\gamma\bar{\delta}}\partial_{\bar{b}}K_{\alpha\bar{\delta}}\partial_{b}K_{\gamma\bar{\beta}},\nonumber \\
& = & \frac{3\mu}{4(\tau^{b})^{3}}(i\omega_{\alpha\bar{\beta}}^{b}-\frac{7}{4}\sqrt{\frac{\tau^{s}}{\tau^{b}}}i\omega_{\alpha\bar{\beta}}^{s}),\label{eq:Rbbnew}\\
& = & \frac{1}{3}K_{b\bar{b}}(K_{\alpha\bar{\beta}}-K'_{\alpha\bar{\beta}}\sqrt{\frac{\tau^{s}}{\tau^{b}}}).\label{eq:Rbbnew2}\end{eqnarray}
Here we have defined \[
K'_{\alpha\bar{\beta}}\equiv\frac{9\mu}{4}\frac{i\omega_{\alpha\bar{\beta}}^{s}}{\tau^{b}},\]
to be compared with $K_{\alpha\bar{\beta}}\sim\frac{3\mu}{\tau^{b}}i\omega_{\alpha\bar{\beta}}^{b}$.
Let us use the above results to calculate soft masses. These are given
by\begin{eqnarray*}
m_{\alpha\bar{\beta}}^{2} & = & V_{class}|_{0}\tilde{K}_{\alpha\bar{\beta}}+m_{3/2}^{2}\tilde{K}_{\alpha\bar{\beta}}-F^{b}F^{\bar{b}}R_{b\bar{b}\alpha\bar{\beta}}\\
& & -2ReF^{b}F^{\bar{s}}R_{b\bar{s}\alpha\bar{\beta}}-F^{s}F^{\bar{s}}R_{s\bar{s}\alpha\bar{\beta}}.\end{eqnarray*}
First let us compute the flavor diagonal part - i.e. we will ignore
the contribution to the matter metric from the harmonic form $\omega^{s}$.
We have \begin{eqnarray*}
F^{b}\bar{F}^{\bar{b}}R_{b\bar{b}\alpha\bar{\beta}} & = & m_{3/2}^{2}(1+\frac{21}{16}\frac{\hat{\xi}}{a\tau^{s}{\cal V}})K_{\alpha\bar{\beta}}\\
2F^{b}\bar{F}^{\bar{s}}R_{b\bar{s}\alpha\bar{\beta}} & = & -\frac{27}{8}\frac{\hat{\xi}}{2a\tau^{s}{\cal V}}m_{3/2}^{2}K_{\alpha\bar{\beta}}\\
F^{s}\bar{F}^{\bar{s}}R_{s\bar{s}\alpha\bar{\beta}} & = & \frac{27}{64}\frac{\hat{\xi}}{2(a\tau^{s})^{2}{\cal V}}m_{3/2}^{2}K_{\alpha\bar{\beta}}\end{eqnarray*}
So we have\[
m_{\alpha\bar{\beta}}^{2}=(V_{class}|_{0}+\frac{3}{8}\frac{\hat{\xi}}{a\tau^{s}{\cal V}}m_{3/2}^{2})\tilde{K}_{\alpha\bar{\beta}}+{\rm flavor\, non-diagonal}\]
Now let us compute the flavor non-diagonal piece. The leading contribution
comes from the extra contribution to $F^{b}\bar{F}^{\bar{b}}R_{b\bar{b}\alpha\bar{\beta}}$
coming from the term proportional to $K'_{\alpha\bar{\beta}}$ in
the expression for the Riemann tensor \eqref{eq:Rbbnew2}. Using $F^{b}\bar{F}^{\bar{b}}K_{b\bar{b}}\sim3m_{3/2}^{2}$
we find this to be\[
\Delta m_{\alpha\bar{\beta}}^{2}=\frac{3}{4}\sqrt{\frac{\tau^{s}}{\tau^{b}}}m_{3/2}^{2}K'_{\alpha\bar{\beta}}.\]
So unless $K'_{\alpha\bar{\beta}}$ is strongly suppressed relative
to $ $$\tilde{K}_{\alpha\bar{\beta}}$ (which means in effect the
suppression at the position of the D3brane(s) of $\omega^{s}$ compared
to $\omega^{b}$) there would be a serious FCNC problem.
There is also an issue with the $\mu$ term (see \eqref{eq:Bclassical})
that needs to be discussed. If the uplift comes mainly from giving
F-terms to the complex structure moduli then from \eqref{eq:FU} it
follows that the $|F_{m}|\lesssim m_{3/2}/(\sqrt{h_{21}}\sqrt{\ln m_{3/2}{\cal V}})$
since in a basis in which the relevant metric is diagonal there are
effectively $h_{21}$ terms in the sum. Now the expression for the
effective $\mu$ term has a contribution $F^{m}\partial_{m}Z$. From
\eqref{eq:Z} we see that this gives the factor $\sqrt{h_{21}}$ in
the upper bound \eqref{eq:Bclassical}.
\section*{Appendix B}
The physical gauge coupling (at the cutoff scale/GUT scale $\Lambda$)
can be written in the following form \citep{Kaplunovsky:1994fg}\citep{ArkaniHamed:1997mj}\citep{deAlwis:2008aq}
\begin{equation}
H_{i}=f_{i}-\frac{3c_{i}}{8\pi^{2}}\tau-\sum_{r}\frac{T_{i}(r)}{4\pi^{2}}\tau_{r}-\frac{T(G_{i})}{4\pi^{2}}\tau_{i}.\label{eq:Hphys}\end{equation}
Here $G_{i}$ is the gauge group and $c_{i}=T(G_{i})-\sum_{r}T_{i}(r)$
\footnote{Note that this is the negative of the coefficient defined in \citep{Kaplunovsky:1994fg}.%
} where $T(G_{i}),\, T_{i}(r)$ are respectively the trace of a squared
generator in the adjoint and the matter representations of the group.
The first term is the classical gauge coupling (say at the scale $\Lambda$).
The second arises from the Weyl anomaly which arises when one does
a chiral rotation from the supergravity frame to the Kaehler-Einstein
frame. The third term comes from an anomaly associated with the field
redefinition $C_{\alpha}\rightarrow e^{\tau_{Z}}C_{\alpha}$ needed
to get canonical normalization of the MSSM fields and the last term
arises from the redefinition of the gauge field pre-potential $V_{i}\rightarrow e^{(\tau_{i}+\bar{\tau}_{i})/2}V_{i}$.
The chiral fields $\tau,\tau_{r},\tau_{i}$ are fixed by the relations\begin{eqnarray}
\tau+\bar{\tau} & = & \frac{1}{3}K|_{{\rm harm}},\label{eq:phi}\\
\tau_{r}+\bar{\tau}_{r} & = & \ln\det\tilde{K}_{\alpha\bar{\beta}}^{(r)},\label{eq:tauZ}\\
\exp[-(\tau_{i}+\bar{\tau}_{i})]|_{{\rm harm}} & = & \frac{1}{2}(H_{i}+\bar{H}_{i}),\label{eq:tauV}\end{eqnarray}
It should be emphasized that (as observed by Kaplunovsky and Louis)
the first relation is precisely the one which accomplishes the transformations
and field redefinitions that are needed to get to the Einstein-Kaehler
frame. These are the same transformations that are done in for example
Wess and Bagger \citep{Wess:1992cp} in component form to get to the
final SUGRA action displayed in Appendix G of that work. The last
relation comes from supersymmetrizing the lowest component relation
$\exp[-(\tau_{i}+\bar{\tau}_{i})]|_{0}=1/g_{phys}^{(i)2}\equiv\Re H_{i}|_{0}$.
To get the physical coupling function at a low scale $\mu$ ( $\ll\Lambda$)
we need to evaluate the right hand sides of \eqref{eq:phi}\eqref{eq:tauZ}\eqref{eq:tauV}
at the scale $\mu$ and make the replacement\begin{equation}
f_{i}\rightarrow f_{i}-\frac{b_{i}}{16\pi^{2}}\ln\frac{\Lambda}{\mu}.\label{eq:fmu}\end{equation}
Here $b_{i}=3T(G_{i})-\sum_{r}T_{i}(r)$ and we used the fact that
$f_{i}$ is only renormalized at one-loop.
Projecting the lowest component of \eqref{eq:Hphys} (with the replacement\eqref{eq:fmu})
and using the relations \eqref{eq:phi}\eqref{eq:tauZ}\eqref{eq:tauV}
gives us (the integrated form of) the NSVZ relation (with SUGRA and
Weyl anomaly corrections)\begin{eqnarray}
\frac{1}{g_{{\rm phys}}^{(i)2}} & = & \Re f_{i}-\frac{b_{i}}{16\pi^{2}}\ln\frac{\Lambda}{\mu}-\frac{c_{i}}{16\pi^{2}}K|_{0}\nonumber \\
& & -\sum_{r}\frac{T_{i}(r)}{8\pi^{2}}\ln\det\tilde{K}_{\alpha\bar{\beta}}^{(r)}|_{0}+\frac{T(G_{i})}{8\pi^{2}}\ln\frac{1}{g_{{\rm phys}}^{(i)2}}.\label{eq:gphys}\end{eqnarray}
On the other hand projecting the F-term of \eqref{eq:Hphys} gives
(after solving for $M_{i}/g_{{\rm phys}}^{(i)2}$)\begin{eqnarray}
\frac{M_{i}}{g_{{\rm phys}}^{(i)2}} & = & \frac{1}{2}(F^{A}\partial_{A}f_{i}-\frac{c_{i}}{8\pi^{2}}F^{A}K_{A}-\sum_{r}\frac{T_{i}(r)}{4\pi^{2}}F^{A}\partial_{A}\ln\det\tilde{K}_{\alpha\bar{\beta}}^{(r)})\nonumber \\
& & \times(1-\frac{T(G_{i})}{8\pi^{2}}g_{{\rm {\rm phys}}}^{(i)2})^{-1}.\label{eq:m/g2}\end{eqnarray}
Let us evaluate this for the theory described in this paper. Since
the gauge coupling function $f_{a}=S$ whose F-term is highly suppressed,
the classical contribution can be ignored compared to the AMSB one
coming from the last two terms in the first parenthesis. To one-loop
(keeping only the large modulus ($T^{b}\equiv T$) contribution since
the other contributions are highly suppressed) we have \begin{equation}
M_{i}=-b_{i}\frac{g^{(i)2}}{16\pi^{2}}m_{3/2}=-b_{i}\frac{\alpha_{i}}{4\pi}m_{3/2}\label{eq:gauginomass'}\end{equation}
To get this we used $F^{T}=-(T+\bar{T})m_{3/2}$, $K_{T}=-3/(T+\bar{T})$
and $\tilde{K}_{\alpha\bar{\beta}}=k_{\alpha\beta}/(T+\bar{T}$).
Also note that there is no direct AMSB contribution to the scalar
masses since the Weyl anomaly just affects the gauge coupling function
- at least at the two derivative level %
\footnote{At the four derivative level of course there are curvature squared
contributions.%
}.
Nevertheless there is a contribution to the scalar masses coming from
`gaugino mediation' (see \citep{Luty:2005sn} for a review). This
comes about because of the contribution of gauginos to the running
of the scalar masses. The relevant equation is (see section 11 of
\citep{Luty:2005sn}) assuming GUT unification of coupling constants
($t\equiv\ln\mu/\Lambda$ and $c(r)$ is the quadratic Casimir in
$r$)\begin{equation}
\frac{dm_{scalar}^{2}}{dt}=-\frac{c(r)}{2\pi^{2}}g_{i}^{2}M_{i}^{2},\label{eq:massRG}\end{equation}
Integrating this using the beta function equations and the RG invariance
of $M_{i}/g_{i}^{2}$,\begin{equation}
m_{scalar}^{2}=\frac{2c(r)}{b_{i}}[\frac{g_{i}^{4}(\mu)}{g_{i}^{4}(\Lambda)}-1]M_{i}^{2}\simeq2c(r)\alpha_{GUT}\ln\frac{\Lambda}{\mu}M_{i}^{2}.\label{eq:mscalargaugino}\end{equation}
Note that in \eqref{eq:massRG}we have only kept the dominant terms.
Also we note that the squared scalar masses generated by this mechanism
are always positive. Furthermore for $\mu$ at the TeV scale and $\Lambda$
at the GUT scale and $\alpha_{GUT}\sim1/25$, the scalar mass is of
the order of the gaugino mass which in turn has a magnitude $m_{3}\sim10^{-2}m_{3/2}$,
$m_{2}\sim m_{1}\sim10^{-3}m_{3/2}$. Thus with $m_{3/2}\sim100TeV$
we have TeV scale gluino and squark masses. These clearly dominate
the classical masses that we obtained even in the most favorable case
with ${\cal V}\sim10^{6}$. These numbers would be somewhat different
in the intermediate string scale case, but the above mechanism i.e.
RG running from this intermediate scale, will still be the dominant
mechanism for generating the scalar masses and the A-term. A detailed
account of the phenomenology in both cases will be presented in \citep{Baer:2009tp}.
\[
\]
\bibliographystyle{apsrev} |
0912.2919 | \section{Introduction}\label{s1}
We consider only simple graphs without loops or multiple edges. Our
terminology and notation will be standard except as indicated, and a good
reference for any undefined terms or notation is~\cite{West01}. For two
graphs~$G,H$ on disjoint vertex sets, we denote their \emph{union} by
$G\cup H$. The \emph{join $G+H$} of~$G$ and~$H$ is the graph formed from
$G\cup H$ by adding all edges between~$V(G)$ and~$V(H)$.
For a positive integer~$n$, an \emph{$n$-sequence} (or just a
\emph{sequence}) is an integer sequence $\pi=(d_1,d_2,\dots,d_n)$, with
$0\le d_j\le n-1$ for all~$j$. In contrast to~\cite{West01}, we will
usually write the sequence in nondecreasing order (and may make this
explicit by writing $\pi=(d_1\le\dots\le d_n)$). We will employ the
standard abbreviated notation for sequences, e.g., $(4,4,4,4,4,5,5,6)$ will
be denoted $4^5\,5^2\,6^1$. If $\pi=(d_1,\dots,d_n)$ and
$\pi'=(d_1',\dots,d_n')$ are two $n$-sequences, we say \emph{$\pi'$
majorizes~$\pi$}, denoted $\pi'\ge\pi$, if $d_j'\ge d_j$ for all~$j$.
A \emph{degree sequence} of a graph is any sequence
$\pi=(d_1,d_2,\dots,d_n)$ consisting of the vertex degrees of the graph. A
sequence~$\pi$ is \emph{graphical} if there exists a graph~$G$ having~$\pi$
as one of its degree sequences, in which case we call~$G$ a
\emph{realization} of~$\pi$. If~$P$ is a graph property (e.g., hamiltonian,
$k$-connected, etc.), we call a graphical sequence~$\pi$
\emph{forcibly~$P$} if every realization of~$\pi$ has property~$P$.
\medskip
Historically, the degree sequence of a graph has been used to provide
sufficient conditions for a graph to have certain properties, such as
hamiltonicity or $k$-connectivity. In particular, sufficient conditions
for~$\pi$ to be forcibly hamiltonian were given by several authors,
culminating in the following theorem of Chv\'atal~\cite{Chvatal72}.
\begin{thm}[\cite{Chvatal72}]\label{thm:chvatal}
\;Let $\pi=(d_1\le\dots\le d_n)$ be a graphical sequence, with $n\ge3$.
If $d_i\le i<\frac12n$ implies $d_{n-i}\ge n-i$, then~$\pi$ is forcibly
hamiltonian.
\end{thm}
Unlike its predecessors, Chv\'atal's theorem has the property that if it
does not guarantee that~$\pi$ is forcibly hamiltonian because the condition
fails for some $i<\frac12n$, then~$\pi$ is majorized by
$\pi'=i^i\,(n-i-1)^{n-2i}\,(n-1)^i$, which has a unique nonhamiltonian
realization $K_i+(\overline{K_i}\cup K_{n-2i})$. As we will see below, this
implies that Chv\'atal's theorem is the strongest of an entire class of
theorems giving sufficient degree conditions for~$\pi$ to be forcibly
hamiltonian.
Sufficient conditions for~$\pi$ to be forcibly $k$-connected were given by
several authors, culminating in the following theorem of
Bondy~\cite{Bondy69} (though the form in which we present it is due to
Boesch~\cite{Boesch74}).
\begin{thm}[\cite{Boesch74,Bondy69}]\label{BB}
\;Let $\pi=(d_1\le\dots\le d_n)$ be a graphical sequence with $n\ge2$,
and let $1\le k\le n-1$. If $d_i\le i+k-2$ implies $d_{n-k+1}\ge n-i$,
for $1\le i\le\frac12(n-k+1)$, then~$\pi$ is forcibly $k$-connected.
\end{thm}
Boesch~\cite{Boesch74} also observed that Theorem~\ref{BB} is the strongest
theorem giving sufficient degree conditions for~$\pi$ to be forcibly
$k$-connected, in exactly the same sense as Theorem~\ref{thm:chvatal}.
\medskip
Let~$\omega(G)$ denote the number of components of a graph~$G$. For
$t\ge0$, we call~$G$ \emph{$t$-tough} if $t\cdot\omega(G-X)\le|X|$, for
every $X\subseteq V(G)$ with $\omega(G-X)>1$. The \emph{toughness} of~$G$,
denoted~$\tau(G)$, is the maximum $t\ge0$ for which $G$ is $t$-tough
(taking $\tau(K_n)=n-1$, for all $n\ge1$). So if~$G$ is not complete, then
$\tau(G)=\min\Bigl\{\dfrac{|X|}{\omega(G-X)}\Bigm|\text{$X\subseteq V(G)$
is a cutset of $G$}\Bigr\}$.
In this paper we consider forcibly $t$-tough theorems, for any $t\ge0$.
When trying to formulate and prove this type of theorem, we encountered
very different behavior in the number of conditions required for a best
possible theorem for the cases $t\ge1$ and $t<1$. In order to describe this
behavior precisely, we need to say what we mean by a `condition' and by a
`best possible theorem'.
\medskip
First note that the conditions in Theorems~\ref{thm:chvatal} can be written
in the form:
\[\text{$d_i\ge i+1$ \ or \ $d_{n-i}\ge n-i$, \ for
$i=1,\ldots,\bigl\lfloor\tfrac12(n-1)\bigr\rfloor$,}\]
and the conditions in Theorem~\ref{BB} can be written in a similar way. We
will use the term `Chv\'atal-type conditions' for such conditions.
Formally, a \emph{Chv\'atal-type condition} for $n$-sequences
$(d_1\le d_2\le\cdots\le d_n)$ is a condition of the form
\[d_{i_1}\ge k_{i_1}\;\vee\;d_{i_2}\ge k_{i_2}\;\vee\;\ldots\;\vee\;
d_{i_r}\ge k_{i_r},\]
where all~$i_j$ and~$k_{i_j}$ are integers, with
$1\le i_1<i_2<\dots<i_r\le n$ and
$1\le k_{i_1}\le k_{i_2}\le\cdots\le k_{i_r}\le n$.
\medskip
A graph property~$P$ is called \emph{increasing} if whenever a graph~$G$
has~$P$, so does every edge-augmented supergraph of~$G$. In particular,
``hamiltonian'', ``$k$-connected'' and ``\mbox{$t$-tough}'' are all
increasing graph properties. In this paper, the term ``graph property''
will always mean an increasing graph property.
Given a graph property~$P$, consider a theorem~$T$ which declares certain
degree sequences to be forcibly~$P$, rendering no decision on the remaining
degree sequences. We call such a theorem~$T$ a \emph{forcibly $P$-theorem}
(or just a \emph{$P$-theorem}, for brevity). Thus Theorem~\ref{thm:chvatal}
would be a forcibly hamiltonian theorem. We call a $P$-theorem~$T$
\emph{monotone} if, for any two degree sequences $\pi,\pi'$, whenever~$T$
declares~$\pi$ forcibly~$P$ and $\pi'\ge\pi$, then~$T$ declares~$\pi'$
forcibly~$P$. We call a $P$-theorem~$T$ \emph{optimal} if whenever~$T$ does
not declare a degree sequence~$\pi$ forcibly~$P$, then~$\pi$ is not
forcibly~$P$; $T$ is \emph{weakly optimal} if for any sequence~$\pi$ (not
necessarily graphical) which~$T$ does not declare forcibly~$P$, $\pi$ is
majorized by a degree sequence which is not forcibly~$P$.
A $P$-theorem which is both monotone and weakly optimal is a best monotone
\mbox{$P$-theorem}, in the following sense.
\begin{thm}\label{wo}
\quad Let $T$,~$T_0$ be monotone $P$-theorems, with~$T_0$ weakly optimal.
If~$T$ declares a degree sequence~$\pi$ to be forcibly~$P$, then so
does~$T_0$.
\end{thm}
\textbf{Proof of Theorem \ref{wo}:} Suppose to the contrary that there
exists a degree sequence~$\pi$ so that~$T$ declares~$\pi$ forcibly~$P$,
but~$T_0$ does not. Since~$T_0$ is weakly optimal, there exists a degree
sequence $\pi'\ge\pi$ which is not forcibly~$P$. This means that also~$T$
will not declare~$\pi'$ forcibly~$P$. But if~$T$ declares~$\pi$
forcibly~$P$, $\pi'\ge\pi$, and~$T$ does not declare~$\pi'$ forcibly~$P$,
then~$T$ is not monotone, a contradiction.\eop
\bigskip
If~$T_0$ is Chv\'atal's hamiltonian theorem (Theorem~\ref{thm:chvatal}),
then~$T_0$ is clearly monotone, and we noted above that~$T_0$ is weakly
optimal. So by Theorem~\ref{wo}, Chv\'atal's theorem is a best monotone
hamiltonian theorem.
\medskip
Our goal in this paper is to consider forcibly $t$-tough theorems, for any
$t\ge0$. In Section~\ref{s2} we first give a best monotone $t$-tough
theorem for $n$-sequences, requiring at most
$\bigl\lfloor\frac12n\bigr\rfloor$ Chv\'atal-type conditions, for any
$t\ge1$. In contrast to this, in Sections~\ref{sub1.1} and~\ref{s2a} we
show that for any integer $k\ge1$, a best monotone $1/k$-tough theorem
contains at least $f(k)\cdot n$ nonredundant Chv\'atal-type conditions,
where~$f(k)$ grows superpolynomially as $k\rightarrow\infty$. A similar
superpolynomial growth in the complexity of the best monotone
$k$-edge-connected theorem in terms of~$k$ was previously noted by
Kriesell~\cite{Kp}.
This superpolynomial complexity of a best monotone $1/k$-tough theorem
suggests the desirability of finding more reasonable $t$-tough theorems,
when $t<1$. In Section~\ref{s3} we give one such theorem. This theorem is a
monotone, though not best monotone, $t$-tough theorem which is valid for
any $t\le1$.
\section{A Best Monotone $t$-Tough Theorem for $t\ge1$}\label{s2}
We first give a best monotone $t$-tough theorem for $t\ge1$.
\begin{thm}\label{tge1}
\quad Let $t\ge1$, $n\ge\lceil t\rceil+2$, and let $\pi=(d_1\le\dots\le
d_n)$ be a graphical sequence. If{
\qitemm{($*t$)}\hspace*{\fill
$d_{\lfloor i/t\rfloor}\ge i+1$ \ or \
$d_{n-i}\ge n-\lfloor i/t\rfloor$, \ for
$t\le i<\dfrac{tn}{(t+1)}$,\hspace*{\fill}
}then~$\pi$ is forcibly $t$-tough.
\end{thm}
Clearly, property~($*t$) in Theorem~\ref{tge1} is monotone. Furthermore,
if~$\pi$ does not satisfy~($*t$) for some~$i$ with $t\le i<tn/(t+1)$,
then~$\pi$ is majorized by
$\pi'=i^{\lfloor i/t\rfloor}\linebreak[1]\,
(n-\lfloor i/t\rfloor-1)^{n-i-\lfloor i/t\rfloor}\,(n-1)^i$, which has the
non-$t$-tough realization
$K_i+\bigl(\overline{K_{\lfloor i/t\rfloor}}\cup
K_{n-i-\lfloor i/t\rfloor}\bigr)$. Thus~($*t$) in Theorem~\ref{tge1} is
also weakly optimal, and so Theorem~\ref{tge1} is best monotone by
Theorem~\ref{wo}. Finally, note that when $t=1$, ($*t$) reduces to
Chv\'atal's hamiltonian condition in Theorem~\ref{thm:chvatal}.
\bigskip
\textbf{Proof of Theorem \ref{tge1}:} Suppose~$\pi$ satisfies~($*t$) for
some $t\ge1$ and $n\ge\lceil t\rceil+2$, but~$\pi$ has a realization~$G$
which is not $t$-tough. Then there exists a set $X\subseteq V(G)$ that is
maximal with respect to $\omega(G-X)\ge2$ and $\dfrac{|X|}{\omega(G-X)}<t$.
Let $x\doteq |X|$ and $w\doteq\omega(G-X)$, so that
$w\ge\lfloor x/t\rfloor+1$. Also, let $H_1,H_2,\dots,H_w$ denote the
components of $G-X$, with $|H_1|\ge|H_2|\ge\cdots\ge|H_w|$, and let
$h_j\doteq|H_j|$ for $j=1,\ldots,w$. By adding edges (if needed) to~$G$, we
may assume $\langle X\rangle$ is complete, and each $\langle H_j\rangle$ is
complete and completely joined to~$X$.
Set $i\doteq x+h_2-1$.
\bigskip
\textbf{Claim 1.}\quad$i\ge t$.
\medskip
\textbf{Proof:} It is enough to show that $x\ge t$. Assume instead that
$x<t$. Define $X'\doteq X\cup\{v\}$, with $v\in H_1$. If $h_1\ge2$, then
\[\frac{|X'|}{\omega(G-X')}\:=\:\frac{x+1}{\omega(G-X)}\:<\:\frac{t+1}2\:
\le\:t,\]
which contradicts the maximality of~$X$. Similarly, if $h_1=1$ and $w\ge3$,
then
\[\frac{|X'|}{\omega(G-X')}\:=\:\frac{x+1}{\omega(G-X)-1}\:<\:\frac{t+1}2\:
\le\:t,\]
also a contradiction. Finally, if $h_1=1$ and $w=2$, then~$G$ is the graph
$K_{n-2}+\overline{K_2}$ with $n-2=x<t$, contradicting
$n\ge\lceil t\rceil+2$.\openeop
\bigskip
\textbf{Claim 2.}\quad$i<\dfrac{tn}{t+1}$
\medskip
\textbf{Proof:} Note that $n=x+h_1+h_2+\cdots+h_w\ge x+2h_2+w-2$. Since
$x<tw$, we obtain
\begin{align*}
i\:&=\:x+h_2-1\:=\:
\frac{tx+x+(t+1)(h_2-1)}{t+1}\\[1mm]
&<\:\frac{t(x+w+(1+1/t)(h_2-1))}{t+1}\:\le\:\frac{t(x+2h_2+w-2)}{t+1}
\:\le\:\frac{tn}{t+1}.
\end{align*}
\vspace*{-9mm}
\openeop
\medskip
\bigskip
By the claims we have $t\le i<\dfrac{tn}{t+1}$. Next note that
\[d_{\lfloor i/t\rfloor}\:=\:d_{\lfloor(x+h_2-1)/t\rfloor}\:\le\:
d_{\lfloor x/t\rfloor+h_2-1}\:\le\:d_{w+h_2-2}\:\le\:d_{(h_2+\cdots+h_w)}
\:=\:x+h_2-1\:=\:i.\]
However, we also have
\begin{align*}
d_{n-i}\:&\le\:d_{n-x}\:=\:x+h_1-1\:=\:n-h_2-(h_3+\cdots+h_w)-1\:\le\:
n-(w+h_2-1)\\[1mm]
&<\:n-\Bigl(\frac{x}{t}+h_2-1\Bigr)\:\le\:
n-\frac{x+h_2-1}{t}\:=\:n-i/t\:\le\:n-\lfloor i/t\rfloor,
\end{align*}
contradicting~($*t$).\eop
\section{The Number of Chv\'atal-Type Conditions in\\ Best Monotone
Theorems}\label{sub1.1}
In this section we provide a theory that allows us to lower bound the
number of degree sequence conditions required in a best monotone
$P$-theorem.
Recall that a \emph{Chv\'atal-type condition} for $n$-sequences
$(d_1\le d_2\le\cdots\le d_n)$ is a condition of the form
\[d_{i_1}\ge k_{i_1}\;\vee\;d_{i_2}\ge k_{i_2}\;\vee\;\ldots\;\vee\;
d_{i_r}\ge k_{i_r},\]
where all~$i_j$ and~$k_{i_j}$ are integers, with
$1\le i_1<i_2<\dots<i_r\le n$ and
$1\le k_{i_1}\le k_{i_2}\le \cdots\le k_{i_r}\le n$. Given an $n$-sequence
$\pi=(k_1\le k_2\le\cdots\le k_n)$, let $C(\pi)$ denote the Chv\'atal-type
condition:
\[d_1\ge k_1+1\;\vee\;d_2\ge k_2+1\;\vee\:\ldots\;\vee\;d_n\ge k_n+1.\]
Intuitively, $C(\pi)$ is the weakest condition that `blocks'~$\pi$. For
instance, if $\pi=2^23^35$, then~$C(\pi)$ is
\begin{equation}\label{eq11.1}
d_1\ge3\;\vee\;d_2\ge3\;\vee\;d_3\ge4\;\vee\;d_4\ge4\;\vee\;d_5\ge4\;
\vee\;d_6\ge6.
\end{equation}
Since $n$-sequences are assumed to be nondecreasing, $d_1\ge3$ implies
$d_2\ge3$, etc. Also, we cannot have $d_i\ge n$, so the condition $d_6\ge6$
is redundant. Hence~\eqref{eq11.1} can be simplified to the equivalent
Chv\'atal-type condition
\begin{equation}\label{eq11.2}
d_2\ge3\;\vee\;d_5\ge4,
\end{equation}
and we use $\eqref{eq11.1}\cong\eqref{eq11.2}$ to denote this equivalence.
Conversely, given a Chv\'atal-type condition~$c$, let~$\Pi(c)$ denote the
minimal $n$-sequence that majorizes all sequences which violate~$c$
($\Pi(c)$ may not be graphical). So if~$c$ is the condition
in~\eqref{eq11.2} and $n=6$, then $\Pi(c)$ is $2^23^35$. Of course,
$\Pi(c)$ itself violates~$c$. Note that~$C$ and~$\Pi$ are inverses: For any
Chv\'atal-type condition~$c$ we have $C(\Pi(c))\cong c$, and for any
$n$-sequence~$\pi$ we have $\Pi(C(\pi))=\pi$.
\medskip
Given a graph property~$P$, we call a Chv\'atal-type degree condition~$c$
\emph{$P$-weakly-optimal} if any sequence~$\pi$ (not necessarily graphical)
which does not satisfy~$c$ is majorized by a degree sequence which is not
forcibly~$P$. In particular, each of the
$\bigl\lfloor\frac12(n-1)\bigr\rfloor$ conditions in Chv\'atal's
hamiltonian theorem is weakly optimal.
\medskip
Next consider the poset whose elements are the graphical sequences of
length~$n$, with the majorization relation $\pi\le\pi'$ as the partial
order relation. We call this poset the \emph{$n$-degree-poset}. Posets of
integer sequences with a different order relation were previously used by
Aigner \& Triesch~\cite{AT} in their work on graphical sequences.
Given a graph property~$P$, consider the set of $n$-vertex graphs without
property~$P$ which are edge-maximal in this regard. The degree sequences of
these edge-maximal, \mbox{non-$P$} graphs induce a subposet of the
$n$-degree-poset, called the \emph{$P$-subposet}. We refer to the maximal
elements of this $P$-subposet as \emph{sinks}, and denote their number by
$s(n,P)$.
\medskip
We first prove the following lemma.
\begin{lem}\label{lem1}
\quad Let~$P$ be a graph property. If a sink~$\pi$ of the $P$-subposet
violates a $P$-weakly-optimal Chv\'atal-type condition~$c$, then
$c\cong C(\pi)$.
\end{lem}
\textbf{Proof:} Since~$\pi$ violates~$c$, $\pi\le\Pi(c)$. Since~$\Pi(c)$
violates~$c$, and~$c$ is $P$-weakly-optimal, there is a sequence
$\pi'\ge\Pi(c)$ such that~$\pi'$ has a non-$P$ realization. But
$\pi'\le\pi''$ for some sink~$\pi''$, giving $\pi\le\Pi(c)\le\pi'\le\pi''$.
Since distinct sinks are incomparable, $\pi=\pi''$. This implies
$\Pi(c)=\pi$, and thus $c\cong C(\Pi(c))\cong C(\pi)$.\eop
\begin{thm}\label{poset}
\quad Let~$P$ be a graph property. Then any \mbox{$P$-theorem} for
$n$-sequences whose hypothesis consists solely of $P$-weakly-optimal
Chv\'atal-type conditions must contain at least $s(n,P)$ such conditions.
\end{thm}
\textbf{Proof:} Consider a $P$-theorem whose hypothesis consists solely of
$P$-weakly-optimal Chv\'atal-type conditions. By Lemma~\ref{lem1}, a
sink~$\pi$ satisfies every Chv\'atal-type condition besides~$C(\pi)$. So
the theorem must include all the Chv\'atal-type conditions~$C(\pi)$,
as~$\pi$ ranges over the $s(n,P)$ sinks.\eop
\bigskip
On the other hand, it is easy to see that if we take the collection of
Chv\'atal-type conditions~$C(\pi)$ for all sinks~$\pi$ in the $P$-subposet,
then this gives a best monotone $P$-theorem.
\medskip
We do not have a comparable result for $P$-theorems if we do not require
the conditions to be $P$-weakly-optimal, let alone if we consider
conditions that are not of Chv\'atal-type. On the other hand, all results
we have discussed so far, and most of the forcibly $P$-theorems we know in
the literature, involve only $P$-weakly-optimal Chv\'atal-type degree
conditions.
\section{Best Monotone $t$-Tough Theorems for $t\le1$}\label{s2a}
Using the terminology from Section~\ref{sub1.1}, it follows that
Theorem~\ref{tge1} gives, for $t\ge1$, a best monotone $t$-tough theorem
using a linear number (in~$n$) of weakly optimal Chv\'atal-type conditions.
On the other hand, we now show that for any integer $k\ge1$, a best
monotone \mbox{$1/k$-tough} theorem for $n$-sequences requires at least
$f(k)\cdot n$ weakly optimal Chv\'atal-type conditions, where~$f(k)$ grows
superpolynomially as $k\rightarrow\infty$. In view of Theorem~\ref{poset},
to prove this assertion it suffices to prove the following lemma.
\begin{lem}\label{superpoly}\quad
Let $k\ge2$ be an integer, and let $n=m(k+1)$ for some integer $m\ge9$.
Then the number of ($1/k$-tough)-subposet sinks in the
$n$-degree-subposet is at least $\dfrac{p(k-1)}{5(k+1)}n$, where~$p$
denotes the integer partition function.
\end{lem}
Recall that the integer partition function~$p(r)$ counts the number of ways
a positive integer~$r$ can be written as a sum of positive integers. Since
$p(r)\sim\dfrac1{4r\sqrt3}e^{\pi\sqrt{2r/3}}$ as $r\rightarrow\infty$
\cite{HR18}, $f(k)=\dfrac{p(k-1)}{5(k+1)}$ grows superpolynomially as
$k\rightarrow\infty$.
\bigskip
\textbf{Proof of Lemma \ref{superpoly}:} Consider the
collection~$\mathcal{C}$ of all connected graphs on~$n$ vertices which are
edge-maximally not-($1/k$-tough). Each $G\in\mathcal{C}$ has the form
$G=K_j+(K_{c_1}\cup\dots\cup K_{c_{kj+1}})$, where $j<n/(k+1)=m$, so that
$1\le j\le m-1$, and $c_1+\dots+c_{kj+1}$ is a partition of $n-j$. Assuming
$c_1\le\dots\le c_{kj+1}$, the degree sequence of~$G$ becomes
$\pi\doteq(c_1+j-1)^{c_1}\,\dots\,(c_{kj+1}+j-1)^{c_{kj+1}}\,(n-1)^j$. Note
that~$\pi$ cannot be majorized by the degrees of any disconnected graph
on~$n$ vertices, since a disconnected graph has no vertex of degree $n-1$.
By a \emph{complete degree} of a degree sequence we mean an entry in the
sequence equal to $n-1$.
Partition the degree sequences of the graphs in~$\mathcal{C}$ into $m-1$
groups, where the sequences in the $j^\text{th}$ group, $1\le j\le m-1$,
are precisely those containing~$j$ complete degrees. We establish two basic
properties of the $j^\text{th}$ group.
\bigskip
\textbf{Claim 1.}\quad\textit{There are exactly
$p_{kj+1}\bigl((k+1)(m-j)-1\bigr)$ sequences in the $j^\text{th}$ group.}
\medskip
Here $p_{\ell}(r)$ denotes the number of partitions of integer~$r$ into at
most~$\ell$ parts, or equivalently the number of partitions of~$r$ with
largest part at most~$\ell$.
\medskip
\textbf{Proof of Claim 1:} Each sequence in the $j^\text{th}$ group
corresponds uniquely to a set of $kj+1$ component sizes which sum to $n-j$.
If we subtract~1 from each of those component sizes, we obtain a
corresponding collection of $kj+1$ integers (some possibly~0) which sum to
$n-j-(kj+1)=(k+1)(m-j)-1$, and which therefore form a partition of
$(k+1)(m-j)-1$ into at most $kj+1$ parts.\openeop
\bigskip
\textbf{Claim 2.}\quad \textit{No sequence in the $j^\text{th}$ group
majorizes another sequence in the $j^\text{th}$ group.}
\medskip
\textbf{Proof:} Suppose the sequences
$\pi\doteq(c_1+j-1)^{c_1}\,\dots\,(c_{kj+1}+j-1)^{c_{kj+1}}\,(n-1)^j$ and
$\pi'\doteq(c'_1+j-1)^{c'_1}\,\dots\,(c'_{kj+1}+j-1)^{c'_{kj+1}}\,(n-1)^j$
are in the $j^\text{th}$ group, with $\pi\ge\pi'$. Deleting the~$j$
complete degrees from each sequence gives sequences
$\sigma\doteq(c_1-1)^{c_1}\,\dots\,(c_{kj+1}-1)^{c_{kj+1}}$ and
$\sigma'\doteq(c'_1-1)^{c'_1}\,\dots\,(c'_{kj+1}-1)^{c'_{kj+1}}$, with
$\sigma\ge\sigma'$.
Let~$m$ be the smallest index with $c_m\ne c'_m$; since $\sigma\ge\sigma'$,
we have $c_m>c'_m$. In particular, $c_1+\dots+c_m>c'_1+\dots+c'_m$. But
$c_1+\dots+c_{kj+1}=c'_1+\dots+c'_{kj+1}=n-j$, and so there exists a
smallest index $\ell>m$ with $c_1+\dots+c_\ell\le c'_1+\dots+c'_\ell$. In
particular, $c_\ell<c'_\ell$. Since
$c'_1+\dots+c'_{\ell-1}<c_1+\dots+c_{\ell-1}<c_1+\dots+c_\ell\le
c_1+\dots+c'_\ell$, we have
$d_{c_1+\dots+c_\ell}=c_\ell-1<c'_\ell-1=d'_{c_1+\dots+c_\ell}$, and thus
$\sigma\ngeq\sigma'$, a contradiction.\openeop
\bigskip
Since $K_j+(K_{c_1}\cup\dots\cup K_{c_{kj+1}})$ has~$n$ vertices,
$K_{c_{kj+1}}$ has at most $n-j-kj$ vertices. This means the largest
possible noncomplete degree in a sequence in the $j^\text{th}$ group is
$j+(n-j-kj-1)=n-kj-1$. Using this observation we can prove the following.
\bigskip
\textbf{Claim 3.}\quad\textit{If a sequence
$\pi=\cdots\,d^{d-j+1}\,(n-1)^j$ in the $j^\text{th}$ group has largest
noncomplete degree $d\ge n-k(j+1)$, then~$\pi$ is not majorized by any
sequence in the $i^\text{th}$ group, for $i\ge j+1$.}
\medskip
In particular, such a~$\pi$ is a sink, since~$\pi$ is certainly not
majorized by another sequence in the $j^\text{th}$ group by Claim~2, nor by
a sequence in groups $1,2,\dots,j-1$, since any such sequence has fewer
than~$j$ complete degrees.
\medskip
\textbf{Proof of Claim 3:} If $d\ge n-k(j+1)$, then the $d+1$ largest
degrees $d^{d-j+1}\,(n-1)^j$ in~$\pi$ could be majorized only by complete
degrees in a sequence in group $i\ge j+1$, since the largest noncomplete
degree in any sequence in group~$i$ is at most $n-ki-1<n-k(j+1)$. There are
only $i\le m-1$ complete degrees in a sequence in group~$i$. On the other
hand, since $j+1\le i<m$, we have $d+1\ge n-k(j+1)+1>m(k+1)-km+1=m+1>m-1$,
a contradiction.\openeop
\bigskip
So by Claim 3, the sequences~$\pi$ in the $j^\text{th}$ group which could
possibly be nonsinks (i.e., majorized by a sequence in group~$i$, for some
$i\ge j+1$), must have largest noncomplete degree at most $n-k(j+1)-1$. So
in a graph $G\in\mathcal{C}$, $G=K_j+(K_{c_1}\cup\dots\cup K_{c_{kj+1}})$,
which realizes a nonsink~$\pi$, each of the~$K_c$'s must have order at most
$(n-k(j+1)-1)-j+1=(k+1)(m-j)-k$. Subtracting~1 from the order of each of
these components gives a sequence of $kj+1$ integers (some possibly~0)
which sum to $(n-j)-(kj+1)=(k+1)(m-j)-1$, and which have largest part at
most $(k+1)(m-j)-k-1=(k+1)(m-j-1)$. Thus there are exactly
$p_{(k+1)(m-j-1)}\bigl((k+1)(m-j)-1)\bigr)$ such sequences, and so there
are at most this many nonsinks in the $j^\text{th}$ group. Setting
$N(j)\doteq (k+1)(m-j)-1$, so that $(k+1)(m-j-1)=N(j)-k$, this becomes at
most $p_{N(j)-k}\bigl(N(j)\bigr)$ nonsinks in the $j^\text{th}$ group of
sequences.
But by Claim~1, there are exactly $p_{kj+1}\bigl(N(j)\bigr)$ sequences in
group~$j$, and so the number of sinks in the $j^\text{th}$ group is at
least $p_{kj+1}\bigl(N(j)\bigr)-p_{N(j)-k}\bigl(N(j)\bigr)$.
Note that $p_{kj+1}(N(j))$ reduces to $p(N(j))$ if $kj+1\ge N(j)$. However,
$kj+1\ge N(j)$ is equivalent to $j\ge\dfrac{(k+1)m-2}{2k+1}$. Since
$k\ge2$, the inequality $j\ge\dfrac{(k+1)m-2}{2k+1}$ holds if
$j\ge\frac35m$. Thus $p_{kj+1}(N(j))=p(N(j))$ holds for $j\ge\frac35m$.
On the other hand, for $j\le m-2$ we can show the following.
\bigskip
\textbf{Claim 4.}\quad\textit{If $j\le m-2$, then
\[p\bigl(N(j)\bigr)-p_{N(j)-k}\bigl(N(j)\bigr)\:=\:
1+p(1)+\dots+p(k-1)\:\ge\:p(k-1).\]
\textbf{Proof:} Note that if $j\le m-2$, then $k<\frac12N(j)$. The left
side of the equality in the claim counts partitions of~$N(j)$ with largest
part at least $N(j)-(k-1)$. The right side counts the same according to the
exact order $N(j)-\ell$, $0\le\ell\le k-1$, of the largest part in the
partition, using that the largest part is unique since
$N(j)-\ell\ge N(j)-(k-1)>\frac12N(j)$.\openeop
\bigskip
Completing the proof of Lemma~\ref{superpoly}, we find that the number of
sinks in the \mbox{($1/k$-tough)}-subposet of the $n$-degree-poset is at
least
\begin{align*}
\sum_{j=\lceil 3m/5\rceil}^{m-2}\bigl[p_{kj+1}\bigl(N(j)\bigr)-
p_{N(j)-k}\bigl(N(j)\bigr)\bigr]\:&=\:
\sum_{j=\lceil3m/5\rceil}^{m-2}\bigl[p\bigl(N(j)\bigr)-
p_{N(j)-k}\bigl(N(j)\bigr)\bigr]\\
&\hskip-15mm\ge\:\sum_{j=\lceil3m/5\rceil}^{m-2}p(k-1)\:\ge\:
\bigl(\tfrac25m-\tfrac95\bigr)p(k-1)\\
&\hskip-30mm=\:\Bigl(\frac{2n}{5(k+1)}-\frac95\Bigr)p(k-1)\:\ge\:
\frac{n}{5(k+1)}p(k-1),
\end{align*}
as asserted, since $n=m(k+1)\ge9(k+1)$ implies
$\dfrac{2n}{5(k+1)}-\dfrac95\ge\dfrac{n}{5(k+1)}$.\eop
\bigskip
Combining Lemma~\ref{superpoly} with Theorem~\ref{poset} gives the promised
superpolynomial growth in the number of weakly optimal Chv\'atal-type
conditions for $1/k$-toughness.
\begin{thm}\label{superthm}\quad
Let $k\ge2$ be an integer, and let $n=m(k+1)$ for some integer $m\ge9$.
Then a best monotone $1/k$-tough theorem for $n$-sequences whose degree
conditions consist solely of weakly optimal Chv\'atal-type conditions
requires at least $\dfrac{p(k-1)n}{5(k+1)}$ such conditions, where $p(r)$
is the integer partition function.
\end{thm}
\section{A Simple $t$-Tough Theorem}\label{s3}
The superpolynomial complexity as $k\rightarrow\infty$ of a best monotone
$1/k$-tough theorem suggests the desirability of finding simple $t$-tough
theorems, when $t<1$. We give such a theorem below. It will again be
convenient to assume at first that $t=1/k$, for some integer $k\ge1$. Note
that the conditions in the theorem are still Chv\'atal-type conditions.
\clearpage
\begin{lem}\label{1/kthm}\quad
Let $k\ge1$ be an integer, $n\ge k+2$, and $\pi=(d_1\le\dots\le d_n)$ a
graphical sequence. If{
\qitemm{(i)}$d_i\ge i-k+2$ \ or \ $d_{n-i+k-1}\ge n-i$, \ for
$k\le i<\frac12(n+k-1)$, and
\qitemm{(ii)} $d_i\ge i$ \ or \ $d_n\ge n-i$, \ for
$1\le i\le\frac12n$,
}then $\pi$ is forcibly $1/k$-tough.
\end{lem}
\textbf{Proof of Lemma~\ref{1/kthm}:} Suppose~$\pi$ has a realization~$G$
which is not $1/k$-tough. By~(ii) and Theorem~\ref{BB}, $G$ is connected.
So we may assume (by adding edges if necessary) that there exists
$X\subseteq V(G)$, with $x\doteq|X|\ge1$, such that $G=K_x+(K_{a_1}\cup
K_{a_2}\cup\dots\cup K_{a_{kx+1}})$, where $1\le a_1\le a_2\le\dots\le
a_{kx+1}$.
Set $i\doteq x+k-2+a_{kx}$.
\bigskip
\textbf{Claim 1.}\quad$k\le i<\frac12(n+k-1)$
\medskip
\textbf{Proof:} The fact that $i\ge k$ follows immediately from the
definition of~$i$. Since $kx-x-k+1=(k-1)(x-1)\ge0$, we have
\begin{equation}\label{eq3}
kx-1\:\ge\:x+k-2.
\end{equation}
This leads to
\begin{align*}
n\:=\:x+\sum_{j=1}^{kx-1}a_j+a_{kx}+a_{kx+1}\:&\ge\:x+kx-1+2a_{kx}\\
&\ge\:2x+k-2+2a_{kx}\:=\:2i-k+2,
\end{align*}
which is equivalent to $i<\frac12(n+k-1)$.\openeop
\bigskip
\textbf{Claim 2.}\quad$d_i\le i-k+1$.
\medskip
\textbf{Proof:} From~\eqref{eq3} we get
\begin{equation}\label{eq4}
i\:=\:x+k-2+a_{kx}\:\le\:kx-1+a_{kx}\:\le\:\sum_{j=1}^{kx}a_j.
\end{equation}
This gives $d_i\le x+(a_{kx}-1)\:=\:i-k+1$.\openeop
\bigskip
\textbf{Claim 3.}\quad$d_{n-i+k-1}<n-i$.
\medskip
\textbf{Proof:} We have
$n-i+k-1=n-x-a_{kx}+1\le\sum\limits_{j=1}^{kx+1}a_j$. Thus, using the
bound~\eqref{eq4} for~$i$,
\[d_{n-i+k-1}\:\le\:x+a_{kx+1}-1\:<\:n-\sum_{j=1}^{kx}a_j\:\le\:n-i.\]
\vspace*{-9mm}
\openeop
\medskip
\bigskip
Claims~1, 2 and~3 together contradict condition~(i), completing the proof
of the lemma\eop
\bigskip
We can extend Lemma~\ref{1/kthm} to arbitrary $t\le1$ by letting
$k=\lfloor1/t\rfloor$.
\begin{thm}\label{simple1}\quad
Let $t\le1$, $n\ge\lfloor 1/t\rfloor+2$, and $\pi=(d_1\le\dots\le d_n)$ a
graphical sequence. If{
\qitemm{(i)}$d_i\ge i-\lfloor 1/t\rfloor+2$ \ or \
$d_{n-i+\lfloor 1/t\rfloor-1}\ge n-i$, \ for
$\lfloor 1/t\rfloor\le i<\frac12\bigl(n+\lfloor 1/t\rfloor-1\bigr)$,
and
\qitemm{(ii)} $d_i\ge i$ \ or \ $d_n\ge n-i$, \ for
$1\le i\le\frac12n$,
}then $\pi$ is forcibly $t$-tough.
\end{thm}
\textbf{Proof:} Set $k=\lfloor 1/t\rfloor\ge1$. If~$\pi$ satisfies
conditions (i),~(ii) in Theorem~\ref{simple1}, then~$\pi$ satisfies
conditions (i),~(ii) in Lemma~\ref{1/kthm}, and so is forcibly $1/k$-tough.
But $k=\lfloor 1/t\rfloor\le1/t$ means $1/k\ge t$, and so~$\pi$ is forcibly
$t$-tough.\eop
\bigskip
In summary, if $\dfrac1{k+1}<t\le\dfrac1k$ for some integer $k\ge1$, then
Theorem~\ref{simple1} declares~$\pi$ forcibly $t$-tough precisely if
Lemma~\ref{1/kthm} declares~$\pi$ forcibly $1/k$-tough.
\bigskip
\textbf{Acknowledgements.}\\
The authors thank two anonymous referees for comments and suggestions that
greatly improved the structure and clarity of the paper. We also thank
Michael Yatauro for providing the short argument for Claim~2 in the proof
of Theorem~\ref{tge1}. |
0912.3303 | \section{Introduction}
In recent years there has been a great deal of interest in
graph $C^*$-algebras and their generalisations (see
\cite{CBMSbk} for a survey). To associate $C^*$-algebras to a
given generalisation of directed graphs, one assigns partial
isometries to the edges of the graph in a way which encodes
connectivity in the graph. One then aims to identify relations
amongst the partial isometries so that the $C^*$-algebra
universal for these relations satisfies a version of the
Cuntz-Krieger uniqueness theorem. For directed graphs, this
theorem states that if every cycle has an entrance, then any
representation of its Cuntz-Krieger algebra which is nonzero on
generators is faithful. In trying to identify appropriate
relations, there is typically some analogue of a left-regular
representation which points to a natural notion of an abstract
representation; in the case of directed graphs, each graph can
be represented on the Hilbert space with orthonormal basis
indexed by finite paths in the graph, and this gives rise to
the notion of a Toeplitz-Cuntz-Krieger family. However, the
universal $C^*$-algebra for such representations is typically
too big to satisfy a version of the Cuntz-Krieger uniqueness
theorem, and one has to identify an additional relation to
correct this.
In \cite{Katsura2006c}, in the much more general context of
Cuntz-Pimsner algebras associated to Hilbert bimodules, Katsura
developed a very elegant solution to this problem. For directed
graphs, his results say that given any Toeplitz-Cuntz-Krieger
$E$-family consisting of nonzero partial isometries, if the
$C^*$-algebra $B$ it generates carries a circle action
compatible with the canonical gauge action on the Toeplitz
algebra, then there is a canonical homomorphism from $B$ onto
the Cuntz-Krieger algebra. We call this a co-universal
property: the Cuntz-Krieger algebra is co-universal for
Toeplitz-Cuntz-Krieger families consisting of nonzero partial
isometries and compatible with the gauge action. Katsura proved
this theorem \emph{a posteriori}: the Cuntz-Krieger algebra had
already been defined in terms of a universal property. However,
he pointed out that this theorem implies that the Cuntz-Krieger
algebra could be \emph{defined} to be the algebra co-universal
for nonzero Cuntz-Krieger families; one would then have to work
to prove that such a co-universal algebra exists.
When every cycle in the graph $E$ has an entrance, it is a
consequence of the Toeplitz-Cuntz-Krieger uniqueness theorem
that every Cuntz-Krieger $E$-family consisting of nonzero
partial isometries is automatically compatible with the gauge
action. In particular, in this case the use of the gauge action
in Katsura's analysis should not be necessary. In this article
we show that this is indeed the case: we present a direct
argument that for an arbitrary directed graph in which every
cycle has an entrance, there exists a $C^*$-algebra which is
co-universal for Toeplitz-Cuntz-Krieger families consisting of
nonzero partial isometries. In particular, we do not first
identify the Cuntz-Krieger relation or the corresponding
universal $C^*$-algebra. We also do not proceed via the
machinery of ideal structure of Toeplitz algebras of directed
graphs, and we do not deal with the gauge-action of the circle
or with an analysis of its fixed-point algebra. Instead we work
directly with the abelian subalgebra generated by range
projections associated to paths in the graph.
\section{Preliminaries}\label{sec:prelims}
A directed graph $E = (E^0, E^1, r, s)$ consists of a countable
set $E^0$ of vertices, a countable set $E^1$ of edges, and maps
$r,s : E^1 \to E^0$ indicating the direction of the edges. We
will follow the conventions of \cite{CBMSbk} so that a path is
a sequence $\alpha = \alpha_1 \alpha_2 \dots \alpha_n$ of edges
such that $s(\alpha_i) = r(\alpha_{i+1})$ for all $i$. We write
$|\alpha|$ for $n$, and if we want to indicate a segment of a
path, we shall denote it $\alpha_{[p,q]} = \alpha_{p+1}
\alpha_{p+2} \dots \alpha_q$. If $n = \infty$ (so that $\alpha$
is actually a right-infinite string), then we call $\alpha$ an
\emph{infinite path}. The range of an infinite path $\alpha =
\alpha_1\alpha_2\dots$ is $r(\alpha) := r(\alpha_1)$.
For $n \in \field{N}$, we write $E^n$ for the collection of paths of
length $n$. We write $E^*$ for the category of finite paths in
$E$ (we regard vertices as paths of length zero), and
$E^\infty$ for the collection of all infinite paths. Given
$\alpha \in E^*$ and $X \subset E^* \cup E^\infty$, we denote
$\{\alpha\mu : \mu \in X, r(\mu) = s(\alpha)\}$ by $\alpha X$.
Given $v \in E^0$, a set $X \subset vE^*$ is said to be
\emph{exhaustive} if, for every $\lambda \in vE^*$ there exists
$\alpha \in X$ such that either $\lambda = \alpha\lambda'$ or
$\alpha = \lambda\alpha'$.
Given a directed graph $E$, a \emph{Toeplitz-Cuntz-Krieger
$E$-family} in a $C^*$-algebra $B$ is a pair $(t,q)$ where $t :
e \mapsto t_e$ assigns to each edge a partial isometry in $B$,
and $q : v \mapsto q_v$ assigns to each vertex a projection in
$B$ such that
\begin{itemize}
\item[(TCK1)] the $q_v$ are mutually orthogonal
\item[(TCK2)] each $t_e^* t_e = q_{s(e)}$, and
\item[(TCK3)] for each $v \in E^0$ and each finite subset $F
\subset vE^1$, we have $q_v \ge \sum_{e \in F} t_e
t^*_e$.
\end{itemize}
There is a $C^*$-algebra $\mathcal T C^*(E)$ generated by a
Toeplitz-Cuntz-Krieger family $(s, p)$ which is universal in
the sense that given any Toeplitz-Cuntz-Krieger family $(t,q)$
in a $C^*$-algebra $B$ there is a homomorphism $\pi_{t,q} : \mathcal T
C^*(E) \to B$ such that $\pi_{t,q}(s_e) = t_e$ and
$\pi_{t,q}(p_v) = q_v$ for all $e \in E^1$ and $v \in E^0$.
Given a path $\alpha \in E^*$ and a Toeplitz-Cuntz-Krieger
$E$-family $(t,q)$, we write $t_\alpha$ for
$t_{\alpha_1}t_{\alpha_2} \dots t_{\alpha_{|\alpha|}}$.
\section{The co-universal algebra}\label{sec:CKUT}
\begin{theorem}\label{thm:main}
Let $E$ be a directed graph, and suppose that every cycle in
$E$ has an entrance. There exists a Toeplitz-Cuntz-Krieger
$E$-family $(\Sap, \Pap)$ consisting of nonzero partial
isometries such that $C^*_{\textrm{min}}(E) := C^*(\Sap, \Pap)$
is co-universal in the sense that given any other
Toeplitz-Cuntz-Krieger $E$ family $(t,q)$ in which each $q_v$
is nonzero, there is a homomorphism $\psi_{t,q} : C^*(t,q) \to
C^*_{\textrm{min}}(E)$ such that $\psi_{t,q}(t_e) =
S^{\mathrm{ap}}_e$ and $\psi_{t,q}(q_v) = \Pap_v$ for all $e
\in E^1$ and $v \in E^0$.
\end{theorem}
Our proof relies on understanding the structure the
$C^*$-algebra generated by the projections $\{t_\lambda
t^*_\lambda : \lambda \in E^*\}$ for a Toeplitz-Cuntz-Krieger
$E$-family $(t,q)$. We begin with the following definition.
\begin{dfn}
Let $E$ be a directed graph. A \emph{boolean representation} of
$E$ in a $C^*$-algebra $B$ is a map $p : \lambda \mapsto
p_\lambda$ from $E^*$ to $B$ such that each $p_\lambda$ is a
projection, and
\[
p_\mu p_\nu
= \begin{cases}
p_\nu & \text{ if $\nu = \mu\nu'$} \\
p_\mu & \text{ if $\mu = \nu\mu'$} \\
0 & \text{ otherwise.}
\end{cases}
\]
\end{dfn}
If $p$ is a boolean representation of $E$, then the $p_\lambda$
commute, so $\clsp\{p_\lambda : \lambda \in E^*\}$ is a
commutative $C^*$-algebra.
\begin{lemma}\label{p nu = sum q nu}
Let $E$ be a directed graph, let $p$ be a boolean
representation of $E$, and fix a finite subset $F\subset E^*$.
For $\mu \in F$, define
\[
q_\mu^F:= p_\mu \prod_{\mu \mu' \in F\setminus\{\mu\}}(p_\mu - p_{\mu\mu'}).
\]
Then the $q_\mu^F$ are mutually orthogonal projections and for
each $\mu \in E^*$,
\begin{equation}\label{eq_p_nu= sum q_nu}
p_\mu = \sum_{\mu\mu' \in F} q_{\mu\mu'}^F
\end{equation}
\begin{proof}
We proceed by induction on $|F|$. If $|F|=1$ then
\eqref{eq_p_nu= sum q_nu} is trivial. Suppose \eqref{eq_p_nu=
sum q_nu} holds whenever $|F|<n$, and fix $F$ with $|F|=n$. Let
$\lambda \in F$ be of maximal length, and let $G = F \setminus
\{\lambda\}$. Then $q_\lambda^F = p_\lambda$, and for $\mu \in
G$,
\[
q_\mu^F
= \begin{cases}
q_\mu^G &\text{if } \lambda \neq \mu\mu'\\
q_\mu^G - q_\mu^G p_\lambda &\text{if } \lambda = \mu\mu'.
\end{cases}
\]
Fix $\mu \in F$. If $\lambda \neq \mu\mu'$, then the inductive
hypothesis implies that $\sum_{\mu\mu' \in F} q_{\mu\mu'}^F =
\sum_{\mu\mu' \in G} q_{\mu\mu'}^G = p_\mu$. If $\lambda =
\mu\mu'$, then
\begin{flalign*}
&&\sum_{\mu\mu' \in F} q_{\mu\mu'}^F &= \sum_{\mu\mu' \in G} (q_\mu^G - q_\mu^G p_\lambda) + q_\lambda^F&\\
&&&= \sum_{\mu\mu' \in G} q_\mu^G - \sum_{\mu\mu' \in G} q_\mu^G p_\lambda + p_\lambda&\\
&&&= p_\mu - p_\mu p_\lambda + p_\lambda \qquad\text{by the inductive hypothesis}&\\
&&&= p_\mu - p_\lambda + p_\lambda \qquad\text{since $\lambda = \mu\mu'$}&\\
&&&= p_\mu. &\qedhere
\end{flalign*}
\end{proof}
\end{lemma}
Given a directed graph $E$, we define the set of
\emph{aperiodic boundary paths} in $E$ by
\
\apbdry{E} := \big\{\lambda \in E^* : |s(\lambda)E^1| \in \{0,\infty\}\big\}
\; \sqcup \;E^\infty \setminus \big\{\lambda\mu^\infty : s(\lambda) = r(\mu) = s(\mu)\big\}.
\]
Let $\mathcal H := \ell^2(\apbdry{E})$, with orthonormal basis
$\{\xi_x : x \in \apbdry{E}\}$, and define $\{\Pap_\lambda :
\lambda \in E^*\} \subset \mathcal B(\mathcal H)$ by\label{pg:apboolean}
\[
\Pap_\lambda \xi_x = \begin{cases} \xi_x &\text{if } x\in\lambda\apbdry{E}\\0 &\text{otherwise.}\end{cases}
\]
Since each $\Pap_\lambda = \operatorname{proj}_{\clsp\{\xi_x :
x \in \lambda\apbdry{E}\}}$, it is straightforward to check
that $\Pap$ is a boolean representation of $E$.
Suppose that every cycle in $E$ has an entrance. We claim that
each $\Pap_\lambda$ is nonzero. Indeed, fix $\lambda \in E^*$.
Since every cycle in $E$ has an entrance, there exists an $x
\in \apbdry{E}$ with $r(x) = s(\lambda)$. Then $\lambda x \in
\apbdry{E}$ so $\Pap_\lambda \xi_{\lambda x} = \xi_{\lambda x}
\neq 0$.
\begin{lemma}\label{faux CK4}
Let $E$ be a directed graph. Let $\lambda \in E^*$, and suppose
that $F \subset s(\lambda) E^*$ is finite and exhaustive. Then
\[
\prod_{\mu \in F}(\Pap_\lambda - \Pap_{\lambda\mu}) = 0.
\]
\begin{proof}
Let $x \in \apbdry{E}$. We seek $\mu \in F$ such that
$(\Pap_\lambda - \Pap_{\lambda\mu})\xi_x = 0$.
If $\Pap_\lambda \xi_x = 0$, then any $\mu \in F$ will suffice,
so suppose that $\Pap_\lambda \xi_x \neq 0$. Then
$x(0,|\lambda|) = \lambda$. If there exists $\mu \in F$ such
that $x = \lambda\mu x'$, then $(\Pap_\lambda -
\Pap_{\lambda\mu})\xi_x = \xi_x - \xi_x = 0$, so we suppose
that $x \not= \lambda\mu x'$ for all $\mu \in F$ and seek a
contradiction. Since $F$ is exhaustive, there exists $\mu \in
F$ such that $\lambda\mu = x \mu'$ for some $\mu'$ of nonzero
length. In particular, $x$ is a finite path and since $\mu'$
has nonzero length, $s(x)E^1 \not=\emptyset$, so $x \in
\apbdry{E}$ forces $|s(x)E^1| = \infty$. But since no initial
segment of $x$ belongs to $\lambda F$, that $F$ is exhaustive
implies that for each $e \in s(x)E^1$, there exists $\mu_e \in
F$ such that $\lambda\mu_e = xe x_e'$ for some $x_e \in E^*$,
and this contradicts that $F$ is a finite set.
\end{proof}
\end{lemma}
\begin{lemma}\label{converse of CK4}
Let $E$ be a directed graph and $p$ be a boolean representation
of $E$ such that $p_\lambda \neq 0$ for each $\lambda \in E^*$.
If $\{\alpha' : \alpha\alpha' \in F\}$ is not exhaustive, then
$q_\alpha^F \neq 0.$
\begin{proof}
Since $\{\alpha' : \alpha\alpha' \in F\}$ is not exhaustive,
there exists $\tau \in E^*$ such that $\tau
\not=\alpha'\alpha''$ and $\alpha' \not= \tau\tau'$ for each
$\alpha'$ such that $\alpha\alpha' \in F$. In particular, each
$p_{\alpha\alpha'}p_{\alpha\tau} = 0$, and hence $q_\alpha^F
p_{\alpha\tau} = p_{\alpha \tau} \neq 0$, whence $q_\alpha^F
\neq 0.$
\end{proof}
\end{lemma}
\begin{prop}\label{prp:diag cu}
Let $E$ be a directed graph and let $p$ be a boolean
representation of $E$ such that $p_\lambda \neq 0$ for each
$\lambda \in E^*$. Then there is a homomorphism $\psi_p :
\clsp\{p_\lambda : \lambda \in E^*\} \to \clsp\{\Pap_\lambda :
\lambda \in E^*\}$ satisfying $\psi_p(p_\lambda) =
\Pap_\lambda$ for all $\lambda \in E^*$. Moreover, $\psi_p$ is
injective if and only if $\prod_{\mu \in F}(p_\lambda -
p_{\lambda\mu}) = 0$ for all $\lambda \in E^*$ and finite
exhaustive $F \subset s(\lambda)E^*$.
\begin{proof}
For the first assertion it suffices to show that for every
finite subset $F$ of $E^*$ and every collection of scalars
$\{a_\lambda : \lambda \in F\}$,
\begin{equation}\label{Pap to p is norm decr}
\Big\| \sum_{\lambda \in F} a_\lambda \Pap_\lambda \Big\| \leq \Big\| \sum_{\lambda \in F} a_\lambda p_\lambda\Big\|
\end{equation}
Fix a finite subset $F \subset E^*$, and for $\alpha \in F$,
define $Q^F_\alpha := \Pap_\alpha \prod_{\alpha\alpha' \in F
\setminus \{\alpha\}} (\Pap_\alpha - \Pap_{\alpha\alpha'})$.
For $\lambda \in E^*$, Lemma~\ref{p nu = sum q nu} gives
\[
\sum_{\lambda \in F} a_\lambda \Pap_\lambda = \sum_{\alpha \in F} \Big(\sum_{\substack{\mu \in F \\ \alpha = \mu\mu'}} a_\mu\Big) Q_\alpha^F,
\qquad\text{and}\qquad
\sum_{\lambda \in F} a_\lambda p_\lambda = \sum_{\alpha \in F} \Big(\sum_{\substack{\mu \in F \\ \alpha = \mu\mu'}} a_\mu\Big) q_\alpha^F.
\]
By Lemmas \ref{faux CK4} and \ref{converse of CK4}, we have
$\{\alpha \in F : Q_\alpha^F \neq 0\} \subset \{\alpha \in F :
q_\alpha^F \neq 0\}.$ Hence
\[
\Big\| \sum_{\lambda \in F} a_\lambda \Pap_\lambda \Big\|
= \max_{Q^F_\alpha \not = 0} \Big| \sum_{\substack{\mu \in F \\ \alpha = \mu\mu'}} a_\mu \Big|
\le \max_{q^F_\alpha \not = 0} \Big| \sum_{\substack{\mu \in F \\ \alpha = \mu\mu'}} a_\mu \Big|
= \Big\| \sum_{\lambda \in F} a_\lambda p_\lambda \Big\|,
\]
and the first assertion follows.
For the second assertion, note that if $\prod_{\mu \in
F}(p_\lambda - p_{\lambda\mu}) = 0$ for all $\lambda \in E^*$
and finite exhaustive $F \subset s(\lambda)E^*$, then for each
finite exhaustive $F$, we have $\{\alpha \in F : Q_\alpha^F
\neq 0\} = \{\alpha \in F : q_\alpha^F \neq 0\}$ so the
calculation above shows that $\psi_p$ is isometric.
\end{proof}
\end{prop}
We now show that the $C^*$-algebra generated by any
Toeplitz-Cuntz-Krieger $E$-family in which all the partial
isometries are nonzero admits a conditional expectation onto
the subalgebra spanned by the range projections $s_\lambda
s^*_\lambda$. Recall that given a Toeplitz-Cuntz-Krieger
$E$-family $(t,q)$, we write $\pi_{t,q}$ for the canonical
homomorphism from $\mathcal T C^*(E)$ to $C^*(t,q)$ induced by the
universal property of the former. We first need a technical
lemma.
\begin{lemma}\label{L rho is a loop}
Suppose $\lambda,\mu,\nu \in E$ satisfy $|\lambda| \geq |\nu| >
|\mu|$ and
\begin{equation}\label{eq:product nonzero}
t_\lambda t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^* \neq 0.
\end{equation}
Then $\lambda = \nu\nu' = \mu\mu'\nu'$ for some $\mu',\nu'$,
and
\[
t_\lambda t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^* = t_\lambda t_{\lambda \rho}^*
\]
for some cycle $\rho \in E$.
\end{lemma}
\begin{proof}
Equation~\ref{eq:product nonzero} forces $t_\lambda t^*_\lambda
t_\mu t^*_\mu \not=0$ and $t_\nu t^*_\nu t_\lambda t^*_\lambda
\not= 0$. Hence $\lambda = \nu\nu'$ and $\lambda = \mu \alpha$
for some $\nu'$ and $\alpha$, and since $|\mu| < |\nu|$, this
forces $\nu = \mu\mu'$ and hence $\lambda = \mu\mu'\nu'$ for
some $\mu'$.
We have
\begin{equation}\label{eq rho is a loop}
0 \neq t_\lambda t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^*
= t_\lambda t_\lambda^* t_\mu (t_{\mu'}^* t_\mu^*) (t_\mu t_{\mu'} t_{\nu'}) t_\lambda^*
= t_\lambda t_\lambda^* t_\mu t_{\nu'} t_\lambda^*,
\end{equation}
forcing $s(\mu) = r(\nu')$. Since $r(\mu') = s(\mu)$ and
$s(\mu') = r(\nu')$, $\mu'$ is a cycle, and has nonzero length
since $|\nu| > |\mu|$. Furthermore, continuing from \eqref{eq
rho is a loop}
\[
0 \neq t_\lambda t_\lambda^* t_\mu t_{\nu'} t_\lambda^*
= (t_\mu t_{\mu'\nu'}) (t_{\mu'\nu'}^* t_\mu^*) t_\mu t_{\nu'} (t_{\nu'}^* t_{\mu\mu'}^*)
= t_\mu t_{\mu'\nu'} t_{\mu'\nu'}^* t_{\nu'} t_{\nu'}^* t_{\mu\mu'}^*,
\]
so $t^*_{\mu'\nu'} t_\nu'$ is nonzero, forcing
\begin{equation}\label{eq:munu=nurho}
\mu' \nu' = \nu' \rho\qquad\text{for some $\rho \in E$.}
\end{equation}
We claim $\rho$ is a cycle. We proceed by induction on
$|\nu'|$. As a base case, suppose that $|\nu'| \le |\mu'|$.
Then $\nu' = \mu'_{[0,|\nu'|]}$, so
\[
\mu' \nu' = \nu' \rho \implies \rho = \mu'_{[|\nu'|,|\mu'|]} \mu'_{[0,|\nu'|]},
\]
whence $r(\rho) = s(\rho) = s(\nu')$.
Now suppose as an inductive hypothesis that $\rho$ is a cycle
whenever~\eqref{eq:munu=nurho} holds with $|\nu'| < n$ for some
$n > |\mu'|$, and fix $\nu'$ with $|\nu'| = n$
satisfying~\eqref{eq:munu=nurho}. In particular, $|\nu'| >
|\mu'|$, so $\mu' =\nu'_{[0,|\mu'|]}$ and $\mu' \nu'_{[|\mu'|,
|\nu'|]} = \nu' = \nu'_{[|\mu'|, |\nu'|]} \rho$. Since
$|\nu'_{[|\mu'|, |\nu'|]}| = |\nu'| - |\mu'| < n$, the
inductive hypothesis now implies that $\rho$ is a cycle.
Finally, from \eqref{eq rho is a loop},
\begin{flalign*}
&&0 \not= t_\lambda t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^*
&= t_\lambda t_\lambda^* t_\mu t_{\nu'} t_\lambda^* &\\
&&&= t_\lambda (t_{\mu'\nu'}^* t_\mu^*) t_\mu t_{\nu'} t_\lambda^*
= t_\lambda t_{\nu' \rho}^* t_{\nu'} t_\lambda^*
= t_\lambda t_\rho^* t_{\nu'}^* t_{\nu'} t_\lambda^* = t_\lambda t_\rho^* t_\lambda^*,&\qedhere
\end{flalign*}
\end{proof}
Given a directed graph $E$ and $e \in E^1$, define $\Sap_e \in \mathcal B(\ell^2(\apbdry{E}))$ by
\[
\Sap_e \xi_x
= \begin{cases}
\xi_{ex} &\text{ if $s(e) = r(x)$}\\
0 &\text{ otherwise.}
\end{cases}
\]
Then with $\{\Pap_v : v \in E^0\}$ as on
page~\pageref{pg:apboolean}, $(\Sap,\Pap)$ is a
Toeplitz-Cuntz-Krieger $E$-family, and $\Sap_\lambda
(\Sap_\lambda)^* = \Pap_\lambda$ for all $\lambda \in E^*$. We
denote $C^*(\Sap,\Pap)$ by $C^*_{\mathrm{min}}(E)$.
\begin{prop}\label{prp:expectation}
Let $E$ be a directed graph, and suppose that every cycle in
$E$ has an entrance. Let $(t,q)$ be a Toeplitz-Cuntz-Krieger
$E$-family. Then $q : \lambda \mapsto t_\lambda t^*_\lambda$ is
a boolean representation of $E$, and there exists a conditional
expectation $\Phi_{t,q} : C^*(t,q) \to \clsp\{q_\lambda :
\lambda \in E^*\}$ satisfying $\Phi_{t,q}(t_\mu t^*_\nu) =
\delta_{\mu,\nu} q_\mu$. In particular we have
\begin{equation}\label{eq:compatible}
\psi_{q} \circ \Phi_{t,q} \circ \pi_{t,q} = \Phi_{\Sap,\Pap} \circ \pi_{\Sap,\Pap}.
\end{equation}
\end{prop}
\begin{proof}
It is standard that the $q_\lambda$ form a boolean
representation of $E$.
\textbf{Claim~1.} Given a finite subset $F$ of $E^*$ and a
collection $\{a_{\mu,\nu} : \mu,\nu \in F\}$ of scalars,
\[
\Big\|\sum_{\mu \in F} a_{\mu,\mu} q_\mu\Big\| \le \Big\|\sum_{\mu,\nu \in F} a_{\mu,\nu} t_\mu t^*_\nu\Big\|.
\]
By enlarging $F$ (and setting the extra scalars equal to zero),
we may assume that $F$ is closed under initial segments in the
sense that if $\mu\nu \in F$ then $\mu \in F$.
For each $\lambda \in F$, let $T^F_\lambda := \{\lambda' \in
s(\lambda)E^* : \lambda\lambda' \in F, |\lambda'| > 0\}$. For
each $\lambda \in F$ such that $T^F_\lambda$ is not exhaustive,
fix a path $\alpha^\lambda$ such that $\alpha^\lambda \not=
\mu\mu'$ and $\mu \not= \alpha^\lambda\alpha'$ for all $\mu \in
T^F_\lambda$. Since every cycle in $E$ has an entrance,
\cite[Lemma~3.7]{CBMSbk} implies that for each $v$ such that $v
= s(\alpha^\lambda)$ for some $\lambda$, there exists $\tau^v
\in v E^*$ such that either: (1) $s(\tau^v) E^1 = \emptyset$;
or (2) $|\tau^v|
> \max\{|\lambda| : \lambda \in F, T^F_\lambda\text{ is not exhaustive}\}$,
and $\tau^v_k \not= \tau^v_{|\tau^v|}$ for all $k < |\tau^v|$.
We write $\tau^\lambda$ for $\tau^{s(\alpha^\lambda)}$.
For each $\lambda \in F$ we define
\[
\phi^F_\lambda := \begin{cases}
q_{\lambda\alpha^\lambda\tau^\lambda} &\text{ if $T^F_\lambda$ is not exhaustive} \\
q^F_\lambda &\text{ otherwise}.
\end{cases}
\]
By definition we have each $\phi^F_\lambda \le q^F_\lambda$,
and since the $q^F_\lambda$ are mutually orthogonal, it follows
that the $\phi^F_\lambda$ are also. Hence
\[
\Big\|\sum_{\mu,\nu \in F} a_{\mu,\nu} t_\mu t^*_\nu\Big\|
\ge \Big\|\sum_{\lambda \in F} \phi^F_\lambda \Big(\sum_{\mu,\nu \in F} a_{\mu,\nu} t_\mu t^*_\nu\Big) \phi^F_\lambda \Big\|.
\]
Fix $\lambda,\mu,\nu \in F$. We claim that
\begin{equation}\label{eq:comp by phi}
\phi^F_\lambda t_\mu t^*_\nu \phi^F_\lambda
= \begin{cases}
\phi^F_\lambda &\text{ if $\mu=\nu$ and $\lambda = \mu\lambda'$ for some $\lambda'$} \\
0 &\text{ otherwise.}
\end{cases}
\end{equation}
To see this, suppose first that $\mu = \nu$. If $\lambda =
\mu\lambda'$ then $\phi^F_\lambda \le t_\lambda t^*_\lambda \le
t_\mu t^*_\mu$ by definition of $\phi^F_\lambda$. If $\lambda
\not= \mu\lambda'$, then either $\mu = \lambda\mu'$ in which
case $\phi^F_\lambda \le (t_\lambda t^*_\lambda -
t_{\lambda\mu'} t^*_{\lambda\mu'}) \perp t_\mu t^*_\mu$, or
else $\mu \not= \lambda\mu'$ in which case $\phi^F_\lambda \le
t_\lambda t^*_\lambda \perp t_\mu t^*_\mu$.
Now suppose that $\mu \not= \nu$; by symmetry under adjoints,
we may assume that $|\mu| < |\nu|$. We must show that
$\phi^F_\lambda t_\mu t^*_\nu \phi^F_\lambda = 0$. Since
$\phi^F_\lambda \le t_\lambda t^*_\lambda$, if $t_\lambda
t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^* = 0$ then we
are done, so we may assume that $t_\lambda t_\lambda^* t_\mu
t_\nu^* t_\lambda t_\lambda^* \not= 0$. Then Lemma~\ref{L rho
is a loop} implies that $\lambda = \nu\nu' = \mu\mu'\nu'$ and
that $t_\lambda t_\lambda^* t_\mu t_\nu^* t_\lambda t_\lambda^*
= t_\lambda t_{\lambda \rho}^*$ for some cycle $\rho \in E$.
Hence $\phi^F_\lambda \le t_\lambda t^*_\lambda$ forces
\[
\phi_\lambda^F t_\mu t^*_\nu \phi_\lambda^F
= \phi_\lambda^F t_\lambda t^*_{\lambda\rho} \phi_\lambda^F.
\]
We consider two cases: $T^F_\lambda$ is exhaustive, or it is
not.
First suppose that $T^F_\lambda$ is exhaustive. Fix $n$ such
that $n|\rho| > \max\{|\lambda'| : \lambda' \in T^F_\lambda\}$.
Then $\rho^n = \lambda' \beta$ for some $\lambda' \in
T_\lambda^F$ and $\beta \in E^*$. In particular, $\lambda' =
\rho^n_{[0,|\lambda'|]},$ forcing $\rho_1 = \lambda'_1$. Since
$\lambda\lambda' \in F$ and $F$ is closed under initial
segments, $\lambda \rho_1 \in F$ and then
\[
t_{\lambda\rho}t^*_{\lambda\rho} \phi_\lambda^F
\le t^*_{\lambda \rho} t^*_{\lambda \rho} (t_\lambda t^*_\lambda - t_{\lambda \rho_1}t^*_{\lambda \rho_1}) = 0,
\]
giving $\phi^F_\lambda s_\mu s^*_\nu \phi^F_\lambda = 0$.
Now suppose that $T^F_\lambda$ is not exhaustive. Then
$\phi^F_\lambda = q_{\lambda\alpha^\lambda\tau^\lambda}$. We
then have
\[
\phi_\lambda^F t_\mu t^*_\nu \phi_\lambda^F
= t_{\lambda\alpha^\lambda\tau^\lambda} t^*_{\mu'\nu'\alpha^\lambda\tau^\lambda}
t_{\nu'\alpha^\lambda\tau^\lambda} t^*_{\lambda\alpha^\lambda\tau^\lambda}.
\]
This is nonzero only if $\mu'\nu'\alpha^\lambda\tau^\lambda =
\nu'\alpha^\lambda\tau^\lambda\zeta$ for some $\zeta$. This is
impossible if $s(\tau^\lambda)E^1 = \emptyset$, so suppose that
$s(\tau^\lambda)E^1 \not= \emptyset$. By choice of
$\tau^\lambda$, we have $|\tau^\lambda|
> |\mu'|$. Let $m = |\nu'\alpha^\lambda\tau^\lambda|$. Then
\[
(\mu'\nu'\alpha^\lambda\tau^\lambda)_m
= \tau^\lambda_{|\tau^\lambda| - |\mu'|}
\not= \tau^\lambda_{|\tau^\lambda|}
= (\nu'\alpha^\lambda\tau^\lambda)_m,
\]
so $\mu'\nu'\alpha^\lambda\tau^\lambda \not=
\nu'\alpha^\lambda\tau^\lambda\zeta$ for all $\zeta$, and hence
$\phi^F_\lambda t_\mu t^*_\nu \phi^F_\lambda = 0$,
establishing~\eqref{eq:comp by phi}.
We now have
\begin{align*}
\Big\|\sum_{\mu \in F} a_{\mu,\mu} q_\mu\Big\|
&= \Big\|\sum_{\mu \in F} \Big(\sum_{n \le |\mu|} a_{\mu_{[0,n]},\mu_{[0,n]}}\Big) q^F_\mu \Big\| \\
&= \max_{\mu \in F} \Big|\sum_{n \le |\mu|} a_{\mu_{[0,n]},\mu_{[0,n]}}\Big| \\
&= \Big\|\sum_{\mu \in F} \Big(\sum_{n \le |\mu|} a_{\mu_{[0,n]},\mu_{[0,n]}}\Big) \phi^F_\mu \Big\| \\
&= \Big\|\sum_{\lambda \in F} \phi^F_\lambda \Big(\sum_{\mu,\nu\in F} a_{\mu,\nu} t_\mu t^*_\nu\Big) \phi^F_\lambda\Big\|
\qquad\text{by~\eqref{eq:comp by phi}}\\
&\le \Big\|\sum_{\mu,\nu \in F} a_{\mu,\nu} t_\mu t^*_\nu\Big\|
\end{align*}
completing the proof of Claim~1.
Claim~1 implies that the formula $t_\mu t^*_\nu \mapsto
\delta_{\mu,\nu} t_\mu t^*_\mu$ extends to a well-defined
linear map $\Phi_{t,q}$ from $C^*(t,q)$ to $\clsp\{q_\lambda :
\lambda \in E^*\}$. This $\Phi_{t,q}$ is a linear idempotent of
norm 1, and hence a conditional expectation (see for example
\cite[Definition~II.6.10.2 and
Theorem~II.6.10.2]{Blackadar2006}).
The final statement is straightforward since the two maps in
question agree on spanning elements.
\end{proof}
\begin{lemma}\label{lem:cu exp faithful}
Let $E$ be a directed graph in which every cycle has an
entrance. Then the expectation $\Phi_{\Sap,\Pap} :
C^*_{\mathrm{min}}(E) \to \clsp\{\Pap_\lambda : \lambda \in
E^*\}$ obtained from Proposition~\ref{prp:expectation} is
faithful on positive elements.
\end{lemma}
\begin{proof}
It suffices to show that for $a \in C^*_{\mathrm{min}}(E)$,
$\Phi_{\Sap,\Pap}(a)$ is equal to the strong-operator limit
$\sum_{x \in \apbdry{E}} \big(a \xi_x | \xi_x) \theta_{\xi_x,
\xi_x}$ for all $a \in C^*_{\mathrm{min}}(E)$, where the
$\xi_x$ are the canonical orthonormal basis for
$\ell^2(\apbdry{E})$ and $\theta_{\xi_x, \xi_x}$ is the
rank-one projection onto $\field{C} \xi_x$. Fix $\mu,\nu \in E^*$ and
$x \in \apbdry{E}$. We have
\[
\big(\Sap_\mu (\Sap_\nu)^* \xi_x \big| \xi_x\big)
= \big((\Sap_\nu)^* \xi_x \big| \big (\Sap_\mu)^*\xi_x\big)
= \begin{cases}
1 &\text{ if $x = \nu y = \mu y$ for some $y$}\\
0 &\text{ otherwise.}
\end{cases}
\]
Since $x \in \apbdry{E}$, we have $y \not= \rho^\infty$ for any
cycle $\rho$, so $\mu y = \nu y$ forces $|\mu| = |\nu|$, and
hence $\mu = \nu$.
Hence
\[
\sum_{x \in \apbdry{E}}
\big(\Sap_\mu (\Sap_\nu)^* \xi_x | \xi_x) \theta_{\xi_x, \xi_x}
= \delta_{\mu,\nu} \proj_{\clsp\{\xi_x : x \in \mu\apbdry{E}\}}
= \delta_{\mu,\nu} \Pap_\mu
\]
as required.
\end{proof}
We now have the tools we need to prove the main theorem.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
We showed that the $\Pap_\lambda$, and in particular the
$\Pap_v$ are nonzero immediately subsequent to their
definition.
Fix a Toeplitz-Cuntz-Krieger $E$-family $(t,q)$ with each $q_v$
nonzero. We will show that $\ker(\pi_{t,q}) \subset
\ker(\pi_{\Sap, \Pap})$ in $\mathcal T C^*(E)$, and hence that
$\pi_{\Sap, \Pap}$ descends to the desired homomorphism
$\psi_{t,q} : C^*(t,q) \to C^*_{\mathrm{min}}(E)$.
Since each $t_\lambda^* t_\lambda = q_{s(\lambda)}$, each
$q_\lambda$ is nonzero, so Proposition~\ref{prp:diag cu}
implies that there is a homomorphism $\psi_q : \clsp\{q_\lambda
: \lambda \in E^*\} \to \clsp\{\Pap_\lambda : \lambda \in
E^*\}$ taking each $q_\lambda$ to $\Pap_\lambda$.
We calculate
\begin{flalign}
&&\pi_{t,q}(a) = 0
&\iff \pi_{t,q}(a^*a) = 0 \nonumber & \\
&&&\implies \psi_q \circ \Phi_{t,q} \circ \pi_{t,q}(a^*a) = 0 \label{eq:key implic} & \\
&&&\iff \Phi_{\Sap,\Pap}(\pi_{\Sap,\Pap}(a^*a)) = 0 \quad\text{ by~\eqref{eq:compatible}} \nonumber & \\
&&&\iff \pi_{\Sap,\Pap}(a^*a) \quad\text{ by Lemma~\ref{lem:cu exp faithful}} \nonumber & \\
&&&\iff \pi_{\Sap,\Pap}(a) = 0. \nonumber &\qedhere
\end{flalign}
\end{proof}
It is, of course, interesting to know when the homomorphism
$\psi_{t,q}$ of Theorem~\ref{thm:main} is injective.
\begin{theorem}
Let $E$ be a directed graph in which every cycle has an
entrance.
\begin{enumerate}
\item\label{it:couniv} If $(t,q)$ is a
Toeplitz-Cuntz-Krieger family with each $q_v$ nonzero,
then the homomorphism $\psi_{t,q}$ of
Theorem~\ref{thm:main} is injective if and only if (a)
$\prod_{\lambda \in F} (q_v - q_\lambda) = 0$ whenever
$v \in E^0$ and $F \subset vE^*$ is finite exhaustive;
and (b) the expectation $\Phi_{t,q}$ is faithful.
\item\label{it:injective} If $\pi$ is homomorphism from
$C^*_{\mathrm{min}}(E)$ to a $C^*$-algebra $C$ such
that each $\pi_{\Pap_v}$ is nonzero, then $\pi$ is
injective.
\end{enumerate}
\end{theorem}
\begin{proof}
(\ref{it:couniv}) Since conditions (a)~and~(b) hold in
$C^*_{\mathrm{min}}(E)$, the ``only if" implication is trivial.
For the ``if" implication, note that given a
Toeplitz-Cuntz-Krieger $E$-family $(t,q)$, we have
$\ker(\pi_{t,q}) = \ker(\pi_{\Sap, \Pap})$ whenever the
implication~\eqref{eq:key implic} is equivalence. Condition~(a)
implies that $\psi_q$ is faithful by the final statement of
Proposition~\ref{prp:diag cu}, and this combined with~(b)
implies that $\psi_q \circ \Phi_{t,q}$ is faithful on positive
elements, giving
\[
\pi_{t,q}(a^*a) = 0 \iff \psi_q \circ \Phi_{t,q} \circ \pi_{t,q}(a^*a) = 0
\]
as required.
(\ref{it:injective}) Define a Toeplitz-Cuntz-Krieger $E$-family
by $t_e := \pi(\Sap_e)$ and $q_v := \pi(\Pap_v)$.
Theorem~\ref{thm:main} supplies a homomorphism $\psi_{t,q} :
C^*(t,q) \to C^*_{\mathrm{min}}(E)$ which is an inverse for
$\pi$.
\end{proof} |
2107.06095 | \section{Introduction}
\par Metal hydrides are compounds that form when certain metals react with hydrogen. The hydriding reaction is reversible: the metal absorbs hydrogen exothermically at certain temperatures and pressures and desorbs hydrogen endothermically under other conditions. The search for new energy storage technologies has led to investigation of metal hydrides for this purpose, including using hydrides as a compact form of hydrogen storage for fuel cells \cite{Tange2011ExperimentalSystem,Khayrullina2018AirApplications,Aruna2020ModelingBed,Cho2013DynamicTank,Panos2010DynamicTank,Georgiadis2009DesignStorage} and for energy storage integrated with concentrated solar power generation \cite{Corgnale2016MetalPlants,Feng2019OptimumStorage,Sheppard2016MetalStorage,SatyaSekhar2012TestsSystem,Nyamsi2018SelectionTargets}. Systems with multiple metal hydride reactors have also been evaluated for heating and cooling applications, where the hydride reactors are used in place of a heat pump \cite{Kang1995ThermalConditioning,Muthukumar2010MetalReview,Nagel1984OperatingAir,Magnetto2006ASystem,Satheesh2010PerformancePump,Satheesh2010SimulationPump,Nie2011MetalDriving}. A common design for metal hydride systems in energy storage, heating, and cooling applications is to have a pair of metal hydride reactors connected so that hydrogen can flow between them. In other words, as hydrogen is desorbed in one reactor, it flows to the other reactor, which absorbs it.
\par When used for energy storage, the two-reactor system does not run continuously but instead absorbs heat from an external source during one half-cycle and releases it during the other half-cycle \cite{Corgnale2016MetalPlants,Sheppard2016MetalStorage,Nyamsi2018SelectionTargets,Manickam2019FutureHydrides,Ward2016TechnicalSystems}. In this application, only one reactor is directly used for energy storage, while the other reactor is used to store the hydrogen released by that reactor \cite{Corgnale2016MetalPlants}. In general, two hydride beds are not required for energy storage systems, since hydrogen released by the metal hydride can be compressed, stored, and released from a pressure vessel \cite{Feng2019OptimumStorage,Sheppard2016MetalStorage,SatyaSekhar2012TestsSystem}. However, a two-reactor design is commonly studied since it allows for more compact hydrogen storage without the need to compress hydrogen to high pressures. The two-reactor metal hydride system is also used in single-stage single-effect air conditioning systems, where the hydride reactions are controlled by changing the heat sources and sinks connected to the reactors \cite{Muthukumar2010MetalReview}. The pair of hydride reactors undergo a two-step cycle. In the first (charging) step, the hydride in one reactor is heated by a high-temperature heat source and desorbs hydrogen that flows to the other reactor. Heat is released as the hydrogen is absorbed in the second reactor, and in turn, the heat is transferred to a heat sink at an intermediate temperature. In the second (discharging) step, the first reactor is connected to the intermediate-temperature heat sink while the second reactor is connected to the environment to which the cooling load needs to be delivered. It absorbs heat from this location, providing the cooling load, as it desorbs hydrogen that then flows to the other reactor \cite{Muthukumar2010MetalReview}. In order for this system to operate as intended, the metal hydrides have to be selected so that in each step, changing the heat source or sink connected to the reactor results in a temperature shift large enough to drive the desorption or absorption reaction \cite{Satheesh2010PerformancePump} respectively.
\par If this system design is used for continuous cooling instead of energy storage, a four-reactor system is necessary, with one reactor pair in each step of the cycle at any given time \cite{Kang1995ThermalConditioning,Muthukumar2010MetalReview,Nagel1984OperatingAir}. An alternative design, which uses a \emph{single pair} of connected reactors, provides a continuous cooling load by using a compressor to move hydrogen between the reactors, thus driving the reaction in each reactor by changing the pressure, instead of the temperature
\cite{Magnetto2006ASystem}. This system design requires additional power for operating the compressor, but allows for more flexibility in choosing the specific hydrides used in the reactors. However, to realize the benefits of this system, specifically to guarantee specific heat transfer rates between the hydride reactors and a circulating fluid, a real-time embedded control strategy for the system is needed.
\par Numerous models of the heat and mass transfer in metal hydride reactors have been developed. The majority of these focus on a single metal hydride reactor, including 1-D models \cite{Choi1990HeatApplications,Abdin2018One-dimensionalMatlabSimulink} and 2-D models \cite{Nakagawa2000NumericalBed,Jemni1999ExperimentalReactor,Brown2008AccurateTank}. More computationally-intensive 3-D models of a hydride reactor, first developed by \citet{Aldas2002ABed}, have been used to analyze the performance benefits of complicated system geometries, such as the addition of fins \cite{Askri2009HeatMmNi4.6Fe0.4} or a tapered bed structure \cite{Feng2019OptimumStorage} to improve heat transfer. With respect to two-reactor systems, modeling has been more limited;~\citet{Kang1995ThermalConditioning} used a 1-D model to examine a metal hydride air conditioning system and \citet{Nyamsi2018SelectionTargets} used a 3-D model to examine a pair of hydride reactors used for energy storage. Moreover, neither of these models considered a pair of hydride reactors with an auxiliary hydrogen compressor.
Similarly, control strategies for metal hydride reactors have focused on single-reactor systems used for hydrogen storage; relevant literature is summarized in Table~\ref{tab:lit_rev}. Since these papers are focused on a single-reactor system for hydrogen storage, the output variable of the controllers is either the hydrogen flow rate or the temperature or pressure of the reactor, with the latter generally being used because of its influence on the reaction rate in the reactor. For instance, while
\citet{Georgiadis2009DesignStorage} and \citet{Panos2010DynamicTank} use outlet temperature as the output variable of the controller, they do so in order to achieve the outlet temperature that will result in the optimal hydrogen release rate. Similarly, the input variables considered in all of these papers, except for \citet{Cho2013DynamicTank}, are those used to control the heat source used to drive the desorption in the reactor, whether that be the flow rate \cite{Georgiadis2009DesignStorage,Panos2010DynamicTank} or temperature \cite{Aruna2020ModelingBed} of a heating fluid, or the voltage of a thermoeletric heater \cite{Nuchkrua2013Neuro-fuzzyReactor}. While similar inputs would be considered for a case where the hydride reactor was being used for energy storage or air conditioning, the output variable for such cases would be the heat transfer to or from the reactor, which is not considered by any of these controllers.
\begin{table}[!htb]
\caption{Summary of the controller types and input and output variables that have previously been considered for single reactor metal hydride systems.}
\begin{center}
\resizebox{\hsize}{!}{\begin{tabular}{p{25 mm} p{30 mm} p{25 mm} p{25 mm}}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
\textbf{Author(s)} & \textbf{Controller} & \textbf{Control} \newline \textbf{Variable} & \textbf{Output} \newline \textbf{Variable} \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
\citet{Georgiadis2009DesignStorage} 2009 & Multi-parametric (mp) MPC & Heating Fluid Flow Rate & Outlet \newline Temperature \\
\citet{Panos2010DynamicTank} 2010 & mp-MPC & Heating Fluid Flow Rate & Outlet \newline Temperature \\
\citet{Cho2013DynamicTank} 2013 & PID & Discharge Valve Opening & Hydrogen Flow Rate \\
\citet{Nuchkrua2013Neuro-fuzzyReactor} 2013 & Neuro-fuzzy PID & Thermoelectric Voltage & Reactor \newline Temperature \\
\citet{Aruna2020ModelingBed} 2020 & Fuzzy PID & Heating \newline Temperature & Reactor \newline Pressure \\
\noalign{\vskip 1mm}
\hline
\end{tabular}}
\end{center}
\label{tab:lit_rev}
\end{table}
\par Different models of the hydride system are also used for controller design. For instance, \citet{Georgiadis2009DesignStorage} use a 2-D model to determine the temperature and pressure variations within the reactor, whereas \citet{Cho2013DynamicTank} use a model that neglects the effect of these variations. The linear models used for control synthesis are obtained using model identification techniques and simulation data from the system model \cite{Georgiadis2009DesignStorage,Panos2010DynamicTank,Aruna2020ModelingBed}. \citet{Aruna2020ModelingBed} examine different methods of converting their nonlinear model into a linear model for use in a controller, and find the best results when using a Box-Jenkins model. All of the methods compared are black-box methods \cite{Aruna2020ModelingBed}; that is, they develop the linear model using only input and output data rather than deriving such a model from first principles. This poses a disadvantage from the perspective of control design because it limits the operation of the controller to a small region around the linearization point.
With respect to two-reactor systems, experimental demonstrations \cite{Nagel1984OperatingAir,Magnetto2006ASystem} used logic-based controllers to operate the system, but did not involve the design of real-time controllers that could achieve specific heat transfer rates at specific times, as would be desired for an energy storage system which is charged and discharged over a longer period of time and may not be charged or discharged continuously. These experimental systems have also used either heat transfer from a fluid or a compressor to drive the system operation, but have not examined how to control a system that uses both. In summary, previous work on control strategies for metal hydride reactors has not considered real-time control for a two-reactor system using both temperature and pressure to drive operation, nor control using a linear model derived from the system dynamics rather than from a black-box model.
\par Therefore, in this paper, we present a dynamic model for a metal hydride energy storage system along with a model predictive control strategy for tracking the desired heat transfer rates in each reactor of a two-reactor metal hydride system. Specifically, in Section~\ref{sec:model}, we present the dynamic model of the metal hydride energy storage system including two metal hydride reactors and a compressor to drive hydrogen flow. Then, we derive the linear state-space form of the model using first principles. The linearized model is validated against the nonlinear one, and we highlight how the ability to update the linearized model enables better agreement between the linear and nonlinear models. The resulting model is then used for control synthesis. Because the linearized model is physics-based and fully parameterized, it can be relinearized around any operating condition in real time, enabling it to be used beyond the nominal operating condition. In Section~\ref{sec:control}, we present the model predictive control (MPC) design for determining the mass flow rates of the circulating fluid and the pressure difference between the hydrogen reactors needed to meet the demanded heat transfer rate in each reactor. Finally, in Section~\ref{sec:results}, we demonstrate the performance of the controller in simulation with a series of case studies. The results indicate that the controller can achieve desired heat transfer rates between hydride beds through control of the mass flow rates of the circulating fluids and the compressor pressure. This controller enables the use of a two-reactor system for energy storage and/or integration with a heat pump.
\section{Model Description}
\label{sec:model}
Here we consider a thermal energy storage system consisting of two interconnected metal-hydride reactors with a compressor to drive hydrogen flow between them, as shown in Figure~\ref{fig:Hyd_Schematic_reactors}. Each reactor is modeled as a shell-and-tube heat exchanger (depicted in Figure~\ref{fig:Hyd_Schematic_shellandtube}) absorbing heat from, and releasing heat to, a circulating fluid that interacts with a heat pump. We first derive a dynamic, nonlinear thermodynamic model of the two-reactor system using first principles. We then linearize the model to obtain a form suitable for control synthesis, and validate the linear model against the nonlinear one.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.9\textwidth}
\includegraphics[width=\linewidth]{figures/System_diagram_top.png}
\caption{Two-reactor metal hydride energy storage system}
\label{fig:Hyd_Schematic_reactors}
\end{subfigure}
\begin{subfigure}{0.9\textwidth}
\includegraphics[width=\linewidth]{figures/System_diagram_bottom.png}
\caption{Cross-sectional view of a reactor. Seven tubes are shown for clarity, while the proposed system has 400 tubes within the shell.}
\label{fig:Hyd_Schematic_shellandtube}
\end{subfigure}
\caption{Schematic of the metal hydride energy storage system. (a) Two metal hydride reactors are connected so that hydrogen can flow between them. A compressor drives hydrogen flow when the thermally-driven pressure gradient is unfavorable. (b) Each reactor is a shell-and-tube heat exchanger with multiple tubes for the circulating fluid flow and a single shell filled with metal hydride powder. To simplify the heat exchanger model, the analyzed control volume consists of a single tube and the surrounding metal hydride as shown.}
\label{fig:Hyd_Schematic}
\end{figure}
\subsection{System Model}
\par Each metal hydride reactor is modeled as a shell-and-tube heat exchanger. The shell is filled with a porous bed of metal hydride through which hydrogen flows. Water glycol circulates in the tubes and exchanges heat with the hydride bed. The heat exchanger is modelled using a control volume consisting of one tube and the shell surrounding it, as shown in Figure~\ref{fig:Hyd_Schematic_shellandtube}, with each other tube and surrounding shell assumed to be the same by symmetry. The perimeter of the heat exchanger is assumed to be insulated, so heat losses from the heat exchanger to ambient are neglected.
\par Within the control volume, it is assumed that the hydrogen in the reactors is an ideal gas, that the hydrogen and hydride in the reactor are in thermal equilibrium ($T_{hyd}=T_{H}$), and that all spatial variation in temperature and pressure are negligible. Given these assumptions, the energy balance for a single control volume is given by
\begin{align}
\left (m_{hyd}c_{hyd} + m_{H}c_{p,H} \right )\frac{dT_{hyd}}{dt} = rm_{hyd}\Delta H + \epsilon \dot{m}_{wg}c_{wg}\left ( T_{wg,in} - T_{hyd} \right) \notag \\
+ \dot{m}_{H,in}c_{p,H}\left ( T_{H,in} - T_{hyd} \right), \label{eq:ebal}
\end{align}
where $m_{hyd}$ and $m_H$ are the mass of the hydride and the hydrogen, respectively; $c_{hyd}$, $c_{p,H}$, and $c_{wg}$ are the specific heat of the hydride, hydrogen, and water glycol, respectively; $T_{hyd}$ is the temperature of the metal hydride in the reactor, $r$ is the reaction rate, $\Delta H$ is the heat of reaction, $\epsilon$ is the heat exchanger effectiveness, $\dot{m}_{wg}$ and $\dot{m}_{H,in}$ are the mass flow rates of water glycol and hydrogen into the reactor, and $T_{wg,in}$ and $T_{H,in}$ are the inlet temperatures of water glycol and hydrogen, respectively. The mass balance for the hydrogen gas is given by
\begin{equation}\label{eq:mbal}
\frac{dm_{H}}{dt} =\ rm_{hyd} + \dot{m}_{H,in}.
\end{equation}
Using the ideal gas law, the pressure of the hydrogen gas, $P_{H}$, can be obtained from the mass of hydrogen as
\begin{equation}\label{eq:igas}
P_{H} =\ \frac{m_{H}R_{H}T_{H}}{V_{H}}.
\end{equation}
In this equation, $R_{H}$ is the gas constant for hydrogen. The volume of hydrogen ($V_{H}$) can be found using the porosity of the metal hydride bed ($\phi$) and the total volume of the hydride section of the heat exchanger ($V_{shell}$) according to
\begin{equation}\label{eq:porosity}
V_{H}=\ \phi V_{shell}.
\end{equation}
Using Equations~\ref{eq:igas} and~\ref{eq:porosity} and the thermal equilibrium assumption ($T_{hyd}$ =$T_{H}$), the mass balance can be re-written in terms of pressure as
\begin{equation}\label{eq:pbal}
\frac{dP_{H}}{dt} =\ \frac{RT_{hyd}}{\phi V_{shell}}\left (rm_{hyd} + \dot{m}_{H,in} \right ).
\end{equation}
The reaction rate, $r$, is the rate of change of the weight fraction of hydrogen stored in the metal hydride, $w$. The equations for the reaction rate and equilibrium pressure, $P_{eq}$, are taken from Voskuilen
\cite{Voskuilen2014ASystems}.
The reaction rate is given by
\begin{equation}\label{eq:rrate}
r =\frac{dw}{dt} = \ \left\{\begin{matrix} C_{A}e^{-\frac{E_{A}}{RT_{hyd}}}ln\left ( \frac{P}{P_{eq}} \right )\left ( w_{max} - w \right ), & P > P_{eq,abs} \\ 0, & P_{eq,des} < P < P_{eq,abs}
\\ C_{A}e^{-\frac{E_{A}}{RT_{hyd}}}ln\left ( \frac{P}{P_{eq}} \right )w, & P < P_{eq,des} .
\end{matrix}\right.
\end{equation}
In these equations, $C_{A}$ and $E_{A}$ are constant material properties of the metal hydride, $w_{max}$ is the maximum weight fraction of the metal hydride, and the equilibrium pressures are given by
\begin{equation}\label{eq:Peq}
P_{eq}=\ P_{atm} e ^{\frac{\mu\left ( w,T \right )}{RT}},
\end{equation}
where the chemical potential, $\mu(w,T)$, is defined as
\begin{equation}\label{eq:mu}
\mu =\ \left\{\begin{matrix}\mu_{\alpha,0} + 2RT_{c}\left (1-2\frac{w}{w_{max}} \right ) + RT ln \left ( \frac{w}{w_{max}-w}\right ), & \ w < w_{\alpha,0}\\
\Delta H\degree - T \Delta S\degree + A \left ( \frac{w}{w_{max}} - 0.5 \right ), & w_{\alpha,0} < w < w_{\beta,0}\\
\mu_{\beta,0} + 2RT_{c}\left (1-2\frac{w}{w_{max}} \right )+ RT ln \left ( \frac{w}{w_{max}-w}\right ), & w > w_{\beta,0}
\end{matrix}\right.
\end{equation}
\noindent In this equation, $\alpha$ and $\beta$ are different phases of the metal hydride. The material properties $\Delta H\degree$ and $\Delta S\degree$ have different values for absorption and desorption, resulting in different equilibrium pressures for absorption and desorption.
\par In Equations~\ref{eq:Peq} and~\ref{eq:mu}, $P_{atm}$ is atmospheric pressure, $w_{\alpha,0}$ and $w_{\beta,0}$ are the weight fractions at which the hydride phase changes, $\mu_{\alpha,0}$ and $\mu_{\beta,0}$ are the chemical potential at $w_{\alpha,0}$ and $w_{\beta,0}$, respectively, and $T_{c}$ and $A$ are the critical temperature and phase interaction energy of the metal hydride, respectively. In the energy balance, the heat exchanger effectiveness is calculated using the expression
\begin{equation}\label{eq:hxeff}
\epsilon =\ 1 - e^{-\frac{hA_{S}}{\dot{m}_{wg}c_{wg}}},
\end{equation}
where $h$ is the heat transfer coefficient and $A_s$ is the surface area of the tube. The heat transfer coefficient is calculated from the Reynolds number, $Re_{D}$, Prandtl number, $Pr$, and thermal conductivity, $k$, of the fluid using the Dittus-Boelter correlation for fully-developed turbulent flow in a circular tube
\cite{Incropera2011FundamentalsEdition}:
\begin{equation}\label{eq:hconv}
Nu =\ \frac{hD_{tube}}{k_{wg}} =\ 0.023 Re_{D}^{4/5} Pr^{n}.
\end{equation}
\noindent Then, the outlet temperature of the circulating fluid, $T_{wg,out}$ can be calculated as
\begin{equation}\label{eq:Tout}
T_{wg,out}=\ T_{wg,in} + \epsilon \left ( T_{hyd} - T_{wg,in} \right ).
\end{equation}
Thus, the heat transfer between the hydride and the water glycol, $\dot{Q}_{hyd \rightarrow wg}$, is given by
\begin{equation}\label{eq:Qdot}
\dot{Q}_{hyd \rightarrow wg}=\ \dot{m}c_{wg}\left ( T_{wg,out}-T_{wg,in} \right ) = \epsilon \dot{m}c_{wg}\left ( T_{hyd}-T_{wg,in} \right ).
\end{equation}
\noindent Together, Equations \ref{eq:ebal} through \ref{eq:Qdot} are used to solve for the state of each reactor. The reactors are coupled through the mass flow rate of hydrogen into the reactor, which is calculated for one reactor as
\begin{equation}\label{eq:mdotH_BtoA}
\dot{m}_{H,A}= \left\{\begin{matrix} \ A_{c, H line}\sqrt{\frac{2 \rho_{H} \left ( P_{B} + \Delta P_{comp} - P_{A} \right )}{K_{loss}}}, & \text{flow from B to A}\\
\ -A_{c, H line}\sqrt{\frac{2 \rho_{H} \left ( P_{A} + \Delta P_{comp} - P_{B} \right )}{K_{loss}}}, & \text{flow from A to B}
\end{matrix}\right.
\end{equation}
and for the other reactor as
\begin{equation}\label{eq:mdotH_B}
\dot{m}_{H,B}=\ - \dot{m}_{H,A}.
\end{equation}
In these equations, $A_{c,H line}$ is the cross-sectional area of the hydrogen line connecting the reactors, $\rho_{H}$ is the density of hydrogen, $\Delta P_{comp}$ is the pressure difference created by the compressor, and $K_{loss}$ is the loss coefficient for flow from one reactor to the other. As shown in Figure~\ref{fig:Hyd_Schematic}, the compressor can drive flow in either direction, depending on which valves are open.
\subsection{Linear State-Space Model}
\par While the governing dynamics of the two-reactor hydride system are nonlinear, the derived model is not well suited for control algorithm synthesis. An alternative approach is to linearize the dynamics about a nominal operating condition and synthesize a controller based on this linearized model. To that end, we linearize the first principles model derived in the previous subsection using a standard state-space representation as given by Equations \eqref{eq:stspc_x} and \eqref{eq:stspc_y} \cite{Ogata2010ModernEngineering}.
\begin{equation} \label{eq:stspc_x}
f(x,u)=\ \frac{dx}{dt}=\ \mathbf{A}\left ( x - x_{0} \right ) + \mathbf{B}\left ( u - u_{0} \right ) + f(x_{0},u_{0})
\end{equation}
\begin{equation} \label{eq:stspc_y}
g(x,u)=\ y =\ \mathbf{C}\left ( x - x_{0} \right ) + \mathbf{D}\left ( u - u_{0} \right ) + g(x_{0},u_{0})
\end{equation}
\par Here, $x$ is the standard state-space notation for the dynamic state vector of the system, $u$ is the control input vector and $y$ is the output vector. For the case in which one reactor, \textit{m}, is absorbing hydrogen while the other, \textit{n}, is desorbing, the state, output, and input variables are defined as follows:
\begin{equation}\label{eq:def_x}
x=\ \begin{bmatrix}T_{hyd,m}
\ P_{H,m}
\ w_{m}
\ T_{n}
\ P_{H,n}
\ w_{hyd,n}
\end{bmatrix}^{T} \enspace ,
\end{equation}
\begin{equation}\label{eq:def_y}
y=\ \begin{bmatrix}\dot{Q}_{hyd \rightarrow wg,m}
\ \dot{Q}_{hyd \rightarrow wg,n}
\end{bmatrix}^{T}\enspace ,
\end{equation}
\begin{equation}\label{eq:def_u}
u=\ \begin{bmatrix}\dot{m}_{wg,m}
\ \dot{m}_{wg,n}
\ \Delta P_{comp}
\ T_{wg,in,m}
\ T_{wg,in,n}
\end{bmatrix}^{T} \enspace .
\end{equation}
Note that in these equations, the reactor where \textit{absorption} occurs is always referred to as reactor $m$, and the reactor where \textit{desorption} occurs as reactor $n$, even though these are different reactors depending on the operating mode of the system. For brevity, the complete linearized model equations are not presented here. However, the authors have made the linearized model code available to the public via Github \cite{Krane2021HydrideModel}.
\section{Linearized Model Validation}
\label{sec:modelval}
In this section, we validate the linearized model against the nonlinear one. We first describe the specific parameterization of the model based on chosen material properties and system dimensions, followed by a comparison of the linearized and nonlinear models through numerical simulations.
\subsection{Model Parameterization}
Here we consider MmNi\textsubscript{4.5}Cr\textsubscript{0.5} as the material in reactor A and LaNi\textsubscript{5} as the material in reactor B. These materials were selected based on the operating temperatures of an air conditioning system with which this storage system could ultimately be integrated. The properties of these materials are taken from the toolbox developed by \citet{Voskuilen2014ASystems} and are summarized in Table \ref{tab:hyd_props}. By deriving the nonlinear model, and its linearization, from first principles, the model can easily be parameterized for any choice of metal hydrides.
\begin{table}[!htb]
\caption{Material properties of MmNi\textsubscript{4.5}Cr\textsubscript{0.5} (Reactor A) and LaNi\textsubscript{5} (Reactor B). Since the density and specific heat of MmNi\textsubscript{4.5}Cr\textsubscript{0.5} are not reported in literature, they are estimated from values for this class of alloys \cite{Voskuilen2014ASystems}.}
\begin{center}
\begin{tabular}{l c c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
\small{$\textbf{\text{Property}}$} & \small{$\textbf{\text{Symbol}}$} & \small{$\textbf{\text{Units}}$} & \small{$\textbf{\text{LaNi\textsubscript{5}}}$} & \small{$\textbf{\text{MmNi\textsubscript{4.5}Cr\textsubscript{0.5}}}$}\\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
\small{Density} & \small{$\rho$} & \small{kg/$\text{m}^3$} & \small{8300} & \small{8200}\\
\small{Specific Heat} & \small{$c$} & \small{J/kg$\cdot$K} & \small{355} & \small{419}\\
\small{Enthalpy of Reaction (abs.)} & \small{$\Delta H_{a}$} & \small{MJ/kg M} & \small{15.46} & \small{11.67}\\
\small{Enthalpy of Reaction (des.)} & \small{$\Delta H_{d}$} & \small{MJ/kg M} & \small{15.95} & \small{12.65}\\
\small{Maximum Weight Fraction} & \small{$w_{max}$} & \small{kg H/kg M} & \small{0.0151} & \small{0.0121} \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:hyd_props}
\end{table}
\par Each hydride reactor, modeled as a shell-and-tube heat exchanger, has the dimensions described in Table \ref{tab:hx_dim}. The different lengths of each hydride reactor are due to their different storage capacities and thus different volumes of hydride required. The pipe for transferring hydrogen between the reactors has a diameter of 1 cm and an assumed overall loss coefficient ($K_{loss}$) of 40.
\begin{table}[!htb]
\caption{Dimensions of the shell-and-tube heat exchangers for the hydride reactors. Only the length varies between the two reactors. Each tube and its surrounding shell represent 1 control volume within the system shown in Figure~\ref{fig:Hyd_Schematic}.}
\begin{center}
\begin{tabular}{l c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{Property}}$ & $\textbf{\text{Value}}$ & $\textbf{\text{Units}}$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
Tube Diameter & 4 & mm \\
Shell Diameter & 7 & mm \\
Number of Tubes & 400 & - \\
Length of Reactor A & 1.77 & m \\
Length of Reactor B & 1.54 & m \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:hx_dim}
\end{table}
\subsection{Simulation Results}
\par To determine the suitability of the linear state-space model for use in controller design, the dynamics predicted by the linear model are compared to those predicted by the nonlinear model for a given set of input conditions. To compare the models, the linear and nonlinear governing equations are simulated in MATLAB using a variable-step solver that uses fifth-order numerical differentiation. We consider two cases. In Case 1, the initial conditions are defined so that hydrogen is desorbed in reactor B and flows to reactor A, where it is absorbed, as shown in Figures~\ref{fig:Case1_lin_perf} and~\ref{fig:Case1_lin_perf_heat}. In Case 2, the initial conditions are defined such that hydrogen is desorbed in reactor A and flows to reactor B where it is absorbed, as shown in Figures~\ref{fig:Case2_lin_perf} and~\ref{fig:Case2_lin_perf_heat}. These two cases, with opposite reactions occurring in the reactors, represent the charging and discharging modes in an energy storage system. Note that which mode is charging and which is discharging will depend on which reactor is being used to deliver the load. In both cases, each of the three control input variables is perturbed for a five-minute period, and the dynamic response of each model is observed. These perturbations to the input variables for both cases are shown in Figure~\ref{fig:Case1&2_inputs}. For both cases, the initial values of the state variables ($x_{0}$), as well as the input and disturbance variables ($u_{0}$), are listed in Table~\ref{tab:in_var}, and each simulation is initialized at the linearization point $(x_0,u_0)$.
\begin{table}[!htb]
\caption{Initial values for the state and input variables. Note that kg H indicates mass of hydrogen and kg M indicates mass of the metal hydride.}
\begin{center}
\begin{tabular}{l c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{Variable}}$ & $\textbf{\text{Case 1}}$ & $\textbf{\text{Case 2}}$ & $\textbf{\text{Units}}$\\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$T_{hyd,A}$ & 6.89 & 6.89 & \degree C \\
$T_{hyd,B}$ & 36.9 & 34.9 & \degree C \\
$P_{H,A}$ & 480 & 290 & kPa \\
$P_{H,B}$ & 290 & 360 & kPa \\
$w_{A}$ & 0.006 & 0.007 & kg H / kg M \\
$w_{B}$ & 0.006 & 0.007 & kg H / kg M \\
$\dot{m}_{wg,A}$ & 0.2 & 0.2 & kg/s \\
$\dot{m}_{wg,B}$ & 0.2 & 0.2 & kg/s \\
$\Delta P_{comp}$ & 210 & 80 & kPa \\
$T_{wg,in,A}$ & 1.89 & 11.9 & \degree C\\
$T_{wg,in,B}$ & 42.9 & 30.9 & \degree C \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:in_var}
\end{table}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_inputs_2.jpg}
\caption{Case 1}
\label{fig:dP_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_inputs_2.jpg}
\caption{Case 2}
\label{fig:mdot_case1}
\end{subfigure}
\caption{Input circulating fluid mass flow rate and compressor pressure difference for (a) Case 1 and (b) Case 2.}
\label{fig:Case1&2_inputs}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_tempA_2.jpg}
\caption{Hydride Temperature in Reactor A}
\label{fig:tempA_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_tempB_2.jpg}
\caption{Hydride Temperature in Reactor B}
\label{fig:tempB_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_pressA_2.jpg}
\caption{Hydrogen Pressure in Reactor A}
\label{fig:pressA_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_pressB_2.jpg}
\caption{Hydrogen Pressure in Reactor B}
\label{fig:pressB_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_weightA_2.jpg}
\caption{Hydrogen Weight Fraction in Reactor A}
\label{fig:weightA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_weightB_2.jpg}
\caption{Hydrogen Weight Fraction in Reactor B}
\label{fig:weightB_case1}
\end{subfigure}
\caption{Comparison of the dynamic state variables for Reactors A and B between the linear model (dashed lines) and the nonlinear model (solid lines) for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case1_lin_perf}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_heatA_3.jpg}
\caption{Reactor A}
\label{fig:heatA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_heatB_3.jpg}
\caption{Reactor B}
\label{fig:heatB_case1}
\end{subfigure}
\caption{Comparison of the heat transfer rates associated with Reactors A and B between the linear model (dashed lines) and the nonlinear model (solid lines) for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case1_lin_perf_heat}
\end{figure}
\par In both cases, the linear model matches the nonlinear model closely. This is quantified using the root mean square error (RMSE) between the state variables predicted by the linear model and the nonlinear model, as shown in Table \ref{tab:rmse_case1}. The normalized RMSE (NRMSE) values were calculated using Equation~\ref{eq:NRMSE}. All variables of the same type (e.g. temperature, pressure, or weight fraction) were normalized against the same value, determined by finding the maximum range over which each variable varies in either reactor (A or B) in either of the two simulation cases (see Equations~\ref{eq:delxA_NMRSE}-~\ref{eq:delxB_NMRSE}, where the subscript $i$ refers to the initial value of a given state variable simulated for a given case study).
\begin{equation}\label{eq:NRMSE}
NMRSE =\ \frac{RMSE}{max(\Delta x_{A},\Delta x_{B})}
\end{equation}
\begin{multline}\label{eq:delxA_NMRSE}
\Delta x_{A} = max \left(max(x_{A} - x_{A,i})_{Case 1},max(x_{A} - x_{A,i})_{Case 2}\right) - \\ min \left(min(x_{A} - x_{A,i})_{Case 1},min(x_{A} - x_{A,i})_{Case 2}\right)
\end{multline}
\begin{multline}\label{eq:delxB_NMRSE}
\Delta x_{B} = max \left(max(x_{B} - x_{B,i})_{Case 1},max(x_{B} - x_{B,i})_{Case 2}\right) - \\ min \left(min(x_{B} - x_{B,i})_{Case 1},min(x_{B} - x_{B,i})_{Case 2}\right)
\end{multline}
\begin{table}[!htb]
\caption{RMSE and NRMSE for all state variables for Case 1.}
\begin{center}
\resizebox{\hsize}{!}{\begin{tabular}{l c c c c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
& & & \textbf{RMSE} & & & \\
$t (\text{min})$ & $T_{hyd,A} (\degree \text{C})$ & $T_{hyd,B} (\degree \text{C})$ & $P_{H,A} (\text{kPa})$ & $P_{H,B} (\text{kPa})$ & $w_{A} (\frac{\text{g H}}{\text{kg M}})$ & $w_{B} (\frac{\text{g H}}{\text{kg M}})$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$0-5$ & 0.095 & 0.248 & 3.08 & 3.70 & 0.0047 & 0.0074 \\
$5-10$ & 0.156 & 0.384 & 0.477 & 6.06 & 0.0060 & 0.017 \\
$10-15$ & 0.094 & 0.354 & 2.66 & 7.23 & 0.0034 & 0.022 \\
$15-20$ & 0.077 & 0.376 & 2.66 & 7.23 & 0.0061 & 0.029 \\
$20-25$ & 0.057 & 0.388 & 0.847 & 6.70 & 0.0082 & 0.036 \\
$25-30$ & 0.041 & 0.389 & 0.622 & 7.04 & 0.010 & 0.042 \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
& & & \textbf{NRMSE} & & & \\
$t (\text{min})$ & $T_{hyd,A}$ & $T_{hyd,B}$ & $P_{H,A}$ & $P_{H,B}$ & $w_{A}$ & $w_{B}$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$0-5$ & 2.42\% & 6.29\% & 2.26\% & 2.72\% & 0.356\% & 0.561\% \\
$5-10$ & 3.96\% & 9.75\% & 0.349\% & 4.44\% & 0.455\% & 1.26\% \\
$10-15$ & 2.37\% & 8.97\% & 1.95\% & 5.30\% & 0.258\% & 1.68\% \\
$15-20$ & 1.94\% & 9.55\% & 0.918\% & 4.58\% & 0.462\% & 2.20\% \\
$20-25$ & 1.45\% & 9.84\% & 0.621\% & 4.92\% & 0.621\% & 2.69\% \\
$25-30$ & 1.05\% & 9.88\% & 0.456\% & 5.16\% & 0.780\% & 3.19\% \\
\noalign{\vskip 1mm}
\hline
\end{tabular}}
\end{center}
\label{tab:rmse_case1}
\end{table}
\par For Case 1, the fastest response in the system is that of the hydrogen pressure (seen in Figures~\ref{fig:pressA_case1} and~\ref{fig:pressB_case1}) to the change in the pressure difference created by the compressor ($\Delta P_{comp}$) at $t=5\min$ and at $t=10\min$. At $t=5\min$, increasing $\Delta P_{comp}$ results in an almost immediate change in the hydrogen pressure of reactor A ($P_{H,A}$) approximately equal to the change in $\Delta P_{comp}$, as seen in Figure \ref{fig:pressA_case1}. To understand why $P_{H,A}$ changes more than $P_{H,B}$, we can revisit the two terms of the pressure balance given in Equation~\ref{eq:pbal} (repeated below for convenience): the absorption or desorption rate (depending on the direction of the reaction) $rm_{hyd}$, and the hydrogen mass flow rate into the reactor, $\dot{m}_{H,in}$.
\begin{equation}\nonumber
\frac{dP_{H}}{dt} =\ \frac{RT_{hyd}}{\phi V_{shell}}\left (rm_{hyd} + \dot{m}_{H,in} \right )
\end{equation}
At this operating condition, $r_{B}$ (the reaction rate on a per unit mass of hydride basis) is more sensitive to changes in pressure than $r_{A}$. When $\Delta P_{comp}$ increases, the magnitude of $\dot{m}_{H,in}$ increases in each reactor. This term becomes much larger than the absorption rate in reactor A or the desorption rate in reactor B. However, the desorption rate in reactor B increases enough to balance the mass flow rate after only a small change in $P_{H,B}$, while it takes a much larger change in $P_{H,A}$ before the absorption rate in reactor A balances the mass flow rate.
Thus, $P_{H,A}$ increases much more than $P_{H,B}$ decreases. This process occurs very quickly; in the initial second after $\Delta P_{comp}$ increases, the driving pressure difference $P_{H,B}+\Delta P_{comp} - P_{H,A}$ decreases by 94.8\%. Thus, it appears as a step change when plotted over a 30-minute time span in Figure \ref{fig:Case1_lin_perf}c. The linear state-space model successfully captures this response, as seen in Figures \ref{fig:pressA_case1} and \ref{fig:pressB_case1}.
\par At $t=10\min$, when $\Delta P_{comp}$ returns to its original value, there is a significant step change in both pressures. The change in $\Delta P_{comp}$ causes hydrogen to flow from reactor A to reactor B, whereas it otherwise flows from reactor B to reactor A in Case 1. Therefore, in both reactors, $rm_{hyd}$ and $\dot{m}_{H,in}$ have the same sign, so $r_{B}m_{hyd,B}$ cannot balance $\dot{m}_{H,in,B}$ as it did before. Once the initial pressure difference has been reduced, the mass flow is reduced to a small value, so the changes in pressure are primarily due to the reaction rates. However, the response of $r_{B}$ to the change in the pressure causes $r_{B}$ to go to zero, as shown in Figure~\ref{fig:rrateB_case1}. As shown in Figure~\ref{fig:rrateA_case1}, however, $r_{A}$ decreases but does not go to zero. This means that for approximately 45 seconds, hydrogen is being absorbed in one reactor but not desorbed in the other. Thus, the pressure in both reactors slowly decreases, since hydrogen is flowing out of reactor B, but the absorption rate in reactor A is larger than the flow rate in. After $\sim$45 seconds, the increasing temperature seen in Figure~\ref{fig:tempB_case1} causes the absorption rate for reactor B to grow larger than $\dot{m}_{H,in,B}$, so $P_{H,B}$ starts increasing. This increase leads to an increased mass flow rate to reactor A, which is larger than the absorption rate in reactor A, resulting in $P_{H,A}$ increasing as well.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_rrateA__with_zoom_2.jpg}
\caption{Reactor A}
\label{fig:rrateA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_rrateB_2.jpg}
\caption{Reactor B}
\label{fig:rrateB_case1}
\end{subfigure}
\caption{Reaction rates in Reactors A and B for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case1_rrate}
\end{figure}
The most significant difference between the linear and nonlinear models can be seen in Figure~\ref{fig:pressB_case1}, where the linear model does not capture the initial spike in $P_{H,B}$ and the decay that follows it, but accurately captures the behavior of the pressure after that time. This is because the reaction rate equation (Equation~\ref{eq:rrate}) is discontinuous when it goes to zero; as shown in Equation~\ref{eq:rrate}. Since the linear model does not include this discontinuity, it fails to capture the behavior of the pressure while the reaction rate is equal to zero. However, once the reaction rate is again nonzero, the linear model again follows the nonlinear model.
\par The response of the hydride temperature in each reactor to the change in $\Delta P_{comp}$ is shown in Figures \ref{fig:tempA_case1} and \ref{fig:tempB_case1}. Here, the increased reaction rate after the increase in $\Delta P_{comp}$ results in an increase in the heat released or absorbed by the reaction, pushing the temperature of the hydrides in both reactors away from the temperature of the circulating fluid in their reactors. This causes the heat transfer rate between the hydride in each reactor and the circulating fluid to increase, as shown in Figure \ref{fig:Case1_lin_perf_heat}. Once these heat transfer rates are approximately equal to the heat transfer from absorption and desorption, the temperatures become close to steady. While the linear model underestimates $T_{hyd,B}$ after the step change, as shown in Figure \ref{fig:tempB_case1} at $t=5$ min, it stays within 10\% of the nonlinear model, and $T_{hyd,A}$ stays within 5\% throughout the simulation.
\par In contrast to the visible changes in pressure and temperature that occur when $\Delta P_{comp}$ changes, there is not a significant change in these variables when either of the mass flow rates are changed. However, as shown in Figure~\ref{fig:Case1_lin_perf_heat}, there is a step change in the heat transfer rate in each reactor when there is a change in the mass flow rate in that reactor. This is because the heat transfer rate is a linear function of the mass flow rate. This change is accurately captured by the linear model, as seen in Figures~\ref{fig:heatA_case1} and \ref{fig:heatB_case1}. All variables stay within 10\% of the nonlinear model throughout the simulation.
\par Finally, the change in the weight fraction over time is different from that of the hydride temperature and pressure because the weight fraction changes continually (except for $w_{B}$ while $r_{B}$ is equal to zero) and does not enter a near-equilibrium state. The dynamics of the weight fraction are controlled by the reaction rate, which is the derivative of the weight fraction. As discussed in regards to the pressure dynamics, the reaction rate in both reactors changes quickly in the seconds after the change in $\Delta P_{comp}$ until $r_{A}m_{hyd,A}$, $r_{B}m_{hyd,B}$, and $\dot{m}_{H,in}$ have approximately equal magnitudes. This quick change in the reaction rate, combined with the very slow change for the rest of the simulation period, results in the weight fraction in each reactor resembling a linear function with a different slope after $t=5\min$, as seen in Figures \ref{fig:weightA_case1} and \ref{fig:weightB_case1}. After $\Delta P_{comp}$ returns to its original value at $t=10\min$, the rate of change of the weight fraction also returns to approximately its original value. For the weight fraction, the error of the linear models stays below 4\% for both reactors as shown in Table \ref{tab:rmse_case1}. Like the pressure, the weight fraction does not respond significantly to changes in the mass flow rates.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_tempA_2.jpg}
\caption{Hydride Temperature in Reactor A}
\label{fig:tempA_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_tempB_2.jpg}
\caption{Hydride Temperature in Reactor B}
\label{fig:tempB_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_pressA_2.jpg}
\caption{Hydrogen Pressure in Reactor A}
\label{fig:pressA_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_pressB_2.jpg}
\caption{Hydrogen Pressure in Reactor B}
\label{fig:pressB_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_weightA_2.jpg}
\caption{Hydrogen Weight Fraction in Reactor A}
\label{fig:weightA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_weightB_2.jpg}
\caption{Hydrogen Weight Fraction in Reactor B}
\label{fig:weightB_case2}
\end{subfigure}
\caption{Comparison of the state variables for Reactors A and B comparing the linear model (dashed lines) to the nonlinear model (solid lines) for Case 2 (hydrogen flowing from reactor B to reactor A).}
\label{fig:Case2_lin_perf}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_heatA_3.jpg}
\caption{Reactor A}
\label{fig:heatA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_heatB_3.jpg}
\caption{Reactor B}
\label{fig:heatB_case2}
\end{subfigure}
\caption{Comparison of the heat transfer rates out of Reactors A and B comparing the linear model (dashed lines) to the nonlinear model (solid lines) for Case 2 (hydrogen flowing from reactor B to reactor A).}
\label{fig:Case2_lin_perf_heat}
\end{figure}
\begin{table}[!htb]
\caption{RMSE and NRMSE for all state variables for Case 2.}
\begin{center}
\resizebox{\hsize}{!}{\begin{tabular}{l c c c c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
& & & \textbf{RMSE} & & & \\
$t (\text{min})$ & $T_{hyd,A} (\degree \text{C})$ & $T_{hyd,B} (\degree \text{C})$ & $P_{H,A} (\text{kPa})$ & $P_{H,B} (\text{kPa})$ & $w_{A} (\frac{\text{g H}}{\text{kg M}})$ & $w_{B} (\frac{\text{g H}}{\text{kg M}})$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$0-5$ & 0.273 & 0.380 & 1.01 & 5.22 & 0.011 & 0.012 \\
$5-10$ & 0.317 & 0.400 & 0.829 & 5.55 & 0.017 & 0.017 \\
$10-15$ & 0.391 & 0.455 & 1.16 & 7.46 & 0.025 & 0.023 \\
$15-20$ & 0.428 & 0.474 & 0.787 & 7.01 & 0.033 & 0.034 \\
$20-25$ & 0.424 & 0.476 & 0.980 & 7.29 & 0.039 & 0.045 \\
$25-30$ & 0.407 & 0.479 & 1.17 & 7.58 & 0.045 & 0.056 \\
\noalign{\vskip 1mm}
\hline
& & & \textbf{NRMSE} & & & \\
$t (\text{min})$ & $T_{hyd,A}$ & $T_{hyd,B}$ & $P_{H,A}$ & $P_{H,B}$ & $w_{A}$ & $w_{B}$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$0-5$ & 6.92\% & 9.66\% & 0.743\% & 3.83\% & 0.811\% & 0.886\% \\
$5-10$ & 8.05\% & 10.2\% & 0.608\% & 4.07\% & 1.30\% & 1.27\% \\
$10-15$ & 9.93\% & 11.6\% & 0.851\% & 5.47\% & 1.89\% & 1.75\% \\
$15-20$ & 10.9\% & 12.0\% & 0.577\% & 5.14\% & 2.46\% & 2.61\% \\
$20-25$ & 10.8\% & 12.1\% & 0.718\% & 5.34\% & 2.95\% & 3.43\% \\
$25-30$ & 10.3\% & 12.1\% & 0.858\% & 5.55\% & 3.42\% & 4.23\% \\
\noalign{\vskip 1mm}
\hline
\end{tabular}}
\end{center}
\label{tab:rmse_case2}
\end{table}
\par For Case 2, despite hydrogen flow in the system moving in the opposite direction, the same trends can be seen as discussed for Case 1. The pressure dynamics are similar, including $r_{B}$ going to zero for some time after $t=10$ min. However, because reactor A is desorbing hydrogen here rather than absorbing it as in Case 1, the pressure in both reactors increases while $r_{B}$ equals zero, and then decreases once absorption starts in reactor B, as seen in Figures~\ref{fig:pressA_case2} and~\ref{fig:pressB_case2}. The same general trends can also be seen in the temperature and weight fraction dynamics, with the only differences being due to the reversal of which reactor is absorbing hydrogen and releasing heat to the circulating fluid, and which is desorbing hydrogen and being heated by the circulating fluid.
\par Overall, the linear model predictions match those of the nonlinear model well in Case 2. As seen in Table~\ref{tab:rmse_case2}, there is a larger error for the hydride temperature states in this case than for Case 1, but the linear model is still accurate within 12.5\% across the entire simulation period. The error for $P_{H,A}$ is lower for this case, remaining within 1\% of the nonlinear model, while the highest error for $P_{H,B}$ is still less than 6\% different from the nonlinear model. The error between the linear and nonlinear model predictions for the weight fraction states follows a similar pattern to Case 1, continually increasing over time while never exceeding 5\%, without any of the changes to the inputs noticeably affecting the rate at which the error increases.
\subsection{Resetting the Linearization Point}
\par While the validation results show that the linear model predicts the system dynamics accurately within approximately 12.5\% of the linearization point, it is expected that during operation, the system will deviate further from a single point. This can adversely affect the controller design if it assumes the dynamics do not vary from a single linear model. One way to address this is to periodically re-linearize the model around the current operating condition. To illustrate how re-linearization affects the accuracy of the linear model, we simulate both the nonlinear and linear models for 120 minutes, beginning with the initial conditions shown in Table~\ref{tab:in_var_relin}. In order to sustain the absorption and desorption reactions in the reactors for 120 minutes, the compressor pressure difference is increased by 10 kPa every 10 minutes. We compare three different cases: 1) simulating the linear model from the same initial conditions without any re-linearization, 2) simulating the linear model beginning with the same initial conditions but then re-linearizing it every 10 minutes, and 3) simulating the linear model beginning with the same initial conditions but then re-linearizing it every 30 minutes. The resulting heat transfer rates for these cases are compared against the nonlinear model simulation in Figure~\ref{fig:Qout_relin_with_no_relin}, and the RMSE for the heat transfer rates is given in Table~\ref{tab:rmse_relin}. It is worth noting that because the linearized model is fully parameterized, re-linearizing is equivalent to updating the model parameters. In other words, re-linearizing does not contribute any significant computational complexity to the numerical simulation.
\begin{table}[!htb]
\caption{Initial values for the state and input variables for comparing performance with and without re-linearizing the model and for the case in which the controller is used for flow from reactor B to reactor A.}
\begin{center}
\begin{tabular}{l c c l c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{Variable}}$ & $\textbf{\text{Value}}$ & $\textbf{\text{Units}}$ & $\textbf{\text{Variable}}$ & $\textbf{\text{Value}}$ & $\textbf{\text{Units}}$\\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$T_{i,A}$ & 6.9 & \degree C & $\dot{m}_{wg,A}$ & 0.2 & kg/s \\
$T_{i,B}$ & 34. & \degree C & $\dot{m}_{wg,B}$ & 0.2 & kg/s \\
$P_{i,A}$ & 290 & kPa & $\Delta P_{comp}$ & 80 & kPa \\
$P_{i,B}$ & 360 & kPa & $T_{wg,in,A}$ & 12 & \degree C\\
$w_{i,A}$ & 0.008 & kg H / kg M & $T_{wg,in,B}$ & 27 & \degree C \\
$w_{i,B}$ & 0.0045 & kg H / kg M & & & \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:in_var_relin}
\end{table}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case2_heatA_larger.jpg}
\caption{Reactor A
\label{fig:heatA_relin}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case2_heatB_larger.jpg}
\caption{Reactor B
\label{fig:heatB_relin}
\end{subfigure}
\caption{Heat transfer rates from the hydride beds to the circulating fluid in each reactor. The performance of the linear models is compared to the nonlinear model (solid black line) for three different cases: one where the model is never re-linearized (dot-dashed pink line), one where it is re-linearized every 10 minutes (dotted blue line), and one where it is re-linearized every 30 minutes (dashed red line). Throughout this study, the compressor pressure difference is increased by 10 kPa every 10 minutes.}
\label{fig:Qout_relin_with_no_relin}
\end{figure}
\begin{table}[!htb]
\caption{RMSE for the heat rates for the 3 cases studied. All values are in kW.}
\begin{center}
\begin{tabular}{l c c c c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
& \multicolumn{2}{c}{No Relin.} & \multicolumn{2}{c}{Every 10 min} & \multicolumn{2}{c}{Every 30 min} \\
$t (\text{min})$ & $Q_{A}$ & $Q_{B}$ & $Q_{A}$ & $Q_{B}$ & $Q_{A}$ & $Q_{B}$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$0-30$ & 0.338 & 0.376 & 0.251 & 0.154 & 0.338 & 0.376 \\
$30-60$ & 0.308 & 0.596 & 0.127 & 0.179 & 0.186 & 0.148 \\
$60-90$ & 0.257 & 0.796 & 0.101 & 0.166 & 0.081 & 0.196 \\
$90-120$ & 0.228 & 0.983 & 0.081 & 0.145 & 0.061 & 0.182 \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:rmse_relin}
\end{table}
\par From Figure \ref{fig:Qout_relin_with_no_relin} we can see that re-linearization does improve the accuracy of the linear model over the duration of the simulation. For both heat transfer rates, both cases with re-linearization match the nonlinear model after re-linearization more closely than the case without re-linearization. It is worth noting that the extent of the nonlinearity of each reactor is not the same. For reactor A, as shown in Figure~\ref{fig:heatA_relin}, even without re-linearization, the linear model matches the nonlinear one on average within 0.35 kW. Furthermore, the linear model does not diverge from the nonlinear model: the error for the last 30 minutes is less than that for the first 30 minutes. For reactor B, as shown in Figure \ref{fig:heatB_relin}, the model with no re-linearization diverges from the nonlinear model almost immediately, with error increasing over time.
\par Comparing the performance of the linear model with re-linearization every 10 minutes to that with re-linearization every 30 minutes, we can see in Table~\ref{tab:rmse_relin} that both stay within 0.2 kW of the nonlinear model after the first 30 minutes. The difference in error between them is small, with more frequent re-linearization never resulting in a reduction in error of more than 0.03 kW once both models have been re-linearized at least once. In general, the frequency with which the parameters of the linear model should be updated will depend on how close the last linearization point is to the ``near-equilibrium'' state in which there is only a slow change in the temperature, pressure, and reaction rate in each reactor. For the operating conditions considered here, we can conclude that re-linearizing every 30 minutes is sufficient for maintaining accuracy; this will be applied to the controller design discussed in the next section.
\section{Controller Design and Synthesis}
\label{sec:control}
In this section, we describe the design of a multi-input-multi-output (MIMO) model predictive controller (MPC) for regulating the operation of the proposed two-reactor metal hydride thermal energy storage system.
\subsection{Model Predictive Control Algorithm}
MPC is a control technique that utilizes a model of the system dynamics to optimize a sequence of control inputs over a specified time horizon by predicting the dynamical response of the system over said horizon based on different sequences of inputs \cite{Camacho2013ModelControl}. At each measurement sample, MPC involves solving an $N$-step ahead online optimization problem to predict the optimal sequence of control inputs $\left[u\left(k\right),\ u\left(k+1\right),\cdots u(k+N)\right]$ that will drive the output sequence $\left[y\left(k\right),\ y\left(k+1\right),\cdots y(k+N)\right]$ toward a desired reference trajectory. At each time in the control sampling period, the MPC problem is solved, and the optimized variables $u\left(k\right)$ are taken as the control input at that sample instant.
Given the nonlinear dynamics of the hydride system, a successive linearization technique \cite{Henson1998NonlinearDirections} is used with a linear MPC design to achieve the desired control objectives. As was shown in Sec.~\ref{sec:modelval}, local linearization of the model matches the nonlinear model within 12.1\% (based upon the validation presented in Section \ref{sec:modelval}) and is therefore adequate for use as a prediction model in a MPC design. We design and implement the MPC as a discrete-time controller. Therefore, the state dynamics and system outputs are discretized and expressed as
\begin{align}
\label{eq:discretemodel}
x\left(k+1\right)&=\mathbf{A}x\left(k\right)+\mathbf{B}u
\left(k\right)+\mathbf{B}_{d}d(k) \\
{y}\left(k\right)&=\mathbf{C}x\left(k\right)+\mathbf{D}u\left(k\right)+\mathbf{D}_{d}d(k) \enspace . \nonumber
\end{align}
In these equations, the input vector has been separated into $u$ and $d$, where $u$ is the set of input variables we can control (mass flow rates and compressor pressure difference), while $d$ is the set of disturbance inputs (circulating fluid inlet temperatures), i.e. those input variables that cannot be controlled . To achieve zero steady-state tracking error, we augment the linear-quadratic regulator (LQR) with integral control as shown in Equation~\eqref{eq:augssmodel2}:
\begin{align}
\label{eq:augssmodel2}
\tilde{x}\left(k+1\right)&=\tilde{\mathbf{A}}\tilde{x}\left(k\right)+\tilde{\mathbf{B}}u\left(k\right)+{\tilde{\mathbf{B}}}_\mathbf{d}\tilde{d}(k) \\
\tilde{y}\left(k\right)&=\tilde{\mathbf{C}}\tilde{x}(k) \enspace , \nonumber
\end{align}
\noindent where
\begin{equation}
\tilde{\mathbf{A}}=\left[\begin{matrix}\mathbf{A}&\mathbf{0}\\\mathbf{C}&\mathbf{0}\\\end{matrix}\right], \tilde{\mathbf{B}}=\left[\begin{matrix}\mathbf{B}\\\mathbf{D}\\\end{matrix}\right],\ {\tilde{\mathbf{B}}}_\mathbf{d}=\left[\begin{matrix}\mathbf{B}_\mathbf{d}&\mathbf{0}\\\mathbf{D}_\mathbf{d}&-\mathbf{I}\\\end{matrix}\right],\ \tilde{\mathbf{C}}=\left[\begin{matrix}\mathbf{0}&\mathbf{I}\\\end{matrix}\right],
\label{eq:augssmodelmatrices}
\end{equation}
\noindent and $ \tilde{x}\left(k\right)=\left[\begin{matrix}x\left(k\right),&x_i\left(k\right)\\\end{matrix}\right]^T$ and $\tilde{d}(k)=\left[\begin{matrix}d\left(k\right)&r\left(k\right)\\\end{matrix}\right]^T$. Based on Equation \eqref{eq:augssmodelmatrices}, the output of the control model simplifies to $\tilde{y}\left(k\right)=x_i\left(k\right)$ where $x_i\left(k+1\right)=\dot{Q}\left(k\right)-r(k)$. Therefore, the MPC is formulated as an error regulation problem to drive $x_i(k)$ to zero, which is equivalent to driving the heat transfer rates $\dot{Q}\left(k\right)$ to the reference values $r(k)$. The MPC problem can be stated as
{\begin{equation}
\begin{aligned}
\min_{\mathbf{U},\mathbf{Y}} \quad & J = \sum_{k=1}^{N} \widetilde{y}\left(k\right)^T\mathbf{Q}\widetilde{y}\left(k\right) + u\left(k\right)^T\mathbf{R}u\left(k\right) \\
\textrm{s.t.} \quad & \widetilde{x}\left(k+1\right) = \widetilde{\mathbf{A}}\widetilde{x}\left(k\right) + \widetilde{\mathbf{B}}u\left(k\right) + \mathbf{\widetilde{B}_d}\widetilde{d}\left(k\right) \enspace \forall \enspace k \in [1,N] \\
& \widetilde{y}\left(k+1\right) = \widetilde{\mathbf{C}}\widetilde{x}\left(k+1\right) \enspace \forall \enspace k \in [1,N] \\
& u_{min} \leq u\left(k\right) \leq u_{max} \enspace \forall \enspace k \in [1,N] \\
& |u\left(k\right) - u\left(k+1\right)| \leq \delta u_{max} \enspace \forall \enspace k \in [1,N-1] \enspace ,
\end{aligned}
\label{eq:mpcproblem}
\end{equation}}
\smallskip
\noindent
\noindent where
{\small
\begin{equation}
\begin{aligned}
& \mathbf{U} = [u\left(k=1\right), \enspace u\left(2\right), \enspace \hdots \enspace, \enspace u\left(N\right)] \\
& \mathbf{Y} = [\widetilde{y}\left(k=1\right), \enspace \widetilde{y}\left(2\right), \enspace \hdots \enspace, \enspace \widetilde{y}\left(N\right)] \enspace ,
\end{aligned} \nonumber
\end{equation}}
\smallskip
\noindent and $\mathbf{Q}$ and $\mathbf{R}$ are positive definite weighting matrices. The constraints ensure (1) adherence to the state dynamics prescribed by the control model given in Equations~\eqref{eq:augssmodel2} and \eqref{eq:augssmodelmatrices}, (2) that the optimal values for the inputs $u$ are within the achievable range of inputs, and (3) that the rate of change in the chosen inputs does not exceed actuator limits.
\subsection{Controller Synthesis}
\par For the simulated case studies, the results of which are presented in Section \ref{sec:results}, the matrices $\mathbf{Q}$ and $\mathbf{R}$ are tuned to penalize tracking error and the magnitude of the control inputs, respectively. The relative magnitude of the weights in these matrices drives the controller performance, and can be motivated by the dynamics of the system itself. One representative linearization of the model, corresponding to Case 1 as described later in Section \ref{sec:results}, is shown in Equations~\eqref{eq:case1_f}~-~\eqref{eq:case1_D}.
In the two-reactor hydride system, the output variables $\dot{Q}_{hyd \rightarrow wg,m}$ and $\dot{Q}_{hyd \rightarrow wg,n}$ are directly affected by the mass flow rate of the circulating fluid in each reactor, $\dot{m}_{wg,m}$ and $\dot{m}_{wg,n}$, respectively, through the $\mathbf{D}$ matrix. Note that these correspond to the first two inputs defined in the input vector $u$. The third input variable, the compressor pressure difference $\Delta P_{comp}$, does not directly affect the output variables; it only indirectly affects them through its effect on the hydride temperature states, $T_{hyd,m}$ and $T_{hyd,n}$. Therefore, in tuning $\mathbf{Q}$ and $\mathbf{R}$, we more heavily penalize $\Delta P_{comp}$, relative to the weights placed on $\dot{m}_{wg,m}$ and $\dot{m}_{wg,n}$, to incentivize the controller to use $\Delta P_{comp}$ to help regulate $\dot{Q}_{hyd \rightarrow wg,m}$ and $\dot{Q}_{hyd \rightarrow wg,n}$. The final weights chosen for $\mathbf{Q}$ and $\mathbf{R}$ are
\begin{equation}\label{eq:Qvals}
\mathbf{Q}=\ \begin{bmatrix}100
& 0 \\ 0 & 100
\end{bmatrix}
\end{equation}
\begin{equation}\label{eq:Rvals}
\mathbf{R}=\ \begin{bmatrix}3\times10^{11}
& 0 & 0 \\ 0 & 3\times10^{11} & 0 \\ 0 & 0 & 1
\end{bmatrix}\enspace .
\end{equation}
We formulate and solve the MPC as a quadratic program using the \emph{quadprog} solver within the MATLAB Optimization Toolbox \cite{TheMathWorksInc.2018MatlabToolbox}. Formulating the problem as a quadratic program permits a computationally-efficient controller synthesis to be obtained for each control sample with little computational overhead.
\begin{equation}
\label{eq:case1_f}
f(x,u) = \frac{d}{dt}\left( \begin{bmatrix} T_{hyd,m} \\ P_{H,m} \\ w_{m} \\ T_{hyd,n} \\ P_{H,n} \\ w_{hyd,n} \end{bmatrix} \right) = \mathbf{A} \begin{bmatrix}T_{hyd,m} \\ P_{H,m} \\ w_{m} \\ T_{hyd,n} \\ P_{H,n} \\ w_{hyd,n}
\end{bmatrix} + \mathbf{B} \begin{bmatrix} \dot{m}_{wg,m} \\ \dot{m}_{wg,n} \\ \Delta P_{comp} \\ T_{wg,in,m} \\ T_{wg,in,n}\end{bmatrix}
\end{equation}
\begin{equation}
\label{eq:case1_A}
\resizebox{0.9\hsize}{!}{$\mathbf{A} = \begin{bmatrix} -8.17\times 10^{-3} & -3.35\times10^{-8} & -8.13 & -9.07\times10^{-5} & 4.09\times10^{-7} & 0 \\ 8840 & -4.38 & 1.11\times10^{7} & 0 & 3.96 & 0 \\ -2.28\times10^{-7} & 1.38\times10^{-11} & -2.93\times10^{-4} & 0 & 0 & 0 \\ 0 & 0 & 0 & -0.216 & 1.83\times10^{-5} & -94.4 \\ 0 & 4.93 & 0 & 2.03\times10^{5} & -22.4 & 9.00\times10^{7} \\ 0 & 0 & 0 & -4.78\times10^{-6} & 4.09\times10^{-10} & -2.12\times10^{-3} \\ \end{bmatrix}$}
\end{equation}
\begin{equation}
\label{eq:case1_B}
\mathbf{B} = \begin{bmatrix} -0.039 & 0 & 4.47\times10^{-7} & 1.68\times10^{-3} & 0 \\ 0 & 0 & 3.91 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0.064 & 0 & 0 & 2.99\times10^{-3} \\ 0 & 0 & -5.00 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
\end{equation}
\begin{equation}
\label{eq:case1_g}
g(x,u) = \begin{bmatrix} \dot{Q}_{hyd \rightarrow wg,m} \\ \dot{Q}_{hyd \rightarrow wg,n} \end{bmatrix} = \mathbf{C} \begin{bmatrix}T_{hyd,m} \\ P_{H,m} \\ w_{m} \\ T_{hyd,n} \\ P_{H,n} \\ w_{hyd,n}
\end{bmatrix} + \mathbf{D} \begin{bmatrix} \dot{m}_{wg,m} \\ \dot{m}_{wg,n} \\ \Delta P_{comp} \\ T_{wg,in,m} \\ T_{wg,in,n}\end{bmatrix}
\end{equation}
\begin{equation}
\label{eq:case1_C}
\mathbf{C} = \begin{bmatrix} 367.1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 436.3 & 0 & 0 \\
\end{bmatrix}
\end{equation}
\begin{equation}
\label{eq:case1_D}
\mathbf{D} = \begin{bmatrix} 7930 & 0 & 0 & -341 & 0 \\ 0 & -9600 & 0 & 0 & -452 \\
\end{bmatrix}
\end{equation}
\section{Results}
\label{sec:results}
In this section, we implement the controller in simulation and demonstrate its performance in regulating the dynamics of the two-reactor metal hydride system through a series of case studies.
\subsection{Reference Tracking}
\par The MPC is designed primarily for referencing tracking, so we first verify its performance in the context of tracking variable heat transfer rates in each reactor. The model is simulated using the same initial conditions as those given in Table~\ref{tab:in_var}. Moreover, the control input variables are bounded based on the values shown in Table~\ref{tab:input_minmax}.
\begin{table}[!htb]
\caption{Upper and lower bounds on the control input variables.}
\begin{center}
\begin{tabular}{l c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{Variable}}$ & $\textbf{\text{Units}}$ & $\textbf{\text{Minimum}}$ & $\textbf{\text{Maximum}}$ \\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$\dot{m}_{A}$ & kg/s & 0 & 0.8 \\
$\dot{m}_{B}$ & kg/s & 0 & 0.8 \\
$\Delta P_{comp}$ & kPa & 0 & 500 \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:input_minmax}
\end{table}
\par The model is simulated for a two-hour period with the desired heat transfer rate setpoints (reference values) changing every 30 minutes and the controller re-linearizing around the current operating point when the setpoint changes. The control input variables are updated at a frequency of 1 Hz. As with the model validation, two cases are considered, representing the two operating modes of the system. In Case 1, the reference value for the heat transfer rate in Reactor A increases over a series of three step changes which are mirrored by step \emph{decreases} in the heat transfer rate in Reactor B. The opposite trends in the references are demonstrated in Case 2. The closed-loop simulation results for Case 1 are shown in Figure~\ref{fig:cntr_charge_changeQ}. The results for Case 2 are given in~\ref{sec:appA}.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotA_controller_heatrates_case1.jpg}
\caption{Circulating Fluid Mass Flow Rate A}
\label{fig:mdotA_cntr_heat_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatA_controller_heatrates_case1_2.jpg}
\caption{Heat Transfer Rate in Reactor A}
\label{fig:heatA_cntr_heat_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotB_controller_heatrates_case1.jpg}
\caption{Circulating Fluid Mass Flow Rate B}
\label{fig:mdotB_cntr_heat_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatB_controller_heatrates_case1_2.jpg}
\caption{Heat Transfer Rate in Reactor B}
\label{fig:heatB_cntr_heat_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dP_controller_heatrates_case1.jpg}
\caption{Compressor Pressure Difference}
\label{fig:dP_cntr_heat_case1}
\end{subfigure}
\caption{Input values set by the controller and output heat transfer rates in the hydride reactors compared to the reference value for the case where hydrogen is being moved from reactor B to reactor A. The model is re-linearized and the reference value changed every 30 minutes.}
\label{fig:cntr_charge_changeQ}
\end{figure}
\par Figure~\ref{fig:cntr_charge_changeQ} shows that the controller successfully tracks the step changes in the heat transfer rates in each reactor, doing so primarily through adjustments in each of the circulating fluid mass flow rates (see Figures~\ref{fig:mdotA_cntr_heat_case1} and~\ref{fig:mdotB_cntr_heat_case1}). This is consistent with the algebraic relationship between mass flow rate and heat transfer rate in each reactor. In addition, the mass flow rates slowly increase over time in between step changes, adjusting to maintain a fixed heat transfer rate while the temperature differential between each reactor and the associated circulating fluid decreases. As shown in Figure~\ref{fig:dP_cntr_heat_case1}, the controller makes less use of the compressor pressure difference to achieve its objectives. While its value does change over time, the difference between the minimum and maximum values of $\Delta P_{comp}$ is only around 1\% of the range of values the controller can set it to, while the difference between the minimum and maximum values of $\dot{m}_{A}$ is approximately 50\% of its range.
\par For both this case and the case given in~\ref{sec:appA}, the ratio of the heat transfer rates in the two reactors are held constant even when the magnitude of the reference (desired) heat transfer rate changes. Specifically, the heat transfer rate in reactor A has the opposite sign and 85\% of the magnitude of that in reactor B. To see how the controller performance changes for a different ratio between the heat transfer rates, we consider a case with the same initial conditions and reference values for $\dot{Q}_{B}$ as defined in Case 1, but with the ratio of the magnitudes of the heat transfer rates set to 1. The control input signals and resulting heat transfer rates, compared to the reference values, are shown in Figure~\ref{fig:cntr_charge_changeQ_1ratio}.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotA_controller_heatrates_case3.jpg}
\caption{Circulating Fluid Mass Flow Rate A}
\label{fig:mdotA_cntr_heat_case3}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatA_controller_heatrates_case3_2.jpg}
\caption{Heat Transfer Rate in Reactor A}
\label{fig:heatA_cntr_heat_case3}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotB_controller_heatrates_case3.jpg}
\caption{Circulating Fluid Mass Flow Rate B}
\label{fig:mdotB_cntr_heat_case3}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatB_controller_heatrates_case3_2.jpg}
\caption{Heat Transfer Rate in Reactor B}
\label{fig:heatB_cntr_heat_case3}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dP_controller_heatrates_case3.jpg}
\caption{Compressor Pressure Difference}
\label{fig:dP_cntr_heat_case3}
\end{subfigure}
\caption{Input values set by the controller and output heat transfer rates in the hydride reactors compared to the reference value for the case where hydrogen is being moved from reactor B to reactor A and the ratio between heat transfer rates is set to 1 instead of 0.85. The model is re-linearized and the reference value changed every 30 minutes.}
\label{fig:cntr_charge_changeQ_1ratio}
\end{figure}
\par A major departure in these results, as compared to those shown in Case 1, is the response of the controller when the circulating fluid mass flow rate through reactor A saturates at its upper bound of 0.8 kg/s as shown at $t=80$ min in Figure ~\ref{fig:mdotA_cntr_heat_case3}. The MPC, recognizing this constraint, begins to increase the pressure differential across the compressor, since this is now the best input variable to use to control the heat transfer rate in Reactor A (see Figure~\ref{fig:dP_cntr_heat_case3}). Figure~\ref{fig:heatA_cntr_heat_case3} shows that the compressor is able to track the reference using the compressor pressure difference, but that the system is slower to respond given that the compressor pressure difference has a dynamic, rather than algebraic, effect on the output. While Reactor A requires a larger mass flow rate over time because the reactor temperature is approaching the circulating fluid temperature, in Reactor B, the reactor temperature diverges from the circulating fluid temperature. Thus, as shown in Figure~\ref{fig:mdotB_cntr_heat_case3}, the mass flow rate required to meet the reference heat transfer rate decreases over time, so much so that that the input mass flow rate values used to achieve the final reference heat transfer rate are similar to the values used to achieve the initial value, albeit the final reference value is much larger.
\par Given the coupling of the dynamics between the two reactors, it is desirable to operate the system in a near-equilibrium state. This avoids the situation in which the compressor pressure difference and the mass flow rate in one reactor are saturated, which would degrade the performance of the controller. Near equilibrium, the absorption or desorption rate in each reactor balances the mass flow rate between them, and the energy transfer from the reaction balances the heat transfer to the circulating fluid in each reactor. It is therefore important to select a ratio between the reference heat transfer rates that allows for near-equilibrium operation if the controller is being used for a full charging or discharging cycle.
\subsection{Disturbance Rejection}
\par To test the controller's ability to mitigate exogenous disturbances, we consider constant heat transfer rate references in each reactor for the same sets of initial conditions described in the previous section, but now change the circulating fluid temperatures every 10 minutes as shown in Figure~\ref{fig:Twg_cntr_dist_case1}. For this case, the model is still re-linearized every 30 minutes. The control input signals and resulting heat transfer rates are compared to their reference values in Figure~\ref{fig:cntr_charge_changeT}.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotA_controller_dist_case1.jpg}
\caption{Circulating Fluid Mass Flow Rate A}
\label{fig:mdotA_cntr_dist_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatA_controller_dist_case1_2.jpg}
\caption{Heat Transfer Rate in Reactor A}
\label{fig:heatA_cntr_dist_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotB_controller_dist_case1.jpg}
\caption{Circulating Fluid Mass Flow Rate B}
\label{fig:mdotB_cntr_dist_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatB_controller_dist_case1_2.jpg}
\caption{Heat Transfer Rate in Reactor B}
\label{fig:heatB_cntr_dist_case1}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dP_controller_dist_case1.jpg}
\caption{Compressor Pressure Difference}
\label{fig:dP_cntr_dist_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dist_in_controller_dist_case1.jpg}
\caption{Circulating Fluid Temperatures}
\label{fig:Twg_cntr_dist_case1}
\end{subfigure}
\caption{Input values set by the controller and output heat transfer rates in the hydride reactors for the case where hydrogen is being moved from reactor B to reactor A, compared to their reference values. The circulating fluid temperature changes every 10 minutes and the model is re-linearized every 30 minutes.}
\label{fig:cntr_charge_changeT}
\end{figure}
\par As shown in Figure~\ref{fig:cntr_charge_changeT}, the controller quickly adjusts to the changes in the disturbance inputs, successfully minimizing regulation error after only a very brief deviation away from the desired heat transfer rate. The first of these spikes can be seen in more detail in Figure~\ref{fig:heat_case1_zoomed}, which shows the $1$-minute period of time around the change in the disturbance inputs. From this, we can see that the controller converges to the reference value within 5-6 seconds. As with the case in which we considered time-varying heat transfer rates, the controller primarily changes the mass flow rates in order to track the reference, but does make some small changes to the compressor pressure difference. The controller is robust to changes in the disturbance inputs despite using a prediction model that is re-linearized less frequently than the disturbance signals change.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/QA_charge_zoomed_in.jpg}
\caption{Heat Transfer Rate A}
\label{fig:heatA_case1_zoomed}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/QB_charge_zoomed_in.jpg}
\caption{Heat Transfer Rate B}
\label{fig:heatB_case1_zoomed}
\end{subfigure}
\caption{Heat transfer rates in the reactors for Case 1 from 9.5 to 10.5 minutes.}
\label{fig:heat_case1_zoomed}
\end{figure}
In a further test of the ability of the controller to mitigate changes in the disturbance inputs, the controller was simulated for the initial conditions given in Case 2, with a larger change in the water glycol temperatures. Results for this case are given in~\ref{sec:appB}.
\section{Conclusion}
\par In this paper we presented a a model predictive controller (MPC) for a two-reactor metal hydride system in which reactions in each hydride bed are driven by heat transfer between the metal hydride and a circulating fluid as well as a compressor moving hydrogen between the reactors. The multivariable controller successfully tracks desired values for the heat transfer between the hydride bed and the circulating fluid in each reactor by controlling the pressure difference produced by the compressor and the mass flow rates of the circulating fluid in each reactor. We derived a first-principles nonlinear dynamic model of the two-reactor system, and then linearized it for the purposes of control design. While the nonlinear model of the reactor neglects any temperature or pressure gradients within the hydride beds, it predicts the evolution of the reactor pressure, temperature, and heat transfer rates to the circulating fluid. By analytically linearizing the nonlinear model, we obtained a parameterized state-space model that could be easily updated for different operating conditions. This was leveraged in the design of the MPC, in which the linear prediction model was periodically re-linearized to minimize differences between its predictions and the dynamics of the nonlinear model. Through a series of simulated case studies, we demonstrated the performance of the controller for both reference tracking and disturbance rejection. The proposed model and multivariable controller enable continuous operation of two-reactor metal hydride systems to regulate heat transfer in a metal hydride energy storage or A/C system. Areas of future research include examining the viability of controlling heat transfer in one reactor while minimizing temperature change in both, as well as testing the controller for use with higher-order models and real hydride systems.
\section*{Acknowledgements}
\par Funding for this project was provided by the Center for High-Performance Buildings at Purdue University.
\section*{Model Availability}
\par The code used to linearize the nonlinear dynamics model is available as Matlab .m files on Github at https://github.com/patrickkrane/hydride-linearization-model.git.
\section{Controller Reference Tracking Results for Simulation Case 2}
\label{sec:appA}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotA_controller_heatrates_case2.jpg}
\caption{Circulating Fluid Mass Flow Rate A}
\label{fig:mdotA_cntr_heat_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatA_controller_heatrates_case2_2.jpg}
\caption{Heat Transfer Rate in Reactor A}
\label{fig:heatA_cntr_heat_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotB_controller_heatrates_case2.jpg}
\caption{Circulating Fluid Mass Flow Rate B}
\label{fig:mdotB_cntr_heat_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatB_controller_heatrates_case2_2.jpg}
\caption{Heat Transfer Rate in Reactor B}
\label{fig:heatB_cntr_heat_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dP_controller_heatrates_case2.jpg}
\caption{Compressor Pressure Difference}
\label{fig:dP_cntr_heat_case2}
\end{subfigure}
\caption{Input values set by the controller and output heat transfer rate in each hydride reactor compared to the reference values for the case in which hydrogen is being moved from reactor A to reactor B. The model is re-linearized and the reference values changed every 30 minutes.}
\label{fig:cntr_discharge_changeQ}
\end{figure}
\par Figures~\ref{fig:heatA_cntr_heat_case2} and~\ref{fig:heatB_cntr_heat_case2} demonstrate that the controller is again able to track the desired heat transfer rates in the reactors. The input values used to achieve this result are shown in Figures~\ref{fig:mdotA_cntr_heat_case2},~\ref{fig:mdotB_cntr_heat_case2},and~\ref{fig:dP_cntr_heat_case2}. As in Case 1, the controller successfully tracks the reference values. From Figures~\ref{fig:mdotA_cntr_heat_case2} and~\ref{fig:mdotB_cntr_heat_case2}, we see that the circulating fluid mass flow rates increase as needed to track the time-varying references. Finally, from Figure~\ref{fig:dP_cntr_heat_case2}, we see that the controller does not use the compressor pressure difference as much as the circulating fluid mass flow rates to achieve the desired tracking performance.
\section{Disturbance Rejection Results for Simulation Case 2}
\label{sec:appB}
\par To further evaluate the disturbance rejection capability of the controller, the change in the disturbance inputs is increased in Case 2. The water glycol temperatures for this case are given in Figure~\ref{fig:Twg_cntr_dist_case2}. The control inputs and the resulting outputs, compared to the reference values, are shown in Figure~\ref{fig:cntr_discharge_changeT}.
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotA_controller_dist_case2.jpg}
\caption{Circulating Fluid Mass Flow Rate A}
\label{fig:mdotA_cntr_dist_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatA_controller_dist_case2_2.jpg}
\caption{Heat Transfer Rate in Reactor A}
\label{fig:heatA_cntr_dist_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/mdotB_controller_dist_case2.jpg}
\caption{Circulating Fluid Mass Flow Rate B}
\label{fig:mdotB_cntr_dist_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/heatB_controller_dist_case2_2.jpg}
\caption{Heat Transfer Rate in Reactor B}
\label{fig:heatB_cntr_dist_case2}
\vspace*{2.5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dP_controller_dist_case2.jpg}
\caption{Compressor Pressure Difference}
\label{fig:dP_cntr_dist_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dist_in_controller_dist_case2.jpg}
\caption{Circulating Fluid Temperatures}
\label{fig:Twg_cntr_dist_case2}
\end{subfigure}
\caption{Input values set by the controller, output heat transfer rates in the hydride reactors, and changing disturbance inputs for the case in which hydrogen is being moved from reactor A to reactor B. The circulating fluid temperature changes every 10 minutes and the model is re-linearized every 30 minutes.}
\label{fig:cntr_discharge_changeT}
\end{figure}
\par As shown in Figure~\ref{fig:cntr_discharge_changeT}, the controller still regulates the heat transfer rates to the desired values with only brief deviations in the regulation error (away from zero) whenever the disturbance inputs change. These results are again achieved primarily by varying the mass flow rates, with only slight use of the compressor. Since the changes in the circulating fluid temperatures are larger in this case, larger changes in the mass flow rates are needed to continue regulating the heat transfer rates to their references, as can be seen by comparing Figures~\ref{fig:mdotA_cntr_dist_case1} and~\ref{fig:mdotB_cntr_dist_case1} to Figures~\ref{fig:mdotA_cntr_dist_case2} and~\ref{fig:mdotB_cntr_dist_case2}.
\subsection{Simulation Results}
\par To determine the suitability of the linear state-space model for use in controller design, the dynamics predicted by the linear model are compared to those predicted by the nonlinear model for a given set of input conditions. To compare the models, the linear and nonlinear governing equations are implemented in Matlab and solved using a variable-step solver for stiff problems that uses a fifth-order numerical differentiation formula. We consider two cases. In Case 1, the initial conditions are set so that hydrogen moves from reactor A to reactor B, and each model is simulated for 10 minutes (600 seconds), as shown in Figures ~\ref{fig:Case1_lin_perf} and ~\ref{fig:Case1_lin_perf_heat}. In Case 2, the initial conditions are defined such that hydrogen flows from reactor B to reactor A, as shown in Figures ~\ref{fig:Case2_lin_perf} and ~\ref{fig:Case2_lin_perf_heat}. In both simulations, the compressor pressure difference is increased by 50 kPa after $t=5$ minutes. The initial values of the state variables ($x_{0}$) and the input and disturbance variables ($u_{0}$) for these two simulations are listed in \ref{tab:in_var}, and each simulation is initialized at the linearization point $(x_0,u_0)$.
\begin{table}[!htb]
\caption{Initial values for the state and input variables}
\begin{center}
\begin{tabular}{l c c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{ Variable}}$ & $\textbf{\text{Case 1}}$ & $\textbf{\text{Case 2}}$ & $\textbf{\text{Units}}$\\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$T_{A}$ & 6.89 & 6.89 & \degree C \\
$T_{B}$ & 37.9 & 32.9 & \degree C \\
$P_{A}$ & 480 & 310 & kPa \\
$P_{B}$ & 280 & 310 & kPa \\
$w_{A}$ & 0.006 & 0.007 & kg H / kg M \\
$w_{B}$ & 0.006 & 0.007 & kg H / kg M \\
$\dot{m}_{A}$ & 0.2 & 0.2 & kg/s \\
$\dot{m}_{B}$ & 0.2 & 0.2 & kg/s \\
$\Delta P_{comp}$ & 210 & 40 & kPa \\
$T_{wg,in,A}$ & 1.89 & 11.9 & \degree C\\
$T_{wg,in,B}$ & 42.9 & 30.9 & \degree C \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:in_var}
\end{table}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_tempA.jpg}
\caption{Temperature in Reactor A}
\label{fig:tempA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_tempB.jpg}
\caption{Temperature in Reactor B}
\label{fig:tempB_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_pressA.jpg}
\caption{Pressure in Reactor A}
\label{fig:pressA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_pressB.jpg}
\caption{Pressure in Reactor B}
\label{fig:pressB_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_weightA.jpg}
\caption{Weight Fraction in Reactor A}
\label{fig:weightA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_weightB.jpg}
\caption{Weight Fraction in Reactor B}
\label{fig:weightB_case1}
\end{subfigure}
\caption{State variables for Reactors A and B comparing the linear model (solid lines) to the nonlinear model (dashed lines) for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case1_lin_perf}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_heatA.jpg}
\caption{Reactor A}
\label{fig:heatA_case1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case1_heatB.jpg}
\caption{Reactor B}
\label{fig:heatB_case1}
\end{subfigure}
\caption{Heat transfer rates out of Reactors A and B comparing the linear model (solid lines) to the nonlinear model (dashed lines) for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case1_lin_perf_heat}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_tempA.jpg}
\caption{Temperature in Reactor A}
\label{fig:tempA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_tempB.jpg}
\caption{Temperature in Reactor B}
\label{fig:tempB_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_pressA.jpg}
\caption{Pressure in Reactor A}
\label{fig:pressA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_pressB.jpg}
\caption{Pressure in Reactor B}
\label{fig:pressB_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_weightA.jpg}
\caption{Weight Fraction in Reactor A}
\label{fig:weightA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_weightB.jpg}
\caption{Weight Fraction in Reactor B}
\label{fig:weightB_case2}
\end{subfigure}
\caption{State variables for Reactors A and B comparing the linear model (solid lines) to the nonlinear model (dashed lines) for Case 2 (hydrogen flowing from reactor B to reactor A).}
\label{fig:Case2_lin_perf}
\end{figure}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_heatA.jpg}
\caption{Reactor A}
\label{fig:heatA_case2}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Case2_heatB.jpg}
\caption{Reactor B}
\label{fig:heatB_case2}
\end{subfigure}
\caption{Heat transfer rates out of Reactors A and B comparing the linear model (solid lines) to the nonlinear model (dashed lines) for Case 1 (hydrogen flowing from reactor A to reactor B).}
\label{fig:Case2_lin_perf_heat}
\end{figure}
\par For case 1, the linear model tracks the nonlinear model very closely, but not for case 2. In case 1, the linear model generally matches the trend for each variable, but differs in the magnitude. In both cases, the increase in the pressure difference produced by the compressor after 5 minutes does not significantly affect the accuracy of the model. In both cases, the accuracy is similar before and after the step change in compressor pressure.
\par Considering the state variables in Figure \ref{fig:Case1_lin_perf}, the fastest response in the system is the response of the reactor pressure to the change in the pressure difference between the reactors (which includes the pressure difference created by the compressor). In this initial study, this occurs at $t=5$ minutes when the compressor pressure difference is increased. It results in an almost immediate step change in the pressure of reactor A approximately equal to the change in the compressor pressure difference, as seen in Figure \ref{fig:pressA_case1}. The change in pressure occurs primarily in reactor A because at this operating condition, the reaction rate in reactor B is more sensitive to changes in pressure than the reaction rate in reactor A. When the compressor pressure difference increases, the mass flow rate between the reactors increases, becoming much larger than the absorption rate in reactor A or the desorption rate in reactor B. Since hydrogen is flowing from reactor B to reactor A, this results in the pressure in reactor A increasing and the pressure in reactor B decreasing. However, since magnitude of the the reaction rate increases much faster in reactor B, the pressure there stops decreasing (due to increased desorption balancing out increased flow into the reactor) sooner than the pressure in reactor A stops increasing, and thus the pressure in reactor A increases much more than the pressure in reactor B decreases, as can be seen by comparing Figure \ref{fig:tempA_case1} to Figure \ref{fig:tempB_case1}. This process occurs in a few seconds- (insert quantifiable metric here)- and therefore appears as a step change when plotted against a 10-minute time span in Figure \ref{fig:Case1_lin_perf}. The linear state-space model successfully captures this response: as seen in Figures \ref{fig:tempA_case1} and \ref{fig:tempB_case1}, the pressure in both reactors tracks the nonlinear model before and after the increase in compressor pressure difference (give RMS numbers)
\par The response of the temperature to the increased compressor pressure difference follows a similar pattern as the pressure, where a change in one term in the governing equations results in a change in the state variable until another term grows large enough to balance out the change in the first term. However, here the dynamics are slower, so the process can be seen in Figures \ref{fig:tempA_case1} and \ref{fig:tempA_case2}. Here, the increased reaction rate after the increase in compressor pressure difference results in an increase in the heat released or absorbed by the reaction, pushing the temperature of the hydrides in both reactors away from the temperature of the circulating fluid in their reactors. As the hydride temperature moves away from the circulating fluid temperature, the heat transfer between the reactor and the circulating fluid increases, as seen in Figure \ref{fig:Case1_lin_perf_heat}. This heat grows until it is approximately equal to the heat transfer from the absorption and desorption reactions, at which point the temperature becomes approximately steady. (give a number to quantify the dynamics here- i.e, time for system to reach some near-equilibrium state). Here again, the linear model is largely successful in tracking the nonlinear model, but does noticeably overestimate the change in temperature in reactor B after the step change, as seen in Figure \ref{fig:tempB_case1}. (add RMS numbers here too)
\par The change in the weight fraction over time is different from that of the temperature and pressure because the weight fraction changes continually and does not go to any near-equilibrium state. The dynamics of the weight fraction are controlled by the reaction rate in each reactor, which is the derivative of the weight fraction in that reactor. As discussed with the pressure dynamics, the reaction rate in both reactors changes quickly in the seconds after the change in compressor pressure difference until the absorption rate in reactor A, the desorption rate in reactor B, and the mass flow between the reactors are approximately equal. This quick change in the reaction rate, with the very slow change for the rest of the time, results in the weight fraction in each reactor resembling a linear function with a different slope after the step change than before, as seen in Figures \ref{fig:weightA_case1} and \ref{fig:weightB_case1}. The linear model follows the nonlinear model in this, with (error numbers)
\par For Case 2, despite hydrogen flow in the system moving in the opposite direction, many of the same trends can be seen as in Case 1. In response to the change in compressor pressure difference, there is a rapid change in pressure, primarily in Reactor A due its reaction rate being less sensitive to pressure, seen in Figures \ref{fig:pressA_case2} and \ref{fig:pressB_case2}. The temperature in both reactors moves away from the circulating fluid temperature, leading to increasing heat rates until these balance out the heat released by the reaction in reactor A and absorbed by the reaction in reactor B. This can be seen in , seen in Figures \ref{fig:tempA_case2}, \ref{fig:tempB_case2}, and \ref{fig:Case2_lin_perf_heat}. Finally the weight fraction, shown in , seen in Figures \ref{fig:weightA_case2} and \ref{fig:weightB_case2}, is approximately linear with respect to time both before and after the increased compressor pressure difference, with the magnitude of the slope increasing after the change due to the change in reaction rates with the change in reactor pressures.
\par However, the linear model does track the nonlinear model as well in this case. (Give numbers).
\subsection{Resetting the Linearization Point}
\par For longer time periods, there is a significant concern about whether the linear model will accurately track the nonlinear model as operating conditions move further away from those at the linearization point. Thus, the linear approximation becomes less accurate. To address this, the linear model can be periodically re-linearized around the current operaing condition. To illustrate how re-linearization affects the accuracy of the linear model, we consider the evolution of the system for 100 minutes starting with the initial conditions shown in Table~\ref{tab:in_var_relin} using both the nonlinear model and the linear model. In order to sustain the absorption and desorption reactions in the reactors for 100 minutes, the compressor pressure difference is increased by 10 kPa every 10 minutes. We compare three different cases: 1) simulating the linear model from the same initial conditions without any re-linearization, 2) simulating the linear model beginning with the same initial conditions but then relinearizing it every 10 minutes, and 3) repeating case 2 but also resetting the state variables to match the nonlinear model at the time of relinearization. The resulting heat transfer rates for these cases are compared against the nonlinear model simulation in Figure~\ref{fig:Qout_relin_with_no_relin}.
\begin{table}[!htb]
\caption{Initial values for the state and input variables for comparing performance with and without re-linearizing the model and for the case where the controller is used for flow from reactor B to reactor A.}
\begin{center}
\begin{tabular}{l c c l c c}
\noalign{\vskip -1.5mm}
\hline
\noalign{\vskip 1mm}
$\textbf{\text{Variable}}$ & $\textbf{\text{Value}}$ & $\textbf{\text{Units}}$ & $\textbf{\text{Variable}}$ & $\textbf{\text{Value}}$ & $\textbf{\text{Units}}$\\
\noalign{\vskip 1mm}
\hline
\noalign{\vskip 1mm}
$T_{i,A}$ & 6.9 & \degree C & $\dot{m}_{A}$ & 0.2 & kg/s \\
$T_{i,B}$ & 37. & \degree C & $\dot{m}_{B}$ & 0.2 & kg/s \\
$P_{i,A}$ & 480 & kPa & $\Delta P_{comp}$ & 210 & kPa \\
$P_{i,B}$ & 280 & kPa & $T_{wg,in,A}$ & 1.9 & \degree C\\
$w_{i,A}$ & 0.004 & kg H / kg M & $T_{wg,in,B}$ & 42. & \degree C \\
$w_{i,B}$ & 0.0095 & kg H / kg M & & & \\
\noalign{\vskip 1mm}
\hline
\end{tabular}
\end{center}
\label{tab:in_var_relin}
\end{table}
\begin{figure}[!htb]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case_heatA.jpg}
\caption{Reactor A: All cases}
\label{fig:heatA_relin}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case_heatA_all_relin.jpg}
\caption{Reactor A: No case without re-linearization}
\label{fig:heatA_relin_all_relin}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case_heatB.jpg}
\caption{Reactor B: All cases}
\label{fig:heatB_relin}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Relin_case_heatB_all_relin.jpg}
\caption{Reactor B: No case without re-linearization}
\label{fig:heatB_relin_all_relin}
\end{subfigure}
\caption{Heat transfer rates from the hydride beds to the circulating fluid. The performance of the linear models is compared to the nonlinear model (solid black line) for three different cases: one where the model is never re-linearized (dot-dashed pink line), one where it is re-linearized every 10 minutes (dotted blue line), and one where it is re-linearized and reset to match the target every 10 minutes (dashed red line). Throughout this study, the compressor pressure difference is increased by 10 kPa every 10 minutes.}
\label{fig:Qout_relin_with_no_relin}
\end{figure}
Without re-linearization, the linear model sharply diverges from the nonlinear model until its output eventually bears no resemblance to it, but the periodic re-linearization prevents this from happening. Both re-linearizing and re-linearizing and resetting to match the nonlinear model result in comparable predictions after $\sim$20 minutes. The linear model diverges within the first 10 minutes, but approaches the nonlinear model after the first re-linearization and continues to agree well for the remainder of the simulation. This indicates that the linear model is less accurate when linearized around the initial condition for this case than when linearized around any subsequent point. This is primarily due to the initial reaction rate in reactor B being an order of magnitude larger than the initial rate in reactor A, whereas at all subsequent linearization points are much closer to being balanced. Not re-linearizing the model does not lead to significant divergence if $T_{i,B}$ is reduced to 35 $\degree$C, since this changes the initial reaction rate to be approximately equal to that in reactor A. Re-linearization is therefore less important for some initial conditions than others. The primary advantage of re-linearization is to avoid a case where the linear model diverges sharply from the nonlinear model because it was linearized around an initial point where the rate of change of mass and energy are much larger than they are for most of the operating time. |
2207.13068 | \section{Introduction}
The problem of existence of outliers\footnote{[which] are also referred to as abnormalities, discordants, deviants, or anomalies in the data mining and statistics literature \cite{Aggarwal_2015}.} or outlier detection ``[have] been recognized for a very long time, certainly since the middle of the eighteenth century. Daniel Bernoulli, writing in 1777 about the combination of astronomical observations, said:
\vspace{5px}
Is it right to hold that the several observations are of the same weight or moment, or equally prone to any and every error? $\ldots$ Is there everywhere the same probability? Such an assertion would be quite absurd, which is undoubtedly the reason why astronomers prefer to reject completely observations which they judge to be too wide of the truth, while retaining the rest and, indeed, assigning to them the same reliability. $\ldots$ I see no way of drawing a dividing line between those that are to be utterly rejected and those that are to be wholly retained; it may even happen that the rejected observation is the one that would have supplied the best correction to the others. Nevertheless, I do not condemn in every case the principle of rejecting one or other of the observations, indeed I approve it, whenever in the course of observation an accident occurs which in itself raises an immediate scruple in the mind of the observer, before he has considered the event and compared it with the other observations. If there is no such reason for dissatisfaction I think each and every observation should be admitted whatever its quality, as long as the observer is conscious that he has taken every care." \cite{Barnett_1978}.
\vspace{5px}
We refer the reader to \cite{Aggarwal_2015} for a detailed conceptual account as the amount of literature on outliers is vast. However, for an inclusive definition of an outlier, we start with a general one given by Grubbs in 1969, ``an outlying observation, or `outlier', may be merely an extreme manifestation of the random variability inherent in the data. ... On the other hand, an outlying observation may be the result of gross deviation from prescribed experimental procedure or an error in calculating or recording the numerical value"\cite{Grubbs_1969}. Hawkins in 1980 defined the concept of an outlier as ``[a]n outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism"\cite{Hawkins_1980}.
Now, before we start making clear what \textit{we} imply by an outlier -- an observation (or a subset of observations) in a set of data `which deviates so much from the remaining data as to arouse suspicions', we admit that ``it is a matter of subjective judgement on the part of the observer whether or not he picks out some observation (or set of observations) for scrutiny"\cite{Barnett_1978}; however, our interest and main lines of inquiry rest in identifying observations which can be characterized as \textit{extreme} in some way. In this paper, therefore, our purpose is to propose and investigate definitions of types of outliers so that we can minimize the number of data/sample-specific parameters in order for the working definition to have qualities that we expect from a mathematical definition to possess as well as to see if (and how) the new notion of outliers relates to some important results in probability theory and statistics; e.g. to the law of large numbers and the extreme value theory.
We first briefly discuss the work of Klebanov et al in \cite{Klebanov_2016} as they recently introduced and analyzed a new formulation for the notion of outliers based on order statistics due to certain drawbacks of the classical (and inherently conceptual) definitions of outliers given in \cite{Grubbs_1969}, \cite{Barnett_1978}, \cite{Hawkins_1980} by letting $X_1,X_2,\ldots,X_n$ be i.i.d. non-negative continuous random variables and denoting the corresponding order statistics by $X_{(1)}\leq X_{(2)}\leq \ldots \leq X_{(n)}$. Then $X_{(n)}$ is said to be an outlier of order $1/\kappa$ if $X_{(n -1)} \leq \kappa X_{(n)}$, where $\kappa \in (0,1)$ is some fixed number depending on the choice of the practitioner. Various properties of this new definition were investigated in their paper \cite{Klebanov_2016}. This new statistic, although giving a different look at the problem, has certain deficiencies.
For example, it applies to only to cases with a single outlier, it is not robust in the sense that a small change in only a few points in the data may make an outlier a standard outcome, and vice versa. We generalizes this definition by considering the entire sample and looking at the ratios of partial sums of order statistics. Under the assumption that outliers are a subset of sample maxima, we consider the ratios of the form: \begin{equation}
\label{eq:intro}
\frac{\sum_{i=1}^m X_{(i)}}{\sum_{i=m+1}^n X_{(i)}}
\end{equation}
for $m \in \set{1, \dots, n}$. We define $X_{m+1}, \dots, X_n$ as outliers when the ratio (\ref{eq:intro}) is greater than a (pre-determined) threshold $\kappa$ value.
Our contributions also include a python library focused on tail index estimation and other computational methods related to heavy-tailed distributions \cite{Markovich_2008} as well as the simulations of our statistic. In \cref{section:def} we formally introduce our statistic. \Cref{section:prelim} comprises some preliminary lemmas for our calculations. In \cref{sec:calculation}, we derive the distribution function for our statistic. In \cref{section:approx} we give a concentration for the error margin in our calculations. In \cref{section:experiments}, we discuss outlier generating models and generate comparative simulations of our proposed statistic with well-known distributions. Finally, we propose an algorithmic method for the selection of the $\kappa$-threshold value and use it for distinguishing between two Pareto tails.
\section{A new notion of outliers}
\label{section:def}
Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables with cumulative distribution function $F$. We may further assume that these are absolutely continuous, and call the common p.d.f. $f$. Let $X_{(1)}, \ldots, X_{(n)}$ be the corresponding (increasing) order statistics. We are to investigate the problem of outliers in a way that the number of $\kappa$-outliers is defined via moving averages as
\begin{equation}
\label{definition}
\mathcal{O}_n \coloneqq n - \min\left\{i: \frac{1}{i} \sum_{j=1}^i |X_{(j)}| < \kappa \frac{1}{n -i} \sum_{j=i+1}^n |X_{(j)}| \right\}
\end{equation}
The definition (\ref{definition}) may also be potentially useful for analyzing time series in a nonparametric way (i.e., without the normality or a similar distributional assumption). Assume that the $X_i$'s are nonnegative. We start by defining the following statistics
\begin{equation}\label{definition2}
T_{k,n}\equiv T_k= \sum_{i=n-k+1}^{n} X_{(i)},\quad 1 \leq k \leq n
\end{equation}
denoting the sum of the top $k\in\{1, \dots, n\}$ order statistics from a sample of size $n$ and
\begin{equation}\label{definition3}
S_{m,n} \equiv S_m = \sum_{i= 1}^{m} X_{(i)},\quad 1 \leq m \leq n
\end{equation}
denoting the sum of the first $m\in\{1, \dots, n\}$ order statistics from the same sample. Subsequently, putting (\ref{definition2}) and (\ref{definition3}) together, we investigate probabilities of the form $\mathbb{P}\left(\frac{1}{m}S_m<\kappa\frac{1}{n-m}T_{n-m}\right)$ where $\kappa\in(0,1)$ is fixed. Furthermore, when $n$ and $m$ are fixed, we may redefine $\kappa\equiv \kappa(n,m)\coloneqq\frac{m}{n-m}\kappa$, for convenience. In particular, this probability admits a closed form expression which we were able to simplify to an explicit formula for certain special cases. In order to compute the probability
$$
\mathbb{P}\left(\frac{S_m}{T_{n-m}}<\kappa\right),
$$
we exploit the Markov Property (MP) that the order statistics possess in order to compute the following conditional probability (which is slightly different than the probability we are interested in).
\begin{equation}\label{definition4}
\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}<\kappa \mid X_{(m)}=u\right)
\end{equation}
This is because the conditional distributions of a subset of the order statistics given another subset satisfy some really structured properties including the MP. For reference, see \cite{Reiss_1989}, \cite{Nevzorov_2001}, \cite{Nagaraja_2003}, \cite{Arnold_2008}. The following three lemmas we state are instrumental in calculating the probability defined above. They are well-known and follow from direct computations, so we omit the proofs.
\section{Preliminaries}
\label{section:prelim}
\begin{lemma} Let $X_{1}, X_{2}, \cdots, X_{n}$ be independent observations from a continuous cdf $F$ with density $f .$ Fix $1 \leq i<j \leq n$. Then, the conditional distribution of $X_{(i)}$ given $X_{(j)}=x$ is the same as the unconditional distribution of the $i$-th order statistic in a sample of size $j-1$ from a new distribution, namely the original $F$ truncated at the right at $x .$ In notation,
$$
f_{X_{(i)} \mid X_{(j)}=x}(u)=\frac{(j-1) !}{(i-1) !(j-1-i) !}\left(\frac{F(u)}{F(x)}\right)^{i-1}\left(1-\frac{F(u)}{F(x)}\right)^{j-1-i} \frac{f(u)}{F(x)}, u<x.
$$\\
\end{lemma}
\begin{lemma} Let $X_{1}, X_{2}, \cdots, X_{n}$ be independent observations from a continuous cdf $F$ with density $f$. Fix $1 \leq i<j \leq n$. Then, the conditional distribution of $X_{(j)}$ given $X_{(i)}=x$ is the same as the unconditional distribution of the $(j-i)$-th order statistic in a sample of size $n-i$ from a new distribution, namely the original $F$ truncated at the left at $x$. In notation,
$$
f_{X_{(j)} \mid X_{(i)}=x}(u)=\frac{(n-i) !}{(j-i-1) !(n-j) !}\left(\frac{F(u)-F(x)}{1-F(x)}\right)^{j-i-1}\left(\frac{1-F(u)}{1-F(x)}\right)^{n-j} \frac{f(u)}{1-F(x)}, u>x .
$$\\
\end{lemma}
\begin{lemma}[Markov Property] Let $X_{1}, X_{2}, \cdots, X_{n}$ be independent observations from a continuous cdf $F$ with density $f$. Fix $1 \leq i<j \leq n$. Then, the conditional distribution of $X_{(j)}$ given $X_{(1)}=x_{1}, X_{(2)}=x_{2}, \cdots, X_{(i)}=x_{i}$ is the same as the conditional distribution of $X_{(j)}$ given $X_{(i)}=x_{i} .$ That is, given $X_{(i)}, X_{(j)}$ is independent of $X_{(1)}, X_{(2)}, \cdots, X_{(i-1)} .$\\
\end{lemma}
\begin{corollary}
Let $X_{1}, X_{2}, \cdots, X_{n}$ be independent observations from a continuous cdf $F$ with density $f$. Then, the conditional distribution of $X_{(1)}, X_{(2)}, \cdots, X_{(n-1)}$ given $X_{(n)}=x$ is the same as the unconditional distribution of the order statistics in a sample of size $n-1$ from a new distribution, namely the original $F$ truncated at the right at $x$. In notation,
$$
f_{X_{(1)}, \ldots, X_{(n-1)} \mid X_{(n)}=x}\left(u_{1}, \cdots, u_{n-1}\right)=(n-1) ! \prod_{i=1}^{n-1} \frac{f\left(u_{i}\right)}{F(x)}, \mbox{ where } u_{1}<\cdots<u_{n-1}<x .
$$
\end{corollary}
A similar result holds for the conditional distribution of $X_{(2)}, X_{(3)}, \cdots, X_{(n)}$ given $X_{(1)}=x$
\begin{corollary}
If $F$ is absolutely continuous, the order statistics, $X_{(1)}, \ldots, X_{(n)}$, form a (discrete time) Markov chain with transition densities:
$$
\begin{array}{r}
f_{i+1 \mid i}(y \mid x)=(n-i)\left(\frac{F(y)-F(x)}{1-F(x)}\right)^{n-i-1} \frac{f(y)}{1-F(x)},
\text { for } y>x ;\quad i=1, \ldots, n-1 .
\end{array}
$$
\end{corollary}
An important consequence of the two corollaries is that when conditioned on the $m$-th order statistic the sums $S_{m-1}$ and $T_{n-m}$ are independent. In particular:
\begin{proposition}\label{prop:1}
Let $F$ be absolutely continuous, then for any $1<k<n,$ the random vectors
$$
\boldsymbol{X}^{(1)}=\left(X_{(1)}, \ldots, X_{(k-1)}\right) \text { and } \boldsymbol{X}^{(2)}=\left(X_{(k+1)}, \ldots, X_{(n)}\right)
$$
are conditionally independent given that $X_{(k)}=x_{k},$ that is to say
$$
\begin{array}{c}
\mathbb{P}\left(\boldsymbol{X}^{(1)} \in B_{1}, \boldsymbol{X}^{(2)} \in B_{2} \mid X_{(k)}=x_{k}\right) = \\
\quad\quad \mathbb{P}\left(\boldsymbol{X}^{(1)} \in B_{1} \mid X_{(k)}=x_{k}\right) \mathbb{P}\left(\boldsymbol{X}^{(2)} \in B_{2} \mid X_{(k)}=x_{k}\right)
\end{array}
$$
for any Borel set $B_{1} \in \mathcal{B}\left(\mathbb{R}^{k-1}\right)$ and $B_{2} \in \mathcal{B}\left(\mathbb{R}^{n-k}\right)$. Furthermore, $$\left[S_{m-1}\mid X_{(m)}=u\right]\text{ and } \left[T_{n-m}\mid X_{(m)}=u\right]\text{ are independent as well}$$
since $S_{m-1}=\sum_{i=1}^{m-1}X_{(i)} \text{ and } T_{n-m}=\sum_{i=m+1}^{n}X_{(i)} $ are linear functions of $\boldsymbol{X}^{(1)} \text{ and } \boldsymbol{X}^{(2)}$, respectively.\\
\end{proposition}
\section{Finite sample statistics}
\label{sec:calculation}
As previous, we let $X_1,X_2,\ldots,X_n$ be i.i.d. non-negative random variables with absolutely continuous distribution function $F$, and call the common p.d.f. $f$. Let $X_{(1)}, \ldots, X_{(n)}$ be the corresponding order statistics.
\subsection{Sum of the top order statistics}
Observe, denoting the distribution function of $X_{(i)}$ by $F_i$, that
\begin{align*}
\mathbb{P}\left(T_{k}<t\right)&=\mathbb{P}\left( \sum_{i=n-k+1}^{n} X_{(i)}<t\right)\\
&=\int \mathbb{P}\left( \sum_{i=n-k+1}^{n} X_{(i)}<t \mid X_{n-k}=u\right) d F_{X_{n-k}}(u)
\end{align*}
We know that for each $n-k<j\leq n$, the conditional distribution of $X_{(j)}$ given $X_{(n-k)}=u$ is formed from an i.i.d. sample of size $k$ having the cdf $G_{u}$ given by
$$
G_{u}(t)=\left\{\begin{array}{ll}
0, & t<u \\
\frac{F(t)-F(u)}{1-F(u)}, & t \geq u .
\end{array}\right.
$$
Hence,
$$
\mathbb{P}\left(T_{k}<t\right)=\int_{0}^{t} G_{u}^{*(k)}(t)d F_{X_{n-k}}(u),
$$
where $G_{u}^{*(k)}$ is the $k$-fold convolution of $G_{u}$.\\
We note that $T_{k}$ is also related to the selection differential, which is a familiar term in the genetics literature \cite{Crow_1970}, \cite{Burrows_1972}, \cite{Falconer_1996} given by
$$
D_{k}=\frac{1}{\sigma}\left(\frac{1}{k} \sum_{j=n-k+1}^{n} X_{(j)}-\mu\right)
$$
where $\mu$ and $\sigma$ are the population mean and standard deviation, respectively, \cite{Nagaraja_1981}.
It also serves as a test statistic for outliers in samples from normal distribution, \cite{Barnett_1978},\cite{Burrows_1972}; additionally, from \cite{Nagaraja_1982}, one can obtain the asymptotic distribution of $D_{k}$ (when $X_i$'s have finite second moment) under suitable centering and scaling if $n\rightarrow \infty$ with $k$ fixed as well as if $k=[np], 0<p<1 \text { and } n \rightarrow \infty$.\\
\textbf{Sum of the first $m$ order statistics: } Following a similar argument as before, we get:
$$
\mathbb{P}\left(S_{m}<t\right)=\int_{0}^{t} H_{u}^{*(m)}(t)d F_{X_{m+1}}(u),
$$
where $ H_{u}^{*(m)}$ is the $m$-fold convolution of the df $H_u$ given by
$$
H_{u}(t)=\left\{\begin{array}{ll}
\frac{F(t)}{F(u)}, & t \leq u\\
0, & t>u.
\end{array}\right.
$$
\subsection{The case of $\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}<\kappa\right)$}
Now, we are ready to compute the probability (\ref{definition4}) in which we were interested in getting a somewhat "nice" expression for
$$
\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}<\kappa \mid X_{(m)}=u\right)
$$
Define for a fixed $m\in\{1,2,\ldots,n-1\}$, $$\boldsymbol{X}^{(1)}=\left(X_{(1)}, \ldots, X_{(m-1)}\right) \text { and } \boldsymbol{X}^{(2)}=\left(X_{(m+1)}, \ldots, X_{(n)}\right).$$
Then, it follows from Proposition \ref{prop:1} that $\left[\boldsymbol{X}^{(1)}\mid X_{(m)}=u\right]$, $\left[\boldsymbol{X}^{(2)}\mid X_{(m)}=u\right]$ and $ \left[S_{m-1}\mid X_{(m)}=u\right]$, $\left[T_{n-m}\mid X_{(m)}=u\right]$ are independent.\\
Note that we have
$$
f_{\boldsymbol{X}^{(1)}\mid X_{(m)}}(\boldsymbol{x}^{(1)})\equiv f_{X_{(1)}, \ldots, X_{(m-1)} \mid X_{(m)}}\left(x_{1}, \cdots, x_{m-1}\right)=(m-1) ! \prod_{i=1}^{m-1} \frac{f\left(x_{i}\right)}{F(u)},
$$
for $ x_{1}<\cdots<x_{m-1}<x_m=u$. Also,\\
$$
f_{\boldsymbol{X}^{(2)}\mid X_{(m)}}(\boldsymbol{x}^{(2)})\equiv f_{X_{(m+1)}, \ldots, X_{(n)} \mid X_{(m)}}\left(x_{m+1}, \cdots, x_{n}\right)=(n-m) ! \prod_{i=m+1}^{n} \frac{f\left(x_{i}\right)}{1-F(u)},
$$
for $u=x_m<x_{m+1}<\cdots<x_{n}$.\\
So, let $R=S_{m-1}/T_{n-m}$, and observe that\\
$$f_{S_{m-1},T_{n-m}\mid X_{(m)}}(t_1,t_2)=f_{S_{m-1}\mid X_{(m)}}(t_1)\cdot f_{T_{n-m}\mid X_{(m)}}(t_2)$$
where
$$
f_{S_{m-1}\mid X_{(m)}}(t_1)= \textit{(m-1)-fold conv. using } f_{\boldsymbol{X}^{(1)}\mid X_{(m)}}
$$
and
$$
f_{T_{n-m}\mid X_{(m)}}(t_2)= \textit{(n-m)-fold conv. using } f_{\boldsymbol{X}^{(2)}\mid X_{(m)}}
$$
Hence,
\begin{align*}
\mathbb{P}\left(R<\kappa \mid X_{(m)}=u\right)&=\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}<\kappa \mid X_{(m)}=u\right)\\
&=\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\left(\int_{0}^{\kappa t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1\right) d t_2\\
&= \int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2) \cdot H_{u}^{*(m-1)}(\kappa t_2) d t_2\\
&=:F_{R\mid m}(\kappa)
\end{align*}
where $ H_{u}^{*(m-1)}$ is the $(m-1)$-fold convolution of the df $H_u$ given by
$$
H_{u}(t)=\left\{\begin{array}{ll}
\frac{F(t)}{F(u)}, & t \leq u\\
0, & t>u,
\end{array}\right.
$$\\
which can be calculated explicitly using $ f_{\boldsymbol{X}^{(1)}\mid X_{(m)}},$ but it is generally hard. (However, we will calculate it when we consider specific distributions, e.g. the exponential distribution.)\\
Differentiating $F_{R\mid m}(\kappa)$ with respect to $\kappa$, we get the pdf $f_{R\mid m}(\kappa)$:
$$
f_{R\mid m}(\kappa)=\frac{d}{d \kappa}\left[\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\left(\int_{0}^{\kappa t_2} f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1\right) d t_2\right]
$$
$$
f_{R\mid m}(\kappa)=\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2) \cdot f_{S_{m-1}\mid X_{(m)}}(\kappa t_2) t_2 d t_2
$$
Therefore,
\begin{align*}
\mathbb{P}\left(R<\kappa\right)&=\int_{0}^{\infty} \mathbb{P}\left(R<\kappa \mid X_{(m)}=u\right)dF_m(u)\\
&=\int_{0}^{\infty}F_{R\mid m}(\kappa)dF_m(u)\\
&=\int_{0}^{\infty}\left(\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\left(\int_{0}^{\kappa t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1\right) d t_2\right)dF_m(u)\\
&=\int_{0}^{\infty}\left(\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\cdot H_{u}^{*(m-1)}(\kappa t_2) d t_2\right)dF_m(u)
\end{align*}
where $dF_m(u)=f_{m}(u)du=\frac{n !}{(m-1) !(n-m) !} F^{m-1}(u)[1-F(u)]^{n-m} f(u)du$.
\subsection{Application: the case of the exponential distribution}
Let the parent distribution be the std. exponential; i.e., $X_i\sim$Exp(1) for each $i=1,2,\ldots,n$. Then, it is well-known that (e.g., \cite{Nagaraja_2003}, \cite{Nevzorov_1984})
$$X_{(i)}\stackrel{d}=\sum_{k=1}^{i}\frac{Z_k}{n-k+1}\hspace{2pt},$$ $ \text{where } Z_k\sim \text{Exp(1)}\text{ for each } k\geq1. \text{ Now, define }\widehat{Z_k}\coloneqq\frac{Z_k}{n-k+1} \text{ so that }\newline \widehat{Z_k}\sim$Exp($\lambda_{k}^{-1}),\hspace{5pt} \lambda_{k}=(n-k+1).$ Hence,
\begin{align*}
T_{n-m}=\sum_{i=m+1}^{n}X_{(i)}&\stackrel{d}=\sum_{i=m+1}^{n}\sum_{k=1}^i \widehat{Z_k}\\
&=(n-m)\sum_{k=1}^{m+1}\widehat{Z_k}+\sum_{k=m+2}^{n}(n-k+1)\widehat{Z_k}\\
&= \sum_{k=1}^{m+1}\beta_k Z_k + W
\end{align*}
$\quad \text{where }\hspace{2pt}\beta_k\coloneqq\frac{n-m}{n-k+1}=(n-m)\lambda_k^{-1}\hspace{2pt}\text{ and }\hspace{2pt} W\sim Gamma(n-m-1,1)$.\\
Now, consider the theorem below by Jasiulewicz, H. and Kordecki, W. to find the pdf of\hspace{1.5pt} $\sum_{k=1}^{m+1}\beta_k Z_k$ in the expansion of $T_{n-m}$.
\begin{theorem}\cite{Jasiulewicz_2003}\label{thm:Jasiulewicz_2003}
Let $X_{1}, \ldots, X_{n}$ be $n$ independent random variables such that every $X_{i}$ has a probability density function $f_{X_{i}}$ given by
$$
f_{X_{i}}(t)\coloneqq\beta_{i} \exp \left(-t \beta_{i}\right) \mathbbm{1}_{(0, \infty)}(t)
$$
for all real number $t,$ where the parameter $\beta_{i}$ is positive for all $i=$ $1,2, \ldots, n$. We suppose that the parameters $\beta_{i}$ are all distinct. Then the sum $S_{n}$ has the following probability density function:
$$
f_{S_{n}}(t)=\sum_{i=1}^{n} \frac{\beta_{1} \ldots \beta_{n}}{\prod_{j=1 \atop j \neq i}^{n}\left(\beta_{j}-\beta_{i}\right)} \exp \left(-t \beta_{i}\right) \mathbbm{1}_{(0, \infty)}(t)
$$
for all $t \in \mathbb{R}$.\\\end{theorem}
So, by Theorem \ref{thm:Jasiulewicz_2003} we find the pdf of $L\coloneqq\sum_{k=1}^{m+1}\beta_k Z_k$ in the expansion of $T_{n-m}$ by noting that the $\beta_i$'s in the theorem correspond to $\frac{\lambda_i}{n-m}$ in our notation. Therefore,
\[
f_L(x) = \left(\frac{\lambda_1}{n-m}\right)\left(\frac{\lambda_2}{n-m}\right)\cdots\left(\frac{\lambda_{m+1}}{n-m}\right)\sum_{i=1}^{m+1}\mlarge{\Psi}_{i,m+1}\cdot e^{-(\frac{\lambda_i}{n-m})x}\hspace{4pt},
\]
where
$$
\mlarge{\Psi}_{i,m+1}^{-1}=\prod_{j=1\atop j\neq i}^{m+1}\left(\frac{\lambda_j-\lambda_i}{n-m}\right)=\frac{1}{(n-m)^m}(\lambda_1-\lambda_i)\cdots(\lambda_{i-1}-\lambda_i)(\lambda_{i+1}-\lambda_i)\cdots(\lambda_{m+1}-\lambda_i)
$$
So,
$$
f_L(x)=\frac{\lambda_1\lambda_2\cdots\lambda_{m+1}}{(n-m)^{m+1}}\cdot(n-m)^m\sum_{i=1}^{m+1}\psi_{i,m+1}\cdot e^{-(\frac{\lambda_i}{n-m})x}\hspace{4pt},
$$
where
$$\psi_{i,m+1}^{-1}=(\lambda_1-\lambda_i)\cdots(\lambda_{i-1}-\lambda_i)(\lambda_{i+1}-\lambda_i)\cdots(\lambda_{m+1}-\lambda_i)$$\\
and $\mlarge{\Psi}=(n-m)^{m}\cdot\psi_{i,m+1}$\\
Then,
$$
f_L(x)=\frac{\lambda_1\lambda_2\cdots\lambda_{m+1}}{n-m}\sum_{i=1}^{m+1}\psi_{i,m+1}\cdot e^{-(\frac{\lambda_i}{n-m})x}
$$
Now, using convolution formula, we can compute $f_{T_{n-m}}$ explicitly:
$$
f_{T_{n-m}}(t)=\int_0^\infty f_L(t-x)f_W(x) dx\hspace{2pt},
$$
$\textit{where }\hspace{2pt} W\sim Gamma(n-m-1,1)$.
\begin{align*}
f_{T_{n-m}}(t)&=\frac{\lambda_1\lambda_2\cdots\lambda_{m+1}}{n-m}\sum_{i=1}^{m+1}\psi_{i,m+1}\int_0^t e^{-\frac{\lambda_i}{n-m}(t-x)}\frac{1}{(n-m-2)!}e^{-x}x^{n-m-2}dx\\
&=\frac{1}{(k-1)!} \frac{\lambda_1\lambda_2\cdots\lambda_{m+1}}{k+1}\sum_{i=1}^{m+1}\psi_{i,m+1}e^{-\frac{\lambda_i}{k+1}t}\int_0^t e^{\left(\frac{\lambda_i}{k+1}-1\right)x}x^{k-1}dx\hspace{2pt},
\end{align*}
where we put $n-m-1\equiv k$ (for convenience\footnote{For a given $k$, one can evaluate the integral:
$$
\int_0^t e^{\left(\frac{\lambda_i}{k+1}-1\right)x}x^{k-1}dx= t^k \left(t-\frac{\lambda_i t}{k+1}\right)^{-k} \left(\Gamma (k)-\Gamma \left(k,t-\frac{\lambda_i t}{k+1}\right)\right)
$$}) and $t>0$.\\
Recall that we wanted to get an expression for $\mathbb{P}\left(R<\kappa\right)\text{, where }R=S_{m-1}/T_{n-m}$:
\begin{align*}
\mathbb{P}\left(R<\kappa\right)&=\int_{0}^{\infty} \mathbb{P}\left(R<\kappa \mid X_{(m)}=u\right)dF_m(u)\\
&=\int_{0}^{\infty}\left(\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\cdot H_{u}^{*(m-1)}(\kappa t_2) d t_2\right)dF_m(u)
\end{align*}
where $dF_m(u)=f_{m}(u)du=\frac{n !}{(m-1) !(n-m) !} (1-e^{-u})^{m-1}e^{-u(n-m)}e^{-u}du$, and $ H_{u}^{*(m-1)}$ is the $(m-1)$-fold convolution of the df $H_u$ given by
$$
H_{u}(t)=\left\{\begin{array}{ll}
\frac{1-e^{-t}}{1-e^{-u}}, & t \leq u\\
0, & t>u.
\end{array}\right.
$$
Hence, an explicit expression for our target probability is available as all the ingredients are ready (up to scaling) to be employed.
\section{Approximating $\mathbb{P}\left(S_m/T_{n-m}\leq\kappa\right)$}
\label{section:approx}
\iffalse
Define
$$M_n(\kappa)\coloneqq\max\{l< n : \frac{\sum_{i=1}^l X_{(i)}}{\sum_{i=l+1}^n X_{(i)}} \leq\kappa\}$$
Suppose $M_n(\kappa)=j$ for some $1\leq j\leq n$, and denote the corresponding event by $\mathcal{M}_j$. Then,
$$\mathcal{M}_j =
\bigg\{\frac{\sum_{i=1}^j X_{(i)}}{\sum_{i=j+1}^n X_{(i)}}\leq\kappa\bigg\}\bigcap \bigg\{\frac{\sum_{i=1}^{j+1} X_{(i)}}{\sum_{i=j+2}^n X_{(i)}}>\kappa\bigg\}
$$
so
$$\mathcal{M}_j\hspace{4pt}\mbox{\LARGE$\subset$}\hspace{4pt}\bigg\{\frac{\sum_{i=1}^{j-1} X_{(i)}}{\sum_{i=j+1}^n X_{(i)}}\leq\kappa\bigg\}\bigcap \bigg\{\frac{\sum_{i=1}^{j+1} X_{(i)}}{\sum_{i=j+2}^n X_{(i)}}>\kappa\bigg\}$$
\vspace{10pt}
\textcolor{purple}{Questions and Comments:}
\textit{For the equation below i tagged with A:
I do not understand how we can assume independence of $\leq \kappa$ and $> \kappa$ parts, which we use in the step below. It doesn't seem to me that we can still rely on Proposition 3.1 when $S_m$ and $T_{n-m}$ are in both equations. My original question was gonna be about how $\mathbb{P}$ at the integral could be zero for some values of $u$ which I thought mattered for -again independence, but I managed to talk myself through it.}
Therefore, we derive the following (obvious) upper bound for $\mathbb{P}\left(\mathcal{M}_j\right)$ as:
\begin{align*}
\mathbb{P}\left(\mathcal{M}_j\right) =& \int \mathbb{P}\left(M_n(\kappa)=j\big|\, X_{(j)}=u\right)dF_j(u)\\
\leq& \int \mathbb{P}\left(\frac{\sum_{i=1}^{j-1} X_{(i)}}{\sum_{i=j+1}^n X_{(i)}}\leq\kappa, \frac{\sum_{i=1}^{j+1} X_{(i)}}{\sum_{i=j+2}^n X_{(i)}}>\kappa\, \big|\, X_{(j)}=u \right)dF_j(u)\tag{A}\\
&=\int\mathbb{P}\left(\frac{\sum_{i=1}^{j-1} X_{(i)}}{\sum_{i=j+1}^n X_{(i)}}\leq\kappa\, \big|\, X_{(j)}=u\right)dF_j(u) \int \mathbb{P}\left(\frac{\sum_{i=1}^{j+1} X_{(i)}}{\sum_{i=j+2}^n X_{(i)}}>\kappa\, \big|\, X_{(j)}=u\right)dF_j(u)
\end{align*}
by the conditional independence (Proposition 3.1). Equivalently, denoting the sum of the first $m\leq n$ order statistics by $S_m = \sum_{i= 1}^{m} X_{(i)}$, and the sum of the top $k\leq n$ order statistics by $T_k= \sum_{i=n-k+1}^{n} X_{(i)}$, we write
\begin{equation}
\mathbb{P}\left(\mathcal{M}_j\right) \leq \int P\left(\frac{S_{j-1}}{T_{n-j}}\leq \kappa \mid X_{(j)}=u\right)dF_j(u) \int P\left(\frac{S_{j+1}}{T_{n-j-1}}>\kappa \mid X_{(j)}=u\right) dF_j(u)
\end{equation}
where the integrals of (5) are easily found as in Section 4.2.
\fi
As usual, for $X_1,X_2,\ldots,X_n$ non-negative i.i.d. random variables with absolutely continuous distribution function $F$, letting
$$
T_{k,n}\equiv T_k= \sum_{i=n-k+1}^{n} X_{(i)},\quad 1 \leq k \leq n
$$ and
$$
S_{m,n} \equiv S_m = \sum_{i= 1}^{m} X_{(i)},\quad 1 \leq k \leq n
$$
observe, for $m=1,2,\ldots,n-1$, that
\begin{align*}
\frac{S_m}{T_{n-m}} &= \frac{S_{m-1}+X_{(m)}}{T_{n-m}}\\
&= \frac{S_{m-1}}{T_{n-m}} + \frac{X_{(m)}}{T_{n-m}}
\end{align*}
where $T_{n-m} = \sum_{i=m+1}^n X_{(i)}$.\\
Note that $$\frac{X_{(m)}}{T_{n-m}} = \frac{X_{(m)}}{\sum_{i=m+1}^n X_{(i)}}\leq \frac{1}{n-m}\quad\textit{ almost surely.}$$
Thus, we have
$$\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}\leq\kappa \right) \leq \mathbb{P}\left(\frac{S_m}{T_{n-m}}\leq\kappa\right) \leq \mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}\leq\kappa+\frac{1}{n-m}\right)$$
\vspace{20pt}
Recall that we have, for $R=S_{m-1}/T_{n-m}$, that
$$f_{S_{m-1},T_{n-m}\mid X_{(m)}}(t_1,t_2)=f_{S_{m-1}\mid X_{(m)}}(t_1)\cdot f_{T_{n-m}\mid X_{(m)}}(t_2)$$
So,
\begin{align*}
\mathbb{P}\left(R<\kappa + \frac{1}{n-m} \mid X_{(m)}=u\right)&=\mathbb{P}\left(\frac{S_{m-1}}{T_{n-m}}<\kappa + \frac{1}{n-m} \mid X_{(m)}=u\right)\\
&=\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\left(\int_{0}^{\left(\kappa+\frac{1}{n-m}\right) t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1\right) d t_2\\
\end{align*}
Now note that
\begin{align}
\int_{0}^{\left(\kappa+\frac{1}{n-m}\right) t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1 &= \int_{0}^{\kappa t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1 + \int_{\kappa t_2}^{\left(\kappa+\frac{1}{n-m}\right) t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1 \nonumber\\
&\leq\int_{0}^{\kappa t_2}f_{S_{m-1}\mid X_{(m)}}(t_1) d t_1 + \frac{t_2}{n-m} \label{eq:concentration1}
\end{align}
Also,
\begin{equation}
\label{eq:concentration2}
\int_{0}^{\infty} f_{T_{n-m}\mid X_{(m)}}(t_2)\frac{t_2}{n-m}dt_2 = \frac{\mathbb{E}[T_{n-m}\mid X_{(m)}]}{n-m}
\end{equation}
Therefore,
\begin{align*}
\mathbb{P}\left(R<\kappa + \frac{1}{n-m} \right) &= \int_0^\infty\mathbb{P}\left(R<\kappa + \frac{1}{n-m} \mid X_{(m)}=u\right)dF_m(u)\\
&\leq \int_0^\infty\left(\mathbb{P}\left(R<\kappa \mid X_{(m)}=u\right) + \frac{\mathbb{E}[T_{n-m}\mid X_{(m)}]}{n-m}\right)dF_m(u) & \mbox{by (\ref{eq:concentration1}) and (\ref{eq:concentration2}}),\\
&= \mathbb{P}\left(R<\kappa\right) + \frac{1}{n-m}\int_0^\infty \mathbb{E}[T_{n-m}\mid X_{(m)}=u] dF_m(u)\\
&= \mathbb{P}\left(R<\kappa\right) + \frac{1}{n-m}\mathbb{E}[T_{n-m}]\\
&= \mathbb{P}\left(R<\kappa\right) + o(1),\quad n\to\infty
\end{align*}
since if $X_i'$s have finite moment, then so is $T_{n-m}$, so that $\frac{\mathbb{E}[T_{n-m}]}{n-m} = o(1), \quad n\to\infty$. So, the rough approximation above gives an error bound vanishing in the order of $1/n$.
\section{Simulations and Experiments}
\label{section:experiments}
Empirically it is important to choose a good value of $\kappa$ which can distinguish between normal and anomalous observations. There are a few points which are of concern. First for a given distribution, a good $\kappa$ value should be able to distinguish between the centre and the tail of the distribution. Informally, $\kappa$ value should be natural to choose. Furthermore, it is important that, for a given distribution, the concentration of potential $\kappa$ values should be tight, potentially depending on the family of the given distribution.
Consequently, it is of importance to know how the statistic $R = \frac{S_{m}}{T_{n-m}}$ is distributed. We expect to see similar values of $\kappa$ on lower quantiles for most distributions. The differences between the $\kappa$ values should increase gradually towards the tail. Finally, at the tail we expect the $\kappa$ values to differ the most, with concentrations dependent on the tail index. From another perspective, when considering the moving averages of our statistic for the sample, we expect to have a sharp increase towards the end of the tail.
We will show simulations of $R$ for a set of distributions and anomaly generating models. As most outlier generator models depend upon the exponential distribution, we will use the exponential distribution for comparison. For anomaly generating model we will use the identified outliers model, in which observations \(X_1, \dots, X_n\) are not i.i.d. but some $k \in \{1, \dots, n\}$ come from a separate distribution. We will consider the simplistic exponential case as described below \cite{Balakrishnan_2019}.
\paragraph{Identified Outliers Model} $X=\left\{X_i,\dots, X_{n-k} : 1-e^{-x / \theta} ; \theta>0\right\}, k$ is known and fixed, say for simplicity let's suppose $k=1$, $X_{1}, \ldots, X_{n-k}$ are independent and the index of the contaminant is also known. If we assume further that the distribution function of the contaminant is
\[
G(x)=F\left(b^{-1} x\right)=1-e^{-(b \theta)^{-1} x}, \quad x \in \mathbb{R}
\]
for some $b \geq 1$, then, without loss of any generality, the joint distribution function of $X_{1}, \ldots, X_{n}$ is given by
\[
F_{\left(X_{1}, \ldots, X_{n}\right)}\left(x_{1}, \ldots, x_{n}\right)
= \left[\prod_{i=1}^{n-1}\left(1-e^{-x_{i} / \theta}\right)\right]
\left(1-e^{-x_{n} / b \theta}\right)\]
for $x>0$, $\theta>0$, $b\geq1$.
The simulations for the identified outliers model. In these sets of simulations we took $k=100$, $\theta=1$ for the original sample, and $b=3$ for outliers.
\paragraph{Exponential Distribution}
The simulations for exponential distribution, $f(x;\theta)= \theta e^{-x\theta}$, with $\theta = 1$.
\paragraph{Half-Normal Distribution} We will also use the absolute value of the normal distribution, the half-normal distribution, as a way to quantify the outliers in a normally distributed sample. The behaviour of moving averages of $R$-statistic for the normal distribution is specifically relevant for any empirical study. As characterizing the tail and cut-off of the normal distribution has many applications. For the simulations of Half-Normal distribution, we will use the standard normal $Z \sim N(0, 1)$ and simulate $Y = \abs{Z}$.
Simulations are generated using the following procedure, first we generate and sort an i.i.d. sample of size $n=1000$. Then we compute the $R$-statistic for some values of $m$ in order to better observe the changes in the behaviour of the statistic. We choose the median, $5^{th}$ percentile, and $95^{th}$ percentile points as indicators. This procedure was repeated $10000$ times. The values obtained are given in the figures \ref{plot:distributions2} below. We also provide in figures \ref{plot:distributions} moving averages of $R$-statistic throughout a sample of $n=1000$.
\begin{figure}[H]
\begin{minipage}{\linewidth}
\centering
\begin{tabular}{cc}
\textbf{Exponential Distribution} &
\textbf{Identified Outliers Model} \\
\includegraphics[width=\textwidth/2]{images/experiments/Hist_m5_expon.png}
&\includegraphics[width=\textwidth/2]{images/experiments/Hist_m5_identified_outliers_model.png} \\
\\
\includegraphics[width=\textwidth/2]{images/experiments/Hist_mmed_expon.png}
&\includegraphics[width=\textwidth/2]{images/experiments/Hist_mmed_halfnorm.png} \\
\includegraphics[width=\textwidth/2]{images/experiments/Hist_m95_expon.png}
&\includegraphics[width=\textwidth/2]{images/experiments/Hist_m95_identified_outliers_model.png} \\
\end{tabular}
\end{minipage}
\caption{Distribution of the $R$ statistic across 10 000 samples for fixed $m=50$, $500$, and $950$.}
\label{plot:distributions2}
\end{figure}
\begin{figure}[H]
\begin{minipage}{\linewidth}
\centering
\begin{tabular}{ccc}
\textbf{Half-Normal Distribution} &
\textbf{Exponential Distribution} &
\textbf{Identified Outliers Model} \\
\includegraphics[width=\textwidth/3]{images/experiments/samples_halfnorm.png}
&\includegraphics[width=\textwidth/3]{images/experiments/samples_expon.png}
&\includegraphics[width=\textwidth/3]{images/experiments/samples_identified_outliers_model.png} \\
\includegraphics[width=\textwidth/3]{images/experiments/stat_halfnorm.png}
&\includegraphics[width=\textwidth/3]{images/experiments/stat_expon.png}
&\includegraphics[width=\textwidth/3]{images/experiments/stat_identified_outliers_model.png} \\
\end{tabular}
\end{minipage}
\caption{Samples of $n$=1000 and corresponding $R$ statistic from exponential distribution and identified outliers model.}
\label{plot:distributions}
\end{figure}
\subsection{Automatic selection of $\kappa$-threshold}
Ideally, we would like to define the cut-off point with respect to the derivatives of the distribution function of our statistic, which would then allow us to define a theoretical cut-off point for any sample. However, there is no established well-defined notion for an elbow point. The literature of elbow detection\footnote{Also called knee or kneecap detection.} is therefore algorithmically focused \cite{Gomes_2018}.
The elbow detection literature is based on the following pointwise definition of the curvature of a function. Most works than define the elbow as the point of maximum curvature, and use algorithmic means to calculate it.
\begin{definition}[Curvature of a function \cite{Kneedle_2011}]
\label{def:curvature}
For any continuous function $f$, there exists a standard closed-form $K_{f}(x)$ that defines the curvature of $f$ at any point as a function of its first and second derivative:
\[K_{f}(x)=\frac{f^{\prime \prime}(x)}{\left(1+f^{\prime}(x)^{2}\right)^{\frac32}}\]
\end{definition}
However, the \cref{def:curvature} of curvature does extend easily to discrete data, instead \cite{Kneedle_2011}, \cite{Tolsa_2000}, and \cite{Gomes_2018} use Menger curvature, which is defined for three points as the curvature of the circle circumscribed by those points. There are also angle-based \cite{Kneedle_2011} and exponentially weighted moving average (EWMA) based methods \cite{Albrecht_2006} which use the differences between successive points and EWMA smoothing to check deviations from arrival times respectfully.
We will use the kneedle detection algorithm for estimating the "elbow point" of our statistic \cite{Kneedle_2011}. The kneedle algorithm uses dynamic first derivative threshold, in combination with he IsoData \cite{Ridler_1978} to find the elbows of a discrete data. It can work on discrete datasets and has a sensitivity parameter which can be fine-tuned for how sensitively a knee is to be detected. While the smaller values of the sensitivity parameter respond to quick change, the larger values are more robust.
In the Figure \ref{plot:distributions3} we revisit our simulation studies and show the simulation studies and show the chosen cut-off points using the kneedle algorithm, choosing the sensitivity parameter as $5.0$.
\begin{figure}[H]
\begin{minipage}{\linewidth}
\centering
\begin{tabular}{ccc}
\textbf{Half-Normal Distribution} & \textbf{Exponential Distribution} &
\textbf{Identified Outliers Model} \\
\includegraphics[width=\textwidth/3]{images/experiments/OrderStatsplot_halfnorm.png}
& \includegraphics[width=\textwidth/3]{images/experiments/OrderStatsplot_expon.png}
&\includegraphics[width=\textwidth/3]{images/experiments/OrderStatsplot_identified_outliers_model.png} \\
\includegraphics[width=\textwidth/3]{images/experiments/Rplot_halfnorm.png}
& \includegraphics[width=\textwidth/3]{images/experiments/Rplot_expon.png}
&\includegraphics[width=\textwidth/3]{images/experiments/Rplot_identified_outliers_model.png} \\
\end{tabular}
\end{minipage}
\caption{Simulation studies in figure \ref{plot:distributions} revisited with $\kappa$-threshold chosen automatically.}
\label{plot:distributions3}
\end{figure}
\subsection{Application II: Distinguishing two Pareto tails}\label{subsection:tails}
Now that we have a method for selecting a $\kappa$ value, we conduct further simulation tests based on distinguishing two Pareto tails in order to show the efficacy of our statistic. Our goal is to find a threshold $\kappa$ beyond which $ >.90$ of the observations belong to the "heavier" tail. We set up our experiment as follows. For given two tail indices $\alpha_1 < \alpha_2$, we first determine a $\kappa$-threshold value for the $\alpha_1$ indexed Pareto distribution. Then we sample a $N = 1000000$ observations from both of the distributions and calculate our statistic across the entire sample. We report the percentage of observations from $\alpha_2$ in the samples above the predetermined threshold. We repeat this experiment a $1000$ times in order to build a confidence interval on our results.
A key problem in our experiments is the selection $\alpha_1$ and $\alpha_2$ values, it has been established that as the two indices get closer it becomes numerically difficult to differentiate between the tails, even if one of the tails are outside of the Levy-stable regime \cite{Weron_2001}. We start our experiments with indices $\alpha_1 = 1.5$ and $\alpha_2 = 2.5$ and move $\alpha_2$ closer every iteration.
The results of the experiments are given in the table \ref{table:pareto_sim} below. When the tail indices are far apart our statistic produces an ideal change point for differentiating between the two tails. It is clear that as the two tail indices get closer our statistic becomes unable to distinguish the difference between the two tails. However, it remains consistent with variance $\sim 10^{-5}$ across all samples.
\begin{table}[H]
\begin{center}
\begin{tabular}{||c c c c c||}
\hline
$\alpha_1$ & $\alpha_2$ & $\kappa$-threshold & Expected percentage of $\alpha_2$ above $\kappa$ & Variance \\ [0.5ex]
\hline\hline
1.5 & 2.5 & 2.745 & 0.95 & $1.88\times10^{-5}$ \\
\hline\hline
1.5 & 2.3 & 2.745 & 0.924 & $4.06\times10^{-5}$ \\
\hline
1.5 & 2.1 & 2.745 & 0.85 & $1\times10^{-4}$ \\
\hline\hline
1.5 & 1.9 & 0.78 & 0.95 & $7.37\times10^{-5}$ \\
\hline
1.5 & 1.7 & 2.745 & 0.65 & $3.74\times 10^{-5}$ \\
\hline
\end{tabular}
\end{center}
\caption{Simulation results of two Pareto tails.}
\label{table:pareto_sim}
\end{table}
\section{Results and Discussions}
In this work we stray away from an all-encompassing definition of an outlier, rather focus on a definition for the one dimensional case in terms of order statistics. In a sense our definition tries to approach the problem from the point of view of investigating data points "that arouse suspicions that it was generated by a different mechanism" \cite{Hawkins_1980}.
There are a few good reasons motivating our decision. Similar to the case in extreme value theory, it is difficult to order random vectors and the limit cases are not as intuitive \cite{DeHaan_2006}. Furthermore, any multivariate definition must take into consideration cases where two or more random variables are not independent. Since the requirement of independence is too stringent for applied studies as conditional outliers are all too common an occurrence in real life \cite{Chandola_2009}.
A particular strong point of our method is the ability to select cut-off points for discrete set of points without the need of a priori information. As a result, our definition is innately compatible with any definition which produces outlier scores. For the multivariate case, since much of the literature is applied, we recommend the reader to first use the method which suits them best. Our definition, together with the $\kappa$-threshold selection, can be used afterwards to select a natural cut-off point.
We calculate the case $S_{m-1}/T_{n-m}$ and approximate for the defined $R$-statistic $S_{m}/T_{n-m}$. From the approximation, we can see that concentration of $R$-statistic depends on the tail of the random variable. Our simulation studies in \ref{section:experiments} also confirmatory. In particular, in \ref{plot:distributions2} it is considerably more difficult to find a cut-off point for the half-normal and exponential distributions compared to the identified outliers model. Nevertheless, we see in section \ref{subsection:tails} that even in the cases where the tail index is not in L\'{e}vy stable region, our statistic still produces results with very small variance with varying degrees of success.
For future work, we may choose the possible cut-off point by using the $R$-statistic on moving blocks first, then selecting the candidate blocks based on results. Finally, we can find the threshold value by looking at the block with the most dramatic increase. Identify potential blocks for the cut-off point is also helpful for the cases when a practitioner wishes to pick the value the $\kappa$-threshold by hand.
\clearpage |
1801.02553 | \section{Introduction}
Millimeter Wave (mmWave) communications are expected to play a vital role in 5G mobile communications, expanding the available spectrum and enabling multi-gigabit services that range from ultra-high definition video, to outdoor mesh networks, to autonomous vehicle platoons and drone communication~\cite{alliance20155g}. Although several works examine channel modeling for mmWave networks~\cite{rappaport2015wideband}, the information theoretic capacity of mmWave relay networks is yet relatively unexplored.
In this paper, we present capacity results for a class of networks that we term 1-2-1 networks that offer a simple yet informative model for mmWave networks.
The inherent characteristic of mmWave communications that our model captures is directivity: mmWave requires beamforming with narrow beams to compensate for high path loss.
To establish a communication link, both the mmWave transmitter and receiver employ antenna arrays that they electronically steer to direct their beams towards each other - we term this a 1-2-1 link, as both nodes need to focus their beams to face each other for the link to be active.
Thus, in 1-2-1 networks, instead of broadcasting or interference, we have coordinated steering of transmit and receive beams to activate different links at each time.
An example of a diamond network with $N=4$ relays is shown in Fig.~\ref{fig:example_ntwk}, where two different states for the configuration of the transmit/receive beams are depicted and the resulting activated links are highlighted.
Our main results are as follows. We consider a source connected to a destination through an arbitrary topology of $N$ relay nodes, and derive a min-cut Linear Program (LP) that outer-bounds the capacity within a constant gap, which only depends on $N$; we then show that its dual is equivalent to a fractional path utilization LP. That is, we show that routing is a capacity achieving strategy (up to a constant gap). Moreover, out of an exponential number
of paths that potentially connect the source to the destination, we show we need to utilize at most $2N+2$ to approximately achieve the capacity. We also prove tighter results for classes of networks. For example,
for the special case of diamond (or one-layer) networks, where the source is connected to the destination through one layer of non-interfering relays as in Fig.~\ref{fig:example_ntwk}, we prove that we can approximately achieve the network capacity by routing information along {\em at most two paths}, independently of the total number $N$ of relays.
As a result, selecting to operate the best path always achieves half the diamond network capacity.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{example_ntwk.pdf}
\caption{1-2-1 network with $N = 4$ relays and two states.}
\vspace{-1em}
\label{fig:example_ntwk}
\end{figure}
Although we believe that 1-2-1 networks capture the essence of mmWave networks and enable to build useful insights on near-optimal information flow algorithms, we recognize that this model makes a number of simplifying assumptions that include: 1) we assume no interference among communication links (a reasonable assumption for relays spaced further apart than the beam width), and 2) we do not take into account the overhead of channel knowledge, and of beam-steering.
\smallskip
\noindent{\bf{Related Work.}} Several studies examine channel modeling for mmWave networks~\cite{rappaport2015wideband,akdeniz2014millimeter}. However, to the best of our knowledge, the information theoretic capacity under optimal scheduling has not been analyzed. Recent studies in networking design communication protocols for mmWave mesh networks~\cite{shokri2015millimeter,niu2015survey}. Closer to this work are perhaps works that examine directional networks in the Gupta\&Kumar framework~\cite{GuptaKumar2000}, however they only look at order arguments for multiple unicast sessions~\cite{yi2003capacity}, and do not consider schedules that arrange for both receiver and transmitter beams to align.
\smallskip
\noindent{\bf{Paper Organization.}}
Section~\ref{sec:model} describes the $N$-relay Gaussian 1-2-1 network and derives a constant gap approximation of its capacity;
Section~\ref{sec:FDHandshakeNet} presents our main results;
Section~\ref{sec:ProofMainTh} contains one of the main proofs of this work
\section{System Model and Capacity Formulation}
\label{sec:model}
With $[n_1:n_2]$ we denote the set of integers from $n_1$ to $n_2 \geq n_1$; $\text{Card}(S)$ is the cardinality of the set $S$; $\emptyset$ is the empty set; $\mathds{1}_{P}$ is the indicator function;
$0^N$ indicates the all-zero vector of length $N$.
We consider an $N$-relay Gaussian {\em{1-2-1}} network where $N$ relays assist the communication between a source node (node~$0$) and a destination node (node~$N+1$).
In particular, in this 1-2-1 network, at any particular time, a node in the network can only direct (beamform) its transmissions towards at most another node.
Similarly, a node can only receive transmissions from at most another node (to which its receiving beam points towards).
Thus, each node $i \in [0:N+1]$ in the network is characterized by two states, namely $S_{i,t}$ and $S_{i,r}$ that represent the node towards which node $i$ is beamforming its transmissions and the node towards which node $i$ is pointing its receiving beam, respectively.
In particular, $\forall i \in [0:N+1]$, we have that
\begin{subequations}
\label{eq:state}
\begin{align}
\begin{array}{ll}
S_{i,t} \subseteq [1:N+1] \backslash \{i \}, & \text{Card}(S_{i,t}) \leq 1,
\\ S_{i,r} \subseteq [0:N] \backslash \{i \}, & \text{Card}(S_{i,r}) \leq 1,
\end{array}
\end{align}
where $S_{0,r} = S_{N+1,t} = \emptyset$ since the source node always transmits and the destination node always receives.
We consider two modes of operation at the relays, namely {\em{Full-Duplex}} (FD) and {\em{Half-Duplex}} (HD).
In FD, relay $i \in [1:N]$ can be simultaneously receiving and transmitting, i.e., we can have both $S_{i,t} \neq \emptyset$ and $S_{i,r} \neq \emptyset$.
In HD, relay $i \in [1:N]$ can either receive or transmit, i.e., if $S_{i,t} \neq \emptyset$, then $S_{i,r} = \emptyset$ and vice versa.
In particular, $\forall i \in [1:N]$, we have that
\begin{align}
\text{Card}(S_{i,t}) \!+\! \text{Card}(S_{i,r}) \! \leq \!\!
\left \{
\begin{array}{ll}
\!\!\!2 & \!\text{if relays operate in FD}
\\
\!\!\!1 & \!\text{if relays operate in HD}
\end{array}\!\!.
\right.
\end{align}
\end{subequations}
We can now write the memoryless channel model for this Gaussian 1-2-1 network. We have that $ \forall j \in [1:N+1]$
\begin{align}
Y_j = Z_j + \sum_{i \in [0:N]\backslash\{j\}} h_{ji} \mathds{1}_{\{i \in S_{j,r}, \ j \in S_{i,t}\}} X_i,
\label{eq:model_1}
\end{align}
where: (i) $S_{i,t}$ and $S_{i,r}$ are defined in~\eqref{eq:state};
(ii) $X_i$ (respectively, $Y_i$) denotes the channel input (respectively, output) at node $i$;
(iii) $h_{ji} \in \mathbb{C}$ represents the complex channel coefficient from node $i$ to node $j$; the channel coefficients are assumed to be constant for the whole transmission duration and known by the network;
(iv) the channel inputs are subject to an individual power constraint, i.e., $\mathbb{E}[|X_k|^2] \leq P,\ k \in [0:N]$;
(v) $Z_j,\ j \in [1:N+1]$ indicates the additive white Gaussian noise at the $j$-th node; noises across the network are assumed to be independent and identically distributed as $\mathcal{CN}(0,1)$.
By using a similar approach as the one proposed in~\cite{KramerAllerton2004}, the
channel model in~\eqref{eq:model_1} can be modified to incorporate the state variables in the channel inputs.
In particular, let the vector $\widehat{X}_i = (S_i,\widebar{X}_i)$ be the input to the channel at node $i \in[0:N]$, where: (i) $S_i = (S_{i,t}, S_{i,r})$ with $S_{i,t}$ and $S_{i,r}$ being defined in~\eqref{eq:state} and (ii)
$\widebar{X}_i \in \mathbb{C}^{N+1}$, with elements $\widebar{X}_i(k)$ defined as
\begin{align}
\widebar{X}_i(k) = X_i \mathds{1}_{\{k \in S_{i,t}\}}.
\label{eq:model_2_inp}
\end{align}
In other words, $\widebar{X}_i$ as a vector is a function of $S_{i,t}$ and the input of the original channel $X_i$.
When node $i$ is not transmitting, i.e., $S_{i,t} = \emptyset$, then $\widebar{X}_i = 0^{N+1}$.
It is not hard to see that the power constraint on $X_i$ extends to $\widebar{X}_i$ since at most one single index appears in the vector (recall that $\text{Card}(S_{i,t}) \leq 1$).
Using this new channel input $\widehat{X}_i$, we can now equivalently rewrite the channel model in~\eqref{eq:model_1} as
\begin{align}
Y_j =
\begin{cases}
h_{j S_{j,r}} \widebar{X}_{S_{j,r}}(j) + Z_j & \text{if}\ \text{Card} (S_{j,r}) = 1\\
0 & \text{otherwise}
\end{cases}.
\label{eq:model_2}
\end{align}
The capacity\footnote{We use standard definitions for codes, achievable rates and capacity.} $\mathsf{C}$ of the network defined in~\eqref{eq:model_2_inp} and~\eqref{eq:model_2} is not known, but can be approximated to within a constant-gap as stated in the following theorem which is proved in Appendix~\ref{app:ConstGap}.
\begin{thm}
The capacity $\mathsf{C}$ of the network defined in~\eqref{eq:model_2_inp} and~\eqref{eq:model_2} can be lower and upper bounded as
\begin{subequations}
\label{eq:constGap}
\begin{align}
&\mathsf{C}_{\rm cs,iid} \leq \mathsf{C} \leq \mathsf{C}_{\rm cs,iid} + \mathsf{GAP},
\\
&\mathsf{C}_{\rm cs,iid} \!=\! \max_{\substack{\lambda_s: \lambda_s \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \! \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}, \label{eq:apprCap}
\\
& \ell_{j,i} = \log\left(1 +P\left| h_{ji}\right|^2\right),
\\
&
\mathsf{GAP} = \mathsf{G}_1 +\mathsf{G}_2 +\mathsf{G}_3 \nonumber
\\& \qquad = (N\!+\!1)\log e \!+\! 2\log(N{+}2) \!+\! N\log(\text{Card}(S_1)), \label{eq:GAP}
\end{align}
where:
(i) $\Omega^c = [0:N+1] \backslash \Omega$;
(ii) $\lambda_s = \mathbb{P}(S_{[0:N+1]} = s)$ is the joint distribution of the states, where $s$ enumerates the possible network states $S_{[0:N+1]}$;
(iii) $\text{Card}(S_1)$ is defined as
\begin{align}
\label{eq:RVStateRelays}
\text{Card}(S_1) = \left \{
\begin{array}{ll}
(N+1)^2 & \text{if relays operate in FD}\\
2N+1 & \text{if relays operate in HD}
\end{array}.
\right.
\end{align}
\end{subequations}
\end{thm}
The variable $\mathsf{GAP}$ in~\eqref{eq:GAP} only depends on the number of relays $N$ and represents the maximum loss incurred by using independent inputs and deterministic schedules at the nodes.
In particular, $\mathsf{G}_1$ represents the beamforming loss due to the use of independent inputs, while $\mathsf{G}_2$ (respectively, $\mathsf{G}_3$) accounts for the loss incurred by using a fixed schedule at the source and destination (respectively, at the relays), as we explain next.
Note that from~\eqref{eq:model_2_inp}, the input at the $i$-th node is also characterized by the random state variable $S_{i,t}$
(which indicates to which node -- if any -- node $i$ is transmitting).
Therefore, information can be conveyed from the source to the destination by randomly switching between
these states.
However, as first highlighted in~\cite{KramerAllerton2004} in the context of the HD relay channel, this random switch can only improve the capacity by a constant, whose maximum value equals the logarithm of the cardinality of the support of the state random variable.
It therefore follows that the capacity can be approximated to within this constant by using a fixed/deterministic schedule at the nodes.
In particular, for the source and destination the cardinality of the support of their state random variable equals $N+2$ (since the source and the destination can only be transmitting to and receiving from at most one node, respectively).
Differently, the cardinality of the support of the state variable at the relays depends on the mode of operation (either FD and HD) and is given by~\eqref{eq:RVStateRelays}.
In other words, $\mathsf{C}_{\rm cs,iid}$ in~\eqref{eq:constGap} -- which can be achieved using QMF as in~\cite{OzgurIT2013} or NNC as in~\cite{NNC} -- is a constant gap away from the capacity $\mathsf{C}$ of the network defined in~\eqref{eq:model_2_inp} and~\eqref{eq:model_2}.
Thus, in the rest of the paper we analyze $\mathsf{C}_{\rm cs,iid}$, which we refer to as the {\em approximate} capacity for the Gaussian 1-2-1 network.
\begin{rem}
\label{rem:RatLinkCap}
{\rm
In the rest of the paper, we assume that the point-to-point link capacities are rational numbers.
In fact, as we prove in what follows, when the capacities $\ell_{j,i} \in \mathbb{R}$, we can always further bound the approximate capacity $\mathsf{C}^{\mathbb{R}}_{\rm cs,iid} $ as
\begin{align}
\mathsf{C}^{\mathbb{Q}}_{\rm cs,iid} \leq \mathsf{C}^{\mathbb{R}}_{\rm cs,iid} \leq \mathsf{C}^{\mathbb{Q}}_{\rm cs,iid} + \epsilon,
\label{eq:RealRat}
\end{align}
where $\epsilon > 0$ and $\mathsf{C}^{\mathbb{Q}}_{\rm cs,iid}$ is the approximate capacity of a network with link capacities $ \hat{\ell}_{j,i}$ such that
\begin{align}
\label{eq:RealCap}
\hat{\ell}_{j,i} \leq \ell_{j,i} \leq \hat{\ell}_{j,i} + \frac{\epsilon}{(N+1)^2}, \ \hat{\ell}_{j,i} \in \mathbb{Q}.
\end{align}
Note that such an assignment always exists since the set of rationals $\mathbb{Q}$ is dense in $\mathbb{R}$.
With this, we have
\begin{align*}
\mathsf{C}^{\mathbb{R}}_{\rm cs,iid} &= \max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}\\
&\stackrel{\eqref{eq:RealCap}}{\leq} \max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \left(\hat{\ell}_{j,i} + \frac{\epsilon}{(N+1)^2} \right) \\
& = \max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}}
\left \{ \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \hat{\ell}_{j,i}
+
\sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right)\frac{\epsilon}{(N+1)^2}
\right \} \\
& \stackrel{\sum_{s} \lambda_s \leq 1}{\leq}
\max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}}
\left \{ \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \hat{\ell}_{j,i}
+
\sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\frac{\epsilon}{(N+1)^2}
\right \} \\
& \leq
\max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}}
\sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \hat{\ell}_{j,i} + \epsilon = \mathsf{C}^{\mathbb{Q}}_{\rm cs,iid} + \epsilon .
\end{align*}
Moreover, we have that
\begin{align*}
\mathsf{C}^{\mathbb{R}}_{\rm cs,iid} &= \max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}\\
&\stackrel{\eqref{eq:RealCap}}{\geq}\max_{\substack{\lambda: \lambda \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \hat{\ell}_{j,i} = \mathsf{C}^{\mathbb{Q}}_{\rm cs,iid}.
\end{align*}
This proves~\eqref{eq:RealRat} and hence, in the rest of the paper, we will assume that the point-to-point link capacities are rational numbers.
}
\end{rem}
\section{Main Results}
\label{sec:FDHandshakeNet}
We here present our main results on Gaussian 1-2-1 networks and discuss their implications.
Our first main result
is that for FD networks, $\mathsf{C}_{\rm cs,iid}$ in~\eqref{eq:apprCap} can be computed as the sum of fractions
of the FD capacity of the paths in the network.
\begin{thm}
\label{thm:MainTh}
For any $N$-relay Gaussian FD 1-2-1 network, we have that
\begin{align}
\label{eq:CapPaths}
\begin{array}{llll}
{\rm P1:} &\mathsf{C}_{\rm cs,iid} = {\rm max} \displaystyle\sum_{p \in \mathcal{P}} x_p \mathsf{C}_p & & \\
& ({\rm P1}a) \ x_p \geq 0 & \forall p \!\in\! \mathcal{P}, & \\
& ({\rm P1}b) \ \displaystyle\sum_{p \in \mathcal{P}_i} x_p f^p_{p.\text{nx}(i),i} \!\leq\! 1 & \forall i \! \in \! [0\!:\!N], & \\
& ({\rm P1}c) \ \displaystyle\sum_{p \in \mathcal{P}_i} x_p f^p_{i,p.\text{pr}(i)} \!\leq\! 1 & \forall i \! \in \! [1\!:\!N\!+\!1], &
\end{array}
\end{align}
where:
(i) $\mathcal{P}$ is the collection of all paths from the source to the destination;
(ii) $\mathcal{P}_i \subseteq \mathcal{P}$ is the collection of paths that pass through node $i \in [0:N+1]$ (clearly, $\mathcal{P}_0=\mathcal{P}_{N+1}=\mathcal{P}$ since all paths pass through the source and the destination);
(iii) $\mathsf{C}_p$ is the FD capacity of the path $p \in \mathcal{P}$, i.e., $\mathsf{C}_p = \min_{(i,j)\in p} \ell_{j,i}$;
(iv) $p.\text{nx}(i)$ (respectively, $p.\text{pr}(i)$) with $i \in [0:N+1]$ is the node following (respectively, preceding) node $i \in [0:N+1]$ in path $p \in \mathcal{P}$ (clearly, $p.\text{pr}(0) = p.\text{nx}(N+1) = \emptyset$);
(v) $f^p_{j,i}$ is the optimal activation time for the link of capacity $\ell_{j, i}$ when the path $p \in \mathcal{P}$, such that $(i,j)\in p$, is operated, i.e.,
\begin{align}
\label{eq:actTime}
f^p_{j,i} = \frac{\mathsf{C}_{p}}{\ell_{j, i}}.
\end{align}
\end{thm}
\begin{proof}
The proof of Theorem~\ref{thm:MainTh} is delegated to Section~\ref{sec:ProofMainTh}.
\end{proof}
In the LP in Theorem~\ref{thm:MainTh},
the variable $x_p$ represents the fraction\footnote{Note that $x_p$ in P1 implicitly satisfies that $x_p \leq 1,\ \forall p \in \mathcal{P}$. This is due to the fact that for any path $p \in \mathcal{P}$, the definition of $f_{j,i}^p$ in \eqref{eq:actTime} implies that at least one constraint in (P1$b$) and (P1$c$) has $f_{j,i}^p = 1$.}
of time the path {$p \in \mathcal{P}$} is utilized in the network.
Moreover, each
of the constraints in $({\rm P1}b)$ (respectively, $({\rm P1}c)$) ensures that a node $i \in [0:N+1]$ - even though it can appear in multiple paths in the network - does not transmit (respectively, receive) for more than 100\% of the time.
Lemma~\ref{lem:paths_and_caps} follows from P1 in Theorem~\ref{thm:MainTh}, and states that, although the number of paths $P$ in general is exponential in the number of relays $N$, we need to use at most a linear number of paths; this can also be translated to a guarantee on the rate that can be achieved when only the best path is operated.
\begin{lem}\label{lem:paths_and_caps}
For any $N$-relay Gaussian FD 1-2-1 relay network, we have the following guarantees:
\begin{enumerate}[(L1)]
\item For {a network with arbitrary topology,} the approximate capacity $\mathsf{C}_{\rm cs,iid}$ can {always} be achieved by activating at most $2N+2$ paths in the network.
\item For {a network with arbitrary topology,} the best path has an FD capacity $\mathsf{C}_1$ such that $\mathsf{C}_1 \geq \frac{1}{2N+2}\mathsf{C}_{\rm cs,iid}$.
\item For a 2-layer relay network with $M = N/2$ relays per layer, the approximate capacity $\mathsf{C}_{\rm cs,iid}$ can be achieved by activating at most $2M+1$ paths in the network.
\item For a 2-layer relay network with $M = N/2$ relays per layer, the best path has an FD capacity $\mathsf{C}_1$ such that $\mathsf{C}_1 \geq \frac{1}{2M+1}\mathsf{C}_{\rm cs,iid}$.
.
\end{enumerate}
\end{lem}
\begin{proof}
~
{\bf Proof of L1:} The LP P1 in~\eqref{eq:CapPaths} is bounded and hence there always exists an optimal corner point. In particular, at any corner point in P1, we have at least $P = \text{Card}(\mathcal{P})$ constraints satisfied with equality among $({\rm P1}a)$, $({\rm P1}b)$ and $({\rm P1}c)$. Therefore, we have at least $P-2N-2$ in $({\rm P1}a)$ satisfied with equality (since $({\rm P1}b)$ and $({\rm P1}c)$ combined represent $2N+2$ constraints). Thus, at least $P-2N-2$ paths are not operated, which proves the statement in L1.
{\bf Proof of L3:} Similar to the proof above for L1, a corner point in the LP P1 has at most $2N+2$ constraints among $({\rm P}1b)$ and $({\rm P}1c)$ satisfied with equality. To prove L3, we need to show that in the case of a 2-layered network and we have $2N+2$ equality satisfying constraints, then at least one of the equations is redundant.Note that for a 2-layer relay network, any path $p$ in the network has four nodes and is written as $0 - p(1) - p(2) - N+1$ where $p(1)$ and $p(2)$ represent the node in the path from layer 1 and layer 2, respectively. We assume that the relays in the first layer are indexed with $[1:N/2]$ and the second layer relays are indexed with $[N/2+:N]$. Thus, for any path, $p(1) \in [1:N/2]$ and $p(2) \in [N/2+1:N]$. Now assume that all constraints $({\rm P}1b)$ and $({\rm P}1c)$ satisfied with equality, then by adding all $({\rm P}1b)$ constraints for $i \in[1:N/2]$ and subtracting from all constraints from $({\rm P}1c)$ for $j \in [N/2+1:N]$, we get
\begin{align*}
{\rm LHS}: \sum_{i =1}^{N/2} ({\rm P}1b)_i - \sum_{j =N/2+1}^{N}({\rm P}1c)_j &=\sum_{i =1}^{N/2} \sum_{p \in \mathcal{P}_i} x_p f^p_{p.\text{nx}(i),i} - \sum_{j =N/2+1}^{N}\sum_{p \in \mathcal{P}_j} x_p f^p_{j,p.\text{pr}(j)}\\
&= \sum_{p \in \mathcal{P}} x_p f^p_{p(2),p(1)} - \sum_{p \in \mathcal{P}} x_p f^p_{p(2),p(1)} = 0\\
{\rm RHS}:\sum_{i =1}^{N/2} ({\rm P}1b)_i - \sum_{j =N/2+1}^{N}({\rm P}1c)_j &= \sum_{i =1}^{N/2} 1 - \sum_{j =N/2+1}^{N} 1 = 0, &
\end{align*}
Thus, when all constraints $({\rm P}1b)$ and $({\rm P}1c)$ are satisfied with equality, at least one of them redundant, which proves the statement L3.
{\bf Proof of L2 and L4:} The proof of L2 follows directly from L1 by considering only the $2N+2$ paths needed to achieve $\mathsf{C}_{\rm cs,iid}$ and picking the path that has the largest FD capacity among them.
In particular, the guarantee in L2 is true for the selected path due to the fact that for any feasible point in P1, $x_p \leq 1,\ \forall p \in \mathcal{P}$. The proof of L4 from L3 follows the same argument used to prove L2 from L1.
\end{proof}
The result L3 in Lemma~\ref{lem:paths_and_caps} suggests that for 1-2-1 networks with particular structures, we can further reduce the number of active paths needed to achieve the approximate capacity.
In particular, we explore this observation in the context of 1-2-1 Gaussian diamond networks operating in FD and HD, through the following two lemmas proved in Appendix~\ref{app:diamond_FD}, Appendix~\ref{app:diamond_HD} and Appendix~\ref{app:Lemma5L3}.
\begin{lem}\label{lem:diamond_1-2-1_ntwk}
\label{thm:MainThDiam}{}
For the $N$-relay Gaussian diamond 1-2-1 network, we can calculate the approximate capacity $\mathsf{C}_{\rm cs,iid}$ as
\begin{align}
\label{eq:CapPaths_diamond}
\begin{array}{llll}
{\rm P1^d:} &\mathsf{C}_{\rm cs,iid} = {\rm max} \sum_{p \in [1:N]} x_p \mathsf{C}_p & & \\
& ({\rm P1}a)^{\rm d} \ {0 \leq x_p \leq 1} & \forall p \in [1{:}N], & \\
& ({\rm P1}b)^{\rm d} \ \sum_{p \in [1:N]}x_p \frac{\mathsf{C}_p}{\ell_{p,0}} \!\leq\! 1, & & \\
& ({\rm P1}c)^{\rm d} \ \sum_{p \in [1:N]} x_p \frac{\mathsf{C}_p}{\ell_{N\!+\!1,p}} \!\leq\! 1, & &
\end{array}
\end{align}
where:
(i) $\mathcal{P}$ is the collection of all paths from the source to the destination;
(ii) $\mathsf{C}_p$ is the capacity of the path $0 \to p \to N+1$ and its value depends on whether the network is operating in FD or HD, namely
\begin{align*}
\mathsf{C}_p = \left \{
\begin{array}{ll}
\min\{ \ell_{p,0}, \ell_{N+1,p}\} & \text{if relays operate in FD}\\
\frac{\ell_{p,0}\ \ell_{N+1,p}}{\ell_{p,0} + \ell_{N+1,p}} & \text{if relays operate in HD}
\end{array}.
\right.
\end{align*}
\end{lem}
\begin{lem}\label{lem:simplification_diamond}
For an $N$-relay Gaussian FD diamond relay network, we have the following guarantees:
\begin{enumerate}[(L1)]
\item If the network is operating in FD, then the approximate capacity $\mathsf{C}_{\rm cs,iid}$ can always be achieved by activating at most $2$ relays in the network, independently of $N$.
\item If the network is operating in HD, then the approximate capacity $\mathsf{C}_{\rm cs,iid}$ can always be achieved by activating at most $3$ relays in the network, independently of $N$.
\item In both FD and HD networks, the best path has a capacity $\mathsf{C}_1$ such that $\mathsf{C}_1 \geq \frac{1}{2}\mathsf{C}_{\rm cs,iid}$; furthermore, this guarantee is tight for both the FD and HD cases, i.e.,
there exists a class of Gaussian diamond 1-2-1 networks such that $\mathsf{C}_1 \leq \frac{1}{2}\mathsf{C}_{\rm cs,iid}$ both for the FD and HD cases.
\end{enumerate}
\end{lem}
The results in L1 and L2 in Lemma~\ref{lem:simplification_diamond} are surprising as they state that, independently of the total number of relays in the network, there always exists a subnetwork of $2$ (in FD) and $3$ (in HD) relays that achieves the full network approximate capacity. Moreover, the guarantee provided by L3 is tight. To see this, consider
$N=2$ and $\ell_{1,0}=\ell_{3,2} = 1$ and $\ell_{3,1}=\ell_{2,0}=X \rightarrow \infty$.
For this network, we have that the approximate capacity is $\mathsf{C}_{\rm cs,iid}=\ell_{1,0}+\ell_{3,2} = 2$, while the capacity of each path (both in FD and HD) is $\mathsf{C}_{1}=\min \left \{ 1,X\right \}= 1$, hence $\mathsf{C}_{1}/\mathsf{C}_{\rm cs,iid}=1/2$.
\section{Proof of Theorem~\ref{thm:MainTh}}
\label{sec:ProofMainTh}
We here prove Theorem~\ref{thm:MainTh}.
We note that for a fixed $\lambda_s$, the inner minimization in~\eqref{eq:apprCap} is the standard min-cut problem over a graph with link capacities given by
\begin{align}
\label{eq:ellMaxFlow}
\ell_{j,i}^{(s)} = \left( \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}.
\end{align}
Since the min-cut problem is the dual for the standard max-flow problem, then we can replace the inner minimization in~\eqref{eq:apprCap} with the max-flow problem over the graph with link capacities defined in~\eqref{eq:ellMaxFlow} to give that
\begin{align}
\label{eq:approx_cap_flow_2}
\text{P2-flow}&{\rm\ :} \ \mathsf{C}_{\rm cs,iid} = \max_{\substack{\lambda_s: \lambda_s \geq 0 \\ \sum_s \lambda_s = 1}} \max \sum_{j =1}^{N+1} F_{j,0}& \nonumber\\
& 0 \leq F_{j,i} \leq \ell_{j,i}^{(s)}& (i,j) \in [0:N]\times[1:N{+}1], \\
& \sum_{j\in[1:N{+}1]\backslash\{i\}} F_{j,i} = \sum_{k\in[0:N]\backslash\{i\}} F_{i,k} & i \in [1:N],\nonumber
\end{align}
where $F_{j,i}$ represents the flow from node $i$ to node $j$.
The max-flow problem can be equivalently written as an LP with path flows instead of link flows, thus,~\eqref{eq:approx_cap_flow_2} can be written as P2 described next by using the path flows representation of the max-flow problem. A variable $F_p$ is used
for the flow through the path $p \in \mathcal{P}$.
\begin{align*}
\begin{array}{llll}
{\rm P2}&{\rm :} \ \mathsf{C}_{\rm cs,iid} = \max \displaystyle\sum_{p \in \mathcal{P}} F_p & & \\
& ({\rm P2}a) \ F_p \geq 0 & \forall p \in \mathcal{P}, & \\
& ({\rm P2}b) \! \! \! \! \displaystyle \sum_{\substack{p \in \mathcal{P},\\(i,j) \in p,\\ j = p.\text{nx}(i)}} \! \! \! \!\!\! F_p {\leq} \ell_{j,i}^{(s)} {=} {\rm eq.}\eqref{eq:ellMaxFlow} & \forall (j,i) {\in} [1{:}N{+}1] \! \times \![0{:}N], & \\
& ({\rm P2}c) \ \sum_{s} \lambda_s \leq 1, & & \\
& ({\rm P2}d) \ \lambda_s \geq 0 & \forall s, &
\end{array}
\end{align*}
The constraints $({\rm P2}b)$ ensure that, for any link from node $i$ to node $j$, the sum of the flows through the paths that use this link does not exceed the link modified capacity $\ell_{j,i}^{(s)}$ in~\eqref{eq:ellMaxFlow}. The constraint $({\rm P2}c)$ is the same constraint on $\lambda_s$ as in~\eqref{eq:apprCap}.
To prove Theorem~\ref{thm:MainTh}, we first show that P2 above is equivalent to the following LP P3, and then prove that P3 is equivalent to P1 in~\eqref{eq:CapPaths}, which completes the proof.
\begin{align*}
\begin{array}{llll}
& {\rm P3:} \ \mathsf{C}_{\rm cs,iid} = \max \sum_{p \in \mathcal{P}} F_p & & \\
& ({\rm P3}a) \ F_p \geq 0 & \forall p \in \mathcal{P}, & \\
& ({\rm P3}b) \! \! \! \! \displaystyle \sum_{\substack{p \in \mathcal{P},\\(i,j) \in p,\\ j = p.\text{nx}(i)}} \! \! \! \! F_p \leq \lambda_{\ell_{j,i}} \ell_{j,i} & \forall (i,j) {\in} [0\!:\!N] \!\times\! [1\!:\!N{+}1], & \\
& ({\rm P3}c) \! \! \! \! \displaystyle \sum_{\substack{j \in [1:N{+}1]\backslash\{i\}}} \! \! \! \! \lambda_{\ell_{j,i}} \leq 1 & \forall i \in [0:N], & \\
& ({\rm P3}d) \! \! \! \! \displaystyle \sum_{\substack{i \in [0:N]\backslash\{j\}}} \! \! \! \! \lambda_{\ell_{j,i}} \leq 1 & \forall j \in [1:N+1], & \\
& ({\rm P3}e) \ \lambda_{\ell_{j,i}} \geq 0 &\forall (i,j) {\in} [0\!:\!N] \!\times\! [1\!:\!N{+}1], &
\end{array}
\end{align*}
where $\lambda_{\ell_{j,i}},(i,j) \in [0:N] \times [1:N+1]$ represents the fraction of time the link of capacity $\ell_{j,i}$ is active.
The constraints $({\rm P3}b)$ are similar to $({\rm P2}b)$ except that the capacity of a link is now
modified
through a multiplication by $\lambda_{\ell_{j,i}}$.
The constraints $({\rm P3}c)$ ensure that, for any transmitting node $i$, the sum of the activation times of its outgoing links is less than 100\% of the time.
Similarly, $({\rm P3}d)$ ensure the same logic for the incoming edges to a receiving node.
We delegate the proof of the equivalence between P3 and P1 to Appendix~\ref{app:P3EqualP1}, and here prove the equivalence between P2 and P3, which is more involved. To do so, we first show that a feasible point in P2 gives a feasible point in P3 with the same objective value. We define the following transformation
\[
\lambda_{\ell_{j,i}} = \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s,
\]
while the variable $F_p$ in P2 is the same as the variable $F_p$ in P3.
By substituting these transformations in the constraints $({\rm P2}a)$-$({\rm P2}d)$, it is not difficult to see that the constructed point (through the transformation) is feasible in P3 and has the same objective value in P3 as the objective value in P2. Thus, the solution of P3 is at least as large as the one of P2.
We now prove the direction from P3 to P2, by showing that every corner point in P3 can be transformed into a feasible point in P2 with the same objective value. To do this, we introduce a visualization for P2 and P3 in terms of bipartite graphs. We divide each node $i \in [0:N+1]$ in the network into two nodes ($i_T$ and $i_R$) representing the transmitting and receiving functions of the node;
note that $0_R = (N+1)_T = \emptyset$ since the source (node $0$) is always transmitting and the destination (node $N+1$) is always receiving.
This gives us the bipartite graph $\mathcal{G}_B = (\mathcal{T},\mathcal{R},\mathcal{E})$, where the vertices $\mathcal{T}$ (respectively, $\mathcal{R}$) are the transmitting modules of our nodes (respectively, $\mathcal{R}$ collects our receiving modules), and we have an edge $(i_T,j_R) \in \mathcal{E}$ for each link in the network.
It is easy to see that a valid state in P2 represents a matching in the bipartite graph $\mathcal{G}_B$.
Furthermore, we can write the program P3 in terms of $\mathcal{G}_B$ by simply renaming all $\lambda_{\ell_{j,i}}$ with $\lambda_{j_R,i_T}$, i.e., the activation time of the edge $(i_T,j_R)$ in $\mathcal{G}_B$.
Starting with a corner point in P3, we can follow the procedure described below to construct a feasible point in P2; note that since all coefficients (i.e., $\ell_{j_R,i_T}$) in P3 are rational (see Remark~\ref{rem:RatLinkCap}), the corner points $\lambda^\star_{\ell_{j_R,i_T}} \in \mathbb{Q},\ \forall (i_T,j_R)$ are rational. The main intuition is to use the fractions $\lambda^\star_{\ell_{j_R,i_T}}$ and our bipartite graph $\mathcal{G}_B$ to construct a bipartite multigraph with edges of unit capacity such that for higher values of $\lambda^\star_{\ell_{j_R,i_T}}$, we will have more parallel edges from the $i_T$-th node to the $j_R$-th node.
In particular, our procedure consists of the three following main steps.\\
{\bf Step 1.}
We multiply all $\lambda^\star_{\ell_{j_R,i_T}}$ by the Least Common Multiple (LCM) $M$ of their denominators (for simplicity, if $\lambda^\star_{\ell_{j_R,i_T}}=0$, then the denominator is set to be one).
We then calculate $n_{j,i} = M \lambda^\star_{\ell_{j_R,i_T}},\ \forall i_T,j_R \in [0:N+1]$, and construct the bipartite multigraph $\mathcal{G}^\bullet_B$ from $\mathcal{G}_B$ that has $n_{j,i}$ parallel edges from node $i_T$ to node $j_R$.\\
{\bf Step 2.} We edge color the multigraph $\mathcal{G}^\bullet_B$.
If any of the parallel edges from node $i_T$ to node $j_R$ are colored with color $c_i$, we say that $c_i$ activates the link $(i_T,j_R)$ in $\mathcal{G}_B$ (or equivalently the link $(i,j)$ in the network).
Note that since $\mathcal{G}^\bullet_B$ is a bipartite graph, then there is an optimal coloring for the graph that uses $\Delta$ colors, where $\Delta$ is the maximum degree of the nodes.
Furthermore, since no two adjacent edges in $\mathcal{G}^\bullet_B$ can have the same color, every individual color represents a matching in the bipartite graph $\mathcal{G}^\bullet_B$ (and by extension the graph $\mathcal{G}_B$).
Each of these matchings represents a state in the network; since there are $\Delta$ colors in total, then we assign to state $s_i,\ i \in [1:\Delta]$ (represented by color $c_i$) an activation time of $w_i = 1/\Delta$. \\
{\bf Step 3.} From Step~2, we know that each color in the network represents a matching in $\mathcal{G}_B$, and hence a state. However, some colors can correspond to the same matching in $\mathcal{G}_B$, i.e., two or more colors activate exactly the same set of links. We combine these colors together into one state by multiplying $1/\Delta$ by the number of times this state is repeated. The remaining unique states represent the feasible states in P2 that were obtained from our optimal $\lambda^\star$ from P3.
We now need to prove that the states that we obtain from the previous steps, in addition to the flows through the paths that we have from P3, indeed give us a feasible point in P2.
Without loss of generality, we will prove that the states generated in Step 2 give a feasible point since combining similar states (Step~3) does not change the total sum of activation times and does not change the amount of time a link is active (which is what we look for in the constraint~$({\rm P2}b)$.
To show the feasibility of the constructed schedule, we first derive a consequence of the fact that $\lambda^\star_{\ell_{j_R,i_T}}$ is feasible in P3.
In particular, we can show the following inequality between the maximum degree $\Delta$ and the constant LCM $M$
\begin{align}\label{eq:Delta_2_M}
\Delta &=\!\max\left[\max_{i \in [0:N]}\left( \sum_{j \in [1:N+1]} \! n_{j,i}\! \right), \max_{j \in [1:N+1]}\left( \sum_{i \in [0:N]} \! n_{j,i} \right) \right] \nonumber\\
&= \!M \max\left[ \max_{i_T \in [0:N]}\left( \sum_{j_R \in [1:N+1]} \lambda^\star_{\ell_{j_R,i_T}} \right), \right. \nonumber
\\ & \left. \quad \qquad \max_{j_R \in [1:N+1] }\left( \sum_{i_T \in [0:N]} \lambda^\star_{\ell_{j_R,i_T}} \right) \right] \stackrel{({\rm P3}c,d)}\leq M,
\end{align}
where the inequality follows from the constraints $({\rm P3}c)$ and $({\rm P3}d)$ in P3.
Using~\eqref{eq:Delta_2_M}, we can now show that the constructed schedule (that uses coloring arguments) is feasible in P2 (note that since $F_p$ is unchanged then the constraint in~$({\rm P2}a)$ is already satisfied due to the constraint in~{$({\rm P3}a)$}). Moreover, we have
\begin{align*}
&({\rm P3}b): \ \forall (j,i), \ \sum_{\substack{p \in \mathcal{P},\ (i,j) \in p,\\ j = p..\text{nx}(i)}} F_p \leq \lambda_{\ell_{j,i}} \ell_{j,i}
= \frac{n_{j,i}}{M} \ell_{j,i}
\\& \hspace{0.3cm} = \left( \displaystyle\sum_{s: (i,j) \in s}\lambda_s\right)\frac{\Delta}{M}\ell_{j,i} \stackrel{\eqref{eq:Delta_2_M}}\leq \left( \displaystyle\sum_{s:(i,j) \in s}\lambda_s\right)\ell_{j,i} \implies ({\rm P2}b)\\
&\text{By construction:} \ \sum_s \lambda_s = \sum_{s} \frac{1}{\Delta} = \Delta \frac{1}{\Delta} = 1 \implies ({\rm P2}c)\\
&\text{By construction:} \ \lambda_s = \frac{1}{\Delta} \geq 0 \implies ({\rm P2}d).
\end{align*}
From the discussion above, it follows that we can map a rational point $\lambda^\star_{\ell_{j_R,i_T}}$ in P3 to a feasible point in P2 that has the same objective function value.
Note that the variables $F_p$ in P3 and P2 are unchanged and therefore, the objective values will remain the same.
Thus, the solution of P2 is at least as large as that of P3, concluding the proof that P2 and P3 are equivalent.
{\begin{rem}
\label{rem:Schedule}
{\rm
The procedure described earlier gives a non-polynomial approach to construct an optimal schedule for the approximate capacity of the Gaussian FD 1-2-1 network.
In Appendix~\ref{app:poly_algorithm}, we provide an algorithm that computes the optimal schedule as well as the approximate capacity in polynomial time (in the number of nodes).
}
\end{rem}
}
\appendices
\section{Constant Gap Capacity Approximation for the Gaussian 1-2-1 Network}
\label{app:ConstGap}
The memoryless model of the channel allows to upper bound the channel capacity $\mathsf{C}$ using the cut-set upper bound $\mathsf{C}_{\rm cs}$ as
\begin{align}
\mathsf{C}_{\rm cs} &= \max_{\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} I(\widehat{X}_\Omega; Y_{\Omega^c} | \widehat{X}_{\Omega^c})\nonumber \\
&= \max_{\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} I(S_\Omega, \widebar{X}_\Omega; Y_{\Omega^c} | S_{\Omega^c}, \widebar{X}_{\Omega^c})\nonumber \\
&= \max_{\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} I(\widebar{X}_\Omega; Y_{\Omega^c} | S_{\Omega}, S_{\Omega^c}, \widebar{X}_{\Omega^c}) + I(S_\Omega; Y_{\Omega^c} | S_{\Omega^c}, \widebar{X}_{\Omega^c})\nonumber \\
&\leq \max_{\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} I(\widebar{X}_\Omega; Y_{\Omega^c} | S_{[0:N+1]}, \widebar{X}_{\Omega^c}) + H(S_\Omega) \nonumber \\
&\stackrel{(a)}\leq \max_{\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} I(\widebar{X}_\Omega; Y_{\Omega^c} | S_{[0:N+1]}, \widebar{X}_{\Omega^c}) + 2\log(N+2) + N\log( \text{Card} (S_1) ) \nonumber\\
&{\stackrel{(b)}=} \max_{\mathbb{P}_{\{S_i\}}(\cdot)} \max_{\mathbb{P}_{\{\widebar{X}_i\}|\{S_i\}}(\cdot)}\ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_s \lambda_s\ I(\widebar{X}_\Omega; Y_{\Omega^c} | S_{[0:N+1]} {=} s, \widebar{X}_{\Omega^c}) {+} 2\log(N+2) {+} N\log( \text{Card} (S_1) ) \nonumber \\
&{\stackrel{(c)}\leq} \max_{\mathbb{P}_{\{S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \max_{\mathbb{P}_{\{\widebar{X}_i\}|\{S_i\}}(\cdot)}\sum_s \lambda_s\ I(\widebar{X}_\Omega; Y_{\Omega^c} | S_{[0:N+1]} {=} s, \widebar{X}_{\Omega^c}) {+} 2\log(N+2) {+} N\log( \text{Card} (S_1) ),
\label{eq:cutset_1}
\end{align}
where:
(i) $\Omega^c = [0:N+1] \backslash \Omega$;
(ii) $\mathbb{P}_{\{\widebar{X}_i,S_i\}}(\cdot)$ is the probability distribution of the channel input $\{(S_i,\widebar{X}_i)\}_{i=0}^{N+1}$;
(iii) $S_{\Omega} = \left \{ S_i | i \in \Omega \right \}$;
(iv) the inequality in $(a)$ is due to the fact that the state variable at the source and destination can take $N+2$ values (since the source can only be transmitting to at most one node and the destination can only be receiving from at most one node), while at each relay the state variable can take $\text{Card}(S_1)$ values, where $\text{Card}(S_1)$ depends on the mode of operation at the relays, namely
\begin{align*}
\text{Card}(S_1) = \left \{
\begin{array}{ll}
(N+1)^2 & \text{if relays operate in FD}\\
2N+1 & \text{if relays operate in HD}
\end{array};
\right.
\end{align*}
(v) in the equality in $(b)$ we use $s$ to enumerate the possible network states $S_{[0:N+1]}$ and we denote with $\lambda_s = \mathbb{P}(S_{[0:N+1]} = s)$ the joint distribution of the states; (vi) the inequality in $(c)$ follows from the max-min inequality.
For a network state $s$, we define the channel matrix $\widehat{H}_s$, where the element $[\widehat{H}_s]_{i,j}$ is defined as
\begin{align}
[\widehat{H}_s]_{i,j} =
\begin{cases}
h_{ij} & \text{if}\ i \in s_{j,t}\ \text{and}\ j \in s_{i,r} \\
0 & \text{otherwise},
\end{cases}
\label{eq:H_s}
\end{align}
where $h_{ij}$ is the channel coefficient of the link from node $j$ to node $i$.
It is not difficult to see that every row (and column) of $\widehat{H}_s$ has at most one non-zero element and thus there exists a permutation matrix $\Pi$ such that $\Pi\widehat{H}_s$ is a diagonal matrix.
Also, let $s^+ = \left \{ i | s_{i,t} \neq \emptyset, \forall i \in [0:N] \right \}$ and $s^- = \left \{ i | s_{i,r} \neq \emptyset, \forall i \in [1:N+1] \right \}$.
With this, we can further simplify the mutual information expression in~\eqref{eq:cutset_1} as follows
\begin{align}
& \max_{\mathbb{P}_{\{\widebar{X}_i\}|\{S_i\}}(\cdot)} \sum_s \lambda_s\ I (\widebar{X}_\Omega; Y_{\Omega^c} | S_{[0:N+1]} {=} s, \widebar{X}_{\Omega^c})\nonumber\\
&\stackrel{(a)}=\max_{\mathbb{P}_{\{\widebar{X}_i\}|\{S_i\}}(\cdot)} \sum_s \lambda_s\ I(\widebar{X}_{s^+, \Omega}; Y_{s^-,\Omega^c} | S_{[0:N+1]} {=} s, \widebar{X}_{\Omega^c}) \nonumber \\
&\stackrel{(b)}= \sum_s \lambda_s \log\det\left(I + \widehat{H}_{s,\Omega}\ K_{s,\Omega}\ \widehat{H}^H_{s,\Omega}\right)\nonumber\\
&=\sum_s \lambda_s \log\det\left(I + \widehat{H}^H_{s,\Omega}\widehat{H}_{s,\Omega}\ K_{s,\Omega}\ \right),
\label{eq:mutual_info_gaussian}
\end{align}
where: (i) we define $\widebar{X}_{s^+,\Omega}$ as $\widebar{X}_{s^+,\Omega} = \left\{\left.\widebar{X}_i(s_{i,t}) \right| i \in \Omega \cap s^+ \right\}$ and $\widebar{Y}_{s^-,\Omega^c}$ as $\widebar{Y}_{s^-,\Omega^c} = \left\{\left.\widebar{Y}_i \right| i \in \Omega^c \cap s^- \right\}$;
(ii) the equality in $(a)$ follows since, given the state $s$, all variables $\widebar{X}_i(j)$, with $j \neq s_{i,t}$, as well as all $Y_i$ with $s_{i,r} =\emptyset$ are deterministic;
(iii) the equality in $(b)$ follows due to the maximization of the mutual information by the Gaussian distribution;
(iv) $\widehat{H}_{s,\Omega}$ is a submatrix of $\widehat{H}_s$ (defined in~\eqref{eq:H_s}) and is defined as $\widehat{H}_{s,\Omega} = [\widehat{H}_s]_{\Omega^c,\Omega}$ and $K_{s,\Omega}$ is the submatrix of the covariance matrix of the random vector $\left[\bar{X}_0(s_{0,t})\ \bar{X}_1(s_{1,t})\dots \bar{X}_N(s_{N,t}) \right]^T$, where the rows and columns are indexed by $\Omega$.
We now further upper bound the Right-Hand Side (RHS) of~\eqref{eq:mutual_info_gaussian} using~\cite[Lemma 1]{NNC}, for any $\gamma \geq e-1$ as follows
\begin{align}
\log\det\left(I + \widehat{H}^H_{s,\Omega}\widehat{H}_{s,\Omega}\ K_{s,\Omega}\right) &\leq \log\det\left(I + \gamma^{-1}P \widehat{H}^H_{s,\Omega}\widehat{H}_{s,\Omega}\right) + |\Omega|\log\alpha(\Omega,s,\gamma) \nonumber \\
&\stackrel{(a)}\leq \log\det\left(I +P\widehat{H}_{s,\Omega}\widehat{H}^H_{s,\Omega}\right) + |\Omega|\log\alpha(\Omega,s,\gamma),
\label{eq:lemma_1}
\end{align}
where the inequality in $(a)$ follows since $\gamma > 1$ and by applying Sylvester's determinant identity and $\alpha(\Omega,s,\gamma)$ is defined based on~\cite[Lemma 1]{NNC} as
\begin{align}
\alpha(\Omega,s,\gamma)=
\begin{cases}
e^{\gamma/e} & \text{if}\ \gamma \leq e \frac{{\rm rank}(H_{s,\Omega})}{{\rm trace}(K_{s,\Omega}/P)} = e \frac{ {\rm rank}(H_{s,\Omega})}{|s^+ \cap \Omega|}\\
\left(\gamma \frac{|s^+ \cap\Omega|}{ {\rm rank}(H_{s,\Omega})}\right)^\frac{ {\rm rank}(H_{s,\Omega})}{|s^+ \cap\Omega|} & \text{otherwise}.
\end{cases}
\end{align}
If we select $\gamma = e$, then we have that
\begin{align}
\alpha(\Omega,s,e)= \left(e \frac{|s^+ \cap\Omega|}{ {\rm rank}(H_{s,\Omega})}\right)^\frac{ {\rm rank}(H_{s,\Omega})}{|s^+ \cap\Omega|} \leq \max_{x \geq 0}\ (e x)^\frac{1}{x} = e.
\label{eq:const_lemma1}
\end{align}
Now, if we substitute \eqref{eq:mutual_info_gaussian}, \eqref{eq:lemma_1} and \eqref{eq:const_lemma1} in \eqref{eq:cutset_1}, we get that
\begin{align}
\label{eq:cutset_final}
\mathsf{C}_{\rm cs} &\leq \max_{\mathbb{P}_{\{S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \left[\sum_{s} \lambda_s \log\det\left(I +P\widehat{H}_{s,\Omega}\widehat{H}^H_{s,\Omega}\right) + |\Omega|\log e\right]{+} 2\log(N{+}2) {+} N\log(\text{Card}(S_1))\nonumber \\
&\leq \underbrace{\max_{\mathbb{P}_{\{S_i\}}(\cdot)} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{s} \lambda_s \log\det\left(I +P\widehat{H}_{s,\Omega}\widehat{H}^H_{s,\Omega}\right)}_{\mathsf{C}_{\rm cs,iid}} + \underbrace{(N+1)\log e + 2\log(N{+}2) {+} N\log(\text{Card}(S_1))}_{\mathsf{GAP}}.
\end{align}
The main observation in~\eqref{eq:cutset_final} is that an i.i.d Gaussian distribution on the inputs and a fixed schedule are within a constant additive gap from the information-theoretic cut-set upper bound on the capacity of the 1-2-1 network. With this, we can argue that $\mathsf{C}_{\rm cs,iid}$ is within a constant gap of the capacity. This is due to the fact that $\mathsf{C}_{\rm cs,iid}$ can be achieved using QMF as in~\cite{OzgurIT2013} or Noisy Network Coding as in~\cite{CardoneIT2014}.
Due to the special structure of the Gaussian 1-2-1 network we can further simplify $\mathsf{C}_{\rm cs,iid}$ by making use of the structure of $\widehat{H}_{s,\Omega}$ in \eqref{eq:cutset_final}.
In particular, recall that, since every row (and column) in $\widehat{H}_{s,\Omega}$ has at most one non-zero element, then there exists a permutation matrix $\Pi_{s,\Omega}$ such that $\Pi_{s,\Omega}\widehat{H}_s$ is a diagonal matrix (not necessarily square). Thus we have
\begin{align}
\label{eq:permuted_channel}
\log\det\left(I +P\widehat{H}_{s,\Omega}\widehat{H}^H_{s,\Omega}\right)
&\stackrel{(a)}= \log\det\left(I +P \Pi_{s,\Omega}\widehat{H}_{s,\Omega}\widehat{H}^H_{s,\Omega}\Pi^T_{s,\Omega}\right)\nonumber\\
&\stackrel{(b)}= \sum_{i =1}^{\min\{|\Omega|,|\Omega^c|\}} \log\left(1 +P \left|[\Pi_{s,\Omega}\widehat{H}_{s,\Omega}]_{i,i}\right|^2\right),
\end{align}
where: (i) the equality in $(a)$ follows since permutation matrices are orthogonal matrices and thus multiplying by them only permutes the singular values of a matrix; (ii) the equality in $(b)$ follows since the permuted channel matrix $\Pi_{s,\Omega}\widehat{H}_{s,\Omega}$ can be represented as a parallel MIMO channel with $\min\{|\Omega|,|\Omega^c|\}$ active links.
We can rewrite the expression in~\eqref{eq:permuted_channel} as
\begin{align}
\log\det\left(I +P\widehat{H}_{s,\Omega}\widehat{H}^T_{s,\Omega}\right) &= \sum_{\substack{(i,j):\\ i \in s^+ \cap \Omega,\ j \in s^- \cap \Omega^c,\\ j \in s_{i,t},\ i \in s_{j,r}} } \log\left(1 +P\left| [\widehat{H}]_{j,i}\right|^2\right)\nonumber \\
&= \sum_{\substack{(i,j):\\ i \in s^+ \cap \Omega,\ j \in s^- \cap \Omega^c,\\ j \in s_{i,t},\ i \in s_{j,r}} } \log\left(1 +P\left| h_{ji}\right|^2\right).
\end{align}
Thus, by letting $\ell_{j,i} = \log\left(1 +P\left| h_{ji}\right|^2\right)$, we arrive at the following expression for $\mathsf{C}_{\rm cs,iid}$
\begin{align}
\label{eq:approx_to_flow}
\mathsf{C}_{\rm cs,iid} &= \max_{\substack{\lambda: \|\lambda\|_1 = 1 \\ \lambda \geq 0}} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{s} \lambda_s \sum_{\substack{(i,j):\\ i \in s^+ \cap \Omega,\ j \in s^- \cap \Omega^c,\\ j \in s_{i,t},\ i \in s_{j,r}} } \ell_{j,i} \nonumber \\
&= \max_{\substack{\lambda: \|\lambda\|_1 = 1 \\ \lambda \geq 0}} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{s} \lambda_s \sum_{(i,j) \in [0:N+1]^2} \mathds{1}_{\{j \in s_{i,t},\ i \in s_{j,r} \}}\mathds{1}_{\{i \in \Omega,\ j \in \Omega^c\}} \ell_{j,i}\nonumber\\
&= \max_{\substack{\lambda: \|\lambda\|_1 = 1 \\ \lambda \geq 0}} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{(i,j) \in [0:N+1]^2}\mathds{1}_{\{i \in \Omega,\ j \in \Omega^c\}} \sum_{s} \lambda_s \mathds{1}_{\{j \in s_{i,t},\ i \in s_{j,r} \}} \ell_{j,i}\nonumber\\
&= \max_{\substack{\lambda: \|\lambda\|_1 = 1 \\ \lambda \geq 0}} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}} \left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}\nonumber \\
&= \max_{\substack{\lambda: \|\lambda\|_1 = 1 \\ \lambda \geq 0}} \ \min_{\Omega \subseteq [1:N] \cup \{0\}} \sum_{\substack{(i,j): i \in \Omega,\\ j \in \Omega^c}}\ell^{(s)}_{j,i},
\end{align}
where $\ell_{j,i}^{(s)}$ is defined as
\[
\ell_{j,i}^{(s)} = \left( \sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}.
\]
This concludes the proof that the capacity $\mathsf{C}$ of the Gaussian 1-2-1 network described in~\eqref{eq:model_2} can be characterized to within a constant gap as expressed in~\eqref{eq:constGap}.
\section{Gaussian FD Diamond 1-2-1 Network: Proof of Lemma~\ref{lem:diamond_1-2-1_ntwk} and Lemma~\ref{lem:simplification_diamond}(L1)}
\label{app:diamond_FD}
In this section, {we prove Lemma~\ref{lem:diamond_1-2-1_ntwk} for FD and Lemma~\ref{lem:simplification_diamond}(L1) by analyzing} the Gaussian FD 1-2-1 {FD} network with a diamond topology. In this network the source communicates with the destination by hopping through one layer of $N$ non-interfering relays.
For this network the LP P1 in~\eqref{eq:CapPaths} can be further simplified by leveraging the two following implications of the sparse diamond topology:
\begin{enumerate}
\item In a Gaussian 1-2-1 diamond network, we have $N$ disjoint paths from the source to the destination, each passing through a different relay. We enumerate these paths with the index $i \in [1:N]$ depending on which relay is in the path. Moreover, each path $i \in [1:N]$ has a FD capacity equal to $\mathsf{C}_i = \min \left \{ \ell_{i,0}, \ell_{N+1,i} \right \}$;
\item {In the Gaussian FD 1-2-1 diamond network, each relay $i \in [1:N]$ appears in only one path from the source to the destination. Thus, when considering constraints $({\rm P1}b)$ and $({\rm P1}c)$ in~\eqref{eq:CapPaths} for $i \in [1:N]$ gives us that
\begin{align}
\label{eq:obs_2_FD_diamond}
x_i \frac{\mathsf{C}_i}{\ell_{i,0}} \leq 1\quad \& \quad x_i \frac{\mathsf{C}_i}{\ell_{N+1,i}} \leq 1.
\end{align}
\item Note that $\mathsf{C}_i = \min \{ \ell_{i,0},\ell_{N+1,i} \}$.
Therefore, one of the coefficients $\mathsf{C}_{i}/{\ell_{i, 0}}$ or $\mathsf{C}_{i}/{\ell_{N+1,i}}$ in \eqref{eq:obs_2_FD_diamond} is equal to $1$. This implies that a feasible solution of ${\rm P1}^d$, has $x_1 \leq 1$ and $x_2 \leq 1$. Therefore, the constraints $x_i \leq 1,\ \forall i \in[1:N]$, albeit redundant, can be added to the LP without reducing the feasibility region.
\item In the Gaussian FD 1-2-1 diamond network, the constraints due to the source and destination nodes, namely $({\rm P1}b)$ for $i=0$ and $({\rm P1}c)$ for $i=N+1$ in~\eqref{eq:CapPaths} gives us that
\begin{align}\label{eq:obs_3_FD_diamond}
\sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{i, 0}} \leq 1,\qquad
\sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{N+1, i}} \leq 1.
\end{align}
Note that the constraints in~\eqref{eq:obs_3_FD_diamond} make the constraints \eqref{eq:obs_2_FD_diamond} redundant.
}
\end{enumerate}
By considering the two implications above, we can readily simplify P1 in~\eqref{eq:CapPaths} for Gaussian FD 1-2-1 networks with a diamond topology as follows
{
\begin{align}
\label{eq:CapPathsDiam}
\begin{array}{llll}
{\rm P1}^d: &\mathsf{C}_{\rm cs,iid} = {\rm max} \displaystyle\sum_{i \in [1:N]} x_i \mathsf{C}_i & & \\
& ({\rm P}1a)^d \ 0 \leq x_i \leq 1 & \forall i \in [1:N], & \\
& ({\rm P}1b)^d \ \displaystyle\sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{i, 0}} \leq 1, & & \\
& ({\rm P}1c)^d \ \displaystyle\sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{N+1, i}} \leq 1, & &
\end{array}
\end{align}
which is the LP we have in Lemma~\ref{lem:diamond_1-2-1_ntwk}.}
{To prove Lemma~\ref{lem:simplification_diamond}(L1), we} observe that for a bounded LP, there always exists an optimal corner point. Furthermore, at any corner point in the LP ${\rm P1}^d$, we have at least $N$ constraints satisfied with equality among $(1a)^d$, $(1b)^d$ and $(1c)^d$. Therefore, we have at least $N-2$ constraints in $(1a)^d$ satisfied with equality that make {linearly independent equations} (since $(1b)^d$ and $(1c)^d$ combined represent only two constraints). {Furthermore, recall that as mentioned earlier all constraints $x_i \leq 1$ are redundant.} Thus, at least $N-2$ relays are turned off (i.e., $x_i = 0$), i.e., at most two relays are sufficient to characterize the approximate capacity of any $N$-relay Gaussian FD 1-2-1 network with a diamond topology.
{
\section{Gaussian HD Diamond 1-2-1 Network: Proof of Lemma~\ref{lem:diamond_1-2-1_ntwk} and Lemma~\ref{lem:simplification_diamond}(L2)}
\label{app:diamond_HD}
We prove Lemma~\ref{lem:diamond_1-2-1_ntwk} for HD diamond networks in the first subsection and later prove Lemma~\ref{lem:simplification_diamond}(L2) in the following subsection.
\subsection{Proof of Lemma~\ref{lem:diamond_1-2-1_ntwk} for an HD diamond network}
Throughout this section, we slightly abuse notation by defining $\ell_{i} = \ell_{i,0}$ and $r_i = \ell_{N+1,i}$. Based on this definition, we can write the approximate capacity expression \eqref{eq:apprCap} as
\begin{align}
\mathsf{C}_{\rm cs,iid} &= \max_{\substack{\lambda_s: \lambda_s \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \! \sum_{i \in \Omega^c}\left( \sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s \right) \ell_{i} + \sum_{i \in \Omega}\left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s \right) r_{i} \nonumber \\
&= \max_{\substack{\lambda_s: \lambda_s \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \! \sum_{i=1}^N \left[\mathds{1}_{\{i \in \Omega^c\}} \left( \sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s \right) \ell_{i} + \mathds{1}_{\{i \in \Omega\}} \left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s \right) r_{i}\right]\nonumber \\
&= \max_{\substack{\lambda_s: \lambda_s \geq 0 \\ \sum_s \lambda_s = 1}} \min_{\Omega \subseteq [1:N] \cup \{0\}} \! \sum_{i=1}^N \min \left\{\left( \sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s \right) \ell_{i},\ \left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s \right) r_{i}\right\}. \label{eq:approx_cap_diamond_rewritten}
\end{align}
Our first directive is to show that the approximate capacity $\mathsf{C}_{\rm cs,iid}$ in \eqref{eq:approx_cap_diamond_rewritten} is equivalent to solving the LP P4
\begin{align}
\label{eq:P4}
{\rm P4 :}\quad {\rm maximize}\quad& \sum_{i = 1}^N \lambda_{\ell_i} \ell_i \nonumber \\
{\rm subject\ to}\quad& ({\rm P4}a)\ \lambda_{\ell_i} \ell_i = \lambda_{r_i} r_i \qquad& \forall i \in [1:N], \nonumber \\
& ({\rm P4}b)\ \sum_{i=1}^N \lambda_{\ell_i} \leq 1,\quad \sum_{i=1}^N \lambda_{r_i} \leq 1, \\
& ({\rm P4}c)\ \lambda_{\ell_i} + \lambda_{r_i} \leq 1\qquad &\forall i \in [1:N], \nonumber
\end{align}
where: (i) $f_i = \lambda_{\ell_i} \ell_i = \lambda_{r_i} r_i$ represents the data flow through the $i$-th relay; (ii) $\lambda_{\ell_i}$ (respectively, $\lambda_{r_i}$) represents the fraction of time in which the link from the source to relay $i$ (respectively, from relay $i$ to the destination) is active.
Note that since the network is operating in HD, then in \eqref{eq:approx_cap_diamond_rewritten}, we have that $|s_{i,t}| + |s_{i,r}| \leq 1,\ \forall i \in [1:N]$ which is captured by the constraint $({\rm P}4c)$ above.
To show the first direction (i.e., a feasible schedule in \eqref{eq:approx_cap_diamond_rewritten} gives a feasible point in the LP P1), we define the following transformation
\begin{align}
\forall i \in [1:N] \ \ : \quad &f_i = \min\left\{\left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right)\ell_i,\ \left(\sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s\right)r_i\right\}\nonumber\\
&\qquad\qquad \lambda_{\ell_i} = \frac{f_i}{\ell_i},\quad \lambda_{r_i} = \frac{f_i}{r_i}.
\label{eq:cs_2_P4}
\end{align}
Using this transformation, we have that
\begin{align*}
\lambda_{\ell_i} \ell_i &= f_i = \lambda_{r_i} r_i,\quad \forall i \in [1:N] &\implies ({\rm P}4a) \\
\sum_{i = 1}^N \lambda_{\ell_i} &= \sum_{i = 1}^N \frac{f_i}{\ell_i} = \sum_{i = 1}^N \frac{1}{\ell_i} \min\left\{\left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right)\ell_i,\ \left(\sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s\right)r_i\right\} \\
&\qquad \qquad \leq \sum_{i=1}^N \frac{1}{\ell_i}\left(\sum_{s} \lambda_s\right) \min\{\ell_i,r_i\}\leq \frac{\min\{\ell_i,r_i\}}{\ell_i} \leq 1&\implies ({\rm P}4b)\\
\sum_{i = 1}^N \lambda_{r_i} &= \sum_{i = 1}^N \frac{f_i}{r_i} = \sum_{i = 1}^N \frac{1}{r_i} \min\left\{\left(\sum_{\substack{ s:\\ j \in s_{i,t},\\ i \in s_{j,r} }} \lambda_s\right)\ell_i,\ \left(\sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s\right)r_i\right\} \\
&\qquad \qquad \leq \sum_{i=1}^N \frac{1}{r_i}\left(\sum_{s} \lambda_s\right) \min\{\ell_i,r_i\}\leq \frac{\min\{\ell_i,r_i\}}{r_i} \leq 1&\implies ({\rm P}4b)\\
\lambda_{\ell_i} {+} \lambda_{r_i} &= \left[\frac{1}{\ell_i} + \frac{1}{r_i}\right] \min\left\{\left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right)\ell_i,\ \left(\sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s\right)r_i\right\}\\
&{=} {\min}\left\{\left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right), \left(\sum_{\substack{ s:\\ (N{+}1) {\in} s_{i,t},\\ i \in s_{N{+}1,r} }} \lambda_s\right)\frac{r_i}{\ell_i}\right\} {+} {\min}\left\{\left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right)\frac{\ell_i}{r_i}, \left(\sum_{\substack{ s:\\ (N{+}1) \in s_{i,t},\\ i \in s_{N{+}1,r} }} \lambda_s\right)\right\}\\
&\leq \left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right) + \left(\sum_{\substack{ s:\\ (N{+}1) \in s_{i,t},\\ i \in s_{N{+}1,r} }} \lambda_s\right) \leq \sum_s \lambda_s = 1&\implies ({\rm P}4c).
\end{align*}
Thus, a feasible schedule in~\eqref{eq:approx_cap_diamond_rewritten} gives a feasible point in the LP P4 in~\eqref{eq:P4}.
Furthermore, by substituting~\eqref{eq:cs_2_P4} in~\eqref{eq:approx_cap_diamond_rewritten}, we get that the rate achieved by the schedule is
\[
\sum_{i=1}^N \min\left\{ \left(\sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s\right) \ell_i \ ,\ \left(\sum_{\substack{ s:\\ (N{+}1) \in s_{i,t},\\ i \in s_{N{+}1,r} }} \lambda_s\right) r_i \right\} = \sum_{i=1}^N f_i = \sum_{i=1}^N \lambda_{\ell_i}\ell_i
\]
which is equal to the objective function value of P4 in~\eqref{eq:P4}.
To prove the opposite direction (i.e., P4 $\to$ \eqref{eq:approx_cap_diamond_rewritten}), we show that we can map an optimal solution in P4 to a feasible point (schedule) in \eqref{eq:approx_cap_diamond_rewritten} with a rate equal to the optimal value of P4.
First, note that the LP P4 has $2N$ variables.
As a result, a corner point in P4, should have at least $N$ constraints from $({\rm P}4b)$, $({\rm P}4c)$ and $({\rm P}4d)$ satisfied with equality (we already have $N$ other equality constraints due to $({\rm P}4a)$.
We now prove an interesting property about optimal corner points in P4 which facilitates our proof.
\begin{prope}
\label{prop:property_optimal_1}
For any optimal corner point $\{\lambda_{\ell_i}^\star,\lambda_{r_i}^\star\}$ in P4, there exists an $i' \in [1:N]$ such that $\lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} =1$.
\end{prope}
\begin{proof}
To prove this property, we are going to consider three cases depending on which constraints are satisfied with equality at an optimal corner point.
\noindent {\bf 1) Both conditions in $({\rm P}4b)$ are not satisfied with equality:} In this case, a corner point has at least $N$ constraints among $({\rm P}4c)$ and $({\rm P}4d)$ satisfied with equality.
It is not difficult to see that, in order for the corner point to be optimal, at least for one $i'$ we have $\lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} =1$, otherwise we have a non-optimal zero value for the objective function.
\smallskip
\noindent{\bf 2) Only one condition in $({\rm P4}b)$ is not satisfied with equality:} In this case, a corner point has at least $N-1$ constraints among $({\rm P4}c)$ and $({\rm P4}d)$ satisfied with equality.
Thus, there exists at most one $i$ such that $0 < \lambda_{\ell_i} + \lambda_{r_i} < 1$.
Additionally, by adding the conditions in $({\rm P4}b)$, we get that
\begin{align}
1 < \sum_{i = 1}^N \left( \lambda_{\ell_i} + \lambda_{r_i} \right) < 2.
\label{eq:prope_1_2}
\end{align}
Thus, there exists one $i'$ such that $\lambda_{\ell_{i'}} + \lambda_{r_{i'}} = 1$, otherwise, we cannot satisfy the lower bound in \eqref{eq:prope_1_2}.
\smallskip
\noindent {\bf 3) Both conditions in $({\rm P4}b)$ are satisfied with equality:} In this case, a corner point has at least $N-2$ constraints among $({\rm P}4c)$ and $({\rm P}4d)$ satisfied with equality.
Thus, there exist at most two $i$ such that $0 < \lambda_{\ell_i} + \lambda_{r_i} < 1$.
Furthermore, adding the constraints in $({\rm P4}b)$ implies that
\begin{align}
\sum_{i = 1}^N \left( \lambda_{\ell_i} + \lambda_{r_i} \right) = 2.
\label{eq:prope_1_1}
\end{align}
The two aforementioned observations imply that all $N-2$ equalities cannot be from $({\rm P}4d)$, otherwise we have that $\sum_{i = 1}^N \left( \lambda_{\ell_i} + \lambda_{r_i} \right) < 2$, which contradicts \eqref{eq:prope_1_1}. Thus, there exists a constraint in $({\rm P}4c)$ that is satisfied with equality, i.e., $\lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} =1$ for some $i' \in [1:N]$.
\end{proof}
We now use Property~\ref{prop:property_optimal_1} to show that, for any optimal point in P4, we can find a feasible schedule in \eqref{eq:approx_cap_diamond_rewritten} that gives a rate equal to the objective function in P4. For an optimal point $(\lambda_{\ell_1}^\star,\lambda_{r_1}^\star,\dots,\lambda_{\ell_N}^\star,\lambda_{r_N}^\star)$, let $i'$ be the index such that $\lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} = 1$ (such an index exists thanks to Property~\ref{prop:property_optimal_1}). Thus, we have the following condition for our optimal point
\begin{align}
\label{eq:properties_optimal_P2}
\lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} &= 1,\nonumber \\
\sum_{i \in [1:N]\backslash \{i'\}} \lambda^\star_{\ell_i} &\leq 1 - \lambda^\star_{\ell_{i'}} =\lambda^\star_{r_{i'}} ,\\
\sum_{i\in [1:N]\backslash \{i'\}} \lambda^\star_{r_i} &\leq 1 - \lambda^\star_{r_{i'}} = \lambda^\star_{\ell_{i'}}.\nonumber
\end{align}
Note that, any state $s$ in the 1-2-1 Gaussian HD diamond network activates at most two links in the network: a link between the source and the $m$-th relay and/or well as the link between the $n$-th relay and the destination. For brevity, in our construction we will denote with $s_{m,n}$ the state that activates the link from the source to the $i$-th relay in the diamond network as well as the links from the $n$-th relay to the destination (If either link is not activated, the corresponding index is $\emptyset$). We also use $\lambda_{s_{m,n}}$ to denote the fraction of time during which this network state is active. Using this notation, we can construct the following schedule from the given optimal point in P4
\begin{align}
\label{eq:definitions_P2_to_approx_cap}
\lambda_{s_{i,i'}} &= \lambda^\star_{\ell_i},& \forall i \in [1:N]\backslash\{i'\},\nonumber \\
\lambda_{s_{\emptyset,i'}} &= \lambda^\star_{r_{i'}} - \sum_{i \in [1:N]\backslash\{i'\}}\lambda^\star_{\ell_i},&\nonumber\\
\lambda_{s_{i',i}} &= \lambda^\star_{r_i},&\forall i \in [1:N]\backslash\{i'\},\\
\lambda_{s_{i',\emptyset}} &= \lambda^\star_{\ell_{i'}} - \sum_{i \in [1:N]\backslash\{i'\}}\lambda^\star_{r_i}.&\nonumber
\end{align}
The activation time of all other states, except those described above, is set to zero.
From~\eqref{eq:properties_optimal_P2}, we know that all values defined in~\eqref{eq:definitions_P2_to_approx_cap} are positive.
We can verify that the generated schedule is feasible, i.e., the sum of all $\lambda$ has to add up to one as follows
\begin{align*}
\sum_s \lambda_s &= \lambda_{s_{0,i'}} + \sum_{i \in [1:N]\backslash\{i'\}} \lambda_{s_{i,i'}} +\sum_{i \in [1:N]\backslash\{i'\}} \lambda_{s_{i',i}} + \lambda_{s_{i',0}} \stackrel{(a)}= \lambda^\star_{\ell_{i'}} + \lambda^\star_{r_{i'}} \stackrel{(b)}= 1,
\end{align*}
where: (i) the equality in $(a)$ follows from the definitions in~\eqref{eq:definitions_P2_to_approx_cap} and (ii) the equality in $(b)$ follows from Property~\ref{prop:property_optimal_1}.
In order to conclude the mapping from P4 to \eqref{eq:approx_cap_diamond_rewritten}, we need to verify that the rate achieved with the constructed schedule in~\eqref{eq:definitions_P2_to_approx_cap} is equal to the optimal value of the LP P4.
From~\eqref{eq:approx_cap_diamond_rewritten}, we get that
\begin{align}
\label{eq:P2_in_approx_cap}
&\sum_{i=1}^N\min \left\{\left( \sum_{\substack{ s:\\ i \in s_{0,t}, 0 \in s_{i,r} }} \lambda_s \right) \ell_{i},\ \left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t}, i \in s_{N+1,r} }} \lambda_s \right) r_{i}\right\}\nonumber \\
&=\sum_{i\in [1:N]\backslash\{i'\}} \min \left\{\left( \sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s \right) \ell_{i},\ \left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s \right) r_{i}\right\}\nonumber + \min \left\{\left( \sum_{\substack{ s:\\ i \in s_{0,t},\\ 0 \in s_{i,r} }} \lambda_s \right) \ell_{i},\ \left( \sum_{\substack{ s:\\ (N+1) \in s_{i,t},\\ i \in s_{N+1,r} }} \lambda_s \right) r_{i}\right\}\nonumber \\
&=\left(\sum_{i\in [1:N]\backslash\{i'\}} \min\left\{ \lambda_{s_{i,i'}} \ell_i \ ,\ \lambda_{s_{i',i}} r_i \right\}\right) + \min\left\{ \left(\sum_{j \in [0:N]\backslash\{i'\}} \lambda_{s_{i',j}}\right) \ell_{i'} \ ,\ \left(\sum_{k \in [0:N]\backslash\{i'\}} \lambda_{s_{k,i'}}\right) r_{i'} \right\}\nonumber \\
&=\left(\sum_{i\in [1:N]\backslash\{i'\}} \min\left\{ \lambda^\star_{\ell_{i}} \ell_i \ ,\ \lambda^\star_{r_{i}} r_i \right\}\right) + \min\left\{\lambda_{\ell_{i'}}\ell_{i'} \ ,\ \lambda_{r_{i'}} r_{i'} \right\} =\sum_{i\in [1:N]} \min\left\{ \lambda^\star_{\ell_{i}} \ell_i \ ,\ \lambda^\star_{r_{i}} r_i \right\}.
\end{align}
Now note that, since the optimal corner point in P4 is feasible in P4, then $\lambda^\star_{\ell_{i}} \ell_i = \lambda^\star_{r_{i}} r_i$, $\forall i \in [1:N]$.
Thus the expression in~\eqref{eq:P2_in_approx_cap} can be rewritten as $\sum_{i=1}^N \lambda^\star_{\ell_i} \ell_i$, which is the optimal objective function value in P4.
Thus, we can now conclude that \eqref{eq:approx_cap_diamond_rewritten} is equivalent to P4.
We are now going to relate the LP P4 discussed above to the LP in Lemma~\ref{lem:diamond_1-2-1_ntwk}.
Recall that for a two hop Half-Duplex path with link capacities $\ell_i$ and $r_i$, the capacity is given by
\[
\mathsf{C}_i = \frac{\ell_i r_i}{\ell_i + r_i}.
\]
Thus, we ca write the LP ${\rm P1^d}$ as
\begin{align}
\label{eq:P3}
{\rm P1^d :}\quad {\rm maximize}\quad& \sum_{i = 1}^N x_i \frac{\ell_i r_i}{\ell_i + r_i}\nonumber \\
{\rm subject\ to}\quad& ({\rm P1}a)^{\rm d}\ 0 \leq x_i \leq 1 \qquad\qquad \forall i \in [1:N], \nonumber \\
& ({\rm P1}b)^{\rm d}\ \sum_{i=1}^N x_i \frac{\ell_i}{\ell_i + r_i} \leq 1,\\
& ({\rm P1}c)^{\rm d}\ \sum_{i=1}^N x_i \frac{r_i}{\ell_i + r_i} \leq 1. \nonumber
\end{align}
We are now going to show that P1 is equivalent to the LP P4 and, as a consequence, it is to the formulation of $\mathsf{C}_{\rm cs,iid}$ in~\eqref{eq:approx_cap_diamond_rewritten}.
To do this, we are going to show how a feasible point in P4 can be transformed into a feasible point in ${\rm P1^d}$ and vice versa.
\begin{enumerate}
\item \underline{P4 $\to$ ${\rm P1^d}$.} Define $x_i$ to be
\begin{align}
x_i = \lambda_{\ell_i} \frac{\ell_i + r_i}{r_i},\quad \forall i \in [1:N].
\label{eq:P2_2_P3}
\end{align}
Using this transformation, we get that the constraints in P4 imply the following
\begin{align*}
({\rm P}4b)\ &: \quad 1 \geq \sum_{i=1}^N\lambda_{\ell_i} = \sum_{i=1}^N x_i \frac{r_i}{\ell_i + r_i}& \implies ({\rm P1}c)^{\rm d}\\
({\rm P}4b)\ &: \quad 1 \geq \sum_{i=1}^N\lambda_{r_i} \stackrel{({\rm P}4a)}= \sum_{i=1}^N\lambda_{\ell_i}\frac{\ell_i}{r_i} = \sum_{i=1}^N x_i \frac{\ell_i}{\ell_i + r_i} &\implies ({\rm P1}b)^{\rm d}\\
({\rm P}4c)\ &: \quad \forall i \in [1:N], \quad 1 \geq \lambda_{\ell_i} + \lambda_{r_i} \stackrel{({\rm P}4a)}= \lambda_{\ell_i} \left(1 + \frac{\ell_i}{r_i} \right) = x_i \frac{r_i}{\ell_i + r_i}\left(1+\frac{\ell_i}{r_i}\right) = x_i & \implies ({\rm P1}a)^{\rm d}\\
({\rm P}4d)\ &: \quad \forall i \in [1:N], \quad 0 \leq \lambda_{\ell_i} \frac{\ell_i + r_i}{r_i} = x_i &\implies ({\rm P1}a)^{\rm d}
\end{align*}
\begin{align*}
(\text{P4 objective function})\ &: \sum_{i=1}^N \lambda_{\ell_i}\ell_i = \sum_{i=1}^N x_i \frac{r_i}{\ell_i + r_i}\ell_i = (\text{${\rm P1^d}$ objective function}).
\end{align*}
Thus for any feasible point in P4, we get a feasible point in ${\rm P1^d}$ using the transformation in \eqref{eq:P2_2_P3} that has the same objective function with the same value as the original point in P4.
\item \underline{${\rm P1^d}$ $\to$ P4.} Define $\lambda_{\ell_i}$ and $\lambda_{r_i}$ to be
\begin{align}
\lambda_{\ell_i} = x_i \frac{r_i}{\ell_i + r_i},\quad \lambda_{r_i} = x_i \frac{\ell_i}{\ell_i + r_i},\quad \forall i \in [1:N].
\label{eq:P3_2_P2}
\end{align}
Note that the transformation above directly implies condition ({\rm P}4a) in P4. Now, we are going to show that the constraints in ${\rm P1^d}$ when applied to~\eqref{eq:P3_2_P2} imply the rest of the constraints in P4 as follows
\begin{align*}
({\rm P1}a)^{\rm d}\ &: \quad 1 \geq x_i = x_i \left( \frac{r_i}{\ell_i + r_i} + \frac{\ell_i}{\ell_i + r_i}\right) = \lambda_{\ell_i} + \lambda_{r_i} &\implies ({\rm P}4c)\\
({\rm P1}a)^{\rm d}\ &: \quad 0 \leq x_i \frac{r_i}{r_i + \ell_i} = \lambda_{\ell_i} &\implies ({\rm P}4d)\\
({\rm P1}b)^{\rm d}\ &: \quad 1 \geq \sum_{i=1}^N x_i \frac{\ell_i}{\ell_i + r_i} = \sum_{i=1}^N\lambda_{r_i} &\implies ({\rm P}4b)\\
({\rm P1}c)^{\rm d}\ &: \quad 1 \geq \sum_{i=1}^N x_i \frac{r_i}{\ell_i + r_i} = \sum_{i=1}^N\lambda_{\ell_i} &\implies ({\rm P}4b)
\end{align*}
\begin{align*}
(\text{${\rm P1^d}$ objective function})\ &: \sum_{i=1}^N x_i \frac{r_i}{\ell_i + r_i}\ell_i = \sum_{i=1}^N \lambda_{\ell_i}\ell_i = (\text{P4 objective function}).
\end{align*}
\end{enumerate}
Thus the two problems ${\rm P1^d}$ and P4 are equivalent. This concludes the proof of Lemma~\ref{lem:diamond_1-2-1_ntwk} for the HD case.
\subsection{Proof of Lemma~\ref{lem:simplification_diamond}(L2) for an HD diamond network}
We first prove the following property of the optimal corner points in the LP ${\rm P1^d}$ in~\eqref{eq:P3}.
\begin{prope}
If we have a 1-2-1 Gaussian HD diamond network, then for any optimal corner point solution of ${\rm P1^d}$, at least one of the constraints in $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ is satisfied with equality.
\end{prope}
\begin{proof}
We are going to prove Property~2 by contradiction.
Note that, since the LP ${\rm P1^d}$ has $N$ variables, then any corner point in ${\rm P1^d}$ has at least $N$ constraints satisfied with equality.
Now, assume that we have an optimal point $(x_1^\star,x_2^\star, \dots, x_N^\star)$ such that neither $({\rm P}1b)^{\rm d}$ nor $({\rm P}1c)^{\rm d}$ is satisfied with equality. This implies that the constraints satisfied with equality are only of the type $({\rm P}1a)^{\rm d}$. Thus, from the constraints in $({\rm P}1a)^{\rm d}$, we have that $x_i^\star \in \{0,1\}, \forall i \in [1:N]$.
Additionally, $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ being strict inequalities implies that $\sum_{i=1}^N x^\star_i < 2$.
Thus, there exists at most one $i'$, such that $x_{i'}^\star = 1$, while $x_j^\star = 0, \forall j \in [1:N]\backslash\{i'\}$.
Now, if we pick some $k \neq i'$ and set $x_k^\star = \varepsilon > 0$ such that both $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ are still satisfied, then we increase the objective function by $\varepsilon \frac{\ell_k r_k}{\ell_k + r_k}$, which contradicts the fact that $(x_1^\star,x_2^\star, \dots, x_N^\star)$ is an optimal solution.
\end{proof}
Now using Property~2, we are going to prove Lemma~\ref{lem:simplification_diamond}(L2) by considering the following two cases: (i) There exists an optimal corner point for which only one of the constraints in $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ is satisfied with equality, and (ii) all optimal corner points have both $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ satisfied with equality.
\begin{enumerate}
\item {\bf An optimal corner point exists with only one among $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ satisfied with equality.}
We denote this optimal corner point as $(x_1^\star,x_2^\star, \dots, x_N^\star)$.
Since only one among $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ is satisfied with equality, then at least $N-1$ constraints of the type $({\rm P}1a)^{\rm d}$ are satisfied with equality.
Also note that, since only one among $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ is satisfied with equality, then this implies that $\sum_{i = 1}^N x_i < 2$.
This implies that, although we have at least $N-1$ constraints in $({\rm P}1a)^{\rm d}$ satisfied with equality, we have at most one $i'$ such that $x_{i'}^\star = 1$.
As a result, at least $N-2$ of the constraints satisfied with equality from $({\rm P}1a)^{\rm d}$ are of the form $x_i = 0$.
This proves that at least $N-2$ relays are not utilized at this optimal corner point, which proves Lemma~\ref{lem:simplification_diamond}(L2) in this case.
\item {\bf All optimal corner points have $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ satisfied with equality.}
Pick an optimal corner point and denote it as $(x_1^\star,x_2^\star, \dots, x_N^\star)$.
Define $\mathcal{F}^\star_x = \{i | 0 < x^\star_i < 1\}$ and $\mathcal{I}_x^\star = \{i | x^\star_i = 1\}$, i.e., the sets of indices of the variables with non-integer and unitary values, respectively.
The fact that both $({\rm P}1b)^{\rm d}$ and $({\rm P}1c)^{\rm d}$ are satisfied with equality implies that $\sum_{i=1}^N x^\star_i = 2$, which implies that $|\mathcal{I}_x^\star| \leq 2$.
Additionally, since we are considering a corner point, then we have that at least $N-2$ constraints of the type $({\rm P}1a)^{\rm d}$ are satisfied with equality.
This implies that $|\mathcal{F}_x^\star| \leq 2$. Note that, if $|\mathcal{F}_x^\star| + |\mathcal{I}_x^\star| \leq 3$ for all optimal corner points, then we have proved Lemma~\ref{lem:simplification_diamond}(L2) for this case. Thus, we now show that the events $\{|\mathcal{F}_x^\star| = 2\}$ and $\{|\mathcal{I}_x^\star|= 2\}$ are mutually exclusive (i.e., disprove the possibility that $|\mathcal{F}_x^\star| + |\mathcal{I}_x^\star| = 4$). This follows by observing the following relation
\begin{align*}
2 = \sum_{i = 1}^N x^\star_i = \sum_{i \in [1:N]\backslash\mathcal{I}_x^\star} x^\star_i + \sum_{i \in \mathcal{I}_x^\star} x^\star_i = \sum_{i \in [1:N]\backslash\mathcal{I}_x^\star} x^\star_i + |\mathcal{I}_x^\star|.
\end{align*}
Thus
\begin{align*}
|\mathcal{I}_x^\star| = 2 \implies \sum_{i \in [1:N]\backslash\mathcal{I}_x^\star} x^\star_i = 0 \implies |\mathcal{F}_x^\star| = 0,
\end{align*}
which proves that the two events are mutually exclusive. This concludes the proof of Lemma~\ref{lem:simplification_diamond}(L2).
\end{enumerate}
\section{Proof of Lemma~\ref{lem:simplification_diamond}(L3)}
\label{app:Lemma5L3}
The proof of Lemma~\ref{lem:simplification_diamond}(L3) for the FD case follows directly from~\ref{lem:simplification_diamond}(L1) by taking only the two paths (relays) needed to achieve the $\mathsf{C}_{\rm cs,iid}$. Without loss generality, we assume that relays 1 and 2 are the relays in question. Then we have using the optimal fractions $x_1^\star$ and $x_2^\star$ that
\[
\mathsf{C}_{\rm cs,iid} = x_1^\star \mathsf{C}_1 + x_2^\star \mathsf{C}_2 \stackrel{({\rm P1}a)^{\rm d}}\leq \mathsf{C}_1 + \mathsf{C}_2,
\]
which proves that either $\mathsf{C}_1$ or $\mathsf{C}_2$ are greater than or equal half $\mathsf{C}_{\rm cs,iid}$.
To prove Lemma~\ref{lem:simplification_diamond}(L3) for the HD case, note that for a HD network $\mathsf{C}_i$ in ${\rm P1}^{\rm d}$ is given by
\[
\mathsf{C}_i = \frac{\ell_{i,0}\ \ell_{N+1,i}}{\ell_{i,0}\ +\ \ell_{N+1,i}}.
\]
Thus, by adding the constraints $({\rm P1}b)^{\rm d}$ and $({\rm P1}c)^{\rm d}$, we have the following implication for any feasible point in ${\rm P1}^{\rm d}$
\begin{align}
2 &\geq \sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{i, 0}} + \sum_{i \in [1:N]} x_i \frac{\mathsf{C}_{i}}{\ell_{N+1,i}} \nonumber \\
&= \sum_{i \in [1:N]} x_i \frac{\ell_{N+1,i}}{\ell_{i,0}+\ell_{N+1,i}} + \sum_{i \in [1:N]} x_i \frac{\ell_{i,0}}{\ell_{i,0}+\ell_{N+1,i}} = \sum_{i \in [1:N]} x_i.
\end{align}
Now, assume without loss of generality that the path through relay 1 has the largest HD approximate capacity.
Then, for any optimal point $x_i^\star$ that solves ${\rm P1}^{\rm d}$ in the HD case, we have
\[
\mathsf{C}_{\rm cs,iid} = \sum_{i \in [1:N]} x^\star_i \mathsf{C}_i \leq \left(\sum_{i \in [1:N]} x^\star_i \right) \mathsf{C}_1 \leq 2 \mathsf{C}_1.
\]
This proves that the approximate capacity of the best path in the network is at least half the of $\mathsf{C}_{\rm cs,iid}$.
}
\section{Equivalence between P3 and P1}
\label{app:P3EqualP1}
In this section, we prove the equivalence between the LPs P1 and P3 (which, as proved in Section~\ref{sec:ProofMainTh} is equivalent to P2), hence concluding the proof of Theorem~\ref{thm:MainTh}. In particular, our proof consists of two steps.
We first show that the LP in P3 is equivalent to the LP P5 below
\begin{align}
\label{eq:P1}
\begin{array}{llll}
{\rm P5 :} &\mathsf{C}_{\rm cs,iid}= {\rm max} \sum_{p \in \mathcal{P}} F_p & & \\
& ({\rm P}5a) \ F_p \geq 0 & \forall p \in \mathcal{P}, &\\
& ({\rm P}5b) \ F_p = \lambda^p_{\ell_{p.\text{nx}(i),i}} \ell_{p.\text{nx}(i),i} & \forall i \in p \backslash \{N+1\}, \forall p \in \mathcal{P}, &\\
& ({\rm P}5c)\ F_p = \lambda^p_{\ell_{i, p.\text{pr}(i)}} \ell_{i,p.\text{pr}(i)} & \forall i \in p \backslash \{0\}, \forall p \in \mathcal{P}, & \\
& ({\rm P}5d) \ \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{p.\text{nx}(i),i}} \leq 1 & \forall i \in [0:N], & \\
& ({\rm P}5e)\ \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{i,p.\text{pr}(i)}} \leq 1 & \forall i \in [1:N+1], &
\end{array}
\end{align}
and then show that P5 is equivalent to P1.
\noindent \underline{P3 $\to$ P5.} For $(i,j) \in p$ such that $j = p.\text{nx}(i)$, define the variable $\lambda_{\ell_{j,i}}^p$ to be
\begin{align}
\lambda_{\ell_{j,i}}^p = \frac{F_p}{\ell_{j,i}}.
\label{eq:P3_2_P4}
\end{align}
Note that, the definition above automatically satisfies the constraints $({\rm P}5a)$, $({\rm P}5b)$ and $({\rm P}5c)$ in P5.
Then, by always using the definition in~\eqref{eq:P3_2_P4}, we can equivalently rewrite the constraint $({\rm P}3b)$ as
\[
({\rm P}3b):\ \sum_{\substack{p \in \mathcal{P},\\(i,j) \in p,\\ j = p.\text{nx}(i)}} \lambda_{\ell_{j,i}}^p \leq \lambda_{\ell_{j,i}} ,\qquad \forall (j,i) \in [1:N+1]\times[0:N].
\]
Now, if we fix $\hat{i} \in [0:N]$ and add the left-hand side and right-hand side of $({\rm P}3b)$ for $(j,i) \in [1:N+1]\times \{\hat{i}\}$, then we get
\begin{align*}
\forall \hat{i} \in [0:N],\qquad &\sum_{j \in [1:N+1]}\sum_{\substack{p \in \mathcal{P},\\(\hat{i},j )\in p,\\ j = p.\text{nx}(\hat{i})}} \lambda_{\ell_{j,\hat{i}}}^p \leq \sum_{j \in [1:N+1]}\lambda_{\ell_{j,\hat{i}}}\\
&\implies \sum_{\substack{p \in \mathcal{P}_{\hat{i}}}} \lambda_{\ell_{p.\text{nx}(\hat{i}),\hat{i}}}^p \leq \sum_{j \in [1:N+1]}\lambda_{\ell_{j,\hat{i}}} \stackrel{({\rm P}3c)}{\leq 1} \implies ({\rm P}5d).
\end{align*}
Similarly, by adding the constraints in $({\rm P}3b)$ for a fixed $\hat{j} \in [1:N+1]$, one can show that, under the transformation in~\eqref{eq:P3_2_P4}, the constraint in $({\rm P}5e)$ is satisfied.
Thus, for any feasible point in P3, we can get a feasible point in P5 using the transformation in~\eqref{eq:P3_2_P4}. Regarding the objective function, note that we did not perform any transformation on the variables $F_p$ from P3 to P5.
It therefore follows that the objective function value achieved in P3 is the same as the one achieved in P5.
\noindent \underline{P5 $\to$ P3.}
Given a feasible point in P5, we define the following variables for each link in the network
\[
\lambda_{\ell_{j,i}} = \sum_{\substack{p \in \mathcal{P},\\(i,j) \in p,\\j = p.\text{nx}(i)}} \lambda^p_{\ell_{j,i}}.
\]
Based on this transformation, we automatically have that $({\rm P}3e)$ is satisfied. Moreover, we have that
\begin{align*}
({\rm P}5a)\ &: \quad \forall p \in \mathcal{P}, \quad 0 \leq F_p &\implies ({\rm P}3a)\\
({\rm P}5d)\ &: \quad \forall i,\quad 1 \geq \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{p.\text{nx}(i),i}} = \sum_{\substack{p \in \mathcal{P},\\ (i,j) \in p}} \lambda_{\ell_{j,i}} = \sum_{j \in [1:N+1]\backslash\{i\}} \lambda_{\ell_{j,i}} &\implies ({\rm P}3c)\\
({\rm P}5e)\ &: \quad \forall i,\quad 1 \geq \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{i,p.\text{pr}(i)}} = \sum_{\substack{p \in \mathcal{P},\\ (j,i) \in p}} \lambda_{\ell_{i,j}} = \sum_{j \in [0:N]\backslash\{i\}} \lambda_{\ell_{i,j}} &\implies ({\rm P}3d)\\
({\rm P}5b)\& ({\rm P}5c)\ &: \quad \sum_{\substack{p \in \mathcal{P},\\ (i,j) \in p,\\ j = p.\text{nx}(i)}} \frac{F_p}{\ell_{j,i}} = \sum_{\substack{p \in \mathcal{P},\\ (i,j) \in p,\\ j = p.\text{nx}(i)}} \lambda^p_{\ell_{j,i}} = \lambda_{\ell_{j,i}} &\implies ({\rm P}3b).
\end{align*}
Furthermore, note that the objective function in P5 and P3 is the same. Thus, a feasible point in P5 can be mapped to a feasible point in P3 with the same objective function value.
In conclusion, the problems P2, P3 and P5 are equivalent. We now show that P5 is equivalent to P1 in Theorem~\ref{thm:MainTh}.
\noindent \underline{P5 $\to$ P1.} Define $x_p$ to be
\begin{align}
x_p = \frac{F_p}{\mathsf{C}_p} ,\quad \forall p \in \mathcal{P}.
\label{eq:P4_2_P1}
\end{align}
Using this transformation, we get that the constraints in P4 imply the following
\begin{align*}
({\rm P}5a)\ &: \quad \forall p \in \mathcal{P}, \quad 0 \leq F_p = x_p \mathsf{C}_p &\implies ({\rm P}1a)\\
({\rm P}5d)\ &: \quad \forall i \in [0:N], \quad 1 \geq \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{p.\text{nx}(i),i}} \stackrel{({\rm P}5b)}{=} \sum_{p \in \mathcal{P}_i} \frac{F_p}{\ell_{p.\text{nx}(i),i}} = \sum_{p \in \mathcal{P}_i} \frac{x_p \mathsf{C}_p}{\ell_{p.\text{nx}(i),i}} \stackrel{\eqref{eq:actTime}} {=} \sum_{p \in \mathcal{P}_i} x_p f^p_{p.\text{nx}(i),i} & \implies ({\rm P}1b)\\
%
({\rm P}5e)\ &: \quad \forall i \in [1:N{+}1], \quad 1 \geq \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{i,p.\text{pr}(i)}} \stackrel{({\rm P}5c)}{=} \sum_{p \in \mathcal{P}_i} \frac{F_p}{\ell_{i,p.\text{pr}(i)}} = \sum_{p \in \mathcal{P}_i} \frac{x_p \mathsf{C}_p}{\ell_{i,p.\text{pr}(i)}} \stackrel{\eqref{eq:actTime}} {=} \sum_{p \in \mathcal{P}_i} x_p f^p_{i,p.\text{pr}(i)} & \implies ({\rm P}1c).
\end{align*}
Moreover, we have that
\begin{align*}
(\text{P5 objective function})\ &: \sum_{p \in \mathcal{P}} F_p= \sum_{p \in \mathcal{P}} x_p \mathsf{C}_p = (\text{P1 objective function}).
\end{align*}
Thus, for any feasible point in P5, we get a feasible point in P1 using the transformation in~\eqref{eq:P4_2_P1} that has the objective function with the same value as the original point in P5.
\noindent
\underline{P1 $\to$ P5.} Define $F_p$, $\lambda^p_{\ell_{p.\text{nx}(i),i}}$ and $ \lambda^p_{\ell_{i, p.\text{pr}(i)}}$ as
\begin{align}
F_p = x_p \mathsf{C}_p, \quad \lambda^p_{\ell_{p.\text{nx}(i),i}} = \frac{x_p \mathsf{C}_p}{\ell_{p.\text{nx}(i),i}} \forall i \in p \backslash\{N+1\}, \quad \lambda^p_{\ell_{i, p.\text{pr}(i)}} = \frac{x_p \mathsf{C}_p}{\ell_{i, p.\text{pr}(i)}} \forall i \in p \backslash\{0\}
\label{eq:P1_2_P4}
\end{align}
that hold $\forall p \in \mathcal{P}$.
Note that the transformation above directly implies conditions $({\rm P}5b)$ and $({\rm P}5c)$ in P5. Now, we are going to show that the constraints in P1 when applied to~\eqref{eq:P1_2_P4} imply the rest of the constraints in P5 as follows
\begin{align*}
({\rm P}1a)\ &: \forall p \in \mathcal{P}, \quad 0 \leq x_p = \frac{F_p}{\mathsf{C}_p} &\implies ({\rm P}5a)\\
({\rm P}1b)\ &: \forall i \in [0:N] \quad 1 \geq \sum_{p \in \mathcal{P}_i} x_p f^p_{p.\text{nx}(i),i} \stackrel{\eqref{eq:actTime}} {=} \sum_{p \in \mathcal{P}_i} \frac{x_p \mathsf{C}_p}{\ell_{p.\text{nx}(i),i}} = \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{p.\text{nx}(i),i}} &\implies ({\rm P}5d)\\
({\rm P}1c)\ &: \forall i \in [1:N+1] \quad 1 \geq \sum_{p \in \mathcal{P}_i} x_p f^p_{i,p.\text{pr}(i)} \stackrel{\eqref{eq:actTime}} {=} \sum_{p \in \mathcal{P}_i} \frac{x_p \mathsf{C}_p}{\ell_{i,p.\text{pr}(i)}} = \sum_{p \in \mathcal{P}_i} \lambda^p_{\ell_{i,p.\text{pr}(i)}} &\implies ({\rm P}5e)
\end{align*}
Moreover, we have that
\begin{align*}
(\text{P1 objective function})\ &: \sum_{p \in \mathcal{P}} x_p \mathsf{C}_p =\sum_{p \in \mathcal{P}} F_p = (\text{P5 objective function}).
\end{align*}
Thus, for any feasible point in P1, we get a feasible point in P5 using the transformation in~\eqref{eq:P1_2_P4} that has the objective function with the same value as the original point in P1.
Thus, the two problems P1 and P5 are equivalent.
In conclusion, the problems P1, P2, P3 and P5 are equivalent. This concludes the proof of Theorem~\ref{thm:MainTh}.
{
\section{A polynomial algorithm to compute the optimal schedule for Gaussian FD 1-2-1 networks}\label{app:poly_algorithm}
In this appendix, we show that for the approximate capacity expression in~\eqref{eq:apprCap}, we can compute the optimal schedume $\lambda^\star$ as well as the value of $\mathsf{C}_{\rm cs,iid}$ in polynomial time.
To start of, we note - as in Section~\ref{sec:ProofMainTh} - that for a fixed $\lambda_s$, the inner minimization in~\eqref{eq:apprCap} is the standard min-cut problem over a graph with link capacities given by
\begin{align}
\label{eq:ellMaxFlow2}
\ell_{j,i}^{(s)} = \left( \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i}.
\end{align}
Thus, we can replace the inner minimization in~\eqref{eq:apprCap} with the max-flow problem over the graph with link capacities defined in~\eqref{eq:ellMaxFlow2} to give the linear program $\rm Pflow_1$, i.e.,
\begin{align*}
\begin{array}{llll}
&{\rm Pflow_1}&{\rm\ :} \ \mathsf{C}_{\rm cs,iid} =\max \displaystyle\sum_{j =1}^{N+1} F_{j,0}&\nonumber\\
& ({\rm Pf}1a)\ & 0 \leq F_{j,i} \leq \left( \displaystyle\sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i} & (i,j) \in [0:N]\times[1:N{+}1], \\
& ({\rm Pf}1b)\ & \displaystyle\sum_{j\in[1:N{+}1]\backslash\{i\}} F_{j,i} = \displaystyle\sum_{k\in[0:N]\backslash\{i\}} F_{i,k} & i \in [1:N], \\
& ({\rm Pf}1c)\ &\displaystyle\sum_s \lambda_s \leq 1, \\
& ({\rm P}1d)\ &\lambda_s \geq 0 & \forall s,
\end{array}
\end{align*}
where $F_{j,i}$ is the flow through the link going from node $i$ to node $j$ and $\lambda_s$ is a state of the 1-2-1 network. A solution to the LP $\rm Pflow_1$ gives us the value $\mathsf{C}_{\rm cs,iid}$ as well as the optimal schedule to achieve the approximate capacity. Unfortunately, $\rm Pflow_1$ has an exponential number of variables $\lambda_s$ and therefore, cannot be solved efficiently in its current form.
Our main goal is to show that $\rm Pflow_1$ can be equivalently written as the LP $Pflow_2$ below.
\begin{align*}
{\rm Pflow_2}&{\rm\ :} \ \mathsf{C}_{\rm cs,iid} = \max \sum_{j =1}^{N+1} F_{j,0}& \nonumber\\
& 0 \leq F_{j,i} \leq \lambda_{\ell_{j,i}} \ell_{j,i} & (i,j) \in [0:N]\times[1:N{+}1], \\
& \sum_{j\in[1:N{+}1]\backslash\{i\}} F_{j,i} = \sum_{k\in[0:N]\backslash\{i\}} F_{i,k} & i \in [1:N], \\
& \displaystyle \sum_{\substack{j \in [1:N{+}1]\backslash\{i\}}} \! \! \! \! \lambda_{\ell_{j,i}} \leq 1 & \forall i \in [0:N], & \\
& \displaystyle \sum_{\substack{i \in [0:N]\backslash\{j\}}} \! \! \! \! \lambda_{\ell_{j,i}} \leq 1 & \forall j \in [1:N+1], & \\
&\lambda_{\ell_{j,i}} \geq 0 &\forall (i,j) {\in} [0\!:\!N] \!\times\! [1\!:\!N{+}1], &
\end{align*}
where $\lambda_{\ell_{j,i}}$ presents the fraction of time during which the links $i \to j$ is active.
Assuming this is true, then we have the following appealing outcomes:
\begin{enumerate}[(a)]
\item Since $\rm Pflow_2$ has a polynomial number of variables and constraints in $N$, then we can compute the value of $\mathsf{C}_{\rm cs,iid}$ in polynomial time in $N$.
\item If the mapping from an optimal point in $\rm Pflow_2$ to an optimal point in $Pflow_1$ can be done in polynomial time, then we have an algorithm to find the optimal schedule of the Gaussian FD 1-2-1 network in polynomial time. This can be done by first solving $\rm Pflow_2$ in polynomial time and then mapping its optimal solution in polynomial time to an optimal schedule in $\rm Pflow_1$.
\end{enumerate}
In what follows, we show that the $\rm Pflow_1$ and $\rm Pflow_2$ are indeed equivalent and the mapping an optimal point in $\rm Pflow_2$ to $\rm Pflow_1$ can be done by a construction that is polynomial in $N$. Note that in $\rm Pflow_1$ and $\rm Pflow_2$, the variables $F_{j,i}$ are the same, therefore we only need to find the mapping between $\{\lambda_s\}$ and $\{\lambda_{\ell_{j,i}}\}$.
\noindent \underline{$\rm Pflow_1 \to Pflow_2$.} Given a feasible point in $\rm Pflow_1$ we define
\[
\lambda_{\ell_{j,i}} = \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s
\]
Using this definition, we have that
\begin{align*}
({\rm Pf}1a)\ &: \forall (i,j) \quad F_{j,i} \leq \left( \displaystyle\sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \lambda_s \right) \ell_{j,i} = \lambda_{\ell_{j,i}} \ell_{j,i} &\implies ({\rm Pf}2a)\\
({\rm Pf}1c)\ &: \forall i \in [0:N] \quad \sum_{j=1}^{N+1} \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \leq \sum_s \lambda_s \leq 1 &\implies ({\rm Pf}2c)\\
({\rm Pf}1c)\ &: \forall j \in [1:N+1] \quad \sum_{i=0}^{N} \sum_{\substack{ s:\\ j \in s_{i,t},\ i \in s_{j,r} }} \leq \sum_s \lambda_s \leq 1 &\implies ({\rm Pf}2d).
\end{align*}
In addition, since the variables $F_{j,i}$ are not changed in the mapping then the new mapped point in $\rm Pflow_2$ has the save objective value as the original point in $\rm Pflow_1$.
\noindent \underline{$\rm Pflow_2 \to Pflow_1$.} Given a feasible point in $\rm Pflow_2$ we would like to construct a set of $\lambda_s$ that represent states in the FD 1-2-1 network, which collectively activate each link $(i,j)$ for at least the fraction dictated by $\lambda_{\ell_{j,i}}$.
To map $\rm Pflow_2$ to $\rm Pflow_1$, we use the same visualization introduced in Section~\ref{sec:ProofMainTh}. In particular, we divide each node $i \in [0:N+1]$ in the network into two vertices ($i_T$ and $i_R$) representing the transmitting and receiving functions of the node;
note that $0_R = (N+1)_T = \emptyset$ since the source (node $0$) is always transmitting and the destination (node $N+1$) is always receiving.
This gives us the bipartite graph $\mathcal{G}_B = (\mathcal{T},\mathcal{R},\mathcal{E})$, where the vertices $\mathcal{T}$ (respectively, $\mathcal{R}$) are the transmitting modules of our nodes (respectively, $\mathcal{R}$ collects our receiving modules), and we have an edge $(i_T,j_R) \in \mathcal{E}$ for each link in the network.
It is easy to see that a valid state in $\rm Pflow_1$ represents a matching in the bipartite graph $\mathcal{G}_B$. A perfect matching in a bipartite graph is represented by a permutation matrix $P$ where the rows of the matrix represent the set of vertices $\mathcal{R}$ and the columns are indexed by the vertices in $\mathcal{T}$.
Furthermore, we can write the feasible point in $\rm Pflow_2$ as a weighted adjacency matrix of the graph $\mathcal{G}_B$. In particular, $\lambda_{\ell_{j,i}}$ represents the weight of the edge connecting vertex $i_T$ to vertex $j_R$.
At this point, we can explicitly express our desired mapping in terms of the bipartite graph $\mathcal{G}_B$: Given a weighted adjacency matrix $L$ (which is filled using a feasible point of $\rm Pflow_2$ as $[L]_{ji} = \lambda_{\ell_{j,i}}$), can we efficiently find a set of permutation matrices $\{P_i\}$ that satisfy
\begin{align}
\label{eq:birkoff-vonneuman}
L \leq \sum_{i=1}^K \varphi_i P_i,\quad \sum_{i}^K \varphi_i = 1, \varphi \geq 0.
\end{align}
In particular, we are interested in a polynomial time approach to find these $P_i$ matrices. To answer this question, we need to observe some interesting property of $L$.
\begin{align*}
\forall (i,j) \in [0:N+1]^2, [L]_{ji} \geq 0,\\
\forall i \in [0:N+1], \sum_{j=0}^{N+1} [L]_{ji} \leq 1,\\
\forall j \in [0:N+1], \sum_{i=0}^{N+1} [L]_{ji} \leq 1.
\end{align*}
Such a matrix $L$ is called a \emph{doubly-substochastic matrix}. The result in~\cite{chang1999service} provides an algorithm that finds a set of permutation matrices satisfying~\eqref{eq:birkoff-vonneuman} for any doubly sub-stochastic matrix in $\mathbb{R}^{N\times N}$. The algorithm runs in $O(N^{4.5})$ time and outputs $N^2 - 2N +2$ permutation matrices. This proves the existence of a mapping from $\rm Pflow_2$ to $\rm Pflow_1$ that can be done in polynomial time.
}
\bibliographystyle{IEEEtran} |
1509.04164 | \section{Introduction}\label{sec:intro}
Angular correlations between jets produced together with heavy particles have been studied actively for a long time, because they can provide important information about the heavy particles~\cite{Plehn:2001nj, DelDuca:2001fn, Hankele:2006ma, Klamke:2007cu, Hagiwara:2009wt, Buckley:2010jv, Hagiwara:2013jp}.
For instance it has been shown that the distribution of the azimuthal angle difference $\Delta \phi = \phi_1^{}-\phi_2^{}$ between two partons in the gluon fusion production of a Higgs boson plus the two partons is very sensitive to a charge-conjugation and parity (CP) property of the Higgs boson~\cite{Plehn:2001nj, DelDuca:2001fn, Hankele:2006ma, Klamke:2007cu, Hagiwara:2009wt}. By observing the $\Delta \phi$ distribution and comparing it with theoretical predictions, we can measure CP violation in the Higgs sector~\cite{Hankele:2006ma, Klamke:2007cu}. \\
In order to read the information of heavy particles from angular correlations between jets produced in association with them, it will be necessary to produce the accurate predictions of observables, such as $\Delta \phi$, which measure the angular correlations. Tree level merging algorithms~\cite{Catani:2001cc, Lonnblad:2001iq, Krauss:2002up, Mrenna:2003if, Lavesson:2005xu, Mangano:2006rw, Alwall:2007fs, Giele:2007di, Alwall:2008qv, Hoeche:2009rj, Hamilton:2009ne, Giele:2011cb, Lonnblad:2011xx}, which combine leading order (LO) cross sections for multi-parton in final state with the parton shower, are nowadays standard tools used for simulating processes including multi-jet in final state. Models of the parton shower base the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation~\cite{Gribov:1972, Altarelli:1977, Dokshitzer:1977} and thus the parton shower guarantees the leading logarithmic (LL) accuracy for the kinematics of produced partons. Therefore the accuracy guaranteed in merging algorithms can be seen as the LO plus LL.
The virtue of merging algorithms is that they can combine LO cross sections smoothly with the LL parton shower so that the dependence on an artificial scale at which they are combined is minimized. \\
A LO multi-parton production cross section predicts angular correlations between the produced partons at the LO accuracy, while the LL parton shower does not have ability to predict angular correlations between any partons. Considering this fact, when our objective is to predict angular correlations between constructed jets, the accuracy minimally required for the kinematics of the jets should be the LO, neither LO+LL nor LL. If the kinematics of a jet is determined or largely influenced by the LL parton shower during a merging procedure, it hardly has the LO accuracy and hence it is not appropriate to use the event containing this jet.
Merging algorithms in the literature potentially have the ambiguity in the accuracy of jets, namely it is not necessarily clear whether the kinematics of a jet constructed by clustering particles in final state after a merging procedure has the LO accuracy or not. This is because their virtue is smooth combination of LO cross sections and the LL parton shower. \\
In ref~\cite{Hagiwara:2015tva}, the azimuthal angle difference $\Delta \phi = \phi_1^{}-\phi_2^{}$ between the two highest transverse momentum $p_T^{}$ jets with a large rapidity separation in the $t\bar{t}$ pair production is studied by using the CKKW-L merging algorithm~\cite{Lonnblad:2001iq, Lavesson:2005xu, Lonnblad:2011xx} with the parton shower model~\cite{Sjostrand:2004ef, Corke:2010yf, Norrbin:2000uu} in PYTHIA8~\cite{Sjostrand:2007gs, Sjostrand:2006za}. There, it is found that the correlation between the 2-jet can be lost in a non-negligible fraction of the events when the LO cross sections for the $t\bar{t}$ plus up to 2-parton are merged with the parton shower, because a jet originating from the parton shower has a higher $p_T^{}$ than one of the two jets originating from the 2-parton of the LO cross section.
This shows one example of the case that the kinematics of a jet is determined or largely influenced by the LL parton shower during a merging procedure.
Although it is also found that the loss of the correlation can be avoided by merging the LO $t\bar{t}+3$-parton cross section additionally, calculating LO cross sections for higher multiplicity is time-consuming, thus we want to avoid it.\\
In this work I construct a new merging algorithm which does guarantee the LO accuracy of angular correlations between jets and hence does not have the above ambiguity.
The Sudakov suppression is calculated in the same way as the MLM~\cite{Mangano:2006rw, Alwall:2007fs} and the $k_{\perp}^{}$-jet MLM~\cite{Alwall:2007fs, Alwall:2008qv} algorithms. The differences from these algorithms are
\begin{itemize}
\item The definition of jets used during a merging procedure is set identical to the one used during physics analyses of jets.
\item The LO $n_{\mathrm{max}}^{}$-parton production cross section, where $n_{\mathrm{max}}^{}$ denotes the maximal number of partons produced by the LO cross section, is not allowed to produce the events which contain more than $n_{\mathrm{max}}^{}$-jet.
\end{itemize}
The first point indicates that $n$-jet events are generated exclusively according to the LO $n$-parton production cross section and furthermore each of the $n$-jet is close to each of the $n$-parton in terms of the jet measure. As a result, angular correlations between the $n$-jet should follow those between the $n$-parton and hence the LO accuracy of them should be robust.
In addition to this, when $n$-jet events are exclusively subjects to a study, only the LO $n$-parton production cross section is needed, thus event generation is efficient. \\
The second point makes it forbidden to produce the events which contain more than $n_{\mathrm{max}}^{}$-jet. If our objective is to predict angular correlations between two jets such as $\Delta \phi$ at the LO accuracy, not only do the two jets have angular correlations between them at the LO accuracy, but all of the additional jets also should have angular correlations with the two jets at the same accuracy, because the additional jets can affect $\Delta \phi$ kinematically. If the kinematics of the additional jets is determined by the LL parton shower, those jets do not have angular correlations with any other jets. In such a case, the accuracy of $\Delta \phi$ will be less than the LO.
Our objective is to predict angular correlations between jets at the LO accuracy, therefore the events which contain more than $n_{\mathrm{max}}^{}$-jet should not be used for analyses. In my algorithm, this is naturally achieved by the above second point. \\
The $t\bar{t}$ production in proton-proton collisions is simulated and the azimuthal angle difference $\Delta \phi = \phi_1^{}-\phi_2^{}$ between the two highest $p_T^{}$ jets with large rapidity separations is studied by using the 2-jet events exclusively. The new algorithm is validated by comparing the $\Delta \phi$ distributions with the predictions of one of well-established algorithms, the CKKW-L algorithm~\cite{Lonnblad:2001iq, Lavesson:2005xu, Lonnblad:2011xx}.
It is observed that the new algorithm at $n_{\mathrm{max}}^{}=2$ produces a consistent result with the CKKW-L algorithm at $n_{\mathrm{max}}^{}=3$. \\
In Section~\ref{sec:formalism}, my formalism of tree level merging algorithms and my notations are introduced. Following these, in Section~\ref{sec:new-algorithms} the new merging algorithm and an event generation procedure according to it are described in detail. In Section~\ref{sec:numerical-study}, the results of the simulation are presented. In Section~\ref{sec:conclusion}, I summarize my findings.
\section{Formalism}\label{sec:formalism}
In this section, my formalism of tree level merging algorithms and my notations are introduced. Following these, the new algorithm is described in Section~\ref{sec:new-algorithms}. This section is largely based on ref.~\cite{Hagiwara:2015tva}. \\
The Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation~\cite{Gribov:1972, Altarelli:1977, Dokshitzer:1977} can be written with the Sudakov form factor in the following form~\cite{Marchesini:1987cf}
\begin{align}
t \frac{d}{dt} \frac{q(x,t)}{\Delta(t)}
&= \int_0^{\epsilon(t)} \frac{dz}{z} \frac{\alpha_s^{}}{2\pi} \hat{P}_{qq}^{}(z) \frac{q(x/z, t)}{\Delta(t)}, \label{dglap-with-Sudakov}
\end{align}
where $\Delta(t)$ denotes the Sudakov form factor
\begin{align}
\Delta(t)=\exp{\biggl(-\int^t_{\mu^2_{}}\frac{dt^{\prime}_{}}{t^{\prime}_{}}\int^{\epsilon(t^{\prime}_{})}_0 dz \frac{\alpha_s^{}}{2\pi}\hat{P}_{qq}^{}(z)\biggr)}. \label{Sudakov-original}
\end{align}
Here only the quark parton distribution function $q(x,t)$ (PDF) and the splitting function $\hat{P}_{qq}^{}(z)$ for $q\to qg$ without the virtual correction are introduced in order to simplify writing. The generalization is, however, simple. By integrating eq.~(\ref{dglap-with-Sudakov}) over $t_{\Lambda}^{}<t <t_X^{}$, it follows that
\begin{align}
\frac{q(x,t_X^{})}{\Delta(t_X^{})}=\frac{q(x,t_{\Lambda}^{})}{\Delta(t_{\Lambda}^{})}
+ \int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt}{t}\int^{\epsilon(t)}_0 d\hat{p}_{qq}^{}(z) \frac{q(x/z,t )}{\Delta(t)},\label{dglap-with-sudakov-integrate}
\end{align}
where a short hand notation is introduced
\begin{align}
d\hat{p}_{qq}^{}(z)= \frac{\alpha_s^{}}{2\pi} \frac{dz}{z} \hat{P}_{qq}^{}(z).\label{short-notation-1}
\end{align}
Using eq.~(\ref{dglap-with-sudakov-integrate}) iteratively and dividing it by $q(x,t_X^{})/\Delta(t_X^{})$, we can obtain the following form of the DGLAP equation~\cite{Marchesini:1987cf}
\begin{align}
1=& \frac{q(x,t_{\Lambda}^{})}{q(x,t_X^{})} \frac{\Delta(t_X^{})}{\Delta(t_{\Lambda}^{})}
+ \int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^{\epsilon(t_1^{})}_0 d\hat{p}_{qq}^{}(z_1^{}) \frac{ q(x/z_1^{},t_{\Lambda}^{} ) }{ q(x,t_X^{}) }
\frac{\Delta(t_X^{})}{\Delta(t_{\Lambda}^{})} \nonumber \\
&+ \int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^{\epsilon(t_1^{})}_0 d\hat{p}_{qq}^{}(z_1^{})
\int^{t_1^{}}_{t_{\Lambda}^{}} \frac{dt_2^{}}{t_2^{}}\int^{\epsilon(t_2^{})}_0 d\hat{p}_{qq}^{}(z_2^{})
\frac{ q\bigl(x/(z_1^{}z_2^{}),t_{\Lambda}^{} \bigr) }{ q(x,t_X^{}) }\frac{\Delta(t_X^{})}{\Delta(t_{\Lambda}^{})} + \cdots. \label{infinite-iteration-DGLAP-with-Sudakov-previous}
\end{align}
The first term of the right hand side (RHS) in the above equation can be seen as the probability of generating no radiation from a quark $q(x)$ in a proton during the scale evolution of the proton between $t_X^{}$ and $t_{\Lambda}^{}$ ($t_X^{}>t_{\Lambda}^{}$). The second term of the RHS represents the integrated probability of generating exclusively one radiation from the quark during the evolution, and so on.
The left hand side (LHS) ensures the probability conservation. \\
I write the no radiation probability from a quark $q(x)$ in a proton during the scale evolution of the proton between $t_1^{}$ and $t_2^{}$ ($t_1^{}>t_2^{}$) in the following form~\cite{Marchesini:1987cf}
\begin{align}
\Pi_q^{}(t_1^{}, t_2^{}; x) = \frac{q(x,t_{2}^{})}{q(x,t_1^{})} \frac{\Delta(t_1^{})}{\Delta(t_{2}^{})}.
\end{align}
Then, eq.~(\ref{infinite-iteration-DGLAP-with-Sudakov-previous}) can be expressed as
\begin{align}
1=&
\Pi_{q}^{}( t_X^{}, t_{\Lambda}^{}; x )+
\int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^{\epsilon(t_1^{})}_0 d\hat{p}_{qq}^{}(z_1^{})\
\Pi_{q}^{}( t_X^{}, t_1^{}; x)\
\frac{q(x/z_1^{}, t_1^{})}{q(x, t_1^{})}\
\Pi_q^{}( t_1^{}, t_{\Lambda}^{}; x/z_1^{} ) \nonumber \\
&
+
\int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^{\epsilon(t_1^{})}_0 d\hat{p}_{qq}^{}(z_1^{})
\int^{t_1^{}}_{t_{\Lambda}^{}} \frac{dt_2^{}}{t_2^{}}\int^{\epsilon(t_2^{})}_0 d\hat{p}_{qq}^{}(z_2^{})
\Pi_{q}^{}( t_X^{}, t_1^{}; x)
\frac{q(x/z_1^{}, t_1^{})}{q(x, t_1^{})}
\Pi_{q}^{}( t_1^{}, t_{2}^{}; x/z_1^{} )
\nonumber \\
&
\hspace{0.3cm}
\times
\frac{q\bigl(x/(z_1^{}z_2^{}), t_2^{}\bigr)}{q(x/z_1^{}, t_2^{})}
\Pi_{q}^{}\bigl( t_2^{}, t_{\Lambda}^{}; x/(z_1^{}z_2^{}) \bigr) \nonumber \\
&+\cdots.\label{infinite-iteration-DGLAP-with-Sudakov}
\end{align}
It is an easy task to find the explicit form of $\Pi_q^{}(t_1^{}, t_2^{}; x)$ from the above equation~\cite{Sjostrand:1985xi}
\begin{align}
\Pi_q^{}(t_1^{}, t_2^{}; x) = \exp{\biggl( -\int^{t_1^{}}_{t_{2}^{}} \frac{dt}{t}\int^{\epsilon(t)}_0 d\hat{p}_{qq}^{}(z)\frac{q(x/z, t)}{q(x, t)} \biggr)}. \label{no-rad-prob-ISR}
\end{align}
Given the scale $t_X^{}$ and the energy fraction $x$ of an incoming quark in a proton, eq.~(\ref{infinite-iteration-DGLAP-with-Sudakov}) allows us to generate radiations from the incoming quark by evolving the proton from the scale $t_X^{}$.
This is known as backward evolution~\cite{Sjostrand:1985xi, Gottschalk:1986bk, Marchesini:1987cf}.
\\
Eq.~(\ref{infinite-iteration-DGLAP-with-Sudakov}) derived from the DGLAP equation concerns only radiation from an incoming quark in a proton i.e. initial state radiation. Here I generalize the equation to the one which can predict radiation from outgoing partons i.e. final state radiation, as well as initial state radiation.
What we should notice for this purpose is that the PDFs play a role in constraining scale evolution of the proton, or in other words constraining radiation from the quark in the proton during scale evolution of the proton.
Hence, when radiation from outgoing partons is concerned, we replace the PDFs with a function which bases the kinematic information of the outgoing partons. The function constrains radiation from the outgoing partons, through the energy and momentum conservation for instance.\\
I let $\{p\}_{X+n}^{}$ denotes a complete specification of an event sample consisting of $X+n$ partons~\footnote{This expression is inspired by refs.~\cite{ Giele:2007di, Giele:2011cb}.}. The information of two incoming partons is implicitly included. Then, I introduce a function for the evolution of a $\{p\}_{X+n}^{}$
\begin{align}
f\bigl(z, t; \{p\}_{X+n}^{} \bigr),\label{intro-const-function}
\end{align}
which constrains the evolution of the $\{p\}_{X+n}^{}$ at the evolution scale $t$ and the energy fraction $z$. By using the constraint function, eq.~(\ref{infinite-iteration-DGLAP-with-Sudakov}) can be generalized to
\begin{align}
1=&
\Pi\bigl( t_X^{}, t_{\Lambda}^{}; \{p\}_X^{} \bigr)+
\int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^1_0 d\hat{p}(z_1^{})\
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr)\
\Pi\bigl( t_1^{}, t_{\Lambda}^{}; \{p\}_{X+1}^{} \bigr) \nonumber \\
&
+
\int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^1_0 d\hat{p}(z_1^{})
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr) \nonumber \\
& \hspace{0.5cm}\times
\int^{t_1^{}}_{t_{\Lambda}^{}} \frac{dt_2^{}}{t_2^{}}\int^1_0 d\hat{p}(z_2^{})
\Pi\bigl( t_1^{}, t_{2}^{}; \{p\}_{X+1}^{} \bigr)
f\bigl(z_2^{}, t_2^{}; \{p\}_{X+1}^{})\
\Pi\bigl( t_2^{}, t_{\Lambda}^{}; \{p\}_{X+2}^{} \bigr) \nonumber \\
& + \cdots,
\label{infinite-iteration-DGLAP-with-Sudakov-general}
\end{align}
and accordingly
\begin{align}
\Pi\bigl(t_1^{}, t_2^{}; \{p\}_{X+n}^{}) = \exp{\biggl( -\int^{t_1^{}}_{t_{2}^{}} \frac{dt}{t}\int^1_0 d\hat{p}(z)f\bigl(z, t; \{p\}_{X+n}^{}\bigr) \biggr)}, \label{no-rad-prob-general}
\end{align}
which is defined as the no radiation probability for a $\{p\}_{X+n}^{}$ as a whole, during the scale evolution of it between $t_1^{}$ and $t_2^{}$ ($t_1^{}>t_2^{}$).
In other words, this is the probability that a $\{p\}_{X+n}^{}$ remains the same during the evolution. The splitting probability has the following form
\begin{align}
d\hat{p}(z)=\frac{dz}{z}\frac{\alpha_s^{}}{2\pi}\hat{P}(z) &\mathrm{\ \ \ \ for\ initial\ state\ radiation},\nonumber \\
d\hat{p}(z)=dz\frac{\alpha_s^{}}{2\pi}\hat{P}(z) &\mathrm{\ \ \ \ for\ final\ state\ radiation}, \label{short-hand-splitting-function}
\end{align}
where the appropriate splitting function(s) should be used for $\hat{P}(z)$ according to a branching process $\{p\}_{X+n}^{} \to \{p\}_{X+(n+1)}^{}$. For initial state radiation, the constraint function $f(z, t; \{p\}_{X+n}^{})$ always includes the PDFs. The soft gluon singularity at $z=1$ in the splitting functions will be avoided by introducing $\theta$ functions in the constraint function. \\
Let us consider a hard process which produces $X$ and express its leading order (LO) cross section as $\sigma(X)$. The DGLAP evolution of the hard process can be expressed by multiplying $\sigma(X)$ by eq.~(\ref{infinite-iteration-DGLAP-with-Sudakov-general}),
\begin{align}
\sigma(X)
&=\sigma(X)\ \Pi\bigl( t_X^{}, t_{\Lambda}^{}; \{p\}_X^{} \bigr) \nonumber \\
&+\sigma(X) \int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^1_0 d\hat{p}(z_1^{})\
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr)\
\Pi\bigl( t_1^{}, t_{\Lambda}^{}; \{p\}_{X+1}^{} \bigr) \nonumber \\
&+\sigma(X) \int^{t_X^{}}_{t_{\Lambda}^{}} \frac{dt_1^{}}{t_1^{}}\int^1_0 d\hat{p}(z_1^{})
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr) \nonumber \\
& \hspace{0.5cm}\times
\int^{t_1^{}}_{t_{\Lambda}^{}} \frac{dt_2^{}}{t_2^{}}\int^1_0 d\hat{p}(z_2^{})
\Pi\bigl( t_1^{}, t_{2}^{}; \{p\}_{X+1}^{} \bigr)
f\bigl(z_2^{}, t_2^{}; \{p\}_{X+1}^{})\
\Pi\bigl( t_2^{}, t_{\Lambda}^{}; \{p\}_{X+2}^{} \bigr) \nonumber \\
& + \cdots.\label{X-DGLAP-evolution}
\end{align}
Tree level merging algorithms are designed as to improve the above DGLAP evolution by replacing the terms constructed by the leading order cross section times the universal radiation probability with the exact LO cross sections for multi-parton in final state~\cite{Catani:2001cc}. In my notations, it proceeds as
\begin{align}
\sigma(X)
&=\sigma(X)\ \Pi\bigl( t_X^{}, t_{\Lambda}^{}; \{p\}_X^{} \bigr) \nonumber \\
&+\sigma\bigl(X+1; \{p\}_{X+1}^{} > t_{\Lambda}^{} \bigr) \
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f^{\prime}_{}\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr)\
\Pi\bigl( t_1^{}, t_{\Lambda}^{}; \{p\}_{X+1}^{} \bigr) \nonumber \\
&+\sigma\bigl(X+2; \{p\}_{X+2}^{} > t_{\Lambda}^{} \bigr) \
\Pi\bigl( t_X^{}, t_1^{}; \{p\}_X^{} \bigr)\
f^{\prime}_{}\bigl(z_1^{}, t_1^{}; \{p\}_X^{} \bigr)\
\Pi\bigl( t_1^{}, t_{2}^{}; \{p\}_{X+1}^{} \bigr) \nonumber \\
&
\hspace{0.5cm}
\times f^{\prime}_{}\bigl(z_2^{}, t_2^{}; \{p\}_{X+1}^{})\
\Pi\bigl( t_2^{}, t_{\Lambda}^{}; \{p\}_{X+2}^{} \bigr)\ \nonumber \\
& + \cdots,\label{improved-DGLAP-1}
\end{align}
The soft and collinear divergences in LO cross sections are regularized in the following way. By using the definition of the evolution variable $t$, calculate the minimum value $t_{\mathrm{min}}^{}$ from a $\{p\}_{X+n}^{}$ and require $t_{\mathrm{min}}^{}>t_{\Lambda}^{}$. This is expressed as $\{p\}_{X+n}^{}>t_{\Lambda}^{}$ in the above equation~\footnote{Here it is assumed that the hard process cross section $\sigma(X)$ is finite everywhere in its phase space.}. Notice that the constraint functions in eq.~(\ref{improved-DGLAP-1}) are different from those in eq.~(\ref{X-DGLAP-evolution}), since some part of the constraint is already included in the exact LO cross section. I call $f^{\prime}_{}(z_{n+1}^{}, t_{n+1}^{}; \{p\}_{X+n}^{})$ a weight function hereafter, since it is used to re-weight an event sample during a merging procedure.
The cut off scale $t_{\Lambda}^{}$ is called merging scale and separates phase space into two regions. Partons in the phase space above the merging scale are produced following exact LO cross sections and those in the phase space below the merging scale are produced following the DGLAP equation. The further evolution of each term in the above equation will be performed between the scale $t_{\Lambda}^{}$ and a lower cutoff scale for complete event generation. In my notations, the term consisting of the cross section $\sigma(X+n)$ in the RHS of the above equation is called the $n$ th term. \\
Below tree level merging algorithms which have been proposed and studied in the literature are briefly reviewed. This information can be useful when I describe my algorithm in Section~\ref{sec:new-algorithms}.
There are fundamentally two different types of merging algorithms~\footnote{In this work I concentrate only on discussing a type of merging algorithms which introduces the merging scale. However, it would be also interesting to study jet angular correlations with other types of algorithms such as \cite{Giele:2011cb}.}. One of the two is constructed to generate events according to the improved DGLAP equation strictly, eq.~(\ref{improved-DGLAP-1}) in my notations, by explicitly calculating the Sudakov form factors. This procedure is necessary in order to minimize the dependence on the merging scale and smoothly combine LO cross sections with the parton shower.
Algorithms of this type include the CKKW with the vetoed parton shower~\cite{Catani:2001cc, Krauss:2002up}, the CKKW with the truncated shower~\cite{Hoeche:2009rj, Hamilton:2009ne} and the CKKW-L~\cite{Lonnblad:2001iq, Lavesson:2005xu, Lonnblad:2011xx} algorithms.
The other of the two types does not calculate the Sudakov form factors explicitly and hence event generation follow the improved DGLAP equation approximately.
Algorithms of this type include the MLM~\cite{Mangano:2006rw, Alwall:2007fs}, the $k_{\perp}^{}$-jet MLM~\cite{Alwall:2007fs, Alwall:2008qv} and the shower $k_{\perp}^{}$~\cite{Alwall:2008qv} algorithms. In the MLM and the $k_{\perp}^{}$-jet MLM algorithms, partons in final state obtained after the complete shower evolution are clustered according to the cone jet algorithm and the exclusive $k_{\perp}^{}$-jet algorithm, respectively. Then, if the number of the constructed jets is not equal to the number of partons produced by a LO cross section or if the constructed jets are not close to the partons in terms of the jet measure, the event is vetoed. In the shower $k_{\perp}^{}$ algorithm, if the scale of the first emission is above the merging scale, then the event is vetoed.
\section{Algorithm}\label{sec:new-algorithms}
In this section, the ideas of the new merging algorithm and an event generation procedure according to the algorithm are described in detail.\\
The new algorithm is constructed with a goal of eliminating the ambiguity in the accuracy of jets which potentially exists in merging algorithms as discussed in Section~\ref{sec:intro}, and hence of guaranteeing the leading order (LO) accuracy of angular correlations between jets. The following idea is strictly implemented in the algorithm:\\
{\it The $n$-jet events are generated exclusively according to the LO $n$-parton production cross section and furthermore each of the $n$-jet is close to each of the $n$-parton in terms of the jet measure.}\\
For an algorithm which implements the above idea, which I call the model A algorithm~\footnote{This is the remnant of the fact that there existed the model B too, which turned out not to work properly and thus is not described in this paper.}, eq.~(\ref{improved-DGLAP-1}) may be written in the following form:
\begin{align}
&\sigma(X)_{\mathrm{Model A}}^{} \nonumber\\
&=\sigma(X)\ \Pi\bigl( t_X^{}, \{p\}_{X}^{} \to \{p\}_{X+0\mathrm{jet}}^{} ; \{p\}_X^{} \bigr) \nonumber \\
&+\sigma\Bigl(X+1; \{p\}_{X+1}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}} \Bigr) \
\Pi\bigl( t_{X+1}^{}, \{p\}_{X+1}^{} \to \{p\}_{X+1\mathrm{jet}}^{} ; \{p\}_{X+1}^{} \bigr) \nonumber \\
&+\sigma\Bigl(X+2; \{p\}_{X+2}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}} \Bigr) \
\Pi\bigl( t_{X+2}^{}, \{p\}_{X+2}^{} \to \{p\}_{X+2\mathrm{jets}}^{} ; \{p\}_{X+2}^{} \bigr) \nonumber \\
& + \cdots \nonumber \\
& + \sigma\Bigl(X+n; \{p\}_{X+n}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}} \Bigr) \
\Pi\bigl( t_{X+n}^{}, \{p\}_{X+n}^{} \to \{p\}_{X+n\mathrm{jets}}^{} ; \{p\}_{X+n}^{} \bigr) \nonumber \\
& + \cdots \nonumber \\
& + \sigma\Bigl(X+n_{\mathrm{max}}^{}; \{p\}_{X+n_{\mathrm{max}}^{}}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}} \Bigr) \
\Pi\bigl( t_{X+n_{\mathrm{max}}^{}}^{}, \{p\}_{X+n_{\mathrm{max}}^{}}^{} \to \{p\}_{X+n_{\mathrm{max}}^{} \mathrm{jets}}^{} ; \{p\}_{X+n_{\mathrm{max}}^{}}^{} \bigr) .\label{improved-DGLAP-2}
\end{align}
Here the weight functions are omitted in order to simplify writing. In the numerical studies in Section~\ref{sec:numerical-study}, they are taken into account.
The expression $\{p\}_{X+n}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}}$ indicates that the soft and collinear divergences in LO cross sections are regularized by the definition of $Q_{\mathrm{cut}}^{\mathrm{ME}}$.
The definition of $Q_{\mathrm{cut}}^{\mathrm{ME}}$ must respect the definition of jets used in analyses, namely a jet clustering algorithm and parameters that the algorithm contains.
Throughout of this work, the anti-$k_T^{}$ algorithm~\cite{Cacciari:2008gp} is used as a jet clustering algorithm, thus I concentrate on discussing the case of using the anti-$k_T^{}$ algorithm.
The anti-$k_T^{}$ algorithm basically contains two parameters which we can choose their values freely, namely the radius parameter $R^{\mathrm{jet}}_{}$ and the lower transverse momentum cutoff on jets $p_{T \mathrm{cut}}^{\mathrm{jet}}$. Hence $Q_{\mathrm{cut}}^{\mathrm{ME}}$ should be defined as
\begin{subequations}\label{ME-cutoff}
\begin{align}
\Delta R_{ij}^{} =\sqrt{(y_i^{}-y_j^{})^2_{}+(\phi_i^{}-\phi_j^{})^2_{}} > R_{\mathrm{cut}}^{\mathrm{ME}},\label{R-para}\\
p_{T i}^{} > p_{T \mathrm{cut}}^{\mathrm{ME}},
\end{align}
\end{subequations}
where $p_{T i}^{}$, $y_i^{}$ and $\phi_i^{}$ are the transverse momentum with respect to the beam, rapidity and azimuthal angle of outgoing parton $i$. Imposing cutoffs on the rapidity of partons in not needed.
In order to avoid missed phase space, the above two parameters must satisfy
\begin{align}
p_{T \mathrm{cut}}^{\mathrm{jet}} \ge p_{T \mathrm{cut}}^{\mathrm{ME}},\ \
R^{\mathrm{jet}}_{} \ge R_{\mathrm{cut}}^{\mathrm{ME}}.\label{mergin-scale-rel}
\end{align}
\\
Let us consider generating an $n$-jet event according to the $n$ th term in eq.~(\ref{improved-DGLAP-2}),
\begin{align}
\sigma\Bigl(X+n; \{p\}_{X+n}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}} \Bigr) \
\Pi\bigl( t_{X+n}^{}, \{p\}_{X+n}^{} \to \{p\}_{X+n\mathrm{jets}}^{} ; \{p\}_{X+n}^{} \bigr).
\end{align}
A parton shower program is executed on an event sample $\{p\}_{X+n}^{}$ generated by the LO cross section $\sigma(X+n; \{p\}_{X+n}^{} > Q_{\mathrm{cut}}^{\mathrm{ME}})$, by setting the shower starting scale to $t_{X+n}^{}$.
Once the shower evolution is performed until the shower cutoff scale, all partons in the final state within a rapidity range $|y|< y_{\mathrm{cut}}^{\mathrm{clus}}$ are clustered to construct jets according to the anti-$k_T^{}$ algorithm with the radius parameter $R^{\mathrm{jet}}_{}$ and the lower $p_T^{}$ cutoff $p_{T \mathrm{cut}}^{\mathrm{jet}}$. If the number of the constructed jets $n_{\mathrm{jet}}^{}$ is not identical to $n$, the event sample is vetoed. If the event sample survives, the distance parameters $\Delta R$ defined in eq.~(\ref{R-para}) between the $n$-jet and the $n$-parton are calculated, and then it is checked whether the following is satisfied between each of the $n$-jet and each of the $n$-parton, or not
\begin{align}
\Delta R_{\mathrm{jet,\ parton}}^{} < C_{\mathrm{match}}^{}\times R^{\mathrm{jet}}_{}. \label{C-match-parameter}
\end{align}
If a jet satisfies the above relation with a parton, it means that the jet is close to the parton in terms of the measure of the anti-$k_T^{}$ algorithm. The jet is called matched with the parton.
If a matching between a jet and a parton is confirmed for all of the $n$-jet and all of the $n$-parton, then the event sample is accepted. Once the event sample is accepted, all partons in the final state are again clustered to construct jets which are defined in the same way as above, but this time only those within a rapidity range $|y|< y_{\mathrm{cut}}^{\mathrm{detect}}$ are clustered. A value for $y_{\mathrm{cut}}^{\mathrm{detect}}$ should reflect actual detectors at experiments and can be different from $y_{\mathrm{cut}}^{\mathrm{clus}}$ which is used during the merging procedure. The following relation should be satisfied between the two
\begin{align}
y_{\mathrm{cut}}^{\mathrm{detect}} \le y_{\mathrm{cut}}^{\mathrm{clus}}.
\end{align}
If values for these two cuts are different, the jets constructed at this stage cannot be identical to those constructed during the merging procedure and thus are not necessarily matched with the $n$-parton in some events. However, if $y_{\mathrm{cut}}^{\mathrm{detect}}$ is large enough so that there is only a small fraction of the events $\{p\}_{X+n}^{}$ above $y_{\mathrm{cut}}^{\mathrm{detect}}$, then the effect is expected to be small. In the numerical studies, I use $y_{\mathrm{cut}}^{\mathrm{detect}}=5$ and $y_{\mathrm{cut}}^{\mathrm{clus}}=7$. The constructed jets will be used for physics analyses. \\
The accepted events have $n$-jet exclusively and each of the $n$-jet is close to each of the $n$-parton produced by the LO cross section $\sigma(X+n)$ in terms of the jet measure. Angular correlations between the $n$-jet should follow those between the $n$-parton. Since angular correlations between the $n$-parton is at the LO accuracy, those between the $n$-jet should also be at the LO accuracy. \\
As is clear, the algorithm calculates the effects of the Sudakov form factors in the same way as the MLM~\cite{Mangano:2006rw, Alwall:2007fs} and the $k_{\perp}^{}$-jet MLM~\cite{Alwall:2007fs, Alwall:2008qv} algorithms~\footnote{See the discussion at the end of Section~\ref{sec:formalism}.}. The differences from these algorithms are
\begin{itemize}
\item The definition of jets used during a merging procedure is set identical to the one used during physics analyses of jets.
\item The LO $n_{\mathrm{max}}^{}$-parton production cross section is not allowed to produce the events which contain more than $n_{\mathrm{max}}^{}$-jet.
\end{itemize}
As to the first point, the MLM and the $k_{\perp}^{}$-jet MLM algorithms use the cone algorithm and the exclusive $k_{\perp}^{}$-jet algorithm, respectively, as a clustering algorithm, and parameters that the clustering algorithm contains are chosen independently of the definition of jets used in physics analyses. In other words, the definition of jets used in physics analyses is nothing to do with a merging procedure in these merging algorithms. In my algorithm, the first point is essential in order to guarantee the LO accuracy of angular correlations between jets. \\
The maximal number $n_{\mathrm{max}}^{}$ of partons produced by the LO cross section is limited. If we follow the treatment of the LO $n_{\mathrm{max}}^{}$-parton production cross section in the MLM and the $k_{\perp}^{}$-jet MLM algorithms, the cross section is allowed to produce the events which contain $n_{\mathrm{max}}^{}$ or more than $n_{\mathrm{max}}^{}$-jet and each of the hardest $n_{\mathrm{max}}^{}$-jet is required to be matched with each of the $n_{\mathrm{max}}^{}$-parton. In this approach, the kinematics of the additional jets i.e. jets softer than the $n_{\mathrm{max}}^{}$-jet in terms of the jet measure is determined by the leading logarithmic (LL) parton shower and hence they do not have angular correlations with any other jets.
Since our objective is to predict angular correlations between jets at the LO accuracy, not only do the jets used for an observable such as $\Delta \phi=\phi_1^{}-\phi_2^{}$ have angular correlations between them at the LO accuracy, but all of the jets including jets not used for the observable should have angular correlations between them at the same accuracy, because even the jets not used for the observable can affect the observable kinematically. If some of the jets not used for the observable are generated by the LL parton shower, then the accuracy of the observable will be less than the LO. Considering this argument, the treatment of the LO $n_{\mathrm{max}}^{}$-parton production cross section in the MLM and the $k_{\perp}^{}$-jet MLM algorithms is not appropriate to our objective. In my algorithm, the second point above guarantees the LO accuracy of angular correlations between jets in all events. \\
The parameter $C_{\mathrm{match}}^{}$ in eq.~(\ref{C-match-parameter}) is an important parameter, since the Sudakov suppression can depend on it.
The implementations of the MLM algorithm in Alpgen~\cite{Mangano:2006rw, Alwall:2007fs} and in MadGraph5\verb|_|aMC@NLO~\cite{Alwall:matching} use $C_{\mathrm{match}}^{}=1.5$.
However there is nothing that uniquely determines $C_{\mathrm{match}}^{}$, thus which should be considered as a tuning parameter and can also depend on a jet clustering algorithm. In my implementation of the new algorithm with the anti-$k_T^{}$ algorithm, $C_{\mathrm{match}}^{}=2.0$ is used. \\
Below I describe a procedure of event generation for completeness. The numerical studies in Section~\ref{sec:numerical-study} are performed according to this.
\begin{enumerate}
\item Generate the event samples for the $X+0, 1, \dots, n_{\mathrm{max}}^{}$-parton production processes at proton-proton (pp) collisions according to the LO cross sections, i.e. $\{p\}_{X}^{}, \{p\}_{X+1}^{}, \cdots, \{p\}_{X+n_{\mathrm{max}}^{}}^{}$. Here the $t\bar{t}$ production is simulated, thus $X=t\bar{t}$.
MadGraph5\verb|_|aMC@NLO\cite{Alwall:2014hca} version 5.2.2.1 is used for this purpose. The soft and collinear singularity is regularized by imposing eq.~(\ref{ME-cutoff}). I use
\begin{align}
R_{\mathrm{cut}}^{\mathrm{ME}}=0.4,\ \ p_{T \mathrm{cut}}^{\mathrm{ME}}=20\ \mathrm{GeV}.\label{ME-cut-value}
\end{align}
A fixed value $t_{\Lambda}^{}$ is used for the scales in the strong couplings and in the parton distribution functions (PDFs). The PDF set CTEQ6L1~\cite{Pumplin:2002vw} is used. Note that, when the $X+n$-jet events are exclusively subjects to a study, only the events $\{p\}_{X+n}^{}$ are needed to be generated.
\item Select an event sample for the $X+n$ partons process, i.e. $\{p\}_{X+n}^{}$, with the probability proportional to its integrated LO cross section obtained in step 1,
\begin{align}
P_n=\frac{\sigma(pp\to X+n)}{\sum_{i=0}^{n_{\mathrm{max}}^{}}\sigma(pp\to X+i)}.
\end{align}
\item Construct a PYTHIA8 parton shower history of the $\{p\}_{X+n}^{}$. The history consists of intermediate events $\{p\}_{X+(n-1)}^{}, \{p\}_{X+(n-2)}^{}, \cdots, \{p\}_{X+i}^{}, \cdots, \{p\}_{X+1}^{}, \{p\}_{X}^{}$ with the clustering scales $t_n^{} < t_{n-1}^{} < \cdots < t_{i+1}^{} < \cdots < t_2^{}< t_1^{}$.
\item Calculate the weight function for the $\{p\}_{X+n}^{}$. Let us define the energy fractions and the parton types of the incoming partons in the $\{p\}_{X+i}^{}$ by $x_1^{(i)}$, $x_2^{(i)}$ and $f_1^{(i)}$, $f_2^{(i)}$, respectively. The weight function for the $\{p\}_{X+i}^{}$ is given by~\cite{Lonnblad:2011xx}
\begin{align}
f^{\prime}_{}\bigl(z_{i+1}^{}, t_{i+1}^{}; \{p\}_{X+i}^{} \bigr)
=
\frac{ \alpha_s^{}( t_{i+1}^{} ) }{ \alpha_s^{}( t_{\Lambda}^{} ) }
\frac{ f_1^{(i)}( x_1^{(i)}, t_i^{} ) }{ f_1^{(i)}( x_1^{(i)}, t_{i+1}^{} ) }
\frac{ f_2^{(i)}( x_2^{(i)}, t_i^{} ) }{ f_2^{(i)}( x_2^{(i)}, t_{i+1}^{} ) }, \label{general-const-function}
\end{align}
from which the total weight function for the $\{p\}_{X+n}^{}$ is
\begin{align}
\prod_{i=0}^{n}f^{\prime}_{}\bigl(z_{i+1}^{}, t_{i+1}^{}; \{p\}_{X+i}^{} \bigr),\label{total-const-function}
\end{align}
where $t_0^{}=t_X^{}$ and $t_{n+1}^{}=t_{\Lambda}^{}$. The PYTHIA8 shower model uses the different strong couplings for initial state radiation and final state radiation. Therefore, when a radiation is classified as the initial state radiation (the final state radiation) with a scale $t_{i+1}^{}$ by the shower history construction, $\alpha_s^{}(m_z^{})=1.37$ $(1.383)$ is used for the factor in eq.~(\ref{general-const-function}).
The scale $t_X^{}$ should be determined from the intermediate event $\{p\}_{X}^{}$. I define it by
\begin{align}
t_X^{}=E_T^{}(t) \times E_T^{}(\bar{t}),\label{tX-define}
\end{align}
where $E_T^2=m^2_{}+p_T^2$. This scale $t_X^{}$ is also used as the renormalization scales of the strong couplings $\alpha_s^2$ for the hard process, namely
\begin{align}
\frac{\alpha_s^2(t_X^{}) }{ \alpha_s^2(t_{\Lambda}^{}) }
\end{align}
is added as a multiplicative factor in the weight function. I use $\alpha_s^{}(m_z^{})=0.13$ in the above factor. Once the weight function is calculated, the $\{p\}_{X+n}^{}$ is re-weighted with the function. However, since the weight function is not bounded above by unity, the upper bound of the weight function must be found at first by calculating the weight function for a large number of events $\{p\}_{X+n}^{}$. The integrated LO cross section obtained in step 1 has to be multiplied by the obtained upper bound of the weight function.
\item Calculate the effects of the Sudakov form factors by using the method described above. I use the parton shower model~\cite{Sjostrand:2004ef, Corke:2010yf, Norrbin:2000uu} in PYTHIA8~\cite{Sjostrand:2007gs, Sjostrand:2006za} version 8186. The default tune of the version 8186, tune 4C~\cite{Corke:2010yf}, is used.
The parameters in the anti-$k_T^{}$ algorithm and the cutoff values on the rapidity of partons are set to
\begin{align}
R^{\mathrm{jet}}_{}=0.4,\ \ p_{T \mathrm{cut}}^{\mathrm{jet}}=30\ \mathrm{GeV},\ \ y_{\mathrm{cut}}^{\mathrm{clus}}=7,\ \ y_{\mathrm{cut}}^{\mathrm{detect}}=5.
\end{align}
I use Fastjet~\cite{Cacciari:2011ma} version $3.1.0$ for executing the anti-$k_T^{}$ algorithm.
The parton shower starting scale, which is the maximal shower evolution scale and is denoted by $t_{X+n}^{}$ in eq.~(\ref{improved-DGLAP-2}), is set to
\begin{align}
t_{X+n}^{} = t_X^{}/4 + t_1^{}/16 + t_2^{}/16 + \cdots + t_n^{}/16.\label{starting-scale}
\end{align}
\item Repeat the above procedures from step 2 to step 5 until a large number of the accepted event samples are obtained.
\end{enumerate}
\section{Numerical studies}\label{sec:numerical-study}
In this section, the model A algorithm is validated by discussing numerical differences between its predictions and the predictions of another merging algorithm. As another merging algorithm, I choose the CKKW-L merging algorithm~\cite{Lonnblad:2001iq, Lavesson:2005xu, Lonnblad:2011xx}, which is one of well-established algorithms~\footnote{In my implementation of the CKKW-L algorithm, the method of phase space separation is different from the original one. This is the one called the CKKW-L+ algorithm in ref.~\cite{Hagiwara:2015tva}. However numerical differences are found small.}.
The top quark pair production process at the 14 TeV LHC is simulated, and then angular correlations between the two highest transverse momentum jets are studied by using the $t\bar{t}+2$-jet events exclusively.
The merging scale in the CKKW-L algorithm is set as eq.~(\ref{ME-cut-value}). \\
As an observable measuring angular correlations between the two hardest jets i.e. the two highest transverse momentum jets, the azimuthal angle difference is chosen
\begin{align}
\Delta \phi = \phi_1^{}-\phi_2^{}.
\end{align}
The events which contain 2-jet are exclusively picked up at first and then if the 2-jet pass the following rapidity cut, $\Delta \phi$ is evaluated by using the 2-jet,
\begin{align}
y_1^{} \times y_2^{} < 0,\ \ \ \ |y_1^{}-y_2^{}|>3.5\ \ \mathrm{or}\ \ 2.5, \label{vbf}
\end{align}
which are often called the vector boson fusion (VBF) cuts. One of the 2-jet which has a positive rapidity is chosen for $\phi_1^{}$ and the other jet which has a negative rapidity is chosen for $\phi_2^{}$.
The studies based on the leading order (LO) $t\bar{t}+2$-parton production cross section~\cite{Buckley:2010jv, Hagiwara:2013jp} have shown that there are strong correlations between the two partons with a large rapidity separation, when the $t\bar{t}$ is near or far away from the threshold of its production~\cite{Hagiwara:2013jp}. Here I do not impose any cuts on the $t\bar{t}$, however.
The rapidity cut on jets is set as $|y| < 4.5$. The $t\bar{t}$ is assumed stable, since the main purpose of this study is to investigate a way of accurately modeling the kinematic activity of jets induced by the hard process. Hence the hadronization after the shower evolution and the multiple interaction in PYTHIA8 are also turned off. \\
\begin{figure}[t]
\centering
\includegraphics[scale=0.54]{pt.pdf}
\caption{\small
The normalized differential cross section as a function of the $p_T^{}$ of the highest $p_T^{}$ jet (left panel) and that of the second highest $p_T^{}$ jet (right panel) in the 2-jet events.
The correspondence between the curves and the merging algorithms is labeled inside the left panel.
}
\label{figure:jet-dist}
\end{figure}
At first, I show the transverse momentum $p_T^{}$ distributions of jets.
In the left and the right panels of Figure~\ref{figure:jet-dist},
the normalized differential cross section as a function of the transverse momentum $p_T^{}$ of the highest $p_T^{}$ jet and that of the second highest $p_T^{}$ jet in the 2-jet events are shown, respectively.
The blue solid curve represents the distribution of the model A algorithm.
The red solid curve and the black dashed curve represent the distributions of the CKKW-L algorithm, the maximal number $n_{\mathrm{max}}^{}$ of partons produced by the LO cross section is $n_{\mathrm{max}}^{}=2$ for the former and $n_{\mathrm{max}}^{}=3$ for the latter. \\
The important difference of the two algorithms is that only the $2$-parton production cross section contributes to the 2-jet events in the model A algorithm, while the $0$, $1$ and $2$-parton production cross sections contribute to the 2-jet events in the CKKW-L algorithm at $n_{\mathrm{max}}^{}=2$ and the $0$, $1$, $2$ and $3$-parton production cross sections contribute to the 2-jet events in the CKKW-L algorithm at $n_{\mathrm{max}}^{}=3$~\footnote{The contribution of the $0$ and $1$-parton cross sections to the the 2-jet events is less than $5\%$, since the merging scale is set as eq.~(\ref{ME-cut-value}).}. \\
The panels show that the model A algorithm produces softer distributions than the CKKW-L algorithm, and that the CKKW-L at $n_{\mathrm{max}}^{}=2$ and the CKKW-L at $n_{\mathrm{max}}^{}=3$ produce consistent distributions.
Since the two algorithms calculate the Sudakov form factors in the very different ways, the differences in the $p_T^{}$ distributions are not unexpected. The model A algorithm contains a tuning parameter $C_{\mathrm{match}}^{}$ in eq.~(\ref{C-match-parameter}). In addition to this, the parton shower starting scale, which is defined as in eq.~(\ref{starting-scale}), is not uniquely determined. Therefore, we may tune the algorithm by using these parameters so that the $p_T^{}$ distributions become consistent with the experimental data, before predicting $\Delta \phi$ distributions. This issue is discussed again in Section~\ref{sec:conclusion}. \\
\begin{figure}[t]
\centering
\includegraphics[scale=0.52]{Dy3p5.pdf}
\includegraphics[scale=0.52]{Dy2p5.pdf}
\caption{\small
The normalized differential cross section as a function of $\Delta \phi$ with the rapidity separation $|y_1^{}-y_2^{}|>3.5$ (upper two panels) and $|y_1^{}-y_2^{}|>2.5$ (lower two panels) in the 2-jet events. The correspondence between the curves and the merging algorithms are shown inside the panels.}
\label{figure:dphi-dist}
\end{figure}
In Figure~\ref{figure:dphi-dist} the $\Delta \phi$ distributions are shown. The normalized differential cross section as a function of $\Delta \phi$ with the rapidity separation $|y_1^{}-y_2^{}|>3.5$ and that as a function of $\Delta \phi$ with the rapidity separation $|y_1^{}-y_2^{}|>2.5$ are shown in the upper two panels and the lower two panels, respectively. The correspondence between the curves and the merging algorithms are shown inside the panels. \\
In the left panels of Figure~\ref{figure:dphi-dist}, the CKKW-L at $n_{\mathrm{max}}^{}=2$ and the CKKW-L at $n_{\mathrm{max}}^{}=3$ are compared. The difference between the two predictions, particularly in the upper left panel, can be explained from a loss of the correlation due to the contribution from the leading logarithmic (LL) parton shower in the CKKW-L at $n_{\mathrm{max}}^{}=2$~\cite{Hagiwara:2015tva}. \\
In the right panels of Figure~\ref{figure:dphi-dist}, the model A algorithm and the CKKW-L at $n_{\mathrm{max}}^{}=3$ are compared.
It is shown that the model A algorithm produces a consistent prediction with the CKKW-L at $n_{\mathrm{max}}^{}=3$, despite that only the LO $t\bar{t}+2$-parton cross section contributes to the events in the model A algorithm.
Since the LO accuracy of $\Delta \phi$ is guaranteed in the model A algorithm, this observation confirms the finding in ref.~\cite{Hagiwara:2015tva} that the LO accuracy of $\Delta \phi$ in the 2-jet events cannot be fully retained due to the contribution from the LL parton shower in the CKKW-L at $n_{\mathrm{max}}^{}=2$ and it can be retained by merging the LO $t\bar{t}+3$-parton cross section additionally i.e. the CKKW-L at $n_{\mathrm{max}}^{}=3$.
It should be worth emphasizing that, in the model A algorithm, the LO $t\bar{t}+3$-parton cross section never contributes the $\Delta \phi$ distributions in Figure~\ref{figure:dphi-dist}, since it contributes exclusively to the 3-jet events.
\section{Summary and discussion}\label{sec:conclusion}
In this paper, a new tree level merging algorithm which guarantees the leading order (LO) accuracy of angular correlations between jets is proposed.
The new algorithm has been constructed with a goal of eliminating the ambiguity in the accuracy of jets, namely it is not necessarily clear whether the kinematics of a jet constructed by clustering partons in final state after a merging procedure has the LO accuracy or not. This ambiguity potentially exists in merging algorithm, because their virtue is smooth combination of LO cross sections and the leading logarithmic (LL) parton shower.
The new algorithm requires that $n$-jet events are generated exclusively according to the LO $n$-parton production cross section and each of the $n$-jet is close to each of the $n$-parton in terms of the jet measure. As a result, angular correlations between the $n$-jet should follow those between the $n$-parton and thus the LO accuracy of them is robust. Furthermore, as long as the $n$-jet events are exclusively under consideration, only the LO $n$-parton production cross section is needed and hence event generation is efficient. \\
The $t\bar{t}$ production in proton-proton collisions is simulated and then the distributions of the azimuthal angle differences $\Delta \phi=\phi_1^{}-\phi_2^{}$ between the two highest transverse momentum $p_T^{}$ jets with large rapidity separations, namely $|y_1^{}-y_2^{}|>3.5$ and $|y_1^{}-y_2^{}|>2.5$ in addition to $y_1^{} \times y_2^{} < 0$ i.e. the VBF cuts, are studied by using the 2-jet events exclusively. The new algorithm is evaluated by comparing its predictions with the predictions of one of well-established merging algorithms, the CKKW-L algorithm. \\
It has been observed that the new algorithm produces a consistent prediction with the CKKW-L at $n_{\mathrm{max}}^{}=3$, where $n_{\mathrm{max}}^{}$ denotes the maximal number of partons produced by the LO cross section, despite that only the LO $t\bar{t}+2$-parton cross section contributes to the events in the mew algorithm.
Since the LO accuracy of $\Delta \phi$ is guaranteed in the new algorithm, this observation confirms the previous finding that the LO accuracy of $\Delta \phi$ in the 2-jet events cannot be fully retained due to contribution from the leading logarithmic (LL) parton shower in the CKKW-L at $n_{\mathrm{max}}^{}=2$ and it can be retained by merging the LO $t\bar{t}+3$-parton cross section additionally i.e. in the CKKW-L at $n_{\mathrm{max}}^{}=3$. \\
In the numerical studies of this work, only the $\Delta \phi$ distributions in the $t\bar{t}+2$-jet events are studied and only the CKKW-L algorithm with the PYTHIA8 parton shower is used as one of other merging algorithms to validate the new algorithm. Therefore, the observation in this work is limited to this situation, obviously.
However, it is always the case that other merging algorithms including the CKKW-L algorithm potentially have the ambiguity in the accuracy of jets, dependently of a process, an observable and a shower model.
In contrast, the new algorithm does not have the ambiguity and always guarantees the LO accuracy of angular correlations between jets, independently of a process, an observable and a shower model. \\
There are basically two parameters which cannot be determined uniquely in the new algorithm, namely $C_{\mathrm{match}}^{}$ and the definition of the parton shower starting scale.
This fact might be seen as a weak point in the algorithm. However merging algorithms and parton shower programs are just models after all.
A promising approach will be to tune the algorithm together with a parton shower model by using these parameters so that the $p_T^{}$ and the rapidity distributions of jets in a given process become consistent with the data at first and then make the predictions of observables measuring angular correlations between the jets.
\section*{Acknowledgments}
The work of J.N. is supported by the Institutional Strategy of the University of T\"ubingen (DFG, ZUK 63). |
1308.4439 | \section{Introduction}
Let ${\bf a}_0,{\bf a}_1,\dots,{\bf a}_N\in{\mathbb Z}^n$ and put ${\bf a}_j = (a_{j1},\dots,a_{jn})$.
For each $j=0,\dots,N$, let $\hat{\bf a}_j = (1,{\bf a}_j)\in{\mathbb Z}^{n+1}$ and put
$A=\{\hat{\bf a}_j\}_{j=0}^N$. We let $x_0,\dots,x_n$ be the coordinates on~${\mathbb R}^{n+1}$, so
that $\hat{\bf a}_{j0} = 1$ while $\hat{\bf a}_{jk} = {\bf a}_{jk}$ for $k=1,\dots,n$. Let $L$ be the
module of relations among the $\hat{\bf a}_j$:
\[ L=\bigg\{l=(l_0,\dots,l_N)\in{\mathbb Z}^{N+1}\mid \sum_{j=0}^N l_j\hat{\bf a}_j = {\bf 0}\bigg\}. \]
We consider the $A$-hypergeometric system with parameter $-\hat{\bf a}_0$. This is the system of
partial differential equations in variables $\lambda_0,\dots,\lambda_N$ consisting of the operators
\[ \Box_l = \prod_{l_i>0} \bigg(\frac{\partial}{\partial \lambda_i}\bigg)^{l_i} - \prod_{l_i<0}
\bigg(\frac{\partial}{\partial \lambda_i}\bigg)^{-l_i} \]
for $l\in L$ and the operators
\[ Z_i = \begin{cases} \sum_{j=0}^N a_{ji}\lambda_j\frac{\partial}{\partial \lambda_j} + a_{0i} &
\text{for $i=1,\dots,n$,} \\
\sum_{j=0}^N \lambda_j\frac{\partial}{\partial \lambda_j} + 1 & \text{for $i=0$.} \end{cases} \]
Let $L_+ = \{ l\in L\mid l_i\geq 0\;\text{for $i=1,\dots,N$}\}.$ Taking $v=(-1,0,\dots,0)$ in
\cite[Eq.~(3.36)]{SST} and applying \cite[Proposition 3.4.13]{SST} shows that this system has a
formal solution $\lambda_0^{-1}\Phi(\lambda)$, where
\begin{equation}
\Phi(\lambda) = \sum_{l\in L_+} \frac{(-1)^{-l_0}(-l_0)!}{l_1!\cdots l_N!}\prod_{i=0}^N \lambda_i^{l_i}.
\end{equation}
Since $\sum_{i=0}^N l_i=0$ for $l\in L$, we can rewrite this as
\begin{equation}
\Phi(\lambda) = \sum_{l\in L_+} \frac{(-1)^{\sum_{i=1}^N l_i} (l_1+\cdots+l_N)!}{l_1!\cdots l_N!}
\prod_{i=1}^N \bigg(\frac{\lambda_i}{\lambda_0}\bigg)^{l_i},
\end{equation}
which shows that the coefficients of this series lie in ${\mathbb Z}$. This implies that for each
prime number $p$, the series $\Phi(\lambda)$ converges $p$-adically on the set
\[ {\mathcal D} = \{(\lambda_0,\dots,\lambda_N)\in\Omega^{N+1}\mid |\lambda_i/\lambda_0|<1
\text{ for }i=1,\dots,N\}, \]
(where $\Omega = $ completion of an algebraic closure of ${\mathbb Q}_p$) and $\Phi(\lambda)$ takes
unit values there. In particular, the ratio $\Phi(\lambda)/\Phi(\lambda^p)$ is an analytic function
on ${\mathcal D}$ and takes unit values there.
{\bf Remark:} Rewrite the series $\Phi(\lambda)$ according to powers of $-\lambda_0$:
\begin{equation}
\Phi(\lambda) = \sum_{k=0}^\infty \bigg(\sum_{\substack{l_1,\dots,l_N\in{\mathbb Z}_{\geq 0}\\
l_1\hat{\bf a}_1+\cdots+l_N\hat{\bf a}_N = k\hat{\bf a}_0}} \binom{k}{l_1,\dots,l_N}\lambda_1^{l_1}\cdots
\lambda_N^{l_N}\bigg)(-\lambda_0)^{-k}.
\end{equation}
It is easy to see that the coefficient of $(-\lambda_0)^{-k}$ in this expression is the coefficient
of $x^{k{\bf a}_0}$ in the Laurent polynomial $\big(\sum_{i=1}^N \lambda_ix^{{\bf a}_i}\big)^k$. In
particular, if ${\bf a}_0 = 0$, then it is the constant term of this Laurent polynomial. The series
$\Phi(\lambda)$ may thus be specialized to the hypergeometric series considered in Samol and van
Straten\cite{SvS}.
Define a truncation of $\Phi(\lambda)$ by
\[ \Phi_1(\lambda) = \sum_{\substack{l\in L_+\\ l_1+\cdots+l_N\leq p-1}} \frac{(-1)^{\sum_{i=1}^N l_i}
(l_1+\cdots+l_N)!}{l_1!\cdots l_N!} \prod_{i=1}^N \bigg(\frac{\lambda_i}{\lambda_0}\bigg)^{l_i} \]
and let
\[ {\mathcal D}_+ = \{(\lambda_0,\dots,\lambda_N)\mid |\lambda_i/\lambda_0|\leq 1\text{ for } i=1,
\dots,N \text{ and }
|\Phi_1(\lambda)|= 1\}. \]
Note that ${\mathcal D}_+$ properly contains ${\mathcal D}$.
Let~$\Delta$ be the convex hull of the set $\{{\bf a}_1,\dots,{\bf a}_N\}$. Our main result is the
following theorem.
\begin{theorem}
Suppose that ${\bf a}_0$ is the unique interior lattice point of $\Delta$. Then for every prime
number $p\neq 2$ the ratio $\Phi(\lambda)/\Phi(\lambda^p)$ extends to an analytic function
on~${\mathcal D}_+$.
\end{theorem}
{\bf Remark:} If we specialize $\lambda_1,\dots,\lambda_N$ to elements of $\Omega$ of absolute value
$\leq 1$, Equation~(1.3) allows us to regard $\Phi$ as a function of $t=-1/\lambda_0$ for $|t|<1$.
By Theorem~1.4, this function of $t$ continues analytically to the region where $|t|\leq 1$ and
$|\Phi_1(t)| =1$ (for $p\neq 2$). When $\lambda_1,\dots,\lambda_N\in{\mathbb Z}_p$, this result on
analytic continuation was proved recently by Mellit and Vlasenko\cite{MV} for all primes $p$. We
believe that the restriction $p\neq 2$ in Theorem 1.4 is an artifact of our method and that that
result is in fact true for all $p$.
{\bf Example:}(the Dwork family): Let $N=n$ and let ${\bf a}_i = (0,\dots,n,\dots,0)$, where the
$n$ occurs in the $i$-th position, for $i=1,\dots,n$. Let ${\bf a}_0 = (1,\dots,1)$. Then
\[ L_+=\{(-nl,l,\dots,l,)\mid l\in{\mathbb Z}_{\geq 0}\} \]
and
\[ \Phi(\lambda) = \sum_{l=0}^{\infty} \frac{(-1)^{nl}(nl)!}{(l!)^n} \bigg(\frac{\lambda_1\cdots
\lambda_n}{\lambda_0^n}\bigg)^l. \]
Then Theorem 1.4 implies that for every prime $p\neq 2$ the ratio $\Phi(\lambda)/\Phi(\lambda^p)$
extends to an analytic function on ${\mathcal D}_+$. We note that J.-D. Yu\cite{Y} has given a
treatment of analytic continuation for the Dwork family using a more cohomological approach.
In \cite{D2}, Dwork gave a contraction mapping argument to prove $p$-adic analytic continuation of a
ratio $G(\lambda)/G(\lambda^p)$ of normalized Bessel functions. In \cite{AS}, we modified Dwork's
approach to avoid computations of $p$-adic cohomology for exponential sums, which allowed us to
greatly extend his analytic continuation result. The proof we give here follows the method of
\cite{AS}.
Special values of the ratio $\Phi(\lambda)/\Phi(\lambda^p)$ are related to a unit root of the zeta
function of the hypersurface $\sum_{i=0}^N \lambda_ix^{{\bf a}_0} = 0$. We hope to return to this
connection in a future article. We believe that the methods of this paper will extend to complete
intersections as well.
For the remainder of this paper we assume that $p\neq 2$. This hypothesis is needed in Section~2 to
define the endomorphism $\alpha^*$ of the space $S$.
\section{Contraction mapping}
We begin by constructing a mapping $\beta$ on a certain space of formal series whose coefficients
are $p$-adic analytic functions. The hypothesis that ${\bf a}_0$ is the unique interior point of
$\Delta$ will imply that $\beta$ is a contraction mapping.
Let $\Omega$ be the completion of an algebraic closure of ${\bf Q}_p$ and put
\[ R = \bigg\{ \xi(\lambda) = \sum_{\nu\in({\bf Z}_{\geq 0})^N} c_\nu\bigg(\frac{\lambda_1}{\lambda_0}
\bigg)^{\nu_1}\cdots \bigg(\frac{\lambda_N}{\lambda_0}\bigg)^{\nu_N}\mid \text{$c_\nu\in\Omega$ and
$\{|c_\nu|\}_\nu$ is bounded}\bigg\} \]
Let $R'$ be the set of functions on ${\mathcal D}_+$ that are uniform limits of sequences of
rational functions in the $\lambda_i/\lambda_0$ that are defined on ${\mathcal D}_+$.
The series in the ring $R$ are convergent and bounded on ${\mathcal D}$ and $R'$ is a subring
of $R$. We define a norm on $R$ by setting
\[ |\xi| = \sup_{\lambda\in{\mathcal D}} |\xi(\lambda)|. \]
Note that for $\xi\in R'$ one has $\sup_{\lambda\in{\mathcal D}}|\xi(\lambda)| =
\sup_{\lambda\in{\mathcal D}_+} |\xi(\lambda)|$. Both $R$ and $R'$ are complete in this norm.
Let $C$ be the real cone generated by the elements of $A$, let $M=C\cap{\mathbb Z}A$, and let
$M^\circ\subset M$ be the subset consisting of those points that do not lie on any face of~$C$. Let
$\pi^{p-1}=-p$ and let $S$ be the $\Omega$-vector space of formal series
\[ S = \bigg\{\xi(\lambda,x) = \sum_{\mu\in M^{\circ}} \xi_\mu(\lambda) ({\pi}\lambda_0)^{-\mu_0}x^{-\mu} \mid
\text{$\xi_\mu(\lambda)\in R$ and $\{|\xi_\mu|\}_\mu$ is bounded}\bigg\}. \]
Let $S'$ be defined analogously with the condition ``$\xi_\mu(\lambda)\in R$'' being replaced by
``$\xi_\mu(\lambda)\in R'$''. Define a norm on $S$ by setting
\[ |\xi(\lambda,x)| = \sup_\mu\{|\xi_\mu|\}. \]
Both $S$ and $S'$ are complete under this norm.
Define $\theta(t) = \exp(\pi(t-t^p)) = \sum_{i=0}^\infty b_it^i$. One has
(Dwork\cite[Sec\-tion~4a)]{D1})
\begin{equation}
{\rm ord}\: b_i\geq \frac{i(p-1)}{p^2}.
\end{equation}
Let
\[ F(\lambda,x) = \prod_{i=0}^N\theta(\lambda_ix^{\hat{\bf a}_i}) = \sum_{\mu\in M} B_\mu(\lambda)x^\mu, \]
where
\[ B_\mu(\lambda) = \sum_{\nu\in({\bf Z}_{\geq 0})^{N+1}} B^{(\mu)}_\nu\lambda^\nu \]
with
\begin{equation}
B^{(\mu)}_\nu = \begin{cases} \prod_{i=0}^N b_{\nu_i} & \text{if $\sum_{i=0}^N \nu_i\hat{\bf a}_i = \mu$,}
\\ 0 & \text{if $\sum_{i=0}^N \nu_i\hat{\bf a}_i\neq\mu$.} \end{cases}
\end{equation}
The equation $\sum_{i=0}^N \nu_i\hat{\bf a}_i = \mu$ has only finitely many solutions $\nu\in({\mathbb
Z}_{\geq 0})^{N+1}$, so $B_\mu$ is a polynomial in the $\lambda_i$. Furthermore, all solutions of this
equation satisfy $\sum_{i=0}^N \nu_i = \mu_0$, so $B_\mu$ is homogeneous of degree $\mu_0$. We thus have
\begin{equation}
B_\mu(\lambda_0,\dots,\lambda_N) = \lambda_0^{\mu_0}B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/
\lambda_0).
\end{equation}
Let $\tilde{\pi}\in\Omega$ satisfy ${\rm ord}\;\tilde{\pi} = (p-1)/p^2$.
\begin{lemma}
One has $B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\in R'$ and
\[ |B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\leq |\tilde{\pi}^{\mu_0}|. \]
\end{lemma}
\begin{proof}
The first assertion is clear since $B_\mu$ is a polynomial. We have
\[ B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)=\sum_{\nu\in({\bf Z}_{\geq 0})^{N+1}}
B^{(\mu)}_\nu(\lambda_1/\lambda_0)^{\nu_1}\cdots(\lambda_N/\lambda_0)^{\nu_N} \]
Using (2.1) and (2.2) gives
\[ {\rm ord}\: B^{(\mu)}_\nu\geq\sum_{i=0}^N {\rm ord}\: b_{\nu_i}\geq \sum_{i=0}^N \frac{\nu_i(p-1)}{p^2}=
\mu_0\frac{p-1}{p^2}, \]
which implies the second assertion of the lemma.
\end{proof}
Using (2.3) we write
\begin{equation}
F(\lambda,x) = \sum_{\mu\in M} {B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\lambda_0^{\mu_0}
x^\mu.
\end{equation}
Let
\[ \xi(\lambda,x) = \sum_{\nu\in M^\circ} \xi_\nu(\lambda)({\pi}\lambda_0)^{-\nu_0}x^{-\nu}\in S. \]
We claim that the product $F(\lambda,x)\xi(\lambda^p,x^p)$ is well-defined as a formal series
in $x$. Formally we have
\[ F(\lambda,x)\xi(\lambda^p,x^p) = \sum_{\rho\in{\mathbb Z}^{n+1}} \zeta_\rho(\lambda)\lambda_0^{-\rho_0}
x^{-\rho}, \]
where
\begin{equation}
\zeta_\rho(\lambda) = \sum_{\substack{\mu\in M,\nu\in M^\circ \\ \mu-p\nu = -\rho}} {\pi}^{-\nu_0}
{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_\nu(\lambda^p).
\end{equation}
By Lemma 2.4, we have
\begin{equation}
|{\pi}^{-\nu_0}B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_\nu(\lambda^p)|\leq
|\tilde{\pi}^{\mu_0}\pi^{-\nu_0}|\cdot|\xi(\lambda,x)|.
\end{equation}
Since $\mu=p\nu-\rho$, we have
\begin{align}
{\rm ord}\;\tilde{\pi}^{\mu_0}\pi^{-\nu_0} &= (p\nu_0-\rho_0)\frac{p-1}{p^2} - \frac{\nu_0}{p-1}
\nonumber \\
&= \nu_0\bigg(\frac{p-1}{p} - \frac{1}{p-1}\bigg) - \rho_0\frac{p-1}{p^2}.
\end{align}
Since $(p-1)/p-1/(p-1)>0$ (we are using here our hypothesis that $p\neq 2$), this shows that
$\tilde{\pi}^{\mu_0}\pi^{-\nu_0}\to 0$ as $\nu\to\infty$, so the series (2.6) converges to an element
of~$R$. The same argument shows that if $\xi(\lambda,x)\in S'$, then the series (2.6) converges to
an element of $R'$.
Let $\gamma^\circ$ be the the truncation map
\[ \gamma^\circ\bigg(\sum_{\rho\in{\mathbb Z}^{n+1}} \zeta_\rho(\lambda)\lambda_0^{-\rho_0}x^{-\rho}\bigg) =
\sum_{\rho\in M^\circ} \zeta_\rho(\lambda)\lambda_0^{-\rho_0}x^{-\rho} \]
and define for $\xi(\lambda,x)\in S$
\begin{align*}
\alpha^*\big(\xi(\lambda,x)\big) &= \gamma^\circ\big(F(\lambda,x)\xi(\lambda^p,x^p)\big) \\
&= \sum_{\rho\in M^\circ}\zeta_\rho(\lambda)\lambda_0^{-\rho_0}x^{-\rho}.
\end{align*}
For $\rho\in M^\circ$ put $\eta_\rho(\lambda) = {\pi}^{\rho_0}\zeta_\rho(\lambda)$, so that
\begin{equation}
\alpha^*(\xi(\lambda,x)) = \sum_{\rho\in M^\circ} \eta_\rho(\lambda)({\pi}\lambda_0)^{-\rho_0}x^{-\rho}
\end{equation}
with (by (2.6))
\begin{equation}
\eta_\rho(\lambda) = \sum_{\substack{\mu\in M,\nu\in M^\circ\\ \mu-p\nu = -\rho}} {\pi}^{\rho_0-\nu_0}
{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_\nu(\lambda^p).
\end{equation}
\begin{proposition}
The map $\alpha^*$ is an endomorphism of $S$ and of $S'$, and for $\xi(\lambda,x)\in S$ we have
\begin{equation}
|\alpha^*(\xi(\lambda,x))|\leq |p|\cdot|\xi(\lambda,x)|.
\end{equation}
\end{proposition}
\begin{proof}
By (2.9), the proposition follows from the estimate
\[ |\eta_\rho(\lambda)|\leq |p|\cdot|\xi(\lambda,x)| \quad\text{for all $\rho\in M^\circ$.} \]
By (2.10), this estimate will follow from the estimate
\begin{equation}
|{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq |p|
\end{equation}
for all $\mu\in M$, $\nu\in M^\circ$, $\mu-p\nu=-\rho$.
Consider first the case $\mu_0\leq 2p-1$.
For $0\leq i\leq p-1$ we have $b_i=\pi^i/i!$, hence $|b_i| = |\pi^i|$. One checks easily that for
$p\leq i\leq 2p-1$ one has $|b_i|\leq |\pi^i|$. This implies by (2.2) that
$|B_\mu(\lambda)|\leq |\pi^{\mu_0}|$, thus
\[ |{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq
|\pi^{\rho_0-\nu_0+\mu_0}|. \]
Since $\mu=p\nu-\rho$ we have $\rho_0-\nu_0+\mu_0 = (p-1)\nu_0$, so
\begin{equation}
|{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq |p|^{\nu_0}.
\end{equation}
Since $\nu_0\geq 1$ for $\nu\in M^\circ$, Eq.~(2.14) implies (2.13) for $\mu_0\leq 2p-1$.
Now consider the case $\mu_0\geq 2p$. Lemma 2.4 implies that
\begin{equation}
|{\pi}^{\rho_0-\nu_0}B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq
|\tilde{\pi}^{\mu_0}\pi^{\rho_0-\nu_0}|.
\end{equation}
We have (using $\mu=p\nu-\rho$)
\begin{align*}
{\rm ord}\;\tilde{\pi}^{\mu_0}\pi^{\rho_0-\nu_0} &= (p\nu_0-\rho_0)\frac{p-1}{p^2} + (\rho_0-\nu_0)
\frac{1}{p-1} \\
&= \nu_0\bigg(\frac{p-1}{p}-\frac{1}{p-1}\bigg) + \rho_0\bigg(\frac{1}{p-1}-\frac{p-1}{p^2}\bigg).
\end{align*}
Since $\rho\in M^\circ$ we have $\rho_0\geq 1$, and since $\mu_0\geq 2p$ and $\mu=p\nu-\rho$ we must
have $\nu_0\geq 3$. It follows that
\begin{equation}
\nu_0\bigg(\frac{p-1}{p}-\frac{1}{p-1}\bigg) + \rho_0\bigg(\frac{1}{p-1}-\frac{p-1}{p^2}\bigg)\geq
3\bigg(\frac{p-1}{p}-\frac{1}{p-1}\bigg) + \bigg(\frac{1}{p-1}-\frac{p-1}{p^2}\bigg).
\end{equation}
This latter expression is $>1$ for $p\geq 5$, hence (2.15) implies (2.13) when $\mu_0\geq 2p$ and
$p\geq 5$.
Finally, suppose that $p=3$ and $\mu_0\geq 2p = 6$. By explicitly computing $b_i$, one checks that
$|b_i| = |\pi^i|$ for $i\leq 8$. This implies by (2.2) that $|B_\mu(\lambda)|\leq |\pi^{\mu_0}|$ for
$\mu_0\leq 8$, so~(2.14) holds in this case and we conclude as before that~(2.13) holds also. For
$i=9,10$, one has $|b_i| = |\pi^{i-4}|$, so $|B_\mu(\lambda)|\leq |\pi^{\mu_0-4}|$ for $\mu_0=9,10$. We
thus have
\[ |{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq
|\pi^{\rho_0-\nu_0+\mu_0-4}| \]
for $\mu_0=9,10$. Since $\mu=p\nu-\rho$, this gives
\[ |{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq |3^{\nu_0-2}|. \]
For $\mu_0=9,10$, $\mu=p\nu-\rho$ implies that $\nu_0\geq 4$, hence $|3^{\nu_0-2}|\leq |3^2|$
and~(2.13) holds.
One computes that $|b_{11}| = |\pi^9|$, so $|B_\mu(\lambda)|\leq |\pi^9|$ for $\mu_0=11$. This gives
(using $\mu=p\nu-\rho$)
\[ |{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq |3^{\nu_0-1}| = |3^3| \]
since $\nu_0\geq 4$ for $\mu_0=11$, so (2.13) holds in this case. Finally, if $\mu_0\geq 12$, then
$\mu=p\nu-\rho$ implies $\nu_0\geq 5$. When $p=3$, the left-hand side of (2.16) is $>1$ when
$\nu_0\geq 5$, so~(2.15) implies~(2.13) in this case.
\end{proof}
{\bf Remark:} It follows from the proof of Proposition 2.11 that equality can hold in (2.13) only if
$\nu_0=1$. And since
\[ {\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\in{\mathbb Q}_p(\pi,
\tilde{\pi})[\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0] \]
and ${\mathbb Q}_p(\pi,\tilde{\pi})$ is a discretely valued field, we conclude that there exists a
rational number $C$, $0<C<1$, such that
\begin{equation}
|{\pi}^{\rho_0-\nu_0}{B}_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq C|p|
\end{equation}
for all $\mu\in M$, $\nu\in M^\circ$, $\mu-p\nu=-\rho$, with $\nu_0>1$.
\begin{lemma}
Suppose ${\bf a}_0$ is the unique interior lattice point of $\Delta$. If
$\xi_{\hat{\bf a}_0}(\lambda)=0$, then $|\alpha^*(\xi(\lambda,x))|\leq C |p|\cdot|\xi(\lambda,x)|$.
\end{lemma}
\begin{proof}
Since ${\bf a}_0$ is the unique interior lattice point of $\Delta$, the point $\nu=\hat{\bf a}_0$ is
the unique element of~$M^\circ$ with $\nu_0=1$. So for $\nu\in M^\circ$, $\nu\neq\hat{\bf a}_0$, we
have $\nu_0\geq 2$. The assertion of the lemma then follows from (2.10) and (2.17).
\end{proof}
We examine the polynomial $B_{(p-1)\hat{\bf a}_0}(\lambda)$ to determine its relation to
$\Phi_1(\lambda)$. Let
\[ V = \bigg\{v=(v_0,\dots,v_N)\in ({\mathbb Z}_{\geq 0})^{N+1} \mid \sum_{i=0}^N v_i\hat{\bf a}_i =
(p-1)\hat{\bf a}_0\bigg\}. \]
From (2.2) we have
\[ B_{(p-1)\hat{\bf a}_0}(\lambda) = \sum_{v\in V} \bigg(\prod_{i=0}^N b_{v_i}\bigg)\lambda_0^{v_0}\cdots
\lambda_N^{v_N}. \]
For $v\in V$ we have $\sum_{i=0}^N v_i = p-1$ so $v_0=(p-1)-v_1-\cdots-v_N$. Furthermore,
$v_i\leq p-1$ for all $i$ so $b_{v_i} = \pi^{v_i}/v_i!$. And since $\pi^{p-1} = -p$, this implies
\[ B_{(p-1)\hat{\bf a}_0}(\lambda) = -p\lambda_0^{p-1}\sum_{v\in V}\frac{(\lambda_1/\lambda_0)^{v_1}\cdots
(\lambda_N/\lambda_0)^{v_N}}{(p-1-v_1-\ldots-v_N)!v_1!\cdots v_N!}. \]
It follows that
\[ p^{-1}B_{(p-1){\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0) = -\sum_{v\in V}
\frac{(\lambda_1/\lambda_0)^{v_1}\cdots (\lambda_N/\lambda_0)^{v_N}}
{(p-1-v_1-\ldots-v_N)!v_1!\cdots v_N!}, \]
a polynomial in the $\lambda_i/\lambda_0$ with $p$-integral coefficients.
\begin{lemma}
$\Phi_1(\lambda)\equiv p^{-1}B_{(p-1){\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0) \pmod{p}$.
\end{lemma}
\begin{proof}
The map $(v_0,\dots,v_N)\mapsto (-v_1-\dots-v_N,v_1,\dots,v_N)$ is a one-to-one correspondence from
$V$ to the elements $l\in L_+$ satisfying $l_1+\cdots+l_N\leq p-1$. The lemma then follows
immediately from the congruence
\[ -\frac{1}{(p-1-m)!}\equiv (-1)^m m!\pmod{p} \quad\text{for $0\leq m\leq p-1$}, \]
which is implied by the congruence $(p-1)!\equiv -1\pmod{p}$.
\end{proof}
\begin{corollary} The polynomial $B_{(p-1){\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)$ is
an invertible element of $R'$ with $|B_{(p-1){\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|
= |p|$.
\end{corollary}
\begin{proof}
The first assertion is an immediate consequence of Lemma 2.19. The assertion about the norm follows
from the fact that all the coefficients are divisible by~$p$ and the constant term equals $-p/(p-1)!$.
\end{proof}
Suppose that ${\bf a}_0$ is the unique interior lattice point of $\Delta$, so that
$\nu=\hat{\bf a}_0$ is the unique element of $M^\circ$ with $\nu_0=1$. From Equation (2.10) we have
\begin{equation}
\begin{split}
\eta_{\hat{\bf a}_0}(\lambda) &= \sum_{\substack{\mu\in M,\nu\in M^\circ\\ \mu=p\nu -\hat{\bf a}_0}} {\pi}^{1-\nu_0}
B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_\nu(\lambda^p) \\
& = B_{(p-1)\hat{\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_{\hat{\bf a}_0}(\lambda^p) \\
& \quad +\sum_{\substack{\mu\in M,\nu\in M^\circ\\ \mu=p\nu -\hat{\bf a}_0\\ \nu_0\geq 2}} {\pi}^{1-\nu_0}
B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)\xi_\nu(\lambda^p).
\end{split}
\end{equation}
\begin{lemma}
Suppose that ${\bf a}_0$ is the unique interior lattice point of $\Delta$.
If $\xi_{\hat{\bf a}_0}(\lambda)$ is an invertible element of $R$ (resp.~$R'$) and
$|\xi_{\hat{\bf a}_0}(\lambda)|= |\xi(\lambda,x)|$, then $\eta_{\hat{\bf a}_0}(\lambda)$ is also an
invertible element of $R$ (resp.~$R'$) and $|\eta(\lambda,x)| = |\eta_{\hat{\bf a}_0}(\lambda)| =
|p|\cdot |\xi_{\hat{\bf a}_0}(\lambda)|$.
\end{lemma}
\begin{proof}
By Corollary 2.20 we have that $B_{(p-1)\hat{\bf a}_0}(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)
\xi_{\hat{\bf a}_0}(\lambda^p)$ is an invertible element of norm $|p|\cdot|\xi_{\hat{\bf a}_0}(\lambda)|$.
By hypothesis, we have
\[ |\xi_\nu(\lambda^p)|\leq |\xi_{\hat{\bf a}_0}(\lambda)|\quad\text{for all $\nu\in M^\circ$.} \]
Equation (2.17) with $\rho = \hat{\bf a}_0$ gives
\begin{equation}
|{\pi}^{1-\nu_0} B_\mu(1,\lambda_1/\lambda_0,\dots,\lambda_N/\lambda_0)|\leq C|p|
\end{equation}
for all $\mu\in M$, $\nu\in M^\circ$, $\mu-p\nu=-\hat{\bf a}_0$, $\nu_0\geq 2$ and some constant $C$,
$0<C<1$.
Equation (2.21) then implies that $\eta_{\hat{\bf a}_0}(\lambda)$ is invertible and that
\[ |\eta_{\hat{\bf a}_0}(\lambda)|=|p|\cdot |\xi_{\hat{\bf a}_0}(\lambda)|. \]
Equation (2.12) then implies that $|\eta(\lambda,x)| = |p|\cdot|\xi_{\hat{\bf a}_0}(\lambda)|$.
\end{proof}
Suppose that ${\bf a}_0$ is the unique interior lattice point of $\Delta$. Put
\[ T = \{\xi(\lambda,x)\in S\mid \text{$\xi_{\hat{\bf a}_0}(\lambda) = 1$ and $|\xi(\lambda,x)| = 1$}\}
\]
and put $T' = T\cap S'$. It follows from Lemma~2.22 that if $\xi(\lambda,x)\in T$, then
$\eta_{\hat{\bf a}_0}(\lambda)$ is invertible. We may thus define for $\xi(\lambda,x)\in T$
\[ \beta(\xi(\lambda,x)) = \frac{\alpha^*(\xi(\lambda,x))}{\eta_{\hat{\bf a}_0}(\lambda)}. \]
Lemma 2.22 also implies that
\[ \bigg|\frac{\alpha^*(\xi(\lambda,x))}{\eta_{\hat{\bf a}_0}(\lambda)}\bigg|= 1, \]
so $\beta(T)\subseteq T$. It is then clear that $\beta(T')\subseteq T'$.
\begin{proposition}
Suppose that ${\bf a}_0$ is the unique interior lattice point of $\Delta$. Then the operator
$\beta$ is a contraction mapping on the complete metric space $T$. More precisely, for $C$ as in
$(2.17)$, if $\xi^{(1)}(\lambda,x),\xi^{(2)}(\lambda,x)\in T$, then
\[ |\beta\big(\xi^{(1)}(\lambda,x)\big)-\beta\big(\xi^{(2)}(\lambda,x)\big)|\leq C|\xi^{(1)}(\lambda,x)-
\xi^{(2)}(\lambda,x)|. \]
\end{proposition}
\begin{proof}
We have (in the obvious notation)
\begin{equation*}
\begin{split}
\beta\big(\xi^{(1)}(\lambda,x)\big)-\beta\big(\xi^{(2)}(\lambda,x)\big) &= \frac{\alpha^*\big(
\xi^{(1)}(\lambda,x)\big)}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)} - \frac{\alpha^*\big(\xi^{(2)}(\lambda,x)
\big)}{\eta^{(2)}_{\hat{\bf a}_0}(\lambda)} \\
&= \frac{\alpha^*\big(\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x)\big)}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)}
\\
& \qquad - \alpha^*\big(\xi^{(2)}(\lambda,x)\big)\frac{\eta^{(1)}_{\hat{\bf a}_0}(\lambda) -
\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}.
\end{split}
\end{equation*}
By Lemmas 2.18 and 2.22 we have
\[ \bigg|\frac{\alpha^*(\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x))}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)}
\bigg| \leq
C|\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x))|. \]
Since $\eta^{(1)}_{\hat{\bf a}_0}(\lambda)-\eta^{(2)}_{\hat{\bf a}_0}(\lambda)$ is the coefficient of
$x^{-\hat{\bf a}_0}$ in $\alpha^*\big(\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x)\big)$, we have
\[ |\eta^{(1)}_{\hat{\bf a}_0}(\lambda)-\eta^{(2)}_{\hat{\bf a}_0}(\lambda)|\leq
|\alpha^*\big(\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x)\big)|\leq C |p|\cdot |\xi^{(1)}(\lambda,x)-
\xi^{(2)}(\lambda,x))| \]
by Lemma 2.18. We have $|\eta^{(1)}_{\hat{\bf a}_0}(\lambda)\eta^{(2)}_{\hat{\bf a}_0}(\lambda)|=|p^2|$ by
Lemma~2.22, so by (2.12)
\[ \bigg| \alpha^*\big(\xi^{(2)}(\lambda,x)\big)\frac{\eta^{(1)}_{\hat{\bf a}_0}(\lambda) -
\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}\bigg|
\leq
C|\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x))|. \]
This establishes the proposition.
\begin{comment}
One has $|\tilde{\pi}^{3(p-1)}/p^2|<1$ for $p\geq 7$, so the proposition is established in that case.
For $p=5$, one has by (2.16) and (2.20) that
\[ |\eta^{(1)}_{\hat{\bf a}_0}(\lambda)-\eta^{(2)}_{\hat{\bf a}_0}(\lambda)|\leq
|\tilde{\pi}^{12}|\cdot |\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x))|. \]
It follows that
\[ \bigg| \alpha^*(\xi^{(2)}(\lambda,x))\frac{\eta^{(1)}_{\hat{\bf a}_0}(\lambda) -
\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}{\eta^{(1)}_{\hat{\bf a}_0}(\lambda)\eta^{(2)}_{\hat{\bf a}_0}(\lambda)}
\bigg|\leq |\tilde{\pi}^{16}/5^2|\cdot
|\xi^{(1)}(\lambda,x)-\xi^{(2)}(\lambda,x))|, \]
and since $|\tilde{\pi}^{16}/5^2|<1$ the proposition is proved for $p=5$ also.
\end{comment}
\end{proof}
By a well-known theorem, Proposition 2.24 implies that $\beta$ has a unique fixed point in $T$. And
since $\beta$ is stable on $T'$, that fixed point must lie in $T'$. This fixed point of $\beta$ is
related to a certain eigenvector of $\alpha^*$. Suppose that $\xi(\lambda,x)\in S$ is an
eigenvector of $\alpha^*$, say,
\[ \alpha^*\big(\xi(\lambda,x)\big) = \kappa\xi(\lambda,x), \]
with $\xi_{\hat{\bf a}_0}(\lambda)$ invertible and $|\xi(\lambda,x)| = |\xi_{\hat{\bf a}_0}(\lambda|$.
Then $\xi(\lambda,x)/\xi_{\hat{\bf a}_0}(\lambda)\in T$.
\begin{lemma}
With the above notation, $\xi(\lambda,x)/\xi_{\hat{\bf a}_0}(\lambda)$ is the unique fixed point
of~$\beta$, hence $\xi(\lambda,x)/\xi_{\hat{\bf a}_0}(\lambda)\in T'$. In particular,
$\xi_\rho(\lambda)/\xi_{\hat{\bf a}_0}(\lambda)\in R'$ for all $\rho\in M^\circ$.
\end{lemma}
\begin{proof}
We have
\begin{equation}
\alpha^*\bigg(\frac{\xi(\lambda,x)}{\xi_{\hat{\bf a}_0}(\lambda)}\biggr) = \frac{\alpha^*\big(
\xi(\lambda,x)\big)}{\xi_{\hat{\bf a}_0}(\lambda^p)} = \bigg(\frac{\kappa\xi_{\hat{\bf a}_0}(\lambda)}
{\xi_{\hat{\bf a}_0}(\lambda^p)}\bigg) \frac{\xi(\lambda,x)}{\xi_{\hat{\bf a}_0}(\lambda)}.
\end{equation}
By the definition of $\beta$, this implies the result.
\end{proof}
\begin{corollary}
With the above notation, $\xi_{\hat{\bf a}_0}(\lambda)/\xi_{\hat{\bf a}_0}(\lambda^p) \in R'$.
\end{corollary}
\begin{proof}
Since $\alpha^*$ is stable on $S'$, Lemma 2.25 implies that the right-hand side of~(2.26) lies in
$S'$. Since the coefficient of $(\pi\lambda_0)^{-1}x^{-\hat{\bf a}_0}$ on the right-hand side of (2.26)
is $\kappa\xi_{\hat{\bf a}_0}(\lambda)/\xi_{\hat{\bf a}_0}(\lambda^p)$, the result follows.
\end{proof}
In the next section we find the fixed point of $\beta$ by finding the corresponding eigenvector of
$\alpha^*$. This eigenvector
will be constructed from solutions of the $A$-hypergeometric system; in particular, we shall have
$\xi_{\hat{\bf a}_0}(\lambda) = \Phi(\lambda)$, so Corollary~2.27 will imply Theorem~1.4.
\section{Fixed point}
We begin with a preliminary calculation. Consider the series
\[ G(\lambda_0,x) = \sum_{l=0}^{\infty} (-1)^l l!(\pi\lambda_0x^{\hat{\bf a}_0})^{-1-l}. \]
It satisfies the equations
\begin{equation}
\gamma_-\bigg(x_i\frac{\partial}{\partial x_i}-\pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0}\bigg)
G(\lambda_0,x) = 0
\end{equation}
for $i=0,1,\dots,n$, where $\gamma_-$ is the operator on series defined by
\[\gamma_-\bigg(\sum_{l=-\infty}^{\infty} c_l(\pi\lambda_0x^{\hat{\bf a}_0})^{-1-l}\bigg) = \sum_{l=0}^{\infty}
c_l(\pi\lambda_0x^{\hat{\bf a}_0})^{-1-l}. \]
It is straightforward to check that the series $\sum_{l=0}^{\infty} c_l(\pi\lambda_0
x^{\hat{\bf a}_0})^{-1-l}$ that satisfy the operators $\gamma_-\circ(x_i\partial/\partial x_i-
\pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0})$ form a one-dimensional space.
Consider the series
\[ H(\lambda_0,x)=\gamma_-\big(\exp\big(\pi(\lambda_0x^{\hat{\bf a}_0}-\lambda_0^px^{p\hat{\bf a}_0})\big)
G(\lambda_0^p,x^p)\big). \]
Formally one has
\[ x_i\frac{\partial}{\partial x_i}-\pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0} = \exp(\pi\lambda_0
x^{\hat{\bf a}_0})\circ x_i\frac{\partial}{\partial x_i}\circ \frac{1}{\exp(\pi\lambda_0x^{\hat{\bf a}_0})} \]
and
\[ \gamma_-\circ\bigg(x_i\frac{\partial}{\partial x_i} - \pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0}
\bigg)\circ \gamma_- =
\gamma_-\circ\bigg(x_i\frac{\partial}{\partial x_i} - \pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0}
\bigg). \]
It follows that
\begin{multline*}
\gamma_-\bigg(x_i\frac{\partial}{\partial x_i} - \pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0}\bigg)
H(\lambda_0,x) = \\
\gamma_-\bigg(\exp(\pi\lambda_0x^{\hat{\bf a}_0})x_i\frac{\partial}{\partial x_i}\bigg(G(\lambda_0^p,x^p)/
\exp(\pi\lambda_0^px^{p\hat{\bf a}_0})\bigg)\bigg).
\end{multline*}
Eq.~(3.1) implies that this latter expression equals 0, i.~e., $H(\lambda_0,x)$ also satisfies the
operators $\gamma_-\circ(x_i\partial/\partial x_i-\pi\lambda_0\hat{\bf a}_{0i}x^{\hat{\bf a}_0})$. Since
the solutions of this operator form a one-dimensional space, we have $H(\lambda_0,x) =
cG(\lambda_0,x)$ for some constant $c$.
We determine the constant $c$ by comparing the coefficients of $(\pi\lambda_0x^{\hat{\bf a}_0})^{-p}$ in
$G(\lambda_0,x)$ and $H(\lambda_0,x)$. The coefficient of $(\pi\lambda_0x^{\hat{\bf a}_0})^{-p}$ in
$G(\lambda_0,x)$ is $(-1)^{p-1}(p-1)!$.
A calculation shows that the coefficient of $(\pi\lambda_0x^{\hat{\bf a}_0})^{-p}$ in $H(\lambda_0,x)$ is
\[ -p\sum_{l=0}^\infty b_{pl}(-1)^ll!\pi^{-l}. \]
By Boyarsky\cite[Eq.~(5.6)]{B} (or see \cite[Lemma~3]{A}) this equals $-p\Gamma_p(p)=(-1)^{p+1} p!$,
where $\Gamma_p$ denotes the $p$-adic gamma function. It follows that the constant $c$ satisfies
the equation
\[ (-1)^{p+1}p!=c(-1)^{p-1}(p-1)!, \]
so $c=p$ and we conclude that
\begin{equation}
H(\lambda_0,x) = p G(\lambda_0,x).
\end{equation}
We now consider the series
\[ \xi(\lambda,x) = \gamma^\circ\bigg(G(\lambda_0,x)\prod_{i=1}^N \exp(\pi\lambda_i
x^{\hat{\bf a}_i})\bigg), \]
where $\gamma^\circ$ is as defined in Section~2. A calculation shows that
\begin{multline*}
G(\lambda_0,x)\prod_{i=1}^N \exp(\pi\lambda_ix^{\hat{a}_i}) = \\
\sum_{\rho\in{\mathbb Z}^{n+1}} \bigg( \sum_{\substack{l_0,\dots,l_N\in{\mathbb Z}_{\geq 0}\\
l_1\hat{\bf a}_1+\cdots+l_N\hat{\bf a}_N-(l_0+1)\hat{\bf a}_0 = -\rho}} \frac{(-1)^{l_0}l_0!
\pi^{-1-l_0+\sum_{i=1}^N l_i}
\lambda_1^{l_1}\cdots\lambda_N^{l_N}\lambda_0^{-l_0-1}}{l_1!\cdots l_N!}\bigg)x^{-\rho}.
\end{multline*}
It follows that we can write $\xi(\lambda,x)$ as
\begin{equation}
\xi(\lambda,x) = \sum_{\rho\in M^\circ} \xi_\rho(\lambda)(\pi\lambda_0)^{-\rho_0}x^{-\rho},
\end{equation}
where
\begin{multline}
\xi_\rho(\lambda) = \\
\sum_{\substack{l_0,\dots,l_N\in{\mathbb Z}_{\geq 0}\\ l_1\hat{\bf a}_1+\cdots+l_N\hat{\bf a}_N-(l_0+1)\hat{\bf a}_0 = -\rho}}
\frac{(-1)^{\rho_0-1+\sum_{i=1}^N l_i}(\rho_0-1+\sum_{i=1}^N l_i)!}{l_1!\cdots l_N!} \prod_{i=1}^N
\bigg(\frac{\lambda_i}{\lambda_0}\bigg)^{l_i}.
\end{multline}
{\bf Remark:} One can check that the series $\lambda_0^{-\rho_0}\xi_\rho(\lambda)$ is a solution of
the $A$-hyper\-geometric system with parameter $-\rho$ (although we shall not make use of that fact
here). Thus the coefficients of powers of $x$ in the series $\xi(\lambda,x)$ form a ($p$-adically
normalized) family of ``contiguous'' $A$-hypergeometric functions.
The series $\xi_\rho(\lambda)$ have coefficients in ${\mathbb Z}$ for all $\rho\in M^\circ$, hence
$\xi(\lambda,x)\in S$ and $|\xi(\lambda,x)|\leq 1$.
Note that $\xi_{\hat{\bf a}_0}(\lambda) = \Phi(\lambda)$, where $\Phi(\lambda)$ is defined by (1.1).
In particular, $\xi_{\hat{\bf a}_0}(\lambda)$ takes unit values on ${\mathcal D}$, hence
$|\xi_{\hat{\bf a}_0}(\lambda)|=1$ and $\xi_{\hat{\bf a}_0}(\lambda)$ is an invertible element of $R$. To
show that $\xi(\lambda,x)$ satisfies the hypothesis of Lemma~2.25, it remains only to establish the
following result.
\begin{lemma}
We have $\alpha^*\big(\xi(\lambda,x)\big) = p\xi(\lambda,x)$.
\end{lemma}
\begin{proof}
From the definitions we have
\[ \alpha^*(\xi(\lambda,x)) = \gamma^\circ\bigg(F(\lambda,x)\cdot\gamma^\circ\bigg(G(\lambda_0^p,x^p)
\prod_{i=1}^N \exp(\pi\lambda_i^px^{p\hat{\bf a}_i})\bigg)\bigg). \]
One checks that $\big(\text{mult.\ by $F(\lambda,x)$}\big)\circ\gamma^\circ = \gamma^\circ\circ
\big(\text{mult.\ by $F(\lambda,x)$}\big)$. Furthermore, by definition we have $F(\lambda,x) =
\prod_{i=0}^N \exp(\pi\lambda_ix^{\hat{\bf a}_i})/\exp(\pi\lambda_i^px^{p\hat{\bf a}_i})$. It follows that
\[ \alpha^*(\xi(\lambda,x)) = \gamma^\circ\bigg(\gamma^\circ\bigg(\frac{\exp(\pi
\lambda_0x^{\hat{\bf a}_0})}{\exp(\pi\lambda_0^px^{p\hat{\bf a}_0})}G(\lambda_0^p,x^p)\bigg)\prod_{i=1}^N
\exp(\pi\lambda_ix^{\hat{\bf a}_i})\bigg). \]
By (3.2) we finally have
\[ \alpha^*(\xi(\lambda,x)) = \gamma^\circ\bigg(pG(\lambda_0,x)\prod_{i=1}^N \exp(\pi\lambda_i
x^{\hat{\bf a}_i})\bigg) = p\xi(\lambda,x). \]
\end{proof}
\begin{proof}[Proof of Theorem $1.4$]
We have shown that the series $\xi(\lambda,x)$ given by (3.3) and~(3.4) satisfies the hypotheses of
Lemma 2.25. Since $\xi_{\hat{\bf a}_0}(\lambda)=\Phi(\lambda)$, Theorem~1.4 follows from Corollary~2.27.
\end{proof} |
2203.06172 | \section*{Acknowledgement}
\label{sec.ack}
We thank Yi Zhu, Hang Zhang, Haichen Shen, Mu Li, and Alexander Smola for their help with this work. This work was partially supported by NSF Award PFI:BIC-1632051 and Amazon AWS Machine Learning Research Award.
\section{A list of standard augmentation space} \label{app:augmentation_space}
\begin{table*}[!h]
\centering
{
\begin{tabular}{@{}l|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{Operation}} &
\multicolumn{1}{c}{\textbf{Magnitude}}
\\
\midrule
Identity & - \\
ShearX & [-0.3, 0.3] \\
ShearY & [-0.3, 0.3] \\
TranslateX & [-0.45, 0.45] \\
TranslateY & [-0.45, 0.45] \\
Rotate & [-30, 30] \\
AutoContrast & - \\
Invert & -\\
Equalize & - \\
Solarize & [0, 256] \\
Posterize & [4, 8] \\
Contrast & [0.1, 1.9] \\
Color & [0.1, 1.9] \\
Brightness & [0.1, 1.9] \\
Sharpness & [0.1, 1.9] \\
Flips & -\\
Cutout & 16 (60) \\
Crop & - \\
\bottomrule
\end{tabular}
}
\caption{List of operations in the search space and the corresponding range of magnitudes in the standard augmentation space. Note that some operations do not use magnitude parameters. We add flip and crop to the search space which were found in the default augmentation pipeline in previous works. Flips operates by randomly flipping the images with $50\%$ probability. In line with previous works, crop denotes pad-and-crop and resize-and-crop transforms for CIFAR10/100 and ImageNet respectively. We set Cutout magnitude to 16 for CIFAR10/100 dataset to be the same as the Cutout in the default augmentation pipeline. We set Cutout magnitude to 60 pixels for ImageNet which is the upper limit of the magnitude used in AA~\citep{cubuk2019autoaugment}.}
\label{tab:magnitude}
\end{table*}
\newpage
\section{The distribution of magnitudes for CIFAR-10/100} \label{app:cifar}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{magnitude_cifar_dist.png}
\vspace{-2mm}
\caption{The distribution of discrete magnitudes of each augmentation transformation in each layer of the policy for CIFAR-10/100. The x-axis represents the discrete magnitudes and the y-axis represents the probability. The magnitude is discretized to $12$ levels with each transformation having its own range. A large absolute value of the magnitude corresponds to high transformation intensity.
Note that we do not show identity, autoContrast, invert, equalize, flips{}, Cutout{} and crop{} because they do not have intensity parameters.}
\vspace{-5mm}
\label{fig:magnitude_cifar}
\end{center}
\end{figure}
\newpage
\section{The distribution of magnitudes for ImageNet} \label{app:imagenet}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{magnitude_imagenet_dist.png}
\vspace{-2mm}
\caption{The distribution of discrete magnitudes of each augmentation transformation in each layer of the policy for ImageNet. The x-axis represents the discrete magnitudes and the y-axis represents the probability. The magnitude is discretized to $12$ levels with each transformation having its own range. A large absolute value of the magnitude corresponds to high transformation intensity.
Note that we do not show identity, autoContrast, invert, equalize, flips{}, Cutout{} and crop{} because they do not have intensity parameters.}
\vspace{-5mm}
\label{fig:magnitude_imagenet}
\end{center}
\end{figure}
\newpage
\section{Hyperparameters for Batch Augmentation} \label{app:BA}
The performance of BA is sensitive to the training settings \citep{fort2021drawing,wightman2021resnet}. Therefore, we conduct a grid search on the learning rate, weight decay and number of epochs for TA and \DAA{} with Batch Augmentation. The best found parameters are summarized in Table~\ref{tab:parameters} in Appendix. We did not tune the hyperparameters of AdvAA~\citep{zhang2019adversarial} since AdvAA claims to be adaptive to the training process.
\begin{table*}[!h]
\centering
\resizebox{0.99\textwidth}{!}
{
\begin{tabular}{@{}l|l|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{Dataset}} &
\multicolumn{1}{c}{\textbf{Augmentation}} &
\multicolumn{1}{c}{\textbf{Model}} &
\multicolumn{1}{c}{\textbf{Batch Size}} &
\multicolumn{1}{c}{\textbf{Learning Rate}} &
\multicolumn{1}{c}{\textbf{Weight Decay}} &
\multicolumn{1}{c}{\textbf{Epoch}}
\\
\midrule
\multirow{2}{*}{CIFAR-10} & TA (Wide) & WRN-28-10 & 128 $\times$ 8 & 0.2 & 0.0005 & 100 \\
& DeepAA & WRN-28-10 & 128 $\times$ 8 & 0.2 & 0.001 & 100 \\
\multirow{2}{*}{CIFAR-100} & TA (Wide) & WRN-28-10 & 128 $\times$ 8 & 0.4 & 0.0005 & 35 \\
& DeepAA & WRN-28-10 & 128 $\times$ 8 & 0.4 & 0.0005 & 35 \\
\bottomrule
\end{tabular}
}
\caption{Model hyperparameters of Batch Augmentation on CIFAR10/100 for TA (Wide) and DeepAA. Learning rate, weight decay and number of epochs are found via grid search.}
\label{tab:parameters}
\end{table*}
\newpage
\section{Comparison of data augmentation policy} \label{app:compare_policy}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{comparison_with_other_augmentation-cropped.pdf}
\vspace{-2mm}
\caption{Comparison of the policy of DeepAA and some publicly available augmentaiotn policy found by other methods including AA, FastAA and DADA on CIFAR-10. Since the compared methods have varied numbers of augmentation layers, we cumulate the probability of each operation over all the augmentation layers. Thus, the cumulative probability can be larger than 1. For AA, Fast AA and DADA, we add additional 1.0 probability to flip, Cutout and Crop, since they are applied by default. In addition, we normalize the magnitude to the range [-5, 5], and use color to distinguish different magnitudes.}\label{fig:compare_policy}
\vspace{-5mm}
\end{center}
\end{figure}
\section{Deep AutoAugment}
\subsection{Overview}
Data augmentation can be viewed as a process of filling missing data points in the dataset with the same data distribution~\citep{hataya2020faster}.
By augmenting a single data point multiple times, we expect the resulting data distribution to be close to the full dataset under a certain type of transformation. For example, by augmenting a single image with proper color jittering, we obtain a batch of augmented images which has similar distribution of lighting conditions as the full dataset. As the distribution of augmented data gets closer to the full dataset, the gradient of the augmented data should be steered towards a batch of original data sampled from the dataset.
In \DAA{}, we formulate the search of the data augmentation policy as a regularized gradient matching problem, which manages to steer the gradient to a batch of original data by augmenting a single image multiple times.
Specifically, we construct the augmented training batch by augmenting a single training data point multiple times following the augmentation policy. We construct a validation batch by sampling a batch of original data from the validation set. We expect that by augmentation, the gradient of augmented training batch can be steered towards the gradient of the validation batch.
To do so, we search for data augmentation that maximizes the cosine similarity between the gradients of the validation data and the augmented training data.
The intuition is that an effective data augmentation should preserve data distribution \citep{chen2020group} where the distribution of the augmented images should align with the distribution of the validation set such that the training gradient direction is close to the validation gradient direction.
Another challenge for augmentation policy search is that the search space can be prohibitively large with deep augmentation layers $(K \geq 5)$. This was not a problem in previous works, where the augmentation policies is shallow ($K \leq 2$). For example, in AutoAugment~\cite{cubuk2019autoaugment}, each sub-policy contains $K=2$ transformations to be applied sequentially, and the search space of AutoAugment contains $16$ image operations and $10$ discrete magnitude levels. The resulting number of combinations of transformations in AutoAugment is roughly $(16\times10)^2=25,600$, which is handled well in previous works. However, when discarding the default augmentation pipeline and searching for data augmentations from scratch, it requires deeper augmentation layers in order to perform well. For a data augmentation with $K=5$ sequentially applied transformations, the number of sub-policies is $(16 \times 10)^5 \approx 10^{11}$, which is prohibitively large for the following two reasons.
First, it becomes less likely to encounter a good policy by exploration as good policies become more sparse on high dimensional search space. Second, the dimension of parameters in the policy also grows with $K$, making it more computational challenging to optimize.
To tackle this challenge, we propose to build up the full data augmentation by progressively stacking augmentation layers, where each augmentation layer is optimized on top of the data distribution transformed by all previous layers. This avoids sampling sub-policies from such a large search space, and the number of parameters of the policy is reduced from $|\mathbb{T}|^{K}$ to $\mathbb{T}$ for each augmentation layer.
\subsection{Search Space}
Let $\mathbb{O}$ denote the set of augmentation operations (\emph{e.g.} identity, rotate, brightness), $m$ denote an operation magnitude in the set $\mathbb{M}$, and $x$ denote an image sampled from the space $\mathcal{X}$. We define the set of transformations as the set of operations with a fixed magnitude as $\mathbb{T}:=\{t| t = o(\cdot~;~m), ~ o \in \mathbb{O} ~\text{and}~ m \in \mathbb{M}\}$. Under this definition, every $t$ is a map $t: \mathcal{X} \rightarrow \mathcal{X}$, and there are $|\mathbb{T}|=|\mathbb{M}|\cdot|\mathbb{O}|$ possible transformations.
In previous works \citep{cubuk2019autoaugment,lim2019fast, li2020differentiable,hataya2020faster}, a data augmentation policy $\mathcal{P}$ consists of several sub-policies. As explained above, the size of candidate sub-policies grows exponentially with depth $K$. Therefore, we propose a practical method that builds up the full data augmentation by progressively stacking augmentation layers. The final data augmentation policy hence consists of $K$ layers of sequentially applied policy $\mathcal{P}=\{\mathcal{P}_1,\cdots, \mathcal{P}_K \}$, where policy $\mathcal{P}_k$ is optimized conditioned on the data distribution augmented by all previous $(k-1)$ layers of policies. Thus we write the policy as a conditional distribution $\mathcal{P}_k:=p_{\theta_k}(n|\{\mathcal{P}_1,\cdots, \mathcal{P}_{k-1} \})$ where $n$ denotes the indices of transformations in $\mathbb{T}$. For the purpose of clarity, we use a simplified notation as $p_{\theta_k}$ to replace $p_{\theta_k}(n|\{\mathcal{P}_1,\cdots, \mathcal{P}_{k-1} \})$.
\begin{comment}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/progressive_aug.png}
\vspace{-2mm}
\caption{\DAA{} builds a multi-layer data augmentation pipeline from scratch by adding augmentation layers one at a time, where each layer is conditioned on the distribution of data transformed by all previous layers.}
\vspace{-5mm}
\label{fig:progressive}
\end{center}
\end{figure}
\end{comment}
\label{sec.approach}
\subsection{Augmentation Policy Search via Regularized Gradient Matching}
Assume that a single data point $x$ is augmented multiple times following the policy $p_\theta$. The resulting average gradient of such augmentation is denoted as $g(x, \theta)$, which is a function of data $x$ and policy parameters $\theta$. Let $v$ denote the gradients of a batch of the original data.
We optimize the policy by maximizing the cosine similarity between the gradients of the augmented data and a batch of the original data as follows:
\vspace{-4mm}
\begin{align}
\theta & = \arg \max_\theta ~ \text{cosineSimilarity}(v,g(x, \theta)) \\ \nonumber
& =
\arg \max_\theta ~ \frac{v^T \cdot {g}(x, \theta)}{\rVert v \rVert \cdot \rVert {g}(x, \theta) \rVert} \;
\vspace{4mm}
\end{align}
where $\rVert \cdot \rVert$ denotes the L2-norm. The parameters of the policy can be updated via gradient ascent:
\begin{align} \label{eq:max_cosine_similarity_orig}
\theta \leftarrow \theta + \eta \nabla_{\theta} ~ \text{cosineSimilarity}(v,g(x,\theta)),
\end{align}
where $\eta$ is the learning rate.
\subsubsection{Policy Search for One layer}
We start with the case where the data augmentation policy only contains a single augmentation layer, \textit{i.e.}, $\mathcal{P}=\{p_{\theta}\}$.
Let $L(x; w)$ denote the classification loss of data point $x$ where $w \in \mathbb{R}^D$ represents the flattened weights of the neural network. Consider applying augmentation on a single data point $x$ following the distribution $p_{\theta}$. The resulting averaged gradient can be calculated analytically by averaging all the possible transformations in $\mathbb{T}$ with the corresponding probability $p(\theta)$:
\begin{align} \label{eq:avg_grad}
g(x; \theta) &= \sum_{n=1}^{|\mathbb{T}|}p_{\theta}(n) \nabla_{w} L(t_n(x); w) \\ \nonumber
&= G(x) \cdot p_{\theta}
\end{align}
where $G(x)=\left[ \nabla_w L(t_1(x); w), \cdots, \nabla_w L(t_{|\mathbb T|}(x); w) \right]$ is a $D \times |\mathbb T|$ Jacobian matrix, and $p_{\theta} = \left[ p_{\theta}(1), \cdots, p_{\theta}(|\mathbb{T}|) \right]^T$ is a $|\mathbb T |$ dimensional categorical distribution. The gradient w.r.t. the cosine similarity in Eq.~(\ref{eq:max_cosine_similarity_orig}) can be derived as:
\begin{align} \label{eq:cosine_gradient}
\nabla_{\theta}~\text{cosineSimilarity}(v,{g(x;\theta)}) &=
\nabla_{\theta} p_{\theta} \cdot r
\end{align}
where
\begin{align} \label{eq:jvp}
r = G(x)^T \left( \frac{v}{\lVert {g}(\theta) \rVert} - \frac{v^T {g}(\theta)}{\lVert {g}(\theta) \rVert^2} \cdot \frac{{g}(\theta)}{\lVert {g}(\theta) \rVert} \right)
\end{align}
which can be interpreted as a reward for each transformation. Therefore, $p_\theta \cdot r$ in Eq.(\ref{eq:cosine_gradient}) represents the average reward under policy $p_\theta$.
\subsubsection{Policy Search for Multiple layers}
The above derivation is based on the assumption that $g(\theta)$ can be computed analytically by Eq.(\ref{eq:avg_grad}). However, when $K \geq 2$, it becomes impractical to compute the average gradient of the augmented data given that the search space dimensionality grows exponentially with $K$. Consequently, we need to average the gradient of all $|\mathbb{T}|^{K}$ possible sub-policies.
To reduce the parameters of the policy to $\mathbb{T}$ for each augmentation layer, we propose to incrementally stack augmentations based on the data distribution transformed by all the previous augmentation layers. Specifically, let $\mathcal{P}=\{\mathcal{P}_1,\cdots, \mathcal{P}_K \}$ denote the $K$-layer policy. The policy $\mathcal{P}_k$ modifies the data distribution on top of the data distribution augmented by the previous $(k-1)$ layers. Therefore, the policy at the $k^{th}$ layer is a distribution $\mathcal{P}_k = p_{\theta_k}(n)$ conditioned on the policies $\{\mathcal{P}_1, \cdots, \mathcal{P}_{k-1}\}$ where each one is a $|\mathbb{T}|$-dimensional categorical distribution. Given that, the Jacobian matrix at the $k^{th}$ layer can be derived by averaging over the previous $(k-1)$ layers of policies as follows:
\begin{equation}
\begin{split}
G(x)^k= \sum_{n_{k-1}=1}^{|\mathbb{T}|} \cdots \sum_{n_1=1}^{|\mathbb{T}|} p_{\theta_{k-1}}(n_{k-1}) \cdots p_{\theta_1}(n_1) [ & \nabla_w L((t_1 \circ t_{n_{k-1}}\cdots \circ t_{n_1})(x); w), \cdots, \\
& \nabla_w L((t_{|\mathbb T|} \circ t_{n_{k-1}} \circ \cdots \circ t_{n_1})(x); w) ]
\end{split}
\end{equation}
where $G^k$ can be estimated via the Monte Carlo method as:
\begin{equation}\label{eq:G_k_monte_carlo}
\begin{split}
\tilde{G}^k(x) = \sum_{\tilde{n}_{k-1} \sim p_{\theta_k}} \cdots \sum_{\tilde{n}_1 \sim p_{\theta_1}} [ & \nabla_w L((t_1 \circ t_{\tilde{n}_{k-1}}\cdots \circ t_{\tilde{n}_1})(x); w), \cdots , \\
& \nabla_w L((t_{|\mathbb T|} \circ t_{\tilde{n}_{k-1}} \circ \cdots \circ t_{\tilde{n}_1})(x); w) ]
\end{split}
\end{equation}
where $\tilde{n}_{k-1} \sim p_{\theta_{k-1}}(n), ~\cdots~, \tilde{n}_{1} \sim p_{\theta_{1}}(n)$.
The average gradient at the $k^{th}$ layer can be estimated by the Monte Carlo method as:
\begin{equation} \label{eq:g_monte_carlo}
\tilde{g}(x;\theta_k) = \sum_{\tilde{n}_k \sim p_{\theta_k}} \cdots \sum_{\tilde{n}_1 \sim p_{\theta_1}} \nabla_{w} L \left((t_{\tilde{n}_k} \circ \cdots \circ t_{\tilde{n}_1})(x); w \right).
\end{equation}
Therefore, the reward at the $k^{th}$ layer is derived as:
\begin{align} \label{eq:jvp}
\tilde{r}^k(x) = \left( \tilde{G}^k(x) \right)^T \left( \frac{v}{\lVert \tilde{g}_k(x;\theta_k) \rVert} - \frac{v^T \tilde{g}_k(x;\theta_k)}{\lVert \tilde{g}_k(x;\theta_k) \rVert^2} \cdot \frac{\tilde{g}_k(x;\theta_k)}{\lVert \tilde{g}_k(x;\theta_k) \rVert} \right).
\end{align}
To prevent the augmentation policy from overfitting, we regularize the optimization by avoiding optimizing towards the direction with high variance. Thus, we penalize the average reward with its standard deviation as
\begin{equation} \label{eq:mean_minus_std}
r^k = E_{x}\{ \tilde{r}^k(x) \} - c \cdot \sqrt{E_{x}\{ (\tilde{r}^k(x) - E_{x}\{ \tilde{r}^k(x) \})^2 \}} \; ,
\end{equation}
where we use 16 randomly sampled images to calculate the expectation. The hyperparameter $c$ controls the degree of regularization, which is set to $1.0$. With such regularization, we prevent the policy from converging to the transformations with high variance.
Therefore the parameters of policy $\mathcal{P}_k$ ($k \geq 2$) can be updated as:
\begin{align} \label{eq:max_cosine_similarity}
\theta \leftarrow \theta_k + \eta \nabla_{\theta_k} ~ \text{cosineSimilarity}(v,g(\theta_k))
\end{align}
where
\begin{align} \label{eq:gradient_cosine_similarity2}
\nabla_{\theta}~\text{cosineSimilarity}(v,{g^k(x;\theta)}) &=
\nabla_{\theta} p_{\theta_k} \cdot r^k.
\end{align}
\section{Conclusion}
\label{sec.conclusion}
\vspace{-2mm}
In this work, we present Deep AutoAugment (\DAA{}), a multi-layer data augmentation search method that finds deep data augmentation policy without using any hand-picked default transformations.
We formulate data augmentation search as a regularized gradient matching problem, which maximizes the gradient similarity between augmented data and original data along the direction with low variance.
Our experimental results show that \DAA{} achieves strong performance without using default augmentations, indicating that regularized gradient matching is an effective search method for data augmentation policies.
\section{Checklist} \label{impact}
\section{Introduction} \label{intro}
\begin{wrapfigure}{t}{3in}
\begin{minipage}{3in}
\centering
\vspace{-6mm}
\resizebox{3in}{!}{\includegraphics{compare_methods-cropped.pdf}}
\vspace{-5mm}
\caption{{\small (A) Existing automated data augmentation methods with shallow augmentation policy followed by hand-picked transformations. (B) DeepAA with deep augmentation policy with no hand-picked transformations.}}
\vspace{-2mm}
\label{fig:prob}
\end{minipage}
\end{wrapfigure}
Data augmentation (DA) is a powerful technique for machine learning since it effectively regularizes the model by increasing the number and the diversity of data points~\citep{goodfellow2016deep, zhang2016understanding}.
A large body of data augmentation transformations has been proposed \citep{inoue2018data, zhang2017mixup, devries2017improved, yun2019cutmix, hendrycks2019augmix,yan2020improve} to improve model performance.
While applying a set of well-designed augmentation transformations could help yield considerable performance enhancement especially in image recognition tasks, manually selecting high-quality augmentation transformations and determining how they should be combined still require strong domain expertise and prior knowledge of the dataset of interest.
With the recent trend of automated machine learning (AutoML), data augmentation search flourishes in the image domain \citep{cubuk2019autoaugment,cubuk2020randaugment,ho2019population,lim2019fast,hataya2020faster,li2020differentiable,liu2021direct}, which yields significant performance improvement over hand-crafted data augmentation methods.
Although data augmentation policies in previous works \citep{cubuk2019autoaugment,cubuk2020randaugment,ho2019population,lim2019fast,hataya2020faster,li2020differentiable} contain multiple transformations applied sequentially, only one or two transformations of each sub-policy are found through searching whereas the rest transformations are hand-picked and applied by default in addition to the found policy (Figure \ref{fig:prob}(A)). From this perspective, we believe that previous automated methods are \textit{not entirely automated} as they are still built upon hand-crafted default augmentations.
In this work, we propose \deepautoaugment{} (\DAA{}), a multi-layer data augmentation search method which aims to remove the need of hand-crafted default transformations (Figure \ref{fig:prob}(B)). \DAA{} fully automates the data augmentation process by searching a deep data augmentation policy on an expanded set of transformations that includes the widely adopted search space and the default transformations (\emph{e.g.} flips{}, Cutout{}, crop{}). We formulate the search of data augmentation policy as a regularized gradient matching problem by maximizing the cosine similarity of the gradients between augmented data and original data with regularization. To avoid exponential growth of dimensionality of the search space when more augmentation layers are used, we incrementally stack augmentation layers based on the data distribution transformed by all the previous augmentation layers.
\begin{comment}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\linewidth]{figures/compare_methods_cropped.pdf}
\vspace{-2mm}
\caption{(A) Existing automated data augmentation methods (shallow augmentation search layers followed by hand-picked transformations). (B) Deep AutoAugment (deep augmentation search stages without hand-picked transformations).}
\vspace{-5mm}
\label{fig:prob}
\end{center}
\end{figure}
\end{comment}
We evaluate the performance of \DAA{} on three datasets -- CIFAR-10, CIFAR-100, and ImageNet -- and compare it with existing automated data augmentation search methods including AutoAugment (AA) \citep{cubuk2019autoaugment}, PBA \citep{ho2019population}, Fast AutoAugment (FastAA) \citep{lim2019fast}, Faster AutoAugment (Faster AA) \citep{hataya2020faster}, DADA \citep{li2020differentiable}, RandAugment (RA) \citep{cubuk2020randaugment}, UniformAugment (UA) \citep{lingchen2020uniformaugment}, TrivialAugment (TA) \citep{muller2021trivialaugment}, and Adversarial AutoAugment (AdvAA) \citep{zhang2019adversarial}.
Our results show that, without any default augmentations, \DAA{} achieves the best performance compared to existing automatic augmentation search methods on CIFAR-10, CIFAR-100 on Wide-ResNet-28-10 and ImageNet on ResNet-50 and ResNet-200 with standard augmentation space and training procedure.
We summarize our main contributions below:
\vspace{-1mm}
\begin{itemize}
\item
We propose \deepautoaugment{} (\DAA{}), a fully automated data augmentation search method that finds a multi-layer data augmentation policy from scratch.
\item
We formulate such multi-layer data augmentation search as a regularized gradient matching problem. We show that maximizing cosine similarity along the direction of low variance is effective for data augmentation search when augmentation layers go deep.
\item
We address the issue of exponential growth of the dimensionality of the search space when more augmentation layers are added by incrementally adding augmentation layers based on the data distribution transformed by all the previous augmentation layers.
\item
Our experiment results show that, without using any default augmentations, \DAA{} achieves stronger performance compared with prior works.
\end{itemize}
\section{Related Work}
\label{sec.related}
\textbf{Automated Data Augmentation.}
Automating data augmentation policy design has recently emerged as a promising paradigm for data augmentation. The pioneer work on automated data augmentation was proposed in AutoAugment \citep{cubuk2019autoaugment}, where the search is performed under reinforcement learning framework. AutoAugment requires to train the neural network repeatedly, which takes thousands of GPU hours to converge. Subsequent works \citep{lim2019fast, li2020differentiable, liu2021direct} aim at reducing the computation cost. Fast AutoAugment \citep{lim2019fast} treats data augmentation as inference time density matching which can be implemented efficiently with Bayesian optimization. Differentiable Automatic Data Augmentation (DADA) \citep{li2020differentiable} further reduces the computation cost through a reparameterized Gumbel-softmax distribution \citep{jang2016categorical}. RandAugment \citep{cubuk2020randaugment} introduces a simplified search space containing two interpretable hyperparameters, which can be optimized simply by grid search. Adversarial AutoAugment (AdvAA) \citep{zhang2019adversarial} searches for the augmentation policy in an adversarial and online manner. It also incorporates the concept of Batch Augmentaiton \citep{berman2019multigrain,hoffer2020augment}, where multiple adversarial policies run in parallel. Although many automated data augmentation methods have been proposed, the use of default augmentations still imposes strong domain knowledge.
\textbf{Gradient Matching}. Our work is also related to gradient matching. In \citep{du2018adapting}, the authors showed that the cosine similarity between the gradients of different tasks provides a signal to detect when an auxiliary loss is helpful to the main loss. In \citep{wang2020optimizing}, the authors proposed to use cosine similarity as the training signal to optimize the data usage via weighting data points. A similar approach was proposed in \citep{muller2021loop}, which uses the gradient inner product as a per-example reward for optimizing data distribution and data augmentation under the reinforcement learning framework. Our approach also utilizes the cosine similarity to guide the data augmentation search. However, our implementation of cosine similarity is different from the above from two aspects: we propose a Jacobian-vector product form to backpropagate through the cosine similarity, which is computational and memory efficient and does not require computing higher order derivative; we also propose a sampling scheme that effectively allows the cosine similarity to increase with added augmentation stages.
\section{Experiments and Analysis}
\textbf{Benchmarks and Baselines.} We evaluate the performance of \DAA{} on three standard benchmarks: CIFAR-10, CIFAR-100, ImageNet, and compare it against a baseline based on standard augmentations (\textit{i.e.}, flip left-righ, pad-and-crop for CIFAR-10/100, and Inception-style preprocesing~\citep{szegedy2015going} for ImageNet) as well as nine existing automatic augmentation methods including (1) AutoAugment (AA) \citep{cubuk2019autoaugment}, (2) PBA \citep{ho2019population}, (3) Fast AutoAugment (Fast AA) \citep{lim2019fast}, (4) Faster AutoAugment \citep{hataya2020faster}, (5) DADA \citep{li2020differentiable}, (6) RandAugment (RA) \citep{cubuk2020randaugment}, (7) UniformAugment (UA) \citep{lingchen2020uniformaugment}, (8) TrivialAugment (TA) \citep{muller2021trivialaugment}, and (9) Adversarial AutoAugment (AdvAA) \citep{zhang2019adversarial}.
\textbf{Search Space.} We set up the operation set $\mathbb{O}$ to include $16$ commonly used operations (identity, shear-x, shear-y, translate-x, translate-y, rotate, solarize, equalize, color, posterize, contrast, brightness, sharpness, autoContrast, invert, Cutout{}) as well as two operations (\textit{i.e.}, flips{} and crop{}) that are used as the default operations in the aforementioned methods.
Among the operations in $\mathbb{O}$, $11$ operations are associated with magnitudes.
We then discretize the range of magnitudes into $12$ uniformly spaced levels and treat each operation with a discrete magnitude as an independent transformation. Therefore, the policy in each layer is a 139-dimensional categorical distribution corresponding to $|\mathbb{T}|=139$ $\{$operation, magnitude$\}$ pairs. The list of operations and the range of magnitudes in the standard augmentation space are summarized in Appendix~\ref{app:augmentation_space}.
\subsection{Performance on CIFAR-10 and CIFAR-100}
\textbf{Policy Search.}
Following \citep{cubuk2019autoaugment}, we conduct the augmentation policy search based on Wide-ResNet-40-2~\citep{zagoruyko2016wide}. We first train the network on a subset of $4,000$ randomly selected samples from CIFAR-10. We then progressively update the policy network parameters $\theta_k ~ (k=1,2,\cdots,K)$ for $512$ iterations for each of the $K$ augmentation layers. We use the Adam optimizer~\citep{kingma2014adam} and set the learning rate to $0.025$ for policy updating.
\textbf{Policy Evaluation.}
Using the publicly available repository of Fast AutoAugment~\citep{lim2019fast}, we evaluate the found augmentation policy on both CIFAR-10 and CIFAR-100 using Wide-ResNet-28-10 and Shake-Shake-2x96d models. The evaluation configurations are kept consistent with that of Fast AutoAugment.
\textbf{Results.}
Table~\ref{tab:cifar} reports the Top-1 test accuracy on CIFAR-10/100 for Wide-ResNet-28-10 and Shake-Shake-2x96d, respectively. The results of \DAA{} are the average of four independent runs with different initializations. We also show the $95\%$ confidence interval of the mean accuracy. As shown, \DAA{} achieves the best performance compared against previous works using the standard augmentation space. \emph{
Note that TA(Wide) uses a wider (stronger) augmentation space on this dataset.}
\begin{table*}[h]
\centering
\resizebox{1.0\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c|c|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{}} & \multicolumn{1}{c}{\textbf{Baseline}} & \multicolumn{1}{c}{\textbf{AA}} &
\multicolumn{1}{c}{\textbf{PBA}} &
\multicolumn{1}{c}{\textbf{FastAA}} &
\multicolumn{1}{c}{\textbf{FasterAA}} &
\multicolumn{1}{c}{\textbf{DADA}} &
\multicolumn{1}{c}{\textbf{RA}} &
\multicolumn{1}{c}{\textbf{UA}} &
\multicolumn{1}{c}{\textbf{TA(RA)}} &
\multicolumn{1}{c}{\textbf{TA(Wide) \footnotemark}} &
\multicolumn{1}{c}{\textbf{\DAA{}}}
\\
\midrule
\textbf{CIFAR-10} & & & & & & & & & & & \\
WRN-28-10 & 96.1 & 97.4 & 97.4 & 97.3 & 97.4 & 97.3 & 97.3 & 97.33 & 97.46 & 97.46 & \textbf{97.56} $\pm$ 0.14 \\
Shake-Shake \footnotesize{(26 2x96d)} & 97.1 & 98.0 & 98.0 & 98.0 & 98.0 & 98.0 & 98.0 & 98.1 & 98.05 & 98.21 & \textbf{98.11} $\pm$ 0.12 \\
\midrule
\textbf{CIFAR-100} & & & & & & & & & & \\
WRN-28-10 & 81.2 & 82.9 & 83.3 & 82.7 & 82.7 & 82.5 & 83.3 & 82.82 & 83.54 & 84.33 & \textbf{84.02} $\pm$ 0.18 \\
Shake-Shake \footnotesize{(26 2x96d)} & 82.9 & 85.7 & 84.7 & 85.1 & 85.0 & 84.7 & - & - & - & 86.19 & \textbf{85.19} $\pm$ 0.28 \\
\bottomrule
\end{tabular}
}
\caption{{\small Top-1 test accuracy on CIFAR-10/100 for Wide-ResNet-28-10 and Shake-Shake-2x96d. The results of \DAA{} are averaged over four independent runs with different initializations. The $95\%$ confidence interval is denoted by $\pm$.}}
\label{tab:cifar}
\end{table*}
\vspace{-3mm}
\subsection{Performance on ImageNet}
\textbf{Policy Search.}
We conduct the augmentation policy search based on ResNet-18~\citep{he2016deep}. We first train the network on a subset of $200,000$ randomly selected samples from ImageNet for $30$ epochs. We then use the same settings as in CIFAR-10 for updating the policy parameters.
\textbf{Policy Evaluation.}
We evaluate the performance of the found augmentation policy on ResNet-50 and ResNet-200 based on the public repository of Fast AutoAugment~\citep{lim2019fast}. The parameters for training are the same as the ones of~\citep{lim2019fast}. In particular, we use step learning rate scheduler with a reduction factor of $0.1$, and we train and evaluate with images of size 224x224.
\footnotetext{On CIFAR-10/100, TA (Wide) uses a wider (stronger) augmentation space, while the other methods including TA (RA) uses the standard augmentation space.}
\textbf{Results.}
The performance on ImageNet is presented in Table~\ref{tab:shakeshake}. As shown, \DAA{} achieves the best performance compared with previous methods without the use of default augmentation pipeline. In particular, \DAA{} performs better on larger models (\emph{i.e.} ResNet-200), as the performance of \DAA{} on ResNet-200 is the best within the $95\%$ confidence interval. \emph{Note that while we train DeepAA using the image resolution (224$\times$224), we report the best results of RA and TA, which are trained with a larger image resolution (244$\times$224) on this dataset.}
\addtocounter{footnote}{-1}
\begin{table*}[h]
\centering
\resizebox{1.0\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{}} & \multicolumn{1}{c}{\textbf{Baseline}} &
\multicolumn{1}{c}{\textbf{AA}} &
\multicolumn{1}{c}{\textbf{Fast AA}} &
\multicolumn{1}{c}{\textbf{Faster AA}} &
\multicolumn{1}{c}{\textbf{DADA}} &
\multicolumn{1}{c}{\textbf{RA}} &
\multicolumn{1}{c}{\textbf{UA}} &
\multicolumn{1}{c}{\textbf{TA(RA)}\footnotemark} &
\multicolumn{1}{c}{\textbf{TA(Wide)\footnotemark}} &
\multicolumn{1}{c}{\textbf{\DAA{}}}
\\
\midrule
ResNet-50 & 76.3 & 77.6 & 77.6 & 76.5 & 77.5 & 77.6 & 77.63 & 77.85 & 78.07 & \textbf{78.30 $\pm$ 0.14} \\
ResNet-200 & 78.5 & 80.0 & 80.6 & - & - & - & 80.4 & - & - & \textbf{81.32 $\pm$ 0.17} \\
\bottomrule
\end{tabular}
}
\caption{{\small Top-1 test accuracy (\%) on ImageNet for ResNet-50 and ResNet-200. The results of \DAA{} are averaged over four independent runs with different initializations. The $95\%$ confidence interval is denoted by $\pm$.}}
\label{tab:shakeshake}
\end{table*}
\vspace{-2mm}
\subsection{Performance with Batch Augmentation}
Batch Augmentation (BA) is a technique that draws multiple augmented instances of the same sample in one mini-batch. It has been shown to be able to improve the generalization performance of the network~\citep{berman2019multigrain,hoffer2020augment}. AdvAA~\citep{zhang2019adversarial} directly searches for the augmentation policy under the BA setting whereas for TA and DeepAA, we apply BA with the same augmentation policy used in Table~\ref{tab:cifar}. Note that since the performance of BA is sensitive to the hyperparameters~\citep{fort2021drawing}, we have conducted a grid search on the hyperparameters of both TA and \DAA{} (details are included in Appendix~\ref{app:BA}).
As shown in Table~\ref{tab:BA}, after tuning the hyperparameters, the performance of TA (Wide) using BA is already better than the reported performance in the original paper. The performance of \DAA{} with BA outperforms that of both AdvAA and TA (Wide) with BA.
\begin{table*}[h]
\centering
\resizebox{0.75\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{} &
\multicolumn{1}{l}{\textbf{AdvAA}} &
\multicolumn{1}{c}{
\begin{tabular}[c]{@{}c@{}}\textbf{TA(Wide)}\\ (original paper) \end{tabular}} &
\multicolumn{1}{c}{
\begin{tabular}[c]{@{}c@{}}\textbf{TA(Wide)} \\ (ours) \end{tabular}} &
\multicolumn{1}{c}{\textbf{\DAA{}}}
\\
\midrule
CIFAR-10 & 98.1 $\pm$ 0.15 & 98.04 $\pm$ 0.06 & 98.06 $\pm$ 0.23 & \textbf{98.21 $\pm$ 0.14} \\
CIFAR-100 & 84.51 $\pm$ 0.18 & 84.62 $\pm$ 0.14 & 85.40 $\pm$ 0.15 & \textbf{85.61 $\pm$ 0.17} \\
\bottomrule
\end{tabular}
}
\caption{{\small Top-1 test accuracy (\%) on CIFAR-10/100 dataset with WRN-28-10 with Batch Augmentation (BA), where eight augmented instances were drawn for each image. The results of \DAA{} are averaged over four independent runs with different initializations. The $95\%$ confidence interval is denoted by $\pm$.}}
\label{tab:BA}
\end{table*}
\addtocounter{footnote}{-1}
\footnotetext{TA (RA) achieves 77.55\% top-1 accuracy with image resolution 224$\times$224.}
\addtocounter{footnote}{1}
\footnotetext{TA (Wide) achieves 77.97\% top-1 accuracy with image resolution 224$\times$224.}
\subsection{Understanding \DAA{}}
\begin{wrapfigure}{t}{2.6in}
\begin{minipage}{2.6in}
\centering
\vspace{-7mm}
\resizebox{2.6in}{!}{\includegraphics{DeepAA_simple.png}}
\vspace{-5mm}
\caption{{\small Top-1 test accuracy (\%) on ImageNet of \DAAsimple{}, \DAA{}, and other automatic augmentation methods on ResNet-50.}}
\vspace{-3mm}
\label{fig:deepaa_default}
\end{minipage}
\end{wrapfigure}
\textbf{Effectiveness of Gradient Matching.}
One uniqueness of \DAA{} is the regularized gradient matching objective.
To examine its effectiveness, we remove the impact coming from multiple augmentation layers, and only conduct search for a single layer of augmentation policy. When evaluating the searched policy, we apply the default augmentation in addition to the searched policy. We refer to this variant as \DAAsimple{}.
Figure~\ref{fig:deepaa_default} compares the Top-1 test accuracy on ImageNet using ResNet-50 between \DAAsimple{}, \DAA{}, and other automatic augmentation methods. While there is $0.22\%$ performance drop compared to \DAA{}, \textit{with a single augmentation layer}, \DAAsimple{} still outperforms other methods and is able to achieve similar performance compared to TA (Wide) but with a standard augmentation space and trains on a smaller image size (224$\times$224 vs 244$\times$224).
\textbf{Policy Search Cost.}
Table~\ref{tab:time} compares the policy search time on CIFAR-10/100 and ImageNet in GPU hours.
\DAA{} has comparable search time as PBA, Fast AA, and RA, but is slower than Faster AA and DADA. Note that Faster AA and DADA relax the discrete search space to a continuous one similar to DARTS~\citep{liu2018darts}. While such relaxation leads to shorter searching time, it inevitably introduces a discrepancy between the true and relaxed augmentation spaces.
\begin{table*}[!h]
\centering
\resizebox{0.8\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{Dataset}} &
\multicolumn{1}{c}{\textbf{AA}} &
\multicolumn{1}{c}{\textbf{PBA}} &
\multicolumn{1}{c}{\textbf{Fast AA}} &
\multicolumn{1}{c}{\textbf{Faster AA}} &
\multicolumn{1}{c}{\textbf{DADA}} &
\multicolumn{1}{c}{\textbf{RA}} &
\multicolumn{1}{c}{\textbf{\DAA{}}}
\\
\midrule
CIFAR-10/100 & 5000 & 5 & 3.5 & 0.23 & 0.1 & 25 & 9 \\
ImageNet & 15000 & - & 450 & 2.3 & 1.3 & 5000 & 96 \\
\bottomrule
\end{tabular}
}
\caption{{\small Policy search time on CIFAR-10/100 and ImageNet in GPU hours.}}
\label{tab:time}
\vspace{-3mm}
\end{table*}
\textbf{Impact of the Number of Augmentation Layers.}
Another uniqueness of \DAA{} is its multi-layer search space that can \textit{go beyond} two layers which existing automatic augmentation methods were designed upon.
We examine the impact of the number of augmentation layers on the performance of \DAA{}.
Table~\ref{tab:cifar_layers} and Table~\ref{tab:imagenet_layers} show the performance on CIFAR-10/100 and ImageNet respectively with increasing number of augmentation layers.
As shown, for CIFAR-10/100, the performance gradually improves when more augmentation layers are added until we reach five layers. The performance does not improve when the sixth layer is added.
For ImageNet, we have similar observation where the performance stops improving when more than five augmentation layers are included.
\vspace{2mm}
\begin{table*}[!h]
\centering
\resizebox{0.95\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{}} &
\multicolumn{1}{c}{\textbf{1 layer}} &
\multicolumn{1}{c}{\textbf{2 layers}} &
\multicolumn{1}{c}{\textbf{3 layers}} &
\multicolumn{1}{c}{\textbf{4 layers}} &
\multicolumn{1}{c}{\textbf{5 layers}} &
\multicolumn{1}{c}{\textbf{6 layers}}
\\
\midrule
CIFAR-10 & 96.3 $\pm$ 0.21 & 96.6 $\pm$ 0.18 & 96.9 $\pm$ 0.12 & 97.4 $\pm$ 0.14 & \textbf{97.56} $\pm$ \textbf{0.14} & \textbf{97.6} $\pm$ \textbf{0.12} \\
CIFAR-100 & 80.9 $\pm$ 0.31 & 81.7 $\pm$ 0.24 & 82.2 $\pm$ 0.21 & 83.7 $\pm$ 0.24 & \textbf{84.02} $\pm$ \textbf{0.18} & \textbf{84.0} $\pm$ \textbf{0.19} \\
\bottomrule
\end{tabular}
}
\caption{{\small Top-1 test accuracy of \DAA{} on CIFAR-10/100 for different numbers of augmentation layers. The results are averaged over 4 independent runs with different initializations with the $95\%$ confidence interval denoted by $\pm$.}}
\label{tab:cifar_layers}
\vspace{-3mm}
\end{table*}
\vspace{1mm}
\begin{table*}[!h]
\centering
\resizebox{0.7\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{}} &
\multicolumn{1}{c}{\textbf{1 layer}} &
\multicolumn{1}{c}{\textbf{3 layers}} &
\multicolumn{1}{c}{\textbf{5 layers}} &
\multicolumn{1}{c}{\textbf{7 layers}}
\\
\midrule
ImageNet & 75.27 $\pm$ 0.19 & 78.18 $\pm$ 0.22 & \textbf{78.30} $\pm$ \textbf{0.14} & \textbf{78.30} $\pm$ \textbf{0.14} \\
\bottomrule
\end{tabular}
}
\caption{{\small Top-1 test accuracy of \DAA{} on ImageNet with ResNet-50 for different numbers of augmentation layers. The results are averaged over 4 independent runs w/ different initializations with the $95\%$ confidence interval denoted by $\pm$.}}
\label{tab:imagenet_layers}
\end{table*}
Figure~\ref{fig:transformation} illustrates the distributions of operations in the policy for CIFAR-10/100 and ImageNet respectively.
As shown in Figure~\ref{fig:transformation}(a), the augmentation of CIFAR-10/100 converges to \textit{identity} transformation at the sixth augmentation layer, which is a natural indication of the end of the augmentation pipeline.
We have similar observation in Figure~\ref{fig:transformation}(b) for ImageNet, where the \textit{identity} transformation dominates in the sixth augmentation layer.
These observations match our results listed in Table~\ref{tab:cifar_layers} and Table~\ref{tab:imagenet_layers}.
We also include the distribution of the magnitude within each operation for CIFAR-10/100 and ImageNet in Appendix~\ref{app:cifar} and Appendix~\ref{app:imagenet}.
\begin{figure}
\vspace{-10mm}
\centering
\includegraphics[width=0.7\linewidth]{operation_dist.png}
\vspace{-2mm}
\caption{{\small The distribution of operations at each layer of the policy for CIFAR-10/100 and ImageNet. The probability of each operation is summed up over all $12$ discrete intensity levels (see Appendix~\ref{app:cifar} and~\ref{app:imagenet}) of the corresponding transformation.}}
\label{fig:transformation}
\vspace{-3mm}
\end{figure}
\begin{comment}
\vspace{1mm}
\textbf{Validity of Optimizing Gradient Matching.}
Finally, we want to demonstrate the validity of optimizing gradient matching for automatic data augmentation as we claim throughout this work.
To do so, we obtain a batch of augmented data by augmenting a single randomly sampled image for 512 times following $k$ layers of augmentation policy. We then sample a batch of 64 original images within the same class as the augmented one. The cosine similarity is calculated between the gradients of the augmented image batch and original image batch. We report the cosine similarity over 64 independent runs.
As shown in Table~\ref{tab:cosine}, for CIFAR-10/100, the value of cosine similarity gradually increases when more augmentation layers are added and converges to $0.771$ when five layers are added. Further adding more layers does not change the cosine similarity value.
For ImageNet, we have the similar observation where the cosine similarity value stops increasing when more than five augmentation layers are included.
These results again match our Top-1 test accuracies listed in Table~\ref{tab:cifar_layers} and Table~\ref{tab:imagenet_layers}.
\begin{table*}[!h]
\centering
\resizebox{1.0\textwidth}{!}
{
\begin{tabular}{@{}l|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{}} &
\multicolumn{1}{c}{\textbf{1 layer}} &
\multicolumn{1}{c}{\textbf{2 layers}} &
\multicolumn{1}{c}{\textbf{3 layers}} &
\multicolumn{1}{c}{\textbf{4 layers}} &
\multicolumn{1}{c}{\textbf{5 layers}} &
\multicolumn{1}{c}{\textbf{6 layers}}
\\
\midrule
\textbf{CIFAR-10/100} & $0.728 \pm 0.019$ & $0.741 \pm 0.017$ & $0.760 \pm 0.016$ & $0.763 \pm 0.015$ & \textbf{0.771} $\pm$ \textbf{0.014} & \textbf{0.771} $\pm$ \textbf{0.014} \\
\textbf{ImageNet} & $0.641 \pm 0.012$ & $0.695 \pm 0.014$ & $0.718 \pm 0.014$ & $0.726 \pm 0.013$ & \textbf{0.730} $\pm$ \textbf{0.013} & \textbf{0.730} $\pm$ \textbf{0.014} \\
\bottomrule
\end{tabular}
}
\caption{The average cosine similarity over the course of searching for CIFAR-10/100 and ImageNet with $75\%$ confidence interval calculated from 64 independent samples.}
\label{tab:cosine}
\end{table*}
\end{comment}
\vspace{1mm}
\textbf{Validity of Optimizing Gradient Matching with Regularization.}
To evaluate the validity of optimizing gradient matching with regularization, we designed a search-free baseline named ``DeepTA''. In DeepTA, we stack multiple layers of TA on the same augmentation space of DeepAA without using default augmentations.
As stated in Eq.(\ref{eq:mean_minus_std}) and Eq.(\ref{eq:gradient_cosine_similarity2}), we explicitly optimize the gradient similarities with the average reward minus its standard deviation.
The first term -- the average reward $E_x\{\tilde{r}^k(x)\}$ -- \emph{encourages the direction of high cosine similarity}.
The second term -- the standard deviation of the reward $\sqrt{E_x\{(\tilde{r}^k(x)-E_x\{\tilde{r}^k(x)\})^2\}}$ -- acts as a regularization that \emph{penalizes the direction with high variance}. These two terms jointly maximize the gradient similarity along the direction with low variance. To illustrate the optimization trajectory, we design two metrics that are closely related to the two terms in Eq.(\ref{eq:mean_minus_std}): the mean value, and the standard deviation of the improvement of gradient similarity. The improvement of gradient similarity is obtained by subtracting the cosine similarity of the original image batch from that of the augmented batch. In our experiment, the mean and standard deviation of the gradient similarity improvement are calculated over 256 independently sampled original images.
\begin{figure}[t]
\begin{subfigure}[b]{0.325\linewidth}
\centering
\includegraphics[width=0.98\linewidth]{mean_cosine.png}
\caption{Mean of the gradient similarity improvement}
\label{fig:mean_traj}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.325\linewidth}
\centering
\includegraphics[width=0.98\linewidth]{std_cosine.png}
\caption{Standard deviation of the gradient similarity improvement}
\label{fig:std_traj}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.325\linewidth}
\centering
\includegraphics[width=0.95\linewidth]{mean_accuracy.png}
\caption{Mean accuracy over different augmentation depth}
\label{fig:acc_traj}
\end{subfigure}
\vspace{1mm}
\caption{{\small Illustration of the search trajectory of DeepAA in comparison with DeepTA on CIFAR-10.}}\label{fig:mean_std_acc}
\vspace{-0mm}
\end{figure}
As shown in Figure~\ref{fig:mean_std_acc}(a), the cosine similarity of DeepTA reaches the peak at the fifth layer, and stacking more layers decreases the cosine similarity. In contrast, for DeepAA, the cosine similarity increases consistently until it converges to \textit{identity} transformation at the sixth layer.
In Figure~\ref{fig:mean_std_acc}(b), the standard deviation of DeepTA significantly increases when stacking more layers. In contrast, in DeepAA, as we optimize the gradient similarity along the direction of low variance, the standard deviation of DeepAA does not grow as fast as DeepTA. In Figure~\ref{fig:mean_std_acc}(c), both DeepAA and DeepTA reach peak performance at the sixth layer, but DeepAA achieves better accuracy compared against DeepTA. Therefore, we empirically show that DeepAA effectively scales up the augmentation depth by increasing cosine similarity along the direction with low variance, leading to better results.
\vspace{0mm}
\textbf{Comparison with Other Policies.}
In Figure \ref{fig:compare_policy} in Appendix~\ref{app:compare_policy}, we compare the policy of \DAA{} with the policy found by other data augmentation search methods including AA, FastAA and DADA. We have three interesting observations:
\begin{itemize}
\item AA, FastAA and DADA assign high probability (over 1.0) on flip, Cutout and crop, as those transformations are hand-picked and applied by default. \DAA{} finds a similar pattern that assigns high probability on flip, Cutout and crop.
\item Unlike AA, which mainly focused on color transformations, \DAA{} has high probability over both spatial and color transformations.
\item FastAA has evenly distributed magnitudes, while DADA has low magnitudes (common issues in DARTS-like method). Interestingly, \DAA{} assigns high probability to the stronger magnitudes.
\end{itemize}
\begin{comment}
\begin{table*}[!h]
\centering
\resizebox{0.95\textwidth}{!}
{
\color{blue}
\begin{tabular}{@{}l|c|c|c|c|c|c|c|c|c@{}}
\toprule
\multicolumn{1}{l}{\textbf{Dataset}} &
\multicolumn{1}{c}{\textbf{Augmentation}} &
\multicolumn{1}{c}{\textbf{1 Layer}} &
\multicolumn{1}{c}{\textbf{2 Layers}} &
\multicolumn{1}{c}{\textbf{3 Layers}} &
\multicolumn{1}{c}{\textbf{4 Layers}} &
\multicolumn{1}{c}{\textbf{5 Layers}} &
\multicolumn{1}{c}{\textbf{6 Layers}} &
\multicolumn{1}{c}{\textbf{7 Layers}} &
\multicolumn{1}{c}{\textbf{8 Layers}}
\\
\midrule
\multirow{2}{*}{CIFAR-10} & DeepTA & 96.3 $\pm$ 0.14 & 96.72 $\pm$ 0.16 & 97.18 $\pm$ 0.07 & 97.19 $\pm$ 0.10 & 07.18 $\pm$ 0.17 & \textbf{97.22 $\pm$ 0.13} & 97.11 $\pm$ 0.15 & 97.06 $\pm$ 0.1 \\
& DeepAA & 96.3 $\pm$ 0.21 & 96.6 $\pm$ 0.18 & 96.9 $\pm$ 0.12 & 97.4 $\pm$ 0.14 & 97.56 $\pm$ 0.14 & 97.6 $\pm$ 0.12 & - & - \\
\multirow{2}{*}{CIFAR-100} & DeepTA & 80.5 $\pm$ 0.17 & 81.94 $\pm$ 0.25 & 82.32 $\pm$ 0.29 & 82.56 $\pm$ 0.26 & 82.60 $\pm$ 0.16 & 82.67 $\pm$ 0.22 & 82.89 $\pm$ 0.21 & 82.62 $\pm$ 0.36 \\
& DeepAA & 80.9 $\pm$ 0.31 & 81.7 $\pm$ 0.24 & 82.2 $\pm$ 0.21 & 83.7 $\pm$ 0.24 & 84.02 $\pm$ 0.18 & 84.0 $\pm$ 0.19 & - & - \\
\bottomrule
\end{tabular}
}
\caption{{\small Comparison between DeepTA and DeepAA across layers.}}
\label{tab:parameters}
\end{table*}
\end{comment}
\begin{comment}
\subsection{Discussion}
We visualize the discovered augmentation policy found on CIFAR-10 data in Figure. \ref{fig:prob}. We found that that the first layer keeps a diverse set of transforms, while the 2-4th layers mainly converge to randCutout, randCrop and randFlip. More interestingly, the discovered policy in 2-4th layers coincides with the widely adopted default augmentation.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{layer_prob.png}
\caption{\textbf{The probability assigned to each transformation over at different layers.} (A) First layer augmentation: the distribution is spreaded across multiple augmentation operations. (B) Second layer: The augmentation converged to randCutout. (C) Third layer: The augmentation converged to randCrop. (D) Fourth layer: The augmentation mainly concentrated to randFlip and randCutout.}
\label{fig:prob}
\end{center}
\end{figure}
In Figure \ref{fig:prob_magnitude}, we show the distribution of magnitude over each operation. It is clear seen that, in the first layer, the magnitudes are high for all operations.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\linewidth]{magnitude_dist.png}
\caption{\textbf{The distribution of magnitude for each operation \footnote{We do not show operatgions that do not have magnitude, which are identity, autoContrast, invert, equalize, randFlp, randCutout and randCrop}.} (A) First layer magnitude: the magnitudes are concentrated on both ends. (B-D) Second, third and fourth layer magnitudes: Except for solarize and posterize, the magnitude are small.}
\label{fig:prob_magnitude}
\end{center}
\end{figure}
\end{comment} |
2004.02268 | \section{Introduction}\label{sec1}\setcounter{equation}{0}
The classical second Borel--Cantelli lemma states that if $\Gam_1,\Gam_2,...$ is a sequence of independent
events such that
\begin{equation}\label{1.1}
\sum_{n=1}^\infty P(\Gam_n)=\infty
\end{equation}
then with probability one infinitely many of events $\Gam_i$ occur, i.e.
\begin{equation}\label{1.2}
\sum_{n=1}^\infty\bbI_{\Gam_n}=\infty\quad\mbox{almost surely (a.s.)}
\end{equation}
where $\bbI_\Gam$ is the indicator of a set (event) $\Gam$.
There is a long list of papers, starting probably with \cite{Le}, providing conditions which replace the
independency by a weaker assumption and still yield (\ref{1.2}) (see, for instance, \cite{DMR} and references
there). On the other hand, it was shown in Theorem 3 of \cite{Ph} that under $\phi$-mixing with a summable coefficient
$\phi$ the condition (\ref{1.1}) yields the stronger version of the second Borel--Cantelli lemma in the form
\begin{equation}\label{1.3}
\frac {S_N}{\cE_N}\to 1\,\,\,\mbox{almost surely (a.s.) as}\,\, N\to\infty
\end{equation}
where $S_N=\sum_{n=1}^N\bbI_{\Gam_n}$ and $\cE_N=\sum_{n=1}^NP(\Gam_n)$.
The same paper \cite{Ph} started another line of research, known now under the name dynamical Borel--Cantelli lemmas,
where (\ref{1.3}) is proved for $S_N=\sum_{n=1}^N\bbI_{\Gam_n}\circ T^n$ where $T$ is a measure preserving transformation
on a probability space $(\Om,P)$ and $\Gam_n,\, n\geq 1$ is a sequence of measurable sets. For such $S_N$'s the convergence
(\ref{1.3}) was proved, in particular, for the Gauss map $Tx=\frac 1x$ (mod 1), $x\in(0,1]$ preserving the Gauss measure
$P(\Gam)=\frac 1{\ln 2}\int_\gam\frac {dx}{1+x}$. This line of research became quite popular in the last two decades.
In particular, \cite{CK} proves (\ref{1.3}) in the dynamical setup considering $T$ being the so called subshift of finite
type on a sequence space where $\Gam_n,\, n\geq 1$ is a sequence of cylinders while another series of papers dealt with
uniformly and non-uniformly hyperbolic dynamical systems as a transformation $T$ and with geometric balls as $\Gam_n$'s
(see, for instance, \cite{Gu}, \cite{HNPV} and references there).
In this paper we consider, in particular, "nonconventional" extensions of some of the above results aiming to prove that under
certain conditions (\ref{1.3}) holds true with $S_N=\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{\Gam_{q_i(n)}})$ and $\cE_N=\sum_{n=1}^NP(\Gam_{q_i(n)})$
where $q_i(n),\, i=1,...,\ell$ functions taking on positive integer values on positive integers and satisfying certain assumptions valid, in particular,
for polynomials with integer coefficients. When $\ell=1$ (conventional setup) the $\phi$-mixing with a summable coefficient $\phi$
suffices for our result, while for $\ell>1$ we have to impose stronger $\psi$-mixing conditions.
In the dynamical systems setup we consider $S_N=\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{C_n^{(i)}}\circ T^{q_i(n)})$ and
$\cE_N=\sum_{n=1}^N\prod_{i=1}^\ell P(C_n^{(i)})$ where $T$ is the left shift on a sequence space $\cA^{\bbZ}$ with a finite or countable
alphabet while $C_n^{(i)},\, i=1,...,\ell,\, n\geq 1$ is a sequence of cylinder sets. As an application we study the asymptotic behaviors of expressions
$M_N=\max_{1\leq n\leq N}(\min_{1\leq i\leq\ell}\Phi_{\tilde\om^{(i)}}\circ T^{q_i(n)})$ where $\Phi_{\tilde\om}(\om)=-\ln(d(\om,\tilde\om))$,
$\om,\tilde\om\in\cA^\bbN$ and $d(\cdot,\cdot)$ is the natural distance on the sequence space.
Our results extend some of the previous work in the
following aspects. First, the strong Borel--Cantelli property in the nonconventional setup $\ell>1$ was not studied
before at all. Secondly, even in the conventional setup $\ell=1$ considering rather general functions $q(n)=q_1(n)$
in place of just $q(n)=n$ seems to be new, as well. Thirdly, we extend for shifts some of the results from \cite{CK}
considering sequence spaces with countable alphabets and $\phi$-mixing invariant measures rather than just subshifts
of finite type with Gibbs measures which are exponentially fast $\psi$-mixing (see \cite{Bo}). This allows to apply our
results, for instance, to Gibbs-Markov maps and to Markov chains with a countable state space satisfying the Doeblin condition
since both examples are exponentially fast $\phi$-mixing, see \cite{MN} and \cite{Br}, respectively.
In the next section we will formulate precisely our setups and assumptions and state our main results. In Section \ref{sec3} we
will prove the strong Borel--Cantelli property for events under the $\phi$-mixing condition in the conventional setup
$\ell=1$ and under $\psi$-mixing condition in the nonconventional setup $\ell>1$. In Sections \ref{sec4} and \ref{sec5} we extend
the strong Borel--Cantelli property to shifts under the $\phi$-mixing when $\ell=1$ and under $\psi$-mixing when $\ell>1$, respectively.
In Section \ref{sec6} we exhibit applications to the asymptotic behaviors of maximums along shifts of logarithmic distance functions while in the last Section \ref{sec7} we apply the strong
Borel--Cantelli property to derive the asymptotics of multiple hitting times of shrinking cylinder
sets.
\section{Preliminaries and main results}\label{sec2}\setcounter{equation}{0}
We start with a probability space $(\Om,\cF,P)$ and a two parameter family of
$\sig$-algebras $\cF_{mn}$ indexed by pairs of integers $-\infty\leq m\leq n\leq\infty$ and
such that $\cF_{mn}\subset\cF_{m'n'}\subset\cF$ if $m'\leq n\leq n'$. Recall that the
$\phi$ and $\psi$ dependence coefficient between two $\sig$-algebras $\cG$ and $\cH$ can be
written in the form
(see \cite{Br}),
\begin{eqnarray}\label{2.1}
&\phi(\cG,\cH)=\sup_{\Gam\in\cG,\,\Del\in\cH}\{|\frac {P(\Gam\cap\Del)}{P(\Gam)}-P(\Del)|,\, P(\Gam)\ne 0\}\\
&=\frac 12\sup\{\| E(g|\cG)-Eg\|_{L^\infty}:\, g\,\,\mbox{is $\cH$-measurable and}\,\,\| g\|_{L^\infty}\leq 1\}
\nonumber\end{eqnarray}
and
\begin{eqnarray}\label{2.2}
&\psi(\cG,\cH)=\sup_{\Gam\in\cG,\,\Del\in\cH}\{|\frac {P(\Gam\cap\Del)}{P(\Gam)P(\Del)}-1|,\, P(\Gam)P(\Del)\ne 0\}\\
&=\frac 12\sup\{\| E(g|\cG)-Eg\|_{L^\infty}:\, g\,\,\mbox{is $\cH$-measurable and}\,\, E|g|\leq 1\},
\nonumber\end{eqnarray}
respectively. The $\phi$-dependence (mixing) and the $\psi$-dependence (mixing) in the family $\cF_{mn}$ is measured
by the coefficients
\begin{equation}\label{2.3}
\phi(k)=\sup_m\phi(\cF_{-\infty,m},\cF_{m+k,\infty})\,\,\mbox{and}\,\,\psi(k)=\sup_m\psi(\cF_{-\infty,m},\cF_{m+k,\infty}),
\end{equation}
respectively, where $k=0,1,2,...$. The probability measure $P$ is called $\phi$-mixing or $\psi$-mixing with respect to the family
of $\sig$-algebras $\cF_{mn}$ if $\phi(n)\to 0$ or $\psi(1)<\infty$ and $\psi(n)\to 0$ as $n\to\infty$, respectively.
Our setup includes also functions $q_1(n),\, q_2(n),...,q_\ell(n)$ with $\ell\geq 1$ taking on nonnegative integer values on
integers $n\geq 0$ and satisfying
\begin{assumption}\label{ass2.1} There exists a constant $K>0$ such that
(i) for any $i\ne j$, $1\leq i,j\leq\ell$ and every integer $k$ the number of integers $n\geq 0$ satisfying at least one of
the equations
\begin{equation}\label{2.4}
q_i(n)-q_j(n)=k\quad\mbox{and}\quad q_i(n)=k
\end{equation}
does not exceed $K$ (when $\ell=1$ only the second equation in (\ref{2.4}) should be taken into account);
(ii) the cardinality of the set $\cN$ of all pairs $n>m\geq 0$ satisfying
\begin{equation}\label{2.5}
\max_{1\leq i\leq\ell}q_i(n)\leq\max_{1\leq i\leq\ell}q_i(m)
\end{equation}
does not exceed $K$.
\end{assumption}
Observe that Assumption \ref{ass2.1} is satisfied if $q_i,\, i=1,...,\ell$ are essentially distinct nonconstant
polynomials (i.e. $|q_i(n)-q_j(n)|\to\infty$ as $n\to\infty$ for any $i\ne j$) with integer coefficients taking on
nonnegative values on nonnegative integers. Indeed, $q_i(n)-q_j(n)$ and $q_i(n)$ are nonconstant polynomials,
and so the number of $n$'s solving one of equations in (\ref{2.4}) is bounded by the degree of the corresponding polynomial.
In order to show that (\ref{2.5}) can hold true in the polynomial case only for finitely many pairs $m<n$ observe that
there exists $n_0\geq 1$ such that all polynomials $q_1(n),\, q_2(n),...,q_\ell(n)$ are strictly increasing on $[n_0,\infty)$.
Hence, if $n>m\geq n_0$ then (\ref{2.5}) cannot hold true. If $0\leq m<n_0$ and $n\geq n_0$ then there exists $n_1\geq n_0$ such
that for all $n\geq n_1$ (\ref{2.5}) cannot hold true, as well. The remaining case $0\leq m<n_0$ and $0\leq n<n_1$ concerns less
than $n_0n_1$ pairs $m<n$.
Next, we will state our result concerning sequences of events. Let $\Gam_1,\Gam_2,...\in\cF$ be a sequence of events and each
$\sig$-algebra $\cF_{mn},\, 1\leq m\leq n<\infty$ be generated by the events $\Gam_m,\Gam_{m+1},...,\Gam_n$. Set also
$\cF_{mn}=\cF_{1n}$ for $-\infty\leq m\leq 0$ and $n\geq 1$, $\cF_{mn}=\{\emptyset,\Om\}$ for $m,n\leq 0$ and $\cF_{m,\infty}=
\sig\{\Gam_m,\Gam_{m+1},...\}$. Set
\begin{equation}\label{2.6}
S_N=\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{\Gam_{q_i(n)}})\quad\mbox{and}\quad\cE_N=\sum_{n=1}^N(\prod_{i=1}^\ell P(\Gam_{q_i(n)})).
\end{equation}
\begin{theorem}\label{thm2.2}
Let $\phi$ and $\psi$ be dependence coefficients defined by (\ref{2.3}) for the above $\sig$-algebras $\cF_{mn}$. Assume that
$\phi(n),\, n\geq 0$ is summable in the case $\ell=1$ and $\psi(n),\, n\geq 0$ is summable in the case $\ell>1$. Suppose that
the functions $q_1(n),...,q_\ell(n)$ satisfy Assumption \ref{ass2.1}(i) and
\begin{equation}\label{2.7}
\cE_N\to\infty\quad\mbox{as}\quad N\to\infty.
\end{equation}
Then, with probability one,
\begin{equation}\label{2.8}
\lim_{N\to\infty}\frac {S_N}{\cE_N}=1\quad\mbox{as}\quad N\to\infty.
\end{equation}
\end{theorem}
Next, we will present our results concerning shifts. Here $\Om=\cA^\bbZ$ is the space of sequences $\om=(...,\om_{-1},\om_0,\om_1,...)$
with terms $\om_i$ from a finite or countable alphabet $\cA$ which is not a singleton with the index $i$ running along integers (or along
natural numbers $\bbN$ which can also be considered requiring very minor modifications). We assume that the basic $\sig$-algebra $\cF$ is
generated by all cylinder sets while the $\sig$-algebras $\cF_{mn},\, n\geq m$ are generated by the cylinder sets of the form
$\{\om=(\om_i)_{-\infty<i<\infty}:\,\om_i=a_i\,\,\mbox{ for}\,\,\, m\leq i\leq n\}$ for some $a_m,a_{m+1},...,a_n\in\cA$. The setup includes also the left shift $T:\Om\to\Om$ acting by $(T\om)_i=\om_{i+1}$ and a $T$-invariant probability measure $P$ on $(\Om,\cF)$, i.e. $P(T^{-1}\Gam)=P(\Gam)$
for any measurable $\Gam\subset\Om$. In this setup $\phi$ and $\psi$-dependence coefficients defined by (\ref{2.3}) will be considered with
respect to the family of $\sig$-algebras $\cF_{mn},\, m\leq n$ defined above. Without loss of generality we assume that the probability of each
1-cylinder $[a]=\{\om=(\om_i)_{i\in\bbZ}:\, \om_0=a$ is positive, i.e. $P([a])>0$ for any $a\in\cA$, and since $\cA$ is not a singleton we have
also that $\sup_{a\in\cA}P([a])<1$.
Each cylinder $C$ is defined on an interval of integers $\La=[l,r]$, $l\leq r$, i.e. $C=\{\om=(\om_i)_{-\infty<i<\infty}:\,\om_i=a_i,\, i=l,l+1,...,r\}$
for some $a_l,...,a_r\in\cA$. Given a constant $D>0$ call an interval of integers $\La_1=[l_1,r_1]$ to be right $D$-nested in the interval of
integers $\La_2=[l_2,r_2]$ if $[l_1,r_1]\subset (-\infty,r_2+D)$, i.e. $r_1<r_2+D$. Such an interval $\La_1$ will be called $D$-nested in $\La_2$
if $[l_1,r_1]\subset (l_2-D,r_2+D)$. The latter notion was used also in \cite{CK}.
Let $C_n^{(j)},\, j=1,...,\ell,\, n=1,2,...$ be a sequence of cylinder sets defined on intervals of integers $\La_n,\, n=1,2,...$ so that $C_n^{(j)},\, j=1,...,\ell$ are defined on $\La_n$ for each $n\geq 1$. Set
\begin{equation}\label{2.9}
S_N=\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{C_n^{(i)}}\circ T^{q_i(n)})\quad\mbox{and}\quad\cE_N=\sum_{n=1}^N
\prod_{i=1}^\ell P(C_n^{(i)}).
\end{equation}
\begin{theorem}\label{thm2.3} Suppose that the functions $q_1(n),...,q_\ell(n)$ satisfy Assumption \ref{ass2.1} and
\begin{equation}\label{2.10}
\cE_N\to\infty\quad\mbox{as}\quad N\to\infty.
\end{equation}
Let $C_n^{(j)},\, j=1,...,\ell,\, n\geq 1$ be a sequence of cylinder sets defined on intervals $\La_n\subset\bbZ$ as
described above and $D>0$ be a constant.
(i) If $\ell=1$ assume that the $\phi$-dependence coefficient is summable and that for all $m<n$ the interval
$\La_m$ is right $D$-nested in $\La_n$. Then, with probability one,
\begin{equation}\label{2.11}
\lim_{N\to\infty}\frac {S_N}{\cE_N}=1\quad\mbox{as}\quad N\to\infty.
\end{equation}
(ii) If $\ell>1$ assume that the $\psi$-dependence coefficient is summable and that for all $m<n$ the interval
$\La_m$ is $D$-nested in $\La_n$. Then with probability one (\ref{2.11}) holds true, as well.
\end{theorem}
As in most papers on the strong Borel--Cantelli property both Theorems \ref{thm2.2} and \ref{thm2.3} rely on the following basic result.
\begin{theorem}\label{thm2.4}
Let $\Gam_1,\Gam_2,...$ be a sequence of events such that for any $N\geq M\geq 1$,
\begin{equation}\label{2.12}
\sum_{m,n=M}^N(P(\Gam_m\cap\Gam_n)-P(\Gam_m)P(\Gam_n))\leq c\sum_{n=M}^NP(\Gam_n)
\end{equation}
where a constant $c>0$ does not depend on $M$ and $N$. Then for each $\ve>0$ almost surely
\begin{equation}\label{2.13}
S_N=\cE_N+O(\cE_N^{1/2}\log^{\frac 32+\ve}\cE_N)
\end{equation}
where
\[
S_N=\sum_{n=1}^N\bbI_{\Gam_n}\quad\mbox{and}\quad\cE_N=\sum_{n=1}^NP(\Gam_n).
\]
In particular, if
\[
\cE_N\to\infty\quad\mbox{as}\quad N\to\infty
\]
then with probability one
\[
\lim_{N\to\infty}\frac {S_N}{\cE_N}=1\quad\mbox{as}\quad N\to\infty.
\]
\end{theorem}
This result (as well as the part of Theorem \ref{thm2.2} for $\ell=1$ and $q_1(n)=n$) appears
already in Theorem 3 from \cite{Ph} and in a slightly more general (analytic) form it is proved as Lemma 10
in \S 7 of Ch.1 from \cite{Sp}. Both sources refer to \cite{Sch} as the origin of this result.
We observe that Theorem \ref{thm2.3} extends Theorem 2.1 from \cite{CK} in several directions. First, for
$\ell=1$ we prove the result for arbitrary $\phi$-mixing probability measures with a summable coefficient
$\phi$ on a shift space with a countable alphabet and not just for subshifts of finite type with Gibbs
measures. Secondly, the case $\ell>1$ and rather general functions $q_i(n)$ in place of just $\ell=1$
and $q_1(n)=n$ were not considered before both in the setups of Theorem \ref{thm2.2} and \ref{thm2.3}.
A direct application of Theorem \ref{thm2.3} yields corresponding strong Borel--Cantelli property for
dynamical systems which have symbolic representations by means of finite or countable partitions, for instance,
hyperbolic dynamical systems (see, for instance, \cite{Bo}) where sequences of cylinders in Theorem \ref{thm2.3}
should be replaced by corresponding sequences of elements of joins of iterates of the partition. By a slight
modification (just by considering cylinder sets defined on intervals of nonnegative integers only) Theorem \ref{thm2.3}
remains valid for one-sided shifts and then it can be applied to noninvertible dynamical systems having a symbolic
representation via their finite or countable partitions such as expanding transformations, the Gauss map of the
interval and more general transformations generated by $f$-expansions (see \cite{He}).
In Section \ref{sec6} we apply Theorem \ref{thm2.3} to some limiting problems obtaining a symbolic version
of results from \cite{HNT} which dealt with dynamical systems on $\bbR^d$ or manifolds and not with shifts. Namely, in the
setup of Theorem \ref{thm2.3} introduce the distance between $\om=(\om_i)_{i\in\bbZ}$ and $\tilde\om=(\tilde\om_i)_{i\in\bbZ}$
from $\Om$ by
\begin{equation}\label{2.14}
d(\om,\tilde\om)=\exp(-\gam\min\{ i\geq 0:\,\om_i\ne\tilde\om_i\,\,\mbox{or}\,\,\om_{-i}\ne\tilde\om_{-i}\}),\,\,\gam>0.
\end{equation}
Set
\begin{eqnarray}\label{2.15}
&\Phi_{\tilde\om}(\om)=-\ln(d(\om,\tilde\om))\,\,\mbox{for}\,\,\om,\tilde\om\in\Om\,\,\mbox{and}\\
&M_{N,\bm{\tilde\om}}(\om)=M_{N,\tilde\om^{(1)},...,\tilde\om^{(\ell)}}=
\max_{1\leq n\leq N}\min_{1\leq i\leq\ell}(\Phi_{\tilde\om^{(i)}}\circ T^{q_i(n)}(\om))\nonumber
\end{eqnarray}
for some fixed $\ell$-tuple $\bm{\tilde\om}=(\tilde\om^{(1)},...,\tilde\om^{(\ell)}),\,\tilde\om^{(i)}\in\Om,\,
i=1,...,\ell$.
\begin{theorem}\label{thm2.5} Assume that the entropy of the partition into 1-cylinders is finite, i.e.
\begin{equation}\label{2.16}
-\sum_{a\in\cA}P([a])\ln P([a])<\infty.
\end{equation}
Then, under the conditions of Theorem \ref{2.3} for almost all $\tilde\om^{(1)},...,\tilde\om^{(\ell)}\in\Om$
with probability one,
\begin{equation}\label{2.17}
\frac {M_{N,\tilde\om^{(1)},...,\tilde\om^{(\ell)}}}{\ln N}\to\frac \gam{2\ell h}\,\,\,\mbox{as}\,\,\, N\to\infty
\end{equation}
where $h$ is the Kolmogorov--Sinai entropy of the shift $T$ on the probability space $(\Om,\cF,P)$ and, as in Theorem \ref{thm2.3},
if $\ell=1$ we assume only $\phi$-mixing with a summable coefficient $\phi$ and if $\ell>1$ we assume $\psi$-mixing with a summable
coefficient $\psi$ (and in both cases $h>0$ by Lemma 3.1 in \cite{KY} and Lemma 3.1 in \cite{KR}).
\end{theorem}
In Section \ref{sec7} we demonstrate another application of Theorem \ref{thm2.3} deriving the
asymptotical behavior of multiple hitting times of shrinking cylinders. Namely, set
\[
\tau_{C_n(\tilde\om)}=\min\{ k\geq 1:\,\prod_{i=1}^\ell\bbI_{C_n(\tilde\om)}\circ T^{q_i(k)}(\om)=1\}
\]
where $\om,\tilde\om\in\Om$ and $C_n(\om)=\{\om=(\om_i)_{i\in\bbZ}\in\Om:\,\om_i=\tilde\om_i$ provided
$|i|\leq n\}$.
\begin{theorem}\label{thm2.6} Assume that (\ref{2.16}) holds true. Then under the conditions of
Theorem \ref{thm2.3} for $P\times P$-almost all pairs $(\om,\tilde\om)\in\Om\times\Om$,
\begin{equation}\label{2.18}
\lim_{n\to\infty}\frac 1n\ln\tau_{C_n(\tilde\om)}(\om)=2\ell h.
\end{equation}
\end{theorem}
We observe that (\ref{2.18}) was proved in \cite{KR} under the $\psi$-mixing assumption assuming
additionally stronger conditions than here while the $\phi$-mixing case was not treated there at all.
The proof of Theorem \ref{thm2.6} here is different from \cite{KR} as it relies on the Borel--Cantelli
lemma and the strong Borel--Cantelli property which is an adaptation to our symbolic (and nonconventional) setup of proofs from \cite{Ga1} and \cite{Ga2}. We note that both Theorem
\ref{thm2.5} and Theorem \ref{thm2.6} remain valid (with essentially the same proof) for one sided
shifts just by deleting 2 in (\ref{2.17}) and (\ref{2.18}).
\section{Proof of Theorem 2.2}\label{sec3}\setcounter{equation}{0}
\subsection{The case $\ell=1$}\label{subsec3.1}
Let $N\geq M$ and fix an $m$ between $M$ and $N$. By Assumption \ref{ass2.1} for each $k$ there
exists at most $K$ of integers $n$ such that $q(n)-q(m)=k$ where $q(n)=q_1(n)$. If $q(n)-q(m)=k\geq 1$
then by the definition of the $\phi$-dependence coefficient
\begin{equation}\label{3.1}
|P(\Gam_{q(m)}\cap\Gam_{q(n)})-P(\Gam_{q(m)})P(\Gam_{q(n)})|\leq\phi(k)P(\Gam_{q(m)}).
\end{equation}
Hence,
\begin{equation}\label{3.2}
\sum_{N\geq n\geq M,\, q(n)>q(m)}
|P(\Gam_{q(m)}\cap\Gam_{q(n)})-P(\Gam_{q(m)})P(\Gam_{q(n)})|\leq KP(\Gam_{q(m)})\sum_{k=1}^\infty\phi(k).
\end{equation}
Since the coefficient $\phi$ is summable and that similar inequalities hold true when $q(m)>q(n)$
we conclude that the condition (\ref{2.12}) of Theorem \ref{thm2.4} is satified with $\Gam_{q(n)}$ in place of
$\Gam_n,\, n=1,2,...$ there, and so (\ref{2.8}) follows in the case $\ell=1$ assuming (\ref{2.7}). \qed
\subsection{The case $\ell>1$}.
We start with the following counting arguments concerning the functions $q_i,\, i=1,...,\ell$ satisfying
Assumption \ref{ass2.1}. Introduce
\[
q(n)=\min_{1\leq i\ne j\leq\ell}|q_i(n)-q_j(n)|.
\]
By Assumption \ref{ass2.1}(i) for each pair $i\ne j$ and any $k$ there exists at most $K$ nonnegative integers
$n$ such that $q_i(n)-q_j(n)=k$, and so
\begin{equation}\label{3.3}
\#\{ n>0:\, q(n)=k\}<K\ell^2
\end{equation}
where $\#$ stands for "the number of ...". We will need also the following semi-metric between integers $k,l>0$,
\[
\del(k,l)=\min_{1\leq i,j\leq\ell}|q_i(k)-q_j(l)|.
\]
It follows from Assumption \ref{ass2.1}(i) that for any integers $m>0$ and $k\geq 0$,
\begin{equation}\label{3.4}
\#\{ n>0:\, \del(m,n)=k\}<2K^2\ell^2.
\end{equation}
Indeed, the number of $m$'s such that $q_j(m)=q_i(n)-k$ for a fixed $i,j,n$ and $k$ does not exceed $K$
by Assumption \ref{ass2.1}(i) and (\ref{3.4}) follows since $1\leq i,j\leq\ell$.
In order to prove Theorem \ref{thm2.2} for $\ell>1$ we will estimate first
\begin{equation}\label{3.5}
|E(X_mX_n)-EX_mEX_n|=|P(\cap_{i=1}^\ell(\Gam_{q_i(m)}\cap\Gam_{q_i(n)}))-P(\cap_{i=1}^\ell\Gam_{q_i(m)})P(\cap_{i=1}^\ell\Gam_{q_i(n)})|
\end{equation}
where $m,n>0$ and $X_k=\prod_{i=1}^\ell\bbI_{\Gam_{q_i(k)}}$. If $\del(m,n)=k\geq 1$ then by Lemma 3.3 in \cite{KR}
and the definition of the $\psi$-dependence coefficient
\begin{equation}\label{3.6}
|E(X_mX_n)-EX_mEX_n|\leq 2^{2\ell+2}\psi(k)(2-(1+\psi(k))^\ell)-2EX_mEX_n
\end{equation}
where we assume, in fact, that $k$ is large enough so that $\psi(k)<2^{1/\ell}-1$. Thus, let $k_0=\min\{ k:\,\psi(k)<2^{1/\ell}-1\}$.
Then by (\ref{3.4}) and (\ref{3.6}),
\begin{equation}\label{3.7}
\sum_{N\geq n\geq M}|E(X_mX_n)-EX_mEX_n|\leq cEX_m
\end{equation}
where
\[
c=2K^2\ell^2\big(1+2^{2\ell+2}(2-(1+\psi(k_0))^\ell)^{-2}\sum_{k=k_0}^\infty\psi(k)\big)
\]
where we took into account that
\[
|EX^2_m-(EX_m)^2|\leq EX_m.
\]
Summing in (\ref{3.7}) in $m$ between $M$ an $N$ we obtain the condition (\ref{2.12}) of Theorem \ref{thm2.4}
with $\cap_{i=1}^\ell\Gam_{q_i(n)}$ in place of $\Gam_n$ there. Hence if
\begin{equation}\label{3.8}
\sum_{n=1}^\infty P(\cap_{i=1}^\ell\Gam_{q_i(n)})=\infty
\end{equation}
then Theorem \ref{thm2.4} yields that with probability one
\begin{equation}\label{3.9}
\frac {S_N}{\tilde\cE_N}\to 1\quad\mbox{as}\quad N\to\infty
\end{equation}
where $\tilde\cE_N=\sum_{n=1}^NP(\cap_{i=1}^\ell\Gam_{q_i(n)})$.
Since we assume (\ref{2.10}) and not (\ref{3.8}), it remains to show that under our conditions,
\begin{equation}\label{3.10}
\frac {\tilde\cE_N}{\cE_N}\to 1\quad\mbox{as}\quad N\to\infty.
\end{equation}
By Lemma 3.2 from \cite{KR} we obtain when $q(n)=k\geq 1$ that
\begin{equation}\label{3.11}
|P(\cap_{i=1}^\ell\Gam_{q_i(n)})-\prod_{i=1}^\ell P(\Gam_{q_i(n)})|\leq ((1+\psi(k))^\ell-1)\prod_{i=1}^\ell P(\Gam_{q_i(n)}).
\end{equation}
For $q(n)=0$ we estimate the left hand side of (\ref{3.11}) just by 1. Hence, by (\ref{3.3}),
\begin{eqnarray}\label{3.12}
&|\tilde\cE_N-\cE_N|\leq K\ell^2+\sum_{n=1,q(n)\geq 1}^N\big(((1+\psi(q(n)))^\ell-1)\prod_{i=1}^\ell P(\Gam_{q_i(n)})\big)\\
&\leq K\ell^2+\sum_{n=1,q(n)\geq 1}^N((1+\psi(q(n)))^\ell-1)\nonumber\\
&\leq K\ell^2+K\ell^2\sum_{n=1}^\infty((1+\psi(q(n)))^\ell-1)\leq C<\infty
\nonumber\end{eqnarray}
for some constant $C>0$, since the coefficient $\psi$ is summable. Dividing (\ref{3.12}) by $\cE_N$ and taking into account (\ref{2.10}) we
obtain (\ref{3.10}) and complete the proof of Theorem \ref{thm2.2}. \qed
\section{Proof of Theorem 2.3($i$)}\label{sec4}\setcounter{equation}{0}
Here $\ell=1$, and so we set $C_n=C_n^{(1)}$ and $q(n)=q_1(n)$.
Consider cylinder sets $C_m$ and $C_n$, $1\leq m<n$ defined on intervals of integers
$\La_m=[l_m,r_m]$ and $\La_n=[l_n,r_n]$ with $\La_m$ right $D$-nested in $\La_n$
implying that $r_m<r_n+D$. Let $k=q(n)-q(m)$. By Assumption \ref{ass2.1}(i)
for each $m$ and $k$ this equality can hold true only for at most $K$ of $n$'s and by Assumption \ref{ass2.1}(ii)
for no more than $K$ of $n$'s we may have $q(n)\leq q(m)$. Next, we can write
\begin{equation}\label{4.1}
r_n+q(n)>r_m+q(m)+k-D.
\end{equation}
Assume first that
\begin{equation}\label{4.2}
l_n+q(n)\leq r_m+q(m)\quad\mbox{and}\quad r_n+q(n)>r_m+q(m).
\end{equation}
Let $C_n=[a_{l_n},a_{l_n+1},...,a_{r_n}]$ and $\hat C_{m,n}=[a_{t_{m,n}},a_{t_{m,n}+1},...,a_{r_n}]$ where we assume that
$r_n>l_n$,
\begin{eqnarray*}
&t_{m,n}=s_{m,n}+[\frac 12(r_n-s_{m,n}+1)]\quad\mbox{and}\\
& a_{m,n}=l_n+(r_m+q(m)-l_n-q(n))+1=r_m+q(m)-q(n)+1.
\end{eqnarray*}
It follows that
\begin{equation}\label{4.3}
r_n-t_{m,n}+1\geq [\frac 12(k-D)]\quad\mbox{and}\quad t_{m,n}+q(n)-r_m+q(m)\geq [\frac 12(k-D)]-1.
\end{equation}
Assuming that $k\geq D+4$ we obtain by the definition of the $\phi$-dependence coefficient that
\begin{eqnarray}\label{4.4}
&P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)\leq P(T^{-q(m)}C_m\cap T^{-q(n)}\hat C_{m,n})\\
&\leq P(C_m)P(\hat C_{m,n})+\phi([\frac 12(k-D)]-1)P(C_m).\nonumber
\end{eqnarray}
To make the estimate (\ref{4.4}) suitable for our purposes we recall that according to Lemma 3.1 in \cite{KY}
there exists $\al>0$ such that any cylinder set $C$ defined on an interval of integers $\La=[l,r]$ satisfies
\begin{equation}\label{4.5}
P(C)\leq e^{-\al(r-l)},
\end{equation}
and so
\begin{equation}\label{4.6}
P(\hat C_{m,n})\leq\exp(-\al([\frac 12(k-D)]-1).
\end{equation}
In addition to (\ref{4.4}) we can write also
\begin{equation}\label{4.7}
P(C_m)P(C_n)\leq e^{-\al(r_n-l_n)}P(C_m)\leq e^{-\al(k-D)}P(C_m)
\end{equation}
where we used that by (\ref{4.1}),
\[
r_n-l_n\geq r_n-s_{m,n}+1=r_n+q(n)-r_m-q(m)>k-D.
\]
Observe that by Assumption \ref{ass2.1} there exists at most $K(D+1)$ of $n$'s for which $q(n)-q(m)=k\leq D$,
and so by (\ref{4.1}) the second inequality in (\ref{4.2}) may fail only for at most $K(D+1)$ of $n$'s. For
such $n$'s we use the trivial estimate
\begin{equation}\label{4.8}
|P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)-P(C_m)P(C_n)|\leq P(C_m).
\end{equation}
Now if
\begin{equation}\label{4.9}
l_n+q(n)>r_m+q(m)
\end{equation}
then by the definition of the $\phi$-dependence coefficient we can write by (\ref{4.1}) that
\begin{eqnarray}\label{4.10}
&|P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)-P(C_m)P(C_n)|\\
&\leq\phi(l_n+q(n)-r_m-q(m))P(C_m)\leq\phi(k-D-(r_n-l_n))P(C_m)\nonumber
\end{eqnarray}
but this may not suffice for our purposes when $r_n-l_n$ is large. In this case we proceed as in
(\ref{4.4}), (\ref{4.6}) and (\ref{4.7}) where we take $\hat C_n=[a_{t_n},a_{t_n+1},...,a_{r_n}]$
with $t_n=l_n+[\frac 12(r_n-l_n)]+1$. Then
\[
t_n+q(n)-r_m-q(m)>[\frac 12(r_n-l_n)]+1\quad\mbox{and}\quad r_n-t_n\geq [\frac 12(r_n-l_n)]-1,
\]
and so
\begin{eqnarray}\label{4.11}
&P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)\leq P(T^{-q(m)}C_m\cap T^{-q(n)}\hat C_n)\\
&\leq P(C_m)P(\hat C_n)+\phi([\frac 12(r_n-l_n)]+1)P(C_m)\nonumber\\
&\leq\big( e^{-\al([\frac 12(r_n-l_n)]-1)}+\phi([\frac 12(r_n-l_n)])\big) P(C_m).\nonumber
\end{eqnarray}
Thus, when (\ref{4.9}) holds true we use (\ref{4.10}) if $r_n-l_n\leq\frac {k-D}2$ and (\ref{4.11})
when $r_n-l_n>\frac {k-D}2$. In both cases we will obtain the estimate
\begin{eqnarray}\label{4.12}
&|P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)-P(C_m)P(C_n)|\\
&\leq \big( e^{-\al([\frac 14(k-D)]-1)}+\phi([\frac 14(k-D)])\big) P(C_m).\nonumber
\end{eqnarray}
Finally, taking into account that $q(n)-q(m)=k\leq D$ can occur only for at most $K(D+1)$ of $n$'s
and for each $k$ the equality $q(n)-q(m)=k$ may hold true for at most $K$ of $n$'s we conclude from
(\ref{4.4}), (\ref{4.6})--(\ref{4.8}), (\ref{4.12}) and from the summability of the coefficient $\phi$
that for any $m=M,M+1,...,N$,
\begin{equation}\label{4.13}
\sum_{n=M}^N|P(T^{-q(m)}C_m\cap T^{-q(n)}C_n)-P(C_m)P(C_n)|\leq cP(C_m)
\end{equation}
for some constant $c>0$ independent of $M$ and $N$. Summing in $m$ between $M$ and $N$ we conclude that
the condition (\ref{2.12}) of Theorem \ref{thm2.4} is satisfied with $\Gam_n=T^{-q(n)}C_n$, and so assuming
(\ref{2.10}) we obtain (\ref{2.11}) completing the proof of Theorem \ref{thm2.3}(i). \qed
\section{Proof of Theorem 2.3($ii$)}\label{sec5}\setcounter{equation}{0}
Observe that if $\del(n,m)=k$, $n>m\geq 0$ and the pair $n,m$ does not belong to the exceptional set
$\cN$ having cardinality at most $K$ then by Assumption \ref{ass2.1}(ii) for some $i_0,j_0\leq\ell$,
\begin{equation}\label{5.1}
q_{j_0}=\max_{1\leq j\leq\ell}q_j(n)\geq q_{i_0}(m)+k=\max_{1\leq i\leq\ell}q_i(m)+k.
\end{equation}
Let $C_m$ and $C_n$ be cylinder sets defined on $\La_m=[l_m,r_m]$ and $\La_n=[l_n,r_n]$, respectively.
Since $C_m$ is $D$-nested in $C_n$, $r_m\leq r_n+D$, and so by (\ref{5.1}),
\begin{equation}\label{5.2}
r_m+q_{i_0}(m)\leq r_n+q_{j_0}(n)-k+D.
\end{equation}
Assume first that
\begin{equation}\label{5.3}
l_n+q_{j_0}(n)\leq r_m+q_{i_0}(m)\quad{and}\quad r_n+q_{j_0}(n)>r_m+q_{i_0}(m).
\end{equation}
Let $C_n=[a_{l_n},a_{l_n+1},...,a_{r_n}]$ and $\hat C_{m,n}=[a_{s_{m,n}},a_{s_{m,n}+1},...,a_{r_n}]$ where
\begin{equation}\label{5.4}
s_{m,n}=l_n+(r_m+q_{i_0}(m)-l_n-q_{j_0})+1=r_m+q_{i_0}(m)-q_{j_0}(n)+1,
\end{equation}
and so $\hat C_{m,n}$ is defined on the interval $[s_{m,n},r_n]$ of the length
\begin{equation}\label{5.5}
r_n-s_{m,n}+1=r_n+q_{j_0}(n)-r_m-q_{i_0}(m)\geq k-D
\end{equation}
where the last inequality follows from (\ref{5.2}). Hence, by the definition of the $\psi$-dependence coefficient
\begin{eqnarray}\label{5.6}
&P\big(\cap_{i=1}^\ell(T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&\leq P\big(\cap_{i=1}^\ell(T^{-q_i(m)}C_m^{(i)}\cap T^{-q_{j_0}(n)}\hat C_{m,n}^{(j_0)})\big)\nonumber\\
&\leq (1+\psi(1))P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P( T^{-q_{j_0}(n)}\hat C_{m,n}^{(j_0)})\nonumber\\
&\leq (1+\psi(1))e^{-\al(k-D)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\nonumber
\end{eqnarray}
where $\hat C_{m,n}^{(j_0)}$ is constructed as above with $C_n=C_n^{(j_0)}$.
We can write also that
\begin{eqnarray}\label{5.7}
&P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})\leq P(C_n^{(1)})P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\\
&\leq e^{-\al(r_n-l_n)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
Since $r_n-l_n\geq r_n-s_{m,n}+1\geq k-D$, it follows that under the condition (\ref{5.3}),
\begin{eqnarray}\label{5.8}
&|P\big(\cap_{i=1}^\ell(T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|\nonumber\\
&\leq (1+\psi(1))e^{-\al(k-D)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
On the other hand, if
\begin{equation}\label{5.9}
l_n+q_{j_0}(n)>r_m+q_{i_0}(m),
\end{equation}
then by the definition of the $\psi$-dependence coefficient we obtain similarly to the above that
\begin{eqnarray}\label{5.10}
&|P\big(\cap_{i=1}^\ell(T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|\nonumber\\
&\leq (1+\psi(l_n+q_{j_0}(n)-r_m-q_{i_0}(m)))P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(C_n^{(1)})\nonumber\\
&\leq (1+\psi(1))e^{-\al(r_n-l_n)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
Let a number $d_0\geq 1$ be such that
\begin{equation}\label{5.11}
\psi(d_0)<2^{1/\ell}-1\quad\mbox{and}\quad k-(r_n-l_n+2D)>d_0.
\end{equation}
Since $r_n-l_n\geq r_m-l_m-2D$ by $D$-nesting, it follows by (\ref{4.5}) and Lemma 3.3 from \cite{KR}
that
\begin{eqnarray}\label{5.12}
&|P\big(\cap_{i=1}^\ell (T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|\nonumber\\
&\leq 2^{2\ell+2}\psi(k-\max(r_n-l_n,r_m-l_m))\nonumber\\
&\times (2-(1+\psi(k-\max(r_n-l_n,r_m-l_m))^\ell)^{-2}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\nonumber\\
&\times P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})\leq 2^{2\ell+2}\psi(k-(r_n-l_n+2D))(2-(1+\psi(d_0))^\ell)^{-2}\nonumber\\
&\times e^{-\al(r_n-l_n)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
Since the cardinality of $\cN$ does not exceed $K$ we have
\begin{eqnarray}\label{5.13}
&\sum_{(n,m)\in\cN}|P\big(\cap_{i=1}^\ell (T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|\nonumber\\
&\leq KP(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
Next, we estimate now the remaining sum
\begin{eqnarray}\label{5.14}
&\sum_{n>m,(n,m)\not\in\cN}|P\big(\cap_{i=1}^\ell (T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|.\nonumber
\end{eqnarray}
For the part of the sum in $n$'s satisfying (\ref{5.3}) we apply the inequality (\ref{5.8}) which yields the contribution
to the total sum estimated using (\ref{3.4}) by
\begin{eqnarray}\label{5.15}
&2K\ell^2(1+\psi(1))P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\sum_{k=0}^\infty e^{-\al(k-D)}\\
&=2K\ell^2e^{\al D}(1+\psi(1))(1-e^{-\al})^{-1}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
For the parts of the sum (\ref{5.14}) which correspond to $n$'s satisfying (\ref{5.9}) but not (\ref{5.11})
we obtain that
\begin{equation}\label{5.17}
e^{-\al(r_n-l_n)}\leq e^{-\al k}e^{-\al(2D-d_0)},
\end{equation}
and so taking into account (\ref{3.4}) the summation in (\ref{5.14}) over $n$'s satisfying (\ref{5.9}) can be
estimated by
\begin{eqnarray}\label{5.18}
&2K\ell^2(1+\psi(1))e^{-\al(2D-d_0)}P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\sum_{k=0}^\infty e^{-\al k}\\
&=2K\ell^2(1+\psi(1))e^{-\al(2D-d_0)}(1-e^{-\al})P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})\nonumber.
\end{eqnarray}
It remains to estimate the part of the sum (\ref{5.14}) which corresponds to $n$'s satisfying (\ref{5.11})
where we use (\ref{5.12}). We observe that
\begin{eqnarray}\label{5.19}
&\psi(k-(r_n-l_n+2D))e^{-\al(r_n-l_n)}\\
&=e^{2\al D}\psi(k-(r_n-l_n+2D))e^{-\al(r_n-l_n+2D)}\nonumber\\
&\leq e^{2\al D}\max(\psi([k/2]), \psi(1)e^{-\al[k/2]})\leq e^{2\al D}(\psi([k/2])+\psi(1)e^{-\al[k/2]})\nonumber
\end{eqnarray}
since either $r_n-l_n+2D\geq k/2$ or $k-(r_n-l_n+2D)\geq k/2$. Both summands in the right hand side of (\ref{5.19})
are summable in $k$ (the first one by the assumption) which gives an estimate for the part of the sum (\ref{5.14})
corresponding to $n$'s satisfying (\ref{5.11}) in the form
\begin{equation}\label{5.20}
cP(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})
\end{equation}
where $c>0$ does not depend on $m$. By estimates (\ref{5.8}), (\ref{5.12}), (\ref{5.13}), (\ref{5.15}) and
(\ref{5.18})--(\ref{5.20}) above we conclude that the whole sum consisting of the part appearing in (\ref{5.13})
plus the part displayed by (\ref{5.14}) can be estimated by the expression (\ref{5.20}) with another constant
$c>0$ independent of $m$. It follows that there exists $\tilde c>0$ such that for all $N>M\geq 1$,
\begin{eqnarray}\label{5.21}
&\sum_{n,m=M}^N|P\big(\cap_{i=1}^\ell (T^{-q_i(m)}C_m^{(i)}\cap T^{-q_i(n)}C_n^{(i)})\big)\\
&-P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})|\nonumber\\
&\leq\tilde c\sum_{m=M}^NP(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)}).\nonumber
\end{eqnarray}
If, in addition,
\begin{equation}\label{5.22}
\sum_{m=1}^\infty P(\cap_{i=1}^\ell T^{-q_i(m)}C_m^{(i)})=\infty
\end{equation}
then by Theorem \ref{thm2.4} we obtain that with probability one
\begin{equation}\label{5.23}
\frac {\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{C_n^{(i)}}\circ T^{q_i(n)})}{\sum_{n=1}^NP(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})}\to 1\,\,\mbox{as}\,\, N\to\infty.
\end{equation}
It remains to show that under the condition (\ref{2.10}) with probability one,
\begin{equation}\label{5.24}
\frac {\sum_{n=1}^NP(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})}{\sum_{n=1}^N\prod_{i=1}^\ell P(C_n^{(i)})}\to 1\quad\mbox{as}\quad N\to\infty.
\end{equation}
Observe again that
\begin{equation}\label{5.25}
P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})\leq P(C_n^{(1)})\leq e^{-\al(r_n-l_n)}.
\end{equation}
Next, we split the sum in the left hand side of (\ref{5.22}) into two sums
\begin{eqnarray*}
&S_1=\sum_{n:\,(r_n-l_n)\leq\frac 2\al\ln n}P(\cap_{i=1}^\ell(T^{-q_i(n)}C_n^{(i)})\\
&\mbox{and}\,\,\,S_2=\sum_{n:\,(r_n-l_n)>\frac 2\al\ln n}P(\cap_{i=1}^\ell(T^{-q_i(n)}C_n^{(i)}).
\end{eqnarray*}
By (\ref{5.25}),
\[
S_2\leq\sum_{n=1}^\infty n^{-2}<\infty\,\,\mbox{and also}\,\,\sum_{n:\,(r_n-l_n)>\frac 2\al\ln n}\prod_{i=1}^\ell P(C_n^{(i)})<\infty.
\]
Hence, it suffices to show that under the condition (\ref{2.10}) with probability one,
\begin{equation}\label{5.26}
\frac {\sum_{n\leq N:\,(r_n-l_n)\leq\frac 2\al\ln n}P(\cap_{i=1}^\ell T^{-q_i(n)}C_n^{(i)})}{\sum_{n\leq N:\,n:\,(r_n-l_n)\leq\frac 2\al\ln n}
\prod_{i=1}^\ell P(C_n^{(i)})}\to 1\quad\mbox{as}\quad N\to\infty.
\end{equation}
Set $q(n)=\min_{i\ne j}|q_i(n)-q_j(n)|$. Observe that by Assumption \ref{ass2.1}(i) for each $k$,
\begin{equation}\label{5.27}
\#\{ n:\, q(n)=k\}\leq K\ell^2.
\end{equation}
Consider first $n$'s satisfying
\begin{equation}\label{5.28}
q(n)\leq r_n-l_n.
\end{equation}
In this case by (\ref{5.25}),
\begin{equation}\label{5.29}
P(\cap^\ell_{i=1}T^{-q_i(n)}C_n^{(i)})\leq e^{-\al q(n)}
\end{equation}
and relying on (\ref{5.27}) we conclude that
\[
\sum_{n:\, q(n)\leq r_n-l_n}P(\cap^\ell_{i=1}T^{-q_i(n)}C_n^{(i)})\leq K\ell^2\sum_{k=0}^\infty e^{-\al k}=K\ell^2(1-e^{-\al})^{-1}
\]
and the same estimate holds true for $\sum_{n:\, q(n)\leq r_n-l_n}\prod_{i=1}^\ell P(C_n^{(i)})$. Hence, the sum over such $n$'s does not influence
the asymptotical behavior in (\ref{5.24}) and (\ref{5.26}) since the denominators there tend to $\infty$.
It remains to consider the sums over $n$'s satisfying
\begin{equation}\label{5.30}
q(n)>r_n-l_n.
\end{equation}
In this case we can apply Lemma 3.2 from \cite{KR} to obtain that
\begin{eqnarray}\label{5.31}
&|P(\cap^\ell_{i=1}T^{-q_i(n)}C_n^{(i)})-\prod_{i=1}^\ell P(C_n^{(i)})|\\
&\leq\big((1+\psi(q(n)-(r_n-l_n))^\ell-1\big)\prod_{i=1}^\ell P(C_n^{(i)})\nonumber\\
&\leq \big((1+\psi(q(n)-(r_n-l_n))^\ell-1\big)e^{-\ell\al(r_n-l_n)}.\nonumber
\end{eqnarray}
Now observe that either $r_n-l_n$ or $q(n)-(r_n-l_n)$ is greater or equal to $\frac 12q(n)$. Denote by $\cN_1$ the set of $n$'s
for which $r_n-l_n\geq \frac 12q(n)$ and by $\cN_2$ the set of $n$'s for which $q(n)-(r_n-l_n)\geq\frac 12q(n)$. Taking into account
(\ref{5.27}) and (\ref{5.30}) we obtain that
\begin{eqnarray}\label{5.32}
&\sum_{n\in\cN_1}\big((1+\psi(q(n)-(r_n-l_n))^\ell-1\big)e^{-\ell\al(r_n-l_n)}\\
&\leq((1+\psi(1))^\ell-1)\sum_{n\in\cN_1}e^{-\frac 12\ell\al q(n)}\nonumber\\
&\leq K\ell^2((1+\psi(1))^\ell-1)\sum_{k=0}^\infty e^{-\frac 12\ell\al k}\nonumber\\
&=K\ell^2((1+\psi(1))^\ell-1)(1-e^{-\frac 12\ell\al})^{-1}<\infty.\nonumber
\end{eqnarray}
Next, taking into account that $\psi(k)$ is summable we see that
\begin{eqnarray}\label{5.33}
&\sum_{n\in\cN_2}\big((1+\psi(q(n)-(r_n-l_n))^\ell-1\big)e^{-\ell\al(r_n-l_n)}\\
&\leq\sum_{n\in\cN_2}\big((1+\psi(\max(1,[\frac 12q(n)]))^\ell-1\big)\nonumber\\
&\leq 2K\ell^2\sum_{k=1}^\infty((1+\psi(k))^\ell-1)=2K\ell^2\sum_{k=1}^\infty\sum_{m=1}^\ell{\ell\choose m}(\psi(k))^m<\infty.\nonumber
\end{eqnarray}
Hence,
\begin{equation}\label{5.34}
|\sum_{n=1}^\infty\big( P(\cap^\ell_{i=1}T^{-q_i(n)}C_n^{(i)})-\prod_{i=1}^\ell P(C_n^{(i)})\big)|<\infty
\end{equation}
and since $\sum_{n=1}^\infty\prod_{i=1}^\ell P(C_n^{(i)})=\infty$, we obtain (\ref{5.26}), and so (\ref{5.24}), as well, completing the proof of
Theorem \ref{thm2.3}(ii). \qed
\section{Asymptotics of maximums of logarithmic distance functions}\label{sec6}\setcounter{equation}{0}
In this section we will prove Theorem \ref{thm2.5}. Let $\tilde\om^{(j)}=(\tilde\om_i^{(j)})_{i\in\bbZ}\in\Om$ and $C_n(\tilde\om^{(j)}),\, j=1,...,\ell,\, n=1,2,...$ be a sequence of cylinder sets such that
\[
C_n(\tilde\om^{(j)})=\{\om=(\om_i)_{i\in\bbZ}\in\Om:\,\om_i=\tilde\om_i^{(j)}\,\,\mbox{ provided}\,\, |i|\leq r_n\}
\]
where $r_n\uparrow\infty$ as $n\uparrow\infty$ is a sequence of integers. Observe that by the Shannon--McMillan--Breiman
theorem (see, for instance, \cite{Pe}) for almost all $\tilde\om\in\Om$,
\begin{equation}\label{6.1}
\lim_{n\to\infty}\frac 1{2r_n}\ln P(C_n(\tilde\om))=-h
\end{equation}
where $h$ is the Kolmogorov--Sinai entropy of the shift $T$ with respect to $P$ since the latter measure is ergodic whether
we assume $\phi$ or $\psi$-mixing.
Now suppose that
\begin{equation}\label{6.2}
\sum_{n=1}^\infty \prod_{i=1}^\ell P(C_n(\tilde\om^{(i)}))<\infty.
\end{equation}
It follows from (\ref{5.33}) that (\ref{6.2}) implies also
\begin{equation}\label{6.3}
\sum_{n=1}^\infty P(\cap_{i=1}^\ell T^{-q_i(n)}C_n(\tilde\om^{(i)}))<\infty
\end{equation}
which is, of course, a tautology if $\ell=1$. It follows from the first Borel--Cantelli lemma that for almost all $\om\in\Om$
only finitely many events $\{ T^{q_i(n)}\om\in C_n(\tilde\om^{(i)}),\, i=1,...,\ell\}$ can occur. But if the latter event does not hold true then
\[
T^{q_j(n)}\not\in C_n(\tilde\om^{(j)})\,\,\,\mbox{for some}\,\,\, 1\leq j\leq\ell,
\]
and so
\begin{equation}\label{6.4}
d(T^{q_j(n)}\om,\tilde\om^{(j)})>e^{-\gam r_n}\,\,\mbox{i.e.}\,\,\Phi_{\tilde\om^{(j)}}(T^{q_j(n)}\om)<\gam r_n
\end{equation}
where the distance $d(\cdot,\cdot)$ and the function $\Phi$ were defined in (\ref{2.14}) and (\ref{2.15}).
It follows that in this case there exists $N_{\bm{\tilde\om}},\,\bm{\tilde\om}=(\tilde\om^{(1)},...,\tilde\om^{(\ell)})$ finite with probability one and such that for
all $N>N_{\bm{\tilde\om}}(\om)$,
\[
M_{N,\bm{\tilde\om}}(\om)<\gam r_N,
\]
where $M_{N,\bm{\tilde\om}}(\om)$ was defined in (\ref{2.15}). Hence,
\begin{equation}\label{6.5}
\limsup_{N\to\infty}\frac {M_{N,\bm{\tilde\om}}}{\ln N}\leq\gam\limsup_{N\to\infty}\frac {r_N}{\ln N}\,\,\mbox{a.s.}
\end{equation}
Next, assume that
\begin{equation}\label{6.6}
\cE_{N,\tilde\om}=\sum_{n=1}^N\prod_{i=1}^\ell P(C_n(\tilde\om^{(i)})\to \infty\,\,\mbox{as}\,\, N\to\infty
\end{equation}
which by (\ref{5.33}) implies also that
\begin{equation}\label{6.7}
\sum_{n=1}^\infty P(\cap_{i=1}^\ell T^{-q_i(n)}C_n(\tilde\om^{(i)}))=\infty.
\end{equation}
Set
\[
L_{n,\bm{\tilde\om}}(\om)=\max\{ m\leq n:\, T^{q_i(m)}\om\in C_m(\tilde\om^{(i)})\,\,\mbox{for}\,\, i=1,...,\ell\}.
\]
It follows from Theorem \ref{thm2.3} that under (\ref{6.6}) for almost all $\om\in\Om$,
\[
L_{n,\bm{\tilde\om}}(\om)\to\infty\,\,\,\mbox{as}\,\,\, n\to\infty.
\]
Observe also that
\begin{equation}\label{6.8}
S_N(\om)=\sum_{n=1}^N(\prod_{i=1}^\ell\bbI_{C_n(\tilde\om^{(i)})}\circ T^{q_i(n)}(\om))=S_{L_{n,\bm{\tilde\om}}(\om)}.
\end{equation}
By (\ref{4.13}), (\ref{5.20}) and (\ref{5.33}) we can use (\ref{2.13}) which yields that for almost all $\om\in\Om$,
\begin{equation}\label{6.9}
0\leq\cE_{N,\bm{\tilde\om}}-\cE_{L_{n,\bm{\tilde\om}},\bm{\tilde\om}}\leq O(\cE_{N,\bm{\tilde\om}}^{1/2}\ln^{\frac 32+\ve}\cE_{N,\bm{\tilde\om}}),
\end{equation}
and so for almost all $\om$,
\begin{equation}\label{6.10}
\lim_{N\to\infty}\frac {\cE_{L_{n,\bm{\tilde\om}}(\om),\bm{\tilde\om}}}{\cE_{N,\bm{\tilde\om}}}=1.
\end{equation}
Next, observe that if $m=L_{n,\bm{\tilde\om}}(\om)$ then for each $i=1,...,\ell$,
\[
d(T^{q_i(m)}\om,\tilde\om^{(i)})\leq e^{-\gam r_m}\,\,\mbox{i.e.}\,\,\Phi_{\tilde\om^{(i)}}(T^{q_i(m)}\om)\geq\gam r_m,
\]
and so $M_{m,\bm{\tilde\om}}(\om)\geq\gam r_m$. It follows that
\begin{equation}\label{6.11}
M_{N,\bm{\tilde\om}}(\om)\geq M_{L_{N,\bm{\tilde\om}}(\om)}(\om)\geq\gam r_{L_{N,\bm{\tilde\om}}(\om)},
\end{equation}
and so
\begin{equation}\label{6.12}
\liminf_{N\to\infty}\frac {M_{N,\bm{\tilde\om}}(\om)}{\ln N}\geq\gam(\liminf_{N\to\infty}\frac {r_N}{\ln N})\liminf_{N\to\infty}
\frac {\ln L_{N,\bm{\tilde\om}}(\om)}{\ln N}.
\end{equation}
Next, in order to complete the proof of Theorem \ref{thm2.5}, we will choose sequences $r_n,\, n=1,2,...$ for appropriate upper
and lower bounds. For the upper bound we will take $r_n=[\frac {1+\del}{2\ell h}\ln n]$ for some $\del>0$. Then by (\ref{6.1}) for almost all $\tilde\om^{(1)},...,\om^{(\ell)}\in\Om$,
\[
\ln\prod_{i=1}^\ell P(C_n(\tilde\om^{(i)}))\sim -(1+\del)\ln n\quad\mbox{as}\quad n\to\infty,
\]
and so the series (\ref{6.2}) converges as needed. Substituting such $r_N$'s to (\ref{6.5}) and letting $\del\to 0$ we obtain
\begin{equation}\label{6.13}
\limsup_{N\to\infty}\frac {M_{N,\bm{\tilde\om}}}{\ln N}\leq\frac \gam{2\ell h}\quad\mbox{a.s.}
\end{equation}
Now we deal with the lower bound choosing $r_n=[\frac {1-\del}{2\ell h}\ln n]$. Then by (\ref{6.1}) for almost all
$\tilde\om^{(1)},...,\om^{(\ell)}\in\Om$ as $n\to\infty$,
\begin{equation}\label{6.14}
\ln\prod_{i=1}^\ell P(C_n(\tilde\om^{(i)}))\sim -(1-\del)\ln n,
\end{equation}
and so the series (\ref{6.6}) diverges as needed. For such $r_N$'s we have that
\begin{equation}\label{6.15}
\liminf_{N\to\infty}\frac {r_N}{\ln N}=\frac {1-\del}{2\ell h}
\end{equation}
and letting $\del\to 0$ the proof of Theorem \ref{thm2.5} will be completed by (\ref{6.12}), (\ref{6.13}) and
(\ref{6.15}) once we show that for
almost all $\om\in\Om$,
\begin{equation}\label{6.16}
\liminf_{N\to\infty}\frac {\ln L_{N,\bm{\tilde\om}}(\om)}{\ln N}=1.
\end{equation}
By (\ref{6.14}) there exists a random variable $n(\bm{\tilde\om})<\infty$ a.s. such that if $n\geq n(\bm{\tilde\om})$ then
\begin{equation}\label{6.17}
n^{-(1-\frac 34\del)}\leq\prod_{i=1}^\ell P(C_n(\tilde\om^{(i)})) \leq n^{-(1-\frac 43\del)}.
\end{equation}
If $L_{N,\tilde\om}(\om)\geq n(\om)$ then we obtain from (\ref{6.9}) and (\ref{6.17}) that
\begin{eqnarray}\label{6.18}
&\frac 4{3\del}(N^{\frac 34\del}-(L_{N,\bm{\tilde\om}}(\om)+1)^{\frac 34\del})\leq\sum_{n=L_{N,\bm{\tilde\om}}(\om)+1}^Nn^{-(1-\frac 34\del)}\\
&\leq O\big( (n(\om)+\sum_{n=n(\om)}^Nn^{-(1-\frac 43\del)})^{1/2}\ln^{\frac 32+\ve}(n(\om)+\sum_{n=n(\om)}^Nn^{-(1-\frac 43\del)})\big)\nonumber\\
&\leq O\big(n(\om)+\frac 3{4\del}N^{\frac 43\del})^{1/2}\ln^{\frac 32+\ve}(n(\om)+\frac 3{4\del}N^{\frac 43\del})\big).\nonumber
\end{eqnarray}
Dividing these inequalities by $N^{\frac 34\del}$, letting $N\to\infty$ and taking into account that $N\geq L_{N,\bm{\tilde\om}}(\om)$ by the definition, we see that
\[
\frac {L_{N,\bm{\tilde\om}}(\om)}N\to 1,\,\,\mbox{and so}\,\, \ln N-\ln L_{N,\tilde\om}(\om)\to 0\,\,\mbox{a.s. as}
\,\, N\to\infty
\]
implying (\ref{6.16}) and completing the proof of Theorem \ref{thm2.5}. \qed
\section{Asymptotics of hitting times}\label{sec7}\setcounter{equation}{0}
In this section we will prove Theorem \ref{thm2.6} deriving first that for $P\times P$-almost all pairs
$(\om,\tilde\om)$,
\begin{equation}\label{7.1}
\liminf_{n\to\infty}\frac 1n\ln\tau_{C_n(\tilde\om)}\geq 2\ell h.
\end{equation}
Let $\la k\leq n\leq\la(k+1)$ for some $\la>0$. Then
\[
\frac {\ln\tau_{C_{\la k}(\tilde\om)}}{\la(k+1)}\leq\frac {\ln\tau_{C_n(\tilde\om)}}{n}\leq
\frac {\ln\tau_{C_{\la( k+1)}(\tilde\om)}}{\la k},
\]
and so
\begin{equation}\label{7.2}
\liminf_{n\to\infty}\frac {\ln\tau_{C_n(\tilde\om)}}{n}=\liminf_{k\to\infty}
\frac {\ln\tau_{C_{\la k}(\tilde\om)}}{\la k}
\end{equation}
where we alert the reader that the definition of the cylinder $C_n\tilde\om)$ here agrees with
the corresponding definition in Section \ref{sec6} provided $r_n=n$ there.
Next, assume that $\la>(2\ell h)^{-1}$ and set
\[
I_k(\tilde\om)=\cup_{j=1}^{e^k}\cap_{i=1}^\ell T^{-q_i(j)}C_{\la k}(\tilde\om).
\]
Then
\begin{equation}\label{7.3}
P(I_k(\tilde\om))\leq\sum_{j=1}^{e^k}P(\cap_{i=1}^\ell T^{-q_i(j)}C_{\la k}(\tilde\om))
\end{equation}
and we are going to show that for $P$-almost all $\tilde\om$,
\begin{equation}\label{7.4}
\sum_{k=1}^\infty\sum_{j=1}^{e^k}P(\cap_{i=1}^\ell T^{-q_i(j)}C_{\la k}(\tilde\om))<\infty.
\end{equation}
Indeed, applying the Shannon-McMillan-Breiman theorem we obtain that for $P$-almost all $\tilde\om$
and each $\ve>0$ there exists $k(\ve,\tilde\om)$ such that if $k\geq k(\ve,\tilde\om)$ then
\begin{equation}\label{7.5}
P(C_{\la k}(\tilde\om))\leq\exp(-k(2\la h-\ve)).
\end{equation}
When $\ell=1$ we employ (\ref{7.5}) for $k\geq k(\ve,\tilde\om)$ and (\ref{4.5}) for
$k< k(\ve,\tilde\om)$ which yields the estimate of the left hand side of (\ref{7.4}) by
\begin{equation}\label{7.6}
\sum_{1\leq k\leq k(\ve,\tilde\om)}e^ke^{-\al(2\la k-1)}+\sum_{k=1}^\infty e^{-k(2\la h-\ve-1)}.
\end{equation}
The first sum in (\ref{7.6}) contains finitely many terms, and so it is bounded, while the
second sum in (\ref{7.6}) is also bounded since $2\la h-\ve>1$ by the choice of $\la$ provided
$\ve>0$ is small enough.
Next, we will deal with the case $\ell>1$. First, recall the notation
$q(n)=\min_{i\ne j}|q_i(n)-q_j(n)|$ and observe that by (\ref{5.6}),
\begin{equation}\label{7.7}
\#\{ n:\, q(n)\leq 2\la k+2\}\leq 2(\la k+1)K\ell^2.
\end{equation}
Now we split the sum in the left hand side of (\ref{7.4}) into two sums
\begin{eqnarray}\label{7.8}
&S_1=\sum^\infty_{k=1}\sum_{j:j\leq e^k,\, q(j)\leq 2\la k+2}P(\cap_{i=1}^\ell T^{-q_i(j)}
C_{\la k}(\tilde\om))\\
&\leq 2K\ell^2\sum_{k=1}^\infty(\la k+1)P(C_{\la k}(\tilde\om))\leq 2K\ell^2\sum_{k=1}^\infty(\la k+1)
e^{-2\al(\la k-1)}<\infty,\nonumber
\end{eqnarray}
where we we use (\ref{4.5}), and
\begin{eqnarray}\label{7.9}
&S_2=\sum^\infty_{k=1}\sum_{j:j\leq e^k,\, q(j)> 2\la k+2}P(\cap_{i=1}^\ell T^{-q_i(j)}
C_{\la k}(\tilde\om))\\
&\leq k(\ve,\tilde\om)e^{k(\ve,\tilde\om)}+\sum^\infty_{k=k(\ve,\tilde\om)}
\sum_{j:j\leq e^k,\, q(j)> 2\la k+2}P(\cap_{i=1}^\ell T^{-q_i(j)}C_{\la k}(\tilde\om)).\nonumber
\end{eqnarray}
If $q(j)>2\la k+2$ and $k\geq k(\ve,\tilde\om)$ then employing Lemma 3.2 from \cite{KR} and
(\ref{7.5}) above we obtain
\begin{equation}\label{7.10}
P(\cap_{i=1}^\ell T^{-q_i(j)}C_{\la k}(\tilde\om))\leq (1+\psi(1))^\ell(P(C_{\la k}(\tilde\om)))^\ell\leq(1+\psi(1))^\ell\exp(-k(2\la h\ell-\ve\ell))
\end{equation}
where $\psi$ is the dependence coefficient from (\ref{2.3}). For $\ve>0$ small enough
$2\la h\ell-\ve\ell>1$ by the choice of $\la$, and so by (\ref{7.9}) and (\ref{7.10}),
\[
S_2\leq k(\ve,\tilde\om)e^{k(\ve,\tilde\om)}+\sum^\infty_{k=1}\exp(-k(2\la h\ell-\ve\ell-1))
<\infty\]
which together with (\ref{7.8}) yields (\ref{7.4}).
Hence, by the (first) Borel--Cantelli lemma there exists $K(\om)=K(\om,\tilde\om)<\infty$ a.s.
such that for all $k\geq K(\om)$ there are no events
\[
T^{q_i(j)}\om\in C_{\al k}(\tilde\om)\,\,\,\mbox{for all}\,\,\, i=1,...,\ell\,\,\,\mbox{and some}
\,\,\, 1\leq j\leq e^k.
\]
It follows that for $P$-almost all $\om$ and $k\geq K(\om)$.
\[
\tau_{C_{\al k}(\tilde\om)}(\om)>e^k.
\]
This together with (\ref{7.2}) yields that for $P\times P$-almost all pairs $(\om,\tilde\om)$,
\begin{equation}\label{7.11}
\liminf_{n\to\infty}\frac {\ln\tau_{C_{n}(\tilde\om)}(\om)}n\geq\la^{-1}.
\end{equation}
Since $\la$ can be chosen arbitrarily close to $(2\ell h)^{-1}$ we obtain (\ref{7.1}).
Next, we will prove that for $P\times P$-almost all pairs $(\om,\tilde\om)$,
\begin{equation}\label{7.12}
\limsup_{n\to\infty}\frac {\ln\tau_{C_n(\tilde\om)}(\om)}n\leq 2\ell h.
\end{equation}
Choose $\ve'>\ve>0$ small and $\be>0$ close to $(2\ell h)^{-1}$ so that
\begin{equation}\label{7.13}
\be(2\ell h+\ve)<1\,\,\mbox{and}\,\, \be(2\ell h+\ve')-\frac {1-\be(2\ell h-\ve)}{1-\be(2\ell h+\ve)}>0
\end{equation}
which implies, in particular, that $\be(2\ell h+\ve')>1$.
Set
\[
\Gam=\{(\om,\tilde\om)\in\Om:\,\limsup_{n\to\infty}\frac {\ln\tau_{C_n(\tilde\om)}(\om)}n>2\ell h+\ve'\}.
\]
If $(\om,\tilde\om)\in\Gam$ then for infinitely many $n$'s,
\begin{equation}\label{7.14}
\tau_{C_{\be\ln n}(\tilde\om)}(\om)>n^{\be(2\ell h+\ve')}.
\end{equation}
For $n$'s satisfying (\ref{7.14}),
\[
\om\not\in\cup_{1\leq j\leq n^{\be(2\ell h+\ve')}}\cap_{1\leq i\leq\ell}T^{-q_i(j)}C_{\be\ln n}
\supset\cup_{n\leq j\leq n^{\be(2\ell h+\ve')}}\cap_{1\leq i\leq\ell}T^{-q_i(j)}C_{\be\ln j}
\]
which implies that there exists a sequence $n_k\to\infty$ as $k\to\infty$ such that
\begin{equation}\label{7.15}
\sum_{1\leq j\leq n_k}\prod_{1\leq i\leq\ell}\bbI_{C_{\be\ln j}(\tilde\om)}\circ T^{q_i(j)}(\om)=
\sum_{1\leq j\leq n_k^{\be(2\ell h+\ve')}}\prod_{1\leq i\leq\ell}\bbI_{C_{\be\ln j}(\tilde\om)}\circ T^{q_i(j)}(\om)
\end{equation}
for each $k$.
By the Shannon--McMillan--Breiman theorem there are $\tilde\Om\subset\Om$ with $P(\tilde\Om)=1$ and
a random variable $J$ finite on $\tilde\Om$ such that for any $j\geq J(\tilde\om)$,
\[
j^{-\be(2h+\frac \ve\ell)}=e^{-(2h+\frac \ve\ell)\be\ln j}\leq P(C_{\be\ln j}(\tilde\om))<
e^{-(2h-\frac \ve\ell)\be\ln j}=j^{-\be(2h-\frac \ve\ell)}.
\]
Hence, there are random variables $k_1$ and $k_2$ such that for all $n$ large enough,
\begin{equation}\label{7.16}
k_1(\tilde\om)n^{1-\be(2\ell h+\ve)}\leq\sum_{j=1}^n\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell\leq k_2(\tilde\om)n^{1-\be(2\ell h-\ve)}.
\end{equation}
It follows that for $(\om,\tilde\om)\in\Gam,\,\tilde\om\in\tilde\Om$ and all $k$ large enough
\begin{eqnarray}\label{7.17}
&\frac {\sum_{1\leq j\leq n_k}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell}
{\sum_{1\leq j\leq n_k^{\be(2\ell h+\ve')}}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell}\\
&\leq\frac {k_2(\tilde\om)}{k_1(\tilde\om)}n_k^{(1-\be(2\ell h-\ve)-\be(2\ell h+\ve')(1-\be(2\ell h+\ve))}\to 0\,\,\mbox{as}\,\, k\to\infty
\nonumber\end{eqnarray}
since by the choice of $\ve,\ve'$ and $\be$,
\begin{eqnarray*}
&(1-\be(2\ell h-\ve)-\be(2\ell h+\ve')(1-\be(2\ell h+\ve))\\
&=(1-\be(2\ell h+\ve)(\frac {1-\be(2\ell h-\ve)}{1-\be(2\ell h+\ve)}-\be(2\ell h+\ve')<0.
\end{eqnarray*}
By (\ref{7.15}) we obtain from (\ref{7.17}) that
\begin{eqnarray}\label{7.18}
&\frac {\sum_{1\leq j\leq n_k}\prod_{1\leq i\leq\ell}\bbI_{C_{\be\ln j}(\tilde\om)}
\circ T^{q_i(j)}(\om)}{\sum_{1\leq j\leq n_k}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell}\\
&\times\frac {\sum_{1\leq j\leq n_k^{\be(2\ell h+\ve')}}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell}
{\sum_{1\leq j\leq n_k^{\be(2\ell h+\ve')}}\prod_{1\leq i\leq\ell}\bbI_{C_{\be\ln j}(\tilde\om)}\circ T^{q_i(j)}(\om)}\to\infty\,\,\,\mbox{as}\,\,\, k\to\infty.\nonumber
\end{eqnarray}
By (\ref{7.16}) for all $\tilde\om\in\tilde\Om$,
\[
\sum_{1\leq j\leq n}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell\to\infty\,\,\,\mbox{as}\,\,\, n\to\infty,
\]
and so by Theorem \ref{thm2.3} for $P$-almost all $\om$,
\[
\frac {\sum_{1\leq j\leq n}\prod_{1\leq i\leq\ell}\bbI_{C_{\be\ln j}(\tilde\om)}
\circ T^{q_i(j)}(\om)}{\sum_{1\leq j\leq n}\big(P(C_{\be\ln j}(\tilde\om))\big)^\ell}
\to 1\,\,\,\mbox{as}\,\,\, n\to\infty.
\]
Thus, (\ref{7.18}) can hold true only for a set of pairs $(\om,\tilde\om)$ having $P\times P$-measure zero, and so $P\times P(\Gam)=0$. Since $\ve$ and $\ve'$ can be chosen arbitrarily close to zero,
(\ref{7.12}) follows for $P\times P$-almost all $(\om,\tilde\om)$, which together with (\ref{7.1}) completes the proof of Theorem \ref{thm2.6}. \qed |
2302.10536 | \section{Introduction}
Emotional voice conversion (EVC) attempts to modify perceived emotion style in a given speech signal to a particular target emotion style without modifying the linguistic content of the speech signal \cite{zhou2022emotional}. Early stage of EVC approaches \cite{tao2006prosody,aihara2014exemplar,luo2016emotional,ming2016deep,gao2018nonparallel,robinson2019sequence} mostly focused on speaker-dependent scenarios where they converted emotion styles for a single speaker. Recently, several works attempt to alter both perceived speaker identity and emotion style of a given source speech signal \cite{kishida2020simultaneous,du2021identity}. These methods have potential applications in various tasks such as movie dubbing, conversational assistance, cross-lingual synthesis, etc. Most of the previous approaches can convert the emotion of a speaker whose emotional data is present either at the time of training \cite{moritani2021stargan,he2021improved,zhou2021limited,choi2020stargan} or testing \cite{qian2021global,zhou2021seen}.
However, collecting emotional voice for target speakers is often expensive, time-consuming, and sometimes impossible. In this paper, we address the problem of converting the emotion of speakers whose only neutral data (data having only neutral emotion) is present by leveraging emotional speech data from other supporting speakers. In particular, the target speakers' emotional voice data is never available during the training and testing, and the proposed model learns to infer emotional voice of the target speakers from other supporting speakers of whom we have emotional data.
For brevity, we call speaker-emotion pairs present in the dataset as seen pairs and pairs having speaker-emotion combinations absent in the dataset as unseen pairs.
Some EVC approaches tackle unseen emotion cases by putting aside an entire category of emotion style during training \cite{zhou2021seen,schnell2021emocat,qian2021global}. \cite{zhou2021seen} achieves the EVC task for an unseen emotion by utilizing a pre-trained emotion classifier for generating emotional embeddings. However, their focus is to convert a voice to a particular unseen emotion class for a speaker who has emotional data for several seen emotion categories. In contrast, our aim is to convert emotion for speakers who do not have any emotional recordings. Some approaches attempt emotion voice conversion for the unseen speakers, i.e., speaker-independent EVC \cite{zhou2020converting,shankar2019multi}, by putting aside a few speakers as a heldout database \cite{zhou2020converting,shankar2019multi}. However, these methods utilize both neutral and emotional data from the seen speakers during training. These approaches do not include data from speakers whose only neutral data is available during the training. In addition, these methods are one-to-one and they cannot convert simultaneously both speaker identity as well as emotional identity. However, our approach is many-to-many and can convert both speaker and emotion identities together. To the best of the authors' knowledge, this is the first attempt to consider the specific case of the EVC task for unseen speaker-emotion pair, which fits in the more realistic dubbing applications of EVC, since we often have neutral data but not the emotional data from the desired target speaker.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.85\linewidth]{StarGANv2_EVC.png}
\caption{Block diagram of the proposed EVC-USEP architecture.}
\label{fig:evc_block}
\vspace{-0.2cm}
\end{figure*}
\indent Among various approaches for the EVC task, autoencoder \cite{gao2018nonparallel,zhou2020converting,schnell2021emocat,qian2021global} and generative adversarial network (GAN)-based approaches \cite{zhou2021seen,he2021improved,starganv2vc} are capable of achieving emotional voice conversion in the case of non-parallel training data.
Recently, a StarGANv2-based \cite{choi2020stargan} approach called StarGANv2-VC has demonstrated excellent effectiveness over different style conversion tasks \cite{starganv2vc}. However, the original StarGANv2-VC is designed to separately perform either speaker conversion for seen target speakers or EVC for seen emotion style and a seen target speaker. In this paper, we first modify the StarGANv2-VC architecture for converting the speaker and emotion styles simultaneously in a unified model by utilizing two encoders for learning speaker style and emotion style embeddings along with dual domain source classifiers for classifying source speaker and the emotion style. We then devise training strategies to achieve EVC for \textit{U}nseen \textit{S}peaker-\textit{E}motion \textit{P}airs (i.e., EVC-USEP) by using emotional data from supporting speakers. In particular, we propose a Virtual Domain Pairing (VDP) training strategy, which randomly generates the combinations of speaker-emotion pairs that are not present in the real data without compromising the min-max game of a discriminator and generator in adversarial training. In particular, a fake-pair masking (FPM) strategy is proposed to ensure that the discriminator does not overfit because of the fake pairs. We refer our proposed system as EVC-USEP throughout the paper.
We experimentally show that EVC-USEP successfully converts the emotion of speakers who do not contain any emotional speech during training and testing. We present the effectiveness of the proposed model against baseline and several ablation studies via objective and subjective evaluations. The key contributions from the paper are summarized below.
{\setlength{\leftmargini}{12pt}
\begin{itemize}
\item We propose EVC for the Unseen speaker-emotion Pairs (EVC-USEP) task, where target speakers do not have any emotional data during both training and testing time.
\item We propose an EVC-USEP network by incorporating two separate encoders and dual domain source classifiers.
\item We further propose Virtual Domain Pairing (VDP) and fake-pair masking (FPM) training strategies for generalizing the model to unseen speaker-emotion pairs.
\item We have also applied the annealing strategy in our proposed architecture which improves the emotion conversion accuracy.
\end{itemize}
}
\section{Related work}
Broadly, EVC approaches can be classified into two categories, parallel and non-parallel approaches. In parallel EVC task, the training data consists of utterances having the same linguistic content that is spoken by the speakers in multiple emotions. Although parallel datasets allow the usage of supervised learning-based methods which are in general easier to train, one of the disadvantages of using such datasets is the unnaturalness of having uncorrelated emotion and linguistic content. Also, collecting parallel dataset is expensive and sometimes not possible. Earlier studies have applied Gaussian mixture model (GMM) \cite{tao2006prosody}, sparse representations \cite{aihara2014exemplar}, deep belief network \cite{luo2016emotional}, sequence-to-sequence \cite{ming2016deep,zhou2021limited,robinson2019sequence} and rule-based approaches \cite{xue2018voice} to achieve parallel EVC task.
In the case of non-parallel EVC, the utterances may not have the same linguistic content across multiple speakers. This allows the utterances to have correlated linguistic content and emotion style. Unsupervised methods, namely, CycleGAN \cite{liu2020emotional,zhou2020transforming,shankar2020non}, and autoencoder-based approaches \cite{gao2018nonparallel,zhou2020converting,schnell2021emocat,qian2021global} have been proposed for non-parallel EVC tasks. These methods are suitable for seen speaker-emotion pairs. Recently, variational autoencoding Wasserstein generative adversarial network (VAW-GAN) based methods have been explored that can be extended to achieve EVC for unseen speakers and unseen emotion cases \cite{zhou2020converting,zhou2021seen}. Apart from this, StarGAN \cite{moritani2021stargan,he2021improved} and StarGANv2-VC \cite{starganv2vc} approaches have also become popular for speaker dependent EVC task.
\section{Proposed method}
\subsection{EVC-USEP}
The proposed EVC-USEP architecture is shown in Fig. \ref{fig:evc_block}. It extends StarGANv2-VC \cite{starganv2vc} by treating the speaker style and emotion styles as individual domains (as shown in Fig. \ref{fig:evc_block}). To this end, we incorporate two style encoders, $S_{sp}$ and $S_{em}$, that separately extract the speaker style embedding $h_{sp}=S_{sp}(R_{sp},y_{sp})$ and the emotion style embedding $h_{em}=S_{em}(R_{em},y_{em})$ from reference spectrograms $R_{sp}$, $R_{em}$ to specify the target speaker and emotion style, respectively. Here, $y_{sp}$ is the speaker domain code and $y_{em}$ is the emotion domain code. The domain codes are used to apply domain specific projections to obtain the style embeddings.
The generator $G$ converts the style of source spectrogram $X$ into the target style specified by the style embeddings as $G(X,h_{f0},h_{sp},\\h_{em})$.
As in the original StarGANv2-VC, we also incorporate the information of the fundamental frequency by conditioning the generator with a pitch embedding $h_{f0}$ extracted by a pre-trained joint detection and classification (JDC) network \cite{jdc} to guide the conversion. Hereafter, we omit $h_{f0}$ for clarity.
Along with the style encoders, we also train two mapping networks for generating speaker and emotion style embeddings from a latent variable sampled from a normal distribution, as done in StarGANv2-VC for the single domain case \cite{starganv2vc}. This enables us to specify the target style without having any references during the inference time.
For adversarial training, we use a domain-specific discriminator $D(.,y)$ which predicts whether a sample is real or converted (fake). Here, $y$ is the concatenation of $(y_{sp},y_{em})$. Also, $y_{src}$ and $y_{trg}$ are for source and target speakers, respectively. It has shared layers for all the domains, followed by a domain-specific linear layer. In addition, we incorporate two separate domain classifiers $C_{sp}$ and $C_{em}$ which classify the source speaker domain and emotion domain of the converted samples, respectively.
We use the following training objectives for optimization.\\
\textbf{Adversarial loss:}
The generator $G$ learns to generate a realistic mel sepctrogram via the domain-specific adversarial loss:
\begin{equation}
\label{eq:adv}
\begin{split}
\mathcal{L}_{adv}= &
\mathbb{E}_{X,y_{src}}[\log D(X,{y_{src}})] + \\ &
\mathbb{E}_{X,y_{trg},h_{sp},h_{em}}[\log (1-D(G(X,h_{sp},h_{em}),y_{trg}))].
\end{split}
\end{equation}
\textbf{Adversarial source classifier loss:} The two source domain classifiers, $C_{sp}, C_{em}$ are trained to classify the source speaker and emotion, respectively, using cross entropy loss $\CE(.)$:
\begin{equation}
\label{eq:advcls}
\begin{split}
\mathcal{L}_{advcls}=&
\mathbb{E}_{X,y_{sp},h_{sp}}[\CE(C_{sp}(G(X,h_{sp},h_{em})),y_{sp})] + \\ &
\mathbb{E}_{X,y_{em},h_{sp}}[\CE(C_{em}(G(X,h_{sp},h_{em})),y_{em})].
\end{split}
\end{equation}
where, $y_{sp}$ and $y_{em}$ are from source domain codes. In contrast, the generator is trained to minimize $\mathcal{L}_{advcls}$ by taking $y_{sp}$ and $y_{em}$ from target domain codes.\\
\textbf{Style reconstruction loss:} Style reconstruction loss is used to ensure that the style codes can be reconstructed from the converted sample.
\begin{equation}
\label{eq:sty}
\begin{split}
\mathcal{L}_{sty}=&\mathbb{E}_{X,y_{sp},h_{sp}}|| h_{sp}-S_{sp}(G(X,h_{sp},h_{em}),y_{sp})||_1\\
& \mathbb{E}_{X,y_{em},h_{em}}|| h_{em}-S_{em}(G(X,h_{sp},h_{em}),y_{em})||_1
\end{split},
\end{equation}
where, $y_{sp}$ and $y_{em}$ are from reference domain codes.\\
\textbf{Style diversification loss:} To ensure the diversity of generated samples with different style embeddings, we modified a style diversification loss to incorporate both speaker and emotion styles. We extract two sets of style embeddings, $(h_{sp},h'_{sp})$ and $(h_{em}, h'_{em})$, from the same domain $y_{sp},y_{em}$ and maximize the L1 distance of samples generated using different combination of style embeddings as
\begin{equation}
\label{eq:ds}
\begin{split}
&\mathcal{L}_{ds}=\mathbb{E}_{X,y_{sp},h_{sp}}|| G(X,h_{{sp}},h_{{em}})-G(X,h_{sp},h'_{em})||_1\\
&+\mathbb{E}_{X,y_{sp},h_{sp}}|| G(X,h_{sp},h_{em})-G(X,h'_{sp},h_{em})||_1\\
&+\mathbb{E}_{X,y_{sp},h_{sp}}|| G(X,h'_{sp},h_{em})-G(X,h'_{sp},h'_{em})||_1.
\end{split}
\end{equation}
\textbf{$F_0$ consistency loss}: To produce $F_0$ consistent converted voices, $F_0$ consistent loss is utilized and it is given by \cite{starganv2vc}:
\begin{equation}
\mathcal{L}_{F0}=\mathbb{E}_{X,h_{sp},h_{em}}[||\hat{F}(X)-\hat{F}(G(X,h_{sp},h_{em}))||_1],
\end{equation}
where $\hat{F}(X)$ provides normalized $F_0$ values from an input mel spectrogram.\\
\textbf{Norm consistency loss:} We utilize norm consistency loss ($\mathcal{L}_{norm}$) in order to preserve speech/silence interval in converted spectrum and it is given by \cite{starganv2vc}:
\begin{equation}
\mathcal{L}_{norm}=\mathbb{E}_{X,h_{sp},h_{em}}\Bigg[\frac{1}{T}\sum_{t=1}^{T}\Big| ||X_{.,t}||- ||G(X,h_{sp},h_{em})_{.,t}|| \Big|\Bigg],
\end{equation}
where $||X_{.,t}||$ is norm of absolute column sum for a mel spectorgram $X$ with total $T$ frames at the $t^{th}$ frame. We propose modifications to $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$, which is discussed in sub-section 3.4. Other losses, such as, the speech consistency loss and cycle consistency loss have been preserved as it is from StarGANv2-VC\cite{starganv2vc}. Finally, the full objective is the weighted sum of all these losses.
\subsection{Virtual Domain Pairing (VDP)}
At the time of inference, we aim at converting the emotion style of the source speaker's voice by providing an emotion style embedding of the target emotion. However, since emotional data is available only for the supporting speakers but not for the target speakers, the model does not generalize well to unseen emotion of target speakers if we train the model with speaker-emotion pairs which exist in the training data. Fortunately, the proposed EVC-USEP architecture enables us to virtually incorporate speaker-emotion pairs that do not exist in the training data. When we sample style embeddings in Eq. \eqref{eq:adv}, \eqref{eq:advcls}, \eqref{eq:sty}, and \eqref{eq:ds}, we independently sample the target speaker domain $y_{sp}$ and the emotion domain $y_{em}$ and compute style embeddings by using either the style encoders or the mapping networks.
This allows us to virtually generate emotional references for target speakers having only neutral data by pairing them with emotional references from supporting speakers. We refer to the proposed sampling strategy as Virtual Domain Pairing (VDP).
\subsection{Fake-Pair Masking (FPM)}
When using the VDP, the discriminator is always trained to predict the fake class for unseen speaker-emotion pairs because for unseen speaker-emotion case, real data is unavailable. Therefore, even if the generated samples sound realistic, the discriminator will collapse, and always predict the converted samples as fake.
In such a case, the generator can try to fool the discriminator by generating a voice that sounds like a seen pair, which limits the conversion capability for unseen speaker-emotion pairs and can promote undesired speaker conversion.
To address this problem, we introduce the FPM strategy. We mask samples of unseen speaker-emotion pairs for the real-fake discriminator training and thus, the discriminator is trained only on mel-spectrograms from seen speaker-emotion pairs. In contrast, the goal of source classifiers is to predict the source domain regardless of real or fake pairs. Thus, domain classifiers are only responsible for converting the emotion and speaker identity of unseen pairs. Hence, they should be less prone to fake pairs and should not suffer from the aforementioned problem. In particular, We have experimentally shown the effectiveness of FPM and VDP strategies by doing the ablation studies reported in section 4.
\subsection{Annealing strategy}
$F_0$ consistency and norm consistency losses ensure that normalized $F_0$ value and speech/silence interval remain same after emotion conversion. However, we hypothesize that $F_0$ and speech/silence regions are emotion dependent and as such should be modified during emotion conversion. Hence, we applied a loss annealing strategy (i.e., we linearly decrease the weights assigned to these losses) with each epoch while training the proposed EVC-USEP model. In addition, we have also trained one model w/o annealing (i.e., given in \cite{starganv2vc}) for comparison and one model w/o $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$ in order to check the effectiveness of the annealing strategy on these losses.
\section{Experiments}
\subsection{Experimental Setup}
\textbf{Hindi Emotional Speech Database:} To demonstrate the ability of our models to modify the emotion of speakers having only neutral speech data, we train our models on a set of target speakers having only neutral voices and a set of supporting speakers having multiple emotions. We have utilized an internally developed 54 hours of Hindi emotional speech corpus for this task. The dataset is recorded by total \textit{9} speakers. We have collected text from stories and categorized these texts in six emotions, namely, Neutral, Happy, Sad, Fear, Surprise, and Angry. Each speaker has recorded 1 hour of data in each emotion. In particular, we have utilized emotional data from \textit{7} speakers and only neutral data from remaining \textit{2} speakers for training with an aim to test the EVC models for the unseen speaker-emotion cases. The Hindi database is downsampled to 24kHz. We randomly split all the utterances in the dataset for training and testing in the ratio of 9:1. \\
\textbf{Model Details:} We train all the networks for 150 epochs, with a batch size of 10. The speaker and emotion domain classifiers are introduced in the training process after 50 epochs. Both source domain classifiers have the same architecture as the real-fake discriminator, which follows the same architecture given in StarGANv2-VC \cite{starganv2vc}. The annealing loss strategy is applied after 50 epochs by decreasing the weightage of the $\mathcal{L}_{F0}$, \& $\mathcal{L}_{norm}$ losses linearly from 5 to 0. The other training details are kept the same as reported in \cite{starganv2vc}.
During inference, we convert the emotion of speech samples taken from 2 speakers, whose only neutral data was utilized during training.
We use a pre-trained Parallel WaveGAN\cite{pwg} to synthesize waveforms from mel-spectrograms as in StarGANv2-VC\cite{starganv2vc}. The mapping network is used for extracting style embedding during inference.\\
\textbf{Baseline Model and Ablation Studies}: To the best of authors` knowledge this is the first attempt to consider the specific case of the EVC task for unseen speaker-emotion pair given only neutral emotional data of the speaker. Among earlier approaches, VAW-GAN-based EVC was proposed to tackle the case of unseen speaker \cite{zhou2020converting}. Thus, for baseline, we have considered training the VAW-GAN-based EVC model while utilizing unseen speaker-emotion pair data \cite{zhou2020converting}. However, the VAW-GAN-based EVC model is limited to converting just the emotion whereas our model can convert the emotion as well as the speaker identity. We evaluate our model on the more challenging task of converting both emotion and speaker identity, but because of the limitations of VAW-GAN we have kept the same source and target speakers during conversion for the case of baseline.
In addition to the comparison with the baseline model, we conducted several ablation studies to verify the effectiveness of the VDP and FPM strategies. For this, we do an ablation study on the VDP and FPM. Moreover, the proposed EVC-USEP architecture has utilized annealing strategies for $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$. Hence, we do an ablation study on the annealing strategy by comparing proposed architecture w/o annealing case and w/o $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$ loss cases.
\begin{table}[t]
\caption{Subjective and objective evaluations results. MOS are shown for quality along with margin of error corresponding to 95\% confidence interval.}
\vspace{-0.3cm}
\label{table:1}
\centering
\resizebox{0.5\textwidth}{!}{%
\begin{tabular}{cccc}
\hline
& Subjective & \multicolumn{2}{c}{Objective} \\
\textbf{Method} & \textbf{Quality} & \textbf{Spk. SIM} & \textbf{Emo. Acc} \\ \hline \hline
VAW-GAN \cite{zhou2020converting} & 1.90 $\pm$ 0.02 & 0.61 & 34.03\\
EVC-USEP (Proposed) & \textbf{4.21 $\pm$ 0.01} & \textbf{0.76} & \textbf{41.50} \\ \hline
w/o VDP & 3.82 $\pm$ 0.02 & 0.72 & 24.83 \\
w/o FPM & 3.92 $\pm$ 0.02 & 0.74 & 37.76 \\
w/o annealing & 3.96 $\pm$ 0.02 & 0.73 & 36.73 \\
w/o $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$ & 3.87 $\pm$ 0.02 & 0.71 & 30.95 \\ \hline
\end{tabular}
}
\vspace{-0.3cm}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Abx.png}
\center
\vspace{-0.6cm}
\caption{ABX Subjective Evaluation for Emotion Similarity.}
\label{fig:abx}
\vspace{-0.6cm}
\end{figure}
\vspace{-0.5cm}
\subsection{Subjective Evaluation}
We have conducted two subjective tests, namely, mean opinion scores (MOS) and ABX test to evaluate the quality of converted voices and evaluation of emotion conversion, respectively. Demo audio samples can be found online\footnote{Available at \href{https://demosamplesites.github.io/EVCUP/}{https://demosamplesites.github.io/EVCUP/}}. Total 14 evaluators (with no known hearing impairments and having age between 18 to 34 years) participated in both subjective tests. We asked evaluators to rate the naturalness of each audio clip on a scale of 1 to 5, where, 1 indicates completely distorted quality and 5 indicates high-quality natural voice. A total of 112 samples are evaluated for each system in the MOS test. The results of the subjective evaluation, as shown in Table \ref{table:1}, conclude that our model outperforms the baseline model in terms of the quality of converted voices. The ablation study also shows that the proposed VDP and FPM strategies are crucial for achieving high-quality emotion conversion for speakers having only neutral data. Moreover, the tests confirm that our proposed idea of applying the annealing strategy to $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$ improves the results compared to w/o annealing and w/o $\mathcal{L}_{F0}$ and $\mathcal{L}_{norm}$ experiments.
\indent To evaluate the emotion conversion, we conducted three ABX tests between 1) baseline VAW-GAN and proposed EVC-USEP 2) w/o VDP model and proposed EVC-USEP and 3) w/o FPM model and proposed EVC-USEP. Given two converted samples, we asked the evaluators to prefer the converted voice which has more similar emotion w.r.t. the provided target emotional reference audio signal. A total of 140 samples are considered for evaluating each system in the ABX test. Result of ABX tests are shown in Fig. \ref{fig:abx}. We can observe that the proposed model has preferred more in the context of emotion conversion w.r.t. the baseline. In addition, the proposed model is preferred more compared to the system w/o VDP and w/o FPM in the context of emotional similarity.
\subsection{Objective Evaluation}
Since our goal is to achieve EVC for speakers having only neutral data, we do not have ground truth for the converted samples. Hence, deterministic evaluation metrics such as spectral distortion-based objectives are not appropriate for the problem we consider. Alternatively, we use an emotion classification network to evaluate the accuracy of emotion conversion. We train the emotion classification model on the Hindi emotional dataset in a similar way to \cite{zhou2021seen}.
Classification accuracy obtained on the converted samples are summarized in Table \ref{table:1}. From Table \ref{table:1}, we observe that accuracy of our proposed model is significantly higher compared to the baseline. The ablation study again confirms the effectiveness of VDP and FPM strategies for unseen speaker-emotion pairs.
The speaker similarity score is used to evaluate speaker similarity of the converted voices. We calculate it by taking the inner product between the speaker embeddings extracted from the converted samples and a reference target speaker's sample \cite{wan2018generalized,Resemblyzer}. Table \ref{table:1} also contains speaker similarity scores for different experiments and we can observe that our proposed model has higher speaker similarity scores.
\section{Conclusion}
We proposed the EVC-USEP architecture to tackle the problem of converting the emotion of target speakers whose only neutral data is present by leveraging emotional speech data from other supporting speakers. We further propose Virtual Domain Pairing and FPM strategies which are shown to be essential for achieving the emotional voice conversion for unseen speaker-emotion pairs task.
From both objective and subjective evaluations, we confirm that the proposed method successfully converts the emotion of the target speakers, outperforming the baselines w.r.t. emotion similarity, speaker similarity, and quality of the converted voices, while achieving decent naturalness. In the future, we would like to further improvise the speaker conversion capability of the proposed model for the unseen target speaker case.
\bibliographystyle{IEEEtran.bst} |
2211.00171 | \section{Introduction}
\label{sec:intro}
Human experience is permeated by emotions. They can guide our attention and influence our information consumption, beliefs, and our interactions~\cite{dukes2021rise, wahl2019emotions}. Deep learning has enabled us to extract affective constructs from natural language~\cite{chochlakis2022label, calvo2015oxford}, allowing emotion recognition from text at scale~\cite{guo2022emotion}.
Nevertheless, the need for better performance across various metrics of interest still exists.
When inferring emotions from text, earlier approaches have utilized emotion lexicons \cite{pennebaker2001linguistic}. These struggle in more realistic settings, because, for example, they do not handle context, like negation. On the other hand, while modern efforts relying on deep learning achieve better performance \cite{chochlakis2022label, baziotis2018ntua}, these data-driven models have to contend with a multitude of biases, such as annotation biases in the data used to train the models.
The needs for emotional inference from text have also diversified. First, the domain of interest can vary greatly between applications, ranging from everyday dialogues \cite{li2017dailydialog} to tweets \cite{mohammad2018semeval}. Secondly, it is desirable for the models to be able to handle multiple languages \cite{mohammad2018semeval}, such as when studying perceptions and reactions to international news stories. Finally, the emotions of interest can differ between applications, and perhaps even the annotation scheme might not be similar. For example, in this work, we analyzed tweets annotated for clusters of emotions, where emotions that co-occur frequently were grouped, in contrast to single emotions in other settings. Hence, transfer learning is hindered by the mismatch.
In this work, our main goal is to achieve transfer of emotion recognition between annotation formats, emotions and languages. Our experimental design examines this step-by-step. First, we leverage pretrained multilingual language models \cite{barbieri2022xlmt, devlin2018bert} to enable knowledge transfer between languages. We also use and extend \verb+Demux+ \cite{chochlakis2022label}, a model that incorporates the labels in its input space to achieve the final classification. Emotions are then embedded in the same space as the language tokens. Emotion word embeddings can facilitate transfer between emotions, as shown in \cite{chochlakis2022label}, so we study whether this can also be achieved in a zero-shot manner. Lastly, we examine how an extension of \verb+Demux+ to clusters can transfer knowledge between different annotation formats by directly performing operations on label embeddings \cite{mikolov2013linguistic}.
Our contributions include the following:
\begin{itemize}
\item We show that multilingual emotion recognition models can be competitive with or even outperform monolingual baselines, and that knowledge can be transferred to new languages.
\item We demonstrate that \verb+Demux+ can inherently transfer knowledge to emotions it has not been trained with.
\item We illustrate that operations on the contextual emotion embeddings of \verb+Demux+ can successfully achieve transfer to novel annotation formats in a zero-shot manner. To the best of our knowledge, we are the first to study this setting.
\item We show that \verb+Demux+ can be critical for flexible emotion recognition in a dynamic environment with ever-changing inference needs, such as the addition and subtraction of emotion types, changes in language, and alterations in the annotation format, e.g., the clustering of different emotions.
\end{itemize}
\section{Related Work}
\label{sec:related}
\subsection{Emotion Recognition}
Earlier works utilized Bag-of-Words algorithms driven by emotion lexicons. For instance, LIWC \cite{pennebaker2001linguistic} is a lexicon that is widely used to perform word counting, while DDR \cite{garten2018dictionaries} extends lexicon-based methods from word counting to computing similarities between words. More recently, deep learning has enabled more accurate extraction of emotion signals from text. Initial efforts have treated the task as single-label, and use a threshold to transform into the desired multi-label output \cite{he2018joint}. LSTMs have been widely used for the task \cite{felbo2017using, baziotis2018ntua} e.g., for SemEval 2018 Task 1 \cite{mohammad2018semeval}, where some also used features from affective lexicons. More recently, Transformers \cite{vaswani2017attention} have dominated the field. \verb+Demux+ and \verb+MEmo+ \cite{chochlakis2022label}, state of the art models in SemEval 2018 Task 1 E-c, prompt BERT-based \cite{devlin2018bert} models in different ways, by including all emotions in the input or \verb+[MASK]+ tokens in language prompts, respectively. They also employ an intra-group correlation loss to further improve performance. Transformers have also been used with other architectures, like Graph Neural Networks~\cite{xu2020emograph}.
\subsection{Multilingual Models \& Emotion Recognition} \label{sec:multilingual-related}
Multilingual transformers attempt to model many languages simultaneously. Normalized sampling from each language is used so that low-resource languages are not severely hindered, which we also adopt. This is achieved, given $\alpha \in [0, 1]$, by transforming the frequency $p_l$ of each language as $p_l^\prime \leftarrow p_l^{\alpha}$ and renormalizing to create the new sampling distribution \cite{lample2019cross}. Note that as $\alpha$ decreases, the distribution becomes more balanced, achieving parity at $\alpha=0$.
BERT~\cite{devlin2018bert} and XLM~\cite{lample2019cross} require preprocessing of languages that do not use spaces to delimit words. Both are trained on 100 languages on Wikipedia, and use $\alpha = 0.7,\ 0.5$ respectively. XLM also uses language embeddings, and incorporates Translation Language Modeling (TLM) as a pretraining technique, which requires parallel data. XLM-R~\cite{conneau2019unsupervised} handles all languages without preprocessing. It decreases $\alpha$ to $0.3$ and switches to the CommonCrawl dataset, which has a more balanced language distribution. It also disposes of language embeddings and TLM. XLM-T~\cite{barbieri2022xlmt} extends XLM-R by finetuning it on tweets to perform multilingual sentiment analysis, as does XLM-EMO~\cite{bianchi2022xlm} for emotion recognition on four emotions.
\subsection{Zero-shot Emotion Recognition}
Very few works explicitly study zero-shot emotion recognition in text with transformer-based models \cite{yin2019benchmarking, plaza2022natural}. They do so by formulating the problem as a \textit{Natural Language Inference} problem, i.e., by creating a different prompt per emotion of interest, and classifying whether each prompt follows from the input sequence (entailment) or not (contradiction). This requires running the model once per emotion, creating a bottleneck for classification. Most similar to ours are earlier approaches that used semantic similarity with emotion word embeddings to classify in a zero-shot manner \cite{sappadla2016using}.
\section{Methodology}
\label{sec:method}
We present the technical details of interest for \verb+Demux+ and our simple extension for it to handle clusters of emotions. Let $E = \{e_i: i\in[n]\}$ be the set of emotions and $C = \{C_i: i\in[m]\}$ be some clustering of $E$ s.t. $n \ge m$, $\cup_{i\in[m]} C_i = [n]$ and $\cap_{i\in[m]} C_i = \emptyset.$
\subsection{Demux}
Let $x$ be an input text sequence. We construct $x^\prime=\text{``}e_1, e_2, \dots, \text{or}\ e_n?\text{''}$ and use a LM $L$ with its corresponding tokenizer $T$:
\begin{equation}
\begin{split}
\tilde{x} = T(x^\prime, x) = (&[CLS], t_{1, 1}, \dots, t_{1, N_1}, \dots,
\\ & t_{n, 1}, \dots, t_{n, N_n}, [SEP], x_1, \dots, x_l),
\end{split}
\end{equation}
where $x_i$ are the tokens from $x$, $t_{i, j}$ the $j$-th subtoken of $e_i$, and \verb+[SEP]+ and \verb+[CLS]+ are special tokens of $T$.
We pass $\tilde{x}$ through $L$ to finally get $\hat{x} = L(\tilde{x})$, where $\hat{x}$ contains one output embedding corresponding to each input token. We denote the output embedding corresponding to $t_{i,j}$ as $\hat{t}_{i, j}\in\mathbb{R}^d$, where $d$ is the feature dimension of $L$. Thereafter, we average-pool the outputs of the subtokens of each emotion, and get the final predictions using a 2-layer neural network, $\text{NN}: \mathbb{R}^d \rightarrow \mathbb{R}$, followed by sigmoid $\sigma$:
\begin{equation}
\forall i \in [n],\quad p(e_i|x) = \sigma(\text{NN}(\frac{\sum_{j=1}^{N_i}\hat{t}_{i, j}}{N_i}))).
\end{equation}
In the case of emotion clusters, $x^\prime$ contains all emotions from every cluster. After the forward pass through $L$, we instead aggregate across all emotions of a cluster and predict, for each cluster:
\begin{equation}
\forall i \in [m],\quad p(C_i|x) = \sigma(\text{NN}(\frac{\sum_{j\in C_i}\sum_{k=1}^{N_j}\hat{t}_{j, k}}{\sum_{j\in C_i}N_j}))).
\end{equation}
Moreover, when using multilingual models, we keep all emotions in English to retain the same prompt, $x^\prime$, across all languages.
\subsection{Correlation-aware Regularization}
To provide extra supervision to the model and enhance its correlation awareness between emotions, we include a label-correlation regularization loss. This loss takes into account the ground-truth labels for each example in its formulation. Therefore, the emotions are split into two groups, the present and the absent emotions based on annotations $y$, $\mathcal{P}$ and $\mathcal{N}$ respectively. We regularize intra-group relationships, meaning we only pick pairs of emotions when they are both in $\mathcal{P}$ or both in $\mathcal{N}$. The formulation is:
\begin{equation}
\begin{split}
\mathcal{L}_{L, \text{intra}}(y, \hat{y}) = \frac{1}{2} & \left[ \frac{1}{|\mathcal{N}|^2 - |\mathcal{N}|} \sum_{(i, j) \in \mathcal{N}^2}^{i > j} e^{\hat{y}_i + \hat{y}_j}\right. +\\
& \left. \frac{1}{|\mathcal{P}|^2 - |\mathcal{P}|} \sum_{(i, j) \in \mathcal{P}^2}^{i > j} e^{-\hat{y}_i-\hat{y}_j} \right], \\
\end{split}
\end{equation}
where $\hat{y}$ is the prediction of the model and subscripts indicate indexing. In this manner, we decrease the distance of pairs of emotions when they have the same gold labels. The denominators simply average the terms. The final loss is a convex combination of the classification and the regularization loss, dictated by hyperparameter $c$:
\begin{equation}
\mathcal{L} = (1 - c)\mathcal{L}_{BCE} + c \mathcal{L}_{L, \text{intra}}.
\end{equation}
\section{Experiments} \label{sec:exp}
\subsection{Datasets}
We use the publicly available SemEval 2018 Task 1 E-c \cite{mohammad2018semeval}, and private data containing tweets from the French elections of 2017.
SemEval 2018 Task 1 E-c (SemEval E-c) contains tweets annotated for 11 emotions in a multilabel setting, namely \textit{anger, anticipation, disgust, fear, joy, love, optimism, pessimism, sadness, surprise,} and \textit{trust}, in \textit{English, Arabic,} and \textit{Spanish}.
The cardinalities are 6838 training, 886 development and 3259 testing for English, 2278 training, 585 development and 1518 testing for Arabic, and 3561 training, 679 development and 2854 testing for Spanish.
We have also machine-translated the English subset of the tweets into French.
\begin{table}[t]
\centering
\begin{tabular}{lrr}
Emotion Cluster & Support in FrE-A & Support in FrE-B \\
\midrule
Admiration & 589 (14.4\%) & 51 (1.1\%) \\
Sarcasm & 395 (9.6\%) & 233 (5.1\%) \\
Anger & 595 (14.5\%) & 279 (6.1\%) \\
Embarrassment & 182 (4.4\%) & 56 (1.2\%) \\
Fear & 271 (6.6\%) & 116 (2.5\%) \\
Joy & 169 (4.1\%) & 40 (0.9\%) \\
Optimism & 501 (12.2\%) & 350 (7.7\%) \\
Pride & 539 (13.1\%) & 92 (2\%) \\
Positive-other & 632 (15.4\%) & 1268 (27.7\%) \\
Negative-other & 542 (13.2\%) & 1426 (31.2\%) \\
\midrule
All tweets & 4102 (100\%) & 4574 (100\%) \\
\end{tabular}
\caption{Support per emotion cluster for our French election datasets.}
\label{tab:fr-datasets}
\end{table}
The French election dataset contains mostly French tweets annotated for 10 clusters of emotions in a multilabel setting, namely \textit{admiration-love, amusement-sarcasm, anger-hate-contempt-disgust, embarrassment-guilt-shame-sadness, fear-pessimism, joy-happiness, optimism-hope, pride (including national pride),} any \textit{other positive}, and any \textit{other negative} emotions. The formulation dictates that if one emotion in the cluster is present, the cluster is considered present. Only cluster-wise annotations are requested.
We have two separate subsets, collected by third parties using keywords related to prominent politicians, agendas and concerns of the French elections, and annotated independently also by third parties. The first (``FrE-A'') contains 4102 tweets that we randomly split into training, development and test sets with ratios 8:1:1. The second subset (``FrE-B'') contains 4574 tweet that we also randomly split with ratios 8:1:1. In Table \ref{tab:fr-datasets}, we present the support for each class. Annotations for FrE-B are a lot sparser.
\subsection{Implementation Details}
We use Python (v3.7.4), PyTorch (v1.11.0) and the \textit{transformers} library (v4.19.2). We use NVIDIA GeForce GTX 1080 Ti.
Given the difference in writing formality on Twitter, we use Twitter-based in addition to general-purpose models. For the former, XLM-T~\cite{barbieri2022xlmt} is our multilingual model, BERTweet~\cite{nguyen2020bertweet} for English, RoBERTuito~\cite{perez2021robertuito} for Spanish, AraBERT~\cite{antoun2020arabert} for Arabic, and BERTweetFR~\cite{guo2021bertweetfr} for French. For the latter, we use multilingual BERT~\cite{devlin2018bert} as our multilingual model, BERT for English and Arabic, and BETO~\cite{CaneteCFP2020} for Spanish.
We retain the hyperparameters used in \cite{chochlakis2022label}, such as the learning rate and its warmup, early stopping, batch size, the convex combination coefficient $c$ ($\alpha$ in \cite{chochlakis2022label}), and the text preprocessor. During multilingual finetuning on SemEval E-c, we sample from different languages with equal probability ($\alpha=0$ in aforementioned normalization in Section \ref{sec:multilingual-related}). We evaluate using the F1 score for individual emotions, and micro F1, macro F1 and Jaccard score (JS) otherwise. We also used Micro F1 for early stopping in FrE-A because we found JS to be unreliable due to the label distribution.
\subsection{Knowledge Transfer between Languages}
\textbf{Multilingual Training and Evaluation}\quad First, we examine how feasible and competitive it is to use multilingual emotion recognition models trained and evaluated on a mixture of languages. To establish this, we first consider training monolingual and multilingual models, pretrained either on tweets or general-purpose text. For multilingual models, we also consider how training and/or evaluating on multiple languages fares with training and/or evaluating on a single language. For example, we compare performance on the Arabic subset when training and performing early stopping solely on Arabic, with the performance when training on all languages simultaneously, but still performing early stopping on Arabic, and other combinations.
We find that training and evaluating on all languages in SemEval E-c has either only slightly negative or strongly positive influence on accuracy for higher-resource languages. Our dev set results are presented in Table \ref{tab:multi-v-mono-v-twitter}. It becomes immediately obvious that models trained on Twitter comfortably outperform their general-purpose alternatives (first and the second row, and third and final row). We also observe that using a dev set of multiple languages not only does not hurt performance, but actually achieves positive knowledge transfer for Spanish, achieving the best JS overall. The monolingual alternatives perform favorably in English and Arabic, with the increase in the former being relatively minor. Overall, since our ultimate goal is to transfer to French tweets, and given the competent or superior performance for Latin-based languages, we do adopt the multilingual model trained and evaluated on multilingual data for pretraining.
\begin{table*}[t]
\centering
\begin{tabular}{c@{\hspace{1.50\tabcolsep}}c@{\hspace{1.50\tabcolsep}}c@{\hspace{ 1.50\tabcolsep}}cccc}
&&&& \multicolumn{3}{c}{\textbf{JS}} \\
\cmidrule(lr){5-7}
\multicolumn{4}{c}{\textbf{Setting}} & En & Es & Ar \\
\cmidrule{1-4} \cmidrule(lr){5-7}
& Monolingual models & & & \rescell{61.0}{0.3} & \rescell{53.2}{0.5} & \rescell{55.1}{0.3} \\
Twitter-based & Monolingual models & & & \rescell{\mathbf{62.0}}{0.4} & \rescell{55.6}{0.3} & \rescell{\mathbf{61.6}}{0.5} \\
& Multilingual models & w/ Multilingual Training & \& Evaluation & \rescell{58.4}{1.0} & \rescell{50.3}{0.4} & \rescell{49.2}{0.0} \\
Twitter-based & Multilingual models && & \rescell{61.6}{0.1} & \rescell{56.8}{1.0} & \rescell{56.5}{0.6} \\
Twitter-based & Multilingual models & w/ Multilingual Training & & \rescell{61.3}{0.4} & \rescell{56.5}{0.6} & \rescell{57.9}{1.0} \\
Twitter-based & Multilingual models & w/ Multilingual Training & \& Evaluation & \rescell{61.3}{0.2} & \rescell{\mathbf{58.0}}{0.7} & \rescell{57.7}{0.4} \\
\end{tabular}
\caption{Comparing Jaccard scores in \textit{SemEval 2018 Task 1 E-c}. The variables we consider are: monolingual or multilingual models, Twitter-based or general-purpose models, monolingual or multilingual training, where the training set is comprised of one or a mixture of all languages, and monolingual or multilingual evaluation, where the evaluation set is comprised of one or a mixture of all languages.}
\label{tab:multi-v-mono-v-twitter}
\end{table*}
\textbf{Transfer to New Languages}\quad In trying to establish how emotion recognition knowledge is transferred to new languages, we conducted experiments with one SemEval E-c language left out during training. Results are shown in Table \ref{tab:lang-zeroshot}. We notice a drop in performance, with all models performing roughly equivalently across all metrics on the new language notwithstanding original performance. In detail, all metrics drop by around $25\%$ in Arabic, $<22\%$ for Spanish, while the drop in English ranges from $18\%$ to $32\%$. Nonetheless, emotion recognition in the new language occurs at a competent level, picking up clear signals despite the noise from the language switch, rendering multilingual emotion recognition models capable of being used with new languages.
\begin{table}[t]
\centering
\begin{tabular}{ccccccc}
&&&& \multicolumn{3}{c}{SemEval 2018 Task 1 E-c} \\
\cmidrule(lr){5-7}
\multicolumn{3}{c}{Train langs} & Eval langs & Mic-F1 & Mac-F1 & JS \\
\toprule
En & Es & & Ar & \rescell{55.8}{0.9} & \rescell{42.4}{2.7} & \rescell{43.3}{1.3} \\
En & Es & Ar & Ar & \rescell{70.2}{0.8} & \rescell{57.4}{1.4} & \rescell{57.9}{1.0} \\
\midrule
En & & Ar & Es & \rescell{55.9}{0.7} & \rescell{45.1}{1.6} & \rescell{44.0}{0.9} \\
En & Es & Ar & Es & \rescell{65.1}{0.6} & \rescell{54.7}{0.8} & \rescell{56.5}{0.6} \\
\midrule
& Es & Ar & En & \rescell{54.6}{0.6} & \rescell{47.0}{1.3} & \rescell{41.9}{0.7} \\
En & Es & Ar & En & \rescell{72.3}{0.3} & \rescell{57.6}{2.0} & \rescell{61.3}{0.4} \\
\end{tabular}
\caption{Leave-one-language-out experiments.}
\label{tab:lang-zeroshot}
\end{table}
\subsection{Knowledge Transfer to New Emotions}
We also assess \verb+Demux+'s ability to perform zero-shot emotion recognition by excluding emotions from SemEval E-c, one at a time. We choose \textit{anger}, \textit{joy}, \textit{pessimism}, and \textit{trust} to capture the change in accuracy across a wide spectrum of performance levels. In particular, \textit{joy} and \textit{trust} are the highest and lowest performing emotions, whereas \textit{anger} and \textit{pessimism} have relatively high and low scores, respectively. Results are presented in Table \ref{tab:emotion-zeroshot}. Again, we do notice a decrease in performance. However, the model can still predict these unseen emotions, especially those with which the original model was competent, whereas \textit{pessimism} deteriorates significantly.
To remedy the decrease in performance, we experimented with freezing word embeddings in an effort to retain the relationships between emotions. We examine two alternatives, freezing the word embedding layer altogether, and freezing only the emotion word embeddings (including the novel emotions). Both techniques are also presented in Table \ref{tab:emotion-zeroshot}. In general, we find this decreases performance for the model, indicating \verb+Demux+ already competently captures relationships across emotions in its input.
\begin{table}[t]
\centering
\begin{tabular}{cccccc}
&& \multicolumn{4}{c}{ SemEval 2018 Task 1 E-c F1} \\
\cmidrule(lr){3-6}
ZS & Frozen & Anger & Joy & Pessimism & Trust \\
\toprule
\ding{55} & - & \rescell{78.6}{0.3} & \rescell{86.0}{0.3} & \rescell{40.5}{1.5} & \rescell{9.9}{3.9} \\
\cmark & - & \rescell{42.5}{11.8} & \rescell{57.8}{3.2} & \rescell{11.7}{0.9} & \rescell{3.6}{3.5} \\
\cmark & Words & \rescell{25.1}{18.2} & \rescell{51.5}{7.3} & \rescell{15.1}{12.9} & \rescell{1.7}{2.0} \\
\cmark & Emos & \rescell{21.5}{20.8} & \rescell{36.2}{14.3} & \rescell{4.3}{4.3} & \rescell{3.7}{4.6} \\
\end{tabular}
\caption{Zero-shot performance of unseen emotions.}
\label{tab:emotion-zeroshot}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{cc@{\hspace{0.95\tabcolsep}}c@{\hspace{0.95\tabcolsep}}ccc}
&&& \textbf{Mic-F1} & \textbf{Mac-F1} & \textbf{JS} \\
\cmidrule(r){4-6}
& \textbf{Pretrained} & \textbf{Finetuned} & \multicolumn{3}{c}{FrE-A} \\
\toprule
&\multicolumn{2}{c}{Most frequent} & $0$ & $0$ & $24.9$ \\
&\multicolumn{2}{c}{Uni. Random} & \rescell{17.6}{0.4} & \rescell{18.0}{0.4} & \rescell{10.0}{0.2} \\
\midrule
\parbox[t]{2mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{XLM-T}}} & \cmark & \ding{55} & \rescell{21.0}{1.2} & \rescell{15.8}{2.0} & \rescell{26.6}{2.2} \\
&\ding{55} & \cmark & \rescell{28.9}{1.4} & \rescell{25.7}{0.7} & \rescell{\mathbf{29.8}}{1.7} \\
&\cmark & \cmark & \rescell{\mathbf{30.5}}{2.5} & \rescell{\mathbf{26.5}}{5.2} & \rescell{29.4}{2.3} \\
\midrule
\parbox[t]{2mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\makecell{BERT\\TweetFr}}}} & \cmark & \ding{55} & \rescell{24.8}{0.7} & \rescell{18.5}{1.4} & \rescell{15.2}{0.6} \\
&\ding{55} & \cmark & \rescell{27.0}{12.5} & \rescell{24.9}{12.8} & \rescell{29.6}{2.6} \\
&\cmark & \cmark & \rescell{\mathbf{34.3}}{1.1} & \rescell{\mathbf{31.4}}{1.7} & \rescell{\mathbf{31.8}}{1.0} \\
\cmidrule(r){4-6}
&&& \multicolumn{3}{c}{FrE-B} \\
\cmidrule(r){4-6}
&\multicolumn{2}{c}{Most frequent} & $0$ & $0$ & $35.0$ \\
&\multicolumn{2}{c}{Uni. Random} & \rescell{15.0}{0.7} & \rescell{12.3}{0.7} & \rescell{8.3}{0.4} \\
\midrule
\parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{90}{XLM-T}}} & \cmark & \ding{55} & \rescell{18.3}{3.7} & \rescell{11.5}{1.9} & \rescell{29.4}{4.6} \\
&\ding{55} & \cmark & \rescell{55.3}{6.5} & \rescell{19.3}{9.9} & \rescell{55.3}{6.5} \\
&\cmark & \cmark & \rescell{\mathbf{62.7}}{0.9} & \rescell{\mathbf{44.4}}{2.8} & \rescell{\mathbf{59.3}}{0.8} \\
\cmidrule{2-6}
&\multicolumn{1}{l}{* \phantom{11} \cmark} & \ding{55} & \rescell{15.9}{2.6} & \rescell{11.3}{1.0} & \rescell{14.8}{3.3} \\
&\multicolumn{1}{l}{* \phantom{11} \cmark} & \cmark & \rescell{61.9}{1.2} & \rescell{39.6}{9.6} & \rescell{58.4}{0.8} \\
\midrule
\parbox[t]{2mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\makecell{BERT\\TweetFr}}}} & \cmark & \ding{55} & \rescell{27.9}{1.9} & \rescell{15.9}{1.1} & \rescell{15.3}{1.4} \\
&\ding{55} & \cmark & \rescell{63.8}{1.4} & \rescell{29.7}{7.6} & \rescell{60.2}{1.7} \\
&\cmark & \cmark & \rescell{\mathbf{66.9}}{0.5} & \rescell{\mathbf{40.3}}{6.0} & \rescell{\mathbf{63.2}}{1.3} \\
\end{tabular}
\caption{Performance in French election data when the models pretrain on \textit{SemEval E-c} and/or finetune on the corresponding dataset. *: maximum probability instead of pooling embeddings in Demux.}
\label{tab:clusters}
\end{table}
\subsection{Knowledge Transfer to New Annotation Format}
Finally, we evaluate if the model can successfully transfer knowledge to a new annotation format. In particular, we transfer from SemEval E-c to FrE-a and FrE-B. The former contains tweets in English, Spanish and Arabic, and annotations for emotions. The latter contain French tweets annotated for clusters. Additionally, the emotions of the latter are not a subset of the former. Therefore, we expect to observe compounding effects from the multiple changes in the setting. To address the language switch, we also use the French translations to train BERTweetFr as a monolingual parallel to \mbox{XLM-T}.
We also show two simple baselines, one that predicts the most frequent label for each emotion, and another that predicts uniformly randomly. Moreover, our experiments include an alternative to averaging emotion embeddings with \verb+Demux+, where we instead select the maximum predicted probability across emotions in a cluster.
Results are presented in Table \ref{tab:clusters}. We consider three different settings for each model and each dataset in order to study how training on SemEval E-c (\textit{Pretrained}) and training on the French election datasets (\textit{Finetuned}) affects performance. In the first row per model and dataset, we see the zero-shot performance on the dataset. The second shows the performance of a model only trained on the dataset. The last row shows performance from a model first trained on SemEval E-c and then on the corresponding dataset.
For FrE-A, zero-shot performance of XLM-T is close to either finetuned performance, indicating the model can successfully transfer despite the complete change of environment. Performance increases for both when training on French data, and when SemEval E-c is also included. The monolingual model performs favorably only when fully supervised, which indicates that translations are not ideal for zero-shot transfer, and overall proves the ability of multilingual models to transfer knowledge to a new language.
Performance is evidently better in FrE-B. Again, XLM-T performs better than random and favorably to BERTweetFr in the zero-shot setting. Finetuning improvements are significant for both models. Lastly, the alternative pooling performs worse than our proposed one.
This speaks to the subjectivity of the annotations, allowing operations on emotion embeddings but not the predictions themselves.
\section{Conclusion}
\label{sec:conc}
In this work, we study how to transfer emotion recognition knowledge to different languages, different emotions, and different annotation formats. We find that multilingual models have the capacity to transfer that kind of knowledge sufficiently well. In order to transfer knowledge between emotions, we leverage \verb+Demux+'s transferability between emotions through word-level associations. We see that the model also inherently performs zero-shot emotion recognition without the need for further changes. Finally, we modify \verb+Demux+ to perform aggregation operations on its label embeddings, and show this can transfer knowledge to novel annotation formats, such as clusters of emotions, even in conjunction with the presence of novel emotions and in a different language. We show that multilingual models pretrained on other languages perform favorably in the zero-shot setting to native models pretrained on machine translations.
\section{Acknowledgements}
{\footnotesize This project was funded in part by DARPA under contract HR001121C0168.}
\bibliographystyle{acm} |
0709.4280 | \section{Introduction}
\begin{defn}\label{def:ca}
Let $G$ be a group. A finite \emph{cellular automaton} on $G$ is a
map $\theta:Q^S\to Q$, where $Q$, the \emph{state set}, is a finite
set, and $S$ is a finite subset of $G$.
\end{defn}
Note that usually $G$ is infinite; much of the theory holds trivially
if $G$ is finite. $S$ could be taken to be a generating set of $G$,
though this is not a necessity.
A cellular automaton should be thought of as a highly regular animal,
composed of many cells labeled by $G$, each in a state $\in Q$. Each
cell ``sees'' its neighbours as defined by $S$, and ``evolves''
according to its neighbours' states.
More formally: a \emph{configuration} is a map $\phi:G\to Q$. The
\emph{evolution} of the automaton $\theta:Q^S\to Q$ is the self-map
$\Theta:Q^G\to Q^G$ on configurations, defined by
\[\Theta(\phi)(x)=\theta(s\mapsto\phi(xs)).\]
Two properties of cellular automata received special attention. Let us
call \emph{patch} the restriction of a configuration to a finite
subset $Y\subseteq G$. On the one hand, there can exist patches that
never appear in the image of $\Theta$. These are called \emph{Garden
of Eden} (GOE), the biblical metaphor expressing the notion of
paradise lost forever.
On the other hand, $\Theta$ can be non-injective in a strong sense:
there can exist patches $\phi'_1\neq\phi'_2\in Q^Y$ such that, however
one extends $\phi'_1$ to a configuration $\phi_1$, if one extends
$\phi'_2$ similarly (i.e.\ in such a way that $\phi_1$ and $\phi_2$
have the same restriction to $G\setminus Y$) then
$\Theta(\phi_1)=\Theta(\phi_2)$. These patches $\phi'_1,\phi'_2$ are
called \emph{Mutually Erasable Patterns} (MEP).
Equivalently\footnote{In the non-trivial direction, let
$\phi_1,\phi_2$ differ on a non-empty finite set $F$; set $Y=F(S\cup
S^{-1})$ and let $\phi'_1,\phi'_2$ be the restriction of
$\phi_1,\phi_2$ to $Y$ respectively.} there are two configurations
$\phi_1,\phi_2$ which differ on a non-empty finite set, with
$\Theta(\phi_1)=\Theta(\phi_2)$. The absence of MEP is sometimes
called \emph{pre-injectivity}.
Cellular automata were initially considered on $G=\Z^n$. Celebrated
theorems by Moore and Myhill~\cites{moore:ca,myhill:ca} prove that, in
this context, a cellular automaton admits GOE if and only if it admits
MEP. This result was generalized by Mach\`\i\ and
Mignosi~\cite{machi-m:ca} to $G$ of subexponential growth, and by
Ceccherini, Mach\`\i\ and Scarabotti~\cite{ceccherini-m-s:ca} to $G$
amenable.
We prove that this last result is essentially optimal, and yields a
characterization of amenable groups:
\begin{thm}\label{thm:main}
Let $G$ be a group. Then the following are equivalent:
\begin{enumerate}
\item the group $G$ is amenable;\label{thm:1}
\item all cellular automata on $G$ that admit MEP also admit GOE.\label{thm:2}
\end{enumerate}
\end{thm}
Schupp had already asked in~\cite{schupp:arrays}*{Question~1} in which
precise class of groups the Moore-Myhill theorem holds.
Ceccherini et al.\ write in~\cite{ceccherini-m-s:ca}\footnote{I changed
slightly their wording to match this paper's}:
\begin{conj}[\cite{ceccherini-m-s:ca}*{Conjecture 6.2}]\label{conj:cms}
Let $G$ be a non-amenable finitely generated group. Then for any
finite and symmetric generating set $S$ for $G$ there exist cellular
automata $\theta_1,\theta_2$ \emph{with that $S$} such that
\begin{itemize}
\item In $\theta_1$ there are MEP but no GOE;
\item In $\theta_2$ there are GOE but no MEP.
\end{itemize}
\end{conj}
As a first step, we will prove Theorem~\ref{thm:main}, in which we
allow ourselves to choose an appropriate subset $S$ of $G$. Next, we
extend a little the construction to answer the first part of
Conjecture~\ref{conj:cms}:
\begin{thm}\label{thm:cms}
Let $G=\langle S\rangle$ be a finitely generated, non-amenable
group. Then there exists a cellular automaton $\theta:Q^S\to Q$ that
has MEP but no GOE.
\end{thm}
We conclude that the property of ``satisfying Moore's theorem'' is
independent of the generating set, a fact which was not obvious
\emph{a priori}.
\section{Proof of Theorem~\ref{thm:main}}
The implication $\eqref{thm:1}\Rightarrow\eqref{thm:2}$ has been
proven by Ceccherini et al.; see
also~\cite{gromov:endomorphisms}*{\S8} for a slicker proof. We prove
the converse.
Let us therefore be given a non-amenable group $G$. Let us also, as a
first step, be given a large enough finite subset $S$ of $G$. Then
there exists a ``bounded propagation $2:1$ compressing vector field''
on $G$: a map $f:G\to G$ such that $f(x)^{-1}x\in S$ and
$\#f^{-1}(x)=2$ for all $x\in G$.
We construct the following automaton $\theta$. Its stateset is
\[Q=S\times\{0,1\}\times S.
\]
Order $S$ in an arbitrary manner, and choose an arbitrary $q_0\in Q$.
Define $\theta:Q^S\to Q$ as follows:
\begin{equation}\label{eq:theta}
\theta(\phi)=\begin{cases}
(p,\alpha,q) & \text{for the minimal pair $s<t$ in $S$ with}
\left\{\!\begin{array}{l} \phi(s)=(s,\alpha,p),\\\phi(t)=(t,\beta,q),
\end{array}\right.\\
q_0 & \text{if no such $s,t$ exist}.
\end{cases}
\end{equation}
\subsection{$\Theta$ is surjective}
Namely, $\theta$ does not admit GOE. Let indeed $\phi$ be any
configuration. We construct a configuration $\psi$ with
$\Theta(\psi)=\phi$.
Consider in turn all $x\in G$; write $\phi(x)=(p,\alpha,q)$, and
$f^{-1}(x)=\{xs,xt\}$ for some $s,t\in S$ ordered as $s<t$. Set then
\begin{equation}\label{eq:psi}
\psi(xs)=(s,\alpha,p),\qquad\psi(xt)=(t,0,q).
\end{equation}
Note that $\psi(z)=(f^{-1}(z)z,*,*)$ for all $z\in G$. Since
$\#f^{-1}(z)=2$ for all $z\in G$, it is clear that, for every $x\in
G$, there are exactly two $s\in S$ such that $\psi(xs)=(s,*,*)$; call
them $s,t$, ordered such that $\psi(xs)=(s,\alpha,p)$ and
$\psi(xt)=(t,0,q)$. Then $\Theta(\psi)(x)=(p,\alpha,q)$, so
$\Theta(\psi)=\phi$.
\subsection{$\Theta$ is not pre-injective}
Namely, $\theta$ admits MEP. Let indeed $\phi:G\to Q$ be any
configuration; then construct $\psi$ following~\eqref{eq:psi}, and
define $\psi'$ as follows. Choose any $y\in G$, write
$\phi(y)=(p,\alpha,q)$, and write $f^{-1}(y)=\{ys,yt\}$ for some
$s,t\in S$, ordered as $s<t$. Define
$\psi':G\to Q$ by
\[\psi'(x)=\begin{cases}
\psi(x) & \text{ if }x\neq yt,\\
(t,1,q) & \text{ if }x=yt.
\end{cases}
\]
Then $\psi$ and $\psi'$ differ only at $yt$; and
$\Theta(\psi)=\Theta(\psi')$ because the value of $\beta$ is unused
in~\eqref{eq:theta}. We conclude that $\theta$ has MEP.
\section{Proof of Theorem~\ref{thm:cms}}
We begin by a new formulation of amenability for finitely generated
groups:
\begin{lem}\label{lem:mn}
Let $G$ be a finitely generated group. The following are equivalent:
\begin{enumerate}
\item the group $G$ is not amenable;
\item for every generating set $S$ of $G$, there exist $m>n\in\N$
and a ``$\mcolonn$ compressing correspondence on $G$ with
propagation $S$''; i.e.\ a function $f:G\times G\to\N$ such that
\begin{align}
\forall y\in G:&\quad\sum_{x\in G}f(x,y)=m,\\
\forall x\in G:&\quad\sum_{y\in G}f(x,y)=n,\\
\forall x,y\in G:&\quad f(x,y)\neq0\Rightarrow y\in xS.
\end{align}
\end{enumerate}
\end{lem}
Note that this definition generalizes the notion of ``$2:1$
compressing vector field'' introduced above.
\begin{proof}
For the forward direction, assuming that $G$ is non-amenable, there
exists a rational $m/n>1$ such that every finite $F\subseteq G$
satisfies
\[\#(FS)\ge m/n\#F.\] Construct the following bipartite oriented
graph: its vertex set is $G\times\{1,\dots,m\}\sqcup
G\times\{-1,\dots,-n\}$. There is an edge from $(g,i)$ to $(gs,-j)$
for all $s\in S$ and all $i\in\{1,\dots,m\},j\in\{1,\dots,m\}$. By
hypothesis, this graph satisfies: every finite $F\subseteq
G\times\{1,\dots,m\}$ has at least $\#F$ neighbours. Since $m>n$ and
multiplication by a generator is a bijection, every finite
$F\subseteq G\times\{-1,\dots,-n\}$ also has at least $\#F$
neighbours.
We now invoke the Hall-Rado theorem~\cite{mirsky:transversal}: if a
bipartite graph is such that every subset of any of the parts has as
many neighbours as its cardinality, then there exists a ``perfect
matching'' --- a subset $I$ of the edge set of the graph such that
every vertex is contained in precisely one edge in $I$. Set then
\[f(x,y)=\#\{(i,j)\in\{1,\dots,m\}\times\{1,\dots,n\}:\;I\text{ contains the edge from }(x,i)\text{ to }(y,-j)\}.
\]
For the backward direction: if $G$ is amenable, then there exists an
invariant measure on $G$, hence on bounded natural-valued functions
on $G$. Let $f$ be a bounded-propagation $\mcolonn$ compressing
correspondence; then
\[m=\sum_{x\in G}\int_{\{x\}\times G}f=\sum_{y\in
G}\int_{G\times\{y\}}f=n,
\]
contradicting $m>n$.
\end{proof}
Let now $G=\langle S\rangle$ be a non-amenable group, and apply
Lemma~\ref{lem:mn} to $G=\langle S^{-1}\rangle$, yielding $m>n\in\N$
and a contracting $\mcolonn$ correspondence $f$. Consider the
following cellular automaton $\theta$, with stateset
\[Q=(S\times\{0,1\}\times S^n)^n.
\]
Choose $q_0\in Q$, and give a total ordering to
$S\times\{1,\dots,n\}$.
Consider $\phi\in Q^S$. To define $\theta(\phi)$, let
$(s_1,k_1)<\dots<(s_m,k_m)$ be the lexicographically minimal sequence
in $(S\times\{1,\dots,n\})^m$ such that
\[\phi(s_j)_{k_j}=(s_j,\alpha_j,t_{j,1},\dots,t_{j,n})\in
S\times\{0,1\}\times S^n\quad\text{ for }j=1,\dots,m.
\]
If no such $s_1,k_1,\dots,s_m,k_m$ exist, set $\theta(\phi)=q_0$;
otherwise, set
\begin{equation}\label{eq:theta2}
\theta(\phi)=((t_{1,1},\alpha_1,t_{2,1},\dots,t_{n+1,1}),\dots,(t_{1,n},\alpha_n,t_{2,n},\dots,t_{n+1,n}))\in Q.
\end{equation}
The same arguments as before apply. Given $\phi:G\to Q$, we construct
$\psi:G\to Q$ such that $\Theta(\psi)=\phi$, as follows. We think of
the co\"ordinates $\psi(x)_k$ of $\psi(x)$ as $n$ ``slots'', initially
all ``free''. By definition, $\#f^{-1}(x)=m$ for all $x\in G$, while
$\#f(x)=n$. Consider in turn all $x\in G$; write
$f^{-1}(x)=\{xs_1,\dots,xs_m\}$, and let
$k_1,\dots,k_m\in\{1,\dots,n\}$ be ``free'' slots in
$\psi(xs_1),\dots,\psi(xs_m)$ respectively. By the definition of $f$,
there always exist sufficiently many free slots.
Mark now these slots as ``occupied''. Reorder $s_1,k_1,\dots,s_m,k_m$
in such a way that $(s_1,k_1,\dots,s_m,k_m)$ is minimal among its $m!$
permutations. Set then
\[\psi(xs_j)_{k_j}=(s_j,\alpha_j,t_{j,1},\dots,t_{j,n})\quad\text{ for }j=1,\dots,m,
\]
where $\alpha_{n+1},\dots,\alpha_m$ are taken to be arbitrary values
(say $0$ for definiteness) and
\[\phi(x)=((t_{1,1},\alpha_1,t_{2,1},\dots,t_{n+1,1}),\dots,(t_{1,n},\alpha_1,t_{2,n},\dots,t_{n+1,n})).\]
Finally, define $\psi$ arbitrarily on slots that are still ``free''.
It is clear that $\Theta(\psi)=\phi$, so $\theta$ does not have GOE.
On the other hand, $\theta$ has MEP as before, because the values of
$\alpha_j$ in~\eqref{eq:theta2} are not used for
$j\in\{n+1,\dots,m\}$.
\section{Remarks}
\subsection{$G$-sets} A cellular automaton could more generally be
defined on a right $G$-set $X$. There is a natural notion of
amenability for $G$-sets, but it is not clear exactly to which extent
Theorem~\ref{thm:main} can be generalized to that setting.
\subsection{Myhill's Theorem} It seems harder to produce
counterexamples to Myhill's theorem (``GOE imply MEP'') for arbitrary
non-amenable groups, although there exists an example on
$C=C_2*C_2*C_2$, due to Muller\footnote{University of Illinois 1976
class notes}. Let us make our task even harder, and restrict
ourselves to linear automata over finite rings (so we assume $Q$ is a
module over a finite ring and the map $\theta:Q^S\to Q$ is linear).
The following approach seems promising.
\begin{conj}[Folklore? I learnt it from V.\ Guba]
Let $G$ be a group. The following are equivalent:
\begin{enumerate}
\item The group $G$ is amenable;\\
\item Let $\K$ be a field. Then $\K G$ admits right
common multiples, i.e.\ for any $\alpha,\beta\in\K G$ there
exist $\gamma,\delta\in\K G$ with
$\alpha\gamma=\beta\delta$ and $(\gamma,\delta)\neq(0,0)$.
\end{enumerate}
\end{conj}
The implication $(1)\Rightarrow(2)$ is easy, and follows from F\o
lner's criterion of amenability by linear algebra.
Assume now the ``hard'' direction of the conjecture. Given $G$
non-amenable, we may then find a finite field $\K$, and
$\alpha,\beta\in\K G$ that do not have a common right multiple.
Set $Q=\K^2$ with basis $(e_1,e_2)$, let $S$ contain the
inverses of the supports of $\alpha$ and $\beta$, and define the
cellular automaton $\theta:Q^S\to Q$ by
\[\theta(\phi)=\sum_{x\in
G}\big(\alpha(x^{-1})\langle\phi(x)|e_1\rangle-\beta(x^{-1})\langle\phi(x)|e_2\rangle,0\big).
\]
Then $\theta$ has GOE, indeed any configuration not in $(\K\times0)^G$
is a GOE. On the other hand, if $\theta$ had MEP, then by linearity we
might as well assume $\Theta(\phi)=0$ for some non-zero
finitely-supported $\phi:G\to Q$. Write $\phi=(\gamma,\delta)$ in
co\"ordinates; then $\Theta(\phi)=0$ gives $\alpha\gamma=\beta\delta$,
showing that $\alpha,\beta$ actually did have a common right multiple.
Muller's example is in fact a special case of this construction, with
\[G=\langle x,y,z|x^2,y^2,z^2\rangle,
\]
$\K=\F[2]$, and $\alpha=x$, $\beta=y+z$.
\begin{bibsection}
\begin{biblist}
\bibselect{bartholdi,math}
\end{biblist}
\end{bibsection}
\end{document}
\emph{A priori}, the implication in~\eqref{thm:2} is ambiguous: should
it hold for a fixed generating set $S$, or for all $S$? We show that
both statements are equivalent:
\begin{lem}
Let $\theta:Q^S\to Q$ be a cellular automaton that admits MEP but
does not admit GOE; let $S'$ be another generating set of $G$.
Then there exists an automaton $\theta':(Q')^{S'}\to Q'$ that also
admits MEP but does not admit GOE.
\end{lem}
\begin{proof}
The finite set $(S')^{-1}S$ is contained in the ball of radius $m-1$
in $S'$, for some $m$ large enough. Set
$Q'=Q^{(S')^{-1}S}\times\Z/m\Z$; for a configuration $\phi:G\to Q'$
write $\phi(x)_1$ and $\phi(x)_2$ respectively for the co\"ordinates
of $\phi(x)$. Fix some $s_0\in S'$. Define the cellular automaton
$\theta'$ by
\[\theta'(\psi)=\begin{cases}
(t\mapsto\theta(s\mapsto\phi(s_0)(s_0^{-1}s)),1) & \text{ if }\phi(s_0)_2=0,\\
(t\mapsto\phi(t_1)_1(t_2\dots t_\ell),n+1) & \text{ if
}\phi(s_0)_2=n\neq0,
\end{cases}
\]
where in the above $t=t_1\dots t_\ell$ is a minimal-length
expression of $t$ as a word over $S'$.
The cellular automaton $\theta'$ behaves as follows: the second
co\"ordinate of its stateset $Q'$ is a cyclic time ticker; at time
$0$, $\theta'$ emulates the automaton $\theta$, while at all other
times it performs bookkeeping.
The first co\"ordinate of the stateset of $\theta'$ is a ``cache''
of the surrounding configuration. At time $1$, its constant value is
the new state of $\theta$; during all subsequent clock ticks, the
configuration values are propagated so that, at time $0$, the first
co\"ordinate actually equals the restriction of the configuration to
its $(S')^{-1}S$-neighbourhood.
Let $\phi$ be a configuration for $\theta$. We produce out of it a
configuration $\phi'$ for $\theta'$, by setting
\[\phi'(x)=(t\mapsto\phi(x),1).\]
Then applying $m$ times $\Theta'$ to $\phi'$ amounts precisely to
applying once $\Theta$ to $\phi$.
Therefore, if $\theta$ has MEP, then so does \emph{a fortiori}
$\theta'$.
Assume now that $\theta$ does not have GOE.
!!! check that $\theta'$ does not have GOE. that's because at other
time ticks the automaton is surjective. !!! fix proof !!!
\end{proof} |
0709.2745 | \section{Introduction}
Many cosmological observations \cite{1,2,3,4}, such as the type Ia
Supernova(SN Ia), Wilkinson Microwave Anisotropy Probe(WMAP), the
Sloan Digital Sky Survey(SDSS) etc., support that our universe is
undergoing an accelerated expansion. This accelerated expansion is
always attributed to
the dark energy with negative pressure which induces repulsive gravity \cite%
{5,6}. Except for the property of negative pressure, we do not know too much
about the components and other properties of the dark energy. What is more,
we have no idea about how the dark energy evolves. Various dark energy
models have been proposed to describe the evolution, all of which must be
constrained by astronomical observations. In these models, the equation of
state $\omega =p/\rho $ plays a key role and can reveal the nature of dark
energy which accelerates the universe, where $p$ and $\rho $ are the
pressure and energy density of the dark energy respectively. Different
equations of state lead to different dynamical changes and may influence the
evolution of our universe. $\omega$ and its time derivative with respect to
Hubble time $\omega ^{^{\prime }}=d\omega /d(\ln a)$(where $a$ is scale
factor) are currently constrained by the distance measurements of SN Ia, and
the current observational data constrain the range of equation of state as $%
-1.38<\omega<-0.82$ \cite{7}. The cosmological constant model with equation
of state $\omega=-1$, which is considered as the most acceptable candidate
for dark energy has been investigated in \cite{8}. The Quintessence scalar
field with $\omega$ in the range from $-1/3$ to $-1$ has been investigated
in \cite{9,10,11}. The Phantom scalar field with $\omega<-1$, which can
introduce the negative kinetic energy and violate the well known energy
condition, has been found in \cite{12}.
In order to get more information on dark energy, it has been widely
discussed from the thermodynamical viewpoints, such as thermodynamics of
dark energy with constant $\omega $ in the range $-1<\omega <-1/3$ \cite{13}%
, $\omega =-1$ in the de Sitter space-time and anti-de Sitter space-time
\cite{14}, $\omega <-1$ in the Phantom field \cite{15,16} and the
generalized chaplygin gas \cite{17} and so on. More discussions on the
thermodynamics of dark energy can be found in \cite{18,19,20,21}.
In this paper, we investigate the thermodynamical properties of an
accelerated expanding universe driven by dark energy with the equation of
state $\omega =\omega _{0}+\omega _{1}z$ and the Hubble parameter $%
H^{2}(z)=H_{0}^{2}[\Omega _{0m}(1+z)^{3}+(1-\Omega _{0m})(1+z)^{3(1+\omega
_{0}-\omega _{1})}e^{3\omega _{1}z}]$ (where $H_{0}$ is the Hubble constant
and $\Omega_{0m}$ is the parameter of matter density). This model is in
agreement with the observations in low redshift, where the parameter $\omega
_{0}=-1.25\pm 0.09$ and $\omega _{1}=1.97_{-0.01}^{+0.08}$ are suggested by
\cite{22}.
Though combining quantum mechanics with general relativity, it was
discovered that a black hole can emit particles. The temperature of a black
hole is proportional to its surface gravity and the entropy is also
proportional to its surface area \cite{23,24}. The Hawking temperature is
given as $T_{H}=\kappa /2\pi $, where $\kappa$ is the surface gravity of a
black hole, and the entropy of a black hole is $S=A/4$, where $A$ is its
surface area. Hawking temperature, black hole entropy, and the black hole
mass satisfy the first law of thermodynamics $dM=TdS$ \cite{25}. If a
charged black hole is rotating, the thermodynamical first law is expressed
as $dM=TdS+\Omega dJ+V_{0}dQ$, where $\Omega$, $J$, $V_{0}$, $Q$ are the
dragged velocity, angular momentum, electric potential and charge of a black
hole respectively. The event horizon of a stationary black hole is
considered as the system boundary \cite{26}, inside which the black hole
should maintain thermodynamical equilibrium. If our universe can be
considered as a thermodynamical system\cite{27,28,29,30}, the
thermodynamical properties of the black hole can be generalized to
space-time enveloped by the apparent horizon or the event horizon. The
thermodynamical properties of the universe may be similar to those of the
black hole, the thermodynamical laws should be satisfied.
This paper is organized as follows: In Sec II, we examine the
thermodynamical first and the second laws on the apparent horizon with $%
\omega =\omega _{0}+\omega _{1}z$. In Sec III, we similarly examine the
thermodynamical laws on the event horizon. In Sec IV, we draw some
conclusions and discussions.
\section{The thermodynamical laws on the apparent horizon}
The Friedmann-Robertson-Walker metric is \cite{31}
\begin{equation}
g_{ab}=\left(
\begin{array}{cc}
-1 & 0 \\
0 & a^{2}/(1-kr^{2})%
\end{array}%
\right). \label{y1}
\end{equation}
If setting
\begin{equation}
\tilde{r}=ar, \label{a4}
\end{equation}%
the dynamical apparent horizon can be determined by the relation%
\begin{equation}
f=g^{ab}\tilde{r}_{,a}\tilde{r}_{,b}=1-(H^{2}+k/a^{2})\tilde{r}^{2}=0,
\label{y2}
\end{equation}
so we can get the apparent horizon radius
\begin{equation}
\tilde{r}_{A}=ar_{A}=1/\sqrt{H^{2}+k/a^{2}}. \label{xx}
\end{equation}
The WMAP data suggest that our universe is spatially flat with $k =0$, so
the apparent horizon is $\tilde{r}_{A}=1/H$ in this case. The surface
gravity on the apparent horizon is
\begin{equation}
\kappa =-f^{^{\prime }}/2\mid _{r=\tilde{r}_{A}}=1/\tilde{r}_{A}, \label{m4}
\end{equation}%
and the temperature is
\begin{equation}
T_{A}=\kappa /2\pi =1/2\pi \tilde{r}_{A}. \label{m6}
\end{equation}
In this paper, we concentrate on a more general model with
\begin{equation}
H^{2}(z)=H_{0}^{2}[\Omega _{0m}(1+z)^{3}+(1-\Omega _{0m})(1+z)^{3(1+\omega
_{0}-\omega _{1})}e^{3\omega _{1}z}]. \label{x2}
\end{equation}
As we just want to know more properties about dark energy, we take $\Omega
_{0m}=0$. And the Hubble parameter becomes
\begin{equation}
H^{2}(z)=H_{0}^{2}(z)(1+z)^{3(1+\omega _{0}-\omega _{1})}e^{3\omega _{1}z}.
\label{x3}
\end{equation}
In this section we use temperature, entropy and energy to examine the first
and the second laws of thermodynamics on the apparent horizon. The
temperature and entropy of the universe on the apparent horizon are $%
T_{A}=1/2\pi \tilde{r}_{A}$ and $S=A/4$ respectively.
When calculating the flux of energy through the area $A=4\pi \tilde{r}%
^{2}_{A}=4\pi /H^{2}$ of the apparent horizon, we treat the horizon as a
static surface, but allowing it to vary slowly with time. The amount of
energy flux \cite{15} crossing the apparent horizon within the time interval
$dt $ is \ \
\begin{eqnarray}
-dE_{A} &=&4\pi r_{A}^{2}T_{ab}k^{a}k^{b}dt=4\pi \tilde{r}_{A}^{2}(\rho +P)dt
\nonumber \\
&=&-\frac{3}{2}H_{0}^{-1}\frac{1+\omega _{0}+\omega _{1}z}{1+z} \nonumber \\
&&\times (1+z)^{-\frac{3}{2}(1+\omega _{0}-\omega _{1})}e^{-\frac{3}{2}%
\omega _{1}z}dz. \label{c7}
\end{eqnarray}%
The entropy of the apparent horizon is
\begin{equation}
S_{A}=A/4=4\pi \tilde{r}^{2}_{A}/4=\pi \tilde{r}^{2}_{A}, \label{x4}
\end{equation}
and its differential form is
\begin{equation}
dS_{A}=2\pi \tilde{r}_{A}d\tilde{r}_{A}. \label{a8}
\end{equation}
Thus we can get
\begin{eqnarray}
T_{A}dS_{A} &=&d\tilde{r}_{A} \nonumber \\
&=&-\frac{3}{2}H_{0}^{-1}\frac{1+\omega _{0}+\omega _{1}z}{1+z} \nonumber \\
&&\times (1+z)^{-\frac{3}{2}(1+\omega _{0}-\omega _{1})}e^{-\frac{3}{2}%
\omega _{1}z}dz. \label{c9}
\end{eqnarray}%
From Eq.(9) and Eq.(12), we obtain the result $-dE_{A}=T_{A}dS_{A}$, so the
first law of thermodynamics on the apparent horizon of this
redshift-dependent model is confirmed.
The entropy of the universe inside the apparent horizon is related to its
energy and pressure through
\begin{equation}
TdS_{I}=dE_{I}+PdV. \label{a10}
\end{equation}%
The energy inside the apparent horizon is $E_{I}=4\pi \tilde{r}^{3}_{A}\rho
/3$ and the volume is $V=4\pi \tilde{r}^{3}_{A}/3$. So we get
\begin{equation}
dS_{I}=\pi \tilde{r}_{A}(1+3\omega )d\tilde{r}_{A}, \label{a11}
\end{equation}%
in which $\omega =\omega _{0}+\omega _{1}z$ and $\tilde{r}_{A}=1/H
=H_{0}^{-1}(1+z)^{-\frac{3}{2}(1+\omega _{0}-\omega _{1})}e^{-\frac{3}{2}%
\omega _{1}z}$.
Thus we can get the derivative of $S_{I} $ with respect to $z$
\begin{eqnarray}
\frac{dS_{I}}{dz} &=&-\frac{3\pi H_{0}^{2}}{2}(1+3\omega _{0}+3\omega _{1}z)
\nonumber \\
&&\times (1+\omega _{0}+\omega _{1}z) \nonumber \\
&&\times (1+z)^{-3(1+\omega _{0}-\omega _{1})-1}e^{-3\omega _{1}z}.
\label{c12}
\end{eqnarray}%
For the entropy of the apparent horizon, we get%
\begin{equation}
S_{A}=\pi r_{A}^{2}=\pi H_{0}^{-2}(1+z)^{-3(1+\omega _{0}-\omega
_{1})}e^{-3\omega _{1}z}, \label{a13}
\end{equation}%
and the derivation of $S_{A}$ with respect to $z $ is
\begin{equation}
\frac{dS_{A}}{dz} =-3\pi H_{0}^{-2}(1+\omega_{0}+\omega_{1}z)
(1+z)^{-3(1+\omega _{0}-\omega _{1})-1}e^{-3\omega _{1}z}. \label{a14}
\end{equation}
The total entropy varies with redshift $z $, so its derivation is
\begin{eqnarray}
\frac{d(S_{I}+S_{A})}{dz} &=&-\frac{3\pi H_{0}^{-2}}{2}(1+\omega _{0}+\omega
_{1}z)^{2} \nonumber \\
&&\times(1+z)^{-3(1+\omega _{0}+\omega _{1}z)-1} \nonumber \\
&&\times e^{-3\omega _{1}z}. \label{m3}
\end{eqnarray}
Considering the relation $a=1/(1+z)$, we get
\begin{eqnarray}
\frac{d(S_{I}+S_{A})}{da} &=&\frac{3\pi H_{0}^{-2}}{2}(1+\omega _{0}+\omega
_{1}z)^{2} \nonumber \\
&&\times (1+z)^{-3(1+\omega _{0}-\omega _{1})+1} \nonumber \\
&&\times e^{-3\omega_{1}z} \geq0 . \label{a15}
\end{eqnarray}
We find that the total entropy of the apparent horizon does not decrease
with time and the second thermodynamical law is satisfied on the apparent
horizon.
\section{The thermodynamical laws on the event horizon}
The event horizon of the universe is the position of the greatest distance
that an particle can reach at a particular cosmic epoch, so the definition
of the event horizon is
\begin{eqnarray}
r_{E} &=&a\int_{t}^{\infty }\frac{dt}{a}=\frac{1}{1+z}\int_{z}^{-1}(-\frac{1%
}{H})dz \nonumber \\
&=&-\frac{1}{1+z}\int_{z}^{-1}H_{0}^{-1}(1+z)^{\frac{-3}{2}(1+\omega
_{0}-\omega _{1})} \nonumber \\
&&\times e^{-\frac{3}{2}\omega _{1}z}dz. \label{c6}
\end{eqnarray}
The energy flux through the event horizon can be similarly expressed as
\begin{eqnarray}
-dE_{E} &=&4\pi r_{E}^{2}\rho (1+\omega )dt=-\frac{3}{2}r_{E}^{2}H_{0}(1+%
\omega _{0}+\omega _{1}z) \nonumber \\
&&\times (1+z)^{\frac{3(1+\omega _{0}+\omega _{1}z)}{2}}e^{\frac{3\omega
_{1}z}{2}}dz. \label{cc6}
\end{eqnarray}%
Meanwhile, we get
\begin{eqnarray}
dE_{E} &=&\frac{3}{2}r_{E}^{2}H_{0}(1+\omega _{0}+\omega _{1}z) \nonumber \\
&&\times (1+z)^{\frac{3(1+\omega _{0}+\omega _{1}z)}{2}}e^{\frac{3\omega
_{1}z}{2}}dz. \label{cc7}
\end{eqnarray}%
The entropy of the event horizon is%
\begin{equation}
dS_{E}=2\pi r_{E}dr_{E}=2\pi r_{E}\frac{dr_{E}}{dz}dz. \label{b2}
\end{equation}%
Using Hawking temperature of the event horizon, we can also obtain \
\begin{equation}
T_{E}dS_{E}=dr_{E}. \label{b3}
\end{equation}
\begin{figure}[!t]
\centerline{\psfig{file=fig1-s.eps,width=3.6in,angle=0}}
\caption{$(T_{E}dS_{E})$, $dE_{E}$ and $(-dE_{E})$ as functions of $z$ with $%
\protect\omega _{0}=-1$ and $\protect\omega _{1}=2$. }
\label{fig1}
\end{figure}
\begin{figure}[!t]
\centerline{\psfig{file=fig2-s.eps,width=3.6in,angle=0}}
\caption{$T_{E}dS_{E}+dE_{E}$ as a function of $z$ with $\protect\omega %
_{0}=-1$ and $\protect\omega _{1}=2$. }
\label{fig2}
\end{figure}
It is shown in Fig.1 and Fig.2 that
\begin{equation}
-dE_{E}\neq T_{E}dS_{E}. \label{m8}
\end{equation}
In the higher redshift, $-dE_{E}$ is in agreement with $T_{E}dS_{E} $, but
in the lower redshift, they are not consistent. Therefore, the first law of
thermodynamics with the usual definition of entropy and temperature on the
event horizon is not always confirmed.
For the entropy of matter and fluids inside the event horizon, we have%
\begin{equation}
TdS_{I}=dE_{I}+PdV. \label{b4}
\end{equation}%
The energy inside the event horizon is $E_{I}=4\pi {r}^{3}_{E}\rho /3$, so
we get
\begin{equation}
dS_{I}=2\pi r_{E}^{4}HdH+3\pi r_{E}^{3}H^{2}(1+\omega_{0}+\omega_{1}z)
dr_{E}. \label{b5}
\end{equation}%
The entropy of the event horizon is $S_{E}=\pi r_{E}^{2}$, so we have
\begin{equation}
dS_{E}=2\pi r_{E}\frac{dr_{E}}{dz}dz. \label{b6}
\end{equation}%
Similarly we get
\begin{eqnarray}
\frac{d(S_{I}+S_{E})}{dz} &=&2\pi r_{E}^{4}H\frac{dH}{dz}+ \nonumber \\
&&\pi \lbrack 3r_{E}^{3}H^{2}(1+\omega_{0}+\omega_{1} )+2r_{E}]\frac{dr_{E}}{%
dz}. \label{b7}
\end{eqnarray}%
\
\begin{figure}[!t]
\centerline{\psfig{file=fig3-s.eps,width=3.6in,angle=0}}
\caption{ $\frac{d(S_{I}+S_{E})}{dz}$ as a function of $z $ with $\protect%
\omega _{0}=-1$ and $\protect\omega _{1}=2$.}
\label{fig3}
\end{figure}
In FIG.3, we also find the total entropy inside the event horizon is not
always negative. Through the relation $a=1/(1+z)$, we can get
\begin{equation}
da=-\frac{1}{(1+z)^{2}}dz. \label{m7}
\end{equation}%
As a result, $d(S_{I}+S_{E})/da$ does not keep positive all the time. The
thermodynamical second law breaks down inside the event horizon.
\begin{figure}[!t]
\centerline{\psfig{file=fig4-s.eps,width=3.6in,angle=0}}
\caption{ $t_{H}d(\ln \tilde{r}_{A})/dz$ and $t_{H}d(\ln r_{E})/dz$ with the
redshift during a Hubble scale with $\protect\omega _{0}=-1$ and $\protect%
\omega _{1}=2$.}
\label{fig4}
\end{figure}
\begin{figure}[!t]
\centerline{\psfig{file=fig5-s.eps,width=3.6in,angle=0}}
\caption{The apparent horizon and event horizon as functions of $z$ with $%
\protect\omega _{0}=-1$ and $\protect\omega _{1}=2$. }
\label{fig5}
\end{figure}
\section{Conclusions and discussions}
\bigskip The Hawking temperature, entropy together with the black hole mass
follow the thermodynamical first law $dM=TdS$, and the Einstein equation can
be derived from the thermodynamical law \cite{30}. As the thermodynamical
laws hold on the event horizon of a stationary black hole, the black hole
keeps thermodynamical equilibrium inside it.
The apparent horizon of a black hole is a two-dimensional surface for which
the outgoing orthogonal null geodesics have zero divergence \cite{32}. A
black hole in asymptotically flat space-time is defined as a region so that
no casual signal (i.e., a signal propagating at a velocity not greater than
light) from light can reach, then the event horizon constitutes the boundary
of a black hole. When the weak energy condition is satisfied, the apparent
horizon either coincides with event horizon or lies inside it. In stationary
black holes, the apparent horizon and the event horizon coincide, such as
Schwarzschild black hole and Kerr-Newman black hole and so on. There always
exists the event horizon located outside the apparent horizon. When a
spherically symmetric black hole becomes non-stationary for some time due to
the matter falling into it, the thermodynamical laws are not fit for a
non-stationary black hole. When a black hole is non-stationary, the apparent
horizon is always not consistent with the event horizon and the
thermodynamical laws may not always hold on the event horizon.
The universe is accelerating and should be not stationary. Due to the heat
through apparent horizon area, we have calculated a variation of the
geometrical entropy on the apparent horizon. We find that the
thermodynamical relation is $-dE=TdS$. In addition, we also examine the
changes of the total entropy with the time $d(S_{I}+S_{A})/da\geq 0$,
therefore, the second law of thermodynamics is evidently confirmed. While
the thermodynamical laws on the event horizon do not hold on, the results
are in agreement with the work of Wang \cite{13}. When the apparent horizon
can be regarded as a thermodynamical system which has associated entropy $%
S=A/4$ with the temperature $T=1/2\pi r$, the differential form of the first
thermodynamical law on the apparent horizon can be written to the
differential form of the Friedmann equation \cite{33}. In a word, the
apparent horizon is a good boundary of keeping thermodynamical properties,
and the universe has a thermal equilibrium inside it. These properties
disappear on the event horizon. Maybe the usual definition of temperature
and entropy are not well-defined , or it is non-equilibrium of
thermodynamics inside the universe.
In FIG.4, in the low redshift, the changing rate of the apparent horizon and
the event horizon is not significant.The thermodynamical description of
horizons \cite{15} will be approximately valid. The apparent horizon always
exists and usually is inside the event horizon. In FIG.5, the event horizon
is outside apparent horizon at higher redshift phase and the universe is
undergoing accelerated expansion. But in this model the redshift becomes
smaller than zero and the equation of state is $\omega <-1,$ the universe
may have a superacceleratd expanding process and the apparent horizon is
outside the event horizon. This model is similar with the Phantom field
which will lead to undesirable future singularity and violate the weak
energy condition.
In summary, it is interesting to investigate the universe from the
thermodynamical viewpoints, although thermodynamical properties need to be
explored more deeply in the future.
\section{Acknowledgments}
Yongping Zhang would like to thank Shiwei Zhou and Xiaokai He for their
valuable suggestions and help. This work is supported by the National
Natural Science Foundation of China(Grant No.10473002),
the National Basic Research Program of China (Grant No. 2003 CB 716302) and the Scientific Research Foundation
for the Returned Overseas Chinese Scholars, State Education Ministry. |
0709.2660 | \section{Introduction}
\label{sec:intro}
Propagation of surface plasmon polaritons (SPPs) in linear periodic
chains (LPC) of metal nanoparticle has been in the focus of
considerable recent
attention~\cite{weber_04_1,simovski_05_1,koenderink_06_1,fung_07_1,park_04_1,citrin_05_1,citrin_06_1,markel_07_2}.
The interest is, in part, motivated by the potential application of
such chains as sub-wavelength plasmonic
waveguides~\cite{quinten_98_1,brongersma_00_1,maier_03_1}. Since
energy in a waveguide can be transported in the form of wave packets,
the dispersion relation becomes of primary importance. In the case of
LPCs, the dispersion relation is a mathematical dependence of the SPP
frequency $\omega$ on its Bloch wave number, $q$. It is important to
emphasize that the elementary excitations (electromagnetic modes) in
LPCs are Bloch waves rather than sinusoidal waves which propagate in
continuous media. Only when the Bloch wave number is sufficiently
small, so that $qh\ll 1$, where $h$ is the chain period, can we
neglect the mathematical distinction between the Bloch and the
sinusoidal waves and treat an LPC as an essentially continuous system.
How strong the above inequality should be is not evident {\em a
priori}; we will comment on this question in the concluding part of
the paper.
Dispersion curves have been previously computed for polarization waves
propagating in periodic chains of Drudean spherical
nanoparticles~\cite{weber_04_1,simovski_05_1,koenderink_06_1,fung_07_1}.
It was found that there exist excitations with frequencies $\omega$
and Bloch wave numbers $q$ such that $\omega < qc_h$, $c_h$ being the
speed of light in the surrounding (host) medium. The phase velocity of
such excitations, $v_p = \omega/q$, is less in magnitude than $c_h$.
SPPs with $\vert v_p\vert < c_h$ are considered to be outside of the
``light cone'' and can propagate along the chain without radiative
losses, similarly to plane waves in homogeneous dielectrics. We will
reserve the term ``SPP'' specifically for this type of excitations. In
a chain of perfectly conducting (lossless) particles, an SPP can
propagate infinitely without decay. Of course, absorptive (Ohmic)
losses in realistic metals always result in exponential spatial decay
of SPPs.
The property of the dispersion curve $\omega(q)$ which is specific to
the case of LPCs made of spherical particles is that it is very
flat~\cite{weber_04_1,simovski_05_1,park_04_1}, with only a weak
dependence of the frequency on the Bloch wave number. The dispersion
curves are particularly flat for SPPs polarized transversely to the
chain. This results in very small group velocities, $v_g \ll c_h$. A
factor $v_g/ c_h \sim 10^{-2}$ is typical. One practically important
consequence of the dispersion curve flatness is a relatively narrow
SPP bandwidth. That is, an SPP can be excited in a chain by a
spatially-localized external source only in a narrow band of
frequencies. This can be expected to significantly limit the potential
application of LPCs as optical waveguides.
Maier {\em et al.} have pointed out that the use of spheroidal rather
than of spherical nanoparticles can result in an increased SPP
bandwidth and, correspondingly, in a longer propagation
distance~\cite{maier_02_1}. In the above work, the bandwidth was
defined as twice the spectral shift of the nearly homogeneous SPP
(characterized by $q=0$) with respect to the plasmon peak of an
isolated nanoparticle and it was assumed that the propagation
distance is proportional to the inverse of the latter. The results
were confirmed by FDTD simulations in an LPC of seven prolate
nanospheroid whose longer axis was perpendicular to the chain.
In this paper, we further investigate the effects of nonsphericity of
the LPC constituents. We compute the dispersion relations in such LPCs
in the dipole approximation. We find that the dispersion curves in
LPCs are dramatically altered by replacing spherical particles with
prolate or oblate spheroids. In particular, the SPP bandwidth can be
significantly increased, in agreement with the results of Maier {\em
et al.}. Here, however, we define the bandwidth as the range of
frequencies in which efficient SPP transport along the chain is
possible. This includes modes with various values of $q$, including
those with $q\sim \pi/h$. We also find that the use of oblate
spheroids (nanodisks) whose shorter semiaxis is parallel to the chain
is even more beneficial as it allows to achieve the desired effect at
relatively modest values of the aspect ratio. The increased bandwidth
is expected to result in a higher maximum bit rate and longer
propagation distances for signals transported by an LPC waveguide.
We further report that at some critical values of the spheroid aspect
ratio, gaps appear in the first Brillouin zone of the lattice.
Propagating SPPs do not exist when the Bloch wave number $q$ is inside
one of such gaps. When $q$ is near the gap edge, a number of
interesting phenomena take place. First, the propagation distance
(the decay length) is dramatically increased, as compared to the same
quantity when $q$ is far from the edge. Second, the dispersion curves
acquire very large positive or negative slopes. In this case,
relatively fast ($c_h - \vert v_g \vert \ll c_h$) and even
superluminal ($\vert v_g \vert > c_h$) wave packet propagation can be
obtained. Note that superluminal group velocity does not contradict
special relativity~\cite{bolda_94_1}. Superluminal wave packets exist
in nature and were observed
experimentally~\cite{wang_00_1,gehring_06_1}.
Theory and numerical simulations presented below are based on the
dipole approximation. The use of this approximation dictates that the
inter-particle spacings are larger than a certain threshold at which
excitation of higher multipole moments in nanospheroids becomes
non-negligible. This has limited the range of chain parameters
considered in this paper. We, however, expect on physical grounds that
using chains with smaller inter-particle separations may be
beneficial. For instance, we expect that the SPP propagation length in
such chains can be increased. However, in order to obtain quantitative
results in that limit, a considerably more complex mathematical
formalism must be used. The latter has been developed by Park and
Stroud~\cite{park_04_1} for chains of spherical nanoparticles in the
quasistatic limit. Unfortunately, generalization of this formalism to
non-spherical particles and beyond the quasistatic approximation
(which we deem to be essential for describing SPP propagation in long
chains, as was confirmed recently in the experiments by Koenderink
{\em et al.}~\cite{koenderink_07_1}) appears to be problematic.
The paper is organized as follows. In the Section~\ref{sec:model}, we
describe and justify the basic model used to simulate Bloch waves and
wave packets in LPCs. In Section~\ref{sec:dispersion}, we compute the
dispersion curves for SPPs in chains of prolate and oblate
nanospheroids. In Section~\ref{sec:l}, we discuss the attenuation of
SPPs due to Ohmic losses in LPCs and, for comparison, in metallic
nanowires. In Section~\ref{sec:packets}, we describe direct numerical
simulations of wave packet propagation in LPCs. Section~\ref{sec:disc}
contains a summary and a discussion of obtained results.
\section{The Basic Model}
\label{sec:model}
Consider a linear periodic chain of identical metallic spheroids with
semiaxes $a$ and $b$ ($a \geq b$). We will discuss below two different
cases. In the first case, the chain is made of prolate spheroids whose
axis of symmetry (which coincides with the longer axis) is
perpendicular to the chain. In the second case, the chain is made of
oblate spheroids whose axis of symmetry (which coincides with the
shorter axis) is parallel to the chain. In both cases, the longer axes
of the spheroids are perpendicular to the chain and the eccentricity,
$e$, is given by
\begin{equation}
\label{ecc_def}
e = \sqrt{1 - (b/a)^2} \ .
\end{equation}
\noindent
We will refer to the ratio $b/a \leq 1$ of the shorter and longer
semiaxes of the spheroids as the aspect ratio.
The spheroids are centered at the points $x_n=hn$, where $n$ is an
integer and $h$ is the chain period. The surface-to-surface
separation of two neighboring spheroids is $\sigma=h-2b$; we require
that $h>2b$ to avoid geometrical intersection of particles. Note that
a stronger condition on the interparticle separation will be imposed
below. The smaller semiaxis $b$ is assumed to be on the order of
$10{\rm nm}$ while $a$ can be up to a few times larger. We will see
that SPPs propagating in such chains have frequencies $\omega$ such
that the corresponding wavelength in the host medium, $\lambda=2\pi
c_h/\omega$, is considerably larger than both $a$ and $b$.
Accordingly, we adopt the dipole approximation.
This choice, however, requires an additional justification. While the
dipole approximation accuracy for electromagnetically interacting
spheroids has not been studied directly, many results are available
for spheres. The most basic and frequently considered example is that
of two electromagnetically-interacting spheres in close proximity of
each other~\cite{ruppin_89_1,mazets_00_1}. In this case, the spatially
inhomogeneous fields scattered by the spheres result in the excitation
of vector spherical harmonics of all orders, even if the spheres are
small compared to the wavelength. More specifically, the electric
field inside each sphere can be expanded into the vector spherical
harmonics with nonzero coefficients appearing in arbitrarily high
orders. In the dipole approximation, only the first-order terms
($l=1$, $m=0,\pm 1$) are retained in this expansion. The accuracy of
this approximation was found to be dramatically affected by
polarization. If the electric field polarization is parallel to the
axis connecting the spheres, the dipole approximation starts to
deviate from the exact solution when $\sigma \approx 0.5R$, $R$ being
the sphere radius, for both dielectric~\cite{ruppin_89_1} and
conducting~\cite{mazets_00_1} spheres. However, if the polarization is
perpendicular to the axis, the dipole approximation yields results
(i.e., the total dipole moment of the spheres~\cite{mazets_00_1}) with
a relative error of only 2\% even in the case $\sigma=0$. A careful
study~\cite{markel_04_3} of the transversely-polarized electromagnetic
modes of finite-length linear chains of interacting spheres has
revealed that the effect of multipole interaction is to slightly shift
and broaden the dipole resonance - an effect hardly observable in most
materials due to the spectral line broadening associated with Ohmic
losses.
Below, we work in the regime when $\sigma = 2b$ ($h=4b$). In the case
of spheres ($b=R$), one could expect the dipole approximation to be
very accurate for such relative separations regardless of
polarization. We can, however, apply a more stringent test and compare
$\sigma$ to the larger semiaxis of the spheroid, $a$. Within the
dipole approximation, the physical effect described below is manifest
for the following aspect ratios. For transversely polarized SPP, the
gap in the first Brillouin zone of the chain lattice appears when $b/a
\lesssim 0.25$ in the case of prolate spheroids and when $b/a \lesssim
0.35$ in the case of oblate spheroids. For longitudinal SPP
polarization, the gap appears when $b/a \lesssim 0.1$ in the case of
prolate spheroids and when $b/a \lesssim 0.25$ in the case of oblate
spheroids.
Consider first the transverse polarization. The aspect ratio
$b/a=0.25$ corresponds to $\sigma/a = 0.5$. In the case of two spheres
of radius $R$ separated by the surface-to-surface distance
$\sigma=0.5R$, the multipole effects are negligible. In fact, even a
smaller aspect ratio of $b/a=0.15$, which is used in the numerical
simulations of transversely-polarized wave packets in
Section~\ref{sec:packets}, corresponds to the relative separation
$\sigma=0.3R$. At this separation, the multipole effects can still be
safely ignored. In the case of the longitudinal polarization, the
aspect ratio which is required to observe the effect in chains of
prolate spheroids is $0.1$ which corresponds to $\sigma = 0.2R$. The
multipole effects in this situation are expected to be significant but
not dramatic. However, in oblate spheroid chains, the required aspect
ratio is $0.25$ which corresponds to $\sigma = 0.5R$. At this relative
separation, the effects of higher multipoles are noticeable but small,
even for the longitudinal polarization.
Finally, a simple physical explanation for the dramatic polarization
dependence of the dipole approximation accuracy is available. When
the polarization is parallel to a chain of spheres, the sphere
surfaces which are adjacent to the "junctions" are similar to usual
capacitors and acquire large and sign-opposite surface charge
densities which, in turn, results in highly non-uniform, strongly
enhanced local fields. This causes excitation of very high multipole
moments. However, for the case of transverse polarization, the surface
charge densities near the junctions are proportional to the
geometrical factor $\cos\theta$ ($\theta$ being the angle between the
polarization vector and the radius vector of a point on the sphere
surface) and are small. Obviously, this consideration holds for
spheroids as well.
We thus conclude that the use of the dipole approximation is well
justified for the purpose of this paper. In the case of transverse SPP
polarization, the approximation accuracy is exceedingly good. For the
longitudinal polarization, the accuracy can be questioned in prolate
spheroid chains but is quite reasonable when oblate spheroids are
used. We will report numerical computation of the dispersion curves
for both prolate and oblate spheroids, in transverse and longitudinal
polarization, and for various aspect ratios of the spheroids
(Section~\ref{sec:dispersion},
Figs.~\ref{fig:disp_prol_ort},\ref{fig:disp_prol_par},\ref{fig:disp_obl_ort},\ref{fig:disp_obl_par}).
However, direct simulation of wave packet propagation
(Sections,\ref{sec:packets},
Figs.~\ref{fig:packet_ort},\ref{fig:packet_par}) is reported only for
the choice of parameters such that the accuracy of the dipole
approximation is not in doubt.
In the dipole approximation, each nanoparticle is characterized by a
(possibly, tensor) dipole polarizability $\alpha(\omega)$ and radiates
as a point dipole. The Cartesian components of the nanoparticle dipole
moments $d_n$ are coupled to each other and to the external electric
field by the coupled-dipole
equation~\cite{markel_07_2,markel_93_1,markel_95_1,markel_05_2}, which
we write here in the frequency domain as
\begin{equation}
\label{CDE}
d_n = \alpha(\omega)\left[E_n^{\rm ext} + \sum_{m\neq n} G_k(x_n,x_m) d_m \right] \ .
\end{equation}
\noindent
Here $E_n^{\rm ext}$ is the external field amplitude at the $n$-th
site, $k=\omega/c_h$ is the wave vector in the host material at the
frequency $\omega$ and $G_k(x,x^{\prime})$ is the appropriate element
of the frequency-domain free-space Green's tensor. In this paper, we
consider both the transverse and the longitudinal polarizations of the
SPP. In the absence of magnetic polarizability of the nanoparticles
(which is assumed), the SPPs with the three orthogonal polarizations
are not electromagnetically coupled to each other. Therefore, each
polarization can be considered separately and the quantities appearing
in Eq.~\eqref{CDE} should be understood as follows: $d_n$ and
$E_n^{\rm ext}$ are projections of the dipole moments and of the
external electric field on the selected polarization axis,
$\alpha(\omega)$ is the appropriate scalar element of the
polarizability tensor and $G_k(x,x^{\prime})$ is defined by
\begin{equation}
\label{G_def_a}
G_k(x,x^{\prime}) = \left\{
\begin{array}{l}
\left(\frac{\displaystyle k^2}{\displaystyle \vert x -
x^{\prime}\vert} + \frac{\displaystyle ik}{\displaystyle \vert x -
x^{\prime}\vert^2} - \frac{\displaystyle 1}{\displaystyle \vert x
- x^{\prime}\vert^3}\right) \exp(ik\vert x - x^{\prime}\vert) \ ,
\ \ \text{transverse polarization} \ , \vspace*{3mm} \\
2\left(-\frac{\displaystyle ik}{\displaystyle \vert x -
x^{\prime}\vert^2} + \frac{\displaystyle 1}{\displaystyle \vert x -
x^{\prime}\vert^3}\right) \exp(ik\vert x -x^{\prime}\vert) \ , \
\ \text{longitudinal polarization} \ .
\end{array} \right.
\end{equation}
\section{The Dispersion Relations}
\label{sec:dispersion}
An SPP mode is an excitation that propagates along the chain without
an external source. Thus, to find the dispersion relation, we seek a
solution to \eqref{CDE} with zero free term, $E_n^{\rm ext}=0$, in the
form $d_n \propto \exp(iq x_n)$, where $q$ is in the first Brillouin
zone of the lattice, $q\in [-\pi/h, \pi/h]$. Substitution of this
ansatz into \eqref{CDE} yields the equation
\begin{equation}
\label{Disp_gen}
\alpha^{-1}(\omega) = h^{-3} S(hk, hq) \ ,
\end{equation}
\noindent
where $S(hk,hq)$ is the dimensionless dipole sum (the dipole
self-energy) of the chain defined by
\begin{equation}
\label{Dip_Sum}
S(\xi, \eta) = \left\{ \begin{array}{l}
2\xi^3{\displaystyle \sum_{n=1}^{\infty}} \left[\frac{\displaystyle 1}{\displaystyle
n\xi} + \frac{\displaystyle i}{\displaystyle (n\xi)^2} -
\frac{\displaystyle 1}{\displaystyle (n\xi)^3}\right]
\exp(i n \xi) \cos( n \eta) \ , \ \ \text{transverse polarization} \
, \vspace*{3mm} \\
4\xi^3{\displaystyle \sum_{n=1}^{\infty}} \left[-\frac{\displaystyle i}{\displaystyle
(n\xi)^2} + \frac{\displaystyle 1}{\displaystyle (n\xi)^3}\right]
\exp(i n \xi) \cos( n \eta) \ , \ \ \text{longitudinal polarization}
\ .
\end{array} \right.
\end{equation}
\noindent
Note that the dipole sum is a function of two dimensionless parameters
$\xi=kh$ and $\eta=qh$ but does not depend on the particle shape and
material properties.
The dispersion relation, i.e., the mathematical dependence of the SPP
frequency $\omega = k c_h$ on its Bloch wave number $q$, can be
obtained by finding all pairs of variables $(\omega,q)$ that satisfy
Eq.~\eqref{Disp_gen}. This can give rise to one or more branches of
the complex function $\omega(q)$. Generally, purely real pairs
$(\omega,q)$ that solve \eqref{Disp_gen} do not exist. One can,
however, consider purely real values of $q$ and seek complex
frequencies, as was done by Koenderink and
Polman~\cite{koenderink_06_1}. The imaginary part of $\omega$ is then
interpreted as the SPP decay rate. Alternative approaches include
numerical computation of discrete modes in a finite
chain~\cite{weber_04_1} and plotting ${\rm Im}[\alpha^{-1} - h^{-3}
S]^{-1}$ as a function of two variables $k$ and $q$ and visually
identifying the points at which this function appears to have a
maximum or a saddle point~\cite{fung_07_1}.
In this paper, we are interested in propagation of wave packets which
are excited as superpositions of oscillations with purely real
frequencies. Since SPPs propagate without radiative losses, their
Bloch wave numbers $q$ are purely real if the chain is made of a
non-absorbing material such as an ideal conductor. However, when Ohmic
losses in realistic metal are accounted for, $q$ acquires an imaginary
part. In what follows, we use two different approaches to computing
the dispersion curve. In Section~\ref{subsec:real_q}, we consider
chains made of ideal (lossless) metal and seek purely real solutions
$\omega(q)$, as was suggested by Simovski~\cite{simovski_05_1}.
Numerically, this is accomplished by finding pairs of real variables
$(\omega,q)$ that satisfy the dispersion equation~\eqref{Disp_gen} by
the method of bisection. Such purely real solutions exist if the
permittivity of metal is taken to be real. Next, in
Section~\ref{subsec:real_q}, we consider realistic metals. Here we
seek pairs $(\omega,q)$ that satisfy the dispersion equation such that
$\omega$ is purely real but $q$ is complex. Numerically, such pairs
are obtained by utilizing the root-finding algorithm implemented in
Wolfram's {\em Mathematica}. For the specific case of silver, we find
that both approaches yield the results which are very close
quantitatively when the dependence of $\omega$ on ${\rm Re}(q)$ is
considered; the first approach, however, provides no information on
SPP attenuation which is governed by ${\rm Im}q$. In both cases, we
seek solutions only in the region ${\rm Re}q > k = \omega/c_h$; as was
discussed in the Introduction, excitations with ${\rm Re}q < k$
experience radiative decay in addition to Ohmic losses and are not
considered in this paper.
\subsection{Dispersion Relations for Ideal Metal}
\label{subsec:real_q}
In this Section we assume that $q$ and $k$ are real and view the
dipole sum $S(kh,qh)$ as a function of two purely real variables. As
the first step, we write the inverse polarizability in
\eqref{Disp_gen} in the form~\cite{markel_92_1}:
\begin{equation}
\label{alpha_total}
\alpha^{-1}(\omega)=\alpha_{\rm LL}^{-1}(\omega)-2ik^3/3 \ ,
\end{equation}
\noindent
where $\alpha_{\rm LL}(\omega)$ is the Lorentz-Lorentz quasistatic
polarizability of nanoparticles and $2ik^3/3=i(2/3)(\omega/c_h)^3$ is
the first non-vanishing radiative correction to the inverse
polarizability. The latter is given by
\begin{equation}
\label{z_spheroid}
\alpha^{-1}_{\rm LL}(\omega) = \frac{4\pi}{\epsilon_h v} \left(\nu +
\frac{\epsilon_h}{\epsilon_m - \epsilon_h} \right) \ ,
\end{equation}
\noindent
where $\nu$ is the appropriate depolarization factor and $v$ is the
spheroid volume~\cite{bohren_book_83}.
In the case of prolate spheroids, the volume is given by
\begin{equation}
\label{v_prol}
v = \frac{4\pi}{3} ab^2
\end{equation}
\noindent
and the three depolarization factors are $\nu_1$ for polarization
along the spheroid axis of symmetry and $\nu_2=\nu_3 = (1-\nu_1)/2$
for two linearly independent transverse polarizations, where
\begin{equation}
\label{nu_def_prol}
\nu_1 = \frac{1 - e^2}{e^2} \left[ -1 + \frac{1}{2e}\ln\frac{1 + e}{1 -
e} \right] \ .
\end{equation}
\noindent
Here $\epsilon_m$ and $\epsilon_h$ are the permittivities of the
(metallic) spheroids and of the host medium, respectively, and $e$ is
the spheroid eccentricity given by formula (\ref{ecc_def}).
For oblate spheroids, the volume is
\begin{equation}
\label{v_obl}
v = \frac{4 \pi}{3} a^2b
\end{equation}
\noindent
and the depolarization factors are $\nu_1=\nu_2$ for two linearly
independent polarizations which are orthogonal to the spheroid axis of
symmetry and $\nu_3 = 1 - 2\nu_1$ for the polarization along the axis
of symmetry, where
\begin{equation}
\label{nu_def_obl}
\nu_1 = \frac{g(e)}{2e^2}\left[\frac{\pi}{2} - \arctan g(e) \right] -
\frac{g^2(e)}{2} \ , \ \ g(e) = \frac{\sqrt{1-e^2}}{e} \ .
\end{equation}
While both permittivities $\epsilon_m$ and $\epsilon_h$ have, in
general, some frequency dependence, here we neglect the dispersion in
the host and assume that $\epsilon_h={\rm const}>0$. Then, for
nanoparticles made of a lossless material, ${\rm Im}(\alpha^{-1}_{\rm
LL}) = 0$. At the same time, if $q>k$, the imaginary part of the
dipole sum is~\cite{markel_07_2,burin_04_1} ${\rm Im}[S(kh,qh)] =
-2(kh)^3/3$ and, in the region of $(k,q)$ which is of interest to us,
${\rm Im}\left[\alpha^{-1} - h^{-3}S\right] = 0$. Therefore, the
imaginary part of Eq.~\eqref{Disp_gen} is satisfied identically and
only its real part needs to be considered. We then utilize
Eqs.~\eqref{alpha_total},\eqref{z_spheroid} and arrive at the
following dispersion equation:
\begin{equation}
\label{Disp_Real_alpha}
\nu + {\rm Re}\left(\frac{\epsilon_h}{\epsilon_m - \epsilon_h} \right) = \epsilon_h
\frac{v}{4\pi h^3} {\rm Re}[S(hk, hq)] \ .
\end{equation}
\noindent
In the case of a lossless metal and a transparent host medium, the
real-part symbol in the left-hand side of \eqref{Disp_Real_alpha} can
be omitted.
\begin{figure}
\centerline{\epsfig{file=fig1_a_compressed.ps,width=8cm,clip=t}}
\vspace*{3mm}
\centerline{\epsfig{file=fig1_b_compressed.ps,width=8cm,clip=t}}
\caption{\label{fig:dipsum}
(color online) Contour plot of the real part of the dipole sum,
${\rm Re}[S(kh, qh)]$ for the transverse (top) and the longitudinal
(bottom) polarizations. Due to the symmetry
$S(\xi,\eta)=S(\xi,-\eta)$, only the positive half of the first
Brillouin zone is shown. Note that, for the transverse
polarization, ${\rm Re}[S(kh, qh)]$ diverges logarithmically
(approaches positive infinity) on the light line $q=k$. Near this
line, the function changes so fast that it is not feasible to depict
it quantitatively using the contour plot; the region of divergence
is schematically shown as a diagonal white line in the top panel.
There is no divergence for the longitudinal polarization.}
\end{figure}
Eq.~\eqref{Disp_Real_alpha} allows one to analyze the relation between
the spheroid aspect ratio and the dispersive properties of the LPCs.
Consider first the SPP polarization which is transverse to the chain.
For this polarization, the dependence of the real part of the dipole
sum $S(kh,qh)$ on its arguments is illustrated in
Fig.~\ref{fig:dipsum}(a) and the relevant depolarization factor is
$\nu_1$ (for both prolate and oblate spheroids). We now notice the
following. In the spectral region where the metal experiences
anomalous dispersion, the second term in the left-hand side of
\eqref{Disp_Real_alpha} is negative, assuming that the host medium is
a transparent dielectric. When the aspect ratio $b/a$ is decreased,
the depolarization factors $\nu_1$ given by Eqs.~(\ref{nu_def_prol})
or (\ref{nu_def_obl}), for prolate and oblate spheroids, respectively,
both approach zero. As a result, the whole left-hand side in
\eqref{Disp_Real_alpha} becomes negative. On the other hand, there
are regions in the $(k,q)$ space in which the right-hand side of
\eqref{Disp_Real_alpha} is strictly positive. Thus, it can be seen
from Fig.~\ref{fig:dipsum}(a) that ${\rm Re}[S(kh, qh)]>0$ if $qh/\pi
\gtrsim 0.5$ and $q>k$. As a result, for sufficiently small ratio
$b/a$, equation \eqref{Disp_Real_alpha} ceases to have real-valued
solutions if $q > q_c^\perp$, where $q_c^\perp$ is the (aspect
ratio-dependent) critical value of $q$ specific to the transverse
polarization. The interval of Bloch wave numbers $q > q_c^\perp$
corresponds to a gap in the first Brillouin zone of the chain lattice
in which SPPs do not exist. It is important to emphasize that the
critical constant $q_c^\perp < \pi/h$ exists only for sufficiently
small ratio $b/a$.
Similar considerations can be applied to the longitudinal SPP
polarization, except that the relevant depolarization coefficient is,
in this case, $\nu_3$. The dependence of ${\rm Re}[S(kh,qh)]$ on its
arguments is illustrated in Fig.~\ref{fig:dipsum}(b). It can be seen
that ${\rm Re}[S(kh,qh)]$ is strictly positive for $qh/\pi \lesssim
0.4$ and $q>k$. Correspondingly, Eq.~\eqref{Disp_Real_alpha} ceases to
have real-valued solutions for $q < q_c^\parallel$ which
defines a gap in the first Brillouin zone of the lattice. Here
$q_c^\parallel$ is the critical Bloch wave number for the parallel
polarization; as in the case of transverse polarization,
$q_c^\parallel$ exists only for sufficiently small aspect ratios and
is aspect ratio-dependent. Note that the depolarization factor $\nu_3$
approaches a finite value rather than zero when the aspect ratio is
decreased. This limit is $1/2$ for prolate and $1$ for oblate
spheroids. Due to this reason, observation of the gap for
longitudinally-polarized SPPs requires a smaller aspect ratio. This
point will be illustrated below by numerical examples.
We now compute the dispersion curves numerically. To solve
Eq.~\eqref{Disp_Real_alpha}, a specific expression for the metal
permittivity $\epsilon_m$ is needed. We use here the Drude formula
\begin{equation}
\label{drude}
\epsilon_m = \epsilon_0 - \frac{\omega_p^2}{\omega(\omega + i\gamma)}
\ ,
\end{equation}
\noindent
where $\epsilon_0$ is the contribution due to interzone
transitions~\cite{kreibig_85_1}, $\omega_p$ is the plasma frequency,
and $\gamma$ is the relaxation constant. In this subsection, we set
$\gamma=0$ to describe a lossless metal (realistic values of $\gamma$
will be used in Sections~\ref{subsec:complex_q} and \ref{sec:packets}
below). We then solve Eq.~\eqref{Disp_Real_alpha} by the method of
bisection to obtain the dispersion curves and the SPP group velocity.
The results are shown in Figs.~\ref{fig:disp_prol_ort} through
\ref{fig:disp_obl_par}. The top panels in these four figures show a
number of dispersion curves computed for different aspect ratios
$b/a$, as labeled, and for different SPP polarizations. The group
velocity of the SPPs, $v_g = \partial \omega/ \partial q$, is plotted
as a function of $q$ in the bottom panels. The values of $\epsilon_h$
and $\epsilon_0$ are taken to be $2.5$ (as in the case of a glassy
medium) and $5.0$ (the experimental value for
silver~\cite{kreibig_85_1}), respectively. The computation does not
depend on the absolute value of the plasma frequency $\omega_p$ but on
the dimensionless parameter $\lambda_p/h$, where $\lambda_p = 2\pi
c_h/\omega_p$ is the wavelength (in the host medium) at the plasma
frequency. This ratio was chosen to be $\lambda_p/h = 3.4$. Finally,
we have set the ratio $h/b=4$ so that the minimum surface-to-surface
separation of two neighboring spheroids, $\sigma$, was equal to $2b$.
As an illustration, for the specific case of silver, we have the
following parameters: the vacuum plasma wavelength is $\lambda_p^{\rm
(vac)} \approx 136 {\rm nm}$ and the corresponding value in the host
medium is $\lambda_p = \lambda_p^{\rm (vac)}/\sqrt{\epsilon_h}\approx
86{\rm nm}$; correspondingly, $h\approx 25{\rm nm}$, $b \approx 6{\rm
nm}$ and $a$ varies from $6 {\rm nm}$ to $60{\rm nm}$. The latter
value was obtained for the smallest aspect ratio used, $b/a=0.1$. It
can be seen that, for all points on the dispersion curves shown in
Figs.~\ref{fig:disp_prol_ort}-\ref{fig:disp_obl_par}, SPP frequencies
are well below the plasma frequency $\omega_p$ and $kh/\pi \ll 1$. The
last inequality is especially strong for the central frequencies
(indicated by horizontal arrows in Figs.~\ref{fig:disp_prol_ort} and
\ref{fig:disp_obl_par}) which were used to simulate wave packet
propagation in Section~\ref{sec:packets} below. This reconfirms the
dipole approximation validity.
\begin{figure}
\centerline{\psfig{file=fig2.ps,width=11cm,bbllx=170,bblly=475,bburx=450,bbury=770,clip=}}
\caption{\label{fig:disp_prol_ort} (color online) Dispersion curves (a) and group
velocity (b) for transversely polarized SPPs in chains built from
prolate spheroids whose axis of symmetry is perpendicular to the
chain, for different spheroid aspect ratios $b/a$ and for fixed
ratios $h/b=4$ and $\lambda_p/h=3.4$. Since the dispersion curves
are symmetric with respect to the $k$-axis, only the positive half
of the first Brillouin zone ($q>0$) is shown. Only data points that
were found numerically are plotted. Theoretically, however, all
dispersion curves in infinite chains start at the point $k=q=0$ and
follow the light line (labeled as $k=q$ in the top panel) for some
range of $q$'s. Horizontal arrows indicate central frequencies of
the Gaussian wave packets whose simulated propagation is illustrated
in Fig.~\ref{fig:packet_ort} below.}
\end{figure}
\begin{figure}
\centerline{\psfig{file=fig3.ps,width=11cm,bbllx=170,bblly=475,bburx=450,bbury=770,clip=}}
\caption{\label{fig:disp_obl_ort} (color online) Same as in
Fig.~\ref{fig:disp_prol_ort} but for a chain made of oblate
spheroids whose axis of symmetry is parallel to the chain and for a
different set of aspect ratios. SPP polarization is orthogonal to
the chain.}
\end{figure}
We now discuss the computed dispersion curves in more detail, starting
with the case of transverse SPP polarization
(Figs.~\ref{fig:disp_prol_ort},\ref{fig:disp_obl_ort}). First, we note
that in infinite, strictly periodic chains, the dispersion curves of
transversely-polarized SPPs start at the point $k=q=0$ and then follow
the light line $k=q$ for some range of $q$. This small-$q$ part of the
dispersion curve is related to the logarithmic divergence of the
dipole sum on the light line~\cite{markel_05_2} and is extremely
difficult to find numerically. Indeed, in order to satisfy
Eq.~\eqref{Disp_Real_alpha}, the points on the small-$q$ section of
the dispersion curve must be specified with exponentially large
numerical precision. This is why the small-$q$ section of the
dispersion curve has not been reported in a number of numerical
investigations~\cite{weber_04_1,simovski_05_1,koenderink_06_1,fung_07_1}.
However, the SPPs with $q\approx k \ll \pi/h$ exist and were observed
in numerical simulations~\cite{markel_07_2}.
In this paper, we do not consider the small-$q$ part of the dispersion
curve but focus on the SPPs for which the ratio $k/q$ is considerably
less than unity. Such modes exist for $qh/\pi \gtrsim 0.2$ for all
four values of $b/a$ shown in
Figs.~\ref{fig:disp_prol_ort},\ref{fig:disp_obl_ort}. The main point
of this paper is that the dispersion curve shape is strongly
influenced by the ratio $b/a$. When $b/a=1.0$, the corresponding
dispersion curve is almost flat (apart from the linear small-$q$
section of the curve). At $b/a=0.5$, the curve begins to bend down
noticeably at larger values of $q$. Finally, when $b/a \lesssim 0.25$
(in the case of prolate spheroids) or when $b/a \lesssim 0.35$ (in the
case of oblate spheroids), the curves cross the $k=0$ axis at the
point $q=q_c^\perp$ and no solutions exist for $q > q_c^\perp$. Near
the critical point, the dispersion curves acquires a very large
negative slope. Note that the corresponding slope is positive in the
$q<0$ part of the first Brillouin zone (which is not shown in the
figures).
From comparison of
Figs.~\ref{fig:disp_prol_ort},\ref{fig:disp_obl_ort}, a conclusion can
be made that the dispersion curves are more sensitive to the aspect
ratio in the case of oblate spheroids. In particular, the gap in the
first Brillouin zone of the chain lattice appears for more moderate
values of $b/a$ if the chain is made of oblate spheroids.
\begin{figure}
\centerline{\psfig{file=fig4.ps,width=11cm,bbllx=170,bblly=475,bburx=450,bbury=770,clip=}}
\caption{\label{fig:disp_prol_par} Same as in
Fig.~\ref{fig:disp_prol_ort} but for longitudinal SPP polarization
and a different set of aspect ratios. Note that, unlike in the case
of transverse polarization, the dispersion curves do not have linear
small-$q$ segments. As stated in the text, only data points that
satisfy $q\geq k$ are plotted, since there are no real-valued
solutions in the region $q<k$.}
\end{figure}
\begin{figure}
\centerline{\psfig{file=fig5.ps,width=11cm,bbllx=170,bblly=475,bburx=450,bbury=770,clip=}}
\caption{\label{fig:disp_obl_par} Same as in
Fig.~\ref{fig:disp_obl_ort} but for longitudinal SPP polarization
and a different set of aspect ratios. Note that, unlike in the case
of transverse polarization, the dispersion curves do not have linear
small-$q$ segments. As stated in the text, only data points that
satisfy $q\geq k$ are plotted, since there are no real-valued
solutions in the region $q<k$. Horizontal arrows indicate central
frequencies of the Gaussian wave packets whose simulated propagation
is illustrated in Fig.~\ref{fig:packet_par} below.}
\end{figure}
Dispersion curves for the longitudinal SPP polarization are shown in
Figs.~\ref{fig:disp_prol_par},\ref{fig:disp_obl_par}. It can be seen
that, for sufficiently small aspect ratios, there exists a critical
value $q_c^\parallel$ such that Eq.~\eqref{Disp_Real_alpha} has no
real-valued solutions for $q<q_c^\parallel$. Thus, SPP propagation in
the chain is only possible for $q$ larger than the critical value
$q_c^\parallel$ (if the latter exists). The group velocity shown in
Figs.~\ref{fig:disp_prol_par}(b),\ref{fig:disp_obl_par}(b) acquires
large positive values in the vicinity of $q_c^\parallel$.
Similarly to the case of transverse SPP polarization, the dispersion
curves are more sensitive to the aspect ratio if the chains are made
of oblate spheroids. Thus, in the case of prolate spheroids, the gap
in the first Brillouin zone of the lattice appears only when $b/a
\lesssim 0.1$. However, if the chain is composed of oblate spheroids,
the gap appears when $b/a \lesssim 0.25$.
\subsection{Dispersion Relations for Realistic Metal}
\label{subsec:complex_q}
The dispersion curves shown in Figs.~\ref{fig:disp_prol_ort} through
\ref{fig:disp_obl_par} contain no information on either the rate or
the direction of SPP spatial decay. However, it can be seen in these
figures that an SPP is characterized by phase and group velocities and
that these can have different signs when projected onto the $x$-axis.
On physical grounds, we expect that wave packets should decay in the
direction of propagation. This imposes certain restrictions on the
signs of the real and the imaginary parts of $q$. Thus, if $v_g v_p <
0$, we expect that ${\rm Re}(q) {\rm Im}(q) < 0$ while if $v_g v_p >
0$, we expect that ${\rm Re}(q) {\rm Im}(q) > 0$.
The above statement can be illustrated by considering the following
thought experiment. Assume that a short Gaussian optical pulse is
injected into the central part of a long chain by a spatially
localized source such as a near-field microscope tip operating in the
illumination mode. The pulse will propagate in the form of two wave
packets in both directions along the chain. If $v_g v_p < 0$, both
wave packets would be composed of Bloch waves whose phase velocities
point towards the source and group velocities point away from the
source. Since the sign of the phase velocity is the same as that of
${\rm Re}(q)$, and since both wave packets should decay in the
direction of propagation, we expect in this case that ${\rm Re}(q)
{\rm Im}(q) < 0$. Similar consideration can be applied to the case
$v_g v_p > 0$.
In the numerical simulations of this subsection, we have verified that
the above analysis is, indeed, correct. To this end, we have
incorporated the Ohmic losses into the model of metal permittivity.
Specifically, we have set the Drude relaxation constant in
Eq.~\eqref{drude} to be $\gamma = 0.002\omega_p$ which is the
experimental value for silver. Note that Eq.~\eqref{Disp_Real_alpha}
can still be satisfied in this case with a purely real pair of
$(\omega,q)$ (the real-part symbol in the left-hand side of the
equation must be retained if $\epsilon_m$ is complex-valued). However,
the imaginary part of the more general Eq.~\eqref{Disp_gen} is no
longer satisfied identically. We, therefore, must seek such pairs of
variables $(\omega,q)$, where $q$ is now complex, that satisfy both
the real and the imaginary parts of Eq.~\eqref{Disp_gen}. This can not
be achieved by the use of the simple bisection algorithm that was
employed in the previous subsection. Instead, we employ here the
Wolfram's {\em Mathematica} root finder to obtain complex roots of Eq.~\eqref{Disp_gen},
$q$, for each real value of $\omega$.
\begin{figure}
\centerline{\psfig{file=fig6_a.eps,width=11cm,clip=t}}
\vspace*{3mm}
\centerline{\psfig{file=fig6_b.eps,width=11cm,clip=t}}
\caption{\label{fig:disp_prol_ar0.15} Dispersion curves plotted parametrically
in the complex $q$-plane for transversely (a) and longitudinally (b)
polarized SPPs in an LPC of prolate spheroids. Parameters:
$\gamma/\omega_p = 0.002$, $b/a=0.15$ and other parameters same as
in Fig.~\ref{fig:disp_prol_ort}. Dots label the values of the
dimensionless parameter $kh/\pi = \omega h /\pi c_h$. In panel (a),
only the points that correspond to the the region $v_p v_g < 0$ are
shown, which corresponds, approximately, to $\vert {\rm Re}(q) \vert
h/\pi \gtrsim 0.17$.}
\end{figure}
\begin{figure}
\centerline{\psfig{file=fig7_a.eps,width=11cm,clip=t}}
\vspace*{3mm}
\centerline{\psfig{file=fig7_b.eps,width=11cm,clip=t}}
\caption{\label{fig:disp_obl_ar0.25} Same as in Fig.~\ref{fig:disp_prol_ar0.15}
for an LPC of oblate spheroids whose aspect ratio is $b/a=0.25$}
\end{figure}
The results of this simulation are plotted parametrically in the
complex $q$-plane in Fig.~\ref{fig:disp_prol_ar0.15} for a chain of
prolate spheroids of the aspect ratio $b/a=0.15$ and in
Fig.~\ref{fig:disp_obl_ar0.25} for a chain of oblate spheroids of the
aspect ratio $b/a=0.25$. Other parameters are the same as in
Fig.~\ref{fig:disp_prol_ort}. As expected, the real and imaginary
parts of the Bloch wave number have opposite signs for the
transversely-polarized SPPs so that the wave packets decay in the
direction of propagation specified by $v_g$, even though the phase
velocity is pointing into the opposite direction. For longitudinally
polarized SPPs the product $v_p v_g$ is always positive and,
correspondingly, ${\rm Im}(q) {\rm Re}(q) > 0$.
An important observation is that when ${\rm Re}q$ approaches one of
its critical values, ${\rm Im}q$ tends to zero. Correspondingly, the
decay length of the SPPs is dramatically increased.
We note that the data shown in Fig.~\ref{fig:disp_prol_ar0.15}
represent essentially the same dispersion curves as the ones plotted
in Fig.~\ref{fig:disp_prol_ort}(a) and \ref{fig:disp_prol_par}(a) for
the case $b/a=0.15$. Similarly, data in
Fig.~\ref{fig:disp_obl_ar0.25} correspond to the dispersion curves
plotted in Fig.~\ref{fig:disp_obl_ort}(a) and
\ref{fig:disp_obl_par}(a) for $b/a=0.25$. The discrepancy between the
curves $\omega[{\rm Re}(q)]$ computed by the two methods is negligibly
small. This further justifies the utility of the simple numerical
method of Section~\ref{subsec:real_q} for computing the dispersion
curves in LPCs.
\section{Attenuation of SPPs and the Decay Length}
\label{sec:l}
The nonzero relaxation constant in Eq.~\eqref{drude} leads to
exponential decay of SPPs even in the absence of radiative losses. The
decay lengths in LPCs is given by $\ell_\textsc{lpc} = 1 / {\rm
Im}(q)$. If $q$ is not very close to the edge of one of the gaps
that were discussed above, $\ell_\textsc{lpc}$ can be computed in the
quasi-particle pole approximation~\cite{markel_07_2} which results in
the following expression:
\begin{equation}
\label{lChain_gen}
\ell_\textsc{lpc}
\approx \frac{1}{-h^2{\rm Im}\left[\alpha_{\rm
LL}^{-1}(\omega)\right]}\left\vert \frac{\partial {\rm
Re}S(\xi,\eta)}{\partial \eta}\right\vert_{\eta= q_0(\xi)h }
\end{equation}
\noindent
where $S(\xi,\eta)$ is the dipole sum given by Eq.~\eqref{Dip_Sum} and
$q_0(\xi)$ is a purely real solution to \eqref{Disp_Real_alpha} for a
given value of the dimensionless parameter $\xi = \omega h / c_h$.
Under the assumptions that $\epsilon_h, {\rm Im}(\epsilon_m) \ll {\rm
Re}(\epsilon_m)$, we have
\begin{equation}
\label{delta_eval}
- {\rm Im} \left[ \alpha_{\rm LL}^{-1} (\omega) \right] \approx 4\pi
\omega\gamma/v \omega_p^2 \ .
\end{equation}
\noindent
As we have seen above, the real-valued solutions to
\eqref{Disp_Real_alpha} satisfy, albeit approximately, the more
general dispersion equation \eqref{Disp_gen}. Therefore, we can
evaluate the derivative in Eq.~\eqref{lChain_gen} at the point
$(\xi,\eta)$ with the understanding that $\xi$ and $\eta$ are purely
real variables that satisfy the approximate dispersion equation
\eqref{Disp_Real_alpha}. We further neglect the terms that are of the
order of $\sim 1/\xi$ and $\sim 1 /\xi^2$ in the square brackets of
the expression \eqref{Dip_Sum} for the dipole sum and arrive at the
following estimate:
\begin{equation}
\label{lChain}
\ell_\textsc{lpc} \simeq h\frac{Av}{2\pi h^3}\frac{\omega_p^2}{\omega
\gamma} \left \vert \sum_{m=1}^\infty
\frac{\cos(\omega h m/c_h )\sin(q h m)}{m^2} \right \vert \ ,
\end{equation}
\noindent
where $A=1$ for the transverse polarization and and $A=2$ for the
longitudinal polarization.
A direct calculation for the transversely-polarized SPP in a
prolate spheroid chain, $b/a=0.15$ and other parameters same as in
Fig.~\ref{fig:disp_prol_ort} yields the decay lengths
$\ell_\textsc{lpc}\simeq 7\mu{\rm m}$ for $\omega = 0.1\omega_p$ and
$\ell_\textsc{lpc}\simeq 15\mu {\rm m}$ for $\omega = 0.05\omega_p$.
Interestingly, the polarization dependence of decay length is
contained solely in the factor $A$. However, the two polarizations
have markedly different dispersion curves and therefore, same pairs of
variables $(\xi,\eta)$ may not be accessible for the two different
polarizations.
It is instructive to compare the SPP decay length in
nanoparticle chains and metallic nanowires. The dispersion equation in
a metallic cylindrical waveguide is~\cite{govyadinov_06_1,chang_07_1}:
\begin{equation}
\label{disp_wire_exact}
\frac{\epsilon_m I_1(\kappa_m R)}{\kappa_m I_0(\kappa_m
R)}+\frac{\epsilon_h K_1(\kappa_h R)}{\kappa_h K_0(\kappa_h R)}=0
\end{equation}
\noindent
where $R$ is the cylinder radius, $I_l(x)$ and $K_l(x)$ are the
modified Bessel functions, $\kappa_{m,h}=\sqrt{q^2-\epsilon_{m,h}
\omega^2/c_h^2}$ and the indices ``$m$'' and ``$h$'' label the
quantities for the metal and for the surrounding dielectric host,
respectively. If the wire is sufficiently thin, we can expand the
Bessel function to the first non-vanishing order in $\kappa_m R$ and
$\kappa_h R$ to obtain the simplified dispersion equation
\begin{equation}
\label{disp_wire_1}
\epsilon_m = \frac{2\epsilon_h}{(\kappa_h R)^2} \frac{1}{{\rm
ln}(\kappa_h R / 2) + C} \ ,
\end{equation}
\noindent
where $C$ is the Euler constant. This equation can be solved
approximately (with logarithmic precision) as
\begin{equation}
\label{kappa_sol}
\kappa_h^2 \approx
\frac{2\epsilon_h}{-\epsilon_m}\frac{1}{\displaystyle \frac{1}{2}{\rm
ln}\frac{-2\epsilon_m}{\epsilon_h} - C} \ .
\end{equation}
\noindent
We then use the Drude formula for $\epsilon_m$, take into account the
fact that the propagation constant ${\rm Re}q$ in a metal nanowire is
much larger than $\sqrt{\epsilon_h}\omega/c$ (which is the wavenumber
in the surrounding medium) and obtain the following estimate for the
decay length:
\begin{equation}
\label{lwire}
\ell_{\rm wire} \sim \frac{\omega_p}{\gamma}\frac{R}{2\sqrt{2\epsilon_h}} \ ,
\end{equation}
\noindent
One additional condition that has been used in deriving the above
estimate is $\omega \gg \gamma$. For a silver nanowire of radius $R =
25{\rm nm}$, the estimate yields $\ell_{\rm wire}\approx 2.8\mu {\rm
m}$.
The following conclusions can be made. While the SPP decay length in
nanowires depends only on the metal and host permittivities and the
waveguide radius, the same quantity in the LPCs can be controlled by
changing the inter-particle separation and the particle dimensions.
Decreasing the inter-particle spacings results in stronger
electromagnetic coupling and larger propagation distances. However,
for sufficiently small values of $h$, the dipole approximation breaks
down and the estimate~\eqref{lChain} becomes invalid. The dimensions
of a nanoparticle provide another set of degrees of freedom in
controlling the decay length in LPCs. Note that the expression
\eqref{lChain} contains an overall factor $v/ 2\pi h^3$ which can be
interpreted as the volume fraction of metal (in a unit cell of the
chain lattice). For an LPC of prolate spheroids, this factor is equal
to $2ab^2/3h^3$. However, for an LPC made of oblate spheroids, the
factor is $2a^2b/3h^3$. It obtains that the propagation distance in
oblate spheroid LPCs is effectively increased by the factor of $a/b>1$
compared to prolate spheroid LPCs. Thus constructing an LPC from thin
nano-disks whose axis of symmetry is parallel to the chain axis may be
beneficial.
\section{Modeling of Wave Packet Propagation}
\label{sec:packets}
\begin{figure}
\centerline{\psfig{file=fig8.ps,width=14cm,bbllx=140,bblly=380,bburx=470,bbury=760,clip=}}
\caption{\label{fig:packet_ort}
(color online) Envelopes of transversely polarized wave packets in a
chain of $N=5000$ prolate nanospheroids with the aspect ratio
$b/a=0.15$ at different moments of time $t$. Spheroids are oriented
so that their axes of symmetry are perpendicular to the chain axis.
Time is measured in the units of $\tau=h/c_h$. Column (a):
$\omega_0=0.1\omega_p$, $v_g\approx 0.57c_h$. Column (b):
$\omega_0=0.05\omega_p$, $v_g\approx 1.16c_h$. Arrows indicate that
the wave packet propagates from right to left after being reflected
from the far end of the chain.}
\end{figure}
We now demonstrate that superluminal wave packets can indeed propagate
in chains with sufficiently small aspect ratios $b/a$. To this end, we
consider a finite chain of $N$ nanoparticles excited by a pulse with
Gaussian temporal profile incident on the first particle in the chain.
In the time domain, the pulse is described by the formula
\begin{equation}
\label{E_inc_t}
E_n^{\rm ext}(t) = \delta_{n1} {\mathcal E} \exp\left[ -i\omega_0 t - (t/\Delta t)^2 \right] \ ,
\end{equation}
\noindent
where ${\mathcal E}$ is an arbitrary amplitude and $\Delta t$ is the pulse
duration. This function has the Fourier transform
\begin{equation}
\label{E_inc_omega}
\tilde{E}_n^{\rm ext}(\omega) = \delta_{n1} \sqrt{\pi} \Delta t {\mathcal E} \exp\left[ -
\frac{(\omega - \omega_0)^2}{(\Delta\omega)^2} \right] \ , \ \
\Delta\omega = \frac{2}{\Delta t} \ .
\end{equation}
\noindent
The numerical procedure is as follows. The above expression for
$\tilde{E}_n^{\rm ext}(\omega)$ is used as the free term in the
right-hand side of Eq.~\eqref{CDE}. The equation is solved numerically
by direct matrix inversion for multiple values of $\omega$ sampled in
a sufficiently large interval and with sufficiently small step to
ensure convergence. This yields a family of numerical solutions
$\tilde{d}_n(\omega)$. The real time quantities $d_n(t)$ are then
obtained by the inverse Fourier transform according to
\begin{equation}
\label{d_n_t}
d_n(t) = \int \tilde{d}_n(\omega) \exp(-i\omega t) \frac{d\omega}{2\pi} \ .
\end{equation}
\noindent
Numerically, this integral is evaluated by the trapezoidal rule.
\begin{figure}
\centerline{\psfig{file=fig9.ps,width=14cm,bbllx=140,bblly=380,bburx=470,bbury=760,clip=}}
\caption{\label{fig:packet_par}
Envelopes of transversely polarized wave packets in a chain of
$N=5000$ oblate nanospheroids with the aspect ratio $b/a=0.25$ at
different moments of time $t$. Spheroids are oriented so that their
axes of symmetry coincide with the chain axis. Time is measured in
the units of $\tau=h/c_h$. Note that the largest time shown in this
figure is twice smaller than the respective quantity in
Fig.~\ref{fig:packet_ort}. Column (a): $\omega_0=0.15\omega_p$,
$v_g\approx 0.77c_h$. Column (b): $\omega_0=0.05\omega_p$,
$v_g\approx 2.57c_h$. Arrows indicate that the wave packet
propagates from right to left after being reflected from the far end
of the chain.}
\end{figure}
We have simulated transversely polarized wave packets in LPCs of
prolate spheroids with the aspect ratio $b/a=0.15$. Longitudinally
polarized SPPs were simulated in chains of oblate spheroids with the
aspect ratio $b/a=0.25$. Simulations were performed in a chain
consisting of $N=5\cdot 10^3$ spheroids; the overall length of the
chain was (given $h=25{\rm nm}$) $L=125\mu{\rm m}$. All parameters
were the same as those used for calculating dispersion curves shown in
Figs.~\ref{fig:disp_prol_ort}, with the only exception that Ohmic
losses in the metal were taken into account by means of using the
nonzero Drude relaxation constant $\gamma = 0.002\omega_p$. Four sets
of simulations have been performed, the first two for the transverse
and the other two for the longitudinal polarization.
In the case of the transverse polarization, two different central
frequencies of the pulse have been used. The first pulse had the
central frequency $\omega_0=0.1\omega_p$ (correspondingly,
$k_0h/\pi=0.06$ where $k_0=\omega_0/c_h$) and the pulse spectral width
was $\Delta\omega = \omega_0/5$. Thus the excitation was relatively
broad-band but very narrow in the time domain: given the experimental
value of $\omega_p$ for silver, the above spectral width corresponds
to $\Delta t \approx 7.2{\rm fsec}$. The second pulse had the central
frequency twice smaller than the first, $\omega_0=0.05\omega_p$, with
the same relative spectral width $\Delta\omega = \omega_0/5$. In the
time domain, this corresponds to $\Delta t \approx 14.2{\rm fsec}$.
The central frequencies of the two pulses are shown in
Fig.~\ref{fig:disp_prol_ort}(a) by the horizontal arrows. Amplitudes
of the obtained wave packets are illustrated in
Fig.~\ref{fig:packet_ort} at different moments of time measured in the
units of $\tau=h/c_h$. The maximum time shown on the plots is
$t=6\,10^3\tau \approx 800{\rm fsec}$.
It can be seen that the wave packet with $\omega_0=0.1\omega_p$
propagates away from the source with the subluminal group velocity
$v_g\approx 0.57c_h$. However, the wave packet with the smaller
central frequency propagates at the speed $v_g\approx 1.16c_h$. These
group velocities are in quantitative agreement with the data shown in
Fig.~\ref{fig:disp_prol_ort}(b). Note that the group velocities can be
evaluated as the slopes of the dispersion curve drawn for $b/a=0.15$
in Fig.~\ref{fig:disp_prol_ort}(a) at the central frequencies
indicated by the horizontal arrows. As expected, the superluminal wave
packet is spreading faster than the subluminal wave packet. This is so
because of the larger value of the second derivative
$\partial^2\omega/\partial^2 q$ at the smaller central frequency. Yet,
near the end of the chain, the duration of the superluminal pulse is
still only $\approx 1{\rm psec}$.
The two sets of simulations for the longitudinal polarization are
shown in Fig.~\ref{fig:packet_par}. Here we have used a chain of
oblate spheroids with the aspect ratio $b/a=0.25$. The central
frequencies of the two pulses were $\omega_0 = 0.25\omega_p$ and
$\omega_0 = 0.1\omega_p$. The pulses' relative spectral widths were
the same as in the case of the transverse polarization, namely, the
pulse spectral width was $\Delta\omega = \omega_0/5$. By tracing the
maximum of each wave packet, we deduce $v_g=0.88c_h$ for
$\omega_0=0.25\omega_p$ and $v_g=2.17c_h$ for $\omega_0=0.1\omega_p$.
This is in full agreement with the group velocities shown in
Fig.~\ref{fig:disp_obl_par}(b).
Finally, note that the decay lengths in each simulation can not be
easily deduced from the time evolution of the maxima of the wave
packets. This is because the propagation is accompanied by both decay
and spreading. The latter takes place even in the absence of Ohmic
losses.
\section{Concluding Remarks}
\label{sec:disc}
In this paper, we have employed the coupled dipole approximation to
compute the dispersion curves and to model propagation of wave packets
of surface plasmon polaritons (SPPs) in linear periodic chains (LPCs)
of metallic nanospheroids. The main novel element of this study, as
compared to the previous work on the
subject~\cite{weber_04_1,simovski_05_1,koenderink_06_1,fung_07_1,park_04_1,citrin_05_1,citrin_06_1,markel_07_2},
is the account of particle nonsphericity.
We have shown that the group velocity, decay length and the bandwidth
of surface plasmon polaritons (SPPs) propagating in linear periodic
chains (LPCs) of metallic nanoparticles can be effectively tuned. The
tunability is achieved by means of varying the nanoparticles aspect
ratio. The decay length can be dramatically increased for Bloch wave
numbers $q$ near the edges of gaps that appear in the first Brillouin
zone of the lattice for sufficiently small aspect ratios. At the same
time, the SPP group velocity is increased up to superluminal values.
By replacing the host medium with vacuum, it is also possible to
excite a wave packet whose group velocity is larger than the speed of
light in vacuum. Such wave packets exist in nature and were observed
experimentally~\cite{wang_00_1,gehring_06_1}.
Comparison of Figs.~\ref{fig:disp_prol_ort} through
\ref{fig:disp_obl_par} reveals that the parameters of two different
LPCs can be tuned so that one LPC supports transversely polarized SPPs
and the other chain supports longitudinally polarized SPPs with {\em
the same electromagnetic frequency}. This fact can be utilized for
guiding the SPPs through corners (ninety-degree turns in an LPC)
and/or for splitting and coupling the SPPs at T-junctions. Another
potentially interesting element of an integrated photonic circuit a
two-segment straight LPC. Assume that, at a given frequency, one
segment can support only transversely-polarized SPPs while the other
segment supports only longitudinally polarized SPPs. At the junction
of the two segments, an externally-manipulated (i.e., by magnetic
field) coupling nanospheroid is placed. When the coupling nanospheroid
makes the angle of either $0$ or $\pi$ with respect to the chain axis,
the two segments are decoupled and do not allow direct transmission of
light pulses. However, if the coupling spheroid is rotated by the
angle of $\pi/4$ with respect to the axis, the transversely-polarized
SPP propagating in the first segment is coupled to the
longitudinally-polarized SPP in the second segment and transmission
along the chain becomes possible. Detailed investigation of these
possibilities will be the subject of future work.
In the case of transverse SPP polarization, the group and phase
velocities of SPPs can be antiparallel. We, however, have found that
the negative group velocity {\em per se} (defined here by the
condition $v_g v_p < 0$) does not necessarily imply superluminal
propagation or a negative time delay as was suggested
previously~\cite{bolda_94_1,dogariu_01_1}. For example, the wave
packet shown in Fig.~\ref{fig:packet_ort}(a) propagates slower than
$c_h$ even though it is composed of waves whose frequencies are in the
negative dispersion region. It is important to realize that the
effects theoretically described in these two references, as well as
the corresponding experimental
observations~\cite{wang_00_1,gehring_06_1}, involve propagation of an
optical pulse from a medium with normal dispersion to a medium with
negative dispersion and the presence of the interface is essential. In
this paper, we are looking at a somewhat different physical situation
when the optical pulse is injected into a waveguide by a predetermined
external source which is located in the near field of the waveguide.
We then observe that the pulse propagates away from the source with
the velocity $\vert v_g \vert$, irrespectively of the sign of the
product $v_g v_p$. Thus the superluminal propagation is obtained when
$\vert v_g \vert > c_h$ but not necessarily when $v_g v_p <0$.
Antiparallel phase and group velocities that we have observed in the
case of transverse SPP polarization deserve a separate discussion. We
believe that this phenomenon can not be interpreted as ``negative
refraction''. The reason is that the LPCs considered in this paper are
essentially discrete objects and can not be described by effective
medium parameters. The elementary excitations that propagate in LPCs
are Bloch waves rather than sinusoidal waves, and this fact should not
be disregarded. The region of negative dispersion shown in
Fig.~\ref{fig:disp_prol_ort}(a) starts at $qh \approx 0.2\pi \approx
0.6$. In general, the chain can be viewed as continuous only when $qh
\ll 1$. The above condition is not satisfied in the region of negative
dispersion. It is, however, not clear {\em a priori}, how strong this
inequality should be for the effective medium approximation to be
valid. In the specific case of LPCs, one can consider the following
argument. We expect the effective medium parameters such as the
permittivity $\epsilon(\omega)$ or the refraction index $n(\omega)$ to
be single-valued functions of their argument. However, for every point
on the negative slope section of the dispersion curves shown in
Fig.~\ref{fig:disp_prol_ort}(a), there is another point on the same
curve with the same frequency but a smaller value of $q$. This second
point is located on the linear, small-$q$ section of the dispersion
curve. Although this small-$q$ section is difficult to find
numerically (and, as a result, is often overlooked), it exists. It is
therefore logical to assume that, if a chain be assigned some
effective medium parameter for a given frequency $\omega$, this
parameter must be computed using the point on the small-$q$ section of
the dispersion curve. The latter exhibits positive (and linear)
dispersion. In the above argument, we have disregarded the possibility
of introducing non-local effective medium parameters which are
characteristic, for example, of chiral media and can result in
negative dispersion~\cite{agranovich_06_1}. However, the physical
object that we consider in this paper is essentially non-chiral.
The authors can be reached at: \\
{\tt algov@seas.upenn.edu} and {\tt vmarkel@mail.med.upenn.edu}.
\bibliographystyle{prsty} |
0709.2600 | \section{Introduction}\label{int}
The approximation of a line by a planar lattice yields a stair climbing pattern.
Consider the averages of a function of a random field along a finite window
moving up the stair climbing pattern.
Under which conditions does this sequence converge, and what are explicit
formulae for the limit?
More formally, let $L_{\lambda,t}(z):=(z,[\lambda z+t])\ (z\in \mathbb{Z})$ be a lattice
approximation of the line with slope $\lambda$ and $y$-intercept $t,$
and let $m\in\mathbb{N}$ be a fixed window size.
Let $P$ be a $\mathbb{Z}^{2}$-indexed random field with values in a set
$\Upsilon,$ i.e., a stationary probability measure on $\Omega:=\Upsilon^{{\mathbb{Z}^{2}}},$
and let ${\mathcal F}$ be the canonical $\sigma$-algebra.
Consider the averages
\begin{equation}\label{Intro:ex.sq}
\frac{1}{n}\sum_{i=0}^{n-1}f \big( \omega(L_{\lambda,t}(i,...,I+m-1)) \big)
\qquad(n\in\mathbb{N})
\end{equation}
of a function $f\in \L^{1}(\Omega,{\mathcal F},P).$
What can we say about $P$-almost sure or $\L^{1}(P)$-convergence
of this sequence?
Averages of this type are similar to the ones that occur in the context of directional
Shannon-MacMillan theorems for lattice random fields (cf.~\cite{Bre01SMcM}).
The situation described above can be represented as a special case of a
more general set up involving skew product transformations.
In the independent component, or {\it base,} we have a measure-preserving
transformation $\t$ on a probability space $(M,{\mathcal B},\mu).$ In the dependent
component, or {\it fibre,} we have a mixing semigroup of measure-preserving
transformations $(\theta_{k})_{k\in K}$ on a probability space $(\Omega,{\mathcal F},P),$
which is linked to the base by a $K$-valued ${\mathcal B}$-measurable function
$\kappa$ on $M.$ The class of skew products considered in this paper is
given by
\begin{eqnarray}\label{Intr:def.skew}
S(t,\omega)=(\t (t),\theta_{\kappa(t)}\omega)\qquad(t\in M,\omega\in\Omega).
\end{eqnarray}
This paper deals with the following two questions: \\
\noindent (A) Is $S$ ergodic or mixing with respect to the product measure?\\
\noindent (B) Do the ergodic averages along
the skew product converge uniformly with respect to the base?\\
Under suitable assumptions, we will answer both questions positively.
Ergodicity, and other mixing properties of various classes of skew products
have been studied by a number of authors, but the above situation does
not fit into any of the settings covered by existing literature.
Kakutani \cite{Kak51} introduced a skew product with a Bernoulli-shift
in the base and an ergodic transformation in the fibre was introduced by .
He showed that the skew product is ergodic if and
only if the transformation in the fibre is ergodic. Other mixing
properties were investigated, e.g., by Meilijson \cite{Mei74},
den Hollander and Keane \cite{dHK86},
and Georgii \cite{Geo97}. Adler and Shields (cf.~\cite{AdS72} and
\cite{AdS74}) considered a translation on the cirle for the fibre.
Anzai \cite{Anz51} introduced skew products of two translations
on the circle, and derived a criterion for ergodicity. Furstenberg
\cite{Fur61} studied unique ergodicity. Zhang \cite{Zha96} investigated
this for a translation on a torus in the fibre.
A torus translations in the base can also be combined with the
translation on $\mathbb{R}$ by the value of a real function of the argument
in the base. Skew products of this type are called real extensions of
torus translations, and they were explored in Oren \cite{Ore83},
Hellekalek and Larcher \cite{HeL86}, \cite{HeL89}, and Pask \cite{Pas90}.
\pagestyle{plain}
Our answer to question (A) is summarized in Theorem \ref{ErgS}.
We prove that, under suitable mixing conditions
for the transformations in the fibre, the properties ergodic, weakly mixing,
and strongly mixing, are passed on from the transformation in the base
to the skew product. As an explicit example we study the case when $P$ is
a random field and $(\theta_{k})_{k\in K}$ is a group of shift transformations.
In this case, the conditions on the fibre transformation can be insured
by assuming tail-triviality for $P$ and a growth condition for ergodic sums of
$\kappa$ along $\t$ (cf.~Corollary \ref{ShiftsSProp}).
Our answer to question (B) is given in Theorem \ref{UnifL}.
The proof combines two different approaches. The first approach explores
uniform convergence theorems in the spirit of Weyl's classical result
for the rotation on the circle. As a little addition to the theorems of Weyl and
Oxtoby we show that the ergodic averages of a continuous and uniquely ergodic
transformation on a real interval convergence uniformly for the class of
Riemann-intergrable functions (cf.~Corollary \ref{UnifR}).
The second approach are techniques developed for ergodic theorems
along subsequences. We extend Blum and Hanson's theorem to the
$d$-parameter case replacing the strict monotonicity condition on the sequence
by a growth condition on the coupling function $\kappa,$ uniformly in $t.$
Combining the two approaches we obtain uniform convergence in the base and
$\L^p$-convergence in the fibre for functions that are continuous with respect
to the base and fulfill an integrability condition in the fibre (cf.~Theorem \ref{UnifL}).
We further derive a variation of this theorem for the class of Riemann-integrable
functions on a real interval (cf.~Corollary \ref{UniLRMak}).
We further derive a result (cf.~Theorem \ref{UnifP}) about uniform convergence
in the base and $P$-almost sure convergence in the fibre, provided the
iterates of the function fulfill an equicontinuity condition.
We conclude the paper by returning to our initial questions about the asymptotics of
(\ref{Intro:ex.sq}).
Let $\mathbb{T}:=[0,1)$ be the circle equipped with the Borel $\sigma$-algebra
${\mathcal B}$ and the Lebesgue measure $\mu.$
For $\lambda\in\mathbb{R}$ define a rotation on $\mathbb{T}$ by
$\t_\lambda(t):= t+\lambda\ {\rm mod}\ 1.$
For $x\in\mathbb{R}$ let $[x]$ be the integer part of $x.$
Define a skew product on the product space $\mathbb{T}\times\Omega$ by
\begin{equation}\label{Intro:ex.sp}
S(t,\omega):=\big(\t_{\lambda}(t),\vartheta_{(1,[\lambda +t])}\omega\big)\qquad (t\in\mathbb{T},\omega\in\Omega).
\end{equation}
We will see that the iterates of $S$ follow the stair climbing pattern $L_{\lambda,t}.$
This allows to rewrite the sequence (\ref{Intro:ex.sq}) as an average of
$f$ along the orbit of $S.$ The convergence of this sequence,
for {\it all} starting levels $t$ of the stair climbing pattern,
is a consequence of the uniform ergodic theorems derived in in Section \ref{ret}
(cf.~Corollary \ref{Unif:ex.staircase}).
{\bf Ouline of the paper: }
In the first section we define mixing properties of semigroups of transformations
along sequences and we introduce a the class of skew products considered in this paper.
In Theorem \ref{ErgS}, we give the result on ergodicity and mixing properties of these
skew products. Finally, we have a closer look on the case when the fibre transformation
is a shift operator for a random field.
In Section \ref{ret} we discuss ergodic theorems for skew products
with ergodic base transformation (cf.~Corollary \ref{Reterg}) and with
periodic base transformations (cf.~Corollary \ref{Retper}). We illustrate the results
with two examples related to the sequence \eqref{Intro:ex.sq}.
In the last section we focus on the main aim of this paper, the uniform
convergence with respect to the base. Depending, among other things,
on the regularity of the function with respect to the base variable, we obtain
different kinds of convergence in the fibre.
Theorem \ref{UnifL} states $\L^p(P)$-convergence provided the function is
continuous with respect to the base variable. Corollary \ref{UniLRMak} is a version
of this for Riemann-integrable functions.
Theorem \ref{UnifP} states $P$-almost sure convergence, provided the functions
fulfill a certain equicontinuity condition.
Corollary \ref{Unif:ex.staircase} brings us back to the original motivation for this paper.
It states the convergence of the sequences \eqref{Intro:ex.sq}.
\section{Mixing properties of a class of skew products}\label{spc}
Let $(\Omega,{\mathcal F},P)$ be a probability space,
and let $(\theta_k)_{k\in\mathbb{N}_{0}^{d}}$ be a {\it d-parameter semigroup} of
measure-preserving transformations on $(\Omega,{\mathcal F},P),$ i.e.,
each of the transformations preserves the measure $P,$ and
$\theta_0=\text{{\rm Id}},$ and
$\theta_k\circ\theta_l=\theta_{k+l}\ \text{for all}\ k,l\in\mathbb{N}_0^d.$
$(\theta_k)_{k\in\mathbb{Z}_{0}^{d}}$ is a {\it d-parameter group} if
$(\theta_k)_{k\in\mathbb{N}_{0}^{d}}$ is a semigroup and
$\theta_{-k}=\theta_{k}^{-1}$ for all $k\in \mathbb{N}^d.$
The following example will be used frequently in our settings.
Let $\sigma_1$ and $\sigma_2$ be two commuting
measure-preserving transformations on $(\Omega,{\mathcal F},P).$ Then
\begin{eqnarray}\label{SegrEx}
\theta_k :=\sigma_1^{k^{(1)}}\circ\sigma_2^{k^{(2)}}
\quad\hbox{for}\quad k=(k^{(1)},k^{(2)})\in\mathbb{N}_0^2
\end{eqnarray}
defines a two-parameter semigroup $(\theta_k)_{k\in \mathbb{N}_0^2}$ of
measure-preserving transformations on $(\Omega,{\mathcal F},P).$
If $\sigma_1$ and $\sigma_2$ are invertible
it extends to a two-parameter group $(\theta_k)_{k\in \mathbb{Z}^2}.$
The constructions extends to $d$-parameters in a obvious way.
Let $K=\mathbb{N}_0^d$ or $K=\mathbb{Z}^d.$ Let $\t$ be a measure-preserving
transformation of a probability space $(M ,{\mathcal B} ,\mu ),$
and assume that $\kappa$ is a ${\mathcal B}$-measurable function
on $M$ with values in $K.$ Then
\begin{eqnarray*}
S(t,\omega)=(\t (t),\theta_{\kappa(t)}\omega)\qquad (t\in M,\omega\in\Omega)
\end{eqnarray*}
defines a skew product on the product space $\ov\Omega:=M\times\Omega.$
In particular, choosing $\kappa\equiv k_0$ for a constant $k_0\in K$
yields the uncoupled product of $\t$ and $\theta_{k_0}.$
Obviously, $S$ is measurable with respect to the product $\sigma$-algebra
$\sigma$-algebra $\ov{\mathcal F}:={\mathcal B}\otimes{\mathcal F},$ and it preserves the product measure
$\ov P:=\mu\otimes P.$
It is easy to see that for all $n,m\in\mathbb{N}_0,$ for all $t\in M,$ and for all $\omega\in\Omega,$
\nopagebreak
\begin{eqnarray}\label{ItS}
S^n (t,\omega) = \big(\t^n (t),\theta_{\kappa_n(t)}\omega\big)
\quad \mbox{and}\quad
\kappa_{n+m}=\kappa_n+\kappa_m\circ\t^n,
\text{ where }
\kappa_n = \ErgSum{\kappa}{\t}.
\end{eqnarray}
Let us know study under which conditions the skew product is ergodic.
Furthermore, as suggested by J. Aaronson, we broaden the question to other
mixing properties. Answers to these questions will be given in the next lemma.
Note that, by a simple projection argument, the ergodicity of $\t$ is necessary
for the ergodicity of $S.$ As we know in the case of an uncoupled product,
the ergodicity of both transformations does not garanty the ergodicity of
the product. However, it can be shown that the product
is ergodic whenever one of the transformations is ergodic and the
other one is weakly mixing (cf.~\cite{Kre85}).
A major ingredient for $S$ for our lemma are assumptions that
bring into play the function $\kappa.$ We make use of two conditions:
\begin{itemize}
\item[{\bf (C1)}] $(\theta_k)_{k\in K}$ is weakly mixing along the sequence
$(\kappa_n(t))_{n\in\mathbb{N}}$
for $\mu$-almost all $t\in M.$
\item[{\bf (C2)}] $(\theta_k)_{k\in K}$ is strongly mixing and
$(\kappa_n(t))_{n\in\mathbb{N}}$ goes to infinity for
$\mu$-almost all $t\in M.$
\end{itemize}
\noindent Note that (C2) implies (C1).
Condition (C2) can be easily verified for lattice approximations of a line,
as discussed in the context of Corollary \ref{Unif:ex.staircase}.
We further need the notion of
{\it weakly mixing along a sequence.} This has been introduced for
transformations by N.\ Friedman (cf.~\cite{Fri83}), and
we extend it to $d$-parameter (semi-)groups.
\begin{definition}\label{SeqMixGr} Let $(k_n)_{n\in\mathbb{N}}$ be a $K$-valued
sequence. A (semi-)group $(\theta_k)_{k\in K}$ of
measure-preserving transformations on $(\Omega,{\mathcal F},P),$
is called {\rm weakly mixing along $(k_n)_{n\in\mathbb{N}}$} with respect to $P,$ if
\begin{eqnarray}\label{knwmixGr}
\frac{1}{n}\sum_{i=0}^{n-1}\big\vert P\big(A\cap\theta^{-1}_{k_i}B\big)
- P\big(A\big)\,P\big(B\big)\big\vert\convinfty{n}0\qquad
\text{for all }A,B\in{\mathcal F}.
\end{eqnarray}
\end{definition}
\begin{lemma}\label{ErgS}
(i) Assume condition (C1). If $\t$ is ergodic w.r.t.~$\mu$
then $S$ is ergodic w.r.t.~$\ov{P}.$
\noindent (ii) Assume condition (C1). If $\t$ is weakly mixing
w.r.t.~$\mu$ then $S$ is weakly mixing w.r.t.\ $\ov{P}.$
\noindent (iii) Assume condition (C2). If $\t$ is strongly mixing
w.r.t.~$\mu$ then $S$ is strongly mixing w.r.t.~$\ov{P}.$
\end{lemma}
\noindent {\bf Proof. }
We give the proof of the first statement here; the remaining proofs are
conducted in a similar fashion. Assume condition (C1) and
let $\t$ be ergodic with respect to $\mu.$
To prove the ergodicity of $S$ we will show that for all bounded
$\ov{\mathcal F}$-measurable functions $F$ and $G$
\begin{eqnarray*}
\frac{1}{n}\sum_{i=1}^n \int_{M\times\W}F\circ S^i(t,\omega)\cdot
G(t,\omega)\,d\ov P \convinfty{n}{\widetilde{\omega}} F\cdot{\widetilde{\omega}} G,
\end{eqnarray*}
where
$$
{\widetilde{\omega}} F =\int_{M\times\W}F(t,\omega)\,d\ov P
\quad\quad\text{and}\quad\quad
{\widetilde{\omega}} G =\int_{M\times\W}G(t,\omega)\,d\ov P.
$$
It is sufficient to show this for functions which are products of
functions on the factors, i.e.,
$F(t,\omega)=f(t)\Phi(\omega)\text{ and }G(t,\omega)=g(t)\Psi(\omega),$
where $f$ and $g$ are bounded ${\mathcal B}$-measurable functions on $M$
and $\Phi$ and $\Psi$ are bounded ${\mathcal F}$-measurable functions on $\Omega.$
(The general case follows by approximation.) For these functions we have
$
{\widetilde{\omega}} F={\widetilde{\omega}} f\cdot{\widetilde{\omega}}\Phi \text{ and }
{\widetilde{\omega}} G={\widetilde{\omega}} g\cdot{\widetilde{\omega}}\Psi,
$
with
$$
{\widetilde{\omega}} f=\int_M f\,d\mu,\ \ {\widetilde{\omega}}\Phi=\int_\Omega \Phi\,dP,\ \
{\widetilde{\omega}} g=\int_M g\,d\mu,\ \ \text{and}\ \ {\widetilde{\omega}}\Psi=\int_\Omega\Psi\,dP,
$$
and we obtain
\begin{align}\nonumber
\bigg\vert \,\frac{1}{n}&\sum_{i=1}^n \int_{M\times\W} F\circ S^i(t,\omega)\cdot
G(t,\omega)\,d\ov P\, -\, {\widetilde{\omega}} F\cdot{\widetilde{\omega}} G\,\bigg\vert\\
\nonumber
\le &
\bigg\vert\, \int_M \frac{1}{n}\sum_{i=1}^n \int_{M\times\W}
f(\t^i(t))g(t)\Phi(\theta_{\kappa_i(t)}\omega)\Psi(\omega)
-f(\t^i(t))g(t){\widetilde{\omega}}\Phi\,{\widetilde{\omega}}\Psi
\, d\ov P\,\bigg\vert\,d\mu\\
\nonumber
& +
\bigg\vert\,\frac{1}{n}\sum_{i=1}^n\bigg(\int_{M}
f(\t^i(t)) g(t)\,d\mu
\,-\,{\widetilde{\omega}} f\cdot{\widetilde{\omega}} g\bigg)\cdot{\widetilde{\omega}}\Phi\cdot{\widetilde{\omega}}\Psi\,\bigg\vert\\
\nonumber
\le &
\|f\|_\infty\|g\|_\infty\, \frac{1}{n}\sum_{i=1}^n\bigg\vert\,
\int_\Omega\Phi(\theta_{\kappa_i(t)}\omega)\Psi(\omega)\,dP
-{\widetilde{\omega}}\Phi\cdot{\widetilde{\omega}}\Psi\,\bigg\vert
+
\bigg\vert\,\frac{1}{n}\sum_{i=1}^n\int_M f\circ\t^i\hspace{0.05mm}\cdot\hspace{0.05mm}
g\,d\mu \,-\,{\widetilde{\omega}} f\cdot{\widetilde{\omega}} g\,\bigg\vert\,\vert\,{\widetilde{\omega}}\Phi
\,\vert\,\vert\,{\widetilde{\omega}}\Psi\,\vert.
\end{align}
This goes to $0,$ because
the averages in the first expression convergence to $0$
for $\mu$-almost all $t$ by condition (C1) and
the second expression converges to $0$ because of the ergodicity of $\t.$
\hfill q.e.d.
\vspace{3mm}
This section concludes with a closer look at the case of shift
transformations on a discrete random field with values in a set $\Upsilon.$
Let $\Omega:=\Upsilon^{\mathbb{Z}^d}.$ For any $J\subseteq\mathbb{Z}^d$ let
${\mathcal F}_J$ denote the $\sigma$-algebra generated
by all projections $\omega\mapsto\omega(j)$ with $j\in J,$
and let ${\mathcal F}:={\mathcal F}_{\mathbb{Z}^d}.$ Denote the coordinates of an elements in $\mathbb{Z}^d$ by
upper indices, and let $\|\,\cdot\,\l$ be its maximum norm.
Consider the {\it shift transformations}
$(\vartheta_v)_{v\in\mathbb{Z}^d}$ on $\Omega,$ i.e.,
$\vartheta_v(\omega)(j):=\omega(j+v)$ $(j\in\mathbb{Z}^d).$
Let $P$ be a {\it random field}, i.e., a measure on $(\Omega,{\mathcal F})$
which is invariant with respect to $\vartheta_v$ for all
$v\in\mathbb{Z}^d.$ The {\it tail field} is the $\sigma$-algebra
$
{\mathcal T}:=\bigcap_{V\subset\mathbb{Z}^d\ {\rm finite}}
{\mathcal F}_{\mathbb{Z}^d\setminus V},
$
and $P$ is called {\it tail-trivial} if it fulfills a $0$-$1$ law on ${\mathcal T}.$
Let $v_1, v_2\in\mathbb{Z}^d.$ As in \eqref{SegrEx},
$\theta_k:=\vartheta_{v_1}^{k^{(1)}}\circ \vartheta_{v_2}^{k^{(2)}}$ $(k\in\mathbb{Z}^d)$
defines a 2-parameter group of measure-preserving transformations. We have
\begin{equation}\label{spc:twoShifts}
\theta_{\kappa_n(t)}=\vartheta_{v_1}^{\kappa_n^{(1)}(t)}\,\circ\,\vartheta_{v_2}^{\kappa_n^{(2)}(t)}
=\vartheta_{
{\kappa_n^{(1)}(t)\,v_1 } \,+\,
{\kappa_n^{(2)}(t)\,v_2 }
}.
\end{equation}
In this situation we have the following
\begin{cor}\label{ShiftsSProp}
Let $v_1$ and $v_2$ be linear independent vectors in $\mathbb{Z}^d.$
Assume that $P$ is tail-trivial and that the sequence
$\big(\|\kappa_n(t)\|\big)_{n\in\mathbb{N}}$ goes to infinity for
$\mu$-almost all $t\in M.$
Then, when $\t$ is ergodic, weakly
mixing or strongly mixing with respect to $\mu,$ $S$ is ergodic,
weakly mixing or strongly mixing with repect to $\ov P,$
respectively.
\end{cor}
\noindent {\bf Proof. } We are going to show condition (C2). Define the boxes
$V_n=\big\{v\in\mathbb{Z}^d\,\big\vert\,\|\,v\,\|\le n\big\}$ $(n\in\mathbb{N}),$ and
let $B\in{\mathcal F}_J,$ for some finite subset $J$ of $\mathbb{Z}^d.$ Then there
is an $m\in\mathbb{N}$ such that $J\subseteq V_m.$ Setting
$m(n):=\kappa_n^{(1)}(t)\,v_1+\kappa_n^{(2)}(t)\,v_2$ we observe
that the translated sets $J-m(n)$ are contained in
$V_{{\widetilde m}(n)}^c,$ where ${\widetilde m}(n)=(m(n)-2m)\vee
0.$ For any $A\in{\mathcal F},$ we obtain
\begin{align}\label{ShiftsSProp1}
\nonumber
\big\vert P\big(A\cap\theta_{\kappa_n(t)}^{-1}B\big)
-P\big(A\big)\,P\big(B\big)\big\vert
& = \big\vert P\big(A\cap
\vartheta_{\kappa_n^{(1)}(t)\,v_1+\kappa_n^{(2)}(t)\,v_2}^{-1}B\big)
-P\big(A\big)\,P\big(B\big)\,\big\vert\\
\nonumber
&\le\sup_{C\in{\mathcal F}_{\mathbb{Z}^d\setminus V_{{\widetilde m}(n)}}}
\big\vert\,P(A\cap C)-P(A)P(C)\,\big\vert.
\end{align}
By the assumptions on $v_1, v_2$ and $\kappa,$
$\|{\widetilde m}(n)\|$ goes to infinity.
By Proposition 7.9 in \cite{Geo88}, tail-triviality is equivalent to {\it short-range correlations,} i.e.,
$
\sup_{C\in{\mathcal F}_{\mathbb{Z}^d\setminus V_n}}
\big\vert\,P(A\cap C)-P(A)P(C)\,\big\vert
$
converges to $0$ as $n$ goes to infinity.
\hfill q.e.d.
\section{Ergodic theorems with skew products}\label{ret}
Applying Birkhoff's ergodic theorems to the skew product
$S$ yields, for any $F\in\L^1(\ov \Omega,\ov {\mathcal F},\ov P),$
\begin{eqnarray}\label{BirkS}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)
\convinfty{n}\ov E[F\vert{\mathcal J}]
\qquad \ov P\text{-almost surely and in }\L^1(\ov P),
\end{eqnarray}
where ${\mathcal J}$ is the $\sigma$-algebra of all $S$-invariant sets in $\ov{\mathcal F}.$
We study this limit more closely for two different cases:
when the transformation $\t$ is ergodic and when it is periodic.
In the ergodic case, combining $\eqref{BirkS}$ and Lemma \ref{ErgS}
immediately yields the following ergodic theorem for the skew product.
\begin{cor}\label{Reterg}
Assume that $\t$ is ergodic with respect to $\mu$ and that the condition
(C1) is fulfilled.
Then for any function $F\in\L^1(\ov\Omega,\ov{\mathcal F},\ov P),$
\begin{eqnarray*}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)
\convinfty{n}\ov E[F]
\qquad \ov P\text{-almost surely and in } \L^1(\ov P).
\end{eqnarray*}
\end{cor}
Now consider the case that $\t$ periodic.
We calculate the iterates of the skew product
and derive an ergodic theorem with an explicit expression for the limit.
\begin{lemma}\label{ItSper}
Assume that $\t$ is periodic with $q\in\mathbb{N}.$ Then for all $j\in\mathbb{Z}$
and all $\nu\in\{0,1,...,q-1\},$
\begin{itemize}
\item[(i)]$\kappa_{jq+\nu}=j\kappa_{q}+\kappa_{\nu},$
\item[(ii)]$\theta_{\kappa_{jq+\nu}(t)},
=\Big(\theta_{\kappa_{q}(t)}\Big)^j\circ\theta_{\kappa_{\nu}(t)}
\qquad\text{for all }t\in M,$
\item[(iii)]
$S^{jq+\nu}(t,\omega)= \big (\t^{\nu}(t),
\theta_{\kappa_{\nu}(t)}\circ(\theta_{\kappa_q(t)})^j\omega\big )
\qquad\text{for all }t\in M \text{ and all }\omega\in\Omega.$
\end{itemize}
\end{lemma}
\noindent {\bf Proof. }
(i) follows from the definition of $\kappa$ using the periodicity of $\t.$
(ii) is an immediate consequence of (i) and the semigroup property of $\theta.$
(iii) follows from (\ref{ItS}), the periodicity of $\t$ and because of (ii). \hfill q.e.d.
\begin{cor}\label{Retper}
Assume $\t$ is periodic with period $q\in\mathbb{N},$ and $F\in\L^1(\ov\Omega,\ov{\mathcal F},\ov P).$
Denote by ${\mathcal J}_t$ the $\sigma$-algebra of
$\theta_{\kappa_q(t)}$-invariant sets in ${\mathcal F}.$ Then
\begin{displaymath}\label{RetperA}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)
\convinfty{n}
\frac{1}{q}\sum_{\nu=0}^{q-1}E\big[F\big(\t^\nu(t),
\theta_{{\kappa_\nu}(t)}(\cdot)\big)\big\vert\, {\mathcal J}_t\big],
\end{displaymath}
for $\mu$-almost all
$t\in M,$ and for $P$-almost all $\omega\in\Omega$ and in $\L^1(P).$
If $\theta_{\kappa_q(t)}$ is ergodic with respect to $P$
for $\mu$-almost all $t\in M,$ then the limit
simplifies to the constant
$ 1/q\sum_{\nu=0}^{q-1} E\big[F\big(\t^\nu(t),\,\cdot\,\big)\big].$
\end{cor}
\noindent {\bf Proof. }
Any $n\in\mathbb{N}$ can be represented as
$n=mq+\nu,$ with $m\in\mathbb{N}$ and $\nu\in\{0,1,...,q-1\},$
and we may break down the ergodic averages to
\begin{displaymath}
A_nF:=\ErgAv{F}{S}=\frac{mq}{mq+\nu}\bigg(
\frac{1}{mq}\sum_{i=0}^{mq-1}F\circ S^i
+\frac{1}{mq}\sum_{i=mq}^{mq+\nu-1}F\circ S^i\bigg).
\end{displaymath}
Since the first factor converges to $1,$ and the second addend within the
brackets converges to $0,$
our question reduces to the study of ergodic limits along the subsequence
$(mq)_{m\in\mathbb{N}}.$ They take the form
$A_{mq}F=1/q\sum_{\nu=0}^{q-1}A_m^{(\nu)}F,$ where
\begin{eqnarray*}
A_m^{(\nu)}F(t,\omega)
:= \frac{1}{m}\sum_{j=0}^{m-1}F\circ S^{jq+\nu}(t,\omega)
=\frac{1}{m}\sum_{j=0}^{m-1}
F\big(\t^{\nu}(t), \theta^{\kappa_{\nu}(t)}\circ(\theta^{\kappa_q(t)})^{\,j}\omega\big).
\end{eqnarray*}
The last equality can be seen by applying Lemma \ref{ItSper}.
For $\mu$-almost all $t\in M,$
the function $f^{(\nu)}_t(\omega):= F\big(\t^\nu(t),\theta_{\kappa_\nu(t)}\omega\big)$ $(\omega\in\Omega)$
is integrable, and applying Birkhoff's ergodic theorem yields
\begin{displaymath}
\lim_{m\to\infty} A_m^{(\nu)}F(t,\cdot)
=E[f^{(\nu)}_t\vert{\mathcal J}_t]
=E\big[F\big(\t^\nu(t),\theta_{\kappa_{\nu}(t)}(\cdot)\big)\vert{\mathcal J}_t]
\qquad\mbox{$P$-almost surely and in $\L^1(P)$}
\end{displaymath}
This implies the first statement of the Corollary. In the ergodic case ${\mathcal J}_t$ is trivial,
and, using the invariance of $P$ under $\theta,$ the last expression reduces to
$E\big[F\big(\t^\nu(t),\,\cdot\,\big)].$
\hfill q.e.d.\vspace{3mm}
We end this section by illustrating the results by two special cases relevant to the
skew product tracing a stair climbing pattern introduced in (\ref{Intro:ex.sp}).
\begin{ex}\label{TorTransl} Circle rotations as base transformations.
{\rm
Let $\mathbb{T}:=[0,1)$ be the circle equipped with the Borel $\sigma$-algebra
${\mathcal B}$ and the Lebesgue measure $\mu.$
For $\lambda\in\mathbb{R}$ define a rotation on $\mathbb{T}$ by
$\t_\lambda(t):= t+\lambda\ {\rm mod}\ 1.$
The rotation preserves the measure $\mu$ and it is continuous.
For rational $\lambda$ the rotation is periodic, for
irrational $\lambda$ it is uniquely ergodic.
The circle can be equipped with a metric $d(s,t):=|s-t|\;(s,t\in\mathbb{T}).$
The metric is rotation invariant.
\noindent
(i) Let $\lambda$ be irrational.
Choose $(\theta_k)_{k\in K}$ and $\kappa$ fulfilling condition (C1).
By Lemma \ref{ErgS}, $S$ is ergodic.
By Corollary \ref{Reterg}, for any integrable function $F$ on
$(\mathbb{T}\times\Omega,{\mathcal B}\otimes{\mathcal F},\mu\otimes P),$
\begin{eqnarray*
\frac{1}{n}\sum_{i=0}^{n-1}F(\md{t+i\lambda},\theta_{\kappa_i(t)}
\omega)\convinfty{n}\int_0^1 E[F(t,\cdot)]\,dt,
\end{eqnarray*}
for $\mu\otimes P$-almost all $(t,\omega)\in\mathbb{T}\times\Omega$
and in $\L^1(\mathbb{T}\times\Omega,{\mathcal B}\otimes{\mathcal F},\mu\otimes P).$
\noindent
(ii) Let $\lambda$ be rational.
There is a unique representation
$\lambda=p/q,$ where $p\in\mathbb{Z}, q\in\mathbb{N},$ $p$ and $q$ have
no common divisor. $\t_\lambda$ is periodic with period $q.$
Furthermore, $\t_\lambda$ respects the partition
$[0,\frac{1}{q}), [\frac{1}{q},\frac{2}{q}),...,[\frac{q-1}{q},1)$ of $\mathbb{T},$
i.e., for every $\nu\in\{1,...,q-1\}$ there is a
$\widetilde\nu\in\{1,...,q-1\}$ such that
$\tau_\lambda([\frac{\nu-1}{q},\frac{\nu}{q}))=[\frac{\widetilde\nu-1}{q},\frac{\widetilde\nu}{q}).$
The limit in Corollary \ref{Retper} is of the form
$ 1/q\sum_{\nu=0}^{q-1}E\big[F\big(\md{t+\nu\lambda},\cdot\big)\big\vert{\mathcal J}_t\big].$
If $P$ is ergodic with respect to $\theta_{\kappa_q(t)}$ then limit simplifies to
$1/q\sum_{\nu=0}^{q-1}E\big[F\big(\md{t+\nu\lambda},\cdot\big)\big].$
}\end{ex}
\begin{ex}\label{ShiftsRet} Shifts as fibre transformations. {\rm
Consider the 2-parameter group define above \eqref{spc:twoShifts}.
\noindent
(i) If $\t$ is ergodic then, for any function $F\in\L^1(\ov \Omega,\ov {\mathcal F},\ov P),$
\begin{eqnarray*}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),
\vartheta_{\kappa_i^{(1)}(t)\,v_1+\kappa_i^{(2)}(t)\,v_2}\big)
\convinfty{n}\ov E[F]
\qquad \ov P\text{-almost surely and in } \L^1(\ov P).
\end{eqnarray*}
\noindent
(ii) If $\t$ is periodic with period $q$ then, for any function $F\in\L^1(\ov \Omega,\ov {\mathcal F},\ov P),$
\begin{eqnarray*}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),
\vartheta_{\kappa_i^{(1)}(t)\,v_1+\kappa_i^{(2)}(t)\,v_2}\omega\big)
\convinfty{n}
\frac{1}{q}\sum_{\nu=0}^{q-1}E\big[F\big(\t^\nu(t),
\vartheta_{\kappa_\nu^{(1)}(t)\,v_1+\kappa_\nu^{(2)}(t)\,v_2}(\cdot)\big)\big\vert\, {\mathcal J}_t\big],
\end{eqnarray*}
for $\mu$-almost all
$t\in M,$ and for $P$-almost all $\omega\in\Omega$ and in $\L^1(P).$
If $\vartheta_{\kappa_q^{(1)}(t)\,v_1+\kappa_q^{(2)}(t)\,v_2}$ is ergodic with respect to $P,$
for $\mu$-almost all $t\in M,$ then the limit simplifies to
$ 1/q\sum_{\nu=0}^{q-1} E\big[F\big(\t^\nu(t),\,\cdot\,\big)\big].$
}\end{ex}
\section{Uniform convergence}\label{unif}
This section addresses the question of {\it sure} convergence
with respect to the first parameter.
In addition to the assumptions at the beginning of Section \ref{spc}
we suppose that $M$ is a compact separable metric space endowed
with metric $d,$ and ${\mathcal B}$ is the Borel $\sigma$-algebra on $M$
for the topology induced by $d.$
Recall that the convergence of ergodic averages need not be
true {\it everywhere},
even if we are in a compact topological space and both the transformation
and the function are continuous.
Which conditions guarantee sure convergence in the
first parameter? We will be asking
a little more than this, namely about {\it uniform} convergence in $t.$
We are interested in results of the type
\begin{equation}\label{unif:conv}
\frac{1}{n}\sum_{i=0}^{n-1}
F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)\convinfty{n}
\ov E[F\vert{\mathcal J}](t,\omega)\qquad\text{uniformly in }t\in M\text{ and in }\L^1(P).
\end{equation}
We further investigate when (\ref{unif:conv})
takes place $P$-almost surely, i.e., for $P$-almost all $\omega\in\Omega,$
\begin{eqnarray}\label{UnifConvP}
\lim_{n\to\infty}\;\sup_{t\in M}\bigg\vert
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)
- \ov E[F\vert{\mathcal J}](t,\omega)\,\bigg\vert=0.
\end{eqnarray}
Again, we consider two different cases: when $\t$ is ergodic
and when it is periodic. In the second case we proceed as in the proof
of Corollary \ref{Retper} and obtain the following uniform version.
\begin{cor}\label{Retperunif}
Let $\t$ be periodic with $q\in\mathbb{N}.$ Assume
$F\in\L^1(\ov\Omega,\ov{\mathcal F},\ov P)$ with $F(t,\cdot)\in\L^1(\Omega,{\mathcal F},P)$
for all $t\in M.$ Then, for $P$-almost all $\omega$ and in $\L^1(P),$
\begin{equation*}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),
\theta_{\kappa_i(t)}\,\cdot\big)\convinfty{n}
\frac{1}{q}\sum_{\nu=0}^{q-1}E\big[F\big(\t^\nu(t),
\theta_{\nu_q(t)}\,\cdot\big)\big\vert{\mathcal J}_t\big]
\qquad\text{uniformly in }t\in M,
\end{equation*}
where ${\mathcal J}_t$ denotes
the $\sigma$-algebra of $\theta_{\kappa_q(t)}$-invariant sets in ${\mathcal F}.$
If $P$ is ergodic with respect to $\theta_{\kappa_q(t)},$ for all $t\in M,$ the limit equals
$1/q\sum_{\nu=0}^{q-1} E\big[F\big(\t^\nu(t),\,\cdot\,\big)\big].$
\end{cor}
The ergodic case is more delicate. We begin with careful
investigations of ergodic theorems on the single spaces,
which will later be combined to derive a result on the product space.
In the base we are dealing with a transformation on a compact and metrizable space.
We now recall and refine some of the existing results about uniform convergence
in this situation.
Motivated by applications in information theory.
we put particular emphasis on extending uniform convergence
results to the class of Riemann-integrable functions.
An example for a function that is Riemann-integrable but not
continuous occurs in the proof of a directional Shannon-MacMillan theorem
for random fields (cf.~\cite{Bre01SMcM}).
The classical example for an ergodic theorem that gives
a statement about uniform convergence is the one by Weyl. In its simplest
form, it says that the averages of a continuous function along
the orbit of an irrational translation on the circle converge uniformly
to the integral of the function.
To prove Weyl's theorem, Krengel (cf.~Theorem 2.6 in
Paragraph 1.2.3 in \cite{Kre85}) uses an Arzela-Ascoli
technique which we will make use of at the end of this section.
Under the assumption that $\t:M\rightarrow M$ is continuous
and that $f$ is a function on $M,$ such that the functions
$ F_n:=1/n\sum_{i=1}^{n-1} F\circ\t^i\ (n\in\mathbb{N})$
are equicontinuous on $M,$ Krengel's theorem states that the
convergence in Birkhoff's ergodic theorem is uniform in $t.$
Together with the following Lemma, it yields Weyl's theorem.
\begin{lemma}\label{EquBed}
Let $f$ be a continous function on $M,$ and
$\t:M\rightarrow M$ Lipschitz-continuous with Lipschitz
constant $c\le 1.$ Then the functions $F_n\ (n\in\mathbb{N})$ defined above are equicontinuous.
\end{lemma}
\noindent {\bf Proof. } We have to show that for
every $\varepsilon>0$ there is a $\delta>0$ such that for all
$n\in\mathbb{N}$ and all $s,t\in M$ with $d(s,t)<\delta,$
$1/n\big\vert\sum_{i=0}^{n-1}f(\t^i(s))-f(\t^i(t))\big\vert
<\varepsilon.$
Fix $\varepsilon>0.$ We will show
that there is a $\delta>0,$ such that
$\vert f(\t^i(s))-f(\t^i(t))\big\vert <\varepsilon$
for all $d(s,t)<\delta.$ Since $M$ is compact, $f$ must be
uniformly continuous, i.e., there is a $\delta>0$ such that
for all $x,y\in M$ with $d(x,y)<\delta$ we have
$|f(x)-f(y)|<\varepsilon.$
By assumption,
$d(\t(s),\t(t))\le c\,d(s,t)$ for all $s,t\in M,$
and therefore, $d(\t^i(s),\t^i(t))\le c^i\,d(s,t)\le d(s,t)$
for all $s,t\in M,$ and for all $i\in\mathbb{N}.$
\hfill q.e.d.\vspace{3mm}
We shall ask whether we could replace the assumption of continuity
of the function $f$ in Weyl's theorem by a weaker condition. It is
certainly not true for all measurable functions, which can be seen
in a simple example: Fix $t_0\in\mathbb{T}.$ Its orbit under $\t$ is the
set ${\cal O}:=\{\t^n(t_0)\vert n\in\mathbb{N}_0\,\}.$
For the function $f:=1_{\cal O},$
the ergodic averages converge to 1, for all $t\in {\cal O,}$
but $\int_0^1 1_{\cal O}(t)\,dt=0.$
This question of uniform convergence has a connection with unique
ergodicity (cf.~e.g., Chapter 4.1.e.\ of \cite{HaK95} or
Theorem 6.19 in \cite{Wal82}). A continuous transformation $\t$ of a
compact metrizable space is called {\rm uniquely ergodic} if it has only
one invariant Borel measure.
It can be shown that this measure must be ergodic, which implies
that the ergodic averages of an integrable function converge
almost surely to a constant.
The Lebesgue measure is the only probability measure on $(\mathbb{T},{\mathcal B}),$
which is invariant with respect to rotations of the circle,
so it must be uniquely ergodic.
Oxtoby extended Weyl's theorem to the situation where $\t$ is a
uniquely ergodic transformation on a compact metric space.
It states uniform convergence of the ergodic averages of a
continuous function.
Note that, conversely, uniform convergence does
not imply the continuity of the function.
(Further conditions for this would be needed, such as
topological transitivity of $\t$ or constancy of the limit.)
Below Theorem 2.7 in Chapter 1 of \cite{Kre85}, Krengel mentions
that Weyl's theorem is sometimes spelled out for
to the class of Riemann-integrable functions.
Actually, it was proved by de Bruijn and Post
\cite{BrP68} that the function is
Riemann-integrable if and only if the convergence is uniform.
This also follows from our next proposition. We ask the following question:
Considering uniform convergence of the ergodic
averages along a continuous transformation on a compact
real interval, can we pass automatically from the class of
continuous functions to the class of functions which are
integrable in the sense of Riemann?
\begin{prop}\label{WeylR}
Let $a,b\in\mathbb{R},$ $a<b,$ $\mu$ a measure on $([a,b],{\mathcal B}([a,b]))$
which is absolutely continuous with respect to Lebesgue measure,
with a continuous density. Let $\t:[a,b]\rightarrow [a,b]$ be
continuous. Assume that for any continuous function $f$ on $[a,b]$
\begin{eqnarray}\label{WeylR1}
\ErgAv{f}{\t}\convinfty{n}\int_a^b f\,d\mu\qquad
\text{uniformly.}
\end{eqnarray}
Then the convergence holds as well for any function which
is integrable in the sense of Riemann.
\end{prop}
\noindent The proof is carried out using a common sandwich argument
(cf.~e.g.,\ in Chapter 4.1.e.\ of \cite{HaK95}). Applying the proposition
to the situation of Oxtoby's Theorem yields
\begin{cor}\label{UnifR} Let $a,b\in\mathbb{R},$
$a<b,$ and $\t:[a,b]\rightarrow [a,b]$ continuous and uniquely
ergodic with invariant measure $\mu.$ Assume that $\mu$ is
absolutely continuous with respect to Lebesgue measure, with a
continuous density. Then, for any function $f$ on $M$ which is
integrable in the sense of Riemann,
\begin{displaymath}
\ErgAv{f}{\t}\convinfty{n}\int_0^1 f\,d\mu\quad\quad\text{uniformly.}
\end{displaymath}
\end{cor}
Now we focus on studying the convergence in the fibre.
Fix $t\in M,$ and define a function on $\Omega$ by $f(\omega):=F(t,\omega).$
This reduces the ergodic averages of the skew product to
$1/n\sum_{i=0}^{n-1}f\big(\theta_{\kappa_i(t)}\omega\big),$
which we can view as a sort of
ergodic average along the subsequence $(\kappa_i(t))_{i\in\mathbb{N}}.$
Recalling a classical result about $\L^p$-convergence of ergodic
averages along subsequences, there is the following characterization.
\begin{thm}\label{BHthm}{\rm (Blum \& Hanson)}
Let $T$ be a transformation on $(\Omega,{\mathcal F}).$ Suppose that $T$ is
invertible and that both, $T$ and $T^{-1}$ preserve $P.$ Then $P$
is strongly mixing with respect to $T$ if and only if for all $p,\
1\le p<\infty,$ every strictly increasing sequence $(m_i)_{i\in
\mathbb{N}}$ of integers, and every function $f\in\L^p(\Omega,{\mathcal F},P),$
\begin{equation*}
\frac{1}{n}\sum_{i=0}^{n-1}f\circ T^{m_i}
\convinfty{n} E[f]\quad\quad\text{in }\L^p(P).
\end{equation*}
\end{thm}
\noindent The key to the proof of Blum and Hanson's theorem is the following
\begin{lemma}\label{BHlem}
Under the assumptions of Theorem \ref{BHthm} and supposing that
$P$ is strongly mixing with respect to $T$ we have for all $A\in{\mathcal F},$
for every strictly increasing sequence $(k_i)_{i\in\mathbb{N}},$
\begin{equation*}
\frac{1}{n}\sum_{i=0}^{n-1}1_A\circ T^{k_i}\convinfty{n}
P(A)\quad\quad\text{ in }\L^2(P).
\end{equation*}
\end{lemma}
Our next step is to carry over Blum and Hanson's theorem to the
case of a $d$-parameter group of transformations $(\theta_k)_{k\in\mathbb{Z}^d},$
at the place of iterates of one transformation $T.$
What we need is a condition on the $\mathbb{Z}^d$-valued
sequence $(k_i)_{i\in \mathbb{N}}$ which replaces the strict
monotonicity imposed on $(m_i)_{i\in\mathbb{N}}.$ With an eye toward later
applications on the product space we generalize the result further
by showing that the $\L^2$-convergence takes place uniformly over a family of functions,
indexed by a set $I.$
Recall that $\|\cdot\|$ denotes the maximum norm in $\mathbb{Z}^d.$
\begin{lemma}\label{BHtyplem}
Assume that $P$ is strongly mixing with respect to
$(\theta_k)_{k\in\mathbb{Z}^d},$ and let $(k_n(t))_{n\in\mathbb{N}}$ $(t\in I)$ be a
family of sequences with values in $\mathbb{Z}^d$ that fulfill, for all $m\in\mathbb{N},$
\begin{equation}\label{BHA}
\lim_{n\to\infty}
\frac{1}{n^2}\sup_{t\in I}\,\big|\big\{1\le i,j\le n\,
\big\vert\,\|k_i(t)-k_j(t)\|\le m\big\}\big|=0.
\end{equation}
Then for all $A\in{\mathcal F},$
\begin{equation*}
\sup_{t\in I}\bigg\|
\frac{1}{n}\sum_{i=0}^{n-1}1_A\circ\theta_{k_i(t)}-
P(A)\bigg\|_{\L^2(P)}^2 \convinfty{n}0.
\end{equation*}
\end{lemma}
\noindent {\bf Proof. }
For every $A\in{\mathcal F}$ and $t\in I$ we obtain by simple calculations,
\begin{align}
\bigg\|\frac{1}{n} & \sum_{i=0}^{n-1} 1_A\circ\theta_{k_i(t)}-P(A)\,
\bigg\|_{\L^2(P)}^
=\int_{\Omega}\frac{1}{n^2}\sum_{i,j=0}^{n-1}(1_A\circ\theta_{k_i(t)}-P(A))
(1_A\circ\theta_{k_j(t)}-P(A))\,dP\nonumber\\
&=\frac{1}{n^2}\sum_{i,j=0}^{n-1}\bigg[\int_{\Omega}\big(
1_A\circ\theta_{k_i(t)}\, 1_A\circ\theta_{k_j(t)}\big)\,dP - P(A)
\int_{\Omega}\big(1_A\circ\theta_{k_i(t)}
+ 1_A\circ\theta_{k_j(t)}\big)\,dP + P(A)^2\bigg]\nonumber\\
\label{BH1}
&=\frac{1}{n^2}\sum_{i,j=0}^{n-1}\Big(P\big(\theta_{k_i(t)}^{-1}A
\cap\theta_{k_j(t)}^{-1}A\big)-P(A)^2\Big)
\le\frac{1}{n^2}\sum_{i,j=0}^{n-1}\Big|
P\big(\theta_{k_i(t)-k_j(t)}^{-1}A\cap A\big) - P(A)^2\Big|.
\end{align}
Fix $\varepsilon>0.$ Due to the mixing condition there is
an $m\in\mathbb{N}$ such that
\begin{equation*}
\Big| P\big(\theta_{k_i(t)-k_j(t)}^{-1}A\cap A\big)
- P(A)^2\Big|
<\frac{\varepsilon}{2}\quad\quad\text{for all }i, j\in\mathbb{N}
\text{ with }\|\,k_i(t)-k_j(t)\|> m,
\end{equation*}
and
\begin{equation*}
\Big| P\big(\theta_k^{-1}A\cap A\big) - P(A)^2\Big|
<\frac{\varepsilon}{2}\quad\quad\text{for all }k\in\mathbb{Z}^2
\text{ with }\|k\|\ge m.
\end{equation*}
By assumption (\ref{BHA}) there is a $n_0\in\mathbb{N}$ such that
\begin{equation*}
\frac{1}{n^2}\sup_{t\in I}\,\big|\big\{1\le i,j\le
n\,\big\vert\,\|k_i(t)-k_j(t)\|\le m\big\}\big|
<\frac{\varepsilon}{2}
\quad\text{ for all }n\ge n_0.
\end{equation*}
Applying the last two inequalities to (\ref{BH1})
yields for all $n\ge n_0$
\begin{align*}
\sup_{t\in I}
& \bigg\|\frac{1}{n^2}\sum_{i=0}^{n-1}
1_A\circ\theta_{k_i(t)} -P(A) \,
\bigg\|_{\L^2(\Omega,{\mathcal F},P)}^2 < \varepsilon,
\end{align*}
\nopagebreak
and the assertion of the lemma follows by letting $\varepsilon$ to $0.$
\hfill q.e.d.\vspace{3mm}
\begin{thm}\label{BHtk}
Assume that $P$ is strongly mixing with
respect to $(\theta_k)_{k\in\mathbb{Z}^d}.$ Let $(k_n(t))_{n\in\mathbb{N}}$ $(t\in I)$ be a
family of sequences with values in $\mathbb{Z}^d,$ for which for all $m\in\mathbb{N},$
\begin{equation*}
\lim_{n\to\infty}\frac{1}{n^2}\,
\sup_{t\in I}\big|\big\{1\le i,j\le
n\,\big\vert\,\|k_i(t)-k_j(t)\|\le m\big\}\big|=0.
\end{equation*}
Then for $1\le p<\infty$ and for any $f\in\L^p(\Omega,{\mathcal F},P),$
\begin{equation}\label{BHtkSta}
\frac{1}{n}\sum_{i=0}^{n-1}f\circ\theta_{k_i(t)}\convinfty{n}E[f]
\quad\text{ in }\L^p(P), \text{ uniformly in }t\in I.
\end{equation}
\end{thm}
\noindent {\bf Proof. } As an immediate consequence of the preceeding lemma,
\eqref{BHtkSta} is true for $p=2$ for any simple function $g$ on $(\Omega,{\mathcal F}).$
By a standard argument (cf., e.g., Lemma 4 in \cite{BlH60}),
this convergence holds as well in $\L^p(P),$ for
$1\le p<\infty.$ Finally, for any function
$f$ in $\L^p(P),$ decomposition into positive and negative parts,
$\L^p(P)$-approximation by simple functions, and monotone
convergence yields (\ref{BHtkSta}).
\hfill q.e.d.\vspace{3mm}
We will now combine the approaches developed separately for the base
and the fibre transformation to derive a result about uniform convergence
in the base and $\L^p$-convergence in the fibre. A crucial ingredient is a condition
that regulates the effects of the coupling sequence $(\kappa_n(t))_{n\in\mathbb{N}}.$
\begin{thm}\label{UnifL}
Let $\t:M\rightarrow M$ be continuous and uniquely ergodic, and
suppose that $P$ is strongly mixing
with respect to the group of transformations $(\theta_k)_{k\in\mathbb{Z}^d}.$ Let
$\kappa: M\rightarrow \mathbb{Z}^d$ be ${\mathcal B}$-measurable such that,
\begin{equation}\label{kappaCond}
\lim_{n\to\infty}\frac{1}{n^2}\,
\sup_{t\in M}\big|\big\{1\le i,j\le
n\,\big\vert\,\|\kappa_i(t)-\kappa_j(t)\|\le
m\big\}\big|=0
\end{equation}
for all $m\in\mathbb{N}.$ Let be $1\le p<\infty.$ Then for every ${\mathcal F}$-measurable function
$F$ on $\ov\Omega$ such that $\sup_{t\in M} \vert\, F(t,\cdot\,)\,\vert$ is in
$\L^{p}(\Omega,{\mathcal F},P)$ and $F(\,\cdot\,,\omega)$ is continuous on $M$
for P-almost every $\omega$,
\begin{equation}\label{UnifLClaim}
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),
\theta_{\kappa_i(t)}\,\cdot\big) \convinfty{n}
\ov E[F]
\qquad\mbox{in }\L^p(P)\mbox{ and uniformly in }t\in M.
\end{equation}
\end{thm}
\noindent {\bf Proof. }
We first prove the theorem in the case when $F$ is the indicator of a
set of the form $U\times A$, where $U$ is the intersection of
finitely many metric balls in $M$ or their complements, and $A\in{\mathcal F}.$
By (\ref{ItS}), the expression
\begin{equation}\label{UnifLDiff}
\bigg\|\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),
\theta_{\kappa_i(t)}\,\cdot\big) -\ov E[F]\bigg\|_{\L^2(P)}
\end{equation}
then transforms to
\begin{equation}\label{BHprdlem1}
\begin{split}
\frac{1}{n^2} & \sum_{i,j=0}^{n-1}\int_{\Omega}1_U(\t^i(t)) 1_U(\t^j(t))
1_A(\theta_{\kappa_i(t)}\omega)1_A(\theta_{\kappa_j(t)}\omega)\,P(d\omega) \\
- & \frac{1}{n^2}\sum_{i,j=0}^{n-1}\mu(U)P(A)\bigg[
\int_{\Omega}1_U(\t^i(t))1_A(\theta_{\kappa_i(t)}\omega)\,P(d\omega)
+ \int_{\Omega}1_U(\t^j(t))
1_A(\theta_{\kappa_j(t)}\omega)\,P(d\omega)\bigg] \\
\vspace{-2mm}
+ &\ \mu(U)^2 P(A)^2.
\end{split}
\end{equation}
By $ P\big(\theta_{\kappa_i(t)}^{-1}A\cap\theta_{\kappa_j(t)}^{-1}A\big) =
P\big(\theta_{\kappa_i(t)-\kappa_j(t)}^{-1}A\cap A\big),$
the first addend equals
\begin{equation*}
\frac{1}{n^2}\sum_{i,j=0}^{n-1}1_U(\t^i(t))1_U(\t^j(t))
P\big(\theta_{\kappa_i(t)-\kappa_j(t)}^{-1}A\cap A\big).
\end{equation*}
It may be replaced by
\begin{equation}\label{BHprdlem2}
\frac{1}{n^2}\sum_{i,j=0}^{n-1}1_U(\t^i(t))1_U(\t^j(t)) P(A)^2
\end{equation}
without affecting the asymptotic behavior of (\ref{BHprdlem1}) as can be seen as follows.
We may bound
\begin{align*}
\bigg\vert&\frac{1}{n^2}\sum_{i,j=0}^{n-1}1_U(\t^i(t))1_U(\t^j(t))
\Big(P\big(\theta_{\kappa_i(t)-\kappa_j(t)}^{-1}A\cap A\big)
-P(A)^2\Big)\bigg\vert\nonumber\\
&\le \frac{1}{n^2}\sum_{i,j=0}^{n-1}
\Big\vert P\big(\theta_{\kappa_i(t)-\kappa_j(t)}^{-1}A\cap A\big)
-P(A)^2\Big\vert.
\end{align*}
Now, we argue as in the second part of the proof of Lemma
\ref{BHtyplem}, replacing the sequence $(k_i)_{i\in\mathbb{N}}$ by
$(\kappa_i(t))_{i\in\mathbb{N}},$ and using assumption (\ref{kappaCond})
instead of (\ref{BHA}). This proves that the difference
created by the change \eqref{BHprdlem2}
converges to $0$ uniformly with respect to $t.$
Since the term in the rectangular brackets in the second addend in
(\ref{BHprdlem1}) equals
$ 1_U(\t^i(t))P(A)+1_U(\t^j(t))P(A),$
the whole expression (\ref{BHprdlem1}) simplifies to
\begin{equation*}
\frac{1}{n^2}\sum_{i,j=0}^{n-1}\Big(
1_U(\t^i(t))1_U(\t^j(t))
-\mu(U)\big(1_U(\t^i(t))+1_U(\t^j(t))\big)
+\mu(U)^2\Big) P(A)^2,
\end{equation*}
which can be further reduced to
$ \bigg(1/n\sum_{i=0}^{n-1}1_U(\t^i(t))-\mu(U)\bigg)^2 P(A)^2.$
Since $\t$ is uniquely ergodic and $\mu(\partial U)=0,$
Corollary 4.1.14 in \cite{HaK95} tells us that the first factor converges
to $0$ uniformly in $t,$
which concludes the first part of the proof.
To pass from $\L^2$-convergence to general $\L^p,$ use again
a standard argument (for instance, Lemma 4 in \cite{BlH60}).
Now we let $F$ be a general function, satisfying the conditions of
the theorem. We need to find for every positive
$\epsilon$ a sequence of metric balls $U_i$ in $M$ and $A_i\in{\mathcal F}$, with
real numbers $a_{i}$ such that, for all $t\in M$,
$\| F(t,\cdot)- I(t,\cdot) \|_{p}< \varepsilon,$ where
$I=\sum_{i=1}^{n} a_{i} 1_{U_{i}\times A_{i}}.$
It will then follow that
\begin{align*}
\bigg\|\frac{1}{n} \sum_{i=0}^{n-1}
F\bigl(\tau_{i}(t),\theta_{\kappa_{i}(t)} \,\cdot\, \bigr) -
\frac{1}{n} \sum_{i=0}^{n-1}
I\bigl(\tau_{i}(t),\theta_{\kappa_{i}(t)} \,\cdot\, \bigr) \bigg\|_{p}
\leq \frac{1}{n} \sum_{i=0}^{n-1}
\bigl\|F\bigl(\tau_{i}(t),\,\cdot\, \bigr) -
I\bigl(\tau_{i}(t), \,\cdot\, \bigr) \bigr\|_{p}
< \varepsilon.
\end{align*}
For $\omega\in\Omega$ and $c>0$, let $\delta(c,\omega)$ be the modulus
of continuity for the function $F(\,\cdot\,,\omega)$.
Define the sets
$$
M_k=\sup_{t\in M}\{\omega\,\vert\, |F(t,\omega)|\le k\}
\quad\text{and}\quad
D_k(\varepsilon)=\{\omega\,\vert\,\delta(1/k,\omega)\le\varepsilon\}
\quad\ (k\in\mathbb{N}).
$$
Then the sequence of functions on $\Omega,$
$
F_{k}(\omega):=\sup_{t\in M}|F|^{p}(t,\omega)\,
1_{D_k(\varepsilon/4)^c\cup M_k^c}(\omega)\ (k\in\mathbb{N}),
$
is bounded by $\sup_{t\in M}|F|^{p}(t,\omega)$, which is integrable, and converges to
$0$ for every $\omega$. By the bounded
convergence theorem, the integral of $F_{k}$ converges to 0 as $k$
goes to infinity. Choose a $k$ such that $\int F_{k}\,P(d\omega)\leq (\varepsilon/2)^{p}$.
Since $M$ is compact, we may find a finite sequence
$t_{1},\dots,t_{r}\in M$ such that the balls of radius $1/k$ around
these centers cover $M$. We also define a sequence of real numbers
$-k-1=s_{0}<\cdots<s_{r'}=k$ such that the difference between any two
successive elements is less than $\varepsilon/8$. Now we define a collection
of sets $U_{i,j}$ and $A_{i,j}$
indexed by $r\times r'$. We start with $U_{i,j}$ as the ball of radius
$1/k$ around $t_{i}$, and then remove the intersections, so that the
$U_{i,j}$ is the same for all $j$, and running through $1\leq i\leq
r$ yields a disjoint cover of $M$. The sets $A_{i,j}$ are defined by
$$
A_{i,j}=\big\{ \omega\,\big\vert\, s_{j-1}<F(t_{i},\omega)\leq s_{j}\big\}
\cap D_k(\varepsilon/8) \cap M_k.
$$
Let $a_{i,j}=s_{j}$. We throw in one additional product set,
$U_{0}=M$ and
$A_{0}=D_k(\varepsilon/8)^c\cup M_k^c$
with $a_{0}=0$, and define the simple function $I(t,\omega)$ as
indicated above.
Then for any $t\in M,$
\begin{align*}
\Bigl \| F(t, & \cdot) -I(t,\cdot)\Bigr\|_{p}
=
\bigg(
\int \big\vert F(t,\omega)-I(t,\omega) \big\vert^{p}
1_{D_k(\varepsilon/8)}\,1_{M_k}\,P(d\omega)
\bigg)^{1/p}
+ \: \bigg(\int F_{k}(\omega)\, P(d\omega)
\bigg)^{1/p}.
\end{align*}
We already assumed (in defining $k$) that the second term is smaller
than $\varepsilon/2$. For every $t$, there is a unique pair $i,j$ such that
$t\in U_{i,j}$ and $\omega\in A_{i,j}$. By construction,
$I(t,\omega)=s_{j}$, so the integrand in the first term is bounded by
$
2^{p}\bigl|F(t_{i},\omega)-F(t,\omega) \bigr|^{p} +
2^{p}\bigl|F(t_{i},\omega)-s_{j} \bigr|^{p} .
$
This in turn is bounded by $2^p(\varepsilon/4)^{p}< (\varepsilon/2)^{p}$, since
$\omega$ is not in $A_{0}$ and $d(t_{i},t)<1/k$, completing the proof.
\hfill q.e.d.\vspace{3mm}
Proposition \ref{WeylR} yields the following version for Riemann-integrable functions.
\begin{cor}\label{UniLRMak}
Let $a,b\in\mathbb{R},$ $a<b,$ and $\t:[a,b]\rightarrow [a,b]$ be continuous
and uniquely ergodic with invariant measure $\mu,$ and
assume that $\mu$ is absolutely continuous
with respect to Lebesgue measure, with continuous density.
For $P$ and $\kappa$ assume the same as in Theorem \ref{UnifL}.
Let $F\in L^p([a,b]\times\Omega,{\mathcal B}\otimes{\mathcal F},\ov P)$
be Riemann-integrable with respect to the first
variable. Then we have
\begin{eqnarray*}
\lim_{n\to\infty}\;\sup_{t\in M}\bigg\|
\frac{1}{n}\sum_{i=0}^{n-1}
F\big(\t^i(t),\theta_{\kappa_i(t)}\cdot\big)
- \ov E[F]\,\bigg\|_{\L^p(P)}=0.
\end{eqnarray*}
\end{cor}
Our next goal is to derive a statement about $P$-almost sure convergence
rather than $\L^1(P)$-convergence in the fibre. Further conditions
on $F$ are needed. $P$-almost sure convergence of ergodic theorems
involving weights or subsequences is a very subtle question (cf., e.g. \cite{BeL85}).
Choosing a function $F$ which is
constant in $\omega$ and considering Lemma \ref{EquBed}
suggests that we need an equicontinuity assumption in $t.$
Note also the additional assumptions that non-empty open sets on $M$
have positive mass under $\mu.$
\begin{thm}\label{UnifP}
Let $\mu$ be a $\t$-invariant measure on $(M,{\mathcal B}),$
such that any non-empty open subset $U$ of $M$ has $\mu(U)>0.$
Let $\t:M\rightarrow M$ be continuous and $F$ a function
on $M\times\Omega,$ for which $F(t,\cdot\,)\in\L^1(P)$
for all $t\in M,$ and the sequence of functions
$$
\Bigg(\frac{1}{n}\sum_{i=0}^{n-1}F(\t^i(\cdot\,),
\theta_{\kappa_i(\cdot\,)}\omega)\Bigg)_{n\in\mathbb{N}}
$$
is equicontinuous on $M,$ for all $\omega\in\Omega.$
Then, for $P$-almost all $\omega\in\Omega,$
\begin{eqnarray}\label{UnifPBeh}
\sup_{t\in M}\bigg\vert
\frac{1}{n}\sum_{i=0}^{n-1}F\big(\t^i(t),\theta_{\kappa_i(t)}\omega\big)
-\ov E[F\vert{\mathcal J}](t,\omega)\,\bigg\vert\convinfty{n}0.
\end{eqnarray}
\end{thm}
\noindent {\bf Proof. }
We may assume without loss of generality that $E[F\vert{\mathcal J}]=0.$
The general case can be reduced to this by subtracting $E[F\vert{\mathcal J}]$
on both sides and making use of the invariance of $E[F\vert{\mathcal J}]$ under $S.$
The first step is to construct a countable dense set $M_1\subset M$
and a set $N_1\subset\Omega$ with $P(N_1)=0$ such that
\begin{eqnarray}\label{UnifP1}
\frac{1}{n}\sum_{i=0}^{n-1}F\circ S^i(t,\omega)\longrightarrow 0
\quad\quad\text{ for all $t\in M_1$ and all }\omega\in\Omega\setminus N_1.
\end{eqnarray}
Since $M$ is compact, the conditions on $F$ assure that
$F\in\L^1(\ov\Omega,\ov{\mathcal F},\ov P),$ and therefore by (\ref{BirkS})
there is a set $M_1\subset M$ with $\mu(M_1)=1$ such that for
any $t\in M_1$ there is a set $N(t)\subset\Omega$ with $P(N(t))=0$ and
\eqref{UnifP1} holds for all $\omega\in\Omega\setminus N(t).$
${\widetilde M}_1$ is dense in $M$ because its complement has measure zero
with respect to $\mu$ and therefore, by assumption,
contains no non-empty open subsets.
Since $M$ is separable we can find a countable dense subset $C\subset M,$
and because ${\widetilde M}_1$ is dense in $M,$ we can approximate any
$x\in C$ by a sequence $(a_j(x))_{j\in\mathbb{N}}$ with $a_j(x)\in {\widetilde M}_1$
for all $j\in\mathbb{N}.$
$M_1:=\bigcup_{x\in C}\bigcup_{j\in\mathbb{N}}a_j(x)$ defines a countable dense subset
of $M,$ and $N_1:=\bigcup_{t\in M_1} N(t)$ defines a subset of
$\Omega,$ which fulfills (\ref{UnifP1}). This completes the first step.
For the next step, choose $s\in M$ and
fix $\varepsilon>0.$ By equicontinuity, there is a set
$N_0\subset\Omega$ with $P(N_0)=0$ and a
$\delta>0$ such that for all $r,t\in M$ with $d(r,t)<\delta$
\begin{eqnarray}\label{UnifP3}
\bigg\vert\frac{1}{n}\sum_{i=0}^{n-1}F\circ S^i(r,\omega)-F\circ
S^i(t,\omega)\bigg\vert
<\frac{\varepsilon}{2}\quad\text{for all }n\in\mathbb{N}\text{ and all }
\omega\in\Omega\setminus N_0.
\end{eqnarray}
Define $N:=N_0\cup N_1$ and fix $\omega\in\Omega\setminus N.$
Since $M_1$ is dense in $M$ we can find a $t\in M_1$ with $d(s,t)<\delta,$
and by (\ref{UnifP1}) there is an $n_1\in\mathbb{N}$ such that
\begin{eqnarray*}
\bigg\vert\ErgAv{F}{S}(t,\omega)\bigg\vert <\frac{\varepsilon}{2}
\quad\quad\text{ for all }n\ge n_1.
\end{eqnarray*}
Combining the last two inequalities leads the desired
\begin{eqnarray*}
\bigg\vert\ErgAv{F}{S}(s,\omega)\bigg\vert <\varepsilon
\quad\quad\text{ for all }n\ge n_1\text{ and all }
\omega\in\Omega\setminus N.
\end{eqnarray*}
For uniform convergence w.r.t.~the first variable,
we use a standard compactness argument. $M$
can be covered by a finite number $m$ of $\delta$-neighborhoods
in $M$, which centers are denoted by $s_1,...,s_m.$ Applying the
reasoning of the last step to each of the $s_1,...,s_m$ we
find $n_0\in \mathbb{N}$ such that
\begin{eqnarray*}
\bigg\vert\ErgAv{F}{S}(s_k,\omega)\bigg\vert <\frac{\varepsilon}{2}
\quad\quad\text{ for all }n\ge n_0, k\in\{1,...,K\},
\text{ and }\omega\in\Omega\setminus N.
\end{eqnarray*}
For an arbitrary $s\in M$ there exists $k\in\{1,...,K\}$ such that
$d(s,s_k)<\delta,$ and by (\ref{UnifP3}) we obtain\begin{eqnarray*}
\frac{1}{n}\bigg\vert\sum_{i=0}^{n-1}F\circ S^i(s,\omega)-F\circ
S^i(s_k,\omega)\bigg\vert
<\frac{\varepsilon}{2}\qquad\text{ for all }n\in\mathbb{N}\text{ and all }
\omega\in\Omega\setminus N.
\end{eqnarray*}
Finally, the convergence (\ref{UnifPBeh}) follows by the last two inequalities.
\hfill q.e.d.
We conclude the paper with an application of our results to the question that originally
motivated this work. Consider the ergodic averages of a function of a random field restricted
to the staircase pattern defined by the approximation of a line by a lattice as in \eqref{Intro:ex.sq}.
We actually the function to be multivariate. In this context, this means that it may depend on
a finite number of such steps.
What can be said about the asymptotic behavior of the ergodic averages of such a function?
Let $P$ be a two-dimensional random field, that is, a probability measure on
$\Omega=\Upsilon^{\mathbb{Z}^2}$ invariant w.r.t.~the group of shift transformations
$(\vartheta_v)_{v\in\mathbb{Z}^2}$ (see just above Corollary \ref{ShiftsSProp} for details).
For a set $U\subseteq \mathbb{Z}^2$ let $\pi_U(\omega):=\omega(U)$ and $P_U:=P\circ \pi_U^{-1}$ defines a
probability distribution on $\Upsilon^{\vert U\vert}.$
Use $[x]$ denote the integer part of a real number $x,$ respectively.
Recall that $L_{\lambda,t}(z)=(z,[\lambda z+t])\ (z\in \mathbb{Z})$ is an approximation of the line with slope
$\lambda$ and $y$-intercept $t$ by elements of the lattice $\mathbb{Z}^{2}.$ For $z_1,z_2\in\mathbb{Z},$
$L_{\lambda,t}(z_1,...,z_2)$ are the $z_1${\it th} to the $z_2${\it th} step.
We will use the short form $P_{\lambda,t,m}:=P_{L_{\lambda,t}(0,...,m-1)}.$
\begin{cor}\label{Unif:ex.staircase}
Let $m\in\mathbb{N}$ and $\lambda\in [0,1].$
Let $f$ be a function on $\Upsilon^m$ which is integrable
w.r.t.~$P_{\lambda,t,m}$ for all $t\in[0,1].$
\noindent (i)
Let $\lambda$ be rational and represented as
$\lambda=p/q$ for $p\in\mathbb{Z}$ and $q\in\mathbb{N}$ with no common divisor.
If $P$ is ergodic w.r.t.~the shift transformations then
\begin{equation*}
\frac{1}{n}\sum_{i=0}^{n-1}f \big( \omega (L_{\lambda,t}(i,...,i+m-1)) \big)
\convinfty{n}
\frac{1}{q}\sum_{\nu=0}^{q-1}
\int_{\Upsilon^m} f(y)\,P_{\lambda,\t^\nu(t),m}(dy)
\end{equation*}
in $\L^1(P),$ for $P$-almost all $\omega\in\Omega,$ and uniformly in $t\in [0,1].$
\noindent (ii)
Let $\lambda$ be irrational. Assume further that
$f \big( \omega (L_{\lambda,t}(0,...,m-1))\big)$ is Riemann-integrable with respect to $t\in [0,1],$
for $P$-almost all $\omega\in\Omega.$
If $P$ is strongly mixing w.r.t.~the shift transformations then
\begin{equation*}
\frac{1}{n}\sum_{i=0}^{n-1}f \big( \omega (L_{\lambda,t}(i,...,i+m-1)) \big)
\convinfty{n}
\int_0^1 \int_{\Upsilon^m} f(y)\,P_{\lambda,t,m}(dy)\,dt
\end{equation*}
in $\L^1(P)$ and uniformly in $t\in [0,1].$
\end{cor}
\noindent {\bf Proof. }
Let $\t_{\lambda}$ be the rotation on the circle defined in Example \ref{TorTransl},
and $S_\lambda$ the skew product defined in \eqref{Intr:def.skew}.
Use $\kappa(t):=(1,[t+\lambda])$ and $\kappa_n$ as defined in \eqref{ItS}.
It is easy to show that
$\kappa_n(t)=L_{\lambda,t}(n)$ for all $n\in\mathbb{N}:$
For $n=1,$ it follows immediately from plugging in the definition,
and it remains to induce the statement from $n$ to $n+1.$
It is obvious for the first coordinate. The second coordinate of $\kappa$
can be written as
$\kappa_{n+1}^{(2)}(t)=\kappa_n^{(2)}(t)+\kappa\circ\tau_\lambda^n(t)
=[n \lambda + t] + [\tau_\lambda^n(t) + \lambda].$
The claim now follows from
$ [\t_{\lambda}^n(t)+\lambda]
= [t+n\lambda - [t+n\lambda] + \lambda]
=-[t+n\lambda] + [t+(n+1)\lambda].$
This implies that the iterates of the skew product are of the form
$S_{\lambda}^n(t,\omega)=\big(\t_\lambda^n(t),\theta_{L_{\lambda,t}(n)}\omega\big)$
$(n\in\mathbb{N})$ capturing the lattice approximation of the line.
Using the second equality in \eqref{ItS}, it also follows that
$L_{\lambda,t}(n+u)=L_{\lambda,t}(n)+L_{\lambda,\t^n(t)}(u)$ for all $n,u\in\mathbb{N}_0.$
We thus get
$ L_{\lambda,t}(i,...,i+m-1) = L_{\lambda,t}(i) + L_{\lambda,\t^i(t)}(0,...,m-1).$
Introducing the function $F_\lambda(t,\omega):=f\big(\omega,L_{\lambda,t}(0,...,m-1)\big)$ we obtain
\begin{align}\label{Ex:RewrittenAs}
\frac{1}{n}\sum_{i=0}^{n-1} & f \big( \omega (L_{\lambda,t}(i,...,i+m-1)) \big) \nonumber \\
& = \frac{1}{n}\sum_{i=0}^{n-1}f \circ \vartheta_{L_{\lambda,t}(i)}
\big( \omega (L_{\lambda,\t^i(t)}(0,...,m-1)) \big)
= \frac{1}{n}\sum_{i=0}^{n-1} F_\lambda\circ S_{\lambda}(t,\omega).
\end{align}
Consider case (i). With the representation $\lambda=p/q$ as above, $\t_\lambda$ is periodic with $q.$
By Corollary \ref{Retperunif} and \eqref{Ex:RewrittenAs}, the averages on the left side in (i)
converge uniformly in $t,$ for $P$-almost all $\omega$ and in $\L^1(P).$
Since $P$ is ergodic with respect to $\theta_{\kappa_q(t)}$ for all $t\in M,$ the limit equals
$1/q \sum_{\nu=0}^{q-1} \int_\Omega F_\lambda\big(\t_\lambda^\nu(t),\,\cdot\,\big)\,P(d\omega)$
which simplifies to the expression on the right hand side of(i).
Consider case (ii). For an irrational $\lambda,$ $\t_{{\lambda}}$ is uniquely ergodic.
Since $\|\kappa_n(t)\|$ (with $\|\,\cdot\,\|$ for the maximum norm) is bounded
from below by $n,$ the sequence tends to infinity as $n$ goes to infinity.
Corollary \ref{ShiftsSProp} with $v_1=(1,0)$ and $v_2=(0,1)$
implies the ergodicity of $S_\lambda.$
It remains to verify condition \eqref{kappaCond}.
The latter easily follows from $\|\kappa_i(t)-\kappa_j(t)\|\ge | i-j |,$ and since
${1}/{n^2}\big\vert\big\{1\le i,j\le n\,\big\vert\,\vert i-j\vert\le m\big\}\big\vert$
converges to $0$ for all $m\in\mathbb{N}.$
Now, Corollary \ref{UniLRMak} applied to \eqref{Ex:RewrittenAs} implies that
the averages in (ii) converge uniformly in $t,$ and in $\L^1(P).$
The limit equals $\int_0^1 \int_\Omega F_\lambda(t,\,\cdot\,)\,P(d\omega)\,dt$ which
simplify to the expression on the right hand side of (ii).
\hfill q.e.d.
Note that $P$-almost everywhere convergence in (ii) can be derived as well, but
requires additional conditions of the form stated in Theorem \ref{UnifP}.
For $m=1$ the limit actually simplifies to the integral of $f$ with respect to
the marginal distribution of $P$ in the origin.
In particular, it is independent of $t$, and it is the same in (i) and (ii).
The simplest interesting case is $m=2.$
Let $P_{\mbox{flat}}:=P\circ\pi_{\{(0,0),(1,0)\}}^{-1}$ denote the marginal distribution
of $P$ on the subset $\{(0,0),(1,0)\}.$ This corresponds to the case where
$L_{\lambda,t}$ does not have a step in $z=1.$
Let $P_{\mbox{step}}:=P\circ\pi_{\{(0,0),(1,1)\}}^{-1}$ denote the marginal distribution
of $P$ on the subset $\{(0,0),(1,1)\}.$ This corresponds to the case where
$L_{\lambda,t}$ has a step in $z=1.$
Then, the limits in both (i) and (ii) of the above corollary are of the form
\begin{equation}
\lambda\int_{\Upsilon^2} f(y)\,P_{\mbox{step}}(dy)
+(1-\lambda)\int_{\Upsilon^2} f(y)\,P_{\mbox{flat}}(dy).
\end{equation}
\nocite{Mil88}
\section*{Acknowledgements}
This work was motivated by a special case of Corollary \ref{Unif:ex.staircase}
which originally occured during my thesis work.
I am grateful to my supervisor Hans F{\"o}llmer for the
rich education he gave me in both ergodic theory and probability theory.
It is a pleasure for me to thank Jon Aaronson
for guiding me toward a broader perspective for Lemma \ref{ErgS}.
David Steinsaltz assisted with an approximation
argument I used in the proof of Theorem \ref{UnifL}, and I thank him for this hint.
Benjamin Weiss and Yuval Peres helped me situate my results
within the field of ergodic theory;
I am grateful to them for the interest they showed in my work.
My thanks are also due to Hans Crauel, Manfred Denker,
Didier Piau, Michael Scheutzow and Masha Saprykina for helpful discussions
on different aspects of ergodic theory.
I would like to express my gratitude to Bill MacKillop for supporting me in
manifold ways during the time I've been working at Queen's University. |
0807.4208 | \section{Introduction}
\label{SEC:introduction}
Clusters of galaxies are a unique probe of the growth and dynamics of
structure in the Universe.
In particular, active mergers of sub-clusters provide a window to the
processes by which massive clusters are assembled.
In these systems, the galaxies and associated dark
matter are essentially collisionless. In contrast, the ionized
intracluster gas, typically at temperatures of $T \sim 10^8$~K,
is strongly interacting and experiences complex dynamics.
In extreme cases, the normally associated dark matter and
intracluster gas can be significantly separated.
The Bullet cluster (1E 0657--56) at $z=0.296$, is a massive cluster
consisting of two
sub-clusters in the process of merging. The smaller sub-cluster or
``bullet" has passed through the larger main cluster.
X-ray observations infer a bow shock velocity of $\sim$4700 km/s~\markcite{markevitch2006}({Markevitch} 2006),
while simulations of the collision yield a substantially lower speed for the
sub-cluster \markcite{springel2007}({Springel} \& {Farrar} 2007).
This collision is perpendicular to the line of sight,
providing an ideal system for studying interacting sub-clusters~\markcite{clowe2006}({Clowe} {et~al.} 2006).
The mass surface density of the Bullet cluster has been measured using
weak and strong gravitational lensing of light from background
galaxies. There are significant angular offsets between the peaks of
the X-ray surface brightness, which trace the baryonic gas through
thermal bremsstrahlung emission,
and the peaks of the lensing surface
density, which are associated with the majority of the mass. The
combined weak and strong lensing analyses of \markcite{bradac2006}{Brada{\v c}} {et~al.} (2006) show
that the main cluster and sub-cluster are separated from their
associated X-ray peaks at $10\,\sigma$ and $6\,\sigma$ significance
respectively. This result has been recognized as providing direct
evidence for the presence of collisionless dark matter in this
system~\markcite{clowe2006}({Clowe} {et~al.} 2006).
The Sunyaev-Zel'dovich effect (SZE) provides an independent probe of the intracluster gas.
In the SZE, a small fraction
($\sim1\%$) of cosmic microwave background (CMB) photons undergo
inverse Compton scattering from intracluster
electrons~\markcite{sunyaev1970,birkinshaw1999}({Sunyaev} \& {Zel'dovich} 1970; {Birkinshaw} 1999). This process distorts
the Planck blackbody spectrum of the CMB and produces a signal
proportional to the gas pressure integrated along the line of sight.
At $150\,$GHz, the SZE produces a temperature decrement with respect to the
unperturbed CMB intensity.
Early detections of the SZE in the Bullet cluster include
work by \markcite{andreani1999}{Andreani} {et~al.} (1999) and \markcite{gomez2003}{Gomez} {et~al.} (2004).
Unlike the X-ray surface brightness, the peak SZE surface brightness for a
given cluster is independent of redshift.
Therefore, the SZE has the potential to be an effective probe of intracluster gas
out to the redshifts at which clusters are assembled.
SZE measurements of galaxy clusters provide
complementary constraints on cluster properties typically derived
from X-ray measurements such as central electron density, core
radius of the intracluster gas, cluster gas mass, and fraction of the total cluster mass
in gas. Since the SZE and X-ray signals are proportional to the line-of-sight integral of
the electron density and electron density squared respectively, SZE results will be less
sensitive to clumping of the intracluster gas.
For all comparisons between SZ and X-ray data, we assume
a $\Lambda$CDM cosmology, with $h = 0.7$, $\Omega_{\rm{m}} = 0.27$, and $\Omega_{\Lambda} = 0.73$.
In this paper, we present a 1\arcmin\ resolution SZE image of the
Bullet cluster at $150\,$GHz made with the APEX-SZ instrument. It is the
first reported scientific result from observations with a large array of
multiplexed superconducting transition-edge sensor bolometers. In
\S\,\ref{SEC:observations}, we discuss the instrument and
observations. Calibration is discussed in \S\,\ref{SEC:calibration}.
In \S\,\ref{SEC:datareduction}, we describe the data
reduction procedure, and in \S\,\ref{SEC:results}, we present the
results of fits to the SZE surface brightness with cluster models,
including mass-weighted electron temperature and gas mass
fraction calculations.
We summarize the conclusions and discuss
future work in \S\,\ref{SEC:conclusions}.
\section{Observations}
\label{SEC:observations}
APEX-SZ is a receiver designed specifically for SZE galaxy cluster
surveys~\markcite{schwan2003,dobbs2006,schwan2008}({Schwan} {et~al.} 2003; {Dobbs} {et~al.} 2006; Schwan {et~al.} 2009, in preparation). It
is mounted on the 12-meter diameter APEX telescope, located on the
Atacama plateau in northern Chile~\markcite{guesten2006}({G{\"u}sten} {et~al.} 2006). The observing
site was chosen for its extremely dry and stable atmospheric
conditions. The mean atmospheric transmittance is frequently better
than 95$\%$ in the APEX-SZ frequency band at
150~GHz~\markcite{peterson2003,chamberlain1995}({Peterson} {et~al.} 2003; {Chamberlain} \& {Bally} 1995). The telescope is capable
of round-the-clock observations.
Three reimaging mirrors in the Cassegrain cabin couple the APEX
telescope to the focal plane of APEX-SZ. We achieve the diffraction
limited performance of the telescope across the entire 0.4\mbox{$^{\circ}$}\
field of view with a mean measured beam full width half maximum (FWHM)
of 58\arcsec, and a measured beam solid angle of 1.5~arcmin$^2$,
including measured sidelobes at the $-14$~dB level.
The APEX-SZ receiver houses a cryogenic focal plane, operating at 0.3~K.
The focal plane contains 330 horn-fed absorber-coupled superconducting
transition-edge sensor
bolometers~\markcite{ref:richards-bolometer-review,lee1996}(Richards 1994; {Lee} {et~al.} 1996),
with 55 detectors on each of six sub-array wafers. Of the 330
detectors, 280 are read out with the current frequency-domain multiplexed
readout hardware. We measure
the median individual pixel Noise Equivalent Power (NEP) to be $10^{-16}
$~W/$\sqrt{{\rm Hz}}$ and the median Noise Equivalent Temperature (NET)
to be $860~\mu$K$_{\rm{CMB}}\sqrt{{\rm s}}$.
The measured optical bandwidth of the receiver is 40\% narrower
than the design goal of 38~GHz, resulting in lower sensitivity than
anticipated.
The large field of view of the APEX-SZ instrument is designed for
surveying large areas of sky. In order to efficiently observe a single
target, we use the circular scan pattern illustrated in
Figure~\ref{FIG:scanpattern}. The circle center is fixed in AZ/EL
coordinates for twenty circular sub-scans, with a total duration of 100
seconds. This choice has a number of important advantages.
The sky signal is modulated so that it appears in the timestream at frequencies
higher than atmospheric drifts and readout $1/f$ noise.
In addition, the circle scan has a moderate continuous acceleration; the lack
of high acceleration turn-arounds makes it possible to achieve a high observing
efficiency.
Approximately $20\%$ of the total observing time is spent moving the telescope
to a new center position before the start of the next scan.
Every bolometer maps a $12\arcmin \times 25\arcmin$ sub-field,
with a combined map field of $36\arcmin \times 48\arcmin$ every 100
seconds.
\begin{figure}[h]\centering
\includegraphics[width=0.45\textwidth]{f1.pdf}
\caption[]{ The 100-second circular drift scan pattern. The solid
line shows the track of the center of the array. One circle has a
period of five seconds. The dashed line shows the instantaneous
field of view of the bolometer array. The $+$ marker indicates the source
position with respect to the scan pattern. The center of the
circles is constant in azimuth and elevation as the source drifts
across the field. The small disk in the lower left indicates the
58\arcsec\ mean FWHM beam for a single bolometer.}
\label{FIG:scanpattern}
\end{figure}
Observations of the Bullet cluster were conducted over a period of
seven days in August 2007, when the cluster was visible between the
hours of 03:00 and 15:00 local time. The weather over this period was
typical for the site, with precipitable water vapor varying between
0.25 and 1.5 mm, and a median atmospheric transmittance of $97\%$. For
the analysis in this paper, 235 scans are used, each scan consisting
of twenty 5-s circular sub-scans, for a total of 6.4 hours of on-source
data.
\section{Calibration}
\label{SEC:calibration}
The response of the receiver to astronomical sources is measured with
daily raster scans of Mars over every bolometer in the
array. For each bolometer, the observations provide a primary flux
calibration and a high signal-to-noise beam profile from which we
determine beam parameters such as size, ellipticity, and position with
respect to the array-center pointing.
Additional observations of RCW57
and RCW38 are used to monitor gain stability, and frequent
observations of bright quasars near the cluster source are used to
monitor pointing stability.
The WMAP satellite has been used to calibrate the brightness
temperature of Mars at $93\,$GHz in five measurement periods spanning
several years \markcite{hill08}({Hill} {et~al.} 2009). The WMAP Mars temperatures are tied to
the CMB dipole moment and are accurate to better than $1.0\%$. The
brightness temperature of Mars changes significantly ($\sim 15\%$) as
a function of its orbit and orientation. We use a version of the Rudy
Model \markcite{rudy87,muhleman91}({Rudy} {et~al.} 1987; {Muhleman} \& {Berge} 1991), that has been updated and maintained
by Bryan
Butler,\footnote{http://www.aoc.nrao.edu/\~{}bbutler/work/mars/model/}
to transfer the WMAP Mars temperature results to the APEX-SZ frequency
band and specific times of our Mars observations.
After applying a constant scaling factor, we find the Rudy model predictions
for the Mars brightness temperature to be in excellent agreement with
the WMAP measurements.
We find that the Rudy model
brightness temperatures at $93\,$GHz are systematically a factor of
$1.052 \pm 0.010$ higher than those measured by WMAP in the five
published observation periods.
In contrast, repeating the same exercise with the thermal model
developed by \markcite{wright76,wright07}{Wright} (1976, 2007), as implemented in the
online JCMT-FLUXES program,\footnote{http://www.jach.hawaii.edu/jac-bin/planetflux.pl}
results in a scaling factor of $1.085\pm 0.043$.
This is consistent with the 10\% rescaling of this model
called for in \markcite{hill08}{Hill} {et~al.} (2009), but the scaling factor exhibits significantly
larger rms scatter than that of the Rudy Model.
We therefore use the WMAP $93\,$GHz calibrated Rudy Model to compute
the Mars brightness temperatures at $150\,$GHz for the specific times
of our Mars observations by reducing the Rudy model $150\,$GHz
temperatures by a factor of 1.052. The Rudy model $93\,$GHz to $150\,$GHz frequency scaling factor
is $1.016 \pm 0.009$ at the times of our Mars observations, and we
adopt the rms scatter in this frequency scaling factor as an estimate of its
uncertainty. Combining the uncertainties in the WMAP Mars calibration,
the WMAP to Rudy model scaling factor at $93\,$GHz, and the Rudy model frequency
scaling factor, we estimate the uncertainty in Mars temperature to be $\pm 1.7\%$.
The measured signals from the calibrators are corrected for atmospheric
opacity, which is measured with a sky dip observation at the beginning and end of each day's
observations. Measured zenith transmittance over the observing period ranged
between 0.92 and 0.98, with a median of 0.97. Based on the
observed temporal variability of
the opacity, drifts in atmospheric opacity between the sky dip and
observation contribute $<0.4\%$ to the overall calibration uncertainty.
After correcting for the atmospheric opacity, we find that the
Mars temperature measured by APEX-SZ varies from the model prediction
by up to $\sim 3\%$ over the course of the observation period.
This gain variation is included as a source of error
in the final calibration uncertainty.
The APEX-SZ observing band center is measured with a Fourier
transform spectrometer to be $152\pm2$ GHz.
The uncertainty in the band center results in a $\pm 1.4\%$ uncertainty
in extrapolation of the Mars based calibration to CMB temperature.
The beam shape, including near sidelobes, is characterized by creating
a beam map from Mars observations, combining the same bolometer
channels that are used to make the science maps. We adjust the
calibration and measured beam size for the small ($\sim 1\%$)
correction due to the $8^{\prime \prime}$ angular size of Mars. We estimate
a fractional uncertainty in the beam solid angle of $\pm 4\%$.
The APEX-SZ detectors operate in a state of strong negative
electrothermal feedback which results in a linear response to changes
in the input optical power.
We have measured the response of the detectors during sky dips
between $90^\circ$ and $30^\circ$ elevation (antenna temperature
difference $\sim13\,$K),
and find no significant deviation from the expected linear response to loading.
We therefore conclude that detector
non-linearity makes a negligible contribution to the calibration
uncertainty.
Slowly changing errors in telescope pointing result in both a
pointing uncertainty and a flux calibration
uncertainty due a broadening of the effective beam pattern.
To measure pointing errors during our
observations, we observe a bright quasar within a few degrees of the
Bullet cluster every 1--2 hours, and apply a pointing correction
as needed.
The typical RMS pointing variations of the APEX telescope between
quasar observations is $\sim 4 \arcsec$.
This pointing uncertainty results in a slightly larger effective
beam for the coadded maps than is measured with the individual calibrator
maps.
The correction to the flux calibration of the coadded maps due to
pointing uncertainty is negligible,
particularly for the observation of extended objects such as the
Bullet cluster. We estimate the pointing uncertainty
in the coadded maps to be $\pm 4\arcsec$ in both
the RA and DEC directions.
The uncertainty in the CMB temperature calibration of the APEX-SZ
maps is summarized in Table~\ref{TBL:cal}.
The combination of all contributions to the calibration uncertainty
described above results in an overall point source flux uncertainty of
$\pm 5.5\%$.
\begin{deluxetable}{ll}
\tablecaption{\label{TBL:cal} APEX-SZ Flux Calibration Uncertainty}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{Uncertainty}}
\startdata
WMAP Mars temperature at $93\,$GHz & $\pm1.0\%$ \\
Rudy model to WMAP scaling factor at $93\,$GHz & $\pm1.0\%$ \\
$93\,$GHz to $150\,$GHz frequency scaling factor & $\pm0.9\%$ \\
Frequency band center & $\pm1.4\%$ \\
Beam solid angle & $\pm4.0\%$ \\
Atmospheric attenuation & $\pm0.4\%$ \\
Gain variation & $\pm3.0\%$ \\\hline
Total & $\pm5.5\%$ \\
\enddata
\end{deluxetable}
\section{Data Reduction}
\label{SEC:datareduction}
The data consist of 280 bolometer timestreams sampled at 100
Hz, telescope pointing data interpolated to the same rate,
housekeeping thermometry data, bolometer bias and readout
configuration data, and other miscellaneous monitoring data. The
fundamental observation unit is a scan comprising twenty 5-s
circular sub-scans in AZ/EL coordinates, allowing the source to drift
through the field of view (FOV), as described in \S\,\ref{SEC:observations} above.
Data reduction consists of cuts to remove
poor-quality data, filtering of $1/f$ and correlated noise due to
atmospheric fluctuations, and binning the bolometer data into maps.
These steps are described in more detail below.
\subsection{Timestream Data Cuts}
Timestream data are first parsed into individual circular sub-scans.
We reject $\sim 7$\% of the data at the beginning and end of the scan
where the telescope deviates from the constant angular velocity
circular pattern.
We reject bolometer channels that are optically or electronically unresponsive, or lack high-quality flux calibration data;
typically, 160--200 of the 280 bolometer channels remain after these
preliminary cuts. The large number of rejected channels is due primarily
to low fabrication yield for two of the six bolometer sub-array wafers.
We reject spikes and step-like glitches caused by cosmic rays or
electrical interference. These are infrequent and occur on time scales
faster than the detector optical time constant. We use a simple
signal-to-noise cut on the data to reject these, since the
timestream is noise dominated even for the $\sim$1~mK
Bullet cluster signal. Step-like glitches are often correlated
across many channels in the array, so we reject data from all channels whenever a
spike or glitch in $\geq 2\%$ of the channels is detected. These cuts result
in a loss of 8\% of the remaining data.
For each circular sub-scan, we also reject channels that have excessive
noise in signal band, resulting in a loss of 19\% of the remaining data.
\subsection{Atmospheric Fluctuation Removal}
\label{SEC:atmremoval}
After the timestream data cuts, fluctuations in atmospheric emission
produce the dominant signal in the raw bolometer timestreams. The
atmospheric signal is highly correlated across the array, which can be
exploited to remove the signal.
Principal component analysis (PCA) has been used by some groups
to reduce the atmospheric signal \markcite{scott2008,laurent2005}(see, e.g., {Scott} {et~al.} 2008; {Laurent} {et~al.} 2005).
However, the effect of PCA filtering on the source is difficult to predict and
is a function of the atmospheric conditions.
We have developed an analysis strategy that reduces the atmospheric signal
through the application of spatial filters that have a constant and well
understood effect on the signals we are attempting to measure.
Atmospheric fluctuation power is expected to follow a Kolmogorov
spatial power spectrum, with most power present on scales larger
than the separation between beams as they pass through the atmosphere,
resulting in an atmospheric signal that is highly correlated across the array.
To reduce these fluctuations, we first remove a polynomial and an elevation dependent
airmass opacity model from each channel's timestream, then remove
a first-order two-dimensional spatial polynomial across the
array for each time-step. This algorithm is described in detail
in the two following subsections. This atmospheric fluctuation removal
strategy requires that both the spatial extent of the scan
pattern and the instantaneous array FOV are larger than the
source. The $6\arcmin$ radius circular scan and the 23$\arcmin$ array
FOV allow us to recover most of the Bullet cluster's flux,
but some extended emission is lost as is described in \S\,\ref{SEC:results}.
\subsubsection{Timestream Atmosphere Removal}\label{subsec:circle}
We observe scan-synchronous signals in the bolometer timestreams due
to elevation-dependent atmospheric emission. The optical path length
$L$ through the atmosphere is proportional to the cosecant of the
elevation angle $\epsilon$, $L \propto \csc(\epsilon)$. The change in
optical path-length is nearly a linear function of elevation angle
over the $6\arcmin$ circular scan radius. In the circular drift scans,
this modulation of the elevation-dependent opacity produces an
approximately sinusoidal modulation in the bolometer timestream. For
each channel in each scan, we simultaneously fit and remove an
atmospheric model consisting of this cosecant function plus an order
20 polynomial (one degree of freedom per circular sub-scan) to remove
slow drifts in the atmospheric opacity and readout $1/f$ noise.
This scheme effectively
removes the common scan-induced atmospheric signal as well as most of
the atmospheric fluctuation power below the frequency of the circular
sub-scan (0.20 Hz), while only modestly affecting the central Bullet
cluster signal.
\subsubsection{Spatially Correlated Atmosphere Removal}
Removing the cosecant-plus-polynomial model
from the timestream data reduces low-frequency
atmospheric fluctuation power, but not higher frequency power
corresponding to smaller spatial scales near those where the cluster
signal occurs. To reduce these fluctuations, the atmosphere can be modeled
as a spatially correlated signal across the array pixel positions on
the sky with a low-order two-dimensional polynomial function. At each
time step, we fit and remove a low-order
two-dimensional polynomial
function across the array, similar to the procedure described in
\markcite{sayers07}Sayers (2007).
The relative gain coefficients for each bolometer channel are
calculated by taking the ratio of each channel's timestream, which is
dominated by correlated atmospheric noise, to a median timestream
signal generated from all channels. With the favorable atmospheric
conditions of these observations, we find that a first order spatial
polynomial (offset and tilt) is adequate to remove most of the atmospheric signal while
preserving the cluster signal.
Bolometer channels with excess uncorrelated noise are
more easily identified after removing the correlated atmospheric
noise component; we reject these noisy channels, then perform the
spatially correlated signal removal a second time.
\subsection{Map-Making}
\label{SEC:mapmaking}
The atmospheric removal algorithms described above act as a high-pass
filter. They suppress signals on scales comparable to the scan length
or the focal plane FOV. The cluster emission can be quite
extended, and therefore the data reduction filtering process attenuates
diffuse flux in the cluster signal and produces small
positive sidelobes around the cluster decrement. The data reduction
pipeline filters can be tailored, within limits, to meet various
scientific objectives. Thus, our primary data products consist of two
different high signal-to-noise maps of the cluster.
For one map, we mask a circular region centered on the cluster source
prior to fitting the timestream and spatial filters described in
\S\,\ref{SEC:atmremoval}, then apply the resulting filter functions to
the entire data set, including the source region. The source-mask
procedure prevents the cluster signal within the masked region from
influencing the baseline fits, and thus reduces attenuation of the
source central decrement and extended emission at the expense of
increasing the contribution of low-frequency noise in the map
center. We choose a source-mask radius of $4.75\arcmin$ as a
compromise between attenuation of diffuse emission and increased map
noise. We use the source-masked map to visually interpret the
morphology and extended emission in the cluster. These results are discussed in \S\,\ref{subsec:tempmap}.
We also produce a map in which we do not mask the source when applying
filters. The non-source-masked
map is used for model parameter estimation because it has higher signal to noise
in the central region of the map. In addition, it is easier to take
into account the effects of the data reduction filters, or transfer
function, on the underlying sky intensity distribution, which is
necessary for comparing the data to the model for parameter
estimation. The fitting procedure and results are described in more detail in
\S\,\ref{subsec:fitting}.
For each of the two maps, the post-cut, filtered timestream data are
binned in angular sky coordinates to create maps. For a given scan, a
map is created from each bolometer channel, applying the channel's
pointing offset and flux calibration.
A coadded scan map is created by combining individual channel maps
with minimum variance weighting in each pixel, using the sample variance
of the conditioned timestream data in the scan.
The final coadded map
is created by combining all scan maps, again with minimum variance
weighting in each pixel. We bin maps at a resolution of 10\arcsec\ to
oversample the beam. The source-masked map that we present
in \S\,\ref{SEC:results} is
convolved with a 1\arcmin\ FWHM Gaussian to smooth noise fluctuations
to the angular size of the beam. However, the radial profiles
presented below and the non-source-masked map used for model
fitting do not include this additional smoothing.
\section{Results}
\label{SEC:results}
\begin{figure*}[p!]\centering
\includegraphics[width=0.7\textwidth]{f2a.pdf}
\includegraphics[width=0.7\textwidth]{f2b.pdf}
\caption[]{ (Top) Temperature map of the Bullet cluster system from
the source-masked data reduction, with scale in CMB temperature
units. The circle in the lower left corner represents the 85\arcsec\
FWHM map resolution which is the result of the instrument beam and
data reduction filter convolved with the 1\arcmin\ FWHM Gaussian
smoothing applied to the map. (Bottom) Difference map made by
multiplying alternate scan maps by $+1$ and $-1$,
respectively, then coadding all scan maps with minimum variance
weighting, in the same manner as was used to produce the temperature
map shown in the top panel. The contour interval is 100~$\mu \rm{K}_{\rm{CMB}}$
in both maps.}
\label{FIG:bulletcoadd}
\end{figure*}
\subsection{Temperature Map}
\label{subsec:tempmap}
Figures \ref{FIG:bulletcoadd}~and~\ref{FIG:bulletcoaddzoom} show the
source-masked temperature map from our observations of the Bullet
cluster. The map has a resolution of 85\arcsec\ FWHM which results from the
combination of the 58\arcsec\ instrumental beam, the data reduction filters, and
a final 1\arcmin\ FWHM Gaussian smoothing of the map. The
source-masked map is shown in order to provide a more accurate representation of the
extended emission and cluster morphology.
The noise in the central region of the source-masked map
is 55~$\mu \rm{K}_{\rm{rms}}$ per 85\arcsec\ FWHM resolution
element.
Near the cluster center, the emission hints at elongation
in the East-West direction, which is along the axis between the main
and sub-cluster gas detected in the X-ray, see \S\,\ref{sec:xray}.
The more extended emission appears to be elongated
in the Northwest-Southeast direction, which is the major axis of the best-fit
elliptical $\beta$ model discussed in \S\,\ref{subsec:fitting}.
Figure~\ref{FIG:bulletcoaddzoom} shows the centroid position of the
best-fit elliptical $\beta$ model, and the position of the dust
obscured, lensed galaxy detected at 270~GHz by \markcite{wilson2008}{Wilson} {et~al.} (2008).
As discussed in \S\,\ref{SEC:source_contributions},
we see no evidence for emission from this source in our 150~GHz map.
Radial profiles for the unsmoothed source-masked and non-source-masked
maps are shown in Figure~\ref{FIG:radial_plots}. The source-masked map
has a signal-to-noise of 10 within the central 1\arcmin\ radius,
compared to 23 for the non-source-masked map, due to the fact that
source-masking allows more large-scale atmospheric fluctuation noise
to remain in the map. However, the source-masked map preserves signal on
larger spatial scales than the non-source-masked map. In both the
source-masked and non-source-masked maps, the sky intensity
distribution is filtered by the instrument beam and data reduction
pipeline described in \S\,\ref{SEC:datareduction}. We do not
renormalize the map amplitudes, since the source is extended and an
assumption would need to be made about the shape of the sky-brightness
distribution to do so. However, in order to accurately estimate
cluster parameters such as the central temperature decrement, a model
for cluster emission must be adopted, and the instrument beam and data
reduction filtering must be taken into account.
\begin{figure}[h!]\centering
\includegraphics[width=0.45\textwidth]{f3.pdf}
\caption[]{ Temperature map detail from
Figure~\ref{FIG:bulletcoadd}, with color scale adjusted to the
limits of the detail region, and a contour interval of 100~$\mu \rm{K}_{\rm{CMB}}$.
The $+$ marker indicates the centroid
position of the best-fit elliptical $\beta$ model, see
\S\,\ref{subsec:fitting}. The $*$ marker indicates the position of
the bright, dust obscured, lensed galaxy detected at 270~GHz by
\markcite{wilson2008}{Wilson} {et~al.} (2008), see \S\,\ref{SEC:source_contributions}.}
\label{FIG:bulletcoaddzoom}
\end{figure}
\subsection{Fit to Elliptical $\beta$ Model}\label{subsec:fitting}
We fit an elliptical $\beta$ model to the non-source-masked temperature map
to allow a straightforward comparison of cluster gas properties
derived from our measurements to those derived from X-ray
observations. In all analyses here we assume the cluster gas is
well-described by an isothermal $\beta$ model, and is in hydrostatic
equilibrium. These assumptions are unphysical in the case of the
Bullet cluster, which is a dynamically complex merging system where
the gas is separated from the rest of the mass~\markcite{clowe2006}({Clowe} {et~al.} 2006).
However, we find that with the sensitivity and spatial
resolution of the observations, these assumptions yield
an adequate description of the observed emission.
We model the three-dimensional radial profile of the
electron density with an isothermal $\beta$ model
\markcite{cavaliere1978}({Cavaliere} \& {Fusco-Femiano} 1978):
\begin{equation}
\label{EQN:betamodeldensity}
n_e(r) = n_{e0}\left( 1 + \frac{r^2}{r_c^2}\right)^{-3\beta/2}.
\end{equation}
Here, $n_{e0}$ is the central electron number density, $r_c$ is the
core radius of the gas distribution, and $\beta$ describes the
power-law index at large radii.
The radial surface temperature
profile of the SZE takes a simple analytic form:
\begin{equation}
\label{EQN:betamodeltemperature}
\Delta T_{\rm{SZ}} = \Delta
T_{0}\left( 1 + \frac{\theta^2}{\theta_c^2} \right)^{(1-3\beta)/2},
\end{equation}
where $\Delta T_{0}$ is the central temperature decrement, and
$\theta_{c} = r_c/D_A$ is the core radius divided by the angular-diameter distance. A similar form exists for the
X-ray surface brightness.
Because of the significant ellipticity in the measured SZE intensity
profile, we generalize the cluster gas model to be a spheroidal rather
than spherical function of the spatial coordinates:
\begin{equation}
\label{EQN:ellipticalbeta}
\Delta T_{\rm{SZ}} = \Delta T_{0}\left(1 + A + B \right)^{(1-3\beta)/2},
\end{equation}
with
$$
A = \frac{(\cos(\Phi)(X - X_{\rm{0}}) + \sin(\Phi)(Y - Y_{\rm{0}}))^2}{\theta_{\rm{c}}^2},
$$
$$
B = \frac{(-\sin(\Phi)(X - X_{\rm{0}}) + \cos(\Phi)(Y - Y_{\rm{0}}))^2}{(\eta \theta_{\rm{c}})^2}.
$$
Here $(X - X_{\rm{0}})$ and $(Y - Y_{\rm{0}})$ are angular offsets on the sky in the RA and
DEC directions, with respect to center positions $X_{\rm{0}}$ and
$Y_{\rm{0}}$. The axial ratio, $\eta$, is the ratio between the minor and major axis
core radii, $\Phi$ is the angle
between the major axis and the RA ($X$) direction.
$\Delta T_{0}$ is given by the gas pressure integrated along the
central line of sight through the cluster:
\begin{equation}
\label{EQN:integratedpressure} \frac{\Delta T_{0}}{T_{\rm{CMB}}} =
\frac{k_{\rm{B}} \sigma_T}{m_e c^2} \int f(x,T_e) n_e(l) T_e(l) dl
\end{equation}
where $x = h \nu / k T$, $f(x,T_e)$ describes the frequency dependence
of the SZE, $\sigma_T$ is the Thomson scattering
cross-section, and $T_{\rm{CMB}} = 2.728$ K.
For all results in this paper, we use the relativistic SZE spectrum $f(x,T_e)$
provided by ~\markcite{nozawa2000}{Nozawa} {et~al.} (2000), and neglect the kinematic effect.
At 150~GHz and $T_e = 13.9$~keV (see \S\,\ref{sec:xray}), this is a 9\% correction to
the non-relativistic value.
To accurately estimate $\beta$ model parameters for the cluster, the
instrument beam and data reduction filters, or transfer function, must
be applied to the model before comparing it with the data \markcite{benson2003,reese00}(see,
e.g., {Benson} {et~al.} 2003; {Reese} {et~al.} 2000). We characterize this transfer function
by creating a map from a simulated point source, convolved with the
instrument beam, and inserted into a noiseless timestream, similar to
the method described in \markcite{scott2008}{Scott} {et~al.} (2008). The point source transfer
function map $\mathcal{K}$ is then convolved with a simulated $\beta$
model cluster map $\mathcal{B}$ to generate a filtered model map
$\mathcal{B^\prime}$, which is a noiseless simulated APEX-SZ
observation of a $\beta$ model cluster. The filtered model map,
$\mathcal{B^\prime}$, is then differenced with the data map,
$\mathcal{M}$, and model parameters are estimated by minimizing a
$\chi^2$ statistic.
Simulating maps of many different cluster models is required for model
parameter fitting. Convolving the cluster model with the point source
transfer function map is much faster than processing each model
through the reduction pipeline. We find that the resulting simulated
maps from both methods agree sufficiently well to have negligible
effect on the parameter estimation results.
We use the unsmoothed non-source-masked map with $10\arcsec$
pixelization described in \S\,\ref{SEC:mapmaking} for all parameter
estimation described below, since this map has lower noise and a more
easily characterized transfer function than the source-masked map
shown in Figure~\ref{FIG:bulletcoadd}. Diffuse cluster emission is
more attenuated in the non-source-masked map, but this
is taken into account using the point source transfer function.
Map noise properties are assessed in the spatial frequency domain
using jackknife noise maps \markcite{sayers07,sayers08}(see Sayers 2007; {Sayers} {et~al.} 2009).
To estimate the noise covariance $C_n$, we assume that the noise is
stationary in the map basis. With this assumption, the Fourier
transform of the noise covariance matrix, $\widetilde{C}_n$, is
diagonal, and the diagonal elements are equal to the noise map power
spectral density (PSD). For each of 500 jackknife noise map
realizations, we find the two-dimensional Fourier transform, then
average over all realizations.
This averaged map PSD is the experimental estimate of the diagonal elements of
$\widetilde{C}_n$.
However, these jackknife maps do not include fluctuations due to the
primary CMB anisotropies. We estimate the CMB signal covariance from
the WMAP5 best-fit power spectrum \markcite{nolta09}({Nolta} {et~al.} 2009) convolved with the point source
transfer function described earlier and add it to the jackknife noise
PSD to determine the total covariance matrix.
We construct a $\chi^2$ statistic for the
model fit using the transform of the filtered $\beta$ model,
$\widetilde{\mathcal{B^\prime}}$, and the transform of the
central $14\arcmin \times 14\arcmin$ portion of the data map
$\widetilde{\mathcal{M}}$ as:
\begin{equation}
\chi^2 = (\widetilde{\mathcal{M}}-\widetilde{\mathcal{B^\prime}})^T \widetilde{C}_n^{-1} (\widetilde{\mathcal{M}}-\widetilde{\mathcal{B^\prime}}).
\end{equation}
Using Markov Chain Monte Carlo (MCMC) methods, the likelihood,
$\mathcal{L} = e^{-\frac{1}{2} \chi^2}$, is sampled in the 7-dimensional
model parameter space and integrated to find
the marginal likelihood distributions of the $\beta$ model parameters.
The model parameter estimates and uncertainties that we report are the
maximum likelihood values and constant-likelihood 68\% confidence
intervals, respectively, of the marginal likelihood distributions.
The above approach to noise covariance estimation is chosen for its
simplicity and because we do not have enough linear combinations of
individual scan maps to fully sample the noise covariance matrix using
jackknife noise maps. But, the method relies on several simplifying
assumptions, including that the bolometer noise is stationary for each
100-s scan, the timestream noise is uncorrelated from scan to scan,
and the map coverage is uniform. Our map coverage is not actually
uniform, but we find through simulations of non-uniform Gaussian
noise maps that the $\chi^2$ statistic is not significantly affected.
In addition, the validity of the approach is tested by inserting
simulated clusters into the real timestream data; the simulated
cluster parameters are accurately recovered within the estimated
uncertainties.
Results of the $\beta$ model parameter estimation are given in
Table~\ref{TBL:betamodelfits}. Due to the degeneracy between the core
radius $\theta_{\rm{c}}$ and $\beta$ parameters, we assume a prior probability
density on $\beta$ of $1.04^{+0.16}_{-0.10}$, which is found from fits to ROSAT X-ray data
by \markcite{ota2004}{Ota} \& {Mitsuda} (2004). \markcite{hallman2007}{Hallman} {et~al.} (2007) find that in hydro/N-body simulations,
$\beta$ derived from fits to SZE profiles is higher than that from
X-ray, with $\beta_{\mathrm{SZ}}/\beta_{\mathrm{X}\mbox{-}\mathrm{ray}} = 1.21 \pm 0.13$ for fits
within $r_{500}$. We do not account for that factor here due to the
significant uncertainty in the X-ray derived $\beta$ value, but we
note that our SZE data prefer a higher value for $\beta$ than
the peak value of the prior. We further discuss this choice of prior in
\S\,\ref{subsec:mwte}.
The best-fit $\beta$ model fits the data well, with a reduced $\chi^2$
value of 1.008, and with 7219 degrees of freedom (DOF) has a probability
to exceed (PTE) of 31.5\%. The difference map between the data map,
$\mathcal{M}$, and the best-fit filtered $\beta$ model,
$\mathcal{B^\prime}$, shows no evidence of residual cluster structure
or point sources.
\begin{deluxetable}{llll}
\tablecaption{\label{TBL:betamodelfits} $\beta$ Model Fit Results}
\tablewidth{0pt}
\tablehead{
\colhead{Parameter} & \colhead{Description} & \colhead{Value} & \colhead{Uncertainty$^a$}
}
\startdata
$X_{\rm{0}}$ & RA centroid position &$06^{\rm h}58^{\rm m}30.86^{\rm s}$ (J2000) & $\pm 7.4\arcsec$ \\
$Y_{\rm{0}}$ & DEC centroid position & $-55\degree56\arcmin46.2\arcsec$ (J2000) & $\pm 7.3\arcsec$ \\
$\Delta T_{\rm{0}}$ & Central temperature decrement & $-771\ \mu$K$_{\rm{CMB}}$ & $\pm 71\ \mu$K$_{\rm{CMB}}$ \\
$y_0$ & Central Comptonization$^b$($T_e = 13.9$~keV) & $3.31 \times 10^{-4}$ & $\pm 0.30 \times 10^{-4}$ \\
$y_0$ & Central Comptonization$^b$($T_e = 10.6$~keV) & $3.24 \times 10^{-4}$ & $\pm 0.30 \times 10^{-4}$ \\
$\theta_{\rm{c}} $ & Core radius & $142 \arcsec$ & $\pm 18 \arcsec$ \\
$\eta$ & Ellipse minor/major core radius ratio & 0.889 & $\pm 0.072$ \\
$\Phi $ & Ellipse orientation angle & $-52\mbox{$^{\circ}$}$ & $\pm 20\mbox{$^{\circ}$}$ \\
$\beta$ & Power-law index & 1.15 & $\pm 0.13$ \\
\enddata
\tablenotetext{a}{Quoted uncertainties are 68\% confidence intervals
in the marginal likelihood distribution for each parameter. The
uncertainty in $\Delta T_{\rm{0}}$ includes a statistical uncertainty
from the fit of $\pm 57\,\mu$K, and a $\pm 5.5\%$ flux calibration
uncertainty. The uncertainties in the centroid parameters $X_{\rm{0}}$
and $Y_{\rm{0}}$ include a $\pm 4\arcsec$ pointing uncertainty
and are given in units of arcseconds on the sky.}
\tablenotetext{b}{Central Comptonization, $y_0$, is a derived parameter,
assuming an electron temperature of 13.9 keV from \markcite{govoni2004}{Govoni} {et~al.} (2004)
and 10.6~keV from \markcite{zhang2006}{Zhang} {et~al.} (2006), an SZE observation frequency of 152~GHz, and
$T_{\rm CMB} = 2.728$~K. It is provided to facilitate comparisons with data at other
wavelengths.}
\end{deluxetable}
Radial profile plots of the best-fit $\beta$ model, $\mathcal{B}$, and
the filtered $\beta$ model map, $\mathcal{B^\prime}$, are shown in
Figure~\ref{FIG:radial_plots}. Also plotted for comparison are the
radially binned data from the unsmoothed non-source-masked map
$\mathcal{M}$, used for model fitting, and the unsmoothed
source-masked map, used to visualize extended emission (without the
1\arcmin\ Gaussian smoothing used in
Figures~\ref{FIG:bulletcoadd}~\&~\ref{FIG:bulletcoaddzoom}).
Uncertainties in both sets of radially binned data are highly
correlated due to large-spatial-scale correlated noise in the
maps. The coincidence of the non-source-masked data ($\mathcal{M}$,
filled circles) and the filtered best-fit $\beta$ model
($\mathcal{B^\prime}$, red solid line) show that the data and best-fit
$\beta$ model are in good agreement. The source-masked map preserves
signal on larger spatial scales than the non-source-masked map, and is
useful for visualizing extended emission on larger spatial
scales. But, as expected, even the source-masked map attenuates signal
on scales exceeding the $4.75^\prime$ radius of the source masking, and thus has
a lower signal amplitude when compared with the unfiltered $\beta$ model
($\mathcal{B}$, blue dashed line).
\begin{figure}[h!]\centering
\includegraphics[width=0.45\textwidth]{f4.pdf}
\caption[]{Radial profile of Sunyaev-Zel'dovich effect in the Bullet
cluster compared to the best-fit $\beta$ model. Points with error
bars are the SZE data binned in 1\arcmin\ radial bins, from the
non-source-masked map, ($\mathcal{M}$, filled circles) and the
source-masked map (open circles). The lines show the radial profile
of the best-fit $\beta$ model, unfiltered ($\mathcal{B}$, blue
dashed line) and after convolving with the instrument beam and
non-source-masked data reduction filters ($\mathcal{B^\prime}$, red
solid line). The non-source-masked map radial profile is reasonably
well fit by the filtered $\beta$ model. The source-masked map
preserves signal on larger spatial scales than the non-source-masked
map, but still attenuates signal on scales exceeding the
$4.75^\prime$ radius of the source masking. The source-masked data
thus have a lower signal amplitude when compared with the unfiltered
$\beta$ model, as expected. See text for details.}
\label{FIG:radial_plots}
\end{figure}
\subsection{Radio and IR source contributions}
\label{SEC:source_contributions}
Radio sources associated with a galaxy cluster and background IR
galaxy sources can have a significant impact on the measurement of the
SZE emission at 150~GHz. We interpret the published results of observations
of the Bullet cluster at other frequencies and conclude that the measured
SZE decrement is not significantly contaminated.
The most important source of potential confusion is a bright, dust obscured,
lensed galaxy in the direction of the Bullet cluster recently reported
by \markcite{wilson2008}{Wilson} {et~al.} (2008). This source has a flux density of $13.5 \pm
0.5$~mJy at an observing frequency of 270~GHz, and is centered at
RA $06^{\rm h}58^{\rm m}37.31^{\rm s}$, DEC $-55\degree57\arcmin1.5\arcsec$ (J2000),
$\approx 56^{\prime \prime}$ to
the east of the measured SZE centroid position, see Figure~\ref{FIG:bulletcoaddzoom}.
Assuming a spectral index
$\alpha=3$, where $S \propto \nu^\alpha$, we expect a flux density of
$1.94\,$mJy at $150\,$GHz corresponding to a temperature increment of
$\Delta T = 38\ \mu{\rm K_{CMB}}$ in the 1.5~arcmin$^2$ APEX-SZ beam solid angle.
This lensed
source is expected to be the dominant contribution to positive flux in
the direction of the cluster and we have repeated the $\beta$ model
fit taking it into account. We first add the source at its
measured position with the predicted 150~GHz flux to the SZE $\beta$
model and repeat the model fit. As expected, including the point
source results in a slightly ($\sim\sigma/3$) deeper decrement;
however, the $\chi^2$ of the model fit slightly increases.
We next allow the flux of the point source to vary along
with the other model parameters and find that values of positive flux
are a poorer fit than no source at all.
Therefore, we have no evidence for significant emission from this source
at 150~GHz. For the results in this paper, we use
cluster model parameters derived from fits that do not include this IR
source.
The Bullet cluster is also associated with a number of relatively
compact radio sources and one of the brightest cluster radio halos yet
discovered. However, these sources are predicted to produce negligible
temperature increments in the APEX-SZ beam when extrapolated to
150~GHz.
\markcite{liang2000}{Liang} {et~al.} (2000) report the detection of eight radio point sources
all of which have steeply falling spectra. Only two of these sources
were detected with ACTA at 8.8~GHz, and they were found to have flux
densities of $3.2 \pm 0.5$~mJy and $3.3 \pm 0.5$~mJy. The spectra of
these sources, measured between $4.9$ and $8.8\,$GHz are falling with
$\alpha = -0.93$ and $\alpha=-1.33$, respectively. Extrapolating to
$150\,$GHz, the flux of these sources are expected to be $0.24$ and
$0.07\,$mJy, corresponding to CMB temperature increments of $\Delta T
= 4.7$ and $1.4\,\mu{\rm K_{CMB}}$ in the 1.5~arcmin$^2$ APEX-SZ
beam solid angle.
The radio halo in the Bullet cluster is very luminous, but has a
characteristically steeply falling spectrum. \markcite{liang2000}{Liang} {et~al.} (2000)
measure the flux and spectra for the two main spatial components of
the halo. At $8.8\,$GHz, they find the two components to have fluxes
of $3.5$ and $0.55\,$mJy, with spectral indices of $\alpha=-1.3$ and
$\alpha=-1.4$, respectively. Extrapolating to $150\,$GHz, the combined flux from the
radio halo is expected to be $\sim 0.1\,$mJy. This emission is spread
over an area comparable to the size of the cluster and therefore
corresponds to a temperature increment $< 1\,\mu{\rm K_{CMB}}$ in the
APEX-SZ beam.
\subsection{Comparison with X-ray Data}\label{sec:xray}
X-ray emission in the ionized intracluster gas is dominated by thermal
bremsstrahlung. The X-ray surface brightness can be written
\begin{equation}
\label{xraysurfacebrightness}
S_{X} = \frac{1}{4 \pi (1+z)^4} \int
n_{e} n_{i} \Lambda_{ei} dl,
\end{equation}
where $n_{e,i}$ are the electron and ion densities in this gas,
$\Lambda_{ei}$ is the X-ray cooling function, and the
integral is taken along the line of sight. The X-ray flux is
proportional to the line-of-sight integral of the square of the
electron density, resulting in emission that is more sensitive
to local density concentrations than the SZE emission.
The Bullet sub-cluster and the bow shock are apparent in the X-ray
surface brightness map shown in Figure~\ref{FIG:bullet-overlay}.
The SZE contour map of the Bullet cluster in
Figures~\ref{FIG:bulletcoadd}~\&~\ref{FIG:bulletcoaddzoom} is
overlaid on an X-ray map and weak lensing
surface mass density reconstruction from
\markcite{clowe2006}{Clowe} {et~al.} (2006).\footnote{Data are publicly available at
http://flamingos.astro.ufl.edu/1e0657/public.html.} The X-ray map is
made from XMM data (observation Id: 0112980201) extracted in the
[0.5-2]~keV band, corresponding to Bullet rest frame energies
where the X-ray cooling function for hot gas is relatively
insensitive to temperature. The map is smoothed with a 12\arcsec\
Gaussian kernel.
The SZE contours do not resolve the sub-cluster. However, an
elongation of the inner contours to the West suggests that a
contribution from it may be detected. The observed SZE map is
consistent with expectations, given the 85\arcsec\ resolution of
the SZE map, the different dependence of the X-ray and
SZE signals on gas density, and the mass and temperature difference between the two
merging components which predict a factor of $\sim10$ lower
integrated pressure from the sub-cluster. There is no evidence in the
SZE contours of a contribution from the lensed sub-mm bright galaxy
discussed in \S\,\ref{SEC:source_contributions}.
\begin{figure}[h]\centering
\includegraphics[width=0.45\textwidth]{f5.pdf}
\caption[]{The SZE map of the Bullet system from this work, in white
contours, overlaid on an X-ray map from XMM observations. The green
contours show the weak lensing surface mass density reconstruction
from \markcite{clowe2006}{Clowe} {et~al.} (2006). The SZE contour interval is 100~$\mu \rm{K}_{\rm{CMB}}$.}
\label{FIG:bullet-overlay}
\end{figure}
\subsection{Mass-Weighted Temperature}\label{subsec:mwte}
The combination of cluster SZE and X-ray measurements can be used to place
constraints on the thermal structure of the intracluster gas.
The SZE intensity is proportional to the product of the electron density and the
electron temperature along the line of sight, see equation~(\ref{EQN:integratedpressure}). Therefore, if
the electron density is known from another measurement, the SZE
can be used to measure a mass-weighted temperature.
For simplicity, we assume here that the intracluster gas is isothermal, but a more detailed
comparison of the SZE surface brightness and projected density could be used to
constrain the thermal structure in the cluster.
We perform this calculation with two different descriptions of the intracluster gas density.
First, we model the spatial distribution of the intracluster gas as a
spherical $\beta$ model following equation~(\ref{EQN:betamodeldensity}).
We use $\beta$ model parameters from \markcite{ota2004}{Ota} \& {Mitsuda} (2004), derived from ROSAT HRI
($\sim 2\arcsec$ resolution) measurements of the inner $6\arcmin$ radius of the
Bullet cluster: $\beta = 1.04^{+0.16}_{-0.10}$, $\theta_c =
112.5^{+15.6}_{-10.4} \arcsec$, and $n_{e0} = 7.2^{+0.3}_{-0.3}
\times 10^{-3}$ cm$^{-3}$.
We construct an X-ray derived SZE surface brightness model from
the $\beta$ model electron surface density profile using
equation~(\ref{EQN:integratedpressure}). To account for $\beta$ model
uncertainties, we incorporate the values and
uncertainties for $\beta$, $\theta_c$, and $n_{e0}$ as independent Gaussian
priors. We then use the analysis method described in
\S\,\ref{subsec:fitting} to minimize $\chi^2$ on the difference
between the X-ray derived SZE model, convolved with the point source
transfer function, and the APEX-SZ non-source-masked data. The free
parameters in the fit are the three $\beta$ model parameters, the mass-weighted electron
temperature $T_{mg}$, and the relative map alignment in RA and DEC.
We find $T_{mg}=11.4 \pm 1.4\,$keV after marginalizing over the other
parameters in the fit and including the SZE
flux calibration uncertainty and the
effect of the APEX-SZ band center frequency uncertainty on the
relativistic SZE spectrum $f(x,T_{mg})$. The reduced $\chi^2$ of the
best fit model is $1.008$ with an associated PTE of $31.3\%$,
indicating that the spherical $\beta$ density model and the assumption
of isothermality produce an acceptable fit to the data.
Given the complex morphology of this merging system, the validity of
the spherical $\beta$ model is questionable. We therefore repeat the
determination of the mass-weighted temperature by directly comparing
X-ray measurements of the projected intracluster gas density with the
measured SZE signal in order to produce a less model-dependent
measurement of the mass-weighted temperature.
We make use of the publicly
available\footnote{http://flamingos.astro.ufl.edu/1e0657/public.html}
electron surface density map derived from Chandra X-ray satellite data presented
in \markcite{clowe2006}{Clowe} {et~al.} (2006).
Using the same analysis as above, and marginalizing over the relative
map alignment parameters, we find mass-weighted electron temperature
$T_{mg} = 10.8 \pm 0.9\,$keV. The fit to the data is again good, with a
reduced $\chi^2= 1.037$ and a PTE of 20.6\%. This is in excellent
agreement with the value $T_{mg}=11.4 \pm 1.4 \,$keV found from the above
$\beta$ model analysis.
Given the complex dynamics in the Bullet cluster, there have been several
studies of the temperature structure
\markcite{finoguenov2005, markevitch2006,andersson2007}(e.g., {Finoguenov}, {B{\"o}hringer}, \& {Zhang} 2005; {Markevitch} 2006; {Andersson}, {Peterson}, \& {Madejski} 2007).
There have also been several published results for the spectroscopic
temperature within annuli about the cluster center.
Chandra data was used by \markcite{govoni2004}{Govoni} {et~al.} (2004) to determine a spectroscopic
X-ray temperature of $T_{spec}=13.9 \pm 0.7\,$keV within $0.75\,$Mpc of the
cluster center.
From the analysis of XMM data within an annulus of $0.14-0.7\,$Mpc
radius, \markcite{zhang2006}{Zhang} {et~al.} (2006) find a temperature of $T_{spec}=10.6\pm 0.2\,$keV.
Analyzing the combination of XMM and RXTE data, \markcite{petrosian06}{Petrosian}, {Madejski}, \& {Luli} (2006), find
$T_{spec}=12.1\pm0.2\,$keV within a radius of $0.95\,$Mpc.
The published X-ray spectroscopic temperatures span a range much larger
than the stated uncertainties in the measurements.
Given the complex thermal structure for the cluster, and the presence of
gas at temperatures corresponding to energies at or above the upper limits
of the Chandra and XMM energy response, the variation in the measured
X-ray spectroscopic temperature is not surprising.
The mass-weighted temperature found with APEX-SZ falls near the lowest of the
reported X-ray spectroscopic temperatures.
However, we do not expect exact agreement between the mass-weighted and
spectroscopic temperatures.
Using Chandra data for a sample of 13 relaxed clusters, \markcite{vikhlinin2006}{Vikhlinin} {et~al.} (2006)
find that, due to the presence of thermal structure in the intracluster gas, the
X-ray spectroscopic temperature is typically a factor of
$T_{spec}/T_{mg}=1.11 \pm 0.06$
larger than the X-ray derived mass-weighted electron temperature.
This is consistent with the simulation results of \markcite{nagai07}{Nagai}, {Vikhlinin}, \& {Kravtsov} (2007) who find
$T_{spec}/T_{mg} \approx 1.14$ for relaxed clusters and
$T_{spec}/T_{mg} \approx 1.12$, with a somewhat larger scatter,
for unrelaxed systems.
Naively applying this correction to the published X-ray spectroscopic
temperatures, we infer results for mass-weighted temperatures that bracket the
APEX-SZ result.
\subsection{Gas Mass Fraction}\label{sec:gmass}
Using the SZE measurements, we construct a model for the
intracluster gas distribution which, when combined with X-ray
measurements, can be used to determine the gas mass, total mass, and
therefore gas mass fraction of the cluster. The gas mass is estimated by
integrating a spheroidal model for the cluster gas, following
\markcite{laroque2006}{LaRoque} {et~al.} (2006).
Several assumptions about the model must be made to estimate the gas
mass. We assume that the cluster gas is isothermal in order to convert
pressure to density. We also assume spheroidal symmetry for the gas
distribution in order to convert the two-dimensional SZE integrated
pressure measurement to a three-dimensional gas distribution.
We consider two simple cases, an oblate spheroid generated by rotation
about the minor axis and a prolate spheroid generated by rotation
about the major axis, where the symmetry axis is in the sky plane.
The gas mass, under these assumptions, becomes:
\begin{eqnarray}
\label{EQN:gasmass}
\lefteqn{M_{\rm{gas}}(r) =
8 \mu_e n_{e0} m_p {D_{A}}^3 \int_0^{r/D_{A}} dX \, dY \, dZ } \nonumber \\
& & \left(1+\left(\frac{X}{\theta_{\rm{c}}}\right)^2+ \right.
\left.\left(\frac{Y}{\eta\theta_{\rm{c}}}\right)^2 +
\left(\frac{Z}{\zeta\theta_{\rm{c}}}\right)^2 \right)^{-3\beta/2}
\end{eqnarray}
where $\mu_e$ is the nucleon/electron ratio, taken to be
1.16~\markcite{grego2001}({Grego} {et~al.} 2001). The factor of eight is due to integrating over
only one octant of the spheroid. The factor $\zeta$ is set to unity in
the case of oblate spheroidal symmetry, while in the case of prolate spheroidal
symmetry, $\zeta$ is set to $\eta$.
The total cluster mass is estimated by assuming hydrostatic
equilibrium and integrating the inferred gas distribution
\markcite{grego2000}({Grego} {et~al.} 2000) to find:
\begin{equation}
\label{EQN:hydroeq}
\rho_{\rm{total}} = -\frac{kT_{\rm{e}}}{4\pi G \mu m_{\rm{p}}} \nabla^2 \ln \rho_{\rm{gas}}.
\end{equation}
Here $\mu$ is the mean molecular weight of the intracluster gas,
which is assumed to be 0.62 \markcite{zhang2006}({Zhang} {et~al.} 2006). Using
equations~(\ref{EQN:gasmass}) \&~(\ref{EQN:hydroeq}), and our model parameters in
Table~\ref{TBL:betamodelfits}, we calculate the gas mass, total
mass, and gas mass fraction for the cluster.
In Table~\ref{TBL:massestimates}, we give these results.
The gas mass fraction results for a prolate gas
distribution model are $\sim3\%$ larger than those for
an oblate model, while the total mass and gas mass are $\lesssim18\%$ larger.
We quote only the oblate spheroidal results.
We calculate our results within two different radii. The first is
the radius of the cluster at which its mean density is equal to 2500
times the critical density at the redshift of the cluster,
$r_{2500}$. The second radius is 1.42~Mpc, which is the same radius
used by \markcite{zhang2006}{Zhang} {et~al.} (2006) for their gas mass fraction calculation.
This will allow for a more direct comparison to their result, and is
also near where our measured SZE radial profile has unity signal to
noise.
For all results, we assume a $\Lambda$CDM cosmology,
with $h = 0.7$, $\Omega_{\rm{m}} = 0.27$, and $\Omega_{\Lambda} =
0.73$.
The results of this analysis are summarized in Table~\ref{TBL:massestimates}.
Under the assumption of a $10.6\,$keV mass-weighted temperature (the lowest of the published
X-ray spectroscopic temperatures and near our mass-weighted temperature results
in \S\,\ref{subsec:mwte}), we find gas mass fractions
$f_g=0.216 \pm 0.031$ and $0.179 \pm 0.036$ within $r_{2500}$ and 1.42~Mpc, respectively.
The fact that the computed gas fraction in the central region significantly
exceeds the cosmic average determined by WMAP5 \markcite{dunkley09}($f_g=0.165\pm 0.009$, {Dunkley} {et~al.} 2009),
and a lower value observed in relaxed clusters
\markcite{vikhlinin09}($f_g \simeq 0.12$, see, e.g., {Vikhlinin} {et~al.} 2009), is likely due to deviations of the
intracluster gas from isothermal hydrostatic equilibrium.
On larger scales, baryon fractions produced for the range of reported
X-ray temperatures
bracket the published results using X-ray and weak lensing data.
\markcite{bradac2006}{Brada{\v c}} {et~al.} (2006) measure a gas
mass fraction $f_g=0.14 \pm 0.03$ by comparing the gas mass
calculated from Chandra X-ray measurements to weak lensing total
mass measurements in a $4.9\arcmin \times 3.2\arcmin$ box roughly
centered around the cluster. \markcite{zhang2006}{Zhang} {et~al.} (2006), measured a gas mass
fraction $f_g=0.161 \pm 0.018$ within a radius of 1.42~Mpc.
Despite the limitations of applying a hydrostatic equilibrium model
to this merging cluster, the APEX-SZ results for the
gas mass fraction are in good agreement with previous work.
\begin{deluxetable}{ccccccc}
\tabletypesize{\tiny}
\tablecaption{\label{TBL:massestimates} Mass Estimates for the Bullet Cluster System}
\tablewidth{0pt}
\tablehead{
\colhead{$T_e (keV)^a$} & \colhead{Mean Overdensity} & \colhead{$r_{\rm{int}}$ (\arcmin)$^b$} & \colhead{$r_{\rm{int}}$ (Mpc)$^c$} & \colhead{Gas Mass Fraction} & \colhead{Gas Mass ($10^{14} M_{\sun}$)} & \colhead{Total Mass ($10^{14} M_{\sun}$)}
}
\startdata
$13.9 \pm 0.7$ & $2506 \pm 233$ & 2.77 & 0.739 & $0.124 \pm 0.022$ & $0.944 \pm 0.105$ & $7.56 \pm 0.70$ \\
$13.9 \pm 0.7$ & $961 \pm 98$ & 5.32 & 1.42 & $0.106 \pm 0.024$ & $2.20 \pm 0.33$ & $20.6 \pm 2.1 $ \\
$10.6 \pm 0.2$ & $2521 \pm 230$ & 2.15 & 0.572 & $0.216 \pm 0.031$ & $0.765 \pm 0.072$ & $3.54 \pm 0.32$ \\
$10.6 \pm 0.2$ & $734 \pm 66$ & 5.32 & 1.42 & $0.179 \pm 0.036$ & $2.83 \pm 0.40$ & $15.7 \pm 1.4 $ \\
\enddata
\tablecomments{Two different isothermal electron temperatures are assumed, in order to bracket the range of X-ray spectroscopic temperatures
reported in the literature. The top two rows assume an isothermal electron temperature of $13.9 \pm 0.7$ keV. The bottom two rows assume an isothermal electron temperature of $10.6 \pm 0.2$ keV. For each electron temperature, we integrate to $r_{2500}$, the radius within which the mean cluster density is 2500
times greater than the critical density at the redshift of the cluster.
For each electron temperature, we also integrate to a fixed radius of 1.42~Mpc, allowing a direct comparison
to results in \markcite{zhang2006}{Zhang} {et~al.} (2006). This radius is also near where our
measured SZE radial profile has unity signal to
noise. For all results, we assume a $\Lambda$CDM cosmology,
with $h = 0.7$, $\Omega_{\rm{m}} = 0.27$, and $\Omega_{\Lambda} =
0.73$. Uncertainties in the gas mass and gas mass fraction include
a $\pm 5.5\%$ SZE flux calibration uncertainty.}
\tablenotetext{a}{Isothermal electron temperature}
\tablenotetext{b}{Angular integration radius.}
\tablenotetext{c}{Physical integration radius.}
\end{deluxetable}
\section{Conclusions}
\label{SEC:conclusions} Measurements of the SZE provide a robust and
independent probe of the intracluster gas properties in galaxy
clusters. The APEX-SZ $150\,$GHz observations detect the Bullet system
with $23\,\sigma$ significance within the central 1\arcmin\ radius of the
SZE centroid position.
We do not expect to see a resolved signal from the Bullet
sub-cluster in the $150\,$GHz 85\arcsec\ FWHM resolution SZE maps,
and no obvious feature, such as a secondary peak, is present.
We expect no significant contamination of the observed SZE decrement
due to radio sources, and there is no evidence for
significant contamination by a known bright lensed dusty galaxy.
We process an elliptical $\beta$ model through the observation
transfer function and fit it to the measured temperature decrement map.
We also measure the cluster mass-weighted
electron temperature and gas mass fraction with the SZE data.
Combining the APEX-SZ map with a map of projected electron surface density from
Chandra X-ray observations, we determine the mass-weighted temperature of the
cluster gas to be $T_{mg}=10.8 \pm 0.9\,$keV.
This value is consistent with the lowest X-ray spectroscopic temperatures reported
for this cluster and should be less sensitive to the details of the cluster
thermal structure.
The derived baryon fraction is also found to be in reasonable agreement with
previous X-ray and weak lensing determinations.
Throughout this work, we make an assumption of isothermal cluster gas.
Clearly, incorporating thermal structure, measured by X-ray
observations, in the analysis of the SZE data would improve the
determination of the gas distribution and gas mass fraction.
Ultimately, a more sophisticated analysis could be implemented that
combines X-ray, SZE, and weak lensing data and relaxes assumptions of
hydrostatic equilibrium between the gas and dark matter components of
the cluster. This is particularly important for a detailed
understanding of actively merging systems such as the Bullet cluster.
\acknowledgments
We thank the staff at the APEX telescope site, led by David
Rabanus and previously by Lars-\AA ke Nyman, for their dedicated and exceptional support. We also thank
LBNL engineers John Joseph and Chinh Vu for their work on the
readout electronics. APEX-SZ is funded by the National Science
Foundation under Grant Nos.\ AST-0138348 \& AST-0709497.
Work at LBNL is supported
by the Director, Office of Science, Office of High Energy and
Nuclear Physics, of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231. Work at McGill is supported by the Natural Sciences
and Engineering Research Council of Canada and the Canadian Institute
for Advanced Research. RK acknowledges partial financial support from
MPG Berkeley-Munich fund. NWH acknowledges support from an Alfred P. Sloan
Research Fellowship. |
0807.3391 | \section{Introduction}
Recently, Pierre Auger observatory published results on correlation
of the highest-energy cosmic rays with the positions of nearby
active galactic nuclei (AGN) \cite{Abraham:2007bb}. Such a
correlation is confirmed by the data of Yakutsk \cite{Ivanov:2008it}
while it is not found in the analysis by HiRes \cite{Abbasi:2008md}.
In the Auger result, the correlation is maximal for the threshold
energy of cosmic rays at $5.7\times 10^{19}$ eV, the maximal
distance of AGN at $71$ Mpc and the maximal angular separation of
cosmic ray events at $\psi=3.2^{\circ}$. Due to increasing efforts on
verifying the Auger result, it is worthwhile to examine the above
correlation from a phenomenological point of view.
Since the angular scale of the observed correlation is a few
degrees, one expects that these cosmic ray particles are
predominantly light nuclei. The effect of GZK attenuations on these
cosmic ray particles \cite{Greisen,ZK} can be described by a
distance scale referred to as ``GZK horizon". By definition, the GZK
horizon associated with a threshold energy $E_{\rm th}$ is the
radius of a spherical region which is centered at the Earth and
produce $90\%$ of UHECR events arriving on Earth with energies above
$E_{\rm th}$.
Assuming a uniform distribution of UHECR sources with identical
cosmic ray luminosity and spectral index \cite{Harari:2006uy}, the
GZK horizon for protons with $E_{\rm th}=57$ EeV is about $200$ Mpc
while the V-C catalog\cite{VC06} used by Pierre Auger for the
correlation study is complete only up to $100$ Mpc. Such a deviation
may arise from non-uniformities of spatial distribution, intrinsic
luminosity and spectral index of local AGN as mentioned in
\cite{Abraham:2007bb}. In addition, the energy calibration also
plays a crucial role since the GZK horizon is highly sensitive to
the threshold energy $E_{\rm th}$. Energy values corresponding to
the dip and the GZK cutoff of UHECR spectrum were used to calibrate
energy scales of different cosmic ray experiments
\cite{Berezinsky:2008qh,Kampert:2008pr}. It has been shown that all
measured UHECR energy spectra can be brought into good agreements by
suitably adjusting the energy scale of each experiment
\cite{Berezinsky:2008qh}. Furthermore, it has been shown that a
different shower energy reconstruction method infers a $30\%$ higher
UHECR energy than that determined by Auger's fluorescence
detector-based shower reconstruction \cite{Engel:2007cm}.
In this presentation, we report our results \cite{Lu:2008wj} on
examining the consistency between Auger's UHECR correlation study
and its spectrum measurement. The impact by the local over-density
of UHECR sources is studied. We also study the energy calibration
effect on the estimation of GZK horizon and the spectrum of UHECR.
Certainly a $20\%-30\%$ upward shift on UHECR energies reduces the
departure of theoretically calculated GZK horizon to the maximum
valid distance of V-C catalog \cite{Abraham:2007bb}. The further
implications of this shift will be studied in fittings to the
shifted Auger spectrum.
\section{GZK horizons and the UHECR spectrum}
\begin{table}[hbt]
\begin{center}
\caption{GZK horizons of UHECR calculated with the local over-density $n(l<30 {\rm
Mpc})/n_0=1, \ 2, \ 4,$ and $10$, and arrival threshold energy
$E_{\rm th}=57$ EeV, $70$ EeV, $80$ EeV and $90$ EeV respectively. The listed numbers are in
units of Mpc.}
\begin{tabular}{|c|c|c|c|c|}\hline
$n(l<30 {\rm
Mpc})/n_0$ & $E_{\rm th}=57 \, {\rm EeV}$ & $E_{\rm th}=70 \, {\rm
EeV}$ & $E_{\rm th}=80 \, {\rm EeV}$ & $E_{\rm th}=90 \, {\rm EeV}$
\\ \hline $1$ & $220$ & $150$ & $115$ & $90$\\
\hline $2$ & $210$ & $140$ & $105$ & $75$\\
\hline $4$ & $195$ & $120$ & $85$ & $60$\\
\hline $10$ & $155$ & $85$ & $50$ & $30$\\
\hline
\end{tabular}
\end{center}
\label{horizon_p3}
\end{table}
GZK horizons corresponding to different local over-densities and
$E_{\rm th}$ are summarized in Table I.
Within the same $E_{\rm th}$, local over-densities up to $n(l<30
{\rm Mpc})/n_0=4$ do not significantly alter GZK horizons. One could
consider possibilities for higher local over-densities. However,
there are no evidences for such over-densities either from
astronomical observations \cite{overdensity} or from fittings to the
measured UHECR spectrum. We note that GZK horizons are rather
sensitive to $E_{\rm th}$. Table I shows that GZK horizons are $\sim
100$ Mpc or less for $E_{\rm th}\geq 80$ EeV.
Fittings to the Auger spectrum have been performed in
\cite{spec_fit}. In our work, we take into account the over-density
of UHECR sources in the distance scale $l\leq 30$ Mpc. The local
over-density of UHECR sources affects the cosmic-ray spectrum at the
highest energy, especially at energies higher than $5\cdot 10^{19}$
eV. Hence the degree of local over-density can be examined through
fittings to the measured UHECR spectrum.
\begin{table}[hbt]
\begin{center}
\caption{The values of total $\chi^2$ from fittings to the Auger
measured UHECR spectrum. Numbers in the parenthesis are $\chi^2$
values from fittings to the $8$ data points in the energy range
$19.05\leq \log_{10}(E/{\rm eV}) \leq 19.75$. The last $4$ data
points record events with energy greater than $71$ EeV. }
\begin{tabular}{|c|c|c|c|c|}\hline
$n(l<30 {\rm
Mpc})/n_0$ & $1$ & $2$ & $4$ & $10$
\\ \hline $\gamma=2.5$ & $14.12 (9.34)$ & $14.61 (9.93)$ & $17.09 (10.50)$ & $28.09 (13.93)$\\
\hline $\gamma=2.6$ & $16.64 (12.28)$ & $15.56 (11.90)$ & $16.01 (11.83)$ & $20.76 (11.67)$\\
\hline
\end{tabular}
\end{center}
\label{chi_square}
\end{table}
\begin{table}[hbt]
\begin{center}
\caption{The total $\chi^2$ values from fittings to the Auger
measured UHECR spectrum with a $30\%$ upward shift on UHECR
energies. Numbers in the parenthesis are $\chi^2$ values from
fittings to the $8$ data points in the energy range $19.16\leq
\log_{10}(E/{\rm eV}) \leq 19.86$. The last $4$ data points record
events with energy greater than $92$ EeV. }
\begin{tabular}{|c|c|c|c|c|}\hline
$n(l<30 {\rm
Mpc})/n_0$ & $1$ & $2$ & $4$ & $10$
\\ \hline $\gamma=2.4$ & $8.65 (4.30)$ & $7.39 (4.67)$ & $10.26 (6.35)$ & $27.31 (13.34)$\\
\hline $\gamma=2.5$ & $ 11.82 (6.16)$ & $8.67 (5.49)$ & $7.78 (5.23)$ & $16.18 (7.39)$\\
\hline
\end{tabular}
\end{center}
\label{chi_square2}
\end{table}
\begin{figure}[hbt]
\begin{center}
$\begin{array}{cc}
\includegraphics*[width=7.0cm]{unshift.eps} &
\includegraphics*[width=7.0cm]{shift.eps}
\end{array}$
\end{center}
\caption{Left and right panels depict fittings to the original and
shifted Auger UHECR spectra respectively with the red, green, blue
and black curves denote models with local over-densities $n(l<30
{\rm Mpc})/n_0=1, \ 2, \ 4,$ and $10$ respectively. Solid and dash
curves correspond to $\gamma=2.6$ and $\gamma=2.5$ respectively. We
take the source evolution parameter $m=3$ throughout the
calculations. } \label{fitting}
\end{figure}
The left paenl in Fig.~\ref{fitting} shows our fittings to the Auger
measured UHECR spectrum with $\gamma=2.5$ and $2.6$ respectively. We
take the red-shift dependence of the source density as
$n(z)=n_0(1+z)^m$ with $m=3$. We have fitted $12$ Auger data points
beginning at the energy $10^{19}$ eV. We make a flux normalization
at $10^{19}$ eV while varying the power index $\gamma$ and the the
degree of local over-density, $n(l<30 {\rm Mpc})/n_0$. Part of
$\chi^2$ values from our fittings are summarized in Table II. We
found that $\gamma=2.5, \ n(l<30 {\rm Mpc})/n_0=1$ gives the
smallest $\chi^2$ value with $\chi^2/{\rm d.o.f.}=1.57$. For the
same $\gamma$, $n(l<30 {\rm Mpc})/n_0=10$ is ruled out at the
significance level $\alpha=0.001$. For $\gamma=2.6$, $n(l<30 {\rm
Mpc})/n_0=10$ is ruled out at the significance level $\alpha=0.02$.
We note that, for both $\gamma=2.5$ and $\gamma=2.6$, the GZK
horizon with $n(l<30 {\rm Mpc})/n_0=10$, $E_{\rm th}=57$ EeV, $m=3$
and $E_{\rm cut}=1000$ EeV is about $155$ Mpc. Since $n(l<30 {\rm
Mpc})/n_0=10$ is clearly disfavored by the spectrum fitting, one
expects a GZK horizon significantly larger than $155$ Mpc for
$E_{\rm th}=57$ EeV.
We next perform fittings to the shifted Auger spectrum. The results
are shown in the right panel in Fig.~\ref{fitting} where the cosmic ray energy is shifted
upward by $30\%$. Part of $\chi^2$ values are summarized in Table
III. The smallest $\chi^2$ value occurs approximately at
$\gamma=2.4$, $n(l<30 {\rm Mpc})/n_0=2$ with $\chi^2/{\rm d.o.f}=
0.82$.
One can see that $\chi^2$ values from current fittings are
considerably smaller than those from fittings to the unshifted
spectrum. Given a significance level $\alpha=0.1$, it is seen that
every local over-density listed in Table III except $n(l<30 {\rm
Mpc})/n_0=10$ is consistent with the measured UHECR spectrum. It is
intriguing to test such local over-densities as will be discussed in
the next section. We note that, with a $30\%$ upward shift of
energies, the cosmic ray events analyzed in Auger's correlation
study would have energies higher than $74$ EeV instead of $57$ EeV.
The GZK horizon corresponding to $E_{\rm th}=74$ EeV is $120$ Mpc
for $n(l<30 {\rm Mpc})/n_0=2$ and $105$ Mpc for $n(l<30 {\rm
Mpc})/n_0=4$.
We have so far confined our discussions at $m=3$. In the literature,
$m$ has been taken as any number between $0$ and $5$. It is
demonstrated that the effect on UHECR spectrum caused by varying $m$
can be compensated by suitably adjusting the power index $\gamma$
\cite{DeMarco:2005ia}. Since GZK horizons are not sensitive to
$\gamma$ and $m$, results from the above analysis also hold for
other $m$'s.
\section{Discussions and conclusions}
We have discussed the effect of local over-density of UHECR sources
on shortening the GZK horizon. The result is summarized in Table I.
It is seen that such an effect is far from sufficient to shorten the
GZK horizon at $E_{\rm th}=57$ EeV to $\sim 100$ Mpc for a local
over-density consistent with the measured UHECR spectrum. With a
$30\%$ energy shift, each cosmic ray event in Auger's correlation
study would have an energy above $74$ EeV instead of $57$ EeV. GZK
horizons corresponding to $E_{\rm th}=74$ EeV then match well with
the maximum valid distance of V-C catalog. Fittings to the shifted
Auger spectrum indicate a possibility for the local over-density of
UHECR sources.
We point out that the local over-density of UHECR sources is
testable in the future cosmic ray astronomy where directions and
distances of UHECR sources can be determined. Table IV shows
percentages of cosmic ray events that come from sources within $30$
Mpc for different values of $E_{\rm th}$ and $n(l<30 {\rm
Mpc})/n_0$. Although these percentages are calculated with
$\gamma=2.4$, $m=3$ and $E_{\rm cut}=1000$ EeV, they are however not
sensitive to these parameters.
\begin{table}[hbt]
\begin{center}
\caption{ Percentages of cosmic ray events originated from sources
within 30 Mpc for different values of $E_{\rm th}$ and $n(l<30 {\rm
Mpc})/n_0$.}
\begin{tabular}{|c|c|c|c|c|}\hline
$n(l<30 {\rm
Mpc})/n_0$ & $E_{\rm th}=57 \, {\rm EeV}$ & $E_{\rm th}=70 \, {\rm
EeV}$ & $E_{\rm th}=80 \, {\rm EeV}$ & $E_{\rm th}=90 \, {\rm EeV}$
\\ \hline $1$ & $0.17$ & $0.27$ & $0.36$ & $0.46$\\
\hline $2$ & $0.30$ & $0.43$ & $0.53$ & $0.63$\\
\hline $4$ & $0.46$ & $0.60$ & $0.70$ & $0.77$\\
\hline $10$ & $0.68$ & $0.79$ & $0.85$ & $0.89$\\
\hline
\end{tabular}
\end{center}
\label{event_dist}
\end{table}
For $E_{\rm th}=57$ EeV and $n(l<30 {\rm Mpc})/n_0=1$, only $17\%$
of cosmic ray events come from sources within $30$ Mpc. For $n(l<30
{\rm Mpc})/n_0=2$ and the same $E_{\rm th}$, $30\%$ of cosmic ray
events are originated from sources in the same region.
In conclusion, we have shown that the deviation of theoretically
calculated GZK horizon to the maximum valid distance of V-C catalog
can not be resolved by merely introducing the local over-density of
UHECR sources. On the other hand, if Auger's energy calibration
indeed underestimates the UHECR energy, such a discrepancy can be
reduced. More importantly, fittings to the shifted Auger spectrum
indicate a possible local over-density of UHECR sources, which is
testable in the future cosmic ray astronomy.
\noindent{\bf Acknowledgements}
We thank A. Huang and K. Reil for helpful discussions. We also thank
F.-Y. Chang, T.-C. Liu and Y.-S. Yeh for assistances in computing.
This work is supported by National Science Council of Taiwan under
the grant number 96-2112-M-009-023-MY3. |
0807.3352 | \section{Introduction}
Rare dark matter halos at high redshifts host interesting
astrophysical objects, especially before or at the end of the
reionization epoch. One example is given by the very first Population~III (PopIII)
stars formed in the universe at $z \gtrsim 40$, which started the metal
enrichment of the interstellar medium and the reionization process
\citep{abel02,san02,bromm04,naoz06}, and possibly also produced
intermediate mass black hole seeds that grow to become
super-massive black holes ($M_{BH}> 10^9 M_{\sun}$) within the first
billion year after the Big Bang \citep{vol03}. Another example are
bright $z \approx 6$ QSOs \citep{fan04}, considered to be hosted in the
most massive dark matter halos at that time \citep{MR05}. The
luminosity of such an object is powered through accretion onto a
supermassive black hole \citep[e.g. see][]{hop06}, which may be the
descendant of one of the first generation of PopIII stars formed in
the universe, at $z \gtrsim 40$ (\citealt{mad01}; see also \citealt{ts07a}).
Many numerical and theoretical investigations have aimed at
characterizing the properties of both PopIII stars
\citep[e.g. see][]{abel02,bromm04,ciar,ree05,gao05,osh07} and of high
redshift QSOs \citep[e.g. see][]{hop06,MR05,dimatteo05,li07}. However,
the rarity of these structures makes a fully self-consistent treatment
even of the formation of the underlying dark matter halos
typically outside the current capabilities of a
standard cosmological simulation, as the dynamic range that needs to
be resolved is too large. The main limitation at high mass resolution is the box size,
$l_{box}$: The enforcement of periodic boundary conditions
limits the power spectrum of density fluctuations to modes with
wavelengths shorter than $l_{box}$. This results in a severe bias on
the measured number density of rare, massive halos, especially
before the epoch of the reionization of the universe when the abundance
of galaxies derived from numerical simulations can be underestimated
by up to an order of magnitude \citep{bar04}. While the number density
of rare objects can be estimated in the context of
Press-Schechter-type modeling \citep{lacey93,mo_white96,she99,bar04},
studying the details of the formation histories of the first galaxies
and QSOs, possibly including hydrodynamics and radiative feedback
processes, requires high-resolution numerical simulation.
Thus, given a simulation box that is large enough to contain one or
more of the high-redshift halos of interest, the problem is to
identify a sub-region that contains one of them for high-resolution
simulation. In passing we note that this challenge is different from
that of developing a simulation code that is able to adaptively
increase the resolution when following the gas collapse that leads to
the formation of the first stars. This has been successfully
implemented with adaptive mesh refinement codes (for example ENZO -
\citealt{bry98}). \citet{gao05} proposed a method to identify rare
structures formed at very high redshift by recursively resimulating
successively smaller, nested, sub-regions of a simulation box at
progressively higher resolutions. Specifically, their method
resimulates the region of the box centered on the most massive dark
matter halo, reidentified at higher redshift and lower mass within the
sub-region at each resimulation step, until the desired resolution is
achieved in a small fraction of the entire simulation box. This
procedure is well-motivated by the fact that number density of massive
halos is increased in regions of large-scale over-density
\citep{bar04}, and thus structure formation in the sub-region on small
scales at early times is accelerated by sitting at the top of an
over-density, an over-density that is known to exist because it collapsed
into the massive halo identified in the previous resimulation
step. The \citet{gao05} method extends high-resolution resimulation
techniques used previously \citep[e.g. see ][]{nav94,whi00}, and has
been used recently by several authors to study the first stars and
QSOs \citep{gao05,ree05,gao07,li07}.
The recursive resimulation method for volume selection is very
effective at identifying a region with a very high-redshift halo when
compared to random selection of a region of equal volume within the
parent simulation box, but still it does not lead to the rarest
high-redshift halos in the volume. This is due to the stochastic
nature of the formation and growth of dark matter halos
\citep[e.g. see][]{PS,bond,she99}. The most massive dark matter halo
in a given volume at some early time does not in general evolve into
the most massive halo at a later time. The probability of the most
massive halo to evolve into the most massive halo, quantified
following \citet{lacey93}, decreases as time passes and the
characteristic mass scale increases. As we demonstrate, the
probability falls far below unity for the evolution of the first stars
and QSOs to the present time. The stochastic nature of growth
histories of dark matter halos is addressed and discussed in
\citet{gao05}, but unfortunately some of the subsequent studies using
their method do not take into account this stochasticity and assume a
very strong correlation between the location of the richest clusters
at $z=0$ and that of rare halos at very high redshift.
In this paper we clarify this issue by presenting some basic, but
often overlooked, results of Gaussian random fields in extended
Press-Schechter theory to quantify the locations at later times of the
first dark matter halos to form in a simulation box. We compare these
analytical and numerical results to those published using recursive
resimulation and highlight the benefits and limitations of that
method. In addition, we propose an alternative method, based on
analysis of the density field in the initial conditions, to select the
sub-region of a simulation box that contains some of the earliest
structure in the box without being biased toward the regions hosting
the largest halos at $z=0$. The paper is organized as follows. In
Sec.~\ref{sec:QSOs} we investigate where the descendants of $z=6$ QSOs
are today, while in Sec.~\ref{sec:PopIII} we extend the result to the
first Population III stars in the Universe. In Sec.~\ref{sec:Ic} we
discuss the implications of our results for the recursive mesh
refinement method and propose a viable
alternative. Sec.~\ref{sec:conc} summarizes and concludes.
\section{Bright $z\approx 6$ QSOs: where are they today?} \label{sec:QSOs}
As a first step toward characterizing the assembly history of rare,
massive, dark matter halos, we consider the link between $z \approx 6$
QSOs (considered to be hosted in the largest protoclusters at that
time, e.g. see \citealt{MR05}) and the largest clusters at $z=0$. For
this we use the publicly available merger trees constructed upon the
Millennium Run \citep{MR05,lemson}. This cosmological simulation has
more than $10^{10}$ dark matter particles within a volume of
$500^3~h^{-3}\mathrm{Mpc}^3$ and has been run using a concordance
$\Lambda CDM$ cosmology based on WMAP year 1 results
\citep{WMAP1}\footnote{Note that a change in the cosmological
parameters, and in particular on the value of $\sigma_8$, the root
mean squared mass fluctuation in a sphere of radius
$8~h^{-1}\mathrm{Mpc}$ extrapolated to $z=0$ using linear theory, does
influence the details of our results, such as the expected mass of a
QSO host halo, but not the essence of the relation between dark matter
halos at different redshifts.}.
From the Millennium Catalog\footnote{http://www.mpa-garching.mpg.de/millennium} we select all dark matter halos with more
than $10^3$ particles (corresponding to a minimum halo mass of $8.6
\cdot 10^{11}~h^{-1} \mathrm{M_{\sun}}$) in the $z=0$ snapshot and all
halos with more than $100$ particles in the $z=6.18$ snapshot. Within
these catalogs we also identify the descendants at $z=0$ of the
$z=6.18$ halos and plot them in Fig.~\ref{fig:MR}. The figure
immediately shows that the descendants of the most massive halos at
$z>6$ live in a variety of environments at $z=0$. Some of the most
massive halos at $z=6.18$ have accreted relatively little mass by
$z=0$ (a factor of a few of their initial mass), while others have
increased their masses by more than hundred times, reaching
$M>10^{15}~h^{-1} \mathrm{M_{\sun}}$. The scatter is substantial. In
particular the descendant of the most massive $z=6.18$ halo in this
simulation has a mass of only $2.2 \cdot 10^{14}~h^{-1}
\mathrm{M_{\sun}}$ at $z=0$, to be compared with the largest cluster
in the box, which has $M=3 \cdot 10^{15}~h^{-1}
\mathrm{M_{\sun}}$. Fig.~\ref{fig:MR} also quantifies the relation
between progenitor and descendant halos in terms of the dimensionless
variable $\nu = \delta_{c}^2 / \sigma^2(M)$ used in extended Press-Schechter
modeling. Here $\sigma^2(M)$ is the variance of density
fluctuations over a scale that contains a mass $M$ and $\delta_c(z)$
is the critical value of an overdensity in linear theory at redshift
$z$.
A complementary picture is given by considering the mass distribution
of the most massive $z=6.18$ progenitor for the $10000$ largest $z=0$
halos, that is with a mass $M \gtrsim 4 \cdot 10^{13} h^{-1}
\mathrm{M_{\sun}}$ (Fig.~\ref{fig:MR1}). Again a considerable scatter
is present in the plot, with only a modest correlation between
descendant and progenitor mass. These results reflect the fact that
there are additional contributions to the density fluctuation power
spectrum over a scale $M_1$ compared to a scale $M_2>M_1$. This can be
also illustrated by random walks generated using a $\Lambda CDM$ power
spectrum (see Fig.~\ref{fig:random_walk}). The typical fluctuations at
small scales greatly exceed those at large scale, so that a walk with
excess power at small scales often has a rather typical amplitude
at large scales, and vice versa.
A quantitative model for this behavior is available in the context of
extended Press-Schechter theory, for example by Eqs. 2.15 and 2.16 in
\citet{lacey93}. Using this formalism one can compute the probability
that a dark matter halo of mass $M_1$ at redshift $z_1$ evolves into a
halo of mass $M>M_2$ at $z_2<z_1$ [$P(M>M_2,z_2|M_1,z_1)$]. This is
shown in Fig.~\ref{fig:EPS}, where we plot the contour lines of $P$
for a $M_1=5 \cdot 10^{12}~h^{-1} \mathrm{M_{\sun}}$ halo at
$z_1=6.18$. The median of the distribution at $z=0$ is $M=4 \cdot
10^{14}~h^{-1} \mathrm{M_{\sun}}$ and at the 68\% confidence level
interval is $M \in [1.8:8.8] \cdot 10^{14}~h^{-1}
\mathrm{M_{\sun}}$. In this respect the Millennium Run is typical as
it lies within the $1 \sigma$ interval. From Fig.~\ref{fig:EPS} one
can also immediately see that it is relatively improbable for the
descendant of the most massive halo at $z=6.18$ to be the most massive
halo at $z=0$. In fact there is only a probability $p<2\%$ that the
mass of the descendant halo is above $2 \cdot 10^{15}~h^{-1}
\mathrm{M_{\sun}}$. The Millennium Run has a volume large enough to
contain $z=0$ halos more massive than $2 \cdot 10^{15}~h^{-1}
\mathrm{M_{\sun}}$ (in fact there are two of them), thus it is
expected that in less than $2\%$ of Millennium-like realizations the
most massive $z \approx 6$ halo is the progenitor of the most massive
$z=0$ halo. As a reference we provide the number density contours from
the \citet{she99} mass function, integrated to obtain the number of
objects above that mass at that redshift in the volume of the
Millennium Run (red dotted lines in Fig.~\ref{fig:EPS}): when $z
\lesssim 2.5$, the upper 2$\sigma$ confidence level contour lies at a
lower mass than the contour for the number density ($n=1$) associated
with the most massive dark matter halo in the volume. This mean that
the typical mass of the descendants progressively shifts toward that
of more common halos as the redshift decreases, as it is also
immediately visible when looking at the redshift evolution of the
value of $\nu$ for the median descendant, which decreases
progressively (small panel of fig.~\ref{fig:EPS}).
\section{The very first PopIII stars: where are they today?} \label{sec:PopIII}
A similar scenario holds for the relation between the location of the
very first PopIII stars, formed in dark matter halos with $M \approx
10^6 \mathrm{M_{\sun}}$ at $z>40$, and that of dark matter halos at
$z=0$. Here there is even less correlation than in the case of $z=6$
QSO halos (see Sec.~\ref{sec:QSOs}) because the halo mass dynamic
range involved is larger and thus there is an additional contribution
to the scatter from modes at small scales in the density fluctuation
power spectrum. Using a simulation volume of $\approx 1~\mathrm{Gpc}^3$,
\citet{ts07a} showed that the remnants of the very first PopIII stars
formed in dark matter halos of mass $10^6~h^{-1} \mathrm{M_{\sun}}$ at
$z>40$ end up at $z=0$ in dark matter halos with a median mass of $3
\cdot 10^{13}~h^{-1} \mathrm{M_{\sun}}$, about two order of magnitude
smaller than the largest halo in the simulation box. Here we confirm
and extend their result using extended Press-Schechter modeling.
Using Eq. 2.15 and 2.16 of \citet{lacey93}, we compute at different
redshifts the probability distribution for the mass of the descendant
of a $10^6~h^{-1} \mathrm{M_{\sun}}$ dark matter halo formed at
$z_1=40$. The results are shown in Fig.~\ref{fig:EPS_PopIII}, where we
plot the median descendant mass versus the redshift $z_2$ (black
line), the $1 \sigma$ confidence level contours (blue lines) and $2
\sigma$ confidence level contours (green lines). These analytic
results agree well with the numerical simulations in \citet{ts07a} and
confirm that at $z=0$ the typical remnant of one of the very first
PopIII halos does not live in a supercluster, but is rather hiding in
a more common group environment. This is, e.g., contrary to the
conclusions ( ``The very oldest stars should be found today in the
central regions of rich galaxy clusters'') that \citet{whi00} drew
from the resimulation of a massive galaxy cluster. Similarly, at $z=6$
the remnant of a typical $z \geq 40$ PopIII star does not live in the
largest dark matter halos of that time, but rather has seeded a dark
matter halo of mass $3-4 \cdot 10^{10}~h^{-1} \mathrm{M_{\sun}}$,
typical for the faint $z\approx 6$ galaxies observed in deep surveys
such as the Hubble Space Telescope Ultra Deep Field \citep{ts08}.
These results have been obtained for very rare PopIII stars formed at
$z>40$. Extended Press-Schechter modeling predicts that the remnants
of more common PopIII stars formed at lower redshift live in
yet lower mass dark matter halos. For example a $10^6~h^{-1} \mathrm{M_{\sun}}$ dark matter halo
formed at $z=20$ has a $z=0$ descendant with median mass $4 \cdot
10^{12}~h^{-1} \mathrm{M_{\sun}}$. More massive, rarer, PopIII halos, with virial
temperature above $10^4$ K, can cool by neutral hydrogen rather than
molecular hydrogen so that their formation is less sensitive to
radiative feedback from other stars \citep[e.g. see][]{bromm04}. These
halos have a typical mass of $\approx 10^{8}~h^{-1} \mathrm{M_{\sun}}$ and if
they form at $z=20$, then their typical descendants are similar to
those of a $z=40$ $10^6 h^{-1} \mathrm{M_{\sun}}$ dark halo (see
Fig.~\ref{fig:EPS_PopIII}).
\section{Generation of Initial Conditions around Rare High-Redshift Halos}\label{sec:Ic}
Given the large scatter in the assembly histories of dark matter halos
is there an optimal way to select a region of a simulation box
centered around a rare high-redshift halo under the constraint of
limited computational resources?
\subsection{Analysis of the recursive resimulation method}
The method introduced by \citet{gao05} certainly presents a very
competitive advantage over a random selection of an equal subvolume
though it identifies a halo with an atypical
accretion history, biased toward having an above average merging rate
and living in an environment that tends to have an overdensity of
nearby halos \citep{bar04}.
To better quantify the properties of halos identified by recursive
resimulation, we use the Monte Carlo method presented in
\citet{ts07a}, which is based on the identification of virialized dark
matter halos as density peaks in the linear density field. This
approach predicts well and without introducing systematic biases the
location and virialization redshift of the very first dark matter
halos when compared to the full non-linear dynamics of the simulation,
even though there are some statistical fluctuations on a halo-to-halo
basis \citep{bond96a,mes07}. For the comparison with \citet{gao05} we
adopt the following parameters from their paper: (i) $\Omega_M=0.3$,
$\Omega_{\lambda}=0.7$, $\Omega_b = 0.04$, $h=0.7$, $\sigma_8=0.9$,
spectral index $n_s=1$; (ii) parent box size edge $l_{box}=479$ Mpc
$h^{-1}$; (iii) largest dark matter halo in the box at $z=0$ of mass
$M_1=8.1 \cdot 10^{14} h^{-1} \mathrm{M_{\sun}}$; (iv) final
high-resolution simulation sphere of $1.25 h^{-1}$~Mpc (with volume
$V_5 = 8.18 h^{-3}$~Mpc$^3$); (v) the most massive halo in the final
resimulation region R5 is $M_{halo_{R5}} = 1.2 \cdot 10^5 h^{-1}
\mathrm{M_{\sun}}$ at $z=48.84$. Based on these assumptions, our
analysis yields the following results:
\begin{itemize}
\item The number density of dark matter halos of mass $M \geq
M_{halo_{R5}}$ is $n(M>M_{halo_{R5}}) = 0.17 h^3 $Mpc$^{-3}$
(Sheth-Tormen mass function) or $n(M>M_{halo_{R5}}) = 0.0025 h^3
$Mpc$^{-3}$ (Press-Schechter mass function) at $z=48.84$, so these
halos are not particularly rare. Specifically, in a random region of
volume $V_5$ at $z=48.84$, the expectation value for the number of
halos more massive than $M_{halo_{R5}}$ is greater than unity ($n(M>
M_{halo_{R5}}) \cdot V_5 = 1.39$) when using the \citet{she99} formula.
However, due to the high bias of the halos, $b \approx 16$, the actual
fraction of empty volumes as given by our Monte Carlo code is
$98.7\%$. Thus the \citet{gao05} method is a substantial improvement
over random selection of a volume $V_5$ within the parent box.
\item The first halo of mass $M_{halo_{R5}} = 1.2 \cdot 10^5 h^{-1}
\mathrm{ M_{\sun}}$ in the \emph{whole} box virializes at $z>62$ at
99\% of confidence level. The possibility that the most massive halo
identified by \citet{gao05} in their final resimulated volume is the
most massive of the whole box at that redshift is ruled out at a
confidence level greater than $1-10^{-7}$ (confidence level limited by
the precision of the numerical integration).
\item The first halo of mass $M_{halo_{R5}}$ that ends up at $z=0$ in
a dark matter halo of mass $M_1=8.1 \cdot 10^{14} h^{-1} \mathrm{
M_{\sun}}$ is formed at $z>50.8$ at 99.9\% of confidence level. Thus
it is unlikely that the \citet{gao05} halo of mass $M_{halo_{R5}}$ at
$z=48.84$ is the most massive progenitor of the cluster that will
contain it at $z=0$.
\end{itemize}
From this analysis it is indeed confirmed that recursive refinement
works well to select a sub-region for resimulation that host a rare
high-z dark matter halo, but still this halo is not one of the very
first of its kind in the box. If the interest is primarily in the
local physics around one of these rare structures, a viable
alternative is the constrained realization method
\citep{hf91,bert01}. A constrained realization does not require a
hierarchy of resimulations, but rather introduces the overdensity in
the initial conditions by construction, so that the realized initial
conditions do not carry information about its rarity or its typical
surrounding environment.
\subsection{Unbiased selection of rare halos: the density field method}
If one is instead interested in selecting one of the very first dark
matter halos in a given simulation volume, with a particular interest
in representative halos, that is halos with unbiased accretion
histories, we propose instead to select the final region of interest
from a density field at uniform resolution. When the dynamic range of
the resimulation is not too large, it is of course possible to
identify the region to be refined directly from the low-resolution
dark matter halo catalogs. If this is not possible, then the
resimulation volume can be identified from a high resolution, but
uniform, refinement of the density field. The idea is the following:
\begin{enumerate}
\item A high-resolution density field is generated over the whole box.
The mass of a field cell is that of the high-$z$ dark
matter halo of interest (e.g. $10^6 h^{-1} \mathrm{ M_{\sun}} $ for a PopIII
halo). A technical, but important, detail is that a top-hat
filter in frequency space must be used when generating this field, as this
guarantees that the mass function obtained from peak analysis matches
that of \citet{PS}.
\item The highest peak in the high-resolution field is
identified. This is the location where one of the first halos at the
mass scale of the density field is formed within the \emph{whole} simulation
box.
\item A region around that peak can now be selected for resimulation,
with additional fluctuations added over the high resolution density
field, for example by using the GRAFIC2 refinement method of
\citet{bert01}. Outside the selected region the high resolution field
can be degraded if necessary to generate more massive particles.
\end{enumerate}
This method does not necessarily require overwhelming computational
resources. For example, to select the largest $z=6$ dark matter halo
in a volume relevant for the study of bright high-$z$ QSOs, a mass
resolution of $\approx 5 \times 10^{11} h^{-1} \mathrm{ M_{\sun}} $ is
more than sufficient, which translates into a $N = 512^3$ density
field grid for a cosmic volume of $\approx (1 \mathrm{Gpc}/h)^3$. Such
a grid only requires 512MB of RAM for storing. Even identifying the
positions of the first PopIII stars in a large volume is not
unrealistic. A $N=2048^3$ density grid requires only 32GB of RAM and
provides a mass resolution of $1.2 \times 10^5 h^{-1}
\mathrm{ M_{\sun}} \equiv M_{halo_{R5}}$ (the first halo in
\citealt{gao05}) over a volume of $(23.74 h^{-1}
\mathrm{Mpc})^3$. Given the number density of such halos---$n \approx
0.11 h^3 \mathrm{Mpc}^{-3}$---about 1471 will have formed on average
in the box by $z=48.84$. In fact, using the Monte Carlo code of
\citet{ts07a} we predict that there is a probability $p< 10^{-3}$ of not
forming one of these halos before $z=48.84$. The median redshift of
formation for the first halo of mass $M_{halo_{R5}}$ is $z=51.6$. With
larger resources one can generate a $4096^3$ density grid over a
volume of $(100 \mathrm{Mpc}/h)^3$, obtaining a mass resolution of $10^6 h^{-1}
\mathrm{ M_{\sun}} $. This requires about 256GB of RAM, easily
available on a parallel computing cluster of moderate size. With a
top-end supercomputer one can use more than $10^{12}$ cells, with a
dynamic range that permits identification of smaller mass halos in the
same volume or an increase the volume of the parent box for the same
assumed PopIII halo mass.
Our proposed density field method has also another advantage: As we
generate a uniform, very high-resolution density grid over the whole
box, aliasing errors in the zoom refinement procedure are less severe
than if one follows the \citet{gao05} recursive refinement method (see
\citealt{bert01} for a detailed analysis of the errors introduced when
large zooming factors are used).
\subsection{Preliminary testing of the density field method}\label{sec:valid}
Detailed analysis and applications of our method to select
high-redshift halos is deferred to a follow-up paper. Here we present
a preliminary testing to assess its performance. For this we consider
a $N=512^3$ cosmological simulation in a box of edge $l_{box}=512
h^{-1} \mathrm{Mpc}$ (details on the simulation are discussed in
\citealt{ts07a}). We construct the halo catalog for a snapshot at
$z=5$ using the HOP halo finder \citep{eis98}, finding that the most
massive halo has a mass of $1.25 \cdot 10^{13} h^{-1}
\mathrm{M_{\sun}}$ ($187$ particles). The density field used to
generate the initial conditions for this $N=512^3$ run is then (i)
downgraded to a $N_{down}=128^3$ grid (where one low resolution cell
corresponds to 64 of the original grid cells) using the \citet{hf91}
constrained realization method and (ii) evolved in linear theory to
$z=5$. The highest density peaks in the low-resolution field are then
identified and their location compared with that of the dark matter
halos identified from the N-body simulation at full resolution. Within
the ten highest peaks (with linear overdensities from $\delta =1.84$
to $\delta = 1.75$), six of them are associated with dark matter halos
with more than $64$ particles, including all the top three
overdensities. The top overdensity is matched to the fourth most
massive halo of the snapshot, with $141$ particles (versus the $187$ of
the most massive). Of the remaining unmatched peaks, two are
associated with halos with less than 64 particles and two appear to be
still in the process of virialization, so the central density of their
groups in the N-body run does not qualify them as halos. The most
massive halo in the box is missed by the density field analysis
because its particles are spread over several adjacent cells in the
downgraded density field. This is an intrinsic limitation of our
method rooted in the use of a fixed grid in the position space, which
is bound to miss some of the the dark matter halos that are off-center
with respect to the spatial location of the grid cells. However this
only leads to miss a random fraction of the rare halos, without an
environmental bias as in the \citet{gao05} method.
\section{Conclusion and Discussion}\label{sec:conc}
By means of both analysis of numerical simulations and of extended
Press-Schechter modeling we investigated the relation between the most
massive dark matter halos at different redshifts. The main conclusion
of this work is that---contrary to expectations from many recent works
(e.g. see \citealt{MR05,ree05,gao05,gao07,li07})---the most massive
halo at a redshift $z_1>z_2$ does not necessarily evolve into the most
massive at $z=z_2$. This is a robust conclusion that can be naturally
understood in the context of growth of dark matter density
perturbations, for example by constructing merger trees through the
\citet{lacey93} model. Rare high-redshift objects, such as the
remnants of the first PopIII stars and QSOs, are not hosted at $z=0$
in the most massive halos, but rather live in a variety of
environments. For example, the typical $z>40$ PopIII star remnant
lives in a dark matter halo that at $z=0$ has a mass of $\approx 2
\cdot 10^{13} h^{-1} \mathrm{M_{\sun}}$, typical for a galaxy group,
and not within rich clusters as claimed by \citet{whi00}. Similarly
the descendant of a typical $z\approx 6$ QSO is not located within the
most massive clusters at $z=0$ as assumed by \citet{li07}.
These conclusions have important consequences on the application of
the recursive simulation method introduced by \citet{gao05} to
identify high-redshift rare halos based on progressive refinement of
regions centered around the most massive $z=0$ cluster. In fact, while
the recursive method is indeed effective at identifying a sub-region
of the simulation with earlier-than-average structure formation, it
finds neither the earliest structures in the box, which are dominated
by small-scale density fluctuations, nor typical early structure, as
it preferentially identifies objects located in the regions with the
highest bias.
These limitations may have only a minor effect when the goal is to
investigate the formation of one rare Population III halo in the
simulation box, as it is done for example in \citet{gao05} and in
\citet{ree05}. However in different physical scenarios it is important
to correctly estimate the rarity of the halo simulated and to assess
how typical their growth histories are. This is critical if additional
physics beyond gravitational interactions is included, such as star
formation and radiative feedback. One such example is the formation of
rare high-redshift QSOs: \citet{li07} use the \citet{gao05} method to
identify at $z\approx 6.5$ ``the most massive halo in a $\approx$ 3
Gpc$^3$ volume'' and then conclude that the QSO formed in this halo
reproduces the properties of observed QSOs with the same number
density. From our analysis in Sec.~\ref{sec:QSOs} it is clear that the
halo identified by \citet{li07} as progenitor of the largest $z=0$
cluster is not likely to be the most massive at $z \approx 6.5$. Thus
other similar or more massive halos are expected to be present at
$z \approx 6.5$ in their $\approx$ 3 Gpc$^3$ simulation volume: in principle
any of these halos could host a bright QSOs, with important
consequences for the comparison between observed and simulated QSO
number densities. In addition, when the goal is to study the
environment in which QSOs live, selecting host halos with the
resimulation method introduces systematic effects in the results
difficult to quantify and correct for, because these halos would have
above-average growth (and merging) histories.
To avoid selecting only the halos with atypical accretion histories
and in an attempt to improve over the identification of some of the
rarest high-redshift halos in a box, we propose instead to select the
initial conditions for high-resolution resimulation based on the
analysis of the linear density field at uniform resolution. The
method, described in Sec.~\ref{sec:Ic}, identifies subregions of the
simulation box with high-redshift halos as those with the highest
peaks in the density field, requiring a mass resolution in the field
comparable to that of the mass of the halos one wishes to select. The
applicability of the method is thus limited only by the size of the
largest density grid that can be accommodated in the available memory,
otherwise requiring only a modest amount of computing time compared to the
\citet{gao05} method. This is because our method bypasses the need of
a series of N-body simulations to be carried out in addition to the
final run.
Unfortunately, our method does not guarantee identification of \emph{the
first} halo on the desired mass scale, as highlighted by some preliminary
testing we presented in Sec.~\ref{sec:valid}. This seems still an
elusive goal. When the density field is defined over a fixed grid,
there is not a perfect match between the halo catalog constructed from
the density field and that obtained by increasing the resolution of
the field and then following the full non-linear dynamics with an
N-body simulation. An extensive validation of our linear density field
initial conditions generation and its application to the formation of the first
bright QSOs will be discussed in a subsequent paper.
\acknowledgements
This work was supported in part by NASA grants JWST IDS NAG5-12458 and
HST-GO10632. We thank the referee for useful comments and
suggestions. We are grateful to Gerard Lemson and to the Millennium
Run collaboration for allowing us to use their merger tree catalogs. |
1805.03277 | \section{Introduction}\label{intro}
One of the most important unsolved problem in Operator Theory is the \emph{Invariant Subspace Problem}: Does every bounded operator on an infinite dimensional, separable, complex Hilbert space have a non-trivial invariant closed subspace? Von Neumann proved the existence of such subspaces for compact operators acting on a separable Hilbert space, a result which was extended by Aronszajn and Smith \cite{AS54} to separable Banach spaces. Lomonosov \cite{L73} greatly increases the class of operators with invariant subspaces by showing that every operator commuting with a compact operator has an invariant subspace. Enflo (\cite{E76, E87}, see also \cite{B85}) constructed the first
example of a bounded operator on a (non-reflexive) Banach space which has no non-trivial invariant subspaces, followed by a construction by Read \cite{R84}. Later Read produced several such examples: strictly singular operators, quasinilpotent operators, and operators acting on $l_1$ (see \cite{R85}, \cite{R97},\cite{R91}). All these examples are on non-reflexive Banach spaces, and the Invariant Subspace Problem is still open for general reflexive Banach spaces. For an overview of the Invariant Subspace Problem see the monographs by Radjavi and Rosenthal \cite{RR03} or the more recent book by Chalendar and Partington \cite{CP11}.
A very important special case for which the Invariant Subspace Problem is still open is that of \emph{quasinilpotent} operators on Hilbert spaces, or, more generally, on reflexive Banach spaces. An operator $T$ is called \emph{quasinilpotent} if $\sigma(T)=\{0\}$, where by $\sigma(T)$ we denote the spectrum of $T$. Substantial work has been devoted over the years to ISP for quasinilpotent operators, in particular on Hilbert spaces. We mention several important papers, without attempting to provide an exhaustive list: Apostol and Voiculescu \cite{AV74}, Herrero \cite{H78}, Foia\c{s} and Pearcy \cite{FP74}, Foia\c{s}, Jung, Ko, and Pearcy,\cite{JKP03, FJKP04, FJKP05}.
In Section \ref{mainsection} we develop a method of investigating invariant subspaces for quasinilpotent operators on complex Banach spaces by examining the resolvent function. In our main result in this section, Theorem \ref{main}, we prove a necessary and sufficient condition for a quasinilpotent operator to have invariant subspaces, a condition which is related to the stability of the spectrum under rank-one perturbations.
Next we examine the existence of invariant \emph{half-spaces} for rank-one perturbations of quasinilpotent operators. By a \emph{half-space} we understand a closed subspace which is both infinite dimensional and infinite codimensional. A method of examining invariant half-spaces for finite rank perturbations was introduced by Androulakis, Popov, Tcaciuc, and Troitsky in \cite{APTT09}, where the authors showed that certain classes of bounded operators have rank-one perturbations which admit invariant half-spaces. In \cite{PT13} Popov and Tcaciuc showed that \emph{every} bounded operator $T$ acting on a reflexive Banach space can be perturbed by a rank-one operator $F$ such that $T+F$ has an invariant half-space. Moreover, when a certain spectral condition is satisfied, $F$ can be chosen to have arbitrarily small norm. Recently these results were extended to general Banach spaces in \cite{T17}. In this line of investigation, Jung, Ko, and Pearcy \cite{JKP17, JKP18} adapted this theory to operators on Hilbert spaces, where the presence of additional structure and specific Hilbert space methods allowed them to prove important results regarding the matricial structure of arbitrary operators on Hilbert spaces. For algebras of operators this type of problems have been studied in \cite{P10}, \cite{MPR13}, and \cite{SW16}. More control on the construction of rank-one perturbation that have invariant half-spaces was achieved in \cite{TW17}. In that paper the authors showed that for any bounded operators $T$ with countable spectrum acting on a Banach space $X$, and for any non-zero $x\in X$, one can find a rank one operator with range span$\{x\}$ such that $T+F$ has an invariant subspace.
In Section \ref{perturb}, we refine the method developed in the previous section to show that \emph{almost all} (in a sense that is made precise in Theorem \ref{main2} below ) rank-one perturbations of quasinilpotent operators have invariant half-spaces.
\section{Invariant subspaces for quasinilpotent operators}\label{mainsection}
For a Banach space $X$, we denote by $\mathcal{B}(X)$ the algebra of all (bounded linear) operators on $X$. When $T\in{\mathcal B}(X)$, we write $\sigma(T)$, $\sigma_p(T)$,$\sigma_{ess}(T)$, and $\rho(T)$ for the spectrum
of~$T$, point spectrum of $T$, the essential point spectrum of $T$, and the resolvent set of~$T$, respectively. The closed span of a set $\{x_n\}_n$ of vectors in $X$ is denoted by $[x_n]$.
For $T\in\mathcal{B}(X)$, the \emph{resolvent} of $T$ is the function $R:\rho(T)\to\mathcal{B}(X)$ defined by $R(z)=(zI-T)^{-1}$. When $\abs{z}>r(T)$, where $r(T)$ is the spectral radius of $T$, the resolvent is given by the Neumann series expansion
$$
R(z)=(zI-T)^{-1}=\sum_{i=0}^{\infty}\frac{T^i}{z^{i+1}}.
$$
In particular, when $T$ is quasinilpotent this expansion holds for all complex numbers $z\neq 0$. The resolvent $R$ is analytic on $\rho(T)$, hence on $\mathbb{C}\setminus\{0\}$ when $T$ is quasinilpotent.
We first prove a simple lemma which gives sufficient and necessary conditions for $\lambda\in\rho(T)$ to be an eigenvalue for some fixed rank-one perturbation.
\begin{lemma} \label{eigen}
Let $X$ be a separable Banach space, $T\in\mathcal{B}(X)$, and $F:=e^*\otimes f$ a rank one operator. Fix $\lambda\in\rho(T)$ and $\alpha\in\mathbb{C}\setminus\{0\}$. Then the following are equivalent:
\begin{enumerate}
\item $e^*((R(\lambda)f)=\alpha^{-1}$.
\item $\lambda\in\sigma_p(T+\alpha F)$.
\end{enumerate}
\end{lemma}
\begin{proof}
i)$\Rightarrow$ ii)
We are going to show that $y:=R(\lambda)f$ is an eigenvector for $T+\alpha F$, corresponding to the eigenvalue $\lambda$. Note that $Ty=\lambda y -f$. Then:
$$
(T+\alpha F)y=Ty+\alpha e^*(y)f=\lambda y -f +\alpha e^*(y)f=\lambda y -f +\alpha \alpha^{-1} f=\lambda y.
$$
ii)$\Rightarrow$ i)
Let $y\in X$ be an eigenvector for $T+\alpha F$ corresponding to $\lambda$. Hence $Ty+\alpha e^*(y)f=\lambda y$. Note that since $\lambda\in\rho(T)$, it follows that $e^*(y)\neq 0$. We have:
$$
Ty+\alpha e^*(y)f=\lambda y \Leftrightarrow (\lambda I-T)y=\alpha e^*(y)f \Leftrightarrow y=\alpha e^*(y)(\lambda I-T)^{-1}f.
$$
Applying $e^*$ to both sides of the last equality, we get that
$$
e^*(y)=\alpha e^*(y)e^*((\lambda I-T)^{-1}f),
$$
and since $e^*(y)\neq 0$ it follows that $e^*(R(\lambda)f)=\alpha^{-1}$.
\end{proof}
\begin{remark}\label{remark1}
Note that when $e^*(R(\lambda)f)=\alpha^{-1}$, from the proof of the previous lemma it follows that $R(\lambda)f$ is an eigenvector for $T+\alpha F$ corresponding to the eigenvalue $\lambda$.
\end{remark}
We are now ready to prove the main theorem of this section.
\begin{theorem}\label{main}
Let $X$ be a separable Banach space and $T\in\mathcal{B}(X)$ a quasinilpotent operator. Then the following are equivalent:
\begin{enumerate}
\item $T$ has an invariant subspace.
\item There exists a rank one operator $F$ such that for any $\alpha\in\mathbb{C}$, $T+\alpha F$ is quasinilpotent.
\item There exists a rank one operator $F$ and $\alpha\in\mathbb{C}$, $\alpha\neq 0$, $\alpha\neq 1$, such that $T+F$ and $T+\alpha F$ are quasinilpotent.
\end{enumerate}
\end{theorem}
\begin{proof}
Note first that since $\sigma_{ess}(T)=\{0\}$, and the essential spectrum is stable under compact perturbations, it follows that for any $\alpha\in\mathbb{C}$ and any rank-one operator $F$, $\sigma_{ess}(T+\alpha F)=\{0\}$. Therefore $\sigma(T+\alpha F)$ is at most countable with $0$ the only accumulation point, and any $\lambda\in\sigma(T+\alpha F)\setminus\{0\}$ is an eigenvalue((see e.g. \cite{AA02}, Corollary 7.49 and 7.50). Hence, the condition that $T+\alpha F$ is quasinilpotent, is equivalent to $\sigma_p(T+\alpha F)\setminus \{0\}=\emptyset$.
i) $\Rightarrow$ ii)
Suppose $Y$ is a non-trivial invariant subspace for $T$. Pick $f\in Y$, and $e^*\in X^*$ such that $e^*(Y)=0$. Let $F$ be the rank one operator defined by $F:=e^*\otimes f$. Then, since $Y$ is $T$-invariant and $f\in Y$, we have that the orbit $(T^nf)$ is contained in $Y$, hence for all $n\in\mathbb{N}$, $e^*(T^nf)=0$. It follows that, for any $z\in\mathbb{C}\setminus\{0\}$ we have:
\begin{equation}\label{zero}
e^*(R(z)f)=e^*\left(\sum_{i=0}^{\infty}\frac{T^i f}{z^{i+1}}\right)=\sum_{i=0}^{\infty}\frac{e^*(T^i f)}{z^{i+1}}=0.
\end{equation}
Fix $\alpha\neq 0$, arbitrary. From (\ref{zero}) and Lemma \ref{eigen} it now follows that for any $z\in\mathbb{C}\setminus\{0\}$ we have that $z\notin\sigma_p(T+\alpha F)$. Therefore, for any $\alpha\neq 0$ we have that $\sigma_p(T+\alpha F)\setminus \{0\}=\emptyset$, hence $T+\alpha F$ is quasinilpotent.
ii) $\Rightarrow$ iii) obvious
iii) $\Rightarrow$ i) We argue by contradiction. Assume that $T$ has no invariant subspaces, and fix $F:=e^*\otimes f$ an arbitrary rank one operator. Since $T$ has no invariant subspaces it follows that $e^*(T^nf)\neq 0$ for infinitely many values of $n$. Indeed, otherwise there exist $k\in\mathbb{N}$ such that $e^*(T^jf)=0$ for all $j\geq k$. However this means that the closed span of $(T^jf)_{j\geq k}$ is contained in the kernel of $e^*$, thus it would be a non-trivial $T$-invariant subspace, contradicting the assumption.
To simplify the notation, we denote by $g:\mathbb{C}\setminus\{0\}\to\mathbb{C}$ the analytic function defined by $g(z)=e^*(R(z)f)$. Note that $g$ has an isolated singularity at $z=0$ and its Laurent series about $z=0$ is
$$
g(z)=\sum_{i=0}^{\infty}\frac{1}{z^{i+1}}e^*(T^if).
$$
Since $e^*(T^nf)\neq 0$ for infinitely many values of $n$, it follows that the Laurent expansion of $g$ will have infinitely many non-zero terms of the form $\frac{1}{z^{i+1}}e^*(T^if)$. Therefore $z=0$ is an isolated \emph{essential} singularity for $g$. From Picard's Great Theorem it follows that $g$ attains any value, with possibly one exception, infinitely often, in any neighbourhood of $z=0$. Hence, for all $\alpha\neq 0$, with possibly one exception, the set $\{z\in\mathbb{C} : g(z)=\alpha^{-1}\}$ is infinite. Note that this set is in fact \emph{countably} infinite, as it is easy to see that in Picard's Theorem the values can be attained at most countably many times. Therefore $\sigma_p(T+\alpha F)\setminus\{0\}=\{z\in\mathbb{C} : g(z)=\alpha^{-1}\}$ is countably infinite for all $\alpha$, with possibly one exception, so $T+\alpha F$ is quasinilpotent for at most one non-zero value $\alpha$. Since $F$ was arbitrary, this contradicts iii), and the implication is proved.
\end{proof}
The techniques employed in the proof of the previous theorem also gives the following characterization of the spectrum of rank-one perturbation of quasinilpotent operators.
\begin{proposition} \label{3options}
Let $X$ be a Banach space, $T\in\mathcal{B}(X)$ a quasinilpotent operator, and $F:=e^*\otimes f$ a rank one operator. Then exactly one of the following three possibilities holds:
\begin{enumerate}
\item For all $\alpha\in\mathbb{C}$, $T+\alpha F$ is quasinilpotent.
\item For all non-zero $\alpha\in\mathbb{C}$, with possibly one exception, $\sigma_p(T+\alpha F)$ is countably infinite.
\item There exists $K\in\mathbb{N}$ such that for all non-zero $\alpha\in\mathbb{C}$, $0<\abs{\sigma_p(T+\alpha F)\setminus\{0\}}<K$.
\end{enumerate}
\end{proposition}
\begin{proof}
From the proof of Theorem \ref{main} the options $(i)$ and $(ii)$ hold when $e^*(T^nf)=0$ for all $n$, and when $e^*(T^nf)\neq 0$ for infinitely many values of $n$, respectively. It remains to examine the case when $e^*(T^nf)\neq 0$ for finitely, non-zero, values of $n$. Let $k>0$ be the smallest natural number such that $e^*(T^kf)\neq 0$ and $e^*(T^jf)= 0$ for all $j>k$. With the notations from Theorem \ref{main} it follows that:
$$
g(z)= e^*(R(z)f)=\sum_{i=0}^{k}\frac{1}{z^{i+1}}e^*(T^if).
$$
Therefore $z=0$ is a pole of order $k+1$ for $g$. In this case it is easy to see that for any $\alpha\neq 0$, the equation $g(z)=\alpha^{-1}$ has at most $k+1$ solutions, hence the cardinality of the non-empty set $\sigma_p(T+\alpha F)\setminus\{0\}$ is at most $k+1$, and $(iii)$ holds.
\end{proof}
We will show in the next example that the second option in Proposition \ref{3options} can indeed hold, and that the one exception is in general unavoidable.
\begin{example} \label{exa}
Let $H$ be a separable Hilbert space, denote by $(e_n)_n$ an orthonormal basis, and define $T\in\mathcal{B}(H)$ to be the weighted shift defined by
$$
Te_n=\frac{1}{n}e_{n+1},\ \text{ for } n=1,2,\dots
$$
It is easy to see that $T$ is a compact quasinilpotent operator. Consider the rank one operator $F\in\mathcal{B}(H)$ defined by $F(x):=\inner{x,f}e_1$, where $f=\sum_{n=1}^{\infty}\frac{1}{n}e_n$. We are going to show that $T-F$ is quasinilpotent, and that for any $\alpha\neq -1$ we have $\sigma_p(T+\alpha F)$ is countably infinite. From the previous considerations this is equivalent to showing that the function $g(z):=\inner{R(z)e_1, f}$ is analytic on $\mathbb{C}\setminus\{0\}$, has an essential
singularity at $0$, and $g(z)\neq -1$ for all $z\in\mathbb{C}\setminus\{0\}$. We have
$$
R(z)e_1=\sum_{n=0}^{\infty}\frac{1}{z^{n+1}}T^ne_1=\sum_{i=0}^{\infty}\frac{1}{z^{n+1}}\frac{1}{n!}e_{i+1}.
$$
Therefore
$$
g(z):=\inner{R(z)e_1, f}=\inner{\sum_{n=0}^{\infty}\frac{1}{z^{n+1}}\frac{1}{n!}e_{n+1},\sum_{n=1}^{\infty}\frac{1}{n}e_n}=
\sum_{n=1}^{\infty}\frac{1}{n!}\frac{1}{z^n}=\exp(1/z)-1.
$$
Clearly $g$ has an essential singularity at $z=0$ and $g(z)\neq -1$ for any $z\in\mathbb{C}\setminus\{0\}$.
\end{example}
\section{Invariant half-spaces for rank one perturbations}\label{perturb}
We next turn our attention to the study of invariant half-spaces of rank-one perturbations of quasinilpotent operators. First recall some standard notations and definitions. A sequence $(x_n)_{n=1}^{\infty}$ in $X$ is called a \emph{basic sequence} if any $x\in[x_n]$ can be written uniquely as $x=\sum_{n=1}^{\infty} a_n x_n$, where the convergence is in norm (see \cite[section 1.a]{LT77} for background on Schauder bases and basic sequences). As $[x_{2n}]\cap[x_{2n+1}]=\{0\}$ it is immediate that $[x_{2n}]$ is of both infinite dimension and infinite codimension in $[x_n]$, thus a half-space, and since every Banach space contains a basic sequence, it follows that every infinite dimensional Banach space contains a half-space.
An important tool that we are going to use is the following criterion of Kadets and Pe{\l}czy\'{n}ski for a subset of Banach space to contain a basic sequence (see, e.g., \cite[Theorem 1.5.6]{AK06})
\begin{theorem}[Kadets, Pe{\l}czy\'{n}ski] \label{criterion}
Let $S$ be a bounded subset of a Banach space $X$ such that $0$ does not belong to the norm closure of $S$. Then the following
are equivalent:
\begin{enumerate}
\item $S$ fails to contain a basic sequence,
\item The weak closure of $S$ is weakly compact and fails to contain $0$.
\end{enumerate}
\end{theorem}
As mentioned in the introduction, Tcaciuc and Wallis proved in \cite{TW17} the following theorem:
\begin{proposition}\cite[Proposition 2.11]{TW17}
Let $X$ be an infinite-dimensional complex Banach space and $T\in\mathcal{B}(X)$ a bounded operator such that $\sigma(T)$ is countable and $\sigma_p(T)=\emptyset$. Then for any nonzero $x\in X$ and any $\varepsilon>0$ there exists $F\in\mathcal{B}(X)$ with $\norm{F}<\varepsilon$ and $\Range(F)=[x]$, and such that $T+F$ admits an invariant half-space.
\end{proposition}
When $X$ is reflexive, a companion result, \cite[Proposition 2.12]{TW17} allows one to (separately) control the kernel of the perturbation. Our main result in this section shows that for quasinilpotent operators we can control \emph{both} the range and the kernel at the same time, in a very strong way: with at most two exceptions, all perturbations by scalar multiples of a fixed rank-one operator have invariant half-spaces. Also note that this results holds in general Banach spaces. No reflexivity condition is needed.
\begin{theorem} \label{main2}
Let $X$ be a separable Banach space and $T\in\mathcal{B}(X)$ a quasinilpotent operator such that $\sigma_p(T)=\sigma_p(T^*)=\emptyset$. Then for any rank one operator $F$, and any non-zero $\alpha\in\mathbb{C}$, with possibly two exceptions, $T+\alpha F$ has an invariant half-space.
\end{theorem}
\begin{proof}
It is easy to check that $\sigma_p(T)=\emptyset$ if and only if $T$ has no non-trivial finite dimensional invariant subspaces, and that $\sigma_p(T^*)=\emptyset$ if and only if $T$ has no non-trivial finite codimensional invariant subspaces. Therefore we can conclude from the hypotheses that any non-trivial invariant subspace of $T$ must be a half-space.
Let $F=e^*\otimes f$ be a rank-one operator, and consider the orbit $(T^nf)$. If there exists $k\in\mathbb{N}$ such that $e^*(T^nf)=0$ for all $n>k$, then $Y:=[T^nf]_{n>k}$ is an invariant subspace for $T$, contained in the kernel of $F$. Therefore $Y$ is a $T$-invariant half-space, and it is also invariant for $T+\alpha F$, for any $\alpha\in\mathbb{C}$.
There remains to consider the situation when $e^*(T^nf)\neq 0$ for infinitely many values of $n$. In this case it follows from the proof of Theorem \ref{main} that for all non-zero $\alpha\in\mathbb{C}$, with possibly one exception, $\sigma_p(T+\alpha F)$ is countably infinite. Moreover, $0$ is the only accumulation point for $\sigma_p(T+\alpha F)$. Denote by $\mathbb{C}_0$ the set of all these values $\alpha$; in other words, $\mathbb{C}_0$ does not contain $0$, and at most one more other value, depending on $F$.
For any $\alpha\in\mathbb{C}_0$, define the set $S_{\alpha}$ as
$$
S_{\alpha}:=\{R(z)f : z\in\sigma_p(T+\alpha F)\setminus\{0\}\}\subseteq X.
$$
Note from Remark \ref{remark1} that $S_\alpha$ is a set of (linearly independent) eigenvectors corresponding to all distinct eigenvalues from $\sigma_p(T+\alpha F)\setminus\{0\}$. For any $z\in\sigma_p(T+\alpha F)\setminus\{0\}$ we have that $e^*(R(z)f)=\alpha^{-1}$, hence $\norm{R(z)f}\geq (|\alpha|\norm{e^*})^{-1}$. That is, for any $\alpha\in\mathbb{C}_0$, $S_{\alpha}$ is bounded away from zero, therefore $0$ does not belong to the norm closure of $S_{\alpha}$. Define the following sets:
\begin{eqnarray}
\nonumber A &:=& \{\alpha\in\mathbb{C}_0 : S_\alpha\ \text{is not bounded}\} \\
\nonumber B &:=& \{\alpha\in\mathbb{C}_0 : S_\alpha\ \text{is bounded and } \overline{S_\alpha}^{w}\ \text{is not } w \text{-compact}\} \\
\nonumber C &:=& \{\alpha\in\mathbb{C}_0 : S_\alpha\ \text{is bounded and } \overline{S_\alpha}^{w}\ \text{is } w \text{-compact}\}.
\end{eqnarray}
Clearly $\mathbb{C}_0=A\cup B\cup C$, and the union is disjoint. We are going to show that for any $\alpha\in A\cup B$, $T+\alpha F$ has an invariant half-space, and that $|C|\leq 1$.
Let first $\alpha\in A$. Denote $\sigma_p(T+\alpha F)=(\lambda_n)_n$, and note that we have that $\lambda_n\to 0$. We are going to show that $S_\alpha$ contains a basic sequence. Since $S_\alpha$ is not bounded, by passing to a subsequence we may assume that $\norm{R(\lambda_n)f}\to\infty$. For any $n\in\mathbb{N}$, denote by $x_n:=R(\lambda_n)f/\norm{R(\lambda_n)f}$, and put $W_\alpha:=\{x_n : n\in\mathbb{N}\}$.
The set $W_\alpha$ is bounded. If $\overline{W_\alpha}^{w}$ is not weakly-compact, we can apply Kadets-Pe{\l}czy\'{n}ski criterion (Theorem \ref{criterion}) to conclude that $W_\alpha$ contains a basic sequence. Therefore, by passing to a subsequence, we can assume that $(x_n)$ is a basic sequence in $X$. Then $Y:=[x_{2n}]$ is a half-space which is invariant for $T+\alpha F$.
If $\overline{W_\alpha}^{w}$ is weakly compact, then it is weakly sequentially compact by the Eberlein-\v{S}mulian theorem, and by passing to a subsequence we can assume that $x_n\stackrel{w}{\longrightarrow} x\in X$. It is easy to see that
\begin{equation}\label{equ}
Tx_n=\lambda_n x_n - \frac{1}{\norm{R(\lambda_n)f}}f.
\end{equation}
Since $Tx_n\stackrel{w^*}{\longrightarrow} Tx$, $\lambda_nx_n\stackrel{w^*}{\longrightarrow} 0$, and $\norm{R(\lambda_n)f}\stackrel{w^*}{\longrightarrow}\infty$, it follows from (\ref{equ}) that $Tx=0$. However $0$ is not an eigenvalue for $T$, so we must have $x=0$. Hence $0\in\overline{W_\alpha}^{w}$, and again by the Kadets-Pe{\l}czy\'{n}ski criterion we have that $W_\alpha$ contains a basic sequence, and we finish up as in the case when $\overline{W_\alpha}^{w}$ is not weakly-compact.
When $\alpha\in B$, therefore $S_\alpha$ is bounded and $\overline{S_\alpha}^{w}$ is not weakly compact, we can again apply the Kadets-Pe{\l}czy\'{n}ski criterion to conclude that $S_\alpha$ contains a basic sequence, and again finish up as before. Therefore, we have shown that for $\alpha\in A\cup B$, $T+\alpha F$ has an invariant half-space. There remains to show that $|C|\leq 1$.
Assume towards a contradiction that there exist $\alpha\neq\beta$ in $C$. Denote by $(\lambda_n)$ and by $(\mu_n)$ the eigenvalues in $\sigma_p(T+\alpha F)\setminus\{0\}$ and $\sigma_p(T+\beta F)\setminus\{0\}$, respectively, and note that both $(\lambda_n)$ and $(\mu_n)$ converge to $0$. For each $n\in\mathbb{N}$, we define $h_n:=R(\lambda_n)f$, and by $k_n:=R(\mu_n)f$. We have
\begin{equation} \label{equ2}
Th_n=\lambda_n h_n-f {\rm \hskip 2cm and \hskip 2cm } Tk_n=\mu_n k_n-f.
\end{equation}
Since $\overline{S_\alpha}^{w}$ and $\overline{S_\beta}^{w}$ are weakly compact, we can assume, by passing to subsequences, that $h_n\stackrel{w}{\longrightarrow} h$ and $k_n\stackrel{w}{\longrightarrow} k$. Note that for any $n\in\mathbb{N}$ we have that
\begin{equation*}
e^*(h_n)=g(\lambda_n)=\alpha^{-1} {\rm \hskip 2cm and \hskip 2cm } e^*(h_n)=g(\mu_n)=\beta^{-1}.
\end{equation*}
Therefore, $e^*(h)=\alpha^{-1}$ and $e^*(k)=\beta^{-1}$, and since $\alpha\neq\beta$, it follows that $h\neq k$. Taking weak limits in (\ref{equ2}), and taking into account that $\lambda_n\to 0$ and $\mu_n\to 0$, we get that $Th=-f$ and $Tk=-f$. Therefore $T(h-k)=0$, and since $h-k\neq 0$ it follows that $0$ is an eigenvalue for $T$, which is a contradiction since $\sigma_p(T)=\emptyset$. It follows that $|C|\leq 1$, and this completes the proof.
\end{proof}
While not explicitly stated in the previous theorem, note that, in particular, we can obtain rank-one perturbations of arbitrarily small norms that have invariant half-spaces. Indeed, since -- for a fixed rank-one $F$ -- almost all perturbations $T+\alpha F$ have invariant half-spaces, for any given $\varepsilon>0$ we can take a \enquote{good} $\alpha<\varepsilon/\norm{F}$. We summarize this in the following corollary:
\begin{corollary}
Let $X$ be a separable Banach space and $T\in\mathcal{B}(X)$ a quasinilpotent operator such that $\sigma_p(T)=\sigma_p(T^*)=\emptyset$. Then for any non-zero $f\in X$, $e^{*}\in X^*$, and $\varepsilon>0$, we can find rank-one $F\in\mathcal{B}(X)$ with $\Range(F)=[f]$, $\ker{F}=\ker{e^*}$, and $\norm{F}<\varepsilon$ such that $T+F$ has an invariant half-space.
\end{corollary}
In the Hilbert space setting we get more specific information about the structure of a quasinilpotent operator.
\begin{corollary} \label{hilbert}
Let $\mathcal{H}$ be a separable Hilbert space and $T\in\mathcal{B}(\mathcal{H})$ a quasinilpotent operator such that $\sigma_p(T)=\sigma_p(T^*)=\emptyset$. Then for any rank one operator $F$, and any non-zero $\alpha\in\mathbb{C}$, with possibly two exceptions, there exists an orthogonal projection of infinite rank and co-rank such that $P^{\perp}TP=\alpha P^{\perp}FP$.
\end{corollary}
\begin{proof}
Fix a rank one operator $F\in\mathcal{B}(\mathcal{H})$. From Theorem \ref{main2} we have that for all non-zero $\alpha\in\mathbb{C}$, with possibly two exceptions, $T-\alpha F$ has an invariant half-space. Fix such an $\alpha\in\mathbb{C}$, let $Y$ be an invariant half-space for $T-\alpha F$, and let $P\in\mathcal{B}(\mathcal{H})$ be the orthogonal projection onto $Y$ (which clearly has infinite rank and co-rank). Since $Y$ is invariant for $T-\alpha F$ it is easy to see that $P^{\perp}(T-\alpha F)P=0$, and the conclusion follows.
\end{proof}
\section{Open questions}
In light of Theorem \ref{main}, the behaviour of the spectrum of a quasinilpotent operator under rank one perturbations is related to a solution for the invariant Subspace Problem for quasinilpotent operators. This suggests a natural and very important open question in this direction.
{\bf Question 1:} Given $T\in\mathcal{B}(X)$ a quasinilpotent operator acting on an infinite dimensional, separable, complex Banach space, can we find a rank-one operator $F$ such that $T+F$ is quasinilpotent?
\vskip .3cm
Note that a positive answer to this question is not sufficient to conclude a positive solution to the Invariant Subspace Problem for quasinilpotent operators by applying Theorem \ref{main}. Indeed, from $iii)$ in Theorem \ref{main} we would need one more perturbation by a scalar multiple of the same rank one operator that is also quasinilpotent. On the other hand, a negative answer to the \emph{Question 1} will provide a counterexample to ISP for quasinilpotent operators. As we mentioned before, Read \cite{R97} already provided such a counterexample on $l_1$, therefore it would be a natural starting point to examine \emph{Question 1} for Read's operator.
The requirement that $F$ has rank one is very important, as we can always find $F$ of rank two such that $T+F$ is quasinilpotent. Indeed, let $N$ be a rank one nilpotent operator (thus $N^2=0$) and put $S:=(I-N)T(I+N)$. Then $S$ is similar to $T$, therefore $S$ is also quasinilpotent, and an easy calculation shows that $T-S$ has rank at most two.
In Section 2 we constructed an example of a quasinilpotent operator $T\in\mathcal{B}(l_2)$, and a rank one operator $F$ such that $T+F$ is quasinilpotent, but $T+\alpha F$ is not, for all $\alpha\neq 0$, $\alpha\neq 1$. Therefore $F$ does not satisfy $iii)$ in Theorem \ref{main}, but of course there are other rank one operators that do, as $T$ in that example has plenty of invariant subspaces. Whether a positive solution to \emph{Question 1} already implies a positive solution to ISP for quasinilpotent operators is another important open question:
{\bf Question 2:} If $T\in\mathcal{B}(X)$ is a quasinilpotent operator with the property that there exists $F\in\mathcal{B}(X)$ rank one such that $T+F$ is quasinilpotent, does $T$ have an invariant subspace?
\vskip .3cm
Note that if \emph{Question 1} has a positive solution for Read's operator, then Read's operator provides a negative answer to \emph{Question 2}.
\vskip .3cm
\emph{Acknowledgments.} It is my pleasure to acknowledge the helpful conversations and feedback provided to me by Heydar Radjavi. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. |
1805.03175 | \section{#2}\label{sec:#1}\vspace{-0.075in}}
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\section*{Acknowledgments}
We thank \ch{the anonymous reviewers of SIGMETRICS
2017 and SAFARI group members for their} feedback. We acknowledge the support of
Google, Intel, \fix{NVIDIA}, Samsung, VMware, and the United States Department
of Energy. This research was supported in part by the ISTC-CC, SRC, and NSF
(grants 1212962 and 1320531). Kevin Chang was supported in part by an SRCEA/Intel
Fellowship.
\balance
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\bibliographystyle{IEEEtranS}
\section{Conclusion}
\changes{ \new{Our SIGMETRICS 2017} paper~\cite{chang-sigmetrics2017} provides the
first experimental study that comprehensively characterizes and analyzes the
behavior of DRAM chips when the supply voltage is reduced below its nominal
value. We demonstrate, \fixIV{using 124 DDR3L DRAM chips}, that the \fix{DRAM}
supply voltage can be reliably reduced to a certain level, beyond which errors
arise within the data. We then experimentally demonstrate the relationship
between the supply voltage and the latency of the fundamental DRAM operations
(activation, restoration, and precharge). \fixIV{We show that bit errors caused
by reduced-voltage operation can be eliminated by increasing the latency of
the three fundamental DRAM operations.} By \fix{changing} the memory
controller configuration to allow for the longer latency of these operations, we
can thus \emph{further} lower the supply voltage without inducing errors in the
data. We also \fixIV{experimentally} characterize the relationship between
reduced supply voltage and error locations, stored data patterns, temperature,
and data retention.
Based on these observations, we propose and evaluate \voltron, a low-cost energy
reduction mechanism that reduces DRAM energy \emph{without} \fix{affecting}
memory data throughput. \voltron reduces the supply voltage for \emph{only} the
DRAM array, while maintaining the nominal voltage for the peripheral circuitry
to continue operating the memory channel at a high frequency. \voltron uses a
new piecewise linear performance model to find the array supply voltage that
maximizes the system energy reduction within a given performance loss target.
\fix{Our experimental evaluations across a wide variety of workloads}
demonstrate that \voltron significantly reduces system energy \fix{consumption}
with only \fix{very} modest performance loss.
We conclude that it is very promising to understand and exploit reduced-voltage
operation in modern DRAM chips. We hope that the experimental characterization,
analysis, and optimization techniques presented in \new{our SIGMETRICS 2017}
paper will enable the development of other new mechanisms that can effectively
\fixIII{exploit the trade-offs between voltage, reliability, and latency in DRAM
to improve system performance, efficiency, and/or reliability. \new{We also
hope that our paper's studies inspire new experimental studies to understand
reduced-voltage operation in other memory technologies, such as NAND flash
memory, PCM, and STT-MRAM.}}}
\section{Exploiting Reduced-Voltage Behavior}
\label{sec:voltron}
Based on the extensive understanding we have developed on reduced-voltage operation
of real DRAM chips, we propose a new mechanism called \emph{\voltron}, which
reduces DRAM energy without sacrificing memory throughput. \voltron exploits the
fundamental observation that reducing the supply voltage to DRAM requires
increasing the latency of the three DRAM operations in order to prevent errors.
Using this observation, the key idea of Voltron is to use a performance model to
determine \fix{by how much to reduce the DRAM supply voltage}, without
introducing errors and without exceeding a user-specified threshold for
performance loss. \voltron consists of two main components: \myitem{i}
\emph{array voltage scaling} and \myitem{ii} \emph{performance-aware voltage
control}.
\subsection{Components of Voltron}
\label{sec:voltron:components}
\noindent\textbf{Array Voltage Scaling.} Unlike prior works, Voltron does
\emph{not} reduce the voltage of the \emph{peripheral circuitry}, which is
responsible for transferring commands and data between the memory controller and
the DRAM chip. If Voltron were to reduce the voltage of the peripheral
circuitry, we would \emph{have to} also reduce the operating frequency of DRAM.
A reduction in the operating frequency reduces the memory data throughput, which
can significantly degrade the performance of applications that require high
memory bandwidth. Instead, Voltron reduces the voltage supplied to \emph{only}
the DRAM array without changing the voltage supplied to the peripheral
circuitry, thereby allowing the DRAM channel to maintain a high frequency while
reducing the power consumption of the DRAM array. To prevent errors from
occurring during reduced-voltage operation, Voltron increases the latency of the
three DRAM operations (activation, restoration, and precharge) based our
observation in \ssecref{long_latency}.
\new{
\noindent\textbf{Performance-Aware Voltage Control.} Array voltage scaling
provides system users with the ability to decrease DRAM array voltage (\varr) to
reduce DRAM power. Employing a lower \varr provides greater power savings, but
at the cost of longer DRAM access latency, which leads to larger performance
degradation. This trade-off varies widely across different applications, as each
application has a different tolerance to the increased memory latency. This
raises the question of how to pick a ``suitable'' array voltage level for
different applications as a system user or designer. For our evaluations, we say that
an array voltage level is suitable if it does not degrade system performance by
more than a user-specified threshold. Our goal is to provide a simple
\fixIII{technique} that can automatically select \fix{a} suitable \varr
\fix{value} for different applications. \changes{To this end, we propose
\emph{performance-aware voltage control}, a \fix{power--performance} management
policy that selects a minimum \varr which satisfies a desired performance
constraint. The key observation is that an application's performance loss (due
to increased memory latency) scales linearly with the application's memory
\fix{demand \fixIII{(e.g., memory intensity)}.} Based on this
\fixIV{empirical} observation \fixIV{we make}, we build a \emph{performance
loss predictor} that leverages a linear model to predict an application's
performance loss based on its characteristics \ch{and the effect of different
voltage level choices} at runtime. Using the
performance loss predictor, Voltron finds a value of \varr that \fix{can keep the}
predicted performance within the user-specified target at runtime.} We refer the
reader to Section 5.2 of our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017} for
more detail and for an evaluation of the performance model \ch{alone}. }
\subsection{Evaluation}
\label{sec:voltron:eval}
We evaluate the system-level energy \ch{and performance} impact of \voltron using
Ramulator~\cite{kim-cal2015, safari-github}, integrated with
McPAT~\cite{mcpat:micro} and DRAMPower~\cite{drampower} for modeling the energy
consumption of both the processor and DRAM. Our workloads consist of 27
benchmarks from SPEC CPU2006~\cite{spec2006} and YCSB~\cite{cooper-socc2010}.
\new{We evaluate Voltron with a target performance loss of 5\%. \voltron
executes the performance-aware voltage control mechanism once every four
million cycles.} We refer the reader to Section~6.1 of our \new{SIGMETRICS
2017 paper~\cite{chang-sigmetrics2017} for more detail on the} system configuration and workloads. We
qualitatively and quantitatively compare Voltron \fixIV{to} \textit{\memdvfs}, a
dynamic DRAM frequency and voltage scaling mechanism proposed by prior
work~\cite{david-icac2011}.
\figref{voltron_perf_e} shows the system energy savings \new{and the system
performance (i.e., weighted
speedup~\cite{eyerman-ieeemicro2008,snavely-asplos2000}) loss} due to
\mbox{\memdvfs} and \voltron, compared to a baseline DRAM with \fix{a supply
voltage of} 1.35V. The graph uses box plots to show the distribution among all
workloads that are categorized as \fixV{either non-memory-intensive or
memory-intensive}. The memory intensity is determined based on the
commonly-used metric MPKI (last-level cache misses per kilo-instruction). We
categorize an application as memory intensive when its MPKI is greater than or
equal to 15. We make two observations.
\figputHS{voltron_perf_e}{1}{\new{Energy (left) and performance (right)} comparison between Voltron
and MemDVFS on non-memory-intensive and memory-intensive workloads. Adapted from
\cite{chang-sigmetrics2017}.}
\new{First, Voltron is effective and saves more energy than MemDVFS.} \memdvfs has
almost zero effect on \fixIV{memory-intensive} workloads. This is because
\memdvfs avoids scaling DRAM frequency \fix{(and hence voltage)} when an
application's memory bandwidth \fixIV{utilization} is above a fixed threshold.
Reducing the frequency can result in a large performance loss since the
\fixIV{memory-intensive} workloads require high memory throughput. As
\fix{memory-intensive} applications have high memory bandwidth consumption that
easily exceeds the \fixIV{fixed} threshold \fixIV{used by \memdvfs}, \memdvfs
\emph{cannot} perform frequency and voltage scaling during most of the execution
time. In contrast, \voltron reduces system energy by 7.0\% on average for
\fix{memory-intensive} workloads. \fix{Thus, we} demonstrate that \voltron is an
effective mechanism that improves \fixIV{system} energy efficiency not only on
\fix{non-memory-intensive} applications, but also \fixIV{(especially)} on
\fix{memory-intensive} workloads where prior work was unable to do so.
\new{\fix{Second, as shown in \figref{voltron_perf_e} (right)}, \voltron consistently
selects a \varr value that satisfies the performance loss bound of 5\% across
all workloads. \voltron incurs \fix{ an average (maximum) performance loss of
2.5\% (4.4\%) and 2.9\% (4.1\%)} for \fixV{non-memory-intensive and
memory-intensive} workloads, respectively. This demonstrates that our
performance model enables \voltron to select a low voltage value that saves
energy while bounding performance loss based on the user's requirement.
Our SIGMETRICS 2017 paper contains extensive performance and energy analysis of
the Voltron mechanism in Sections 6.2 to 6.8~\cite{chang-sigmetrics2017}. In
particular, we show that if we exploit spatial locality of errors
(\ssecref{locality}), we can improve the performance benefits of Voltron,
\ch{reducing the average performance loss for memory-intensive workloads
to 1.8\%}
(\ch{see} Section 6.5 of our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017}). We refer the reader to these sections for
a detailed evaluation of Voltron.
}
\section{Motivation}
In a wide range of modern computing systems, spanning from warehouse-scale data
centers to mobile platforms, energy consumption is a first-order
concern\new{~\cite{hoelzle-book2009,ibm-power7,deng-asplos2011,mutlu-imw2013,mudge-computer2001}}.
In these systems, the energy consumed by the DRAM-based main memory system
constitutes a significant fraction of the total energy. For example,
\fix{experimental} studies of production systems have shown that DRAM consumes
40\% of the total energy in servers~\cite{hoelzle-book2009,ibmpower7-hpca}
\fixII{and 40\%} of the total power in graphics cards~\cite{paul-isca2015}.
Improvements in manufacturing process technology have allowed DRAM vendors to
lower the DRAM supply voltage conservatively, which reduces some of the DRAM
energy consumption~\cite{jedec-ddr3l, jedec-lpddr3, jedec-lpddr4}. In this work, we would like
to reduce DRAM energy by \emph{further reducing DRAM supply voltage}. Vendors
choose a conservatively high supply voltage, to provide a \emph{guardband} that
allows DRAM chips with worst-case process variation to operate without errors
\fixIV{under the worst-case operating conditions}~\cite{david-icac2011}. The
exact amount of supply voltage guardband varies across chips, and lowering the
voltage below the guardband can result in erroneous or even undefined
behavior~\cite{chang-sigmetrics2017}.
Therefore, we need to understand how DRAM chips behave during reduced-voltage
operation. To our knowledge, no previously published work examines the effect of
using a wide range of different supply voltage values on the reliability,
latency, and retention characteristics of DRAM chips.
\changes{\textbf{Our goal} in our \new{SIGMETRICS 2017} paper~\cite{chang-sigmetrics2017}
is to \myitem{i}~characterize and understand the relationship between supply
voltage reduction and various characteristics of DRAM, including DRAM
reliability, latency, and data retention; and \myitem{ii}~use the insights
derived from this characterization and understanding to design a new mechanism
that can aggressively lower the supply voltage to reduce DRAM energy consumption
while keeping performance loss under a bound.}
To this end, we build an FPGA-based testing platform based on
SoftMC~\cite{hassan-hpca2017}
that allows us to tune the DRAM
supply voltage and change DRAM timing parameters (i.e., the amount of time the
memory controller waits for a DRAM operation to complete). We perform an
experimental study on 124 real 4Gb DDR3L (low-voltage) DRAM chips manufactured
recently (between 2014 and 2016) by three major DRAM vendors. Our extensive
experimental characterization yields four major observations on how DRAM
\highlight{latency, reliability, and data retention are} affected by reduced
voltage.
Based on our experimental observations, we propose a new low-cost DRAM energy
\fix{reduction} mechanism called \emph{Voltron}. The key idea of
Voltron is to use a performance model to determine \fixIV{by} how much we can
reduce the DRAM array voltage at runtime without introducing errors and
without exceeding a user-specified threshold for \fixII{acceptable}
performance loss.
\section{Related Work}
\response{To our knowledge, this is the \fixIII{first work}} to \myitem{i}
experimentally \changes{characterize the reliability and performance of modern
low-power DRAM chips under different supply voltages}, and \myitem{ii}
\fixIV{introduce} a new mechanism that reduces DRAM energy while retaining high
memory \fixIV{data} throughput by \fixIV{adjusting} the DRAM array voltage. We
briefly discuss other prior work in DRAM energy reduction.
\paratitle{DRAM Frequency and Voltage Scaling} Many prior works propose to
reduce DRAM energy by adjusting the memory channel frequency and/or the DRAM
supply voltage dynamically. Deng et al.~\cite{deng-asplos2011} propose MemScale,
which scales the frequency of DRAM at runtime based on a performance predictor
of an in-order processor. Other work focuses on developing management policies
to improve system energy efficiency by coordinating DRAM \emph{DFS} with DVFS on
the CPU~\cite{deng-micro2012, deng-islped2012, begum-iiswc2015} or
GPU~\cite{paul-isca2015}. In addition to frequency scaling, David et
al.~\cite{david-icac2011} propose to scale the DRAM supply voltage along with
the memory channel frequency, based on the memory bandwidth utilization of
applications.
In contrast to all these works, our work focuses on a detailed experimental
characterization of real DRAM chips as the supply voltage varies. Our study
provides fundamental observations for potential mechanisms that can mitigate
DRAM and system energy consumption. Furthermore, frequency scaling hurts memory
throughput, and thus significantly degrades \fixIV{the} system performance of
\fixIV{especially} memory-intensive workloads (see \new{Section 2.4 in our
SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017}} for
our quantitative analysis). We demonstrate the importance \fix{and benefits}
of \fix{exploiting} our \fix{experimental} observations by proposing \voltron,
one \fixIV{new example} mechanism that uses our observations to reduce DRAM and
system energy without sacrificing memory throughput.
\paratitle{Low-Power Modes for DRAM} Modern DRAM chips support various low-power
standby modes. \fix{Entering and exiting these modes incurs some amount of
latency, which delays memory requests that must be serviced.} To increase the
opportunities to exploit these low-power modes, several prior works propose
mechanisms that increase periods of memory idleness through data placement
(e.g.,~\cite{lebeck-asplos2000, fan-islped2001}) and memory traffic reshaping
(e.g.,~\cite{aggarwal-hpca2008, bi-hpca2010, amin-islped2010, lyuh-dac2004,
diniz-isca2007}). Exploiting low-power modes is orthogonal to our work on
studying the impact of \fixIV{reduced-voltage operation in DRAM}. Furthermore,
low-power modes have a smaller effect on memory-intensive workloads, which
exhibit little idleness in memory accesses, \fixIV{whereas, as we \new{show in
Section~\ref{sec:voltron:eval}}, our mechanism is especially effective on memory-intensive
workloads.}
\paratitle{Low-Power DDR DRAM Chips} Low-power DDR (LPDDR)~\cite{jedec-lpddr3,
jedec-lpddr4, patel-isca2017} is a specific type of DRAM that is optimized for
low-power systems like mobile devices. To reduce power consumption, LPDDRx
(currently in its 4th generation) employs a few major design changes that differ
from conventional DDRx chips. First, LPDDRx uses a low-voltage swing I/O
interface that consumes 40\% less I/O power than DDR4
DRAM~\cite{choi-memcon2013}. Second, it supports additional low-power modes with
a lower supply voltage. Since the LPDDRx array design remains the same as DDRx,
our observations on the correlation between access latency and array voltage are
applicable to LPDDRx DRAM \fixIV{as well}. \voltron, our proposal, can provide
significant benefits in LPDDRx, \fixIII{since array} energy consumption is
significantly \emph{higher} than the energy consumption of \fixIII{peripheral
circuitry in LPDDRx chips}~\cite{choi-memcon2013}. \changes{We leave the
detailed evaluation of LPDDRx chips for future work since our current
experimental platform is not capable of evaluating them. \new{\ch{Two recent
experimental works~\cite{patel-isca2017, kim-hpca2018} examine} the retention time behavior
of LPDDRx chips and find it to be similar to DDRx chips.}}
\paratitle{Low-Power DRAM Architectures} \fixIV{Prior} works
(e.g.,~\cite{udipi-isca2010, zhang-isca2014, cooper-balis-ieeemicro2010,
chatterjee-hpca2017}) propose to modify the DRAM chip architecture to reduce
the \act power by activating only a fraction of a row instead of the entire row.
Another common technique, called sub-ranking or \new{mini-ranks}, reduces dynamic DRAM power by
accessing data from a subset of chips from a DRAM module~\cite{zheng-micro2008,
yoon-isca2011, ware-iccd2006}. \fix{A couple of} prior
works~\cite{malladi-isca2012, yoon-isca2012} propose DRAM module architectures
that integrate many low-frequency LPDDR chips to enable DRAM power reduction.
These proposed changes to DRAM chips or DIMMs are orthogonal to our work.
\fix{ \paratitle{Reducing Refresh Power} In modern DRAM chips, although
different DRAM cells have widely different retention times~\cite{liu-isca2013,
kim-edl2009, patel-isca2017}, memory controllers conservatively refresh
\emph{all} of the cells based on the retention time of a small fraction of weak
cells, which have the longest retention time out of all of the cells. To reduce
DRAM refresh power, many prior works (e.g., \new{\cite{liu-isca2012,
agrawal-hpca2014, qureshi-dsn2015, liu-isca2013, baek-tc2014,
khan-micro2017, mutlu-date2017, superfri,
venkatesan-hpca2006,bhati-isca2015,lin-iccd2012,ohsawa-islped1998,
patel-isca2017, khan-sigmetrics2014, khan-dsn2016, khan-cal2016}}) propose
mechanisms to reduce unnecessary refresh operations, and, thus, refresh power,
by characterizing the retention time profile (i.e., the \fixIV{data} retention
behavior of each cell) within the DRAM chips. However, these techniques do not
reduce the power of \emph{other} DRAM operations, and these prior works do
\emph{not} provide an experimental characterization of the effect of reduced
voltage level\fixIV{s} on \fixIV{data} retention time.
\paratitle{Improving DRAM Energy Efficiency by Reducing Latency or Improving
Parallelism} \fixIV{Various} prior works (e.g.,\new{~\cite{hassan-hpca2017,
lee-hpca2015, chang-hpca2014, lee-thesis2016, lee-hpca2013, hassan-hpca2016,
seshadri-micro2017, seshadri-micro2015, mutlu-imw2013,
lee-sigmetrics2017,kim-isca2012, lee-taco2016,
seshadri-micro2013, chang-hpca2016, lee-pact2015}}) improve DRAM energy
efficiency by reducing the execution time through techniques that reduce the
DRAM access latency or improve parallelism between memory requests. These
mechanisms are orthogonal to ours, because they do \fixIV{not reduce} the
voltage level of DRAM. }
\new{
\paratitle{Improving Energy Efficiency by Processing in Memory} Various prior
works~\cite{ahn-isca2015,ahn-isca2015-2,7056040,7429299,guo-wondp14,592312,
seshadri-cal2015,mai-isca2000,draper-ics2002,
seshadri-micro2015,hsieh-iccd2016,hsieh-isca2016,
amirali-cal2016, stone-1970, fraguela-2003,375174,808425,
4115697,694774,sura-2015,zhang-2014,akin-isca2015,
babarinsa-2015,7446059,6844483,pattnaik-pact2016, seshadri-thesis2016,
seshadri-micro2017, chang-hpca2016,kim-apbc2018,hashemi-isca2016, ami-asplos2018} examine
processing in memory to improve energy efficiency. Our analyses and techniques
can be combined with these works to enable low-voltage operation in
processing-in-memory engines.
}
\fixIII{\paratitle{Experimental Studies of DRAM Chips} Recent works
\fixIV{experimentally investigate various} reliability, \fixIV{data} retention,
and latency characteristics of modern DRAM chips~\cite{liu-isca2012,
liu-isca2013, kim-isca2014, chang-sigmetrics2016, lee-hpca2015,
lee-sigmetrics2017, chandrasekar-date2014, khan-sigmetrics2014,
jung-memsys2016, jung-patmos2016,
hassan-hpca2017,khan-dsn2016,patel-isca2017,lee-thesis2016,kim-thesis,
meza-dsn2015, schroeder-sigmetrics2009, sridharan-asplos2015,
sridharan-sc2012} \new{usually using FPGA-based DRAM testing infrastructures,
like SoftMC~\cite{hassan-hpca2017}, or using large-scale data from the field}.
None of these works study these characteristics under reduced-voltage operation,
which we do in this paper.}
\fixIII{\paratitle{Reduced-Voltage Operation in SRAM Caches} Prior works propose
different techniques to enable SRAM caches to operate under reduced voltage
levels
(e.g.,~\cite{alameldeen-isca2011,alameldeen-tc2011,wilkerson-ieeemicro2009,chishti-micro2009,roberts-dsd2007,wilkerson-isca2008}).
These works are orthogonal to our experimental study because we focus on
\fixIV{understanding and enabling reduced-voltage operation in} DRAM, which is a
\fixIV{significantly} different memory technology than SRAM.}
\section{Characterization of DRAM Under \\ Reduced Supply Voltage}
\label{sec:results}
In this section, we briefly summarize our four major observations from our
detailed experimental characterization of 31 commodity DRAM modules, also called
DIMMs, from three vendors, when \changes{the DIMMs} operate under \fix{reduced}
supply voltage (i.e., below the nominal voltage level of 1.35V). Each DIMM
comprises 4 DDR3L DRAM chips, totaling to 124 chips for 31 DIMMs. Each chip has
a 4Gb density. Thus, \fixII{each} of our DIMMs \fixII{has} a 2GB capacity.
\tabref{dimm_list} describes the relevant information about \fixIV{the} tested
DIMMs. For a complete discussion on all of our observations and experimental
methodology, we refer the reader to our \new{SIGMETRICS 2017}
paper~\cite{chang-sigmetrics2017}.
\input{sec/tables/dimm_table}
\subsection{DRAM Reliability as Voltage Decreases}
We first study the reliability of DRAM chips under low voltage, which was not
studied by prior works on DRAM voltage scaling
\fix{(e.g.,~\cite{david-icac2011}; see Section 4 for a detailed discussion of
these works).} \figref{dimm_errors_all} shows the fraction of cache lines that
\changes{experience} at least 1~bit of error (i.e., \emph{1~bit flip}) in each
DIMM \changes{(represented by each curve)}, \fixIV{categorized based on vendor.}
\figputHW{dimm_errors_all}{The fraction of erroneous cache lines in each DIMM as
we reduce the supply voltage, with a fixed latency. Reproduced from
\cite{chang-sigmetrics2017}.}
We observe that we can reliably access data when DRAM supply voltage is lowered
below the nominal voltage level, {\em until a certain voltage value}, \vmin,
which is the minimum voltage level at which no bit errors occur. Furthermore, we
find that we can reduce the voltage below \vmin to attain further energy
savings, but that errors start occurring in some of the data read from memory.
However, not all cache lines exhibit errors for all supply voltage values below
\vmin. Instead, the number of erroneous cache lines \changes{for each DIMM}
increases as we reduce the voltage further below \vmin. Specifically, Vendor A's
DIMMs \fix{experience a} near-exponential increase \fix{in} errors as \fix{the}
supply voltage reduces below \vmin. This is mainly due to the
\new{\emph{manufacturing process}~\cite{lee-hpca2015} and \emph{architectural
variation}~\cite{lee-sigmetrics2017}}, which introduces strength and size
variation across the different DRAM cells within a chip.
\new{We make two major conclusions: \myitem{i} the variation of errors due
to reduced-voltage operation across vendors is very significant; and
\myitem{ii} in most cases, there is a significant margin in the voltage
specification, i.e., \vmin for each chip is significantly lower than the
manufacturer-specified supply voltage value.}
\subsection{Longer Access Latency Mitigates \\ Voltage-Induced Errors}
\label{ssec:long_latency}
We observe that while reducing the voltage below \highlight{\vmin introduces bit
errors} in the data, we can prevent these errors if we increase the timing
parameters of three major DRAM operations, i.e., activation, restoration, and
precharge\new{~\cite{hassan-hpca2016,chang-sigmetrics2016,chang-sigmetrics2017,lee-hpca2015,lee-sigmetrics2017}}.\footnote{\new{We refer the reader to our prior
works~\cite{chang-sigmetrics2016, kim-isca2012, lee-hpca2013,
lee-hpca2015, kim-micro2010, kim-hpca2010,chang-hpca2016, hassan-hpca2016,
chang-sigmetrics2017, lee-sigmetrics2017, lee-taco2016, lee-pact2015,
liu-isca2012, liu-isca2013, patel-isca2017, chang-hpca2014,
seshadri-micro2013, seshadri-micro2017, hassan-hpca2017, kim-cal2015,
kim-isca2014, kim-hpca2018} for a detailed background on
DRAM.}}
When the supply voltage is reduced, the DRAM cell capacitor charge takes a
longer time to change, thereby causing these DRAM operations to become slower to
complete. Errors are introduced into the data when the memory controller does
\emph{not} account for this \highlight{slowdown in the DRAM operations}. We
find that if the memory controller allocates extra time for these operations to
finish when the supply voltage is below \vmin, errors no longer occur. We
\highlight{validate, analyze,} and explain this behavior using SPICE simulation
of a detailed circuit-level model, which we have \ch{openly} released
online~\cite{safari-github}. \new{Sections~4.1 and 4.2 of our SIGMETRICS 2017
paper~\cite{chang-sigmetrics2017} provide our extensive circuit-level analyses,
validated using data from real DRAM chips. }
\subsection{Spatial Locality of Errors}
\label{ssec:locality}
While reducing the supply voltage induces errors when the DRAM latency is \emph{not}
long enough, we also show that \new{\emph{not}} all DRAM locations experience errors at all
supply voltage levels. To understand the locality of the errors induced by a low
supply voltage, we show the probability of each DRAM row in a DIMM
\fix{experiencing} at least one bit of error across all experiments.
\figref{locality_C} shows the probability of each row \fix{experiencing} at
least a one-bit error due to reduced voltage in the two representative DIMMs.
For each DIMM, we choose the supply voltage \new{at which} errors start appearing (i.e.,
the voltage \fix{level} one step below \vmin), and we do \emph{not} increase the
DRAM access latency \fixIV{(i.e., \new{keep it at} 10ns for both \trcd and \trp,
\new{which are the activation and precharge timing parameters, respectively})}. The x-axis
and y-axis indicate the bank number and row number (in thousands), respectively.
Our tested DIMMs are divided into eight banks, and each bank consists of
32K~rows of cells. Additional results showing the error locations at different voltage
\fix{levels} are in our \new{SIGMETRICS 2017} paper~\cite{chang-sigmetrics2017}.
\input{sec/subplots/loc}
The major observation is that when only a small number of errors occur due to
reduced supply voltage, these errors tend to \emph{cluster} physically in
certain \emph{regions} of a DRAM chip, as opposed to being randomly distributed
throughout the chip.\footnote{\new{We believe this observation is due to both
process and architectural variation across different regions in the DRAM
chip.}} This observation implies that when we reduce the supply voltage to
the DRAM array, we need to increase the fundamental operation latencies for
\emph{only} the regions where errors can occur.
\subsection{Impact on Refresh Rate}
Commodity DRAM chips guarantee that all cells can safely retain data for 64ms,
after which the cells are \emph{refreshed} to replenish charge that leaks out of
the capacitors\new{~\cite{liu-isca2012,liu-isca2013,chang-hpca2014}}. We observe that
the effect of \fixIV{the supply voltage} on retention times is \emph{not}
statistically significant. Even when we reduce the supply voltage from 1.35V to
1.15V (i.e., a 15\% reduction), the rate at which charge leaks from the
capacitors is so slow that no data is lost during the 64ms refresh interval at
both 20$\celsius$ and 70$\celsius$. Therefore, we conclude that using a reduced
supply voltage does not require any changes to the standard refresh interval at
20$\celsius$ and 70$\celsius$. \new{Detailed results are in Section 4.6 of our
SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017}.}
\subsection{Other Experimental Observations}
We refer the reader to our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017} for
more details on the other two key observations. First, \new{we find that} the
most commonly-used ECC scheme, SECDED~\cite{luo-dsn2014, kang14,
sridharan-asplos2015}, is unlikely to alleviate errors induced by a low supply
voltage. \new{This is because lowering voltage increases the fraction of data that
contains more than two bits of errors, exceeding the one-bit correction
capability of SECDED (see Section 4.4 of our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017})}. Second,
temperature affects the reliable access latency at low supply voltage levels and
the effect is very vendor-dependent (see Section 4.5 of our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017}).
\new{Out of the three major vendors whose DIMMs we evaluate, DIMMs from
two vendors require \ch{longer activation and
precharge latencies} to operate reliably at high temperature under low supply
voltage. The main reason is that DRAM chips become slower at higher
temperature~\cite{lee-sigmetrics2017, lee-hpca2015,chandrasekar-date2014}.}
\section{Significance}
\new{Our SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017} presents a new set of
detailed experimental characterization and \ch{analyses} on the
voltage-latency-reliability trade-offs in modern DRAM chips. In this section, we
describe the potential impact that our study can bring to the research community
and industry.}
\subsection{Potential Industry Impact}
We believe our experimental characterization results and proposed mechanism can
have significant impact in fast-growing data centers \new{as well as mobile
systems}, where DRAM power consumption is growing due to higher demand for
memory capacity for certain types of service (e.g., memcached). To reduce the
energy and power consumed by DRAM, DRAM manufacturers have been decreasing the
supply voltage of DRAM chips with newer DRAM standards (e.g., DDR4) or
low-voltage variants of DDR, such as LPDDR4 (Low-Power DDR4) and DDR3L (DDR3
Low-voltage). However, the supply voltage reduction has been conservative with
each new DDR standard, which takes years to be adopted by the vendors and the
market. For example, since the release of DDR3L (1.35V) in 2010, the supply
voltage has reduced by \emph{only} 11\% with the latest DDR4 standard (1.2V)
released in 2014. Furthermore, since the release of DDR4 in 2014, the supply
voltage for most commodity DDR4 chips has remained at 1.2V. As a result, further
reducing DRAM supply voltage below the standard voltage, \new{as we do in our
SIGMETRICS 2017 paper~\cite{chang-sigmetrics2017}}, can be a \new{very} effective way
of reducing DRAM power consumption. However, to do so, we need to \new{carefully
and rigorously} understand how DRAM
chips behave under reduced-voltage operation.
To enable the development of new mechanisms that leverage reduce-voltage
operation in DRAM, we provide \new{the first set of} comprehensive experimental
results on the effect of using a wide range of different supply voltage values
on the reliability, latency, and retention characteristics of DRAM chips. In
this work, we demonstrate how we can use our experimental data to design a new
mechanism, Voltron (\secref{voltron}), which reduces DRAM energy consumption
through voltage reduction. Therefore, we believe that understanding and
leveraging reduced-voltage operation will help industry improve the energy
efficiency of memory subsystems.
\subsection{Potential Research Impact}
Our paper sheds \new{new} light on the feasibility of enabling reduced-voltage
operation in manufactured DRAM chips. One important research question that our
work raises is \textit{how do modern DRAM chips behave under a wide range of
supply voltage levels?} Existing systems are limited to a few DRAM power
states, which prevent DRAM from serving memory accesses when it enters a
low-power state. However, in our work, we show that it is possible to operate
commodity DRAM chips under a wide range of supply voltage levels while still
being able to serve memory accesses under a different set of trade-offs. To
facilitate further research initiative to exploit reduced-voltage operation in
DRAM chips, we have open-sourced our characterization results, FPGA-based
testing platform~\cite{hassan-hpca2017}, and DRAM SPICE circuit model (for
validation) in our GitHub repository~\cite{safari-github}. We believe that these
tools can be extended for other research objectives besides studying voltage
reduction in DRAM. \new{One potential direction is to leverage our results to
design mechanisms that reduce DRAM latency by operating DRAM at a higher supply
voltage. }
\subsection{Applicability to Other Memory Technologies}
\new{We believe the high-level ideas of our work can be \ch{leveraged} in the context
of other memory technologies, such as NAND flash memory~\cite{cai-ieee2017, cai-ieeearxiv2017, cai-bookchapter2017},
PCM\ch{~\cite{lee-isca2009, lee-ieeemicro2010, qureshi-isca2009,
yoon-taco2014,lee-cacm2010,qureshi-micro2009,yoon-iccd2012, meza-weed2013}},
\mbox{STT-MRAM}\ch{~\cite{ku-ispass2013,guo-isca2009,chang-hpca2013, meza-weed2013,
naeimi-itj2013}},
RRAM~\cite{wong-ieee2012}, \ch{or hybrid memory
systems~\cite{meza-weed2013, qureshi-isca2009, yoon-iccd2012, ramos-ics2011,
zhang-pact2009, li-cluster2017, yu-micro2017, jiang-hpca2010, phadke-date2011,
agarwal-asplos2017, dulloor-eurosys2016, pena-cluster2014, bock-iccd2016,
gai-hpcc2016, liu-iccd2016}}. A recent work on NAND flash memory, for example,
proposes reducing the pass-through voltage~\cite{cai-dsn2015,cai-ieee2017, cai-ieeearxiv2017, cai-bookchapter2017} to
reduce read disturb errors, which in turn saves energy. We refer the reader to
past works on NAND flash memory for a more detailed analysis of
reliability-voltage
trade-offs~\cite{cai-dsn2015,cai-hpca2015,cai-hpca2017,cai-ieee2017, cai-ieeearxiv2017, cai-bookchapter2017}. We hope
our work inspires characterization and understanding of reduced-voltage
operation in other memory technologies, with the goal of enabling a more
energy-efficient system design.}
\ignore{Another key impact of our paper is that we demonstrate the effectiveness
of utilizing real-world characterization to design new optimization techniques
for memory subsystems. Given that a diverse set of new memory technologies
(e.g., STT-RAM) are becoming promising substrates for disrupting the once
clear-cut memory hierarchy, we believe that characterizing these new memory
technologies is a new and important research direction.
We hope that the experimental characterization, analysis, and optimization
techniques presented in this paper will enable the development of other new
mechanisms that can effectively exploit the trade-offs between voltage,
reliability, and latency in DRAM to improve system performance, efficiency,
and/or reliability.
} |
2106.03832 | \section{Introduction}
Let $\mathcal{H}:=(E_t,\{1,-1\}^t)$ be the oriented matroid on its {\em ground set\/}~$E_t:=[t]$ $:=[1,t]:=\{1,\ldots,t\}$, where~$t\geq 3$, and with its set of {\em topes\/} $\{1,-1\}^t$. This oriented matroid is realizable as the {\em arrangement\/} of {\em coordinate hyperplanes\/} in the real Euclidean space~$\mathbb{R}^t\supset\{1,-1\}^t$ of row vectors, see~\cite[Example~4.1.4]{BLSWZ}.
See, e.g.,~\cite{BK,Bo,BS,DRS,K,S,Z} on {\em oriented matroids}.
Each of the $2^t$ {\em maximal covectors\/} $T:=(T(1),\ldots,T(t))\in\{1,-1\}^t$ of~$\mathcal{H}$ can be regarded as the {\em characteristic tope\/} of the {\em negative part\/} $T^-:=\{e\in E_t\colon T(e)=-1\}$. Conversely, given an arbitrary subset $A\subseteq E_t$, we define the {\em characteristic tope\/} of~$A$ to be the {\em reorientation\/}~${}_{-A}\mathrm{T}^{(+)}$ of the {\em positive tope\/}~$\mathrm{T}^{(+)}:=(1,\ldots,1)$ on the subset~$A$; recall that~$({}_{-A}\mathrm{T}^{(+)})^-:=A$.
Let~$\boldsymbol{H}(t,2)$ denote the {\em hypercube graph\/} of {\em topes\/} of the oriented matroid~$\mathcal{H}$, that is, the {\em vertex\/} set of the graph~$\boldsymbol{H}(t,2)$ is the set~$\{1.-1\}^t$, and the {\em edges\/} of~$\boldsymbol{H}(t,2)$ are the
pairs $\{T',T''\}\subset\{1,-1\}^t$, such that $|\{e\in E_t\colon T'(e)\neq T''(e)\}|=1$.
Let $\boldsymbol{R}:=(R^0,R^1,\ldots,R^{2t-1},R^0)$ be a distinguished {\em symmetric cycle\/} in the graph~$\boldsymbol{H}(t,2)$, where
\begin{equation}
\label{eq:12}
\begin{split}
R^0:\!&=\mathrm{T}^{(+)}\; ,\\
R^s:\!&={}_{-[s]}R^0\; ,\ \ \ 1\leq s\leq t-1\; ,\\
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:13}
R^{t+k}:=-R^k\; ,\ \ \ 0\leq k\leq t-1\; .
\end{equation}
For any vertex $T\in\{1,-1\}^t$ of the
graph~$\boldsymbol{H}(t,2)$, there exists a {\em unique inclusion-minimal\/} subset
\begin{equation}
\label{eq:37}
\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathrm{V}(\boldsymbol{R}):=(R^0,R^1,\ldots,R^{2t-1})
\end{equation}
of the vertex sequence $\mathrm{V}(\boldsymbol{R})$ of the cycle~$\boldsymbol{R}$,
such that
\begin{equation}
\label{eq:32}
T=\sum_{Q\in\boldsymbol{Q}(T,\boldsymbol{R})}Q\; ,
\end{equation}
see~\cite[\S{}11.1]{M-PROM}. This subset $\boldsymbol{Q}(T,\boldsymbol{R})\subset\mathbb{R}^t$ is {\em linearly independent}, and it contains an {\em odd\/} number~$\mathfrak{q}(T):=\mathfrak{q}(T,\boldsymbol{R}):=|\boldsymbol{Q}(T,\boldsymbol{R})|$ of topes.
In fact, the linear algebraic decomposition~(\ref{eq:32}) is just a way to describe a particular mechanism of {\em majority voting}.
Let~$\boldsymbol{\sigma}(e)$ denote the $e$th standard unit vector
of the space~$\mathbb{R}^t$, $e\in[t]$. The bijections
\begin{align}
\label{eq:8}
\{1,-1\}^t&\to\{0,1\}^t\colon & T&\mapsto\tfrac{1}{2}(\mathrm{T}^{(+)}-T)\; ,\\
\intertext{and}
\label{eq:9}
\{0,1\}^t&\to\{1,-1\}^t\colon & \widetilde{T}&\mapsto\mathrm{T}^{(+)}-2\widetilde{T}\; ,
\end{align}
between the vertex set $\{1,-1\}^t$ of the hypercube graph~$\boldsymbol{H}(t,2)$ and the vertex set $\{0,1\}^t$ of the hypercube graph~$\widetilde{\boldsymbol{H}}(t,2)$ allow us to associate with the symmetric cycle~$\boldsymbol{R}$ in the graph~$\boldsymbol{H}(t,2)$ a symmetric cycle~$\widetilde{\boldsymbol{R}}:=(\widetilde{R}^0,\widetilde{R}^1,\ldots,\widetilde{R}^{2t-1},\widetilde{R}^0)$ in the graph~$\widetilde{\boldsymbol{H}}(t,2)$, where
\begin{equation*}
\begin{split}
\widetilde{R}^0:\!&=(0,\ldots,0)\; ,\\
\widetilde{R}^s:\!&=\sum\nolimits_{e\in[s]}\boldsymbol{\sigma}(e)\; ,\ \ \ 1\leq s\leq t-1\; ,
\end{split}
\end{equation*}
and
\begin{equation*}
\widetilde{R}^{t+k}:=\mathrm{T}^{(+)}-\widetilde{R}^k\; ,\ \ \ 0\leq k\leq t-1\; .
\end{equation*}
For any vertex~$\widetilde{T}$ of the hypercube graph~$\widetilde{\boldsymbol{H}}(t,2)$, let us define a subset~$\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})\subset\mathrm{V}(\widetilde{\boldsymbol{R}}):=
(\widetilde{R}^0,\widetilde{R}^1,\ldots,\widetilde{R}^{2t-1})$ indirectly, via the mapping
\begin{equation*}
\widetilde{T}\; \overset{(\ref{eq:9})}{\mapsto}\; T\; ,
\end{equation*}
and via the bijection
\begin{equation*}
\boldsymbol{Q}(T,\boldsymbol{R})\; \xrightarrow{(\ref{eq:8})}\;
\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})\; .
\end{equation*}
Involving the quantity~$\mathfrak{q}(\widetilde{T}):=\mathfrak{q}(\widetilde{T},\widetilde{\boldsymbol{R}})
:=|\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})|=\mathfrak{q}(T)$,
we can write down the decomposition
\begin{equation}
\label{eq:69}
\widetilde{T}=-\tfrac{1}{2}(\mathfrak{q}(\widetilde{T})-1)\cdot\mathrm{T}^{(+)}
+\sum_{\substack{\widetilde{Q}\in\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})\colon\\
\widetilde{Q}\neq(0,\ldots,0)=:\widetilde{R}^0}}\widetilde{Q}\; ,
\end{equation}
that describes yet another mechanism of {\em majority voting},\footnote{$\quad$Let $\mathbf{2}^{[t]}$ denote the {\em power set\/} (i.e., the family of all subsets) of $E_t$. Recall that maps
\begin{align*}
f\colon\{0,1\}^t\;&\to\;\mathbb{R}\; , & -\tfrac{1}{2}(\mathfrak{q}(\widetilde{T})-1)\cdot\mathrm{T}^{(+)}
+\sum_{\substack{\widetilde{Q}\in\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})\colon\\
\widetilde{Q}\neq(0,\ldots,0)=:\widetilde{R}^0}}\widetilde{Q}\;=\;\widetilde{T}\,\;&\mapsto\,\; f(\widetilde{T})\; ,\\
\intertext{and}
g\colon\{1,-1\}^t\;&\to\;\mathbb{R}\; , & \sum_{Q\in\boldsymbol{Q}(T,\boldsymbol{R})}Q\;=\;T\,\;&\mapsto\,\; g(T)\; ,
\end{align*}
are {\em pseudo-Boolean functions}. Maps $\mathfrak{f}\colon\mathbf{2}^{[t]}\to\mathbb{R}$ are {\em set functions}. Maps $f\colon\{0,1\}^t\to\{0,1\}$, and $g\colon\{1,-1\}^t\to\{1,-1\}$ are {\em Boolean functions}.
}
but this decomposition has no essential meaning from the linear algebraic viewpoint, since the set~$\widetilde{Q}\in\widetilde{\boldsymbol{Q}}(\widetilde{T},\widetilde{\boldsymbol{R}})$ can contain the origin~$(0,\ldots,0)=:\widetilde{R}^0$
of the space~$\mathbb{R}^t$, which should be omitted in calculations.
For the topes $T\in\{1,-1\}^t$ of the oriented matroid $\mathcal{H}$, we define topes $\ro(T)\in\{1,-1\}^t$ by\footnote{
$\quad$$\ro(T)$ means the {\em relabeled opposite\/} of~$T$.}
\begin{equation}
\label{eq:6}
\ro(T):=-T\,\overline{\mathbf{U}}(t)\; ,
\end{equation}
where $\overline{\mathbf{U}}(t)$ denotes\footnote{$\quad$In~\cite[Sect.~2.1]{M-PROM} the similar notation $\mathbf{U}(m)$ was used to denote the backward identity matrix of order $(m+1)$ whose rows and columns were indexed starting with zero.} the {\em backward identity matrix\/} (with the rows and columns indexed starting with $1$) of order $t$ whose $(i,j)$th
entry is the Kronecker delta~$\delta_{i+j,t+1}$.
For vertices $\widetilde{T}$ of the discrete hypercube $\{0,1\}^t$, the counterparts of topes $\ro(T)$ of the oriented matroid~$\mathcal{H}$ are vertices $\rn(\widetilde{T})\in\{0,1\}^t$, defined by\footnote{$\quad$$\rn(\widetilde{T})$ means the {\em relabeled negation\/} of~$\widetilde{T}$.}
\begin{equation}
\label{eq:35}
\rn(\widetilde{T}):=\mathrm{T}^{(+)}-\widetilde{T}\,\overline{\mathbf{U}}(t)\; .
\end{equation}
For example, suppose{\small
\begin{align*}
T:\!&=(1,-1,\phantom{-}1, -1,-1)\in\{1,-1\}^5\; ,\\
\widetilde{T}:\!&=(0,\phantom{-}1,\phantom{-}0, \phantom{-}1,\phantom{-}1)\in\{0,1\}^5\; .\\
\intertext{\normalsize{Then we have}}
\ro(T)&=(1,\phantom{-}1, -1,\phantom{-}1,-1)\; ,\\
\rn(\widetilde{T})&=(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1)\; .
\end{align*}
}
\noindent$\bullet$ In the first part of the paper we compare the decompositions~$\boldsymbol{Q}(T,\boldsymbol{R})$ and~$\boldsymbol{Q}(\ro(T),\boldsymbol{R})$ of topes $T$ and $\ro(T)$ with respect to the symmetric cycle~$\boldsymbol{R}$ in the graph~$\boldsymbol{H}(t,2)$, defined by~(\ref{eq:12})(\ref{eq:13}).
Our interest in considering {\em relabeled opposites\/}~$\ro(T)$ and
{\em relabeled negations\/}~$\rn(\widetilde{T})$ lies in their application to combined {\em blocking/voting-models\/} of increasing families of sets and of clutters. We study those impractical $2^t$-dimensional vector models in order to gain a better understanding of the structure of families.
Recall that a family $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}\subset\mathbf{2}^{[t]}$ of subsets\footnote{$\quad$We denote by $\hat{0}$ the {\em empty subset\/} of the ground set~$E_t$, and we let $\emptyset$ denote the {\em empty family} containing no sets.
Given a family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$, such that $\emptyset\neq\mathcal{F}\not\ni\hat{0}$, the set $E_t:=[t]$ is the {\em ground set\/} of $\mathcal{F}$, while the set $\mathrm{V}(\mathcal{F}):=\bigcup_{F\in\mathcal{F}}F\subseteq E_t$ is
the {\em vertex set\/} of $\mathcal{F}$.
The families $\emptyset$ and $\{\hat{0}\}$ are the two {\em trivial clutters\/} on the ground set~$E_t$. The other clutters on $E_t$ are {\em nontrivial}.
} of the ground set~$E_t$ is called a {\em clutter}\footnote{$\quad$Or {\em Sperner family}, {\em antichain}, {\em simple hypergraph}.} if {\em no set\/} $A_i$ from $\mathcal{A}$ contains another set~$A_j$.
Given a family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$, we let $\bmin\mathcal{F}$ denote the clutter composed of the {\em inclusion-minimal\/} sets in $\mathcal{F}$.
We say that a family of subsets $\mathcal{F}\subseteq\mathbf{2}^{[t]}$
is an {\em increasing family}\footnote{$\quad$Or {\em up-set}, {\em upward-closed family of sets}, {\em filter of sets}.} if the following implications hold:
\begin{equation*}
A\in\mathcal{F},\ \ \mathbf{2}^{[t]}\ni B\supset A\ \ \ \Longrightarrow\ \ \ B\in\mathcal{F}\; .
\end{equation*}
If $C\subseteq E_t$, then the family $\{C\}^{\triangledown}:=\{D\subseteq E_t\colon D\supseteq C\}$ is called the {\em principal\/} increasing family generated by the {\em one-member\/} clutter $\{C\}$. Conversely, an increasing family~$\mathcal{F}\subseteq\mathbf{2}^{[t]}$ is said to be {\em principal\/} if\footnote{$\quad$We denote by $|\!\cdot\!|$ the cardinality of a set, and we denote by $\#\,\cdot$ the number of sets in a family.} $\#\bmin\mathcal{F}=1$.
Given an arbitrary nonempty family $\mathcal{C}\subseteq\mathbf{2}^{[t]}$, we
denote by~$\mathcal{C}^{\triangledown}$ the {\em increasing family\/} on $E_t$, generated by~$\mathcal{C}$:
\begin{equation*}
\mathcal{C}^{\triangledown}:=\bigcup\nolimits_{C\in\mathcal{C}}\{C\}^{\triangledown}
=\bigcup\nolimits_{C\in\bmin\mathcal{C}}\{C\}^{\triangledown}\; .
\end{equation*}
``Decreasing'' constructs are defined in the obvious similar way.\footnote{
$\quad$For a family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$, we use the notation $\bmax\mathcal{F}$ to denote the clutter composed of the {\em inclusion-maximal\/} sets in $\mathcal{F}$.
A family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$ is said to be
a {\em decreasing family\/} (or
{\em down-set}, {\em downward-closed family of sets}, {\em ideal of sets}) if the following implications hold:
\begin{equation*}
B\in\mathcal{F},\ \ A\subset B\ \ \ \Longrightarrow\ \ \ A\in\mathcal{F}\; .
\end{equation*}
If $\emptyset\neq\mathcal{F}\neq\{\hat{0}\}$, then this decreasing family is the {\em abstract simplicial complex\/} on its {\em vertex set\/} $\bigcup_{M\in\bmax\mathcal{F}}M$, with the {\em facet\/} family~$\bmax\mathcal{F}$.
If $D\subseteq E_t$, then the family $\{D\}^{\vartriangle}:=\{C\colon C\subseteq D\}$ is called the {\em principal\/} decreasing family generated by the {\em one-member\/} clutter $\{D\}$. Conversely, a decreasing family~$\mathcal{F}\subseteq\mathbf{2}^{[t]}$ is said to be {\em principal\/} if $\#\bmax\mathcal{F}=1$.
Given an arbitrary nonempty family $\mathcal{D}\subseteq\mathbf{2}^{[t]}$, we
denote by~$\mathcal{D}^{\vartriangle}$ the {\em decreasing family\/} on $E_t$, generated by~$\mathcal{D}$:
\begin{equation*}
\mathcal{D}^{\vartriangle}:=\bigcup\nolimits_{D\in\mathcal{D}}\{D\}^{\vartriangle}=\bigcup\nolimits_{D\in\bmax\mathcal{D}}\{D\}^{\vartriangle}\; .
\end{equation*}
}
The duality philosophy behind clutters and increasing families is that any clutter is the {\em blocker}\footnote{$\quad$
\vspace{-5.5mm}
\begin{quote}
{\sl I enjoyed working with Ray} [{\sl{}Fulkerson}] {\sl and I coined the terms ``clutter'' and~``blocker''.}
\hfill Jack~Edmonds~\cite[p.~201]{50Y}
\end{quote}
} of a unique clutter, and any increasing family is the family of {\em blocking sets\/} of a unique clutter.
We often meet in the literature the {\em free distributive lattice\/} of antichains in the Boolean lattice of subsets of a finite nonempty set, ordered by containment of the corresponding generated {\em order ideals}, but an intrinsically related construct, the {\em free distributive lattice\/} of those antichains ordered by containment of the corresponding generated {\em order filters\/} has greater discrete mathematical expressiveness, because the latter lattice can be interpreted as the {\em lattice\/} of {\em blockers}, for which the {\em blocker map\/} is its anti-automorphism.\footnote{$\quad$For this paper, we chose the language of {\em power sets}, {\em clutters}, and {\em increasing\/} and {\em decreasing families}. A parallel exposition could be presented in poset-theoretic terms of {\em Boolean lattices}, {\em antichains}, and {\em order filters\/} and {\em ideals}.}
Recall that a subset $B\subseteq E_t$ is called a {\em blocking set\footnote{$\quad$Or
{\em transversal}, {\em hitting set}, {\em vertex cover} (or {\em node cover}), {\em system of representatives}.}\/} of a subset family~\mbox{$\mathcal{F}\subset\mathbf{2}^{[t]}$,} where~$\emptyset\neq\mathcal{F}\not\ni\hat{0}$, if we have
\begin{equation*}
|B\cap F|>0\; ,
\end{equation*}
for each set $F\in\mathcal{F}$. The {\em blocker}\footnote{$\quad$Or {\em blocking hypergraph} (or {\em transversal hypergraph}), {\em blocking clutter}, {\em dual clutter}, {\em Alexander dual clutter}.} $\mathfrak{B}(\mathcal{F})$ of the family~$\mathcal{F}$ is
the family of all {\em inclusion-minimal blocking sets\/} of $\mathcal{F}$; note that we have
$\mathfrak{B}(\mathcal{F})=\mathfrak{B}(\bmin\mathcal{F})$. The notation~$\mathfrak{B}(\mathcal{F})^{\triangledown}$ just means the increasing family of all blocking sets of the family~$\mathcal{F}$.
For a nonempty family of subsets $\mathcal{F}\subseteq\mathbf{2}^{[t]}$,
we define a family\footnote{$\quad$Given a nonempty family of subsets $\mathcal{F}\subseteq\mathbf{2}^{[t]}$,
we define a family of complements~$\mathcal{F}^{\bot}$ by $\mathcal{F}^{\bot}:=\{F^{\bot}\colon F\in\mathcal{F}\}$, where $F^{\bot}:=
\mathrm{V}(\mathcal{F})-F$.
}
of complements $\mathcal{F}^{\compl}$ by $\mathcal{F}^{\compl}:=\{F^{\compl}\colon F\in\mathcal{F}\}$, where $F^{\compl}:=E_t-F$.
Given a nontrivial clutter $\mathcal{A}\subset\mathbf{2}^{[t]}$, one associates with $\mathcal{A}$ the four
extensively studied partitions of the power set of the ground set $E_t$:
\begin{align}
\label{eq:22}
\mathbf{2}^{[t]}&=\mathcal{A}^{\triangledown}\ \dot{\cup}\ (\mathfrak{B}(\mathcal{A})^{\compl})^{\vartriangle}\; ,\\
\nonumber
\mathbf{2}^{[t]}&=\mathcal{A}^{\vartriangle}\ \dot{\cup}\ \mathfrak{B}(\mathcal{A}^{\compl})^{\triangledown}\; ,\\
\nonumber
\mathbf{2}^{[t]}&=\mathfrak{B}(\mathcal{A})^{\triangledown}\ \dot{\cup}\ (\mathcal{A}^{\compl})^{\vartriangle}\; ,\\
\intertext{and}
\nonumber
\mathbf{2}^{[t]}&=\mathfrak{B}(\mathcal{A})^{\vartriangle}\ \dot{\cup}\ \mathfrak{B}(\mathfrak{B}(\mathcal{A})^{\compl})^{\triangledown}\; .
\end{align}
\noindent$\bullet$ In the second part of the paper we arrange the subsets of the ground set~$E_t$ in linear order. We then turn to the so-called {\em characteristic vectors\/} $\boldsymbol{\gamma}(\mathcal{F})\in\{0,1\}^{2^t}$ of subset families~$\mathcal{F}\subseteq\mathbf{2}^{[t]}$. If~$\mathcal{A}\subset\mathbf{2}^{[t]}$ is a nontrivial clutter on~$E_t$, then relation~(\ref{eq:22}) reformulated in the form (cf.~(\ref{eq:35}))
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
=\mathrm{T}_{2^t}^{(+)}-\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\; ,
\end{equation*}
where $\mathrm{T}_{2^t}^{(+)}:=(1,\ldots,1)$ is the $2^t$-dimensional row vector of all $1$'s, provides us with the characteristic vector
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})=\rn(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))
\end{equation*}
of the increasing family of blocking sets~$\mathfrak{B}(\mathcal{A})^{\triangledown}$ of the clutter~$\mathcal{A}$.
\noindent$\bullet$ In the third part of the paper we mention a blocking/voting-connection of the characteristic vectors $\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})$ and $\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})$
with the decompositions of the corresponding characteristic topes of the increasing families~$\mathcal{A}^{\triangledown}$ and~$\mathfrak{B}(\mathcal{A})^{\triangledown}$ with respect to a distinguished symmetric cycle in the hypercube graph~$\boldsymbol{H}(2^t,2)$, which is analogous to the cycle~(\ref{eq:12})(\ref{eq:13}) in the graph~$\boldsymbol{H}(t,2)$.
\part*{Decomposing}
\section{Topes, their relabeled opposites, and decompositions}
In this section we consider vertices $T$ of the discrete hypercube~$\{1,-1\}^t$, their relabeled opposites~$\ro(T)$ defined by~(\ref{eq:6}), and we discuss basic properties of the decompositions $\boldsymbol{Q}(T,\boldsymbol{R})$ and $\boldsymbol{Q}(\ro(T),\boldsymbol{R})$ of the topes~$T$ and~$\ro(T)$ with respect to the distinguished symmetric cycle~$\boldsymbol{R}:=(R^0,R^1,\ldots,R^{2t-1},R^0)$ in the graph~$\boldsymbol{H}(t,2)$, defined by~(\ref{eq:12})(\ref{eq:13}).
\noindent$\bullet$ Definitions~(\ref{eq:6}) and~(\ref{eq:35}) determine the maps
\begin{align}
\label{eq:10}
\{1,-1\}^t &\to \{1,-1\}^t: & T &\mapsto \ro(T):=\phantom{\mathrm{T}^{(+)}}-T\,\overline{\mathbf{U}}(t)\; ,\\
\label{eq:11}
\{0,1\}^t &\to \{0,1\}^t: & \widetilde{T} &\mapsto \rn(\widetilde{T})
:=\mathrm{T}^{(+)}-\widetilde{T}\,\overline{\mathbf{U}}(t)\; ,
\end{align}
and since we deal with the standard one-to-one correspondences between the vertex sets of the discrete hypercubes $\{1,-1\}^t$ and $\{0,1\}^t$, established by means of the maps~(\ref{eq:8}) and~(\ref{eq:9}), we mention the mappings
\begin{align}
\nonumber
\{1,-1\}^t\ni \ro(T)\ &\overset{(\ref{eq:8})}{\mapsto}\ \rn(\widetilde{T})=\tfrac{1}{2}(\mathrm{T}^{(+)}+T\,\overline{\mathbf{U}}(t))\in\{0,1\}^t\; ,\\
\intertext{and}
\label{eq:36}
\{0,1\}^t\ni\rn(\widetilde{T})\ &\overset{(\ref{eq:9})}{\mapsto}\ \ro(T)=-\mathrm{T}^{(+)}+2\widetilde{T}\,\overline{\mathbf{U}}(t)\in\{1,-1\}^t\; .
\end{align}
\noindent$\bullet$ Of course, the maps~(\ref{eq:10}) and~(\ref{eq:11}) are both {\em involutions}:
\begin{equation*}
\{1,-1\}^t\ni\ro(\ro(T))=T\; ,\ \ \ \text{and}\ \ \ \ \{0,1\}^t\ni\rn(\rn(\widetilde{T}))=\widetilde{T}\; .
\end{equation*}
\noindent$\bullet$ Given a vector $\boldsymbol{z}:=(z_1,\ldots,z_t)\in\mathbb{R}^t$, we denote its {\em support} $\{e\in E_t\colon z_e\neq 0\}$
by~$\supp(\boldsymbol{z})$. For a vertex~$\widetilde{T}$ of the discrete hypercube~$\{0,1\}^t$, we let~$\hwt(\widetilde{T})$ denote its {\em Hamming weight}: $\hwt(\widetilde{T}):=|\supp(\widetilde{T})|$.
Note that we have
\begin{align*}
\{1,-1\}^t\ni \ro(T)=T\ \ \ &\Longleftrightarrow\ \ \ -T=T\,\overline{\mathbf{U}}(t)\; ;\\
\ro(T)=T\ \ \ &\,\Longrightarrow\ \ \ |T^-|=\tfrac{t}{2}\; .\\
\intertext{We also have}
\{0,1\}^t\ni\rn(\widetilde{T})=\widetilde{T}\ \ \ &\Longleftrightarrow\ \ \ \mathrm{T}^{(+)}-\widetilde{T}=\widetilde{T}\,\overline{\mathbf{U}}(t)\; ;\\
\rn(\widetilde{T})=\widetilde{T}\ \ \ &\,\Longrightarrow\ \ \
\hwt(\widetilde{T})
=\tfrac{t}{2}\; .
\end{align*}
Thus, if $t$ is {\em odd}, then we always have $\ro(T)\neq T$, and $\rn(\widetilde{T})\neq\widetilde{T}$.
\newpage
\noindent$\bullet$ Let $\langle\cdot,\cdot\rangle$ denote the standard scalar product on the space~$\mathbb{R}^t$.
For vertices~$\widetilde{T}\in\{0,1\}^t$ and $T:={}_{-\supp(\widetilde{T})}\mathrm{T}^{(+)}\in\{1,-1\}^t$,
we have
\begin{multline*}
\langle T,\ro(T)\rangle=-T\,\overline{\mathbf{U}}(t)T^{\top}
=-\sum_{e\in[t]}T(e)T(t-e+1)
\\
=\begin{cases}
-1-2\sum_{e\in[(t-1)/2]}T(e)T(t-e+1)\; , & \text{if $t$ is odd},\\
\quad\vspace{-3mm}\\
\phantom{-1}-2\sum_{e\in[t/2]}T(e)T(t-e+1)\; , & \text{if $t$ is even};
\end{cases}
\end{multline*}
\begin{multline*}
\langle\widetilde{T},\rn(\widetilde{T})\rangle:=\langle\widetilde{T},\mathrm{T}^{(+)}-\widetilde{T}\,\overline{\mathbf{U}}(t)\rangle
=\hwt(\widetilde{T})-\sum_{e\in[t]}\widetilde{T}(e)\widetilde{T}(t-e+1)\\
=\hwt(\widetilde{T})-\begin{cases}
\widetilde{T}((t+1)/2)+2\sum_{e\in[(t-1)/2]}\widetilde{T}(e)\widetilde{T}(t-e+1)\; , & \text{if $t$ is odd},\\
\quad\vspace{-3mm}\\
\phantom{T((t+1)/2)+}\ \, 2\sum_{e\in[t/2]}\widetilde{T}(e)\widetilde{T}(t-e+1)\; , & \text{if $t$ is even}.
\end{cases}
\end{multline*}
\noindent$\bullet$ Given two {\em words\/} $X,Y\in\{-1,0,1\}^t$, we let $d(X,Y):=|\{e\in E_t\colon X(e)\neq Y(e)\}|$ denote the {\em Hamming distance\/} between them.\footnote{$\quad$If $X$ and $Y$ are {\em topes}, then one says that $d(X,Y)$ is the {\em graph distance}.}
Since the equal distances $d(T,\ro(T))=d(\widetilde{T},\rn(\widetilde{T}))$ can be calculated with the help of the formulas (see~(\ref{eq:9}) and~(\ref{eq:36}))
\begin{align*}
d(T,\ro(T))&=\tfrac{1}{2}\bigl(t-\langle T,\ro(T)\rangle\bigr)\; ,\\
d(\widetilde{T},\rn(\widetilde{T}))&=\tfrac{1}{2}\bigl(t-\langle \mathrm{T}^{(+)}-2\widetilde{T},-\mathrm{T}^{(+)}+2\widetilde{T}\,\overline{\mathbf{U}}(t)\rangle\bigr)\; ,
\end{align*}
we see that
\begin{align*}
d(T,\ro(T))&=\tfrac{1}{2}\bigl(t+T\,\overline{\mathbf{U}}(t)T^{\top}\bigr)
=\tfrac{1}{2}\bigl(t+\sum_{e\in[t]}T(e)T(t-e+1)\bigr)\\
&=\frac{t}{2}+
\begin{cases}
\frac{1}{2}+\sum_{e\in[(t-1)/2]}T(e)T(t-e+1)\; , & \text{if $t$ is odd},\\
\quad\vspace{-3mm}\\
\phantom{\frac{1}{2}+}\ \sum_{e\in[t/2]}T(e)T(t-e+1)\; , & \text{if $t$ is even},
\end{cases}
\end{align*}
and
\begin{multline*}
d(\widetilde{T},\rn(\widetilde{T}))
=t-2\cdot\hwt(\widetilde{T})+2\sum_{e\in[t]}\widetilde{T}(e)\widetilde{T}(t-e+1)\\
=t-2\cdot\hwt(\widetilde{T})
\\+2\cdot\begin{cases}
\widetilde{T}((t+1)/2)+2\sum_{e\in[(t-1)/2]}\widetilde{T}(e)\widetilde{T}(t-e+1)\; , & \text{if $t$ is odd},\\
\quad\vspace{-3mm}\\
\phantom{T((t+1)/2)+}\ \, 2\sum_{e\in[t/2]}\widetilde{T}(e)\widetilde{T}(t-e+1)\; , & \text{if $t$ is even}.
\end{cases}
\end{multline*}
\noindent$\bullet$ Suppose that $4|t$ (i.e., $t$ is divisible by $4$). Note that
\begin{align*}
\langle T,\ro(T)\rangle=0\ \ \ &\Longleftrightarrow\ \ \
\sum_{e\in[t/2]}T(e)T(t-e+1)=0\; ;\\
\langle T,\ro(T)\rangle=0\ \ \ &\Longleftrightarrow\ \ \
\sum_{e\in[t/2]}\widetilde{T}(e)\widetilde{T}(t-e+1)=\frac{4\cdot\hwt(\widetilde{T})-t}{8}\; .
\end{align*}
\noindent$\bullet$ Considering the restriction of the map~(\ref{eq:10})
to the vertex set $\mathrm{V}(\boldsymbol{R})$ of the symmetric cycle $\boldsymbol{R}$ in the hypercube graph~$\boldsymbol{H}(t,2)$, defined by~(\ref{eq:12})(\ref{eq:13}), we have the mappings
\begin{equation*}
R^i\ \overset{(\ref{eq:10})}{\mapsto}\ \ro(R^i)=R^{(3t-i)\hspace{-1.5mm}\mod{2t}}
=\begin{cases}
R^{t-i}, & \text{if $0\leq i\leq t$}\; ,\\
\quad\vspace{-3mm}\\
R^{3t-i}, & \text{if $t+1\leq i\leq 2t-1$}\; .
\end{cases}
\end{equation*}
If $t$ is {\em even}, then the following implication holds:
\begin{equation*}
R^i\in\mathrm{V}(\boldsymbol{R}),\ \ \ro(R^i)=R^i\ \ \ \Longrightarrow\ \ \ i\in\{\tfrac{t}{2},\tfrac{3t}{2}\}\; .
\end{equation*}
\begin{remark} Let $\boldsymbol{R}$ be the symmetric cycle in the hypercube graph~$\boldsymbol{H}(t,2)$, defined by~{\rm(\ref{eq:12})(\ref{eq:13})}.
Given a vertex $T\in\{1,-1\}^t$ of $\boldsymbol{H}(t,2)$,
suppose that
\begin{equation*}
(R^0,R^1,\ldots,R^{2t-1})=:\mathrm{V}(\boldsymbol{R})\supset\boldsymbol{Q}(T,\boldsymbol{R})=(R^{i_0},R^{i_1},\ldots,R^{i_{\mathfrak{q}(T)-1}})\; ,
\end{equation*}
for some indices $i_0<i_1<\cdots<\mathfrak{q}(T)-1$.
\begin{itemize}
\item[\rm(i)]
We have
\begin{equation*}
\boldsymbol{Q}(\ro(T),\boldsymbol{R})=(R^{(3t-i_0)\hspace{-1.5mm}\mod{2t}},R^{(3t-i_1)\hspace{-1.5mm}\mod{2t}},\ldots,
R^{(3t-i_{\mathfrak{q}(T)-1})\hspace{-1.5mm}\mod{2t}})\; ,
\end{equation*}
or, in other words,
\begin{equation*}
\boldsymbol{Q}(\ro(T),\boldsymbol{R})=\{R^{(3t-i)\hspace{-1.5mm}\mod{2t}}\colon R^i\in\boldsymbol{Q}(T,\boldsymbol{R})\}\; .
\end{equation*}
\item[\rm(ii)] If $t$ is {\em even}, then
\begin{equation*}
\ro(T)=T\ \ \ \Longleftrightarrow\ \ \
\Bigl(Q\in\boldsymbol{Q}(T,\boldsymbol{R}) \Longrightarrow \ro(Q)\in\boldsymbol{Q}(T,\boldsymbol{R})\Bigr)\; .
\end{equation*}
Note that the following implication holds:
\begin{equation*}
\ro(T)=T\ \ \ \Longrightarrow\ \ \
|\{R^{t/2},R^{3t/2}\}\cap\boldsymbol{Q}(T,\boldsymbol{R})|=1\; .
\end{equation*}
\end{itemize}
\end{remark}
\noindent$\bullet$ Recall that for any vertex $T\in\{1,-1\}^t$ of the hypercube graph~$\boldsymbol{H}(t,2)$ with its distinguished symmetric cycle~$\boldsymbol{R}$ defined by~{\rm(\ref{eq:12})(\ref{eq:13})}, there exists a unique row vector $\boldsymbol{x}:=\boldsymbol{x}(T):=\boldsymbol{x}(T,\boldsymbol{R}):=(x_1,\ldots,x_t)\in\{-1,0,1\}^t$ such that
\begin{equation*}
T=\sum_{i\in[t]}x_i\cdot R^{i-1}=\boldsymbol{x}\mathbf{M}\; ,
\end{equation*}
where
\begin{equation*}
\mathbf{M}:=\mathbf{M}(\boldsymbol{R}):=\left(\begin{smallmatrix}
R^0\\
R^1\vspace{-2mm}\\
\vdots\\
R^{t-1}
\end{smallmatrix}
\right)\; .
\end{equation*}
In other words, the inclusion-minimal linearly independent set $\boldsymbol{Q}(T,\boldsymbol{R})$ of odd cardinality, given in~(\ref{eq:37})(\ref{eq:32}), is described as
\begin{equation*}
\boldsymbol{Q}(T,\boldsymbol{R})=\{x_i\cdot R^{i-1}\colon x_i\neq 0\}\; .
\end{equation*}
Recall that if $x_e\neq 0$ for some $e\in E_t$, then $x_e=T(e)$.
We will now give an explicit description of decompositions $\boldsymbol{Q}(T,\boldsymbol{R})$ and $\boldsymbol{Q}(\ro(T),\boldsymbol{R})$ via the corresponding ``$\boldsymbol{x}$-vectors''.
Let $\overline{\mathbf{T}}(t)$ denote\footnote{$\quad$
In~\cite[Sect.~2.1]{M-PROM} the similar notation $\mathbf{T}(m)$ was used to denote the forward shift matrix of order $(m+1)$ whose rows and columns were indexed starting with zero.
}
the {\em forward shift matrix\/} of order $t$ whose $(i,j)$th entry is $\delta_{j-i,1}$.
\begin{proposition}\cite[Prop.~2.4, extended]{M-SC-II} \label{th:10}
Let $\boldsymbol{R}$ be the symmetric cycle in the hypercube graph~$\boldsymbol{H}(t,2)$, defined by~{\rm(\ref{eq:12})(\ref{eq:13})}.
Let $A$ be a nonempty subset of the ground set $E_t$, regarded as a disjoint union
\begin{equation*}
A=[i_1,j_1]\;\dot\cup\;[i_2,j_2]\;\dot\cup\;\cdots\;\dot\cup\;[i_{\varrho-1},j_{\varrho-1}]\;\dot\cup\;[i_{\varrho},j_{\varrho}]
\end{equation*}
of intervals such that
\begin{equation*}
j_1+2\leq i_2,\ \ j_2+2\leq i_3,\ \ \ldots,\ \
j_{\varrho-2}+2\leq i_{\varrho-1},\ \ j_{\varrho-1}+2\leq i_{\varrho}\; ,
\end{equation*}
for some $\varrho:=\varrho(A)$.
\begin{itemize}
\item[\rm(i)]
{\rm(a)} If $\{1,t\}\cap A=\{\underset{\overset{\uparrow}{i_1}}{1}\}$, then
we have
\begin{align*}
|\boldsymbol{Q}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})|&=
2\varrho-1\; ,\\
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})&=
\sum_{1\leq k\leq\varrho}\boldsymbol{\sigma}(j_k+1)-\sum_{2\leq \ell\leq\varrho}\boldsymbol{\sigma}(i_{\ell})
\; .
\end{align*}
{\rm(b)} Since
\begin{multline*}
\{t-e+1\colon e\in E_t-A\}=[\underset{\overset{\uparrow}{i_1}}{1},t-j_{\varrho}]\;\dot\cup\;[t-i_{\varrho}+2,t-j_{\varrho-1}]\\
\dot\cup\;\cdots\;\dot\cup\;[t-i_3+2,t-j_2]
\;\dot\cup\;[t-i_2+2,t-j_1]\; ,
\end{multline*}
and $\{1,t\}\cap\{t-e+1\colon e\in E_t-A\}=\{1\}$, we see that
\begin{align*}
|\boldsymbol{Q}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})|&=
2\varrho-1\; ,\\
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})&=
\sum_{1\leq k\leq\varrho}\boldsymbol{\sigma}(t-j_k+1)-\sum_{2\leq \ell\leq\varrho}\boldsymbol{\sigma}(t-i_{\ell}+2)\; .
\end{align*}
{\rm(c)} Note that
\begin{equation*}
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})=
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\item[\rm(ii)]
{\rm(a)} If $\{1,t\}\cap A=\{\underset{\overset{\uparrow}{i_1}}{1},\underset{\overset{\uparrow}{j_{\varrho}}}{t}\}$, then
\begin{align*}
|\boldsymbol{Q}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})|&=2\varrho-1\; ,\\
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})&=-\boldsymbol{\sigma}(1)+\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(j_k+1)-
\sum_{2\leq\ell \leq\varrho}\boldsymbol{\sigma}(i_{\ell})
\; .
\end{align*}
{\rm(b)} Since
\begin{multline*}
\{t-e+1\colon e\in E_t-A\}=[t-i_{\varrho}+2,t-j_{\varrho-1}]\;\dot\cup\;[t-i_{\varrho-1}+2,t-j_{\varrho-2}]\\
\dot\cup\;\cdots\;\dot\cup\;[t-i_3+2,t-j_2]
\;\dot\cup\;[t-i_2+2,t-j_1]\; ,
\end{multline*}
and $|\{1,t\}\cap\{t-e+1\colon e\in E_t-A\}|=0$, we have
\begin{align*}
|\boldsymbol{Q}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})|&=
2\varrho-1\; ,\\
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})&=\boldsymbol{\sigma}(1)+
\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(t-j_k+1)-\sum_{2\leq \ell\leq\varrho}\boldsymbol{\sigma}(t-i_{\ell}+2)\; .
\end{align*}
{\rm(c)} Note that
\begin{equation*}
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})=
\boldsymbol{\sigma}(1)+\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\item[\rm(iii)]
{\rm(a)} If $|\{1,t\}\cap A|=0$, then
\begin{align*}
|\boldsymbol{Q}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})|&=2\varrho\;+1\; ,\\
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})&=\boldsymbol{\sigma}(1)\;+\sum_{1\leq k\leq\varrho}\boldsymbol{\sigma}(j_k+1)\;-
\sum_{1\leq\ell \leq\varrho}\boldsymbol{\sigma}(i_{\ell})
\; .
\end{align*}
{\rm(b)} Since
\begin{multline*}
\{t-e+1\colon e\in E_t-A\}=[1,t-j_{\varrho}]\;\dot\cup\;[t-i_{\varrho}+2,t-j_{\varrho-1}]\\
\dot\cup\;\cdots\;\dot\cup\;[t-i_2+2,t-j_1]
\;\dot\cup\;[t-i_1+2,t]\; ,
\end{multline*}
and $\{1,t\}\cap\{t-e+1\colon e\in E_t-A\}=\{1,t\}$, we have
\begin{align*}
|\boldsymbol{Q}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})|&=
2\varrho+1\; ,\\
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})&=
-\boldsymbol{\sigma}(1)+\sum_{1\leq k\leq\varrho}\boldsymbol{\sigma}(t-j_k+1)-\sum_{1\leq \ell\leq\varrho}\boldsymbol{\sigma}(t-i_{\ell}+2)\; .
\end{align*}
{\rm(c)} Note that
\begin{equation*}
\boldsymbol{x}(-({}_{-A}\mathrm{T}^{(+)})\cdot\overline{\mathbf{U}}(t),\boldsymbol{R})=
-\boldsymbol{\sigma}(1)+
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\item[\rm(iv)]
{\rm(a)} If $\{1,t\}\cap A=\{\underset{\overset{\uparrow}{j_{\varrho}}}{t}\}$, then
\begin{align*}
|\boldsymbol{Q}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})|&=2\varrho-1\; ,\\
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})&=
\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(j_k+1)
-\sum_{1\leq \ell\leq\varrho}\boldsymbol{\sigma}(i_{\ell})
\; .
\end{align*}
{\rm(b)} Since
\begin{multline*}
\{t-e+1\colon e\in E_t-A\}=[t-i_{\varrho}+2,t-j_{\varrho-1}]\;\dot\cup\;[t-i_{\varrho-1}+2,t-j_{\varrho-2}]\\
\dot\cup\;\cdots\;\dot\cup\;[t-i_2+2,t-j_1]
\;\dot\cup\;[t-i_1+2,\underset{\overset{\uparrow}{j_{\varrho}}}{t}]\; ,
\end{multline*}
and $\{1,t\}\cap\{t-e+1\colon e\in E_t-A\}=\{t\}$, we see that
\begin{align*}
|\boldsymbol{Q}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})|&=
2\varrho-1\; ,\\
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})&=
\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(t-j_k+1)-\sum_{1\leq \ell\leq\varrho}\boldsymbol{\sigma}(t-i_{\ell}+2)\; .
\end{align*}
{\rm(c)} Note that
\begin{equation*}
\boldsymbol{x}(\ro({}_{-A}\mathrm{T}^{(+)}),\boldsymbol{R})=
\boldsymbol{x}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\end{itemize}
\end{proposition}
\begin{corollary}\label{th:11}
Let $\boldsymbol{R}$ be the symmetric cycle in the hypercube graph~$\boldsymbol{H}(t,2)$, defined by~{\rm(\ref{eq:12})(\ref{eq:13})}.
For any vertex $T\in\{1,-1\}^t$ of the graph~$\boldsymbol{H}(t,2)$ we have
\begin{equation*}
\mathfrak{q}(\ro(T)):=|\supp(\boldsymbol{x}(\ro(T),\boldsymbol{R}))|=
|\supp(\boldsymbol{x}(T,\boldsymbol{R}))|=:\mathfrak{q}(T)\; .
\end{equation*}
\begin{itemize}
\item[\rm(i)]
If\/ $|T^-\cap\{1,t\}|=1$, then
\begin{equation*}
\boldsymbol{x}(\ro(T))=
\boldsymbol{x}(T)\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\item[\rm(ii)]
If\/ $|T^-\cap\{1,t\}|=2$, then
\begin{equation*}
\boldsymbol{x}(\ro(T))=
\boldsymbol{\sigma}(1)+\boldsymbol{x}(T)\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\item[\rm(iii)]
If\/ $|T^-\cap\{1,t\}|=0$, then
\begin{equation*}
\boldsymbol{x}(\ro(T))=
-\boldsymbol{\sigma}(1)+
\boldsymbol{x}(T)\cdot\overline{\mathbf{U}}(t)\cdot
\overline{\mathbf{T}}(t)\; .
\end{equation*}
\end{itemize}
\end{corollary}
\part*{Blocking}
Blocking sets and the blockers of set families
(families are often regarded as the {\em hyperedge\/} families of {\em hypergraphs}) are discussed, e.g., in the monographs~\cite{Berge89,Bretto,CCZ,Co,CH,CFKLMPPS,FLSZ,G,GJ,GLS,HHH,HY-Total,HY,Jukna-E2,M-PROM,Mirsky,NI,NW,SchU,Sch-C,V}
and in the works~\cite{AN-2017,AH-2017,Alon-1990,AKB-2021,ABCS-2021,BS-2005,Barat-2021,vBS-2020,BH-2004,BFLMSCh-2019,BFSch-2021,BM-2009,Bou-2013,
BKMN-2016,BHT-2012,BRT-2021,ChHuZ-2013,ChMcD-1992,CCGK-2016,CFM-1991,D-2011,DNU-2020,EG-1995,EGT-2003,EMG-2008,Elbassioni-2008,FHNS-2020,
FKh-1996,F-1988,G-DV-L-2017,HBC-2007,KLMN-2014,Karp-1972,KS-2005,KhBEG-2006,LS-2008,LJ-2003,MU-2014,NP-Kao, PQ-2012,Reiter-1987,Y-2018}.
\noindent$\bullet$ Let
\begin{equation*}
\mho([t]):=\{\mathcal{A}\subset\mathbf{2}^{[t]}\colon\ \mathcal{A}\,=\,\bmin\mathcal{A}\,=\,\bmax\mathcal{A}\}
\end{equation*}
denote the family of clutters on the ground set~$E_t$.
The map
\begin{equation}
\label{eq:38}
\mho([t])\to\mho([t])\; ,\ \ \ \mathcal{A}\mapsto\mathfrak{B}(\mathcal{A})\; ,
\end{equation}
is called the {\em blocker map\/} on clutters~\cite{CFM-1991}.
\noindent$\bullet$ If the (\!{\em{}abstract simplicial}) {\em complex\/}~$\Delta:=(\mathfrak{B}(\mathcal{A})^{\compl})^{\vartriangle}$ appearing in~(\ref{eq:27}) and the {\em complex\/}~$\Delta^{\!\!\vee}:=\{F^{\compl}\colon F\in\mathcal{A}^{\triangledown}\}$ both have the same vertex set~$E_t$, then the complex $\Delta^{\!\!\vee}$ is called the {\em Alexander dual\/} of the complex~$\Delta$; see, e.g.,~\cite{V} and~\cite{BT-2009} on combinatorial {\em Alexander duality}.
\noindent$\bullet$ Given a clutter~$\mathcal{A}$, the quantity
\begin{equation*}
\tau(\mathcal{A}):=\min\{|B|\colon B\in\mathfrak{B}(\mathcal{A})\}
\end{equation*}
is called the {\em transversal number}\footnote{$\quad$Or {\em covering number}, {\em vertex cover number}, {\em blocking number}.} of $\mathcal{A}$.
\noindent$\bullet$ Recall a classical result in combinatorial optimization: For any clutter $\mathcal{A}$
we have
\begin{equation*}
\mathfrak{B}(\mathfrak{B}(\mathcal{A}))=\mathcal{A}\; ,
\end{equation*}
see~\cite{EF-1970,I-1958,La-1966,Leh-164}.
\noindent$\bullet$ For a nontrivial clutter~$\mathcal{A}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$, we have
\begin{equation}
\label{eq:27}
\#\mathcal{A}^{\triangledown}\; +\; \#\mathfrak{B}(\mathcal{A})^{\triangledown}=2^t\; .
\end{equation}
More precisely, for any $s$, where $0\leq s\leq t$, we have
\begin{equation}
\label{eq:28}
\#(\mathfrak{B}(\mathcal{A})^{\triangledown}\cap\tbinom{E_t}{s})\; +\; \#(\mathcal{A}^{\triangledown}\cap\tbinom{E_t}{t-s})
=\tbinom{t}{s}\; ,
\end{equation}
where
\begin{equation*}
\tbinom{E_t}{s}:=\{F\subseteq E_t\colon |F|=s\}
\end{equation*}
is the {\em complete $s$-uniform clutter\/} on the vertex set $E_t$.
The increasing families~$\mathcal{A}^{\triangledown}$ and~$\mathfrak{B}(\mathcal{A})^{\triangledown}$ are {\em comparable by containment}: either we have
\begin{equation*}
\mathfrak{B}(\mathcal{A})^{\triangledown}\subseteq\mathcal{A}^{\triangledown}\; ,
\ \ \ \text{or\/}\ \ \ \
\mathfrak{B}(\mathcal{A})^{\triangledown}\supseteq\mathcal{A}^{\triangledown}\; .
\end{equation*}
The following implications hold:
\begin{gather*}
\mathfrak{B}(\mathcal{A})^{\triangledown}\subsetneqq \mathcal{A}^{\triangledown}\ \ \Longleftrightarrow\ \ \#\mathcal{A}^{\triangledown}>2^{t-1}\; ;
\\
\mathfrak{B}(\mathcal{A})^{\triangledown}\supsetneqq \mathcal{A}^{\triangledown}\ \ \Longleftrightarrow\ \ \#\mathcal{A}^{\triangledown}<2^{t-1}\; .
\end{gather*}
Note also that the following implications hold:
\begin{gather*}
\#\mathcal{A}^{\triangledown}>2^{t-1}\ \ \ \Longrightarrow\ \ \ \min\{|A|\colon A\in\mathcal{A}\}\leq\min\{|B|\colon B\in\mathfrak{B}(\mathcal{A})\}\; ;
\\
\#\mathcal{A}^{\triangledown}<2^{t-1}\ \ \ \Longrightarrow\ \ \ \min\{|A|\colon A\in\mathcal{A}\}\geq\min\{|B|\colon B\in\mathfrak{B}(\mathcal{A})\}\; .
\end{gather*}
\noindent$\bullet$ A clutter $\mathcal{A}$
is called {\em self-dual\/}~\cite[Ch.~9]{Jukna-E2}\cite[\S{}5.7]{M-PROM} or {\em identically self-blocking\/}~\cite{ACL-2019,AP-2018} if
\begin{equation*}
\mathfrak{B}(\mathcal{A})=\mathcal{A}\; ;
\end{equation*}
see also the early reference~\cite[\S{}2.1]{Berge89}.
In other words, the self-dual clutters~$\mathcal{A}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$ are the {\em fixed points\/} of the {\em blocker map\/}~(\ref{eq:38}); for each of them we also have
\begin{equation*}
\mathfrak{B}(\mathcal{A})^{\triangledown}=\mathcal{A}^{\triangledown}\; .
\end{equation*}
As noted in~\cite[Cor.~5.28(i)]{M-PROM}, one criterion for a {\em clutter\/} $\mathcal{A}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$ to be {\em self-dual\/} is as follows:
\begin{equation*}
\mathfrak{B}(\mathcal{A})=\mathcal{A}\ \ \Longleftrightarrow\ \ \#\mathcal{A}^{\triangledown}=2^{t-1}\; .
\end{equation*}
\noindent$\bullet$ Let~$X$ be a subset of the ground set~$E_t$. Given a nontrivial clutter~$\mathcal{A}$ on~$E_t$, its {\em deletion\/}
$\mathcal{A}\,\backslash\,X$ is defined to be the clutter
\begin{equation*}
\mathcal{A}\,\backslash\,X:=\{A\in\mathcal{A}\colon |A\cap X|=0\}\; .
\end{equation*}
The {\em contraction\/} $\mathcal{A}\,/X$ is defined to be the clutter
\begin{equation*}
\mathcal{A}\,/X:=\bmin\{A-X\colon A\in\mathcal{A}\}\; .
\end{equation*}
A classical result in combinatorial optimization is as follows:
\begin{equation*}
\mathfrak{B}(\mathcal{A})\,\backslash\,X=\mathfrak{B}(\mathcal{A}\,/X)\; ,\ \ \ \ \ \text{and}\ \ \ \ \ \
\mathfrak{B}(\mathcal{A})\,/X=\mathfrak{B}(\mathcal{A}\,\backslash\,X)\; ,
\end{equation*}
see~\cite{S-1975}.
We also have
\begin{equation*}
(\mathfrak{B}(\mathcal{A})\,\backslash\,X)^{\triangledown}=\mathfrak{B}(\mathcal{A}\,/X)^{\triangledown}\subseteq
\mathfrak{B}(\mathcal{A})^{\triangledown}
\subseteq
(\mathfrak{B}(\mathcal{A})\,/X)^{\triangledown}=\mathfrak{B}(\mathcal{A}\,\backslash\,X)^{\triangledown}\; ,
\end{equation*}
cf.~\cite[Eq.\,(5.4)]{M-PROM}. Further,
\begin{align*}
\#(\mathcal{A}\,\backslash\,X)^{\triangledown}\ +\ \#(\mathfrak{B}(\mathcal{A})\,/X)^{\triangledown}\ &=\ 2^t\; ,\\
\#(\mathcal{A}\,/X)^{\triangledown}\ +\ \#(\mathfrak{B}(\mathcal{A})\,\backslash\,X)^{\triangledown}\ &=\ 2^t\; ,
\end{align*}
see~\cite[Cor.~5.28(ii)]{M-PROM}. More precisely, for any $s$, where $0\leq s\leq t$, we have
\begin{align*}
\#\bigl((\mathfrak{B}(\mathcal{A})\,/X)^{\triangledown}\cap\tbinom{E_t}{s}\bigr)\ +\
\#\bigl((\mathcal{A}\,\backslash\,X)^{\triangledown}\cap\tbinom{E_t}{t-s}\bigr)\ &=\ \tbinom{t}{s}\; ,\\
\#\bigl((\mathfrak{B}(\mathcal{A})\,\backslash\,X)^{\triangledown}\cap\tbinom{E_t}{s}\bigr)\ +\
\#\bigl((\mathcal{A}\,/X)^{\triangledown}\cap\tbinom{E_t}{t-s}\bigr)\ &=\ \tbinom{t}{s}\; .
\end{align*}
\noindent$\bullet$ Let $p$ be a rational number such that $0\leq p<1$. Given a nontrivial clutter $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$, a subset $B\subseteq E_t$ is called a {\em $p$-committee}\footnote{$\quad$By convention, a $\tfrac{1}{2}$-committee of a clutter $\mathcal{A}$ is called its {\em committee}.} of the clutter $\mathcal{A}$, if we have
\begin{equation*}
|B\cap A_i|>p\cdot|B|\; ,
\end{equation*}
for each $i\in[\alpha]$. The {\em $0$-committees\/} of the clutter~$\mathcal{A}$ are its {\em blocking sets}.
\noindent$\bullet$ For a nontrivial clutter $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$, we have
\begin{equation*}
\#\bigl(\mathfrak{B}(\mathcal{A})^{\triangledown}\cap\tbinom{E_t}{k}\bigr)
=\tbinom{t}{k}+\sum_{S\subseteq[\alpha]\colon |S|>0}(-1)^{|S|}\cdot\tbinom{t-|\bigcup_{s\in S}A_s|}{k}
\; ,\ \ \ \ \ 1\leq k\leq t\; .
\end{equation*}
Several ways to count the blocking $k$-sets of clutters are mentioned in~\cite{M-PROM}.
\section{Increasing families of blocking sets, and blockers: Set covering problems}
In this section we recall
the set covering problem(s); see, e.g.,~\cite[Sect.~2.4]{CCZ} and~\cite[Ch.~1]{Co}.
Let ${\boldsymbol{\chi}}(A):=(\chi_1(A),\ldots,\chi_t(A))\in\{0,1\}^t$ denote the familiar row {\em characteristic vector\/} of a subset $A$ of the ground set $E_t$, defined for each element~$j\in E_t$ by
\begin{equation*}
\chi_j(A):={\small\begin{cases}
1\; , & \text{if $j\in A$},\\
0\; , & \text{if $j\not\in A$}.
\end{cases}
}
\end{equation*}
If $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}\subset\mathbf{2}^{[t]}$ is a nontrivial clutter on
$E_t$, then
\begin{equation}
\label{eq:2}
\boldsymbol{A}:=\boldsymbol{A}(\mathcal{A}):=\left(\begin{smallmatrix}
{\boldsymbol{\chi}}(A_1)\vspace{-2mm}\\ \vdots\\ {\boldsymbol{\chi}}(A_{\alpha})
\end{smallmatrix}
\right)
\end{equation}
is its {\em incidence matrix}.
Consider\footnote{$\quad$We will denote by~$\mathbbm{1}$ and~$\mathbbm{2}$ the $\alpha$-dimensional column vectors $\left(\begin{smallmatrix}1\vspace{-2mm}\\ \vdots\\ 1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}2\vspace{-2mm}\\ \vdots\\ 2\end{smallmatrix}\right)$, respectively.}
the {\em set covering collection}
\begin{equation}
\label{eq:24}
\widetilde{\boldsymbol{\mathcal{S}}}:=\widetilde{\boldsymbol{\mathcal{S}}}{}^{\mathtt{C}}(\boldsymbol{A}):=\bigl\{\tilde{\mathbf{z}}\in\{0,1\}^t\colon \boldsymbol{A}\tilde{\mathbf{z}}^{\top}\geq\mathbbm{1}\bigr\}\; ,
\end{equation}
which is the collection of {\em characteristic vectors\/} of the {\em blocking sets\/} of the clutter~$\mathcal{A}$, that is,
\begin{equation*}
\widetilde{\boldsymbol{\mathcal{S}}}=
\bigl\{\boldsymbol{\chi}(B)\colon B\in\mathfrak{B}(\mathcal{A})^{\triangledown}\bigr\}\; , \ \ \text{and}\ \ \ \
\mathfrak{B}(\mathcal{A})^{\triangledown}=
\bigl\{\supp(\tilde{\boldsymbol{z}})\colon \tilde{\boldsymbol{z}}\in \widetilde{\boldsymbol{\mathcal{S}}}\bigr\}
\; .
\end{equation*}
The latter expression just rephrases the convention according to which the {\em supports\/} of the vectors in the collection~$\widetilde{\boldsymbol{\mathcal{S}}}\subset\{0,1\}^t$ are the {\em blocking sets\/} of the clutter~$\mathcal{A}$.
Let us redefine the collection
\begin{equation*}
\widetilde{\boldsymbol{\mathcal{S}}}:=
\Bigl\{\tilde{\mathbf{z}}\in\{0,1\}^t\colon \left(\begin{smallmatrix}
\boldsymbol{\chi}(A_1)\vspace{-2mm}\\ \vdots\\ \boldsymbol{\chi}(A_{\alpha})
\end{smallmatrix}
\right)\tilde{\mathbf{z}}^{\top}\geq\mathbbm{1}\Bigr\}
\end{equation*}
as
\begin{equation}
\label{eq:23}
\widetilde{\boldsymbol{\mathcal{S}}}:=\biggl\{\tfrac{1}{2}(\mathrm{T}^{(+)}-\mathbf{z})\in\{0,1\}^t\colon \left(\begin{smallmatrix}
\frac{1}{2}(\mathrm{T}^{(+)}-T^1)\vspace{-2mm}\\ \vdots\\ \frac{1}{2}(\mathrm{T}^{(+)}-T^{\alpha})
\end{smallmatrix}
\right)\cdot\tfrac{1}{2}(\mathrm{T}^{(+)}-\mathbf{z})^{\top}\geq\mathbbm{1}\biggr\}\; ,
\end{equation}
where the vertices $T^i$ of the discrete hypercube~$\{1,-1\}^t$ and the vector of unknowns $\mathbf{z}\in\{1,-1\}^t$ are given by
\begin{align*}
T^i:={}_{-A_i}\mathrm{T}^{(+)}&=\mathrm{T}^{(+)}-2\boldsymbol{\chi}(A_i)
\; ,\ \ \ i\in[\alpha]\; ,\\
\intertext{and}
\mathbf{z}:\!&=\mathrm{T}^{(+)}-2\tilde{\mathbf{z}}\; .
\end{align*}
Let us now associate with the collection~$\widetilde{\boldsymbol{\mathcal{S}}}\subset\{0,1\}^t$, described in~(\ref{eq:23}), a collection~\mbox{$\boldsymbol{\mathcal{S}}:=\boldsymbol{\mathcal{S}}{}^{\mathtt{C}}(\boldsymbol{A})\subset\{1,-1\}^t$,} defined by
\begin{equation*}
\boldsymbol{\mathcal{S}}:=
\biggl\{\mathbf{z}\in\{1,-1\}^t\colon \boldsymbol{A}\mathbf{z}^{\top}\leq
\left(\begin{smallmatrix}
|(T^1)^-|\vspace{-2mm}\\ \vdots\\ |(T^{\alpha})^-|
\end{smallmatrix}
\right)
-2\cdot\mathbbm{1}
\biggr\}\; ,
\end{equation*}
that is, the collection
\begin{equation}
\label{eq:25}
\boldsymbol{\mathcal{S}}:=
\biggl\{\mathbf{z}\in\{1,-1\}^t\colon \boldsymbol{A}\mathbf{z}^{\top}\leq
\left(\begin{smallmatrix}
|A_1|\vspace{-2mm}\\ \vdots\\ |A_{\alpha}|
\end{smallmatrix}
\right)
-\mathbbm{2}
\biggr\}\; .
\end{equation}
We have defined the twin collections $\widetilde{\boldsymbol{\mathcal{S}}}\subset\{0,1\}^t$ and $\boldsymbol{\mathcal{S}}\subset\{1,-1\}^t$, given in~(\ref{eq:24}) and~(\ref{eq:25}), respectively, that are equipped with the {\em bijections\/}
$\widetilde{\boldsymbol{\mathcal{S}}}\to\boldsymbol{\mathcal{S}}\colon$ \mbox{$\widetilde{T}\mapsto\mathrm{T}^{(+)}-2\widetilde{T}$,} and
$\boldsymbol{\mathcal{S}}\to\widetilde{\boldsymbol{\mathcal{S}}}\colon$ $T\mapsto\tfrac{1}{2}(\mathrm{T}^{(+)}-T)$; see Example~\ref{th:3}.
\begin{example}
\label{th:3}
Consider the clutter $\mathcal{A}:=\{A_1,A_{\alpha:=2}\}:=\bigl\{\{1,2\},\{2,3\}\bigr\}$, on the ground set $E_{t:=3}:=\{1,2,3\}$, with its incidence matrix
\begin{equation*}
\boldsymbol{A}:=\boldsymbol{A}(\mathcal{A})=\left(
\begin{smallmatrix}
1&1&0\\
0&1&1
\end{smallmatrix}
\right)\; .
\end{equation*}
The {\em set covering $\{0,1\}$-collection}
\begin{equation*}
\begin{split}
\widetilde{\boldsymbol{\mathcal{S}}}:\!&=
\bigl\{\tilde{\mathbf{z}}\in\{0,1\}^t\colon \boldsymbol{A}\tilde{\mathbf{z}}^{\top}\geq\mathbbm{1}\bigr\}\\
&=\bigl\{\tilde{\mathbf{z}}\in\{0,1\}^3\colon \left(
\begin{smallmatrix}
1&1&0\\
0&1&1
\end{smallmatrix}
\right)\tilde{\mathbf{z}}^{\top}\geq\left(
\begin{smallmatrix}
1\\
1
\end{smallmatrix}
\right)\bigr\}
\end{split}
\end{equation*}
is the collection
\begin{equation*}
\begin{split}
\widetilde{\boldsymbol{\mathcal{S}}}&=
\bigl\{
\left(\begin{smallmatrix}0&1&0\end{smallmatrix}\right),
\left(\begin{smallmatrix}1&1&0\end{smallmatrix}\right),
\left(\begin{smallmatrix}1&0&1\end{smallmatrix}\right),
\left(\begin{smallmatrix}0&1&1\end{smallmatrix}\right),
\left(\begin{smallmatrix}1&1&1\end{smallmatrix}\right)\bigr\}\\
&=
\bigl\{\boldsymbol{\chi}(\{2\}),\boldsymbol{\chi}(\{1,2\}),\boldsymbol{\chi}(\{1,3\}),
\boldsymbol{\chi}(\{2,3\}),\boldsymbol{\chi}(\{1,2,3\})\bigr\}\; .
\end{split}
\end{equation*}
The {\em set covering $\{1,-1\}$-collection}
\begin{equation*}
\begin{split}
\boldsymbol{\mathcal{S}}:\!&=
\biggl\{\mathbf{z}\in\{1,-1\}^t\colon
\boldsymbol{A}\mathbf{z}^{\top}
\leq
\left(\begin{smallmatrix}
|A_1|\vspace{-2mm}\\ \vdots\\ |A_{\alpha}|
\end{smallmatrix}
\right)
-\mathbbm{2}
\biggr\}
\\
&=
\biggl\{\mathbf{z}\in\{1,-1\}^3\colon
\left(\begin{smallmatrix}
1&1&0\\ 0&1&1
\end{smallmatrix}
\right)\mathbf{z}^{\top}
\leq
\left(\begin{smallmatrix}
2\\ 2
\end{smallmatrix}
\right)
-\mathbbm{2}
=\left(\begin{smallmatrix}
0\\ 0
\end{smallmatrix}
\right)
\biggr\}
\end{split}
\end{equation*}
is the collection
\begin{equation*}
\begin{split}
\boldsymbol{\mathcal{S}}&=
\bigl\{
\left(\begin{smallmatrix}1& -1& 1\end{smallmatrix}\right),
\left(\begin{smallmatrix}-1& -1& 1\end{smallmatrix}\right),
\left(\begin{smallmatrix}-1& 1& -1\end{smallmatrix}\right),
\left(\begin{smallmatrix}1& -1& -1\end{smallmatrix}\right),
\left(\begin{smallmatrix}-1& -1& -1\end{smallmatrix}\right)
\bigr\}
\\
&=\bigl\{{}_{-\{2\}}\mathrm{T}^{(+)}, {}_{-\{1,2\}}\mathrm{T}^{(+)}, {}_{-\{1,3\}}\mathrm{T}^{(+)},
{}_{-\{2,3\}}\mathrm{T}^{(+)}, {}_{-\{1,2,3\}}\mathrm{T}^{(+)}\bigr\}\; .
\end{split}
\end{equation*}
\end{example}
\noindent{}$\bullet$ Let $\boldsymbol{w}\in\mathbb{R}^t$ be a row vector of {\em nonnegative\/} weights. The {\em set covering\/} problems are
\begin{equation*}
\min\{\boldsymbol{w}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}\}
=\min\bigl\{\boldsymbol{w}\cdot\tfrac{1}{2}(\mathrm{T}^{(+)}-\mathbf{z})^{\top}\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
\; .
\end{equation*}
\noindent$\bullet$ Suppose $\boldsymbol{w}:=\mathrm{T}^{(+)}$, and consider the (\!{\em unweighted}) set covering problems
\begin{multline*}
\tau(\mathcal{A}):=\min\{\hwt(\tilde{\mathbf{z}})\colon \tilde{\mathbf{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}\}=\min\{\mathrm{T}^{(+)}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}\}
\\=
\min\bigl\{\mathrm{T}^{(+)}\cdot\tfrac{1}{2}(\mathrm{T}^{(+)}-\mathbf{z})^{\top}\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
=\min\bigl\{\tfrac{1}{2}(t-\underbrace{\mathrm{T}^{(+)}\mathbf{z}^{\top}}_{t-2|\mathbf{z}^-|})
\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
\\=
\min\bigl\{|\mathbf{z}^-|\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}=:\tau(\mathcal{A})\; ,
\end{multline*}
that is, the problem
\begin{equation}
\label{eq:29}
\underbrace{
\min\{\mathrm{T}^{(+)}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}\}
}_{
\tau(\mathcal{A}):=\min\{\hwt(\tilde{\mathbf{z}})\colon \tilde{\mathbf{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}\}
}
:=\min\{\mathrm{T}^{(+)}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}
\in\{0,1\}^t,\
\boldsymbol{A}\tilde{\mathbf{z}}\geq\boldsymbol{1}\}\; ,
\end{equation}
and the problem
\begin{gather}
\nonumber
\underbrace{
\tfrac{1}{2}\min\bigl\{t-\mathrm{T}^{(+)}\mathbf{z}^{\top}
\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
}_{
\tau(\mathcal{A}):=\min\bigl\{|\mathbf{z}^-|\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
}
\\
\nonumber
:=
\frac{1}{2}\min\biggl\{t-\mathrm{T}^{(+)}\mathbf{z}^{\top}
\colon \mathbf{z}\in\{1,-1\}^t,\
\boldsymbol{A}
\mathbf{z}^{\top}
\leq
\left(\begin{smallmatrix}
|A_1|\vspace{-2mm}\\ \vdots\\ |A_{\alpha}|
\end{smallmatrix}
\right)
-\mathbbm{2}
\biggr\}
\\
\nonumber
=\frac{1}{2}\cdot\!\left(t-\max\biggl\{\mathrm{T}^{(+)}\mathbf{z}^{\top}
\colon \mathbf{z}\in\{1,-1\}^t,\
\boldsymbol{A}
\mathbf{z}^{\top}
\leq
\left(\begin{smallmatrix}
|A_1|\vspace{-2mm}\\ \vdots\\ |A_{\alpha}|
\end{smallmatrix}
\right)
-\mathbbm{2}
\biggr\}\right)
\\
\label{eq:30}
=:\underbrace{
\tfrac{1}{2}\cdot\left(t-\max\bigl\{\mathrm{T}^{(+)}\mathbf{z}^{\top}
\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}\right)
}_{
\tau(\mathcal{A}):=\min\bigl\{|\mathbf{z}^-|\colon \mathbf{z}\in\boldsymbol{\mathcal{S}}\bigr\}
}
\; .
\end{gather}
For vectors $\tilde{\boldsymbol{z}}\in\widetilde{\boldsymbol{\mathcal{S}}}$ and $\boldsymbol{z}\in\boldsymbol{\mathcal{S}}$, where $\tilde{\boldsymbol{z}}:=\tfrac{1}{2}(\mathrm{T}^{(+)}-\boldsymbol{z})$, we have the inclusions
\begin{align*}
\tilde{\boldsymbol{z}}&\in
\Arg\min\{\mathrm{T}^{(+)}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}
\in \widetilde{\boldsymbol{\mathcal{S}}}\}\; ,\\
\boldsymbol{z}&\in
\Arg\max\{\mathrm{T}^{(+)}\mathbf{z}^{\top}\colon \mathbf{z}
\in \boldsymbol{\mathcal{S}}\}\; ,
\end{align*}
that is, $\tilde{\boldsymbol{z}}$ and~$\boldsymbol{z}$ provide the solution
to the problems~(\ref{eq:29}) and~(\ref{eq:30}), respectively, if and only if the member
\begin{equation*}
B:=\supp(\tilde{\boldsymbol{z}})=\boldsymbol{z}^-\in\mathfrak{B}(\mathcal{A})
\end{equation*}
of the blocker of the clutter $\mathcal{A}$ has the {\em minimum\/} cardinality
\begin{equation*}
|B|=\tau(\mathcal{A})\; .
\end{equation*}
\noindent$\bullet$ We conclude this section by noting that the rows of incidence matrices~$\boldsymbol{A}$, as well as the vectors in the set covering collections $\widetilde{\boldsymbol{\mathcal{S}}}
\subset\{0,1\}^t$ and $\boldsymbol{\mathcal{S}}\subset\{1,-1\}^t$, admit their decompositions with respect to symmetric cycles in the corresponding hypercube graphs~$\widetilde{\boldsymbol{H}}(t,2)$ and~$\boldsymbol{H}(t,2)$.
\section{Families of subsets of the ground set $E_t$: Characteristic vectors and characteristic topes}
The {\em generation\/} of fundamental combinatorial objects is extensively treated in~\cite{Knuth-4A}.
\noindent$\bullet$ Consider the family $\tbinom{E_t}{s}$, for some $s$, where $0\leq s\leq t$. We denote this family of all $s$-subsets $L^s_j\subseteq E_t$, ordered {\em lexicographically}, by $\overrightarrow{\tbinom{E_t}{s}}=:(L^s_1,\ldots,L^s_{\binom{t}{s}})$.
For an $s$-uniform clutter $\mathcal{G}:=\{G_1,\ldots,G_k\}\subseteq \tbinom{E_t}{s}$, we define its row {\em characteristic vector\/} $\boldsymbol{\gamma}^{(s)}(\mathcal{G}):=(\gamma^{(s)}_1(\mathcal{G}),\ldots,\gamma^{(s)}_{\binom{t}{s}}(\mathcal{G}))\in\{0,1\}^{\binom{t}{s}}$ in the familiar way: for each $j$, where~$1\leq j\leq\tbinom{t}{s}$, we set
\begin{equation*}
\gamma^{(s)}_j(\mathcal{G}):=\begin{cases}
1\; , & \text{if $\overrightarrow{\tbinom{E_t}{s}}\ni L^s_j\in \mathcal{G}$},\\
\quad\vspace{-3mm}\\
0\; , & \text{if $\overrightarrow{\tbinom{E_t}{s}}\ni L^s_j\not\in \mathcal{G}$};
\end{cases}
\end{equation*}
see~(\ref{eq:14})--(\ref{eq:15}) in Example~\ref{th:1}.
Now, given an arbitrary family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$, we set
\begin{equation*}
\boldsymbol{\gamma}^{(s)}(\mathcal{F}):=\boldsymbol{\gamma}^{(s)}(\mathcal{F}\cap\tbinom{E_t}{s})\; ,\ \ \ 0\leq s\leq t\; ,
\end{equation*}
and in a natural way\footnote{$\quad$Indeed, this is the
most popular ordering of all subsets of a nonempty finite set, the {\em lexicographic ordering subordinated to cardinality}, that is,
considering smaller subsets first, and in case of equal cardinality, the lexicographic
ordering, see~\cite[p.~77]{Gr}.} we define the {\em characteristic vector\/}
$\boldsymbol{\gamma}(\mathcal{F})
:=(\gamma_1(\mathcal{F}),\ldots,$ $\gamma_{2^t}(\mathcal{F}))\in\{0,1\}^{2^t}$ of the family~$\mathcal{F}$ to be the {\em concatenation\/}
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{F})
:=\boldsymbol{\gamma}^{(0)}(\mathcal{F})\;\centerdot\;
\boldsymbol{\gamma}^{(1)}(\mathcal{F})
\;\centerdot\;\cdots
\;\centerdot\;
\boldsymbol{\gamma}^{(t-1)}(\mathcal{F})
\;\centerdot\;
\boldsymbol{\gamma}^{(t)}(\mathcal{F})\; ;
\end{equation*}
see~(\ref{eq:16})--(\ref{eq:17}).
\noindent$\bullet$ The characteristic vector $\boldsymbol{\gamma}(\mathbf{2}^{[t]})=\mathrm{T}_{2^t}^{(+)}$, whose components are all $1$'s, describes the linearly ordered {\em power set\/} $\mathbf{2}^{[t]}$
of the ground set $E_t$; see~(\ref{eq:20}).
\noindent$\bullet$ The Hamming weights $\hwt(\boldsymbol{\gamma}^{(s)}(\mathcal{F}))$ of the vectors $\boldsymbol{\gamma}^{(s)}(\mathcal{F})$, $0\leq s\leq t$, are the components $f_s(\mathcal{F};t)$ of the so-called {\em long $f$-vectors\/} $\boldsymbol{f}(\mathcal{F};t)$ associated with families $\mathcal{F}\subseteq\mathbf{2}^{[t]}$, see~\cite[Sect.~2.1]{M-PROM}.
\noindent$\bullet$ If $\mathcal{F}'\subseteq\mathbf{2}^{[t]}$ and~$\mathcal{F}''\subseteq\mathbf{2}^{[t]}$ are families of subsets of the ground set~$E_t$, then we will use the {\em componentwise product\/} of their characteristic vectors
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{F}')\ast\boldsymbol{\gamma}(\mathcal{F}'')
:=\bigl(\gamma_1(\mathcal{F}')\cdot\gamma_1(\mathcal{F}''),\ldots,\gamma_{2^t}(\mathcal{F}')\cdot\gamma_{2^t}(\mathcal{F}'')\bigr)
\in\{0,1\}^{2^t}
\end{equation*}
to describe\footnote{$\quad$ The notation {\tiny$\sideset{}{^\ast}\prod$} will be used to denote the {\em componentwise product\/} of several vectors.} the {\em intersection\/} of these families:
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{F}'\cap\mathcal{F}'')=\boldsymbol{\gamma}(\mathcal{F}')\ast\boldsymbol{\gamma}(\mathcal{F}'')\; .
\end{equation*}
\noindent$\bullet$ Let $\varGamma(k)$ denote the subset $A\subseteq E_t$, for which the characteristic vector
of the corresponding
one-member clutter $\{A\}$ on $E_t$
by convention is the $k$th standard unit vector~$\boldsymbol{\sigma}(k)$ of the space~$\mathbb{R}^{2^t}$; we thus use the map
\begin{equation*}
\varGamma: [2^t]\to\mathbf{2}^{[t]}\; ,\ \ \ \ \ k\mapsto A\colon\ \boldsymbol{\gamma}(\{A\})=\boldsymbol{\sigma}(k)\in\{0,1\}^{2^t}\; ;
\end{equation*}
see~(\ref{eq:18})--(\ref{eq:19}).
Conversely, we denote by $\varGamma^{-1}(A)$, where $A\subseteq E_t$, the position number $k$ such that the vector~$\boldsymbol{\sigma}(k)$ is the characteristic vector of the
one-member clutter $\{A\}$ on $E_t$:
\begin{equation*}
\varGamma^{-1}: \mathbf{2}^{[t]}\to [2^t]\; ,\ \ \ \ \ A\mapsto k\colon\ \boldsymbol{\sigma}(k)=\boldsymbol{\gamma}(\{A\})\in\{0,1\}^{2^t}\; ;
\end{equation*}
see~(\ref{eq:18})--(\ref{eq:19}).
By construction, we have the implications
\begin{gather}
\label{eq:21}
\ell',\ell''\in [2^t]\; ,\ \ \ell'<\ell''\ \ \ \Longrightarrow\ \ \ |\varGamma(\ell')|\leq|\varGamma(\ell'')|\; ;\\
\nonumber
A,B\in\mathbf{2}^{[t]}\; ,\ \ |A|<|B|\ \ \ \Longrightarrow\ \ \ \varGamma^{-1}(A)<\varGamma^{-1}(B)\; ,
\end{gather}
and, in particular,
\begin{equation*}
A,B\in\mathbf{2}^{[t]}\; ,\ \ A\subsetneqq B\ \ \ \Longrightarrow\ \ \ \varGamma^{-1}(A)<\varGamma^{-1}(B)\; .
\end{equation*}
Note also that for any index $\ell\in[2^t]$, the disjoint union
\begin{equation*}
\varGamma(\ell)\;\,\dot\cup\;\,\varGamma(2^t-\ell+1)=E_t
\end{equation*}
is a {\em partition\/} of the ground set.
\begin{example}
\label{th:1}
Suppose $t:=3$, and $E_t=\{1,2,3\}$. We have
{\small
\begin{align}
\label{eq:14}
\boldsymbol{\gamma}^{(0)}(\tbinom{E_t}{0}):=\boldsymbol{\gamma}^{(0)}(\{\hat{0}\})&=(1)\in\{0,1\}^{\binom{t}{0}}\; ,\\
\boldsymbol{\gamma}^{(1)}(\tbinom{E_t}{1}):=\boldsymbol{\gamma}^{(1)}(\{\{1\},\{2\},\{3\}\})&=(1,1,1)\in\{0,1\}^{\binom{t}{1}}\; ,\\
\boldsymbol{\gamma}^{(2)}(\tbinom{E_t}{2}):=\boldsymbol{\gamma}^{(2)}(\{\{1,2\},\{1,3\},\{2,3\}\})&=(1,1,1)\in\{0,1\}^{\binom{t}{2}}\; ,\\
\boldsymbol{\gamma}^{(t)}(\tbinom{E_t}{t}):=\boldsymbol{\gamma}^{(t)}(\{\{1,2,3\}\})&=(1)\in\{0,1\}^{\binom{t}{t}}\; ,\\
\boldsymbol{\gamma}^{(1)}(\{\{2\}\})
&=(0,1,0)\in\{0,1\}^{\binom{t}{1}}\; ,\\
\boldsymbol{\gamma}^{(2)}(\{\{1,2\},\{2,3\}\})
&=(1,0,1)\in\{0,1\}^{\binom{t}{2}}\; ,\\
\label{eq:15}
\boldsymbol{\gamma}^{(2)}(\{\{1,3\}\})
&=(0,1,0)\in\{0,1\}^{\binom{t}{2}}\; , \\
\intertext{\normalsize{and}}
\label{eq:16}
\boldsymbol{\gamma}(\emptyset)&=(0,0,0,0,0,0,0,0)\in\{0,1\}^{2^t}\; ,\\
\boldsymbol{\gamma}(\tbinom{E_t}{0})
&=(1,0,0,0,0,0,0,0)
\; ,\\
\boldsymbol{\gamma}(\tbinom{E_t}{1})
&=(0,1,1,1,0,0,0,0)
\; ,\\
\boldsymbol{\gamma}(\tbinom{E_t}{2})
&=(0,0,0,0,1,1,1,0)
\; ,\\
\boldsymbol{\gamma}(\tbinom{E_t}{t})
&=(0,0,0,0,0,0,0,1)
\; ,\\
\label{eq:20}
\boldsymbol{\gamma}(\mathbf{2}^{[t]})=\mathrm{T}_{2^t}^{(+)}:\!&=(1,1,1,1,1,1,1,1)
\; .
\intertext{\label{page:1}\normalsize{If $\mathcal{A}:=\{A_1,A_2\}$ and $\mathcal{B}:=\{B_1,B_2\}$ are clutters on $E_t$, where $A_1:=\{1,2\}$, \mbox{$A_2:=\{2,3\}$,}
$B_1:=\{1,3\}$, $B_2:=\{2\}$, and $\mathcal{B}=\mathfrak{B}(\mathcal{A})$, then we have
}
}
\boldsymbol{\gamma}(\mathcal{A}):=\boldsymbol{\gamma}(\{A_1,A_2\}):=\boldsymbol{\gamma}(\{\{1,2\},\{2,3\}\})&=(0,0,0,0,1,0,1,0)\in\{0,1\}^{2^t}\; ,\\
\boldsymbol{\gamma}(\mathcal{B}):=\boldsymbol{\gamma}(\{B_1,B_2\}):=\boldsymbol{\gamma}(\{\{1,3\},\{2\}\})&=(0,0,1,0,0,1,0,0)
\; ,\\
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}):=\boldsymbol{\gamma}(\{\{1,2\},\{2,3\}\}^{\triangledown})&=(0,0,0,0,1,0,1,1)
\; ,\\
\boldsymbol{\gamma}(\mathcal{B}^{\triangledown}):=\boldsymbol{\gamma}(\{\{1,3\},\{2\}\}^{\triangledown})&=(0,0,1,0,1,1,1,1)
\; ,\\
\boldsymbol{\gamma}(\{A_1\}^{\triangledown}):=\boldsymbol{\gamma}(\{\{1,2\}\}^{\triangledown})&=(0,0,0,0,1,0,0,1)
\; ,\\
\boldsymbol{\gamma}(\{A_2\}^{\triangledown}):=\boldsymbol{\gamma}(\{\{2,3\}\}^{\triangledown})&=(0,0,0,0,0,0,1,1)
\; ,\\
\boldsymbol{\gamma}(\{B_1\}^{\triangledown}):=\boldsymbol{\gamma}(\{\{1,3\}\}^{\triangledown})&=(0,0,0,0,0,1,0,1)
\; ,\\
\label{eq:17}
\boldsymbol{\gamma}(\{B_2\}^{\triangledown}):=\boldsymbol{\gamma}(\{\{2\}\}^{\triangledown})&=(0,0,1,0,1,0,1,1)
\; .
\end{align}
}
We have
{\small
\begin{align}
\label{eq:18}
\varGamma(3)&=\{2\}\; , & \boldsymbol{\gamma}(\{\{2\}\})&=\boldsymbol{\sigma}(3)\in\{0,1\}^{2^t}\; , & \varGamma^{-1}(\{2\})&=3\; ,\\
\varGamma(6)&=\{1,3\}\; , & \boldsymbol{\gamma}(\{\{1,3\}\})&=\boldsymbol{\sigma}(6)\in\{0,1\}^{2^t}\; , & \varGamma^{-1}(\{1,3\})&=6\; , \\
\label{eq:19}
\varGamma(2^t)&=E_t\; , & \boldsymbol{\gamma}(\{E_t\})&=\boldsymbol{\sigma}(2^t)\in\{0,1\}^{2^t}\; , & \varGamma^{-1}(E_t)&=2^t\; .
\end{align}
}
\end{example}
\noindent$\bullet$ Given a family $\mathcal{F}\subseteq\mathbf{2}^{[t]}$ of subsets of the ground set $E_t$, we call the tope
\begin{equation}
\label{eq:68}
T_{\mathcal{F}}:={}_{-\supp(\boldsymbol{\gamma}(\mathcal{F}))}\mathrm{T}_{2^t}^{(+)}=\mathrm{T}_{2^t}^{(+)}-2\boldsymbol{\gamma}(\mathcal{F})
\end{equation}
of the oriented matroid $\mathcal{H}_{2^t}:=(E_{2^t},\{1,-1\}^{2^t})$ the {\em characteristic tope\/} of the family $\mathcal{F}$;
see Example~\ref{th:2}.
If we let $\mathrm{T}^{(+)}_{\binom{t}{s}}:=(1,\ldots,1)\in\mathbb{R}^{\binom{t}{s}}$ denote the $\tbinom{t}{s}$-dimensional row vector
of all~$1$'s, $0\leq s\leq t$, then
\begin{equation*}
T^{(s)}_{\mathcal{F}}:={}_{-\supp(\boldsymbol{\gamma}^{(s)}(\mathcal{F}))}\mathrm{T}_{\binom{t}{s}}^{(+)}
=\mathrm{T}_{\binom{t}{s}}^{(+)}-2\boldsymbol{\gamma}^{(s)}(\mathcal{F})\; , \ \ \ 0\leq s\leq t\; .
\end{equation*}
\begin{example}
\label{th:2}
Suppose $t:=3$ and $E_t=\{1,2,3\}$. If
$\mathcal{A}$ and
$\mathcal{B}=\mathfrak{B}(\mathcal{A})$ are clutters on the ground set~$E_t$, mentioned in~Example~{\rm\ref{th:1}} on page~{\rm\pageref{page:1}}, then we have
{\small
\begin{align*}
\boldsymbol{\gamma}(\mathcal{A})
&=
(0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0)
\in\{0,1\}^{2^t}\; ,\\
T_{\mathcal{A}}:\!&=
(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,-1,\phantom{-}1)\in\{1,-1\}^{2^t}\; ;\\
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})
&=(0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1)\; ,\\
T_{\mathcal{A}^{\triangledown}}:\!&=(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,-1,-1)\; ;\\
\boldsymbol{\gamma}(\mathcal{B})
&=(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0)\; ,\\
T_{\mathcal{B}}
:\!&=(1,\phantom{-}1,-1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,\phantom{-}1)\; ;\\
\boldsymbol{\gamma}(\mathcal{B}^{\triangledown})
&=(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1)\; ,\\
T_{\mathcal{B}^{\triangledown}}
:\!&=(1,\phantom{-}1,-1,\phantom{-}1,-1,-1,-1,-1)
\; .
\end{align*}
}
\end{example}
\section{Increasing families of blocking sets, and blockers: Characteristic vectors and characteristic topes}
In this section, we begin with somewhat sophisticated restatements of the simple basic observations~(\ref{eq:22}), (\ref{eq:27}) and~(\ref{eq:28}) on set families in terms of their characteristic vectors.
\noindent$\bullet$
For a nontrivial clutter $\mathcal{A}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$, we have
\begin{gather*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
\ast\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})
\\
=\rn(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))
\ast\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}):=
\bigl(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\bigr)
\ast\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})
\\
=\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
\ast\rn(\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown}))
:=
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
\ast\bigl(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\bigr)
\\=
\begin{cases}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\; , & \text{if $\#\mathcal{A}^{\triangledown}<2^{t-1}$,}\\
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})\; , & \text{if $\#\mathcal{A}^{\triangledown}>2^{t-1}$,}\\
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})
=\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})\; , & \text{if $\#\mathcal{A}^{\triangledown}=2^{t-1}$.}
\end{cases}
\end{gather*}
\begin{remark}
\label{th:4}
Let $\mathcal{A}\subset\mathbf{2}^{[t]}$ be a nontrivial clutter on the ground set~$E_t$. We have
\begin{gather}
\nonumber
\gamma_1(\mathfrak{B}(\mathcal{A})^{\triangledown})=0\; ,\ \ \ \text{and}\ \ \ \ \gamma_{2^t}(\mathfrak{B}(\mathcal{A})^{\triangledown})=1\; ;\\
\label{eq:3}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})=\rn(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))
:=\mathrm{T}^{(+)}_{2^t}-\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\; ;\\
\label{eq:86}
T_{\mathfrak{B}(\mathcal{A})^{\triangledown}}=\ro(T_{\mathcal{A}^{\triangledown}})
:=-T_{\mathcal{A}^{\triangledown}}\cdot\overline{\mathbf{U}}(2^t)\; ;\\
\nonumber
\underbrace{
\hwt(\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown}))
}_{\#\mathfrak{B}(\mathcal{A})^{\triangledown}}
\; +\;
\underbrace{
\hwt(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))
}_{\#\mathcal{A}^{\triangledown}}
=2^t\; ;\\
\nonumber
\underbrace{
|(T_{\mathfrak{B}(\mathcal{A})^{\triangledown}})^-|
}_{\#\mathfrak{B}(\mathcal{A})^{\triangledown}}
\; +\;
\underbrace{
|(T_{\mathcal{A}^{\triangledown}})^-|
}_{\#\mathcal{A}^{\triangledown}}
=2^t\; ;\\
\label{eq:26}
\boldsymbol{\gamma}^{(s)}(\mathfrak{B}(\mathcal{A})^{\triangledown})=\rn(\boldsymbol{\gamma}^{(t-s)}(\mathcal{A}^{\triangledown}))
:=\mathrm{T}^{(+)}_{\binom{t}{s}}-\boldsymbol{\gamma}^{(t-s)}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(\tbinom{t}{s})\; ,
\ \ \ 0\leq s\leq t\; ;\\
\label{eq:87}
T^{(s)}_{\mathfrak{B}(\mathcal{A})^{\triangledown}}=\ro(T^{(t-s)}_{\mathcal{A}^{\triangledown}})
:=-T^{(t-s)}_{\mathcal{A}^{\triangledown}}\cdot\overline{\mathbf{U}}(\tbinom{t}{s})\; ,\ \ \ 0\leq s\leq t\; ;\\
\nonumber
\underbrace{
\hwt(\boldsymbol{\gamma}^{(s)}(\mathfrak{B}(\mathcal{A})^{\triangledown}))
}_{\#(\mathfrak{B}(\mathcal{A})^{\triangledown}\cap\binom{E_t}{s})}
\; +\;
\underbrace{
\hwt(\boldsymbol{\gamma}^{(t-s)}(\mathcal{A}^{\triangledown}))
}_{\#(\mathcal{A}^{\triangledown}\cap\binom{E_t}{t-s})}
=\tbinom{t}{s}\; , \ \ \ 0\leq s\leq t\; ;\\
\nonumber
\underbrace{
|(T^{(s)}_{\mathfrak{B}(\mathcal{A})^{\triangledown}})^-|
}_{\#(\mathfrak{B}(\mathcal{A})^{\triangledown}\cap\binom{E_t}{s})}
+\;
\underbrace{
|(T^{(t-s)}_{\mathcal{A}^{\triangledown}})^-|
}_{\#(\mathcal{A}^{\triangledown}\cap\binom{E_t}{t-s})}
=\tbinom{t}{s}\; , \ \ \ 0\leq s\leq t\; .
\end{gather}
\end{remark}
In addition to~(\ref{eq:3}) and~(\ref{eq:86}), relations~(\ref{eq:26}) and~(\ref{eq:87}) imply that
\begin{gather*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})=
\underbrace{
\rn(\boldsymbol{\gamma}^{(t)}(\mathcal{A}^{\triangledown}))
}_{(0)}
\;\centerdot\;
\rn(\boldsymbol{\gamma}^{(t-1)}(\mathcal{A}^{\triangledown}))
\;\centerdot\; \cdots
\;\centerdot\;
\rn(\boldsymbol{\gamma}^{(1)}(\mathcal{A}^{\triangledown}))
\;\centerdot\;
\underbrace{
\rn(\boldsymbol{\gamma}^{(0)}(\mathcal{A}^{\triangledown}))
}_{(1)}
\; ,\\
T_{\mathfrak{B}(\mathcal{A})}^{\triangledown}=
\underbrace{
\ro(T^{(t)}_{\mathcal{A}^{\triangledown}})
}_{(1)}
\;\centerdot\;
\ro(T^{(t-1)}_{\mathcal{A}^{\triangledown}})
\;\centerdot\; \cdots
\;\centerdot\;
\ro(T^{(1)}_{\mathcal{A}^{\triangledown}})
\;\centerdot\;
\underbrace{
\ro(T^{(0)}_{\mathcal{A}^{\triangledown}})
}_{(-1)}\; .
\end{gather*}
\noindent$\bullet$ In view of~(\ref{eq:21}), if
\begin{equation*}
\ell^{\star}:=\min\supp(\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown}))
=\min\,(T_{\mathfrak{B}(\mathcal{A})^{\triangledown}})^-
\; ,
\end{equation*}
then the member $\varGamma(\ell^{\star})$ of the blocker $\mathfrak{B}(\mathcal{A})$ of a nontrivial clutter $\mathcal{A}\subset\mathbf{2}^{[t]}$ is a blocking set of {\em minimum\/} cardinality for~$\mathcal{A}$, that is,
the vectors~$\boldsymbol{\chi}(\varGamma(\ell^{\star}))$ and ${}_{-\varGamma(\ell^{\star})}\mathrm{T}^{(+)}$ provide
the solution
$|\varGamma(\ell^{\star})|=\tau(\mathcal{A})$
to the set covering problems~(\ref{eq:29}) and~(\ref{eq:30}), respectively:
\begin{align*}
\boldsymbol{\chi}(\varGamma(\ell^{\star}))&\in
\Arg\min\{\mathrm{T}^{(+)}\tilde{\mathbf{z}}^{\top}\colon \tilde{\mathbf{z}}
\in \widetilde{\boldsymbol{\mathcal{S}}}\}\; ,\\
{}_{-\varGamma(\ell^{\star})}\mathrm{T}^{(+)}&\in
\Arg\max\{\mathrm{T}^{(+)}\mathbf{z}^{\top}\colon \mathbf{z}
\in \boldsymbol{\mathcal{S}}\}\; .
\end{align*}
\subsection{A clutter $\{\{a\}\}$} $\quad$ \label{section:1}
Let $\{\{a\}\}$ be a (nontrivial) clutter on the ground set $E_t$, whose only member is a {\em one-element\/} subset $\{a\}\subset E_t$.
\subsubsection{The principal increasing family of blocking sets $\mathfrak{B}(\{\{a\}\})^{\triangledown}=\{\{a\}\}^{\triangledown}$} $\quad$
\noindent$\bullet$ The {\em increasing family\/} of {\em blocking sets\/} $\mathfrak{B}(\{\{a\}\})^{\triangledown}$ of the {\em self-dual\/} clutter~$\{\{a\}\}$ coincides with the principal increasing family~$\{\{a\}\}^{\triangledown}$.
\noindent$\bullet$ We will use the notation $\widetilde{\boldsymbol{\mathfrak{a}}}(a):=\widetilde{\boldsymbol{\mathfrak{a}}}(a;2^t)$ and $\boldsymbol{\mathfrak{a}}(a):=\boldsymbol{\mathfrak{a}}(a;2^t)$ to denote the characteristic vector and the characteristic tope, respectively, that are associated with the principal increasing family~$\{\{a\}\}^{\triangledown}=\mathfrak{B}(\{\{a\}\})^{\triangledown}$\,:
\begin{gather*}
\widetilde{\boldsymbol{\mathfrak{a}}}(a):=\boldsymbol{\gamma}(\{\{a\}\}^{\triangledown})=
\boldsymbol{\gamma}(\mathfrak{B}(\{\{a\}\})^{\triangledown})
\in\{0,1\}^{2^t}\; ,
\\
\boldsymbol{\mathfrak{a}}(a):=T_{\{\{a\}\}^{\triangledown}}=T_{\mathfrak{B}(\{\{a\}\})^{\triangledown}}
\in\{1,-1\}^{2^t}\; .
\end{gather*}
We have
\begin{equation*}
\widetilde{\boldsymbol{\mathfrak{a}}}(a)=
\underbrace{
(0)
}_{\widetilde{\boldsymbol{\mathfrak{a}}}^{(0)}(a)}
\;\centerdot\;
\underbrace{
\boldsymbol{\chi}(\{a\})
}_{\widetilde{\boldsymbol{\mathfrak{a}}}^{(1)}(a)}
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(2)}(\{\{a\}\}^{\triangledown})
}_{\widetilde{\boldsymbol{\mathfrak{a}}}^{(2)}(a)}
\;\centerdot\;
\cdots
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(t-1)}(\{\{a\}\}^{\triangledown})
}_{\widetilde{\boldsymbol{\mathfrak{a}}}^{(t-1)}(a)}
\;\centerdot\;
\underbrace{
(1)
}_{\widetilde{\boldsymbol{\mathfrak{a}}}^{(t)}(a)}\; ,
\end{equation*}
see~(\ref{eq:39}), (\ref{eq:40}) and~(\ref{eq:41}) in~Example~\ref{th:16};
\begin{equation*}
\boldsymbol{\mathfrak{a}}(a)=
\underbrace{
(1)
}_{\boldsymbol{\mathfrak{a}}^{(0)}(a)}
\;\centerdot\;
\underbrace{
{}_{-\{a\}}\mathrm{T}^{(+)}
}_{\boldsymbol{\mathfrak{a}}^{(1)}(a)}
\;\centerdot\;
\underbrace{
T_{\{\{a\}\}^{\triangledown}}^{(2)}
}_{\boldsymbol{\mathfrak{a}}^{(2)}(a)}
\;\centerdot\;
\cdots
\;\centerdot\;
\underbrace{
T_{\{\{a\}\}^{\triangledown}}^{(t-1)}
}_{\boldsymbol{\mathfrak{a}}^{(t-1)}(a)}
\;\centerdot\;
\underbrace{
(-1)
}_{\boldsymbol{\mathfrak{a}}^{(t)}(a)}
\; ,
\end{equation*}
see~(\ref{eq:42}), (\ref{eq:43}) and~(\ref{eq:44}).
\begin{remark}[see~Remark~{\rm\ref{th:4}}, and cf.~Remark~{\rm\ref{th:13}}]
\label{th:12}
Note that
\begin{gather}
\label{eq:1}
\widetilde{\boldsymbol{\mathfrak{a}}}(a)=\rn(\widetilde{\boldsymbol{\mathfrak{a}}}(a))\; ,\ \ \ \text{and}\ \ \ \
\boldsymbol{\mathfrak{a}}(a)=\ro(\boldsymbol{\mathfrak{a}}(a))\; ;\\
\nonumber
\hwt(\widetilde{\boldsymbol{\mathfrak{a}}}(a))=|\boldsymbol{\mathfrak{a}}(a)^-|=
\#\{\{a\}\}^{\triangledown}=\#\mathfrak{B}(\{\{a\}\})^{\triangledown}=2^{t-1}\; ;\\
\nonumber
\widetilde{\boldsymbol{\mathfrak{a}}}^{(s)}(a)=\rn(\widetilde{\boldsymbol{\mathfrak{a}}}^{(t-s)}(a))\; ,\ \ \ \text{and}\ \ \ \
\boldsymbol{\mathfrak{a}}^{(s)}(a)=\ro(\boldsymbol{\mathfrak{a}}^{(t-s)}(a))\; ,\ \ \ 0\leq s\leq t\; ;\\
\nonumber
\hwt(\widetilde{\boldsymbol{\mathfrak{a}}}^{(s)}(a))=
|\boldsymbol{\mathfrak{a}}^{(s)}(a)^-|=\tbinom{t-1}{s-1}\; ,\ \ \ 0\leq s\leq t\; .
\end{gather}
\end{remark}
\subsubsection{The blocker $\mathfrak{B}(\{\{a\}\})=\{\{a\}\}$} $\quad$
\noindent$\bullet$ The {\em blocker\/} $\mathfrak{B}(\{\{a\}\})$ coincides with the self-dual clutter $\{\{a\}\}$.
\noindent$\bullet$ We associate with the clutter $\mathfrak{B}(\{\{a\}\})=\{\{a\}\}$ its characteristic vector~$\boldsymbol{\gamma}(\{\{a\}\})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{a\}\}))\in\{0,1\}^{2^t}$ and its characteristic tope~\mbox{$T_{\{\{a\}\}}$} $=T_{\mathfrak{B}(\{\{a\}\})}\in\{1,-1\}^{2^t}$,
where
\begin{equation*}
\boldsymbol{\gamma}(\{\{a\}\})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{a\}\}))=
(0)\;\centerdot\;\boldsymbol{\chi}(\{a\})
\;\centerdot\;(0,\ldots,0)\;\centerdot\;\cdots\;\centerdot\;(0)\; ,
\end{equation*}
see~(\ref{eq:45}), (\ref{eq:46}) and~(\ref{eq:47}) in~Example~\ref{th:16}.
\subsubsection{More on the principal increasing family~$\mathfrak{B}(\{\{a\}\})^{\triangledown}=\{\{a\}\}^{\triangledown}$} $\quad$
In view of~(\ref{eq:1}), we can make the following observation:
\begin{remark}[cf.~Remark~\ref{th:15}] \label{th:14}
For any element $a\in E_t$, we have
\begin{equation*}
\begin{split}
\min\{i\in E_{2^t}\colon T_{\{\{a\}\}^{\triangledown}}(i)=-1\}:\!&=\min\{i\in E_{2^t}\colon \gamma_i(\{\{a\}\}^{\triangledown})=1\}\\
:\!&=\varGamma^{-1}(\{a\})=1+a\; ,\\
\max\{i\in E_{2^t}\colon T_{\{\{a\}\}^{\triangledown}}(i)=1\}:\!&=\max\{i\in E_{2^t}\colon \gamma_i(\{\{a\}\}^{\triangledown})=0\}\\
&=2^t-\min\{a\}=2^t-a\; ;\\
\underbrace{\min\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{\{a\}\})^{\triangledown}}(j)=-1\}}_{
\min\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{\{a\}\})}(j)=-1\}
}
:\!&=
\underbrace{
\min\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{\{a\}\})^{\triangledown})=1\}
}_{
\min\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{\{a\}\}))=1\}
}
\\
&=1+\min\{a\}=1+a\; ,\\
\max\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{\{a\}\})^{\triangledown}}(j)=1\}:\!&=\max\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{\{a\}\})^{\triangledown})=0\}\\
&=1+2^t-\varGamma^{-1}(\{a\})=2^t-a\; .
\end{split}
\end{equation*}
\end{remark}
\noindent$\bullet$
We have
\begin{equation*}
\{\{a\}\}^{\triangledown}\ \dot{\cup}\ \{E_t-\{a\}\}^{\vartriangle}=\mathbf{2}^{[t]}\; .
\end{equation*}
Let us denote by $\widetilde{\boldsymbol{\mathfrak{c}}}(a):=\widetilde{\boldsymbol{\mathfrak{c}}}(a;2^t)$
and~$\boldsymbol{\mathfrak{c}}(a):=\boldsymbol{\mathfrak{c}}(a;2^t)$ the characteristic vector and the characteristic tope, respectively, of the principal decreasing family~$\{E_t-\{a\}\}^{\vartriangle}$:
\begin{equation*}
\widetilde{\boldsymbol{\mathfrak{c}}}(a):=\boldsymbol{\gamma}(\{E_t-\{a\}\}^{\vartriangle})\in\{0,1\}^{2^t}\; ,
\end{equation*}
see~(\ref{eq:48}), (\ref{eq:49}) and~(\ref{eq:50});
\begin{equation*}
\boldsymbol{\mathfrak{c}}(a):=T_{\{E_t-\{a\}\}^{\vartriangle}}\in\{1,-1\}^{2^t}\; ,
\end{equation*}
see~(\ref{eq:51}), (\ref{eq:52}) and~(\ref{eq:53}).
We have
\begin{gather*}
\widetilde{\boldsymbol{\mathfrak{c}}}(a)=\mathrm{T}_{2^t}^{(+)}-\widetilde{\boldsymbol{\mathfrak{a}}}(a)=
\widetilde{\boldsymbol{\mathfrak{a}}}(a)\cdot\overline{\mathbf{U}}(2^t)\; ,\\
\boldsymbol{\mathfrak{c}}(a)=-\boldsymbol{\mathfrak{a}}(a)=\boldsymbol{\mathfrak{a}}(a)\cdot\overline{\mathbf{U}}(2^t)\; .
\end{gather*}
\noindent$\bullet$ For any two-element subset $\{i,j\}\subset E_t$ of the ground set, we have
\begin{equation*}
\#(\{\{i\}\}^{\triangledown}\cap\{\{j\}\}^{\triangledown})
=\#\{\{i,j\}\}^{\triangledown}=2^{t-2}\; ,
\end{equation*}
and
\begin{equation*}
\#(\;\underbrace{
(\mathbf{2}^{[t]}-\{\{i\}\}^{\triangledown})
}_{
\{E_t-\{i\}\}^{\vartriangle}
}
\,\cap\,
\underbrace{
(\mathbf{2}^{[t]}-\{\{j\}\}^{\triangledown})
}_{
\{E_t-\{j\}\}^{\vartriangle}
}
\;)
=\#\{E_t-\{i,j\}\}^{\vartriangle}=2^{t-2}\; .
\end{equation*}
Thus, if $i$ and $j$ are elements of the ground set $E_t$, and $i\neq j$, then we have
\begin{equation*}
\begin{split}
d\bigl(\widetilde{\boldsymbol{\mathfrak{a}}}(i),\widetilde{\boldsymbol{\mathfrak{a}}}(j)\bigr)
=d\bigl(\boldsymbol{\mathfrak{a}}(i),\boldsymbol{\mathfrak{a}}(j)\bigr)
&=d\bigl(\widetilde{\boldsymbol{\mathfrak{c}}}(i),\widetilde{\boldsymbol{\mathfrak{c}}}(j)\bigr)
=d\bigl(\boldsymbol{\mathfrak{c}}(i),\boldsymbol{\mathfrak{c}}(j)\bigr)
\\
&=2^{t-1}\; .
\end{split}
\end{equation*}
\begin{remark}
For any two elements $i$ and $j$ of the ground set~$E_t$ we have
\begin{equation*}
\bigl\langle \boldsymbol{\mathfrak{a}}(i),\boldsymbol{\mathfrak{a}}(j)\bigr\rangle=
\bigl\langle \boldsymbol{\mathfrak{c}}(i),\boldsymbol{\mathfrak{c}}(j)\bigr\rangle=
\delta_{i,j}\cdot 2^t\; .
\end{equation*}
In other words, the sequences of\/ $t$ row vectors
\begin{equation*}
\bigl(\,\tfrac{1}{\sqrt{2^{t}}}\cdot
\boldsymbol{\mathfrak{a}}(1),\; \tfrac{1}{\sqrt{2^{t}}}\cdot
\boldsymbol{\mathfrak{a}}(2),\;\ldots,\;\tfrac{1}{\sqrt{2^{t}}}\cdot
\boldsymbol{\mathfrak{a}}(t)\bigr)\subset\mathbb{R}^{2^t}
\end{equation*}
and
\begin{equation*}
\bigl(\,\tfrac{1}{\sqrt{2^{t}}}\cdot
\boldsymbol{\mathfrak{c}}(t),\;\tfrac{1}{\sqrt{2^{t}}}\cdot \boldsymbol{\mathfrak{c}}(t-1),\;\ldots,\;\tfrac{1}{\sqrt{2^{t}}}\cdot
\boldsymbol{\mathfrak{c}}(1)\bigr)\subset\mathbb{R}^{2^t}
\end{equation*}
are both {\em orthonormal}.
\end{remark}
\begin{example}\label{th:16}
Suppose $t:=3$, and $E_t=\{1,2,3\}$. We have
{\small
\begin{align}
\label{eq:39}
\widetilde{\boldsymbol{\mathfrak{a}}}(1):=\boldsymbol{\gamma}(\{\{1\}\}^{\triangledown})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{1\}\})^{\triangledown})
&=(\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1)\in\{0,1\}^{2^t}\; ,\\
\label{eq:42}
\boldsymbol{\mathfrak{a}}(1):=T_{\{\{1\}\}^{\triangledown}}=T_{\mathfrak{B}(\{\{1\}\})^{\triangledown}}
&=(\phantom{-}1,-1,\phantom{-}1,\phantom{-}1,-1,-1,\phantom{-}1,-1)\in\{1,-1\}^{2^t}\; ,
\\
\label{eq:48}
\widetilde{\boldsymbol{\mathfrak{c}}}(1):=\boldsymbol{\gamma}(\{E_t-\{1\}\}^{\vartriangle})
&=(\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0)
\; ,\\
\label{eq:51}
\boldsymbol{\mathfrak{c}}(1):=T_{\{E_t-\{1\}\}^{\vartriangle}}
&=(-1,\phantom{-}1,-1,-1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1)
\; ,
\\
\label{eq:45}
\boldsymbol{\gamma}(\{\{1\}\})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{1\}\}))
&=(\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:40}
\widetilde{\boldsymbol{\mathfrak{a}}}(2):=\boldsymbol{\gamma}(\{\{2\}\}^{\triangledown})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{2\}\})^{\triangledown})
&=(\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:43}
\boldsymbol{\mathfrak{a}}(2):=T_{\{\{2\}\}^{\triangledown}}=T_{\mathfrak{B}(\{\{2\}\})^{\triangledown}}
&=(\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,-1,\phantom{-}1,-1,-1)
\; ,\\
\label{eq:49}
\widetilde{\boldsymbol{\mathfrak{c}}}(2):=\boldsymbol{\gamma}(\{E_t-\{2\}\}^{\vartriangle})
&=(\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:52}
\boldsymbol{\mathfrak{c}}(2):=T_{\{E_t-\{2\}\}^{\vartriangle}}
&=(-1,-1,\phantom{-}1,-1,\phantom{-}1,-1,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:46}
\boldsymbol{\gamma}(\{\{2\}\})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{2\}\}))
&=(\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:41}
\widetilde{\boldsymbol{\mathfrak{a}}}(t):=\boldsymbol{\gamma}(\{\{t\}\}^{\triangledown})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{t\}\})^{\triangledown})
&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:44}
\boldsymbol{\mathfrak{a}}(t):=T_{\{\{t\}\}^{\triangledown}}=T_{\mathfrak{B}(\{\{t\}\})^{\triangledown}}
&=(\phantom{-}1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,-1,-1,-1)
\; ,\\
\label{eq:50}
\widetilde{\boldsymbol{\mathfrak{c}}}(t):=\boldsymbol{\gamma}(\{E_t-\{t\}\}^{\vartriangle})
&=(\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:53}
\boldsymbol{\mathfrak{c}}(t):=T_{\{E_t-\{t\}\}^{\vartriangle}}
&=(-1,-1,-1,\phantom{-}1,-1,\phantom{-}1,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:47}
\boldsymbol{\gamma}(\{\{t\}\})=\boldsymbol{\gamma}(\mathfrak{B}(\{\{t\}\}))
&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:60}
\boldsymbol{\gamma}(\{\{1,2\}\})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:54}
\boldsymbol{\gamma}(\{\{1,2\}\}^{\triangledown})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1)
\; ,\\
\label{eq:63}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{1,2\}\}))&=(\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:57}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{1,2\}\})^{\triangledown})&=(\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:61}
\boldsymbol{\gamma}(\{\{1,t\}\})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:55}
\boldsymbol{\gamma}(\{\{1,t\}\}^{\triangledown})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1)
\; ,\\
\label{eq:64}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{1,t\}\}))&=(\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:58}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{1,t\}\})^{\triangledown})&=(\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:62}
\boldsymbol{\gamma}(\{\{2,t\}\})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0)
\; ,\\
\label{eq:56}
\boldsymbol{\gamma}(\{\{2,t\}\}^{\triangledown})&=(\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1)
\; ,\\
\label{eq:65}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{2,t\}\}))&=(\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:59}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{2,t\}\})^{\triangledown})&=(\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1)
\; .
\end{align}
}
\end{example}
\noindent$\bullet$ Given a nontrivial clutter~$\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}\subset\mathbf{2}^{[t]}$ on the ground set~$E_t$, such that $\mathcal{A}\neq\{E_t\}$, we have
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathcal{A}):\!&=\sum_{i\in[\alpha]}
\boldsymbol{\sigma}(\varGamma^{-1}(A_i))
=\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\ast\boldsymbol{\gamma}(\mathcal{A}^{\vartriangle})
=\sum_{i\in[\alpha]}\bigl(\boldsymbol{\gamma}(\{A_i\}^{\triangledown})\ast\boldsymbol{\gamma}(\{A_i\}^{\vartriangle})\bigr)
\\
&=\sum_{i\in[\alpha]}\Bigl(\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)
\;\ast\sideset{}{^\ast}\prod_{c^i\in E_t-A_i}\widetilde{\boldsymbol{\mathfrak{c}}}(c^i)\Bigr)
\\
&=\sum_{i\in[\alpha]}\Bigl(\sideset{}{^\ast}\prod_{a^i\in A_i}
\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)
\;
\ast\Bigl(\Bigl(\sideset{}{^\ast}\prod_{c^i\in E_t-A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(c^i)\Bigr)\cdot\overline{\mathbf{U}}(2^t)\Bigr)
\Bigr)
\\
&=\sum_{i\in[\alpha]}\Bigl(\Bigl(\Bigl(\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{c}}}(a^i)\Bigr)\cdot\overline{\mathbf{U}}(2^t)\Bigr)
\;\ast\sideset{}{^\ast}\prod_{c^i\in E_t-A_i}\widetilde{\boldsymbol{\mathfrak{c}}}(c^i)\Bigr)
\; .
\end{split}
\end{equation*}
\subsection{A clutter $\{A\}$} $\quad$ \label{section:2}
Let $\{A\}$ be a (nontrivial) clutter on the ground set~$E_t$, whose only member is a {\em nonempty subset} $A\subseteq E_t$.
\subsubsection{The increasing family of blocking sets $\mathfrak{B}(\{A\})^{\triangledown}=\{\{a\}\colon a\in A\}^{\triangledown}$} $\quad$
\noindent$\bullet$ The {\em family\/} of {\em blocking sets\/} $\mathfrak{B}(\{A\})^{\triangledown}$ of the clutter $\{A\}$ is the increasing family~$\{\{a\}\colon a\in A\}^{\triangledown}$.
We have
\begin{equation*}
\{A\}^{\triangledown}=\bigcap_{a\in A}\{\{a\}\}^{\triangledown}\; , \ \ \ \text{and}\ \ \ \
\mathfrak{B}(\{A\})^{\triangledown}=\bigcup_{a\in A}\{\{a\}\}^{\triangledown}\; .
\end{equation*}
Let us associate with the increasing families $\{A\}^{\triangledown}$ and $\mathfrak{B}(\{A\})^{\triangledown}$ their characteristic vectors
$\boldsymbol{\gamma}(\{A\}^{\triangledown})\in\{0,1\}^{2^t}$ and~$\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})\in\{0,1\}^{2^t}$,
and their characteristic topes~$T_{\{A\}^{\triangledown}}\in\{1,-1\}^{2^t}$ and~$T_{\mathfrak{B}(\{A\})^{\triangledown}}\in\{1,-1\}^{2^t}$,
where
\begin{multline*}
\boldsymbol{\gamma}(\{A\}^{\triangledown})=\boldsymbol{\gamma}\Bigl(\bigcap_{a\in A}\{\{a\}\}^{\triangledown}\Bigr)
\\
=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\{A\}^{\triangledown})
}
\;\centerdot\;\cdots\;\centerdot\;
\underbrace{
(0,\ldots,0)
}_{
\boldsymbol{\gamma}^{(|A|-1)}(\{A\}^{\triangledown})
}
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(|A|)}(\{A\})
}_{
\boldsymbol{\gamma}^{(|A|)}(\{A\}^{\triangledown})
}
\\
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(|A|+1)}\Bigl(\bigcap_{a\in A}\{\{a\}\}^{\triangledown}\Bigr)
}_{
\boldsymbol{\gamma}^{(|A|+1)}(\{A\}^{\triangledown})
}
\;\centerdot\;
\cdots
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(t-1)}\Bigl(\bigcap_{a\in A}\{\{a\}\}^{\triangledown}\Bigr)
}_{
\boldsymbol{\gamma}^{(t-1)}(\{A\}^{\triangledown})
}
\;\centerdot\;
\underbrace{
(1)
}_{
\boldsymbol{\gamma}^{(t)}(\{A\}^{\triangledown})
}
\; ,
\end{multline*}
see~(\ref{eq:54}), (\ref{eq:55}) and~(\ref{eq:56}) in Example~\ref{th:16},
and
\begin{multline*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})=\boldsymbol{\gamma}\Bigl(\bigcup_{a\in A}\{\{a\}\}^{\triangledown}\Bigr)
\\=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathfrak{B}(\{A\})^{\triangledown})
}
\;\centerdot\;
\underbrace{
\boldsymbol{\chi}(A)
}_{\boldsymbol{\gamma}^{(1)}(\mathfrak{B}(\{A\})^{\triangledown})
}
\\
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(2)}(\bigcup_{a\in A}\{\{a\}\}^{\triangledown})
}_{
\boldsymbol{\gamma}^{(2)}(\mathfrak{B}(\{A\})^{\triangledown})
}
\;\centerdot\;
\cdots
\;\centerdot\;
\underbrace{
\boldsymbol{\gamma}^{(t-1)}(\bigcup_{a\in A}\{\{a\}\}^{\triangledown})
}_{
\boldsymbol{\gamma}^{(t-1)}(\mathfrak{B}(\{A\})^{\triangledown})
}
\;\centerdot\;
\underbrace{
(1)
}_{
\boldsymbol{\gamma}^{(t)}(\mathfrak{B}(\{A\})^{\triangledown})
}
\; ,
\end{multline*}
see~(\ref{eq:57}), (\ref{eq:58}) and~(\ref{eq:59}).
\begin{remark}[see~Remark~{\rm\ref{th:4}}, and cf.~Remark~{\rm\ref{th:12}}]
\label{th:13}
Note that
\begin{gather*}
\nonumber
\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})=\rn(\boldsymbol{\gamma}(\{A\}^{\triangledown}))
:=\mathrm{T}^{(+)}_{2^t}-\boldsymbol{\gamma}(\{A\}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\; ;\\
\nonumber
T_{\mathfrak{B}(\{A\})^{\triangledown}}=\ro(T_{\{A\}^{\triangledown}})
:=-T_{\{A\}^{\triangledown}}\cdot\overline{\mathbf{U}}(2^t)\; ;\\
\nonumber
\underbrace{
\hwt(\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown}))
}_{\#\mathfrak{B}(\{A\})^{\triangledown}}=\underbrace{
|(T_{\mathfrak{B}(\{A\})^{\triangledown}})^-|
}_{\#\mathfrak{B}(\{A\})^{\triangledown}}=2^t-2^{t-|A|}\; ;\\
\underbrace{
\hwt(\boldsymbol{\gamma}(\{A\}^{\triangledown}))
}_{\#\{A\}^{\triangledown}}=
\underbrace{
|(T_{\{A\}^{\triangledown}})^-|
}_{\#\{A\}^{\triangledown}}
=2^{t-|A|}\; ;\\
\boldsymbol{\gamma}^{(s)}(\mathfrak{B}(\{A\})^{\triangledown})=\rn(\boldsymbol{\gamma}^{(t-s)}(\{A\}^{\triangledown}))
:=\mathrm{T}^{(+)}_{\binom{t}{s}}-\boldsymbol{\gamma}^{(t-s)}(\{A\}^{\triangledown})\cdot\overline{\mathbf{U}}(\tbinom{t}{s})\; ,
\ \ \ 0\leq s\leq t\; ;\\
\nonumber
T^{(s)}_{\mathfrak{B}(\{A\})^{\triangledown}}=\ro(T^{(t-s)}_{\{A\}^{\triangledown}})
:=-T^{(t-s)}_{\{A\}^{\triangledown}}\cdot\overline{\mathbf{U}}(\tbinom{t}{s})\; ,\ \ \ 0\leq s\leq t\; ;\\
\nonumber
\underbrace{
\hwt(\boldsymbol{\gamma}^{(s)}(\mathfrak{B}(\{A\})^{\triangledown}))
}_{\#(\mathfrak{B}(\{A\})^{\triangledown}\cap\binom{E_t}{s})}\; =
\underbrace{
|(T^{(s)}_{\mathfrak{B}(\{A\})^{\triangledown}})^-|
}_{\#(\mathfrak{B}(\{A\})^{\triangledown}\cap\binom{E_t}{s})}
=\tbinom{t}{s}-\tbinom{t-|A|}{s}\; ,\ \ \ 0\leq s\leq t;\\
\underbrace{
\hwt(\boldsymbol{\gamma}^{(t-s)}(\{A\}^{\triangledown}))
}_{\#(\{A\}^{\triangledown}\cap\binom{E_t}{t-s})}\;
=\underbrace{
|(T^{(t-s)}_{\{A\}^{\triangledown}})^-|
}_{\#(\{A\}^{\triangledown}\cap\binom{E_t}{t-s})}
=\tbinom{t-|A|}{s}\; , \ \ \ 0\leq s\leq t\; .
\end{gather*}
\end{remark}
\subsubsection{The blocker $\mathfrak{B}(\{A\})=\{\{a\}\colon a\in A\}$} $\quad$
\noindent$\bullet$ The {\em blocker\/}
of the clutter $\{A\}$ is the clutter
\begin{equation*}
\mathfrak{B}(\{A\})=\{\{a\}\colon a\in A\}\; .
\end{equation*}
Thus, $\#\mathfrak{B}(\{A\})=|A|$, and the members of the blocker $\mathfrak{B}(\{A\})$ are the one-element subsets of the set $A$.
\noindent$\bullet$ We associate with the clutters $\{A\}$ and $\mathfrak{B}(\{A\})$ their characteristic vectors~$\boldsymbol{\gamma}(\{A\})\in\{0,1\}^{2^t}$ and $\boldsymbol{\gamma}(\mathfrak{B}(\{A\}))\in\{0,1\}^{2^t}$, and their characteristic topes~$T_{\{A\}}\in\{1,-1\}^{2^t}$ and $T_{\mathfrak{B}(\{A\})}\in\{1,-1\}^{2^t}$, where
\begin{equation*}
\boldsymbol{\gamma}(\{A\})=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\{A\})
}
\;\centerdot\;\cdots\;\centerdot\;
\underbrace{
(0,\ldots,0)
}_{
\boldsymbol{\gamma}^{(|A|-1)}(\{A\})
}
\;\centerdot\;
\boldsymbol{\gamma}^{(|A|)}(\{A\})
\;\centerdot\;
\underbrace{
(0,\ldots,0)
}_{
\boldsymbol{\gamma}^{(|A|+1)}(\{A\})
}
\centerdot\;\cdots\;\centerdot\;
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(t)}(\{A\})
}
\; ,
\end{equation*}
see~(\ref{eq:60}), (\ref{eq:61}) and~(\ref{eq:62}) in Example~\ref{th:16}, and
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\}))
=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathfrak{B}(\{A\}))
}
\;\centerdot\;
\underbrace{
\boldsymbol{\chi}(A)
}_{
\boldsymbol{\gamma}^{(1)}(\mathfrak{B}(\{A\}))
}
\;\centerdot\;
\underbrace{
(0,\ldots,0)
}_{
\boldsymbol{\gamma}^{(2)}(\mathfrak{B}(\{A\}))
}
\;\centerdot\;\cdots\;\centerdot\;
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(t)}(\mathfrak{B}(\{A\}))
}
\; ,
\end{equation*}
see~(\ref{eq:63}), (\ref{eq:64}) and~(\ref{eq:65}).
\subsubsection{More on the increasing families $\{A\}^{\triangledown}$ and\/ $\mathfrak{B}(\{A\})^{\triangledown}$} $\quad$
We can make the following observation:
\begin{remark}[cf.~Remark~\ref{th:14}] \label{th:15}
For a nonempty subset $A\subseteq E_t$, we have
\begin{equation*}
\begin{split}
\min\{i\in E_{2^t}\colon T_{\{A\}^{\triangledown}}(i)=-1\}:\!&=\min\{i\in E_{2^t}\colon \gamma_i(\{A\}^{\triangledown})=1\}\\
:\!&=\varGamma^{-1}(A)\; ,\\
\max\{i\in E_{2^t}\colon T_{\{A\}^{\triangledown}}(i)=1\}:\!&=\max\{i\in E_{2^t}\colon \gamma_i(\{A\}^{\triangledown})=0\}\\
&=2^t-\min A\; ;\\
\underbrace{\min\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{A\})^{\triangledown}}(j)=-1\}}_{
\min\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{A\})}(j)=-1\}
}
:\!&=
\underbrace{
\min\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{A\})^{\triangledown})=1\}
}_{
\min\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{A\}))=1\}
}
\\
&=1+\min A\; ,\\
\max\{j\in E_{2^t}\colon T_{\mathfrak{B}(\{A\})^{\triangledown}}(j)=1\}:\!&=\max\{j\in E_{2^t}\colon \gamma_j(\mathfrak{B}(\{A\})^{\triangledown})=0\}\\
&=1+2^t-\varGamma^{-1}(A)\; .
\end{split}
\end{equation*}
\end{remark}
\noindent$\bullet$ Recall that the partition
\begin{equation*}
\{A\}^{\triangledown}\;\dot{\cup}\; (\mathfrak{B}(\{A\})^{\compl})^{\vartriangle}=\mathbf{2}^{[t]}
\end{equation*}
implies that
\begin{equation*}
\mathfrak{B}(\{A\})^{\triangledown}=\{D^{\compl}\colon D\in\mathbf{2}^{[t]}-\{A\}^{\triangledown}\}\; .
\end{equation*}
\noindent$\bullet$
Note that
\begin{gather*}
\boldsymbol{\gamma}(\{A\}^{\triangledown})=\sideset{}{^\ast}\prod_{a\in A}\boldsymbol{\gamma}(\{\{a\}\}^{\triangledown})
=:\sideset{}{^\ast}\prod_{a\in A}\widetilde{\boldsymbol{\mathfrak{a}}}(a)
\\
=\sideset{}{^\ast}\prod_{a\in A}\bigl(\mathrm{T}_{2^t}^{(+)}-\widetilde{\boldsymbol{\mathfrak{c}}}(a)\bigr)
=\Bigl(\sideset{}{^\ast}\prod_{a\in A}\widetilde{\boldsymbol{\mathfrak{c}}}(a)\Bigr)\cdot\overline{\mathbf{U}}(2^t)\; ,
\end{gather*}
and recall that
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})=\rn(\boldsymbol{\gamma}(\{A\}^{\triangledown}))\; .
\end{equation*}
\begin{remark}
\label{th:8}
For a nonempty subset $A\subseteq E_t$, we have{\rm:}
\begin{itemize}
\item[\rm(i)]
\begin{equation*}
\boldsymbol{\gamma}(\{A\}^{\triangledown})=\sideset{}{^\ast}\prod_{a\in A}\widetilde{\boldsymbol{\mathfrak{a}}}(a)\; .
\end{equation*}
\item[\rm(ii)]
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})
=\mathrm{T}_{2^t}^{(+)}
-\Bigl(\sideset{}{^\ast}\prod_{a\in A}\widetilde{\boldsymbol{\mathfrak{a}}}(a)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
\; .
\end{equation*}
\end{itemize}
\end{remark}
\subsection{A clutter $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$} $\quad$ \label{section:3}
Let $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$ be a nontrivial clutter on the ground set $E_t$.
\subsubsection{The increasing family of blocking sets $\mathfrak{B}(\mathcal{A})^{\triangledown}$} $\quad$
\noindent$\bullet$ See~Remark~\ref{th:4}, and note that
\begin{align*}
\mathcal{A}^{\triangledown}&=
\bigcup_{k\in[\alpha]}\{A_k\}^{\triangledown}=
\bigcup_{k\in[\alpha]}\; \bigcap_{a^k\in A_k}\{\{a^k\}\}^{\triangledown}\; ,\\
\intertext{and}
\mathfrak{B}(\mathcal{A})^{\triangledown}&=
\bigcap_{k\in[\alpha]}\mathfrak{B}(\{A_k\})^{\triangledown}=
\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown}\; .
\end{align*}
\noindent$\bullet$ We associate with the increasing families $\mathcal{A}^{\triangledown}$ and $\mathfrak{B}(\mathcal{A})^{\triangledown}$ their characteristic vectors~$\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\in\{0,1\}^{2^t}$ and~$\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})\in\{0,1\}^{2^t}$, and their characteristic topes~$T_{\mathcal{A}^{\triangledown}}\in\{1,-1\}^{2^t}$ and~$T_{\mathfrak{B}(\mathcal{A})^{\triangledown}}
\in\{1,-1\}^{2^t}$,
where
\begin{multline*}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})=\boldsymbol{\gamma}\Bigl(\bigcup_{k\in[\alpha]}\; \bigcap_{a^k\in A_k}\{\{a^k\}\}^{\triangledown}\Bigr)
\\
=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathcal{A}^{\triangledown})
}
\;\centerdot\;
\boldsymbol{\gamma}^{(1)}(\bigcup_{k\in[\alpha]}\; \bigcap_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})
\;\centerdot\;\cdots\;\centerdot\;
\boldsymbol{\gamma}^{(t-1)}(\bigcup_{k\in[\alpha]}\; \bigcap_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})
\;\centerdot\;
\underbrace{
(1)
}_{
\boldsymbol{\gamma}^{(t)}(\mathcal{A}^{\triangledown})
}
\; ,
\end{multline*}
and
\begin{multline*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})=\boldsymbol{\gamma}\Bigl(\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown}\Bigr)
\\=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathfrak{B}(\mathcal{A})^{\triangledown})
}
\;\centerdot\;\boldsymbol{\gamma}^{(1)}(\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})
\;\centerdot\;\cdots\;\centerdot\;
\boldsymbol{\gamma}^{(t-1)}(\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})
\;\centerdot\;
\underbrace{
(1)
}_{
\boldsymbol{\gamma}^{(t)}(\mathfrak{B}(\mathcal{A})^{\triangledown})
}
\; .
\end{multline*}
\subsubsection{The blocker~$\mathfrak{B}(\mathcal{A})$} $\quad$
\noindent$\bullet$ The {\em blocker\/}
of the clutter $\mathcal{A}$ is the clutter
\begin{multline*}
\mathfrak{B}(\mathcal{A})=\bmin\bigcap_{k\in[\alpha]}\mathfrak{B}(\{A_k\})^{\triangledown}
\\=\bmin\bigcap_{k\in[\alpha]}\{\{a^k\}\colon a^k\in A_k\}^{\triangledown}=
\bmin\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown}
\; .
\end{multline*}
\noindent$\bullet$ We associate with the clutters $\mathcal{A}$ and $\mathfrak{B}(\mathcal{A})$ their characteristic vectors $\boldsymbol{\gamma}(\mathcal{A})\in\{0,1\}^{2^t}$ and $\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A}))\in\{0,1\}^{2^t}$, and their characteristic topes \mbox{$T_{\mathcal{A}}\in\{1,-1\}^{2^t}$} and $T_{\mathfrak{B}(\mathcal{A})}\in\{1,-1\}^{2^t}$,
where
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{A}):=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathcal{A})
}
\;\centerdot\;
\boldsymbol{\gamma}^{(1)}(\mathcal{A})
\;\centerdot\;\cdots\;\centerdot\;
\boldsymbol{\gamma}^{(t)}(\mathcal{A})\; ,
\end{equation*}
and
\begin{multline*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A}))=
\underbrace{
(0)
}_{
\boldsymbol{\gamma}^{(0)}(\mathfrak{B}(\mathcal{A}))
}
\;\centerdot\;\boldsymbol{\gamma}^{(1)}(\bmin\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})
\\
\;\centerdot\;\cdots\;\centerdot\;
\boldsymbol{\gamma}^{(t)}(\bmin\bigcap_{k\in[\alpha]}\;\bigcup_{a^k\in A_k}\{\{a^k\}\}^{\triangledown})\; .
\end{multline*}
\subsubsection{More on the increasing families $\mathcal{A}^{\triangledown}$ and\/ $\mathfrak{B}(\mathcal{A})^{\triangledown}$} $\quad$
\noindent$\bullet$ Recall that we have
\begin{equation*}
\mathcal{A}^{\triangledown}\;\dot{\cup}\; (\mathfrak{B}(\mathcal{A})^{\compl})^{\vartriangle}=\mathbf{2}^{[t]}\; ,
\end{equation*}
that is,
\begin{equation*}
\mathfrak{B}(\mathcal{A})^{\triangledown}=\{D^{\compl}\colon D\in\mathbf{2}^{[t]}-\mathcal{A}^{\triangledown}\}\; .
\end{equation*}
\noindent$\bullet$ According to~Remark~\ref{th:8}(ii), we have
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})&=\sideset{}{^\ast}\prod_{i\in[\alpha]}\boldsymbol{\gamma}(\mathfrak{B}(\{A_i\})^{\triangledown})
=\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}_{2^t}^{(+)}
-\Bigl(\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\cdot\overline{\mathbf{U}}(2^t)\Bigr)\\
&=\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
\; .
\end{split}
\end{equation*}
Since
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})=\rn(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))
:=\mathrm{T}_{2^t}^{(+)}
-\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\; ,
\end{equation*}
by~(\ref{eq:3}), we have
\begin{equation*}
\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}_{2^t}^{(+)}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
=\mathrm{T}_{2^t}^{(+)}
-\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)\; ,
\end{equation*}
that is,
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})\cdot\overline{\mathbf{U}}(2^t)=
\mathrm{T}_{2^t}^{(+)}-
\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}_{2^t}^{(+)}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
\; .
\end{equation*}
\begin{theorem} \label{th:5} If $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$ is a nontrivial clutter on the ground set~$E_t$, then we have{\rm:}
\begin{itemize}
\item[\rm(i)]
\begin{equation}
\label{eq:5}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})=
\mathrm{T}_{2^t}^{(+)}-
\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\; .
\end{equation}
\item[\rm(ii)]
\begin{equation}
\label{eq:4}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
=\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
\; .
\end{equation}
\end{itemize}
\end{theorem}
\begin{example}\label{page:3} \label{th:17}
Suppose $t:=3$, and $E_t=\{1,2,3\}$.
We have in our hands the characteristic vectors
{\small
\begin{align*}
\widetilde{\boldsymbol{\mathfrak{a}}}(1):=\widetilde{\boldsymbol{\mathfrak{a}}}(1;2^t):=\boldsymbol{\gamma}(\{\{1\}\}^{\triangledown})
&=(0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1)\in\{0,1\}^{2^t}\; ,\\
\widetilde{\boldsymbol{\mathfrak{a}}}(2):=\widetilde{\boldsymbol{\mathfrak{a}}}(2;2^t):=\boldsymbol{\gamma}(\{\{2\}\}^{\triangledown})
&=(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1)\; ,\\
\widetilde{\boldsymbol{\mathfrak{a}}}(3):=\widetilde{\boldsymbol{\mathfrak{a}}}(3;2^t):=\boldsymbol{\gamma}(\{\{3\}\}^{\triangledown})
&=(0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1)\; ,
\end{align*}
}
$\quad$\vspace{-5mm}\\
associated with the principal increasing families that are generated by the clutters~$\{\{a\}\}$, for the elements $a\in E_t$ of the ground set.
We are given the clutter~$\mathcal{A}:=\{A_1,A_2\}$ on the ground set~$E_t$, where $A_1:=\{1,2\}$ and $A_2:=\{2,3\}$,
and we want to know the characteristic vector~$\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})$ of the increasing family of blocking sets~$\mathfrak{B}(\mathcal{A})^{\triangledown}$ of the clutter~$\mathcal{A}$.
Turning to Theorem~{\rm\ref{th:5}(ii)}, we see that
{\small
\begin{align*}
\sideset{}{^\ast}\prod_{a^1\in A_1}\widetilde{\boldsymbol{\mathfrak{a}}}(a^1):=\sideset{}{^\ast}\prod_{a^1\in \{1,2\}}\widetilde{\boldsymbol{\mathfrak{a}}}(a^1)
=\quad &(0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1)\\
\ast\, &(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1)\\
=\quad &(0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0,\phantom{-}1)\; ,
\\
\sideset{}{^\ast}\prod_{a^2\in A_2}\widetilde{\boldsymbol{\mathfrak{a}}}(a^2):=
\sideset{}{^\ast}\prod_{a^2\in\{2,3\}}\widetilde{\boldsymbol{\mathfrak{a}}}(a^2)
=\quad &(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1)\\
\ast\, &(0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1)\\
=\quad&(0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}0,\phantom{-}1,\phantom{-}1)\; ;\\
\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^1\in A_1}\widetilde{\boldsymbol{\mathfrak{a}}}(a^1)
=\quad &(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0)\; ,\\
\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^2\in A_2}\widetilde{\boldsymbol{\mathfrak{a}}}(a^2)
=\quad &(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}0)\; ;\\
\sideset{}{^\ast}\prod_{i\in[2]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)
=\quad &(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}0)\\
\ast\, &(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}0)\\
=\quad &(1,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}0)\; ,\\
\intertext{\normalsize{and finally}}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
=\Bigl(\sideset{}{^\ast}\prod_{i\in[2]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\widetilde{\boldsymbol{\mathfrak{a}}}(a^i)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
=\quad &(0,\phantom{-}0,\phantom{-}1,\phantom{-}0,\phantom{-}1,\phantom{-}1,\phantom{-}1,\phantom{-}1)\; .
\end{align*}
}
In Example~{\rm\ref{th:7}} on page~{\rm\pageref{page:2}}, we will attempt to extract from the above vector~$\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})$
the characteristic vector~$\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A}))$ of the blocker~$\mathfrak{B}(\mathcal{A})$.
\end{example}
\subsubsection{The characteristic vector of the subfamily of inclusion-minimal sets $\bmin\mathcal{F}$ in a family~$\mathcal{F}$} $\quad$
Suppose we are given the characteristic vector $\boldsymbol{\gamma}(\mathcal{F})$ of a
nonempty
family~$\mathcal{F}\subset\mathbf{2}^{[t]}$ of subsets of the ground set $E_t$, such that $\mathcal{F}\not\ni\hat{0}$. We can read off the position numbers of all the inclusion-minimal sets in the family~$\mathcal{F}$ in the following straightforward way (see~Example~\ref{th:7} on page~\pageref{page:2}):
\begin{algorithm}
\label{th:6}
$\quad$ {\em
\begin{itemize}
\item[\tt Input:] The char.-vector $\boldsymbol{\gamma}(\mathcal{F})$ of a family~$\mathcal{F}\subset\mathbf{2}^{[t]}$, such that
$\emptyset\neq\mathcal{F}\not\ni\hat{0}$.
\item[\tt Output:] \noindent{}A set~$M$ is the set $\supp(\boldsymbol{\gamma}(\bmin\mathcal{F}))$ of position numbers of the members of the clutter~$\bmin\mathcal{F}$;
\noindent{}a vector $\boldsymbol{\beta}$ is the char.-vector $\boldsymbol{\gamma}(\bmin\mathcal{F})$ of the clutter~$\bmin\mathcal{F}$
(\!{\em this data is optional});
\noindent{}a family $\mathcal{B}$ is the clutter~$\bmin\mathcal{F}$ (\!{\em this data is optional}).
\end{itemize}
\begin{itemize}
\item[\rm(0).] \textsf{Define}
$\boldsymbol{\phi}\in\{0,1\}^{2^t}$, and \textsf{store}\ \ $\boldsymbol{\phi}\, \leftarrow\, \boldsymbol{\gamma}(\mathcal{F})$;
\noindent{}\textsf{define} $\boldsymbol{\beta}\in\{0,1\}^{2^t}$, and \textsf{store} $\boldsymbol{\beta}\, \leftarrow\, (0,\ldots,0)$;\hspace{0.2cm}\text{\% {\em this action is optional}}
\noindent{}\textsf{define} $\mathcal{B}\subset\mathbf{2}^{[t]}$, and \textsf{store} $\mathcal{B}\, \leftarrow\, \emptyset$;\hspace{2.25cm}\text{\% {\em this action is optional}}
\noindent{}\textsf{define} $M\subset[2^t]$, and \textsf{store} $M\, \leftarrow\, \hat{0}$;
\noindent{}\textsf{define} $m\in\mathbb{N}$, and \textsf{store} $m\, \leftarrow\, 0$;
\noindent{}\textsf{define} $B\in\mathbf{2}^{[t]}$, and \textsf{store} $B\, \leftarrow\, \hat{0}$.
\item[\rm(1).] \textsf{If} $|\supp(\boldsymbol{\phi})|=0$, \textsf{then} \textsf{go to} Step~{\rm(3)},
\noindent{}\textsf{else} \textsf{go to} Step~{\rm(2)}.
\item[\rm(2).] \textsf{Store} $m\, \leftarrow\, \min\supp(\boldsymbol{\phi})$,
and \textsf{store} $M\, \leftarrow\, M\,\dot{\cup}\,\{m\}$,
and \textsf{store} $B\, \leftarrow\, \varGamma(m)$,
and \textsf{store} $\mathcal{B}\, \leftarrow\, \mathcal{B}\,\,\dot{\cup}\, \{B\}$;\hspace{3cm}\text{\% {\em this action is optional}}
\noindent{}\textsf{store} $\boldsymbol{\beta}\, \leftarrow\, \boldsymbol{\beta}+\boldsymbol{\sigma}(m)$;\hspace{4.05cm}\text{\% {\em this action is optional}}
\noindent{}\textsf{If} $|\supp(\boldsymbol{\phi})|=1$, \textsf{then go to} Step~(3),
\noindent{}\textsf{else store} $\boldsymbol{\phi}\, \leftarrow\,
\boldsymbol{\phi}\,-\,\boldsymbol{\phi}\,\ast\underbrace{
\sideset{}{^\ast}\prod_{e\in B}\widetilde{\boldsymbol{\mathfrak{a}}}(e)
}_{
\boldsymbol{\gamma}(\{B\}^{\triangledown})
}\
$.
\noindent{}\textsf{Go to} Step~{\rm(1)}.
\item[\rm(3).] \textsf{Stop}.
\end{itemize}
}
\end{algorithm}
\subsubsection{More on the blocker $\mathfrak{B}(\mathcal{A})$} $\quad$
If we know (see, e.g.,~Theorem~\ref{th:5}(ii)) the characteristic vector~$\boldsymbol{\gamma}(\mathcal{F})$ of the increasing family of blocking sets~$\mathcal{F}:=\mathfrak{B}(\mathcal{A})^{\triangledown}$ of a clutter~\mbox{$\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$} on the ground set $E_t$, then a description of the blocker $\bmin\mathcal{F}:=\mathfrak{B}(\mathcal{A})$ can be obtained by an application of Algorithm~\ref{th:6} to the vector~$\boldsymbol{\gamma}(\mathcal{F})$; see~Example~\ref{th:7}.
\begin{example}\label{page:2}
\label{th:7}
Suppose $t:=3$, and $E_t=\{1,2,3\}$. Note that
\begin{align*}
\widetilde{\boldsymbol{\mathfrak{a}}}(2):=\widetilde{\boldsymbol{\mathfrak{a}}}(2;2^t)
:=\boldsymbol{\gamma}(\{\{2\}\}^{\triangledown})
&=(0,0,1,0,1,0,1,1)\in\{0,1\}^{2^t}\; .\\
\intertext{We are given the characteristic vector}
\boldsymbol{\gamma}(\mathcal{F}):\!&=(0,0,1,0,1,1,1,1)\in\{0,1\}^{2^t}
\end{align*}
of the increasing family of blocking sets~$\mathcal{F}:=\mathfrak{B}(\mathcal{A})^{\triangledown}$ of the clutter~$\mathcal{A}:=\{\{1,2\},\{2,3\}\}$
on the ground set~$E_t$; see, e.g.,~Example~{\rm\ref{th:17}} on page~{\rm\pageref{page:3}}. In order
to find a description of the clutter~$\bmin\mathcal{F}:=\mathfrak{B}(\mathcal{A})$, let us apply~Algorithm~{\rm\ref{th:6}} to
the vector~$\boldsymbol{\gamma}(\mathcal{F})${\rm:}
{\small
\begin{align*}
\boldsymbol{\phi}\ &\leftarrow\ \boldsymbol{\gamma}(\mathcal{F}):=(0,0,1,0,1,1,1,1)\; ;\\
|\supp(\boldsymbol{\phi})|\ &>\ 0\; ;\\
m\ &\leftarrow\ \underbrace{\min\supp((0,0,1,0,1,1,1,1))}_{3}\; ,\\
M\ \leftarrow\ \underbrace{\hat{0}\;\dot\cup\;\{3\}}_{\{3\}}\; ,\ \ \ \ \
B\ \leftarrow\ \underbrace{\varGamma(3)}_{\{2\}}\; ,\ \ \ \ \
\mathcal{B}\ &\leftarrow\ \underbrace{\emptyset\;\dot\cup\; \{2\}}_{\{\{2\}\}}\; ;\\
\boldsymbol{\beta}\ &\leftarrow\ \underbrace{(0,0,0,0,0,0,0,0)+\boldsymbol{\sigma}(3)}_{
(0,0,1,0,0,0,0,0)
}\; ;\\
\boldsymbol{\phi}\ &\leftarrow\ \underbrace{(0,0,1,0,1,1,1,1)-(0,0,1,0,1,1,1,1)\ast
\widetilde{\boldsymbol{\mathfrak{a}}}(2)}_{
(0,0,0,0,0,1,0,0)
}\; ;\\
|\supp(\boldsymbol{\phi})|\ &>\ 0\; ;\\
m\ &\leftarrow\ \underbrace{\min\supp((0,0,0,0,0,1,0,0))}_{6}\; ,\\
M\ \leftarrow\ \underbrace{\{3\}\;\dot\cup\;\{6\}}_{\{3,6\}}\; ,\ \ \ \ \
B\ \leftarrow\ \underbrace{\varGamma(6)}_{\{1,3\}}\; ,\ \ \ \ \
\mathcal{B}\ &\leftarrow\ \underbrace{\{\{2\}\}\;\dot\cup\; \{1,3\}}_{\{\{2\},\{1,3\}\}}\; ;\\
\boldsymbol{\beta}\ &\leftarrow\ \underbrace{(0,0,1,0,0,0,0,0)+\boldsymbol{\sigma}(6)}_{
(0,0,1,0,0,1,0,0)
}\; ;\\
|\supp(\boldsymbol{\phi})|\ &=\ 1\; ;\\
\text{\em\textsf{Stop.}} & \quad
\end{align*}
}
We see that the set~$\supp(\boldsymbol{\gamma}(\bmin\mathcal{F}))=:M$ of the position numbers of the members of the blocker~$\mathfrak{B}(\mathcal{A})=:\bmin\mathcal{F}$ is the set $\{3,6\}$.
The characteristic vector $\boldsymbol{\gamma}(\bmin\mathcal{F})=:\boldsymbol{\beta}$ of the blocker~$\mathfrak{B}(\mathcal{A})=:\bmin\mathcal{F}$ is the vector~$(0,0,1,0,0,1,0,0)$.
The blocker~$\mathfrak{B}(\mathcal{A})=:\bmin\mathcal{F}=:\mathcal{B}$ of the clutter~$\mathcal{A}:=\{\{1,2\},\{2,3\}\}$ is the clutter~$\{\{2\},\{1,3\}\}$.
\end{example}
\part*{Blocking\,/\,Voting}
\section{Decompositions of the characteristic topes and of the characteristic vectors of families}
\noindent$\bullet$ The vertices $R^i\in\{1,-1\}^t$ of the symmetric cycle~$\boldsymbol{R}$ in the hypercube graph~$\boldsymbol{H}(t,2)$, given in~(\ref{eq:12})(\ref{eq:13}), are just simply defined and useful decomposition components of topes of the oriented matroid~$\mathcal{H}:=(E_t,\{1,-1\}^t)$.
In the context of the combinatorics of finite sets, the vertices $R^i\in\{1,-1\}^{2^t}$ of a distinguished {\em symmetric cycle\/}
\begin{equation*}
\boldsymbol{R}:=(R^0,R^1,\ldots,R^{2\cdot 2^t-1},R^0)
\end{equation*}
in the hypercube graph of topes~$\boldsymbol{H}(2^t,2)$ of the oriented matroid~$\mathcal{H}_{2^t}:=(E_{2^t},\{1,-1\}^{2^t})$, where
\begin{equation}
\label{eq:66}
\begin{split}
R^0:\!&=\mathrm{T}_{2^t}^{(+)}\; ,\\
R^s:\!&={}_{-[s]}R^0\; ,\ \ \ 1\leq s\leq 2^t-1\; ,\\
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:67}
R^{2^t+k}:=-R^k\; ,\ \ \ 0\leq k\leq 2^t-1\; ,
\end{equation}
have an additional meaning:
\begin{remark} Let $\boldsymbol{R}$ be the symmetric cycle in the tope graph of the oriented matroid~$\mathcal{H}_{2^t}:=(E_{2^t},\{1,-1\}^{2^t})$, defined by~{\rm(\ref{eq:66})(\ref{eq:67})}.
\begin{itemize}
\item[\rm(i)] The vertex $R^0:=\mathrm{T}_{2^t}^{(+)}\in\mathrm{V}(\boldsymbol{R})$ is the {\em characteristic tope}
$T_{\emptyset}$ of the {\em empty family} $\emptyset$ on the ground set~$E_t$.
\noindent{}The vertex $R^{2^t}:=-\mathrm{T}_{2^t}^{(+)}\in\mathrm{V}(\boldsymbol{R})$ is the {\em characteristic tope}~$T_{\mathbf{2}^{[t]}}$ of the {\em power set\/} $\mathbf{2}^{[t]}$ of the set~$E_t$.
\item[\rm(ii)] If $1\leq i\leq 2^t-1$, then the vertex $R^i\in\mathrm{V}(\boldsymbol{R})$ is the {\em characteristic tope} $T_{\mathcal{F}}$ of a {\em decreasing family} $\mathcal{F}$ of subsets of the ground set~$E_t$. In other words, the family $\mathcal{F}$ is a particular abstract simplicial complex, when $1<i\leq 2^t-1$.
Either the subfamily $\bmax\mathcal{F}$ is an $s$-uniform clutter, where $s:=|\varGamma(\max (T_{\mathcal{F}})^{-})|$, or we have $\{|F|\colon F\in\bmax\mathcal{F}\}=\{s,s-1\}$. Indeed, we have
\begin{equation*}
\bmax\mathcal{F}=\underbrace{(\mathcal{F}\cap\tbinom{E_t}{s})}_{(\bmax\mathcal{F})\,\cap\,\binom{E_t}{s}}\; \dot\cup\ \;
\underbrace{\Bigl(\tbinom{E_t}{s-1}
-(\mathcal{F}\cap\tbinom{E_t}{s})^{\vartriangle}\Bigr)}_{(\bmax\mathcal{F})\,\cap\,\binom{E_t}{s-1}}\; .
\end{equation*}
\item[\rm(iii)] If $2^t+1\leq i\leq 2\cdot 2^t-1$, then the vertex $R^i\in\mathrm{V}(\boldsymbol{R})$ is the {\em characteristic tope} $T_{\mathcal{F}}$ of an {\em increasing family} $\mathcal{F}$ of subsets of the ground set~$E_t$.
Either the subfamily $\bmin\mathcal{F}$ is an $s$-uniform clutter, where $s:=|\varGamma(\min (T_{\mathcal{F}})^{-})|$, or we have $\{|F|\colon F\in\bmin\mathcal{F}\}=\{s,s+1\}$. We have
\begin{equation*}
\bmin\mathcal{F}=\underbrace{(\mathcal{F}\cap\tbinom{E_t}{s})}_{(\bmin\mathcal{F})\,\cap\,\binom{E_t}{s}}\; \dot\cup\ \;
\underbrace{\Bigl(\tbinom{E_t}{s+1}
-(\mathcal{F}\cap\tbinom{E_t}{s})^{\triangledown}\Bigr)}_{(\bmin\mathcal{F})\,\cap\,\binom{E_t}{s+1}}\; .
\end{equation*}
If $i=3\cdot 2^{t-1}$, then the clutter~$\bmin\mathcal{F}$ is {\em self-dual}.
\end{itemize}
\end{remark}
\noindent$\bullet$ A distinguished symmetric cycle~$\widetilde{\boldsymbol{R}}:=(\widetilde{R}^0,\widetilde{R}^1,\ldots,\widetilde{R}^{2\cdot 2^t-1},\widetilde{R}^0)$ in the hypercube graph~$\widetilde{\boldsymbol{H}}(2^t,2)$ on the vertex set $\{0,1\}^{2^t}$ is defined\footnote{$\quad$Here $\boldsymbol{\sigma}(e)$ is the $e$th standard unit vector of the space~$\mathbb{R}^{2^t}$.} as follows:
\begin{equation*}
\begin{split}
\widetilde{R}^0:\!&=(0,\ldots,0)\; ,\\
\widetilde{R}^s:\!&=\sum\nolimits_{e\in[s]}\boldsymbol{\sigma}(e)\; ,\ \ \ 1\leq s\leq 2^t-1\; ,
\end{split}
\end{equation*}
and
\begin{equation*}
\widetilde{R}^{2^t+k}:=\mathrm{T}_{2^t}^{(+)}-\widetilde{R}^k\; ,\ \ \ 0\leq k\leq 2^t-1\; .
\end{equation*}
We let~$\mathrm{V}(\widetilde{\boldsymbol{R}}):=(\widetilde{R}^0,\widetilde{R}^1,\ldots,\widetilde{R}^{2\cdot 2^t-1})$ denote the vertex sequence of the cycle~$\widetilde{\boldsymbol{R}}$.
\noindent$\bullet$ Let $\mathcal{F}\subset\mathbf{2}^{[t]}$ be a family of subsets of the ground set~$E_t$, $\emptyset\neq\mathcal{F}\not\ni\hat{0}$.
As earlier, we associate with the family $\mathcal{F}$ its characteristic tope $T_{\mathcal{F}}\in\{1,-1\}^{2^t}$, defined by~(\ref{eq:68}).
Recall that there exists a unique inclusion-minimal subset
\begin{equation*}
\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R})\subset\mathrm{V}(\boldsymbol{R}):=(R^0,R^1,\ldots,R^{2\cdot 2^t-1})
\end{equation*}
of the vertex sequence $\mathrm{V}(\boldsymbol{R})$ of the cycle~$\boldsymbol{R}$, defined by~(\ref{eq:66})(\ref{eq:67}), such that
\begin{equation*}
T_{\mathcal{F}}=\sum_{Q\in\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R})} Q\; .
\end{equation*}
In other words, there exists a unique row vector~$\boldsymbol{x}:=\boldsymbol{x}(T_{\mathcal{F}}):=\boldsymbol{x}(T_{\mathcal{F}},\boldsymbol{R}):=(x_1,\ldots,x_{2^t})
\in\{-1,0,1\}^{2^t}$, such that
\begin{equation}
\label{eq:70}
T_{\mathcal{F}}=\sum_{i\in[2^t]}x_i\cdot R^{i-1}=\boldsymbol{x}\mathbf{M}\; ,
\end{equation}
where
\begin{equation}
\label{eq:73}
\mathbf{M}:=\mathbf{M}(\boldsymbol{R}):=
\left(
\begin{smallmatrix}
R^0\\
R^1 \vspace{-1.5mm}\\
\vdots \vspace{0.5mm}\\
R^{2^t-1}
\end{smallmatrix}
\right)\; .
\end{equation}
One can see this matrix (in the case $t:=3$) in Example~\ref{th:18} on page~\pageref{page:4}. Thus, we have
\begin{equation*}
\boldsymbol{x}=T_{\mathcal{F}}\cdot\mathbf{M}^{-1}\; ,
\end{equation*}
and
\begin{equation*}
\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R}):=\{x_i\cdot R^{i-1}\colon x_i\neq 0\}\; .
\end{equation*}
We use the notation $\mathfrak{q}(T_{\mathcal{F}}):=\mathfrak{q}(T_{\mathcal{F}},\boldsymbol{R}):=|\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R})|$ to denote the cardinality of the set~$\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R})$.
\noindent$\bullet$ Let us consider the subset
\begin{equation*}
\widetilde{\boldsymbol{Q}}(\underbrace{
\boldsymbol{\gamma}(\mathcal{F})
}_{\frac{1}{2}(\mathrm{T}_{2^t}^{(+)}-T_{\mathcal{F}})},
\widetilde{\boldsymbol{R}}):=
\{\tfrac{1}{2}(\mathrm{T}_{2^t}^{(+)}-Q)\colon \; Q\in\boldsymbol{Q}(T_{\mathcal{F}},\boldsymbol{R})\}\subset\mathrm{V}(\widetilde{\boldsymbol{R}})
\; ,
\end{equation*}
and let us use the notation $\mathfrak{q}(\boldsymbol{\gamma}(\mathcal{F})):=\mathfrak{q}(\boldsymbol{\gamma}(\mathcal{F}),\widetilde{\boldsymbol{R}})
:=|\widetilde{\boldsymbol{Q}}(\boldsymbol{\gamma}(\mathcal{F}),
\widetilde{\boldsymbol{R}})|=\mathfrak{q}(T_{\mathcal{F}})$ to denote its cardinality.
In analogy with~(\ref{eq:69}), we have
\begin{equation}
\label{eq:31}
\boldsymbol{\gamma}(\mathcal{F})=-\tfrac{1}{2}\bigl(\mathfrak{q}(\boldsymbol{\gamma}(\mathcal{F}))
-1\bigr)\cdot\mathrm{T}_{2^t}^{(+)} +
\sum_{
\substack{\widetilde{Q}\in\widetilde{\boldsymbol{Q}}(\boldsymbol{\gamma}(\mathcal{F}),\widetilde{\boldsymbol{R}})\colon\\
\widetilde{Q}\neq
(0,\ldots,0)=:\widetilde{R}^0
}
}
\widetilde{Q}\; .
\end{equation}
\noindent$\bullet$ Let $\mathcal{A}\subset\mathbf{2}^{[t]}$ be a nontrivial clutter on the ground set~$E_t$, and let $\mathcal{B}:=\mathfrak{B}(\mathcal{A})$ be its blocker.
We associate with the families $\mathcal{A}^{\triangledown}$, $\mathcal{B}^{\triangledown}$, $\mathcal{A}$ and $\mathcal{B}$ their characteristic topes $T_{\mathcal{A}^{\triangledown}}$, $T_{\mathcal{B}^{\triangledown}}$, $T_{\mathcal{A}}$, $T_{\mathcal{B}}\in\{1,-1\}^{2^t}$, and their characteristic vectors $\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})$, $\boldsymbol{\gamma}(\mathcal{B}^{\triangledown})$, $\boldsymbol{\gamma}(\mathcal{A})$, $\boldsymbol{\gamma}(\mathcal{B})\in\{0,1\}^{2^t}$. See~(\ref{eq:74})--(\ref{eq:81}) in~Example~\ref{th:9}.
\begin{example} \label{th:9}
Suppose $t:=3$, and $E_t=\{1,2,3\}$. Let $\boldsymbol{R}$ by the symmetric cycle in the hypercube graph~$\boldsymbol{H}(2^t,2)$ on the vertex set $\{1,-1\}^{2^t}$, defined by~{\rm(\ref{eq:66})(\ref{eq:67})}.
We are given the {\em blocking pair}\! of {\em clutters\/} $\mathcal{A}:=\{\{1,2\},\{2,3\}\}$ and $\mathcal{B}:=\mathfrak{B}(\mathcal{A})=\{\{1,3\},\{2\}\}$ on the ground set~$E_t$.
The families $\mathcal{A}^{\triangledown}$, $\mathcal{B}^{\triangledown}$, $\mathcal{A}$ and $\mathcal{B}$ are described by their
characteristic topes
{\small
\begin{align}
\label{eq:74}
T_{\mathcal{A}^{\triangledown}}:\!&=(1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,-1,-1)\in\{1,-1\}^{2^t}\; ,\\
\label{eq:75}
T_{\mathcal{B}^{\triangledown}}:\!&=(1,\phantom{+}1,-1,\phantom{+}1,-1,-1,-1,-1)\; ,\\
\label{eq:76}
T_{\mathcal{A}}:\!&=(1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,-1,\phantom{+}1)\; ,\\
\label{eq:77}
T_{\mathcal{B}}:\!&=(1,\phantom{+}1,-1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,\phantom{+}1)\; ,\\
\intertext{\normalsize{and by their characteristic vectors}}
\label{eq:78}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}):\!&=(0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}1)\in\{0,1\}^{2^t}\; ,\\
\label{eq:79}
\boldsymbol{\gamma}(\mathcal{B}^{\triangledown})
:\!&=(0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\; ,\\
\label{eq:80}
\boldsymbol{\gamma}(\mathcal{A}):\!&=(0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}0)\; ,\\
\label{eq:81}
\boldsymbol{\gamma}(\mathcal{B}):\!&=(0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}0)\; .
\end{align}
}
Turning to decompositions of the form~{\rm(\ref{eq:70})}, we see that
{\small
\begin{align}
\label{eq:82}
\boldsymbol{x}(T_{\mathcal{A}^{\triangledown}}
)&=(0,\phantom{-}0,\phantom{-}0,\phantom{-}0,-1,\phantom{-}1,-1,\phantom{-}0)\in\{-1,0,1\}^{2^t}\; ,\\
\label{eq:83}
\boldsymbol{x}(T_{\mathcal{B}^{\triangledown}}
)&=(0,\phantom{-}0,-1,\phantom{-}1,-1,\phantom{-}0,\phantom{-}0,\phantom{-}0)
\; ,\\
\label{eq:84}
\boldsymbol{x}(T_{\mathcal{A}}
)&=(1,\phantom{-}0,\phantom{-}0,\phantom{-}0,-1,\phantom{-}1,-1,\phantom{-}1)
\; ,\\
\label{eq:85}
\boldsymbol{x}(T_{\mathcal{B}}
)&=(1,\phantom{-}0,-1,\phantom{-}1,\phantom{-}0,-1,\phantom{-}1,\phantom{-}0)
\; .
\end{align}
}
Thus, we have the decompositions:
{\small
\begin{equation*}
\begin{split}
T_{\mathcal{A}^{\triangledown}}:=\phantom{-}&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,-1,-1)
=-\underbrace{R^4}_{-R^{12}}+\;R^5-\underbrace{R^6}_{-R^{14}}\\
=-\,&(-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1)=R^5+R^{12}+R^{14}\\
=\phantom{=}&(-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1)\; ,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
T_{\mathcal{B}^{\triangledown}}:=\phantom{-}&(\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,-1,-1,-1,-1)
=-\underbrace{R^2}_{-R^{10}}+\;R^3-\underbrace{R^4}_{-R^{12}}\\
=-\,&(-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)=R^3+R^{10}+R^{12}\\
=\phantom{=}&(-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,-1,-1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1,-1,-1)\; ,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
T_{\mathcal{A}}:=\phantom{-}\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,-1,\phantom{+}1)
=\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}-\underbrace{R^4}_{-R^{12}}+\;R^5-\underbrace{R^6}_{-R^{14}}+\;R^7\\
=\phantom{+}&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,-1,-1,\phantom{+}1)=\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}+R^5+R^7+R^{12}+R^{14}\\
=\phantom{+}&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,-1,-1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1)\; ,
\end{split}
\end{equation*} }
\noindent{}and{\small
\begin{equation*}
\begin{split}
T_{\mathcal{B}}:=\phantom{-}&(\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,\phantom{+}1,-1,\phantom{+}1,\phantom{+}1)
=\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}-\underbrace{R^2}_{-R^{10}}+\;R^3-\underbrace{R^5}_{-R^{13}}+\;R^6\\
=\phantom{+}&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}-\,&(-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1)=\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}+R^3+R^6+R^{10}+R^{13}\\
=\phantom{+}&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(-1,-1,-1,-1,-1,-1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,-1,-1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,-1,-1,-1)\; .
\end{split}
\end{equation*}
}
Relations of the form~{\rm(\ref{eq:31})} imply that
{\small
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}):=\phantom{+}
&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}1)
=-\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^{5}+\widetilde{R}^{12}+\widetilde{R}^{14}\\
=\phantom{+}\,&(-1,-1,-1,-1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1)\; ,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathcal{B}^{\triangledown})
:=\phantom{-}&(\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)
=-\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^3+\widetilde{R}^{10}+\widetilde{R}^{12}\\
=\phantom{+}&(-1,-1,-1,-1,-1,-1,-1,-1)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\; ,
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathcal{A}):=\phantom{-}&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}1,\phantom{+}0)
=-2\mathrm{T}_{2^t}^{(+)}+\underbrace{\widetilde{R}^0}_{(0,\ldots,0)}+\widetilde{R}^5+\widetilde{R}^7+\widetilde{R}^{12}+\widetilde{R}^{14}\\
=-\,&2\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^5+\widetilde{R}^7+\widetilde{R}^{12}+\widetilde{R}^{14}\\
=\phantom{-\,}&(-2,-2,-2,-2,-2,-2,-2,-2)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1)\; ,
\end{split}
\end{equation*} }
\noindent{}and{\small
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathcal{B}):=\phantom{-}&(\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}0,\phantom{+}0)
=-2\mathrm{T}_{2^t}^{(+)}+\underbrace{\widetilde{R}^0}_{(0,\ldots,0)}+\widetilde{R}^3+\widetilde{R}^6+\widetilde{R}^{10}+\widetilde{R}^{13}\\
=-\,&2\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^3+\widetilde{R}^6+\widetilde{R}^{10}+\widetilde{R}^{13}\\
=\phantom{+\,}&(-2,-2,-2,-2,-2,-2,-2,-2)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}0,\phantom{+}0)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1,\phantom{+}1)\\
\phantom{=}+\,&(\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}0,\phantom{+}1,\phantom{+}1,\phantom{+}1)\; .
\end{split}
\end{equation*}
}
\end{example}
\noindent$\bullet$ Corollary~\ref{th:11}(i) and Proposition~\ref{th:10}(iv), restated in dimensionality~$2^t$, suggest the following:
\begin{theorem} Let $\boldsymbol{R}$ be the symmetric cycle in the hypercube graph $\boldsymbol{H}(2^t,2)$ on the vertex set~$\{1,-1\}^{2^t}$, defined by~{\rm(\ref{eq:66})(\ref{eq:67})}.
Let $\mathcal{A}\subset\mathbf{2}^{[t]}$ be a nontrivial clutter on the ground set~$E_t$, and let\/ $\mathcal{B}:=\mathfrak{B}(\mathcal{A})$ be its blocker. Since the characteristic topes of the increasing families~$\mathcal{A}^{\triangledown}$ and $\mathcal{B}^{\triangledown}$ obey the relation
\begin{equation*}
T_{\mathcal{B}^{\triangledown}}=\ro(T_{\mathcal{A}^{\triangledown}})\; ,
\end{equation*}
we have{\rm:}
\begin{itemize}
\item[\rm(i)]
\begin{equation*}
\mathfrak{q}(T_{\mathcal{B}^{\triangledown}}):=|\boldsymbol{Q}(T_{\mathcal{B}^{\triangledown}},\boldsymbol{R})|=
|\boldsymbol{Q}(T_{\mathcal{A}^{\triangledown}},\boldsymbol{R})|=:\mathfrak{q}(T_{\mathcal{A}^{\triangledown}})\; ,
\end{equation*}
and
\begin{equation*}
\boldsymbol{x}(T_{\mathcal{B}^{\triangledown}}
)=\boldsymbol{x}(T_{\mathcal{A}^{\triangledown}}
)
\cdot\overline{\mathbf{U}}(2^t)\cdot\overline{\mathbf{T}}(2^t)\; .
\end{equation*}
\item[\rm(ii)]
Suppose that the subset $(T_{\mathcal{A}^{\triangledown}})^-=\supp(\boldsymbol{\gamma}(\mathcal{A}^{\triangledown}))\subset E_{2^t}$ is a disjoint union
\begin{equation*}
[i_1,j_1]\;\dot\cup\;[i_2,j_2]\;\dot\cup\;\cdots\;\dot\cup\;[i_{\varrho-1},j_{\varrho-1}]\;\dot\cup\;[i_{\varrho},j_{\varrho}]
\end{equation*}
of intervals such that
\begin{equation*}
j_1+2\leq i_2,\ \ j_2+2\leq i_3,\ \ \ldots,\ \
j_{\varrho-2}+2\leq i_{\varrho-1},\ \ j_{\varrho-1}+2\leq i_{\varrho}\; ,
\end{equation*}
for some $\varrho$. We have
\begin{equation*}
\mathfrak{q}(T_{\mathcal{B}^{\triangledown}})=\mathfrak{q}(T_{\mathcal{A}^{\triangledown}})=2\varrho-1\; ;\\
\end{equation*}
\begin{align*}
\boldsymbol{x}(T_{\mathcal{A}^{\triangledown}})&=\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(j_k+1)-\sum_{1\leq \ell\leq\varrho}\boldsymbol{\sigma}(i_{\ell})\; ,\\
\intertext{and}
\boldsymbol{x}(T_{\mathcal{B}^{\triangledown}})&=
\sum_{1\leq k\leq\varrho-1}\boldsymbol{\sigma}(2^t-j_k+1)-\sum_{1\leq \ell\leq\varrho}\boldsymbol{\sigma}(2^t-i_{\ell}+2)\; .
\end{align*}
\end{itemize}
\end{theorem}
See~expressions~(\ref{eq:74})(\ref{eq:75}) and~(\ref{eq:82})(\ref{eq:83}) in~Example~\ref{th:9}.
\subsection{A clutter $\{\{a\}\}$} $\quad$
As earlier (in Section~\ref{section:1}), let $\{\{a\}\}$ be a clutter on the ground set~$E_t$, whose only member is a {\em one-element\/} subset $\{a\}\subset E_t$.
\noindent$\bullet$
Let us associate with the characteristic tope $\boldsymbol{\mathfrak{a}}(a):=T_{\{\{a\}\}^{\triangledown}}$ of the principal increasing family~$\{\{a\}\}^{\triangledown}$ the row vector $\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a)):=\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a),\boldsymbol{R})$ \mbox{$\in\{-1,0,1\}^{2^t}$,} described in~(\ref{eq:70}), where $\boldsymbol{R}$ is the symmetric cycle in the hypercube graph $\boldsymbol{H}(2^t,2)$, defined by~(\ref{eq:66})(\ref{eq:67}). Recall that
\begin{equation}
\label{eq:72}
\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a))=\boldsymbol{\mathfrak{a}}(a)\cdot\mathbf{M}^{-1}\; ,
\end{equation}
where the matrix~$\mathbf{M}$ is defined by~$(\ref{eq:73})$, and
\begin{equation*}
\boldsymbol{Q}(\boldsymbol{\mathfrak{a}}(a),\boldsymbol{R}):=\{x_i\cdot R^{i-1}\colon
x_i\neq 0\}\; ,\ \ \ \text{and}\ \ \ \boldsymbol{\mathfrak{a}}(a)=\sum_{Q\in\boldsymbol{Q}(\boldsymbol{\mathfrak{a}}(a),\boldsymbol{R})}Q\; ,
\end{equation*}
see~Example~\ref{th:18}.
\begin{example}
\label{th:18}
Suppose $t:=3$, and~$E_t=\{1,2,3\}$.
The characteristic topes associated with the principal increasing families that are generated by the clutters~$\{\{a\}\}$, for the elements $a\in E_t$ of the ground set, are as follows{\rm:}
{\small
\begin{align*}
\boldsymbol{\mathfrak{a}}(1):=\boldsymbol{\mathfrak{a}}(1;2^t):=T_{\{\{1\}\}^{\triangledown}}
&=(1,-1,\phantom{-}1,\phantom{-}1,-1,-1,\phantom{-}1,-1)\in\{1,-1\}^{2^t}\; ,\\
\boldsymbol{\mathfrak{a}}(2):=\boldsymbol{\mathfrak{a}}(2;2^t):=T_{\{\{2\}\}^{\triangledown}}
&=(1,\phantom{-}1,-1,\phantom{-}1,-1,\phantom{-}1,-1,-1)\; ,\\
\boldsymbol{\mathfrak{a}}(3):=\boldsymbol{\mathfrak{a}}(3;2^t):=T_{\{\{3\}\}^{\triangledown}}
&=(1,\phantom{-}1,\phantom{-}1,-1,\phantom{-}1,-1,-1,-1)\; .\\
\intertext{\normalsize{The corresponding ``$\boldsymbol{x}$-vectors'', given in~{\rm(\ref{eq:72})} for the symmetric cycle~$\boldsymbol{R}$ in the hypercube graph~$\boldsymbol{H}(2^t,2)$ defined by~{\rm(\ref{eq:66})(\ref{eq:67})}, are{\rm:}}}
\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(1)):=\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(1),\boldsymbol{R})
&=(0,-1,\phantom{-}1,\phantom{-}0,-1,\phantom{-}0,\phantom{-}1,-1)\in\{-1,0,1\}^{2^t}\; ,\\
\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(2)):=\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(2),\boldsymbol{R})
&=(0,\phantom{-}0,-1,\phantom{-}1,-1,\phantom{-}1,-1,\phantom{-}0)\; ,\\
\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(3)):=\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(3),\boldsymbol{R})
&=(0,\phantom{-}0,\phantom{-}0,-1,\phantom{-}1,-1,\phantom{-}0,\phantom{-}0)\; .
\end{align*}
}
Indeed, we see that
\begin{equation*}
\begin{split}
\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(2))\cdot\mathbf{M}
&=
\left(
\begin{smallmatrix}
0&\phantom{-}0&-1&\phantom{-}1&-1&\phantom{-}1&-1&\phantom{-}0
\end{smallmatrix}
\right)
\cdot
\left(
\begin{smallmatrix}
\phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& -1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& -1& -1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& -1& -1& -1& \phantom{-}1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& -1& -1& -1& -1& \phantom{-}1& \phantom{-}1& \phantom{-}1\\
-1& -1& -1& -1& -1& -1& \phantom{-}1& \phantom{-}1\\
-1& -1& -1& -1& -1& -1& -1& \phantom{-}1
\end{smallmatrix}
\right)\\
&=
\left(
\begin{smallmatrix}
1&\phantom{-}1&-1&\phantom{-}1&-1&\phantom{-}1&-1&-1
\end{smallmatrix}
\right)=:\boldsymbol{\mathfrak{a}}(2)\; .
\end{split}
\end{equation*}
\label{page:4}
\end{example}
\noindent$\bullet$
For the row vector $\boldsymbol{y}(1+a):=\boldsymbol{y}(1+a;2^t)\in\{-1,0,1\}^{2^t}$, defined by
\begin{equation*}
\boldsymbol{y}(1+a):=\boldsymbol{x}({}_{-\{1+a\}}\mathrm{T}_{2^t}^{(+)}
)=:\boldsymbol{x}(T_{\{\{a\}\}}
)\; ,
\end{equation*}
we have (see~\cite[Sect.~2]{M-SC-II}):
\begin{equation*}
\boldsymbol{y}(1+a)=\boldsymbol{\sigma}(1)-\boldsymbol{\sigma}(1+a)+\boldsymbol{\sigma}(2+a)\; .
\end{equation*}
In other words,
\begin{equation*}
\boldsymbol{Q}(T_{\{\{a\}\}},\boldsymbol{R})=\{\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}\, ,R^{1+a},R^{2^t+a}\}\; ,
\end{equation*}
and
\begin{equation*}
T_{\mathfrak{B}(\{\{a\}\})}=T_{\{\{a\}\}}=
\mathrm{T}_{2^t}^{(+)}+R^{1+a}+R^{2^t+a}\; .
\end{equation*}
Equivalently,
\begin{equation*}
\widetilde{\boldsymbol{Q}}(\boldsymbol{\gamma}(\{\{a\}\}),\widetilde{\boldsymbol{R}})=\{\underbrace{\widetilde{R}^0}_{(0,\ldots,0)},
\widetilde{R}^{1+a},\widetilde{R}^{2^t+a}\}\; ,
\end{equation*}
and
\begin{equation*}
\begin{split}
\boldsymbol{\gamma}(\mathfrak{B}(\{\{a\}\}))=\boldsymbol{\gamma}(\{\{a\}\})&=-\tfrac{1}{2}(3-1)\cdot\mathrm{T}_{2^t}^{(+)}
+\widetilde{R}^{1+a}+\widetilde{R}^{2^t+a}\\ &=-\mathrm{T}_{2^t}^{(+)}
+\widetilde{R}^{1+a}+\widetilde{R}^{2^t+a}\; .
\end{split}
\end{equation*}
\subsection{A clutter $\{A\}$} $\quad$
As in Section~\ref{section:2}, let $\{A\}$ be a clutter on the ground set~$E_t$, whose only member is a {\em nonempty subset} $A\subseteq E_t$.
\noindent$\bullet$ Dealing with the symmetric cycle~$\boldsymbol{R}$ in the hypercube graph~$\boldsymbol{H}(2^t,2)$, defined by~(\ref{eq:66})(\ref{eq:67}), with the matrix~$\mathbf{M}$ given in~(\ref{eq:73}), and with ``$\boldsymbol{x}$-vectors'' described in~(\ref{eq:70}), for the row vector
\begin{equation}
\label{eq:71}
\boldsymbol{y}(\varGamma^{-1}(A)):=\boldsymbol{x}({}_{-\{\varGamma^{-1}(A)\}}\mathrm{T}_{2^t}^{(+)})
=:\boldsymbol{x}(T_{\{A\}})=T_{\{A\}}\cdot\mathbf{M}^{-1}\in\{-1,0,1\}^{2^t}\; ,
\end{equation}
we have (see~\cite[Sect.~2]{M-SC-II}):
\begin{equation*}
\boldsymbol{y}(\varGamma^{-1}(A))=
\begin{cases}
\phantom{-}\boldsymbol{\sigma}(1)-\boldsymbol{\sigma}(\varGamma^{-1}(A))+\boldsymbol{\sigma}(1+\varGamma^{-1}(A))\; , & \text{if $A\neq E_t$},\\
\quad\vspace{-3mm}\\
-\boldsymbol{\sigma}(2^t)\; , & \text{if $A=E_t$}.
\end{cases}
\end{equation*}
In other words,
\begin{equation*}
\boldsymbol{Q}(T_{\{A\}},\boldsymbol{R})=
\begin{cases}
\{\,\underbrace{R^0}_{\mathrm{T}_{2^t}^{(+)}}\, ,R^{\varGamma^{-1}(A)},R^{2^t+\varGamma^{-1}(A)-1}\}\; , & \text{if $A\neq E_t$},\\
\quad\vspace{-3mm}\\
\{R^{2\cdot 2^t -1}\}\; , & \text{if $A=E_t$},
\end{cases}
\end{equation*}
and
\begin{equation*}
T_{\{A\}}=
\begin{cases}
\mathrm{T}_{2^t}^{(+)}+R^{\varGamma^{-1}(A)}+R^{2^t+\varGamma^{-1}(A)-1}\; , & \text{if $A\neq E_t$},\\
\quad\vspace{-3mm}\\
R^{2\cdot 2^t -1}\; , & \text{if $A=E_t$}.
\end{cases}
\end{equation*}
Equivalently,
\begin{equation*}
\widetilde{\boldsymbol{Q}}(\boldsymbol{\gamma}(\{A\}),\widetilde{\boldsymbol{R}})=
\begin{cases}
\{\underbrace{\widetilde{R}^0}_{(0,\ldots,0)},\widetilde{R}^{\varGamma^{-1}(A)},\widetilde{R}^{2^t+\varGamma^{-1}(A)-1}\}\; , & \text{if $A\neq E_t$},\\
\quad\vspace{-3mm}\\
\{\widetilde{R}^{2\cdot 2^t -1}\}\; , & \text{if $A=E_t$},
\end{cases}
\end{equation*}
and
\begin{equation*}
\boldsymbol{\gamma}(\{A\})=
\begin{cases}
\phantom{=}-\tfrac{1}{2}(3-1)\cdot\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^{\varGamma^{-1}(A)}+\widetilde{R}^{2^t+\varGamma^{-1}(A)-1}\\
=-\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^{\varGamma^{-1}(A)}+\widetilde{R}^{2^t+\varGamma^{-1}(A)-1}\; , & \text{if $A\neq E_t$},\\
\quad\vspace{-3mm}\\
\phantom{=-}\widetilde{R}^{2\cdot 2^t -1}\; , & \text{if $A=E_t$}.
\end{cases}
\end{equation*}
\noindent$\bullet$
Recall that $\boldsymbol{\gamma}(\{A\}^{\triangledown})=\sideset{}{^\ast}\prod_{a\in A}\widetilde{\boldsymbol{\mathfrak{a}}}(a)$, and $\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})=\rn(\boldsymbol{\gamma}(\{A\}^{\triangledown}))$.
\begin{remark}[cf.~Remark~\ref{th:8}]
For a nonempty subset~$A\subseteq E_t$, we have
\begin{itemize}
\item[\rm(i)]
\begin{equation*}
\boldsymbol{\gamma}(\{A\}^{\triangledown})=\sideset{}{^\ast}\prod_{a\in A}\bigl(\tfrac{1}{2}\bigr(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a))
\cdot\mathbf{M}\bigr)\bigr)\; .
\end{equation*}
\item[\rm(ii)]
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\})^{\triangledown})
=\mathrm{T}_{2^t}^{(+)}
-\Bigl(\sideset{}{^\ast}\prod_{a\in A}\bigl(\tfrac{1}{2}\bigl(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a))\cdot\mathbf{M}\bigr)\bigr)\Bigr)
\cdot\overline{\mathbf{U}}(2^t)\; .
\end{equation*}
\end{itemize}
\end{remark}
\noindent$\bullet$ Since the {\em blocker\/} of the clutter $\{A\}$ is the clutter $\mathfrak{B}(\{A\})=\{\{a\}\colon a\in A\}$, and
$\boldsymbol{\gamma}(\mathfrak{B}(\{A\}))=\sum_{a\in A}\boldsymbol{\gamma}(\{\{a\}\})$, we have
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\}))=\sum_{a\in A}(-\mathrm{T}_{2^t}^{(+)}+\widetilde{R}^{1+a}+\widetilde{R}^{2^t+a})\; ,
\end{equation*}
that is,
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\{A\}))=-|A|\cdot\mathrm{T}_{2^t}^{(+)}+\sum_{a\in A}(\widetilde{R}^{1+a}+\widetilde{R}^{2^t+a})\; .
\end{equation*}
\subsection{A clutter $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$} $\quad$
As in Section~\ref{section:3}, let $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$ be a nontrivial clutter on the ground set $E_t$.
\noindent$\bullet$ In analogy with~\cite[Rem.~2.2]{M-SC-II}, dealing with the symmetric cycle~$\boldsymbol{R}$ in the hypercube graph~$\boldsymbol{H}(2^t,2)$, defined by~(\ref{eq:66})(\ref{eq:67}), with the matrix~$\mathbf{M}$ given in~(\ref{eq:73}), with ``$\boldsymbol{x}$-vectors'' described in~(\ref{eq:70}), and with ``$\boldsymbol{y}$-vectors'' given in~(\ref{eq:71}), we have
\begin{equation*}
\boldsymbol{x}(T_{\mathcal{A}})=(1-\#\mathcal{A})\cdot\boldsymbol{\sigma}(1)+\sum_{A\in\mathcal{A}}\boldsymbol{y}(\varGamma^{-1}(A))\; ,
\end{equation*}
that is,
\begin{equation*}
\boldsymbol{x}(T_{\mathcal{A}})=
\begin{cases}
\phantom{-}\boldsymbol{\sigma}(1)+\sum_{A\in\mathcal{A}}(-\boldsymbol{\sigma}(\varGamma^{-1}(A))+\boldsymbol{\sigma}(1+\varGamma^{-1}(A)))\; , & \text{if $\mathcal{A}\neq\{E_t\}$},\\
\quad\vspace{-3mm}\\
-\boldsymbol{\sigma}(2^t)\; , & \text{if $\mathcal{A}=\{E_t\}$},
\end{cases}
\end{equation*}
or
\begin{equation*}
T_{\mathcal{A}}=
\begin{cases}
\phantom{-}\mathrm{T}_{2^t}^{(+)}+\sum_{A\in\mathcal{A}}(R^{\varGamma^{-1}(A)}+R^{2^t+\varGamma^{-1}(A)-1})\; , & \text{if $\mathcal{A}\neq \{E_t\}$},\\
\quad\vspace{-3mm}\\
\phantom{-}R^{2\cdot 2^t -1}\; , & \text{if $\mathcal{A}= \{E_t\}$}.
\end{cases}
\end{equation*}
We also have
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{A})=
\begin{cases}
-(\#\mathcal{A})\cdot\mathrm{T}_{2^t}^{(+)}+\sum_{A\in\mathcal{A}}(\widetilde{R}^{\varGamma^{-1}(A)}+\widetilde{R}^{2^t+\varGamma^{-1}(A)-1})\; , & \text{if $\mathcal{A}\neq \{E_t\}$},\\
\quad\vspace{-3mm}\\
\phantom{-}\widetilde{R}^{2\cdot 2^t -1}\; , & \text{if $\mathcal{A}= \{E_t\}$}.
\end{cases}
\end{equation*}
\noindent$\bullet$ Theorem~\ref{th:5} can be accompanied with the following statement:
\begin{corollary}
If $\mathcal{A}:=\{A_1,\ldots,A_{\alpha}\}$ is a nontrivial clutter on the ground set~$E_t$, then we have{\rm:}
\begin{itemize}
\item[\rm(i)]
\begin{equation*}
\boldsymbol{\gamma}(\mathcal{A}^{\triangledown})=
\mathrm{T}_{2^t}^{(+)}-
\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\bigl(\tfrac{1}{2}\bigl(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a^i))
\cdot\mathbf{M}\bigr)\bigr)\Bigr)\Bigr)\; .
\end{equation*}
\item[\rm(ii)]
\begin{equation*}
\boldsymbol{\gamma}(\mathfrak{B}(\mathcal{A})^{\triangledown})
=\Bigl(\sideset{}{^\ast}\prod_{i\in[\alpha]}\Bigl(\mathrm{T}^{(+)}_{2^t}
-\sideset{}{^\ast}\prod_{a^i\in A_i}\bigl(\tfrac{1}{2}\bigl(\mathrm{T}_{2^t}^{(+)}-\boldsymbol{x}(\boldsymbol{\mathfrak{a}}(a^i))
\cdot\mathbf{M}\bigr)\bigr)\Bigr)\Bigr)\cdot\overline{\mathbf{U}}(2^t)
\; .
\end{equation*}
\end{itemize}
\end{corollary} |
2106.03849 | \section{Introduction}
\label{sec:intro}
The problem of \emph{unsupervised visual scene understanding} has become an increasingly central topic in machine learning \cite{malik_2015,inbooksceneunderstanding}. The attention is merited by potential gains to reasoning, autonomous navigation, and myriad tasks. However, within the current literature, different studies frame the problem in different ways. One approach aims to decompose images into component objects and object features, supporting (among other things) generation of alternative data that permits insertion, deletion, or repositioning of individual objects \citep{burgess2019monet, greff2019multi, engelcke2020genesis, lin2020space}. Another approach aims at a very different form of decomposition---between allocentric scene structure and a variable viewpoint---supporting generation of views of a scene from new vantage points \cite{eslami2018gqn, sitzmann2019srn, mildenhall2020nerf} and, if not supplied as input, estimation of camera pose \cite{Cadena16tro-SLAMfuture}. Although there is work pursuing both of these approaches concurrently in the supervised setting \cite{xu2019mid, nanbo2020mulmon, chen2020object}, very few previous studies have approached the combined challenge in the unsupervised case. In this work, we introduce the Sequence-Integrating Multi-Object Net (SIMONe{}), a model which pursues that goal of object-level and viewpoint-level scene decomposition and synthesis without supervision. SIMONe\ is designed to handle these challenges without privileged information concerning camera pose, and in dynamic scenes.
Given a video of a scene our model is able to decouple scene structure from viewpoint information (see Figure \ref{fig:simonet_headline}). To do so, it utilizes video-based cues, and a structured latent space which separates time-invariant per-object features from time-varying global features. These features are inferred using a transformer-based network which integrates information jointly across space and time.
Second, our method seeks to summarize objects' dynamics. It learns to disentangle not only static object attributes (and their 2D spatial masks), but also object trajectories, without any prior notion of these objects, from videos alone. The learnt trajectory features are temporally abstract and per object; they are captured independently of the dynamics of camera pose, which being a global property, is captured in the model’s per-frame (time-varying) latents.\footnote{Animated figures are at \href{https://sites.google.com/view/simone-scene-understanding/}{https://sites.google.com/view/simone-scene-understanding/}.}
\begin{figure}[t]
\centering
\includegraphics[width=0.85\columnwidth]{figures/headline_plot_final_seq5.png}
\caption{\textbf{Decomposition (A):} SIMONe{} factorizes a scene sequence $\mathbf{X}$ into \textit{scene content} (``object latents,'' constant across the sequence) and \textit{view/global content} (``frame latents,'' one per frame) without supervision. Its spatio-temporal attention-based inference naturally allows stable object tracking (e.g. the green sphere is assigned the same segment across frames). \textbf{Recomposition (B):} Object latents of a given sequence $\mathbf{X}$ can be recomposed with the frame latents of a different (i.i.d.) sequence $\mathbf{X}'$ to generate a consistent rendering of the same scene (i.e. objects and their properties, relative arrangements, and segmentation assignments) from entirely different viewpoints. Notice that both camera pose and lighting are transferred, as evidenced by the wall corners in the background and the shadows of the green sphere.
}
\label{fig:simonet_headline}
\end{figure}
Our model thus advances the state of the art in unsupervised, object-centric scene understanding by satisfying the following desiderata: \begin{enumerate*}[label=\textbf{(\arabic*})]
\item decomposition of multi-object scenes from RGB videos alone;
\item handling of changing camera pose, and simultaneous inference of scene contents and viewpoint from correlated views (i.e. sequential observations of a moving agent);
\item learning of structure across diverse scene instances (i.e. procedurally sampled contents);
\item object representations which summarize static object attributes like color or shape, view-dissociated properties like position or size, as well as time-abstracted trajectory features like direction of motion;
\item no explicit assumptions of 3D geometry, no explicit dynamics model, no specialized renderer, and few a priori modeling assumptions about the objects being studied; and
\item simple, scalable modules (for inference and rendering) to enable large-scale use.
\end{enumerate*}
\section{Related Work}
\label{sec:related_work}
Given the multifaceted problem it tackles, SIMONe{} connects across several areas of prior work. We describe its nearest neighbors from three scene understanding domains below:
\textbf{Scene decomposition models.}
\begin{enumerate*}[label=\textbf{(\arabic*})]
\item Our work builds on a recent surge of interest in unsupervised scene decomposition and understanding, especially using slot structure to capture the objects in a scene \cite{greff2020binding}. One line of work closely related to SIMONe{} includes methods like \cite{burgess2019monet, greff2019multi, engelcke2020genesis, lin2020space, locatello2020sa}, which all share SIMONe{}'s Gaussian mixture pixel likelihood model. While these prior methods handled only static scenes, more recent work \cite{creswell2021oat, veerapaneni2020entity, zablotskaia2020unsupervised, nanbo2020mulmon} has extended them to videos with promising results. Nevertheless, these approaches have no mechanism or inductive bias to separate view information from scene contents. Moreover, many of them are conditioned on extra inputs like the actions of an agent/camera to simplify inference.
\item Another family of decomposition models originated with Attend, Infer, Repeat (AIR) \cite{eslami2016air}. AIR's recurrent attention mechanism does split images into components with separate appearance and pose latents each. Later work \cite{lin2020space, kosiorek2018sqair, crawford2019esilot, crawford2019spair, jiang2019scalor} also extended the model to videos. Despite their structured latents, these models do not learn to distill object appearance into a time-invariant representation (as their appearance and pose latents are free to vary as a function of time). They also require separate object discovery and propagation modules to handle appearing/disappearing objects. In contrast, SIMONe{} processes a full sequence of images using spatio-temporal attention and produces a single time-invariant latent for each object, hence requiring no explicit transition model or discovery/propagation modules.
\end{enumerate*}
\textbf{Multi-view scene rendering models.} Models which assume viewpoint information for each image like GQN \cite{eslami2018gqn}, SRNs \cite{sitzmann2019srn}, and NeRF \cite{mildenhall2020nerf} have shown impressive success at learning implicit scene representations and generating novel views from different viewpoints. Recent work \cite{pumarola2020dnerf, park2020nerfies, li2020neuralsceneflow, du2020neuralradianceflow} has further extended these models to videos using deformation fields to model changes in scene geometry over time. In contrast to SIMONe{}, these models can achieve photorealistic reconstructions by assuming camera parameters (viewpoint information). To allow a direct comparison, we use a view-supervised version of SIMONe{} in Section \ref{sec:results_view_supervised}. There is also recent work \cite{lin2021barf, wang2021nerfmm} that relaxes the known viewpoint constraint, but they still model single scenes at a time, which prevents them from exploiting regularities over multiple scenes. A more recent line of work \cite{kosiorek2021nerfvae, trevithick2020grf, yu2020pixelnerf} explored amortizing inference by mapping from a given set of images to scene latents, but they cannot handle videos yet. Note that all of these models treat the whole scene as a single entity and avoid decomposing it into objects. One exception here is \cite{niemeyer2020giraffe}, which represents objects with separate pose and appearance latents.
However, this model is purely generative and cannot infer object latents from a given scene. Another exception is \cite{chen2020object}, which can in fact infer object representations, but nevertheless depends on view supervision.
\textbf{Simultaneous localization and mapping.} The problem of inferring scene representations in a novel environment by exploring it (rather than assuming given views and viewpoint information) is well studied in robotics and vision \cite{Cadena16tro-SLAMfuture}. Classic SLAM techniques often rely on EM \cite{slam1,slam2} or particle filters \cite{10.5555/1566899.1566949} to infer viewpoint and scene contents jointly. While our problem is slightly simpler (we can leverage shared structure across scene instances; certain elements such as the shape of the room are held constant; and we use offline data rather than active exploration), our approach of using a factorized variational posterior provides a learning-based solution to the same computational problem. Our simplified setting is perhaps justified by our unsupervised take on the problem. On the other hand, we don't assume simplifications which may be common in robotics practice (e.g. known camera properties like field of view; or the use of multiple cameras or depth sensors). Most popular SLAM benchmarks \cite{Burri25012016, Geiger2012CVPR, sturm12iros} are on unstructured 3D scenes and hence it was not straightforward for us to compare directly to classic methods. But an encouraging point of overlap is that object-centric SLAM formulations \cite{xu2019mid} as well as learning-based solutions \cite{9047170, geng2020unsupervised} are active topics of research. Our work could open new avenues in object-centric scene mapping without supervision.
\section{Model}
SIMONe{} is a variational auto-encoder \cite{kingma2019introduction} consisting of an inference network (encoder) which infers latent variables from a given input sequence, and a generative process (decoder) which decodes these latents back into pixels. Using the Evidence Lower Bound (ELBO), the model is trained to minimize a pixel reconstruction loss and latent compression KL loss. Crucially, SIMONe\ relies on a factorized latent space which enforces a separation of static object attributes from global, dynamic properties such as camera pose. We introduce our latent factorization and generative process in Section \ref{sec:model_latent_structure}. Then in Section \ref{sec:model_inference}, we describe how the latents can be inferred using a transformer-based encoder, significantly simplifying the (recurrent or autoregressive) architectures used in prior work. Finally, we fully specify the training scheme in Section \ref{sec:model_loss}.
\subsection{Latent Structure and Generative Process}
\label{sec:model_latent_structure}
Our model aims to capture the structure of a scene, observed as a sequence of images from multiple viewpoints (often along a smooth camera trajectory, though this is not a requirement). Like many recently proposed object-centric models we choose to represent the scene as a set of $K$ \emph{object} latent variables $\mathbf{O} \coloneqq \{\mathbf{o}_k\}_{k=1}^K$. These are invariant by construction across all frames in the sequence (i.e. their distribution is constant through time, and expected to summarize information across the whole sequence). We also introduce $T$ \emph{frame} latents $\mathbf{F} \coloneqq \{\mathbf{f}_t\}_{t=1}^T$, one for each frame in the sequence, that capture time-varying information. Note that by choosing this factorization we reduce the number of required latent variables from $K \cdot T$ to $K + T$. The latent prior $p(\mathbf{O}, \mathbf{F})=\prod_k \mathcal{N}(\mathbf{o}_k \mid \mathbf{0, I})\prod_t \mathcal{N}(\mathbf{f}_t \mid \mathbf{0, I}) $ is a unit spherical Gaussian, assuming and enforcing independence between object latents, frame latents, and their feature dimensions.
Given the latent variables, we assume all pixels and all frames to be independent. Each pixel is modeled as a Gaussian mixture with $K$ components. The mixture weights for pixel $\mathbf{x}_{t, i}$ capture which component $k$ ``explains'' that pixel ($1 \le i \le HW$). The mixture logits $\hat{m}_{k, t, i}$ and RGB (reconstruction) means $\bm{\mu}_{k, t, i}$ are computed for every component $k$ at a specific time-step $t$ and specific pixel location $\mathbf{l}_i$ using a decoder $\mathcal{D_{\theta}}$:
\begin{align}
\hat{m}_{k,t,i}, \bm{\mu}_{k,t,i} &= \mathcal{D_{\theta}}(\mathbf{o}_k, \mathbf{f}_t ; \mathbf{l}_i, t) \\
p(\mathbf{x}_{t, i} \mid \mathbf{o}_1, ..., \mathbf{o}_K, \mathbf{f}_t;t,\mathbf{l}_i) &= \sum_k m_{k, t, i}\mathcal{N}(\mathbf{x}_{t,i} \mid \bm{\mu}_{k, t, i}; \sigma_x) \label{eq:pixel_likelihood}
\end{align}
We decode each pixel independently, "querying" our \emph{pixel-wise} decoder using the sampled latents, coordinates $\mathbf{l}_i \in [-1, 1]^2$ of the pixel, and time-step $t \in [0, 1)$ being decoded as inputs. The decoder's architecture is an MLP or 1x1 CNN. (See Appendix \ref{app:decoder} for the exact parameterization as well as a diagram of the generative process). By constraining the decoder to work on individual pixels, we can use a subset of pixels as training targets (as opposed to full images; this is elaborated in Section \ref{sec:model_loss}). Once they are decoded, we obtain the mixture weights $m_{k,t,i}$ by taking the softmax of the logits across the $K$ components: $m_{k,t,i}=\texttt{softmax}_k(\hat{m}_{k,t,i})$. Equation \ref{eq:pixel_likelihood} specifies the full pixel likelihood, where $\sigma_x$ is a scalar hyperparameter.
\subsection{Inference}
\label{sec:model_inference}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{figures/inference_architecture_fixed.png}
\caption{\textbf{Architecture} of the SIMONe{} inference network $\mathcal{E_{\phi}}$. The transformers integrate information jointly across space and time to infer (the posterior parameters of) the object and frame latents.}
\label{fig:inference
\end{figure}
Given a sequence of frames $\mathbf{X} \coloneqq \{\mathbf{x}_t\}_{t=1}^T$ we now wish to infer the corresponding object latents $\mathbf{O}$ and frame latents $\mathbf{F}$. The exact posterior distribution $p(\mathbf{O}, \mathbf{F} \mid \mathbf{X})$ is intractable so we resort to using a Gaussian approximate posterior $q(\mathbf{O}, \mathbf{F} \mid \mathbf{X})$. The approximate posterior is parameterized as the output of an inference (encoder) network $\mathcal{E_{\phi}}(\mathbf{X})$ which outputs the mean and (diagonal) log scale for all latent variables given the input sequence.
SIMONe{}'s inference network is based on the principle that spatio-temporal data can be processed \emph{jointly} across space and time using transformers. Beyond an initial step, we don't need the translation invariance of a CNN, which forces spatial features to interact gradually via a widening receptive field. Nor do we need the temporal invariance of an RNN which forces sequential processing. Instead, we let feature maps interact simultaneously across the cross-product of space and time. See Figure \ref{fig:inference} for an overview of our encoder architecture implementing this.
Concretely, each frame $\mathbf{x}_t$ in the sequence is passed through a CNN which outputs $IJ$ spatial feature maps at each time-step (containing $C$ channels each). $IJ$ can be larger than the number of object latents $K$. (For all results in the paper, we set $I$ and $J$ to 8 each, and $K=16$.) The rest of the inference network consists of two transformers $\mathcal{T}_1$ and $\mathcal{T}_2$. $\mathcal{T}_1$ takes in all $TIJ$ feature maps. Each feature map attends to all others as described. $\mathcal{T}_1$ outputs $TIJ$ transformed feature maps. When $IJ > K$, we apply a spatial pool to reduce the number of slots to $TK$ (see Appendix \ref{app:encoder} for details). These slots serve as the input to $\mathcal{T}_2$, which produces an equal number of output slots. Both transformers use absolute (rather than relative) positional embeddings, but these are 3D to denote the spatio-temporal position of each slot. We denote the output of $\mathcal{T}_2$ as $\mathbf{\hat{e}}_{k, t}$. This intermediate output is aggregated along separate axes (and passed through MLPs) to obtain $T$ frame and $K$ object posterior parameters respectively. Specifically, $\bm{\lambda}_{\mathbf{o}_k} = \textrm{mlp}_o(1/T\sum_t \mathbf{\hat{e}}_{k, t})$ while $\bm{\lambda}_{\mathbf{f}_t} = \textrm{mlp}_f(1/K\sum_k \mathbf{\hat{e}}_{k, t})$. Using these posterior parameters we can sample the object latents $\mathbf{o}_k \sim \mathcal{N}(\bm{\lambda}^{\mu}_{\mathbf{o}_k}, \exp(\bm{\lambda}^{\sigma}_{\mathbf{o}_k}) \mathbb{1})$, and the frame latents $\mathbf{f}_t \sim \mathcal{N}(\bm{\lambda}^{\mu}_{\mathbf{f}_t}, \exp(\bm{\lambda}^{\sigma}_{\mathbf{f}_t}) \mathbb{1})$.
\subsection{Loss and Training}
\label{sec:model_loss}
The model is trained end to end by minimizing the following negative-ELBO derivative:
\begin{equation}
\begin{aligned}
\frac{-\alpha}{T_d H_d W_d} \sum_{t=1}^{T_d} \sum_{i=1}^{H_d W_d} \log p(\mathbf{x}_{t, i} \mid \mathbf{o}_1, ..., \mathbf{o}_K, \mathbf{f}_t;t,\mathbf{l}_i) &+ \frac{\beta_o}{K} \sum_k D_{KL}\infdivx{q(\mathbf{o}_k \mid \mathbf{X} )}{p(\mathbf{o}_k)} \\
&+ \frac{\beta_f}{T} \sum_t D_{KL}\infdivx{q(\mathbf{f}_t \mid \mathbf{X} )}{p(\mathbf{f}_t)}
\end{aligned}
\end{equation}
We normalize the data log-likelihood by the number of decoded pixels $(T_d H_d W_d)$ to allow for decoding fewer than all input pixels ($THW$). This helps scale the size of the decoder (without reducing the learning signal, due to the correlations prevalent between adjacent pixels). Normalizing by $1/T_d H_d W_d$ ensures consistent learning dynamics regardless of the choice of how many pixels are decoded. $\alpha$ is generally set to 1, but available to tweak in case the scale of $\beta_o$ and $\beta_f$ is too small to be numerically stable. Unless explicitly mentioned, we set $\beta_o = \beta_f$. See Appendix \ref{app:our_model} for details.
\section{Comparative Evaluation}
\label{sec:comparative_eval}
To evaluate the model we focus on two tasks: novel view synthesis and video instance segmentation. On the first task (Section \ref{sec:results_view_supervised}), we highlight the benefit of view information when it is provided as ground truth to a simplified version of our model (denoted ``SIMONe-VS'' for view supervised), as well as baseline models like GQN \cite{eslami2018gqn} and NeRF-VAE \cite{kosiorek2021nerf}. On the second task (Section \ref{sec:results_segmentation_performance}), we deploy the fully unsupervised version of our model; we showcase not only the possibility of inferring viewpoint from data, but also its benefit to extracting object-level structure in comparison to methods like MONet \cite{burgess2019monet}, Slot Attention \cite{locatello2020sa}, and Sequential IODINE \cite{greff2019multi}.
Our results are based on three procedurally generated video datasets of multi-object scenes. In increasing order of difficulty, they are: \textbf{Objects Room 9} \cite{multiobjectdatasets19}, \textbf{CATER} (moving camera) \cite{Girdhar2020CATER}, and \textbf{Playroom} \cite{abramson2020imitating}. These were chosen to meet a number of criteria: we wanted at least 9-10 objects per scene (there can be fewer in view, or as many as 30 in the case of Playroom). We wanted a moving camera with a randomized initial position (the only exception is CATER, where the camera moves rapidly but is initialized at a fixed position to help localization). We wanted ground-truth object masks to evaluate our results quantitatively. We also wanted richness in terms of lighting, texture, object attributes, and other procedurally sampled elements. Finally, we wanted independently moving objects in one dataset (to evaluate trajectory disentangling and temporal abstraction), and unpredictable camera trajectories in another (the Playroom dataset is sampled using an arbitrary agent policy, so the agent is not always moving). Details on all datasets are in Appendix \ref{app:datasets}.
\subsection{View synthesis (with viewpoint supervision)}
\label{sec:results_view_supervised}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/view_interpolation_final_main_text.png}
\caption{\textbf{Comparison of scene representation and view synthesis capabilities} between SIMONe-VS, NeRF-VAE, and GQN. All models partially observe a procedurally generated Playroom from a given sequence of frames (we visualize 4 of the 16 input frames fed to the models). Then, we decode novel views on a circular trajectory around the room, with the yaw linearly spaced in $[-\pi, \pi]$. NeRF-VAE retains very little object structure, while GQN hallucinates content. SIMONe-VS can produce fine reconstructions of objects that it observes even partially or at a distance (such as the bed or shelves in the scene). SIMONe-VS also segments the scene as a bonus. See Appendix \ref{app:view_supervised_kl_comparison} for similar plots from different scenes/input sequences.}
\label{fig:view_interpolations
\end{figure}
We first motivate view-invariant object representations by considering the case when ground-truth camera pose is provided to our model (a simplified variant we call ``SIMONe-VS''). In this scenario, we don't infer any frame latents. Rather, the encoder and decoder are conditioned on the viewpoint directly. This \emph{view-supervised} setting is similar to models like GQN and NeRF which represent the contents of a scene implicitly and can be queried in different directions.
We compare three such models on view synthesis in the Playroom. The models are provided a set of 16 consecutive frames as context, partially revealing a generated room. Having inferred a scene representation from the input context, the models are then tasked with generating unobserved views of the scene. This extrapolation task is performed without any retraining, and tests the coherence of the models' inferred representations. The task is challenging given the compositional structure of each Playroom scene, as well as the variation across scenes (the color, position, size, and choice of all objects are procedurally sampled per scene; only the L-shaped layout of the room is shared across scenes in the dataset). Because each model is trained on and learns to represent many Playroom instances, NeRF itself is not directly suitable for the task. It needs to be retrained on each scene, whereas we want to infer the specifics of any given room at evaluation time. NeRF-VAE addresses this issue and makes it directly comparable to our model.
To set up the comparison, we first trained SIMONe-VS and evaluated its log-likelihood on Playroom sequences. Then, we trained GQN and NeRF-VAE using constrained optimization (GECO \cite{rezende2018taming}) to achieve roughly the same log likelihood per pixel. See Appendix \ref{app:view_supervised_kl_comparison} for a comparison of the models in terms of the reconstruction-compression trade-off.
Qualitatively, the models show vast differences in their perceived structure (see Figure \ref{fig:view_interpolations}). NeRF-VAE blurs out nearly all objects in the scene but understands the geometry of the room and is able to infer wall color. GQN produces more detailed reconstructions, but overfits to particular views and does not interpolate smoothly. SIMONe-VS on the other hand finely reproduces the object structure of the room. Even when it observes objects at a distance or up close, it places and sizes them correctly in totally novel views. This makes SIMONe-VS a powerful choice over NeRF-VAE and GQN-style models when the priority is to capture scene structure across diverse examples.
\subsection{Instance segmentation (fully unsupervised)}
\label{sec:results_segmentation_performance}
Having shown the benefit of view information to inferring scene structure in the Section \ref{sec:results_view_supervised}, we now turn to the added challenge of inferring viewpoint directly and simultaneously with scene contents (without any supervision).
We compare SIMONe\ to a range of competitive but viewpoint-unaware scene decomposition approaches. First, we train two static-frame models: MONet and Slot Attention. MONet uses a similar generative process and training loss to our model, achieving segmentation by modeling the scene as a spatial mixture of components, and achieving disentangled representations using a $\beta$-weighted KL information bottleneck. On the other hand, it uses a deterministic, recurrent attention network to infer object masks. Slot Attention is a transformer-based autoencoding model which focuses on segmentation performance rather than representation learning. Finally, we also compare against Sequential IODINE (``S-IODINE''), which applies a refinement network to amortize inference over time, separating objects by processing them in parallel.
It also uses a $\beta$-weighted KL loss to disentangle object representations. Note that S-IODINE is a simplified version of OP3 \cite{veerapaneni2020entity}, which additionally attempts to model (pairwise) object dynamics using an agent's actions as inputs. SIMONe\ and S-IODINE both avoid relying on this privileged information.
Table \ref{tab:segmentation_performance_comparison} contains a quantitative comparison of segmentation performance across these models, while Figure \ref{fig:segmentations_qualitative} shows qualitative results.
\begin{table}[t]
\centering
\footnotesize
\begin{tabular}{llll|llll}
\toprule
& \multicolumn{3}{c|}{Static ARI-F} & \multicolumn{4}{c}{Video ARI-F} \\
\multicolumn{1}{c}{} & MONet & SA & S-IODINE & MONet & SA & S-IODINE & SIMONe \\
\midrule
Objects Room 9 & 0.886 & 0.784 & 0.695 & 0.865 & 0.066 & 0.673 & 0.936 \\
& {\scriptsize ($\pm$0.061)} & {\scriptsize ($\pm$0.138)} & {\scriptsize ($\pm$0.007)} & {\scriptsize ($\pm$0.007)} & {\scriptsize ($\pm$0.014)} & {\scriptsize ($\pm$0.0.002)} & {\scriptsize ($\pm$0.010)}\\
CATER & 0.937 & 0.923 & 0.728 & 0.412 & 0.073 & 0.668 & 0.918 \\
& {\scriptsize ($\pm$0.004)} & {\scriptsize ($\pm$0.076)} & {\scriptsize ($\pm$0.032)} & {\scriptsize ($\pm$0.012)} & {\scriptsize ($\pm$0.006)} & {\scriptsize ($\pm$0.033)} & {\scriptsize ($\pm$0.036)}\\
Playroom & 0.647 & 0.653 & 0.439 & 0.442 & 0.059 & 0.356 & 0.800 \\
& {\scriptsize ($\pm$0.012)} & {\scriptsize ($\pm$0.024)} & {\scriptsize ($\pm$0.009)} & {\scriptsize ($\pm$0.010)} & {\scriptsize ($\pm$0.002)} & {\scriptsize ($\pm$0.006)} & {\scriptsize ($\pm$0.043)}\\ \bottomrule
\end{tabular}
\vspace{5pt}
\caption{\textbf{SIMONe{} segmentation performance} (in terms of Adjusted Rand Index for foreground objects, ARI-F) compared to state-of-the-art unsupervised baselines: two static-frame models (MONet and Slot Attention, SA) and a video model (S-IODINE). We calculate static and video ARI-F scores separately. For static ARI-F, we evaluate the models per still image. For video ARI-F, we evaluate the models across space and time, taking an object's full trajectory as a single class. The video ARI-F thus penalizes models (especially Slot Attention) which fail to track objects stably. We report the mean and standard deviation of scores across 5 random seeds in each case.
\label{tab:segmentation_performance_comparison}
}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.82\columnwidth]{figures/segmentations_qualitative.png}
\caption{\textbf{Segmentations} and reconstructions produced by SIMONe{} on CATER and Playroom. SIMONe{} copes well with clutter and different-sized objects. It learns to use object motion as a segmentation signal on CATER, evident from the fact that an object’s shadow is correctly assigned to that object’s segment as it moves. This is true even when there's multiple shadows per object (due to multiple lights in the scene). SIMONe{} also overcomes color-based cues to segment two-toned objects such as beds in the Playroom as single objects. See Appendix \ref{app:segmentation_comparison} to compare with baseline models.}
\label{fig:segmentations_qualitative}
\end{figure}
\vspace{-5pt}
\section{Analysis}
We take a closer look at the representations learnt by our model by decoding them in various ways.
First, we manipulate latent attributes individually to assess the interpretability of object representations visually in Section \ref{sec:traversals}. Next, we exploit SIMONe{}'s latent factorization to render views of a given scene using the camera trajectory of a different input sequence. These cross-over visualizations help identify how the model encodes object dynamics in Section \ref{sec:crosssovers}. Finally, we measure the predictability of ground-truth camera dynamics and object dynamics from the two types of latents in Section \ref{sec:predictions_from_latents}. These analyses use a single, fully unsupervised model per dataset.
\vspace{-5pt}
\subsection{Latent attribute traversals}
\label{sec:traversals}
We visualize the object representations learnt by SIMONe{} on Playroom to highlight their disentanglement, across latent attributes and across object slots, in Figure \ref{fig:obj_latent_traversals}. We seed all latents using a given input sequence, then manipulate one object latent attribute at a time by adding fixed offsets.
Note that object position and size are well disentangled in each direction. Aided by the extraction of view-specific information in the frame latents, SIMONe{} also learns object features corresponding to identity. The decoder nevertheless obeys the biases in the dataset–for instance, shelves will slide along a wall when their position latent is traversed. The rubber duck does not morph into a chest of drawers because those are always located against a wall. This further suggests a well-structured latent representation, which the decoder can adapt to.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\columnwidth]{figures/traversals_horizontal_noarrows.png}
\caption{\textbf{Object attributes learnt by SIMONe{}.} In each row, we manipulate a particular object latent attribute for an arbitrary target object (circled in red) in two scenes. This reveals the attributes' relationship to interpretable object characteristics like color, size, position and identity.}
\label{fig:obj_latent_traversals}\vspace{-5pt}
\end{figure}
\vspace{-5pt}
\subsection{Object and frame latent cross-overs}
\label{sec:crosssovers}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/temporal_abstraction_main_text.png}
\caption{\textbf{Separation of object trajectories from camera trajectories. Left:} When encoding a sequence with consistent (i.i.d.) object dynamics, this information is extracted in the object latents and is unaffected by changing frame latents (see green cone). \textbf{Right:} Movement events are sequenced correctly; object relative positions also remain consistent (see pattern of shadows on the floor circled in yellow). See Appendix~\ref{app:temporal_abstraction} for cross-over plots showing more object trajectories.}
\label{fig:temporal_abstraction}\vspace{-5pt}
\end{figure}
We expect SIMONe to encode object trajectories and camera trajectories independently of each other. In fact, each object's trajectory should be summarized in its own time-invariant latent code.
To examine this, we recompose object latents from one sequence with frame latents from other sequences in the CATER dataset. The result, in Figure \ref{fig:temporal_abstraction}, is that we can observe object motion trajectories from multiple camera trajectories.
Note the consistency of relative object positions (at any time-step) from all camera angles. In the single moving object case, its motion could in fact be interpreted as a time-varying global property of the scene. Despite this challenge, SIMONe{} is able to encode the object's motion as desired in its specific time-invariant code. In Section \ref{sec:predictions_from_latents}, we further confirm that object trajectories are summarized in the object latents, which can be queried with time to recover allocentric object positions.
\vspace{-5pt}
\subsection{Camera pose and object trajectory prediction}
\label{sec:predictions_from_latents}
\begin{table}[b!]
\centering
\begin{tabular}{@{}llll@{}}
\toprule
& Linear($\mathbf{f}_t$) & MLP($\mathbf{f}_t$) & MLP($\mathbf{o}_1$, ..., $\mathbf{o}_K$) \\ \midrule
Camera location & $0.832 \pm 0.0$ & $0.949 \pm 0.002$ & $0.044 \pm 0.026$ \\
Camera orientation (Rodrigues) & $0.800 \pm 0.0$ & $0.946 \pm 0.002$ & $0.292 \pm 0.025$ \\ \bottomrule
\end{tabular}
\vspace{10pt}
\caption{\textbf{Decoding camera pose.} We show that ground-truth camera location or orientation is predictable from the corresponding frame latent, but cannot be predicted from all object latents put together. We report the test $R^2$ score across 5 independently trained decoders per input type.}
\label{tab:predicting_camera_pose
\end{table}
\begin{table}[b!]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lllll@{}}
\toprule
& MLP($\mathbf{o}_k$) & MLP($\mathbf{o}_k, t$) & MLP($\mathbf{o}_k, \mathbf{f}_t, t$) & MLP($\{\mathbf{o}_j: j \ne k\}$) \\ \midrule
Trained on all objects & $0.710 \pm 0.006$ & $0.871 \pm 0.006$ & $0.876 \pm 0.003$ & $-0.062 \pm 0.006$\\
Trained on moving objects & $0.724 \pm 0.007$ & $0.894 \pm 0.004$ & $0.898 \pm 0.005$ & $-0.022 \pm 0.025$ \\
\bottomrule
\end{tabular}%
}
\vspace{10pt}
\caption{\textbf{Decoding object trajectories.} We test MLP decoders on predicting allocentric object positions (of moving object in unseen scenes) based on the following inputs: (a) the corresponding object latent, (b) the timestep as well, and (c) the frame latent corresponding to that timestep as well, and (d) remaining object latents from the scene (not pertaining to the object of interest). The decoders were trained on arbitrary objects or a subset containing moving objects only. We report the test $R^2$ score across 5 independently trained decoders per input type.}
\label{tab:sa_predicting_obj_trajectories
\end{table}
We assessed SIMONe{}'s frame latents by decoding the true camera position and orientation from them. We trained linear and MLP regressors to predict the camera pose at time $t$ from the corresponding frame latent $\mathbf{f}_t$ on a subset of CATER sequences. We also trained an MLP on the time-invariant object latents $\mathbf{o}_{1:K}$ for the same task. We evaluated these decoders on held-out data. Table \ref{tab:predicting_camera_pose} shows that frame latents describe the viewpoint almost perfectly.
We also assessed if the object latents contain precise information about allocentric object positions (to be clear, position information is not provided in any form while training SIMONe{}). Table~\ref{tab:sa_predicting_obj_trajectories} shows that the correct object latent is predictive of the allocentric position of a dynamic object (when queried along with the timestep). Adding the frame latent does not provide more information, and using the ``wrong'' objects (from the same scene) is completely uninformative. To perform this analysis, we needed to align SIMONe's inferred objects with the ground-truth set of objects. We used the Hungarian matching algorithm on the MSE of inferred object masks and ground-truth object masks to perform the alignment. Given SIMONe's disentangling of object dynamics, its time-abstracted object representations could prove helpful for a variety of downstream tasks (e.g. ``catch the flying ball!'').
Taken together, Table~\ref{tab:predicting_camera_pose} and Table~\ref{tab:sa_predicting_obj_trajectories} show the separation of information that is achieved between the object and frame latents, helping assert our two central aims of view-invariant and temporally abstracted object representations.
\section{Discussion and Future Work}
\label{sec:discussion}
\textbf{Scalability.} The transformer-based inference network in SIMONe\ makes it amenable to processing arbitrarily large videos, just as transformer-based language models can process long text. SIMONe{} could be trained on windows of consecutive frames sampled from larger videos (aka "chunks"). For inference over a full video, one could add memory slots which carry information over time from one window to the next. Applying SIMONe{} on sliding windows of frames also presents the opportunity to amortize inference at any given time-step if the windows are partially overlapping (so the model could observe every given frame as part of two or more sequences). Our use of the standard transformer architecture also makes SIMONe\ amenable to performance improvements via alternative implementations.
\textbf{Limitations.} \begin{enumerate*}[label=\textbf{(\arabic*})]
\item SIMONe\ cannot generate novel videos (e.g. sample a natural camera trajectory via consecutive frame latents) in its current version. This could be addressed in a similar fashion to the way GENESIS~\cite{engelcke2020genesis} built on MONet~\cite{burgess2019monet}–it should be possible (e.g. using recurrent networks) to learn conditional priors for objects in a scene and for successive frame latents, which would make SIMONe\ fully generative.
\item We see another possible limitation arising from our strict latent factorization. We have shown that temporally abstracted object features can predict object trajectories when queried by time. This can cover a lot of interesting cases (even multiple object-level "events" over time), but will start to break as object trajectories get more stochastic (i.e. objects transition considerably/chaotically through time). We leave it to future work to explore how temporal abstraction can be combined with explicit per-step dynamics modeling in those cases. For simpler settings, our approach to encoding object trajectories (distilling them across time) is surprisingly effective.
\end{enumerate*}
\section{Conclusion}
We've presented SIMONe{}, a latent variable model which separates the time-invariant, object-level properties of a scene video from the time-varying, global properties. Our choice of scalable modules such as transformers for inference, and a pixel-wise decoder, allow the model to extract this information effectively.
SIMONe\ can learn the common structure across a variety of procedurally instantiated scenes. This enables it to recognize and generalize to novel scene instances from a handful of correlated views, as we showcased via 360-degree view traversal in the view-supervised setting. More significantly, SIMONe{} can learn to infer the two sets of latent variables jointly without supervision. Aided by cross-frame spatio-temporal attention, it achieves state-of-the-art segmentation performance on complex 3D scenes. Our latent factorization (and information bottleneck pressures) further help with learning meaningful object representations. SIMONe{} can not only separate static object attributes (like size and position), but it can also separate the dynamics of different objects (as time-invariant localized properties) from global changes in view.
We have discussed how the model can be applied to much longer videos in the future. It also has potential for applications in robotics (e.g. sim-to-real transfer) and reinforcement learning, where view-invariant object information (and summarizing their dynamics) could dramatically improve how agents reason about objects.
\section*{Acknowledgements}
We thank Michael Bloesch, Markus Wulfmeier, Arunkumar Byravan, Claudio Fantacci, and Yusuf Aytar for valuable discussions on the purview of our work. We are also grateful for David Ding's support on the CATER dataset. The authors received no specific funding for this work.
\medskip
{\small
\bibliographystyle{unsrtnat} |
2011.02603 | \section{Introduction}
It is the threshold theorem\cite{Shor-FT-1996,Steane-FT-1997,
Gottesman-FT-1998,Dennis-Kitaev-Landahl-Preskill-2002,
Knill-FT-2003,*Knill-2004B,
*Aliferis-Gottesman-Preskill-2006,*Reichardt-2009,
Katzgraber-Bombin-MartinDelgado-2009} that makes large-scale quantum
computation feasible, at least in theory. Related is the notion of
quantum channel capacity $R_Q$, such that for any rational $R<R_Q$,
there exists a quantum error correcting code (QECC) with rate $R$
which can be used to suppress the logical error probability to any
chosen (arbitrarily small) level, but not for $R>R_Q$. Here the code
rate $R\equiv k/n$ is the ratio of the number $k$ of the logical
(encoded) qubits to the length $n$ of the code. The precise value of
the capacity is not known for most quantum channels of interest,
except for the \emph{quantum erasure channel} with qubit erasure
probability $p$, in which case $R_Q=\min(0,1-2p)$, see
Ref.~\onlinecite{Bennett-DiVincenzo-Smolin-1997}.
In practice, it is often easier to deal with the threshold \emph{error
probability} for a given family (infinite sequence) of QECCs with
certain asymptotic code rate $R$. Depending on the nature of the
quantum channel in question, the threshold error probability may be
related to the location of a thermodynamical phase transition in
certain spin model associated with the codes. In particular, for a
family of qubit toric codes on transitive graphs locally isomorphic to
a regular euclidean or hyperbolic tiling
$\mathcal{H}$
under independent
$Z$ Pauli errors, the decoding threshold is upper bounded by the
position of the multicritical point located at the Nishimori line of
the Ising model on
$\mathcal{H}$, see
Refs.~\onlinecite{Dennis-Kitaev-Landahl-Preskill-2002
Kubica-etal-color-2017,Jiang-Kovalev-Dumer-Pryadko-2018}. It is
widely believed that the two thresholds coincide, at least for the
euclidean tilings like the infinite square lattice and square-lattice
toric codes. With a slightly more general model of independent
$X$/$Z$ Pauli errors, the threshold is the minimum of the
corresponding thresholds for each error type which can be computed
independently.
A special case is the relation between quantum erasure errors and
percolation\cite{Delfosse-Zemor-2010,Delfosse-Zemor-2012
Delfosse-Zemor-2014}. An erasure is formed by rendering inoperable
all qubits in a known randomly selected set. Information loss happens
when erasure covers a logical operator of the code. For certain code
families, and for qubit erasure probability
$p$ sufficiently small,
$p<p_E$, the probability to cover a codeword may go to zero as the
code length
$n$ is increased to infinity. The corresponding threshold value
$p_E$ is called the \emph{erasure threshold} associated with the
chosen code family or code sequence. With a
Calderbank-Shor-Steane (CSS)
code\cite{Calderbank-Shor-1996,Steane-1996}, one may consider the
erasure thresholds for $X$ and
$Z$ logical operators separately, so that the conventional erasure
threshold becomes $p_E=\min(p_E^{X},p_E^{Z})$.
The link between erasure and percolation thresholds is especially
simple in the case of toric/surface\cite{kitaev-anyons
Bravyi-Kitaev-1998,Freedman-Meyer-1998
Dennis-Kitaev-Landahl-Preskill-2002,Delfosse-Iyer-Poulin-2016} and
related quantum cycle codes\cite{Zemor-2009} where qubits are labeled
by the edges of a graph and, by convention, $Z$ logical operators are
supported on $1$-chains in certain equivalence classes, e.g., those
connecting two opposite boundaries of a rectangular region, or wrapped
around a torus.
Then, the erasure threshold $p_E^Z$ coincides with the discrete
version of the homological percolation
transition\cite{Bobrowski-Skraba-2020-euler,Bobrowski-Skraba-2020} for
$1$-chains. It is also known that for square-lattice toric code the
erasure threshold $p_E^Z$
coincides\cite{Stace-Barrett-Doherty-2009,Fujii-Tokunaga-2012} with
the edge percolation threshold, $p_E^Z=p_{\rm c}(\mathbb{Z}^2)=1/2$. On the
other hand, for a family of hyperbolic surface codes based on a given
infinite graph $\mathcal{H}$, a regular tiling on the hyperbolic
plane, we only know that the erasure threshold is upper
bounded\cite{Delfosse-Zemor-2010,Delfosse-Zemor-2012,Delfosse-Zemor-2014}
by the percolation threshold on ${\cal H}$, $p_E\le p_{\rm c}({\cal H})$.
Surely, the erasure and the percolation thresholds cannot always
coincide. Indeed, percolation threshold is associated with the
formation of an infinite cluster; it is defined on an infinite graph,
while quantum codes are finite. Further, erasure threshold is not a
bulk quantity, as it can be rendered zero by removing a vanishingly
small fraction of well-selected qubits. Similarly, many different
finite graphs can be associated with a given infinite graph
${\cal H}$, and it is not at all clear that the erasure threshold
should remain the same independent of the details.
The goal of this work is to quantify the relation between edge
percolation and the stability of quantum cycle codes (QCCs) to erasure
errors. Specifically, we consider sequences of finite graphs
$\mathcal{G}_t=(\mathcal{V}_t,\mathcal{E}_t)$, $t\in \mathbb{N}$, with
a common infinite covering graph $\mathcal{H}$, and use the covering
map $f_t:\mathcal{H}\to \mathcal{G}_t$ to identify homologically
non-trivial cycles on $\mathcal{G}_t$. The distance
$d_t\equiv d_{Z,t}$ of the corresponding quantum code (the smallest
length of a non-trivial cycle) necessarily diverges with $t$ when the
sequence converges weakly to $\mathcal{H}$. First, we show that it is
the scaling of $d_{t}$ with the logarithm of the code block length,
$n_t\equiv |\mathcal{E}_t|$, that determines the location of the
$Z$-erasure threshold, or the $1$-chain lower erasure threshold
$p_E^Z\equiv p_E^0$, the point above which the probability of an
open homologically non-trivial $1$-cycle remains non-zero in the limit
of arbitrarily large graphs $\mathcal{G}_t$. Roughly, with
sublogarithmic distance scaling, $d_{t}/\ln n_t\to 0$ as $t\to\infty$,
$p_E^0=0$. On the other hand, with superlogarithmic distance
scaling, $d_{t}/\ln n_t\to \infty$, $p_E^0$ coincides with the edge
percolation threshold $p_{\rm c}(\mathcal{H})$, so that for
$p<p_{\rm c}(\mathcal{H})$, probability to find an open homologically
non-trivial $1$-cycle be asymptotically zero. We also give an example
of a graph family with logarithmic distance scaling, where the
inequality in the upper bound is strict, $p_E^0< p_{\rm c}({\cal H})$,
and give numerical evidence that for some regular tilings of the
hyperbolic plane, erasure threshold is strictly below the percolation
threshold, $p_E^0<p_{\rm c}(\mathcal{H})$.
Second, the distance $d_t$ grows at most logarithmically with $n_t$
when $\mathcal{H}$ is non-amenable, which is also a necessary
requirement to have a finite asymptotic code rate $k_t/n_t\to R>0$,
where $k_t$ is the number of encoded qubits. For such a graph
sequence, we define a pair of thermodynamical homological transitions,
$p_H^0$ and $p_H^1$, which characterize singularities in the
\emph{erasure rate}, asymptotic ratio of the expected homology rank of
the open subgraph and the number of edges $n_t$. Namely, erasure rate
is zero for $p<p_H^0$, it saturates at
$R$
for $p>p_H^1$, and it takes intermediate values in the interval
$p_H^0<p<p_H^1$ (subsequence construction may be needed in this regime
to achieve convergence). We prove that $p_H^1-p_H^0>R$, and, if
$\mathcal{H}$ and its dual, $\widetilde{\cal H}$, is a pair of
transitive planar graphs, we show that $p_H^0=p_{\rm c}(\mathcal{H})$ and
$p_H^1=1-p_{\rm c}(\widetilde{\cal H})$; the latter point coincides with the
uniqueness threshold $p_{\rm u}(\mathcal{H})$ on the original graph. We
also conjecture that the two homological transitions coincide with the
percolation and the uniqueness thresholds, respectively, for any
non-amenable (quasi)transitive graph, $p_H^0=p_{\rm c}(\mathcal{H})$ and
$p_H^1=p_{\rm u}(\mathcal{H})$.
The outline of the paper is as follows. In Sec.~\ref{sec:notations}
we give the necessary notations. We present our analytical results in
Sec.~\ref{sec:analytical} and numerical results in
Sec.~\ref{sec:numeric}, with the proofs collected in the Appendix.
In section \ref{sec:conclusions} we give the conclusions and discuss
some related open questions.
\section{Definitions}
\label{sec:notations}
\subsection{Classical binary and quantum CSS codes}
A linear binary code with parameters $[n,k,d]$ is a vector space
$\mathcal{C}\subseteq \mathbb{F}_2^n$ of length-$n$ binary strings of
dimension $k$, where the minimum distance $d$ is the smallest Hamming
weight of a non-zero vector in $\mathcal{C}$. Such a code
$\mathcal{C}\equiv \mathcal{C}_G$ can be specified in terms of a
generator matrix $G$ whose rows are the basis vectors, or in terms of
a parity check matrix $H$,
$\mathcal{C}\equiv \mathcal{C}_H^\perp=\{c\in\mathbb{F}_2^n:
Hc^T=0\}$, where $\mathcal{C}_H^\perp$ denotes the space dual
(orthogonal) to $\mathcal{C}_H$. A generator matrix and a parity
check matrix of any length-$n$ code satisfy
\begin{equation}
GH^T=0,\quad
\rank G+\rank H=n;
\label{eq:dual-matrices}
\end{equation}
such matrices are called \emph{mutually dual}.
If $I\subset \{1,\ldots,n\}$ is a set of bit indices, for any vector
$b\in\mathbb{F}_2^{n}$, we denote $b[I]$ the corresponding
\emph{punctured} vector with positions outside of $I$ dropped.
Similarly, $G[I]$ (with columns outside of $I$ dropped) generates the
code $\mathcal{C}_G$ \emph{punctured} to $I$, denoted
${\cal C}_G[I]\equiv {\cal C}_{G[I]}$. A \emph{shortened} code is
formed similarly, except by puncturing only the vectors supported
inside $I$,
$$
\text{$\mathcal{C}$ shortened to $I$} = \left\{c[I]: \;c\in
\mathcal{C}\; \wedge\;\supp(c)\subseteq I\right\}.
$$
We use ${G}_I$ to denote a generating matrix of the code ${\cal C}_G$
shortened to $I$. If $G$ and $H$ is a pair of mutually dual
binary matrices, see Eq.~(\ref{eq:dual-matrices}), then $H_I$ is
a parity check matrix of the punctured code $\mathcal{C}_{G}[I]$,
and\cite{MS-book}
\begin{equation}
\label{eq:puncture-shortening-rank}
\rank G[I]+\rank H_I=|I|,
\end{equation}
i.e., matrices $G[I]$ and $H_I$ are mutually dual. In
addition, if $\overline I=\{1,2,\ldots,n\}\setminus I$ is the
complement of $I$, then
\begin{equation}
\label{eq:dual-puncture-rank}
\rank G[\overline I]+\rank G_I=\rank G.
\end{equation}
For the present purposes, it is sufficient that an $n$-qubit quantum
CSS code $\mathcal{Q}=\css(G_X,G_Z)$ can be specified in terms of two
$n$-column binary \emph{stabilizer generator matrices} with mutually
orthogonal rows, $G_XG_Z^T=0$. It is isomorphic to a direct sum of
two quotient spaces, $\mathcal{Q}=\mathcal{Q}_X\oplus\mathcal{Q}_Z$,
where $\mathcal{Q}_X=\mathcal{C}_{G_Z}^\perp/\mathcal{C}_{G_X}$ and
$\mathcal{Q}_Z=\mathcal{C}_{G_X}^\perp/\mathcal{C}_{G_Z}$. Vectors in
$\mathcal{Q}_X$ and $\mathcal{Q}_Z$, respectively, are also called
$X$- and $Z$-logical operators. Explicitly, $\mathcal{Q}_X$ is formed
by vectors in $\mathcal{C}_{G_Z}^\perp$, with any two vectors that
differ by an element of $\mathcal{C}_{G_X}$ identified (notice that
$\mathcal{C}_{G_X}\subset\mathcal{C}_{G_Z}^\perp$). Such a pair of
vectors $c'=c+\alpha G_X$ that differ by a linear combination of the
rows of $G_X$ are called mutually degenerate; we write $c'\simeq c$.
The second half of the code, $\mathcal{Q}_Z$, is defined similarly,
with the two generator matrices interchanged. For such $Z$-like
vectors, the degeneracy is defined in terms of the rows of $G_Z$.
The distances $d_X$ and $d_Z$ of a CSS code are the minimum weights of
non-trivial vectors in $\mathcal{Q}_X$ and $\mathcal{Q}_Z$,
respectively, e.g.,
$d_X= \min\{\wgt c:c\in \mathcal{C}_{G_Z}^\perp\setminus
\mathcal{C}_{G_X}\}$. Any minimum-weight codeword is always
\emph{irreducible}, that is, it cannot be written as a sum of two
vectors with disjoint supports, one of them being a
codeword\cite{Dumer-Kovalev-Pryadko-bnd-2015}. The conventional
distance, the minimum weight of a logical operator in $\mathcal{Q}$,
is $d=\min(d_X,d_Z)$. The dimension $k$ of a CSS code is the
dimension of the vector space $\mathcal{Q}_X$ (it is the same as the
dimension of $\mathcal{Q}_Z$), the number of linearly independent and
mutually non-degenerate vectors that can be used to form a basis of
$\mathcal{Q}_X$. For a length-$n$ code with stabilizer generator
matrices $G_X$ and $G_Z$,
\begin{equation}
\label{eq:CSS-k}
k=n-\rank G_X-\rank G_Z.
\end{equation}
The parameters of a quantum CSS code are commonly written as
$[[n,k,(d_X,d_Z)]]$ or just $[[n,k,d]]$.
Any CSS code formed by matrices $G_X$ and $G_Z$ of respective
dimensions $r_X\times n$ and $r_Z\times n$ also defines a binary chain
complex with three non-trivial vector spaces,
\begin{equation}
\mathcal{A}:\ldots \leftarrow\{0\}\stackrel{\partial_0}\leftarrow\mathcal{A}_0\stackrel{\partial_1}\leftarrow
\mathcal{A}_1\stackrel{\partial_2}\leftarrow\mathcal{A}_2\stackrel{\partial_3}\leftarrow\{0\}\leftarrow \ldots,\label{eq:3-chain}
\end{equation}
where the spaces $\mathcal{A}_i=\mathbb{F}_2^{a_i}$ have dimensions
$a_0=r_X$, $a_1=n$, and $a_2=r_Z$, and the non-trivial boundary
operators are expressed in terms of the generator matrices
$\partial_1=G_X$, $\partial_2=G_Z^T$. This guarantees the defining
property of a chain complex, $\partial_i\partial_{i+1}=0$,
$i\in\mathbb{Z}$. Then, the code $\mathcal{Q}_Z$ is defined
identically to the first homology group
$H_1(\mathcal{A})= \ker(\partial_{1})/\im(\partial_{2})$, where
elements of $\im(\partial_{2})$ called \emph{cycles} are linear
combinations of the columns of $\partial_2={G}_Z^T$, while elements of
$\ker(\partial_1)$ called \emph{boundaries} are vectors orthogonal to
the rows of $\partial_1={G}_X$. The other definitions also match. In
particular, the dimension $k$ of the quantum code is the rank of the
first homology group, $k=\rank H_1(\mathcal{A})$, while the definition
of the homological distance $d_1(\mathcal{A})$ matches that of $d_Z$.
The other code, $\mathcal{Q}_X$, corresponds to the co-homology group defined
in the co-chain complex $\widetilde{\mathcal{A}}$ formed similarly but
with the two matrices interchanged.
Let us now consider the structure of the homology group where the
space ${\cal A}_1$ is restricted so that only components with indices
in the index set $I\subset \{1,2,\ldots,n\}$ be non-zero.
Respectively, the spaces $\ker\partial_1={\cal C}_{G_X}^\perp$ and
$\im \partial_2={\cal C}_{G_Z}$ should be replaced with the
corresponding reduced spaces. The result is isomorphic to a chain
complex ${\cal A}'_I$ where the two boundary operators
are obtained by puncturing and shortening, respectively:
$\partial_1'=G_X[I]$ and $\partial_2'=(G_Z)_I^T$. The
dimension of thus defined restricted homology group is given b
\begin{equation}
\label{eq:rank-H1-restricted}
k_I'\equiv \rank H_1({\cal A}'_I)=|I|-\rank G_X[I]-\rank (G_Z)_I.
\end{equation}
Using Eq.~(\ref{eq:dual-puncture-rank}), we also
get\cite{Delfosse-Zemor-2012}
\begin{equation}
\label{eq:rank-H1-restricted-two}
k_I'=|I|-\rank G_Z-\rank G_X[I]+\rank G_Z[\overline I].
\end{equation}
The corresponding result for the rank $\tilde k_{I}'$ of the
restricted co-homology group can be found by exchanging the matrices
$G_X$ and $G_Z$; this gives the duality relation
\begin{equation}
\label{eq:rank-H1-restricted-duality}
k_I'+\tilde k_{\overline I}'=k.
\end{equation}
\subsection{Graphs, cycles, and cycle codes}
We consider only simple graphs with no loops or multiple edges. A
graph ${\cal G}=({\cal V}, {\cal E})$ is specified by its sets of
vertices ${{\cal V}}\equiv {{\cal V}}_{\cal G}$, also called sites,
and edges ${{\cal E}}\equiv {{\cal E}}_{\cal G}$.
Each edge $e\in {\cal E}$ is a set of two vertices, $e=\{u,v\}$; it
can also be denoted with a wave, $u\sim v$. For every vertex
$v\in {\cal V}$, its degree $\deg(v)$ is the number of edges that
include $v$. An infinite graph $\mathcal{G}$ is called quasi-transitive if
there is a finite subset $\mathcal{V}_0\subset\mathcal{V}_{\cal G}$
of its vertices, such that for every vertex $v\in\mathcal{V}$ there is
an automorphism (symmetry) of $\mathcal{G}$ mapping $v$ to an element
of $\mathcal{V}_0$. A transitive graph is a quasi-transitive graph
where the subset $\mathcal{V}_0$ of vertex classes contains only one
element. All vertices in a transitive graph have the same degree.
We say that vertices $u$ and $v$ are connected on ${\cal G}$ if there
is a path $P\equiv P(u_0,u_\ell)$ between $u\equiv u_0$ and
$v\equiv u_\ell$, a set of edges which can be ordered and oriented to
form a \emph{walk}, a sequence of vertices starting with $u$ and
ending with $v$, with each directed edge in $P$ matching the
corresponding pair of neighboring vertices in the sequence,
\begin{equation}
\label{eq:path}
P(u_0,u_\ell)= \{u_0\sim u_1, u_1\sim
u_2,\ldots, u_{\ell-1}
\sim u_\ell\}\subseteq {\cal E}.
\end{equation}
We call such a path open if $u_0\neq u_\ell$, and closed otherwise.
The path is called self-avoiding (simple) if $u_i\neq u_j$ for any
$0\le i<j\le \ell$, except that $u_0$ and $u_\ell$ coincide if the
path is closed.
The length of the path is the number of edges in the set,
$\ell=|{P}|$. The distance $d(u,v)$ between vertices $u$ and $v$ is
the smallest length of a path between them. Given a vertex
$v\in {\cal V}$ and a natural $r\in\mathbb{N}$, a ball
$\mathcal{B}(v,r;\mathcal{G})$ is the subgraph of ${\cal G}$ induced
by the vertices $u\in {\cal V}$ such that $d(v,u)\le r$.
The edge boundary $\partial\mathcal{U}$ of a set of
vertices $\mathcal{U}\subseteq \mathcal{V}$ is the set of edges
connecting $\mathcal{U}$ and its complement
$\overline {\cal U}\equiv {\cal V}\setminus {\cal U}$. Given an
exponent $\alpha\le 1$, we define the isoperimetric constant of a
graph,
\begin{equation}
\label{eq:edge-expander}
b_\alpha=\inf_{\emptyset\neq \mathcal{U}\subsetneq\mathcal{V}, |{\cal U}|\neq\infty}
{|\partial \mathcal{U}|\over
\left[\min\left(\left|\mathcal{U}\right|, \left|\overline{\cal
U}\right|\right)\right]^\alpha}.
\end{equation}
For an infinite graph, or a set of finite graphs that includes graphs
of arbitrarily large size, particularly important is the largest
$\alpha$ such that the corresponding $b_\alpha>0$. Such a graph (or
graph family) is called an $\alpha$-expander; when $\alpha<1$, the
related parameter $\delta\equiv (1-\alpha)^{-1}$ is called the
isoperimetric dimension. Isoperimetric dimension of any regular
$D$-dimensional lattice is $\delta=D$. When $\alpha=1$, the
isoperimetric constant $b_1$ of a graph $\mathcal{G}$ is called its
Cheeger constant, $h(\mathcal{G})=b_1$. An infinite graph with a
non-zero Cheeger constant is called non-amenable.
A set of edges $C\subseteq {\cal E}$ is called a cycle if the degree
of each vertex in the subgraph induced by $C$, ${\cal G}'=({\cal V},C)$, is
even. The set of all cycles on a graph ${\cal G}$, with the symmetric
difference defined as
$A \oplus B\equiv (A\setminus B)\cup (B\setminus A)$ used as the group
operation, forms an abelian group, the \emph{cycle group} of
${\cal G}$, denoted $\mathcal{C}({\cal G})$. Clearly, a closed path
is a cycle. A simple cycle is a self-avoiding closed path.
A graph ${\cal H}$ is called a covering graph of ${\cal G}$ if there
is a function $f$ mapping ${\cal V}_{\cal H}$ onto
${\cal V}_{\cal G}$, such that an edge $(u, v)\in{{\cal E}}_{\cal H}$
is mapped to the edge
$\biglb(f(u), f(v)\bigrb)\in{{\cal E}}_{\cal G}$, with an additional
property that $f$ be invertible in the vicinity of each vertex, i.e.,
for a given vertex $u'\in {{\cal V}}_{\cal H}$ and an edge
$(f(u'), v)\in{{\cal E}}_{\cal G}$, there must be a unique edge
$(u', v')\in{{\cal E}}_{\cal H}$ such that $f(v')=v$. As a result,
given a path $P$ connecting vertices $u$ and $v$ on ${\cal G}$ and a
vertex $u'\in {\cal V}_{\cal H}$ such that $f(u')=u$, there is a
unique path $P'$ on ${\cal H}$, the lift of $P$, such that $f$ maps
the sequence of vertices $u_0'\equiv u$, $u_1'$, $u_2'$, \ldots\ in
$P'$ to that in $P$. To simplify the notations, we will in some cases
write a covering map as a map between the graphs,
$f:\mathcal{H}\to \mathcal{G}$.
A set of vertices $u'$ with the same covering map image $u$,
$f(u')=u$, is called the \emph{fiber} of $u$. A lift of a closed path
starting and ending with $u$ is either a closed path, or an open path
connecting two different vertices in the fiber of $u$. We call a
simple cycle on ${\cal G}$ homologically trivial if all its lifts are
simple cycles (of the same length). A cycle on ${\cal G}$ is trivial
if it is a union of edge-disjoint homologically trivial simple cycles.
The set of trivial cycles on ${\cal G}$, with ``$\oplus$'' used for
group operation, is a subgroup of the cycle group on ${\cal G}$. We
denote such a group $\mathcal{C}_0({\cal H};f)$. The corresponding
group quotient,
$H_1(f)\equiv\mathcal{C}(\mathcal{G})/\mathcal{C}_0(\mathcal{H};f)$,
is the (first) homology group associated with the map $f$; its
elements are equivalence classes formed by sets of cycles whose
elements differ by an addition of a trivial cycle. Namely, cycles $C$
and $C'$ are equivalent, $C'\simeq C$, if $C'=C\oplus C_0$, with
$C_0\in\mathcal{C}_0(\mathcal{H};f)$.
The cycle space of a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$
with $n=|\mathcal{E}|$ edges can be defined algebraically in terms of
the vertex-edge incidence matrix $J\equiv J_{\cal G}$. Namely, it
is isomorphic to the binary code
$\mathcal{C}_{J}^\perp\subset\mathbb{F}_2^n$ whose parity check matrix
is the incidence matrix $J$,
$\mathcal{C}(\mathcal{G})\cong\mathcal{C}_J^\perp$. On the other
hand, the code $\mathcal{C}_J$ generated by the incidence matrix is
isomorphic to the cut space of the graph. Elements of the cut space
are edge boundaries $\partial \mathcal{U}$ of different partitions
defined by sets of vertices $\mathcal{U}\subset\mathcal{V}$.
In principle, any set $\mathcal{C}'\subset \mathcal{C}(\mathcal{G})$
of cycles on $\mathcal{G}$ can be used to construct a binary matrix
$K$ with the rows orthogonal to $J$, $JK^T=0$; the code
$\mathcal{C}_K\subset\mathbb{F}_2^n$ is isomorphic to the subspace of
the cycle space generated by elements of $\mathcal{C}'$. In
particular, given the covering map $f:\mathcal{H}\to \mathcal{G}$,
such a matrix $K$ can be constructed using a basis set of
homologically trivial cycles $\mathcal{C}_0(\mathcal{H};f)$. Thus,
such a covering map has a chain complex (\ref{eq:3-chain}) associated
with it, where $\mathcal{A}_0$, $\mathcal{A}_1$, and $\mathcal{A}_2$
are spaces generated by sets of vertices, edges, and homologically
trivial cycles, respectively. In particular, the support $\supp(a)$
of any vector $a\in{\cal A}_1$ corresponds to a set of edges. The
boundary operators are given by the constructed matrices
$\partial_1=J$, $\partial_2=K^T$. Equivalently, the same matrices can
be used to define a stabilizer code $\css(J,K)$ with generators
$G_X=J$ and $G_Z=K$. We will denote such a \emph{quantum cycle code}
associated with the covering map $f:{\cal H}\to {\cal G}$ as
$\mathcal{Q}(\mathcal{H};f)$. The length of the code is
$n=|\mathcal{E}|$, the number of encoded qubits
$k=\rank H_1(\mathcal{A})$ is the rank of the first homology group
associated with covering map, and the distances $d_Z$, $d_X$,
respectively, are the homological distances $d_1(\mathcal{A})$,
$d_1(\widetilde{\cal A})$ associated with the chain complex
$\mathcal{A}$ and the co-chain complex $\widetilde{\cal A}$.
Given a graph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ and a subgroup
$\Gamma$ of its automorphism group $\aut(\mathcal{H})$, consider the
partition of $\mathcal{V}$ induced by $\Gamma$, where a pair of
vertices $u$, $v$ are in the same class iff there is an element
$g\in\Gamma$ such that $g(u)=v$. Then, the \emph{quotient graph}
$\mathcal{G}=\mathcal{H}/\Gamma$ has the vertex set given by the set
of vertex classes, with an edge between any two classes which contain
a pair of vertices connected by an edge from $\mathcal{E}$. A graph
quotient $\mathcal{H}/\Gamma$ is covered by $\mathcal{H}$ if no
neighboring vertices fall into the same class. When $\mathcal{H}$ is
infinite, a finite quotient graph $\mathcal{H}/\Gamma$ is obtained
if the subgroup $\Gamma$ has a finite index; in such a case
$\mathcal{H}$ must be quasitransitive.
\subsection{Percolation transitions}
\label{sec:percolation}
We only consider Bernoulli edge percolation, where each edge
$e\in {\cal E}$ of a graph ${\cal H}=({\cal V},{\cal E})$ is
independently labeled as open or closed, with probabilities $p$ and
$1-p$, respectively. We are focusing on the subgraph $[{\cal H}]_p$
remaining after removal of all closed edges; connected components of
$[{\cal H}]_p$ are called clusters. For a given $v\in {\cal V}$, the
cluster which contains $v$ is denoted
$\mathcal{K}_v\subseteq [{\cal H}]_p$. If $\mathcal{K}_v$ is
infinite, for some $v$, we say that percolation occurs.
Three observables are usually associated with percolation: the
probability that vertex $v$ is in an infinite cluster
\begin{equation}
\theta_v\equiv
\theta_v({\cal H},p)=\Pp(|\mathcal{K}_v|=\infty),
\label{eq:theta-v}
\end{equation}
the
connectivity function,
\begin{equation}
\label{eq:connectivity}
\tau_{u,v}\equiv
\tau_{u,v}({\cal H},p)=\Pp\bigl(u\in{\cal K}_v\bigr),
\end{equation}
the probability that vertices $u$ and $v$ are in the same cluster, and
the local cluster susceptibility,
\begin{equation}
\chi_v\equiv
\chi_v({\cal H},p)=\Ep(|\mathcal{K}_v|),
\label{eq:chi-v}
\end{equation}
the expected size of the cluster connected to $v$. Equivalently,
cluster susceptibility can be defined as the sum of probabilities for
individual vertices to be in the same cluster as $v$, i.e., as a sum of
connectivities,
\begin{equation}
\label{eq:chi-v-alt}
\chi_v=\sum_{u\in{\cal V}}\tau_{v,u}.
\end{equation}
The critical propability $p_{\rm c}$, the percolation threshold, is
associated with the formation of an infinite cluster. There is no
percolation, $\theta_v=0$, for $p<p_{\rm c}$, but $\theta_v > 0$ for
$p>p_{\rm c}$. An equivalent definition is based on the existence of an
infinite cluster anywhere on $[{\cal H}]_p$: the probability of
finding such a cluster is zero at $p<p_{\rm c}$, and one at $p>p_{\rm c}$, see,
e.g., Theorem (1.11) in Ref.~\onlinecite{Grimmett-percolation-book}
(the same proof works for any infinite connected graph).
Similarly, the critical probability $p_T$ is associated with
divergence of site susceptibilities: $\chi_v$ is finite for $p<p_T$
but not for $p>p_T$. Again, in a connected graph, this definition does not
depend on the choice of $v\in{\cal V}$. If percolation occurs (i.e.,
with probability $\theta_v>0$, $|{\cal K}_v|=\infty$), then clearly
$\chi_v=\infty$. This implies $p_{\rm c}\ge p_T$. The reverse is known to
be true for percolation on quasi-transitive
graphs\cite{Menshikov-1986,Menshikov-Sidorenko-eng-1987}:
$\chi_v=\infty$ can only happen inside or on the boundary of the
percolation phase. Thus, for a quasi-transitive graph, $p_{\rm c}=p_T$.
An important question is the number of infinite clusters on
$[{\cal H}]_p$, in particular, whether an infinite cluster is unique.
For infinite quasi-transitive graphs, there are only three
possibilities: (\textbf{a}) almost surely there are no infinite
clusters; (\textbf{b}) there are infinitely many infinite clusters;
and (\textbf{c}) there is only one infinite
cluster\cite{Benjamini-Schramm-1996
Haggstrom-Jonasson-2006,Hofstad-2010}. A third critical
probability, $p_{\rm u}$, is associated with the number of infinite
clusters. Most generally, we expect $p_T\le p_{\rm c}\le p_{\rm u}$. For a
quasi-transitive graph, one has\cite{Hofstad-2010}
\begin{equation}
\label{eq:thresholds}
0<p_T=p_{\rm c}\le p_{\rm u}.
\end{equation}
Here, $p_{\rm u}$ is the uniqueness threshold, such that there can be only
one infinite cluster for $p>p_{\rm u}$, whereas for $p<p_{\rm u}$, the number of
infinite clusters may be zero, or infinite. For an amenable
quasitransitive graph, $p_{\rm c}=p_{\rm u}$
\cite{Aizenman-Kesten-Newman-1987,Burton-1989,Kesten-2002}; it was
conjectured by Benjamini and Schramm\cite{Benjamini-Schramm-1996} that
$p_{\rm c}<p_{\rm u}$ for non-amenable quasi-transitive graphs. Among other
examples, the conjecture has been recently verified for a large class
of Gromov-hyperbolic graphs\cite{Hutchcroft-hyp-2019}.
In order for the uniqueness threshold to be non-trivial, $p_{\rm u}<1$, the
graph $\mathcal{H}$ has to have only one end. That is, it can not be
separated into two or more infinite components by removing a finite
number of edges.
In addition to uniqueness of the infinite cluster, the same threshold
$p_{\rm u}$ can be characterized in terms of the connectivity
function\cite{Tang-2019}. Namely,
$\inf_{u,v\in \mathcal{V}}\tau_{u,v}(p)>0$ for $p>p_{\rm u}$ and it is
zero for $p<p_{\rm u}$. Further, for planar transitive graphs, the
uniqueness threshold is related to the percolation threshold on the
dual graph,
\begin{equation}
p_{\rm u}(\mathcal{H})=1-p_{\rm c}(\widetilde{\cal H}),
\label{eq:planar-duality}
\end{equation}
see the proof of Theorem 7.1 in
Ref.~\onlinecite{Haggstrom-Jonasson-2006}. In the case of planar
amenable graphs where $p_{\rm c}(\mathcal{H})=p_{\rm u}(\mathcal{H})$,
the duality (\ref{eq:planar-duality}) is between the two percolation
transitions\cite{Grimmett-percolation-book}.
\section{Homology-changing transitions}
\label{sec:analytical}
\subsection{Weakly converging sequences of graphs with a common cover}
Consider a finite graph
$\mathcal{G}=(\mathcal{V}_{\cal G},\mathcal{E}_{\cal G})$ covered by
an infinite graph $\mathcal{H}=(\mathcal{V},\mathcal{E})$. While the
graph $\mathcal{H}$ needs not be quasi-transitive, the set of vertex
degrees of $\mathcal{H}$ is finite and matches that of $\mathcal{G}$;
in particular, the two graphs have the same maximal degree
$\Delta_{\rm max}$. The covering map
$f:\mathcal{V}\to \mathcal{V}_{\cal G}$ also defines a quantum cycle
code $\mathcal{Q}(\mathcal{H};f)$ with parameters
$[[n,k,(d_{X},d_{Z})]]$, where $n=|\mathcal{E}_{\cal G}|$ the number
of edges in $\mathcal{G}$, and $k=\rank H_1(f)$ the dimension of the
first homology group associated with the map $f$. We are particularly
interested in the case where the graphs $\mathcal{G}$ and
$\mathcal{H}$ look identically on some scale. Formally, this is
formulated in terms of the injectivity radius, defined as the largest
integer $r_f$ such that the map $f$ is one-to-one in any ball
$\mathcal{B}(v,r_f;\mathcal{H})$. Necessarily, for any covering map
$f$, the injectivity radius $r_f\ge1$. We start by giving lower
bounds for the distances $d_X$, $d_Z$ in terms of the injectivity
radius.
First, an injectivity radius $r_f$ implies that no two vertices
located at distance $r_f$ or smaller from any vertex on $\mathcal{H}$
map to the same vertex on $\mathcal{G}$. On the other hand, any simple cycle
$C\subset \mathcal{G}$ of length $\ell$ is for sure covered by a ball
of radius $r=\lceil \ell/2\rceil$ centered on a vertex in $C$. This
gives (formal proofs are given in the Appendix):
\begin{restatable}{lemma}{dZbnd}
\label{th:dZ-bnd}
Consider a finite graph $\mathcal{G}$ covered by an infinite graph
$\mathcal{H}$, with the injectivity radius $r_f$. Then the minimum
weight $d_Z$ of a non-trivial cycle on $\mathcal{G}$ satisfies the
inequality $ 2r_f+1\le d_Z\le 2r_f+3$.
\end{restatable}
Second, the minimum distance $d_{X}$ is the minimum size of a
homologically non-trivial \emph{co-cycle}, a set of edges on
$\mathcal{G}$ which has even overlap with any homologically trivial
cycle, but is not a cut of $\mathcal{G}$. A lower bound for $d_{X}$
requires some additional assumptions:
\begin{restatable}{lemma}{dXbnd}
\label{th:dX-bnd}
Consider a finite graph $\mathcal{G}$ covered by an infinite
one-ended graph $\mathcal{H}$, with the injectivity radius $r_f$.
Assume that the cycle group of $\mathcal{H}$ can be generated by
cycles of weight not exceeding $\omega\ge3$. Then, the minimum
weight of a non-trivial co-cycle on ${\cal G}$ satisfies the
inequality $d_X> r_f/\omega$.
\end{restatable}
In addition, it will be important that for any covering map
$f:\mathcal{H}\to \mathcal{G}$, the vertices of $\mathcal{G}$ can be
lifted in such a way that they induce a connected subgraph of
$\mathcal{H}$, just as a square-lattice torus with periodic boundary
conditions becomes a rectangular piece of the square lattice after
cutting two rows of edges.
\begin{restatable}{lemma}{FlatSub}
\label{th:flat-sub}
Let $\mathcal{G}$ be a finite connected graph, $\mathcal{H}$ its
cover with the covering map
$f:\mathcal{V}\to \mathcal{V}_{\cal G}$ and the
injectivity radius $r_f$. For any $v'\in\mathcal{V}$ let
$v\equiv f(v')\in\mathcal{V}_{\cal G}$ be its image. Then there
exists a set of vertices $\mathcal{V}_f\subset \mathcal{V}$
which contains a unique representative from the fiber of every
vertex of $\mathcal{V}_{\cal G}$, such that the subgraph
$\mathcal{H}_f\subset \mathcal{H}$ induced by $\mathcal{V}_f$ be
connected and contains the ball $\mathcal{B}(v',r_f;\mathcal{H})$.
\end{restatable}
In the following, we consider not a single graph $\mathcal{G}$, but a
sequence $(\mathcal{G}_t)_{t\in\mathbb{N}}$ of finite graphs
$\mathcal{G}_t=(\mathcal{V}_t,\mathcal{E}_t)$ sharing an infinite
connected covering graph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, with
the covering maps $f_t:\mathcal{V}\to \mathcal{V}_t$. If the
corresponding sequence of injectivity radii $r_t\equiv r_{f_t}$
diverges, we say that the sequence $({\cal G}_t)_t$ weakly converges
to $\mathcal{H}$. Such a convergent sequence can be constructed, e.g., as a
sequence of finite quotients of the graph $\mathcal{H}$ with respect
to a sequence of subgroups of its symmetry group, which requires
$\mathcal{H}$ to be quasitransitive. We do not know whether
quasitransitivity of $\mathcal{H}$ is necessary to have a sequence of
finite graphs covered by $\mathcal{H}$ and weakly convergent to
$\mathcal{H}$. By this reason, in the following, we specify
(quasi)transitivity only when necessary for the corresponding proof.
Given a graph sequence with a common covering graph ${\cal H}$, we use
$\mathcal{Q}_t$ to denote the CSS code with parameters
$[[n_t,k_t,(d_{Xt},d_{Zt})]]$ associated with the covering map $f_t$.
We also denote the ``flattened'' subgraphs from Lemma
\ref{th:flat-sub} as
${\cal H}_t\equiv \mathcal{H}_{f_t}\subset \mathcal{H}$. When the
sequence $(r_t)_t$ diverges, we can always construct a subsequence
$(t_s)_s$, $t_{s+1}>t_s$, such that the corresponding sequence of
graphs $(\mathcal{H}_{t_s})_s$ be increasing,
$\mathcal{H}_{t_{s+1}}\subsetneq \mathcal{H}_{t_s}$. To this end, it
is sufficient to take $r_{t_{s+1}}>n_{t_s}$, regardless of the
particular spanning trees used in the construction of the graphs
$\mathcal{H}_t$.
\subsection{Homology erasure thresholds}
Coming back to percolation, let $H_1(f_t,p)$ denote the first homology
group formed by classes of homologically non-trivial cycles on the
open subgraph $[{\cal G}_t]_p$. We will consider several observables
that quantify the changes in homology in the open subgraphs at large
$t$ as the probability $p$ is increased. The first two, defined by
analogy with corresponding quantities for $1$-cycle proliferation in
continuum percolation\cite{Bobrowski-Skraba-2020}, are designed to
detect any changes in homology compared to the empty graphs at $p=0$,
and the graphs with all edges present at $p=1$. Respectively, we
define the probability that a homologically non-trivial cycle exists
in the open subgraph,
\begin{equation}
\label{eq:prob-E}
{\bf P}_E(t,p)\equiv \Pp\biglb(\rank H_1(f_t,p)\neq 0\bigrb),
\end{equation}
and the probability that not all homologically non-trivial cycles are
covered in the open subgraph,
\begin{equation}
\label{eq:prob-A}
{\bf P}_A(t,p)\equiv \Pp\biglb(\rank H_1(f_t,p)\neq k_t\bigrb).
\end{equation}
Equivalently, ${\bf P}_A(t,p)$ is the probability that the open
subgraph at $\bar p=1-p$ covers a homologically non-trivial co-cycle.
In terms of the associated CSS code $\mathcal{Q}_t$, ${\bf P}_E(t,p)$
and ${\bf P}_A(t,1-p)$ are the erasure probabilities for a $Z$- and an
$X$-type codeword, respectively. These quantities do not necessarily
characterize bulk phase(s), as they may be sensitive to the state of a
sublinear number of edges.
As $p$ is increasing from $0$ to $1$, ${\bf P}_E(t,p)$ is monotonously
increasing from $0$ to $1$ while ${\bf P}_A(t,p)$ is monotonously
decreasing from $1$ to $0$. Thus, a version of the subsequence
construction can be used to ensure the existence of their $t\to\infty$
limits almost everywhere on the interval $p\in[0,1]$. Instead, we
define the (lower) cycle erasure threshold for any given graph
sequence,
\begin{equation}
\label{eq:pE0}
p_E^0=\sup\left\{p\in [0,1]: \lim_{t\to\infty}{\bf P}_E(t,p)=0\right\}.
\end{equation}
Because of monotonicity of
${\bf P}_E(t,p)$ as a function of $p$, a zero limit at some $p=p_0>0$
ensures the limit exists and remains the same everywhere on the
interval $p\in [0,p_0]$. Further, the absence of convergence of the
sequence ${\bf P}_E(t,p)$ at some $p=p_1$ implies that the superior
and the inferior limits at $t\to\infty$ must be different, which, in
turn, implies the existence of a subsequence convergent to the
non-zero limit given by $\limsup_{t\to\infty}{\bf P}_E(t,p_1)>0$.
Similarly, we define the upper cycle erasure threshold
\begin{equation}
\label{eq:pE1}
p_E^1=\inf\left\{p\in [0,1]: \lim_{t\to\infty}{\bf P}_A(t,p)=0\right\},
\end{equation}
as the smallest $p$ such that open subgraphs preserve the full-rank
homology group with probability approaching one in the limit of the
sequence.
Existence of a homologically non-trivial cycle not covered by open
edges implies that closed edges must cover a conjugate codeword, a
non-trivial co-cycle. The related threshold on an infinite graph can
be interpreted in terms of a transition dual to percolation,
proliferation of the boundaries at the complementary edge
configuration, with all closed edges replaced by open edges, and v.v.,
so that the open edge probability becomes $\bar p=1-p$. On a locally
planar graph, like a tiling of a two-dimensional manifold, the dual
transition maps to the usual percolation on the dual graph.
We also notice that the usual erasure threshold $p_E$ for a family (or
a sequence) of quantum codes corresponds to a non-zero probability of
an \emph{erasure}, a configuration where a codeword is covered by
erased qubits. For a CSS code, this implies a non-zero probability
that either an $X$- or a $Z$-type codeword be covered. For codes
${\cal Q}_t$ associated with covering maps $f_t:{\cal H}\to{\cal G}_t$ in
the sequence $(\mathcal{G}_t)_{t\in\mathbb{N}}$, the conventional
erasure threshold can be found in terms of the thresholds for cycles
and co-cycles,
\begin{equation}
\label{eq:pE}
p_E=\min(p_E^0,1-p_E^1).
\end{equation}
The following lower bound constructed using a Peierls-style counting
argument is adapted from
Ref.~\onlinecite{Dumer-Kovalev-Pryadko-bnd-2015}:
\begin{restatable}{statement}{thPeierlsOne}
\label{th:peierls-ineq}
Consider a sequence of finite graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ with a common covering graph
$\mathcal{H}$. Let $\Delta_{\rm max}$ be the maximum degree of
$\mathcal{H}$, and assume that for some $t_0>0$, the injectivity
radius $r_t$ associated with the maps
$f_t:\mathcal{H}\to\mathcal{G}_t$ at $t\ge t_0$ scales at least
logarithmically with the number of edges $n_t$, $r_t\ge A\ln n_t$,
with some $A>0$. The cycle erasure threshold for the corresponding
sequence of CSS codes $(\mathcal{Q}_t)_{t\in\mathbb{N}}$ satisfies
the lower bound $p_E^0\ge e^{-1/(2 A)}/(\Delta_{\rm max}-1)$.
\end{restatable}
It follows from the fact that $\mathcal{Q}_t=\css(J_t,K_t)$, where
$J_t$ is the vertex-edge incidence matrix of $\mathcal{G}_t$, with row
weights given by the vertex degrees, and Lemma \ref{th:dZ-bnd}.
We would like to ensure that the conventional erasure threshold
(\ref{eq:pE}) also be non-trivial, which requires that $p_E^1<1$. To
construct such an upper bound, which becomes a lower bound in terms of
$\bar p=1-p$ in the dual representation, it is
sufficient\cite{Dumer-Kovalev-Pryadko-bnd-2015} that rows of the
trivial-cycle--edge adjacency matrix $K_t$ have bounded weights, and
that the distance $d_{Xt}$ diverges logarithmically or faster with
$n_t$. Notice that here we do not rely on Lemma \ref{th:dX-bnd} which
gives a rather weak lower bound for the distance but, instead,
directly assume desired scaling of the minimum weight $ d_{Xt}$ of a
non-trivial co-cycle with $n_t$. We have
\begin{restatable}{statement}{thPeierlsTwo}
\label{th:peierls-ineq-two}
Consider a sequence of finite graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ with a common covering graph
$\mathcal{H}$, with the cycle group ${\cal C}(\mathcal{H})$
generated by cycles of weight not exceeding $\omega>1$. Further,
assume that the minimum weight $d_{Xt}\equiv d_X({\cal H};f_t)$ of a
non-trivial co-cycle associated with the map
$f_t:\mathcal{H}\to\mathcal{G}_t$ grows at least logarithmically
with the number of edges $n_t$, $d_{Xt}\ge A'\ln n_t$, for
sufficiently large $t\ge t_0'$ and some $A'>0$. The upper erasure
threshold for the corresponding sequence of CSS codes
$(\mathcal{Q}_t)_{t\in\mathbb{N}}$ satisfies the bound
$1-p_E^1\ge e^{-1/A'}/(\omega-1)$.
\end{restatable}
Let us now relate the cycle erasure threshold $p_E^0$ with the bulk
percolation threshold. Most generally, it serves as an upper bound:
\begin{restatable}{theorem}{thErasurePercol}
\label{th:pE-easy}
Consider a sequence of finite graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ covered by an infinite graph
$\mathcal{H}$. Then,
$p_E^0\le p_{\rm c}(\mathcal{H})$.
\end{restatable}
This includes the case where the sequence of the injectivity radii
remains bounded (no weak convergence to ${\cal H})$, in which case,
obviously, $p_E^0=0$. More precise results for $p_E^0$ are available
with additional assumptions, including the scaling of the injectivity
radius with the logarithm of the graph size:
\begin{restatable}{theorem}{pElogA}
\label{th:pE-logA}
Consider a sequence of finite transitive graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ covered by an infinite graph
$\mathcal{H}$. If the homological distance $d_{Zt}$ scales
sublogarithmically with graph size,
$\displaystyle\lim_{t\to\infty}{d_{Zt}\over\ln n_{t}\strut}=0$, then
$p_E^0=0$.
\end{restatable}
\begin{restatable}{theorem}{pElogB}
\label{th:pE-logB}
Consider a sequence of finite graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ covered by an infinite
quasi-transitive graph $\mathcal{H}$. If the injectivity radius
scales superlogarithmically with the graph size,
$\displaystyle\lim_{t\to\infty}{r_t\over\ln n_t}=\infty$, then
$p_E^0=p_{\rm c}$.
\end{restatable}
Information about the other threshold,
$p_E^1$, can be obtained in the planar case with the help of duality:
\begin{restatable}{corollary}{pEplanar}
\label{th:pE-planar}
Let ${\cal H}$ and $\widetilde{\cal H}$ be a pair of mutually dual
infinite quasitransitive planar graphs. Consider a sequence of
finite graphs $({\cal G}_t)_{t\in\mathbb{N}}$ weakly convergent to
${\cal H}$, a cover of the graphs in the sequence. Then,
\begin{enumerate}[label={\em(\textbf{\roman*})}]
\item
$p_E^1\ge 1-p_{\rm c}(\widetilde{\cal
H})$. In addition, \smallskip
\item if the graphs $\mathcal{G}_t$ in the sequence are transitive,
$t\in\mathbb{N}$, and the injectivity radius grows
sublogarithmically with the graph size, then $p_E^1=1$;
\item if the injectivity radius grows superlogarithmically, then
$p_E^1=1-p_{\rm c}(\widetilde{\cal H})$.
\end{enumerate}
\end{restatable}
Notice that for a superlogarithmic scaling of the injectivity radius,
the graph must be amenable, in which case
$p_{\rm u}(\mathcal{H})=p_{\rm c}(\mathcal{H})$. We also believe that
under conditions of the Corollary, the duality gives
$p_{\rm u}(\mathcal{H})=1-p_{\rm c}(\widetilde{\cal H})$, see
Eq.~(\ref{eq:planar-duality}), although we only found the proof for
the case where the graph $\mathcal{H}$ is
transitive\cite{Haggstrom-Jonasson-2006}. Whenever such a duality
relation holds, the upper cycle erasure threshold is bounded below by
the uniqueness threshold, $p_E^1\ge p_{\rm u}(\mathcal{H})$; with
superlogarithmic scaling of the injectivity radius, the sequence of
thresholds collapses to a single point,
$p_E^0=p_E^1=p_{\rm c}(\mathcal{H})=p_{\rm u}(\mathcal{H})$.
These results leave out an important case of percolation with
logarithmic distance scaling. It is easy to see that logarithmic
distance scaling does not necessarily imply that
$p_E^0$ and $p_{\rm c}(\mathcal{H})$ be equal:
\begin{example}[Anisotropic square-lattice toric codes]
\label{ex:anisotropic-square}
Consider a sequence of tori
$\mathcal{G}_t=\mathcal{T}_{L_x(t),L_y(t)}$ obtained from the
infinite square lattice
$\mathcal{H}$ by identifying the vertices at distances
$L_x(t)$ and $L_y(t)$ along the edges in $x$ and
$y$ directions, respectively. For some
$A>0$, consider the scaling $L_x(t)=t$,
$L_y(t)=e^{t/A}/(2t)$. This gives $d_{Zt}=t$ and
$n_t=e^{t/A}$, so that $d_{Zt}=A\ln
n_t$. The cycle erasure threshold
$p_E^0$ for this graph sequence satisfies $e^{-1/A}/3<p_E^0\le
e^{-1/A}$.
\end{example}
The upper bound follows from considering
$L_y(t)$ independent non-trivial cycles of length
$t$, while the lower bound is given by Statement
\ref{th:peierls-ineq}. In comparison, for edge percolation on
infinite square lattice, $p_{\rm c}=1/2$.
In addition to Example \ref{ex:anisotropic-square}, in
Sec.~\ref{sec:numeric} we give numerical evidence that
$p_E^0<p_{\rm c}(\mathcal{H})$ for several families of hyperbolic codes based
on regular $\{f,d\}$ tilings of the hyperbolic plane (here $2df>d+f$;
these are known to have a finite asymptotic rate $R=1-2/d-2/f$).
\subsection{Erasure rate thresholds}
Logarithmic scaling of the minimum distance $d_{Zt}$ associated with
the first homology group is the largest one may hope for in the
important case when the covering graph $\mathcal{H}$ is
non-amenable. We specifically focus on the case of a graph sequence
with \emph{extensive homology rank} scaling, i.e., where the
associated codes have an asymptotically finite rate,
$R\equiv \lim_{t\to\infty}k_t/n_t>0$. For such graph sequences, we
also consider the expected dimension of the erased subspace per edge,
or the \emph{erasure rate},
\begin{equation}
\label{eq:mean-erased}
{\bf R}_E(t,p)\equiv
n_t^{-1}\,
\Ep\biglb(\rank H_1(f_t,p)\bigrb).
\end{equation}
Analogous quantity was analyzed in detail by Delfosse and
Z{\'e}mor\cite{Delfosse-Zemor-2012}. Unlike the probabilities
${\bf P}_E$ and ${\bf P}_A$, the erasure rate ${\bf R}_E$ is a bulk
quantity which can be used to define a thermodynamical transition in
the usual sense. For any $t\in\mathbb{N}$, the erasure rate
${\bf R}_E(t,p)$ is a monotonously increasing function of
$p\in [0,1]$, bounded by the values at the ends of the interval,
\begin{equation}
0\le {\bf R}_E(t,p)\le R_t\equiv k_t/n_t\le 1.
\label{eq:bounds-RE}
\end{equation}
Let us now consider the thresholds associated with the erasure rate
(\ref{eq:mean-erased}). We define the lower $p_H^0$ and the upper
$p_H^1$ critical points as the values of $p$ where ${\bf R}_E(t,p)$ in
the limit of large $t$ starts to deviate from $0$ and from $R$,
respectively:
\begin{eqnarray}
\label{eq:pH0-def}
p_H^0&=&\sup\{p\in[0,1]: \lim_{t\to\infty}{\bf R}_E(t,p)=0\},\\
\label{eq:pH1-def}
p_H^1&=&\inf\{p\in[0,1]: \lim_{t\to\infty}{\bf R}_E(t,p)=R\}.
\end{eqnarray}
We call these, respectively, the lower and the upper {\em
homological\/} thresholds. Evidently,
$p_E^0\le p_H^0\le p_H^1\le p_E^1$. The critical point
$p_H^0$ was discussed in
Refs.~\onlinecite{Delfosse-Zemor-2010,Delfosse-Zemor-2012}. Our first
result, an analogue of the corresponding inequality for the Ising
model, Eq.~(34) in Ref.~\onlinecite{Jiang-Kovalev-Dumer-Pryadko-2018},
gives a lower bound on the gap between the two homological
thresholds:
\begin{restatable}{theorem}{pHdiff}
\label{th:pH-diff}
Consider a sequence of finite graphs
$(\mathcal{G}_t)_{t\in\mathbb{N}}$ weakly convergent to an infinite
graph $\mathcal{H}$, a cover of the graphs in the sequence, with
rate-$R$ extensive homology rank. Then there is a finite gap between the two homological
thresholds,
\begin{equation}
\label{eq:pH-diff}
p_H^1-p_H^0\ge R.
\end{equation}
\end{restatable}
Second, we prove an ``easy'' inequality relating the lower homological
threshold with the percolation threshold on the covering graph:
\begin{restatable}{theorem}{pHeasybounds}
\label{th:pH-easy-bounds}
For a sequence of finite graphs $(\mathcal{G}_t)_{t\in\mathbb{N}}$
weakly convergent to an infinite graph $\mathcal{H}$, a cover of the
graphs in the sequence with extensive homology rank,
$ p_{\rm c}(\mathcal{H})\le p_H^0 $.
\end{restatable}
The remaining analytical result is obtained with the help of the usual
duality between locally planar graphs, and is therefore limited to
planar graphs $\mathcal{H}$:
\begin{restatable}{theorem}{PHplanar}
\label{th:pH-planar}
Let ${\cal H}$ and $\widetilde{\cal H}$ be a pair of infinite
mutually dual transitive planar graphs. Consider a
sequence of finite graphs $({\cal G}_t)_{t\in\mathbb{N}}$ weakly
convergent to ${\cal H}$, a cover of the graphs in the sequence with
extensive homology rank. Then,
\begin{equation}
({\bf i})\ p_H^0=p_{\rm c}(\mathcal{H}),\quad
({\bf ii})\ p_H^1= 1-p_{\rm c}(\widetilde{\cal H})=p_{\rm u}({\cal H}).
\label{eq:pH-planar-equality}
\end{equation}
\end{restatable}
This is an easy consequence of two previous results: the expression
\cite{Delfosse-Zemor-2012} for the expected homology rate in terms of
the average inverse cluster sizes on the graph and its dual, and the
exponential decay\cite{Antunovic-Veselic-2008,Hermon-Hutchcroft-2019}
of the size of finite clusters away from the percolation point on
transitive
graphs.
Notice that in Theorem \ref{th:pH-planar}, the lower and the higher
homological thresholds, respectively, are actually associated with the
percolation and the uniqueness thresholds on the infinite graph
$\mathcal{H}$. We believe this is not a coincidence, and put forward
\begin{conjecture}
Consider a sequence of finite graphs $({\cal G}_t)_{t\in\mathbb{N}}$
weakly convergent to a quasitransitive infinite graph
${\cal H}$, a cover of the graphs in the sequence with extensive
homology rank. Then,
\begin{equation}
({\bf i})\ p_H^0=p_{\rm c}(\mathcal{H}),\quad
({\bf ii})\ p_H^1=p_{\rm u}({\cal H}).
\label{eq:pH-pc-pu-equality}
\end{equation}
\end{conjecture}
Such a result makes sense, since neither the percolation nor the
uniqueness thresholds can be seen locally, by examining a finite
subgraph of ${\cal H}$. Similarly, the homological transitions
require changes in cycles of length exceeding the injectivity radius,
which diverges without a bound.
\section{Numerical results for locally planar hyperbolic codes}
\label{sec:numeric}
In addition to analytical results, we also evaluated the erasure and
the percolation thresholds numerically for several families of planar
hyperbolic codes, as well as for a planar euclidean family of square
lattice toric codes. Each family corresponds to a particular infinite
graph $\mathcal{H}_{f,d}$, regular tiling of the hyperbolic or
euclidean plane, parameterized by the Schl\"afli symbol $\{f,d\}$,
with $2/d+2/f\le 1$. In such a graph, $d$ identical $f$-gons meet in
each vertex. The finite graphs are
constructed\cite{Siran-2001,Sausset-Tarjus-2007} as finite quotients
of the corresponding graph $\mathcal{H}_{f,d}$ with respect to
subgroups of the symmetry group.
The parameters of the graphs used in the calculations are listed in
Tab.~\ref{tab:graphs}, where $\{f,d\}$ is the Schl\"afli symbol of the
corresponding tiling, $n$ is the number of edges, and $d_Z$ and $d_X$,
respectively, are the distances of the corresponding CSS codes. The
smaller graphs with $n<10^3$ edges are from
N.~P.~Breuckmann\cite{Breuckmann-thesis-2017}. We generated the
remaining graphs with a custom \texttt{GAP}\cite{GAP4} program, which
constructs coset tables of freely presented groups obtained from the
infinite van Dyck group $D(d,f,2)=\langle a,b|a^d,b^f,(ab)^2\rangle$
[here $a$ and $b$ are group generators, while the remaining arguments
are \emph{relators} which correspond to imposed conditions,
$a^d=b^f=(ab)^2=1$] by adding one more relator obtained as a pseudo
random string of generators to obtain a suitable finite group
${\cal D}$, a quotient of the original infinite group $D(d,f,2)$.
Then, the vertices, edges, and faces are enumerated by the right
cosets with respect to the subgroups $\langle a\rangle$,
$\langle ab\rangle$, and $\langle b\rangle$, respectively. The
vertex-edge and face-edge incidence matrices $J$ and $K$ are obtained
from the coset tables. Namely, non-zero matrix elements are in the
positions where the corresponding pair of cosets share an element.
Finally, the distance $d_Z$ of the CSS code $\css(J,K)$ was computed
using the covering set algorithm, which has the advantage of being
extremely fast when distance is
small\cite{Dumer-Kovalev-Pryadko-2014,Dumer-Kovalev-Pryadko-IEEE-2017},
and additionally verified by comparing the number of cycles through a
given vertex on the finite graph $\mathcal{G}$ and on a sufficiently
large subgraph of the infinite covering graph $\mathcal{H}_{f,d}$ (or
the corresponding dual graphs in the case of $d_X$).
\begin{turnpage}
\begin{table*}[hp]
\resizebox{\textheight}{!}{
\begin{tabular}{c|c|ccccccccccccccccccccccccccccccccc
&$n$ & 720 & 864 & 2448 & 6144 & 8640 & 18144 & 18216 & 19584 & 23760
& 24360 & 24576 & 25308 & 25920 & 27360 & 29760 & 31200 & 32256
& 32928 & 35280 & 36288 & 38880 & 40320 & 41040 & 46080 & 46656 \\
\hline $\{4,6\}$& $d_Z$
&\textbf{8} & \grayout{8} & \grayout{8} & \textbf{10} & \grayout{8}
&\grayout{10}&\grayout{10}&\grayout{8}&\grayout{10}&\grayout{10}&
10 & 10 & 10 & 10 & 10 & 10 & 10 &\grayout{8}& 10 & 10 & 10 & 10
&\grayout{8}& 10 &\textbf{12}\\
$\{6,4\}$ & $d_X$
&\grayout{8}&\textbf{10}
&\textbf{12} &\grayout{12}&\textbf{14} & 12 & 14
& 14 & 12 & 14
&\grayout{12}& 14 & 10 &\grayout{12}& 14 & 12 & 14 & 14
&\grayout{10}&\grayout{14}&\grayout{12}& 14 & 10 & 12 & 12
\\ \hline & $n$ &
46800 & 48576 & 50616 & 51888 & 52416 & 58800 & 62400 \\ \cline{1-9} $\{4,6\}$&$d_Z$ & 8 &
10 & 10 & 11 & 10 & 11 & 12\\ $\{6,4\}$&$d_X$ & 12 & 14 & 14 & 14 &\grayout{12}& 15
&\grayout{12}\\ \hline\hline
& $n$ & 504 & 648 & 768 & 864 & 1080 &
1224 & 1944 & 2016 & 2448 & 2592 & 3072 & 3240 & 4032 & 4320 & 5616 &
5832 & 6000 & 6072 & 6144 & 7344 & 14880 & 16848 & 18216 & 25944 &
32256\\ \cline{1-27} $\{3,8\}$&$d_Z$ &\textbf{6}
&\grayout{6}&\grayout{6}&\grayout{6}&\grayout{6}& 6 & 6 &\textbf{8} &
8 & 6 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8
&\textbf{9}&\textbf{10} & 9 & 10\\ $\{8,3\}$&$d_X$
&\grayout{14}& 14 &\textbf{16} & 12 & 16 & 15
&\grayout{12}&\textbf{18}& 16
&\grayout{18}&\grayout{18}&\grayout{18}&\grayout{18}&\grayout{16}& 16
& 18 &\textbf{22}& 20 &\grayout{18}& 18 & 22 & 20 &\textbf{24}
&\grayout{21}&\grayout{24}\\
\hline\hline
& $n$ & 660 & 1800 & 1920 & 3420 & 4860 &
5760 & 7440 & 9600 & 10240 & 11520 & 12180 & 14880 & 17100 & 19200 &
23040 & 29400 & 34440 & 37500 & 38880 & 43200 & 57600 & 58240 & 58800
& 60900 & 61440\\
\cline{1-27}$\{4,5\}$& $d_Z
&\grayout{8}&\textbf{10}
&\grayout{10}&\grayout{10}& \textbf{12} & 10 & 12 & 12 & 12 & 10 & 12
& 12 &\textbf{14} & 12 & 12 & 14 & 14 & 14 & 12 & 14 & 10 & 13 & 14 &
12 & 12\\ $\{5,4\}$& $d_X$
&\textbf{10} &\grayout{10}& \textbf{12}
&\textbf{14} &\grayout{12}& 10 & 14 & 14 & 14 & 10 & 14 & 14
&\textbf{16} & 14 & 15 &\textbf{17} & 16 & 16 & 16 & 16 & 10 & 15
&\textbf{18} & 16 & 12\\
\hline\hline
\cline{1-25}
& $n$ & 900 & 4800 &
9600 & 9720 & 10800 & 11520 & 14400 & 15360 & 17220 & 18750 & 19440 &
19800 & 21600 & 29120 & 29400 & 30450 & 38880 & 40960 & 51330 & 52800
& 56730 & 58240\\ \cline{1-24} $\{5,5\}$&$d_Z,d_X$
& \textbf{8} & \textbf{10} &
10 & 10 & 9 & 8 & 8 & \textbf{11} & 10 & 10 & 10 & 10 & 11 &
\textbf{12} & 10 & 12 & 10 & 12 & 12 & 10 & 11 & 12\\ \hline\hline
& $n$ & 546 & 672 & 4914 & 5376 & 6090 & 17220 & 19866\\ \cline{1-9}
$\{3,7\}$& $d_Z$ &\textbf{7}&\textbf{8} &\textbf{10}&\textbf{12}& 10 & 12 &
12\\$\{7,3\}$& $d_X$ & 14&\textbf{16}&\textbf{24}& 24 &\grayout{20}&\textbf{28}& 26
\end{tabular
}
\caption{Parameters of the hyperbolic graphs used to calculate
critical $p$ values. Given a Schl\"afli symbol
$\{f,d\}$, finite graphs
$\mathcal{G}$ and their dual graphs $\widetilde{\cal
G}$ are parameterized by the number of edges
$n$; they have $2n/d$ and
$2n/f$ vertices, respectively, and the first homology groups of
rank $k=2+(1-2/d-2/f)n$. Distances $d_Z$ and
$d_X$ are the lengths of the shortest homologically non-trivial
cycles on $\mathcal{G}$ and $\widetilde{\cal
G}$, respectively. Numbers in bold indicate the smallest graph
found with such a distance; only such graphs were used for
calculating the cycle erasure threshold
$p_E^0$, see Figs.~\ref{fig:pE-hyp55} and ~\ref{fig:pE-hyp73}.
Percolation transition critical point $p_{\rm
c}$ on the infinite hyperbolic graph $\mathcal{H}_{f,d}$ (same
as the giant cluster transition) was calculated using all graphs
in the corresponding family, with the exception of graphs whose
distances are shown in gray.}
\label{tab:graphs}
\end{table*}
\end{turnpage}
To analyze percolation, we used a version of the Newman--Ziff
(NZ) Monte Carlo algorithm\cite{Newman-Ziff-alg-2001}. The original
version of the algorithm simultaneously draws from a sequence of
canonical ensembles with $x=1$, $2$, $\ldots$ open edges, by starting
with all closed edges and randomly adding one open edge at a time,
with the acceleration due to a lower cost of statistics update. To
find the rank $k'$ of the first homology group associated with the
open subgraph, we used the formula
\begin{equation}
\label{eq:k-K-tK}
k'=x-|\mathcal{V}|+|\mathcal{K}'|-|\overline{\mathcal{K}}'|+1,
\end{equation}
where $x\equiv |\mathcal{E}'|$ is the number of open edges, and
$|\mathcal{K}'|$ and $|\overline{\cal K}'|$, respectively, are the
numbers of connected components in the open subgraph of $\mathcal{G}$
and in the closed subgraph of the corresponding dual graph
$\widetilde{\cal G}$. Eq.~(\ref{eq:k-K-tK}) is a consequence of
Eq.~(\ref{eq:rank-H1-restricted-two}). It can also be derived with
the help of the cycle rank Euler formula and the fact that any trivial
open cycle is a cut for the corresponding dual graph. Respectively,
in our version of the NZ algorithm, we simultaneously evolve a pair of
dual subgraphs, starting with all closed edges on $\mathcal{G}$ and
all open edges on $\widetilde{\cal G}$, and adding an open edge to
$\mathcal{G}'$ and removing the corresponding open edge from
$\widetilde{\cal G}'$ at each step. In addition, for each set of
average quantities $A_x$ computed in the canonical ensemble with
$x\in\{0,\ldots,n\}$ edges open, we calculated the corresponding
grand-canonical quantity
\begin{equation}
\label{eq:grand-canonical}
A_p=\sum_{x=0}^{n}{n\choose x }p^x (1-p)^{n-x}A_x.
\end{equation}
For the sake of numerical efficiency, we restricted the summation to
terms with $|x-p n|<M\sqrt{ n p(1-p)} $, with $M=10^2$. We verified
that the results do not change when $M$ is increased by a factor of
two. For each graph, we run $10^6$ Newman--Ziff sweeps and saved the
grand-canonical averages of observables for $10^3$ values of $p$ with
intervals of $\Delta p=10^{-3}$. In addition, to get an independent
estimate of the errors, each threshold calculation was repeated three
times.
A de-facto standard way for estimating the erasure threshold $p_E^0$
is the crossing point method. The method is based on the expectation
that the block error probability is asymptotically zero for any
$p<p_E^0$ and is equal to one for $p>p_E^0$, with the crossover region
small for large codes. Respectively, when the erasure probability
found numerically for several graphs is plotted as a function of $p$,
the corresponding lines are expected to cross in a single point, which
is identified as the pseudothreshold.
This works well for codes with power-law distance scaling. An example
is shown in Fig.~\ref{fig:PE-4-4}, where the homological error
probability (\ref{eq:prob-E}) for several square lattice toric codes
with parameters $[[2d^2,2,d]]$ and $d$ ranging from $60$ to $220$ is
plotted as a function of the open edge probability $p$. Visually, a
beautiful crossing point close to $p_E^0=1/2$ is observed. To find
the corresponding erasure pseudothreshold, the data was fitted
collectively with polynomials of $\xi\equiv p-p^0$. The polynomials
had different coefficients for different graphs, except the zeroth
order coefficient used to find the ordinate of the crossing point.
With the fit range $0.49 \le p\le 0.51$, the degree of the polynomials
was adjusted by hand to minimize the standard deviation of $p^0$, the
abscissa of the crossing point extracted from the data. For the
square-lattice graphs, using 6th degree polynomials, we obtained
$p^0(\{4,4\})=0.500004\pm 0.000002$, very close to the square lattice
percolation threshold $p_{\rm c}(\{4,4\})=1/2$, as expected from
Refs.~\onlinecite{Stace-Barrett-Doherty-2009,Fujii-Tokunaga-2012} and
Theorem \ref{th:pE-logB}. The corresponding linear terms $A_{1n}$ (the
derivative at the crossing point) have a power law $A_{1n}=b n^\alpha$
scaling (not shown), with the exponent $\alpha=0.375\pm 0.003$,
consistent with the expectation of a sharp threshold in the
large-$n$ limit.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{PEC44}
\caption{(Color online) Finding the erasure pseudothreshold for
square lattice toric codes. Symbols show the homological error
probability (\ref{eq:prob-E}) evaluated numerically for different
graphs labeled by the number of edges $n=2d^2$, with distances $d$
ranging from $60$ to $220$, plotted as a function of open edge
probability $p$. As expected, beautiful crossing point very close
to $p_E^0=1/2$ is observed. Lines are the polynomials
$f_n(\xi)=A_0+A_{1n}\xi+A_{2n}\xi^2+\ldots$ of $\xi=p-p^0$ of
degree 6 obtained by fitting the data collectively in the range
$0.49\le p\le 0.51$. The vertical dashed line indicates the
square lattice percolation threshold $p_{\rm c}=1/2$. }
\label{fig:PE-4-4}
\end{figure}
We used a similar technique to process the homological error
probability data for hyperbolic graphs. A sample of the corresponding
plots is shown in the top portions of Figs.~\ref{fig:pE-hyp55} and
\ref{fig:pE-hyp73}. These plots have two significant differences with
that in Fig.~\ref{fig:PE-4-4}. First, the crossing points are
significantly below the percolation transitions indicated by the
vertical dashed lines. Second, despite smaller scales, the
convergence near the crossing crossing points does not look as nice.
Empirically, deviations in the position of the curves are associated
with the differences in the ratio $\ln n/d$, cf.~the bounds in
Statements \ref{th:peierls-ineq}, \ref{th:peierls-ineq-two} and
Example \ref{ex:anisotropic-square}. To reduce the corresponding
errors, in the calculation of the erasure thresholds we only used the
``optimal'' graphs, the smallest graphs with the corresponding
distances; such graphs are indicated in Tab.~\ref{tab:graphs} with the
distance shown in bold.
Yet, using only the optimal graphs was not sufficient to completely
eliminate the finite-size variation. Much better crossing points are
obtained by introducing a vertical shift $B\, \ln n/d$, where $B$ is
an additional global fit parameter (see bottom plots in
Figs.~\ref{fig:pE-hyp55} and \ref{fig:pE-hyp73}).
\begin{figure}[htbp]
\centering
~~\,\includegraphics[width=0.6\columnwidth]{PEC55}\\[0.15em]
\includegraphics[width=0.62\columnwidth]{PEy55}
\caption{(Color online) Top: as in Fig.~\ref{fig:PE-4-4} but
for the hyperbolic code family $\{5,5\}$. The green arrow
indicates the position of the crossing point found by the fit; it
is significantly below the percolation threshold for the
corresponding infinite lattice (vertical red dashed line). In
addition, the data for the graph with $n=15\,350$ is shifted
upward, which we associate with a slightly smaller ratio
$d/\ln n$, see Fig.~\ref{fig:dist}. This is verified in the
bottom plot, where an additional vertical shift proportional to
$\ln n/d$ is added, which substantially improves the convergence
at the crossing point.}
\label{fig:pE-hyp55}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{dist-n}
\caption{Homological distance $d\equiv d_Z$ associated with
non-trivial cycles for optimal graphs in $\{7,3\}$ and $\{5,5\}$
families vs.\ the graph size $n$ (number of edges) with the
logarithmic scale. Numbers also indicate the graph sizes.
Smaller relative distances $d/\ln n$ result in larger erasure
probabilities in Figs.~\ref{fig:pE-hyp55} and \ref{fig:pE-hyp73}
(top); this can be compensated to some extend by using the
correction term as in bottom plots in Figs.~\ref{fig:pE-hyp55} and
\ref{fig:pE-hyp73}.}
\label{fig:dist}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{PEC73}\\[0.15em]
\includegraphics[width=0.6\columnwidth]{PEy73}
\caption{(Color online) As in Fig.~\ref{fig:pE-hyp55} but for the
hyperbolic code family $\{7,3\}$. The convergence at the crossing
point is much better than that in Fig.~\ref{fig:pE-hyp55} (top),
which we associate with substantially higher ratios $d/\ln n$ for
the graphs in this family. Bottom plot: addition of the
additional vertical shift $B\ln n/d$ causes a substantial shift of
the crossing point position without visibly improving the
convergence.}
\label{fig:pE-hyp73}
\end{figure}
In comparison, the crossing point method does not work for measuring
the location of the homological transition $p_H^0$, even though the
variation between the graphs is not expected to matter that much here.
Main reason for the difference is that the erasure rate
(\ref{eq:mean-erased}) retains a finite slope in the infinite graph
limit, which makes the crossing point analysis unreliable.
To check for spurious errors, in our simulations we have also measured
the conventional percolation characteristics, in particular, the
average sizes $S_j=\langle {\cal K}_j\rangle $, $j=1,2,3$ of the three
largest clusters. We have used several finite-size scaling techniques
to extract the location of the percolation transition which coincides
with the giant-cluster transition, see Theorem 1.3 in
Ref.~\onlinecite{Benjamini-Nachmias-Peres-2011}. All techniques,
including the cluster-size ratio
technique\cite{Margolina-Herrmann-Stauffer-1982,daSilva-Lyra-Viswanathan-2002},
give transition points in a reasonable agreement with the values
expected from invasion percolation simulations in
Ref.~\onlinecite{Mertens-Moore-2017}. For hyperbolic graphs
$\mathcal{H}_{f,d}$ with $(f+d)/fd<2$, we found that the most accurate
values of $p_{\rm c}$ are found using the technique based on the
expectation of cluster size scaling similar to that for random
graphs\cite{Kozma-Nachmias-2009,Heydenreich-vanderHofstad-book-2017},
$S_j\propto n^{2/3}$ near $p_{\rm c}$, with the critical region of
width $\Delta p\sim n^{-1/3}$. Respectively, when interpolated values
of $p$ such that the expected size of the largest cluster satisfies
$S_1(p)=\omega n^{2/3}$ are plotted for $\omega\in\{1/4,1/2,1\}$ as a
function of $x\equiv n^{-1/3}$, the data for graphs with different $n$
fit nicely, and can be extrapolated to $x=0$ (infinite graph size)
using polynomial fits, see Fig.~\ref{fig:ns23}. Notice that while
this technique works well for hyperbolic graphs and for random graphs,
in our simulations it failed dramatically for the planar $\{4,4\}$
graph family, as can be seen from the corresponding value of
$p_c^{(2/3)}$ in Tab.~\ref{tab:pc}.
\begin{table*}[htp]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{r|l|ldl|ldl|ldl}
$\{f,d\}$ & $p_{\rm c}$
&$p_c^{(2/3)}$&\text{$n_\sigma^{(2/3)}$}°&$p_E^{(C)}$&\text{$n_\sigma^{(C)}$}°&$p_E^{\rm (shift)}$&\text{$n_\sigma^{\rm (shift)}$}& $B$ \\ \hline
$\{$3,7$\}$ & 0.1993505(5) & 0.1999(8) & 0.7 & 2& 0.1941(2) & -26 & 6&0.19318(9) &$-69$& 0.081(5)\\
$\{$7,3$\}$ & 0.5305246(8) & 0.5320(5) & 3.0 & 2& 0.52109(8) & -120 &4&0.52042(5)&$-200$& 0.087(2)\\
$\{$3,8$\}$ & 0.1601555(2) & 0.160(2) & -0.08& 3&0.1519(4) & -21 & 7& 0.1524(1)&$-78$& 0.26(1)\\
$\{$8,3$\}$ & 0.5136441(4) & 0.513(2) & -0.3 & 3&0.5032(2) & -52 & 6&0.5026(1) &$-110$& 0.32(3)\\
$\{$4,5$\}$ & 0.2689195(3) & 0.2695(6) & 1.0 & 2&0.2581(2) & -54 & 5& 0.2547(2) &$-71$& 0.306(8) \\
$\{$5,4$\}$ & 0.3512228(3) & 0.3519(7) & 1.0 & 2&0.3415(4) & -24 & 2&0.3412(4) &$-25$& 0.18(9) \\
$\{$4,6$\}$ & 0.20714787(9) & 0.2076(2) & 2.3& 1&0.19564(4) & -290 &3& 0.1949(3) &$-41$& -0.08(3) \\
$\{$6,4$\}$ & 0.3389049(2) & 0.3395(1) & 6.0 & 1&0.3271(4) & -29 & 2&0.3275(3) &$-38$& 0.14(4) \\
$\{$5,5$\}$ & 0.25416087(3) & 0.2545(7) & 0.5& 2&0.2437(4) & -26 &5& 0.2453(2) &$-44$& 0.88(6) \\
\hline
$\{$4,4$\}$ & 1/2 & {\bf 0.4897(3)} &
-34 &3& 0.500004(2) & 2 &6& 0.499992(6) & -1.3 & 0.003(1) \\
$\{\infty,5\}$ & 1/4 & 0.2500(2) & 0&2 & \text{--} & \text{--} & \text{--} & \text{--} & \text{--} &\text{--}
\end{tabular
}
\caption{Critical $p$ values found for graph families characterized by
Schl\"afli symbols $\{f,d\}$, where $f=\infty$ stands for random
graphs of degree $d$. Here $p_c$ is the percolation threshold
(using invasion percolation data from
Ref.~\onlinecite{Mertens-Moore-2017}, or exact values where known),
$p_c^{(2/3)}$ is the percolation threshold using random-graph-like
cluster size scaling (see Fig.~\ref{fig:ns23}), $p_E^{(C)}$ is the
cycle erasure pseudothreshold obtained using the crossing point
method (see Fig.~\ref{fig:PE-4-4} and top plots in
Figs.~\ref{fig:pE-hyp55} and \ref{fig:pE-hyp73}), and
$p_E^{\rm (shift)}$ is the same from the crossing point with an
additional shift as in the bottom plots in Figs.~\ref{fig:pE-hyp55}
and \ref{fig:pE-hyp73}, with $B$ the shift coefficient. Numbers in
the parenthesis indicate the standard deviation $\sigma$ in the
units of the last significant digit, so that, e.g.,
$0.14(4)\equiv 0.14\pm0.04$. Values of
$n_\sigma\equiv (p-p_{\rm c})/\sigma$ give the ``number of sigmas''
for the deviation of the corresponding critical value found from the
invasion percolation or exact threshold value if available. Numbers
in columns labeled ``deg'' give the degrees of the polynomials used
to interpolate the data; polynomials of the same degrees were used
to obtain $p_E^{(C)}$ and $p_E^{\rm (shift)}$.}
\label{tab:pc}
\end{table*}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{Ns23-55}\\
\includegraphics[width=0.6\columnwidth]{Ns23-73}\\
\includegraphics[width=0.6\columnwidth]{Ns23-r5}
\caption{(Color online) Using the random-graph-like scaling for locating the
percolation transition for hyperbolic graphs
in the $\{5,5\}$ (top) and $\{7,3\}$ (middle) families, and for
degree-$5$ random graphs (bottom). Values of the open bond
probability $p$ where the expected size of the largest cluster
equals $\omega n^{2/3}$ are plotted as a function of $n^{-1/3}$,
for values of $\omega$ as indicated. Here $n$ is the number of
edges in the graph. The lines intersect close to the percolation
transition point, as indicated by horizontal dashed lines.
}
\label{fig:ns23}
\end{figure}
The obtained critical values $p_c^{(2/3)}$, $p_E^{(C)}$, and
$p_E^{\rm (shift)}$ for different graph families are summarized in
Tab.~\ref{tab:pc}, where they are compared with the corresponding
percolation thresholds from Ref.~\onlinecite{Mertens-Moore-2017}
obtained from invasion percolation simulations, or exact values where
available. Numerical data indicates that the erasure
(pseudo)threshold is substantially below $p_c$ for hyperbolic graphs
with logarithmic distance scaling, with the variation of the ratio
$\ln n/d$ having a significant effect on the quality of the crossing
point. In contrast, for graphs from the $\{4,4\}$ family where
$d\propto n^{1/2}$, the cycle erasure (pseudo)threshold is very close
to the bulk percolation threshold, as generally expected from
Refs.~\onlinecite{Stace-Barrett-Doherty-2009,Fujii-Tokunaga-2012} and
Theorem \ref{th:pE-logB}.
Our results also indicate that for expander graphs, most accurate
results for percolation transition critical point are obtained using
the random-graph-like scaling, see Fig.~\ref{fig:ns23}, although this
technique is not at all applicable when the limiting graph is a tiling
of the euclidean plane. Detailed comparison of the performance of
different extrapolation methods for percolation transition critical
point for various amenable and non-amenable graph families will be
published elsewhere.
\section{Conclusions}
\label{sec:conclusions}
In this work we focused on critical points associated with
homology-changing percolation transitions in a sequence of finite
graphs weakly convergent to an infinite graph $\mathcal{H}$, a
covering graph of the graphs in the sequence. We also quantified the
relation between these critical points and edge percolation threshold
on $\mathcal{H}$.
The position of the homological $1$-cycle erasure threshold $p_E^0$ is
governed by the scaling of the homological distance $d$ with $\log n$,
where $d$ is the size of a smallest non-trivial cycle and $n$ is the
graph size (number of edges). Generally,
$p_E^0\le p_{\rm c}(\mathcal{H})$, where the equality is reached for
superlogarithmic distance scaling, while $p_E^0=0$ is expected for
sublogarithmic distance scaling. In the case of logarithmic distance
scaling where the quantity $d/\ln n$ remains bounded away from 0 and
from infinity, the cycle erasure threshold $p_E^0$ remains strictly
positive as long as $\mathcal{H}$ is a bounded-degree graph, and we
expect $p_E^0$ to be strictly below $p_{\rm c}$.
For an amenable graph $\mathcal{H}$ with a finite isoperimetric
dimension, an easy upper bound on the distance can be constructed by
considering a ball with the radius equal to the injectivity radius,
giving a power-law scaling of the distance with $n$. Generically, we
expect that a sequence of covering maps with superlogarithmic distance
scaling can be constructed when such a graph is quasitransitive,
resulting in $p_E^0=p_{\rm c}(\mathcal{H})$. In particular, this is
the case for any periodic lattice in dimension $D>1$, since covering
maps can be constructed by using periodic boundary conditions along
each axis.
On the other hand, logarithmic scaling of the distance is the most one
can expect when $\mathcal{H}$ is non-amenable. For such a graph the
uniqueness threshold is expected\cite{Haggstrom-Jonasson-2006} to be
strictly higher than the percolation threshold,
$\Delta p \equiv p_{\rm u}(\mathcal{H})-p_{\rm c}(\mathcal{H})>0$,
which gives a non-trivial upper bound for the asymptotic rate,
$R\le\Delta p$, where $R>0$ corresponds to an extensive scaling of the
homology rank associated with non-trivial $1$-cycles. For any graph
sequence with $R>0$, we also introduced a pair of homological
thresholds $p_H^0$ and $p_H^1$, associated with the points where
asymptotic erasure rate (\ref{eq:mean-erased}) deviates from the
values at $p=0$ and $p=1$, respectively. Generally,
$p_H^0\ge p_{\rm c}$; for planar transitive graphs we proved
$p_H^0=p_{\rm c}(\mathcal{H})$ and $p_H^1=p_{\rm u}(\mathcal{H})$. We
conjecture this to be the case more generally.
A number of open questions remain. First, related to the sequences of
finite graphs both weakly convergent to an infinite graph
$\mathcal{H}$, and covered by $\mathcal{H}$. What are the properties
of $\mathcal{H}$ necessary for such a sequence to exist, in
particular, is it necessary that $\mathcal{H}$ be quasi-transitive?
Second, is it true that with a logarithmic distance scaling, the
strict inequality holds $p_E^0<p_c(\mathcal{H})$?
Finally, an important open question is to what extent present results can be
extended to other models, in particular, Ising and, more generally,
$q$-state Potts model on various graphs. Indeed, successful decoding
probability in qubit quantum LDPC codes can be mapped to ratios of
partition functions of associated random-bond Ising
models\cite{Dennis-Kitaev-Landahl-Preskill-2002
Kovalev-Pryadko-SG-2015,Jiang-Kovalev-Dumer-Pryadko-2018}. In the
clean (no-disorder) limit, these can be rewritten in terms of
Fortuin-Kasteleyn (FK) random-cluster models. For such a model with
$q\ge1$, Hutchcroft \cite{Hutchcroft-2019} has recently proved the
exponential decay of cluster size distribution in the subscritical
regime. In particular, this could help fixing the location of the
boundary of the decodable region for certain families of graph-based
quantum CSS codes in the weak-noise limit.
\acknowledgements
\medskip\noindent\textbf{Acknowledgment:}
This work was supported in part by the NSF Division of Physics via
grant No.\ 1820939. |
2201.04146 | \section{Introduction}\label{sec:intro}
Gas giant planets have been found to reside in many extrasolar planetary systems. The diversity in their sizes, masses, orbits, compositions, and formation pathways has been the subject of numerous studies. However, selection biases often cloud our understanding. For instance, the sensitivity of the transit method wanes for planets on orbits beyond a few tenths of an au owing to the inverse relation between transit probability and semi-major axis. Consequently, the vast majority of known exoplanets with $\gtrsim$1~au orbits have been discovered via Doppler spectroscopy \citep[e.g.,][]{Mayor2011,Fulton2021}. Planet mass can be inferred from time series radial velocity (RV) observations but, without a transit, the planet radius and thereby bulk density remains unknown.
A critical component of the bulk composition for giant planets is the total mass of elements heavier than H and He \citep[e.g.,][]{Guillot2006,Miller2011,Thorngren2016,Teske2019}. This property is inferred using structural evolution models along with the measured planetary mass, radius, stellar age, and incident flux \citep[e.g.,][]{Thorngren2019a}. Numerous theoretical planet formation studies have found that the correlation between the total mass of heavy elements in giant planets---or, similarly, the metal enrichment relative to the host star---and planet mass is a useful tracer of planet formation processes \citep[e.g.,][]{Mordasini2014,Hasegawa2018,Ginzburg2020,Shibata2020}. Probing this giant planet mass--metallicity correlation along a third axis, in orbital properties (i.e., period or separation), could prove informative. Yet, previous efforts have simply not had a large enough sample size of giant planets on orbits wider than a few 0.1~au to conduct such an investigation \citep{Miller2011,Thorngren2016}.
Transit surveys such as the {\it Kepler}\ mission \citep{Borucki2010,Thompson2018} and the Transiting Exoplanet Survey Satellite (TESS) mission \citep{Ricker2015} occasionally detect giant planet candidates with orbital periods of 100 to 1,000 days \citep[e.g.,][]{Wang2015b,ForemanMackey2016b,Osborn2016,Uehara2016,Herman2019,Kawahara2019,Eisner2021}. These candidates warrant close scrutiny, at least based on the $\sim$50\% false-positive rate for giant planets with similar orbits found for the {\it Kepler}\ mission \citep{Santerne2014,Dalba2020b}. Following vetting, long-term RV monitoring is often required to measure the planet mass, which then enables modeling of the bulk metallicity \citep[e.g.,][]{Beichman2016,Dubber2019,Santerne2019,Dalba2021a,Dalba2021c}. {TESS}\ planets with orbital periods on the order of 100~days or more will typically be detected as single-transit events owing to the observational strategy of the TESS mission \citep[e.g.,][]{LaCourse2018,Villanueva2019,Diaz2020,Cooke2021}. In this case, the RV monitoring is also necessary to determine the orbital period.
Here, we describe the discovery and investigation of a 24-hour single-transit event observed for HD~238894, hereafter referred to as {TOI-2180~b}. The host star, {TOI-2180}\ (TIC~298663873), is a bright ($V=9.2$), slightly evolved G5 star. The confirmation of {TOI-2180~b}\ is notable relative to the current sample of TESS exoplanets, including those identified as single-transits, as it is the first to surpass an orbital period of 100~days\footnote{According to the list of confirmed TESS exoplanets on the NASA Exoplanet Archive (\url{https://exoplanetarchive.ipac.caltech.edu/index.html}) as of 2021 September 2.}. In Section~\ref{sec:obs}, we describe the {TESS}\ observations of {TOI-2180}\ along with the follow-up photometric, imaging, and spectroscopic data sets. In Section~\ref{sec:model}, we discuss the consistency of the transit and RV data, which are then jointly modeled to determine the {TOI-2180}\ system properties. We find that {TOI-2180~b}\ is a {2.7~$M_{\rm J}$} giant planet on a {261~day} orbit with an orbital eccentricity of {0.368}. In Section~\ref{sec:teers}, we describe a global effort to detect an additional transit of {TOI-2180~b}\ from the ground. Although this effort failed to detect the transit, the nondetections provided a substantial improvement in the precision on the orbital period. In Section~\ref{sec:metal}, we determine the bulk heavy-element mass of {TOI-2180~b}. In Section~\ref{sec:drift}, we use the long-term acceleration of {TOI-2180}\ to place limits on the properties of a distant massive companion. In Section~\ref{sec:disc}, we place {TOI-2180~b}\ in the context of other transiting giant planets and discuss the prospects for future characterization of the system. Lastly, in Section~\ref{sec:concl}, we summarize our work.
\section{Observations}\label{sec:obs}
\subsection{TESS Photometry}\label{sec:tess}
TESS observed {TOI-2180}\ (TIC~298663873) at 2-min (fast) cadence for the entirety of Cycle~2 of its primary mission (Sectors~14--26) and in Sectors~40 and 41 of its extended mission. The image data were processed by the Science Processing Operations Center (SPOC) at NASA Ames Research Center \citep{Jenkins2016} to extract photometry from this target. A search for transiting planets failed to find a transit signature with two or more transits. Only a single transit event was observed in Sector~19 (2019 November 28 through 2019 December 22). This Sector~19 single-transit was first identified by citizen scientists with the light curve processing software \texttt{LcTools} \citep{Schmitt2019}, leading to early commencement of our RV follow-up campaign. In May 2020, the Planet Hunters TESS collaboration \citep{Eisner2020} announced this star as a Community TESS Object of Interest (cTOI). In August 2020, its disposition was elevated to TOI \citep{Guerrero2021}.
We downloaded the Sector~19, 2-min cadence light curve of {TOI-2180}\ from the Mikulski Archive for Space Telescopes (MAST) using the \texttt{lightkurve} package \citep{Lightkurve2018}. We used the pre-search data conditioning simple aperture photometry (PDCSAP) flux for which most of the high frequency noise was removed \citep{Stumpe2012,Stumpe2014,Smith2012}. The PDCSAP light curve exhibited low frequency noise features that we removed using a Savitzky--Golay filter after masking the clear transit event. The raw and flattened light curves, centered on the $\sim$0.5\% transit of {TOI-2180~b}, are shown in Figure~\ref{fig:tess}.
\begin{figure}
\centering
\begin{tabular}{c}
\includegraphics[width=\columnwidth]{raw_pdc_v1.pdf} \\
\includegraphics[width=\columnwidth]{transits_v1.pdf}
\end{tabular}
\caption{The single-transit of {TOI-2180~b}\ observed by TESS with short cadence. {\it Top:} unflattened PDCSAP flux and the trend from the Savitsky--Golay filter. {\it Bottom:} flattened light curve. The red points are individual exposures (gray points) binned by a factor of 40. The blue line shows the best fit transit model.}
\label{fig:tess}
\end{figure}
Under the assumption of a circular central transit, \citet[][Equation~19]{Winn2010b} showed how the transit duration ($T$) and the stellar bulk density ($\rho_{\star}$) can give an estimate of the orbital period ($P$) following
\begin{equation}\label{eq:P}
T \approx 13\;\mathrm{hr} \; \left (\frac{P}{1\; \mathrm{yr}} \right )^{1/3} \left (\frac{\rho_{\star}}{\rho_{\sun}} \right )^{-1/3}
\end{equation}
\noindent where $P$ has units of years and $\rho_{\sun}$ is the solar bulk density. For $T=24$~hr and $\rho_{\star}=0.335$~g~cm$^{-3}$ from the TESS Input Catalog \citep[TIC;][]{Stassun2019}, Equation~\ref{eq:P} gives $P\approx547\pm111$~days assuming reasonable uncertainty in $T$ and $\rho_{\star}$. Combined with the nondetection of a matching transit event in the other Cycle 2 sectors, this possible orbital period suggested that {TOI-2180~b}\ was likely to be one of only a few long-period planets predicted to be detected by TESS through single-transit events \citep{Villanueva2019}.
\subsection{Speckle Imaging}\label{sec:imaging}
We acquired a high-resolution speckle image of {TOI-2180}\ to search for nearby neighboring stars that might indicate the false-positive nature of the TESS single-transit event. We observed {TOI-2180}\ on 2020 June 6 using the `Alopeke
speckle instrument on the Gemini-North telescope\footnote{\url{https://www.gemini.edu/instrumentation/alopeke-zorro}} located on Maunakea in Hawai`i. `Alopeke acquires an image of the star in a blue (562~nm) and red (832~nm) band simultaneously. From these images, we derived contrast curves that show the limiting magnitude difference ($\Delta m$) in each band as a function of angular separation \citep{Howell2011}. As shown in Figure~\ref{fig:speckle}, we achieved a $\sim$5~mag contrast at 0$\farcs$1 and a nondetection of sources with $\Delta m$ = 5--8.5 within 1$\farcs$2 of {TOI-2180}. Given the distance to {TOI-2180}\ ({116~pc}; Table~\ref{tab:stellar}), 0$\farcs$1 projected separation corresponds to {$\sim$12~au}. Although we cannot rule out a scenario whereby a luminous companion star was near conjunction with {TOI-2180}\ at the time of our speckle observation, we assume in what follows that {TOI-2180}\ is likely a single star. The speckle imaging nondetection suggests that any massive object in this system other than {TOI-2180~b}\ is likely to be a late M star or a substellar object.
The conclusion of {TOI-2180}\ being a single star is further supported by the renormalized unit weight error (RUWE) as determined by Gaia Data Release (DR) 2 \citep{Gaia2018}. The RUWE value for {TOI-2180}\ is 1.01, which is typical of a single star \citep[e.g.,][]{Belokurov2020}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{speckle.png}
\caption{Limiting magnitudes ($\Delta m$) for the nondetection of a neighboring star based on the speckle imaging from `Alopeke. The inset shows the speckle image at 832~nm.}
\label{fig:speckle}
\end{figure}
\subsection{Spectroscopy}\label{sec:spec}
Immediately following the discovery of the TESS single-transit of {TOI-2180}, we began a Doppler monitoring campaign with the Automated Planet Finder (APF) telescope at Lick Observatory \citep{Radovan2014,Vogt2014} as part of the TESS-Keck Survey (TKS). TKS is a collaborative effort between the University of California, the California Institute of Technology, the University of Hawai`i,
and NASA aimed at providing multi-site RV follow-up for many of the planetary systems discovered by TESS \citep[e.g.,][]{Dai2020,Dalba2020a,Chontos2021,Lubin2021,Rubenzahl2021,Weiss2021}. The APF uses the Levy Spectrograph, a high-resolution ($R\approx$ 114,000) slit-fed optical echelle spectrometer \citep{Radovan2010} that is ideal for bright (V$\le$9) stars such as {TOI-2180}\ \citep{Burt2015}. The starlight passes through a heated iodine gas cell that allows for precise wavelength calibration and instrument profile tracking. The precise RV is inferred from each spectrum using a forward modeling procedure \citep{Butler1996,Fulton2015a}. Table~\ref{tab:rvs} lists the RVs collected in our campaign.
\begin{deluxetable}{cccc}
\tabletypesize{\scriptsize}
\tablecaption{RV Measurements of {TOI-2180} \label{tab:rvs}}
\tablehead{
\colhead{BJD$_{\rm TDB}$} &
\colhead{RV (m s$^{-1}$)} &
\colhead{$S_{\rm HK}$$^a$} &
\colhead{Tel.}}
\startdata
2458888.063868 & $-16.1\pm3.9$ & $0.1304\pm0.0020$ & APF\\
2458894.911472 & $-29.1\pm4.5$ & $0.1476\pm0.0020$ & APF\\
2458899.027868 & $-36.3\pm3.8$ & $0.1463\pm0.0020$ & APF\\
2458906.015644 & $-43.6\pm3.2$ & $0.1093\pm0.0020$ & APF\\
2458914.011097 & $-38.9\pm3.3$ & $0.1506\pm0.0020$ & APF\\
2458954.765221 & $-59.0\pm4.0$ & $0.1468\pm0.0020$ & APF\\
2458961.864505 & $-49.7\pm4.3$ & $0.1323\pm0.0020$ & APF\\
2458964.879814 & $-53.3\pm3.1$ & $0.1298\pm0.0020$ & APF\\
2458965.811833 & $-47.5\pm4.2$ & $0.1363\pm0.0020$ & APF\\
2458966.879965 & $-65.9\pm5.5$ & $0.1204\pm0.0020$ & APF\\
\enddata
\tablenotetext{a}{The $S_{\rm HK}$ values from APF and Keck data have different zero-points.}
\tablenotetext{}{This is a representative subset of the full data set. The full table will be made available in machine readable format.}
\end{deluxetable}
In August 2020, Lick Observatory and the APF shut down owing to nearby wildfires. To maintain our coverage of the emerging Keplerian RV signal, we temporarily conducted observations of {TOI-2180}\ using the High Resolution Echelle Spectrometer \citep[HIRES;][]{Vogt1994} on the Keck~I telescope at W. M. Keck Observatory. The reduction and analysis procedure for Keck-HIRES spectra is broadly similar to that for data from the APF \citep[e.g.,][]{Howard2010}. The full version of Table~\ref{tab:rvs} also contains the Keck-HIRES RVs of {TOI-2180}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{rv_v1.pdf}
\caption{RV measurements of {TOI-2180}. {\it Panel a:} the RV time series after subtraction of an offset velocity from each data set. Transit windows are shown in green. {\it Panel b:} the residuals between the RV time series and best fit planet model not including acceleration terms. A quadratic trend is visible on top of the Keplerian signal from {TOI-2180~b}. {\it Panel c:} the phase-folded RVs after removal of the acceleration terms. A phase of 0 corresponds to conjunction (transit).}
\label{fig:rv}
\end{figure*}
The time series RVs from both telescopes are shown in the top panel of Figure~\ref{fig:rv}. Our {1.5~yr} baseline captured {two} periods of a 261~day, eccentric ($e\approx0.4$) Keplerian signal as well as a long-term acceleration. In Section~\ref{sec:model} we will discuss the consistency of the RV data with the single-transit from TESS and conduct a joint modeling of the system parameters.
Each RV measurement is accompanied by an estimate of stellar chromospheric activity approximated as the $S_{\rm HK}$ index \citep{Isaacson2010}. The indicators calculated for each instrument have different zero points. The Pearson correlation coefficient between the $S_{\rm HK}$ indices and the RVs for APF and Keck are {$-0.10$ (from 84 data points)} and {$0.68$ (from 15 data points)}, respectively. Although the Keck data demonstrate a positive correlation significant to {3.3$\sigma$}, the much larger APF data set shows no correlation {($<1\sigma$)}. Furthermore, we do not detect significant periodicity in the $S_{\rm HK}$ time series. Both of these findings suggest that the RV signals are not affected, or only minimally so, by stellar activity.
The extraction of the APF and Keck RVs rely on a high signal-to-noise template spectrum of {TOI-2180}\ acquired with Keck-HIRES without the iodine cell \citep{Butler1996}. We also used this spectrum for a basic spectroscopic analysis of {TOI-2180}\ with \texttt{SpecMatch} \citep{Petigura2015,Petigura2017b}. This analysis indicated that {TOI-2180}\ has an effective temperature of $T_{\rm eff} = 5739\pm100$~K, a surface gravity of $\log{g} = 4.00\pm0.10$, and a relative iron abundance (metallicity proxy) of [Fe/H] = 0.25$\pm$0.06~dex consistent with a slightly evolved mid-G type star. These values are in close agreement with those listed in the TESS Input Catalog \citep{Stassun2019}.
\subsection{Ground-based Photometry}
The single-transit detection for {TOI-2180}\ by TESS and the subsequent RV follow-up effort allowed us to plan a ground-based photometry campaign with the goal of detecting another transit of {TOI-2180~b}. This campaign occurred in late August and early September of 2020 around the time of the first transit of {TOI-2180~b}\ since the one observed by TESS. We acquired 55 photometric data sets of {TOI-2180}\ comprised of $\sim$20,000 individual exposures spanning 11~days and 14 sites. Contributors to this campaign consisted of a mix of professional and amateur astronomers.
The quality of the ground-based telescope data sets varied widely. Some data achieved precision sufficient to rule out ingress or egress events for the relatively shallow (0.5\%) transit of {TOI-2180~b}. Other data showed correlated noise or systematic errors several times this magnitude. We leave a detailed discussion of the individual observing sites for Appendix~\ref{app:ground}. In Section~\ref{sec:teers}, we will discuss our treatment of the ground-based data as a whole and our procedure for using it to refine the transit ephemeris of {TOI-2180~b}.
\section{Modeling}\label{sec:model}
\subsection{Consistency between the Transit and the RV Data}\label{sec:tran_rv}
A single-transit event only places a weak constraint on orbital period even under various simplifying assumptions \citep[e.g.,][]{Yee2008,Yao2021}. We must therefore assess whether the same object caused the single-transit and Doppler reflex motion of the host star.
In Section~\ref{sec:tess}, we estimated that the duration of the single-transit of {TOI-2180~b}\ and the stellar density listed in the TIC corresponded to a 547-day orbital period assuming a circular, central transit. However, the RV measurements of {TOI-2180}\ suggest a shorter period that has significant eccentricity. Depending on the argument of periastron ($\omega_{\star}$), orbital eccentricity can lead to shorter or longer transit duration compared to that from a circular orbit \citep[e.g.,][]{Kane2012}, which can bias the orbital period estimation from a single-transit. Orbital eccentricity also increases the transit probability \citep{Kane2007}, which is inherently low for objects with au-scale orbital distances.
We can account for eccentricity in our estimation of the orbital period to good approximation by including a factor of $\sqrt{1-e^2}/(1+e\sin{\omega_{\star}})$ on Equation~\ref{eq:P} \citep{Winn2010b}. We determined $e$ and $\omega_{\star}$ by conducting a Keplerian model fit to the APF and Keck RVs using the \texttt{RadVel} modeling toolkit\footnote{\url{https://radvel.readthedocs.io/}} \citep{Fulton2018}. In this fit, we applied a normal prior on the time of conjunction (BJD$_{\rm TDB} = 2458830.79\pm0.05$) using the transit timing from the Planet Hunters TESS characterization of the single-transit \citep{Eisner2021}. The fit converged quickly and we found that $e = 0.367\pm0.0074$, $\omega_{\star} = -0.76\pm0.023$~rad, and $P = 260.68 \pm 0.54$~days. This argument of periastron corresponds to a time of periastron that is $70.0\pm1.2$~days prior to transit. Therefore, at the time of transit, the orbital velocity of {TOI-2180~b}\ is decreasing.
Reevaluating Equation~\ref{eq:P} with the factor to account for an eccentric orbit gives $283\pm59$~days, which is consistent with the period of the Keplerian signal in the RVs. Considering the uncertainty introduced by the transit impact parameter could further explain the difference between this orbital period estimate and the observed {261}~day period. This result demonstrates self-consistency with our assumption of a T$_0$ value in the \texttt{RadVel} fit. Therefore, we continue our analysis with the implicit assumption that the single-transit in the photometry and the Keplerian signal in the RVs can be ascribed to the same planet: {TOI-2180~b}.
\subsection{Comprehensive System Modeling}\label{sec:EFv2}
We modeled the stellar and planetary parameters for the {TOI-2180}\ system using the \texttt{EXOFASTv2} modeling suite \citep{Eastman2013,Eastman2019}. We include the TESS single-transit photometry, all of the RVs from Keck-HIRES and APF-Levy, and archival broadband photometry of {TOI-2180}\ from Gaia DR 2 \citep{Gaia2018}, the Two Micron All Sky Survey \citep{Cutri2003}, and the Wide-field Infrared Survey Explorer \citep{Cutri2014}. \texttt{EXOFASTv2} computes spectral energy distributions from MIST models and the Gaia parallax measurements and fits them to the archival photometry to infer the properties of the host star. We placed normal priors on $T_{\rm eff}$ and [Fe/H] based on the \texttt{SpecMatch} analysis of the high S/N template spectrum of {TOI-2180}\ (Section~\ref{sec:spec}). The width of the $T_{\rm eff}$ prior was inflated to 115~K (2\%) and a noise floor of 2\% was applied to bolometric flux used in the SED determination to account for systematic uncertainties inherent in the MIST models \citep{Tayar2020}. We also placed a uniform prior on extinction ($A_{\rm V}\le0.1376$) using the galactic dust maps of \citet{Schlafly2011} and a normal prior on parallax ($\varpi = 8.597\pm0.017$~mas) using the Gaia DR3 measurement corrected for the zero point offset \citep{Lindegren2021,Gaia2021}. These priors are summarized at the top of Table~\ref{tab:stellar}.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecaption{Median values and 68\% confidence intervals for the stellar parameters for {TOI-2180}. \label{tab:stellar}}
\tablehead{\colhead{~~~Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\multicolumn{2}{l}{Informative Priors:}& \smallskip\\
~~~~$T_{\rm eff}$ &Effective Temperature (K) & $\mathcal{N}(5739,115)$\\
~~~~$[{\rm Fe/H}]$ &Metallicity (dex) & $\mathcal{N}(0.25,0.06)$\\
~~~~$\varpi$ &Parallax (mas) & $\mathcal{N}(8.597,0.017)$\\
~~~~$A_V$ &V-band extinction (mag) & $\mathcal{U}(0,0.1376)$\\
\smallskip\\\multicolumn{2}{l}{Stellar Parameters:}&\smallskip\\
~~~~$M_*$ &Mass (\msun) &$1.111^{+0.047}_{-0.046}$\\
~~~~$R_*$ &Radius (\rsun) &$1.636^{+0.033}_{-0.029}$\\
~~~~$L_*$ &Luminosity (\lsun) &$2.544^{+0.091}_{-0.093}$\\
~~~~$F_{Bol}$ &Bolometric Flux (cgs) &$6.01\times10^{-9}$$^{+2.1\times10^{-10}}_{-2.2\times10^{-10}}$\\
~~~~$\rho_*$ &Density (g~cm$^{-3}$) &$0.359^{+0.015}_{-0.016}$\\
~~~~$\log{g}$ &Surface gravity (cgs) &$4.057^{+0.015}_{-0.016}$\\
~~~~$T_{\rm eff}$ &Effective Temperature (K) &$5695^{+58}_{-60}$\\
~~~~$[{\rm Fe/H}]$ &Metallicity (dex) &$0.253\pm0.057$\\
~~~~$[{\rm Fe/H}]_{0}$ &Initial Metallicity$^{a}$ (dex) &$0.269^{+0.055}_{-0.054}$\\
~~~~${\rm Age}$ &Age (Gyr) &$8.1^{+1.5}_{-1.3}$\\
~~~~$EEP$ &Equal Evolutionary Phase$^{b}$ &$452.1^{+3.9}_{-5.0}$\\
~~~~$A_V$ &V-band extinction (mag) &$0.077^{+0.041}_{-0.048}$\\
~~~~$\sigma_{SED}$ &SED photometry error scaling &$0.69^{+0.24}_{-0.16}$\\
~~~~$\varpi$ &Parallax (mas) &$8.597\pm0.017$\\
~~~~$d$ &Distance (pc) &$116.32\pm0.23$\\
\enddata
\tablenotetext{}{See Table~3 in \citet{Eastman2019} for a detailed description of all parameters and all default (non-informative) priors beyond those specified here. $\mathcal{N}(a,b)$ denotes a normal distribution with mean $a$ and variance $b^2$. $\mathcal{U}(a,b)$ denotes a uniform distribution over the interval [$a$,$b$].}
\tablenotetext{a}{Initial metallicity is that of the star when it formed.}
\tablenotetext{b}{Corresponds to static points in a star's evolutionary history. See Section~2 of \citet{Dotter2016}.}
\end{deluxetable}
We gauged convergence of the \texttt{EXOFASTv2} fit by the number of independent draws \citep{Ford2006b}, which exceeded 1,000, and the Gelman--Rubin statistic \citep{Gelman1992}, which was smaller than 1.01, for every fitted parameter. From the fit parameters, numerous properties of {TOI-2180~b}\ were derived. Table~\ref{tab:planet} lists all relevant planetary parameters for {TOI-2180~b}. The derived planetary radius and transit depth assume no oblateness \citep[e.g.,][]{Seager2002} and no system of rings \citep[e.g.,][]{Barnes2004,Akinsanmi2018}, although {TOI-2180~b}\ could plausibly have both.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecaption{Median values and 68\% confidence interval of the planet parameters for {TOI-2180~b}. \label{tab:planet}}
\tablehead{\colhead{~~~Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\multicolumn{2}{l}{Planetary Parameters:}&\smallskip\\
~~~~$P$ &Period$^{a}$ (d) &$260.79^{+0.59}_{-0.58}$\\
~~~~$R_P$ &Radius (\rj) &$1.010^{+0.022}_{-0.019}$\\
~~~~$M_P$ &Mass (\mj) &$2.755^{+0.087}_{-0.081}$\\
~~~~$T_C$ &Time of conjunction (\bjdtdb) &$2458830.7652\pm0.0010$\\
~~~~$a$ &Semi-major axis (au) &$0.828\pm0.012$\\
~~~~$i$ &Inclination (degrees) &$89.955^{+0.032}_{-0.044}$\\
~~~~$e$ &Eccentricity &$0.3683\pm0.0073$\\
~~~~$\omega_*$ &Argument of Periastron (degrees) &$-43.8\pm1.3$\\
~~~~$T_{eq}$ &Equilibrium temperature$^{b}$ (K) &$348.0^{+3.3}_{-3.6}$\\
~~~~$\tau_{\rm circ}$ &Tidal circularization timescale (Gyr) &$54600000^{+3800000}_{-4500000}$\\
~~~~$K$ &RV semi-amplitude (m~s$^{-1}$) &$87.75^{+0.98}_{-0.99}$\\
~~~~$\dot \gamma$ &RV slope$^{c}$ (m~s$^{-1}$~day$^{-1}$) &$-0.1205\pm0.0043$\\
~~~~$\ddot \gamma$ &RV quadratic term$^{c}$ (m~s$^{-1}$~day$^{-2}$) &$0.000214^{+0.000039}_{-0.000038}$\\
~~~~$\delta_{\rm TESS}$ &Transit depth in TESS band &$0.004766^{+0.000078}_{-0.000076}$\\
~~~~$\tau$ &Ingress/egress transit duration (d) &$0.06044^{+0.0019}_{-0.00063}$\\
~~~~$T_{14}$ &Total transit duration (d) &$1.0040^{+0.0032}_{-0.0031}$\\
~~~~$b$ &Transit Impact parameter &$0.100^{+0.095}_{-0.070}$\\
~~~~$\rho_P$ &Density (g~cm$^{-3}$) &$3.32^{+0.14}_{-0.16}$\\
~~~~$logg_P$ &Surface gravity (cgs) &$3.827^{+0.012}_{-0.015}$\\
~~~~$\fave$ &Incident Flux (\fluxcgs) &$0.00442\pm0.00018$\\
~~~~$T_P$ &Time of Periastron (\bjdtdb) &$2458760.7\pm1.3$\\
~~~~$T_S$ &Time of eclipse (\bjdtdb) &$2458745.41^{+1.00}_{-1.0}$\\
~~~~$b_S$ &Eclipse impact parameter &$0.060^{+0.056}_{-0.042}$\\
~~~~$\tau_S$ &Ingress/egress eclipse duration (days) &$0.03593^{+0.00098}_{-0.00088}$\\
~~~~$T_{S,14}$ &Total eclipse duration (days) &$0.599\pm0.013$\\
~~~~$\delta_{S,2.5\mu m}$ &Blackbody eclipse depth at 2.5$\mu$m (ppm) &$0.00237^{+0.00034}_{-0.00032}$\\
~~~~$\delta_{S,5.0\mu m}$ &Blackbody eclipse depth at 5.0$\mu$m (ppm) &$1.54\pm0.10$\\
~~~~$\delta_{S,7.5\mu m}$ &Blackbody eclipse depth at 7.5$\mu$m (ppm) &$11.29^{+0.49}_{-0.50}$\\
\smallskip\\\multicolumn{2}{l}{TESS Parameters:}& \smallskip\\
~~~~$u_{1}$ &linear limb-darkening coefficient &$0.316\pm0.029$\\
~~~~$u_{2}$ &quadratic limb-darkening coefficient &$0.248\pm0.044$\\
\smallskip\\\multicolumn{2}{l}{APF-Levy Parameters:}& \smallskip\\
~~~~$\gamma_{\rm rel}$ &Relative RV Offset$^{c}$ (m~s$^{-1}$) &$-30.0^{+1.2}_{-1.3}$\\
~~~~$\sigma_J$ &RV Jitter (m~s$^{-1}$) &$4.16^{+0.67}_{-0.62}$\\
\smallskip\\\multicolumn{2}{l}{Keck-HIRES Parameters:}& \smallskip\\
~~~~$\gamma_{\rm rel}$ &Relative RV Offset$^{c}$ (m~s$^{-1}$) &$-12.7^{+1.4}_{-1.5}$\\
~~~~$\sigma_J$ &RV Jitter (m~s$^{-1}$) &$4.86^{+1.3}_{-0.93}$\\
\enddata
\label{tab:HD238894.}
\tablenotetext{}{See Table 3 in \citet{Eastman2019} for a detailed description of all parameters and all default (non-informative) priors.}
\tablenotetext{a}{This orbital period is derived from the full posterior from the \texttt{EXOFASTv2} fit. See Section~\ref{sec:teers} for a description of the likely orbital period values (\red{$260.18^{+0.19}_{-0.30}$}~days or \red{$261.76^{+0.29}_{-0.16}$}~days) after the ground-based photometric transit recovery campaign for {TOI-2180}.}
\tablenotetext{b}{Assumes a Jupiter-like Bond albedo (0.34) and perfect heat redistribution.}
\tablenotetext{c}{Reference epoch = 2459157.439110}
\end{deluxetable}
The TESS data and the best fit transit model are shown in Figure~\ref{fig:tess}. All RV data and the best fit RV model are shown in Figure~\ref{fig:rv}. We included a quadratic function of time in the fit to account for the force from an additional planet or star on {TOI-2180}. Both coefficients in the quadratic function are greater than {$5\sigma$} discrepant from zero, suggesting that there is indeed a long-term variation in the RVs. Even with more than a {500}~day baseline of observations, we do not sample enough of the long-term signal to determine its cause. The lack of correlation between the RVs and the $S_{\rm HK}$ activity indicators disfavors stellar activity as the true explanation (Section~\ref{sec:spec}). Instead, we suggest that another massive object is orbiting {TOI-2180}\ (Section~\ref{sec:drift}).
Owing to the well sampled time series of precise RVs, the posterior for orbital period for the single-transit {TOI-2180~b}\ has a standard deviation below 1~day ({0.23\%}) that is well characterized by a normal distribution. In the following section, we will further constrain the orbital period of {TOI-2180~b}\ based on ground based photometry acquired near the timing of an additional transit.
\section{Ephemeris Refinement from Ground-based Observations}\label{sec:teers}
With each new RV we acquired of {TOI-2180}, we calculated the timing of the next transit of {TOI-2180~b}. The second transit (i.e., the first transit to occur since the TESS single-transit) occurred at some point in late August or early September of 2020. At that time, the $2\sigma$ transit window---which was almost entirely determined by the uncertainty on orbital period---was approximately 10~days wide\footnote{Note that the uncertainty for orbital period reported in Table~\ref{tab:planet} is informed by the full RV data set and is therefore much smaller than the corresponding estimate in August 2020.}.
We attempted the formidable task of observing this 24-hour-long, relatively shallow (0.5\%) transit by planning a global ground-based photometry campaign. In total, {15} telescopes acquired {55} data sets containing over {20,000} individual exposures of {TOI-2180}\ spanning 11~days. These included professional observatories, amateur observatories, and two portable digital {\it eVscope} telescopes. Each data set was processed with standard differential aperture photometry using background stars as references. Basic information about each telescope is provided in Appendix~\ref{app:ground}, a summary of each observation is listed in Table~\ref{tab:ground_obs}, and all data are plotted in Figure~\ref{fig:all_ground}.
None of the observations provided a conclusive detection of a {TOI-2180~b}\ transit. As has been the case for other attempts to detect transits of long-period planets \citep[e.g.,][]{Winn2009,Dalba2016,Dalba2019c}, any single observation could only observe out-of-transit baseline and ingress or egress at best. This photometric signature can be easily mimicked by flux variations between the target and reference stars as the airmass changed. On the other hand, owing to the duration of the transit and the difficulty of absolute flux calibration at the precision of the transit depth, distinguishing a fully in-transit observation from a fully out-of-transit one is challenging. Also, the $\sim$0.5\% transit depth was on the order of the noise floor for many of the sites. All of these factors contributed to the nondetection.
However, despite the lack of an obvious transit detection, we developed a straightforward method to refine the orbital period of {TOI-2180~b}\ by simultaneously searching all data sets for times where ingress or egress \emph{did not} occur to high statistical significance. Ruling out these times rules out chunks of the orbital period posterior. Conversely, at times when we cannot determine if ingress or egress occurred---or if ingress or egress even appears to be favored by the data---we do not rule out those portions of the posterior.
We began with the relative light curves for each data set, which were the aperture flux values of {TOI-2180}\ divided by the aperture flux of one or more reference stars. We sigma-clipped the relative light curves for 4$\sigma$ outliers and normalized each to its median. So far, we had not attempted to remove flux variations due to airmass. Our first task was to remove data sets with high scatter to avoid introducing spurious results in the ephemeris refinement. Although even imprecise data contain useful information, we found that many of the high-scatter data sets contained time correlated noise or systematic noise features. Instead of attempting to correct for these noise properties, we chose to exclude the data set entirely. For the purpose of this procedure, we fit and subtracted an airmass model
\begin{equation}\label{eq:am}
F^{\prime}(t) = c_1 e^{-c_2 X(t)}
\end{equation}
\noindent where $F^{\prime}$ is the flux variation owing to a varying airmass $X$, both of which are functions of time $t$. The model had two fitted coefficients $c_1$ and $c_2$, which can have any finite value. After this airmass correction, data sets for which the standard deviation exceeded the transit depth (0.5\%) were removed (gray points in Figure~\ref{fig:all_ground}). We list this standard deviation for each data set in Table~\ref{tab:ground_obs}.
Next, we established a fine linear grid of mid-transit ($T_0$) times that spanned fourth contact at the time of the very first flux measurement (BJD$_{\rm TDB} = 2459085.674949$) to first contact at the time of our very last flux measurement (BJD$_{\rm TDB} = 2459096.875975$). These $T_0$ values were used to generate transit models that were compared with the data. Each value of $T_0$ maps to a unique value of orbital period.
Then, we iterated over each $T_0$ value and each data set conducting the following procedure. A transit model was generated at that $T_0$ using the \texttt{batman} package \citep{Kreidberg2015b}. All other transit parameters were fixed at the values derived from the \texttt{EXOFASTv2} model (Table~\ref{tab:planet}) except for the quadratic limb darkening coefficients, which were drawn from the look-up tables of \citet{Claret2011} according to filter. We subtracted this transit model from the relative light curve that was sigma-clipped and normalized to its median. Note that the light curve in this case had not been corrected for airmass variations, which were therefore still present after the subtraction. The model-subtracted light curve was then fit to Equation~\ref{eq:am} with a basic least-squares algorithm that minimized the $\chi^2$ statistic.
Consider the logic behind this procedure. If the airmass model is a good fit to the light curve after the transit model is subtracted (i.e., low $\chi^2$ value), then we failed to rule out the transit at that time. We also cannot confidently claim that we have detected the transit, since the good fit to the airmass model could be coincidental. In this case, the probability of this transit time is still described by the posterior from the RV and transit joint fit. Conversely, if the airmass model is a poor fit to the light curve after the transit model is subtracted (i.e., high $\chi^2$ value), then we can be confident that the transit did not occur at that time.
Each $T_0$ for each data set yielded its own minimum $\chi^2$ value from which we calculated the corresponding log likelihood value (i.e., the maximum of the likelihood function) and Bayesian information criterion \citep[BIC;][]{Schwarz1978}. We then summed the BIC values across data sets to produce a total BIC as a function of $T_0$ and hence orbital period.
We also repeated our entire procedure and calculated the same metrics assuming that none of the observations sampled the transit. In this ``no transit'' scenario, the transit occurred either before or after the observations or fell entirely within a data gap.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{teers_results_v1.pdf}
\caption{Summary of the transit ephemeris refinement analysis. Positive values of $\Delta$BIC indicate mid-transit times that are disfavored relative to a nondetection. We consider mid-transit times with $\Delta$BIC$\ge10$ (gray regions) to be ruled out by the ground-based photometry. Negative values of $\Delta$BIC indicate times where the transit model provided a good fit. However, this could be coincidental and we do not interpret large negative $\Delta$BIC as evidence of the transit. The $1\sigma$ and $3\sigma$ ranges for mid-transit time ($T_C$) from the final ephemeris (Table~\ref{tab:planet}) are shown as vertical lines. Each mid-transit time corresponds to a specific orbital period of {TOI-2180~b}, as indicated at the top of the figure.}
\label{fig:teers}
\end{figure}
In Figure~\ref{fig:teers}, we show the difference in BIC values ($\Delta$BIC) between the transit and ``no transit'' scenarios as a function of mid-transit time. The earliest and latest values of $\Delta$BIC approach zero as expected. Values of $T_0$ with substantially negative $\Delta$BIC primarily correspond to times when ingress and/or egress occurred during data gaps and the relatively flat ground-based photometry mimics the basin of a transit. On the other hand, values of $T_0$ with substantially positive $\Delta$BIC correspond to ingress and/or egress lining up with ground-based photometry that clearly does not contain such features. We adopted a $\Delta$BIC threshold of 10---corresponding to ``very strong`` evidence against the transit model \citep{Kass1995}---to determine which values of $T_0$ we could rule out (Figure~\ref{fig:teers}, gray regions) and mapped this refinement back to orbital period. The resulting trimmed posterior for orbital period is shown in Figure~\ref{fig:trimmed_posterior}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{trimmed_posterior_v1.pdf}
\caption{Orbital period posterior for {TOI-2180~b}. The light blue shows the full posterior while the dark blue shows the region that is still allowed after the transit ephemeris refinement using the ground-based photometry, which rules out the most likely orbital periods (around the mean and median) inferred from the comprehensive \texttt{EXOFASTv2} analysis.}
\label{fig:trimmed_posterior}
\end{figure}
The ground-based transit detection campaign served to broadly divide the normal posterior for orbital period into two smaller, non-Gaussian groups while ruling out the median and most likely values of the original distribution. Described by their median and 68\% credible intervals, the two possible orbital periods are {$260.18^{+0.19}_{-0.30}$}~days and {$261.76^{+0.29}_{-0.16}$}~days.
To assess the efficiency of the ground-based photometry campaign, we consider the duty cycle between exposure time and orbital period posterior time ruled out. Multiplying the number of exposures by their respective exposure times (Table~\ref{tab:ground_obs}) for all observations (regardless of whether they were excluded from this analysis) yields {4.4}~days. The aforementioned $\Delta$BIC threshold of 10 rules out {3.8}~days worth of orbital period posterior space. Therefore, the duty cycle of the campaign was {86\%} (i.e., for every 1~hour of exposure time, we ruled out {52~minutes} of orbital period). Ideally, a campaign like this would detect the transit and the duty cycle would be less important. However, this metric can be recalculated for any future single-transit detection campaigns that yield nondetections or proposals to conduct such campaigns to compare strategies and assess effectiveness.
\subsection{Prospects for Future Transit Detection}\label{sec:future_tran}
Despite the ephemeris refinement for {TOI-2180~b}\ conducted here, future characterization of this planet will be challenging until another transit is detected. The third transit of {TOI-2180~b}\ (assuming the TESS transit as the first) occurred sometime in mid-May 2021 and was not observed. The fourth transit is predicted to occur in late January or early February 2022. Specifically, our predictions following the trimmed orbital period posterior are {19:19:30 UTC 31 January 2022} and {13:05:05 UTC 5 February 2022}. The uncertainties on these predictions are asymmetric but roughly on the order of {a day} or less ($1\sigma$).
Fortunately, TESS is expected to re-observe {TOI-2180}\ in Sector~48 of the extended mission beginning around 28 January 2022\footnote{According to the Web TESS Viewing Tool (\url{https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py}) accessed 2021 September 4.}. We therefore predict that TESS will observe another transit of {TOI-2180~b}\ at that time. Barring data gaps, if another transit is not seen by TESS in Sector 48, then only a small (relatively unlikely) tail of the orbital period posterior distribution would be consistent with the original single-transit and the RVs.
A second transit detection would drastically reduce the uncertainty on the orbital period and preserve the transit ephemeris for years into the future. However, some giant planets on 100--1,000~day orbits are known to exhibit day-long timing variations from transit-to-transit \citep[e.g.,][]{Wang2015b}. A third transit would need to be observed to explore the existence of transit timing variations \citep[e.g.,][]{Dalba2019c}.
\section{Bulk Heavy-element Analysis}\label{sec:metal}
{TOI-2180~b}\ is one of a small but growing collection of valuable giant transiting exoplanets on $\sim$au-scale orbits with precisely measured masses and radii \citep[e.g.,][]{Dubber2019,Chachan2021,Dalba2021a,Dalba2021c}. With these two properties, we can infer bulk heavy-element mass and metallicity relative to stellar to better understand their structure and formation history.
Following \citet{Thorngren2019a}, we generated one-dimensional spherically symmetric giant planet structure models with a rock/ice core, a convective envelope consisting of rock, ice, and H/He, and a radiative atmosphere interpolated from the \citet{Fortney2007} grid. Along with the mass, radius, and age for {TOI-2180~b}\ (drawn from the posteriors of the \texttt{EXOFASTv2} fit), the models recovered the total mass of heavy elements and thereby bulk metallicity needed to explain the planet's size. We find that the bulk metallicity for {TOI-2180~b}\ is {$Z_p = 0.12\pm0.03$}, which corresponds to {105~$M_{\earth}$} of heavy elements. We can approximate the stellar metallicity using the iron abundance following $Z_{\star} = 0.142 \times 10^{\rm [Fe/H]}$, which gives {$Z_{\star} = 0.0254\pm0.0033$.} This sets the metal enrichment ($Z_p/Z_{\star}$) at {$4.7\pm1.3$}.
The metallicity enrichment of {TOI-2180~b}\ is consistent with the core accretion theory of giant planet formation \citep{Pollack1996} with late-stage accretion of icy planetesimals, which can explain enrichment by a factor of a few to a dozen \citep[e.g.,][]{Gautier2001,Mousis2009}. An alternate theory to late stage accretion---the mergers of planetary cores during the gas accretion phase \citep[e.g.,][]{Ginzburg2020}---can explain enrichment factors between 1.5--10 for a {2.7~$M_{\rm J}$} planet, making it a viable formation pathway as well.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{zpzs_mp_v3.pdf}
\caption{Giant planet mass--metallicity correlation shown with the \citet{Thorngren2016} sample (circles), {TOI-2180~b}\ (square), and two recently published {\it Kepler}\ long-period giant planets (triangles): Kepler-1514~b ($P\approx218$~days) and Kepler-1704~b ($P\approx989$~days). Blue and red indicate orbital periods below and above 100~days, respectively. The dashed lines and shaded $1\sigma$ regions indicate separate regressions to these two sets of planets. The slopes of these fits are discrepant to $2.6\sigma$, hinting that the mass--metallicity correlation for giant planets may be dependent upon orbital properties.}
\label{fig:zpzs}
\end{figure}
In Figure~\ref{fig:zpzs}, we plot the mass and metal enrichment of {TOI-2180~b}\ relative to the \citet{Thorngren2016} sample of giant exoplanets. We also include two other recently published high mass, giant exoplanets on long-period orbits: Kepler-1514~b \citep[$P\approx218$~days;][]{Dalba2021a} and Kepler-1704~b \citep[$P\approx989$~days;][]{Dalba2021c}. The metal enrichment of {TOI-2180~b}\ relative to its host star is consistent with other giant planets with similar mass and falls near to the best fit line for all of the \citet{Thorngren2016} sample.
The \citet{Thorngren2016} exoplanets are plotted in Figure~\ref{fig:zpzs} as circles. The points are given different colors based on orbital period. The two {\it Kepler}\ planets are shown as triangles and {TOI-2180~b}\ is shown as a square. Although each of these planets individually is fully consistent with the \citet{Thorngren2016} mass--metallicity correlation, there is possibly a subtle difference in the slope of the relation for the longer-period planets. For lower mass planets, enrichment appears to be higher than average for the longer-period objects and vice versa for the higher mass planets including {TOI-2180~b}.
We explored this possibility quantitatively by separating the planets in Figure~\ref{fig:zpzs} into short-period ($P<100$~days) and long-period ($P\ge100$~days) groups. The median orbital periods in the two groups were 10~days and 223~days. {TOI-2180~b}\ and the {\it Kepler}\ planets are also given the corresponding $P>100$~days color. Although 100~days is a somewhat arbitrary separation value, it possibly separates planets that experienced different formation histories. We conducted orthogonal distance regression fits, which include uncertainties on both the explanatory and response variables, to both sets of planet masses and metal enrichments in logspace. The resulting power-law fits are drawn as dashed lines in Figure~\ref{fig:zpzs}. The $1\sigma$ uncertainty regions are also shown. The fits for the short and long-period planets are ($10.4\pm1.6$)$M^{(-0.372\pm0.084)}$ and (14.6$\pm$2.9)$M^{(-0.81\pm0.14)}$, respectively. The slopes in these fits are inconsistent at $2.6\sigma$.
The mass--metallicity correlation derived for planets with orbital periods below 100~days is fully consistent with that measured by \citet{Thorngren2016}. On the other hand, the small set of long-period planets that includes {TOI-2180~b}\ produces a notably steeper correlation. We refrain from placing too much emphasis on this finding owing to the small number of data points and moderate statistical significance. The addition of any number of additional long-period giant planets would be elucidating. If this trend is real, though, it suggests that the current orbital properties of giant planets trace different heavy-element accretion mechanisms such as pebble or planetesimal \citep{Hasegawa2018}. A statistically robust analysis of the mass--metallicity correlation is warranted, but we leave such an analysis to future work.
For a solar system comparison, the {\it Galileo Entry Probe} measured volatile gases in Jupiter's atmosphere and identified enrichment of 2--6 for several heavy elements and noble gasses \citep{Wong2004}. More recently, the {\it Juno} spacecraft measured the equatorial water abundance on Jupiter to be 1--5 times the protosolar value \citep{Li2020}. The comparison between our enrichment measurement of {TOI-2180~b}\ and these measurements at Jupiter comes with caveats. For instance, the Jupiter enrichment is derived from its equatorial oxygen abundance while the exoplanet enrichment is a model dependent bulk value. The direct comparison of these two qualities may be problematic. However, the main point is that these values are all of a similar order of magnitude.
\section{Analysis of RV Drift}\label{sec:drift}
We characterized the trend and curvature in TOI-2180's RV time series using the technique described in \citet{Lubin2021}. Because TOI-2180 is not in the \textit{Hipparcos} catalog \citep{ESA1997}, we could not calculate its astrometric acceleration using the \textit{Hipparcos--Gaia} Catalog of Accelerations \citep{Brandt2018}. Instead we analyzed only the RV data, which leaves a degeneracy between companion mass and semi-major axis.
We sampled $10^8$ synthetic companion models, each comprising a mass $M_P$, semi-major axis $a$, eccentricity $e$, inclination $i$, argument of periastron $\omega$, and mean anomaly $M$. For each model companion, we calculated the trend ($\dot{\gamma}$) and curvature ($\ddot{\gamma}$) that such a companion would produce. We then calculated the model likelihood given the measured trend and curvature in Table \ref{tab:planet}, and marginalized over $\{e, i, \omega, M\}$ by binning the likelihoods in $a$-$M_P$ space. Figure \ref{fig:RV_drift} shows the joint posterior distribution for TOI-2180's companion.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RV_drift.png}
\caption{The set of $(a, M_P)$ models consistent with the measured RV trend and curvature at the 1$\sigma$ (dark) and 2$\sigma$ (light) levels. We rule out semi-major axes corresponding to orbital periods shorter than the observing baseline, as well as companion masses too low to produce the minimum RV amplitude.}
\label{fig:RV_drift}
\end{figure}
If located within a few au, the object is likely to be planetary and more massive than $\sim$2~$M_{\rm J}$. Past roughly 8~au, however, the object transitions to the substellar regime. We truncate the upper mass boundary of Figure~\ref{fig:RV_drift} at a few tenths of a solar mass at which point we would have expected to detect such a companion in the speckle imaging (Section~\ref{sec:imaging}). For a 13~$M_{\rm J}$ object, if we extend the RV monitoring by an additional $\sim$3.5~years, we will have sampled between 25\% and 100\% of the full orbit by period. At that point, we should be more capable of distinguishing between planetary and non-planetary scenarios.
\section{Discussion}\label{sec:disc}
With an equilibrium temperature of {348 K} (assuming a Jupiter-like Bond albedo, see Table~\ref{tab:planet}), {TOI-2180~b}\ qualifies as a temperate Jupiter that occupies an interesting region of parameter space. It exists within 1--3~au where the giant planet occurrence rate increases \citep[e.g.,][]{Wittenmyer2020,Fernandes2019,Fulton2021}, but it is not so close to its star that its orbit has been tidally circularized (e.g., following a high eccentricity migration pathway). This means that its orbital properties may contain information about previous migration. {TOI-2180~b}\ is warmer than Jupiter and Saturn, but it receives a weak enough irradiation to not be inflated. Planets like this are useful laboratories for models of interior and atmospheric structure and planet formation.
{TOI-2180~b}\ stands out among other transiting temperate Jupiters for two main reasons: its long orbital period and its host star's favorable brightness. In Figure~\ref{fig:MR}, we put {TOI-2180~b}\ in context with other giant planets ($M_P>0.2$~$M_{\rm J}$) with orbital periods greater than 20~days (to exclude hot Jupiters) with measured mass and radius that did not have a controversial flag\footnote{According to the NASA Exoplanet Archive, accessed 2021 September 5}. The {\it Kepler}\ mission has so-far proven most successful at discovering planets in this region of parameter space \citep[e.g.,][]{Wang2015b,Uehara2016,ForemanMackey2016b,Kawahara2019}. As a result, many of the planets in Figure~\ref{fig:MR} are sufficiently faint that follow-up opportunities to further characterize their systems are very limited. TESS, however, is slowly beginning to populate the long-period giant planet parameter space with systems orbiting bright host stars that are more amenable to follow-up characterization \citep[e.g.,][]{Eisner2020}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mass_radius_v1.pdf}
\caption{The subset of long-period ($P>20$~days), giant ($M_P>0.2$~$M_{\rm J}$) exoplanets with measured mass and radii sized to show their hosts' $V$ band magnitudes. {TOI-2180~b}\ is labeled and drawn with a dashed line. Of the four planets with the brightest host stars, {TOI-2180~b}\ has the longest period by far demonstrating how it is a valuable extension of the parameter space of temperate Jupiters.}
\label{fig:MR}
\end{figure}
As a temperate giant planet, the atmosphere of {TOI-2180~b}\ represents a stepping stone between well characterized hot Jupiters and the cold solar system gas giant planets. At {348~K}, we might expect to find non-equilibrium carbon chemistry that relies heavily on vertical transport \citep{Fortney2020}. Moreover, an atmospheric metallicity measurement of {TOI-2180~b}\ would be an ideal comparison to the now known stellar and bulk planetary metallicity. Perhaps the atmospheric metallicity will be lower than the bulk value since some amount of heavy elements are likely sequestered into a core. However, evidence from solar system observations suggests that giant planets may instead have interior regions of inward-decreasing metallicity \citep[e.g.,][]{Wahl2017,Guillot2018,Debras2019}. These theories could possibly be explored with transmission spectroscopy, which is suspected to probe CH$_4$ and the byproducts of disequilibrium chemistry and photolysis in long-period giant planets \citep{Dalba2015,Fortney2020}. However, owing to its high surface gravity and an unfavorable planet-star radius ratio, {TOI-2180~b}\ is likely not amenable to transmission spectroscopy. The atmospheric scale height of {TOI-2180~b}\ is only {$\sim$23~km}, which corresponds to a transit depth of a few parts per million (PPM) and a transmission spectroscopy metric of {7} \citep{Kempton2018}. Similarly, the near-infrared depth of the secondary eclipse is likely to be on the order of or less than 10~ppm (Table~\ref{tab:planet}). Any future endeavor to identify favorable targets for temperate Jupiter atmospheric characterization should examine whether the brightness of {TOI-2180}\ compensates for the difficulty introduced by its large stellar radius and the high surface gravity of {TOI-2180~b}.
Other avenues of follow-up characterization for the {TOI-2180}\ system are more promising. Continued RV monitoring of {TOI-2180}\ would eventually capture a sufficient fraction of the long-term acceleration to infer the properties of the outer companion. This RV trend could also be interpreted jointly with imaging and astrometric data to further constrain the outer object's orbit and inclination \citep[e.g.,][]{Crepp2012,Wittrock2016,Kane2019b,Brandt2021,Dalba2021b}. Be it a star or substellar object, it could have influenced the evolution and migration of {TOI-2180~b}, including eccentricity excitation through Kozai-Lidov oscillations \citep[e.g.,][]{Wu2003,Fabrycky2007,Naoz2012}. However, we need not invoke secular interactions \citep[e.g.,][]{Wu2011} or planet-planet scattering \citep[e.g.,][]{Rasio1996} to explain the moderate eccentricity ({$e\approx0.37$}) of {TOI-2180~b}. \citet{Debras2021} argued that disk cavity migration could explain warm Jupiter eccentricity up to $\sim$0.4. Additional modeling of the formation and dynamical evolution of the objects in the {TOI-2180}\ system, with comparison to the bulk metallicity of {TOI-2180~b}, would be illuminating.
The degeneracy between disk migration and secular interactions could possibly be broken with a measurement of the stellar obliquity via the Rossiter-McLaughlin (RM) effect \citep{Rossiter1924,McLaughlin1924}. Migration via interactions with a distant massive object would cause both orbits to be misaligned with the stellar spin. Alternatively, if it migrated in the disk and the disk is assumed to have been aligned with the stellar equator, we would expect little obliquity. This assumption may be problematic, though, as disks can be tilted such that they yield misaligned planets under disk migration \citep[e.g.,][]{Spalding2016b}. Nonetheless, the \texttt{SpecMatch} analysis of the {TOI-2180}\ spectrum returned a low rotational velocity of $v\sin{i} = 2.2\pm1.0$~km~s$^{-1}$. The largest possible amplitude of the RM effect would therefore be $\sim$9~m~s$^{-1}$ \citep{Winn2010b}, which is well within the capabilities of next-generation precise RV facilities such as MAROON-X \citep{Seifahrt2018} or the Keck Planet Finder \citep{Gibson2016}. The RM effect would also be detectable from either Keck-HIRES or APF-Levy, which have average internal RV precisions of 1.2~m~s$^{-1}$ and 3.6~m~s$^{-1}$, respectively. Achieving the proper timing for such an experiment given the 24~hr transit is challenging, though, and may not be feasible for several years. The scientific benefit of an RM detection for this system, and for such a long-period planet in general, further motivates the need to refine the transit ephemeris.
Lastly, we briefly consider {TOI-2180~b}\ as a host for exomoons. The TESS transit offered no evidence to suggest that an exomoon is present, but the stellar brightness, the long-period planetary orbit, and the (possibly) gentle migration history warrant raise the possibility of exomoon detection in this system. {TOI-2180}\ is sufficiently bright that the precision of a single-transit observation would likely reach the noise floors of NIRISS and NIRSpec \citep[several tens of PPM;][]{Greene2016,Batalha2017b}. A Ganymede-size moon would produce a $\sim$5~PPM transit, which would not be detectable. Alternatively, an Earth-size moon would yield a $\sim$30~PPM occultation, which is more reasonable. As more transits of {TOI-2180~b}\ are observed by TESS or any other facility, we recommend that the community conduct photodynamical modeling to test for variations in transit timing and duration that might indicate the presence or lack of an exomoon \citep[e.g.,][]{kipping2012b,Heller2014b,Kipping2021}.
\section{Summary}\label{sec:concl}
Single-transit events are the primary avenue to discovering exoplanets with orbital periods longer than approximately a month in TESS photometry. Here, we describe the follow-up effort surrounding a 24~hr long single-transit of {TOI-2180}\ (Figure~\ref{fig:tess}), a slightly evolved mid G type star, in Sector 19 data from TESS (Section \ref{sec:tess}). Citizen scientists identified the transit event shortly after the data became public, allowing a Doppler monitoring campaign with the APF telescope to begin immediately (Section~\ref{sec:spec}). After nearly two years of RV observations with the APF and Keck telescopes (Figure~\ref{fig:rv}), we determined that {TOI-2180~b}---a {2.8}~$M_{\rm J}$ giant planet on a {260.79}~day, eccentric ({0.368}) orbit---was the cause of the single-transit event (Section~\ref{sec:tran_rv}).
{TOI-2180~b}\ is a member of a rare but growing sample of valuable \emph{transiting} giant exoplanets with orbital periods in the hundreds of days. We conduct a thorough comprehensive fit to the transit and RV data to infer the stellar and planetary properties of this system (Section~\ref{sec:EFv2}). Our precise and regularly sampled RVs refine the ephemeris of {TOI-2180~b}, and we attempt to detect a second transit through 11~days of photometric observations with ground-based telescopes situated over three continents (Figure~\ref{fig:all_ground}). Although we do not detect a transit in these data, we develop a straightforward method to combine the orbital period posterior from the fit to the single-transit and RVs with the extensive collection of ground-based data sets (Section~\ref{sec:teers}). This analysis substantially refines the orbital period of {TOI-2180~b}\ by eliminating a substantial fraction of the most likely posterior solutions (Figure~\ref{fig:trimmed_posterior}), leaving the prediction that TESS will likely detect the transit of {TOI-2180~b}\ in January or February 2022 (Section~\ref{sec:future_tran}).
With a measured mass and radius for {TOI-2180~b}, we infer the bulk heavy-element content and metallicity relative to stellar from interior structure models (Section~\ref{sec:metal}). {TOI-2180~b}\ likely has {over 100~$M_{\earth}$} of heavy elements in its envelope and interior and is enriched relative to its host star by a factor of {$4.7\pm1.3$}. We place {TOI-2180~b}\ in context of the mass--metallicity correlation for giant planets in Figure~\ref{fig:zpzs}. Along with a few other recently characterized exoplanets on several hundred day orbital periods, {TOI-2180~b}\ suggests at 2.6$\sigma$ confidence that the relation between metal enrichment (relative to stellar) and mass for giant planets is dependent on orbital properties. We leave further analysis of this possibility to future work.
Lastly, we place the discovery of {TOI-2180~b}\ in context of other temperate giant planets with known mass and radius (Section~\ref{sec:disc}, Figure~\ref{fig:MR}). It is a poor candidate for transmission spectroscopy owing to its high surface gravity and the {1.6~$R_{\sun}$} radius of its host star. Still, this system is a promising target for continued RV monitoring and stellar obliquity measurement to test theories of how giant planets migrate within the occurrence rate increase near 1~au but not so close as to the become hot Jupiters. {TOI-2180~b}\ also remains as an excellent candidate for exomoon investigations since the host star brightness ($V=9.2$; $J=8.0$) makes it amenable to incredibly precise space-based photometry in the future. \\ \\
\section*{Acknowledgements}
The authors recognize and acknowledge the cultural role and reverence that the summit of Maunakea has within the indigenous Hawaiian community. We are deeply grateful to have the opportunity to conduct observations from this mountain.
The authors would like to thank the anonymous referee for helpful comments that improved this paper. We also thank Jack Lissauer for interesting conversations about giant planets that guided some of our analysis. We thank Ken and Gloria Levy, who supported the construction of the Levy Spectrometer on the Automated Planet Finder. We thank the University of California and Google for supporting Lick Observatory and the UCO staff for their dedicated work scheduling and operating the telescopes of Lick Observatory. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
This paper includes data collected by the {TESS}\ mission. Funding for the {TESS}\ mission is provided by the NASA's Science Mission Directorate. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.
P. D. is supported by a National Science Foundation (NSF) Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1903811. E.A.P. acknowledges the support of the Alfred P. Sloan Foundation. L.M.W. is supported by the Beatrice Watson Parrent Fellowship and NASA ADAP Grant 80NSSC19K0597. A.C. is supported by the NSF Graduate Research Fellowship, grant No. DGE 1842402. D.H. acknowledges support from the Alfred P. Sloan Foundation, the National Aeronautics and Space Administration (80NSSC21K0652), and the National Science Foundation (AST-1717000). I.J.M.C. acknowledges support from the NSF through grant AST-1824644. R.A.R. is supported by the NSF Graduate Research Fellowship, grant No. DGE 1745301. C. D. D. acknowledges the support of the Hellman Family Faculty Fund, the Alfred P. Sloan Foundation, the David \& Lucile Packard Foundation, and the National Aeronautics and Space Administration via the TESS Guest Investigator Program (80NSSC18K1583). J.M.A.M. is supported by the NSF Graduate Research Fellowship, grant No. DGE-1842400. J.M.A.M. also acknowledges the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant No. 1829740, the Brinson Foundation, and the Moore Foundation; his participation in the program has benefited this work. T.F. acknowledges support from the University of California President's Postdoctoral Fellowship Program. N.E. acknowledges support from the UK Science and Technology Facilities Council (STFC) under grant code ST/R505006/1. D. D. acknowledges support from the TESS Guest Investigator Program grant 80NSSC19K1727 and NASA Exoplanet Research Program grant 18-2XRP18\_2-0136. M. R. is supported by the National Science Foundation Graduate Research Fellowship Program under Grant Number DGE-1752134.
P. D. thanks the Walmart of Yucca Valley, California for frequent and prolonged use of its wireless internet to upload hundreds of gigabytes of data.
\facilities{Keck:I (HIRES), Automated Planet Finder (Levy), TESS, Gemini:Gillett (`Alopeke)}, LCOGT\\
\vspace{5mm}
\software{
\texttt{batman} \citep{Kreidberg2015b}, \\
\texttt{EXOFASTv2} \citep{Eastman2013,Eastman2017,Eastman2019},
\texttt{lightkurve} \citep{Lightkurve2018},
\texttt{RadVel} \citep{Fulton2018}
\texttt{SpecMatch} \citep{Petigura2015,Petigura2017b},
\texttt{AstroImageJ} \citep{Collins2017}
\texttt{LcTools} \citep{Schmitt2019}
} \\
\section{Introduction}\label{sec:intro}
Gas giant planets have been found to reside in many extrasolar planetary systems. The diversity in their sizes, masses, orbits, compositions, and formation pathways has been the subject of numerous studies. However, selection biases often cloud our understanding. For instance, the sensitivity of the transit method wanes for planets on orbits beyond a few tenths of an au owing to the inverse relation between transit probability and semi-major axis. Consequently, the vast majority of known exoplanets with $\gtrsim$1~au orbits have been discovered via Doppler spectroscopy \citep[e.g.,][]{Mayor2011,Fulton2021}. Planet mass can be inferred from time series radial velocity (RV) observations but, without a transit, the planet radius and thereby bulk density remains unknown.
A critical component of the bulk composition for giant planets is the total mass of elements heavier than H and He \citep[e.g.,][]{Guillot2006,Miller2011,Thorngren2016,Teske2019}. This property is inferred using structural evolution models along with the measured planetary mass, radius, stellar age, and incident flux \citep[e.g.,][]{Thorngren2019a}. Numerous theoretical planet formation studies have found that the correlation between the total mass of heavy elements in giant planets---or, similarly, the metal enrichment relative to the host star---and planet mass is a useful tracer of planet formation processes \citep[e.g.,][]{Mordasini2014,Hasegawa2018,Ginzburg2020,Shibata2020}. Probing this giant planet mass--metallicity correlation along a third axis, in orbital properties (i.e., period or separation), could prove informative. Yet, previous efforts have simply not had a large enough sample size of giant planets on orbits wider than a few 0.1~au to conduct such an investigation \citep{Miller2011,Thorngren2016}.
Transit surveys such as the {\it Kepler}\ mission \citep{Borucki2010,Thompson2018} and the Transiting Exoplanet Survey Satellite (TESS) mission \citep{Ricker2015} occasionally detect giant planet candidates with orbital periods of 100 to 1,000 days \citep[e.g.,][]{Wang2015b,ForemanMackey2016b,Osborn2016,Uehara2016,Herman2019,Kawahara2019,Eisner2021}. These candidates warrant close scrutiny, at least based on the $\sim$50\% false-positive rate for giant planets with similar orbits found for the {\it Kepler}\ mission \citep{Santerne2014,Dalba2020b}. Following vetting, long-term RV monitoring is often required to measure the planet mass, which then enables modeling of the bulk metallicity \citep[e.g.,][]{Beichman2016,Dubber2019,Santerne2019,Dalba2021a,Dalba2021c}. {TESS}\ planets with orbital periods on the order of 100~days or more will typically be detected as single-transit events owing to the observational strategy of the TESS mission \citep[e.g.,][]{LaCourse2018,Villanueva2019,Diaz2020,Cooke2021}. In this case, the RV monitoring is also necessary to determine the orbital period.
Here, we describe the discovery and investigation of a 24-hour single-transit event observed for HD~238894, hereafter referred to as {TOI-2180~b}. The host star, {TOI-2180}\ (TIC~298663873), is a bright ($V=9.2$), slightly evolved G5 star. The confirmation of {TOI-2180~b}\ is notable relative to the current sample of TESS exoplanets, including those identified as single-transits, as it is the first to surpass an orbital period of 100~days\footnote{According to the list of confirmed TESS exoplanets on the NASA Exoplanet Archive (\url{https://exoplanetarchive.ipac.caltech.edu/index.html}) as of 2021 September 2.}. In Section~\ref{sec:obs}, we describe the {TESS}\ observations of {TOI-2180}\ along with the follow-up photometric, imaging, and spectroscopic data sets. In Section~\ref{sec:model}, we discuss the consistency of the transit and RV data, which are then jointly modeled to determine the {TOI-2180}\ system properties. We find that {TOI-2180~b}\ is a {2.7~$M_{\rm J}$} giant planet on a {261~day} orbit with an orbital eccentricity of {0.368}. In Section~\ref{sec:teers}, we describe a global effort to detect an additional transit of {TOI-2180~b}\ from the ground. Although this effort failed to detect the transit, the nondetections provided a substantial improvement in the precision on the orbital period. In Section~\ref{sec:metal}, we determine the bulk heavy-element mass of {TOI-2180~b}. In Section~\ref{sec:drift}, we use the long-term acceleration of {TOI-2180}\ to place limits on the properties of a distant massive companion. In Section~\ref{sec:disc}, we place {TOI-2180~b}\ in the context of other transiting giant planets and discuss the prospects for future characterization of the system. Lastly, in Section~\ref{sec:concl}, we summarize our work.
\section{Observations}\label{sec:obs}
\subsection{TESS Photometry}\label{sec:tess}
TESS observed {TOI-2180}\ (TIC~298663873) at 2-min (fast) cadence for the entirety of Cycle~2 of its primary mission (Sectors~14--26) and in Sectors~40 and 41 of its extended mission. The image data were processed by the Science Processing Operations Center (SPOC) at NASA Ames Research Center \citep{Jenkins2016} to extract photometry from this target. A search for transiting planets failed to find a transit signature with two or more transits. Only a single transit event was observed in Sector~19 (2019 November 28 through 2019 December 22). This Sector~19 single-transit was first identified by citizen scientists with the light curve processing software \texttt{LcTools} \citep{Schmitt2019}, leading to early commencement of our RV follow-up campaign. In May 2020, the Planet Hunters TESS collaboration \citep{Eisner2020} announced this star as a Community TESS Object of Interest (cTOI). In August 2020, its disposition was elevated to TOI \citep{Guerrero2021}.
We downloaded the Sector~19, 2-min cadence light curve of {TOI-2180}\ from the Mikulski Archive for Space Telescopes (MAST) using the \texttt{lightkurve} package \citep{Lightkurve2018}. We used the pre-search data conditioning simple aperture photometry (PDCSAP) flux for which most of the high frequency noise was removed \citep{Stumpe2012,Stumpe2014,Smith2012}. The PDCSAP light curve exhibited low frequency noise features that we removed using a Savitzky--Golay filter after masking the clear transit event. The raw and flattened light curves, centered on the $\sim$0.5\% transit of {TOI-2180~b}, are shown in Figure~\ref{fig:tess}.
\begin{figure}
\centering
\begin{tabular}{c}
\includegraphics[width=\columnwidth]{raw_pdc_v1.pdf} \\
\includegraphics[width=\columnwidth]{transits_v1.pdf}
\end{tabular}
\caption{The single-transit of {TOI-2180~b}\ observed by TESS with short cadence. {\it Top:} unflattened PDCSAP flux and the trend from the Savitsky--Golay filter. {\it Bottom:} flattened light curve. The red points are individual exposures (gray points) binned by a factor of 40. The blue line shows the best fit transit model.}
\label{fig:tess}
\end{figure}
Under the assumption of a circular central transit, \citet[][Equation~19]{Winn2010b} showed how the transit duration ($T$) and the stellar bulk density ($\rho_{\star}$) can give an estimate of the orbital period ($P$) following
\begin{equation}\label{eq:P}
T \approx 13\;\mathrm{hr} \; \left (\frac{P}{1\; \mathrm{yr}} \right )^{1/3} \left (\frac{\rho_{\star}}{\rho_{\sun}} \right )^{-1/3}
\end{equation}
\noindent where $P$ has units of years and $\rho_{\sun}$ is the solar bulk density. For $T=24$~hr and $\rho_{\star}=0.335$~g~cm$^{-3}$ from the TESS Input Catalog \citep[TIC;][]{Stassun2019}, Equation~\ref{eq:P} gives $P\approx547\pm111$~days assuming reasonable uncertainty in $T$ and $\rho_{\star}$. Combined with the nondetection of a matching transit event in the other Cycle 2 sectors, this possible orbital period suggested that {TOI-2180~b}\ was likely to be one of only a few long-period planets predicted to be detected by TESS through single-transit events \citep{Villanueva2019}.
\subsection{Speckle Imaging}\label{sec:imaging}
We acquired a high-resolution speckle image of {TOI-2180}\ to search for nearby neighboring stars that might indicate the false-positive nature of the TESS single-transit event. We observed {TOI-2180}\ on 2020 June 6 using the `Alopeke
speckle instrument on the Gemini-North telescope\footnote{\url{https://www.gemini.edu/instrumentation/alopeke-zorro}} located on Maunakea in Hawai`i. `Alopeke acquires an image of the star in a blue (562~nm) and red (832~nm) band simultaneously. From these images, we derived contrast curves that show the limiting magnitude difference ($\Delta m$) in each band as a function of angular separation \citep{Howell2011}. As shown in Figure~\ref{fig:speckle}, we achieved a $\sim$5~mag contrast at 0$\farcs$1 and a nondetection of sources with $\Delta m$ = 5--8.5 within 1$\farcs$2 of {TOI-2180}. Given the distance to {TOI-2180}\ ({116~pc}; Table~\ref{tab:stellar}), 0$\farcs$1 projected separation corresponds to {$\sim$12~au}. Although we cannot rule out a scenario whereby a luminous companion star was near conjunction with {TOI-2180}\ at the time of our speckle observation, we assume in what follows that {TOI-2180}\ is likely a single star. The speckle imaging nondetection suggests that any massive object in this system other than {TOI-2180~b}\ is likely to be a late M star or a substellar object.
The conclusion of {TOI-2180}\ being a single star is further supported by the renormalized unit weight error (RUWE) as determined by Gaia Data Release (DR) 2 \citep{Gaia2018}. The RUWE value for {TOI-2180}\ is 1.01, which is typical of a single star \citep[e.g.,][]{Belokurov2020}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{speckle.png}
\caption{Limiting magnitudes ($\Delta m$) for the nondetection of a neighboring star based on the speckle imaging from `Alopeke. The inset shows the speckle image at 832~nm.}
\label{fig:speckle}
\end{figure}
\subsection{Spectroscopy}\label{sec:spec}
Immediately following the discovery of the TESS single-transit of {TOI-2180}, we began a Doppler monitoring campaign with the Automated Planet Finder (APF) telescope at Lick Observatory \citep{Radovan2014,Vogt2014} as part of the TESS-Keck Survey (TKS). TKS is a collaborative effort between the University of California, the California Institute of Technology, the University of Hawai`i,
and NASA aimed at providing multi-site RV follow-up for many of the planetary systems discovered by TESS \citep[e.g.,][]{Dai2020,Dalba2020a,Chontos2021,Lubin2021,Rubenzahl2021,Weiss2021}. The APF uses the Levy Spectrograph, a high-resolution ($R\approx$ 114,000) slit-fed optical echelle spectrometer \citep{Radovan2010} that is ideal for bright (V$\le$9) stars such as {TOI-2180}\ \citep{Burt2015}. The starlight passes through a heated iodine gas cell that allows for precise wavelength calibration and instrument profile tracking. The precise RV is inferred from each spectrum using a forward modeling procedure \citep{Butler1996,Fulton2015a}. Table~\ref{tab:rvs} lists the RVs collected in our campaign.
\begin{deluxetable}{cccc}
\tabletypesize{\scriptsize}
\tablecaption{RV Measurements of {TOI-2180} \label{tab:rvs}}
\tablehead{
\colhead{BJD$_{\rm TDB}$} &
\colhead{RV (m s$^{-1}$)} &
\colhead{$S_{\rm HK}$$^a$} &
\colhead{Tel.}}
\startdata
2458888.063868 & $-16.1\pm3.9$ & $0.1304\pm0.0020$ & APF\\
2458894.911472 & $-29.1\pm4.5$ & $0.1476\pm0.0020$ & APF\\
2458899.027868 & $-36.3\pm3.8$ & $0.1463\pm0.0020$ & APF\\
2458906.015644 & $-43.6\pm3.2$ & $0.1093\pm0.0020$ & APF\\
2458914.011097 & $-38.9\pm3.3$ & $0.1506\pm0.0020$ & APF\\
2458954.765221 & $-59.0\pm4.0$ & $0.1468\pm0.0020$ & APF\\
2458961.864505 & $-49.7\pm4.3$ & $0.1323\pm0.0020$ & APF\\
2458964.879814 & $-53.3\pm3.1$ & $0.1298\pm0.0020$ & APF\\
2458965.811833 & $-47.5\pm4.2$ & $0.1363\pm0.0020$ & APF\\
2458966.879965 & $-65.9\pm5.5$ & $0.1204\pm0.0020$ & APF\\
\enddata
\tablenotetext{a}{The $S_{\rm HK}$ values from APF and Keck data have different zero-points.}
\tablenotetext{}{This is a representative subset of the full data set. The full table will be made available in machine readable format.}
\end{deluxetable}
In August 2020, Lick Observatory and the APF shut down owing to nearby wildfires. To maintain our coverage of the emerging Keplerian RV signal, we temporarily conducted observations of {TOI-2180}\ using the High Resolution Echelle Spectrometer \citep[HIRES;][]{Vogt1994} on the Keck~I telescope at W. M. Keck Observatory. The reduction and analysis procedure for Keck-HIRES spectra is broadly similar to that for data from the APF \citep[e.g.,][]{Howard2010}. The full version of Table~\ref{tab:rvs} also contains the Keck-HIRES RVs of {TOI-2180}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{rv_v1.pdf}
\caption{RV measurements of {TOI-2180}. {\it Panel a:} the RV time series after subtraction of an offset velocity from each data set. Transit windows are shown in green. {\it Panel b:} the residuals between the RV time series and best fit planet model not including acceleration terms. A quadratic trend is visible on top of the Keplerian signal from {TOI-2180~b}. {\it Panel c:} the phase-folded RVs after removal of the acceleration terms. A phase of 0 corresponds to conjunction (transit).}
\label{fig:rv}
\end{figure*}
The time series RVs from both telescopes are shown in the top panel of Figure~\ref{fig:rv}. Our {1.5~yr} baseline captured {two} periods of a 261~day, eccentric ($e\approx0.4$) Keplerian signal as well as a long-term acceleration. In Section~\ref{sec:model} we will discuss the consistency of the RV data with the single-transit from TESS and conduct a joint modeling of the system parameters.
Each RV measurement is accompanied by an estimate of stellar chromospheric activity approximated as the $S_{\rm HK}$ index \citep{Isaacson2010}. The indicators calculated for each instrument have different zero points. The Pearson correlation coefficient between the $S_{\rm HK}$ indices and the RVs for APF and Keck are {$-0.10$ (from 84 data points)} and {$0.68$ (from 15 data points)}, respectively. Although the Keck data demonstrate a positive correlation significant to {3.3$\sigma$}, the much larger APF data set shows no correlation {($<1\sigma$)}. Furthermore, we do not detect significant periodicity in the $S_{\rm HK}$ time series. Both of these findings suggest that the RV signals are not affected, or only minimally so, by stellar activity.
The extraction of the APF and Keck RVs rely on a high signal-to-noise template spectrum of {TOI-2180}\ acquired with Keck-HIRES without the iodine cell \citep{Butler1996}. We also used this spectrum for a basic spectroscopic analysis of {TOI-2180}\ with \texttt{SpecMatch} \citep{Petigura2015,Petigura2017b}. This analysis indicated that {TOI-2180}\ has an effective temperature of $T_{\rm eff} = 5739\pm100$~K, a surface gravity of $\log{g} = 4.00\pm0.10$, and a relative iron abundance (metallicity proxy) of [Fe/H] = 0.25$\pm$0.06~dex consistent with a slightly evolved mid-G type star. These values are in close agreement with those listed in the TESS Input Catalog \citep{Stassun2019}.
\subsection{Ground-based Photometry}
The single-transit detection for {TOI-2180}\ by TESS and the subsequent RV follow-up effort allowed us to plan a ground-based photometry campaign with the goal of detecting another transit of {TOI-2180~b}. This campaign occurred in late August and early September of 2020 around the time of the first transit of {TOI-2180~b}\ since the one observed by TESS. We acquired 55 photometric data sets of {TOI-2180}\ comprised of $\sim$20,000 individual exposures spanning 11~days and 14 sites. Contributors to this campaign consisted of a mix of professional and amateur astronomers.
The quality of the ground-based telescope data sets varied widely. Some data achieved precision sufficient to rule out ingress or egress events for the relatively shallow (0.5\%) transit of {TOI-2180~b}. Other data showed correlated noise or systematic errors several times this magnitude. We leave a detailed discussion of the individual observing sites for Appendix~\ref{app:ground}. In Section~\ref{sec:teers}, we will discuss our treatment of the ground-based data as a whole and our procedure for using it to refine the transit ephemeris of {TOI-2180~b}.
\section{Modeling}\label{sec:model}
\subsection{Consistency between the Transit and the RV Data}\label{sec:tran_rv}
A single-transit event only places a weak constraint on orbital period even under various simplifying assumptions \citep[e.g.,][]{Yee2008,Yao2021}. We must therefore assess whether the same object caused the single-transit and Doppler reflex motion of the host star.
In Section~\ref{sec:tess}, we estimated that the duration of the single-transit of {TOI-2180~b}\ and the stellar density listed in the TIC corresponded to a 547-day orbital period assuming a circular, central transit. However, the RV measurements of {TOI-2180}\ suggest a shorter period that has significant eccentricity. Depending on the argument of periastron ($\omega_{\star}$), orbital eccentricity can lead to shorter or longer transit duration compared to that from a circular orbit \citep[e.g.,][]{Kane2012}, which can bias the orbital period estimation from a single-transit. Orbital eccentricity also increases the transit probability \citep{Kane2007}, which is inherently low for objects with au-scale orbital distances.
We can account for eccentricity in our estimation of the orbital period to good approximation by including a factor of $\sqrt{1-e^2}/(1+e\sin{\omega_{\star}})$ on Equation~\ref{eq:P} \citep{Winn2010b}. We determined $e$ and $\omega_{\star}$ by conducting a Keplerian model fit to the APF and Keck RVs using the \texttt{RadVel} modeling toolkit\footnote{\url{https://radvel.readthedocs.io/}} \citep{Fulton2018}. In this fit, we applied a normal prior on the time of conjunction (BJD$_{\rm TDB} = 2458830.79\pm0.05$) using the transit timing from the Planet Hunters TESS characterization of the single-transit \citep{Eisner2021}. The fit converged quickly and we found that $e = 0.367\pm0.0074$, $\omega_{\star} = -0.76\pm0.023$~rad, and $P = 260.68 \pm 0.54$~days. This argument of periastron corresponds to a time of periastron that is $70.0\pm1.2$~days prior to transit. Therefore, at the time of transit, the orbital velocity of {TOI-2180~b}\ is decreasing.
Reevaluating Equation~\ref{eq:P} with the factor to account for an eccentric orbit gives $283\pm59$~days, which is consistent with the period of the Keplerian signal in the RVs. Considering the uncertainty introduced by the transit impact parameter could further explain the difference between this orbital period estimate and the observed {261}~day period. This result demonstrates self-consistency with our assumption of a T$_0$ value in the \texttt{RadVel} fit. Therefore, we continue our analysis with the implicit assumption that the single-transit in the photometry and the Keplerian signal in the RVs can be ascribed to the same planet: {TOI-2180~b}.
\subsection{Comprehensive System Modeling}\label{sec:EFv2}
We modeled the stellar and planetary parameters for the {TOI-2180}\ system using the \texttt{EXOFASTv2} modeling suite \citep{Eastman2013,Eastman2019}. We include the TESS single-transit photometry, all of the RVs from Keck-HIRES and APF-Levy, and archival broadband photometry of {TOI-2180}\ from Gaia DR 2 \citep{Gaia2018}, the Two Micron All Sky Survey \citep{Cutri2003}, and the Wide-field Infrared Survey Explorer \citep{Cutri2014}. \texttt{EXOFASTv2} computes spectral energy distributions from MIST models and the Gaia parallax measurements and fits them to the archival photometry to infer the properties of the host star. We placed normal priors on $T_{\rm eff}$ and [Fe/H] based on the \texttt{SpecMatch} analysis of the high S/N template spectrum of {TOI-2180}\ (Section~\ref{sec:spec}). The width of the $T_{\rm eff}$ prior was inflated to 115~K (2\%) and a noise floor of 2\% was applied to bolometric flux used in the SED determination to account for systematic uncertainties inherent in the MIST models \citep{Tayar2020}. We also placed a uniform prior on extinction ($A_{\rm V}\le0.1376$) using the galactic dust maps of \citet{Schlafly2011} and a normal prior on parallax ($\varpi = 8.597\pm0.017$~mas) using the Gaia DR3 measurement corrected for the zero point offset \citep{Lindegren2021,Gaia2021}. These priors are summarized at the top of Table~\ref{tab:stellar}.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecaption{Median values and 68\% confidence intervals for the stellar parameters for {TOI-2180}. \label{tab:stellar}}
\tablehead{\colhead{~~~Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\multicolumn{2}{l}{Informative Priors:}& \smallskip\\
~~~~$T_{\rm eff}$ &Effective Temperature (K) & $\mathcal{N}(5739,115)$\\
~~~~$[{\rm Fe/H}]$ &Metallicity (dex) & $\mathcal{N}(0.25,0.06)$\\
~~~~$\varpi$ &Parallax (mas) & $\mathcal{N}(8.597,0.017)$\\
~~~~$A_V$ &V-band extinction (mag) & $\mathcal{U}(0,0.1376)$\\
\smallskip\\\multicolumn{2}{l}{Stellar Parameters:}&\smallskip\\
~~~~$M_*$ &Mass (\msun) &$1.111^{+0.047}_{-0.046}$\\
~~~~$R_*$ &Radius (\rsun) &$1.636^{+0.033}_{-0.029}$\\
~~~~$L_*$ &Luminosity (\lsun) &$2.544^{+0.091}_{-0.093}$\\
~~~~$F_{Bol}$ &Bolometric Flux (cgs) &$6.01\times10^{-9}$$^{+2.1\times10^{-10}}_{-2.2\times10^{-10}}$\\
~~~~$\rho_*$ &Density (g~cm$^{-3}$) &$0.359^{+0.015}_{-0.016}$\\
~~~~$\log{g}$ &Surface gravity (cgs) &$4.057^{+0.015}_{-0.016}$\\
~~~~$T_{\rm eff}$ &Effective Temperature (K) &$5695^{+58}_{-60}$\\
~~~~$[{\rm Fe/H}]$ &Metallicity (dex) &$0.253\pm0.057$\\
~~~~$[{\rm Fe/H}]_{0}$ &Initial Metallicity$^{a}$ (dex) &$0.269^{+0.055}_{-0.054}$\\
~~~~${\rm Age}$ &Age (Gyr) &$8.1^{+1.5}_{-1.3}$\\
~~~~$EEP$ &Equal Evolutionary Phase$^{b}$ &$452.1^{+3.9}_{-5.0}$\\
~~~~$A_V$ &V-band extinction (mag) &$0.077^{+0.041}_{-0.048}$\\
~~~~$\sigma_{SED}$ &SED photometry error scaling &$0.69^{+0.24}_{-0.16}$\\
~~~~$\varpi$ &Parallax (mas) &$8.597\pm0.017$\\
~~~~$d$ &Distance (pc) &$116.32\pm0.23$\\
\enddata
\tablenotetext{}{See Table~3 in \citet{Eastman2019} for a detailed description of all parameters and all default (non-informative) priors beyond those specified here. $\mathcal{N}(a,b)$ denotes a normal distribution with mean $a$ and variance $b^2$. $\mathcal{U}(a,b)$ denotes a uniform distribution over the interval [$a$,$b$].}
\tablenotetext{a}{Initial metallicity is that of the star when it formed.}
\tablenotetext{b}{Corresponds to static points in a star's evolutionary history. See Section~2 of \citet{Dotter2016}.}
\end{deluxetable}
We gauged convergence of the \texttt{EXOFASTv2} fit by the number of independent draws \citep{Ford2006b}, which exceeded 1,000, and the Gelman--Rubin statistic \citep{Gelman1992}, which was smaller than 1.01, for every fitted parameter. From the fit parameters, numerous properties of {TOI-2180~b}\ were derived. Table~\ref{tab:planet} lists all relevant planetary parameters for {TOI-2180~b}. The derived planetary radius and transit depth assume no oblateness \citep[e.g.,][]{Seager2002} and no system of rings \citep[e.g.,][]{Barnes2004,Akinsanmi2018}, although {TOI-2180~b}\ could plausibly have both.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecaption{Median values and 68\% confidence interval of the planet parameters for {TOI-2180~b}. \label{tab:planet}}
\tablehead{\colhead{~~~Parameter} & \colhead{Units} & \colhead{Values}}
\startdata
\multicolumn{2}{l}{Planetary Parameters:}&\smallskip\\
~~~~$P$ &Period$^{a}$ (d) &$260.79^{+0.59}_{-0.58}$\\
~~~~$R_P$ &Radius (\rj) &$1.010^{+0.022}_{-0.019}$\\
~~~~$M_P$ &Mass (\mj) &$2.755^{+0.087}_{-0.081}$\\
~~~~$T_C$ &Time of conjunction (\bjdtdb) &$2458830.7652\pm0.0010$\\
~~~~$a$ &Semi-major axis (au) &$0.828\pm0.012$\\
~~~~$i$ &Inclination (degrees) &$89.955^{+0.032}_{-0.044}$\\
~~~~$e$ &Eccentricity &$0.3683\pm0.0073$\\
~~~~$\omega_*$ &Argument of Periastron (degrees) &$-43.8\pm1.3$\\
~~~~$T_{eq}$ &Equilibrium temperature$^{b}$ (K) &$348.0^{+3.3}_{-3.6}$\\
~~~~$\tau_{\rm circ}$ &Tidal circularization timescale (Gyr) &$54600000^{+3800000}_{-4500000}$\\
~~~~$K$ &RV semi-amplitude (m~s$^{-1}$) &$87.75^{+0.98}_{-0.99}$\\
~~~~$\dot \gamma$ &RV slope$^{c}$ (m~s$^{-1}$~day$^{-1}$) &$-0.1205\pm0.0043$\\
~~~~$\ddot \gamma$ &RV quadratic term$^{c}$ (m~s$^{-1}$~day$^{-2}$) &$0.000214^{+0.000039}_{-0.000038}$\\
~~~~$\delta_{\rm TESS}$ &Transit depth in TESS band &$0.004766^{+0.000078}_{-0.000076}$\\
~~~~$\tau$ &Ingress/egress transit duration (d) &$0.06044^{+0.0019}_{-0.00063}$\\
~~~~$T_{14}$ &Total transit duration (d) &$1.0040^{+0.0032}_{-0.0031}$\\
~~~~$b$ &Transit Impact parameter &$0.100^{+0.095}_{-0.070}$\\
~~~~$\rho_P$ &Density (g~cm$^{-3}$) &$3.32^{+0.14}_{-0.16}$\\
~~~~$logg_P$ &Surface gravity (cgs) &$3.827^{+0.012}_{-0.015}$\\
~~~~$\fave$ &Incident Flux (\fluxcgs) &$0.00442\pm0.00018$\\
~~~~$T_P$ &Time of Periastron (\bjdtdb) &$2458760.7\pm1.3$\\
~~~~$T_S$ &Time of eclipse (\bjdtdb) &$2458745.41^{+1.00}_{-1.0}$\\
~~~~$b_S$ &Eclipse impact parameter &$0.060^{+0.056}_{-0.042}$\\
~~~~$\tau_S$ &Ingress/egress eclipse duration (days) &$0.03593^{+0.00098}_{-0.00088}$\\
~~~~$T_{S,14}$ &Total eclipse duration (days) &$0.599\pm0.013$\\
~~~~$\delta_{S,2.5\mu m}$ &Blackbody eclipse depth at 2.5$\mu$m (ppm) &$0.00237^{+0.00034}_{-0.00032}$\\
~~~~$\delta_{S,5.0\mu m}$ &Blackbody eclipse depth at 5.0$\mu$m (ppm) &$1.54\pm0.10$\\
~~~~$\delta_{S,7.5\mu m}$ &Blackbody eclipse depth at 7.5$\mu$m (ppm) &$11.29^{+0.49}_{-0.50}$\\
\smallskip\\\multicolumn{2}{l}{TESS Parameters:}& \smallskip\\
~~~~$u_{1}$ &linear limb-darkening coefficient &$0.316\pm0.029$\\
~~~~$u_{2}$ &quadratic limb-darkening coefficient &$0.248\pm0.044$\\
\smallskip\\\multicolumn{2}{l}{APF-Levy Parameters:}& \smallskip\\
~~~~$\gamma_{\rm rel}$ &Relative RV Offset$^{c}$ (m~s$^{-1}$) &$-30.0^{+1.2}_{-1.3}$\\
~~~~$\sigma_J$ &RV Jitter (m~s$^{-1}$) &$4.16^{+0.67}_{-0.62}$\\
\smallskip\\\multicolumn{2}{l}{Keck-HIRES Parameters:}& \smallskip\\
~~~~$\gamma_{\rm rel}$ &Relative RV Offset$^{c}$ (m~s$^{-1}$) &$-12.7^{+1.4}_{-1.5}$\\
~~~~$\sigma_J$ &RV Jitter (m~s$^{-1}$) &$4.86^{+1.3}_{-0.93}$\\
\enddata
\label{tab:HD238894.}
\tablenotetext{}{See Table 3 in \citet{Eastman2019} for a detailed description of all parameters and all default (non-informative) priors.}
\tablenotetext{a}{This orbital period is derived from the full posterior from the \texttt{EXOFASTv2} fit. See Section~\ref{sec:teers} for a description of the likely orbital period values ({$260.18^{+0.19}_{-0.30}$}~days or {$261.76^{+0.29}_{-0.16}$}~days) after the ground-based photometric transit recovery campaign for {TOI-2180}.}
\tablenotetext{b}{Assumes a Jupiter-like Bond albedo (0.34) and perfect heat redistribution.}
\tablenotetext{c}{Reference epoch = 2459157.439110}
\end{deluxetable}
The TESS data and the best fit transit model are shown in Figure~\ref{fig:tess}. All RV data and the best fit RV model are shown in Figure~\ref{fig:rv}. We included a quadratic function of time in the fit to account for the force from an additional planet or star on {TOI-2180}. Both coefficients in the quadratic function are greater than {$5\sigma$} discrepant from zero, suggesting that there is indeed a long-term variation in the RVs. Even with more than a {500}~day baseline of observations, we do not sample enough of the long-term signal to determine its cause. The lack of correlation between the RVs and the $S_{\rm HK}$ activity indicators disfavors stellar activity as the true explanation (Section~\ref{sec:spec}). Instead, we suggest that another massive object is orbiting {TOI-2180}\ (Section~\ref{sec:drift}).
Owing to the well sampled time series of precise RVs, the posterior for orbital period for the single-transit {TOI-2180~b}\ has a standard deviation below 1~day ({0.23\%}) that is well characterized by a normal distribution. In the following section, we will further constrain the orbital period of {TOI-2180~b}\ based on ground based photometry acquired near the timing of an additional transit.
\section{Ephemeris Refinement from Ground-based Observations}\label{sec:teers}
With each new RV we acquired of {TOI-2180}, we calculated the timing of the next transit of {TOI-2180~b}. The second transit (i.e., the first transit to occur since the TESS single-transit) occurred at some point in late August or early September of 2020. At that time, the $2\sigma$ transit window---which was almost entirely determined by the uncertainty on orbital period---was approximately 10~days wide\footnote{Note that the uncertainty for orbital period reported in Table~\ref{tab:planet} is informed by the full RV data set and is therefore much smaller than the corresponding estimate in August 2020.}.
We attempted the formidable task of observing this 24-hour-long, relatively shallow (0.5\%) transit by planning a global ground-based photometry campaign. In total, {15} telescopes acquired {55} data sets containing over {20,000} individual exposures of {TOI-2180}\ spanning 11~days. These included professional observatories, amateur observatories, and two portable digital {\it eVscope} telescopes. Each data set was processed with standard differential aperture photometry using background stars as references. Basic information about each telescope is provided in Appendix~\ref{app:ground}, a summary of each observation is listed in Table~\ref{tab:ground_obs}, and all data are plotted in Figure~\ref{fig:all_ground}.
None of the observations provided a conclusive detection of a {TOI-2180~b}\ transit. As has been the case for other attempts to detect transits of long-period planets \citep[e.g.,][]{Winn2009,Dalba2016,Dalba2019c}, any single observation could only observe out-of-transit baseline and ingress or egress at best. This photometric signature can be easily mimicked by flux variations between the target and reference stars as the airmass changed. On the other hand, owing to the duration of the transit and the difficulty of absolute flux calibration at the precision of the transit depth, distinguishing a fully in-transit observation from a fully out-of-transit one is challenging. Also, the $\sim$0.5\% transit depth was on the order of the noise floor for many of the sites. All of these factors contributed to the nondetection.
However, despite the lack of an obvious transit detection, we developed a straightforward method to refine the orbital period of {TOI-2180~b}\ by simultaneously searching all data sets for times where ingress or egress \emph{did not} occur to high statistical significance. Ruling out these times rules out chunks of the orbital period posterior. Conversely, at times when we cannot determine if ingress or egress occurred---or if ingress or egress even appears to be favored by the data---we do not rule out those portions of the posterior.
We began with the relative light curves for each data set, which were the aperture flux values of {TOI-2180}\ divided by the aperture flux of one or more reference stars. We sigma-clipped the relative light curves for 4$\sigma$ outliers and normalized each to its median. So far, we had not attempted to remove flux variations due to airmass. Our first task was to remove data sets with high scatter to avoid introducing spurious results in the ephemeris refinement. Although even imprecise data contain useful information, we found that many of the high-scatter data sets contained time correlated noise or systematic noise features. Instead of attempting to correct for these noise properties, we chose to exclude the data set entirely. For the purpose of this procedure, we fit and subtracted an airmass model
\begin{equation}\label{eq:am}
F^{\prime}(t) = c_1 e^{-c_2 X(t)}
\end{equation}
\noindent where $F^{\prime}$ is the flux variation owing to a varying airmass $X$, both of which are functions of time $t$. The model had two fitted coefficients $c_1$ and $c_2$, which can have any finite value. After this airmass correction, data sets for which the standard deviation exceeded the transit depth (0.5\%) were removed (gray points in Figure~\ref{fig:all_ground}). We list this standard deviation for each data set in Table~\ref{tab:ground_obs}.
Next, we established a fine linear grid of mid-transit ($T_0$) times that spanned fourth contact at the time of the very first flux measurement (BJD$_{\rm TDB} = 2459085.674949$) to first contact at the time of our very last flux measurement (BJD$_{\rm TDB} = 2459096.875975$). These $T_0$ values were used to generate transit models that were compared with the data. Each value of $T_0$ maps to a unique value of orbital period.
Then, we iterated over each $T_0$ value and each data set conducting the following procedure. A transit model was generated at that $T_0$ using the \texttt{batman} package \citep{Kreidberg2015b}. All other transit parameters were fixed at the values derived from the \texttt{EXOFASTv2} model (Table~\ref{tab:planet}) except for the quadratic limb darkening coefficients, which were drawn from the look-up tables of \citet{Claret2011} according to filter. We subtracted this transit model from the relative light curve that was sigma-clipped and normalized to its median. Note that the light curve in this case had not been corrected for airmass variations, which were therefore still present after the subtraction. The model-subtracted light curve was then fit to Equation~\ref{eq:am} with a basic least-squares algorithm that minimized the $\chi^2$ statistic.
Consider the logic behind this procedure. If the airmass model is a good fit to the light curve after the transit model is subtracted (i.e., low $\chi^2$ value), then we failed to rule out the transit at that time. We also cannot confidently claim that we have detected the transit, since the good fit to the airmass model could be coincidental. In this case, the probability of this transit time is still described by the posterior from the RV and transit joint fit. Conversely, if the airmass model is a poor fit to the light curve after the transit model is subtracted (i.e., high $\chi^2$ value), then we can be confident that the transit did not occur at that time.
Each $T_0$ for each data set yielded its own minimum $\chi^2$ value from which we calculated the corresponding log likelihood value (i.e., the maximum of the likelihood function) and Bayesian information criterion \citep[BIC;][]{Schwarz1978}. We then summed the BIC values across data sets to produce a total BIC as a function of $T_0$ and hence orbital period.
We also repeated our entire procedure and calculated the same metrics assuming that none of the observations sampled the transit. In this ``no transit'' scenario, the transit occurred either before or after the observations or fell entirely within a data gap.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{teers_results_v1.pdf}
\caption{Summary of the transit ephemeris refinement analysis. Positive values of $\Delta$BIC indicate mid-transit times that are disfavored relative to a nondetection. We consider mid-transit times with $\Delta$BIC$\ge10$ (gray regions) to be ruled out by the ground-based photometry. Negative values of $\Delta$BIC indicate times where the transit model provided a good fit. However, this could be coincidental and we do not interpret large negative $\Delta$BIC as evidence of the transit. The $1\sigma$ and $3\sigma$ ranges for mid-transit time ($T_C$) from the final ephemeris (Table~\ref{tab:planet}) are shown as vertical lines. Each mid-transit time corresponds to a specific orbital period of {TOI-2180~b}, as indicated at the top of the figure.}
\label{fig:teers}
\end{figure}
In Figure~\ref{fig:teers}, we show the difference in BIC values ($\Delta$BIC) between the transit and ``no transit'' scenarios as a function of mid-transit time. The earliest and latest values of $\Delta$BIC approach zero as expected. Values of $T_0$ with substantially negative $\Delta$BIC primarily correspond to times when ingress and/or egress occurred during data gaps and the relatively flat ground-based photometry mimics the basin of a transit. On the other hand, values of $T_0$ with substantially positive $\Delta$BIC correspond to ingress and/or egress lining up with ground-based photometry that clearly does not contain such features. We adopted a $\Delta$BIC threshold of 10---corresponding to ``very strong`` evidence against the transit model \citep{Kass1995}---to determine which values of $T_0$ we could rule out (Figure~\ref{fig:teers}, gray regions) and mapped this refinement back to orbital period. The resulting trimmed posterior for orbital period is shown in Figure~\ref{fig:trimmed_posterior}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{trimmed_posterior_v1.pdf}
\caption{Orbital period posterior for {TOI-2180~b}. The light blue shows the full posterior while the dark blue shows the region that is still allowed after the transit ephemeris refinement using the ground-based photometry, which rules out the most likely orbital periods (around the mean and median) inferred from the comprehensive \texttt{EXOFASTv2} analysis.}
\label{fig:trimmed_posterior}
\end{figure}
The ground-based transit detection campaign served to broadly divide the normal posterior for orbital period into two smaller, non-Gaussian groups while ruling out the median and most likely values of the original distribution. Described by their median and 68\% credible intervals, the two possible orbital periods are {$260.18^{+0.19}_{-0.30}$}~days and {$261.76^{+0.29}_{-0.16}$}~days.
To assess the efficiency of the ground-based photometry campaign, we consider the duty cycle between exposure time and orbital period posterior time ruled out. Multiplying the number of exposures by their respective exposure times (Table~\ref{tab:ground_obs}) for all observations (regardless of whether they were excluded from this analysis) yields {4.4}~days. The aforementioned $\Delta$BIC threshold of 10 rules out {3.8}~days worth of orbital period posterior space. Therefore, the duty cycle of the campaign was {86\%} (i.e., for every 1~hour of exposure time, we ruled out {52~minutes} of orbital period). Ideally, a campaign like this would detect the transit and the duty cycle would be less important. However, this metric can be recalculated for any future single-transit detection campaigns that yield nondetections or proposals to conduct such campaigns to compare strategies and assess effectiveness.
\subsection{Prospects for Future Transit Detection}\label{sec:future_tran}
Despite the ephemeris refinement for {TOI-2180~b}\ conducted here, future characterization of this planet will be challenging until another transit is detected. The third transit of {TOI-2180~b}\ (assuming the TESS transit as the first) occurred sometime in mid-May 2021 and was not observed. The fourth transit is predicted to occur in late January or early February 2022. Specifically, our predictions following the trimmed orbital period posterior are {19:19:30 UTC 31 January 2022} and {13:05:05 UTC 5 February 2022}. The uncertainties on these predictions are asymmetric but roughly on the order of {a day} or less ($1\sigma$).
Fortunately, TESS is expected to re-observe {TOI-2180}\ in Sector~48 of the extended mission beginning around 28 January 2022\footnote{According to the Web TESS Viewing Tool (\url{https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py}) accessed 2021 September 4.}. We therefore predict that TESS will observe another transit of {TOI-2180~b}\ at that time. Barring data gaps, if another transit is not seen by TESS in Sector 48, then only a small (relatively unlikely) tail of the orbital period posterior distribution would be consistent with the original single-transit and the RVs.
A second transit detection would drastically reduce the uncertainty on the orbital period and preserve the transit ephemeris for years into the future. However, some giant planets on 100--1,000~day orbits are known to exhibit day-long timing variations from transit-to-transit \citep[e.g.,][]{Wang2015b}. A third transit would need to be observed to explore the existence of transit timing variations \citep[e.g.,][]{Dalba2019c}.
\section{Bulk Heavy-element Analysis}\label{sec:metal}
{TOI-2180~b}\ is one of a small but growing collection of valuable giant transiting exoplanets on $\sim$au-scale orbits with precisely measured masses and radii \citep[e.g.,][]{Dubber2019,Chachan2021,Dalba2021a,Dalba2021c}. With these two properties, we can infer bulk heavy-element mass and metallicity relative to stellar to better understand their structure and formation history.
Following \citet{Thorngren2019a}, we generated one-dimensional spherically symmetric giant planet structure models with a rock/ice core, a convective envelope consisting of rock, ice, and H/He, and a radiative atmosphere interpolated from the \citet{Fortney2007} grid. Along with the mass, radius, and age for {TOI-2180~b}\ (drawn from the posteriors of the \texttt{EXOFASTv2} fit), the models recovered the total mass of heavy elements and thereby bulk metallicity needed to explain the planet's size. We find that the bulk metallicity for {TOI-2180~b}\ is {$Z_p = 0.12\pm0.03$}, which corresponds to {105~$M_{\earth}$} of heavy elements. We can approximate the stellar metallicity using the iron abundance following $Z_{\star} = 0.142 \times 10^{\rm [Fe/H]}$, which gives {$Z_{\star} = 0.0254\pm0.0033$.} This sets the metal enrichment ($Z_p/Z_{\star}$) at {$4.7\pm1.3$}.
The metallicity enrichment of {TOI-2180~b}\ is consistent with the core accretion theory of giant planet formation \citep{Pollack1996} with late-stage accretion of icy planetesimals, which can explain enrichment by a factor of a few to a dozen \citep[e.g.,][]{Gautier2001,Mousis2009}. An alternate theory to late stage accretion---the mergers of planetary cores during the gas accretion phase \citep[e.g.,][]{Ginzburg2020}---can explain enrichment factors between 1.5--10 for a {2.7~$M_{\rm J}$} planet, making it a viable formation pathway as well.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{zpzs_mp_v3.pdf}
\caption{Giant planet mass--metallicity correlation shown with the \citet{Thorngren2016} sample (circles), {TOI-2180~b}\ (square), and two recently published {\it Kepler}\ long-period giant planets (triangles): Kepler-1514~b ($P\approx218$~days) and Kepler-1704~b ($P\approx989$~days). Blue and red indicate orbital periods below and above 100~days, respectively. The dashed lines and shaded $1\sigma$ regions indicate separate regressions to these two sets of planets. The slopes of these fits are discrepant to $2.6\sigma$, hinting that the mass--metallicity correlation for giant planets may be dependent upon orbital properties.}
\label{fig:zpzs}
\end{figure}
In Figure~\ref{fig:zpzs}, we plot the mass and metal enrichment of {TOI-2180~b}\ relative to the \citet{Thorngren2016} sample of giant exoplanets. We also include two other recently published high mass, giant exoplanets on long-period orbits: Kepler-1514~b \citep[$P\approx218$~days;][]{Dalba2021a} and Kepler-1704~b \citep[$P\approx989$~days;][]{Dalba2021c}. The metal enrichment of {TOI-2180~b}\ relative to its host star is consistent with other giant planets with similar mass and falls near to the best fit line for all of the \citet{Thorngren2016} sample.
The \citet{Thorngren2016} exoplanets are plotted in Figure~\ref{fig:zpzs} as circles. The points are given different colors based on orbital period. The two {\it Kepler}\ planets are shown as triangles and {TOI-2180~b}\ is shown as a square. Although each of these planets individually is fully consistent with the \citet{Thorngren2016} mass--metallicity correlation, there is possibly a subtle difference in the slope of the relation for the longer-period planets. For lower mass planets, enrichment appears to be higher than average for the longer-period objects and vice versa for the higher mass planets including {TOI-2180~b}.
We explored this possibility quantitatively by separating the planets in Figure~\ref{fig:zpzs} into short-period ($P<100$~days) and long-period ($P\ge100$~days) groups. The median orbital periods in the two groups were 10~days and 223~days. {TOI-2180~b}\ and the {\it Kepler}\ planets are also given the corresponding $P>100$~days color. Although 100~days is a somewhat arbitrary separation value, it possibly separates planets that experienced different formation histories. We conducted orthogonal distance regression fits, which include uncertainties on both the explanatory and response variables, to both sets of planet masses and metal enrichments in logspace. The resulting power-law fits are drawn as dashed lines in Figure~\ref{fig:zpzs}. The $1\sigma$ uncertainty regions are also shown. The fits for the short and long-period planets are ($10.4\pm1.6$)$M^{(-0.372\pm0.084)}$ and (14.6$\pm$2.9)$M^{(-0.81\pm0.14)}$, respectively. The slopes in these fits are inconsistent at $2.6\sigma$.
The mass--metallicity correlation derived for planets with orbital periods below 100~days is fully consistent with that measured by \citet{Thorngren2016}. On the other hand, the small set of long-period planets that includes {TOI-2180~b}\ produces a notably steeper correlation. We refrain from placing too much emphasis on this finding owing to the small number of data points and moderate statistical significance. The addition of any number of additional long-period giant planets would be elucidating. If this trend is real, though, it suggests that the current orbital properties of giant planets trace different heavy-element accretion mechanisms such as pebble or planetesimal \citep{Hasegawa2018}. A statistically robust analysis of the mass--metallicity correlation is warranted, but we leave such an analysis to future work.
For a solar system comparison, the {\it Galileo Entry Probe} measured volatile gases in Jupiter's atmosphere and identified enrichment of 2--6 for several heavy elements and noble gasses \citep{Wong2004}. More recently, the {\it Juno} spacecraft measured the equatorial water abundance on Jupiter to be 1--5 times the protosolar value \citep{Li2020}. The comparison between our enrichment measurement of {TOI-2180~b}\ and these measurements at Jupiter comes with caveats. For instance, the Jupiter enrichment is derived from its equatorial oxygen abundance while the exoplanet enrichment is a model dependent bulk value. The direct comparison of these two qualities may be problematic. However, the main point is that these values are all of a similar order of magnitude.
\section{Analysis of RV Drift}\label{sec:drift}
We characterized the trend and curvature in TOI-2180's RV time series using the technique described in \citet{Lubin2021}. Because TOI-2180 is not in the \textit{Hipparcos} catalog \citep{ESA1997}, we could not calculate its astrometric acceleration using the \textit{Hipparcos--Gaia} Catalog of Accelerations \citep{Brandt2018}. Instead we analyzed only the RV data, which leaves a degeneracy between companion mass and semi-major axis.
We sampled $10^8$ synthetic companion models, each comprising a mass $M_P$, semi-major axis $a$, eccentricity $e$, inclination $i$, argument of periastron $\omega$, and mean anomaly $M$. For each model companion, we calculated the trend ($\dot{\gamma}$) and curvature ($\ddot{\gamma}$) that such a companion would produce. We then calculated the model likelihood given the measured trend and curvature in Table \ref{tab:planet}, and marginalized over $\{e, i, \omega, M\}$ by binning the likelihoods in $a$-$M_P$ space. Figure \ref{fig:RV_drift} shows the joint posterior distribution for TOI-2180's companion.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RV_drift.png}
\caption{The set of $(a, M_P)$ models consistent with the measured RV trend and curvature at the 1$\sigma$ (dark) and 2$\sigma$ (light) levels. We rule out semi-major axes corresponding to orbital periods shorter than the observing baseline, as well as companion masses too low to produce the minimum RV amplitude.}
\label{fig:RV_drift}
\end{figure}
If located within a few au, the object is likely to be planetary and more massive than $\sim$2~$M_{\rm J}$. Past roughly 8~au, however, the object transitions to the substellar regime. We truncate the upper mass boundary of Figure~\ref{fig:RV_drift} at a few tenths of a solar mass at which point we would have expected to detect such a companion in the speckle imaging (Section~\ref{sec:imaging}). For a 13~$M_{\rm J}$ object, if we extend the RV monitoring by an additional $\sim$3.5~years, we will have sampled between 25\% and 100\% of the full orbit by period. At that point, we should be more capable of distinguishing between planetary and non-planetary scenarios.
\section{Discussion}\label{sec:disc}
With an equilibrium temperature of {348 K} (assuming a Jupiter-like Bond albedo, see Table~\ref{tab:planet}), {TOI-2180~b}\ qualifies as a temperate Jupiter that occupies an interesting region of parameter space. It exists within 1--3~au where the giant planet occurrence rate increases \citep[e.g.,][]{Wittenmyer2020,Fernandes2019,Fulton2021}, but it is not so close to its star that its orbit has been tidally circularized (e.g., following a high eccentricity migration pathway). This means that its orbital properties may contain information about previous migration. {TOI-2180~b}\ is warmer than Jupiter and Saturn, but it receives a weak enough irradiation to not be inflated. Planets like this are useful laboratories for models of interior and atmospheric structure and planet formation.
{TOI-2180~b}\ stands out among other transiting temperate Jupiters for two main reasons: its long orbital period and its host star's favorable brightness. In Figure~\ref{fig:MR}, we put {TOI-2180~b}\ in context with other giant planets ($M_P>0.2$~$M_{\rm J}$) with orbital periods greater than 20~days (to exclude hot Jupiters) with measured mass and radius that did not have a controversial flag\footnote{According to the NASA Exoplanet Archive, accessed 2021 September 5}. The {\it Kepler}\ mission has so-far proven most successful at discovering planets in this region of parameter space \citep[e.g.,][]{Wang2015b,Uehara2016,ForemanMackey2016b,Kawahara2019}. As a result, many of the planets in Figure~\ref{fig:MR} are sufficiently faint that follow-up opportunities to further characterize their systems are very limited. TESS, however, is slowly beginning to populate the long-period giant planet parameter space with systems orbiting bright host stars that are more amenable to follow-up characterization \citep[e.g.,][]{Eisner2020}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mass_radius_v1.pdf}
\caption{The subset of long-period ($P>20$~days), giant ($M_P>0.2$~$M_{\rm J}$) exoplanets with measured mass and radii sized to show their hosts' $V$ band magnitudes. {TOI-2180~b}\ is labeled and drawn with a dashed line. Of the four planets with the brightest host stars, {TOI-2180~b}\ has the longest period by far demonstrating how it is a valuable extension of the parameter space of temperate Jupiters.}
\label{fig:MR}
\end{figure}
As a temperate giant planet, the atmosphere of {TOI-2180~b}\ represents a stepping stone between well characterized hot Jupiters and the cold solar system gas giant planets. At {348~K}, we might expect to find non-equilibrium carbon chemistry that relies heavily on vertical transport \citep{Fortney2020}. Moreover, an atmospheric metallicity measurement of {TOI-2180~b}\ would be an ideal comparison to the now known stellar and bulk planetary metallicity. Perhaps the atmospheric metallicity will be lower than the bulk value since some amount of heavy elements are likely sequestered into a core. However, evidence from solar system observations suggests that giant planets may instead have interior regions of inward-decreasing metallicity \citep[e.g.,][]{Wahl2017,Guillot2018,Debras2019}. These theories could possibly be explored with transmission spectroscopy, which is suspected to probe CH$_4$ and the byproducts of disequilibrium chemistry and photolysis in long-period giant planets \citep{Dalba2015,Fortney2020}. However, owing to its high surface gravity and an unfavorable planet-star radius ratio, {TOI-2180~b}\ is likely not amenable to transmission spectroscopy. The atmospheric scale height of {TOI-2180~b}\ is only {$\sim$23~km}, which corresponds to a transit depth of a few parts per million (PPM) and a transmission spectroscopy metric of {7} \citep{Kempton2018}. Similarly, the near-infrared depth of the secondary eclipse is likely to be on the order of or less than 10~ppm (Table~\ref{tab:planet}). Any future endeavor to identify favorable targets for temperate Jupiter atmospheric characterization should examine whether the brightness of {TOI-2180}\ compensates for the difficulty introduced by its large stellar radius and the high surface gravity of {TOI-2180~b}.
Other avenues of follow-up characterization for the {TOI-2180}\ system are more promising. Continued RV monitoring of {TOI-2180}\ would eventually capture a sufficient fraction of the long-term acceleration to infer the properties of the outer companion. This RV trend could also be interpreted jointly with imaging and astrometric data to further constrain the outer object's orbit and inclination \citep[e.g.,][]{Crepp2012,Wittrock2016,Kane2019b,Brandt2021,Dalba2021b}. Be it a star or substellar object, it could have influenced the evolution and migration of {TOI-2180~b}, including eccentricity excitation through Kozai-Lidov oscillations \citep[e.g.,][]{Wu2003,Fabrycky2007,Naoz2012}. However, we need not invoke secular interactions \citep[e.g.,][]{Wu2011} or planet-planet scattering \citep[e.g.,][]{Rasio1996} to explain the moderate eccentricity ({$e\approx0.37$}) of {TOI-2180~b}. \citet{Debras2021} argued that disk cavity migration could explain warm Jupiter eccentricity up to $\sim$0.4. Additional modeling of the formation and dynamical evolution of the objects in the {TOI-2180}\ system, with comparison to the bulk metallicity of {TOI-2180~b}, would be illuminating.
The degeneracy between disk migration and secular interactions could possibly be broken with a measurement of the stellar obliquity via the Rossiter-McLaughlin (RM) effect \citep{Rossiter1924,McLaughlin1924}. Migration via interactions with a distant massive object would cause both orbits to be misaligned with the stellar spin. Alternatively, if it migrated in the disk and the disk is assumed to have been aligned with the stellar equator, we would expect little obliquity. This assumption may be problematic, though, as disks can be tilted such that they yield misaligned planets under disk migration \citep[e.g.,][]{Spalding2016b}. Nonetheless, the \texttt{SpecMatch} analysis of the {TOI-2180}\ spectrum returned a low rotational velocity of $v\sin{i} = 2.2\pm1.0$~km~s$^{-1}$. The largest possible amplitude of the RM effect would therefore be $\sim$9~m~s$^{-1}$ \citep{Winn2010b}, which is well within the capabilities of next-generation precise RV facilities such as MAROON-X \citep{Seifahrt2018} or the Keck Planet Finder \citep{Gibson2016}. The RM effect would also be detectable from either Keck-HIRES or APF-Levy, which have average internal RV precisions of 1.2~m~s$^{-1}$ and 3.6~m~s$^{-1}$, respectively. Achieving the proper timing for such an experiment given the 24~hr transit is challenging, though, and may not be feasible for several years. The scientific benefit of an RM detection for this system, and for such a long-period planet in general, further motivates the need to refine the transit ephemeris.
Lastly, we briefly consider {TOI-2180~b}\ as a host for exomoons. The TESS transit offered no evidence to suggest that an exomoon is present, but the stellar brightness, the long-period planetary orbit, and the (possibly) gentle migration history warrant raise the possibility of exomoon detection in this system. {TOI-2180}\ is sufficiently bright that the precision of a single-transit observation would likely reach the noise floors of NIRISS and NIRSpec \citep[several tens of PPM;][]{Greene2016,Batalha2017b}. A Ganymede-size moon would produce a $\sim$5~PPM transit, which would not be detectable. Alternatively, an Earth-size moon would yield a $\sim$30~PPM occultation, which is more reasonable. As more transits of {TOI-2180~b}\ are observed by TESS or any other facility, we recommend that the community conduct photodynamical modeling to test for variations in transit timing and duration that might indicate the presence or lack of an exomoon \citep[e.g.,][]{kipping2012b,Heller2014b,Kipping2021}.
\section{Summary}\label{sec:concl}
Single-transit events are the primary avenue to discovering exoplanets with orbital periods longer than approximately a month in TESS photometry. Here, we describe the follow-up effort surrounding a 24~hr long single-transit of {TOI-2180}\ (Figure~\ref{fig:tess}), a slightly evolved mid G type star, in Sector 19 data from TESS (Section \ref{sec:tess}). Citizen scientists identified the transit event shortly after the data became public, allowing a Doppler monitoring campaign with the APF telescope to begin immediately (Section~\ref{sec:spec}). After nearly two years of RV observations with the APF and Keck telescopes (Figure~\ref{fig:rv}), we determined that {TOI-2180~b}---a {2.8}~$M_{\rm J}$ giant planet on a {260.79}~day, eccentric ({0.368}) orbit---was the cause of the single-transit event (Section~\ref{sec:tran_rv}).
{TOI-2180~b}\ is a member of a rare but growing sample of valuable \emph{transiting} giant exoplanets with orbital periods in the hundreds of days. We conduct a thorough comprehensive fit to the transit and RV data to infer the stellar and planetary properties of this system (Section~\ref{sec:EFv2}). Our precise and regularly sampled RVs refine the ephemeris of {TOI-2180~b}, and we attempt to detect a second transit through 11~days of photometric observations with ground-based telescopes situated over three continents (Figure~\ref{fig:all_ground}). Although we do not detect a transit in these data, we develop a straightforward method to combine the orbital period posterior from the fit to the single-transit and RVs with the extensive collection of ground-based data sets (Section~\ref{sec:teers}). This analysis substantially refines the orbital period of {TOI-2180~b}\ by eliminating a substantial fraction of the most likely posterior solutions (Figure~\ref{fig:trimmed_posterior}), leaving the prediction that TESS will likely detect the transit of {TOI-2180~b}\ in January or February 2022 (Section~\ref{sec:future_tran}).
With a measured mass and radius for {TOI-2180~b}, we infer the bulk heavy-element content and metallicity relative to stellar from interior structure models (Section~\ref{sec:metal}). {TOI-2180~b}\ likely has {over 100~$M_{\earth}$} of heavy elements in its envelope and interior and is enriched relative to its host star by a factor of {$4.7\pm1.3$}. We place {TOI-2180~b}\ in context of the mass--metallicity correlation for giant planets in Figure~\ref{fig:zpzs}. Along with a few other recently characterized exoplanets on several hundred day orbital periods, {TOI-2180~b}\ suggests at 2.6$\sigma$ confidence that the relation between metal enrichment (relative to stellar) and mass for giant planets is dependent on orbital properties. We leave further analysis of this possibility to future work.
Lastly, we place the discovery of {TOI-2180~b}\ in context of other temperate giant planets with known mass and radius (Section~\ref{sec:disc}, Figure~\ref{fig:MR}). It is a poor candidate for transmission spectroscopy owing to its high surface gravity and the {1.6~$R_{\sun}$} radius of its host star. Still, this system is a promising target for continued RV monitoring and stellar obliquity measurement to test theories of how giant planets migrate within the occurrence rate increase near 1~au but not so close as to the become hot Jupiters. {TOI-2180~b}\ also remains as an excellent candidate for exomoon investigations since the host star brightness ($V=9.2$; $J=8.0$) makes it amenable to incredibly precise space-based photometry in the future. \\ \\
\section*{Acknowledgements}
The authors recognize and acknowledge the cultural role and reverence that the summit of Maunakea has within the indigenous Hawaiian community. We are deeply grateful to have the opportunity to conduct observations from this mountain.
The authors would like to thank the anonymous referee for helpful comments that improved this paper. We also thank Jack Lissauer for interesting conversations about giant planets that guided some of our analysis. We thank Ken and Gloria Levy, who supported the construction of the Levy Spectrometer on the Automated Planet Finder. We thank the University of California and Google for supporting Lick Observatory and the UCO staff for their dedicated work scheduling and operating the telescopes of Lick Observatory. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
This paper includes data collected by the {TESS}\ mission. Funding for the {TESS}\ mission is provided by the NASA's Science Mission Directorate. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.
P. D. is supported by a National Science Foundation (NSF) Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1903811. E.A.P. acknowledges the support of the Alfred P. Sloan Foundation. L.M.W. is supported by the Beatrice Watson Parrent Fellowship and NASA ADAP Grant 80NSSC19K0597. A.C. is supported by the NSF Graduate Research Fellowship, grant No. DGE 1842402. D.H. acknowledges support from the Alfred P. Sloan Foundation, the National Aeronautics and Space Administration (80NSSC21K0652), and the National Science Foundation (AST-1717000). I.J.M.C. acknowledges support from the NSF through grant AST-1824644. R.A.R. is supported by the NSF Graduate Research Fellowship, grant No. DGE 1745301. C. D. D. acknowledges the support of the Hellman Family Faculty Fund, the Alfred P. Sloan Foundation, the David \& Lucile Packard Foundation, and the National Aeronautics and Space Administration via the TESS Guest Investigator Program (80NSSC18K1583). J.M.A.M. is supported by the NSF Graduate Research Fellowship, grant No. DGE-1842400. J.M.A.M. also acknowledges the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant No. 1829740, the Brinson Foundation, and the Moore Foundation; his participation in the program has benefited this work. T.F. acknowledges support from the University of California President's Postdoctoral Fellowship Program. N.E. acknowledges support from the UK Science and Technology Facilities Council (STFC) under grant code ST/R505006/1. D. D. acknowledges support from the TESS Guest Investigator Program grant 80NSSC19K1727 and NASA Exoplanet Research Program grant 18-2XRP18\_2-0136. M. R. is supported by the National Science Foundation Graduate Research Fellowship Program under Grant Number DGE-1752134.
P. D. thanks the Walmart of Yucca Valley, California for frequent and prolonged use of its wireless internet to upload hundreds of gigabytes of data.
\facilities{Keck:I (HIRES), Automated Planet Finder (Levy), TESS, Gemini:Gillett (`Alopeke)}, LCOGT\\
\vspace{5mm}
\software{
\texttt{batman} \citep{Kreidberg2015b}, \\
\texttt{EXOFASTv2} \citep{Eastman2013,Eastman2017,Eastman2019},
\texttt{lightkurve} \citep{Lightkurve2018},
\texttt{RadVel} \citep{Fulton2018}
\texttt{SpecMatch} \citep{Petigura2015,Petigura2017b},
\texttt{AstroImageJ} \citep{Collins2017}
\texttt{LcTools} \citep{Schmitt2019}
} \\ |
1912.04434 | \section{Introduction}
\label{sec:Introduction}
According to the principle of the equilibrium thermodynamics, a
quasi-static adiabatic cycle is trivial, in the sense that the initial
and final states are identical~\cite{Callen}.
Once we
slightly relieve the quasi-static adiabatic condition,
or
the
thermodynamic condition,
however,
a cyclic operation may transform a stationary
state into another one.
A promising system to realize such stationary state transformations in a
many-body, nearly thermodynamic setting is cold atoms. This is because
it is possible to manipulate the system with its
quantum coherence intact, as it can be well-isolated from
environmental degrees of freedom.
An example~\cite{Yonezawa-PRA-87-062113} can be found in the
Lieb-Liniger model~\cite{Lieb-PR-130-15,Guan-IJMPB-28-1430015}, which
describes Bose particles confined in a one-dimensional ring. A
stationary state of the Lieb-Liniger model is delivered to another
stationary state through a cyclic operation where the strength of
interparticle interaction is adiabatically increased except at a
point: Once the interparticle interaction becomes infinitely repulsive
to make the Tonks-Girardeau regime~\cite{Girardeau-JMP-1-516},
the interparticle interaction
strength
is
flipped
to infinitely attractive to
form the super-Tonks-Girardeau
regime~\cite{Astrakharchik-PRL-95-190407,Haller-Science-325-1224}. During
the cyclic operation, the stationary state is smoothly deformed even
at the
strength flipping
point, where normalizable stationary states are ensured
to be kept unchanged.
The state transformations in the Lieb-Liniger model is an example of exotic
quantum holonomy in adiabatic cycles of microscopic, non-thermal
systems: Although one may expect that
an adiabatic cycle
brings no change in
stationary states up to a phase factor,
it
may
transform a stationary state into
another,
for example,
in Floquet systems
through the winding of
quasienergy~\cite{Tanaka-PRL-98-160407,Kitagawa-PRB-82-235114,Zhou-PRB-94-075443}
and Hamiltonian systems with level
crossings~\cite{Cheon-PLA-374-144,Spurrier-PRA-97-043603}.
A similar, but distinct concept to the exotic quantum holonomy,
Wilczek-Zee's holonomy, where an adiabatic cycle offers a
transformation of degenerated stationary
states~\cite{Wilczek-PRL-52-2111,Bohm-GPQS-2003,Chruscinski-GPCQM-2004},
is
also
utilized to control quantum
states~\cite{Zanardi-PLA-264-94,Tanimura-PLA-325-199,Sjoeqvist-NJP-14-103035}.
We note that the adiabatic state transformation in the Lieb-Liniger
model~\cite{Yonezawa-PRA-87-062113} heavily depends on the
particularity
of
the model.
The number of
particles is required to be specified precisely. Also, the theoretical
argument in Ref.~\cite{Yonezawa-PRA-87-062113}
depends
essentially
on
the solvability of the
system.
Hence it seems
difficult to
extend this result to other quantum many-body systems.
In this paper, we examine
an
adiabatic state transformation in a
simple quantum many-body system, which might be extensible to various
ways. We here examine cyclic operations on Bose particles confined in
a
quasi one-dimensional box
trap~\cite{Meyrath-PRA-71-041604,Batchelor-JPA-38-7787,%
Syrwid-PRA-96-043602,Sciacca-PRA-95-013628}
to
generate a
nonequilibrium stationary state
from the ground state.
Applying an
almost%
-%
adiabatic cycle
that involves
a quench of the strength
of
a sharp impurity
potential,
we obtain a
population-inverted state, in which the Bosons occupy a single-particle
excited state.
The population inversion
has been
utilized to achieve negative
temperature~\cite{Purcell-PR-81-279}, for example, and is related to
dark solitons in the studies of Bose-Einstein
condensates~\cite{Dum-PRL-80-2972,Damski-PRA-65-013604}.
Our starting point is the analysis of cyclic operations for
noninteracting Bose
particles~\cite{Kasumie-PRA-93-042105,Tanaka-NJP-18-045023}. We
define a cycle using a sharp impurity potential, which is often
utilized to manipulate condensates both
experimentally and
theoretically~\cite{Piazza-PRA-80-021601,Karkuszewski-PRA-63-061601,Frantzeskakis-PRA-66-053608,Hans-PRA-92-013627,Ramanathan-PRL-106-130401,Kunimi-PRA-100-063617}.
In order to incorporate the interparticle interaction, we suppose that
the system is described by the time-dependent Gross-Pitaevskii
equation. We
show that the interparticle interaction can
significantly modify the almost-adiabatic processes due to the
appearance of bifurcations in the solution of time-independent
Gross-Pitaevskii
equation
such as
the
swallowtail structure~\cite{Wu-PRA-61-023402,Damski-PRA-65-013604,%
Liu-PRA-66-023404,Mueller-PRA-66-063603,Kunimi-PRA-91-053608,%
PerezObiol-arXiv-1907.04574}.
The plan of this paper is the following. In Sec.~2, we introduce a
cyclic operation
that involves
a flip of the potential strength
which is analogous to
the cyclic operation
introduced in Ref.~\cite{Kasumie-PRA-93-042105} for non-interacting
systems.
In Sec.~3, we numerically examine the cyclic operation in the repulsively
interacting Bose
particles
using the Gross-Pitaevskii equation,
where it is shown that the
strong
nonlinearity
invalidates
the population
inversion.
In Sec.~4, a theoretical interpretation for the
numerical result
is shown.
Sec.~5 concludes this paper with
summary and outlook. Appendix~\ref{sec:Bogoliubov} offers
details of the linear stability analysis (the Bogoliubov analysis)
at the
quench
point.
\section{Cycle for non-interacting particles}
We look at a cyclic operation for particles confined in a
quasi one-dimensional
boxed trap
with
an impurity
potential. We
illustrate how this cycle works for non-interacting
particles~\cite{Kasumie-PRA-93-042105}, which offers a basis
for examining
the case of interacting Bosons.
We assume that a particle is described by the one-dimensional
time-dependent Schr\"odinger equation
\begin{equation}
\label{eq:defTDS0}
i\hbar\pdfrac{}{t}\Psi(x,t)
= -\frac{\hbar^2}{2M}\pdfrac{^2}{x^2}\Psi(x,t)
\end{equation}
under the boundary condition $\Psi(0,t)=\Psi(L,t)=0$, where $M$ is the
mass of the particle, and $L$ is the size of the box trap. In the
following, we assume $\hbar=1$, $M=1/2$ and $L=2\pi$.
After the system is prepared to be in a stationary state, a
sharp impurity potential
is placed at $x=X$ to realize cyclic operations, where the
strength $v$
is varied.
We assume that the
impurity
potential
$V_{\rm i}(x; v)$ is
described by the
Dirac's
delta function:
\begin{equation}
\label{eq:defVi}
V_{\rm i}(x; v) = v\delta\left(x - X\right)
.
\end{equation}
There are two building blocks of the cyclic operation: One is the smooth
and monotonic variations of parameter: $C_{\rm s}(v', v'')$,
which denotes the variation of $v$ from $v'$ to $v''$,
while keeping the value of $X$.
The other is the discontinuous operation $C_{\rm d}(v',v'')$ in
which the value of $v$ is
changed
from $v'$ to $v''$, which resembles
the process that is utilized to create the super-Tonks-Girardeau gas
from the Tonks-Girardeau gas~\cite{Haller-Science-325-1224}.
We define the almost-adiabatic cyclic operation $C(X)$
(see, Fig.~\ref{fig:paths_Xv}), which
involves a
quench
of the
impurity
potential placed
at $X$. This cycle is a succession of three operations
$C_{\rm s}(0, \infty)$, $C_{\rm d}(\infty, -\infty)$ and
$C_{\rm s}(-\infty, 0)$.
\begin{figure}[hbt]
\centering
\includegraphics[%
height=3.5cm%
]{figure1}
\caption{%
The closed path $C(X)$ in $(x,v)$-plane. Throughout the cycle,
the value of $X$ is fixed.
At the initial point, we assume $v=0$.
To close the cycle, $v_1=\infty$ and $v_2=-\infty$ are
assumed. In numerical experiments, we assume that
the values of $v_1$ and $-v_2$ are large, but finite.
}
\label{fig:paths_Xv}
\end{figure}
We show that an initial stationary state $\Psi_n(x)$ can be
transformed
to another stationary state after the completion of a cycle.
Here we assume that the parameters are varied adiabatically during the smooth
operations.
Hence the system is governed by the adiabatic theorem~\cite{Kato-JPSJ-5-435}.
Since the relevant eigenenergy and eigenfunction are continuous during
the
quench,
the parametric dependence of eigenenergies tells us the final
stationary state for the almost-adiabatic cycle $C(X)$.
We depict an example of the parametric evolution of eigenenergies in
Fig.~\ref{fig:linearCq}.
After the completion of the almost-adiabatic cycle $C(X)$,
the final state is $\Psi_{n+1}(x)$, as long as
$X$ does not coincide with the node of the initial wavefunction
$\Psi_n(x)$. This is because the
eigenenergies are increased monotonically during
$C_{\rm s}(0, \infty)$ and $C_{\rm s}(-\infty, 0)$ as $v$ is
increased monotonically~\cite{Kasumie-PRA-93-042105,Simon-CPAM-39-75},
and are continuous at the quench process
$C_{\rm d}(\infty, -\infty)$~\cite{Kasumie-PRA-93-042105}.
\begin{figure}[hbt]
\centering
\includegraphics[%
width=7.0cm%
]{figure2}
\caption{%
(Color online) Parametric evolution of eigenenergies along
$C(X=0.42L)$, which connects $n$-th and $(n+1)$-th eigenenergies. The
horizontal axis is linear in $\tan^{-1} v$.
}
\label{fig:linearCq}
\end{figure}
Let us extend the above argument to the non-interacting Bose
particles. Suppose that all particles occupy the single-particle
ground state $\Psi_1(x)$. After the completion of the almost adiabatic
operation $C(X)$, all particles occupy the first excited state
$\Psi_2(x)$. Namely, a population
inverted
state can be created from the ground state through the almost
adiabatic operation $C(X)$.
Although the almost-adiabatic cycle $C(X)$ induce exotic changes
as shown above,
the cycle involves several idealizations.
In particular,
we need to take into account the effect of interparticle interaction.
In
the following sections, we scrutinize the almost-adiabatic cyclic operation $C(X)$ for interacting Bose particles.
\section{%
Cycle for
an interacting Bose system}
We here examine the almost-adiabatic cycle $C(X)$ in a many-body
setting. We assume that Bose particles are confined in the
quasi one-dimensional
box
and the interparticle interaction is
repulsive.
First, we
examine
the parametric evolution of stationary states along
the cycle, which suggests that the population inversion is indeed
possible if the interparticle interaction is weak.
Also, it
is
shown that the population inversion breaks down when
the interparticle interaction is strong enough.
Second, we
numerically integrate
the time evolution equation to
confirm the picture obtained through the parametric evolution of the
stationary states.
We
provide
a theoretical explanation based on a perturbation theory to
these observations in the next section.
We assume that the Bose particles are described by the time-dependent
one-dimensional Gross-Pitaevskii equation
\begin{equation}
\label{eq:defGP0}
i\pdfrac{}{t}\Psi(x,t)
= -\pdfrac{^2}{x^2}\Psi(x,t) + g|\Psi(x,t)|^2\Psi(x,t)
,
\end{equation}
where $\hbar=1$, $M=1/2$ and $L=2\pi$ are assumed, and $g \ge 0$
represents the strength of the effective interparticle interaction.
We impose the boundary condition $\Psi(0,t)=\Psi(L,t)=0$ and the
normalization condition $\int_0^L|\Psi(x,t)|^2dx=1$. Let $E_n(g)$ and
$\Psi_n(x; g)$ ($n=1,2,\ldots$) denote the $n$-th eigenenergy
(chemical potential) and the corresponding stationary state for
Eq.~\eref{eq:defGP0}. We suppose that the system is initially in
$n$-th stationary state $\Psi_n(x; g)$.
During the cycle,
we impose
the sharp
impurity
potential (Eq.~\eref{eq:defVi})
to Eq.~\eref{eq:defGP0}
and vary
its
strength
$v$
slowly except at the quench point.
To find the stationary states of the system where the position $X$ and
strength $v$ of the
impurity
potential~(Eq.~\eref{eq:defVi}) are ``frozen'',
we examine the time-independent Gross-Pitaevskii equation with
the
impurity
potential
\begin{equation}
\label{eq:tiGP}
\left(-\pdfrac{^2}{x^2} + g|\Psi(x)|^2
+ V_{\rm i}(x; v)\right)\Psi(x)
= E\Psi(x)
.
\end{equation}
Examples of parametric evolution of $n$-th
eigenenergy
along the cycle
$C(X)$ are shown in Fig.~\ref{fig:nCyE}. Corresponding parametric
evolution of stationary states are shown in
Figs.~\ref{fig:nCyPsi_weak} and~\ref{fig:nCyPsi}.
\begin{figure}[hbt]
\centering
\includegraphics[width=4.1cm]{figure3a}\hfill%
\includegraphics[width=4.1cm]{figure3b}\\
\includegraphics[width=4.1cm]{figure3c}\hfill%
\includegraphics[width=4.1cm]{figure3d}
%
\caption{
(Color online)
Parametric evolution of the ground and first excited eigenenergies
(chemical potentials) along $C(X=0.42L)$.
The horizontal axis is linear in $\tan^{-1} v$.
%
(a) The eigenenergies form smooth and monotonic curves at $g=1$,
which are similar to the case $g=0$.
(b) A tiny loop around $|v|=\infty$ appears in the ground energy curve at $g=2$.
(c) The loop of the ground energy grows at $g=3$, while the derivative of the first excited energy seems to be discontinuous at $|v|=\infty$.
(d) There are two noticeable loops at $g=4$.
%
Blue dotted lines are the eigenenergies estimated by the two mode approximation (see, Sec.~4) of the ground branch at $|v|=\infty$.
}
\label{fig:nCyE}
\end{figure}
When the interparticle interaction $g$ is small enough, the quasi-adiabatic
cyclic operation induces
the population inversion.
The parametric evolution of eigenenergy along the cycle connects the
initial eigenenergy $E_{1}(g)$ with $E_{2}(g)$ (see,
Fig.~\ref{fig:nCyE}~(a)). The connection is equivalent to the case of
noninteracting particles (see, Fig.~\ref{fig:linearCq}). This allows
us to infer the parametric evolution of eigenfunction whose initial
condition is the ground state $\Psi_1(x; g)$ of Eq.~\eref{eq:defGP0},
i.e., at $g=0$ (see, Fig.~\ref{fig:nCyPsi_weak} (A)).
While the stationary state is nodeless during
the strength of the sharp
impurity potential
$v$ is positive and finite,
the state becomes localized at a side of the
impurity
as $v$ become
larger (Fig.~\ref{fig:nCyPsi_weak} (B)). Immediately before the
quench,
i.e, $v=\infty$,
the localization completes~\cite{GeaBanacloche-AJP-70-307}.
The state is
unchanged during
the flip of the potential strength from
$v=\infty$ to
$v=-\infty$~\cite{Kasumie-PRA-93-042105}. As $v$ is
slightly increased from $-\infty$, the stationary state extends to the
other side of the
impurity
to produce a node (Fig.~\ref{fig:nCyPsi_weak}
(C)). The resultant stationary state has a single node while $v$ is finite
(Fig.~\ref{fig:nCyPsi_weak} (D)). This is the reason why the
destination of the stationary state at the end of the cycle is the
first excited state $\Psi_2(x;g)$.
\begin{figure}[htb]
\centering
\includegraphics[%
width=4.3cm%
]{figure4a}%
\hfill
\includegraphics[%
width=4.3cm%
]{figure4b}
\includegraphics[%
width=4.3cm%
]{figure4c}%
\hfill
\includegraphics[%
width=4.3cm%
]{figure4d}
\caption{
(Color online)
Parametric evolution of stationary state $\Psi(x)$ of
weakly interacting Bose particles along the cycle $C(X = 0.42L)$ and
$g=1$.
The corresponding points in $(g, E)$-plane is shown in
Fig.~\ref{fig:nCyE}~(a).
(A) $\Psi(x)$ is the ground state initially (i.e., $v=0$);
(B) and (C) correspond to the case immediately before and after the quench.
(D) the final state is the first excited state.
The phase of $\Psi(x)$ is chosen so that
$\Psi(x)$ is positive at right hand side of the
sharp impurity.
}
\label{fig:nCyPsi_weak}
\end{figure}
On the other hand, when the interparticle interaction is large,
discrepancies from the linear case become significant. In particular,
as is seen in Figs.\ref{fig:nCyE} (c) and (d),
the parametric evolution of a stationary energy involves a loop structure
that emanates from the quench point
$|v|=\infty$
in $C(X)$.
We note that
loop structures are
often observed in the studies of time-independent Gross-Pitaevskii
equation~\cite{Wu-PRA-61-023402,Damski-PRA-65-013604,Liu-PRA-66-023404,Mueller-PRA-66-063603,Kunimi-PRA-91-053608}.
The corresponding parametric evolution of the stationary state along
the loop is explained in Fig.~\ref{fig:nCyPsi}.
In the vicinity of the quench, the wavefunction extends to both sides of
the sharp impurity potential,
which is a distinctive feature of the case with stronger
interparticle interaction (${\rm B}_{+}$). Across the
quench point
the wavefunction
smoothly connect to the lower branch of the loop to acquire two nodes
(${\rm B}_{-}$). As $v$ increased, the lower branch arrives the
extremum point (C) to connect the uppermost branch, where the
wavefunction localizes at a side of the
impurity potential,
where the stationary
state mimics the one in the linear system. As $v$ decreased to follow the
uppermost branch, the stationary state become localized at a side of
the impurity
(${\rm D}_{-}$). At the quench point in the uppermost branch, the
localization completes. Across the quench, the number of the nodes of
the stationary decreases
from $2$ (${\rm D}_{-}$) to $1$ (${\rm D}_{+}$).
Then the uppermost branch arrives another extremum
point (E), where the state delocalize again to connect the final
branch at (F), which smoothly connects the first excited state $\Psi_2(x;g)$,
see (G).
We expect that the loop structure disturbs
the adiabatic
evolution, as reported in Refs.~\cite{Wu-PRA-61-023402,Liu-PRA-66-023404},
and thus hinders the population inversion. This is because the
stationary state is transformed into a non-stationary state when
the adiabatic time evolution has to departs from the parametric evolution
by having a loop.
To clarify whether the adiabatic time evolution along $C(X)$
really occurs, we
numerically
examine the linear stability of the stationary
states in $C(X)$ by diagonalizing
the Bogoliubov equation~\cite{Bogolubov-JPhysUSSR-11-23,Fetter-AP-70-67}
corresponding to the stationary solutions.
Also, an analytical study on the linear stability for the quench point
is shown in Appendix~\ref{sec:Bogoliubov}.
When the interparticle interaction is small (see, Figs.~\ref{fig:nCyE}~(a)
and \ref{fig:nCyPsi_weak}), we find that the stationary states are linearly
stable along $C(X)$. Hence we may expect that the adiabatic time evolution
remains intact for the weakly interacting case.
Meanwhile, when the interparticle interaction is larger to form
the loop structure as shown in
Figs.~\ref{fig:nCyE}~(d) and \ref{fig:nCyPsi},
we find that
the stationary state is linearly stable within the intervals
from $v=0$ through $v=\pm\infty$, i.e., from ${\rm A}$ to ${\rm B}$
and from ${\rm G}$ to ${\rm F}$ in Fig.~\ref{fig:nCyE}~(d),
and the uppermost branch of the loop
(from ${\rm C}$ to ${\rm E}$).
On the other hand, the stationary state is linearly
unstable at the lower part of the loop
(i.e, from ${\rm B}$ to ${\rm C}$ and ${\rm E}$ to ${\rm F}$).
The result of the linear stability analysis for stronger interparticle
interaction suggests that
the adiabatic time evolution whose initial condition is
the ground state $\Psi_1$ remains intact until the quench point,
i.e., within the interval ${\rm A}$ to ${\rm B}$.
After the system passes the quench point,
the adiabatic time evolution breaks down during
the interval ${\rm B}$ to ${\rm C}$ due to the linear instability.
We note that the emergence of the unstable stationary state at the lower
branch of the loop structure in the Brillouin zone is reported in
Refs.~\cite{Wu-NJP-5-104,Danshita-PRA-75-033612}.
We
also
note that this is a distinctive point from
the instabilities in
the conventional studies~\cite{Wu-PRA-61-023402,Liu-PRA-66-023404}
of adiabatic time evolution along the loop
structure, where the linearly unstable region appears only at the uppermost
of the loop structure.
After the point ${\rm C}$, the adiabatic time evolution is impossible since
the stationary solution cannot be adiabatically extended
anymore~\cite{Wu-PRA-61-023402,Liu-PRA-66-023404}.
\begin{figure}[hbt]
\centering
\includegraphics[%
width=2.5c
]{figure5a}
\includegraphics[%
width=2.5c
]{figure5bp}
\includegraphics[%
width=2.5c
]{figure5bm}
\includegraphics[%
width=2.5c
]{figure5c}
\includegraphics[%
width=2.5c
]{figure5dm}
\includegraphics[%
width=2.5c
]{figure5dp}
\includegraphics[%
width=2.5c
]{figure5e}
\includegraphics[%
width=2.5c
]{figure5f}
\includegraphics[%
width=2.5c
]{figure5g}
%
\caption{
(Color online) Parametric evolution of stationary state $\Psi(x)$
along an eigenenergy loop associated with the cycle $C(X = 0.42L)$ at
$g=4$, which correspond to the case shown in Fig.~\ref{fig:nCyE}~(d).
The loop connects the initial point A and the final point G of the
cycle $C(X)$. Around the quench points $B$ and $D$, the sign of $v$ is
indicated by the suffix $\pm$.
}
\label{fig:nCyPsi}
\end{figure}
We test the scenario above through numerical integration of the
time-dependent Gross-Pitaevskii equation along the almost-adiabatic
cycle $C(X)$.
We show our numerical result for various values of $X$ in
Fig~\ref{fig:pop2}, where the initial condition is prepared to be in
the ground state $\Psi_1(x;g)$ of Eq.~\eref{eq:defGP0}. We
numerically evaluate the fidelity for the population inversion
$|\bracket{\Psi_2(g)}{\Psi}|^2$, where $\Psi$ is a state after the
completion of the cycle. Since the finial states may not be
stationary, we depict the time-average of the fidelity probability
after the completion of the cycles.
From Fig~\ref{fig:pop2}, we conclude that the population inversion
fails if the value of $g$ exceeds a critical value $g_{\rm c}$, which
depends on the position of the
impurity
potential $X$. Moreover, when
we restrict the case $0 < X < L/2$, $g_{\rm c}$ becomes larger as $X$
become smaller.
We make a remark on the integration of time-dependent Gross-Pitaevskii
equation along $C(X)$, where we introduce an approximation
for the quench of
impurity potential.
We keep $v$, the strength of the
impurity potential.
finite.
Namely, $v$ is increased from $0$ to $v_{\rm max}$ with a finite
velocity $dv/dt$ during the first process $C_{\rm s}(0, v_{\rm max})$. At the
quench, $v$ is suddenly changed from
$v_{\rm max}$ to $-v_{\rm max}$. Then, during $C_{\rm s}(-v_{\rm max}, 0)$, the
value of $v$ is increased from $-v_{\rm max}$ to $0$ with non-zero
velocity $dv/dt$.
Although this induces a tiny nonadiabatic error during the quench, as
is seen from Fig.~\ref{fig:pop2}, the error is far less important
than the nonlinear effect.
\begin{figure}[htb]
\centering
\includegraphics[%
width=8.66c
]{figure6}
\caption{
(Color online)
Population inversion probability from the ground state through
$C(X)$.
Vertical dashed lines indicate the critical point predicted by the
two-mode approximation (see, Sec.~4).
}
\label{fig:pop2}
\end{figure}
\section{Two-mode approximation
at the
quench
point}
We discuss our numerical results
in the previous section
with an approximate
theory. In particular, we would like to clarify the reason why the
population inversion breaks down as the strength of the interparticle
interaction becomes stronger (Fig.~\ref{fig:pop2}).
A key ingredient
must be the emergence of the loop structure in
$(g,E)$-plane (Fig.~\ref{fig:nCyE}). To identify the loop structure,
we examine the quench point
$|v|=\infty$, because a loop emanates from a point at $|v|=\infty$ in
$(g,E)$-plane. This analysis allows us to infer the loop structure, as
long as the loop is small enough.
In the following,
we utilize a two-mode approximation. Namely, we assume that the
stationary wavefunction $\Psi(x)$ is a superposition of two
eigenfunctions $\psi_j(x)$ ($j=0,1$) of the noninteracting
system. Since the
infinitely strong impurity
divide the box completely~\cite{GeaBanacloche-AJP-70-307}, as
suggested in Fig.~\ref{fig:nCyE}, we utilize the unperturbed
eigenfunction $\psi_j(x)$ that is localized at the left or right side
of the
impurity.
For example, to examine the stationary states that are associated with
the ground state at the initial point of the cycle,
we assume that $\psi_0(x)$ and $\psi_1(x)$
describe the ground state of a single particle confined within the right
and left boxes, respectively, i.e,
\begin{align}
\psi_0(x)
&=
\begin{cases}
0 & \text{for $0 \le x \le X$}
\\
\sqrt{\frac{2}{L-X}} \sin\frac{\pi(x-X)}{L-X}
&\text{for $X \le x \le L$}
\end{cases}
,
\\
\psi_1(x)
&=
\begin{cases}
\sqrt{\frac{2}{X}} \sin\frac{\pi x}{X}
&\text{for $0 \le x \le X$}
\\
0 & \text{for $X \le x \le L$}
\end{cases}
,
\end{align}
whose eigenenergies are
$E_0=E_{\rm g}/r'^2$ and $E_1=E_{\rm g}/r^2$, respectively,
where $r=X/L$, $r'=(L-X)/L$, and $E_{\rm g}= \hbar^2\pi^2/(2ML^2)$.
In the following, we assume $0 < X < L/2$, which
implies $E_0 < E_1$, i.e., the ground state $\psi_0(x)$ under the
presence of the
infinitely strong impurity
localizes at the right side of the impurity.
From the two-mode assumption that $\Psi(x,t) =
\Psi_0(t)\psi_0(x)+\Psi_1(t)\psi_1(x)$ satisfies the time-dependent
Gross-Pitaevskii equation, we obtain the time-evolution equation for
the amplitudes $\Psi_j(t)$ ($j=0,1$):
\begin{equation}
i\frac{d}{dt} \Psi_j(t)
= E_j \Psi_j(t)
+ g\int_0^L\psi^*_j(x)\left|\Psi(x,t)\right|^2\Psi(x,t) dx
.
\end{equation}
Because $\psi_0(x)$ and $\psi_1(x)$ have no overlap in the position
space, i.e., $\psi_0(x)\psi_1(x)=0$ holds, and are real, we obtain
\begin{equation}
i\frac{d}{dt} \Psi_j
= E_j\Psi_j + g\int_0^Ldx\left\{\psi_j(x)\right\}^4
\left|\Psi_j\right|^2\Psi_j
.
\end{equation}
Hence the nonlinear Scr\"odinger equation for $\Psi_j$ is
\begin{equation}
\label{eq:nls2}
i\frac{d}{dt} \Psi_j
= \left(E_j + g c_j|\Psi_j|^2\right)\Psi_j
,
\end{equation}
where
$c_0\equiv 3g/\{2(L-X)\}$
and
$c_1\equiv 3g/(2X)$.
We also impose the normalization condition
$|\Psi_0|^2+|\Psi_1|^2=1$.
The stationary solutions of Eq.~\eref{eq:nls2} are classified into
two groups. First, there are two localized solutions $(\Psi_0,
\Psi_1) = (1,0)$ and $(0,1)$, whose eigenenergies are
\begin{gather}
\label{eq:Elocalized}
E_0(g)
= E_0 + g c_
,\quad\text{and}\quad
E_1(g)
= E_1 + g c_
,
\end{gather}
respectively.
Second, the other two solutions $\Psi_{\pm}$ are
\begin{equation}
\label{eq:nls2c}
\begin{bmatrix}
\Psi_{\pm,0} \\ \Psi_{\pm,1}
\end{bmatrix}
=
\begin{bmatrix}
\sqrt{r' - (E_0-E_1)({2Lrr'})/(3g)}
\\
\pm\sqrt{r + (E_0-E_1)({2Lrr'})/(3g)}
\end{bmatrix}
,
\end{equation}
which are delocalized to the both side of the impurity.
The corresponding
eigenenergies are doubly degenerate
\begin{equation}
\label{eq:Edeloc}
E_{\pm}(g) = r' E_0 + r E_1 + \frac{3g}{2L}
.
\end{equation}
We explain the condition that the stationary states Eq.~\eref{eq:nls2c} are
physical, i.e.,
$0\le |\Psi_0|^2, |\Psi_1|^2 \le 1$ holds. First, we introduce the
critical points
\begin{align}
g_0 &= \frac{2(L-X)}{3}(E_1-E_0)\\
g_1 &= -\frac{2X}{3}(E_1-E_0)
,
\end{align}
where $|\Psi_j|^2=1$ holds if $g=g_j$ ($j=0,1$).
Since we assume $X<L/2$,
the physical condition for the stationary solution Eq.~\eref{eq:nls2c} is
summarized as $g\ge g_0$ or $g\le g_1$.
Also, as we restrict the case that the interparticle interaction is repulsive,
i.e., $g\ge 0$, the delocalized solutions
Eq.~\eref{eq:nls2c}
exist
only when $g\ge g_0$.
From the linear stability analysis (Bogoliubov analysis)
of Eq.~\eref{eq:nls2},
whose details are shown in Section~\ref{sec:Bogoliubov},
we find that
the localized solutions $(\Psi_0, \Psi_1) = (1,0)$ and $(0,1)$ are stable.
On the other hand, the delocalized solutions Eq.~\eref{eq:nls2c} are
marginally stable.
With the two mode approximation at $|v|=\infty$, we recapitulate the
parametric evolution of stationary states and eigenenergies along
$C(X)$. We assume that the system is initially in the ground state
$\Psi_1(x;g)$.
First, we revisit the case that the interparticle interaction is
weak enough, i.e., $g$ is smaller than the critical value $g_0$.
The stationary state
is localized at $|v|=\infty$, which is consistent with
Fig.~\ref{fig:nCyPsi_weak}. The corresponding estimation of
eigenenergy at $|v|=\infty$ is given by Eq.~\eref{eq:Elocalized}, which is
indicated in Fig.~\ref{fig:nCyE}.
Also, the stationary state at $|v|=\infty$ is stable, according to the
linear stability analysis. Hence the stationary state must be stable during
$C(X)$, which is also consistent with the numerical result that $C(X)$
induces the population inversion for smaller $g$
(Fig.~\ref{fig:pop2}).
Second, when the strength of the interparticle interaction $g$ exceeds the
critical value $g_0$, the localized and delocalized stationary
solutions coexist at the quench point: the degenerated eigenenergies of the
delocalized solutions $E_{\pm}(g)$ (Eq.~\eref{eq:Edeloc})
is lower than the one of the localized
solution (Eq.~\eref{eq:Elocalized}), as shown in Fig.~\ref{fig:nCyE}.
The delocalized nodeless solution $\Psi_+$
(Fig~\ref{fig:nCyPsi} ${\rm B}_{\pm}$) is connected with
the initial ground state $\Psi_1$ through $C_{\rm s}(0,\infty)$,
the former half of the whole cycle. After $\Psi_+$ is evolved
along the loop, the stationary solution arrives at the localized solution
$(\Psi_0, \Psi_1)=(1,0)$ at $|v|=\infty$
(see, Fig~\ref{fig:nCyPsi} ${\rm D}_{\pm}$).
After the completion of the
loop, the stationary solution becomes the another delocalized solution
$\Psi_-$ with a node (Fig~\ref{fig:nCyPsi}~${\rm F}$).
The latter half of the cycle $C_{\rm s}(-\infty,0)$
smoothly connects $\Psi_-$ with the first excited state $\Psi_2$. In this
sense, the parametric evolution of the stationary solution smoothly
deforms the ground state into the first excited state.
Meanwhile the adiabatic population inversion is hindered by the presence
of the loop structure due to the instability of the stationary state
at the lower branch of the loop, as explained in the previous section.
In Fig.~\ref{fig:pop2}, we indicate the
critical interparticle interaction strength $g_0$, which is estimated
by the two-mode approximation for each value of $X$ by vertical lines.
Hence we conclude that the two-mode approximation qualitatively describes
the breakdown of the population inversion.
We depict how the eigenenergies at the quench point depend on the
position $X$ of the sharp impurity in Fig.~\ref{fig:Cy_r_vs_E}, using
the two-mode approximation. This helps us to understand the cycle
$C(X)$ for a given value of $g$. For example, at $X=0.42 L$, the
ground branch connected with a loop whose section at $v=|\infty|$ has
two delocalized and a localized stationary states, which predicts the
breakdown of the population inversion to the first excited
states. Also, the first (the second) excited state at the initial
point of the cycle is connected with a localized stationary state,
which suggests that the population inversion to the second (the third)
excited state may be possible.
\begin{figure}[htb]
\centering
\includegraphics[%
width=7.0cm%
]{figure7}%
\hspace*{1.00cm}%
\caption{
(Color online)
$X$-dependence
of stationary energies at the quench $|v|=\infty$ and $g=2$
under the two-mode approximation.
The solid (dashed) lines correspond to the stable stationary states localized at the right (left) side of the
infinitely strong impurity.
The dotted lines correspond to the delocalized stationary states that are marginally stable.
}
\label{fig:Cy_r_vs_E}
\end{figure}
\section{Summary and outlook}
We have shown that the adiabatic
cyclic operation
with a
quench
$C(X)$ induces the population inversion of Bose
particles
described by the Gross-Pitaevskii equation
confined in a
quasi one-dimensional
box, if the Bose
particles are initially prepared to be in the ground state and the
strength $g$ of the interparticle interaction is weak enough. An
estimation of the critical value of $g$ where the population inversion
is broken is also shown. We find that these results are consistent
with our numerical investigation through the time-dependent
Gross-Pitaevskii equation.
We note that the time evolution generated by the almost-adiabatic
operation $C(X)$ can confine the system within a family of
stationary states in the weak interaction regime. Namely, the
adiabatic time evolution can be realized in spite of the presence of
the flip of the potential strength at the quench point.
This ``adiabatic'' cycle converts the ground state of Bose particles
into a nonequilibrium statonary state.
The present study offers an
example
of the subtle difference between
the adiabatic processes in thermodynamic systems and non-thermal,
mechanical systems.
We
believe
that the present result offers an experimentally feasible
method to produce a nonequilibrium stationary state. Application of
the acceleration of the adiabatic scheme to the present procedure
(e.g.,
Ref.~\cite{MartinezGaraot-PRL-111-213001,Masuda-PRL-113-063003})
should be also interesting.
We
note
that the preparation of condensates in a
quasi one-dimensional
box trap is experimentally achieved in
Ref.~\cite{Meyrath-PRA-71-041604}, which motives theoretical studies,
e.g., on solitonic excitations~\cite{Sciacca-PRA-95-013628}.
We also remark that the box trap may be useful to investigate quantum
information
processing through,
e.g. the Szilard engine in the quantum
regime~\cite{Kim-PRL-106-070401,Sodal-PRA-99-022121}. To extend these
works to many-body settings, our analysis on the
infinitely strong impurity
should
be applicable.
\section*{Acknowledgments}
A.T. wishes to thank Axel P\'erez-Obiol and Masaya Kunimi for
discussions. This work has been supported by the Grant-in-Aid for
Scientific Research of MEXT, Japan (Grants No. 17K05584). |
2007.16086 | \section{Introduction}
This paper deals with flat metric defined by Abelian differentials on compact Riemann surfaces (\emph{translation surfaces}).
For a translation surface, we define the \textit{relative systole} $\mathrm{Sys}(S)$ to be the length of the shortest saddle connection of $S$.
A sequence of area one translation surfaces $(S_n)_{n\in \mathbb{N}}$ in a stratum of the moduli space of translation surfaces leaves any compact set if and only if $\mathrm{Sys}(S_n)\to 0$.
The set of translation surfaces with short relative systole and compactification issues of strata are related to dynamics and counting problems on translation surfaces and have been widely studied in the last 30 years (see for instance \cite{KMS,EMZ,EKZ}).
Here, we are interested in the opposite problem: we study surfaces that are ``far'' from the boundary. In \cite{BG}, we have characterized global maxima for $Sys$ and we have shown that each stratum of genus greater than or equal to 3 contains local but non global maxima for the function $\mathrm{Sys}$. The constructed surfaces in \cite{BG} are not in the hyperelliptic connected components.
In this paper, we prove that there are no such local maxima in hyperelliptic connected components (Theorem~\ref{th:cc:hyp} in the text), while they exist in every other connected component (Theorem~\ref{th:cc:spin} in the text). This gives us the following characterization.
\begin{thmNN}[Main Theorem]
Let $\mathcal{C}$ be a connected component of a stratum of area one surfaces. The relative systole fonction on $\mathcal{C}$ admits a local maximum that is not a global maximum if and only if $\mathcal{C}$ is not hyperelliptic.
\end{thmNN}
Note that our notion of relative systole is different from the ``true systole'' \emph{i.e.} shortest closed curve that has been studied by Judge and Parlier in \cite{JP}. In the rest of the paper, for simplicity, if not mentioned otherwise, the term ``systole'' will mean ``relative systole''.
\section{Background}
\subsection{Translation surfaces}
A \emph{translation surface} is a (real, compact, connected) genus $g$ surface $S $ with a translation atlas \emph{i.e.} a triple $(S,\mathcal U,\Sigma)$ such that $\Sigma$ (whose elements are called {\em singularities}) is a finite subset of $S$ and $\mathcal U = \{(U_i,z_i)\}$ is an atlas of $S \setminus \Sigma$ whose transition maps are translations of $\mathbb{C}\simeq \mathbb{R}^2$. We will require that for each $s\in \Sigma$, there is a neighborhood of $s$ isometric to a Euclidean cone whose total angle is a multiple of $2\pi$. One can show that the holomorphic structure on $S\setminus \Sigma$ extends to $S$ and that the holomorphic 1-form $\omega=dz_i$ extends to a holomorphic $1-$form on $S$ where $\Sigma$ corresponds to the zeroes of $\omega$ and maybe some marked points. We usually call $\omega$ an \emph{Abelian differential}. A zero of $\omega$ of order $k$ corresponds to a singularity of angle $(k+1)2\pi$. By a slight abuse of notation, we authorize the order of a zero to be 0, in this case it corresponds to a regular marked point.
A \emph{saddle connection} is a geodesic segment joining two singularities (possibly the same) and with no singularity in its interior. Integrating $\omega$ along the saddle connection we get a complex number. Considered as a planar vector, this complex number represents the affine holonomy vector of the saddle connection. In particular, its Euclidean length is the modulus of its holonomy vector.
For $g \geq 1$, we define the moduli space of Abelian differentials $\mathcal{H}_g$ as the moduli space of pairs $(X,\omega)$ where $X$ is a genus $g$ (compact, connected) Riemann surface and $\omega$ non-zero holomorphic $1-$form defined on $X$. The term moduli space means that we identify the points $(X,\omega)$ and $(X',\omega')$ if there exists an analytic isomorphism $f:X \rightarrow X'$ such that
$f^*\omega'=\omega$.
One can also see a translation surface obtained from a polygon (or a finite union of polygons) whose sides come by pairs, and for each pair, the corresponding segments are parallel and of the same length. These parallel sides are glued together by translation and we assume that this identification preserves the natural orientation of the polygons. In this context, two translation surfaces are identified in the moduli space of Abelian differentials if and only if the corresponding polygons can be obtained from each other by cutting and gluing and preserving the identifications.
The moduli space of Abelian differentials is stratified by the combinatorics of the zeroes; we will denote by $\mathcal{H}(k_1,\ldots ,k_r)$ the stratum of $\mathcal{H}_g$ consisting of (classes of) pairs $(X,\omega)$ such that $\omega$ has exactly $r$ zeroes, of order $k_1,\dots ,k_r$ respectively. It is well known that this space is (Hausdorff) complex analytic. We have the classical Gauss--Bonnet formula $\sum_i k_i=2g-2$, where $g$ is the genus of the underlying surfaces. We often restrict to the subset $\mathcal{H}_1(k_1,\dots ,k_r)$ of \emph{area one} surfaces.
Local coordinates for a stratum of Abelian differentials are obtained by integrating the holomorphic 1--form along a basis of the relative homology $H_1(S,\Sigma;\mathbb{Z})$, where $\Sigma$ denotes the set of conical singularities of $S$.
\subsection{Connected component of strata}\label{background:cc}
Here, we recall the Kontsevich--Zorich classification the connected components of the strata of Abelian differentials \cite{KoZo}.
A translation surface $(X,\omega)$ is \emph{hyperelliptic} if the underlying Riemann surface is hyperelliptic, \emph{i.e.} there is an involution $\tau$ such that $X/\tau$ is the Riemann sphere. In this case $\omega$ satisfies $\tau^*\omega=-\omega$. A connected component of a stratum is said to be \emph{hyperelliptic} if it consists only of hyperelliptic translation surfaces (note that a connected component which is not hyperelliptic might contain some hyperelliptic translation surfaces).
Let $\gamma$ be a simple closed smooth curve parametrized by the arc length on a translation surface that avoids the singularities. Then $t\to \gamma'(t)$ defines a map from $\mathbb{S}^1$ to $\mathbb{S}^1$. We denote by $Ind(\gamma)$ the index of this map. Assume that the translation surface $S$ has only even degree singularities $S\in \mathcal{H}(2k_1,\dots ,2k_r)$. Let $(a_i,b_i)_{i\in \{1,\dots ,g\}}$ be a collection of simple closed curves as above and representing a symplectic basis of the homology of $S$. Then
$$\sum_{i=1}^g (ind(a_i)+1)(ind(b_i)+1) \mod 2$$
is an invariant of connected component ans is called the \emph{parity of the spin structure} (see \cite{KoZo} for details).
Here is a reformulation of the classification of connected component of strata by Kontsevich--Zorich (Theorem~1 and Theorem~2 of \cite{KoZo}).
\begin{theorem}[Kontsevich--Zorich]
Let $\mathcal{H}=\mathcal{H}(k_1,\dots ,k_r)$ be a stratum of genus $g\geq 2$ translation surfaces.
\begin{itemize}
\item The stratum $\mathcal{H}$ contains a hyperelliptic connected component if and only if $\mathcal{H}=\mathcal{H}(2g-2)$ or $\mathcal{H}=\mathcal{H}(g-1,g-1)$. In this case there is only one hyperelliptic component. In genus two, any stratum is connected (and hyperelliptic).
\item If there exists $i$ such that $k_i$ is odd, or if $g=3$, then there exists a unique nonhyperelliptic connected component.
\item If $g\geq 4$ and, for all $i$, $k_i$ is even, then there are exactly two nonhyperelliptic connected components distinguished by the parity of the spin structure.
\end{itemize}
\end{theorem}
The following lemma is classical and will be useful in the next section.
\begin{lemma}\label{lemme:hyp}
Let $S$ be a translation surface in a hyperelliptic connected component and let $\gamma$ be a saddle connection. Then $\gamma$ and $\tau(\gamma)$ are homologous.
\end{lemma}
\section{Hyperelliptic connected component}
In this section, we prove the first part of the Main Theorem.
\begin{theorem}\label{th:cc:hyp}
Let $\mathcal{C}$ be a hyperelliptic connected component of the moduli space of Abelian differentials.
Let $S\in \mathcal{C}$ be a local maximum of the relative systole function $Sys$. Then $S$ is a global maximum for $Sys$ in $\mathcal{C}$.
\end{theorem}
The proof uses the following technical lemma. We postpone its proof to the end of the section.
\begin{lemma}\label{lem:decrease:area:disk}
Let $D$ be a translation surface that is topologically a disk and whose boundary consists of $n$-saddle connections (a ``$n$-gon'') with $n\geq 4$. We assume that all boundary saddle connections are of length greater than or equal to 1. Then, we can continuously deform $D$ so that its area decreases and the boundary saddle connections of length 1 remain of length 1.
\end{lemma}
\begin{proof} [Proof of Theorem~\ref{th:cc:hyp}]
Let $S\in\mathcal{C}$ be a translation surface that such that $Sys(S)$ is not a global maximum. We use the same normalization as in \cite{BG}: after rescaling the surface we assume that $Sys(S)$ equals 1, and we will continuously deform $S$ so that $Sys(S)$ remains 1 and $\mathrm{Area}(S)$ decreases.
Let $\gamma_1,\dots ,\gamma_r$ be the set of saddle connections realizing the systole.
Recall that $\gamma_1,\dots ,\gamma_r$ are sides of the Delaunay triangulation and
that global maxima correspond to surfaces which Delaunay cells are only equilateral triangles (see Lemma~3.1 and Theorem~3.3 in \cite{BG}).
Let $C_1,\dots ,C_k$ be the connected components of $S\backslash \cup_i \gamma_i$. Up to renumbering we can assume that $C_1$ is not a triangle. We consider $\tau(C_1)$, where $\tau$ is the hyperelliptic involution. We study the two possible cases whether $\tau(C_1)$ equals $C_1$ or not. Note that $C_1$ does not contain any singularity in its interior since there are at most two singularities in $S$ and if there are two singularities $P_1,P_2$ we must have $\tau(P_1)=P_2$.
{\bf Case 1.} We first assume that $\tau(C_1)\neq C_1$. Since the hyperelliptic involution preserves $\cup_i \gamma_i$, we have (up to renumbering) $\tau(C_1)=C_2$.
We observe that $C_1$ has only one boundary component. Indeed, suppose that there are more than one such components and consider a saddle connection $\eta$ in $C_1$ that joins a singularity of one boundary component to a singularity of another boundary component. Then $\tau(\eta)$ is a curve in $C_2$ and must be homologous to $\eta$ by Lemma~\ref{lemme:hyp}. But $C_1\backslash \eta$ is connected, and hence $S\backslash (\eta\cup \tau(\eta))$ is connected, which is a contradiction. Therefore $C_1$ is a disk because it embeds in $S/\tau$ which is a sphere.
Since the boundary of $C_1$ consists of at least 4 saddle connections of length 1, by Lemma~\ref{lem:decrease:area:disk}, we can continuously decrease its area while keeping the boundary saddle connections of length 1.
This continuous deformation of $C_1$ leads to the following area decreasing continuous deformation of $S$:
The component $C_2$ is deformed in a symmetric way as $C_1$.
For each saddle connection $\gamma$ in the boundary of $C_1$, the components of $S\backslash (\gamma\cup \tau(\gamma))$ correspond to components of the complementary of $[\gamma]$ in the quotient sphere $S/\tau$. Since $[C_1]=[C_2]$, then $C_1,C_2$ are in the same connected component of $S\backslash (\gamma\cup \tau(\gamma))$. We denote by $D_\gamma$ the other component. Note that $D_{\gamma}$ is empty if $\gamma=\tau(\gamma)$. We observe that if $\gamma_1,\gamma_2$ are two distinct saddle connections in the boundary of $C_1$, then $D_{\gamma_1}$ and $D_{\gamma_2}$ are disjoint. In particular, we can rotate all such components $D_{\gamma}$ independently in a compatible way with the deformation of $C_1,C_2$ and we glue these components in the natural way. The area of each $C_i$ decreases while the area of the $D_\gamma$ remains constant. Therefore the total area of the surface decreases.
\medskip
{\bf Case 2.} Now we assume that $\tau(C_1)=C_1$.
We claim that we can cut $C_1$ along saddle connections and obtain two discs $A$ and $B$ such that $\tau(A)=B$ and for each saddle connection $\gamma$ in the boundary of $A$, either $\gamma$ is of length 1, or $\tau(\gamma)=\gamma$ (equivalently, $\gamma$ is also a boundary saddle connection of $B$).
To prove the claim, we first consider the Delaunay cells of $S$. Recall that the shortest geodesics (hence the boundary saddle connections of $C_1$) are sides of the Delaunay cells (see \cite{BG}, Lemma~3.1). This induces a decomposition of $C_1$ into Delaunay cells, and this decomposition is preserved by the involution $\tau$ because of the uniqueness of the Delaunay cell decomposition. We define a Delaunay subdivision $\mathcal{D}$ in the following way: for each Delaunay cell $d$, if $\tau(d)\neq d$, then $d,\tau(d) \in \mathcal{D}$. If $\tau(d)=d$ and since $d$ is cyclic it can be cut by a diagonal into two polygons $d'$ and $d''=\tau(d')$, then $d',d''\in \mathcal{D}$.
Now we use the following algorithm:
\begin{itemize}
\item We start from a pair $d_0,\tau(d_0)$ in $\mathcal{D}$ and let $A_0=d_0$ and $B_0=\tau(d_0)$.
\item Suppose we have constructed the disks $A_k$ and $B_k$ such that $\tau(A_k)=B_k$ and $A_k, B_k$ are union of elements in $\mathcal{D}$.
If $A_k\cup B_k\neq C_1$, there exists an element $d_{k+1}\in \mathcal{D}$ adjacent to $A_k$ along a saddle connection $\gamma_k$ (and $\tau(d_{k+1})\in \mathcal{D}$ is adjacent to $B_k$ along $\tau(\gamma_k)$). We define $A_{k+1}$ by gluing $A_k$ and $d_{k+1}$ along $\gamma_k$. Note that $\gamma_k$ is the only saddle connection in the common boundary of $A_{k}$ and $d_{k+1}$ because otherwise $S\backslash (\gamma_k\cup \tau(\gamma_k))$ would be connected which is impossible in the hyperelliptic connected component.
If $A_k\cup B_k=C_1$, we define $A=A_k$, $B=B_k$.
\end{itemize}
The boundary of the disk $A$ consists of $n\geq 3$ saddle connections of lengths at least 1. If $n\geq 4$, then from Lemma~\ref{lem:decrease:area:disk}, it can be continuously deformed so that the area decreases and the boundary saddle connections of length 1 remain of length one. Otherwise $A$ is a triangle but cannot be a equilateral triangle, hence it can also be deformed as above.
We deform $B$ in a symmetric way. Note that $A$ and $B$ are directly glued together in $C_1$ along the boundary saddle connections of lengths greater than one.
Therefore the possible changes of these saddle connections are not a problem. The deformation of $S\backslash C_1$ is treated as in the previous case.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:decrease:area:disk}]
The sum of the boundary angles (coming from the intersection of two consecutive boundary saddle connections) of $D$ equals $(n-2)\pi$. Therefore $D$ has boundary angles smaller than $\pi$. If such a boundary angle has a corresponding boundary saddle connection which is of length greater than 1, then by slightly changing its length we can decrease the area of the corresponding triangle and hence of $D$.
So we can assume that for each boundary angle smaller than $\pi$, the two adjacent saddle connections are of length 1.
We claim that we can find two consecutive angles such that one is smaller than $\pi$ and the other is smaller than $2\pi$ (note that since $D$ is not necessarily embedded in the plane, it can have boundary angles greater than $2\pi$). Indeed, consider the sequence of consecutive boundary angles of $D$. If each time an angle is smaller than $\pi$, the following one is greater than or equal to $2\pi$, then the global sum will be greater than $n\pi$, which is not possible.
Now we consider the 3 consecutive saddle connections corresponding to these two angles, and see them as a broken line on the plane. We close this line by adding a segment $t$ to obtain a quadrilateral $\mathcal{Q}$ (that can be also crossed). Without loss of generality, we can assume that $t$ is horizontal. We have
$$\mathrm{Area}(D)=\mathrm{Area}(D_0)+\mathrm{Area}_{alg}(\mathcal{Q})$$
where $D_0$ is the translation surface obtained by ``replacing'' the broken line by $t$ (see Figure~\ref{fig:D:et:Q}). Here $\mathrm{Area}_{alg}(\mathcal{Q})$ means that the part of $\mathcal{Q}$ below the segment $t$ is counted negatively.
{\bf Claim:} We can deform continuously $\mathcal{Q}$ without changing the lengths of its sides so that $\mathrm{Area}_{alg}(\mathcal{Q})$ decreases.
Denote by $MNPQ$ the quadrilateral $\mathcal{Q}$ and by $a,b,c,d$ the lengths of the sides of $\mathcal{Q}$ with $a$ being the length of the segment $t=MN$. Denote by $\alpha$ the oriented angle from $MN$ to $MQ$, and by $\gamma$ its opposite angle in $\mathcal{Q}$ (\emph{i.e.} the angle from $PQ$ to $PN$). Without loss of generality we assume that $b=c=1$, $d\geq 1$ and $0<\gamma<\pi$ (in fact we must have $\gamma>\pi/3$ otherwise there would be a smallest saddle connection). We also have $-\pi<\alpha<\pi$. Also, the sides $NP$ and $QM$ do not intersect since it would imply intersecting boundary saddle connections in $D$ (see Figure~\ref{fig:D:et:Q}).
\begin{figure}[htb]
\begin{tikzpicture}[scale=0.8]
\coordinate (z1) at (2,2);
\coordinate (z2) at (2,1);
\coordinate (z3) at (1,-2) ;
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\draw (0,0) node {\tiny $\bullet$} node[below] {\small $M$} --++ (z1) node[midway,above] {$d$} node {\tiny $\bullet$} node[above] {\small $Q$} --++ (z2) node[midway,above] {$c$} node {\tiny $\bullet$} node[above] {\small $P$}--++ (z3) node[midway,above] {$b$} node {\tiny $\bullet$} node[below] {\small $N$};
\draw [dashed] (-3,0)--++(3,0);
\draw [dashed] ($(z1)+(z2)+(z3)$)--++(3,0);
\draw [dotted] (0,0) --($(z1)+(z2)+(z3)$) node[midway,above] {$a$};
\coordinate (z1) at (2,-1);
\coordinate (z2) at (1,2);
\coordinate (z3) at (2,-1) ;
\fill [fill=gray!20] (-3,-5)--++(3,0) --++ (z1) --++ (z2) --++ (z3) --++ (3,0) --++(0,-3) --++(-11,0) node[near start, above] {$D$}--cycle;
\draw (0,-5) node {\tiny $\bullet$} node[below] {\small $M$} --++ (z1) node[midway,below] {$d$} node {\tiny $\bullet$} node[below] {\small $Q$}
--++ (z2) node[midway,above] {$c$} node {\tiny $\bullet$} node[above] {\small $P$}
--++ (z3) node[midway ,above] {$b$} node {\tiny $\bullet$} node[below] {\small $N$};
\draw [dashed] (-3,-5)--++(3,0);
\draw [dashed] ($(0,-5)+(z1)+(z2)+(z3)$)--++(3,0);
\draw [dotted] (0,-5) --++($(z1)+(z2)+(z3)$) node[near end,below] {$a$};
\coordinate (z1) at (-35:3);
\coordinate (z2) at (30:2);
\coordinate (z3) at (10:2) ;
\fill [fill=gray!20] (-3,-10)--++(2.42,0) --++ (z1) --++ (z2) --++ (z3) --++ (2.42,0) --++(0,-3) --++(-11.04,0) node[near start, above] {$D$}--cycle;
\draw (-0.58,-10) node {\tiny $\bullet$} node[below] {\small $M$} --++ (z1) node[midway,below] {$d$} node {\tiny $\bullet$} node[below] {\small $Q$}
--++ (z2) node[midway,above] {$c$} node {\tiny $\bullet$} node[above] {\small $P$}
--++ (z3) node[midway ,below] {$b$} node {\tiny $\bullet$} node[below] {\small $N$};
\draw [dashed] (-3,-10)--++(2.42,0);
\draw [dashed] ($(-0.58,-10)+(z1)+(z2)+(z3)$)--++(2.42,0);
\draw [dotted] (-0.58,-10) --++($(z1)+(z2)+(z3)$) node[midway,above] {$a$};
\end{tikzpicture}
\caption{The disk $D$ and the quadrilateral $\mathcal{Q}$ in 3 configurations.}
\label{fig:D:et:Q}
\end{figure}
Denote $K=\mathrm{Area}_{alg}(\mathcal{Q})$. We compute $K$ by adding the (algebraic) area of the triangles $MNQ$ and $NPQ$. We obtain
\begin{equation}
K=\frac{1}{2}\left(ad\sin(\alpha)+bc\sin(\gamma)\right).
\end{equation}
The expression of the length of $NQ$ gives the second equality:
\begin{equation}
a^2+d^2-2ad\cos(\alpha)=b^2+c^2-2bc\cos(\gamma).
\end{equation}
These two equations imply the Bretschneider's formula for $\mathcal{Q}$:
\begin{equation}
K^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos(\frac{\alpha+\gamma}{2}),
\end{equation}
where $s=\frac{a+b+c+d}{2}$.
From now on, we fix $a,b,c,d$ and study the variations of the area with respect to $\alpha,\gamma$. Equation (2) implies that $\gamma$ depends differentially on
$\alpha$. Hence we can write $K=K(\alpha)$. We need to prove that $K'(\alpha)\neq 0$ or $K(\alpha)$ is a strict local maximum (note that $\alpha$ varies in an open set). We have:
$$(K^2)'(\alpha)=abcd(1+\gamma'(\alpha))\sin(\frac{\alpha+\gamma}{2})\cos(\frac{\alpha+\gamma}{2}).$$
We assume that $K'(\alpha)=0$, hence $(K^2)'(\alpha)=0$, hence we are in one of the following three cases:
\begin{enumerate}
\item $\sin(\frac{\alpha+\gamma}{2})=0$. The conditions $-\pi<\alpha<\pi$ and $0<\gamma<\pi$ imply $\alpha=-\gamma<0$. Hence the quadrilateral $\mathcal{Q}$ have self-intersections. Since the sides $NP$ and $QM$ do not intersect, the sides $MN$ and $PQ$ intersect.
The condition $\alpha=-\gamma$ implies that the points $M,N,P,Q$ are cocyclic, and since $b=c=1$ we must have $d< 1$ which is a contradiction.
\item $\cos(\frac{\alpha+\gamma}{2})=0$. Then $\alpha+\gamma=\pi$, and therefore $\alpha>0$ and hence $K>0$. From $(2)$ and $(3)$, we have a strict local maximum for $K^2$ and therefore for $K$.
\item $\gamma'(\alpha)=-1$. By differentiating $(2)$ and using $(1)$, we see that $K=0$, hence $\mathcal{Q}$ has a self-intersection $I=MN\cap PQ$. By differentiating $(1)$ and using (2), we obtain $K'(\alpha)=0=\frac{1}{2}(\frac{a^2+d^2}{2}-1)$, hence $a^2+d^2=2$. Since $d\geq 1$, we have $a\leq 1 \leq d$. However, triangle inequalities for $INP$ and $IMQ$ give $a+c>d+b$ and hence $a>d$, which is a contradiction.
\end{enumerate}
\end{proof}
\section{Nonhyperelliptic connected components}
In this section, we prove the second part of the Main Theorem.
\begin{theorem}\label{th:cc:spin}
Each nonhyperelliptic connected component of each stratum of area one surfaces with no marked points contains local maxima of the function $\mathrm{Sys}$ that are not global.
\end{theorem}
The proof is a refinement of the proof of Theorem~4.7 in \cite{BG}.
We will need the following lemma, which is a refinement of Lemma~3.2 (2) in \cite{BG}.
\begin{lemma}\label{lemme:sc:indice}
Let $\mathcal{C}\subset \mathcal{H}(2k_1,\dots ,2k_r)$ be a connected component of a stratum of abelian differentials with $2k_1,\dots ,2k_r\geq 0$. There exists a surface $S\in \mathcal{C}$ realizing the global maximum for the systole function, and such that there exists a shortest saddle connection $\gamma$ joining a singularity of degree $2k_1$ to itself and $\mathrm{Ind}([\gamma])=0$.
\end{lemma}
\begin{proof}
We do as in the proof of Lemma~3.2 in \cite{BG}. There exists a square tiled surface in $\mathcal{C}$ with singularities on each corner of the squares as in Figure~\ref{squares:to:triangles}, and we can assume that the top left horizontal segment identifies with the bottom left horizontal segment (see Figure~\ref{squares:to:triangles}). After a suitable transformation as in the figure, we obtain the required surface.
\begin{figure}[htb]
\begin{tikzpicture}[scale=0.8]
\coordinate (v0) at (1,0);
\coordinate (v1) at (0.5,0.866);
\coordinate (v2) at ($(v0)-(v1)$);
\coordinate (u) at (0,1);
\coordinate (v3) at ($-1*(v0)$);
\coordinate (v4) at ($-1*(v1)$);
\draw (0,0) --++ (v0) node[midway,below] {\small $1$} node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (u) node{$\bullet$} --++ (v3) node{$\bullet$} --++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node[midway,above] {\small $1$} node{$\bullet$} --++ (0,-1) node{$\bullet$};
\draw (0,-3) --++ (v0) node[midway,below]{\small $1$} node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v0) node{$\bullet$} --++ (v1) node{$\bullet$} --++ (v3) node{$\bullet$} --++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node{$\bullet$}--++ (v3) node[midway,above] {\small $1$} node{$\bullet$} --++ (v4) node[midway, left] {$\gamma$} node{$\bullet$};
\draw ($(0,-3)+(v1)$)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2);
\draw[very thick,->] (4,-0.5)--++(0,-1);
\end{tikzpicture}
\caption{A global maximum with a closed shortest saddle connection $\gamma$ sastisfying $\mathrm{Ind}([\gamma])=0$}
\label{squares:to:triangles}
\end{figure}
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:cc:spin}]
In Theorem~4.7 in \cite{BG} we have already constructed examples in each genus $g\geq 3$ stratum. By Theorem~\ref{th:cc:hyp} each such example is in a nonhyperelliptic component. So it remains to construct new examples only in strata with more than one nonhyperelliptic connected component.
From the Theorem of Kontsevich--Zorich stated in Section~\ref{background:cc}, there is more than one nonhyperelliptic connected component only for genus $g\geq 4$ strata with only even degree singularities and in this case there are two nonhyperelliptic components distinguished by the parity of the spin structure.
In Figure~\ref{fig:examples} we give surfaces $S_{2,0}\in \mathcal{H}(2,0)$ and $S_{2,0,0}\in \mathcal{H}(2,0,0)$ that are local but nonglobal maxima for the systole function.
\begin{figure}[htb]
\begin{tikzpicture}[scale=1.3]
\coordinate (v0) at (1,0);
\coordinate (v1) at (0.5,0.866);
\coordinate (A) at (0,0);
\coordinate (B) at (4,0);
\draw (A) --++ (v0) --++ (v1) --++ ($(v1)-(v0)$) --++ ($-1*(v0)$)--++ ($-1*(v1)$) node[midway,left] {$2$}--++ ($(v0)-(v1)$) node[midway,left] {$3$};
\draw (A) --++ ($(v0)-(v1)$) node[midway,left] {$4$} --++ (v1) node[midway,right] {$1$} --++ (v0) node[midway,below] {$5$} --++ ($(v1)-(v0)$) node[midway,right] {$4$}--++ (v1) node[midway,right] {$2$} --++ ($-1*(v0)$) node[midway,above] {$5$} --++ ($(v1)-(v0)$) node[midway,right] {$3$} --++ ($-1*(v1)$) node[midway,left] {$1$};
\draw ($(A)+(0.5,-1.5)$) node {$S_{2,0}\in \mathcal{H}(2,0)$};
\draw (B) --++ (v0) --++ (v1) --++ ($(v1)-(v0)$) --++ ($-1*(v0)$)--++ ($-1*(v1)$) node[midway,left] {$2$}--++ ($(v0)-(v1)$) node[midway,left] {$3$};
\draw (B) --++ ($(v0)-(v1)$) node[midway,left] {$4$} --++ (v1) node[midway,right] {$1$} --++ (v0) node[midway,below] {$5$} --++ ($(v1)-(v0)$) --++ (v1) --++ ($-1*(v0)$) node[midway,above] {$5$} --++ ($(v1)-(v0)$) node[midway,right] {$3$} --++ ($-1*(v1)$) node[midway,left] {$1$};
\draw ($(B)+(v0)+(v1)$) --++ (v0) --++ ($(v1)-(v0)$) node[midway,right] {$4$};
\draw ($(B)+(v0)+(v0)$) --++ (v1) node[midway,right] {$2$};
\draw ($(B)+(0.5,-1.5)$) node {$S_{2,0,0}\in \mathcal{H}(2,0,0)$};
\end{tikzpicture}
\caption{Local but nonglobal maxima in $\mathcal{H}(2,0)$ and $\mathcal{H}(2,0,0)$}
\label{fig:examples}
\end{figure}
We consider the following construction: start from the surface $S_{2,0}$ and a surface $M$ that is a global maximum for $Sys$ in $\mathcal{H}(2k_1,\dots ,2k_r)$. There exists a shortest saddle connection $\gamma_1$ in $S_{2,0}$ joining the two singularities. By Lemma~\ref{lemme:sc:indice}, we can assume that there exists a shortest saddle connection $\gamma_2$ in $M$ joining the singularity of degree $2k_1$ to itself and such that $\mathrm{Ind}([\gamma_2])=0$. We can assume that $\gamma_1,\gamma_2$ are vertical and of the same length. Now we glue the two surfaces by the following classical surgery: cut the two surfaces along $\gamma_1$ and $\gamma_2$, and glue the left side of $\gamma_1$ with the right side of $\gamma_2$ and the right side of $\gamma_1$ with the left side of $\gamma_2$. We get a surface $S$ in $\mathcal{H}(2k_1+4,2k_2,\dots ,2k_r)$ that satisfies the hypothesis of Theorem~4.1 in \cite{BG} and hence is a local but nonglobal maximum. By Theorem~\ref{th:cc:hyp}, the surface $S$ is necessarily in a nonhyperelliptic component.
We compute $\mathrm{Spin}(S)$: we choose a symplectic basis $(a_i,b_i)_{i}$ of $H_1(M,\mathbb{Z})$ such that $[\gamma_2]=a_1$. Then a simple computation gives
\begin{equation}\label{eq:spin}
\mathrm{Spin}(S)=\mathrm{Spin}(S_{0,2})+\mathrm{Spin}(M)+\mathrm{Ind}(a_1)+1 \mod 2. \end{equation}
Since $\mathrm{Ind}(a_1)=0$, we have
$$\mathrm{Spin}(S)=\mathrm{Spin}(S_{0,2})+\mathrm{Spin}(M)+1 \mod 2. $$
When $\sum_{i}2 k_i\geq 4$, we can prescribe any value of $\mathrm{Spin}(M)$ by choosing $M$ in a suitable component and in this way we can obtain any possible value for $\mathrm{Spin}(S)$. Note that this is also true for $M\in \mathcal{H}(4)$ or $M\in \mathcal{H}(2,2)$. Indeed, in these strata there are two components, the hyperelliptic one and the nonhyperelliptic one, and the spin structure distinguishes them (see \cite{KoZo}, Theorem~2 and Corollary~5)
By this construction, we obtain a local but non global maximum for $Sys$ in any (nonhyperelliptic) connected component of any stratum $\mathcal{H}(2n_1,\dots ,2n_r)$ for $r\geq 1$, as soon as $\sum_i 2n_i\geq 8$ and $2n_j\geq 4$ for at least one $j\in \{1,\dots ,r\}$.
We do an analogous construction as above starting from $S_{2,0,0}$ (see Figure~\ref{fig:examples}) and $M\in \mathcal{H}(0,2^r)$ with $\gamma_1\in S_{2,0,0}$ joining the two marked points, and $\gamma_2\in M$ joining the marked point to itself. We obtain a local but nonglobal maximum in $\mathcal{H}(2^{r+2})$. For $r\geq 2$ we can choose the spin structure of $M$ and thus get $S$ in any nonhyperelliptic component of $\mathcal{H}(2^{r+2})$. Note that for $r=1$, we get $S\in \mathcal{H}(2,2,2)$ with odd spin structure.
\begin{figure}[htb]
\begin{tikzpicture}[scale=1]
\coordinate (v0) at (1,0);
\coordinate (v1) at (0.5,0.866);
\coordinate (v2) at ($(v0)-(v1)$);
\coordinate (u) at (0,1);
\coordinate (v3) at ($-1*(v0)$);
\coordinate (v4) at ($-1*(v1)$);
\draw (0,0) --++ (v0) node[midway,below]{$b$} node{$\bullet$} --++ (v0) node[midway,below]{$c$} node{$\bullet$} --++ (v0) node[midway,below]{$d$} node{$\bullet$} --++ (v1) node[midway,right]{$a$} node{$\bullet$} --++ (v3) node[midway,above]{$c$} node{$\bullet$} --++ (v3) node[midway,above]{$d$} node{$\bullet$}--++ (v3) node[midway,above]{$b$} node{$\bullet$}--++ (v4) node[midway, left] {$a$} node{$\bullet$};
\draw ($(0,0)+(v1)$)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2);
\draw (6,0) --++ (v0) node[midway,below]{$b$} node{$\bullet$} --++ (v0) node[midway,below]{$c$} node{$\circ$} --++ (v0) node[midway,below]{$d$} node{$\bullet$} --++ (v0) node[midway,below]{$e$} node{$\circ$} --++ (v1) node[midway,right]{$a$} node{$\bullet$} --++ (v3) node[midway,above]{$b$} node{$\circ$} --++ (v3) node[midway,above]{$c$} node{$\bullet$} --++(v3) node[midway,above]{$d$} node{$\circ$}--++ (v3) node[midway,above]{$e$} node{$\bullet$}--++ (v4) node[midway, left] {$a$} node{$\circ$};
\draw ($(6,0)+(v1)$)--++(v2)--++(v1)--++(v2)--++(v1)--++(v2) --++(v1)--++(v2);
\draw (1.5,-1) node {$S_{2}\in \mathcal{H}(2)$};
\draw (8,-1) node {$S_{1,1}\in \mathcal{H}(1,1)$};
\end{tikzpicture}
\caption{Global maxima in $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$}
\label{fig:h2:h11}
\end{figure}
There remain the following cases:
\begin{itemize}
\item $\mathcal{H}(6)$. We do the same construction as above starting from $S_{2,0}$ and $M\in \mathcal{H}(2)$. We consider for $M\in \mathcal{H}(2)$ the surface $S_2$ in Figure~\ref{fig:h2:h11}. We see that $[a],[b]$ in this figure have different indices $\mod 2$. Hence choosing $\gamma_2=a$ or $\gamma_2=b$ gives surfaces with different Spin structure (see Equation (\ref{eq:spin})).
\item $\mathcal{H}(4,2)$. We do the same as for $\mathcal{H}(6)$, starting from $S_{2,0,0}$ and $M=S_2$.
\item The even component of $\mathcal{H}(2,2,2)$. We do the same construction but starting from $S_{2,0,0}$ and $M\in \mathcal{H}(1,1)$ the surface $S_{1,1}$ in Figure~\ref{fig:h2:h11}. We consider $\gamma_2=a$ (joining the two singularities of degree 1). By a direct computation, we see that the above construction gives a surface $S\in \mathcal{H}(2,2,2)$ with $\mathrm{Spin}(S)=0 \mod 2$.
\end{itemize}
\end{proof} |
2007.16160 | \section{The multiplayer triangle game and formal statement of Corollary~1}
\label{SM1}
\subsection{Multiplayer triangle game}
\label{N_Player}
In this section we formally state the win conditions for the multiplayer triangle game. Consider a scenario where $n$ players located around a cycle are enumerated as players 0 through $n-1$ in a clockwise manner. Suppose that three players with labels $\alpha$, $\beta$, and $\gamma$ with $\alpha<\beta<\gamma$ are given one bit each from a three-bit input string $\mathbf{x} = (x_\alpha, x_\beta, x_\gamma)\in\{0,1\}^3$ drawn uniformly at random. Each player is tasked to fill in the row or column of their respective table with a three-bit output string $\mathbf{y}_j$ depending on if their received input is 0 or 1, respectively. All players other than $\alpha$, $\beta$, and $\gamma$ receive input 0. We may depict players $\alpha$, $\beta$, and $\gamma$ as residing at the top, lower right, and lower left corners of a triangle, respectively. If we denote the output bit string as a function of the input, i.e. $\mathbf{y}_j = \mathbf{y}_j(x_j)$, we can write $\mathbf{y}_j(0) = (a_j, b_j, c_j)$ and $\mathbf{y}_j(1) = (d_j, b_j, e_j)$ in correspondence with the table. The players are said to win the game whenever the joint output $\{\mathbf{y}_j\}_{j=0}^{n-1}$ forms a string of even parity (i.e. $\sum_{j=0}^{n-1} |\mathbf{y}_j|=0 \textrm{ mod }2$, where $|\cdot|$ denotes the Hamming weight) and the following equations hold
\begin{subequations}
\begin{align}
a_\alpha + a_\beta + a_\gamma &= a_R + a_L + a_B \label{N_Tri_1}\\
b_\alpha + b_\beta + b_\gamma &= b_R + b_L + b_B \label{N_Tri_2}\\
d_\alpha + e_\beta + a_\gamma &= 1+c_R + a_B + a_L\label{N_Tri_3}\\
d_\beta + e_\gamma + a_\alpha &= 1+c_B + a_L + a_R \label{N_Tri_4}\\
d_\gamma + e_\alpha + a_\beta &= 1+c_L + a_R + a_B \label{N_Tri_5},
\end{align}
\end{subequations}
where using joint indices for the right (R) edge, bottom (B) edge, and left (L) edge of the triangle, we define for $\sigma\in\{a,b,c\}$,
\begin{subequations}
\begin{align}
\sigma_R &= \sum_{j=\alpha+1}^{\beta-1} \sigma_j \label{R}\\
\sigma_B &= \sum_{j=\beta+1}^{\gamma-1} \sigma_j \label{B}\\
\sigma_L &= \sum_{j=\gamma+1}^{\alpha-1} \sigma_j. \label{L}
\end{align}
\end{subequations}
Notice that Eqs.~(\ref{N_Tri_1})-(\ref{N_Tri_5}) are identical to the win conditions of the triangle game up to some dependence on a ``correction string" given by the outputs of the additional $n-3$ players. Thus the dependence of the win condition on the correction string does not affect the overall contradictory nature of the system of equations. Indeed, summing Eqs.~(\ref{N_Tri_1})-(\ref{N_Tri_5}) returns $\sum_{j\in\{\alpha,\beta,\gamma\}} (d_j + b_j + e_j) + \sum_{\Sigma\in\{R,B,L\}} (a_\Sigma + b_{\Sigma} + c_\Sigma) =1$, implying that the total parity of all player's outputs for input $(1,1,1)$ cannot have even parity. Thus, no classical strategy where players
do not communicate can win the multiplayer triangle game with probability greater than 7/8.
For this multiplayer game, even classical strategies assisted by geometrically local communication between players fail to win with probability greater than $7/8$. Consider communication-assisted classical strategies in which players $\alpha$, $\beta$, and $\gamma$ communicate with mutually exclusive parties. With this restriction the outputs of the players are restricted to be affine boolean functions of the input. Furthermore, suppose that players $\alpha$, $\beta$, and $\gamma$ communicate only with players on adjacent edges (i.e. player $\alpha$ cannot communicate with players in $B$, player $\beta$ cannot communicate with players in $L$, and player $\gamma$ cannot communicate with players in $R$). In this case the affine boolean function for the collective outputs of players on each edge of the triangle are independent of the opposite input (i.e. $\sigma_R = \sigma_R(x_\alpha,x_\beta)$, $\sigma_B = \sigma_B(x_\beta,x_\gamma)$, and $\sigma_L = \sigma_L(x_\alpha,x_\gamma)$). Substituting these Boolean functions into Eqs.~(\ref{N_Tri_1})-(\ref{N_Tri_5}) reveals that the system of equations are still contradictory.
If the communication can be geometrically nonlocal, the classical strategy can be perfect. For example, the strategy attributed to the boolean functions; $c_R(\mathbf{x}) = x_\alpha + x_\gamma$, $c_B(\mathbf{x}) = x_\alpha + x_\beta$, $c_L(\mathbf{x}) = x_\beta + x_\gamma$, and all other outputs being 0, satisfies the win conditions. To see how quantum strategies can surpass even nonlocal communication-assisted classical strategies, it is imperative to move to a 2D scenario on a lattice in which there are many possible cycles between the three players. For communication-assisted classical strategies with a constant number of rounds of communication between at most $K$ players at a time, it becomes increasingly likely that there will be a cycle in the lattice satisfying the locality conditions that ensure failure of the classical strategy. We now elaborate on this 2D setting, which gives a worst-case unconditional separation between constant-depth classical and quantum circuits \cite{bravyi2018quantum}.
\subsection{Physical setting for demonstrating unconditional separation}
For completeness we first restate the informal version of Cor.~\ref{SPTO_Advantage}
\begin{SPTO_Advantage}
$(\mathrm{Informal})$ Consider a relation problem given by the multiplayer triangle game embedded on arbitrary cycles in a 2D lattice. A quantum device that can prepare a 1D SPTO ground state with string order parameters greater than 1/3 on the arbitrary cycles and perform the fixed-point protocol of Theorem~2 solves the problem with probability greater than 7/8 on all inputs exponentially faster than all classical Boolean circuits.
\label{SPTO_Advantage}
\end{SPTO_Advantage}
Here we expound upon the setting for the computational separation in Cor.~\ref{SPTO_Advantage}. Consider a black-box device taking inputs and producing outputs in correspondence with vertices in some connected two-dimensional lattice graph $\mathcal{G}=(V,E)$ that can be deformed such that its vertices lie on a grid of size $N\times N$. The device is then tasked to solve a relation problem called the 2D multiplayer triangle problem, which is an embedding of the multiplayer triangle game into the graph $\mathcal{G}$. We define the relation problem as follows.
\fbox{\parbox{\textwidth}{
\begin{2D_Triangle}
An input to the problem is provided as a tuple $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)$. $\alpha$, $\beta$, and $\gamma$ are three arbitrary vertices in the 2D lattice graph $\mathcal{G}$ of size $N \times N$. $\mathbf{x}\in\{0,1\}^{N^2}$ is any binary string of length $N^2$ and Hamming weight $|\mathbf{x}|\leq 3$, where each bit $x_v$ in the string corresponds to a vertex $v$ in $\mathcal{G}$, and the only possible nonzero bits in $\mathbf{x}$ are $x_\alpha$, $x_\beta$, and $x_\gamma$. $\Gamma$ is a cycle in $\mathcal{G}$ connecting vertices $\alpha$, $\beta$, and $\gamma$ by paths $\Gamma_{\alpha\beta}$, $\Gamma_{\beta\gamma}$, and $\Gamma_{\gamma\alpha}$. An output is a $3N^2$-bit string $\mathbf{y}\in\{0,1\}^{3N^2}$ composed of three-bit strings $(y_{0,v},y_{1,v},y_{2,v})$ corresponding to the output of each vertex $v$. A solution of the problem is a string $\mathbf{y} \in\{0,1\}^{3N^2}$ of even parity satisfying the following linear equations.
\renewcommand{\labelenumi}{$\mathrm{(\Roman{enumi})}$}
\begin{enumerate}
\item \label{One} For all $\mathbf{x}\in\{0,1\}^{N^2}$, $\sum_{v\in \Gamma} y_{1,v} = 0$.
\item \label{Two} If $(x_\alpha,x_\beta,x_\gamma)=(0,0,0)$, then $\sum_{v\in\Gamma} y_{0,v} = 0$ and $\sum_{v\in\Gamma} y_{2,v} =0$.
\item \label{Three} If $(x_\alpha,x_\beta,x_\gamma)=(1,1,0)$, then $y_{0,\alpha} + y_{2,\beta} + y_{0,\gamma} = 1 + \sum_{v\in \Gamma_{\alpha\beta}} y_{2,v} + \sum_{v\in \Gamma_{\beta\gamma}\cup \Gamma_{\gamma\alpha}} y_{0,v}$.
\item \label{Four} If $(x_\alpha,x_\beta,x_\gamma)=(0,1,1)$, then $y_{0,\alpha} + y_{0,\beta} + y_{2,\gamma} = 1 + \sum_{v\in \Gamma_{\beta\gamma}} y_{2,v} + \sum_{v\in \Gamma_{\alpha\beta}\cup \Gamma_{\gamma\alpha}} y_{0,v}$.
\item \label{Five} If $(x_\alpha,x_\beta,x_\gamma)=(1,0,1)$, then $y_{2,\alpha} + y_{0,\beta} + y_{0,\gamma} = 1 + \sum_{v\in \Gamma_{\gamma\alpha}} y_{2,v} + \sum_{v\in \Gamma_{\alpha\beta}\cup \Gamma_{\beta\gamma}} y_{0,v}$.
\end{enumerate}
\end{2D_Triangle}
}}
Note that each input specifies an instance of the multiplayer triangle game on the cycle $\Gamma\subset\mathcal{G}$. The vertices $\alpha$, $\beta$, and $\gamma$ correspond to the players at the corners of the triangle and each vertices' output $(y_{0,v},y_{1,v},y_{2,v})$ around the cycle corresponds to the three-bit output string $\mathbf{y}_j$ of player $j$. Furthermore, the defining linear relations are equivalent to Eqs.~(\ref{N_Tri_1})-(\ref{N_Tri_5}).
A key insight of \cite{bravyi2018quantum} is that communication-assisted classical strategies for multiplayer non-local games are in one to one correspondence with classical Boolean circuits. The analogy goes as follows.
\begin{itemize}
\item[(1)] Each player is represented by a spatial block of polynomially many ``wires" in the circuit. The inputs to these wires can be initialized to be the player's given input $x_j$, an ancillary input $0$, or a random bit drawn from some distribution.
\item[(2)] Communication between $K$ players is represented in the circuit by a gate with fan-in $K$ and unbounded fan-out. When $K$ players $j_1,\cdot\cdot\cdot,j_K$ communicate they share their inputs $x_{j_1},\cdot\cdot\cdot,x_{j_K}$ with each other. With this knowledge, each player may compute arbitrary Boolean functions $f(x_{j_1},\cdot\cdot\cdot,x_{j_K})$, of those inputs. The gate computes this Boolean function. The inputs may be reused as many times as desired due to the unbounded fan-out.
\item[(3)] Each player's output $\mathbf{y}_j$ is represented by the output of three chosen wires in their respective spatial block of wires.
\end{itemize}
The device is a \textit{constant-depth classical device} or \textit{$\mathsf{NC}^0$-device} if internally it can perform classical circuits of constant-depth consisting of arbitrarily nonlocal gates with bounded fan-in at most $K$. Likewise, the device is a \textit{constant-depth quantum device} or $\mathsf{QNC}^0$-\textit{device} if internally it can perform constant-depth quantum circuits consisting of quantum gates with bounded fan-in at most $K$. As summarized in the formal version of Cor.~1 below, a constant-depth quantum device can solve the relation problem with probability greater than $7/8$ for all possible inputs. Indeed, it is possible to prepare fixed-point SPTO ground states in constant depth when the symmetry $G$ is disregarded \cite{chen2010local}. Generic states in the SPTO phase can be prepared well approximately in the constant-depth circuit that simulates symmetric, quasi-adiabatic continuation of Hamiltonians from the fixed-point \cite{hastings2005quasiadiabatic}. Thus, the advantageous quantum strategy is achieved by first constructing a 1D SPTO ground state with sufficiently large string order parameter $\langle\mathcal{S}\rangle>1/3$ and appropriate symmetry group---for example, $\mathbb{Z}_2\times\mathbb{Z}_2$---and then implementing the respective fixed-point protocol as in Thm.~2. In other words, each player holds $l$ constituent particles of the 1D SPTO chain and measures in one of the horizontal or vertical context of the table
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.275\linewidth]{Triangle_Square}}}
\label{Tri_Sqr}
\end{align}
depending on if their received input is 0 or 1, respectively.
This quantum strategy can be implemented by constant-depth quantum circuits that are also geometrically local with respect to the graph $\mathcal{G}$. However, we will prove in the following that a constant-depth classical device cannot solve this relation problem with probability greater than $7/8$ on all inputs.
Notice that the set of all inputs $\mathcal{I}= \{(\alpha,\beta,\gamma,\mathbf{x},\Gamma)\}$ has size $|\mathcal{I}|=O(2^{N^2})$, since there are an exponential number of cycles in any 2D lattice graph. However, to demonstrate the quantum computational advantage, we only need to check a polynomially large subset of provably hard instances $\mathcal{I}_{\textrm{Hard}}\subset \mathcal{I}$ of size $|\mathcal{I}_{\textrm{Hard}}| = O(N^{16/3})$. An exact description of $\mathcal{I}_{\textrm{Hard}}$ and the proof of its polynomial size are given in the next section. It will be shown that any classical circuit that solves the problem with probability greater than $7/8$ on this subset of inputs must have depth $D\geq \frac{2}{5}\log_K(N)$ (i.e. depth that is logarithmic in the system size). This demonstrates an unconditional exponential separation between the power of quantum and classical circuits. In particular, it implies a worst-case separation of computational complexity classes $\mathsf{NC}^0\subsetneq\mathsf{QNC}^0$.
\begin{figure}
\includegraphics[width=\linewidth]{Advantage_Setting}
\caption{Setting for the unconditional exponential separation between classical and quantum circuits. For a fixed lattice graph $\mathcal{G}$, depicted here as the grid graph, a blackbox device is tasked to solve the 2D multiplayer triangle problem on inputs drawn from $\mathcal{I}_\textrm{Hard}$. Each input is a tuple $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)$. $\alpha$, $\beta$, and $\gamma$ are three vertices depicted by the shaded red squares chosen from three distinct boxes, outlined in red. $\mathbf{x}\in\{0,1\}^{N^2}$ is a binary string of Hamming weight $|\mathbf{x}|\leq 3$ whose nonzero inputs correspond to the vertices $\alpha$, $\beta$, and $\gamma$. $\Gamma$ is a cycle connecting $\alpha$, $\beta$, and $\gamma$, depicted in blue. A randomly generated input is given to a constant-depth quantum device that implements the 1D SPTO strategy for the multiplayer triangle game on the cycle $\Gamma$. The performance of the quantum device is compared against all classical strategies, which can implement classical circuits consisting of AND, XOR, and NOT gates (denoted $\otimes$, $\oplus$, and $\neg$) with arbitrary locality and fan-in at most $K$. The device produces an output, which consists of a three-bit string for each vertex in $\mathcal{G}$. The outputs are then checked against the linear relations defining the solution set. Repeating this many times for many different inputs, the success probability for each input is obtained. If for each input in $\mathcal{I}_{\textrm{Hard}}$ the success probability is greater than $7/8$, the constant-depth quantum device outperforms exponentially all classical circuits as the latter needs at least logarithmic depth to match.}
\label{Advantage_Setting}
\end{figure}
The protocol by which the unconditional separation can be demonstrated is depicted in Fig.~\ref{Advantage_Setting}. For a fixed lattice graph $\mathcal{G}$, generate a random input drawn uniformly from the subset of hard instances, $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)\in\mathcal{I}_{\textrm{Hard}}$. Given the input, the device implements the quantum strategy for the multiplayer triangle game on the cycle $\Gamma$ and outputs a three-bit string for each vertex in $\mathcal{G}$. The outputs are then checked against the linear relations (\ref{One})-(\ref{Five}) and the result is recorded as a success or failure. This procedure is repeated many times to build up statistics for the success probability for each input. If for all inputs the success probability is greater than $7/8$, then the computational separation has been demonstrated.
\subsection{Proof of unconditional separation}
\label{Proof_of_separation}
In this section we give a formal proof of Cor.~\ref{SPTO_Advantage} from the main text. Given Theorem~2, the formal proof does not need any radically new ideas beyond the results of \cite{bravyi2018quantum, coudron2018trading, gall2018average}.
As described above, communication-assisted classical strategies for the multiplayer triangle game that simply compute affine Boolean functions with geometrically restricted dependence succeed with probability no greater than $7/8$ (i.e. they must fail for one or more inputs). We first recast this condition for failure in terms of correlations generated by a classical circuit, which are conveniently expressed via \textit{light cones}.
\begin{Definition}
\cite{bravyi2018quantum, gall2018average} Let $\mathcal{C}$ be a classical circuit with inputs indexed by set $\mathbf{X}$ and outputs indexed by set $\mathbf{Y}$. Given an input $x\in \mathbf{X}$, the forward light cone of $x$, denoted $L^+_\mathcal{C}(x)$, is the set of all output bits that depend on $x$. Similarly, the backward light cone of an output $y\in \mathbf{Y}$, denoted $L^-_{\mathcal{C}}(y)$, is the set of all inputs it depends on.
\end{Definition}
With this definition we can recast the conditions for failure in terms of conditions on the light cones of the inputs.
\fbox{\parbox{\textwidth}{
\begin{Failure}
For a given input $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)$ for the 2D multiplayer triangle problem, a classical circuit $\mathcal{C}$ will fail to solve the problem for at least one string $\mathbf{x}$ whenever the following conditions hold.
\begin{itemize}
\item[$\mathrm{(I)}$] The forward light cones of the possible nonzero inputs $x_\alpha$, $x_\beta$, and $x_\gamma$ are pairwise disjoint.
\begin{align}
L_{\mathcal{C}}^+(x_\alpha)\cap L_{\mathcal{C}}^+(x_\beta) = L_{\mathcal{C}}^+(x_\alpha)\cap L_{\mathcal{C}}^+(x_\gamma) = L_{\mathcal{C}}^+(x_\beta)\cap L_{\mathcal{C}}^+(x_\gamma) = \varnothing. \label{Fail1}
\end{align}
\item[$\mathrm{(II)}$] Let $\Gamma_{ab}$ denote the direct path within the cycle going from vertex $a$ to vertex $b$. The forward light cones of each possibly nonzero input are disjoint from the outputs on the opposite edge. I.e.
\begin{subequations}
\begin{align}
L_{\mathcal{C}}^+(x_\alpha)\cap\Gamma_{\beta\gamma} &= \varnothing \label{Fail2}\\
L_{\mathcal{C}}^+(x_\beta)\cap\Gamma_{\alpha\gamma} &= \varnothing \label{Fail3}\\
L_{\mathcal{C}}^+(x_\gamma)\cap\Gamma_{\alpha\beta} &= \varnothing \label{Fail4}.
\end{align}
\end{subequations}
\end{itemize}
\end{Failure}
}}
We now show that for sufficiently large $N$ any classical circuit of depth $D<\frac{2}{5}\log_K(N)$ will satisfy the above failure conditions for at least one input $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)\in\mathcal{I}_{\textrm{Hard}}$.
\begin{Proposition}
\cite{bravyi2018quantum, gall2018average} For any classical circuit $\mathcal{C}$ of depth $D$ consisting of gates with fan-in at most $K$, all outputs $y\in \mathbf{Y}$ have backwards light cones of bounded size $|L_{\mathcal{C}}^-(y)|\leq K^D$.
\label{Prop1}
\end{Proposition}
\textit{Proof}: For each layer of gates each output is correlated with at most $K$ inputs. Therefore, after $D$ layers of gates each output is correlated with at most $K^D$ inputs. $\square$
If the depth of the circuit satisfies $D<\frac{2}{5}\log_K(N)$, we find $|L_{\mathcal{C}}^-(y)|\leq N^{2/5}$. Let $V_{\textrm{Big}}=\{x\in\mathbf{X}~|~|L^+(x)|\geq N^{1/2}\}$ and $V_{\textrm{Small}}=\{x\in\mathbf{X}~|~|L^+(x)|<N^{1/2}\}$.
\begin{Proposition}
\cite{bravyi2018quantum, gall2018average} $|V_{\mathrm{Big}}|<N^{19/10}$.
\label{Prop2}
\end{Proposition}
\textit{Proof}: Consider the interaction graph $\mathcal{G}_{\textrm{int},\mathcal{C}}$ of the circuit $\mathcal{C}$ defined as the bipartite graph with vertex set $V_{\textrm{int}}=\mathbf{X}\cup\mathbf{Y}$ and edge set $E_{\textrm{int}}$ where $(x,y)\in E_{\textrm{int}}$ iff $x\in L_{\mathcal{C}}^-(y)$. $|E_{\textrm{int}}|$ satisfies, $|V_{\textrm{Big}}| N^{1/2}<|E_{\textrm{int}}|<N^2 N^{2/5}$ which implies that $|V_{\textrm{Big}}|<N^{19/10}$ so $|V_{\textrm{Small}}|=\Omega(N^2)$ (i.e. most forward light cones are small). $\square$
Now consider the lattice graph $\mathcal{G}=(V,E)$ over which the classical circuit $\mathcal{C}$ takes inputs and generates outputs. This graph can be partitioned into $N^{2/3}$ disjoint connected sets containing $N^{4/3}$ vertices with diameter $\Theta(N^{2/3})$, where the diameter is defined with respect to the Euclidean distance on the grid, referred to as neighborhoods. For each vertex $v\in V$ let $\textrm{Nbhd}(v)$ denote the unique neighborhood to which that vertex belongs. Also define three connected regions $\mathcal{U},\mathcal{V},\mathcal{W}\subset V_{\textrm{Small}}$, each containing $\left\lfloor \frac{N}{3}\right\rfloor^2$ vertices, that are well separated from each other in that $\textrm{dist}(\mathcal{U},\mathcal{V}),\textrm{dist}(\mathcal{U},\mathcal{W}),\textrm{dist}(\mathcal{V},\mathcal{W})\geq \left\lfloor\frac{N}{3}\right\rfloor$. Furthermore, the diameter of these sets is $\textrm{diam}(\mathcal{U}),\textrm{diam}(\mathcal{V}),\textrm{diam}(\mathcal{W})=\Theta(N)$.
The following proposition gives a bound on the probability that a randomly selected triple of vertices $(\alpha,\beta,\gamma)$ with $\alpha\in\mathcal{U}$, $\beta\in\mathcal{V}$, and $\gamma\in\mathcal{W}$ have forward light cones that intersect each other's neighborhoods.
\begin{Proposition}
\cite{bravyi2018quantum, gall2018average} Let $(\alpha,\beta,\gamma)$ be a randomly selected triple of vertices with $\alpha\in\mathcal{U}$, $\beta\in\mathcal{V}$, and $\gamma\in\mathcal{W}$. Consider the following set of light cone conditions,
\begin{subequations}
\begin{align}
&L_{\mathcal{C}}^+(x_\alpha)\cap\mathrm{Nbdh}(\beta) = \varnothing \textrm{ and } L_{\mathcal{C}}^+(x_\alpha)\cap\mathrm{Nbdh}(\gamma) = \varnothing \label{Nbdh_1}\\
&L_{\mathcal{C}}^+(x_\beta)\cap\mathrm{Nbdh}(\alpha) = \varnothing \textrm{ and } L_{\mathcal{C}}^+(x_\beta)\cap\mathrm{Nbdh}(\gamma) = \varnothing \label{Nbdh_2}\\
&L_{\mathcal{C}}^+(x_\gamma)\cap\mathrm{Nbdh}(\alpha) = \varnothing \textrm{ and } L_{\mathcal{C}}^+(x_\gamma)\cap\mathrm{Nbdh}(\beta) = \varnothing.\label{Nbdh_3}
\end{align}
\end{subequations}
The probability that any one of the following conditions fails to hold is $O(N^{-1/6})$. Therefore, for sufficiently large $N$ there will be at least one triple for which Eqs.~(\ref{Nbdh_1})-(\ref{Nbdh_3}) are satisfied.
\label{Prop3}
\end{Proposition}
\textit{Proof}: Without loss of generality consider vertex $\alpha\in\mathcal{U}$ and its corresponding input $x_\alpha$. Since $\alpha\in V_{\textrm{Small}}$ its forward light cone $L_\mathcal{C}^+(x_\alpha)$ intersects at most $N^{1/2}$ different neighborhoods. Since each neighborhood contains $N^{4/3}$ vertices, the total number of vertices $v\in V$ satisfying $L_\mathcal{C}^+(x_\alpha)\cap\textrm{Nbdh}(v)\neq\varnothing$ is $O(N^{11/6})$. Since the total number of vertices in $\mathcal{V}$ is $O(N^2)$ we have that $\textrm{pr}[L_{\mathcal{C}}^+(x_\alpha)\cap\mathrm{Nbdh}(\beta) \neq \varnothing]= O(N^{-1/6})$. By the union bound the probability that any one of Eqs.~(\ref{Nbdh_1})-(\ref{Nbdh_3}) is not satisfied is $O(N^{-1/6})$. $\square$
The following proposition deals with the likelihood that the failure condition (I), Eq.~(\ref{Fail1}), is satisfied for some input.
\begin{Proposition}
\cite{bravyi2018quantum, gall2018average} Let $(\alpha,\beta,\gamma)$ be a triple of vertices as described above. The probability that any one of the equalities in Eq.~(\ref{Fail1}) fails to hold is $O(N^{-3})$. Therefore, for sufficiently large $N$ there will be at least one triple for which Eq.~(\ref{Fail1}) is satisfied.
\label{Prop4}
\end{Proposition}
\textit{Proof}: Without loss of generality consider vertices $\alpha$ and $\beta$. The probability that a vertex $v\in V$ lies in $L_\mathcal{C}^+(x_\alpha)$ is $O(N^{-3/2})$. Thus, $\textrm{pr}[v\in L_\mathcal{C}^+(x_\alpha) \textrm{ and } v\in L_\mathcal{C}^+(x_\beta) ] = O(N^{-3})$. By the union bound the probability that the forward light cones of any pair of vertices from the triple $(\alpha,\beta,\gamma)$ intersect is $O(N^{-3})$. $\square$
The following proposition deals with the likelihood that the failure condition (II), Eqs.~(\ref{Fail2})-(\ref{Fail4}), are satisfied for some input.
\begin{Proposition}
\cite{bravyi2018quantum, gall2018average} For the triple $(\alpha,\beta,\gamma)$ described above, construct a random cycle $\Gamma$ formed by three direct paths between each pair of vertices $\Gamma_{\alpha\beta}$, $\Gamma_{\beta\gamma}$, and $\Gamma_{\alpha\gamma}$. The probability that Eqs.~(\ref{Fail2})-(\ref{Fail4}) are not satisfied is $O(N^{-1/6})$. Therefore, for sufficiently large $N$ there will be at least one cycle $\Gamma$ for which Eqs.~(\ref{Fail2})-(\ref{Fail4}) are satisfied.
\label{Prop5}
\end{Proposition}
\textit{Proof}: Without loss of generality consider the path $\Gamma_{\alpha\beta}$. Since the triple $(\alpha,\beta,\gamma)$ satisfies Eqs.~(\ref{Nbdh_1})-(\ref{Nbdh_3}) we need only consider the section of the path $\Gamma_{\alpha\beta}'=\Gamma_{\alpha\beta}\backslash (\textrm{Nbdh}(\alpha)\cup\textrm{Nbdh}(\beta))$, which extends from the boundary of $\textrm{Nbdh}(\alpha)$ to the boundary of $\textrm{Nbdh}(\beta)$. We can construct $O(N^{2/3})$ disjoint paths $\Gamma_{\alpha\beta}'$, which emerge from half of the points on the circumference of $\textrm{Nbdh}(\alpha)$ and $\textrm{Nbdh}(\beta)$. Since $|L_\mathcal{C}^+(x_\gamma)|<N^{1/2}$ it can at most intersect $N^{1/2}$ such paths $\Gamma_{\alpha\beta}'$. Thus for a randomly chosen path $\Gamma_{\alpha\beta}$, $\textrm{pr}[L_\mathcal{C}^+(x_\gamma)\cap\Gamma_{\alpha\beta}\neq\varnothing]=O(N^{-1/6})$. By the union bound the probability that any one of Eqs.~(\ref{Fail2})-(\ref{Fail4}) are not satisfied is $O(N^{-1/6})$. $\square$
Propositions \ref{Prop1}--\ref{Prop5} define a subset $\mathcal{I}_{\textrm{Hard}}\subset\mathcal{I}$ of the input set that are provably hard for constant depth classical circuits. We now show that $\mathcal{I}_{\textrm{Hard}}$ has size growing polynomially in the grid size.
\begin{Proposition}
$|\mathcal{I}_{\textrm{Hard}}|=O(N^{16/3})$.
\label{Prop6}
\end{Proposition}
\textit{Proof:}. Divide $\mathcal{G}$ into $N^{2/3}$ disjoint contiguous regions (neighborhoods) each containing $N^{4/3}$ vertices and each having diameter $O(N^{2/3})$. For example, one could use ``boxes" of size $N^{2/3}\times N^{2/3}$ in the $N\times N$ square grid as in Ref.~\cite{bravyi2018quantum}. To obtain $\alpha$, $\beta$, and $\gamma$ choose three vertices from three distinct neighborhoods (there are $O(N^{10/3})$ such choices). $\mathbf{x}$ is obtained by choosing from one of the eight possible strings with appropriate support. Finally, for each pair of neighborhoods choose one path connecting them from a set of $O(N^{2/3})$ non-intersecting paths between the boundary of each. For each choice of three paths we can connect their endpoints to the nearest vertex $\alpha$, $\beta$, or $\gamma$ via a path within each neighborhood to create a cycle $\Gamma$. There are $O(N^{2})$ such cycles. Therefore, subset of hard instances has polynomial size $|\mathcal{I}_{\textrm{Hard}}|=O(N^{16/3})$. $\square$
\begin{Cor}
$\mathrm{(Formal)}$ Consider the 2D multiplayer triangle problem on a 2D lattice graph $\mathcal{G}$ of size $N \times N$. Consider a constant-depth quantum device with two capabilities (i) to prepare a generic 1D SPTO ground state on arbitrary cycles in $\mathcal{G}$ with a string order parameter greater than $\frac{1}{3}$, and (ii) to measure sites in the contexts of Eq.~(\ref{Tri_Sqr}), performing the fixed-point protocol of Thm.~2. This quantum device outputs a solution of the 2D multiplayer triangle problem with probability higher than $\frac{7}{8}$ for all the inputs, which any classical circuit of depth $D<\frac{2}{5}\log_K(N)$ cannot do.
\end{Cor}
\textit{Proof}: By Props.~\ref{Prop1}-\ref{Prop5}, for any classical circuit consisting of gates with fan-in $K$ and depth $D<\frac{2}{5}\log_{K}(N)$ there exists at least one input $(\alpha,\beta,\gamma,\mathbf{x},\Gamma)$ satisfying the failure conditions of Eq.~(\ref{Fail1}) and Eqs.~(\ref{Fail2})-(\ref{Fail4}). Therefore, such a classical circuit cannot succeed on all inputs with probability greater than $7/8$. On the other hand, a constant-depth quantum device with the above capability can. Therefore, the constant-depth quantum device outperforms all such sub-logarithmic depth classical circuits. $\square$
\section{Determination of the boundary operators $V^L(g)$ and $V^R(g)$}
\label{SM2}
In this section we will review matrix product state representations of 1D SPTO ground states and use them to construct the fixed-point boundary operators, $V^L(g)$ and $V^R(g)$, used in the text. We will use these explicit expressions to prove various properties of the boundary operators from the main text. Namely, we show the following properties of the fixed-point boundary operators
\begin{align}
V^R(g)V^R(h) &= \omega(g,h) V^R(gh) \label{R_Proj} ,\\
V^L(g)V^L(h) &= \omega(g,h)^* V^L(gh) \label{L_Proj} ,\\
V^R(g)V^L(h) &= V^L(h)V^R(g) \label{RL_Com} ,\\
V^R(g)V^L(g)&= u(g)^{\otimes l} \label{RL_Sym}.
\end{align}
Furthermore, for a fixed-point ground state $|\psi\rangle$
\begin{align}
S_{[j,k]}(g)|\psi\rangle &= \left(V^L_j(g)\otimes V^R_k(g)\right)U_{[j,k]}(g)|\psi\rangle = |\psi\rangle \label{Trunc_Sym} \\
T_{[j,k]}^{(g,h)} |\psi\rangle &= V_{k}^R(g) U(h) V_{j}^L(g) U_{[j,k]}(g)|\psi\rangle = \Omega(g,h) |\psi\rangle. \label{Twist_Phase_Expectation}
\end{align}
\subsection{Matrix product states and tensor network notation}
A matrix product state (MPS) representation of a quantum state expresses the amplitude of each basis vector as the trace of a product of matrices. In particular we write,
\begin{align}
|\psi\rangle = \sum_{j_0,...,j_{N-1}} \textrm{Tr}\left(A^{(j_{N-1})}\cdot\cdot\cdot A^{(j_0)}\right) |j_0, ..., j_{N-1}\rangle
\end{align}
for uniform many-body states on periodic boundary conditions, where $A^{(j_k)}\in L(\mathbb{C}^D)$ is a linear operator on the vector space $\mathbb{C}^D$ (i.e. a complex $D\times D$ matrix). The vector space, $\mathbb{C}^D$, on which the matrices act is referred to as the virtual space. Its dimension is called the bond dimension of the MPS.
An MPS can seen as a contraction of many 3-index tensors,
\begin{align}
\mathcal{A} = \sum_{j=0}^{d-1}\sum_{k,k'=0}^{D-1} A^{(j)}_{k,k'} |j;k\rangle\langle k'|.
\end{align}
When studying MPS we will appeal to the diagrammatic tensor network notation for preforming calculations. In tensor network notation multi-index objects are denoted as boxes with lines emerging from them. Each line represents an index that takes values in the corresponding indexing set. Connecting two lines denotes a contraction over those indices. In tensor network notation the MPS is expressed as,
\begin{align}
|\psi\rangle &= \vcenter{\hbox{\includegraphics[width=0.3\linewidth]{MPS_1}}}.
\end{align}
In the diagrammatic representation we have labeled the lines representing the virtual space with arrows directed in a clockwise manner. These arrows are used to depict the order in which multiplication should be performed. Here we will work with the convention that, multiplication is to be performed in the opposite order as denoted by the arrow. For instance,
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.13\linewidth]{Mult_1}}} &= \vcenter{\hbox{\includegraphics[width=0.08\linewidth]{Mult_2}}}.
\end{align}
For a good review of tensor network notation see \cite{bridgeman2017hand}.
\subsection{Matrix product states and symmetry-protected topological order}
1D SPTO phases can be classified using MPS according their symmetry properties \cite{schuch2011classifying}. Consider symmetry groups that carry a uniform \textit{on-site} unitary representation $U(g) = u(g)^{\otimes N}$, i.e. represented in tensor product on the many-body Hilbert space $\mathcal{H} = (\mathbb{C}^d)^{\otimes N}$. The on-site representation $u(g)$ maps each component of the MPS tensor $A^{(j)}$ to $\sum_{k}u(g)_{jk}A^{(k)}$. Since the state is invariant under the global symmetry, the transformed MPS tensors must be related to the original ones by a ``gauge transformation" \cite{perez2007matrix}
\begin{align}
\sum_{k=0}^{d-1}u(g)_{jk}A^{(k)} = V(g)^\dagger A^{(j)}V(g).
\end{align}
In tensor network notation we say that the on-site representation $u(g)$ can be \textit{pushed through} each local MPS tensor $\mathcal{A}$ to yield another representation $V(g)$, which acts via conjugation on the virtual space. In tensor network notation this is written,
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.3\linewidth]{MPS_Prep}}}. \label{MPS_Prep}
\end{align}
Symmetric MPS represent 1D SPTO ground states whenever the unitaries $V(g)$ form a nontrivial \textit{projective representation} as described in the text.
Algebraically, a projective representation of $G$ is a map $V:G\rightarrow GL(\mathbb{V})$, for some vector space $\mathbb{V}$,
that obeys the group multiplication law up to a $G$-dependent phase, i.e.,
\begin{align}
V(g)V(h) = \omega(g,h)V(gh),
\end{align}
where $\omega(g,h)\in U(1)$ is called a 2-cocycle.
Associativity is assured by the 2-cocycle condition, $\omega(a,b)\omega(ab,c) = \omega(a,bc)\omega(b,c)$ $\forall a,b,c\in G$. A projective representation is said to be nontrivial whenever the 2-cocycle cannot be removed by a multiplicative phase, $d\beta(g,h) = \beta(g)\beta(h)/\beta(gh)$, known as a 2-coboundary. This defines an equivalence relation on 2-cocycles where $\omega\sim\omega'$ iff $\omega'=(d\beta)\omega$.
The equivalence classes, denoted $[\omega]$, form a multiplicative group called the second group cohomology, $H^2(G,U(1))$. 1D SPTO phases are in one to one correspondence with elements of $H^2(G,U(1))$ \cite{chen2011classification}.
\subsection{The boundary operators in the MPS representation}
The operators $V^L(g)$ and $V^R(g)$ introduced in the text send the state resulting from a truncated symmetry transformation back to the ground state as described in Eq.~(\ref{Trunc_Sym}). The same action can be achieved in the virtual space by placing $V(g)^\dagger$ and $V(g)$ on the virtual indices at the left and right endpoints of the truncated symmetry operator. We thus expect that $V^L(g)$ and $V^R(g)$ are operators that can be pushed through to the virtual space where they act as $V(g)^\dagger$ and $V(g)$ on the right or left bonds, respectively.
Consider the MPS tensor as a map from the virtual space to the physical space $\mathcal{A}: \mathbb{C}^D\otimes\mathbb{C}^D\rightarrow \mathbb{C}^d$, this map is called the MPS projector. Given any operator on the virtual space $W$ there is an operator $P$ on the physical space such that $P\mathcal{A} = \mathcal{A}W$ whenever the map $\mathcal{A}$ is injective. Indeed, the injectivity of $\mathcal{A}$ implies there is a left inverse $\mathcal{A}^{-1}$. It follows that $P = \mathcal{A} W \mathcal{A}^{-1}$.
Some MPS projectors $\mathcal{A}$ may not have a left inverse simply because the dimension $d$ of the physical space is simply too small for the map $\mathcal{A}$ to be injective (i.e. $d<D^2$). Such tensors can be made injective by blocking some number of sites $l$ to form a new tensor $\mathcal{A}^l$ where,
\begin{align}
\mathcal{A}^l = \sum_{j_1,\cdot\cdot\cdot,j_l = 0}^{d-1} \sum_{k,k'=0}^{D-1} \left(\sum_{r_1,\cdot\cdot\cdot,r_{l-1} = 0}^{D-1}A^{(j_{l})}_{k,r_{l-1}}\cdot\cdot\cdot A^{(j_2)}_{r_2,r_1}A^{(j_1)}_{r_1,k'} \right) |j_1,\cdot\cdot\cdot,j_l;k\rangle\langle k'|.
\end{align}
In tensor network notation this is simply written,
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.4\linewidth]{Injective_Block_Def}}}.
\end{align}
The thick line on the lefthand side indicates that the physical space dimension is larger. The left inverse property is written,
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.09\linewidth]{Inverse_Relation_L}}} &= \vcenter{\hbox{\includegraphics[width=0.06\linewidth]{Inverse_Relation_R}}}. \label{L_Inv_Def}
\end{align}
An MPS whose tensors form injective maps after blocking some number of sites $l$ are called \textit{injective MPS}. The number of sites to be blocked $l$ is called the \textit{injectivity length}. It turns out that injectivity is both a necessary and sufficient condition for an MPS to be the unique ground state of its associated parent Hamiltonian. Thus, MPS representations of 1D SPTO ground states on periodic boundary conditions are always injective MPS. Furthermore, the projective unitaries can always be found, given the MPS, as,
\begin{align}
V^L(g) &= \mathcal{A}^l(\mathbbm{1}\otimes V(g)^\dagger)({\mathcal{A}^l})^{-1} \\
V^R(g) &= \mathcal{A}^l(V(g)\otimes\mathbbm{1})({\mathcal{A}^l})^{-1}.
\end{align}
In tensor network notation these are,
\begin{align}
V^L(g) &= \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{V_L_TNN}}} \label{VL_TNN} \\
V^R(g) &= \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{V_R_TNN}}} \label{VR_TNN} .
\end{align}
These operators have nontrivial support on at most $l$ sites. In tensor network notation, their action on the MPS can be written,
\begin{align}
&\vcenter{\hbox{\includegraphics[width=0.25\linewidth]{V_L_Def}}} \label{V_L_Def} \\
&\vcenter{\hbox{\includegraphics[width=0.24\linewidth]{V_R_Def} \label{V_R_Def}}}.
\end{align}
Notice that the support of the fixed-point boundary operators used in the main text $m$ is simply the injectivity length of the fixed-point MPS.
The fixed-point boundary operators play a predominant role in the main text. The fixed-point state is the MPS obtained under quantum state renormalization group (RG) implemented on the \textit{transfer matrix}. The transfer matrix of a MPS generated by MPS tensor $\mathcal{A} = \sum_{j=0}^{d-1} A^{(j)}\otimes|j\rangle$ is defined as
$\mathcal{E}_{\mathcal{A}}= \sum_{j=0}^{d-1} A^{(j)}\otimes {A^{(j)}}^*.$
This object is also called the ``identity transfer matrix" and will later be denoted as $\mathcal{E}_{\mathbbm{1}}$ for convenience. The quantum state RG is then implemented by blocking some number $M>1$ transfer matrices and finding a new MPS $\mathcal{A}'$ with equivalent transfer matrix. In other words, one RG step is implemented by solving for $\mathcal{A}'$ in the equation $\mathcal{E}_{\mathcal{A}'} = \mathcal{E}_{\mathcal{A}}^M$.
The fixed-point obtained under quantum state RG implemented on a given MPS $\mathcal{A}$ can be understood in terms of the left and right eigenvectors of $\mathcal{A}$. Injective MPS have unique left and right eigenvectors with largest eigenvalue one when written in their so-called canonical form \cite{perez2007matrix}. The second largest eigenvalue $\lambda_2$ determines the rate of exponential decay of two point correlation functions, which is characterized by the \textit{correlation length} $\xi = 1/\ln(1/\lambda_2)$. The fixed-point MPS thus has correlation length $\xi=0$. If the left and right eigenvectors are $|R_\mathcal{A}\rangle$ and $|L_\mathcal{A}\rangle$, the fixed-point transfer matrix is $\mathcal{E}_{\mathcal{A}_\textrm{Fix}} = |L_\mathcal{A}\rangle\langle R_{\mathcal{A}}|$. Thus up to unitary transformation $({\mathcal{A}_{\textrm{Fix}}^l})^{-1}={\mathcal{A}_{\textrm{Fix}}^{l}}^\dagger$ and the fixed point boundary operators are,
\begin{align}
V^L(g) &= \mathcal{A}_{\textrm{Fix}}^l(\mathbbm{1}\otimes V(g)^\dagger){{\mathcal{A}_{\textrm{Fix}}^l}}^\dagger\\
V^R(g) &= \mathcal{A}_{\textrm{Fix}}^l(V(g)\otimes\mathbbm{1}){{\mathcal{A}_{\textrm{Fix}}^l}}^\dagger.
\end{align}
The fixed-point boundary operators are thus guaranteed to be unitary and can thus be measured via a projective value measure (PVM).
\subsection{Properties of the boundary operators in the MPS representation}
We now prove each expression.
\begin{itemize}
\item We first prove Eq.~(\ref{Trunc_Sym}). This follows simply from Eqs.~(\ref{MPS_Prep}), (\ref{V_L_Def}), and (\ref{V_R_Def}). Observe that in tensor network notation we have,
\begin{align}
\left(V^L_j(g)\otimes V^R_k(g)\right) U_{[j,k]}(g)|\psi\rangle &=~~ \vcenter{\hbox{\includegraphics[width=0.35\linewidth]{Trunc_Sym_1}}} \label{TS_Der_1} \\
&=~~ \vcenter{\hbox{\includegraphics[width=0.55\linewidth]{Trunc_Sym_2}}} \label{TS_Der_2} \\
&=~~ \vcenter{\hbox{\includegraphics[width=0.35\linewidth]{Trunc_Sym_3}}} \label{TS_Der_3}\\
&= |\psi\rangle.~~~~~\square \label{TS_Der_4}
\end{align}
In Eq.~(\ref{TS_Der_1}) we have written the expression in tensor network notation using Eqs.~(\ref{VL_TNN}) and (\ref{VR_TNN}). In Eq.~(\ref{TS_Der_2}) we utilized Eq.~({\ref{L_Inv_Def}}) and (\ref{MPS_Prep}) to push though $V^L(g)$, $V^R(g)$, and each $u(g)$ to the virtual space where each unitary $V(g)$ cancels with its inverse giving Eq.(\ref{TS_Der_3}), which is simply the state $|\psi\rangle$.
\item Next, we prove Eq.~(\ref{R_Proj}). This follows from a direct calculation. In tensor network notation,
\begin{align}
V^R(g)V^R(h) = \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{R_Proj_1}}} = \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{R_Proj_2}}} = \omega(g,h)\vcenter{\hbox{\includegraphics[width=0.1\linewidth]{R_Proj_3}}} = \omega(g,h)V^R(gh).
\end{align}
\item Next, we prove Eq.~(\ref{L_Proj}). This follows from a direct calculation. In tensor network notation,
\begin{align}
V^L(h)V^L(g) &= \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{L_Proj_1}}} = \vcenter{\hbox{\includegraphics[width=0.1\linewidth]{L_Proj_2}}} =\frac{1}{\omega(g,h)}\vcenter{\hbox{\includegraphics[width=0.1\linewidth]{L_Proj_3}}} = \frac{1}{\omega(g,h)}V^L(gh).
\end{align}
\item Next, we prove Eq.~(\ref{RL_Com}). This follows from a direct calculation. In tensor network notation,
\begin{align}
V^R(g)V^L(h) &= \vcenter{\hbox{\includegraphics[width=0.13\linewidth]{RL_Com_1}}} = \vcenter{\hbox{\includegraphics[width=0.13\linewidth]{RL_Com_2}}} = \vcenter{\hbox{\includegraphics[width=0.13\linewidth]{RL_Com_3}}} = V^L(h) V^R(g).
\end{align}
\item Next, we prove Eq.~(\ref{RL_Sym}). This follows from a direct calculation. In tensor network notation,
\begin{align}
V^R(g)V^L(g) &= \vcenter{\hbox{\includegraphics[width=0.13\linewidth]{RL_Sym_1}}} = \vcenter{\hbox{\includegraphics[width=0.13\linewidth]{RL_Sym_2}}} = \vcenter{\hbox{\includegraphics[width=0.085\linewidth]{RL_Sym_3}}}.
\end{align}
The left inverse also satisfies $\mathcal{A}^l(\mathcal{A}^l)^{-1} = \Pi_{\textrm{Range}(\mathcal{A})}$ where $\Pi_{\textrm{Range}(\mathcal{A})}$ is the projector onto the range of $\mathcal{A}$. Thus,
\begin{align}
\vcenter{\hbox{\includegraphics[width=0.085\linewidth]{Proj_Red}}} = \Pi_{\textrm{Range}(\mathcal{A})}.
\end{align}
Since $|\psi\rangle$ is the MPS generated by $\mathcal{A}$, we have $|\psi\rangle\in\textrm{Range}(\mathcal{A})$. Hence $V^R(g)V^L(g) = u(g)^{\otimes l}$. $\square$
\item Finally, we prove Eq.~(\ref{Twist_Phase_Expectation}). This follows from a direct calculation. In tensor network notation,
\begin{align}
V_j^R(g)U(h)V_k^L(g) U_{[j,k]}(g) |\psi\rangle &= \vcenter{\hbox{\includegraphics[width=0.48\linewidth]{Twist_1}}} \label{Twist_Der_1} \\
&= \vcenter{\hbox{\includegraphics[width=0.55\linewidth]{Twist_2}}} \label{Twist_Der_2} \\
&= \vcenter{\hbox{\includegraphics[width=0.47\linewidth]{Twist_3}}} \label{Twist_Der_3} \\
&= \vcenter{\hbox{\includegraphics[width=0.4\linewidth]{Twist_4}}} \label{Twist_Der_4} \\
&= \Omega(g,h) |\psi\rangle.
\end{align}
It then follows that $\langle \psi| T^{(g,h)}_{[j,k]}|\psi\rangle = \Omega(g,h)$. $\square$
\end{itemize}
\section{Fixed-point protocol at the AKLT point}
\label{SM3}
The AKLT state is the ground state of a 2-local spin-1 Hamiltonian of antiferromagnetic Heisenberg-type interactions
\begin{align}
H_{\mathrm{AKLT}} = \sum_{j=0}^{n-1} \mathbf{S}_j\cdot\mathbf{S}_{j+1} + \frac{1}{3}\left(\mathbf{S}_j\cdot\mathbf{S}_{j+1}\right)^2.
\end{align}
Here $\mathbf{S}_j = (S_j^x,S_j^y,S_j^z)$ where $S_j^\mu$ is the $\mu\in\{x,y,z\}$ spin component operator for the $j^{\textrm{th}}$ spin-1 particle. All subscripts indexing sites are taken modulo $n$. Upon noting that each Hamiltonian term is simply a projector onto the spin-2 subspace of the two spin-1 Hilbert space---recall that $3\otimes3 = 1\oplus3\oplus5$---up to a rescaling by $2/3$, it is clear that the AKLT state only has support on the singlet and triplet (i.e. spin-0 and spin-1) subspaces.
The AKLT state has an MPS representation generated by the following three-index tensor,
\begin{align}
\mathcal{A}_{\textrm{AKLT}} = \frac{1}{\sqrt{3}} \sum_{\mu\in\{x,y,z\}} \sigma_\mu \otimes |\mu\rangle.
\end{align}
Here $\sigma_\mu$ denotes the Pauli matrices and $|\mu\rangle$ denotes the spin-1 cartesian basis defined by $S^\mu|\mu\rangle = 0$ for each ${\mu\in\{x,y,z\}}$. In the eigenbasis of $S^z$ (i.e. $\{|-1\rangle,|0\rangle,|+1\rangle\}$) the cartesian basis vectors are
\begin{subequations}
\begin{align}
|x\rangle &= \frac{-|+1\rangle + |-1\rangle}{\sqrt{2}} \\
|y\rangle &= i\frac{|+1\rangle + |-1\rangle}{\sqrt{2}} \\
|z\rangle &= |0\rangle.
\end{align}
\end{subequations}
The AKLT state has a $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry with onsite representation given by $u(\mu) = \exp(i\pi S^\mu)$ for each $\mu\in\{x,y,z\}$. It is easily checked that the action of $u(x)$, $u(y)$, or $u(z)$ on the physical index of $\mathcal{A}_\mathrm{AKLT}$ is equivalent to conjugation by $\sigma_x$, $\sigma_y$, or $\sigma_z$ on the virtual indices, respectively. The Pauli matrices form a projective representation of $\mathbb{Z}_2\times\mathbb{Z}_2$ and thus the AKLT state has 1D $\mathbb{Z}_2\times\mathbb{Z}_2$ SPTO.
This MPS has transfer matrix $\mathcal{E}_{\mathcal{A}_{\textrm{AKLT}}}=(\sigma_x\otimes\sigma_x - \sigma_y\otimes\sigma_y + \sigma_z\otimes\sigma_z)/3$, which is diagonal in the Bell-basis $\{|\phi^+\rangle, |\phi^-\rangle, |\psi^+\rangle, |\psi^-\rangle\}$ giving $\mathcal{E}_{\mathcal{A}_{\textrm{AKLT}}} = \textrm{diag}(1,1/3,1/3,1/3)$.
Since the correlation length for this MPS is $\xi = 1/\ln(3)$, it resides outside the fixed-point of the $\mathbb{Z}_2\times\mathbb{Z}_2$ SPTO phase.
This MPS is injective, with injectivity length 2. The two site MPS tensor is given by,
\begin{align}
\mathcal{A}_{\textrm{AKLT}}^2 = \frac{1}{\sqrt{3}}\mathbbm{1}\otimes|\tilde{e}\rangle + \sqrt{\frac{2}{9}}\sum_{\mu\in\{x,y,z\}} \sigma_\mu \otimes|\tilde{\mu}\rangle.
\label{AKLT_Two_Site}
\end{align}
Where $|\tilde{e}\rangle = \frac{1}{\sqrt{3}}\sum_{\nu\in\{x,y,z\}} |\nu\nu\rangle$ and $|\tilde{\mu}\rangle = \frac{i}{\sqrt{2}}\sum_{\nu,\gamma\in\{x,y,z\}}\epsilon_{\mu\nu\gamma} |\nu\gamma\rangle$. Notice that $|\tilde{e}\rangle$ is the singlet for two spin-1 particles. Furthermore, $|\tilde{\mu}\rangle$ for $\mu\in\{x,y,z\}$ span the two spin-1 triplet. Furthermore, these form the cartesian basis in the sense that $(S_1^\mu + S_2^\mu)|\tilde{\mu}\rangle = 0$. It is easily checked that the two site transfer matrix is simply the square of the single site transfer matrix, $\mathcal{E}_{\mathcal{A}_{\textrm{AKLT}}^2} = \mathcal{E}_{\mathcal{A}_{\textrm{AKLT}}}^2 = \textrm{diag}(1,1/9,1/9,1/9)$.
The fixed-point MPS approached by the AKLT state under RG flows has a transfer matrix $\lim_{M\rightarrow\infty}\mathcal{E}_{\mathcal{A}_{\textrm{AKLT}}}^M = \textrm{diag}(1,0,0,0)$. The fixed-point MPS with this transfer matrix is
\begin{align}
\mathcal{A}_\textrm{AKLT, Fix}^2 = \frac{1}{2}\mathbbm{1}\otimes|\tilde{e}\rangle + \frac{1}{2}\sum_{\mu\in\{x,y,z\}} \sigma_\mu \otimes|\tilde{\mu}\rangle.
\end{align}
Applying the local isometry $\Pi = |++\rangle\langle\tilde{e}| + |-+\rangle\langle\tilde{z}| + |+-\rangle\langle\tilde{x}| - i|--\rangle\langle\tilde{y}|$ to the physical degrees of freedom gives an MPS tensor that generates the 1D cluster state. We remark that this mapping replaces each pair of spin-1 particles with a pair of qubits. The fixed-point boundary operators for $\mu\in\{x,y,z\}$ are thus equivalent to those given for the 1D cluster state in the main text under the aforementioned isometry. These are given as
\begin{subequations}
\begin{align}
{V}^L(\mu) &= |\tilde{\mu}\rangle\langle\tilde{e}| + |\tilde{e}\rangle\langle\tilde{\mu}| - i\sum_{\nu\gamma}\epsilon_{\mu\nu\gamma}|\tilde{\nu}\rangle\langle\tilde{\gamma}| \label{VL_AKLT}\\
{V}^R(\mu) &= |\tilde{\mu}\rangle\langle\tilde{e}| + |\tilde{e}\rangle\langle\tilde{\mu}| + i\sum_{\nu\gamma}\epsilon_{\mu\nu\gamma} |\tilde{\nu}\rangle\langle\tilde{\gamma}\label{VR_AKLT}|,
\end{align}
\end{subequations}
which are unitary and Hermitian. Furthermore, $V^R(e)=V^L(e)=\mathbbm{1}$.
Explicitly for $\mu = z$ we have,
\begin{subequations}
\begin{align}
{V}^L(z) &= |\tilde{z}\rangle\langle\tilde{e}| + |\tilde{e}\rangle\langle\tilde{z}| - i |\tilde{x}\rangle\langle\tilde{y}| +i |\tilde{y}\rangle\langle\tilde{x}| \label{VLz}\\
{V}^R(z) &= |\tilde{z}\rangle\langle\tilde{e}| + |\tilde{e}\rangle\langle\tilde{z}| + i |\tilde{x}\rangle\langle\tilde{y}| - i |\tilde{y}\rangle\langle\tilde{x}|\label{VRz}.
\end{align}
\end{subequations}
It can be checked that the above fixed-point boundary operators can be expressed in terms of spin-1 observables as
\begin{subequations}
\begin{align}
V^L(\mu) &= \frac{1}{\sqrt{24}}\left\{\frac{(S^2-4)}{2},S_1^\mu - S_2^\mu \right\} + \frac{1}{2}\left\{\frac{S^2(6-S^2)}{8}, S_1^\mu + S_2^\mu\right\} \label{VL_AKLT_spin}\\
V^R(\mu) &= \frac{1}{\sqrt{24}}\left\{\frac{(S^2-4)}{2},S_1^\mu - S_2^\mu \right\} - \frac{1}{2}\left\{\frac{S^2(6-S^2)}{8}, S_1^\mu + S_2^\mu\right\} \label{VR_AKLT_spin}.
\end{align}
\end{subequations}
Here $S^2 = (\mathbf{S}_1 + \mathbf{S}_2)^2$ and $\{\cdot,\cdot\}$ denotes the anticommutator. We remark that for MPS calculations it is convenient to work with Eqs.~(\ref{VL_AKLT})--(\ref{VR_AKLT}); however, Eqs.~(\ref{VL_AKLT_spin})--(\ref{VR_AKLT_spin}) inform how the boundary operators can be measured in an experiment.
Expectation values of strings of local operators can be conveniently calculated in the MPS formalism via transfer matrices. For an MPS generated by tensor $\mathcal{A} = \sum_{j=0}^{d-1} A^{(j)} \otimes|j\rangle$ and for any single-site operator $O\in L(\mathbb{C}^d)$, the transfer matrix $\mathcal{E}_O$ is defined as,
\begin{align}
\mathcal{E}_O = \sum_{j,k=0}^{d-1} \langle j|O|k\rangle A^{(k)}\otimes{A^{(j)}}^*.
\end{align}
The expected value of $O$ in the state $|\psi\rangle$, consisting of $N$ sites, can be written,
\begin{align}
\langle\psi|O|\psi\rangle = \textrm{Tr}(\mathcal{E}_\mathbbm{1}^{N-1}\mathcal{E}_O)/ \textrm{Tr}(\mathcal{E}_\mathbbm{1}^{N}).
\end{align}
To compute the expected value of the string order parameter of Eq.~(\ref{Trunc_Sym}) at the AKLT point, we must analyze the transfer matrices $\mathcal{E}_{u(g)}$, $\mathcal{E}_{V^L(g)}$, and $\mathcal{E}_{V^R(g)}$. Notice that $\mathcal{E}_{V^L(g)}$ and $\mathcal{E}_{V^R(g)}$ are obtained from the two-site tensor in Eq.~(\ref{AKLT_Two_Site}). Taking $g=z$, these transfer matrices are written,
\begin{subequations}
\begin{align}
\mathcal{E}_{u(z)} &=
\begin{pmatrix}
-\frac{1}{3} & 0 & 0 & 0 \\
0 & -\frac{1}{3} & 0 & 0 \\
0 & 0 & -\frac{1}{3} & 0 \\
0 & 0 & 0 & 1
\end{pmatrix} ,\\
\mathcal{E}_{{V}^L(z)} &=
\begin{pmatrix}
0 & 0 & 0 & \frac{2}{3}\left( \sqrt{\frac{2}{3}}-\frac{2}{3}\right) \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\frac{2}{3}\left( \sqrt{\frac{2}{3}}+\frac{2}{3}\right) & 0 & 0 & 0
\end{pmatrix} ,
\\
\mathcal{E}_{{V}^R(z)} &=
\begin{pmatrix}
0 & 0 & 0 & \frac{2}{3}\left( \sqrt{\frac{2}{3}}+\frac{2}{3}\right) \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\frac{2}{3}\left( \sqrt{\frac{2}{3}}-\frac{2}{3}\right) & 0 & 0 & 0
\end{pmatrix}.
\end{align}
\end{subequations}
The expected value string order parameter of Eq.~(\ref{Trunc_Sym}) can then be written as,
\begin{align}
\langle S_{[j,k]}(z)\rangle_{\textrm{AKLT}} &= \textrm{Tr}\left(\mathcal{E}_\mathbbm{1}^{N-|j-k|-4}\mathcal{E}_{V^R(z)}\mathcal{E}_{u(z)}^{|j-k|} \mathcal{E}_{V^L(z)}\right) \nonumber \\
&= \frac{4}{9}\left( \sqrt{\frac{2}{3}} + \frac{2}{3} \right)^2 + \frac{4}{9}\left( \sqrt{\frac{2}{3}} - \frac{2}{3} \right)^2\left(-\frac{1}{3} \right)^{N-4}.
\end{align}
The MPS tensor is symmetric with respect to permutations of $\{x,y,z\}$ and thus the value is the same for any nontrivial group element $g\in\{x,y,z\}$. Thus, the AKLT state on an even number of sites, $N>4$, has $\frac{4}{9}\left( \sqrt{\frac{2}{3}} + \frac{2}{3} \right)^2 < \langle S_{[j,k]}(g)\rangle_{\textrm{AKLT}} < \frac{80}{81}$. Since the AKLT state possesses the $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, this calculation carries over for the twist operator since by Eq.~(\ref{RL_Sym}), $\langle T_{[j,k]}^{(g,h)}\rangle_{\textrm{AKLT}} = \Omega(g,h) \langle S_{[j,k]}(g)\rangle_{\textrm{AKLT}} $. Five of eight instances of the triangle game require measurement of the symmetry---which is perfect at the AKLT point---and the other three require measurement of the twist operator. Therefore, the average success probability for the SPTO strategy applied at the AKLT point is,
\begin{align}
\textrm{pr}(\textrm{win}|\textrm{AKLT}) &= \frac{13}{16} + \frac{1}{12}\left(\sqrt{\frac{2}{3}}+\frac{2}{3}\right)^2 + \frac{1}{12} \left(\sqrt{\frac{2}{3}}-\frac{2}{3}\right)^2\left(-\frac{1}{3}\right)^{N-4} \nonumber \\
& \stackrel{N\rightarrow\infty}{\longrightarrow} \frac{13}{16} + \frac{1}{12}\left(\sqrt{\frac{2}{3}}+\frac{2}{3}\right)^2 \approx 0.996.
\end{align} |
1512.01800 | \section{INTRODUCTION}
Point-contact (PC) spectroscopy of the electron- phonon interaction (EPI) in normal metals \cite{Yanson1} enables direct measurement of the spectral function of EPI under the condition that the inelastic electron mean-free path length ${{l}_{\varepsilon }}$ is greater than the dimensions of the contact and the temperature is quite low ($kT\ll \hbar {{\omega }_{\max }}$, where ${{\omega }_{\max }}$ is the limiting phonon frequency).
The purpose of this work is to obtain quantitative information about the EPI in tantalum - determine the absolute intensity of the PC EPI function ${{g}_{_{pc}}}(\omega )$ and the parameter ${{\lambda }_{pc}}$ and also to determine more accurately the form of the PC EPI function. A new method was employed to prepare pure $Ta-Ta$ homocontacts and $Ta-Cu$ and $Ta-Au$ heterocontacts, which enabled us to obtain high-intensity spectra. The value of the EEP parameter ${{\lambda }_{pc}}$ in a pure homocontact is 6.5 times greater than the value obtained in the preceding work \cite{Rybal'chenko}. It is demonstrated that quantitative information can be obtained about the EPI of tantalum in $Ta-Cu$ heterocontacts. It was found that the contributions of copper and gold to the PC spectrum of the heterocontact are not noticeable though the form of the PC spectra of $Ta$ is different in homo- and heterocontacts. The reasons for these phenomena are established. It is shown that heterocontacts enable the determination of the ratio of the Fermi velocities of the corresponding electrodes.
\section{EXPERIMENTAL PROCEDURE}
A tantalum single-crystal with the resistance ratio ${{{\rho }_{300}}}/{{{\rho }_{5}}\sim{\ }20}\;$, a copper single-crystal with the resistance ratio ${{{\rho }_{300}}}/{{{\rho }_{4.2}}\sim{\ }1000}\;$ and a gold polycrystal with approximately the same value of ${{{\rho }_{300}}}/{{{\rho }_{4.2}}}\;$ were employed as the material for the electrodes. The electron momentum mean-free path length in tantalum at liquid-helium temperature, determined by the impurities, equals about 900 {\AA} for our samples. In the estimate $\rho l=0.59\cdot {{10}^{-11}}\Omega \cdot c{{m}^{2}}$ \cite{Ryazanov} and ${{\rho }_{273}}=12.6\cdot {{10}^{-6}}\Omega \cdot cm$\ \cite{Startsev} were used. Electrodes with dimensions of $1.5\times 1.5\times 10\ m{{m}^{3}}$ were cut out by the electric-spark method and subjected to chemical polishing in a mixture of concentrated acids. For tantalum, the mixture consisted of $\text{HF}\ \text{:}\ \text{HN}{{\text{O}}_{\text{3}}}\text{:}\ \text{HCl}{{\text{O}}_{\text{4}}}$, taken in equal volume ratios, while for copper the mixture consisted of $\text{HN}{{\text{O}}_{\text{3}}}\ \text{:}\ {{\text{H}}_{\text{3}}}\text{P}{{\text{O}}_{\text{4}}}\ \text{:}\ \text{C}{{\text{H}}_{\text{3}}}\text{COOH}$ in the volume ratio 2:1:1. The gold electrodes were chemically treated in aqua regia. The electrodes were then flushed in distilled water, dried, and mounted in a clamping device.
The contacts were created by the modified shear method \cite{Yanson1}, and in so doing, in order to obtain stable pure contacts, the following procedure was employed: A reference current equal to $10-50\ \mu A$ was fixed with completely shunted samples, after which the shunts were switched off and a contact was obtained with a resistance from several hundreds to several thousands of ohms, most often with a dependence R(V) of a nonmetallic type. After this the current was increased in steps with the help of a resistance box connected in series with the point contact. The initial resistance of the box was much greater than the resistance of the point contact. The voltage on the point contact in this case, as a rule, varied insignificantly and equaled $500\pm 200\ mV$. When the required resistance was achieved the contact was held under that current for several minutes. The contacts prepared in this manner were much cleaner than those prepared by the standard procedure \cite{Yanson1} and were distinguished by high mechanical and electric stability.
We note that the quality of the spectra was significantly higher than those which we obtained previously\cite{Rybal'chenko}. At the same time, together with a general increase in the intensity of the spectrum, the relative intensity of the peak increases significantly. This can be explained by the decrease in the degree of deformation of the material in the region of the contact \cite{Bostok} when it is prepared. The critical fields required for suppressing superconductivity also dropped by a factor of 3-5. With the new method for forming the contact with bias voltages of several hundreds of millivolts, the contact was in the normal regime. The temperature at the center of the contact reaches the softening point of the metal and, unlike impulsive breakdown, is maintained at this level for quite a long time, so as to order the crystal lattice of the metal in the region near the contact. Additional purification of this region apparently also occurs at the same time by the electric transport method. We note that for electroforming the heating in the $Ta-Cu$ heterocontacts is not symmetric: only the tantalum edge is heated. This is linked with the fact that for high bias voltages the flight of the electrons from the copper side remains close to ballistic. This is indirectly confirmed by the change in the differential resistance - for $Cu-Cu$ homocontacts with bias voltages of the order of 300-350 $meV$ it increases by 20-40\% compared with $eV\sim30\ meV$, while for $Ta-Cu$ heterocontacts and $Ta-Ta$ homocontacts it more than doubles for such bias voltages.
Since the short-circuiting begins most often with a resistance of several kiloohms, while the final resistances of the point contact equal, as a rule, not more than several tens of ohms, the shunting of the short circuit by the tunneling gap of the reference oxide by the comparison resistance is eliminated. Multiple contacts are also unlikely to form under conditions of electroforming. Another advantage of this method is the possibility of "growing" short-circuits with the required resistance with an accuracy of several percent. The same method was also employed to prepare $Ta-Au$ point contacts, i.e., the method is in principle also applicable to metals which are mutually soluble (copper and tantalum are not mutually soluble in the solid and liquid phases).
Some of the $Ta-Ta$, $Ta-Cu$, and $Ta-Au$ point contacts were prepared by the standard shear method. The optimal resistances of the point contacts, prepared by electro-forming, turned out to be several times higher than the corresponding values for point contacts prepared by the standard shear method. This could be attributable to the difference in the form of the short-circuits formed: In the shear method the short-circuit apparently forms along cracks in the reference oxide, and in this case its form is described best by the model of a prolate ellipse. For electroforming, the form of the short circuit is probably closer to a circular opening. Measurements were performed in an intermediate cryostat with a capillary, structurally analogous to that described in Ref.\cite{Engen}, in the temperature range 1.4-6 K. For measurements of the PC characteristics in the normal state the superconductivity was destroyed either with a magnetic field from a superconducting solenoid or by raising the temperature above ${{T}_{c}}$ for tantalum.
\section{EXPERIMENTAL RESULTS AND DISCUSSION}
For a detailed investigation the point contacts were selected beforehand according to quality criteria described in Ref.\cite{Yanson1} Because the electron energy mean-free path length at energies close to the Debye energy is quite short, the best results were obtained for comparatively high-resistance samples, whose resistance fell into the range from 20 to $100\ \Omega$. The typical PC spectra ${{V}_{2}}(eV)\sim{\ }{{{d}^{2}}V}/{d{{I}^{2}}}\;$ are presented in Fig. \ref{Fig1}.
\begin{figure}[]
\includegraphics[width=8cm,angle=0]{Fig1.pdf}
\caption[]{PC EPI spectra ${{V}_{2}}\sim{\ }{{{d}^{2}}V}/{d{{I}^{2}}}\;$ in the N state under conditions of electroforming:\\
1) $Ta-Ta$, $R=50\ \Omega $, ${{V}_{1}}(0)=300\ \mu V$, $V_{2}^{\max }=0.355\ \mu V$, $T=4.8\ K$, $H=0$;\\
2) $Ta-Ta$, $R=78\ \Omega $, ${{V}_{1}}(0)=336\ \mu V$, $V_{2}^{\max }=0.401\ \mu V$, $T=4.6\ K$, $H=0$;\\
3) $Ta-Ta$, $R=64\ \Omega $, ${{V}_{1}}(0)=347\ \mu V$, $V_{2}^{\max }=0.532\ \mu V$, $T=4.6\ K$, $H=0$;\\
4) $Ta-Cu$, $R=80\ \Omega $, ${{V}_{1}}(0)=492\ \mu V$, $V_{2}^{\max }=0.787\ \mu V$, $T=1.88\ K$, $H=3\ kOe$;\\
5) $Ta-Cu$, $R=73\ \Omega $, ${{V}_{1}}(0)=612\ \mu V$, $V_{2}^{\max }=0.594\ \mu V$, $T=1.65\ K$, $H=5.6\ kOe$;\\
6) $Ta-Cu$, $R=73\ \Omega $, ${{V}_{1}}(0)=497\ \mu V$, $V_{2}^{\max }=0.874\ \mu V$, $T=1.72\ K$, $H=2.4\ kOe$.}
\label{Fig1}
\end{figure}
\begin{figure}[]
\includegraphics[width=5cm,angle=0]{Fig2.pdf}
\caption[]{PC EPI function in Ta:\\
1) $Ta-Ta$, from 2 (Fig. \ref{Fig1}), $\lambda _{pc}^{f}=0.764$, $g_{pc}^{c\ \max }=0.500$;\\
2) $Ta-Ta$, from 3 (Fig. \ref{Fig1}),$\lambda _{pc}^{f}=0.882$, $g_{pc}^{c\ \max }=0.622$;\\
3) $Ta-Ta$, from 1 (Fig. \ref{Fig1}), $\lambda _{pc}^{f}=0.768$, $g_{pc}^{c\ \max }=0.476$;\\
4) $Ta-Cu$; from 6 (Fig. \ref{Fig1}), $\lambda _{pc}^{f}=0.770$, $g_{pc}^{c\ \max }=0.570$;\\
5) $Ta-Cu$, spectrum not presented, $\lambda _{pc}^{f}=0.696$, \\$g_{pc}^{c\ \max }=0.446$;\\
6) $Ta-Cu$, from 4 (Fig. \ref{Fig1}),$\lambda _{pc}^{f}=0.820$, $g_{pc}^{c\ \max }=0.554$;\\
7) from\cite{Shen2}, $\lambda =0.69,\quad {{g}_{\max }}=0.594$.}
\label{Fig2}
\end{figure}
The curves 1-3 refer to $Ta-Ta$ homocontacts, while the curves 4-6 refer to $Ta-Cu$ contacts. The orientation of the contact axis relative to the axis of the crystal lattice of tantalum was not monitored and was random. For this reason the position of the phonon peaks could have varied along the energy scale from 11.5 to 12.5 $meV$ and from 17.5 to 18 $meV$. In many spectra a soft mode at 7-7.5 $meV$ as well as a break near 3 $meV$ are clearly visible. In spectra measured at low temperature, a feature in the region of 15 $meV$ is clearly visible. Figure \ref{Fig2} shows the PC EPI functions of tantalum, reconstructed from these spectra. The absolute values of the EPI functions $q{{g}_{pc}}(\omega )$ and the parameter ${{\lambda }_{pc}}$ were calculated in the free-electron model for contacts in the form of a circular opening \cite{Kulik}:
\begin{equation} \label{eq__1}
{{\lambda }_{pc}}=2\int{{{g}_{pc}}(\omega ){d\omega }/{\omega }\;}
\end{equation}
\begin{equation} \label{eq__2}
\begin{matrix}
{{g}_{pc}}(\omega )=\frac{3\hbar }{2\sqrt{2}e}{{v}_{F}}\frac{{{{\tilde{V}}}_{2}}}{V_{{{1.0}^{d}}}^{2}}= \\
=0.6988\cdot {{10}^{-8}}v\ [cm/\sec ]\frac{{{{\tilde{V}}}_{2}}[V]}{V_{1.0}^{2}[V]d[nm]}. \\
\end{matrix}
\end{equation}
Here the voltage of the second harmonic of the modulating signal ${{V}_{2}}(eV)\sim{\ }{{{d}^{2}}V}/{d{{I}^{2}}}\;$; ${{V}_{1.0}}$ is the value of the modulating voltage with zero bias; $d={{\left( {16\rho l}/{3\pi {{R}_{0}}}\; \right)}^{{1}/{2}\;}}$ is the diameter of the contact;
\begin{equation} \label{eq__3}
\rho l={{p}_{F}}/n{{e}^{2}}=\frac{3{{\pi }^{2}}\hbar }{k_{F}^{2}{{e}^{2}}}=\frac{1.66\cdot {{10}^{4}}}{{{\left\{ {{k}_{F}}\left( c{{m}^{-1}} \right) \right\}}^{2}}}\ \left[ \Omega \cdot c{{m}^{2}} \right];
\end{equation}
\begin{equation} \label{eq__4}
d=\frac{4}{e{{k}_{F}}}{{\left( \frac{\pi \hbar }{{{R}_{0}}} \right)}^{{1}/{2}\;}}=44.49\frac{{{10}^{8}}{{\left( R\left[ \Omega \right] \right)}^{-1/2}}}{{{k}_{F}}\left[ c{{m}^{-1}} \right]}\left[ nm \right].
\end{equation}
In the free-electron model \cite{Harrison}\\
\begin{equation} \label{eq__5}
{{k}_{F}}={{\left( {3{{\pi }^{2}}z}/{\Omega }\; \right)}^{{1}/{3}\;}},
\end{equation}
where $z$ is the number of conduction electrons per unit cell; $\Omega $ is the volume of the unit cell. For a $bbc$ lattice
\begin{equation} \label{eq__6}
\Omega ={{{a}^{3}}}/{2}\;,
\end{equation}
where $a$ is the lattice constant. In the free-electron approximation the true wave functions are approximated by smooth pseudowave functions. The greatest differences are observed in this case in the region of the atomic core which in simple metals is small and occupies about 10\% of the volume. In transport phenomena, in particular, electric conduction, the free-electron approximation in metals such as, for example, copper and gold, "works" very well. In the VA subgroup for $V$, $Nb$, and $Ta$ over filled shells with the configuration of argon, krypton, and xenon, respectively, each atom has five valence electrons. Because of the small number of electrons filling the d bands, the Fermi level intersects them, and for this reason the band structure of these metals is very complicated near the Fermi surface. All metals of the subgroup are uncompensated with a total number of carriers equal to one hole per atom \cite{Startsev}. Therefore, for tantalum, like for niobium in Ref.\cite{Ajhcroft}, we take \emph{z}=1, in the free-electron approximation. At the same time, taking into account the fact that a=3.296 {\AA} \cite{Startsev}, we obtain ${{k}_{F}}=1.183\cdot {{10}^{8}}\ c{{m}^{-1}}$, ${{V}_{F}}=1.396\cdot {{10}^{8}}\,cm/\sec $, $\rho l=0.834\cdot {{10}^{-11}}\ \Omega \cdot c{{m}^{2}}$, $d=37.6{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}\ \left[ nm \right]$. In this approximation the maximum value obtained in our experiments on homocontacts ${{\lambda }_{pc}}=0.88$, which is greater than the EPI parameter $\lambda =0.73$, measured with the help of the tunnel effect \cite{Wolf}, but is less than $\lambda =1.17$, calculated from first principles \cite{Pishche}. We shall evaluate the expected contribution of copper to the spectrum of the heterocontact $Ta-Cu$ for the symmetric geometry. It follows from Ref.\cite{Pishche} that
\begin{equation} \label{eq__7}
L={{{\left( \frac{1}{R}\frac{dR}{dv} \right)}_{1}}}\Big/{{{\left( \frac{1}{R}\frac{dR}{dv} \right)}_{2}}=\frac{{{v}_{{{F}_{2}}}}}{{{v}_{{{F}_{1}}}}}}\;{{\left( \frac{{{p}_{{{F}_{2}}}}}{{{p}_{{{F}_{1}}}}} \right)}^{2}}\frac{g_{pc}^{(1)}}{g_{pc}^{(2)}}.
\end{equation}
Since in Ref.\cite{Shekhter} the analogous relation was obtained for model metals, which have PC EPI functions of the same form and absolute intensity but with different Fermi velocities and momenta, the cofactor ${g_{pc}^{(1)}}/{g_{pc}^{(2)}}\;$ does not occur there. Equation (\ref{eq__7}) does not agree with the expression which would be obtained in an analysis of the relative intensity of the spectra of homocontacts with the same diameter. The additional factor ${{({{p}_{{{F}_{1}}}}/{{p}_{{{F}_{1}}}})}^{2}}$ arises because the point contact form factors of metals with different momenta in heterocontacts do not equal one another and correspondingly differ from the form factors of the homocontact. In the metal with high ${{p}_{F}}$, the relative phase volume of states filled in a nonequilibrium manner is smaller because some of the electron trajectories are reflected from the interface. As shown in Ref.\cite{Shekhter},
${{\left( {{{p}_{{{F}_{1}}}}}/{{{p}_{{{F}_{2}}}}}\; \right)}^{2}}=\left\langle {{{K}_{1}}}/{{{K}_{2}}}\; \right\rangle $ where $\left\langle {{K}_{2}} \right\rangle $ is the angle-averaged form factor of the s-th metal in the heterocontact. The formula (\ref{eq__7}) was obtained for a heterocontact formed by dirty metals, but, since, according to Ref.\cite{Shekhter}, the ratio of the intensities of the spectra of the metals is independent of the electron momentum mean-free path length and remains correct in the regime close to the ballistic regime\footnote {Private remark by R. L. Shekhter}, we shall use it in what follows, making the assumption that our contacts are clean.
Our maximum absolute intensity of the PC EPI function of tantalum in homocontacts in the free-electron approximation was $g_{pc}^{Ta}{{\,}_{\max }}=0.621$; for copper in the same approximation $g_{pc}^{Cu}{{\,}_{\max }}=0.241$ [13], $v_{F}^{Cu}=1.57\cdot {{10}^{8}}\,cm/\sec $, $k_{F}^{Cu}=1.36\cdot {{10}^{8}}\,c{{m}^{-1}}$, and then
\begin{equation} \label{eq__8}
{{L}_{c}}={{\left[\left(\frac{1}{R}\frac{dR}{dv}\right)_{Cu}^{\max}\Big/\left(\frac{1}{R}\frac{dR}{dv}\right)_{Ta}^{\max}\right]}_{c}}\simeq0.255.
\end{equation}
We shall go beyond the free-electron approximation and determine this ratio employing the electronic parameters of tantalum and copper, which were determined based on the experimental data. Taking into account the fact that
\begin{equation} \label{eq__9}
\rho l=3{{\left( 2N(0){{v}_{F}}{{e}^{2}} \right)}^{-1}};\ \ \gamma ={}^{2}/{}_{3}{{\pi }^{2}}k_{B}^{2}N(0)\left( 1+\lambda \right),
\end{equation}
we obtain
\begin{equation} \label{eq__10}
\begin{matrix}
\rho l{{v}_{F}}=\frac{3}{2N(0){{e}^{3}}}={{\left( \frac{\pi {{k}_{B}}}{e} \right)}^{2}}\frac{1+\lambda }{\gamma }= \\
=0.7347\frac{\left( 1+\lambda \right)}{\gamma \left[ erg\cdot c{{m}^{-3}}\cdot {{K}^{-2}} \right]}\left[ \Omega \cdot c{{m}^{3}}\cdot {{\sec }^{-1}} \right] \\
\end{matrix}.
\end{equation}
Here N(0) is the single-spin unrenormalized electron density of states at the Fermi level and $\gamma $ is the coefficient of electronic thermal conductivity. Using ${{\gamma }_{Ta}}=4.36\,mJ\cdot mol{{e}^{-1}}{{K}^{-2}}=4\cdot {{10}^{3}}erg\cdot c{{m}^{-3}}{{K}^{-2}}$\cite{Startsev}, $\rho {{l}_{Ta}}=0.59\cdot {{10}^{-11}}\Omega \cdot c{{m}^{2}}$\cite{Ryazanov}, $\lambda =0.65$, we obtain
\begin{equation} \label{eq__11}
\begin{matrix}
{{\left( \rho l{{v}_{F}} \right)}_{Ta}}=303\cdot {{10}^{-6}}\Omega \cdot c{{m}^{3}}\cdot {{\sec }^{-1}};\ \\
v_{F}^{Ta}=0.51\cdot {{10}^{+8}}cm/\sec. \\
\end{matrix}
\end{equation}
This quantity is identical to the value of $\left\langle v_{F}^{2} \right\rangle _{Nb}^{{1}/{2}\;}$, determined in Ref.\cite{Mattheiss}, and is quite close to $\left\langle v_{F}^{2} \right\rangle _{Nb}^{{1}/{2}\;}=0.63\cdot {{10}^{8}}cm/\sec $ from Ref.\cite{Alekseevskii}, which is the true consequence of the fact that niobium and tantalum are electronic analogs; their Fermi energies are virtually identical, and the Fermi surfaces are very close \cite{Mattheiss}.
Because of the very complicated Fermi surface in tantalum the use of the formula \eqref{eq__3} for determining ${{k}_{F}}$ will lead to a large error. To evaluate $k_{{{F}_{e}}}^{Ta}$ we shall employ our value of $v_{{{F}_{e}}}^{Ta}$ and the results of Ref.\cite{Halloran}, in which the ratio of the effective electron mass, averaged over the Fermi surface, to the free-electron value ${{{m}^{*}}}/{{{m}_{0}}=1.85}\;$ was determined from the de Haas-van Alphen effect. Then
\begin{equation} \label{eq__12}
k_{{{F}_{e}}}^{Ta}={{m}^{*}}v_{{{F}_{e}}}^{Ta}{{\hbar }^{-1}}=0.82\cdot {{10}^{-8}}c{{m}^{-1}}.
\end{equation}
The Fermi surface of copper is not very different from a spherical surface, so that the formula \eqref{eq__3} and the experimental value $\rho {{l}_{Cu}}0.53\cdot {{10}^{-11}}\ \Omega \cdot c{{m}^{2}}$ can be employed without making any special error \cite{Gniwek}. Substituting the numerical value of $\rho {{l}_{Cu}}$ into the formula \eqref{eq__3} we obtain
\begin{equation} \label{eq__13}
k_{{{F}_{e}}}^{Cu}=1.48\cdot {{10}^{-8}}c{{m}^{-1}}.
\end{equation}
We can now find the absolute intensity of the PC EPI function ${{g}_{pc}}$ and the parameter ${{\lambda }_{pc}}$ using the experimental data. The transfer factor from the free-electron to the experimental values for tantalum equals
\begin{equation}
\label{eq__14}
\begin{matrix}
{{k}_{t}}=\lambda _{pc}^{f}/\lambda _{pc}^{e}=g_{pc}^{f}/g_{pc}^{e}=v_{F}^{f}{{d}^{e}}/v_{F}^{e}{{d}^{f}}= \\
=v_{F}^{f}k_{F}^{f}/v_{F}^{e}k_{F}^{e}=3.87. \\
\end{matrix}
\end{equation}
From here ${{\lambda }_{pc}}$=0.227. The corresponding formulas for determining the diameter of tantalum and copper homocontacts have the form
\begin{equation} \label{eq__15}
\begin{matrix}
d_{Ta}^{e}=54.26{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}\left[ nm \right]; \\
d_{Cu}^{e}=30{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}\left[ nm \right]. \\
\end{matrix}
\end{equation}
The contribution of copper to the spectrum of the heterocontact, using the experimental electronic parameter s, is less than in the free-electron approximation. It follows from the formulas \eqref{eq__7} and \eqref{eq__14} that
\begin{equation} \label{eq__16}
{{{L}_{c}}}/{{{L}_{e}}={{\left( {k_{F}^{Ta}}/{k_{F}^{Cu}}\; \right)}_{f}}}\;{{\left( {k_{F}^{Cu}}/{k_{F}^{Ta}}\; \right)}_{e}}=1.57.
\end{equation}
Here the expected relative contribution of copper to the spectrum of the heterocontact, using the electronic parameters of tantalum and copper determined from the experimental data, is given by
\begin{equation} \label{eq__17}
{{L}_{e}}=0.162.
\end{equation}
For what follows, we note that the maximum absolute intensity of the PC EPI function of copper with the experimental value for ${{k}_{F}}$ (see the formula \eqref{eq__14}) equals
\begin{equation} \label{eq__18}
g_{pc\max }^{Cu}=0.262.
\end{equation}
The point-contact EPI function ${{g}_{pc}}(\omega )$ is related to Eliashberg's thermodynamic function $g(\omega )$, determined in experiments on the elastic tunneling effect in superconductors and the transport EPI function ${{g}_{tr}}(\omega )$, which is determined from precision measurements of the temperature dependence of the resistivity, owing to scattering by phonons. Only the factor in the integrand - the form factor, determined by the scattering angle - changes. The role of electron scattering by large angles increases in the sequence $g\to {{g}_{tr}}\to {{g}_{pc}}$. The transport EPI parameter satisfies the relation
\begin{equation} \label{eq__19}
{{\lambda }_{tr}}=\left( {\hbar }/{2\pi }\; \right)\left( {{{v}_{F}}}/{\rho l{{k}_{B}}}\; \right)\partial \rho /\partial T,
\end{equation}
where $\partial \rho /\partial T$ is the slope of the temperature dependence of the resistivity at high temperatures$\left( T>{{\Theta }_{D}} \right)$. Using the experimental values for $\rho l$ and ${{v}_{F}}$ for tantalum we obtain ${{\lambda }_{tr}}=0.456$ \cite{Yanson1}. If, however, $\rho l$ and ${{v}_{F}}$, calculated in the free-electron approximation, are employed, then we obtain ${{\lambda }_{tr}}=0.854$. Since the transport EPI function ${{g}_{tr}}$ occupies an intermediate position between the PC EPI function ${{g}_{pc}}$ and Eliashberg's thermodynamic function $g$, with the use of the experimental electronic parameters, the relation ${{\lambda }_{pc}}=0.227<{{\lambda }_{tr}}=0.456<\lambda =0.73$ holds.
We shall now determine the absolute intensity of the partial spectrum of tantalum in a symmetric heterocontact. The resistance of a heterocontact in the presence of a $\delta $ function barrier at the boundary between the metals equals \cite{Shekhter}
\begin{equation} \label{eq__20}
R_{het}^{-1}=\frac{{{e}^{2}}S{{S}_{F}}}{{{\left( 2\pi \hbar \right)}^{3}}}{{\left\langle \alpha D(\alpha ) \right\rangle }_{\begin{smallmatrix}
1,2 \\
{{v}_{z}}>0
\end{smallmatrix}}}.
\end{equation}
Here $\left\langle ... \right\rangle {{v}_{z}}>0$ denotes averaging over the Fermi surfaces of metals 1 and 2, respectively, under the condition ${{v}_{z}}>0$; S is the area of the contact; ${{S}_{F}}$ is the area of the Fermi surface; $\alpha ={{{v}_{z}}}/{{{v}_{F}}=\cos \theta }\;$; and, $D$ is the transmission coefficient of the boundary. The quantity $R_{het}^{-1}$ is independent of the metal over which the averaging is performed, and it is independent of the electron dispersion law, i.e.,
\begin{equation} \label{eq__21}
{{\left\{ {{S}_{F}}\left\langle \alpha D(\alpha ) \right\rangle \right\}}_{1}}={{\left\{ {{S}_{F}}\left\langle \alpha D(\alpha ) \right\rangle \right\}}_{2}}.
\end{equation}
Taking into account the fact that
\begin{equation} \label{eq__22}
\frac{{{\left( 2\pi \hbar \right)}^{3}}}{{{e}^{2}}S{{S}_{F}}{{\left\langle \alpha \right\rangle }_{{{v}_{z}}>0}}}=\frac{16\pi \hbar }{{{e}^{2}}k_{F}^{2}{{d}^{2}}}={{R}_{0}},
\end{equation}
where ${{R}_{0}}$ is the resistance of the homocontact, we obtain ${{R}_{het}}={{R}_{0}}\left( {{{\left\langle \alpha \right\rangle }_{{{v}_{z}}>0}}}/{{{\left\langle \alpha D(\alpha ) \right\rangle }_{{{v}_{z}}>0}}}\; \right)$. For a spherical Fermi surface
\begin{equation} \label{eq__23}
\begin{matrix}
{{\left\langle \alpha \right\rangle }_{{{v}_{z}}>0}}=1/2;\quad {{\left\langle \alpha D(\alpha ) \right\rangle }_{{{v}_{z}}>0}}=\int\limits_{0}^{1}{\alpha D(\alpha )}d\alpha ; \\
R_{het}^{-1}=2R_{0}^{-1}\int\limits_{0}^{1}{\alpha D(\alpha )}d\alpha. \\
\end{matrix}
\end{equation}
The expression for the form factors of the metals in heterocontacts was calculated in Ref.\cite{Shekhter}:
\begin{equation} \label{eq__24}
{{K}_{s}}\left( \mathbf{p},\mathbf{{p}'} \right)=\frac{{{D}_{12}}({{\mathbf{p}}_{\mathbf{s}}}=\mathbf{p}){{D}_{12}}({{\mathbf{p}}_{\mathbf{s}}}=\mathbf{{p}'})}{4\left\langle \alpha D(\alpha ) \right\rangle }{{K}_{0s}}\left( \mathbf{p},\mathbf{{p}'} \right).
\end{equation}
Here ${{K}_{s}}$ denotes ${{K}_{Ta}}$ or ${{K}_{Cu}}$; ${{K}_{0s}}\left( \mathbf{p},\mathbf{{p}'} \right)$ are the form factors of the pure homocontacts, which are identical for both metals in the clean limit $\left\langle {{K}_{0}} \right\rangle ={1}/{4}\;$ in the model of an opening. For direct contact between the metals we have \cite{Shekhter}
\begin{equation}
\label{eq__25}
D=\frac{4{{v}_{z1}}{{v}_{z2}}}{{{\left( {{v}_{z1}}+{{v}_{z2}} \right)}^{2}}},
\end{equation}
where ${{v}_{z1}}={{v}_{F1}}\cos {{\theta }_{1}}$; ${{v}_{z2}}={{v}_{F2}}\cos {{\theta }_{2}}$; from the law of conservation of momentum $p_\parallel ={{p}_{F1}}\sin {{\theta }_{1}}={{p}_{F2}}\sin {{\theta }_{2}}$. We denote ${{{p}_{F1}}}/{{{p}_{F2}}=b}\;$; ${{{v}_{F1}}}/{{{v}_{F2}}=c}\;$; $\cos {{\theta }_{1}}={{\alpha }_{1}}$; $\cos {{\theta }_{2}}={{\alpha }_{2}}$. We assume for definiteness that $b<1$. As a result we obtain
\begin{equation}
\label{eq__26}
{{\alpha }_{1}}={{b}^{-1}}{{\left( \alpha _{2}^{2}+{{b}^{2}}-1 \right)}^{{1}/{2}\;}};\ {{\alpha }_{2}}=b{{\left( \alpha _{1}^{2}+{{b}^{-2}}-1 \right)}^{{1}/{2}\;}}.
\end{equation}
The transmission coefficients at each edge can be written in the form
\begin{equation}
\label{eq__27}
D\left( {{\alpha }_{1}} \right)=\frac{4b{{\alpha }_{1}}{{\left( \alpha _{1}^{2}+{{b}^{-2}}-1 \right)}^{{1}/{2}\;}}}{c{{\left[ {{\alpha }_{1}}+\left( b/c \right){{\left( \alpha _{1}^{2}+{{b}^{-2}}-1 \right)}^{{1}/{2}\;}} \right]}^{2}}};
\end{equation}
\begin{equation}
\label{eq__28}
D\left( {{\alpha }_{2}} \right)=\frac{4c{{\alpha }_{1}}{{\left( \alpha _{2}^{2}+{{b}^{2}}-1 \right)}^{{1}/{2}\;}}}{b{{\left[ {{\alpha }_{2}}+\left( c/b \right){{\left( \alpha _{2}^{2}+{{b}^{2}}-1 \right)}^{{1}/{2}\;}} \right]}^{2}}}.
\end{equation}
We shall solve the problem from the side of the tantalum edge. In this case
\begin{equation} \label{eq__29}
2{{\left\langle {{\alpha }_{1}}D\left( {{\alpha }_{1}} \right) \right\rangle }_{{{v}_{z}}>0}}=2\int\limits_{0}^{1}{{{\alpha }_{1}}D\left( {{\alpha }_{1}} \right)d{{\alpha }_{1}}}=0.597.
\end{equation}
Here we assume that $b={p_{F}^{Ta}}/{p_{F}^{Cu}=0.554}\;$; $c={v_{F}^{Ta}}/{v_{F}^{Cu}=0.325}\;$. For the diameter of the $Ta-Cu$ heterocontact we have from \eqref{eq__22} and \eqref{eq__23}
\begin{equation} \label{eq__30}
{{d}_{het}}={{d}_{Ta}}{{\left[ 2{{\left\langle {{\alpha }_{1}}D({{\alpha }_{1}}) \right\rangle }_{{{v}_{z}}>0}} \right]}^{{-1}/{2}\;}}=70.2{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}[nm],
\end{equation}
where $d_{Ta}$ is the diameter of the tantalum homocontact with the same resistance (see \eqref{eq__15}). The problem can also be solved from the copper side:
\begin{equation} \label{eq__31}
\begin{matrix}
2{{\left\langle {{\alpha }_{2}}D({{\alpha }_{2}}) \right\rangle }_{{{v}_{z}}>0}}=2\int\limits_{\sqrt{1-{{b}^{2}}}}^{1}{{{\alpha }_{2}}D({{\alpha }_{2}})}d{{\alpha }_{2}}\equiv \\
\equiv 2{{b}^{2}}{{\left\langle {{\alpha }_{1}}D({{\alpha }_{1}}) \right\rangle }_{{{v}_{z}}>0}}=0.1833. \\
\end{matrix}
\end{equation}
Here the integration is performed from $\sqrt{1-{{b}^{2}}}$, because of the fact that for ${{\alpha }_{2}}\ll \sqrt{1-{{b}^{2}}}$ the quantity $D\left( {{\alpha }_{2}} \right)=0$, since total internal reflection of the electrons occurs. The diameter of the heterocontact in this case equals
\begin{equation} \label{eq__32}
\begin{matrix}
{{d}_{het}}={{d}_{Cu}}{{b}^{-1}}{{\left[ 2{{\left\langle {{\alpha }_{1}}D({{\alpha }_{1}}) \right\rangle }_{{{v}_{z}}>0}} \right]}^{-1/2\;}}= \\
=70.2{{\left( R\left[ \Omega \right] \right)}^{-1/2\;}}[nm], \\
\end{matrix}
\end{equation}
i.e., the same result is obtained as in the calculation from the tantalum side. (We note that if the diameter of the tantalum heterocontact is determined from Sharvin's formula
\[{{d}_{Ta}}={{\left( \frac{16\rho l}{3\pi } \right)}^{{1}/{2}\;}}{{R}^{{-1}/{2}\;}}=31.65{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}[nm],\]
then we obtain
\[{{d}_{het}}=41.1{{\left( R\left[ \Omega \right] \right)}^{{-1}/{2}\;}}[nm],\]
i.e., the calculation from the copper side yields a result which is different from the calculation on the tantalum side. The fact that the simple relation \eqref{eq__21} does not hold is apparently linked with the presence of several groups of carriers in the tantalum, and this makes it necessary to generalize the relation \eqref{eq__21} to this case.)
To determine the averaged form factors of tantalum and copper in the heterocontact in the ballistic state, the numerator in the formula \eqref{eq__24} must first be calculated:
\[{{A}_{S=Ta,}}\ {{B}_{S=Cu}}={{\left\langle {{D}_{12}}({{\mathbf{p}}_{\mathbf{s}}}=\mathbf{p}){{D}_{12}}({{\mathbf{p}}_{\mathbf{s}}}=\mathbf{{p}'}){{K}_{0s}}(\mathbf{p},\mathbf{{p}'}) \right\rangle }_{{{v}_{z}}>0}}\]
In the model of an opening \cite{Shekhter}
\begin{equation} \label{eq__33}
{{K}_{0s}}={{K}_{0}}(\mathbf{p},\mathbf{{p}'})=\frac{\left| {{v}_{z}}{{v}_{z}}^{\prime } \right|\theta \left( -{{v}_{z}}{{v}_{z}}^{\prime } \right)}{\left| {{v}_{z}}\mathbf{{v}'}-{{v}_{z}}\mathbf{v} \right|}
\end{equation}
($\theta $ is the Heaviside function).
Decomposing the numerator of the formula into components and transforming it for a spherical Fermi function we obtain
\begin{equation} \label{eq__34}
A=\frac{32}{\pi }\int\limits_{0}^{1}{dx}\int\limits_{0}^{1}{dy\frac{xD\left( {{\alpha }_{1}}=x \right)D\left( {{\alpha }_{1}}=y \right)K(k)}{{{(n+m)}^{2}}}},\ y>x;
\end{equation}
\begin{equation} \label{eq__35}
\small{B=\frac{32}{\pi }\int\limits_{\sqrt{1-{{b}^{2}}}}^{1}{dx}\int\limits_{\sqrt{1-{{b}^{2}}}}^{1}{dy\frac{xD\left( {{\alpha }_{2}}=x \right)D\left( {{\alpha }_{2}}=y \right)K(k)}{{{(n+m)}^{2}}}},\ y>x.}
\end{equation}
Here
\[\begin{matrix}
K(k)=\int\limits_{0}^{{\pi }/{2}\;}{\frac{d\varphi }{\sqrt{1-{{k}^{2}}{{\sin }^{2}}\varphi }},\quad k={{\left( \frac{n-m}{n+m} \right)}^{2}};} \\
n={{\left( 1-{{x}^{2}} \right)}^{{1}/{4}\;}}+{{\left( 1-\frac{{{x}^{2}}}{{{y}^{2}}} \right)}^{{1}/{4}\;}};\ \\
m=8\left\{ {{\left[ \left( 1-{{x}^{2}} \right)\left( 1-\frac{{{x}^{2}}}{{{y}^{2}}} \right) \right]}^{{1}/{4}\;}} \right.\times \\
\times {{\left. \left[ {{\left( 1-{{x}^{2}} \right)}^{{1}/{2}\;}}+{{\left( 1-\frac{{{x}^{2}}}{{{y}^{2}}} \right)}^{{1}/{2}\;}} \right] \right\}}^{{1}/{4}\;}} \\
\end{matrix}\].
For $x>y$ symmetric integrands are obtained, in which x and y are interchanged. For this reason it is sufficient to calculate the integrals for $x>y$ and multiply the result obtained by 2. In the formulas presented this multiplication has already been performed. Since the series $K(k)$ converges very rapidly, for the calculations we can take $K(k)={\pi }/{2}\;$. A somewhat different form of $B$, obtained by substitution of variables, is more convenient for calculations:
\begin{equation} \label{eq__36}
\begin{matrix}
B=\frac{32{{b}^{3}}}{\pi }\int\limits_{0}^{1}{dx}\int\limits_{0}^{1}{dy\frac{xyD\left( {{\alpha }_{1}}=x \right)D\left( {{\alpha }_{1}}=y \right)K({{k}_{1}})}{{{\left( {{y}^{2}}-{{b}^{-2}}-1 \right)}^{{1}/{2}\;}}{{({{n}_{1}}+{{m}_{1}})}^{2}}}},\ y>x; \\
{{k}_{1}}={{\left( \frac{{{n}_{1}}-{{m}_{1}}}{{{n}_{1}}+{{m}_{1}}} \right)}^{2}}; \\
{{n}_{1}}={{\left[ {{b}^{2}}\left( 1-{{x}^{2}} \right) \right]}^{{1}/{4}\;}}+{{\left( \frac{{{y}^{2}}-{{x}^{2}}}{{{y}^{2}}+{{b}^{-2}}+1} \right)}^{{1}/{4}\;}}; \\
{{m}_{1}}=\left\{ 8{{\left[ \frac{{{b}^{2}}\left( 1-{{x}^{2}} \right)\left( {{y}^{2}}-{{x}^{2}} \right)}{{{y}^{2}}-{{b}^{-2}}-1} \right]}^{{1}/{4}\;}} \right.\times \\
{{\left. \times \left[ {{\left( {{b}^{2}}\left( 1-{{x}^{2}} \right) \right)}^{{1}/{2}\;}}+{{\left( \frac{{{y}^{2}}-{{x}^{2}}}{{{y}^{2}}+{{b}^{-2}}-1} \right)}^{{1}/{2}\;}} \right] \right\}}^{^{^{{1}/{4}\;}}}}. \\
\end{matrix}
\end{equation}
Substituting the numerical values into \eqref{eq__34} and \eqref{eq__36} and integrating, we obtain
\begin{equation} \label{eq__37}
A=0.088;\quad B=0.0137.
\end{equation}
The averaged form factors for a heterocontact in the model of a clean opening satisfy the expression
\begin{equation} \label{eq__38}
{\left\langle {{K}_{Ta}} \right\rangle }/{\left\langle {{K}_{0}} \right\rangle }\;=A\left\langle {{\alpha }_{1}}D\left( {{\alpha }_{1}} \right) \right\rangle _{{{v}_{z}}>0}^{-1}=0.296;
\end{equation}
\begin{equation} \label{eq__39}
\begin{matrix}
{\left\langle {{K}_{Cu}} \right\rangle }/{\left\langle {{K}_{0}} \right\rangle }\;=B\left\langle {{\alpha }_{2}}D\left( {{\alpha }_{2}} \right) \right\rangle _{{{v}_{z}}>0}^{-1}= \\
=B{{b}^{-2}}\left\langle {{\alpha }_{1}}D\left( {{\alpha }_{1}} \right) \right\rangle _{{{v}_{z}}>0}^{-1}=0.150. \\
\end{matrix}
\end{equation}
Since the contribution of copper to the spectrum of the heterocontact is not noticeable, it may be assumed without introducing any special error that the entire spectrum is determined by the tantalum edge. In this case the more accurate value of the intensity of the PC EPI function of tantalum in the hererocontact equals
\begin{equation} \label{eq__40}
g_{pc}^{het}={{A}^{-1}}\left\langle {{\alpha }_{1}}D\left( {{\alpha }_{1}} \right) \right\rangle \left( {{{d}_{Ta}}}/{{{d}_{het}}}\; \right){{g}_{pc}},
\end{equation}
Where ${g}_{pc}$ is the PC EPI function of the $Ta-Ta$ homocontact with the same resistance (see \eqref{eq__3} and \eqref{eq__4}). From here it is necessary to calculate correction factor
\begin{equation} \label{eq__41}
\begin{matrix}
{{k}_{korr}}=\lambda _{pc}^{het}/\lambda _{pc}^{f}=\ g_{pc}^{het}/g_{pc}^{f}=\ \\
=\sqrt{2}{{A}^{-1}}\left( v_{F}^{e}k_{F}^{e}/v_{F}^{f}k_{F}^{f}\ \right)\left\langle {{\alpha }_{1}}D\left( {{\alpha }_{1}} \right) \right\rangle _{{{v}_{z}}>0}^{3/2\;}=0.674 \\
\end{matrix}
\end{equation}
Here $g_{pc}^{f}$ and $\lambda _{pc}^{f}$ are the absolute intensity of the PC EPI function and the parameter $\lambda $, calculated in the free-electron approximation for a $Ta-Ta$ homocontact with the same resistance.
We shall calculate the ratio of the averaged form factors for a heterocontact in the ballistic regime. As follows from the formulas \eqref{eq__38} and \eqref{eq__39},
\begin{equation} \label{eq__42}
{\left\langle {{K}_{Cu}} \right\rangle }/{\left\langle {{K}_{Ta}} \right\rangle =B/{{b}^{2}}A}\;=0.506.
\end{equation}
This quantity is greater than ${{b}^{2}}=0.307$, which is presented for a dirty heterocontact\footnote {According to a remark by R.I. Shekhter, this is attributable to the strong angular dependence of the form factors, which accentuate the scattering at large angles. In the presence of an atomically smooth boundary between the mewls, ${\left\langle {{K}_{Cu}} \right\rangle }/{\left\langle {{K}_{Ta}} \right\rangle }\;\simeq b$. If the transit regime of the electrons moving through the point contact remains ballistic, but the boundary between the metals is rough and scatters electrons diffusely, then ${\left\langle {{K}_{Cu}} \right\rangle }/{\left\langle {{K}_{Ta}} \right\rangle }\;\simeq {{b}^{2}}$.}
From here the partial contributions of $C$u and $Ta$ to the spectrum in the ballistic regime can be determined more accurately (compare with \eqref{eq__7}):
\begin{equation} \label{eq__43}
L=v_{F}^{Ta}g_{pc}^{Cu}{\left\langle {{K}_{Cu}} \right\rangle }/{v_{F}^{Cu}}\;g_{pc}^{Ta}\left\langle {{K}_{Ta}} \right\rangle .
\end{equation}
Taking this into account we can put the formula \eqref{eq__16} into a different form:
\begin{equation} \label{eq__44}
\frac{{{L}_{f}}}{{{L}_{e}}}={{\left( \frac{k_{F}^{Cu}}{k_{F}^{Ta}} \right)}_{f}}{{\left( \frac{k_{F}^{Ta}}{k_{F}^{Cu}} \right)}_{e}}{{\left( \frac{\left\langle {{K}_{Cu}} \right\rangle }{\left\langle {{K}_{Ta}} \right\rangle } \right)}_{f}}{{\left( \frac{\left\langle {{K}_{Ta}} \right\rangle }{\left\langle {{K}_{Cu}} \right\rangle } \right)}_{e}}\simeq 1.041.
\end{equation}
Table I shows the parameters of homo- and heterocontacts in the free-electron approximation and for two values of the Fermi velocity of tantalum.
\begin{table*}[]
\caption[]{}
\begin{tabular}{|p{1in}|p{1.3in}|p{1.3in}|p{1.5in}|} \hline
Parameter & $v_{F} =1.369\cdot 10^{8} \, cm/s$; $k_{F} =1.183\cdot 10^{8} \, cm/s$ & $v_{F} =0.51\cdot 10^{8} \, cm/s$; $k_{F} =0.82\cdot 10^{8} \, cm/s$ & $v_{F} =0.24\cdot 10^{8} \, cm/s$ \cite{Shen1};\newline $k_{F} =0.348\cdot10^{8} \, cm/s$ \\ \hline
b & 0.872 & 0.554 & 0.259 \\ \hline
c & 0.872 & 0.325 & 0.153 \\ \hline
$\left\langle \alpha _{1} D\left(\alpha _{1} \right)\right\rangle _{v_{z} >0} $ & 0.479 & 0.299 & 0.168 \\ \hline
${\left\langle K_{Ta} \right\rangle \mathord{\left/ {\vphantom {\left\langle K_{Ta} \right\rangle \left\langle K_{0} \right\rangle }} \right. \kern-\nulldelimiterspace} \left\langle K_{0} \right\rangle } $ & 0.468 & 0.296 & 0.168 \\ \hline
${\left\langle K_{Cu} \right\rangle \mathord{\left/ {\vphantom {\left\langle K_{Cu} \right\rangle \left\langle K_{0} \right\rangle }} \right. \kern-\nulldelimiterspace} \left\langle K_{0} \right\rangle } $ & 0.387 & 0.150 & 0.039 \\ \hline
${\left\langle K_{Cu} \right\rangle \mathord{\left/ {\vphantom {\left\langle K_{Cu} \right\rangle \left\langle K_{Ta} \right\rangle }} \right. \kern-\nulldelimiterspace} \left\langle K_{Ta} \right\rangle } $ & 0.827 & 0.506 & 0.234 \\ \hline
$d_{{\rm homo}} ,nm$ & $37.6\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ & $54.26\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ & $115.86\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ \\ \hline
$d_{{\rm het}} ,nm$ & $38.4\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ & $70.2\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ & $200.1\left(R\left[\Omega \right]\right)^{{-1\mathord{\left/ {\vphantom {-1 2}} \right. \kern-\nulldelimiterspace} 2} } $ \\ \hline
$k_{n} $ & 1 & 3.87 & 17.57 \\ \hline
$g_{pc}^{\max } $ & 0.622 & 0.16 & 0.035 \\ \hline
$k_{korr} $ & 2.088 & 0.674 & 0.196 \\ \hline
$g_{pc\, \max }^{het} $ & 1.157 & 0.374 & 0.109 \\ \hline
$L$ & 0.262 & 0.269 & 0.268 \\ \hline
$L\left(g_{pc\, \max }^{het} \right)$ & 0.14 & 0.115 & 0.086 \\ \hline
$\lambda _{pc}^{{\rm homo}} $ & 0.88 & 0.227 & 0.05 \\ \hline
$\lambda _{pc}^{{\rm het}} $ & 1.71 & 0.553 & 0.161 \\ \hline
$\lambda _{tr} $ & 0.854 & 0.456 & 0.215 \\ \hline
$J$ & 1.236 & 0.575 & 0.259 \\ \hline
\end{tabular}
\label{Table1}
\end{table*}
The value $v_{F}^{Ta}=0.2\text{4}\cdot \text{1}{{0}^{8}}cm/\sec $ is taken from Ref. \cite{Shen1}. The corresponding wave vector ${{k}_{F}}=0.38\text{4}\cdot \text{1}{{0}^{8}}cm/\sec $ is determined analogously to the preceding wave vector (see formula \eqref{eq__12}) using ${m^*}/{{{m}_{0}}=1.85}\;$ from Ref. \cite{Halloran}. The expected contribution of copper to the spectrum of the heterocontact was evaluated using the formula \eqref{eq__43}. In this case the values of $L$ corresponded to the use of the absolute intensity of the PC EPI function, obtained for $Ta-Ta$ homocontacts, and the values $L\left( g_{pc}^{het} \right)$ corresponded to the use of the more accurate absolute intensity of the PC EPI function of tantalum in $Ta-Cu$ heterocontacts. Although it follows from Table I that the contribution of copper to the spectrum of the heterocontact in the estimates must exceed 25\%, the contribution of copper is not noticeable in the spectra of heterocontacts. At the same time, if the absolute intensity of the PC EPI function ${{g}_{pc}}$ and the parameter ${{\lambda }_{pc}}$ for tantalum in the $Ta-Cu$ heterocontacts are calculated just as for $Ta-Ta$ homocontacts with the same resistance, we obtain values that are close to those obtained for homocontacts. Thus, for the best heterocontact in the free-electron approximation, $g_{pc}^{\max f}=0.\text{554}$ and $\lambda _{pc}^{f}=0.\text{82}$. From here there follow two possibilities: 1) the maximum intensity of the spectrum is not reached in the homocontacts; 2) in the existing technology of forming the contact, an unsymmetric heterocontact is formed, and tantalum occupies most of the effective volume of phonon generation near the constriction. We shall study both possibilities separately. In the analysis we shall assume that $v_{F}^{Ta}=0.\text{51}\cdot \text{1}{{0}^{\text{8}}}\text{cm}/\text{sec}$.
\section{SYMMETRIC CONTACT}
In this case the maximum corrected intensity of the PC EPI and the relative contribution of copper to the spectrum $g_{pc\max }^{het}=0.\text{374}$, copper relative contribution to the spectrum $L\left( g_{pc}^{het} \right)=0.115$, i.e. it is only somewhat greater than 10\% and indeed should not be noticeable. Figure \ref{Fig3} shows an example of the addition of the PC EPI functions of tantalum and copper homocontacts with the relative intensity $g_{pc\max }^{Cu}=0.1g_{pc\max }^{Ta}$.
\begin{figure}[]
\includegraphics[width=8cm,angle=0]{Fig3.pdf}
\caption[]{Comparison of the PC EPI functions of $Ta$ in homo- and heterocontacts: \\1) $Ta-Ta$, curve 2 from Fig. \ref{Fig2}; \\2) PC EPI function of $Cu$ from Ref. \cite{Yanson2}; \\3) the summed curve $g_{p{{c}_{_{^{Cu}}}}}^{\max }=0.1g_{p{{c}_{Ta}}}^{\max }$; \\4) $Ta-Cu$, curve 6 from Fig.\ref{Fig2}.}
\label{Fig3}
\end{figure}
As one can see the contribution of copper to the total spectrum is not noticeable, though the form of the total spectrum differs from the spectrum of the heterocontact - in the case of heterocontacts the relative intensity of the $L$ peak is almost 40\% higher.
One possible reason for this strong difference in absolute intensities and the form of Ta spectra in homo- and heterocontacts could be the following.
The difference in the absolute intensities of PC EPI functions and Eliashberg's thermodynamic EPI function is linked with the different form factors - in the case of the thermodynamic function every electron scattering event is effective, and for the PC EPI function large-angle scattering processes make the main contribution. This is usually also linked with the decrease in the relative intensity of the high-frequency maximum in the PC spectra \cite{Lee}. In a heterocontact the tangential component of the electron momentum is conserved across the interface. For this reason any electron that has undergone inelastic scattering in tantalum and whose $z$ component of the momentum has been reversed will return into the copper, being deflected after refraction at the boundary toward the $z$ axis, which increases the contribution of small-angle scattering processes in $Ta$ to the "return current", which determines the inelastic correction to the current through the point contact. (We note that the maximum angle of incidence of the electrons from the copper side, when total internal reflection does not yet occur, equals ${{33}^{\circ }}$ with respect to the $z$ axis.) At the same time the form factor for the metal with the lower value of ${{p}_{F}}(Ta)$ also approaches the form factor of Eliashberg's thermodynamic function, while the absolute intensities of the PC function ${{g}_{pc}}$ and the EPI parameter ${{\lambda }_{pc}}$ become closer to the corresponding values of the thermodynamic function. This is also indicated by the closer form of the PC EPI function of the heterocontact (Fig. \ref{Fig2}, curves 4-6) and the thermodynamic EPI function (Fig. \ref{Fig2}, curve 7).
We note that the PC spectrum of the metal with the large value of ${{p}_{F}}$ in the heterocontact undergoes opposite changes, owing to the fact that the scattering by angles $\theta <{{\theta }_{\min }}$ does not contribute to the return current (see Fig. \ref{Fig2} in Ref. \cite{Baranger}) and thereby accentuates the effect of the $K$ factor, which emphasizes the scattering by large angles. Therefore if we could separate out the low intensity contribution to the PC spectrum from the metal with large $p_F$, we would see that its form differs even more strongly from Eliashberg's EPI function than the form of the PC EPI function obtained from the spectrum of the homocontact of this metal. At the same time, as one can see from Table I, a physically more reliable result for $\lambda _{pc}^{het}$ is obtained using $v_{F}^{Ta}=0.\text{51}\cdot \text{1}{{0}^{\text{8}}}\text{cm}/\text{sec}$.
\section{UNSYMMETRIC HETEROCONTACT}
The magnitude of the excess current when there is no geometric asymmetry and additional scatterers at the boundary in a clean heterocontact of the $S-c-N$ type must equal \cite{Zaitsev}
\begin{equation} \label{eq__45}
\small{{I}_{exc}}=\frac{\Delta }{{{R}_{N}}}J,\ J=\frac{1}{2\left\langle \alpha D \right\rangle }{{\left\langle \alpha \frac{{{D}^{2}}}{R}\left[ 1-\frac{{{D}^{2}}arth\sqrt{R}}{\sqrt{R}(1+R)} \right] \right\rangle }_{{{v}_{z}}>0}},
\end{equation}
where $R=1-D$. Substituting $D\left( {{\alpha }_{1}} \right)$ into the formula for $J$ and integrating the expression obtained we obtain $J=0.575$. Thus for $Ta-Cu$ symmetric heterocontacts ${{I}_{exc}}=0.575\,{\Delta }/{{{R}_{N}}}\;$. In the limit $D\to 1$ we have $J=\text{4}/\text{3}$. For clean $S-c-S$ homocontacts in the presence of a semitransparent barrier between the metals D does not depend on $\alpha$. In this case
\begin{equation} \label{eq__46}
{{I}_{exc}}=\frac{2\Delta }{{{R}_{N}}}J,\ J=\frac{D}{2R}\left[ 1-\frac{{{D}^{2}}arth\sqrt{R}}{\sqrt{R}(1+R)} \right].
\end{equation}
The deviation of $D$ from 1 affects the magnitude of the excess current much more strongly than it affects the intensity of the spectrum. Thus the excess current for $D=0.7$ is half of the excess current for $D=1$, while the intensity of the spectrum in this case equals ${{D}^{{1}/{2}\;}}=0.84$ of the starting intensity.
In our experiments there was no correlation between the absolute intensity of the spectrum and the magnitude of the excess current, observed in Refs. \cite{Yanson3} and \cite{Khotkevich} for tin contacts. The superconducting characteristics turned out to be more sensitive to contamination than the spectral characteristics. The magnitude of the excess current in the contacts with an intense spectrum in the N state could vary from 0 up to 0.8 of the theoretical value for a clean $S-c-S$ contact, but the critical Josephson current was almost always absent. This indicates the presence of depairing centers in the plane of the constriction. In heterocontacts with an intense PC spectrum the parameter $J$ could vary from 0 to 1.1. From here it follows that at least for $J\gtrsim 0.6$ a geometric asymmetry of the heterocontact existed and $Ta$ filled most of the volume near the constriction, and in addition in this asymmetric heterocontact for $J$ close to 1.1 the ballistic regime was realized.
In this case it is obvious that the absolute intensity of the PC function ${{g}_{pc}}$ and the EPI parameter ${{\lambda }_{pc}}$, calculated for the $Ta-Cu$ heterocontact, are close, just like for a $Ta-Ta$ heterocontact with the same resistance and the same characteristics calculated for $Ta-Ta$ homocontacts. The reason for the fact that copper does not contribute to the spectrum is also understandable. The decrease in the relative intensity of the $L$ peak in the homocontact compared with the heterocontacts could be linked with the heating of the contact for energies close to the Debye energies. The typical resistances of our contacts equal $20-100\ \Omega$, which corresponds to diameters of 120-50 {\AA}. If the energy mean-free path length is evaluated from the relation ${{l}_{\varepsilon }}={{v}_{F}}{{\tau }_{\varepsilon }}$ (here $\tau _{\varepsilon }^{-1}=2\pi {{\hbar }^{-1}}\int\limits_{0}^{\varepsilon }{g(\omega )d\omega }$) is the energy relaxation time and $g(\omega )$ is Eliashberg's thermodynamic function, given, e.g., in the tables of Ref. \cite{Rowell} then for $\omega ={{\omega }_{D}}$ we have ${{l}_{\varepsilon }}=146{\AA}$, i.e., contacts with the lowest resistances at energies close to the Debye energies must be in a state close to the thermal state, when spectroscopy is impossible. At the same time, although the background for the high-resistance contacts is high (50-70\%) and the intensity of the $L$ peak is lowered appreciably, it is nonetheless quite well resolved. This indicates that it is incorrect to use the relation indicated above for determining the energy relaxation time, since this relation is predicated on the equilibrium electron distribution function. A substantially nonequilibrium electron distribution function is realized in the point contact near the constriction, and although the energy electron mean-free path length is much greater than the diameter of the contact, it is still not large enough, which increases the background and decreases the intensity of the $L$ peak. It is much easier to satisfy the condition ${{l}_{\varepsilon }}\gg d$ with the use of copper as the counterelectrode, since there are no restrictions from the side of the copper edge and the condition ${{l}_{\varepsilon }}\gg r$ ($r=d/2$ is the radius of the contact) must be satisfied. The decrease in the background level in heterocontacts is also explained by the lower heating of the contact owing to the much better thermal conductivity of the copper edge, which efficiently removes heat.
The two possibilities, however, are not mutually exclusive. Both symmetric and unsymmetric heterocontacts with a dominant contribution of tantalum are apparently realized. At the same time, as the interface between the metals moves away from the plane of the geometric symmetry of the contact, the effective volume of phonon generation in tantalum increases and the effect of the $\delta $ function barrier between the metals owing to the difference in the Fermi velocities decreases. At the same time, the effect of the form factor, which is determined by the difference in the Fermi momenta and the intensity of the PC EPI function of the metal with the smaller value of ${{\rho }_{F}}$, approaching Eliashberg's thermodynamic function, also decreases. In $Ta-Cu$ heterocontacts, the mutual effect of these factors is largely compensated, and for this reason the contribution of copper to the spectrum is not noticeable, while the intensity of the spectra does not change much as the symmetry of the contact changes. The magnitude of the excess current, however, changes at the same time over a wide range. To check this proposition, we prepared a series of $Ta-Ta$, $Ta-Cu$, and $Ta-Au$ contacts using the standard shear method, with which a geometrically symmetric contact is most likely to be formed. As in the case of the electroforming of contacts, however, no contribution from copper and gold to the spectrum was observed in heterocontacts. The absolute intensity of the spectra for $Ta-Ta$ homocontacts and $Ta-Cu$ heterocontacts in this case was much lower than with electroforming, and turned out to be close to the absolute intensity of the spectra which we obtained previously \cite{Rybal'chenko}. In the case of $Ta-Au$ point contacts, the absolute intensity of the PC spectra obtained for electrically formed point contacts turned out to be only slightly higher than for point contacts prepared by the standard shear method \cite{Yanson1}. This is apparently explained by the mutual solubility of $T$a and $Au$.
Figure \ref{Fig4} shows typical PC spectra of $Ta-Cu$ and $Ta-Au$ for contacts prepared by different methods.
\begin{figure}[t]
\includegraphics[width=8cm,angle=0]{Fig4.pdf}
\caption[]{Check of the effect of the technology employed to fabricate the point contact on the form of the PC EPI spectrum ${{V}_{2}}\sim{\ }{dV}/{d{{I}^{2}}}\;$, for electroforming (1) and using the shear method (2, 3): \\1) $Ta-Au$, $R=70\ \Omega $, ${{V}_{1}}(0)=710\ \mu V$, $v_{2}^{\max }=0.6\,\mu V$, $T=4.8\,K,\ H=0$;
\\2) $Ta-Au$, $R=5\ \Omega $, ${{V}_{1}}(0)=545\ \mu V$, $v_{2}^{\max }=0.57\,\mu V$, $T=5\,K,\ H=0$;
\\3) $Ta-Cu$, $R=30\ \Omega $, ${{V}_{1}}(0)=754\ \mu V$, $v_{2}^{\max }=0.54\,\mu V$, $T=4.8\,K,\ H=0$.}
\label{Fig4}
\end{figure}
All values of $g_{pc}^{f}$ and $\lambda _{pc}^{f}$ in the caption in Fig. \ref{Fig2} correspond to the free-electron approximation, and in addition in heterocontacts the values of $g_{pc}^{f}$ and $\lambda _{pc}^{f}$ were calculated just as for homocontacts with the same resistance.
All estimates made above must be treated with care. Since $d$ metals have two groups of valence electrons, in different experiments one or another group of electrons can make the dominant contribution. For example, in niobium, which is the electronic analog of tantalum, the Fermi velocity, determined in optical measurements, $v_{F}^{Nb}=0.9\text{4}\cdot \text{1}{{0}^{8}}cm/\sec $ (Ref. \cite{Mash}). In Ref. \cite{Blonder}, in order to make the computational results agree with experiment for $Cu-Nb$ point contacts, the value $v_{F}^{Nb}=0.897\cdot \text{1}{{0}^{8}}cm/\sec $ was employed. For different values of ${{v}_{F}}$ and ${{p}_{F}}$ the values of $J$ and $D$ will be different.
\section{CONCLUSIONS}
In conclusion we shall list the basic results obtained in this work.
A new procedure was developed for preparing clean point contacts.
\begin{enumerate}
\item {The EPI parameter ${{\lambda }_{pc}}$ and the absolute intensity of the PC function ${{g}_{pc}}(\omega )$ for tantalum were determined, and the exact form of the PC function was found.}
\item {The PC characteristics of a symmetric heterocontact were calculated numerically for the first time in this work: the absolute intensity of the EPI function ${{g}_{pc}}(\omega )$ and the parameter ${{\lambda }_{pc}}$. At the same time, an expression taking into account the degree to which the electronic parameters of each electrode affect the form factor of the heterocontact was derived, which makes it possible to evaluate the relative contribution of the specific metal to the resulting spectrum. The calculation is based on the results of the theory of Shekhter and Kulik \cite{Shekhter}, and the case of the ballistic contacts in the approximation of a spherical Fermi surface was examined.}
\item {It was established that the contribution of the second metal to the PC spectra of the heterocontacts $Ta-Cu$ and $Ta-Au$ is not noticeable, while the intensities of the PC spectra remain virtually constant as the geometric symmetry of the contacts changes. At the same time the forms of the spectra in homo- and heterocontacts are different. An explanation of these phenomena with the help of the compensation effect was proposed.}
\item {The effective mass approximation was used for the first time to calculate the PC characteristics, since the free-electron approximation obviously leads to incorrect results.}
\item {A new procedure was developed for preparing clean point contacts.}
\end{enumerate}
We thank V.M. Kirzhner, O.I. Shklyarevskii, and V.V.~Khotkevich for assistance in the computer calculations.
\section{NOTATION}
Here ${{v}_{F}}$ and ${{p}_{F}}$ are the Fermi velocity and Fermi momentum, respectively; $g$ and $\lambda $ are the EPI function and parameter, respectively; ${{k}_{F}}$ is the Fermi wave vector; $d$ is the diameter of the contact; $a$ is the lattice constant; $\Omega $ is the volume of the unit cell; $L$ is the relative intensity of the partial spectra of a heterocontact; $\gamma $ is the electronic heat capacity; and, $D$ is the coefficient of electron transit through the boundary in a heterocontact. |
2004.09634 | \section{Introduction}
We study the motion of particles subject to a covariant mechanical friction force (MFF) caused by the presence of a material medium. In general, in the presence of any force, a charged particle emits radiation, a result obtained by Larmor considering properties of Maxwell\rq s equations. Emitted radiation complements the MFF as an induced radiation friction force (RFF). MFF will be used as an insightful model to learn how to accommodate the dynamics of radiation reaction force, {\it i.e.\/} radiation reaction (RR).
To the best of our knowledge, all prior covariant studies of RR employed the Lorentz force (LF) due to externally prescribed electromagnetic (EM) fields. Considered in the context of a Lorentz-Maxwell system of dynamical equations, it is well known that RR is an unsolved problem. The advantage of our approach is that we can focus on a better understanding of the effect of the radiation reaction on the mechanically accelerated particle, without need to reconcile the LF with Maxwell field dynamics.
We choose a MFF force which reduces to the familiar form of Newton\rq s friction force in the non-relativistic limit in the case of linear relativistic motion. In Sect.~\ref{sec:covfriction} we find that the relativistic generalization of Newtonian friction has a unique form used also in the study of Brownian motion~\cite{Dunkel:2005}. For constant MFF we evaluate stopping distance, rapidity shift, and stopping power, which provide background for the later study of motion including RFF.
In this work we introduce, as a mathematical tool for making RR consistent with special relativity, a matter warping model along the particle path by particle acceleration and parameterized by the proper time of the particle. Matter warping is not a field as it is known only along the path of an accelerated particle. When borrowing tools of differential geometry we therefore prefer to speak of a warped matter metric rather than a curved space-time metric. There have been earlier efforts to modify the space-time metric for accelerated particles spearheaded by Caianiello~\cite{Caianiello:1989wm}, whose work was driven by the postulate of a maximal proper acceleration. For a recent review see~\cite{Torrome:2018zck}. Our objective is clearly very different even though some of the methods are similar as we explore the physical environment of a resistive medium and in particular its warping due to accelerated motion.
To summarize the theoretical advantages of using the material medium:
\begin{enumerate}
\item Unlike in the space-time empty of matter (vacuum) case, the presence of a material medium provides a reference frame against which we measure particle motion. Hence the covariant form of MFF depends on the particle 4-velocity as well as the 4-velocity of the medium.
\item When a particle experiences energy loss due to RFF, this occurs in the model always at the expense of the well-defined relative motion with respect to the medium.
\item The specific form of the LF does not enter and thus the inconsistency between the EM field equations and the description of charged particle motion subject to EM-force, see discussion on p. 745 in Jackson~\cite{Jackson:1999}, is not introduced. We tacitly employ the Maxwell field equations when characterizing the magnitude of radiative energy loss for an accelerated charged particle as it is well known from Larmor\rq s work.
\item A metric warped by particle acceleration within a material medium can have the simple interpretation as being due to local material response to particle motion.
\end{enumerate}
This study is constrained to exploring RR for accelerated charged particle motion within matter only. Our exploration is limited to warping along the particle worldline due to acceleration. Although we use similar mathematical methods as in the case of curved space-time we don't actually claim that the space-time is curved. We expect that path warping method can help advance the generalization of our approach to the case of accelerated charged particle motion in vacuum, which motivates the introduction of this method in this work. However, understanding of space-time geometry warping outside the particle worldline maybe also required. Such a new theoretical framework is beyond our current scope, has not yet ben formulated and is not needed here to advance the understanding of RR we develop.
The path-warped method description of RR avoids the introduction of higher order derivatives into the equations of motion, see discussion on p. 393 in Panofsky-Phillips~\cite{Panofsky:1962}. The Lorentz-Abraham-Dirac (LAD) equation\rq s higher order derivative term was introduced to assure orthogonality of the equation of motion with respect to 4-velocity. This term leads to causality challenges and runaway solutions. There are different interpretations of this term: In some derivations of RR this term is introduced ad-hoc as necessary to assure constancy of the speed of light $u^2 =$ const~\cite{Rohrlich:1965cl,Barut:1980aj}; There is an effort to derive it based on the Lorentz-force of the regularized self-field, see for example the work in Ref.~\cite{Bild:2019dlu} which is further developing Dirac\rq s derivation of LAD~\cite{Dirac:1938nz}, we return to this issue below in Section~\ref{ssec:LADfric}. This controversial term does not appear in our formulation.
In view of these difficulties, Landau-Lifshitz~\cite{LL:1962} proposed an iterative scheme using dynamical equations to eliminate higher derivatives. We show in Sect.~\ref{sec:LL} that for a particle decelerated in the medium the Landau-Lifshitz-like model of radiation friction predicts non-physical behavior for particle motion. This alone demonstrates the need to find another method to incorporate RR into in-medium particle dynamics. The validity for both LAD and LL description is restricted to the classical domain of particle behavior \cite{DiPiazza:2008}.
In Sect.~\ref{sec:modmetr} we show that the \lq pure\rq\ Larmor-RR term proportional to particle 4-velocity can be a natural consequence of a suitable matter warping along the particle path. To achieve this we characterize matter warping by an explicit dependence of the metric on the particle path and its acceleration. This naturally satisfies the requirement discussed by Langevin~\cite{Langevin:1911} that \lq\lq being accelerated\rq\rq\ marks the body in a distinct way in that the magnitude of time dilation depends on the history of acceleration. We note that Langevin\rq s remarks do not depend on the particle moving only in vacuum, they retain in full their meaning for motion in resistive material medium as well. We will return to the more difficult case of motion in vacuum in the follow-up work.
The present reformulation of RR contributes as well to a better understanding of LAD, which has been interpreted as the interaction of the charged particle with its own radiation field~\cite{Bild:2019dlu}. However, both classical and quantum particles do not move within their own Coulomb fields. With this in mind, we posit that such particles should not be allowed to move under the influence of their own radiation fields as well. It is a textbook exercise, see sect.\;29.4 in Ref\;\cite{Rafelski:2017}, to show that a charged accelerated particle in its instantaneous co-moving frame generates both the Coulomb field and the radiative field and there is no relative motion with an observer required to establish this. This is because, unlike velocity, acceleration has an absolute meaning, and only in the instantaneous co-moving frame does the acceleration 4-vector have the pure space-like format $a^\mu=(0,\vec a)$. Use of matter warping naturally prevents the particle from being accelerated by its own radiation field.
In Sect.~\ref{sec:solution} we implement the numerical solution for the equations of motion and compare it to the motion without radiation friction. We show that in the limit of high rapidity and/or mechanical friction strength the radiation friction loss is at most matching the mechanical friction energy loss. We briefly consider application of such a model for high energy particle collisions.
\section{Friction force in medium}\label{sec:covfriction}
In this section we describe motion of a particle under the influence of a covariant friction force in a resistive medium. The form of the force is such that it reduces to the Newtonian friction in the non-relativistic limit. We derive the expressions for stopping distance, rapidity shift and loss of energy and momentum. This force is present for both neutral and charged particles, and the energy loss manifests itself in general as heat dissipation.
\subsection{Covariant equation of motion}
The covariant equation of motion and the friction force are given by
\begin{equation}\label{eq:motion}
\dot{u}^\mu = \frac{1}{m}\mathcal{F}^\mu, \quad \mathcal{F}^\mu = \frac{r}{c}P^\mu_{\ \ \nu}\eta^\nu\,,
\end{equation}
where $r$ is the strength of the friction and $P^\mu_{\ \ \nu}$ is the projector on the orthogonal direction to 4-velocity $u^\mu$ of the particle
\begin{equation}
P^\mu_{\ \ \nu} = \delta^\mu_\nu - \frac{u^\mu u_\nu}{c^2}\,.
\end{equation}
This ensures that our friction force is automatically orthogonal to the 4-velocity. Finally, we denote the 4-velocity of the medium as $\eta^\nu$. For general choice of the 4-velocities
\begin{equation}
\eta^\mu = (\gamma_\text{M}c,\gamma_\text{M} \pmb{v}_\text{M}), \quad u^\mu = (\gamma c, \gamma \pmb{v}),
\end{equation}
\begin{equation}
\eta \cdot u = \gamma_\text{M} \gamma c^2 (1 - \pmb{\beta}_\text{M} \cdot \pmb{\beta}\;,
\end{equation}
the zeroth and spatial components of the equation of the motion \req{motion} read
\begin{align}
\label{eq:friction0th}\gamma \frac{d\gamma}{dt} &= \frac{r}{mc}\gamma_\text{M}(1 - \gamma^2(1-\pmb{\beta}_\text{M}\cdot \pmb{\beta}))\;,\\
\gamma \frac{d\gamma\pmb{\beta}}{dt} &= \frac{r}{mc}\gamma_\text{M}(\pmb{\beta}_\text{M} - \gamma^2(1-\pmb{\beta}_\text{M} \cdot \pmb{\beta})\pmb{\beta})\,.
\end{align}
The energy balance, given by \req{friction0th}, is overall negative when
\begin{equation}
\pmb{\beta}_\text{M} \cdot \pmb{\beta} < \beta^2\,,
\end{equation}
which means the particle loses energy due to friction. In the opposite case the medium is moving faster than the particle and the particle is being accelerated to match the velocity of the medium. When $\pmb{\beta}_\text{M} = \pmb{\beta}$ the particle reaches an equilibrium state of rest with respect to the medium and the friction force completely disappears.
Let\rq s explore further the behavior of the friction force in the rest frame of the medium when $\pmb{\beta}_\text{M} = 0$, $\gamma_\text{M} = 1$ or in 4-vector notation
\begin{equation}
\eta^\mu = (c, 0, 0,0), \quad \eta \cdot u = \gamma c^2\;.
\end{equation}
In this case the components of \req{motion} are
\begin{align}
\label{eq:zerothrfm}\gamma \frac{d\gamma}{dt} &= \frac{r}{mc} (1-\gamma^2)\,,\\
\label{eq:spatialrfm}\frac{d}{dt}(\gamma \pmb{\beta}) &= - \frac{r}{mc} \gamma \pmb{\beta}\,.
\end{align}
We can always orient our coordinate system so that the initial velocity of the particle coincides with one of the coordinate axes. Consider that the particle enters the medium in the $x$-direction. The perpendicular velocity then remains zero for the duration of the particle\rq s travel and the motion is entirely one-dimensional. In terms of rapidity $y$ satisfying
\begin{equation}\label{eq:rapidity}
\gamma = \cosh y, \quad \gamma \beta = \sinh y, \quad \beta = \tanh y\,,
\end{equation}
we can re-write both equations \req{zerothrfm} and \req{spatialrfm} as
\begin{equation}\label{eq:dydtnoRR}
\frac{dy(t)}{dt} = - \frac{r}{mc}\tanh y(t)\,,
\end{equation}
which has a solution for initial rapidity $y(0) = y_0$
\begin{equation}\label{eq:ytnoRR}
y(t) = \text{Arcsinh}\left(\sinh(y_0)\exp\left(-\frac{r}{mc}t \right)\right)\,.
\end{equation}
Note that if the velocity of the particle with respect to the medium is small, $\beta \ll 1$, the equation of motion \req{spatialrfm} becomes
\begin{equation}
m \frac{dv}{dt} = - \frac{r}{c} v \,,
\end{equation}
which is a familiar Newtonian friction force linearly proportional to velocity. In order to account for friction with more complicated behavior than linear dependence on velocity we need to replace the constant $r$ with a function of relative velocity $r(\eta\cdot u)$, which is manifestly a Lorentz scalar. In such case the solution \req{ytnoRR} would have to be replaced by a numerical solution of \req{dydtnoRR} with a specific function $r(y)$. The friction strength $r$ is also in general a function of the medium density.
\subsection{Distance traveled and rapidity shift}
Rewriting the solution $y(t)$ in \req{ytnoRR} as
\begin{equation}
\sinh y(t) = \sinh y_0 \exp\left(-\frac{r}{mc}t\right)\,,
\end{equation}
and using $\sinh y = \gamma \beta$, we can find solution for $\beta(t)$ as
\begin{equation}
\beta(t) = \frac{\gamma_0\beta_0}{\sqrt{\exp\left(\frac{2r}{mc}t\right) + \gamma_0^2\beta_0^2}}\,.
\end{equation}
The distance traveled in the rest frame of the medium is given by the integral
\begin{align}
x(t) =& x_0 + \int_0^t \beta(t')cdt' \nonumber\\[10pt]
=& x_0 + \int_0^t \frac{\gamma_0\beta_0}{\sqrt{\exp\left(\frac{2r}{mc}t'\right) + \gamma_0^2\beta_0^2}}cdt'\,,
\end{align}
which can be evaluated as
\begin{align}
x(t) = &x_0 + \frac{mc^2}{r}\left(\text{Arctanh}(\beta_0) - \text{Arctanh}\left(\beta(t)\right)\right)\nonumber\\[5pt]
\label{eq:xtnoRR}
=& x_0 + \frac{mc^2}{r}(y_0 - y(t))\;.
\end{align}
The total distance traveled $D$ until the particle comes to a stop $y(t_s) = 0$ at time $t_s$ is simply
\begin{equation}\label{eq:stopdist}
D = \left. (x(t) - x_0)\right|_{y(t_s) = 0} = \frac{mc^2}{r}y_0\,.
\end{equation}
By inverting \req{xtnoRR} we can obtain $y$ as a function of $x$
\begin{equation}\label{eq:yx}
y(x) = y_0 - \frac{r}{mc^2}(x - x_0)\,.
\end{equation}
The rapidity shift per change in distance $dy/dx$ is therefore a constant
\begin{equation}\label{eq:stoppowy}
\frac{dy}{dx} = - \frac{r}{mc^2}\,.
\end{equation}
This is the mechanical rapidity shift caused by the medium\rq s resistance. In the case of charged particle motion we also need to account for the additional radiation rapidity shift effect. The description of this contribution is the main focus of Sect.~\ref{sec:LL} and Sect.~\ref{sec:modmetr}.
\subsection{Energy and momentum loss}
We now can consider the stopping power in terms of the change of energy $E = mc^2\gamma$ per unit of distance
\begin{equation}\label{eq:dydx}
\frac{dy}{dx} = \frac{d\text{Arccosh}(\gamma)}{dx} = \frac{1}{\sqrt{\gamma^2-1}}\frac{d\gamma}{dx} = \frac{1}{mc^2\sinh y}\frac{dE}{dx}\,.
\end{equation}
Therefore by substituting \req{dydx} into \req{stoppowy}, we obtain
\begin{equation}\label{eq:dEdx}
\frac{dE}{dx} = - r \sinh y(x)\,.
\end{equation}
Clearly, when rapidity reaches zero in the rest frame of the medium, the particle energy stops changing as expected. Another way to write this expression uses
\begin{equation}
\sinh y = \gamma \beta = \frac{p}{mc}\,,
\end{equation}
where $p = m\gamma v$ is particle\rq s momentum. Then
\begin{equation}
\frac{dE}{dx} = - \frac{r}{mc} p\,,
\end{equation}
and conversely using the relativistic expression for energy $E = \sqrt{m^2c^4 + p^2c^2}$
\begin{equation}
\frac{dp}{dx} = - \frac{r}{mc^3}E\,.
\end{equation}
Finally, energy and momentum can be expressed as
\begin{align}
E = mc^2 \gamma &= mc^2 \cosh y\,,\\
p = mc\gamma\beta &= mc \sinh y\,,
\end{align}
which we can evaluate either as a function of laboratory time or position, using the solutions for rapidity $y(t)$ \req{ytnoRR} and $y(x)$ \req{yx}, respectively.
\section{Radiation friction}\label{sec:LL}
This section introduces: a) the \lq standard model\rq\ of RFF, the Lorentz-Abraham-Dirac (LAD) equations of motion for description of RR; and b) Landau-Lifshitz reduction of LAD differential order. We apply this procedure to our problem of the charged particle moving in a resistive medium. We present Landau-Lifshitz-like (LLL) equations of motion for both Newtonian friction and friction with strength generally dependent on $\eta\cdot u$ and discuss their behavior. We show that the LLL model leads to non-physical behavior for the particle motion.
\subsection{LAD radiation friction in vacuum}\label{ssec:LADfric}
The unresolved question of the consistent description of accelerated charged particle motion including its radiation and radiation friction is now well over a century old. Indeed, the power radiated by such particle was first described by Larmor at the end of the 19th century~\cite{Larmor:1897rad}. In a covariant form
\begin{equation}\label{eq:larmor}
P = m \tau_0 \dot{u}^2\,,
\end{equation}
where the characteristic time $\tau_0$ reads
\begin{equation}
\tau_0 = \frac{2}{3} \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r mc^3} = \frac{2}{3} \frac{\alpha\hbar}{mc^2\varepsilon_r}
\approx 6.26 \times 10^{-24}\text{ s}\,,
\end{equation}
for an electron in vacuum, where $\alpha\approx1/137.036$, $\varepsilon_r = 1$. If we add a corresponding momentum change to \req{motion} the equation of motion reads
\begin{equation}\label{eq:larmorfirst}
\dot{u}^\mu \stackrel{?}{=} \frac{1}{m}\mathcal{F}^\mu + \tau_0 \dot{u}^2 \frac{u^\mu}{c^2}.
\end{equation}
We see that this expression does not preserve $u^2 = c^2$ because $\dot{u} \cdot u \neq 0$.
Further work by Abraham~\cite{Abraham:1903cl}, Dirac~\cite{Dirac:1938nz}, and Lorentz~\cite{Lorentz:1952th} resulted in the formulation of the LAD equation for accelerated charged particle motion
\begin{equation}\label{eq:LAD}
\dot{u}^\mu = \frac{1}{m}\mathcal{F}^\mu + \tau_0 \left(\ddot{u}^\mu + \dot{u}^2 \frac{u^\mu}{c^2}\right)\,,
\end{equation}
where the term proportional to the second derivative of 4-velocity is the so-called Schott term.
We consider this term controversial as briefly outlined in the introduction. In some very well known LAD derivations it is added ad-hoc to ensure $u^2 =$ const~\cite{Rohrlich:1965cl,Barut:1980aj}. However, search to justify this terms presence is offered in derivations based on the Lorentz-force of the regularized self-field~\cite{Bild:2019dlu}. This derivation clarifies that to justify the Schott term one must posit that a particle can experience its own self-force. This cannot be the case if classical dynamics arises in a limiting process of quantum physics, where such a self-force for matter particles (fermions) is not possible (see \cite{Merzbacher:1970}, p.533).
The Shott term is the origin of the theoretically unwelcome second order proper-time derivative of the 4-velocity. This leads to a number of unresolved issues with initial conditions, causality, run-away and pre-accelerated solutions~\cite{Spohn:2004, Poisson:1999}. This field of study remains very active till this day, with recent publications exploring particles of finite extent and their point limit~\cite{Gralla:2009} and improving as noted the methods explored in~\cite{Bild:2019dlu}. Lastly, and of fundamental importance, the Schott term is the critical obstacle in many failed efforts to find a variational principle formulation of RR -- in absence of such a formulation the charged particle dynamics with RR lacks conservation laws required in a complete and consistent description.
Currently there are two main approaches aiming to resolve the issue of the LAD formulation. The first is to impose appropriate boundary and asymptotic conditions on the solution so that the non-physical solutions are discarded~\cite{Plass:1961}. This approach is difficult to implement for problems which require numerical solutions. Instead we will compare our results with a second approach of the Landau-Lifshitz (LL) model~\cite{LL:1962} which approximates the LAD equation by iterating the acceleration due to external force and expanding into powers of the parameter $\tau_0$. This approach reduces the order of the equation of motion and thus resolves the known issues of the LAD formulation at the expense of approximation of the the full radiation reaction.
Spohn~\cite{Spohn:2000} showed that LAD restricted onto a physical manifold produces the LL series, so both approaches lead in certain environment to the same dynamics. However, this is not the case in the study of electron stopping by a frontal light plane wave, see Ref.\cite{Hadad:2010mt}. This occurs because a traveling light wave front creates a quasi-material edge for an incoming particle. This resembles, but is not exactly the same as, the case of the material medium we look at next.
\subsection{Landau-Lifshitz-like RR in medium}
\subsubsection{Constant material friction}
For our system in the zeroth order in $\tau_0$, the acceleration is given by the external force
\begin{equation}
\dot{u}^\mu_{(0)} = \frac{r}{mc} P^\mu_{\ \ \nu}\eta^\nu\,.
\end{equation}
By substituting this expression to the radiation friction term in \req{LAD} we obtain for the second derivative of 4-velocity
\begin{multline}
\ddot{u}^\mu_{(0)} = \frac{r}{mc}\dot{P}^\mu_{\ \ \nu}\eta^\nu = \frac{r}{mc}\left(-\frac{\dot{u}^\mu_{(0)} (\eta \cdot u)}{c^2} - \frac{u^\mu (\dot{u}_{(0)} \cdot \eta)}{c^2}\right)\\
= - \frac{r^2}{m^2c^4}\left(P^\mu_{\ \ \nu}\eta^\nu (\eta \cdot u) + u^\mu(\eta \cdot P \cdot \eta)\right)\,,
\end{multline}
and for the square of acceleration in the zeroth order in $\tau_0$
\begin{equation}
\dot{u}^2_{(0)} = \frac{r^2}{m^2c^2}(\eta \cdot P \cdot P \cdot \eta) = \frac{r^2}{m^2c^2}(\eta \cdot P \cdot \eta)\,,
\end{equation}
because of the property of the projector $P^2 = P$. We see that the Larmor term cancels with one of the two terms arising from the Schott term and the final Landau-Lifsthitz-like (LLL) equation of motion reads
\begin{equation}
\dot{u}_{(1)}^\mu = \frac{r}{mc} P^\mu_{\ \ \nu}\eta^\nu - \tau_0 \frac{r^2}{m^2c^4} (\eta \cdot u) P^\mu_{\ \ \nu}\eta^\nu\,.
\end{equation}
The zeroth component of this equation in the rest frame of the medium is
\begin{equation}
\gamma \frac{d\gamma}{dt} = \frac{r}{mc}(1-\gamma^2)-\tau_0 \frac{r^2}{m^2c^2}\gamma(1-\gamma^2)\,,
\end{equation}
and in terms of rapidity \req{rapidity} we have
\begin{equation}\label{eq:dydtLL}
\frac{dy}{dt} = - \frac{r}{mc}\tanh y + \tau_0 \frac{r^2}{m^2c^2}\sinh y\,.
\end{equation}
In the second term of \req{dydtLL} we see a reversal in the sign. The effect of radiation friction is then, up to the first power in $\tau_0$, to increase the energy of the particle experiencing deceleration in a medium. Moreover, the rapidity has to satisfy
\begin{equation}\label{eq:LLcondition}
y < \text{arccosh}\left(\frac{mc}{\tau_0 r} \right)\,,
\end{equation}
otherwise the radiation friction overpowers the mechanical friction. Teitelboim et al. \cite{Teitelboim:1980} argue that both, LAD equation and its LLL reduction are unjustified when the radiation friction force is comparable to the driving external force. Violation of this condition in the medium results in a runaway solution which is clearly an unacceptable behavior. As the Lorentz force is not relevant in the material friction case, the incompatibility must originate more fundamentally with the LAD extension. \\
\subsubsection{Variable material friction in LLL approach}
The derivation above assumes that the radiation friction $r$ is constant. If we evaluate LLL model for non-constant $r$ as a function of $\eta\cdot u$, then the covariant equation of motion up to first order in $\tau_0$ is
\begin{multline}
\dot{u}^\mu_{(1)} = \frac{r}{mc} P^\mu_{\ \ \nu}\eta^\nu +\\
+ \tau_0 \frac{r}{m^2c^2}\left(-r \frac{\eta \cdot u}{c^2} + \frac{dr}{d(\eta \cdot u)}(\eta \cdot P \cdot \eta)\right) P^\mu_{\ \ \nu}\eta^\nu\,,
\end{multline}
and the zeroth component in the rest frame of the medium is
\begin{equation}
\gamma \frac{d\gamma}{dt} = \frac{r}{mc}(1-\gamma^2) + \tau_0 \frac{r}{m^2c^2}\left(-r\gamma + \frac{dr}{d\gamma}(1-\gamma^2)\right)(1-\gamma^2)\,.
\end{equation}
Such LLL friction term has a chance of having negative contribution to energy if
\begin{equation}\label{eq:inequality}
\frac{dr}{d\gamma} < - \frac{r\gamma}{\gamma^2 -1}\,.
\end{equation}
This cannot happen in the non-relativistic limit, because $dr/d\gamma$ would have to go to minus infinity. The terms exactly cancel when
\begin{equation}\label{eq:zeroRR}
r \propto \frac{1}{\sqrt{\gamma^2 -1}} = \frac{1}{\gamma\beta} = \frac{mc}{p}\,.
\end{equation}
In such a case there is no radiation friction according to the LLL approach. We introduce this example of mechanical friction with the friction coefficient $r$ depending inversely on momentum to present the singular case when radiation friction force disappears completely. For more realistic models of mechanical friction, when the coefficient $r$ grows with momentum, the inequality in \req{inequality} shows that the LLL terms add energy to the system. We conclude that the conventional LAD radiation reaction combined with LLL reduction of the order of differentiation produces an unacceptable description of radiation friction in a material medium, even when the friction strength is an arbitrary function of relative velocity.
\section{Matter warping}\label{sec:modmetr}
Here we propose an alternative radiation friction model for the case of motion in matter medium. We show that formally we can introduce radiation friction in medium through matter warping while keeping the form of the covariant Larmor formula. This allows us to formulate the dynamics without higher order derivatives and with self-consistent formula for the magnitude of acceleration. As the driving force we take the covariant mechanical friction force. We establish equations of motion for such a system in the warped matter model and evaluate stopping power.
\subsection{General considerations}
As already noted in the introduction, our warped path formulation is not requiring exploration of space-time beyond the particle path: We start with the equation of motion with only the Larmor term present
\begin{equation}\label{eq:curvedlarmor}
\dot{u}^\mu = \frac{1}{m}\mathcal{F}^\mu + \tau_0 \dot{u}^2 \frac{u^\mu}{c^2}\;.
\end{equation}
This specific form of the equation of motion is our choice of collective medium response model to accelerated charged particle motion, guided by the visual similarity with \req{larmorfirst}, mathematical tractability, and interpretability of the results {\it i.e.} mathematical simplicity and beauty. Instead of adding a second order derivative Schott term we assume path-warped metric allowing us to impose the condition
\begin{equation}\label{eq:u2=c2}
u^2 \equiv g_{\mu\nu}u^\mu u^\nu = c^2
\end{equation}
i.e. the 4-force remains orthogonal to 4-velocity. To see this we multiply \req{curvedlarmor} by $g_{\mu\nu}u^\nu$ to obtain
\begin{equation}\label{eq:larmordotuu}
\dot{u} \cdot u = \tau_0 \dot{u}^2\,.
\end{equation}
If we use this identity in \req{curvedlarmor} we derive an expression which is explicitly orthogonal to $u_\mu$
\begin{equation}\label{eq:motion_modmetr}
\dot{u}^\mu - (\dot{u} \cdot u) \frac{u^\mu}{c^2} = \frac{1}{m}\mathcal{F}^\mu\,,
\end{equation}
suggesting that the covariant friction is proportional to the product of 4-velocity and 4-acceleration $\dot{u} \cdot u$, which is normally zero. Upon differentiating the condition $u^2 = c^2$ \req{u2=c2} with respect to proper time we obtain
\begin{equation}\label{eq:dotuu}
\dot{u} \cdot u = - \frac{1}{2} \frac{dg_{\mu\nu}}{d\tau} u^\mu u^\nu\,,
\end{equation}
which is a condition for components of the metric. \req{curvedlarmor}--\req{dotuu} provide a consistent characterization of the magnitude of acceleration $\dot{u}^2$. The square of this expression reads
\begin{equation}
\dot{u}^2 = \frac{1}{m^2}\mathcal{F}^2 + \tau_0^2 \frac{\dot{u}^4}{c^2}\;,
\end{equation}
which is a quadratic equation for $\dot{u}^2$ with solutions
\begin{equation}
\dot{u}^2 = \frac{c^2}{2\tau_0^2}\left(1 \pm \sqrt{1-4\frac{\tau_0^2}{c^2}\frac{\mathcal{F}^2}{m^2}}\right)\,.
\end{equation}
We take the minus sign as the physical solution, because it reduces in the limit $\tau_0 \rightarrow 0$ to the usual expression $\dot{u}^2 = \mathcal{F}^2/m^2$. This expression can be further simplified to
\begin{equation}\label{eq:dotu2}
\dot{u}^2 = \frac{2\mathcal{F}^2/m^2}{1+\sqrt{1-4\frac{\tau_0^2}{c^2}\frac{\mathcal{F}^2}{m^2}}}\,.
\end{equation}
It is worth noting that as $\mathcal{F}^2 \rightarrow 0$ then also $\dot{u}^2 \rightarrow 0$ and conversely as $\mathcal{F}^2 \rightarrow -\infty$ the growth of $\dot{u}^2$ is damped.
\subsection{Specific warped matter model}
In order to avoid mixing the spatial and time components we will assume that the metric is diagonal and in our 1D situation we choose a parametrization
\begin{equation}
g_{\mu\nu} = \text{diag}(f_0^2,-f^2,-1,-1)\,.
\end{equation}
In the following we will suppress the two trivial degrees of freedom. The proper time of the particle is by definition
\begin{equation}
d\tau = \frac{1}{c}\sqrt{g_{\mu\nu}\frac{dx^\mu}{dt}\frac{dx^\nu}{dt}} dt= \sqrt{f_0^2 - \frac{f^2}{c^2}\left(\frac{dx}{dt}\right)^2}dt\,,
\end{equation}
where we took the position 4-vector as $x^\mu = (ct,x)$. We can then perform a coordinate transformation from $dt$ and $dx$ to measurable quantities
\begin{equation}\label{eq:transformation}
dt_\text{lab} = f_0 dt, \quad dx_\text{lab} = f dx\;,
\end{equation}
which simplifies the increment of proper time to
\begin{equation}
d\tau = \sqrt{1-\frac{1}{c^2}\left(\frac{dx_\text{lab}}{dt_\text{lab}}\right)^2}dt_\text{lab}\,.
\end{equation}
If we define the true physical velocity of the particle as
\begin{equation}
v \equiv \frac{dx_\text{lab}}{dt_\text{lab}} = \frac{f}{f_0}\frac{dx}{dt}\,,
\end{equation}
we can write the gamma factor in the usual form
\begin{equation}
\gamma = \frac{dt_\text{lab}}{d\tau} = \frac{1}{\sqrt{1-v^2/c^2}}\,.
\end{equation}
Note that the transformation \req{transformation} is a transformation to flat space coordinates, as can be seen by evaluating
\begin{equation}
ds^2 = g_{\mu\nu}dx^\mu dx^\nu = c^2dt_\text{lab}^2 - dx_\text{lab}^2\,.
\end{equation}
The coordinate 4-velocity $u^\mu$ that enters our equation of motion is given by
\begin{equation}\label{eq:umu}
u^\mu \equiv \frac{dx^\mu}{d\tau} = \gamma \left(\frac{dct}{dt_\text{lab}}, \frac{dx}{dt_\text{lab}} \right) = \gamma \left(\frac{c}{f_0}, \frac{v}{f}\right)\;.
\end{equation}
Here $u^\mu$ is the quantity in terms of which the equation of motion is formulated, thus it is tempting to look at it as the actual 4-velocity. However, a more robust theoretical framework is needed to give $u^\mu$ a physical meaning and this will be required for the solution of the vacuum case. This expression satisfies $u^2 = c^2$ \req{u2=c2} as expected and energy of the particle is given by the usual expression
\begin{equation}
E = \gamma m c^2\,.
\end{equation}
For the 4-velocity of the medium we can write analogically
\begin{equation}
\eta^\mu = \gamma_\text{M} \left(\frac{c}{f_0},\frac{v_\text{M}}{f}\right)\,,
\end{equation}
preserving $\eta^2 = c^2$. Therefore the RHS of the equation of motion \req{motion_modmetr} in the rest frame of the medium reads
\begin{align}
\frac{1}{m}\mathcal{F}^\mu = \frac{r}{mc}P^\mu_{\ \ \nu}\eta^\nu =& \frac{r}{mc}\left(\eta^\mu - \frac{u^\mu}{c^2}(\eta \cdot u)\right) \nonumber\\
\label{eq:force}
=& \frac{r}{mc}\left(\frac{c}{f_0}(1-\gamma^2), - \frac{\gamma^2}{f}v\right)\,,
\end{align}
which is equivalent to rescaling the zeroth component of the force by $f_0$ and spatial component by $f$. Note that $r$ can be in general a function of $\eta \cdot u$ to account for more complicated mechanical friction than Newtonian friction, but this fact does not modify our derivation and holds true throughout the current section (Sec.~\ref{sec:modmetr}). Finally, we can evaluate the dot product $\dot{u}\cdot u$ using \req{dotuu}
\begin{equation}\label{eq:dotuueval}
\dot{u}\cdot u = -\frac{1}{2}\frac{df_0^2}{d\tau}\frac{\gamma^2c^2}{f_0^2} + \frac{1}{2}\frac{df^2}{d\tau}\frac{\gamma^2v^2}{f^2} \equiv -A \gamma^2 c^2 + B \gamma^2 v^2\,,
\end{equation}
where we denoted
\begin{equation}
A \equiv \frac{d}{d\tau}\ln f_0, \quad B \equiv \frac{d}{d\tau} \ln f\,.
\end{equation}
Now we are prepared to establish the equations of motion.
\subsection{Radiation energy loss in our model}
If we substitute the 4-velocity \req{umu}, the force \req{force}, and the dot product \req{dotuueval} to the equation of motion \req{motion_modmetr} we obtain for the zeroth component
\begin{equation}\label{eq:zeroth1}
\frac{d}{d\tau}\left(\frac{\gamma}{f_0}\right) + (A\gamma^2c^2-B\gamma^2v^2)\frac{\gamma}{f_0 c^2} = \frac{r}{mcf_0}(1-\gamma^2)\,.
\end{equation}
The first term can be further expanded
\begin{equation}
\frac{d}{d\tau}\left(\frac{\gamma}{f_0}\right) = \frac{\gamma}{f_0}\frac{d\gamma}{dt_\text{lab}} - \frac{1}{f_0^2}\gamma \frac{df_0}{d\tau} = \frac{\gamma}{f_0}\frac{d\gamma}{dt_\text{lab}} - \frac{\gamma}{f_0} A\,.
\end{equation}
Finally, by substituting back to \req{zeroth1} and multiplying by $f_0/\gamma$ we have an expression for the change in gamma factor
\begin{equation}\label{eq:zeroth2}
\frac{d\gamma}{dt_\text{lab}} = \frac{r}{mc\gamma}(1-\gamma^2) + A(1-\gamma^2) + B\gamma^2\beta^2\,.
\end{equation}
Similarly for the spatial component of \req{motion_modmetr}
\begin{equation}\label{eq:spatial1}
\frac{d}{d\tau}\left(\frac{\gamma v}{f}\right) + (A\gamma^2c^2-B\gamma^2v^2)\frac{\gamma v}{f c^2} = - \frac{r\gamma^2}{mf}v\,.
\end{equation}
For the first term in \req{spatial1} we evaluate the derivative
\begin{align}
\frac{d}{d\tau}\left(\frac{\gamma v}{f}\right) =& \gamma \frac{d\gamma}{dt_\text{lab}} \frac{v}{f} - \frac{1}{f^2} \frac{df}{d\tau} \gamma v + \frac{\gamma^2}{f}\frac{dv}{dt_\text{lab}}\nonumber\\
=& \gamma \frac{d\gamma}{dt_\text{lab}}\frac{v}{f} - \frac{\gamma}{f}B v + \frac{\gamma^2}{f}\frac{dv}{dt_\text{lab}}\,.
\end{align}
If we use \req{zeroth2} for change in $\gamma$ and substitute back to \req{spatial1} then after several cancellations we obtain
\begin{equation}
\frac{\gamma^2}{f}\frac{dv}{dt_\text{lab}} = - \frac{r}{mc}\frac{v}{f} - (A-B)\frac{\gamma v}{f}\;,
\end{equation}
which if we multiply by $f/\gamma^2 c$ becomes
\begin{equation}
\frac{d\beta}{dt_\text{lab}} = - \frac{r}{mc}\frac{\beta}{\gamma^2} - (A-B)\frac{\beta}{\gamma}\,.
\end{equation}
We can use identity $1 + \gamma^2\beta^2 = \gamma^2$ to further simplify the zeroth part \req{zeroth2} and write the components of the covariant equation of motion in a final form
\begin{align}
\label{eq:zerothfinal}\frac{d\gamma}{dt_\text{lab}} &= \frac{r}{mc\gamma}(1-\gamma^2) + (A-B)(1-\gamma^2)\,,\\
\label{eq:spatialfinal}\frac{d\beta}{dt_\text{lab}} &= - \frac{r}{mc}\frac{\beta}{\gamma^2} - (A-B)\frac{\beta}{\gamma}\,,
\end{align}
which are mutually equivalent. Although the metric $g_{\mu\nu}$ is specified by two unknowns, $A$ and $B$, the dynamics of the particle motion depends only on their difference
\begin{equation}
A - B = \frac{d}{d\tau}\ln \frac{f_0}{f}\,,
\end{equation}
which means that any arbitrary factor re-scaling the whole metric does not change the motion. Therefore we can set either $A$ or $B$ to zero and evaluate the other without any loss of generality.
\subsection{Stopping power and limiting cases}
We assume that the more general equation of motion \req{motion_modmetr} is in the form of \req{curvedlarmor} where the friction term is given by the Larmor formula. Combining expressions for $\dot{u} \cdot u$ in \req{larmordotuu} and \req{dotuu} we can equate
\begin{equation}
\tau_0 \dot{u}^2 = - \frac{1}{2} \frac{dg_{\mu\nu}}{d\tau} u^\mu u^\nu\,,
\end{equation}
where the self-consistent magnitude of acceleration $\dot{u}^2$ was evaluated in \req{dotu2} and the right hand side is given in our metric as \req{dotuueval}
\begin{equation}
\frac{2\tau_0\mathcal{F}^2/m^2}{1+\sqrt{1-4\frac{\tau_0^2}{c^2}\frac{\mathcal{F}^2}{m^2}}} = -A \gamma^2 c^2 + B \gamma^2 v^2\,.
\end{equation}
The square of the external force can be computed using equation \req{force}
\begin{equation}
\frac{1}{m^2}\mathcal{F}^2 =\frac{1}{m^2}g_{\mu\nu}\mathcal{F}^\mu\mathcal{F}^\nu = \frac{r^2}{m^2}(1-\gamma^2)\,.
\end{equation}
Notice that this expression does not depend on the metric and is equal to the square of the external force in the flat space-time. As discussed above we can set $A = 0$ and evaluate $B$
\begin{equation}
B = \frac{-2\tau_0 \frac{r^2}{m^2c^2}}{1 + \sqrt{1+4\frac{\tau_0^2}{c^2}\frac{r^2}{m^2}\gamma^2\beta^2}}\,,
\end{equation}
where we used the identity $1-\gamma^2 = -\gamma^2\beta^2$. Using this solution in the equations of motion \req{zerothfinal} and \req{spatialfinal} yields
\begin{align}
\label{eq:zerothspecific}
\frac{d\gamma}{dt_\text{lab}} &= \frac{r}{mc\gamma}(1-\gamma^2) + \frac{2\tau_0 \frac{r^2}{m^2c^2}(1-\gamma^2)}{1 + \sqrt{1+4\frac{\tau_0^2}{c^2}\frac{r^2}{m^2}\gamma^2\beta^2}}\;,\\
\label{eq:spatialspecific}
\frac{d\beta}{dt_\text{lab}} &= - \frac{r}{mc}\frac{\beta}{\gamma^2} - \frac{2\tau_0 \frac{r^2}{m^2c^2}}{1 + \sqrt{1+4\frac{\tau_0^2}{c^2}\frac{r^2}{m^2}\gamma^2\beta^2}} \frac{\beta}{\gamma}\,.
\end{align}
From \req{zerothspecific} we can calculate the radiation energy loss in powers of $\tau_0$
\begin{equation}
\left. \frac{dE}{dt_\text{lab}}\right|_\text{RF} = mc^2 \left. \frac{d\gamma}{dt_\text{lab}}\right|_\text{RF} = \tau_0 \frac{r^2}{m}(1-\gamma^2) + O(\tau_0^3)\;,
\end{equation}
which matches the covariant Larmor energy loss formula \req{larmor} with 4-acceleration given purely by the external force.
In terms of stopping power $dE/dx$ we can convert \req{zerothspecific} to
\begin{equation}
\frac{dE}{dx_\text{lab}} = \left. \frac{dE}{dx_\text{lab}}\right|_\text{M} - \frac{\frac{2\tau_0}{mc} \left(\left.\frac{dE}{dx_\text{lab}}\right|_\text{M}\right)^2\coth y}{1 + \sqrt{1 + 4 \frac{\tau_0^2}{m^2c^2}\left(\left.\frac{dE}{dx_\text{lab}}\right|_\text{M}\right)^2}}\,,
\end{equation}
where the $dE/dx_\text{lab}|_\text{M}$ is the stopping power caused by the medium friction given by \req{dEdx}. The two important limiting cases are determined by critical mechanical stopping power
\begin{equation}\label{eq:criticalstoppow}
\left. \frac{dE}{dx_\text{lab}}\right|_\text{crit} \equiv \frac{mc^2}{c\tau_0} = \frac{ 3}{ 2}\; \frac{(mc^2)^2}{\alpha \hbar c} \approx 0.27 \varepsilon_r \text{ MeV/fm}\,,
\end{equation}
where the value given is for an electron in an environment with relative permittivity $\varepsilon_r$.
If mechanical stopping power is much higher than the critical stopping power the radiation friction part of the stopping power is approximately
\begin{equation}\label{eq:supercritical}
\left. \frac{dE}{dx_\text{lab}}\right|_\text{RF} \approx \left. \frac{dE}{dx_\text{lab}}\right|_\text{M} \coth y\,.
\end{equation}
In the opposite case, when mechanical stopping power is much less than the critical stopping power,
\begin{equation}
\left. \frac{dE}{dx_\text{lab}}\right|_\text{RF} \approx \left. \frac{dE}{dx_\text{lab}}\right|_\text{M} \left(\frac{\left.\frac{dE}{dx_\text{lab}}\right|_\text{M}}{\left.\frac{dE}{dx_\text{lab}}\right|_\text{crit}}\right) \coth y\,.\
\end{equation}
Note that if $y \rightarrow 0$ then the stopping power in medium also goes to zero as $\sinh y$, so the expression is well behaved.
\section{Motion examples}\label{sec:solution}
With the dynamics developed in the previous section (Sect.~\ref{sec:modmetr}) we can evaluate the motion of the radiating charged particles and compare to the LLL model (Sect.~\ref{sec:LL}) and to the motion without any radiation friction (Sect.~\ref{sec:covfriction}). This section presents numerical solutions for the particle motion in each of the three situations with the underlying Newtonian mechanical friction force. We show that our model unlike the LLL model increases, as expected, the energy loss due to radiation friction. This additional energy loss can at most match the mechanical friction loss in the medium in the limit of high rapidity or friction strength. Finally, we discuss possible experimental applications of our model for high energy particle collisions.
\subsection{Solution of dynamical equations}
\begin{figure}
\centerline{\includegraphics[width=1.08\linewidth]{y_t_log.pdf}}
\caption{\label{fig:y_t} Rapidity as a function of time for our model, LLL model and analytical solution without radiation friction. Initial condition $y(0) = 10^{-3}$ and $\tilde{r} =10^{-3}$.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=1.1\linewidth]{deltadydx.pdf}}
\caption{\label{fig:delta} Relative change of the stopping power (see \req{dydxwithRR}) if we consider radiation friction for selected friction strengths $\tilde{r}$ and range of rapidities $y$.}
\end{figure}
Let us define a unitless friction strength
\begin{equation}
\tilde{r} \equiv \frac{r\tau_0}{mc}=\frac{r}{\left. {dE}/{dx_\text{lab}}\right|_\text{crit}} \,,
\end{equation}
then we can write the equation of motion \req{spatialspecific} as
\begin{equation}
\frac{d\beta}{dt_\text{lab}/\tau_0} = - \tilde{r} \frac{\beta}{\gamma^2} - \frac{2\tilde{r}^2}{1+\sqrt{1+4\gamma^2\beta^2\tilde{r}^2}}\frac{\beta}{\gamma}\,.
\end{equation}
Switching to rapidity \req{rapidity} we obtain
\begin{equation}\label{eq:dydt}
\frac{dy}{dt_\text{lab}/\tau_0} = - \tilde{r}\tanh y - \frac{2\tilde{r}^2}{1+\sqrt{1+4\tilde{r}^2\sinh^2 y}} \sinh y.
\end{equation}
For comparison the LLL model \req{dydtLL} gives us an equation for rapidity
\begin{equation}
\frac{dy}{dt_\text{lab}/\tau_0} = - \tilde{r}\tanh y + 2\tilde{r}^2\sinh y.
\end{equation}
These expressions are suitable for numerical analysis. Let us consider motion with initial rapidity $y_0 = 7$ and $\tilde{r} = 10^{-3}$ to demonstrate the character of the solution. Figure \ref{fig:y_t} shows rapidity as a function of time for both our model and the LLL model as propagated by the RK4 integration scheme. Note that the condition \req{LLcondition} is satisfied to prevent runaway solutions in the LLL model. The dashed line indicates the analytical solution without any radiation friction \req{ytnoRR}. We see that in the LLL model, the particle decelerates more slowly than without radiation friction and in our model faster. The behavior of our model should match our intuition as the \lq correct\rq\ physical behavior where both sources of friction impede the particle\rq s motion.
Stopping power can be evaluated by diving the whole expression \req{dydt} with $\beta = \tanh y$ because $dx_\text{lab} = \beta c dt_\text{lab}$
\begin{align}
\frac{dy}{dx_\text{lab}/c\tau_0} =& -\tilde{r}\left(1 + \frac{2\tilde{r}\cosh y}{1+\sqrt{1+4\tilde{r}^2\sinh^2 y}} \right)\nonumber \\[5pt]
\label{eq:dydxwithRR}
\equiv& \tilde{r}(1 + \Delta)\,,
\end{align}
where distance is measured in units of $c\tau_0$ and $\Delta$ is the relative change from the case without any radiation friction. Fig. \ref{fig:delta} shows possible values for $\Delta$ for selected unitless friction strengths $\tilde{r}$ and range of rapidities $y$. We see that the radiation friction at most doubles the stopping power $dy/dx_\text{lab}$.
With our choice of parameters the stopping distance $D$ is in the case without radiation friction given by \req{stopdist}, which in unitless quantities reads
\begin{equation}
\frac{D}{c\tau_0} = \frac{y_0}{\tilde{r}} = 7 \times 10^3\,.
\end{equation}
As can be seen from the trajectories in Fig. \ref{fig:x_t} with radiation friction present, the particle in our model stops in a significantly shorter distance and for the LLL model it travels further. Figure~\ref{fig:deltaspec} demonstrates that initially the stopping power is doubled for high rapidities and as particle slows down the radiation friction contributes less and less.
\begin{figure}
\centerline{\includegraphics[width=1.05\linewidth]{x_t.pdf}}
\caption{\label{fig:x_t} Distance traveled by the particle as a function of time for initial rapidity $y_0 = 7$ and friction strength $\tilde{r} = 10^{-3}$. We show results with and without radiation friction and for the LLL model. Dotted line marks the stopping distance without radiation friction.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=1.1\linewidth]{delta.pdf}}
\caption{\label{fig:deltaspec} Relative change of the stopping power (see equation \req{dydxwithRR}) as a function of distance for initial rapidity $y_0 = 10$ and friction strength $\tilde{r} = 10^{-3}$.}
\end{figure}
\subsection{Experimental verification}
From the expression for the critical stopping power \req{criticalstoppow} we see that a very high energy loss is needed to reach significant radiation friction. However, the typical stopping power of proton beams in a material medium is on the order of $1-100$ MeV/cm~\cite{Berger:1999} and this applies also to many other particles. This value is many orders of magnitude too small to induce sizable radiation energy production. This value is of course dependent on the density of the medium and varies with the energy of the particle. But even at the high end of the range, such stopping power is only a $10^{-11}$ fraction of the critical stopping power.
We conclude that in normal materials with atomic structure, RR induced by the material stopping power is negligible, hence RR induced by MFF is negligible too. However, microscopically the particle motion is also experiencing Coulomb scattering off atomic nuclei of the medium with high accelerations, which at each scattering event contribute to the bremsstrahlung~\cite{Seltzer:1985}. In this work we do not consider these microscopic processes as we do not want to deal with EM forces presently.
Our RR effect is highly relevant in the experimental environment of quark-gluon plasma, particularly in application to parton jet quenching processes. In the case of fast moving quarks the continuous medium covariant friction model is justified because quark-gluon plasma is successfully modeled as a fluid, beginning with the seminal work of Bjorken~\cite{Bjorken:1982qr}.
The value for the critical stopping power \req{criticalstoppow} for up- and down-quarks is
\begin{align}
\left. \frac{dE}{dx_\text{lab}}\right|_\text{crit} (u) &= 11.4 \varepsilon_r \text{ MeV/fm},\\
\left. \frac{dE}{dx_\text{lab}}\right|_\text{crit} (d) &= 204 \varepsilon_r \text{ MeV/fm},
\end{align}
where we used for the mass of the up-quark $m_u = 2.2$ MeV/$c^2$, and of the down-quark $m_d = 4.7$ MeV/$c^2$, respectively. Since the up-quark has a higher fractional charge and a lower mass, the required critical mechanical stopping power is significantly lower when compared to the down-quark. Electrically charged quarks approaching critical mechanical friction would then emit significant amount of (soft) electromagnetic radiation. In addition there is the possibility of superluminal Cherenkov radiation, and a further small contribution of acoustic wave production.
According to work of Baier et.al~\cite{Baier:2000} collisional stopping power of light quarks in a quark-gluon plasma at $T = 0.25$ GeV is on the order of $200$--$300$\,MeV/fm. Additional contributions arise from gluon-emission friction and other strong field effects. Certainly, the up-quark mechanical friction is supercritical leading to a significant EM radiation energy emission according to \req{supercritical}. A full study of EM emissivity by quark jets would require incorporation of electrical permittivity of quark-gluon plasma and is well beyond our current scope. Excess soft photon emission in QGP has already attracted attention of the heavy-ion community. For a full discussion of relevant work see Section 3.3 in Ref.~\cite{Adamova:2019vkf}, most recent experimental results are found in Ref.\cite{Belogianni:2002}. Further effort to explore this effect in the context of our theoretical framework is warranted.
\section{Future work and conclusions}
In the forthcoming work we hope to investigate other, less material environments. A study of EM interaction has, as we have mentioned, the challenge of reconciling the form of the Lorentz force with that of Maxwell\rq s equations. Therefore a training non-material problem in study of RR is an exploration of charged particle dynamics in presence of an external scalar force field. In this case the field dynamics providing the Larmor RR term is decoupled from the force form, which is non-material and thus reaching beyond the case considered in this work. We plan to return to this example after a short delay.
A further forthcoming training exercise is the study of RR for particles under influence of an externally prescribed constant electromagnetic field. There is in particular a very good reason to take a second look at the constant field case: The LAD or LL format of RR may not correctly describe the physical reality of a constant EM field. For example, a well known prediction of the LL model is that a particle linearly accelerated by an electric field (so-called hyperbolic motion) does not feel any radiation friction and yet produces radiation. This is due to the contribution of the Schott term in the equation of motion, which in this special case exactly balances out the Larmor term, a situation that is subject to ongoing discussion~\cite{Eriksen:2000,Eriksen:2002}.
Even though a novel RR force patterned after this work requires establishment of consistency between the Maxwell field equations and the equation governing the particle motion we are optimistic that our proposed new ideas, the warped path approach, may succeed. What encourages us to pursue constant fields is that for an observer, for whom the energy-momentum tensor $T^\mu_{\ \ \nu}$ of the constant external electromagnetic field is diagonal, we obtain the metric
\begin{equation}
g_{\mu\nu}(\tau) = \eta_{\mu\alpha}\exp\left(2\frac{\tau_0 e^2}{m^2}T^\alpha_{\ \ \nu}\tau \right)\,,
\end{equation}
which exactly reproduces to the first order in $\tau_0$ the Landau-Lifshitz format of the equations of motion.
Another particularly interesting RR study involves the motion of particles in plane wave fields. The well studied case of electron interaction with a light wave edge is here of particular interest~\cite{Hadad:2010mt}. This case was explored using the LL approach, and critical acceleration effects were demonstrated. A self-consistent warped path-metric formulation of RR could lead to directly verifiable experimental outcomes using present day experimental pulsed laser facilities.
Another possible extension we would like to pursue is the development of a variational principle for RR force using the here proposed warped path model. Unlike LAD or LL models where a variational principle was never established, the warped path formulation has a better chance of arising from a specific covariant action since the resultant equations of motion do not contain higher order derivative terms.
{\bf To conclude:}
We have shown that it is possible to describe mechanically decelerated particle energy loss due to radiation friction without introducing the Schott force term and instead we proposed warped matter modification along the particle's path. Our approach resolves well known contradictions: For example, the Landau-Lifshitz-like procedure predicts that particles would gain energy, presumably due to interaction with its own radiation field~\cite{Bild:2019dlu}. The warped path approach does not introduce such an interaction and the total energy loss remains consistent with the Larmor radiation energy loss formula. Therefore a conceptual advantage of our proposed reformulation of RR is that we do not need to provide an interpretation of the causality difficulties created by the contorted derivation and implementation of LAD. There is no self-acceleration without external force possible in our approach.
We have shown that when solved consistently, radiative fields are given by particle acceleration due to both external force and radiation friction. The prediction of such a self-consistent calculation is that the radiation friction at most doubles the \lq mechanical\rq\ energy loss, see Figure \ref{fig:delta}. This result is intuitively and theoretically satisfactory and it can have some interesting experimental consequences awaiting study in parton jet quenching processes in quark-gluon plasma.
\acknowledgements{Martin Formanek and Johann Rafelski would like to express their gratitude to Dr. Tam\'as Bir\'o and Dr. P\'eter L\'evai for their hospitality during the summer 2019 at the Wigner Research Centre for Physics, Budapest and 2019 Balaton Workshop when part of this work was conducted. Johann Rafelski was a Fulbright Fellow during this period. |
2007.15821 | \section{Introduction}
A tensor is a multi-dimensional array that can effectively capture the complex multidimensional features. A Boolean tensor is a tensor that assumes binary values endowed with the Boolean algebra. Boolean tensor has been widely adopted in many fields, including dynamic networks, knowledge graphs, recommendation system, spatial-temporal data etc \cite{wan2019mebf,rukat2018probabilistic,hore2016tensor,zhou2013tensor,bi2018multilayer}. Tensor decomposition is a powerful tool in extracting meaningful latent structures in the data, for which the popular CANDECOMP/PARAFAC (CP) decomposition is a generalization of the matrix singular value decomposition to tensor \cite{carroll1970analysis}. However, these algorithms are not directly usable for Boolean tensors. In this study, we focus on Boolean tensor decomposition (BTD) under similar framework to the CP decomposition.
As illustrated in Figure \ref{fig:BTD}, BTD factorizes a binary tensor \(\mathscr{X}\) as the Boolean sum of multiple rank 1 tensors. In cases when the error distribution of the data is hard to model, BTD applied to binarized data can retrieve more desirable patterns with better interpretation than regular tensor decomposition \cite{miettinen2009matrix,rukat2017bayesian}. This is probably due to the robustness of logic representation of BTD. BTD is an NP-hard problem \cite{miettinen2009matrix}. Existing BTD methods suffers from low efficiency due to high space/time complexity,
and particularly, most BTD algorithms adopted a least square updating approach with substantially high computational cost~\cite{miettinen2010sparse,park2017fast}. This has hindered their application to neither large scale datasets, such as social network or genomics data, or tensors of high-order.
We proposed an efficient BTD algorithm motivated by the geometric underpinning of rank-1 tensor bases, namely GETF (Geometric Expansion for all-order Tensor Factorization). To the best of our knowledge, GETF is the first algorithm that can efficiently deal with all-order Boolean tensor decomposition with an \(O(n)\) complexity, where \(n\) represents the total number of entries in a tensor. Supported by rigorous theoretical analysis, GETF solves the BTD problem via sequentially identifying the fibers that most likely coincides with a rank-1 tensor component. Our synthetic and real-world data based experiments validated the high accuracy of GETF and its drastically improved efficiency compared with existing methods, in addition to its potential utilization on large scale or high order data, such as complex relational or spatial-temporal data. The key contributions of this study include: (1) Our proposed GETF is the first method capable of all-order Boolean tensor decomposition; (2) GETF has substantially increased accuracy in identifying true rank-1 patterns, with less than a tenth of the computational cost compared with state-of-the-art methods; (3) we provided thorough theoretical foundations for the geometric properties for the BTD problem.
\section{Preliminaries}
\subsection{Notations}
Notations in this study follow those in \cite{kolda2009tensor}. We denote the \textit{order} of a tensor as \(k\), which is also called \textit{ways} or \textit{modes}. Scalar value, vector, matrix, and higher order tensor are represented as lowercase character \(x\), bold lowercase character \(\textbf{x}\), uppercase character \(X\), and Euler script \(\mathscr{X}\), respectively. Super script with mark \(\times\) indicates the size and dimension of a vector, matrix or tensor while subscript specifies an entry. Specifically, a \(k\)-order tensor is denoted as \(\mathscr{X}^{m_1\times m_2...\times m_k}\) and the entry of position \(i_1,i_2,…,i_k\) is represented as \(\mathscr{X}_{i_1i_2…i_k}\). For a 3-order tensor, we denote its fibers as \(\mathscr{X}_{:i_2i_3 }\), \(\mathscr{X}_{i_1:i_3 }\) or \(\mathscr{X}_{i_1i_2: }\) and its slices \(\mathscr{X}_{i_1::}\), \(\mathscr{X}_{:i_2:}\), \(\mathscr{X}_{::i_3}\). For a \(k\)-order tensor, we denote its mode-\(p\) fiber as \(\mathscr{X}_{i_1...i_{p-1}:i_{p+1}...i_k}\) with all indices fixed except for \(i_p\). \(||\mathscr{X}||\) represents the norm of a tensor, and \(|\mathscr{X}|\) the \(L_1\) norm in particular. The basic Boolean operations include \(\wedge(and, 1\wedge1=1,1\wedge0=0,0\wedge0=0)\), \(\vee(or,1\vee1=1,1\vee0=1,0\vee0=0)\), and \(\neg(not,\neg1=0,\neg0=1)\). Boolean entry-wise sum, subtraction and product of two matrices are denoted as \(A\oplus B=A\vee B\), \(A\ominus B=(A\wedge\neg B)\vee(\neg A\wedge B)\) and \(A\circledast B=A\wedge B\). The outer Boolean product in this paper is considered as the addition of rank-1 tensors, which in general follows the scope of canonical polyadic (CP) decomposition \cite{carroll1970analysis}. Specifically, a three-order Rank-1 tensor can be represented as the Boolean outer product of three vectors, i.e. \(\mathscr{X}^{m_1\times m_2\times m_3}=\textbf{a}^{m_1}\otimes\textbf{b}^{m_2 }\otimes\textbf{c}^{m_3}\). Similarly, for higher order tensor, a \(k\)-order tensor \(\mathscr{X}^{m_1\times m_2 ... \times m_k}\) of rank \(l\) is the outer product of \(A^{m_1\times l,1},A^{m_2\times l,2},..., A^{m_k\times l,k}\), i.e. \(\mathscr{X}^{m_1\times m_2...\times m_k}=\vee_{j=1}^l (A_{:j}^{m_1\times l,1}\otimes A_{:j}^{m_2 \times l,2}...\otimes A_{:j}^{m_k\times l,k})\) and \(\mathscr{X}_{i_1i_2,...,i_k}=\vee_{j=1}^l (A_{i_1j}^{m_1\times l,1}\wedge A_{i_2j}^{m_2\times l,2}...A_{i_kj}^{m_k\times l,k})\), \(j=1...l\) represents the rank-1 tensor components of a rank \(l\) CP decomposition of \(\mathscr{X}\). In this paper, we denote \(A^{m_i\times l,i},\, i=1...k\) as the pattern matrix of the \(i\)th order of \(\mathscr{X}\), its \(j\)th column \(A_{:j}^{m_i\times l, i}\) as the \(j\)th pattern fiber of the \(i\)th order, and \(A_{:j}^{m_1\times l,1}\otimes A_{:j}^{m_2 \times l,2}...\otimes A_{:j}^{m_k\times l,k}\) as the $j$-th rank-1 tensor pattern.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{BTD.png}
\caption{Boolean tensor decomposition}
\label{fig:BTD}
\end{figure}
\subsection{Problem Statement}
As illustrated in Figure \ref{fig:BTD}, for a binary \(k\)-order tensor \(\mathscr{X}\in\{0,1\}^{m_1\times m_2 ... \times m_k}\) and a convergence criteria parameter \(\tau\), the binary tensor decomposition problem is to identify low rank binary pattern matrices \(A^{m_1\times l,1*}, A^{m_2\times l,2*},...A^{m_k\times l,k*}\), the outer product of which best fit \(\mathscr{X}\), where \(A^{m_1\times l,1*},A^{m_2\times l,2*},...,A^{m_k\times l,k*}\) are matrices of \(l\) columns. In other words, \((A^{m_1\times l,1*},A^{m_2\times l,2*},...A^{m_k\times l,k*})=argmin_{A^1,A^2,...,A^k}(\gamma(A^{m_1\times l,1},A^{m_2\times l,2},...,A^{m_k\times l,k}; \mathscr{X})|\tau)\) Here \(\gamma(A^{m_1\times l,1}, A^{m_2\times l,2},...A^{m_k\times l,k}; \mathscr{X})\) is the cost function. In general, \(\gamma\) is defined to the reconstruction error \(\gamma(A^{m_1\times l,1*},...A^{m_k\times l,k*}; \mathscr{X})=||\mathscr{X}\ominus(A^{m_1\times l,1*}\otimes...\otimes A^{m_k\times l,k*})||_{L_p}\), and $p$ is usually set to be 1.
\subsection{Related work}
In order of difficulty, Boolean tensor decomposition consists of three major tasks, Boolean matrix factorization (BMF, \(k=2\)) \cite{stockmeyer1975set}, three-way Boolean tensor decomposition (BTD, \(k=3\)) and higher order Boolean tensor decomposition (HBTD, \(k>3\)) \cite{leenen1999indclas}. All of them are NP hard \cite{miettinen2009matrix}. Numerous heuristic solutions for the BMF and BTD problems have been developed in the past two decades \cite{miettinen2010sparse,miettinen2008discrete,miettinen2011boolean,miettinen2014mdl4bmf,erdos2013walk,karaev2015getting,lucchese2010mining,lucchese2013unifying}.
For BMF, the ASSO algorithm is the first heuristic BMF approach that finds binary patterns embedded within row-wise correlation matrix \cite{miettinen2008discrete}. On another account, PANDA \cite{lucchese2010mining} sequentially retrieves the significant patterns under current (residual) binary matrix amid noise. Recently, BMF algorithms using Bayesian framework have been proposed \cite{rukat2017bayesian}. The latent probability distribution adopted by Message Passing (MP) achieved top performance among all the state-of-the-art methods for BMF \cite{ravanbakhsh2016boolean}.
For BTD, Miettinen et al thoroughly defined the BTD problem (\(k=3\)) in 2011 \cite{miettinen2011boolean}, and proposed the use of least square update as a heuristic solution. To solve the scalability issue with the least square update, they later extended developed Walk'N'Merge, which applies random walk over a graph in identifying dense blocks as proxies of rank 1 tensors \cite{erdos2013walk}. Despite the increase of scalability, Walk'N'Merge tends to pick up many small patterns, the addition of which doesn't necessarily decrease the loss function by much. The DBTF algorithm introduced by Park et al. is a parallel distributed implementation of alternative least square update based on Khatri-Rao matrix product \cite{park2017fast}. Though DBTF reduced the high computational cost, its space complexity increases exponentially with the increase of tensor orders due to the khatri-Rao product operation. Recently, Tammo et al. proposed a probabilistic solution to BTD, called Logistical Or Machine (LOM), with improved fitting accuracy, robustness to noises, and acceptable computational complexity \cite{rukat2018probabilistic}. However, the high number of iterations it takes to achieve convergence of the likelihood function makes LOM prohibitive to large data analysis. Most importantly, to the best of our knowledge, none of the existing algorithms are designed to handle the HBTD problem for higher order tensors.
\section{GETF Algorithm and Analysis}
GETF identifies the rank-1 patterns sequentially: it first extracts one pattern from the tensor; and the subsequent patterns will be extracted sequentially from the residual tensor after removing the preceding patterns. We first derive the theoretical foundation of GETF. We show that the geometric property of the largest rank-1 pattern in a binary matrix developed in \cite{wan2019mebf} can be naturally extended to higher order tensor. We demonstrated the true pattern fiber of the largest pattern can be effectively distinguished from fibers of overlapped patterns or errors by reordering the tensor to maximize its overlap with a left-triangular-like tensor. Based on this idea, the most likely pattern fibers can be directly identified by a newly develop geometric folding approach that circumvents heuristic greedy searching or alternative least square based optimization.
\subsection{Theoretical Analysis}
We first give necessary definitions of the slice, re-order and sum operations on a \(k\) order tensor and an theoretical analysis of the property of a left-triangular-like (LTL) tensor.
\begin{Def}
(\(p\)-order slice). The \(p\)-order slice of a tensor \(\mathscr{X}^{m_1\times...\times m_k}\) indexed by \(\mathbb{P}\) is defined by \(\mathscr{X}_{i_1,...,i_k}\), where \(i_k\) is a fixed value \(\in\{1,...,m_k\}\) if \(k\in\Bar{\mathbb{P}}\), and \(i_k\) is unfixed \((i_k=:)\) if \(k\in\mathbb{P}\), here \(p=|\mathbb{P}|\) and \(\Bar{\mathbb{P}}=\{1,...,k\}\setminus\mathbb{P}\). Specifically, we denote a \(|\mathbb{P}|\) order slice of \(\mathscr{X}^{m_1\times...\times m_k}\) with the index set \(|\mathbb{P}|\) unfixed as \(\mathscr{X}^{m_1\times...\times m_k}_{\mathbb{P},I_{\Bar{\mathbb{P}}}}\) or \(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}}\), in which \(\mathbb{P}\) is the unfixed index set and \(I_{\Bar{\mathbb{P}}}\) are fixed indices.
\end{Def}
\begin{Def}
(Index Reordering Transformation, IRT). The index reordering transformation (IRT) transforms a tensor \(\mathscr{X}^{m_1\times,...\times m_k}\) to \(\Tilde{\mathscr{X}}=\mathscr{X}_{P_1,P_2,...,P_k}\), where \(P_1,...,P_k\) are any permutation of the index sets, \(\{1,...,m_1\},...,\{1,...,m_k\}\).
\end{Def}
\begin{Def}
(Tensor slice sum). The tensor slice sum of a \(k\)-order tensor \(\mathscr{X}^{m_1\times...\times m_k}\) with respect to the index set \(\mathbb{P}\) is defined as \(T_{sum}(\mathscr{X},\mathbb{P})\triangleq \sum_{i_1=1}^{m_{i_1}}...\sum_{i_{|\mathbb{P}|}=1}^{m_{i_{|\mathbb{P}|}}}\mathscr{X}_{:...:i_1:...:i_{|\mathbb{P}|}:....:},i_1,...,i_{|\mathbb{P}|}\in\mathbb{P}\). \(T_{sum}(\mathscr{X},\mathbb{P})\) results in a \(k-|\mathbb{P}|\) order tensor.
\end{Def}
\begin{Def}
(p-left-triangular-like, p-LTL). A \(k\)-order tensor \(\mathscr{X}^{m_1\times...\times m_k}\) is called p-LTL, if any of its \(p\)-order slice, \(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}}\), \(\mathbb{P}\subset\{1,...,k\}\) and \(|\mathbb{P}|=p\), and \(\forall j\in \mathbb{P}, 1\leq j_1<j_2\leq m_j,\; T_{sum}(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}},\mathbb{P}\setminus \{j\})_{j_1}\leq T_{sum}(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}},\mathbb{P}\setminus\{j\})_{j_2}\).
\end{Def}
\begin{Def}
(flat 2-LTL), A \(k\)-order 2-LTL tensor \(\mathscr{X}^{m_1\times...\times m_k}\) is called flat 2-LTL within an error range \(\epsilon\), if any of its \(2\)-order slice, \(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}}\), \(\mathbb{P}\subset\{1,...,k\}\) and \(|\mathbb{P}|=p\), and \(\forall j\in \mathbb{P}, 1\leq j_1<j_2\leq m_j,\; |T_{sum}(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}},\mathbb{P}\setminus\{j\})_{j_1}+ T_{sum}(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}},\mathbb{P}\setminus\{j\})_{j_2}-2T_{sum}(\mathscr{X}_{\mathbb{P},I_{\Bar{\mathbb{P}}}},\mathbb{P}\setminus\{j\})_{(j_1+j_2)/2}|<\epsilon\).
\end{Def}
The \(\textbf{Definition 5}\) indicates the tensor sum of over any \(2\)-order slice of a flat 2-LTL tensor is close enough to a linear function with the largest error less than \(\epsilon\). Figure \ref{fig:lemma2}A,C illustrate two examples of flat 2-\(LTL\) matrix and 2-\(LTL\) 3-order tensor. By the definition, the non-right angle side of a flat 2-\(LTL\) \(k\)-order tensor is close to a \(k-1\) dimension plane, which is specifically called as the \(\textbf{k-1 dimension plane}\) of the flat 2-\(LTL\) tensor in the rest part of this paper.
\begin{Lem}[Geometric segmenting of a flat 2-\(LTL\) tensor]
Assume \(\mathscr{X}\) is a \(k\)-order flat 2-\(LTL\) tensor and \(\mathscr{X}\) has none zero fibers. Then the largest rank-1 subarray in \(\mathscr{X}\) is seeded where one of the pattern fibers is paralleled with the fiber that anchored on the \(\nicefrac{1}{k}\) segmenting point (entry \(\{\nicefrac{|m_1|}{k},\nicefrac{|m_2|}{k},...,\nicefrac{|m_k|}{k}\}\)) along the sides of the right angle.
\end{Lem}
\begin{Lem} (Geometric perspective in seeding the largest rank-1 pattern)
For a \(k\) order tensor \(\mathscr{X}\) sparse enough and a given tensor size threshold \(\lambda\), if its largest rank-1 pattern tensor is larger than \(\lambda\), the IRT that reorders \(\mathscr{X}\) into a (k-1)-\(LTL\) tensor reorders the largest rank-1 pattern to a consecutive block, which maximize the size of the connected solid shape overlapped with the \(k-1\) dimension plane over a flat 2-\(LTL\) tensor larger than \(\lambda\).
\end{Lem}
\begin{Lem}
If a \(k\)-order tensor \(\mathscr{X}^{m_1\times...\times m_k}\) can be transformed into a p-LTL tensor by IRT, the p-LTL tensor is unique, p=2,...,k-1.
\end{Lem}
\begin{Lem}
If a \(k\)-order tensor is p-\(LTL\), then it is x-\(LTL\), for all the x=p,p+1,...,k.
\end{Lem}
Detailed proofs of the \(\textbf{Lemma\ 1-4}\) are given in APPENDIX. \(\textbf{Lemma\ 1}\) and \(\textbf{2}\) reflect our geometric perspective in identifying the largest rank-1 pattern and seeding the most likely pattern fibers. Specifically, \(\textbf{Lemma\ 1}\) suggests the optimal position of the fiber that is most likely the pattern fiber of the largest rank-1 pattern tensor under a flat 2-\(LTL\) tensor. Figure \ref{fig:lemma2}B,D illustrate the position (yellow dash lines) of the most likely pattern fibers in the flat 2-\(LTL\) matrix and \(3\)-order tensor. It is noteworthy that the (k-1)-\(LTL\) tensor must exists for a \(k\)-order tensor, which can be simply derived by reordering the indices of each tensor order \({j}\) by the decreasing order of \(T_{sum}(\mathscr{X},\{1,...,k\}\setminus\{j\})\). However, not all \(k\) order tensor can be transformed into a 2-\(LTL\) tensor via IRT when \(k>2\). A (k-1)-\(LTL\) tensor with only one rank-1 pattern tensor is 2-\(LTL\). Intuitively, the left bottom corner of a \(k\)-order (k-1)-\(LTL\) tensor of the largest rank-1 pattern is also 2-\(LTL\) (Figure \ref{fig:lemma2}D). However, the existence of multiple rank-1 patterns, overlaps among patterns and errors limit the 2-\(LTL\) property of left bottom corner of its (k-1)-\(LTL\) tensor. \(\textbf{Lemma\ 2}\) suggests the indices of the largest rank-1 pattern form the largest overlap between the (k-1)-\(LTL\) IRT and the the \(k-1\) dimension plane over a flat 2-\(LTL\) tensor. Based on this property, the largest rank-1 pattern and its most likely fiber can be seeded without heuristic greedy search or likelihood optimization that can substantially improve the computational efficiency. \(\textbf{Lemma\ 3}\) and \(\textbf{4}\) suggest that the (k-1)-\(LTL\) tensor is the IRT of \(\mathscr{X}\) that is closest to a 2-\(LTL\) tensor. Hence how close the intersect between a (k-1)-\(LTL\) tensor and a 2-\(LTL\) sub tensor is to a 2-\(LTL\) tensor, can reflect if the optimal pattern fiber position derived in \(\textbf{Lemma\ 1}\) fits to the 2-\(LTL\) sub tensor region of the (k-1)-\(LTL\) tensor.
\begin{figure*}
\centering
\begin{minipage}{.4\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Lemma2.png}
\captionof{figure}{Optimal rank 1 subarray}
\label{fig:lemma2}
\end{minipage}%
\begin{minipage}{.6\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{k-1LTL.png}
\captionof{figure}{Suboptimal subarray for \(k-1\) LTL tensor}
\label{fig:k-1}
\end{minipage}
\end{figure*}
\subsection{GETF algorithm}
Based on the geometric property of the largest rank-1 pattern and its most likely pattern fibers, we developed an efficient BTD and HBTD algorithm—GETF, by iteratively reconstructing the to-be-decomposed tensor into a \(k-1\) LTL tensor and identifying the largest rank-1 pattern. The main algorithm of GETF is formed by the iteration of the following five steps. \textit{Step 1}: For a given tensor \(\mathscr{X}^{m_1\times m_2...\times m_k}\), in each iteration, GETF first reorders the indices of the current tensor into a (k-1)-\(LTL\) tensor by IRT (Figure \ref{fig:k-1}A,D); \textit{Step 2}:
GETF utilizes \textbf{2\_LTL\_projection} algorithm to identify the flat 2-\(LTL\) tensor that maximizes the overlapped region between its \(k-1\) dimension plane and current (k-1)-\(LTL\) tensor (Figure \ref{fig:k-1}B,E); \textit{Step 3}: A \textbf{Pattern\_fiber\_finding} algorithm is applied to identify the most likely pattern fiber of the overlap region of the 2-\(LTL\) tensor and the (k-1)-\(LTL\) tensor, i.e., the largest rank-1 pattern (Figure \ref{fig:Basis}); \textit{Step 4}: A \textbf{Geometric\_folding} algorithm is applied to reconstruct the rank-1 tensor component from the identified pattern fiber that best fit the current to-be-decomposed tensor (Figure \ref{fig:geo}); and \textit{Step 5}: Remove the identified rank-1 tensor component from the current to-be-decomposed tensor (Figure \ref{fig:k-1}C,F). The inputs of GETF include the to-be-decomposed tensor \(\mathscr{X}\), a noise tolerance threshold \(t\) parameter, a convergence criterion \(\tau\) and a pattern fiber searching indicator \(Exha\).
\begin{algorithm}
\SetAlgoLined
\textbf{Inputs:} \(\mathscr{X}\in\{0,1\}^{m_1\times m_2...\times m_k}\), \(t\in(0,1)\),\(\tau\), \(Exha\in\{0,1\}\) \par
\textbf{Outputs:} \(A^{1*}\in\{0,1\}^{m_1\times l}\), \(A^{2*}\in\{0,1\}^{m_2\times l}\), ... \(A^{k*}\in\{0,1\}^{m_k\times l}\) \par
\(GETF(\mathscr{X}, t, \tau, Exha)\):\par
\(\mathscr{X}^{\text{Residual}}\leftarrow \mathscr{X}\),
\(A^1\leftarrow NULL\),..., \(A^k\leftarrow NULL\)\par
\(\Omega \leftarrow\) Generate set of directions for geometric-folding(\(k,Exha)\) \par
\While{\(!\tau\)}{
\(\gamma_0\leftarrow inf\), \(\textbf{a}^{1*}\leftarrow NULL\),..., \(\textbf{a}^{k*}\leftarrow NULL\) \par
\For{each direction \(\textbf{o}\) in \(\Omega\)}{
\(\mathscr{X}^{2-LTL}\leftarrow \textbf{2\_LTL\_projection}(\mathscr{X})\)\\
\(Pattern\, fiber^*\leftarrow \textbf{Pattern\_fiber\_finding}(\mathscr{X}^{2-LTL},\textbf{o})\)\\
\((\textbf{a}^1,...,\textbf{a}^k)\leftarrow \textbf{Geometric\_folding}(\mathscr{X}^{Residual}, Pattern fiber^*,\textbf{o},t)\)
\uIf{\(\gamma(\textbf{a}^1,...,\textbf{a}^k|\mathscr{X})<\gamma_0\)}{
\((\textbf{a}^{1*},...,\textbf{a}^{k*})\leftarrow (\textbf{a}^1,...,\textbf{a}^k)\);
\(\gamma_0\leftarrow \gamma(\textbf{a}^1,...,\textbf{a}^k|\mathscr{X})\)
}
}
\uIf{\(\gamma_0\neq inf\)}{
\(\mathscr{X}^{Residual}_{i_1i_2...i_k}\leftarrow0\) when \((\textbf{a}^{1*}\otimes \textbf{a}^{2*}...\otimes \textbf{a}^{k*})_{i_1i_2...i_k}=1\)\\
\(A^{j*}\leftarrow append(A^{j*},\textbf{a}^{j*}), j\in\{1,2,...,k\}\)
}
}
\caption{GETF}
\end{algorithm}
Details of the GETF and its sub algorithms are given in APPENDIX. In \textbf{Algorithm 1}, \textbf{o} represents a direction of geometric folding, which is a permutation of \(\{1,...,k\}\). The \textbf{2\_LTL\_projection} utilizes a project function and a scoring function to identify the flat 2-\(LTL\) tensor that maximizes the solid overlapped region between its \(k-1\) dimension plane and a (k-1)-\(LTL\) tensor. The \textbf{Pattern\_fiber\_finding} and \textbf{Geometric\_folding} algorithm are described below. Noted, there are \(k\) directions of pattern fibers and \(k!\) combinations of the orders in identifying them from a \(k\)-order tensor or reconstructing a rank-1 pattern from them. Empirically, to avoid duplicated computations, we tested conducting \(k\) times of geometric folding is sufficient to identify the fibers and reconstruct the suboptimal rank-1 pattern. GETF also provides options to set the
rounds and noise tolerance level of geometric folding in expanding a pattern fiber via adjusting the parameters \(Exha\) and \(t\).
\subsection{Pattern fiber finding}
The \textbf{Pattern\_fiber\_finding} algorithm is developed based on \textbf{Lemma 1}. Its input include a \(k\)-order tensor and a direction vector. Even the input is the entry-wise product of a flat 2-\(LTL\) tensor and the largest rank-1 pattern in a (k-1)-\(LTL\) tensor, it may still not be 2-\(LTL\) due to the existence of errors. We propose a recursive algorithm that recurrently re-orders an order of the input tensor and reports the coordinate of the pattern fiber on this order (See details in APPENDIX). The output is the position of the pattern fiber.
\begin{wrapfigure}{L}{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{Basis_find.png}
\caption{Illustration of finding pattern fiber in 3D}
\label{fig:Basis}
\end{wrapfigure}
Figure \ref{fig:Basis} illustrates the pattern fiber finding approach for a 3-order flat 2-\(LTL\) tensor \(\mathscr{X}^{m_1\times m_2\times m_3}\). To identify the coordinates of the yellow colored pattern fiber with unfixed index of the 1st order \(\mathscr{X}_{:i_2i_3}\) (Figure \ref{fig:Basis}A), its coordinate of the 2nd order is anchored on the 1/3 segmenting point of \(T_{sum}(\mathscr{X},\{2\})\), denoted as \(i_2\) (Figure \ref{fig:Basis}B), and its coordinate of the 3rd order is on the 1/2 segmenting point of \(T_{sum}(\mathscr{X}^{m_1\times m_2 \times m_3}_{:i_2:},\{2\})\) (Figure \ref{fig:Basis}C).
\subsection{Geometric folding}
\begin{wrapfigure}{R}{0.65\textwidth}
\centering
\includegraphics[width=\linewidth]{geo_fold.png}
\caption{Illustration of Geometric\_folding in 3D}
\label{fig:geo}
\end{wrapfigure}
The geometric folding approach is to reconstruct the rank-1 tensor pattern best fit \(\mathscr{X}\) from the pattern fiber identified by the \textbf{Pattern\_fiber\_finding} algorithm (see details in APPENDIX). For a \(k\)-order tensor \(\mathscr{X}\) and the identified position of pattern fiber, the pattern fiber is denote as \(\mathscr{X}_{:i_2^0...i_k^0}\) (Figure \ref{fig:geo}A). The algorithm computes the inner product between \(\mathscr{X}_{:i_2^0...i_k^0}\) and each fiber \(\mathscr{X}_{:i_2...i_k}\) to generate a new \(k-1\) order tensor \(\mathscr{H}^{m_2\times...\times m_k}\), \(\mathscr{H}_{i_2...i_m}=\sum_{j=1}^{m_1}\mathscr{X}_{ji_2^0...i_k^0}\wedge \mathscr{X}_{ji_2...i_k}\) (Figure \ref{fig:geo}B). This new tensor is further discretized based on a user defined noise tolerance level and generates a new binary \(k\)-1 order tensor \(\mathscr{X}'^{m_2\times m_3...\times m_k}\) (Figure \ref{fig:geo}C). This approach is called as geometric folding of a \(k\)-order tensor into a \(k\)-1 order tensor based on the pattern fiber \(\mathscr{X}_{:i_2^0...i_k^0}\). This approach will be iteratively conducted to fold the \(k\)-way tensor into a 2 dimensional matrix with \(k\)-2 rounds of \textbf{Pattern\_fiber\_finding} and \textbf{Geometric\_folding} and identifies \(k\)-2 pattern fibers. The pattern fibers of the last 2 dimensional will be identified as a BMF problem by using MEBF \cite{wan2019mebf}. The output of \textbf{Geometric\_folding} is the set of \(k\) pattern fibers of a rank-1 tensor (Figure \ref{fig:geo}E).
\subsection{Complexity analysis}
Assume \(k\)-order tensor has \(n=m^k\) entries.
The computation of \textbf{2\_LTL\_projection} is fixed based on its screening range, which is smaller than \(O(m^k)\).
The computation of each \textbf{Pattern\_fiber\_finding} is \(\frac{m^{k+1}-m}{m-1}+kmlog(m)\). \textbf{Geometric\_folding} is a loop algorithm consisted of additions and \textbf{Pattern\_fiber\_finding}. The computation for \textbf{Geometric\_folding} to fold a \(k\)-order tensor takes \(\frac{m^{k+2}-m^2}{(m-1)^2}-\frac{km}{m-1}+\frac{k(k+1)}{m-1}+\frac{k(k+1)}{2}mlog(m)\) computations. \(GETF\) conducts \(k\) times \(Geometric\_folding\) in each iteration to extract the suboptimal rank-1 tensor, by which, the overall computing cost on each iteration is \(k(\frac{m^{k+2}-m^2}{(m-1)^2}-\frac{km}{m-1}+\frac{k(k+1)}{m-1}+\frac{k(k+1)}{2}mlog(m))\sim O(m^k)\). Hence \(GETF\) is an \(O(m^k)=O(n)\) complexity algorithm.
\subsection{Discussion}
\textbf{Lemma\ 1}, \textbf{3}, and \textbf{4} are mathematically rigorous while \textbf{Lemma\ 2} is relatively descriptive due to the errors and level of overlaps among pattern tensors cannot be generally formulated, especially in a high order tensor. However, our derivations in APPENDIX reflects the geometric property described in \textbf{Lemma\ 2} stands for most of the tensors whose pattern tensors are not heavily overlapped. The advantage of GETF is significant. The computational cost of the IRT and identification of the flat 2-\(LTL\) tensor mostly cross the largest pattern are all \(O(n)\), where \(n\) is the tensor size. The property of the position of the most likely pattern fiber enables circumventing heuristic greedy search or optimization for seeding the largest rank-1 pattern. Due to the heuristic consideration of the algorithm, we focused on the method performance and robustness evaluation on an extensive set of synthetic data to demonstrate GETF is robust for high order tensor decomposition with different level of overlapped patterns and errors, followed by the applications on real-world datasets.
\section{Experimental Results on Synthetic Datasets}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{simulation.png}
\caption{Performance analysis on simulated data}
\label{fig:simu}
\end{figure*}
We simulated 4 scenarios each for tensor orders $k=2,3,4,5$ that corresponds to BMF, BTD, 4-order HBTD and 5-order HBTD: (1) low density tensor without error, (2) low density tensor with error, (3) high density tensor without error and (4) high density tensor with error. Detailed experiment setup is listed in APPENDIX. We compared GETF with MP on BMF and LOM on BTD settings, which according to recent reviews, are the best performing algorithms for BMF and BTD problems respectively. The evaluation focus on two metrics, time consumption and reconstruction error \cite{rukat2017bayesian,ravanbakhsh2016boolean}. For 4-order and 5-order HBTD, we conducted GETF only as there is no competing algorithm.
GETF significantly outperformed MP in reconstruction error (Figure \ref{fig:simu}A,B) and time consumption (Figure \ref{fig:simu}C) for all the four scenarios. This is also true when comparing to LOM except for the high density with high noise case, where GETF and LOM performed comparatively in terms of reconstruction error (Figure \ref{fig:simu}G,H,I). We also evaluated each algorithm on different data scale in supplementary materials. GETF maintains better performance with over 10 times faster in computational speed. Figure \ref{fig:simu} D-F,J-L show the capability of GETF on decomposing high order tensor data. Notably, the reconstruction error curve of GETF flattened after reaching the true number of components (Figure \ref{fig:simu}A,B,D,E,G,H,J,K), which is 5, suggesting its high accuracy in identifying true number of patterns. The error bar stands for standard derivation of time consumption in Figure \ref{fig:simu} C,F,I,L. Importantly, when the tensor order increases, its size would increase exponentially. The high memory cost remains a challenge for higher order tensors, for which an O(n) algorithm like GETF is extremely desirable. GETF showed consistent performance in the scenarios with or without noise. For a 5-way tensor with more than \(3*10^8\) elements, GETF completed the task in less than 1 minute. Overall, our experiments on synthetic datasets advocated the efficiency and robustness of GETF for the data with different tensor orders, data sizes, signal densities and noise levels.
\section{Experimental Results on Real-world Datasets}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Real.png}
\caption{Real data benchmark and applications}
\label{fig:real}
\end{figure}
We applied GETF on two real-world datasets, the Chicago crime record data\footnote{Chicago crime records downloaded on March 1st, 2020 from https://data.cityofchicago.org/Public-Safety}, and a breast cancer spatial-transcriptomics data \footnote{Breast cancer spatial transcriptomics data is retrieved from https://www.spatialresearch.org/resources-published-datasets/doi-10-1126science-aaf2403/}, which represents two scenarios with relatively lower and higher noise.
We retrieved the crime records in Chicago from 2001 to 2019 and organized them into a 4D tensor, with the four dimensions representing: 436 regions, 365 dates, 19 years and two crime categories (severe, and non-severe), respectively, i.e., \(\mathscr{X}^{436\times 365\times 19 \times2}\). An entry in the tensor has value 1 if there exists the crime category in the region on the date of the year. We first benchmark the performance of GETF and LOM on a 3D slice, \(\mathscr{X}_{:::1}\in\{0,1\}^{436\times 365 \times 19}\). GETF showed clear advantage over LOM with faster decline in reconstruction error. GETF plateured after the first two patterns, while it is more than eight for LOM (Figure \ref{fig:real}B). We applied GETF only to the 4D tensor, and used the top two patterns to reconstruct the original tensor, $\mathscr{X^*}$. To look for the crime date pattern, the crime index of a region defined as the total days of a year with crime occurrences in the region. We show that $\mathscr{X^*}$ is able to denoise the data and tease out the date patterns. As visualized in Figure \ref{fig:real}C, red indicates regions of high crime index, while blue for low crime index. Clearly, the GETF reconstructed tensor is able to distinguish the two regions. However, such a clear separation is not possible on the original tensor (Figure \ref{fig:real}D). Next we examine the validity of the two regions with an outsider factor, regional crime counts, defined as the total number of crimes from 2001 to 2019 for that region. From Figure \ref{fig:real}E, we could see that the regions with higher crime index according to GETF indeed corresponds to regions of higher regional crime coutns, and vice versa. In summary, we show that GETF is able to reveal the overall crime patterns by denoising the original tensor.
The breast cancer spatial transcriptomics dataset \cite{staahl2016visualization,wu2019deep}, as in Figure \ref{fig:real}F, was collected on a 3D coordinates with 1020 cell positions (\(x\times y\times z=15\times 17\times 4\)), each of which has expression values of 13360 genes, i.e., \(\mathscr{X}^{13360\times 15\times 17 \times 4}\). The tensor was first binarized, and it takes value 1 if the expression of the gene is larger than zero. We again benchmarked the performance of GETF and LOM on a 3D slice, \(\mathscr{X}_{:::1}\). LOM failed to generate any useful information seen from the non-decreasing reconstruction error, possibly because of the high noise of the transcriptomics data. On the other hand, GETF manage to derive patterns gradually (Figure \ref{fig:real}I). We applied GETF only to the 4D tensor, and among the top 10 patterns, we analyzed two extremest patterns: one the most sparse (red) and the other the most dense (blue) (Figure \ref{fig:real}F). The sparse pattern has 24 cell positions all expressing 232 genes (\(232\times 4\times 4\times 2\)), the dense pattern has 90 cells positions expressing 40 genes (\(40\times 15\times 3\times 2\)). A lower dimensional embedding of the 114 cells using UMAP \cite{mcinnes2018umap} clearly demonstrated them to be two distinct clusters (Figure \ref{fig:real}J). We also conducted functional annotations using gene ontology enrichment analysis for the genes of the two patterns. Figure \ref{fig:real}K,L showed the $-log(p)$ of the top 5 pathway enriched by the genes in each pattern, assessed by hypergeometric test. It implies that genes in the most dense pattern maintains the vibrancy of the cancer by showing strong activities in transcription and translation; while genes in the most sparse pattern maintains the tissue structure and suppress anti-tumor immune effect. Our analysis demonstrated that the GETF is able to reveal the complicated but integrated spatial structure of breast cancer tissue with different functionalities.
\section{Conclusion and Future Work}
In this paper, we proposed GETF as the first efficient method for the all-way Boolean tensor decomposition problem. We provided rigorous theoretical analysis on the validity of GETF and conducted experiments on both synthetic and real-world datasets to demonstrate its effectiveness and computational efficiency. In the future, to enable the integration of prior knowledge, we plan to enhance GETF with constrained optimization techniques and we believe it can be beneficial for broader applications that desire a better geometric interpretation of the hidden structures.
\section{Broader Impact}
GETF is a Boolean tensor factorization algorithm, which provides a solution to a fundamental mathematical problem. Hence we consider it is not with a significant subjective negative impact to the society. The structure of binary data naturally encodes the structure of subspace clusters in the data structure. We consider the efficient BTD and HBTD capability led by GETF enables the seeding of patterns for subspace clustering identification or disentangled representation learning, for the data with unknown subspace structure, such as recommendation of different item classes to customers with unknown groups or biomedical data of different patient classes. As we have demonstrated the high computational efficiency of GETF grants the capability to analyze large or high order tensor data, another field can be potentially benefited by GETF is the inference made to the spatial-temporal data collected from mobile sensors. The high efficiency of GETF enable a possible implementation on smart phones for a real-time inference of the data collected from the phones or other multi-modal personal wearable sensors. |
0905.0946 | \section{Introduction}
We prove that any two birational Mori fibre spaces are connected by a sequence of
elementary transformations, known as Sarkisov links:
\begin{theorem}\label{t_main} Suppose that $\phi\colon\map X.S.$ and $\psi\colon\map
Y.T.$ are two Mori fibre spaces with $\mathbb{Q}$-factorial terminal singularities.
Then $X$ and $Y$ are birational if and only if they are related by a sequence of Sarkisov
links.
\end{theorem}
Recall the following:
\begin{conjecture}\label{c_mori} Let $(Z,\Phi)$ be a kawamata log terminal pair.
Then we may run $f\colon\rmap Z.X.$ the $(K_Z+\Phi)$-MMP such that either
\begin{enumerate}
\item $(X,\Delta)$ is a log terminal model, that is $K_X+\Delta$ is nef, or
\item there is a Mori fibre space $\phi\colon\map X.S.$, that is $\rho(X/S)=1$ and
$-(K_X+\Delta)$ is $\phi$-ample,
\end{enumerate}
where $\Delta=f_*\Phi$.
\end{conjecture}
We will refer to the log terminal model $X$ and the Mori fibre space $\phi$ as the output
of the $(K_Z+\Phi)$-MMP. If $h\colon\rmap Z.X.$ is any sequence of divisorial
contractions and flips for the $(K_Z+\Phi)$-MMP then we say that $h$ is the result of
running the $(K_Z+\Phi)$-MMP. In other words if $h$ is the result of running the
$(K_Z+\Phi)$-MMP then $X$ does not have to be either a log terminal model or a Mori fibre
space.
By \cite{BCHM06} the only unknown case of \eqref{c_mori} is when $K_Z+\Phi$ is
pseudo-effective but neither $\Phi$ nor $K_Z+\Phi$ is big. Unfortunately the output is
not unique in either case. We will call two Mori fibre spaces $\phi\colon\map X.S.$ and
$\psi\colon\map Y.T.$ \textit{Sarkisov related} if $X$ and $Y$ are outcomes of running the
$(K_Z+\Phi)$-MMP, for the same $\mathbb{Q}$-factorial kawamata log terminal pair
$(Z,\Phi)$. This defines a category, which we call the Sarkisov category, whose objects
are Mori fibre spaces and whose morphisms are the induced birational maps $\rmap X.Y.$
between two Sarkisov related Mori fibre spaces. Our goal is to show that every morphism
in this category is a product of Sarkisov links. In particular a Sarkisov link should
connect two Sarkisov related Mori fibre spaces.
\begin{theorem}\label{t_sarkisov} If $\phi\colon\map X.S.$ and $\psi\colon\map Y.T.$
are two Sarkisov related Mori fibres spaces then the induced birational map
$\sigma\colon\rmap X.Y.$ is a composition of Sarkisov links.
\end{theorem}
Note that if $X$ and $Y$ are birational and have $\mathbb{Q}$-factorial terminal
singularities, then $\phi$ and $\psi$ are automatically the outcome of running the
$K_Z$-MMP for some projective variety $Z$, so that \eqref{t_main} is an easy
consequence of \eqref{t_sarkisov}.
It is proved in \cite{BCHM06} that the number of log terminal models is finite if either $\Phi$
or $K_Z+\Phi$ is big, and it is conjectured that in general the number of log terminal models
is finite up to birational automorphisms. Moreover Kawamata, see \cite{Kawamata07}, has
proved:
\begin{theorem}\label{t_minimal} Suppose that $\sigma\colon\rmap X.Y.$ is a birational
map between two $\mathbb{Q}$-factorial varieties which is an isomorphism in codimension
one.
If $K_X+\Delta$ and $K_Y+\Gamma$ are kawamata log terminal and nef and $\Gamma$ is the
strict transform of $\Delta$ then $\sigma$ is the composition of $(K_X+\Delta)$-flops.
\end{theorem}
Note that if the pairs $(X,\Delta)$ and $(Y,\Gamma)$ both have $\mathbb{Q}$-factorial
terminal singularities then the birational map $\sigma$ is automatically an isomorphism in
codimension one.
We recall the definition of a Sarkisov link. Suppose that $\phi\colon\map X.S.$ and
$\psi\colon\map Y.T.$ are two Mori fibre spaces. A Sarkisov link $\sigma\colon\rmap X.Y.$
between $\phi$ and $\psi$ is one of four types:
\begin{align*}
&\begin{diagram}
& \text{I} & \\
X' & \rDashto & Y \\
\dTo & & \dTo_{\psi}\\
X & & T \\
\dTo^{\phi} & \ldTo & \\
S & &
\end{diagram}
&&
\begin{diagram}
& \text{II} & \\
X' & \rDashto & Y' \\
\dTo & & \dTo \\
X & & Y \\
\dTo^{\phi}& & \dTo_{\psi} \\
S & = & T
\end{diagram}
&&
\begin{diagram}
& \text{III} & \\
X & \rDashto & Y'\\
\dTo^{\phi} & & \dTo \\
S & & Y \\
& \rdTo & \dTo_{\psi} \\
& & T
\end{diagram}
&&
\begin{diagram}
& & \text{IV} & & \\
X & & \rDashto & & Y \\
\dTo^{\phi} & & & & \dTo_{\psi} \\
S & & & & T \\
& \rdTo & & \ldTo & \\
& & R. & &
\end{diagram}
\end{align*}
There is a divisor $\Xi$ on the space $L$ on the top left (be it $L=X$ or $L=X'$) such
that $K_L+\Xi$ is kawamata log terminal and numerically trivial over the base (be it $S$,
$T$, or $R$). Every arrow which is not horizontal is an extremal contraction. If the
target is $X$ or $Y$ it is a divisorial contraction. The horizontal dotted arrows are
compositions of $(K_L+\Xi)$-flops. Links of type IV break into two types, IV${}_m$ and
IV${}_s$. For a link of type IV${}_m$ both $s$ and $t$ are Mori fibre spaces. For a link
of type IV${}_s$ both $s$ and $t$ are small birational contractions. In this case $R$ is
not $\mathbb{Q}$-factorial; for every other type of link all varieties are
$\mathbb{Q}$-factorial. Note that there is an induced birational map $\sigma\colon\rmap
X.Y.$ but not necessarily a rational map between $S$ and $T$.
The Sarkisov program has its origin in the birational classification of ruled surfaces.
A link of type I corresponds to the diagram
\begin{diagram}
\Hz 1. & = & \Hz 1. \\
\dTo & & \dTo_{\psi}\\
\pr 2. & & \pr 1. \\
\dTo^{\phi} & \ldTo & \\
\text{pt.} & &
\end{diagram}
Note that there are no flops for surfaces so the top horizontal map is always the
identity. The top vertical arrow on the left is the blow up of a point in $\pr 2.$ and
$\psi$ is the natural map given by the pencil of lines. A link of type III is the same
diagram, reflected in a vertical line,
\begin{diagram}
\Hz 1. & = & \Hz 1. \\
\dTo^{\phi} & & \dTo \\
\pr 1. & & \pr 2. \\
& \rdTo & \dTo_{\psi} \\
& & \text{pt.}
\end{diagram}
A link of type II corresponds to the classical elementary transformation between ruled
surfaces,
\begin{diagram}
X' & = & Y' \\
\dTo & & \dTo \\
X & & Y \\
\dTo^{\phi} & & \dTo_{\psi} \\
S & = & T. \\
\end{diagram}
The birational map $\map X'.X.$ blows up a point in one fibre and the birational map $\map
Y'.Y.$ blows down the old fibre. Finally a link of type IV corresponds to switching
between the two ways to project $\pr 1.\times \pr 1.$ down to $\pr 1.$,
$$
\begin{diagram}
\pr 1.\times\pr 1. & &= & & \pr 1.\times \pr 1. \\
\dTo^{\phi} & & & & \dTo_{\psi} \\
\pr 1. & & & & \pr 1. \\
& \rdTo & & \ldTo & \\
& & \text{pt.} & &
\end{diagram}
$$
It is a fun exercise to factor the classical Cremona transformation $\sigma\colon\rmap
{\pr 2.}.{\pr 2.}.$, $\map [X:Y:Z].[X^{-1}:Y^{-1}:Z^{-1}].$ into a product of Sarkisov
links. Indeed one can use the Sarkisov program to give a very clean proof that the
birational automorphism of $\pr 2.$ is generated by this birational map $\sigma$ and
$\operatorname{PGL}(3)$. More generally the Sarkisov program can sometimes be used to calculate the
birational automorphism group of Mori fibre spaces, especially Fano varieties. With this
said, note that the following problem seems quite hard:
\begin{question}\label{q_three} What are generators of the birational automorphism group
of $\pr 3.$?
\end{question}
Note that a link of type IV${}_s$ only occurs in dimension four or more. For an example
of a link of type IV${}_s$ simply take $\rmap S.T.$ to be a flop between threefolds, let
$\map S.R.$ be the base of the flop and let $X=S\times \pr 1.$ and $Y=T\times \pr 1.$ with
the obvious maps down to $S$ and $T$. It is conceivable that one can factor a link of
type IV${}_s$ into links of type I and III. However given any positive integer $k$ it is
easy to write down examples of links of type IV which cannot be factored into fewer than
$k$ links of type I, II or III.
Let us now turn to a description of the proof of \eqref{t_sarkisov}. The proof is based
on the original ideas of the Sarkisov program (as explained by Corti and Reid
\cite{Corti95}; see also \cite{BM97a}). We are given a birational map $\sigma\colon\rmap
X.Y.$ and the objective is to factor $\sigma$ into a product of Sarkisov links. In the
original proof one keeps track of some subtle invariants and the idea is to prove:
\begin{itemize}
\item the first Sarkisov link $\sigma_1$ exists,
\item if one chooses $\sigma_1$ appropriately then the invariants improve, and
\item the invariants cannot increase infinitely often.
\end{itemize}
Sarkisov links arise naturally if one plays the $2$-ray game. If the relative Picard
number is two then there are only two rays to contract and this gives a natural way to
order the steps of the minimal model program. One interesting feature of the original
proof is that it is a little tricky to prove the existence of the first Sarkisov link,
even if we assume existence and termination of flips. In the original proof one picks a
linear system on $Y$ and pulls it back to $X$. There are then three invariants to keep
track of; the singularities of the linear system on $X$, as measured by the canonical
threshold, the number of divisors of log discrepancy one (after rescaling to the canonical
threshold) and the pseudo-effective threshold. Even for threefolds it is very hard to
establish that these invariants satisfy the ascending chain condition.
Our approach is quite different. We don't consider any linear systems nor do we try to
keep track of any invariants. Instead we use one of the main results of \cite{BCHM06},
namely finiteness of ample models for kawamata log terminal pairs $(Z,A+B)$. Here $A$ is
a fixed ample $\mathbb{Q}$-divisor and $B$ ranges over a finite dimensional affine space
of Weil divisors. The closure of the set of divisors $B$ with the same ample model is a
disjoint union of finitely many polytopes and the union of all of these polytopes
corresponds to divisors in the effective cone.
Now if the space of Weil divisors spans the N\'eron-Severi group then one can read off
which ample model admits a contraction to another ample model from the combinatorics of
the polytopes, \eqref{t_polytope}. Further this property is preserved on taking a general
two dimensional slice, \eqref{c_polytope}. Sarkisov links then correspond to points on
the boundary of the effective cone which are contained in more than two polytopes,
\eqref{t_two}. To obtain the required factorisation it suffices to simply traverse the
boundary. In other words instead of considering the closed cone of curves and playing the
$2$-ray game we look at the dual picture of Weil divisors and we work inside a carefully
chosen two dimensional affine space. The details of the correct choice of underlying
affine space are contained in \S 4.
To illustrate some of these ideas, let us consider an easy case. Let $S$ be the blow up
of $\pr 2.$ at two points. Then $S$ is a toric surface and there are five invariant
divisors. The two exceptional divisors, $E_1$ and $E_2$, the strict transform $L$ of the
line which meets $E_1$ and $E_2$, and finally the strict transform $L_1$ and $L_2$ of two
lines, one of which meets $E_1$ and one of which meets $E_2$. Then the cone of effective
divisors is spanned by the invariant divisors and according to \cite{HK00} the polytopes
we are looking for are obtained by considering the chamber decomposition given by the
invariant divisors. Since $L_1=L+E_1$ and $L_2=L+E_2$ the cone of effective divisors is
spanned by $L$, $E_1$ and $E_2$. Since $-K_S$ is ample, we can pick an ample
$\mathbb{Q}$-divisor $A$ such that $K_S+A \sim_{\mathbb{Q}} 0$ and $K_S+A+E_1+E_2+L$ is
divisorially log terminal. Let $V$ be the real vector space of Weil divisors spanned by
$E_1$, $E_2$ and $L$. In this case projecting $\mathcal{L}_A(V)$ from the origin we get
\includegraphics[bbllx=146,bblly=500,bburx=415,bbury=662]{pic1}
We have labelled each polytope by the corresponding model. Imagine going around the
boundary clockwise, starting just before the point corresponding to $L$. The point $L$
corresponds to a Sarkisov link of type IV${}_m$, the point $L+E_2$ a link of type II, the
point $E_2$ a link of type III, the point $E_1$ a link of type I and the point $L+E_1$
another link of type II.
\section{Notation and conventions}
\label{s-notation}
We work over the field of complex numbers $\mathbb{C}$. An $\mathbb{R}$-Cartier divisor
$D$ on a variety $X$ is \textit{nef} if $D\cdot C\geq 0$ for any curve $C\subset X$. We
say that two $\mathbb{R}$-divisors $D_1$, $D_2$ are $\mathbb{R}$-linearly equivalent
($D_1\sim _{\mathbb{R}} D_2$) if $D_1-D_2=\sum r_i(f_i)$ where $r_i\in \mathbb{R}$ and
$f_i$ are rational functions on $X$. We say that an $\mathbb{R}$-Weil divisor $D$ is
\textit{big} if we may find an ample $\mathbb{R}$-divisor $A$ and an $\mathbb{R}$-divisor
$B\geq 0$, such that $D \sim _{\mathbb{R}} A+B$. A divisor $D$ is
\textit{pseudo-effective}, if for any ample divisor $A$ and any rational number $\epsilon
>0$, the divisor $D+\epsilon A$ is big. If $A$ is a $\mathbb Q$-divisor, we say that $A$
is a \textit{general ample $\mathbb{Q}$-divisor} if $A$ is ample and there is a
sufficiently divisible integer $m>0$ such that $mA$ is very ample and $mA\in |mA|$ is very
general.
A \textit{log pair} $(X,\Delta)$ is a normal variety $X$ and an $\mathbb{R}$-Weil divisor
$\Delta\geq 0$ such that $K_X+\Delta$ is $\mathbb{R}$-Cartier. We say that a log pair
$(X,\Delta)$ is \textit{log smooth}, if $X$ is smooth and the support of $\Delta$ is a
divisor with global normal crossings. A projective birational morphism $g\colon \map
Y.X.$ is a \textit{log resolution} of the pair $(X,\Delta )$ if $Y$ is smooth and the
strict transform $\Gamma$ of $\Delta$ union the exceptional set $E$ of $g$ is a divisor
with normal crossings support. If we write
$$
K_Y+\Gamma+E=g^*(K_X +\Delta)+\sum a_iE_i,
$$
where $E=\sum E_i$ is the sum of the exceptional divisors then the log discrepancy
$a(E_i,X,\Delta)$ of $E_i$ is $a_i$. By convention the log discrepancy of any divisor $B$
which is not exceptional is $1-b$, where $b$ is the coefficient of $B$ in $\Delta$. The
log discrepancy $a$ is the infinimum of the log discrepancy of any divisor.
A pair $(X,\Delta)$ is \textit{kawamata log terminal} if $a>0$. We say that the pair
$(X,\Delta)$ is \textit{log canonical} if $a\geq 0$. We say that the pair $(X,\Delta)$ is
\textit{terminal} if the log discrepancy of any exceptional divisor is greater than one.
We say that a rational map $\phi\colon\rmap X.Y.$ is a \textit{rational contraction} if
there is a resolution $p\colon\map W.X.$ and $q\colon\map W.Y.$ of $\phi$ such that $p$
and $q$ are contraction morphisms and $p$ is birational. We say that $\phi$ is a
\textit{birational contraction} if $q$ is in addition birational and every $p$-exceptional
divisor is $q$-exceptional. If in addition $\phi^{-1}$ is also a birational contraction,
we say that $\phi$ is a \textit{small birational map}. We refer the reader to
\cite{BCHM06} for the definitions of negative and non-positive rational contractions
and of log terminal models.
If $\mathcal{C}$ is a closed convex in a finite dimensional real vector space then
$\mathcal{C}^*$ denotes the dual convex set in the dual real vector space.
\section{The combinatorics of ample models}
We fix some notation. $Z$ is a smooth projective variety, $V$ is a finite dimensional
affine subspace of the real vector space $\operatorname{WDiv}_{\mathbb{R}}(Z)$ of Weil divisors on $Z$,
which is defined over the rationals, and $A\geq 0$ is an ample $\mathbb{Q}$-divisor on
$Z$. We suppose that there is an element $\Theta_0$ of $\mathcal{L}_A(V)$ such that
$K_Z+\Theta_0$ is big and kawamata log terminal.
We recall some definitions and notation from \cite{BCHM06}:
\begin{definition}\label{d_ample} Let $D$ be an $\mathbb{R}$-divisor on $Z$.
We say that $f\colon\rmap Z.X.$ is the \textbf{ample model} of $D$, if $f$ is a rational
contraction, $X$ is a normal projective variety and there is an ample divisor $H$ on $X$
such that if $p\colon\map W.Z.$ and $q\colon\map W.X.$ resolve $f$ and we write
$p^*D\sim_{\mathbb{R}}q^*H+E$, then $E\geq 0$ and for every $B\sim_{\mathbb{R}} p^*D$ if
$B\geq 0$ then $B\geq E$.
\end{definition}
Note that if $f$ is birational then $q_*E=0$.
\begin{definition}\label{d_cones} Let
\begin{align*}
V_A &=\{\,\Theta \,|\, \Theta=A+B, B\in V \,\}, \\
\mathcal{L}_A(V)&=\{\,\Theta=A+B\in V_A \,|\, \text{$K_Z+\Theta$ is log canonical and $B\geq 0$} \,\}, \\
\mathcal{E}_A(V) &=\{\,\Theta\in \mathcal{L}_A(V) \,|\, \text{$K_Z+\Theta$ is pseudo-effective} \,\}.
\end{align*}
Given a rational contraction $f\colon\rmap Z.X.$, define
$$
\mathcal{A}_{A,f}(V)=\{\, \Theta\in \mathcal E_A(V) \,|\, \text{$f$ is the ample model of $(Z,\Theta)$} \,\}.
$$
\end{definition}
In addition, let $\mathcal{C}_{A,f}(V)$ denote the closure of $\mathcal{A}_{A,f}(V)$.
\begin{theorem}\label{t_polytope} There are finitely many $1\leq i\leq m$ rational
contractions $f_i\colon\rmap Z.X_i.$ with the following properties:
\begin{enumerate}
\item $\displaystyle{\{\, \mathcal{A}_i=\mathcal{A}_{A,f_i}\,|\, 1\leq i\leq m \,\}}$ is a partition
of $\mathcal{E}_{A}(V)$. $\mathcal{A}_i$ is a finite union of interiors of rational
polytopes. If $f_i$ is birational then $\mathcal{C}_i=\mathcal{C}_{A,f_i}$ is a rational
polytope.
\item If $1\leq i\leq m$ and $1\leq j\leq m$ are two indices such that
$\mathcal{A}_j\cap\mathcal{C}_i\neq\varnothing$ then there is a contraction morphism
$f_{i,j}\colon\map X_i.X_j.$ and a factorisation $f_j=f_{i,j}\circ f_i$.
\end{enumerate}
Now suppose in addition that $V$ spans the N\'eron-Severi group of $Z$.
\begin{enumerate}
\setcounter{enumi}{2}
\item Pick $1\leq i\leq m$ such that a connected component $\mathcal{C}$ of
$\mathcal{C}_i$ intersects the interior of $\mathcal{L}_A(V)$. The following are
equivalent
\begin{itemize}
\item $\mathcal{C}$ spans $V$.
\item If $\Theta\in \mathcal{A}_i\cap \mathcal{C}$ then $f_i$ is a log terminal model of
$K_Z+\Theta$.
\item $f_i$ is birational and $X_i$ is $\mathbb{Q}$-factorial.
\end{itemize}
\item If $1\leq i\leq m$ and $1\leq j \leq m$ are two indices such that $\mathcal{C}_i$
spans $V$ and $\Theta$ is a general point of $\mathcal{A}_j\cap \mathcal{C}_i$ which is
also a point of the interior of $\mathcal{L}_A(V)$ then $\mathcal{C}_i$ and $\ccone
X_i/X_j.^*\times \mathbb{R}^k$ are locally isomorphic in a neighbourhood of $\Theta$, for
some $k\geq 0$. Further the relative Picard number of $f_{i,j}\colon\map X_i.X_j.$ is
equal to the difference in the dimensions of $\mathcal{C}_i$ and $\mathcal{C}_j\cap
\mathcal{C}_i$.
\end{enumerate}
\end{theorem}
\begin{proof} (1) is proved in \cite{BCHM06}.
Pick $\Theta\in\mathcal{A}_j\cap\mathcal{C}_i$ and $\Theta'\in\mathcal{A}_i$ so that
$$
\Theta_t=\Theta+t(\Theta'-\Theta)\in \mathcal{A}_i \qquad \text{if} \qquad t\in (0,1].
$$
By finiteness of log terminal models, cf. \cite{BCHM06}, we may find a positive constant
$\delta>0$ and a birational contraction $f\colon\rmap Z.X.$ which is a log terminal model
of $K_Z+\Theta_t$ for $t\in (0,\delta]$. Replacing $\Theta'=\Theta_1$ by
$\Theta_{\delta}$ we may assume that $\delta=1$. If we set
$$
\Delta_t=f_*\Theta_t,
$$
then $K_X+\Delta_t$ is kawamata log terminal and nef, and $f$ is $K_Z+\Theta_t$
non-positive for $t\in [0,1]$. As $\Delta_t$ is big the base point free theorem implies
that $K_X+\Delta_t$ is semiample and so there is an induced contraction morphism
$g_i\colon\map X.X_i.$ together with ample divisors $H_{1/2}$ and $H_1$ such that
$$
K_X+\Delta_{1/2}=g_i^*H_{1/2} \qquad \text{and} \qquad K_X+\Delta_1=g_i^*H_1.
$$
If we set
$$
H_t=(2t-1)H_1+2(1-t)H_{1/2},
$$
then
\begin{align*}
K_X+\Delta_t &= (2t-1)(K_X+\Delta_1)+2(1-t)(K_X+\Delta_{1/2}) \\
&= (2t-1)g_i^*H_1+2(1-t)g_i^*H_{1/2} \\
&= g_i^*H_t,
\end{align*}
for all $t\in [0,1]$. As $K_X+\Delta_0$ is semiample, it follows that $H_0$ is semiample
and the associated contraction $f_{i,j}\colon\map X_i.X_j.$ is the required morphism.
This is (2).
Now suppose that $V$ spans the N\'eron-Severi group of $Z$. Suppose that $\mathcal{C}$
spans $V$. Pick $\Theta$ in the interior of $\mathcal{C}\cap \mathcal{A}_i$. Let
$f\colon\rmap Z.X.$ be a log terminal model of $K_Z+\Theta$. It is proved in
\cite{BCHM06} that $f=f_j$ for some index $1\leq j\leq m$ and that $\Theta\in
\mathcal{C}_j$. But then $\mathcal{A}_i\cap \mathcal{A}_j\neq\varnothing$ so
that $i=j$.
If $f_i$ is a log terminal model of $K_Z+\Theta$ then $f_i$ is birational and $X_i$ is
$\mathbb{Q}$-factorial.
Finally suppose that $f_i$ is birational and $X_i$ is $\mathbb{Q}$-factorial. Fix
$\Theta\in \mathcal{A}_i$. Pick any divisor $B\in V$ such that $-B$ is ample
$K_{X_i}+f_{i*}(\Theta+B)$ is ample and $\Theta+B\in \mathcal{L}_A(V)$. Then $f_i$ is
$(K_Z+\Theta+B)$-negative and so $\Theta+B\in \mathcal{A}_i$. But then $\mathcal{C}_i$
spans $V$. This is (3).
We now prove (4). Let $f=f_i$ and $X=X_i$. As $\mathcal{C}_i$ spans $V$, (3) implies
that $f$ is birational and $X$ is $\mathbb{Q}$-factorial so that $f$ is a
$\mathbb{Q}$-factorial weak log canonical model of $K_Z+\Theta$. Suppose that $\llist
E.k.$ are the divisors contracted by $f$. Pick $B_i\in V$ numerically equivalent to
$E_i$. If we let $E_0=\sum E_i$ and $B_0=\sum B_i$ then $E_0$ and $B_0$ are numerically
equivalent. As $\Theta$ belongs to the interior of $\mathcal{L}_A(V)$ we may find
$\delta>0$ such that $K_Z+\Theta+\delta E_0$ and $K_Z+\Theta+\delta B_0$ are both kawamata
log terminal. Then $f$ is $(K_Z+\Theta+\delta E_0)$-negative and so $f$ is a log terminal
model of $K_Z+\Theta+\delta E_0$ and $f_j$ is the ample model of $K_Z+\Theta+\delta E_0$.
But then $f$ is also a log terminal model of $K_Z+\Theta+\delta B_0$ and $f_j$ is also the
ample model of $K_Z+\Theta+\delta B_0$. In particular $\Theta+\delta B_0\in
\mathcal{A}_j\cap \mathcal{C}_i$. As we are supposing that $\Theta$ is general in
$\mathcal{A}_j\cap \mathcal{C}_i$, in fact $f$ must be a log terminal model of
$K_Z+\Theta$. In particular $f$ is $(K_Z+\Theta)$-negative.
Pick $\epsilon>0$ such that if $\Xi\in V$ and $\|\Xi-\Theta\|<\epsilon$ then $\Xi$ belongs
to the interior of $\mathcal{L}_A(V)$ and $f$ is $(K_Z+\Xi)$-negative. Then the condition
that $\Xi\in \mathcal{C}_i$ is simply the condition that $K_X+\Delta=f_*(K_Z+\Xi)$ is nef.
Let $W$ be the affine suspace of $\operatorname{WDiv}_{\mathbb{R}}(X)$ given by pushing forward the
elements of $V$ and let
$$
\mathcal{N}=\{\, \Delta\in W \,|\, \text{$K_X+\Delta$ is nef} \,\}.
$$
Given $(\llist a.k.)\in \mathbb{R}^k$ let $B=\sum a_iB_i$ and $E=\sum a_iE_i$. If
$\|B\|<\epsilon$ then, as $\Xi+B$ is numerically equivalent to $\Xi+E$, $K_X+\Delta\in
\mathcal{N}$ if and only if $K_X+\Delta+f_*B\in \mathcal{N}$. In particular
$\mathcal{C}_i$ is locally isomorphic to $\mathcal{N}\times \mathbb{R}^k$.
But since $f_j$ is the ample model of $K_Z+\Theta$, in fact we can choose $\epsilon$
sufficiently small so that $K_X+\Delta$ is nef if and only if $K_X+\Delta$ is nef over
$X_j$, see \S 3 of \cite{BCHM06}. There is a surjective affine linear map from
$W$ to the space of Weil divisors on $X$ modulo numerical equivalence over $X_j$
and this induces an isomorphism
$$
\mathcal{N}\simeq \ccone X/X_j.^*\times \mathbb{R}^l,
$$
in a neighbourhood of $f_*\Theta$.
Note that $K_X+f_*\Theta$ is numerically trivial over $X_j$. As $f_*\Theta$ is big and
$K_X+f_*\Theta$ is kawamata log terminal we may find an ample $\mathbb{Q}$-divisor $A'$
and a divisor $B'\geq 0$ such that
$$
K_X+A'+B' \sim_{\mathbb{R}} K_X+f_*\Theta,
$$
is kawamata log terminal. But then
$$
-(K_X+B') \sim_{\mathbb{R}} -(K_X+\Delta')+A',
$$
is ample over $X_j$. Hence $f_{ij}\colon\map X.X_j.$ is a Fano fibration and
so by the cone theorem
$$
\rho(X_i/X_j)=\operatorname{dim} \mathcal{N}.
$$
This is (4). \end{proof}
\begin{corollary}\label{c_polytope} If $V$ spans the N\'eron-Severi group of $Z$ then
there is a Zariski dense open subset $U$ of the Grassmannian $G(\alpha,V)$ of real affine
subspaces of dimension $\alpha$ such that if $[W]\in U$ and it is defined over the
rationals then $W$ satisfies (1-4) of \eqref{t_polytope}.
\end{corollary}
\begin{proof} Let $U\subset G(\alpha,V)$ be the set of real affine subspaces $W$ of $V$ of
dimension $\alpha$, which contain no face of any $\mathcal{C}_i$ or $\mathcal{L}_A(V)$.
In particular the interior of $\mathcal{L}_A(W)$ is contained in the interior of
$\mathcal{L}_A(V)$. \eqref{t_polytope} implies that (1-2) always hold for $W$ and (1-4)
hold for $V$ and so (3) and (4) clearly hold for $W\in U$. \end{proof}
From now on in this section we assume that $V$ has dimension two and satisfies (1-4) of
\eqref{t_polytope}.
\begin{lemma}\label{l_easy-cases} Let $f\colon\rmap Z.X.$ and $g\colon\rmap Z.Y.$
be two rational contractions such that $\mathcal{C}_{A,f}$ is two dimensional and
$\mathcal{O}=\mathcal{C}_{A,f}\cap \mathcal{C}_{A,g}$ is one dimensional. Assume that
$\rho(X)\geq \rho(Y)$ and that $\mathcal{O}$ is not contained in the boundary of
$\mathcal{L}_A(V)$. Let $\Theta$ be an interior point of $\mathcal{O}$ and let
$\Delta=f_*\Theta$.
Then there is a rational contraction $\pi\colon\rmap X.Y.$ which factors $g=\pi\circ f$
and either
\begin{enumerate}
\item $\rho(X)=\rho(Y)+1$ and $\pi$ is a $(K_X+\Delta)$-trivial morphism, in which case,
either
\begin{enumerate}
\item $\pi$ is birational and $\mathcal{O}$ is not contained in the boundary of
$\mathcal{E}_A(V)$, in which case, either
\begin{enumerate}
\item $\pi$ is a divisorial contraction and $\mathcal{O}\neq\mathcal{C}_{A,g}$, or
\item $\pi$ is a small contraction and $\mathcal{O}=\mathcal{C}_{A,g}$, or
\end{enumerate}
\item $\pi$ is a Mori fibre space and $\mathcal{O}=\mathcal{C}_{A,g}$ is contained in the
boundary of $\mathcal{E}_A(V)$, or
\end{enumerate}
\item $\rho(X)=\rho(Y)$, in which case, $\pi$ is a $(K_X+\Delta)$-flop and
$\mathcal{O}\neq \mathcal{C}_{A,g}$ is not contained in the boundary of
$\mathcal{E}_A(V)$.
\end{enumerate}
\end{lemma}
\begin{proof} By assumption $f$ is birational and $X$ is $\mathbb{Q}$-factorial. Let
$h\colon\rmap Z.W.$ be the ample model corresponding to $K_Z+\Theta$. Since $\Theta$ is
not a point of the boundary of $\mathcal{L}_A(V)$ if $\Theta$ belongs to the boundary of
$\mathcal{E}_A(V)$ then $K_Z+\Theta$ is not big and so $h$ is not birational. As
$\mathcal{O}$ is a subset of both $\mathcal{C}_{A,f}$ and $\mathcal{C}_{A,g}$ there are
morphisms $p\colon\map X.W.$ and $q\colon\map Y.W.$ of relative Picard number at most one.
There are therefore only two possibilities:
\begin{enumerate}
\item $\rho(X)=\rho(Y)+1$, or
\item $\rho(X)=\rho(Y)$.
\end{enumerate}
Suppose we are in case (1). Then $q$ is the identity and $\pi=p\colon\map X.Y.$ is a
contraction morphism such that $g=\pi\circ f$. Suppose that $\pi$ is birational. Then
$h$ is birational and $\mathcal{O}$ is not contained in the boundary of
$\mathcal{E}_A(V)$. If $\pi$ is divisorial then $Y$ is $\mathbb{Q}$-factorial and so
$\mathcal{O}\neq\mathcal{C}_{A,g}$. If $\pi$ is a small contraction then $Y$ is not
$\mathbb{Q}$-factorial and so $\mathcal{C}_{A,g}=\mathcal{O}$ is one dimensional. If
$\pi$ is a Mori fibre space then $\mathcal{O}$ is contained in the boundary of
$\mathcal{E}_A(V)$ and $\mathcal{O}=\mathcal{C}_{A,g}$.
Now suppose we are in case (2). By what we have already proved $\rho(X/W)=\rho(Y/W)=1$.
$p$ and $q$ are not divisorial contractions as $\mathcal{O}$ is one dimensional. $p$ and
$q$ are not Mori fibre spaces as $\mathcal{O}$ cannot be contained in the boundary of
$\mathcal{E}_A(V)$. Hence $p$ and $q$ are small and the rest is clear. \end{proof}
\begin{lemma}\label{l_negative} Let $f\colon\rmap W.X.$ be a birational contraction between
projective $\mathbb{Q}$-factorial varieties. Suppose that $(W,\Theta)$ and $(W,\Phi)$
are both kawamata log terminal.
If $f$ is the ample model of $K_W+\Theta$ and $\Theta-\Phi$ is ample then $f$ is the
result of running the $(K_W+\Phi)$-MMP.
\end{lemma}
\begin{proof} By assumption we may find an ample divisor $H$ on $W$ such that $K_W+\Phi+H$
is kawamata log terminal and ample and a positive real number $t<1$ such that $tH
\sim_{\mathbb{R}} \Theta-\Phi$. Note that $f$ is the ample model of $K_W+\Phi+tH$. Pick
any $s<t$ sufficiently close to $t$ so that $f$ is $(K_W+\Phi+sH)$-negative and yet $f$ is
still the ample model of $K_W+\Phi+sH$. Then $f$ is the unique log terminal model of
$K_W+\Phi+sH$. In particular if we run the $(K_W+\Phi)$-MMP with scaling of $H$ then,
when the value of the scalar is $s$, the induced rational map is $f$. \end{proof}
We now adopt some more notation for the rest of this section. Let $\Theta=A+B$ be a point
of the boundary of $\mathcal{E}_A(V)$ in the interior of $\mathcal{L}_A(V)$. Enumerate
$\llist \mathcal{T}.k.$ the polytopes $\mathcal{C}_i$ of dimension two which contain
$\Theta$. Possibly re-ordering we may assume that the intersections $\mathcal{O}_0$ and
$\mathcal{O}_k$ of $\mathcal{T}_1$ and $\mathcal{T}_k$ with the boundary of
$\mathcal{E}_A(V)$ and $\mathcal{O}_i=\mathcal{T}_i\cap \mathcal{T}_{i+1}$ are all one
dimensional. Let $f_i\colon\rmap Z.X_i.$ be the rational contractions associated to
$\mathcal{T}_i$ and $g_i\colon\rmap Z.S_i.$ be the rational contractions associated to
$\mathcal{O}_i$. Set $f=f_1\colon\rmap Z.X.$, $g=f_k\colon\rmap Z.Y.$, $X'=X_2$,
$Y'=X_{k-1}$. Let $\phi\colon\map X.S=S_0.$, $\psi\colon\map Y.T=S_k.$ be the induced
morphisms and let $\rmap Z.R.$ be the ample model of $K_Z+\Theta$.
\includegraphics[bbllx=90,bblly=550,bburx=487,bbury=666]{pic2}
\begin{theorem}\label{t_two} Suppose $\Phi$ is any divisor such that $K_Z+\Phi$ is
kawamata log terminal and $\Theta-\Phi$ is ample.
Then $\phi$ and $\psi$ are two Mori fibre spaces which are outputs of the $(K_Z+\Phi)$-MMP
which are connected by a Sarkisov link if $\Theta$ is contained in more than two
polytopes.
\end{theorem}
\begin{proof} We assume for simplicity of notation that $k\geq 3$. The case $k\leq 2$ is
similar and we omit it. The incidence relations between the corresponding polytopes yield
a commutative heptagon,
$$
\begin{diagram}
X' & & \rDashto & & Y' \\
\dDashto^p & & & & \dDashto_q \\
X & & & & Y \\
\dTo^{\phi} & & & & \dTo_{\psi} \\
S & & & & T \\
& \rdTo(2,2)_s & & \ldTo(2,2)_t & \\
& & R & &
\end{diagram}
$$
where $p$ and $q$ are birational maps. $\phi$ and $\psi$ are Mori fibre spaces by
\eqref{l_easy-cases}. Pick $\Theta_1$ and $\Theta_k$ in the interior of $\mathcal{T}_1$
and $\mathcal{T}_k$ sufficiently close to $\Theta$ so that $\Theta_1-\Phi$ and
$\Theta_k-\Phi$ are ample. As $X$ and $Y$ are $\mathbb{Q}$-factorial, \eqref{l_negative}
implies that $\phi$ and $\psi$ are possible outcomes of the $(K_Z+\Phi)$-MMP. Let
$\Delta=f_*\Theta$. Then $K_X+\Delta$ is numerically trivial over $R$.
Note that there are contraction morphisms $\map X_i.R.$ and that $\rho(X_i/R)\leq 2$. If
$\rho(X_i/R)=1$ then $\map X_i.R.$ is a Mori fibre space. By \eqref{t_polytope} there is
facet of $\mathcal{T}_i$ which is contained in the boundary of $\mathcal{E}_A(V)$ and so
$i=1$ or $k$. Thus $\rmap X_i.X_{i+1}.$ is a flop, $1<i<k-1$. Since $\rho(X'/R)=2$ it
follows that either $p$ is a divisorial contraction and $s$ is the identity or $p$ is a
flop and $s$ is not the identity. We have a similar dichotomy for $q\colon\rmap Y'.Y.$
and $t\colon\map T.R.$.
There are then four cases. If $s$ and $t$ are the identity then $p$ and $q$ are
divisorial extractions and we have a link of type II.
If $s$ is the identity and $t$ is not then $p$ is a divisorial extraction and $q$ is a
flop and we have a link of type I. Similarly if $t$ is the identity and $s$ is not then
$q$ is a divisorial extraction and $p$ is a flop and we have a link of type III.
Finally suppose neither $s$ nor $t$ is the identity. Then both $p$ and $q$ are flops.
Suppose that $s$ is a divisorial contraction. Let $F$ be the divisor contracted by $s$
and let $E$ be its inverse image in $X$. Since $\phi$ has relative Picard number one
$\phi^*(F)=mE$, for some positive integer $m$. Then $K_X+\Delta+\delta E$ is kawamata log
terminal for any $\delta>0$ sufficiently small and $E=\mathbf{B}(K_X+\Delta+\delta E/R)$.
If we run the $(K_X+\Delta+\delta E)$-MMP over $R$ then we end with a birational
contraction $\rmap X.W.$, which is a Mori fibre space over $R$. Since $\rho(X/R)=2$,
$W=Y$ and we have a link of type III, a contradiction. Similarly $t$ is never a
divisorial contraction. If $s$ is a Mori fibre space then $R$ is $\mathbb{Q}$-factorial
and so $t$ must be a Mori fibre space as well. This is a link of type IV${}_m$. If $s$
is small then $R$ is not $\mathbb{Q}$-factorial and so $t$ is small as well. Thus we have
a link of type IV${}_s$. \end{proof}
\section{Proof of \eqref{t_sarkisov} }
\begin{lemma}\label{l_perturb} Let $\phi\colon\map X.S.$ and $\psi\colon\map Y.T.$ be two
Sarkisov related Mori fibre spaces corresponding to two $\mathbb{Q}$-factorial kawamata
log terminal projective varieties $(X,\Delta)$ and $(Y,\Gamma)$.
Then we may find a smooth projective variety $Z$, two birational contractions
$f\colon\rmap Z.X.$ and $g\colon\rmap Z.Y.$, a kawamata log terminal pair $(Z,\Phi)$, an
ample $\mathbb{Q}$-divisor $A$ on $Z$ and a two dimensional rational affine subspace $V$
of $\operatorname{WDiv}_{\mathbb{R}}(Z)$ such that
\begin{enumerate}
\item if $\Theta\in \mathcal{L}_A(V)$ then $\Theta-\Phi$ is ample,
\item $\mathcal{A}_{A,\phi\circ f}$ and $\mathcal{A}_{A,\psi\circ g}$ are not
contained in the boundary of $\mathcal{L}_A(V)$,
\item $V$ satisfies (1-4) of \eqref{t_polytope},
\item $\mathcal{C}_{A,f}$ and $\mathcal{C}_{A,g}$ are two dimensional, and
\item $\mathcal{C}_{A,\phi\circ f}$ and $\mathcal{C}_{A,\psi\circ g}$ are one
dimensional.
\end{enumerate}
\end{lemma}
\begin{proof} By assumption we may find a $\mathbb{Q}$-factorial kawamata log terminal
pair $(Z,\Phi)$ such that $f\colon\rmap Z.X.$ and $g\colon\rmap Z.Y.$ are both outcomes of
the $(K_Z+\Phi)$-MMP.
Let $p\colon\map W.Z.$ be any log resolution of $(Z,\Phi)$ which resolves the
indeterminancy of $f$ and $g$. We may write
$$
K_W+\Psi=p^*(K_Z+\Phi)+E',
$$
where $E'\geq 0$ and $\Psi\geq 0$ have no common components, $E'$ is exceptional and
$p_*\Psi=\Phi$. Pick $-E$ ample over $Z$ with support equal to the full exceptional locus
such that $K_W+\Psi+E$ is kawamata log terminal. As $p$ is $(K_W+\Psi+E)$-negative,
$K_Z+\Phi$ is kawamata log terminal and $Z$ is $\mathbb{Q}$-factorial, the
$(K_W+\Psi+E)$-MMP over $Z$ terminates with the pair $(Z,\Phi)$ by \eqref{l_negative}.
Replacing $(Z,\Phi)$ with $(W,\Psi+E)$, we may assume that $(Z,\Phi)$ is log smooth and
$f$ and $g$ are morphisms.
Pick general ample $\mathbb{Q}$-divisors $A, \llist H.k.$ on $Z$ such that $\llist H.k.$
generate the N\'eron-Severi group of $Z$. Let
$$
H=A+\alist H.+.k..
$$
Pick sufficiently ample divisors $C$ on $S$ and $D$ on $T$ such that
$$
-(K_X+\Delta)+\phi^*C \qquad \text{and} \qquad -(K_Y+\Gamma)+\psi^*D,
$$
are both ample. Pick a rational number $0<\delta<1$ such that
$$
-(K_X+\Delta+\delta f_*H)+\phi^*C \qquad \text{and} \qquad -(K_Y+\Gamma+\delta g_*H)+\psi^*D,
$$
are both ample and $K_Z+\Phi+\delta H$ is both $f$ and $g$-negative. Replacing $H$ by
$\delta H$ we may assume that $\delta=1$. Now pick a $\mathbb{Q}$-divisor $\Phi_0\leq
\Phi$ such that $A+(\Phi_0-\Phi)$,
$$
-(K_X+f_*\Phi_0+ f_*H)+\phi^*C \quad \text{and} \quad -(K_Y+g_*\Phi_0+ g_*H)+\psi^*D,
$$
are all ample and $K_Z+\Phi_0+ H$ is both $f$ and $g$-negative.
Pick general ample $\mathbb{Q}$-divisors $F_1\geq 0$ and $G_1\geq 0$
$$
F_1\sim_{\mathbb{Q}} -(K_X+f_*\Phi_0+ f_*H)+\phi^*C \quad \text{and} \quad G_1 \sim_{\mathbb{Q}} -(K_Y+g_*\Phi_0+ g_*H)+\psi^*D.
$$
Then
$$
K_Z+\Phi_0+ H+F+G,
$$
is kawamata log terminal, where $F=f^*F_1$ and $G=g^*G_1$.
Let $V_0$ be the affine subspace of $\operatorname{WDiv}_{\mathbb{R}}(Z)$ which is the translate by
$\Phi_0$ of the vector subspace spanned by $\llist H.k.,F,G$. Suppose that $\Theta=A+B\in
\mathcal{L}_A(V_0)$. Then
$$
\Theta-\Phi=(A+\Phi_0-\Phi)+(B-\Phi_0),
$$
is ample, as $B-\Phi_0$ is nef by definition of $V_0$. Note that $\Phi_0+F+H\in
\mathcal{A}_{A,\phi\circ f}(V_0)$, $\Phi_0+G+H\in \mathcal{A}_{A,\psi\circ g}(V_0)$, and
$f$, respectively $g$, is a weak log canonical model of $K_Z+\Phi_0+F+H$, respectively
$K_Z+\Phi_0+G+H$. \eqref{t_polytope} implies that $V_0$ satisfies (1-4) of
\eqref{t_polytope}.
Since $\llist H.k.$ generate the N\'eron-Severi group of $Z$ we may find constants $\llist
h.k.$ such that $G$ is numerically equivalent to $\sum h_iH_i$. Then $\Phi_0+F+\delta
G+H-\delta(\sum h_iH_i)$ is numerically equivalent to $\Phi_0+F+H$ and if $\delta>0$ is
small enough $\Phi_0+F+\delta G+H-\sum \delta h_iH_i\in \mathcal{L}_A(V_0)$. Thus
$\mathcal{A}_{A,\phi\circ f}(V_0)$ is not contained in the boundary of
$\mathcal{L}_A(V_0)$. Similarly $\mathcal{A}_{A,\psi\circ g}(V_0)$ is not contained in
the boundary of $\mathcal{L}_A(V_0)$. In particular $\mathcal{A}_{A,f}(V_0)$ and
$\mathcal{A}_{A,g}(V_0)$ span $V_0$ and $\mathcal{A}_{A,\phi\circ f}(V_0)$ and
$\mathcal{A}_{A,\psi\circ g}(V_0)$ span affine hyperplanes of $V_0$, since
$\rho(X/S)=\rho(Y/T)=1$.
Let $V_1$ be the translate by $\Phi_0$ of the two dimensional vector space spanned by
$F+H-A$ and $F+G-A$. Let $V$ be a small general perturbation of $V_1$, which is defined
over the rationals. Then (2) holds. (1) holds, as it holds for any two dimensional
subspace of $V_0$, (3) holds by \eqref{c_polytope} and this implies that (4) and (5)
hold. \end{proof}
\begin{proof}[Proof of \eqref{t_sarkisov}] Pick $(Z,\Phi)$, $A$ and $V$ given by
\eqref{l_perturb}. Pick points $\Theta_0\in \mathcal{A}_{A,\phi\circ f}(V)$ and
$\Theta_1\in\mathcal{A}_{A,\psi\circ g}(V)$ belonging to the interior of
$\mathcal{L}_A(V)$. As $V$ is two dimensional, removing $\Theta_0$ and $\Theta_1$ divides
the boundary of $\mathcal{E}_A(V)$ into two parts. The part which consists entirely of
divisors which are not big is contained in the interior of $\mathcal{L}_A(V)$. Consider
tracing this boundary from $\Theta_0$ to $\Theta_1$. Then there are finitely many $2\leq
i\leq l$ points $\Theta_i$ which are contained in more than two polytopes
$\mathcal{C}_{A,f_i}(V)$. \eqref{t_two} implies that for each such point there is a
Sarkisov link $\sigma_i\colon\rmap X_i.Y_i.$ and $\sigma$ is the composition of these
links. \end{proof}
\bibliographystyle{/home/mckernan/Jewel/Tex/hamsplain} |
0905.1129 | \section{Introduction}
Repetitions in words have been studied since the beginning of the
previous century \cite{thueI,thue}. Recently, there has been much
interest in repetitions with fractional exponent
\cite{brandenburg,carpi,dejean,longfrac,krieger,mignosi}. For
rational $1<r\le 2$, a {\bf fractional $r$-power} is a non-empty
word $w=pe$ such that $e$ is the prefix of $p$ of length
$(r-1)|p|$. We call $e$ the {\bf excess} of the repetition.
We also say that $r$ is the {\bf exponent} of the repetition $pe$.
For example, $010$ is a $3/2$-power, with excess 0. A basic problem
is that of identifying the repetitive threshold for each alphabet
size $n>1$:
\begin{quote}
What is the infimum of $r$ such that an
infinite sequence on $n$ letters exists, not containing any factor
of exponent greater than $r$?
\end{quote}
This infimum is called the {\bf repetitive threshold} of an $n$-letter
alphabet and is denoted by $RT(n)$. Dejean's conjecture \cite{dejean}
is that
$$
RT(n)=
\begin{cases}
7/4,&n=3\\
7/5,&n=4\\
n/(n-1),&n\ne 3,4.
\end{cases}
$$
Thue, Dejean and Pansiot, respectively \cite{thue,dejean,pansiot},
established the values $RT(2)$, $RT(3)$, $RT(4)$. Moulin Ollagnier
\cite{ollagnier} verified Dejean's conjecture for $5\le n\le 11$,
and Mohammad-Noori and Currie \cite{morteza} proved the conjecture
for $12\le n\le 14$. Recently, Carpi \cite{carpi} showed that Dejean's
conjecture holds for $n\ge 33$. The present authors strengthened Carpi's
construction to show that Dejean's conjecture holds for $n\ge
27$ \cite{currie,archiv}. In this note we show that in fact
Dejean's conjecture holds for $n\ge 2$. We will freely assume the usual notions of combinatorics on
words as set forth in, for example, \cite{acw}.
\section{Morphisms}\label{morphisms}
Given previous work, it remains only to show that Dejean's conjecture holds for $15\le n\le 26$. This follows from the fact that the following morphisms are `convenient' in the sense of \cite{ollagnier}.
To make our exposition self-contained, we demonstrate in the remainder
of this paper how these morphisms are used to prove Dejean's conjecture
for $15\le n\le 26.$ We introduce several simplifications
and one correction to the work of Moulin Ollagnier \cite{ollagnier}.
\tiny
\begin{eqnarray*}h_{15}(0) &=& 01101101011011011011010101101010110110110110110101101101\\
h_{15}(1) &=& 10101011011011010110101101101011010110110110101101010101\\
\\
h_{16}(0) &=& 101010110110110101101101101010101010101010110101010101010101\\
h_{16}(1) &=& 011010110110110110101101101010110110101010110101010101010101\\
\\
h_{17}(0) &=& 1010101010101010101101101011011010101010101011010110110110101101\\
h_{17}(1) &=& 1010101010101010101101101101101010101010110110110110110110110110\\
\\
h_{18}(0) &=& 10101010101101101101010110110101011011011010101011010110110101010101\\
h_{18}(1) &=& 01101010101101101010110110110101011011011010101011010101010101010101\\
\\
h_{19}(0) &=& 101010101010101010101101101010110110101010101010101101011010110110101101\\
h_{19}(1) &=& 101010101010101010101101101011011010101010101011011011011010110110110110\\
\\
h_{20}(0) &=& 1010101010101010101011011011010101010101011011011011011010110101011011010101\\
h_{20}(1) &=& 1010101010101010101011011011010101101101011011011011010110110101011011010110\\
\\
h_{21}(0) &=& 101010101010101010101011011010101010101101101010101010110110110110101010101010101101\\
h_{21}(1) &=& 101010101010101010101011011010101010110110101010101010110101101010101010101010110110\\
\\
h_{22}(0) &=& 101010101010101010101011010101010101010101010101101101101101101011011011011011010101\\
h_{22}(1) &=& 101010101010101010101011010101010101011011010101101101101101011011011011011011010110\\
\\
h_{23}(0) &=& 1010101010101010101010101010101010101011011010110110110110101011010110110110110110101101\\
h_{23}(1) &=& 1010101010101010101010101010101010101101101010110110110110110110110110110110110110110110\\
\\
h_{24}(0) &=& 10101010101010101010101011010101101101010101010101011010101010110110101101101101011011010101\\
h_{24}(1) &=& 10101010101010101010101011010101101101010110110101011010101010110101101101101101011011010110\\
\\
h_{25}(0) &=& 101010101010101010101010101011011010101011011010110110110101101101011010101010101011011010110110\\
h_{25}(1) &=& 101010101010101010101010101011011010101010110110110110110101101101101101101010101011011010101101\\
\\
h_{26}(0) &=& 1010101010101010101010101011010101010101101101010101010110110110101101011010110110110110110110110101\\
h_{26}(1) &=& 1010101010101010101010101011010101010101101101101101010110110110101101010110110110110110110110110110
\end{eqnarray*}
\normalsize
We remark that the last letter of $h_n(0)$ is different from the last letter of $h_n(1)$ in each case.
We also note that for each $n$, $|h_n(0)| = 4n-4$, except for $n=21$ where we have $|h_n(0)|=4n$.
\section{Maximal repetitions}
For each $h_n$ of Section~\ref{morphisms}, word 011 is a factor of $h_n(0)$ and $110$ is a factor of $h_n(1)$. It follows that $|h_n^m(1)|$ becomes arbitrarily large as $m$ increases, and that every factor of $h_n^\omega(0)$ is a factor of $h_n^m(1)$ for some $m$.
Let an occurrence of $v$ in $h_n^\omega(0)$ be written $h_n^\omega(0) = xv{\bf y}$. Suppose that $v$ has period $q$. We can
write $x = x'x''$, ${\bf y}=y'{\bf y''}$ such that $x''vy'$ has
period $q$, and $|x^{\prime\prime}vy'|$ is maximal. This is possible since every factor
of $h_n^\omega(0)$ is a factor of $h_n^m(1)$ for some $m$, and word 1 has two distinct left
extensions 01 and 11, and two distinct right extensions 10 and 11.
We refer to $x''vy'$ as the {\bf maximal period $q$ extension} of the occurrence $xv{\bf y}$ of $v$.
\section{Pansiot encoding}
Fix $n\ge 2.$ Let $\Sigma_n = \{1, 2, \ldots, n\}$. Let
$v\in\Sigma_n^*$ have length $m\ge n-1$, and write $v=v_1v_2\cdots
v_m$, $v_i\in \Sigma_n$. In the case where every factor of $v$ of
length $n-1$ contains $n-1$ distinct letters, we define the {\bf
Pansiot encoding of $v$} to be the word $b(v) = b_1b_2\cdots
b_{m-(n-1)}$ where for $1\le i\le m-n+1$
$$b_i=\left\{\begin{array}{ll}0,&v_i=v_{i+n-1}\\1,&\mbox{otherwise. }
\end{array}\right.$$
We can recover $v$ from $b(v)$ and $v_1v_2\ldots v_{n-1}$. We see
that $v$ has period $q$ if and only if $b(v)$ does. The exponent
$|v|/q$ of $v$ corresponds to an exponent ${\displaystyle
{|v|-n+1\over q}}$ of $b(v)$.
Let $S_n$ denote the symmetric group on $\Sigma_n$ with identity
{\tt id} and left multiplication, i.e.,
$$(fg)(i) = f(g(i)) \mbox{ for }f,g\in S_n,i\in\Sigma_n.$$
Let $\sigma:\{0,1\}^*\rightarrow S_n$ be the semigroup
homomorphism generated by
\begin{eqnarray*}
\sigma(0)&=&\left(\begin{array}{cccccc} 1&2&\cdots&(n-2)&(n-1)&n\\
2&3&\cdots&(n-1)&1&n
\end{array}\right)\\
\sigma(1)&=&\left(\begin{array}{cccccc} 1&2&\cdots&(n-2)&(n-1)&n\\
2&3&\cdots&(n-1)&n&1
\end{array}\right)
\end{eqnarray*}
One proves by induction that
\begin{eqnarray}\label{kernel}
\sigma(b(v))&=&\left(\begin{array}{cccccc} 1&2&\cdots&(n-2)&(n-1)&n\\
v_{m-n+2}&v_{m-n+3}&\cdots&v_{m-1}&v_m&\hat{v}
\end{array}\right)
\end{eqnarray}
where $\hat{v}$ is the unique element of
$\Sigma \setminus \{v_m,v_{m-1},\ldots,v_{m-n+2}\}$.
Suppose that $PE\in\Sigma_n^*$ is a repetition of period $q=|P|>0$ with
$|E|\ge n-1$. It follows from (\ref{kernel}) that
$\sigma(b(P))={\tt id}$; i.e.\ that $P$ is in the kernel of
$\sigma$. We refer to $b(PE)$ as a {\bf kernel repetition} of
period $q$. Conversely, if $u\in\Sigma_n^*$ and $b(u)$ is a kernel
repetition of period $q$, then we may write $u=PE=EP'$ for some
words $P,P',E$ where $|P|=|P'|=q$.
Suppose that for a morphism $h:\{0,1\}^*\rightarrow \{0,1\}^*$ there is a $\tau\in S_n$ such that
$$
\begin{array}{lcl}\tau\cdot\sigma(h(0))\cdot\tau^{-1}&=&\sigma(0)\\
\tau\cdot\sigma(h(1))\cdot\tau^{-1}&=&\sigma(1)
\end{array}
$$
In this case we say that $h$ satisfies the `algebraic condition'.
\section{Kernel repetitions with markable excess}\label{markable}
Let a uniform morphism $h:\{0,1\}^*\rightarrow \{0,1\}^*$ be given. Let $|h(0)|=r>0$. A word $v\in\{0,1\}^*$ is {\bf markable} (with respect to $h$) if whenever $h(X)xv$ and $h(Y)yv$ are prefixes of $h^\omega(0)$ with $|x|,|y|<r$, then $x = y$. If a word is markable, its extensions are markable.
Let $U$ be the set of length 2 factors of $h^\omega(0)$. A word $v\in\{0,1\}^*$ is {\bf 2-markable} (with respect to $h$) if whenever
\begin{enumerate}
\item $u$, $u'\in U$,
\item $h(X)xv$ is a prefix of $h(u)$ with $|x|<r$, and
\item $h(Y)yv$ is a prefix of $h(u')$ with $|y|<r$,
\end{enumerate}
then $x = y$.
If $|v|= r$ and $v$ is a factor of $h^\omega(0)$, then $v$ is a factor of $h(u)$, some $u\in U$. It follows that if $v$ is 2-markable, then $v$ is markable. For each $n$, if $h=h_n$, we find $U=\{01,10,11\}$. It follows that all length $r$ factors $v$ are factors of $h(0110)$. A finite check shows that
if $|v|= r$ and $v$ is a factor of $h^\omega(0)$, then $v$ is 2-markable, hence markable.
Let $n$ be fixed, $15\le n\le 26$ and let $h=h_n$. One checks that
$h$ satisfies the algebraic condition. Suppose that $v=pe$ is a
kernel repetition with period $q=|p|$, where $h^\omega(0)=xv{\bf y}$.
Notice that every length $q$ factor of $pe$ is conjugate to $p$,
by the periodicity of $pe$. It follows that every length $q$
factor of $pe$ lies in the kernel of $\sigma$. Suppose that the
excess $e$ of $v$ is markable. Let $V= x^{\prime\prime}vy'$ be the
maximal period $q$ extension of the occurrence $xv{\bf y}$ of $v$. Write
$x=Xx'$, ${\bf y}=y'{\bf Y}$, so that $h^\omega(0)=XV{\bf Y}$. Write $V=PE=EP'$
where $|P|=q.$ Since $E$ is an extension of $e$, $E$ is markable.
Write $X=h(\chi)\chi'$ where $|\chi'|<r$ and write $XP =
h(\gamma)\gamma'$ where $|\gamma'|<r$. It follows from the
markability of $E$ that $\chi'=\gamma'$. Then the maximality of
$V$ yields $|\chi'|=|\gamma'|=0$. We may thus write $X=h(\chi)$,
$E=h(\eta)\eta'$, with $|\eta'|<r$. By the maximality of $V$, word
$\eta'$ must be the longest common prefix of $h(0)$ and $h(1)$.
Since $E$ is a prefix and suffix of $PE$ and $E$ is markable, we
know that $r$ divides $|P|$. In total then, we may write
$XPE=h(\chi\pi\eta)\eta'$ where $h(\pi)=P$, and $\eta$ is a prefix
of $\pi$. Also, since $h$ satisfies the algebraic condition,
$\sigma(\pi)={\tt id}$. Thus $\pi\eta$ is a kernel repetition in
$h^\omega(0)$. We see that $|PE|=r|\pi\eta|+|\eta'|$.
The maximality of $V$ implies that $\pi\eta$ is maximal with
respect to having period $|\pi|$. This means that if $\eta$ is
markable, we can repeat the foregoing construction. Eventually we
obtain a kernel repetition $\mathcal{P}\mathcal{E}$ with
non-markable excess $\mathcal{E}$. If it takes $s$ steps to arrive
at $\mathcal{P}\mathcal{E}$ then we find that
$|PE|=r^s|\mathcal{P}\mathcal{E}|+|\eta'|\sum_{i=0}^{s-1}r^{i}$
and $|P|=r^s|\mathcal{P}|$.
\section{Main result}
Let $n$ be fixed, $15 \leq n \leq 26$ and let $h = h_n$.
Suppose that $u_1$ is a factor of $h^\omega(0)$ with $|u_1|=\ell.$
Extending $u_1$ by a suffix of length at most $r-1$, and a prefix
of length at most $r-1$, we obtain a word $h(u_2)$, some factor
$u_2$ of $h^\omega(0)$, where $|u_2|\le \lfloor
(\ell+2(r-1))/r\rfloor$. Repeating the argument, we find that
$u_1$ is a factor of $h^2(u_3)$, some factor $u_3$ of
$h^\omega(0)$ where
\begin{equation}\label{iterate}
|u_3|\le \left\lfloor
\frac{\left\lfloor(\ell+2(r-1))/r\right\rfloor+2(r-1)}{r}
\right\rfloor.
\end{equation}
Define $$I(\ell,r) = \left\lfloor
\frac{\left\lfloor(\ell+2(r-1))/r\right\rfloor+2(r-1)}{r}
\right\rfloor.$$
Let ${\bf w}$ be the $\omega$-word over $\Sigma_n$ with prefix
$123\cdots (n-1)$ and Pansiot encoding $b({\bf w}) = h^\omega(0)$.
We will show that ${\bf w}$ contains no $\left({n\over
n-1}\right)^+$-powers. Suppose to the contrary that $pe$ is a
repetition in ${\bf w}$ with $|pe|/|p|
> n/(n-1)$ and $e$ a prefix of $p$.
First suppose that $|e|\geq(n-1)$. Let $PE=b(pe)$. Then $PE$ is a
kernel repetition. Let $\eta'$ be the longest common prefix of
$h(0)$ and $h(1)$. As in the previous section, replacing $pe$ and
$PE$ by longer repetitions of period $|P|$ if necessary, we may
assume that $h^\omega(0)$ contains a kernel repetition
$\mathcal{P}\mathcal{E}$ with non-markable excess $\mathcal{E}$
such that
$|PE|=r^s|\mathcal{P}\mathcal{E}|+|\eta'|\sum_{i=0}^{s-1}r^{i}$
and $|P|=r^s|\mathcal{P}|$.
We find that
\begin{eqnarray*}
1+{1\over n-1}&=&{n\over
n-1}\\
&<&{|pe|\over|p|}\\
&=&{|PE|+n-1\over|P|}\\
&=&{r^s|\mathcal{P}\mathcal{E}|+|\eta'|\sum_{i=0}^{s-1}r^{i}+n-1\over
r^s|\mathcal{P}|}\\
&=&{r^s|\mathcal{P}|+r^s|\mathcal{E}|\over
r^s|\mathcal{P}|}+{|\eta'|\sum_{i=1}^{s}r^{-i}\over
|\mathcal{P}|}+{n-1\over
r^s|\mathcal{P}|}\\
&<&1+{1 \over |\mathcal{P}|} \left(|\mathcal{E}|+|\eta'|{r\over
r-1}+n-1\right)
\end{eqnarray*}
so that
\begin{eqnarray*}
|\mathcal{P}|&<&(n-1)\left({|\mathcal{E}|}
+{|\eta'|} {r\over r-1}+{n-1}\right)\\
\end{eqnarray*}
and
\begin{eqnarray*}
|\mathcal{P}\mathcal{E}|&<&|\mathcal{E}|+(n-1)\left({|\mathcal{E}|}
+{|\eta'|} {r\over r-1}+{n-1}\right)\\
&\le&r+(n-1)\left(r
+(r-1) {r\over r-1}+{n-1}\right)\\
&\le&4n +(n-1)(9n-1)\\
&=&9n^2-6n+1.
\end{eqnarray*}
We use that $|\mathcal{E}|<r$ (since all factors of $h^\omega(0)$
of length $r$ or greater are markable) and $r\le 4n$ (as observed
in Section~\ref{morphisms}). Finally, since $\eta'$ is a proper
prefix of $h(0)$, $|\eta'|<r$.
One verifies that $I(9n^2-6n+1,r)=2$. Since every length
2 factor of $h^\omega(0)$ is a factor of $0110$, word $b(PE)$ must
be a factor of $h^2(0110)$. Let $v$ be the word of $\Sigma_n$ with
prefix $123\cdots (n-1)$ and Pansiot encoding $h^2(0110)$. Since
$b(PE)$ is a kernel repetition, word $v$ contains a repetition
$\hat{p}\hat{e}$ with $|\hat{e}|\ge n-1$. However, a computer
search shows that $v$ contains no such repetition.
We conclude that $|e|\le n-2$. In this case,
\begin{eqnarray*}
{n\over n-1}<{|pe|\over |p|}
&\implies&|e|n>|pe|\\
&\implies&(n-2)n-(n-1)>|b(pe)|\\
&\implies&n^2-3n+1>|b(pe)|
\end{eqnarray*}
However, $n^2-3n+1<9n^2-6n+1$, so that again $b(pe)$ must be a
factor of $h^2(0110)$, and $v$, defined as in the previous case,
must contain a $\left({n\over n-1}\right)^+$-power. However, a
computer search shows that word $v$ is $\left({n\over
n-1}\right)^+$-power free.
We have proved the following:
\noindent{\bf Main Result:} Let ${\bf w}$ be the word over
$\Sigma_n$ with prefix $123\cdots (n-1)$ and Pansiot encoding
$b({\bf w}) = h^\omega(0)$. Word ${\bf w}$ contains no
$\left({n\over n-1}\right)^+$-powers.
\section{Final Remarks}
Our result builds on that of \cite{ollagnier}, but uses somewhat
simpler arguments, taking advantage of properties of our specific
morphisms. In addition, we have specified bounds for the various
computer checks, rather than invoking mere decidability.
A large simplification results from the fact that our morphisms
give binary words with no kernel repetitions at all (even of small
exponent). When moving from $PE$ to $\pi\eta$ in
Section~\ref{markable} one can give the relationship between the
exponents of these two kernel repetitions.
$${|PE|\over|P|}={|\pi\eta|\over |\pi|}+{|\eta'|\over r|\pi|}.$$
If it takes $s$ steps to arrive from repetition $PE$ to a
repetition $\pi\eta$ with non-markable excess, then the exponents
differ by $${|\eta'|\over|\pi|}\sum_{i=1}^sr^{-i}.$$ In the
notation of \cite{ollagnier}, $PE$ corresponds to
$\mu^s(\pi,\eta)$, and has the largest exponent among the
$\mu^i(\pi,\eta)$, $0\le i\le s$. Unfortunately, \cite{ollagnier}
is marred by getting this backward, saying that for uniform
morphisms the largest exponent occurs either for $i = 0$ or for
$i=1$!
In fact, for the morphisms given for $n=5,6,7$, $\eta'$ is empty, so the aforementioned
reversal has no effect. However, for $8\le n\le 11$, $\eta'$ is
non-empty, and a more complicated check than indicated in
\cite{ollagnier} is necessary to ensure that the given
constructions work. Happily, they do indeed work, as a more
careful check shows.
Finally, we mention a few points regarding the search strategy for
finding morphisms. The second step of the strategy
indicated in \cite{ollagnier} calls for enumerating all candidate morphisms
of short enough length. A priori, this involves enumerating all binary
words of length at most $r$ which are Pansiot encodings of $\left({n\over n-1}\right)^+$-free words over $\Sigma_n$. Initially this was
part of our strategy. Unfortunately, our experience supports the conjecture in
\cite{shur}, that the number of these words grows approximately as $1.24^r$
(independently of $n$.)
For successive $r$ values we looked at all possible pairs $\langle h(0), h(1)\rangle$ such that $|h(0)|,|h(1)|\le r$ where $h(0),h(1)$ were Pansiot encodings of $\left({n\over n-1}\right)^+$-free words and satisfied
the algebraic condition; this allowed us to
verify the claim of \cite{ollagnier} that the morphisms presented therein for $5\le n\le 11$ are shortest possible `convenient morphisms'; the uniforms are all uniform, with
lengths around $4n-4$ in each case. However, storing all legal Pansiot encodings up to length $4n-4$
fills up a laptop with
2G RAM at around $n=15$. Therefore, our search program had to
migrate to computers with more and more RAM, simply to store Pansiot encodings. On the plus side,
we found a great number of `convenient morphisms' for $12\le n\le 17$, not just the ones presented in this paper.
To find morphisms for $n$ up to $26$ (and indeed for various other higher values of $n$) we adopted a different strategy. Using backtracking, we found legal
Pansiot encodings of length exactly $r=4n-4$ (or $r=4n$, in the case $n=21$), but only saved encodings $v$ for which the permutation $\sigma(v)$ was an $r$-cycle
(and thus a candidate for $h(1)$) or an $(r-1)$-cycle (and thus a candidate for $h(0))$. As soon as a candidate for $h(i)$ was found, it was tested together with
each previously found candidate for $h(1-i)$ to see whether a `convenient morphism' could be formed, in which case the search terminated. This search used very little memory, and
terminated quickly. For $n=26$, our $C^{++}$ code found the morphism in just over 6 hours.
\section{Acknowledgments}
We would like to thank Dr.\ Randy Kobes for facilitating access to
computational resources.
Some of the calculations were performed on the WestGrid high performance
computing system (\url{www.westgrid.ca}).
We have recently been informed that Dr.\ Micha\"el Rao has also announced a proof
of Dejean's conjecture. |
0905.0402 | \section{Introduction}
The $X(3872)$ was discovered at BELLE \cite{Choi:2003ue} and then later also
observed at ... and ...
Its decay into $J/\psi \pi \pi$ has been observed in \cite{Choi:2003ue,Abulencia:2005zc}
as well as into $J/\psi \pi \pi$ \cite{Abe:2005ix}. The quantum numbers
have been investigated in \cite{Abulencia:2006ma}, concluding that it must
correspond to $J^P=1^{++}$ or $J^P=2^{-+}$. Observed only on a neutral charge
state it is assumed to have isospin $I=0$. Its decay into $J/\psi \eta$ has
been investigated in \cite{Aubert:2004fc} but only an upper bound has been
found. Should this indicate that this decay is forbidden the $C$-parity $= -1$ of
the $X(3872)$ would be ruled out. It could also mean that in the particular
reaction of \cite{Aubert:2004fc} the $X(3872)$ was necessarily produced with
positive $C$-parity, without ruling out the possibility of a nearby state with
negative $C$-parity. The existence of two nearly degenerate $X(3872)$ states
appears in some theoretical models \cite{Terasaki:2007uv,danielaxial}. The most
popular view about the nature of this resonance is that it is made of $D
\bar{D}^*$ \cite{danielaxial,Liu:2008fh,Liu:2007bf,Dong:2008gb} y mcuhas mas, a
recent review can be seen in \cite{Liu:2008du}. One of the problems faced by
these models is the large ratio for
J/psi pi pi / J/psi pi pi pi
If the resonance has positive C-parity the numerator can go via $\rho J/\psi $ as
supported by the experiment \cite{}. However, the $X(3872)$ state has $I=0$ and
then isospin is violated. On the contrary the denominator can go through
$\omega J/\psi $ as supported by experiment \cite{}, in which case there is no
violation of isospin. That the ratio is so large in spite of the violation of
isospin found a plausible explanation in \cite{Swanson:2003tb}, where the state
was supposed to be largely $D^0 \bar{D}^{*0}$ but with some coupling to both
$\omega J/\psi $ and $\rho J/\psi $. Even if the coupling to $\rho J/\psi $ is
small, as expected from symmetry breaking, the larger phase space for
$\rho J/\psi $ decay than for $\omega J/\psi $, because of the large width of the
$\rho$ can account for the large ratio. Although other charged
$D \bar{D}^*$ components can appear in the wave function, the neutral charge
component is preferred since it is the one much closest to threshold and hence
should have the largest weight. The idea is intuitive and extended, see
\cite{braaten}. The idea on the dominance of the neutral component is worth
pursuing. Indeed, in \cite{danielaxial} where a dynamical theory for the
generation of the $X(3872)$ resonance based on the hidden gauge approach for the
vector meson interaction was done, isospin symmetry was kept and the masses of
the $D$ and $ \bar{D}^*$ mesons were taken equal. The fact that the binding
energy for the $D^0 \bar{D}^{*0}$ is so small advises to revise the model to
account for the mass differences with the charged $D \bar{D}^*$, which can induce isospin breaking and a
dominance of the $D^0 \bar{D}^{*0}$ in the wave function. In the present paper
we will face this problem and will discuss qualitatively as well as quantitatively,
the limit of zero binding energy, with interesting results. We will also revisit
the interpretation of the reaction reaction production de D D* \cite{}
showing that it gives support to the existence of the $X(3872)$ as a
$D \bar{D}^*$ narrow state of positive parity, without ruling out the possible
existence of a wider one of negative parity. We discuss the decay modes of the
negative C parity state and speculate on where could it be found and the
possible difficulties in the observation.
\section*{Acknowledgments}
This work is
partly supported by DGICYT contract number
FIS2006-03438. We acknowledge the support of the European Community-Research
Infrastructure Integrating Activity
"Study of Strongly Interacting Matter" (acronym HadronPhysics2, Grant Agreement
n. 227431) under the Seventh Framework Programme of EU.
Work supported in part by DFG (SFB/TR 16, ``Subnuclear Structure of Matter'').
\section{Introduction}
The $X(3872)$ was discovered at Belle \cite{belledisc} and then later was also
observed at CDFII and D0 collaborations and BaBar \cite{conf1,conf2,conf3}.
In all these experiments the $X$ has been discovered and observed in the decay channel
$J/\psi\pi^+\pi^-$. There is strong evidence that the dipion generated in this decay
channel comes from a $\rho$ meson \cite{cdf2pi}. Later on also the decays of the $X$ into
$J/\psi\pi^+\pi^-\pi^0$ and $J/\psi\gamma$ have been observed \cite{bellegj}, this latter
decay channel indicating that the C-parity of the $X$ is positive. The quantum numbers of the $X(3872)$
have been investigated in \cite{xqn}, concluding that it must
correspond to $J^P=1^{++}$ or $J^P=2^{-+}$. Observed only on a neutral charge
state it is assumed to have isospin $I=0$. Its decay into $J/\psi \eta$ has
been investigated in \cite{nojeta} but only an upper bound has been
found. The non observation of the decay $J/\psi\eta$ is a further evidence of the positive C-parity of the $X$.
It could also mean that in the particular
reaction of \cite{nojeta} the $X(3872)$ was necessarily produced with
positive $C$-parity, without ruling out the possibility of a nearby state with
negative $C$-parity. The existence of two nearly degenerate $X(3872)$ states
appears in some theoretical models \cite{Terasaki:2007uv,danielaxial}. The most
popular view about the nature of this resonance is that it is made of $D
\bar{D}^*$ \cite{danielaxial,Liu:2008fh,Liu:2007bf,Dong:2008gb,Swanson:2003tb,gutsch1,gutsch2}, a
recent review can be seen in \cite{Liu:2008du}. One of the problems faced by
these models is the large ratio for
\begin{eqnarray}
\frac{{\cal B}(X\rightarrow J/\psi\pi^+\pi^-\pi^0)}{{\cal B}(X\rightarrow J/\psi\pi^+\pi^-)}&=&1.0\pm0.4\pm0.3\textrm{ .}
\end{eqnarray}
Indeed, since the resonance has positive C-parity the denominator can go via $J/\psi\rho$ as
supported by the experiment \cite{cdf2pi}. However, the $X(3872)$ state has $I=0$ and
then isospin is violated. On the contrary the numerator can go through
$J/\psi\omega$ as supported by experiment \cite{bellegj}, in which case there is no
violation of isospin. The fact that the ratio is so large in spite of the violation of
isospin found a plausible explanation in \cite{Swanson:2003tb,gutsche}, where the state
was supposed to be largely $D^0 \bar{D}^{*0}$ but with some coupling to both
$J/\psi\omega$ and $J/\psi\rho$. Even if the coupling to $J/\psi\rho$ is
small, as expected from isospin symmetry breaking, the larger phase space for
$J/\psi\rho$ decay than for $J/\psi\omega$, because of the large width of the
$\rho$, can account for the large ratio. Although other charged
$D \bar{D}^*$ components can appear in the wave function, the neutral charge
component is preferred since it is the one closest to threshold and hence
should have the largest weight. The idea is intuitive and widely accepted, see
\cite{braaten}. The idea on the dominance of the neutral component is worth
pursuing. Indeed. In \cite{danielaxial}, where a dynamical theory for the
generation of the $X(3872)$ resonance based on the hidden gauge approach for the
vector-meson interaction was done, isospin symmetry was kept and the masses of
the charged and neutral $D$ mesons were taken equal. The fact that the binding
energy for the $D^0 \bar{D}^{*0}$ is so small advises to revise the model to
account for the mass differences with the charged $D \bar{D}^*$, which can induce isospin breaking and a
dominance of the $D^0 \bar{D}^{*0}$ in the wave function. In the present paper
we will face this problem and will discuss qualitatively as well as quantitatively,
the limit of zero binding energy, with interesting results. We will also revisit
the interpretation of the reaction production of $D\bar D^*$ \cite{expcross}
showing that it gives support to the existence of the $X(3872)$ as a
$D \bar{D}^*$ narrow state of positive parity, without ruling out the possible
existence of a broader one of negative parity. We discuss the decay modes of the
negative C-parity state and speculate on where could it be found and the
possible difficulties in its observation.
The present work should also be looked at with some perspective. Although most of the work
for the $X(3872)$ has concentrated on the molecular $D\bar D^*$ picture, as we have mentioned,
since the mass of is so close to the threshold, there are other pictures proposed to describe it
in terms of quarks, tetraquarks and other dynamical pictures (see \cite{qq1,qq2,qq3} for reviews).
The molecular picture of some meson resonances is catching up, particularly when it comes to interpret
states that do not fit clearly in a $q\bar q$ picture. This is the case of the $X$, $Y$ and $Z$ resonances
recently discovered at the $B$-factories for which there are also other structures proposed and about which
there is an intensive debate (see \cite{olsen1} for a recent overview on the subject).
This work is organized as follows: in the next section we shortly explain our phenomenological
Lagrangian and present our framework to generate dynamically resonances from the interaction
of pseudoscalars with vector-mesons. In the same section we show results with and without isospin
violation and we also show what happens when the binding energy goes to zero. In section III we
comment on the two C-parity states that our model generates and in section IV we make our
final remarks and conclusions.
\section{Theoretical Model}
The framework that we describe here is explained in more details in \cite{danielaxial} and references therein.
We want to study the interaction of a pseudoscalar with a vector-meson that, in s-wave, has the quantum numbers
of an axial: $1^+$.
The starting point of our model consists of the fields belonging to the 15-plet and a singlet of $SU(4)$ describing the
pseudoscalar and vector-mesons:
\begin{widetext}
\begin{eqnarray}
\Phi&=&\left(
\begin{array}{cccc}
\frac{\eta }{\sqrt{3}}+\frac{\pi^0}{\sqrt{2}}+\frac{\eta'
}{\sqrt{6}} & \pi ^+ & K^+ & \overline{D}^0 \\& & & \\
\pi ^- & \frac{\eta }{\sqrt{3}}-\frac{\pi
^0}{\sqrt{2}}+\frac{\eta'}{\sqrt{6}} & K^0 & D^- \\& & & \\
K^- & \overline{K}^0 & \sqrt{\frac{2}{3}} \eta'-\frac{\eta
}{\sqrt{3}} & {D_s}^- \\& & & \\
D^0 & D^+ & {D_s}^+ & \eta _c
\end{array}
\right) \\
\cal{V}_\mu&=&\left( \begin{array}{cccc}
{\rho_\mu^0 \over \sqrt{2}}+{\omega_\mu \over \sqrt{2}} & \rho^+_\mu & K^{*+}_\mu & \bar D^{*0}_\mu \\ & & & \\
\rho^{*-}_\mu & {-\rho^0_\mu \over \sqrt{2}}+{\omega_\mu \over \sqrt{2}} & K^{*0}_\mu & D^{*-}_\mu \\& & & \\
K^{*-}_\mu & \bar K^{*0}_\mu & \phi_\mu & D_{s\mu}^{*-} \\& & & \\
D^{*0}_\mu & D^{*+}_\mu & D_{s\mu}^{*+} & J/\psi_\mu \\ \end{array} \right).
\end{eqnarray}
\end{widetext}
Note that these fields differ from those used in \cite{danielaxial} because of the inclusion of $\eta$-$\eta'$
and $\omega$-$\phi$ mixing.
For each one of these fields a current is defined:
\begin{eqnarray}
J_\mu&=&(\partial_\mu \Phi)\Phi-\Phi\partial_\mu\Phi \\
\cal{J}_\mu&=&(\partial_\mu \cal{V}_\nu)\cal{V}^\nu-\cal{V}_\nu\partial_\mu \cal{V}^\nu.
\end{eqnarray}
The Lagrangian is constructed by coupling these currents:
\begin{eqnarray}
{\cal L}_{PPVV}&=&-{1\over 4f^2}Tr\left(J_\mu\cal{J}^\mu\right). \label{lag}
\end{eqnarray}
In the way it is constructed this Lagrangian is $SU(4)$ symmetric, but we know that $SU(4)$ symmetry is badly
broken in nature. To take this into account we will break the $SU(4)$ symmetry of the Lagrangian in the following way:
Assuming vector-meson dominance we recognize that the interaction behind our Lagrangian is the exchange of a vector
meson in between the two hadronic currents. If the initial and final pseudoscalars (and vector-mesons), in a given process,
have different charm quantum number, it means that the vector-meson exchanged in such a process is a charmed meson,
and hence a heavy one. In these cases we suppress the term in the Lagrangian containing such processes by a factor $\gamma=m_L^2/m_H^2$ where $m_L$ is the typical value of a light vector-meson mass (800 MeV) and $m_H$ the typical value of the heavy vector-meson mass (2050 MeV). We also
suppress, in the interaction of $D$-mesons the amount of the interaction which is driven by a $J/\psi$ exchange by the factor
$\psi=m_L^2/m_{J/\psi}^2$. Another source of symmetry breaking will be the meson decay constant $f$ appearing in the Lagrangian.
For light mesons we use $f=f_\pi=93$ MeV but for heavy ones $f=f_D=165$ MeV.
So, for a given process $(P(p)V(k))_i\rightarrow (P'(p')V'(k'))_j$ we have the amplitude:
\begin{eqnarray}
{\cal M}_{ij}(s,t,u)&=&-{\xi_{ij}\over4f_i f_j}(s-u)\epsilon . \epsilon ' \label{ampli}
\end{eqnarray}
where $s$ and $u$ are the usual Mandelstam variables, $f_i$ is the pseudoscalar $i$ meson decay constant, $\epsilon$ are the vector-meson polarization vectors and $i$, $j$ refer to the initial and final channels in the coupled channel space. The coefficient matrices $\xi_{ij}$ can be directed calculated from the Lagrangian of eq. (\ref{lag}) in charge basis. The $\xi_{ij}$ coefficients in charge basis are given in the appendix.
The amplitude in eq. (\ref{ampli}) is projected in s-wave and plugged into the scattering equation for the coupled channels:
\begin{eqnarray}
T&=&V+VGT. \label{bseq}
\end{eqnarray}
In this equation $G$ is a diagonal matrix with each one of its elements given by the loop function for each channel in the coupled channel space. For channel $i$ with mesons of masses $m_1$ and $m_2$ $G_{ii}$ is given by:
\begin{eqnarray}
G_{ii}&=&{1 \over 16\pi ^2}\biggr( \alpha _i+Log{m_1^2 \over \mu ^2}+{m_2^2-m_1^2+s\over 2s}
Log{m_2^2 \over m_1^2}\nonumber\\
&+ &{p\over \sqrt{s}}\Big( Log{s-m_2^2+m_1^2+2p\sqrt{s} \over -s+m_2^2-m_1^2+
2p\sqrt{s}}\nonumber\\
&+&Log{s+m_2^2-m_1^2+2p\sqrt{s} \over -s-m_2^2+m_1^2+ 2p\sqrt{s}}\Big)\biggr) \label{loop}
\end{eqnarray}
where $p$ is the three momentum of the two mesons in the center of mass frame. The two parameters $\mu$ and $\alpha$ are not independent, we fix $\mu$=1500 MeV and change $\alpha$ to fit our results within reasonable values in the natural range \cite{hyodo}. We actually use two $\alpha$ as free parameters, one for loops with only light mesons in the channel $i$, we call $\alpha_L$ and set it to $\alpha_L$=-0.8, the value used in \cite{danielaxial} to fit the low lying axial resonances. For loops with heavy particles we use $\alpha_H$ and we comment, in what follows, the effects of changing this parameter. Note that just the combination $\alpha-Log(\mu^2)$ is the free parameter in $G$ of eq. (\ref{loop}). The parameter $\mu$ is there to set a scale of energies, it can be chosen arbitrarily and hence is not related with the cut off in three momentum which can be alternatively used to evaluate $G_{ii}$ from the loop function of two meson propagators. Once $\mu$ is fixed then there is a relationship between $\alpha$ of eq. (\ref{loop}) and an equivalent cut off \cite{ollerloop}. In any case, the equivalence of the cut off method and the dimensional regularization of eq. (\ref{loop}) in a certain region of energies requires the cut off to be reasonably bigger than the on shell three momenta of the intermediate states, which is fulfilled in the energy regime that we study here. In the present case, the $D\bar D^*$ states, with 20 MeV above threshold have a three momentum around 195 MeV, while the cut offs are of the order of 650-850 MeV, as we shall see later on.
The imaginary part of the loop function ensures that the T-matrix is unitary, and since this imaginary part is known, it is possible to do an analytic continuation for going from the first Riemann sheet to the second one. Possible physical states (resonances) are identified as poles in the T-matrix calculated in the second Riemann sheet for the channels which have the threshold below the resonance mass.
\subsection{Isospin Symmetric Case}
If we set the masses of all mesons belonging to a same isospin multiplet to a common value, our results will be isospin symmetric.
Moreover we can consider the transformation under C-parity of pseudoscalar and vector-mesons in order to construct C-parity
symmetric states:
\begin{eqnarray}
\hat{C} P &=&\bar P \\
\hat{C} V &=&-\bar V
\end{eqnarray}
States like $D\bar D^*$ and $\bar D D^*$ mix up to form a positive and a negative C-parity state. The same happens for the kaons and the $D_s$ mesons. If one writes the $\xi_{ij}$ coefficients that appear in the amplitude of eq. (\ref{ampli}) in C-parity basis for all two meson states with quantum numbers C=0 and S=0, the coupled channel space splits into two, a positive and a negative C-parity part. These coefficients are given in the appendix.
In charge basis, for positive C-parity one has the following channels: $\bar K^{*0}K^0-c.c.$, $\rho^+\pi^--c.c.$, $\bar D^{*0}D^0-c.c.$, $D^{*+}D^--c.c.$, $D_s^{*+}D_s^--c.c.$ and $K^{*+}K^--c.c.$. While for negative C-parity the channels are: $\rho^+\pi^-+c.c.$, $K^{*+}K^-+c.c.$, $\rho^0\pi^0$, $\omega\pi^0$, $\phi\pi^0$, $\rho^0\eta$, $\rho^0\eta\prime$, $\bar K^{*0}K^0+c.c.$, $D^{*+}D^-+c.c.$, $\bar D^{*0}D^0+c.c.$, $\rho^0\eta_c$, $J/\psi\pi^0$, $\omega\eta$, $\phi\eta$, $\omega\eta\prime$, $\phi\eta\prime$, $\omega\eta_c$, $\phi\eta_c$, $J/\psi\eta$, $J/\psi\eta\prime$, $D_s^{*+}D_s^-+c.c.$ and $J/\psi\eta_c$.
For the masses of the mesons we use the following values:
$m_\pi$=137.5 MeV, $m_K$=496 MeV, $m_\eta$=548 MeV, $m_D$=1867.5 MeV, $m_{D_s}$=1968 MeV, $m_{\eta_c}$=2980 MeV, $m_{\eta\prime}$=958 MeV, $m_\rho$=775 MeV, $m_{K^*}$=894 MeV, $m_\omega$=783 MeV, $m_\phi$=1019 MeV, $m_{D^*}$=2008.5 MeV, $m_{D_s^*}$=2112 MeV and $m_{J/\psi}$=3097 MeV.
If we set $\alpha_H$=-1.34, which is equivalent to a cut-off of 830 MeV in the three momentum, we get two poles with opposite C-parity, the positive one at 3866 MeV with a width smaller than 1 MeV and the negative one at (3875-25$i$) MeV, which means a width around 50 MeV. The poles appear in isospin I=0, as we determine from combining the charge states into definite isospin states. Now while increasing the value of $\alpha_H$ (lowering the cut-off) the poles approach the threshold (at 3876 MeV in the isospin symmetric case). The negative C-parity pole touches the threshold for $\alpha_H$ values bigger that -1.33 (cut-off of 820 MeV), while the positive C-parity one reaches the threshold for $\alpha_H$ around -1.185 (cut-off equivalent to 660 MeV). Once the pole crosses the threshold it does not appear in the second Riemann sheet, it is no longer a resonance, but becomes a virtual state. Yet a peak can be seen in the cross section of some channels, but can not be identified as a pole in the second Riemann sheet of the T-matrix.
\subsection{Isospin Breaking}
In order to investigate the isospin breaking we define the following quantities:
$\Delta m_\pi$=2.5 MeV, $\Delta m_K$=-2 MeV, $\Delta m_D$=2.5 MeV, $\Delta m_{K^*}$=-2 MeV and $\Delta m_{D^*}$=1.5 MeV. In this way the masses of the members of a multiplet split: for the charged members of a multiplet the mass will be equal to $m+\Delta m$ while for the neutral members it will be $m-\Delta m$.
Now there are two $\bar D D^*$ thresholds nearby, the neutral one at 3872 MeV and the charged one at 3880 MeV. The $X(3872)$ state is a very weakly $D^0 \bar D^{*0}$ bound state and the fact that the binding energy is much smaller than the difference between these two thresholds could reflect itself in a large isospin violation in observables.
For simplicity let us consider, for the moment, a toy model with only two channels, with neutral and charged $D$ and $D^*$ mesons. In this model we assume the potential $V$ to be a 2x2 matrix:
\begin{eqnarray}
V&=&\left(\begin{tabular}{cc} $v$ & $v$ \\ $v$ &$v$ \end{tabular}\right),
\end{eqnarray}
with $v$ constant, which indeed is very close to the real one in a small range of energies.
In this case the solution of the scattering equation (\ref{bseq}) is:
\begin{eqnarray}
T&=&\frac{V}{1-vG_{11}-vG_{22}}
\end{eqnarray}
where $G_{11}$ and $G_{22}$ are the loop function calculated for channels 1 and 2 respectively. If there is a pole at $s$=$s_R$ we can expand $T$ close to this pole as:
\begin{eqnarray}
T_{ij}&=&\frac{g_ig_j}{s-s_R}
\end{eqnarray}
where $g_i$ is the coupling of the pole to the channel $i$. The product $g_ig_j$ is the residue at the pole and can be calculated with:
\begin{eqnarray}
\lim_{s\rightarrow s_R} (s-s_R)T_{ij}&=&\lim_{s\rightarrow s_R} (s-s_R) \frac{V_{ij}}{1-vG_{11}-vG_{22}}
\end{eqnarray}
We can apply the l'H\^opital rule to this expression and we get:
\begin{eqnarray}
\lim_{s\rightarrow s_R} (s-s_R)T_{ij}&=& \frac{V_{ij}}{-v( \frac{dG_{11}}{ds}+\frac{dG_{22}}{ds})} \label{lhopt}
\end{eqnarray}
If one has a resonance lying right at the threshold of channel 1 the couplings $g_i$ will be zero, since the derivative of the loop function $G_{11}$, in the denominator of eq. (\ref{lhopt}) is infinity at threshold. This is a general property which has its roots in basic Quantum Mechanics as shown in \cite{jnieves}. In figure \ref{fig1} we show plots of the real part of the loop function for the neutral and charged $D$ meson channels.
It is interesting to note that eq. (\ref{lhopt}) for just one channel is the method used to get couplings of bound states to their building blocks in studies \cite{gutsche,hana0} of dynamically generated states following the method of the compositness condition of Weinberg \cite{weinberg,efimov}.
We will come back to this issue again with the realistic model. Now, in what follows, the arguments used do not require the toy model any longer.
\begin{figure}
\begin{center}
\includegraphics[width=5cm,angle=-90]{loops.ps}
\caption{Loops} \label{fig1}
\end{center}
\end{figure}
Suppose that the $X(3872)$ decays through the diagram in figure \ref{fig2}. In this figure the $D$ mesons can be either charged or neutral. For the isospin I=1 state with the $\rho$ meson in the final state, the diagrams with neutral $D$ mesons interfere destructively with those with charged $D$ mesons, while in the $\omega$ case they sum up. If the vertices have the same strength for $\rho$ and $\omega$ production (this is the case in the framework of the hidden gauge formalism \cite{hidden1,hidden2,raquelhidden}) the ratio of the amplitudes will be given by the ratio of the difference between the charged and neutral loops divided by the sum of the loops:
\begin{eqnarray}
R_{\rho/\omega}&=&\left(\frac{G_{11}-G_{22}}{{G_{11}+G_{22}}}\right)^2 \label{ratio}
\end{eqnarray}
\begin{figure}[h]
\begin{center}
\includegraphics[width=5cm,angle=-0]{xdecay.eps}
\caption{$X$ decay} \label{fig2}
\end{center}
\end{figure}
In the isospin symmetric case, the charged and neutral loops are equal, because these loops depend only on the masses, and therefore this ratio would be zero because the $\rho$ contribution would vanish (no isospin violation).
Actually the decays $X\rightarrow J/\psi\rho$ and $X\rightarrow J/\psi\omega$ are not allowed because of phase-space, for $\rho$ and $\omega$ with fixed masses, but can occur when their mass distribution is considered and will be seen in the decays $X\rightarrow J/\psi\pi\pi$ and $X\rightarrow J/\psi\pi\pi\pi$ respectively, where the two and three pion states are the result of the decays of the $\rho$ and $\omega$. Hence to measure the ratio of the $X$ decaying to two and three pions plus a $J/\psi$ one has to multiply the expression in (\ref{ratio}) by the ratio of the phase-space available for the decay of a $\rho$ to two pions divided by the phase-space for the decay of a $\omega$ into three pions:
\begin{widetext}
\begin{eqnarray}
\frac{{\cal B}(X\rightarrow J/\psi\pi\pi )}{{\cal B}(X\rightarrow J/\psi\pi\pi\pi )}&=&\left(\frac{G_{11}-G_{22}}{{G_{11}+G_{22}}}\right)^2 \frac{\int_0^{\infty } q \mathcal{S}\left(s,m_{\rho },\Gamma _{\rho }\right) \theta \left(m_X-m_{J/\psi }-\sqrt{s}\right) \,
ds}{\int_0^{\infty } q \mathcal{S}\left(s,m_{\omega },\Gamma _{\omega }\right) \theta \left(m_X-m_{J/\psi }-\sqrt{s}\right) \, ds}
\frac{{\cal B}_\rho}{{\cal B}_\omega} \label{branchtot}
\end{eqnarray}
\end{widetext}
where ${{\cal B}_\rho}$ and ${{\cal B}_\rho}$ are the branching fractions of $\rho$ decaying into two pions ($\sim$ 100 \%) and $\omega$ decaying into three pions ($\sim$ 89 \%), $\theta(y)$ is the Heaviside theta function and $\mathcal{S}\left(s,m,\Gamma\right)$ is the spectral function of the mesons given by:
\begin{eqnarray}
\mathcal{S}\left(s,m,\Gamma\right)&=&-\frac{1}{\pi} Im\left(\frac{1}{s-m^2+i \Gamma m}\right)
\end{eqnarray}
From the expression in eq. (\ref{ratio}) one observes that the isospin violation in the decay of the $X$ will be proportional to the square of the difference between the loops with charged and neutral $D$ mesons. Moreover if one looks at figure \ref{fig1} one sees that this difference is maximal at the threshold of the $D^0\bar D^{*0}$, such that the closer the resonance is to that threshold (the smaller the binding energy) the bigger is the isospin violation in the decay of the $X$. If the $X$ is right over the threshold, the value of $R_{\rho/\omega}$, with the loops calculated with dimensional regularization for $\rho$ and $\omega$ fixed masses, is:
\begin{eqnarray}
R_{\rho/\omega}&=&0.032
\end{eqnarray}
This is a measure of the isospin violation in the decay of the $X$, which is only about 3\% in spite of the fact that we have chosen the conditions to maximize it. This ratio is of the same order of magnitude as the one obtained in \cite{gutsch2} (see eq. (36) of this paper). However, even this small isospin breaking can lead to sizable values of the ratio of eq. (\ref{branchtot}) when one takes into account the mass distributions of the $\rho$ and $\omega$, which provide different effective phase-spaces in this two possible $X$ decays. Thus, using eq. (\ref{branchtot}), which considers explicitly the $\rho$ and $\omega$ mass distributions, we find the branching ratio:
\begin{eqnarray}
\frac{{\cal B}(X\rightarrow J/\psi\pi^+\pi^-\pi^0 )}{{\cal B}(X\rightarrow J/\psi\pi^+\pi^- )}&=&1.4 \label{res14}
\end{eqnarray}
which is compatible with the value $1.0\pm0.4$ from experiment \cite{bellegj}.
\section{The two C-parity states}
There are six channels with charm and strangeness equal to zero and positive C-parity. We show in table \ref{tab1} the couplings of the pole obtained solving the scattering equation for these channels.
\begin{table}
\begin{center}
\caption{Couplings of the pole at (3871.6-$i$0.001) MeV to the channels ($\alpha_H$=-1.27 here).} \label{tab1}
\begin{tabular}{c|c}
\hline
Channel & $|g_{R\rightarrow PV}|$ [MeV] \\
\hline
\hline
& \\
$\pi^-\rho^+-c.c.$ & 1.4 \\
\hline
& \\
$K^- \bar K^{*+}-c.c.$ & 8.7 \\
\hline
& \\
$K^0 \bar K^{*0}-c.c.$ & 7.4 \\
\hline
& \\
$D^- \bar D^{*+}-c.c.$ & 2982 \\
\hline
& \\
$D^0 \bar D^{*0}-c.c.$ & 3005 \\
\hline
& \\
$D_s^- \bar D_s^{*+}-c.c.$ & 2818 \\
\hline
\end{tabular}
\end{center}
\end{table}
One can see in table 1 that, although there is some isospin violation in the couplings, it is very small, less than 1 \%. One might think that if the binding energy is much smaller than the difference between the neutral and charged thresholds (8 MeV), the resonance will be mostly dominated by the neutral channel, the one closest to the threshold. The binding energy in the case of the pole in Table \ref{tab1} is 0.4 MeV. As we mentioned, in the limit that the binding energy goes to zero, the couplings should all vanish. We show in figure \ref{plotcoup1} that, indeed, the coupling of the $X$ to $D^0\bar D^{*0}$goes to zero for small binding energies, and in figure \ref{plotdiff} we show that even though the difference between the neutral and charged couplings grows for small binding energies, they are of the same order of magnitude. The wave function of the $X(3872)$ is, thus, very close to the isospin I=0 combination of $D^0\bar D^{*0}-c.c.$ and $D^-\bar D^{*+}-c.c.$ and has a sizable fraction of the $D_s^-D_s^{*+}-c.c.$ state.
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=5cm,angle=-90]{plotcoup.ps} \\
\includegraphics[width=5cm,angle=-90]{plotlog.ps}
\end{tabular}
\caption{Coupling of the $X$ to the $D^0\bar D^{*0}$ channel for different biding energies.} \label{plotcoup1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=5cm,angle=-90]{plotdiff.ps}
\caption{Difference of the coupling of the $X$ with neutral and charged $D\bar D^*$ channels.} \label{plotdiff}
\end{center}
\end{figure}
From figure \ref{plotdiff} we notice that indeed the isospin violation in the couplings of the $X$ to the $D\bar D^*$ channels is bigger for small binding energies, but it reaches a maximum of about 1.4\% which is a very small value. We can go back to the argument that lead to eq. (\ref{ratio}) and the only difference would be that the $G_{11}$ and $G_{22}$ functions would be multiplied by the $D^0\bar D^{*0}-c.c.$ and $D^+\bar D^{*-}$ couplings from table \ref{tab1}, which barely affect the results obtained in eq. (\ref{res14}), since the differences in the couplings are much smaller than those between $G_{11}$ and $G_{22}$. Although a $D^0\bar D^{*0}+c.c.$ state is proposed for the $X(3872)$ in \cite{braaten2}, a formalism accounting for the charged component of this resonance is also presented in \cite{braaten3}. Our results would correspond to taking a value of the parameter $\gamma_1$ much bigger than $\kappa_1(0)$ in size in eq. (39b) of \cite{braaten3}. However, no claims for any particular value of $\gamma_1$ are made in \cite{braaten3}, where only the formalism is presented.
As we already mentioned, we find a second state with negative C-parity. Some of the 22 channels with negative C-parity
have isospin I=0 to which the resonance can decay. There are also pure isospin I=1 channels but, although the generated resonance is an isospin I=0 state, these isospin I=1 channels will couple to it since we are considering here some amount of isospin violation coming from the different masses of charged and neutral members of a same isospin multiplet.
For values of $\alpha_H$ similar to those used in the generation of the $X(3872)$ ($\alpha_H$-1.27), the pole with negative C-parity is in the wrong Riemann sheet, but its effects can still be seen in the cross sections of some channels. We show in figure \ref{figcross} the $|T|^2$ plots of some channels. By taking smaller values of $\alpha_H$ (around -1.36) one can recover a pole below threshold with $\sqrt s$=(3871.4-i26.2) MeV. In our previous work of \cite{danielaxial} this state was narrower. The reason for its relative big width in the present work is the inclusion of the $\eta$-$\eta\prime$ mixing. As was explained in \cite{withzou} the hidden charm dynamically generated states that we obtain are $SU(3)$ singlets and if one considers only the mathematical $\eta_8$ without its mixing with a $SU(3)$ singlet state $\eta_1$, the open channels for the resonance to decay are $SU(3)$ octets and are therefore suppressed. Only when considering also this singlet and hence the physical $\eta$ and $\eta\prime$ states the open channels acquire a $SU(3)$ singlet component to which the resonance strongly couples.
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=2.5cm,angle=-90]{cross1.ps} & \includegraphics[width=2.5cm,angle=-90]{cross2.ps} \\
\includegraphics[width=2.5cm,angle=-90]{cross3.ps} & \includegraphics[width=2.5cm,angle=-90]{cross4.ps}
\end{tabular}
\caption{The $|T|^2$ plot for some of the negative C-parity channels.} \label{figcross}
\end{center}
\end{figure}
The channels shown in figure \ref{figcross} are those where there is phase-space available for the resonance to decay and to which it couples most strongly.
Next we want to compare the results obtained from our approach with the experiment data from \cite{expcross} for $B\rightarrow KD^0\bar D^{*0}$. For this we follow the approach of \cite{hanhart,danielaxial} where one shows that the experimental data for $d\Gamma/dM_{inv}(D \bar D^{*})$ are proportional to $p|T_{D\bar D^*\rightarrow D\bar D^*}|^2$ where $p$ is the center of mass momentum of the $D$ meson with $M_{inv}$ invariant mass.
We show in figure \ref{crossexp} plots of the $d\Gamma/dM_{inv}$ for the channels $D^0\bar D^{*0}\pm c.c.$ and the pure $D^0\bar D^{*0}$ and compare it with experimental data from \cite{expcross}.
\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=5cm,angle=-90]{crossdd3.ps} \\
\includegraphics[width=5cm,angle=-90]{crossdd1.ps} \\
\includegraphics[width=5cm,angle=-90]{crossdd2.ps}
\end{tabular}
\caption{The $p|T|^2$, where $p$ is the three momentum of the $D$ mesons and $E$ is the energy above threshold, compared to data from \cite{expcross}. $\alpha=-$ 1.27 here.} \label{crossexp}
\end{center}
\end{figure}
In the plots of figure \ref{crossexp} the theoretical curves have been normalized to fit the experimental data. One can clearly see that the positive C-parity state alone describes the data while the negative C-parity state alone does not describes it. However this experiment can not determine whether the $D^0\bar D^{*0}$ (together with the $\bar D^0D^{*0}$ summed incoherently) comes from a given C-parity. In the lower plot of figure \ref{crossexp} we evaluate the differential cross section for the state $D^0\bar D^{*0}$, which has contribution from both C-parity states. As we can see in the figure, the results obtained are also in agreement with the data and, hence, in spite of the results in the middle plot of figure \ref{crossexp}, the experimental data does not rule out the existence of the negative C-parity state.
The former discussions put the two states that we predict in a perspective concerning the $D^0\bar D^{*0}$ production experiment. In what follows we are going to do a more subtle exercise to bring some light into a current discussion on whether the combination of the data on $X\rightarrow J/\psi \pi \pi$ and $X\rightarrow D^0\bar D^{*0}$ reactions determine if the state $X(3872)$ is a bound state or a virtual one. In what follows we are going to consider only the contribution from the positive C-parity state. In \cite{braaten2} a slightly bound state is preferred, although a virtual state is not ruled out, while in \cite{hanhart} a virtual state is claimed. With our detailed description of coupled channels, our approach is in a favorable position to get into the debate and bring new information. Yet, to do so one needs to introduce two new elements into consideration: the width of the $D^{*0}$ meson and the smearing of the results with experimental resolution. This was claimed to be relevant in \cite{voloshin2} and \cite{braaten2}. We have considered this by taking for the $D^{*0}$ width $\Gamma_{D^{*0}}$=65 KeV as in \cite{braaten2} and the experimental resolution $\Delta E$=2.5 MeV. The consideration of the width of the $D^{*0}$ is taken into account by folding the $D^0\bar D^{*0}$ loop function $G$ with the spectral function of the $D^*$ meson,
\begin{eqnarray}
{\cal S}(\tilde{M})&=&\left({-1\over\pi}\right) Im{1\over \tilde{M}^2-M_{D^*}^2+iM_{D^*}\Gamma_{D^*}}, \label{spectral}
\end{eqnarray}
in the calculation of the T-matrix, as done in eq. (20) of \cite{danielaxial}. On the other hand the result for $\frac{d\Gamma}{dM_{inv}}$ is folded with the mass distribution of the $D^*$ of eq. (\ref{spectral}), since in the phase-space the three momentum $q$ of the $D^0\bar D^*$ system appears as a factor and this three momentum depends on the mass of the $D^*$. The final result is folded by a Gaussian distribution with a width of 2.5 MeV to simulate the experimental resolution. In this way one gets strength below the nominal threshold of $D^0\bar D^{*0}$ for the decay of the $X(3872)$ into $D^0\bar D^{*0}$.
With these considerations we change slightly the $\alpha$ parameter which governs whether we obtain a bound state or a slightly unbound, virtual state. We normalize the two invariant mass distributions to the experimental data. The shapes alone tell us which option is preferable.
In figures \ref{figcomp1} and \ref{figcomp2} we show the results for the different values of $\alpha$. To the left we have the results for $X\rightarrow J/\psi \pi \pi$ and to the right those for $X\rightarrow D^0\bar D^{*0}$. What we see is that the effect of the convolution with the $D^*$ width and the experimental resolution is important, as claimed in \cite{voloshin2, braaten2} and help us make a choice of the preferred situation. At simple eye view, corroborated by a $\chi^2$ evaluation, see table \ref{tabchi2}, the preferred combined solution corresponds to $\alpha=-$1.23 for which we have a slightly unbound, virtual state. This is the preferred solution in \cite{hanhart}, also not ruled out in \cite{braaten2}.
\begin{widetext}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm,angle=-90]{crossp2a122.ps} & \includegraphics[width=6cm,angle=-90]{crossa122.ps} \\
\includegraphics[width=6cm,angle=-90]{crossp2a123.ps} & \includegraphics[width=6cm,angle=-90]{crossa123.ps} \\
\includegraphics[width=6cm,angle=-90]{crossp2a124.ps} & \includegraphics[width=6cm,angle=-90]{crossa124.ps} \\
\end{tabular}
\caption{Theoretical results compared to data from \cite{expcross}. The smeared points are calculated from the theoretical curve by folding it with a Gaussian, simulating the experimental resolution.} \label{figcomp1}
\end{figure}
\end{widetext}
\begin{widetext}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=6cm,angle=-90]{crossp2a125.ps} & \includegraphics[width=6cm,angle=-90]{crossa125.ps} \\
\includegraphics[width=6cm,angle=-90]{crossp2a126.ps} & \includegraphics[width=6cm,angle=-90]{crossa126.ps} \\
\includegraphics[width=6cm,angle=-90]{crossp2a127.ps} & \includegraphics[width=6cm,angle=-90]{crossa127.ps}
\end{tabular}
\caption{Theoretical results compared to data from \cite{expcross}.} \label{figcomp2}
\end{figure}
\end{widetext}
\begin{table}
\caption{$\chi^2$ values for the fits of $J/\psi \pi \pi$ and $D^0\bar D^{*0}$ production. The column to the right show the average value between the two.} \label{tabchi2}
\begin{tabular}{c||c|c|c}
\hline
& $\chi^2$ & $\chi^2$ & \\
$\alpha$ & for & for & $\bar \chi^2$ \\
& $J/\psi \pi \pi$ & $D^0\bar D^{*0}$ & \\
\hline
-1.22 & 1.83 & 0.49 & 1.16 \\
-1.23 & 1.26 & 0.50 & 0.88 \\
-1.24 & 0.87 & 0.93 & 0.90 \\
-1.25 & 0.72 & 2.77 & 1.74 \\
-1.26 & 0.92 & 17.96 & 9.44 \\
-1.27 & 0.92 & 20.31 & 10.61 \\
\hline
\end{tabular}
\end{table}
\section{Conclusion and outlook}
There is strong evidence that the $X(3872)$ state has $J^{PC}$ quantum numbers equal to $1^{++}$. It is very tempting to associate this state with a s-wave $D^0\bar D^{0*}$ molecular state if one takes into account the fact that the observed mass for this state is close to this threshold.
Using a phenomenological Lagrangian and an unitarization scheme for solving the scattering equation in couple channels, for all possible pairs of pseudoscalar and vector-mesons with zero charge and strangeness one obtains a pole with positive C-parity which can be associated with the $X(3872)$. Apart from this pole one also obtains strong attraction in the channels with negative C-parity, indicating the possible existence of a new state.
Experimentally the decays of the $X(3872)$ into $J/\psi$ with two and three pions have been measured to be of the same order of magnitude, suggesting a huge isospin violation in the decays of the $X$ state. We have studied here the effects of isospin violation in the decays of the $X$. The couplings of the $X$ to charged and neutral $D$ mesons are very similar, with at most 1.4\% of isospin violation, in our model. As a consequence of that, once one considers the decay of $X$ to $J/\psi\rho$ or $J/\psi\omega$ pairs as going through $D\bar D^*$ loops the $J/\psi\rho$ production should be suppressed in relation to the $J/\psi\omega$ by about a factor 30. If one considers the decays of the $X$ to $J/\psi$ with two and three pions as going through these channels respectively, one has also to take into account the phase-space available for each decay which, due to the $\rho$ width, is much bigger for this channel than for the $\omega$ channel, compensating the factor 30 suppressing the decays of the $X$ to $J/\psi\rho$. Moreover, the fact that the $X$ is observed in these decay modes indicates a non negligible coupling of the $X$ to vector-vector states, since it is visible in these channels ($J/\psi\rho$ and $J/\psi\omega$) even with the small phase-space available. This brought other authors to consider those channels as possible extra building blocks of the $X$ \cite{Swanson:2003tb,gutsche}, but we showed that we could explain the experimental phenomenology with dominant pseudoscalar-vector components.
The prediction of a negative C-parity state is peculiar to $D\bar D^*$ and would not appear as a vector-vector molecule. We predict this state with a framework which describes many low lying axial states and also most of the already observed axial charmed resonances. The comparison of our theoretical invariant mass distribution for $D\bar D^*$ production with data from Belle shows that one can not rule out the existence of this state and our model shows that the channels to which this resonance couples mostly are $\eta\phi$, $\eta\omega$, $\eta\prime\omega$ and $\eta_c\omega$. We also made predictions for observables where the negative C-parity state should in principle be seen and we hope these results stimulate experimental efforts in this direction.
Finally we made an investigation of the shapes of the $X(3872)$ distributions in the $J/\psi \pi \pi$ and $D^0\bar D^{*0}$ decays and found that they favor a slightly unbound, virtual state for this resonance.
\section*{Acknowledgments}
We would like to thank discussions and encouragement from E. Swanson, E. Braaten, Y. Dong and E. Lyubovitskij.
This work is
partly supported by DGICYT contract number
FIS2006-03438. We acknowledge the support of the European Community-Research
Infrastructure Integrating Activity
"Study of Strongly Interacting Matter" (acronym HadronPhysics2, Grant Agreement
n. 227431) under the Seventh Framework Programme of EU.
Work supported in part by DFG (SFB/TR 16, "Subnuclear Structure of Matter").
\newpage
\newpage |
2207.08114 | \section{Introduction}
\IEEEPARstart{G}{lobally}, as of March 2022, more than 452 million confirmed cases of COVID-19 have been reported to WHO, including more than 6 million deaths.
Especially, the new wave of epidemics caused by the Delta and Omicron variants of COVID that broke out in India, South Korea, and Hong Kong from 2021 is more contagious, the global epidemic situation still cannot be relaxed.
As a rapid and large-scale COVID-19 testing method, the RT-PCR testing has been widely adopted worldwide, but its false negative rate is as high as 17\% to 25.5\%, which is only suitable for preliminary screening.
In clinical practice, to make a definite diagnosis of suspected cases and determine an appropriate treatment plan, lung imaging interpretation by ultrasound \cite{boundariesatten,Ultrasound}, X-rays \cite{r6,r9,r10} or computed tomography (CT) \cite{r17,wang2020weakly,infnet,AnamNet,COPLENet,r20,DBLP:journals/tim/Roy22,2022SSA} is an indispensable link.
However, different imaging devices have their own advantages. Ultrasound can propagate in a certain direction and penetrate objects. Based on the principle that ultrasonic waves generate echoes, we can collect and display such echoes on the screen through instruments to understand the internal structure of objects, as shown in the first row of Fig. \ref{fig1}(a). X-ray examination uses the penetrating power of X-rays that travel through the body to a detector on the other side. Due to the different absorption of X-rays by bones, muscle tissues and air, an image with different gray levels from black to white is formed, as shown in the second row of Fig. \ref{fig1}(a). Computed tomography (CT) scans a certain part of the human body with X-ray beam to obtain a cross-sectional or three-dimensional image of the part being examined, which can clearly display the organs and structures, as shown in the third row of Fig. \ref{fig1}(a) and Fig. \ref{fig1}(b). Comparing these three imaging techniques, ultrasound can detect the consolidation of lung tissue, and X-ray has a good ability to show the presence of large areas of infected areas. However, these are based on the fact that the virus infected by the patient has developed to a certain stage. By contrast, the chest CT scans show clearer structures, have better sensitivity and accuracy than traditional chest X-rays and ultrasound waves, and can detect subtle lesions in the early stages of lung infection. Based on the characteristics of different data, ultrasound data and X-ray data are more suitable for follow-up observation of patients with pulmonary infection after diagnosis, and their convenience and low cost have great advantages. However, CT images show the irreplaceable role of the lung cavity in the definite diagnosis and early diagnosis, which has been accepted and adopted by the international medical community.
However, problems such as the surge of patients, the shortage of professional doctors, and the huge workload can easily delay diagnosis and even cause misjudgment. To this end, artificial intelligence technology can be used to automatically interpret the CT images, and provide auxiliary diagnosis information such as infection area classification and segmentation, thereby reducing the workload of doctors and improving the accuracy of screening. From this point, correct segmentation of the COVID-19 infection regions is critical for timely treatment of patients, and the research work in this paper takes this as a starting point to achieve accurate and efficient COVID-19 lung infection segmentation. Therefore, in the research sense, our proposed method in this paper uses artificial intelligence technology for COVID-19 infection segmentation, which can improve the segmentation accuracy and reduce the work pressure of radiologists.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig1.pdf}}
\caption{(a) Some images taken in the lung with three different imaging devices. (b) The COVID-19 infected lung CT images, where the ground-glass opacity, consolidation and pleural effusion are marked in blue, yellow and green respectively. (c) Visual examples of different COVID-19 infection segmentation methods.}
\label{fig1}
\end{figure}
In recent years, the vigorous development of deep learning has greatly promoted the development of computer vision-related fields \cite{crm2020tc,crmbridgenet,crmunderwater,crmDPANet,crmglnet,crmRRNet,crm2022rsi,crmDBLP:journals/spl/HuJCGS21,crmDBLP:journals/tip/LiAHCGR21,crmDBLP:journals/tip/WenYZCSZZBD21,crmDBLP:journals/tmm/MaoJCGSK22,crmCoADNet,crmACMMM20-1,crm-nc,crm2019tgrs,crm-acmmm,crmijcai20,crmgcl2019tip,crmGCPANet}. Among them, medical image segmentation algorithms have made great progress and achieved a qualitative leap in performance, such as lung nodules segmentation \cite{intro-4wang2017central}, breast-region segmentation \cite{DBLP:journals/tim/PramanikGBN20}, brain and brain-tumor segmentation \cite{intro-5cherukuri2017learning}, Brain Image Synthesis \cite{crmbrain}, polyp segmentation \cite{crmpolyp}, COVID-19 infection detection \cite{DBLP:journals/tim/DairiHS22}, COVID-19 forecasting \cite{DBLP:journals/tim/SharmaKMR21}, and COVID-19 infection segmentation \cite{DBLP:journals/tim/Roy22,infnet} \textit{etc}. However, differences in imaging equipment and disease characteristics make it difficult to use a unified segmentation model for different diseases.
Researchers often need to design some unique modules to better achieve lesion area segmentation. Observing an example shown in Fig. \ref{fig1}, we can summarize three difficult problems that the COVID-19 lung infection segmentation models need to solve:
\begin{enumerate}[(1)]
\item The infection regions of COVID-19 are very scattered, with many isolated areas of various sizes, which are very challenging for complete detection. Moreover, the sizes of the infected areas varies greatly, which makes it more difficult to accurately detect the infected areas of different scales. In addition, the boundary of the infection region is not easy to segment accurately and sharply. To this end, we can seek a solution from the perspective of encoder features containing relatively rich and effective information, which can be used to guide the feature learning in the decoder stage. Concretely, we propose an Attention-Guided Global Context (AGGC) module to select the most valuable information from encoder features for decoder, where the spatial attention and boundary attention are used to provide more accurate important spatial and boundary guidance information, and the global context modeling unit is used to model the global context dependence and constrain the generation of more complete segmentation result.
\item The infected areas have many detailed boundaries, and more attention needs to be paid to the clearness and sharpness of the boundaries in the segmentation results.
From a clinical point of view, the boundary of the infected area is of great significance for diagnosis. However, due to the pooling or up-sampling/down-sampling operation, the CNN-based method may blur the boundaries of the segmentation result. To generate a clearer and sharper boundary, we introduce boundary guidance in the proposed network.
In addition to introducing the supervised learning of the boundary map, we also treat the generated boundary map as an attention weight to highlight the encoder features in the AGGC module and allow the boundary constraint to play a greater role in the network.
\item For the COVID-19 infected lung CT image, although the processed data is only limited to the image of the lung area, the background regions are relatively complex and there are still a lot of interferences, such as inflamed areas that are not infected by COVID-19. Therefore, it is crucial to effectively suppress background noises. Considering that the high-level features including more semantic and category attributes can be used to suppress complex backgrounds, we design a Semantic Guidance (SG) unit to aggregate multi-scale high-level features and formulate a semantic guidance map to refine the decoder features. Moreover, in order to alleviate the information loss from multiple sampling operations and maintain the correlation between feature maps of adjacent scales, the multi-scale feature fusion is unified on the intermediate resolution to generate the semantic guidance map.
\end{enumerate}
In summary, we propose an automatic COVID-19 lung infection segmentation network with the encoder-decoder structure, equipped with three Boundary-Context-Semantic Reconstruction (BCSR) blocks. Although the encoder-decoder architecture is a common structure in medical image segmentation, we make a delicate design in it to cope with the special characteristics of the COVID-19 infection segmentation task, such as relatively scattered infected regions, more boundary details, complex backgrounds and noisy interferences. First, in order to address the problem of inaccurate and incomplete detection caused by the scattered infected regions, we design an AGGC module to select the most valuable information from encoder features for decoder, which highlights the important spatial and boundary locations in an attention manner, and models the global context dependence for the complete segmentation. Second, in order to achieve the sharp boundaries of the final segmentation result, we introduce the BA unit by adding the boundary supervised learning and boundary map refinement. Third, in order to suppress the background interferences, a SG unit is designed to aggregate multi-scale high-level features and formulate a semantic guidance map to refine the decoder features. All modules cooperate with each other to make our network achieve the competitive performance. As shown in Fig. \ref{fig1}(c), our proposed method has advantages in detection accuracy, completeness, clarity and sharpness.
The main contributions of this paper are as follows:
\begin{itemize}
\item An end-to-end network, named BCS-Net, is proposed to achieve automatic COVID-19 lung infection segmentation from CT images, which models the boundary constraint, context relationship, and semantic guidance. Moreover, our network achieves the competitive performance on the publicly available dataset both qualitatively and quantitatively.
\item An AGGC module is designed to filter the most valuable encoder information for decoder, which highlights the important spatial and boundary locations in an attention manner, and models the global context dependence for the complete segmentation.
\item A SG unit is proposed to aggregate multi-scale high-level features and generate the semantic guidance map to refine the decoder features, in which the multi-scale fusion is unified on the intermediate resolution to alleviate the information loss and maintain the correlation between adjacent scales.
\end{itemize}
\section{RELATED WORK}
\begin{figure*}[!t]
\centerline{\includegraphics[width=1\textwidth]{framework.pdf}}
\caption{Illustration of the overall framework of proposed network. The input CT slice image is first embedded into the backbone with four layers to extract the multi-level features, then we utilize the stacked Boundary-Context-Semantic Reconstruction (BCSR) blocks to progressively reconstruct the segmentation results.
The BCSR block consists of Attention-Guided Global Context (AGGC) module, Semantic Guidance (SG) unit, Boundary Attention (BA) unit. The framework finally produces five prediction maps, in which the $S_2$ map is the final segmentation result.
}
\label{fig2}
\end{figure*}
In this section, we will briefly introduce some related works in COVID-19 diagnosis based on deep learning for CXR and CT images, such as classification, segmentation.
There exist lots of deep learning-based segmentation methods, such as Mask R-CNN \cite{maskrcnn}, YOLACT \cite{YOLACT}, Deeplab \cite{deeplab}, FCN \cite{FCN}, \textit{etc}, which have made great achievements in segmentation tasks of various scenes and can be used for the medical segmentation task. But there are some differences between medical images and traditional ordinary images. Taking the COVID-19-infected lung CT image as an example, the areas to be segmented are usually scattered, with more details, complex backgrounds, and more interfering noises. Therefore, after the emergence of the U-Net \cite{unet}, it has gradually become the infrastructure for medical image segmentation and even general image segmentation tasks. Its success lies in the fact that a symmetric encoder-decoder structure with skip connections can better and more comprehensively utilize features at different levels, thereby generating more task-discriminative representations and improving performance.
Due to the high sensitivity and clarity of CT images, most of the current diagnosis of COVID-19 is based on CT images, including segmentation \cite{r17,wang2020weakly,infnet,AnamNet,COPLENet,r20}.
In practice, segmenting the key area from chest CT can provide useful information for medical staffs to diagnose the COVID-19, including GGO (Ground Glass Opacity) and consolidation. So far, many COVID-19 lung infection segmentation methods from CT Images based on deep learning have been proposed, and promising performance has been obtained.
Zhou \textit{et al}. \cite{r17} proposed an automated segmentation network by integrating the spatial and channel attention mechanisms.
Fan \textit{et al}. \cite{infnet} designed the parallel partial decoder, reverse attention, and edge-attention to boost the segmentation accuracy, and also provided a semi-supervised framework to alleviate the shortage of labeled data.
Paluru \textit{et al}. \cite{AnamNet} proposed an anamorphic depth embedding-based lightweight CNN to segment anomalies in COVID-19 chest CT images, which incorporates fully convolutional anamorphic depth blocks with depth-wise squeezing and stretching after the down-sampling and up-sampling operations.
Wang \textit{et al}. \cite{COPLENet} proposed a noise-robust learning framework based on self-ensembling of CNNs.
Han \textit{et al}. \cite{r20} proposed an attention-based deep 3D multiple instance learning (AD3D-MIL) to achieve the goal of accurate and interpretable screening of COVID-19 from chest CT images.
In these previous works, some ingenious and effective structures were designed, which achieved good performance and brought us a lot of inspiration. However, the existing work does not fully consider the following two points: (1) the loss of semantic information caused by oversampling in multi-layer feature fusion; (2) the correlation between contextual information and semantic information plays an important role in complete segmentation. For the first issue, we use the middle scale as the output criterion in the SG unit to generate the semantic guidance mask, which avoids excessive up- and down-sampling of a single layer in the multi-layer feature fusion. As for the second issue, we design an AGGC module to select the most valuable information from encoder features for decoder, which can not only highlight the important spatial and boundary locations, but also perceive the global correlation between different areas.
\section{PROPOSED METHOD}
\subsection{Overview}
Fig. \ref{fig2} illustrates the overall framework of our proposed BCS-Net to achieve the COVID-19 lung infection segmentation from CT images. Our network follows an encoder-decoder architecture in an end-to-end manner, in which the backbone extractor \cite{p1-gao2019res2net} in the encoder stage aims to extract the top-down multi-level features, and the decoder stage includes three progressively Boundary-Context-Semantic Reconstruction (BCSR) blocks to learn the segmentation-related features and generate the segmentation mask. In each BCSR block, we jointly consider the global semantic guidance, context dependence modeling, and boundary attention refinement to provide more sufficient supplementary information for feature decoding. Specifically, in order to deal with the problem of scattered lesion location and irregular shape, we design an attention-guided global context (AGGC) module, which selects the most valuable encoder propagation features by performing attention and context modeling on the encoder features of the corresponding decoder layer. As shown in Fig. \ref{fig3}, the corresponding encoder features and boundary attention map generated by the Boundary Attention (BA) unit are fed into the AGGC module. The encoder features are first refined by the spatial attention and boundary attention to highlight the important spatial locations and boundary details. Then, the Global Context Modeling (GCM) unit is used to correlate the different locations to consistently detect the lesion regions and generate the enhanced encoder features. With the guidance of the enhanced encoder features, the initial decoder features of the current level are generated by combining with the previous decoder features generated by the previous BCSR block. Furthermore, considering the importance of high-level semantic information for suppressing irrelevant background noises, we design a Semantic Guidance (SG) unit to aggregate multi-scale high-level features and formulate a semantic guidance map, which is further combined with the decoder features by means of residual connection to obtain the corresponding final decoder features:
\begin{equation}
F^i=\delta(conv(\bar{F}^i+\bar{F}^i\odot S_s))
\end{equation}
where $\bar{F}^i=concat(F^{i+1}, f^i_{aggc})$, $concat(\cdot)$ is channel-wise concatenation operation, $f^i_{aggc}$ are the output features generated by the AGGC module, $S_s$ denotes the semantic guidance map generated by the SG unit, $\odot$ represents element-wise multiplication, $\delta(\cdot)$ denotes the ReLU activation, and $conv(\cdot)$ is a customized convolutional block.
At each BCSR block, we use a convolutional layer with the kernel size of $1\times1$ to produce the corresponding segmentation map $S_i$:
\begin{equation}
S_i=\sigma(conv_{1\times 1}(F^i))
\end{equation}
where $conv_{1\times1}$ denotes a convolutional layer with the kernel size of $1\times1$, and $\sigma(\cdot)$ is the Sigmoid activation. Our network produces five side-output maps $(S_b,S_s,S_4,S_3,S_2)$, where the output $S_2$ of the last BCSR block is used as the final infection segmentation mask.
\subsection{Attention-Guided Global Context Module}
The clinical practice and data analysis have demonstrated that the pneumonia lesion areas, including the COVID-19, have no specific distribution characteristics and are generally scattered. Moreover, lung opacities in the CT image are inhomogenous, and there is no clear center or boundary. These undoubtedly pose great challenges to the segmentation model, including: (1) The scattered lesion areas can easily lead to missed detection or incomplete detection; (2) The boundary of the lesion area is not easy to segment accurately and sharply. To address these issues, we introduce more effective and valuable short-connection encoder as guidance. The evolution of encoder features is specifically manifested in two aspects: First, we use spatial attention and boundary attention to filter the features in terms of highlighting the important spatial locations and boundary details. Second, we capture the context dependence of different regions which constrains the generation of more complete and accurate segmentation result. The overall architecture is illustrated in Fig. \ref{fig3}. The AGGC module can not only ensure the integrity of the features of the local lesion area, but also perceive the global correlation between different areas. The specific process is as follows.
Although the encoder features can provide rich details and other useful information for the decoder, it also has much redundancy.
Therefore, the spatial attention \cite{p2-woo2018cbam} is introduced to modify the encoder features from the spatial dimension, focusing on highlighting the important spatial positions. Specifically, we first apply the average-pooling and max-pooling operations along the channel axis on the input features, and then concatenate them to generate an efficient feature descriptor. Finally, a convolution layer with the kernel size of $3\times 3$ followed by a sigmoid function is used to generate a spatial attention map $A^i_s$:
\begin{equation}
A^i_s=\sigma(conv_{3\times3}(concat(avepool(f^i),maxpool(f^i))))
\end{equation}
where $avepool$ and $maxpool$ denote the average-pooling and max-pooling along the channel axis, respectively. With the spatial attention map, the initial encoder features $f^i$ can be refined as spatial-enhanced features $f^i_s$ via residual connection:
\begin{equation}
f^i_s=A^i_s\odot f^i+f^i
\end{equation}
where $\odot$ represents element-wise multiplication.
The visualization of spatial-enhanced features is shown in Fig. \ref{fig5}(c). It can be seen that the spatial-enhanced features have a high response in the infected area to be segmented, and a low response in the irrelevant background area, thus achieving the effect of highlighting important spatial locations and suppressing interferences.
As mentioned above, segmenting the lesion areas with clear and definite boundary is very important for diagnosis. In order to maintain the sharpness and clarity of the boundary in each decoding stage, we directly use the boundary map $S_b$ generated by the BA unit to emphasize the important boundary details. To highlight the boundary locations without losing other important information, we still use the residual connection for integration:
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig3.pdf}}
\caption{Illustration of the proposed AGGC module, integrating the important locations, boundary details, and global context. ``SA'' is the spatial attention unit, ``$S_b$'' denotes boundary attention map generated by the BA unit, ``GCM'' is the global context modeling unit that models the global context dependence, and ``G'' is the global context-aware feature map.}
\label{fig3}
\end{figure}
\begin{equation}\label{Eq5}
f^i_b=S_b\odot f^i_s+f^i_s
\end{equation}
where $f^i_b$ represents the boundary-enhanced features, and $S_b$ is the boundary map generated by the BA unit.
In addition to highlighting the important location and boundary of the initial encoder features, we also need to solve the problem of incomplete detection caused by scattered lesions. Therefore, modeling the dependencies between different pixels is an effective solution. In this way, if an infected area is segmented, other scattered interference areas highly related to it may also be located. To this end, we introduce the global context modeling (GCM) unit to achieve feature alignment and mutual enhancement between features by modeling the dependencies between different locations in the feature map. Following \cite{zhang2020dense}, we first learn the global context-aware feature map $G^i$ by measuring the influence of the embedding features of all other spatial locations on the current location:
\begin{equation}
\begin{aligned}
&G^i = \bigtriangledown (\bigtriangleup (\widetilde{f^i_{b}} \otimes \omega^i ))\\
&= \bigtriangledown (\bigtriangleup (\widetilde{f^i_{b}})\otimes Norm(((\bigtriangleup (\widetilde{f^i_{b}}))^T \otimes \bigtriangleup (\widetilde{f^i_{b}}))^T))
\end{aligned}
\end{equation}
where $\widetilde{f^i_{b}}$~are the normalized boundary-enhanced features, $\bigtriangleup(\cdot)$ reshapes a matrix of $\mathbb{R}^{D_{1}\times D_{2}\times D_{3}}$ into $\mathbb{R}^{D_{1}\times D_{23}}$, $\bigtriangledown(\cdot)$ is the inverse operator of $\bigtriangleup(\cdot)$, $\otimes$ is the matrix multiplication, $\omega^i$ is the global context relationship map, and $Norm$ is the normalization operation. Then, the $G^i$ that encodes the global context relationship is used to further refine the boundary-enhanced features in a residual connection manner:
\begin{equation}
f^i_{aggc}=\varsigma \cdot (G^i\odot f^i_b)+f^i_b
\end{equation}
where $\varsigma$ is a learnable weight parameter that controls the contribution of the global contextual information.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig5.pdf}}
\caption{(a) CT Image. (b) Infection segmentation ground truth. (c) Visualization of spatial-enhanced features. (d) Visualization of boundary attention map.}
\label{fig5}
\end{figure}
The entire AGGC model adopts a progressive enhancement structure, which not only highlights important spatial locations and boundary details, but also encodes the context dependencies between different spatial positions. The operation sequence of these three enhancement strategies is spatial attention, boundary attention, and global context. The reasons for this design are as follows: For encoder features with rich information, reducing redundancy is a primary task, otherwise the continued existence of redundancy in the later learning process will weaken the expression of features; Then, in order to ensure that the boundary details are well preserved and highlighted, we apply the boundary attention to spatial-enhanced features; Finally, calculating the global context dependence on relatively clear and distinct features after spatial and boundary enhancements can further strengthen the discrimination of features. This process can obtain encoder features with prominent spatial positions, highlighted boundary details, and comprehensive global context, which can then be used as the guidance for decoder features.
\subsection{Boundary Attention Unit}
For the medical image segmentation task, especially the COVID-19 segmentation where the lesion areas are scattered throughout the image, it is particularly important to clearly and accurately outline the boundaries of the lesion areas for later diagnosis.
If only the semantic information of each pixel is considered and the boundary information is ignored in the segmentation process, this may cause the boundary of the segmentation result to be blurred and not sharp enough.
Inspired by \cite{zhang2019Etnet,infnet}, we introduce the boundary constraint into our approach by utilizing the boundary attention unit, as shown in Fig. \ref{fig4}. On the one hand, we also introduce the boundary supervised learning to the network. In this way, the network can gradually learn the features of generating a clear boundary map.
Specifically, the low-level features $f^1$ are fed into three convolution layers with different kernel sizes, and then a sigmoid function is employed to generate the final boundary attention map:
\begin{equation}
S_b=\sigma(conv_{1\times1}(conv_{3\times3}(conv_{1\times1}(f^1))))
\end{equation}
where $f^1$ are the encoder features of the first layer, $conv_{n \times n}$ denotes the convolution layer with the kernel size of $n\times n$.
The learning process of boundary attention map is a supervised learning manner, which can guarantee the effectiveness and pertinence of learning. As shown in Fig. \ref{fig5}(d), the boundary attention map $S_b$ effectively highlights the boundary positions of the infected regions to be segmented, which can be further used to refine the high-level features in the AGGC module.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig4.pdf}}
\caption{Illustration of the proposed Boundary Attention (BA) unit, where $S_b$ denotes boundary map.}
\label{fig4}
\end{figure}
In addition to generating boundary map through label supervised learning, we also treat the generated boundary map as an attention weight to highlight the encoder features as introduced in Eq. (\ref{Eq5}). Different from the work \cite{infnet}, we did not use the boundary features to perform feature enhancement. Instead, we adopt a method similar to spatial attention, treat the generated boundary map as an attention map, and use the residual connection for feature enhancement. In this way, boundary constraints can be introduced more intuitively and efficiently, and at the same time, unnecessary information redundancy in the feature space can be reduced. This boundary attention map will be used to enhance all the high-level features (\textit{i}.\textit{e}., $f^2$, $f^3$, $f^4$).
\subsection{Semantic Guidance Unit}
The features learned by different encoder layers include different information. For example, the low-level features highlight the details of the scene such as the texture and boundary, while the high-level features tend to learn the semantic and category attributes of the scene. For the segmentation task of COVID-19 lesion areas, there are relatively more background interferences, and how to effectively suppress them is particularly important. Considering the importance of high-level semantic information for suppressing irrelevant background interference, we design a Semantic Guidance (SG) unit to aggregate multi-scale high-level features and formulate a semantic guidance map to refine the decoder features.
For the feature fusion of different scales, the correlation between feature maps of adjacent scales on the feature pyramid is the strongest. However, in the previous multi-scale feature fusion methods, some features are usually up-sampled or down-sampled multiple times, which will lose the semantic information of the original features and even cause semantic ambiguity. Based on this, we try our best to avoid the worthless and unadvisable multiple cross-scale sampling of single layer. As illustrated in Fig. \ref{fig2}, the spatial resolution of high-level features ${f^2,f^3,f^4}$ are unified on the intermediate resolution of the third encoder layer, which not only avoids the information loss due to multiple sampling, but also preserves the correlation between adjacent-scale features to the utmost extent. To be more specific, the encoder features $f^2$ and $f^4$ are $\times2$ down-sampled and $\times2$ up-sampled to the resolution of features $f^3$ respectively, and then they are fused together to obtain a semantic guidance map through the addition and convolution operations. The process can be formulated as:
\begin{equation}
S_s=conv(down_2(f^2)+f^3+up_2(f^4))
\end{equation}
where $up_2$ and $down_2$ are the $\times 2$ spatial up-sampling and down-sampling, respectively.
The learned semantic map $S_s$ encodes the multi-level semantic information, which is served as a global guidance in BCSR block. Since we choose the middle layer as the scale standard of feature output, the semantic map $S_s$ only needs to be down-sampled once in the first BCSR block and up-sampled once in the last BCSR block, respectively. It effectively avoids the multiple cross-level sampling of $S_s$ and retains the feature information of $S_s$ to the maximum extent.
\subsection{Loss Function}
For the supervision of network, the overall loss function involves two parts: (1) The segmentation loss on three decoder outputs (\textit{i}.\textit{e}., $S_2$, $S_3$, and $S_4$) and the semantic loss on the semantic map $S_s$ via weighted IOU loss and weighted BCE loss. Inspired by \cite{l1-qin2019basnet, l2-wei2020f3net}, the weighted IOU loss provides effective global (image level) supervision, and the weighted BCE loss provides effective local (pixel level) supervision. Combining them, accurate segmentation result can be obtained from the image level and pixel level, which is defined as:
\begin{equation}
l_{s} = \sum_{k=\{S_2,S_3,S_4,S_s\}}(l^k_{wiou}+l^k_{wbce})
\end{equation}
\begin{equation}
\begin{aligned}
& l^k_{wbce} = \frac{\sum_{x = 1}^{w}\sum_{y = 1}^{h}-G_{xy}\log(P^k_{xy})\ast (1+\varepsilon_{xy})}{\sum_{x = 1}^{w}\sum_{y = 1}^{h}\varepsilon_{xy}}\\
& - \frac{\sum_{x = 1}^{w}\sum_{y = 1}^{h}(1-G_{xy})\log(1-P^k_{xy})\ast (1+\varepsilon_{xy})}{\sum_{x = 1}^{w}\sum_{y = 1}^{h}\varepsilon_{xy}}
\end{aligned}
\end{equation}
\begin{equation}
l^k_{wiou} =1 - \frac{\sum_{x = 1}^{w}\sum_{y = 1}^{h}(P^k \cap G)\ast (1+\varepsilon_{xy})}{\sum_{x=1}^{w}\sum_{y = 1}^{h}(P^k \cup G)\ast (1+\varepsilon_{xy})}
\end{equation}
where $l^k_{wiou}$ and $l^k_{wbce}$ denote the weighted IOU loss and weighted BCE loss on different supervision types respectively, $P^k$ is the prediction map (\textit{i}.\textit{e}., $S_2$, $S_3$, $S_4$, $S_s$), $G$ is the segmentation ground truth, $w$ and $h$ are the width and height of the input image, $(x, y)$ denotes the coordinate of each pixel in image, and $\varepsilon$ is a weight hyperparameter.
(2) The boundary loss $l^b_{bce}$ on the boundary attention map $S_b$ via standard BCE loss, which is defined as:
\begin{equation}
l^b_{bce} = \sum_{x=1}^{w}\sum_{y=1}^{h}-G^b_{xy}\log(P^b_{xy})-(1-G^b_{xy})\log(1-P^b_{xy})
\end{equation}
where $G^b$ is the boundary ground truth, and $P^b$ is the predicted boundary map.
Therefore, the overall loss function can be defined as:
\begin{equation}
\iota_{total} = l_{s}+l^b_{bce}
\end{equation}
\section{EXPERIMENTS}
\label{sec:guidelines}
\subsection{Datasets and Evaluation Metrics}
At present, there are few public COVID-19 lung CT datasets for infection segmentation.
In order to have relatively sufficient samples for training, we merged the two public datasets \cite{segdata1,segdata2} to obtain $1018$ high-quality CT images, and further divided them into $719$ training images and $300$ testing images.
For each CT slice, it contains the corresponding infection mask, and the infection edge obtained from infection mask using the Canny operator.
Note that, the COVID-19 classification aims to determine whether lung CT images are infected with COVID-19, and the datasets used should contain normal and abnormal data. In contrast, the purpose of COVID-19 infection segmentation is to locate the infected area accurately and completely, so the default option is that the infected area is present in every CT image of the used dataset. In other words, consistent with existing methods \cite{2022SSA,infnet,COPLENet}, all images in the dataset \cite{segdata1,segdata2} we use are COVID-19 infection data.
\begin{figure}[!t]
\centerline{\includegraphics[width=\columnwidth]{fig7.pdf}}
\caption{The training loss curve.}
\label{fig7}
\end{figure}
Following \cite{infnet}, six metrics are employed for quantitative evaluation, including Precision ($Prec$), Recall ($Recall$), Dice Similarity Coefficient ($DSC$) \cite{2020Lung}, Structure Measure ($S_m$) \cite{S}, Enhance-alignment Measure ($E_\phi$) \cite{E}, and Mean Absolute Error ($MAE$) \cite{crm2019tcsvt}.
The Precision ($Prec$) calculates the ratio of true positive cases in the results predicted as positive cases, and the Recall ($Recall$) score only focuses on the rate that the real label is correctly predicted, which can be computed as:
\begin{equation}
Prec = \frac{TP}{TP + FP}, ~~~Recall = \frac{TP}{TP + FN}
\end{equation}
where $TP$, $FP$, and $FN$ are the true positive, false positive, and false negative, respectively.
Dice Similarity Coefficient ($DSC$) is a spatial overlap index and a reproducibility validation metric, which evaluates the overlap ratio between prediction map ($P$) and ground truth map ($G$):
\begin{equation}
DSC = \frac{2\left\lvert P\bigcap G\right\rvert }{\left\lvert P \right\rvert + \left\lvert G \right\rvert }
\end{equation}
Structure Measure ($S_m$) calculates the similarity between prediction map and ground-truth mask, including a region-aware structural similarity measure $S_o$ and an object-aware structural similarity measure $S_r$, which is
calculated as:
\begin{equation}
S_m = \alpha \cdot S_o(P, G) + (1 - \alpha) \cdot S_r(P, G)
\end{equation}
where $\alpha \in [0,1]$, we set $\alpha = 0.5$ in our implementation suggested in the original paper.
Enhance-alignment Measure ($E_\phi$) captures global statistics and local pixel matching information to account for both pixel-level and image-level properties, which is defined as:
\begin{equation}
E_\phi = \frac{1}{w \times h}\sum_{x = 1}^{w}\sum_{y = 1}^{h}\phi (x, y)
\end{equation}
where $h$ and $w$ are the height and the width of the input image, and $(x,y)$ denotes the coordinate of each pixel in prediction map and ground truth.
\begin{figure*}[!t]
\centerline{\includegraphics[width=1\textwidth]{vis.pdf}}
\caption{Some visual examples of different methods. (a) Input image. (b) Ground truth. (c)-(l) Results of our method, UNet \cite{unet}, Resnet34-UNet \cite{unet}, UNet++ \cite{unet++}, Attention-UNet \cite{attentionunet}, CE-Net \cite{Cenet}, FCN \cite{FCN}, Inf-Net \cite{infnet}, AnamNet \cite{AnamNet}, and CopleNet \cite{COPLENet}.
}
\label{fig6}
\end{figure*}
Mean Absolute Error ($MAE$) measures the errors between prediction map and ground truth:
\begin{equation}
MAE = \frac{1}{w \times h}\sum_{x = 1}^{w}\sum_{y = 1}^{h}\left\lvert P(x, y) - G(x, y)\right\rvert
\end{equation}
For these six measurements, in addition to the MAE score, all other indicators are that the larger the value, the better the performance.
\subsection{Training Strategies and Implementation Details}
We implement the proposed network via the PyTorch toolbox and is accelerated by an NVIDIA GeForce RTX $2080$Ti GPU. We also implement our network by using the MindSpore Lite tool\footnote{\url{https://www.mindspore.cn/}}.
The Res2Net-50 \cite{Res2Net} pretrained on ImageNet \cite{imagenet} is employed as the backbone feature extractor in the experiment.
We have two points in particular to note: (1) The fundamental reason we choose Res2Net-50 as the backbone network is for fair comparison, so we follow the setting of Inf-Net \cite{infnet}. In addition, its technical advantages (\textit{e}.\textit{g}., hierarchical residual-like connections within one single residual block, multi-scale features at a granular level, and increased range of receptive fields) and low computational cost are also the reasons for our choice. (2) The way we load the pretrained ImageNet data is consistent with existing work \cite{infnet,DBLP:conf/wacv/LaradjiRMLLKP0N21}. Specifically, it is well known that CT images are grayscale, and when we load the CT image, we still use the RGB three-channel method, but the corresponding pixel values in the three channels are the same.
In order to computational efficiency, we resize all the training and testing images to $352\times352$.
The batch size is $8$ and the training process converges until $200$ iterations. We use Adam \cite{adam} as a learning rate optimizer with betas of $(0.9, 0.999)$, and the learning rate is set to $3e^{-4}$.
The training loss curve of our network is shown in Fig. \ref{fig7}.
The average inference time of our method is $0.036$ second ($27$ FPS) for processing an image with the size of $352\times352$, the GFLOPs is $28.02$, and the parameters are $44,822,580$. The code and results of our BCS-Net can be found from the link of \url{https://github.com/rmcong/BCS-Net-TIM22}.
\subsection{Comparison with State-of-the-art Method}\label{formats}
We compare our method with some state-of-the-art methods, including five classical segmentation models (UNet \cite{unet}, UNet++ \cite{unet++}, Attention-UNet \cite{attentionunet}, Resnet34-UNet \cite{unet}, CE-Net \cite{Cenet}, and FCN \cite{FCN}), and three state-of-the-art segmentation methods for COVID-19 (Inf-Net \cite{infnet}, CopleNet \cite{COPLENet}, and AnamNet \cite{AnamNet}).
In order to ensure the fairness of the experiment, all the comparison methods are retrained on the same dataset under the default parameter settings.
\subsubsection{Qualitative Comparison}
Fig. \ref{fig6} shows the qualitative comparisons of our model with the other state-of-the-art methods.
It can be seen that the results of our method have greater advantages in terms of detection accuracy, completeness, and sharpness.
For example, in the first image, the general medical image segmentation networks (\textit{e}.\textit{g}., UNet \cite{unet}, UNet++ \cite{unet++}, Attention-UNet \cite{attentionunet}, Resnet34-UNet \cite{unet}) often cannot effectively suppress the interference of background regions, such as the areas between the two lungs. In contrast, the detection results of the COVID-19 segmentation network are better, but the existing methods (\textit{e}.\textit{g}., Inf-Net \cite{infnet}, CopleNet \cite{COPLENet}, and AnamNet \cite{AnamNet}) cannot completely suppress these interferences. For example, the area under the left lung is not effectively suppressed. Our proposed method has better performance in these aspects, and has a stronger ability to detect details. The last image also confirms that our proposed method can accurately detect infected areas and suppress irrelevant background areas. In addition, our method has more complete structure and sharper boundaries. In the third image, the existing methods (\textit{e}.\textit{g}., Inf-Net \cite{infnet}, CopleNet \cite{COPLENet}, and AnamNet \cite{AnamNet}) cannot completely and continuously detect the infected regions at the bottom of the left lung, while our method can detect them clearly, accurately, and completely. Moreover, compared with the existing methods (\textit{e}.\textit{g}., UNet++ \cite{unet++}, CopleNet \cite{COPLENet}, Attention-UNet \cite{attentionunet}), our results have clearer boundaries, which are very important for doctors to diagnose and treat. In general, our method exhibits competitive visual effects in both the segmentation ability of the infected regions and the suppression ability of the background regions.
\begin{table}
\centering
\caption{Quantitative Comparisons with Different Methods. The best performance is marked in bold.}
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Methods&$DSC \uparrow$&$Prec \uparrow$&$Recall \uparrow$&$S_m \uparrow$&$E_\phi \uparrow$&$MAE \downarrow$\\
\hline
UNet&0.777&0.804&0.814&0.862&0.917&0.020\\
\hline
UNet++&0.771&0.836&0.780&0.867&0.906&0.021\\
\hline
Attention\_UNet&0.746&0.818&0.768&0.853&0.913&0.021\\
\hline
Resnet34\_UNet&0.720&0.702&0.836&0.812&0.873&0.030\\
\hline
FCN&0.800&0.839&0.791&0.855&0.949&0.020\\
\hline
CE-Net&0.818&0.834&0.824&0.854&0.960&0.017\\
\hline
AnamNet&0.775&0.831&0.776&0.856&0.920&0.021\\
\hline
CopleNet&0.816&\textbf{0.850}&0.821&0.874&0.944&0.016\\
\hline
Inf-Net&0.828&0.831&0.846&0.877&0.963&0.016\\
\hline
BCS-Net [ours]&\textbf{0.844}&0.841&\textbf{0.861}&\textbf{0.880}&\textbf{0.972}&\textbf{0.015}\\
\hline
\end{tabular}
}
\label{table:1}
\end{table}
\subsubsection{Quantitative Comparison}
The quantitative comparisons are reported in Table \ref{table:1}. In addition to the Precision score, our proposed BCS-Net achieves the best performance in other five measurements on the testing dataset.
Compared with the classical segmentation models (\textit{e}.\textit{g}., UNet \cite{unet}, UNet++ \cite{unet++}, Attention-UNet \cite{attentionunet}, Resnet34-UNet \cite{unet}), the well-designed methods for COVID-19 are more competitive. For example, compared with the UNet \cite{unet}, the percentage gain of the Inf-Net \cite{infnet} reaches 6.6\% for $DSC$ score, and the percentage gain of our method is 8.6\%. For the $E_\phi$, the performance improvement is more obvious. Specifically, the performance of Inf-Net \cite{infnet} is increased by 5.7\% compared with the UNet++ \cite{unet++}, and the performance is increased by 9.0\% compared with the Resnet34-UNet \cite{unet}.
For the infection segmentation method for COVID-19, our proposed BCS-Net achieves better performance. For example, compared with the \emph{second best} method, the percentage gain of $DSC$ is 1.9\%, the $Recall$ score is 1.8\%, and the $E_\phi$ score achieves 1.0\%. In general, our detection effect is superior in quantitative measurements.
\subsection{Ablation Study}
We conduct several ablation experiments to validate the performance and effectiveness of two contributed components of our proposed model, including Attention-Guided Global Context (AGGC) module and Semantic Guidance (SG) unit. The results are given in Table \ref{table:2} and Fig. \ref{fig8}.
\begin{table}
\centering
\caption{Quantitative evaluation results of different modules used in our proposed model.}
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
No. &Variations&$DSC \uparrow$&$Prec \uparrow$&$Recall \uparrow$&$S_m \uparrow$&$E_\phi \uparrow$&$MAE \downarrow$\\
\hline
1& BCS-Net&0.844&0.841&0.861&0.880&0.972&0.015\\
\hline
2& w/o AGGC&0.838&0.835&0.856&0.874&0.970&0.015\\
\hline
3& w/o SG&0.837&0.840&0.850&0.872&0.969&0.016\\
\hline
4& \makecell[*l]{w/o SG \\ w/ PPD} &0.836&0.838&0.849&0.872&0.969&0.016\\
\hline
\end{tabular}
}
\label{table:2}
\end{table}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=\columnwidth]{ab.pdf}}
\caption{Visual comparison of BCS-Net variants equipped with different modules. (a) Images. (b) Ground truth. (c) BCS-Net. (d) w/o AGGC. (d) w/o SG. (f) w/o SG w/ PPD.}
\label{fig8}
\end{figure}
Firstly, we verify the effect of AGGC module on the whole network. The main function of the AGGC module is to strengthen the important spatial position information and correlate the dependencies of different pixels, thereby ensuring a more accurate and complete detection.
We remove the AGGC module, including the guidance of spatial attention and boundary attention, and the global context awareness. In other words, in the BCSR block, the decoding features of the previous layer will not enter the AGGC module but will be passed directly backwards.
Compared with the No.1 and No.2 in Table \ref{table:2}, without the AGGC module, the five indicators fell, and the MAE score remained flat.
As shown in Fig. \ref{fig8}(d), after removing the AGGC module, there are unsuppressed interference noises in the upper left corner, upper right corner and lower right corner of the first image, and the boundaries are blurred. In the second image, the small lesions on the upper right after removing the AGGC module are not detected, while in the third image, there are obvious false detections in the left region. These all demonstrate the effectiveness of our AGGC module.
Secondly, in order to discuss the role of the SG unit, we design two ablation experiments. One is that we directly delete the SG unit, that is, without introducing global semantic information guidance. The other is that we replace the proposed SG unit with the PPD module in Inf-Net \cite{infnet}. The related results are reported in No.3 and No.4 of Table \ref{table:2}.
Compared with the full model, all indicators will be reduced after removing or replacing the SG unit.
In addition, we also found that the complex PPD model has three indicators (\textit{i}.\textit{e}., $DSC$, $Prec$, and $Recall$) that it is not as good as removing the SG unit directly. Comparing the SG and PPD modules of introducing global semantic guidance (\textit{i}.\textit{e}., No.1 and No.4), we can see that the designed SG model achieves better performance.
After removing the SG module, the upper left and upper right regions in the first image of Fig. \ref{fig8}(e), the upper left region of the second image, and the upper right region of the third image have some irrelevant interferences that are not effectively suppressed. Replacing the SG module with the PPD module also does not improve the performance, and both the left region and the upper right background region in the third image are incorrectly detected as infected regions. Both ablation experiments illustrate the role of the SG module designed in this paper.
\begin{table}
\centering
\caption{The performance comparisons of outputs $S_1$, $S_2$ and $S_3$.}
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
No. &Output&$DSC \uparrow$&$Prec \uparrow$&$Recall \uparrow$&$S_m \uparrow$&$E_\phi \uparrow$&$MAE \downarrow$\\
\hline
1& $S_1$&0.839&0.851&0.846&0.867&0.965&0.017\\
\hline
2& $S_3$&0.802&0.776&0.855&0.853&0.954&0.017\\
\hline
3& $S_2$[ours]&0.844&0.841&0.861&0.880&0.972&0.015\\
\hline
\end{tabular}
}
\label{table:3}
\end{table}
The final output of the network is derived from features at the second decoding stage mainly based on the following two points. First, we design the AGGC module to make full use of the features of the first encoder layer ($f^1$) to supplement boundary information for high-level encoder features ($f^2$, $f^3$ and $f^4$). In implementation, the features $f^1$ are combined with $f^2$, $f^3$ and $f^4$ through the AGGC module respectively. Under such a model framework, we do not perform the AGGC module in the last decoding layer, so the network does not set the output of $S_1$. Second, as we all know, generating the final map at a lower resolution from a higher decoding stage will consume less computational resources, which is beneficial for computational efficiency. Moreover, it has been experimentally verified that its performance will not be obviously degraded. Table \ref{table:3} presents the final segmentation results obtained by different decoding layers. It can be found that in general, $S_2$ achieves better performance than $S_1$ and $S_3$, which also illustrates the effectiveness of our setup. Based on these observations and consideration, the output from the second decoding layer is used as the final segmentation result.
\subsection{Failure Cases and Future Work}
Compared with other methods, our algorithm achieves a more complete structure and more accurate details, but there is still a certain gap with segmentation ground truth, which we need to solve in the future. Some failure cases are shown in Fig. \ref{fig9}. It can be seen that the segmentation results of our method can maintain the structural integrity in the case of a large area of infection in the whole lung, but there will be some wrongly detected holes, as shown in the left image. In addition, our method also misses some extremely dispersed infected areas. For example, in the right image, many infected areas on the left are not detected, and some scattered infected areas on the right are mistakenly merged together. In the future, we can start from the following aspects to further improve performance. On the one hand, some popular and powerful backbones can be used in place of our basic feature extractor, such as ViT \cite{dosovitskiy2020image} and Swin Transformer \cite{liu2021swin}, which can be used to model long-term dependencies in data through self-attention mechanism. On the other hand, the growth of training data can effectively promote the improvement of model performance and robustness. Therefore, in the future, we call on global scientific research institutions to open source relevant data in accordance with relevant regulations, so as to further promote the development and performance improvement of this field.
\section{Conclusion}
This paper focuses on COVID-19 lung infection segmentation task and proposes an end-to-end framework dubbed as BCS-Net to achieve this.
The proposed network follows an encoder-decoder architecture equipped with Boundary-Context-Semantic Reconstruction (BCSR) blocks, which considers the boundary refinement, context modeling, and semantic constraint jointly.
The attention-guided global context (AGGC) module is designed to select the most valuable encoder features from the perspective of important spatial/boundary locations and context dependence.
Moreover, we design a new semantic guidance (SG) unit to provide the semantic guidance and suppress the background noises by aggregating multi-scale high-level features at the intermediate resolution.
Extensive experiments and ablation studies demonstrate the effectiveness of the proposed BCS-Net architecture.
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=1\columnwidth]{fig9.pdf}}
\caption{Failure cases of our BCS-Net.}
\label{fig9}
\end{figure}
\par
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
{
\bibliographystyle{IEEEtran} |
0807.1458 | \section{Introduction}
Rumours are an important form of social communications, and their
spreading plays a significant role in a variety of human affairs. The
spread of rumours can shape the public opinion in a country
\cite{galam}, greatly impact financial markets \cite{kimmel,kosfeld}
and cause panic in a society during wars and epidemics outbreaks. The
information content of rumours can range from simple gossip to
advanced propaganda and marketing material. Rumour-like mechanisms
form the basis for the phenomena of viral marketing, where companies
exploit social networks of their customers on the Internet in order to
promote their products via the so-called `word-of-email' and
`word-of-web' \cite{viral1_domingos}. Finally, rumor-mongering forms
the basis for an important class of communication protocols, called
gossip algorithms, which are used for large-scale information
dissemination on the Internet, and in peer-to-peer file sharing
applications \cite{gossip_demers,gossip2_review}.
Rumours can be viewed as an ``infection of the mind'', and their
spreading shows an interesting resemblance to that of epidemics.
However, unlike epidemic spreading quantitative models and
investigations of rumour spreading dynamics have been rather
limited. An standard model of rumour spreading, was introduced many
years ago by Daley and Kendall \cite{DK1,book1_epidemic}.
The Daley-Kendall (DK) model and its variants, such
as the Maki-Thompson (MK) model \cite{Maki}, have been used
extensively in the past for quantitative studies of rumour spreading
\cite{DK2,sudbury,pittel,noymer}. In the DK model a closed and
homogeneously mixed population is subdivided into three groups, those
who are ignorant of the rumour, those who have heard it and actively
spread it, and those who have heard the rumour but have ceased to
spread it. These groups are called ignorants, spreaders and stiflers,
respectively. The rumour is propagated through the population by {\em
pair-wise} contacts between spreaders and others in the population,
following the law of mass action. Any spreader involved in a pair-wise
meeting attempts to 'infect' the other individual with the rumour. In
the case this other individual is an ignorant, it becomes a
spreader. In the other two cases, either one or both of those involved
in the meeting learn that the rumour is `known' and decided not to
tell the rumour anymore, thereby turning into stiflers
\cite{book1_epidemic}. In the Maki-Thompson variant of the above
model the rumour is spread by {\em directed} contacts of the spreaders
with others in the population. Furthermore, when a spreader contacts
another spreader only the {\em initiating} spreader becomes a
stifler.
An important shortcoming of the above class of models
is that they either do not take into account
the topology of the underlying social interaction networks along which
rumours spread (by assuming a homogeneously mixing population), or use
highly simplified models of the topology \cite{sudbury,pittel}. While
such simple models may adequately describe the spreading process in
small-scale social networks, via the word-of-mouth, they become highly
inadequate when applied to the spreading of rumours in large social
interaction networks, in particular those which are mediated by the
Internet. Such networks, which include email networks
\cite{email_newman,email_ebel,p2pfang}, social networking sites
\cite{soc_nets1} and instant messaging networks \cite{instant}
typically number in tens of thousands to millions of nodes. The
topology of such large social networks shows highly complex
connectivity patterns. In particular, they are often characterized by
a highly right-skewed degree distribution, implying the presence of a
statistically significant number of nodes in these networks with a
very large number of social connections
\cite{email_newman,email_ebel,soc_nets1,instant}.
A number of recent studies have shown that introducing the complex
topology of the social networks along which a rumour spreads can
greatly impact the dynamics of the above models. Zanette performed
simulations of the deterministic Maki-Thompson model on both static
\cite{rumour1_zanette} and dynamic \cite{rumour2_zanette} small-world
networks. His studies showed that on small-world networks with
varying network randomness the model exhibits a critical transition
between a regime where the rumour ``dies'' in a small neighborhood of
its origin, and a regime where it spreads over a finite fraction of
the whole population. Moreno {\it et al.} studied the stochastic
version of the MK model on scale-free networks, by means of Monte
Carlo simulations \cite{rumour1_maziar}, and numerical solution of a
set of mean-field equations \cite{rumour2_maziar}. These studies
revealed a complex interplay between the network topology and the
rules of the rumour model and highlighted the great impact of network
heterogeneity on the dynamics of rumour spreading. However, the scope
of these studies were limited to {\it uncorrelated} networks. An
important characteristic of social networks is the presence of
assortative degree correlations, i.e. the degrees of adjacent vertices
is positively correlated \cite{review_newman,
review_yamir,email_newman, email_ebel}. Furthermore the mean-field
equations used in \cite{rumour2_maziar} were postulated without a
derivation.
In this paper we make several contributions to the study of rumour
dynamics on complex social networks. First of all, we introduce a
new model of rumour spreading on complex networks which, in
comparison with previous models, provides a more realistic
description of this process. Our model unifies the MK model of
rumour spreading with the Susceptible-Infected-Removed (SIR) model
of epidemics, and has both of these models as it limiting cases.
Secondly, we describe a formulation of this model on networks in
terms of Interacting Markov Chains (IMC) \cite{imc1}, and use this
framework to derive, from first-principles, mean-field equations
for the dynamics of rumour spreading on complex networks with
arbitrary degree correlations. Finally, we use approximate
analytical and exact numerical solutions of these equations to
examine both the steady-state and the time-dependent behavior of the
model on several models of social networks: homogeneous networks,
Erd\H os-R\'enyi (ER) random graphs, uncorrelated scale-free
networks and scale-free networks with assortative degree
correlations.
We find that, as a function of the rumour spreading rate, our
model shows a new critical behavior on networks with bounded degree
fluctuations, such as random graphs, and that this behavior is
absent in scale-free networks with unbounded fluctuations in node
degree distribution. Furthermore, the initial spreading rate of a
rumour is much higher on scale-free networks as compared to random
graphs. This spreading rate is further increased when assortative
degree correlations are introduced. The final fraction of nodes
which ever hears a rumour (we call this the final size of a rumour),
however, depends on an interplay between the model parameters and
degree-degree correlations. Our findings are relevant to a number
of rumour-like processes taking place on complex social networks.
These include the spreading of chain emails and Internet hoaxes,
viral advertising and large-scale data dissemination in computer and
communication networks via the so-called gossip protocols
\cite{gossip_demers}.
The rest of this paper is organized as follows. In Section 2 we
describe our rumour model. In section 3 a formulation of the
model within the framework of Interactive Markov Chains is given, and
the corresponding mean-field equations are derived. In section 4
we present analytical results for the steady-state behavior of our
model for both homogeneous and inhomogeneous social networks.
This is followed in section 5
by numerical investigations of the steady-state and dynamics of the
model on several models of social networks: the ER random graph, the
uncorrelated scale-free networks and the assortatively
correlated SF networks. We close this paper in section 6 with conclusions.
\section{A general model for rumour dynamics on social networks}
The spread of rumours is a complex socio-psychological process. An
adequate modeling of this process requires both a correct description
of the underlying social networks along which rumours spread and a
quantitative formulation of various behavioural mechanisms that
motivate individuals to participate in the spread of a rumour. The
model described below is an attempt to formalize and simplify these
behavioral mechanisms in terms of a set of simple but plausible rules.
Our model is defined in the following way. We consider a population
consisting of $N$ individuals which, with respect to the rumour, are
subdivided into ignorants, spreaders and stiflers. Following Maki and
Thompson \cite{Maki}, we assume that the rumour spreads by {\it
directed} contact of the spreaders with others in the
population. However, these contacts can only take takes place along
the links of an undirected social interaction network $G=(V,E)$, where
$V$ and $E$ denote the vertices and the edges of the network,
respectively. The contacts between the spreaders and the rest of the
population are governed by the following set of rules
\begin{itemize}
\item
Whenever a spreader contacts an ignorant, the ignorant
becomes an spreader at a rate $\lambda$.
\item
When a spreader contacts another spreader or a stifler the
{\it initiating} spreader becomes a stifler at a rate $\alpha$.
\end{itemize}
In the above, the first rule models the tendency of individuals to
accept a rumour only with a certain probability which, loosely
speaking, depends on the urgency or credibility of a rumour. The
second rule, on the other hand, models the tendency of individuals
to lose interest in spreading a rumour when they learn, through
contacts with others, that the rumour has become stale news, or is
false. In both the Daley-Kendall and the Maki-Thompson rumour
models, and their variants, stifling is the only mechanism that
results in cessation of rumour spreading. In reality, however,
cessation can occur also purely as a result of spreaders forgetting
to tell the rumour, or their disinclination to spread the rumour
anymore. Following a suggestion in \cite{book1_epidemic}, we take
this important mechanism into account by assuming that individuals
may also cease spreading a rumour spontaneously (i.e. without the
need for any contact) at a rate $\delta$. The spreading process
starts with one (or more) element(s) becoming informed of a rumour
and terminates when no spreaders are left in the population.
\section{Interactive Markov chain mean-field equations}
We can describe the dynamics of the above model on a network within
the framework of the Interactive Markov Chains (IMC). The IMC was
originally introduced as a means for modelling social processes
involving many interacting actors (or agents) \cite{imc1}. More
recently they have been applied to the dynamics of cascading
failures in electrical power networks \cite{imc2}, and the spread of
malicious software (malware) on the Internet \cite{imc3}. An IMC
consists of $N$ interacting nodes, with each node having a state
that evolves in time according to an internal Markov chain. Unlike
conventional Markov chains, however, the corresponding internal
transition probabilities depend not only on the current state of a
node itself, but also on the states of all the nodes to which it is
connected. The overall system evolves according to a global Markov
Chain whose state space dimension is the product of states
describing each node. When dealing with large networks, the
exponential growth in the state space renders direct numerical
solution of the IMCs extremely difficult, and one has to resort to
either Monte Carlo simulations or approximate solutions. In the
case of rumour model, each internal Markov chain can be in one of
the three states: ignorant, spreader or stifler. For this case we
derive below a set of coupled rate equations which describe on a
mean-field level the dynamics of the IMC. We note that a similar
mean-field approach may also be applicable to other dynamical
processes on networks which can be described within the IMC
framework.
Consider now a node $j$
which is in the ignorant state at time $t$. We denote with $p^j_{ii}$
the probability that this node stays in the ignorant state in
the time interval $[t,t+\Delta t]$, and with
$p^j_{is}=1-p^j_{ii}$ the probability that it makes a transition to
the spreader state. It then follows that
\begin{equation}
p_{ii}^{j}= (1-\Delta t \lambda)^{g},
\end{equation}
where $g=g(t)$ denotes the number of neighbors of node $j$
which are in the spreader state at time $t$. In order
to progress, we shall coarse-grain the micro-dynamics of the system by
classifying nodes in our network according to their degree
and taking statistical average of the above transition probability
over degree classes.
Assuming that a node $j$ has $k$ links, $g$ can be considered
as an stochastic variable which has the following binomial distribution:
\begin{equation}
\Pi(g,t)
=\binom{k}{g}\theta(k,t)^{g}(1-\theta(k,t))^{k-g},
\end{equation}
where $\theta(k,t)$ is the probability at time $t$
that an edge emanating from
an ignorant node with $k$ links points to a spreader node. This
quantity can be written as
\begin{equation}
\theta(k,t)= \sum_{k'} P(k'|k)P(s_{k'}|i_k) \approx
\sum_{k'} P(k'|k)\rho^s(k',t) .
\end{equation}
In this equation $P(k'|k)$ is
the degree-degree correlation function, $P(s_{k'}|i_k)$ the
conditional probability that a node with $k'$ links is in the spreader
state, given that it is connected to an ignorant node with degree $k$,
and $\rho^s(k',t)$ is the density of spreader nodes at time
$t$ which belong to connectivity class $k$.
In the above equation the
final approximation is obtained by ignoring dynamic correlations
between the states of neighboring nodes.
The transition probability $\bar{p}_{ii}(k,t)$ averaged over all possible
values of $g$ is then given by:
\begin{eqnarray}
\bar{p}_{ii}(k,t) & = &
\sum_{g=0}^k \binom{k}{g}
(1-\lambda\Delta t )^g\theta(k,t)^{g}(1-\theta(k,t))^{k-g} \nonumber \\
& = &
\left( 1-\lambda \Delta t \sum_{k'}P(k'|k)\rho^s(k',t)\right)^k.
\end{eqnarray}
In a similar fashion we can derive an expression for the
probability
$\bar{p}_{ss}(k,t)$ that a spreader node which has
$k$ links stays in this state in the interval
$[t,t+\Delta t]$. In this case, however, we also need
to compute the expected value of the number of stifler neighbors
of the node at time $t$. Following steps similar to the previous paragraphs
we obtain
\\
\begin{eqnarray}
\bar{p}_{ss}(k,t) &=&
\left( 1-\alpha \Delta t
\sum_{k'}P(k'|k)(\rho^s(k',t)+\rho^r(k',t))\right)^k \nonumber \\
&\times& (1-\delta \Delta t).
\end{eqnarray}
The corresponding probability for a transition from the spreader
to the stifler state, $\bar{p}_{sr}(k,t)$ is given by $
\bar{p}_{sr}(k,t)=1-\bar{p}_{ss}(k,t)$.
The above transition probabilities can be used to set up a system of
coupled Chapman-Kolmogorov equations for the probability distributions
of the population of ignorants, spreaders and stiflers within each
connectivity class. However, ignoring fluctuations around expected
values we can also obtain a set of deterministic rate equations for
the expected values of these quantities.
Denote with $I(k,t), S(k,t), R(k,t)$ the expected values of the
populations of nodes
belonging to connectivity class $k$ which at time $t$ are in the ignorant,
spreader or stifler state, respectively.
The event that an ignorant node in class $k$
will make a transition to the spreader state during $[t,t+\Delta t]$
is a Bernoulli random variable with probability
$(1-p_{ii}(k,t))$ of success. As a sum of i.i.d random Bernoulli
variables, the total number of successful transitions in this time
interval has a binomial distribution, with an expected value given by
$I(k,t)(1-p_{ii}(k,t))$. Hence the rate of change in the expected value
of the population of ignorant nodes
belonging to class $k$ is given by
\begin{eqnarray}
I(k,t+\Delta t) &=& I(k,t)-I(k,t)\nonumber \\
&\times & \left[1-\left(1-\lambda \Delta t \sum_{k'}\rho^s(k',t)P(k'|k)
\right)^k \right]
\end{eqnarray}
Similarly, we can write the corresponding
rate of change in the population of spreaders and
stiflers as follows
\begin{eqnarray}
S(k,t+\Delta t) &=&S(k,t) + I(k,t)\left
[1-
\left(1-\lambda \Delta t \sum_{k'}\rho^s(k',t)P(k'|k)\right)^k\right]
\nonumber \\
&-&
S(k,t)\left[1-
\left(1-\alpha \Delta t
\sum_{k'}(\rho^s(k',t)+\rho^r(k',t))P(k'|k)\right)
^k\right] \nonumber \\
&-&
\delta S(k,t)
\end{eqnarray}
\begin{eqnarray}
R(k,t+\Delta t)&=&R(k,t) \nonumber \\
&+&
S(k,t)
\left[1-
\left(1-\alpha \Delta t
\sum_{k'}(\rho^s(k',t)+\rho^r(k',t))P(k'|k)\right)^k\right]
\nonumber \\
&+&
\delta S(k,t)
\end{eqnarray}
In the above equations $\rho^i(k,t),\rho^s(k,t)$, and $\rho^r(k,t)$
are the fraction of nodes belonging to class $k$ which are in the
ignorant, spreader and stifler states, respectively.
These quantities satisfy the normalization
condition $\rho^{i}(k,t)+\rho^s(k,t)+\rho^r(k,t)=1$.
In the limit $\Delta t \rightarrow 0$ we obtain:
\begin{equation} \frac{\partial
\rho^i(k,t)}{\partial t}=-k \lambda
\rho^i(k,t)\sum_{k'}\rho^s(k',t)P(k'|k)
\label{ceq1}
\end{equation}
\begin{eqnarray}
\frac{\partial \rho^s(k,t)}{\partial t} &=&
k \lambda \rho^i(k,t) \sum_{k'}\rho^s(k',t)P(k'|k)
\nonumber \\
&-&
k\alpha \rho^s(k,t)
\sum_{k'}(\rho^s(k',t)+\rho^r(k',t))P(k'|k) - \delta \rho^s(k,t).
\label{ceq2}
\end{eqnarray}
\begin{eqnarray}
\frac{\partial \rho^r(k,t)}{\partial t} &=&
k \alpha \rho^s(k,t)
\sum_{k'}(\rho^s(k',t)+\rho^r(k',t))P(k'|k) + \delta \rho^s(k,t).
\label{ceq3}
\end{eqnarray}
For future reference we note here that information on the underlying network
is incorporated in the above equations solely via the degree-degree
correlation function. Thus in our analytical and numerical
studies reported in the next section we do not need to generate any
actual network. All that is required is either an analytical expression
for $P(k'|k)$ or a numerical representation of this quantity.
\section{Analysis}
\subsection{Homogeneous networks}
In order to understand some basic features of our rumour model we first
consider the case of homogeneous networks, in which degree
fluctuations are very small and there are no degree correlations.
In this case the rumour equations become:
\begin{eqnarray}
\frac{d\rho^i(t)}{dt} & = &-\lambda \bar{k}\rho^i(t)\rho^s(t) \\
\frac {d\rho^s(t)}{dt} & = & \lambda \bar{k}\rho^i(t)\rho^s(t)
-\alpha \bar{k} \rho^s(t)(\rho^s(t)+\rho^r(t)) \nonumber \\
& - & \delta \rho^s(t)
\\
\frac{d\rho^r(t)}{dt} & =& \alpha\bar{k}\rho^s(t)(\rho^s(t)+
\rho^r(t)) +\delta \rho^s(t),
\end{eqnarray}
where $\bar{k}$ denotes the constant degree distribution of the network (or the
average value for networks in which the probability of finding a node with a
different connectivity decays exponentially fast).
The above system of equations can be integrated analytically using an
standard approach. In the infinite time limit, when
$\rho^s(\infty)=0$, we obtain the following transcendal equation for
$R=\rho^r(\infty)$, the final fraction of nodes which ever hear the
rumour (we call this the final rumour size) \begin{equation} R=1-e^{ -\epsilon R}
\label{threshold-hom} \end{equation} where \begin{equation} \epsilon=
\frac{(\alpha+\lambda)\bar{k}}{\delta+\alpha\bar{k}}. \end{equation} Eq.
(\ref{threshold-hom}) admits a non-zero solution only if
$\epsilon>1$. For $\delta \neq 0$ this condition is fulfilled
provided \begin{equation} \frac{\lambda}{\delta}\bar{k} >1 , \end{equation} which is
precisely the same threshold condition as found in the SIR model of
epidemic spreading on homogeneous networks
\cite{sir1_yamir,sir1_ml}. On the other hand, in the special case
$\delta=0$ (i.e when the forgetting mechanism is absent)
$\epsilon=1+\lambda/\alpha>1$, and so Eq. (14) always admits a
non-zero solution, in agreement with the result in
\cite{rumour2_maziar}.
The above result shows, however, that the presence of a forgetting
mechanism results in the appearance of a finite threshold in the
rumour spreading rate below which rumours cannot spread in homogeneous
networks. Furthermore, the value of the threshold is {\em independent}
of $\alpha$ (i.e. the stifling mechanism), and is the same as that for
the SIR model of epidemics on such networks. This result can be
understood by noting that in the above equations the terms
corresponding to the stifling mechanism are of second order in
$\rho^s$, while the terms corresponding to the forgetting mechanism
are only of first order in this quantity. Thus in the initial state of
the spreading process, where $\rho^s\approx 0$ and $\rho^r\approx0$,
the effect of stifling is negligible relative to that of forgetting,
and the dynamics of the model reduces to that of the SIR model.
\subsection{Inhomogeneous networks}
Next we consider uncorrelated inhomogeneous networks. In such networks
the degree-degree correlations can be written as
\cite{book_vespignani}:
\begin{equation}
P(k'|k)= q(k')=\frac{k'P(k')}{\langle k \rangle},
\label{uncorr}
\end{equation}
where $P(k')$ is the degree distribution and $\langle k \rangle $ is
the average degree. In this case the dynamic of rumour spreading is
describe by Eqs. (\ref{ceq1}-\ref{ceq3}). Eq. (\ref{ceq1}) can be
integrated exactly to yield:
\begin{equation}
\rho^i(k,t)=\rho^i(k,0) e^{-\lambda k\phi(t)},
\end{equation}
where $\rho^i(k,0)$ is the initial density of ignorant nodes
with connectivity $k$, and we have introduced the auxiliary function
\begin{equation}
\phi(t)=\sum_k q(k) \int_0^t
\rho^s(k,t')dt'\equiv \int_0^t \avg{\rho^s(k,t')}dt'.
\end{equation}
In the above equation and hereafter we use the shorthand notation
\begin{equation}
\avg{O(k)}=\sum_k q(k) O(k)
\end{equation}
with
\begin{equation}
q(k)=\frac{kP(k)}{\langle k \rangle}.
\end{equation}
In order to obtain an expression for the final size of the rumour, $R$,
it is more convenient to work with $\phi$. Assuming an homogeneous
initial distribution of ignorants, $\rho^i(k,0)=\rho^i_0$,
we can obtain a differential equation for this quantity by multiplying
Eq. (\ref{ceq2}) with $q(k)$ and summing over $k$. This yields after some
elementary manipulations:
\begin{eqnarray}
\frac{d\phi}{dt} = 1-\avg{e^{-\lambda k\phi}}) -
\delta \phi - \alpha \int_0^t
\left[1-\avg{e^{-\lambda k\phi(t')}}\right] \avg{k\rho^s(k,t')}dt',
\label{phi2}
\end{eqnarray}
where, without loss of generality, we have also put $\rho^i_0\approx
1$.
In the limit $t\rightarrow \infty$ we have
$\frac{d\phi}{dt}=0$, and Eq. (\ref{phi2}) becomes:
\begin{eqnarray}
0 = 1-\avg{e^{-\lambda k\phi_\infty}}
- \delta\phi_\infty -\alpha\int_0^\infty
\left[1-\avg{e^{-\lambda k\phi(t')}}\right] \avg{k\rho^s(k,t')}dt',
\nonumber \\
\label{phiinf}
\end{eqnarray}
where $\phi_\infty=\lim_{t\rightarrow \infty} \phi(t)$.
For $\alpha=0$ Eq. (\ref{phiinf}) can be solved explicitly to
obtain $\Phi_{\infty}$ \cite{sir1_yamir}. For $\alpha\neq0$ we solve
(\ref{phiinf}) to leading order in $\alpha$. Integrating Eq.
(\ref{ceq2}) to zero order in $\alpha$ we obtain \begin{equation}
\rho^s(k,t)=1-e^{-\lambda k\phi}-\delta \int_0^t e^{\delta(t-t')}
\left[1-e^{-\lambda k\phi(t')}\right]dt'+O(\alpha). \end{equation} Close to the
critical threshold both $\phi(t)$ and $\phi_\infty$ are small.
Writing $\phi(t)=\phi_\infty f(t)$, where $f(t)$ is a finite
function, and working to leading order in $\phi_\infty$, we obtain
\begin{equation} \rho^s(k,t)\simeq -\delta \lambda k \phi_\infty \int_0^t
e^{\delta(t-t')}f(t') dt'+O(\phi_\infty^2)+O(\alpha) \end{equation} Inserting
this in Eq. (\ref{phiinf}) and expanding the exponential to the
relevant order in $\phi_\infty$ we find
\begin{eqnarray}
0 = \phi_\infty\left[\lambda\avg{k}-\delta-
\lambda^2\avg{k^2}(1/2+\alpha\avg{k}I)\phi_\infty\right]
+ O(\alpha^2)+O(\phi_\infty^3)
\end{eqnarray}
where $I$ is a finite and positive-defined integral.
The non-trivial solution of this
equation is given by:
\begin{equation}
\phi_\infty=\frac{\lambda\avg{k}-\delta}
{\lambda^2\avg{k^2}(\frac{1}{2}+\alpha I\avg{k})}.
\end{equation}
Noting that $\avg{k}=\langle k^2 \rangle/\langle k \rangle$
and $\avg{k^2}=\langle k^3 \rangle /\langle k \rangle $
we obtain:
\begin{equation}
\phi_\infty=\frac{ 2\langle k \rangle
( \frac{\langle k^2 \rangle}{\langle k \rangle}\lambda -\delta) }
{\lambda^2 \langle k^3 \rangle (1+ 2\alpha I
\frac{ \langle k^2 \rangle}{\langle k \rangle})}.
\label{phi}
\end{equation}
This yields a positive value for $\phi_{\infty}$ provided that
\begin{equation}
\frac{\lambda}{\delta} \geq \frac{\langle k \rangle}{\langle k^2 \rangle}.
\label{threshold}
\end{equation}
Thus, to leading order in $\alpha$, the critical
rumour threshold is independent of the stifling mechanism
and is the same as for the SIR model \cite{sir1_yamir,sir1_ml}.
In particular, for $\delta=1$ the critical rumour
spreading threshold is given by
$\lambda_c= \langle k \rangle/\langle k^2 \rangle$, and
Eq. (\ref{phi}) simplifies to:
\begin{equation}
\phi_\infty=\frac{ 2\langle k \rangle (\lambda -\lambda_c)}
{\lambda^2 \langle k^3 \rangle(\lambda_c+2\alpha I)}.
\end{equation}
Finally, $R$ is given by
\begin{equation}
R=\sum_kP(k)(1-e^{-\lambda k\phi_\infty}),
\label{exp}
\end{equation}
The solution to the above equation depends on the form of $P(k)$. In
particular, for homogeneous networks where all the moments of the
degree distribution are bounded, we can expand the exponential in
Eq. (\ref{exp}) to obtain
\begin{eqnarray}
R & \approx & \sum_k P(k) \lambda k \phi_\infty =
\frac{ 2\langle k \rangle^2 (\lambda -\lambda_c)}
{\lambda \langle k^3 \rangle(\lambda_c+2\alpha I)},
\end{eqnarray}
which shows that $R \sim (\lambda-\lambda_c)$ in the
vicinity of the rumour threshold. For heterogeneous networks, one must
solve the equation for $P(k)$. This can be done for example, as for
the SIR model \cite{sir1_yamir}.
\section{Numerical results}
\subsection{Random graphs and uncorrelated scale-free networks}
We consider first {\it uncorrelated}
networks, for which the degree-degree correlations are given by
Eq. (\ref{uncorr} ). We shall consider two classes of such networks.
The first class is the Erd\H os-R\'enyi random networks, which
for large $N$
have a Poisson degree distribution:
\begin{equation}
P(k)=e^{-k}\frac{\langle k \rangle ^k}{k!}.
\end{equation}
The above degree distribution peaks at an average value $\langle k \rangle$
and shows small fluctuations around this value. The second class we consider
are scale-free networks which have a power law degree distribution:
\begin{equation}
P(k)=
\begin{cases}
Ak^{-\gamma}
& k_{\text{min}}\leq k
\\
0 & \text{otherwise.}
\end{cases}
\end{equation}
In the above equation $k_{\text{min}}$
is the minimum degree of the networks and $A$
is a normalization constant. Recent studies of social networks on the
Internet indicates that many of these networks show highly
right-skewed degree distributions, which could often be modelled by a
power-law degree distribution \cite{email_ebel,soc_nets1,instant}.
For $2\leq \gamma \leq 3$ the variance of the above degree
distribution becomes infinite in the limit of infinite system size
while the average degree distribution remains finite.
We shall consider henceforth SF networks with $\gamma=3$.
Our studies of uncorrelated networks were performed using the above
forms of $P(k)$ to represent ER and SF networks, respectively. The
size of the networks considered was $N=10^5$ and $N=10^6$, and the
average degree was fixed at $ \langle k \rangle =7$. For each network
considered we generated a sequence of $N$ random integers distributed
according to its degree distribution. The coupled set of differential
equation (\ref{ceq1}-\ref{ceq3}) were then solved numerically using an
standard finite difference scheme, and numerical convergence with
respect to the step size was checked numerically. In the following
and throughout the paper all calculations reported are performed by
starting the rumour from a randomly chosen initial spreader, and
averaging the results over 300 runs with different initial spreaders.
The calculations reported below were performed for networks consisting
of $N=10^6$ nodes.
In our first set of calculations we set $\delta=1$ and investigate
the dynamics as a function of the rumour spreading rate $\lambda$
and the stifling rate $\alpha$.
First we focus on the impact of network topology
on the final size of a rumour, $R$, which for inhomogeneous networks
is obtained from
\begin{equation}
R=\sum_k\rho^r(k,t_{\infty}),
\end{equation}
where $t_\infty$ denotes a sufficiently long time at which
the spreading process has reached its steady state
(i.e. no spreaders are left in the network). In Fig. 1 $R$
corresponding to the ER network is
plotted as a function of $\lambda$, and for
several different values of the stifling parameter $\alpha$.
It can be seen that in this network $R$ exhibits a critical
threshold $\lambda_c$ below which a rumour cannot spread in the
network. Furthermore, just as in the case of homogeneous networks,
the value of the threshold does not depend on $\alpha$, and is
at $\lambda_c=0.12507$. This value is in excellent agreement
with the analytical results obtained in the previous section.
We also verified numerically that, in agreement with our
analytical findings, the behaviour of $R$ in the vicinity of the critical
point can, be described with the form
\begin{equation}
R \sim A(\lambda-\lambda_c),
\end{equation}
where $A=A(\alpha)$ is a smooth and monotonically decreasing
function of $\alpha$. The results are shown in Fig. 2 where $R$ is plotted as
function of $\lambda$ for a range of values of $\alpha$, together
with the corresponding fits.
Next we turn to our results for the SF network.
In Fig. 3 results for $R$ in this network are shown. In
this case we also observe the presence of an $\alpha$-independent
rumour threshold,
albeit for much smaller spreading rates than for the ER network.
We have verified numerically that in this case the threshold is
approached with zero slope, as can also be gleaned from Fig. 3.
Since the value of the threshold is independent of $\alpha$,
we can use the well-known result
for the SIR model (the $\alpha=0$ case) to conclude that in the
limit of infinite system size the threshold seen in the
SF network will approach zero. It is therefore not an
intrinsic property of rumour spreading on this network.
In order to further analyze the behavior of $R$ in SF networks, we
have numerically fitted our results to the stretched exponential
form, \begin{equation} R \sim \exp(-C/\lambda), \end{equation} with $C$ depending only
weakly on $\alpha$. This form was found to describe the epidemic
prevalence in both the SIS and the SIR model of epidemics
\cite{sis1_vesp,sir1_yamir}. The results are displayed in Fig. 4,
and they clearly indicate that the stretched exponential form also
nicely describes the behavior of $R$ in our rumour model. This
result provides further support for our conjecture that the general
rumour model does not exhibit any threshold behavior on SF networks
(at least in the limit of infinite systems size).
In addition to investigating the impact of network topology on the
steady-state properties of the model, it is of great interest to
understand how the time-dependent behavior of the model is affected by
topology. In Figs. 5 and 6 we display, as representative examples,
time evolution of, respectively, the total fractions of stiflers and
spreaders, in both networks for $\lambda=1$ and two sets of values of
the cessation parameters: $\{\delta=1,\alpha=0 \}$, and $\{\delta=0,
\alpha=1\}$. The first set of parameters corresponds to a spreading
process in which cessation results purely from spontaneous forgetting
of a rumour by spreaders, or their disinclination to spread the rumour
any further. The second set corresponds to a scenario where
individuals keep spreading the rumour until they become stiflers due
to their contacts with other spreaders or stiflers in the network. As
can be seen in Fig. 5, in the first scenario the initial spreading
rate of a rumour on the SF network is much faster than on the ER
network. In fact, we find that the time required for the rumour to
reach $50\%$ of nodes in the ER random graph is nearly twice as long
as the corresponding time on the SF networks. This large difference
in the spreading rate is due to the presence of highly connected nodes
(social hubs) in the SF network, whose presence greatly speeds up the
spreading of a rumour. We note that in this scenario not only a
rumour spreads initially faster on SF networks, but it also reaches a
higher fraction of nodes at the end of the spreading process.
It can be seen from Figs, 5 and 6 that in the second spreading scenario
(i.e. when stifling is the only mechanism for cessation) the
initial spreading rate on the SF network is, once again, higher than
on the ER network. However, unlike the previous situation, the ultimate
size of the rumour is higher on the ER network.
This behavior is due to the conflicting
roles that hubs play when the stifling mechanism is switched
on. Initially the presence of hubs speeds up the spreading but once
they turn into stiflers they also effectively impede further spreading of the
rumour.
\subsection{Assortatively correlated scale-free networks}
Recent studies have revealed that social networks display
assortative degree correlations, implying that highly connected
vertices preferably connect to vertices which are also highly
connected \cite{review_newman}. In order to study the impact of
such correlations on the dynamics of our model, we make use of the
following ansatz for the degree-degree correlation function
\cite{vazquez1}
\begin{equation}
P(k'|k) = (1-\beta)q(k')+\beta \delta_{kk'}; \; \; \; \; \; \; \;
(0\leq\beta<1).
\end{equation}
The above form allows us to study
in a controlled way the impact of
degree correlations on the spreading of rumour.
Using the above degree-degree correlation function we numerically
solved Eqs. (\ref{ceq1}-\ref{ceq3}) for a SF network characterized by
$\gamma=3$ and $<k>=7$. The network size was fixed at $N=100,000$, and
we used two values for the correlation parameter: $\beta=0.2$ and
$\beta=0.4$. Fig. 7 displays $R$ as a function of $\lambda$, and for
$\alpha=0.5,0.75,1$ (the value of $\delta$ was fixed at $1$).
It can be seen that below $\lambda\approx 0.5$ a rumour will reach a
somewhat smaller fraction of nodes on the correlated networks than on
the uncorrelated ones. However for larger values of $\lambda$ this
behavior reverses, and the final size of the rumour in assortatively
correlated networks shows a higher value than in the uncorrelated
network. We thus conclude that the qualitative impact of degree
correlations on the final size of a rumour depends very much on the
rumour spreading rate.
Finally, we investigate the effect of assortative correlations on the
speed of rumour spreading. In Fig. 8 we show our results for the time
evolution of the total fraction of spreaders, $S(t)$, in scale-free
networks consisting of $N=100,000$ nodes and for a correlation
strength ranging from $\beta=0$ to $\beta=0.4$. In these calculations
the value of $\lambda$ was fixed at $1$, and we considered two values
of $\alpha$: 0,1. It can be seen that the initial rate at which a
rumour spreads increases with an increase in the strength of
assortative correlations regardless of the value of $\alpha$.
However, for $\alpha=1$ the rumour also dies out faster when such
correlations are stronger.
\section{Conclusions}
In this paper we introduced a general model of rumour spreading on
complex networks. Unlike previous rumour models, our model
incorporates two distinct mechanisms that cause cessation of a rumour,
stifling and forgetting. We used an Interactive Markov Chain
formulation of the model to derive deterministic mean-field equations
for the dynamics of the model on complex networks. Using these
equations, we investigated analytically and numerically the behavior
of the model on Erd\H os-R\'enyi random graphs and scale-free networks
with exponent $\gamma=3$. The critical behavior, the dynamics and the
stationary state of our model on these networks are significantly
different from the results obtained for the dynamics of simpler rumour
models on complex networks
\cite{rumour1_zanette,rumour2_zanette,rumour1_maziar,rumour2_maziar}.
In particular, our results show the presence of a critical threshold
in the rumour spreading rate below which a rumour cannot spread in ER
networks. The value of this threshold was found to be independent of
the stifling mechanism, and to be the same as the critical infection
rate of the SIR epidemic model. Such a threshold is also present in
the finite-size SF networks we studied, albeit at a much smaller
value. However in SF networks this threshold is reached with a zero
slope and its value becomes vanishingly small in the limit of infinite
network size. We also found the initial rate of spreading of a rumour
to be much higher on scale-free networks than on ER random graphs. An
effect which is caused by the presence of hubs in these networks,
which efficiently disseminate a rumour once they become informed. Our
results show that SF networks are prone to the spreading of rumours,
just as they are to the spreading of infections.
Finally, we used a local ansatz for the degree-degree correlation
function in order to numerically investigate the impact of assortative
degree correlations on the speed of rumour spreading on SF networks.
These correlations were found to speed up the initial rate of
spreading in SF networks. However, their impact on the final fraction
of nodes which hear a rumour depends very much on the rate of rumour
spreading.
In the present work we assumed the underlying network to be static,
i.e. a time-independent network topology. In reality, however, many
social and communication networks are highly dynamic. An example of
such time-dependent social networks is Internet chatrooms, where
individuals continuously make new social contacts and break old
ones. Modelling spreading processes on such dynamic networks is highly
challenging, in particular when the time scale at which network
topology changes becomes comparable with the time scale of the process
dynamics. We aim to tackle this highly interesting problem in future
work.
M.~N. acknowledges the Abdus Salam International Centre for
Theoretical Physics (ICTP) for a visiting fellowship during which some
amendments to this work were made. Y. M thanks the support of DGA
through a mobility grant (MI06/2005) and BT for hospitality. Y.~ M.
is supported by MEC through the Ram\'{o}n y Cajal Program. This work
was supported by BT and the Spanish DGICYT Projects
FIS2004-05073-C04-01 and FIS2005-00337. |
0807.0413 | \section{\label{sec:intr}Introduction}
In superfluid helium, vortices form when the helium is rotated rapidly or
when there is turbulence \cite{Vine,tilleybook}. Though such vortices are similar
to the vortices that make up a vortex street behind the wings of an airplane
or to the funnel clouds of tornadoes, they are only an Angstrom or two across \cite{Guyon-book}.
A more essential difference is that the vortices in a superfluid do not
need a constant source of energy to survive. In fact, a vortex is long-lived
because the strength of its flow is fixed by the quantization of angular momentum. Thus, the dissipative mechanisms of a conventional fluid are absent.
In this article, we focus on forces that the vortices experience as a result of
geometric constraints, with an emphasis on those encountered in
thin layers of liquid helium wetting a \emph{curved}
substrate with spatially varying Gaussian curvature. As a result of the
broken translational invariance of the underlying curved space, the energy of a single vortex with circulation quantum number $n_{i}$
at position ${\bf u}_{i}$ includes both a divergent term
and a position dependent self-energy, $E_s({\bf u}_i)$, given by \cite{swiss}
\begin{equation}
E_s({\bf u}_i)=- \pi K n_i^2 U_G({\bf u}_i) ,
\label{eq:ecurv-s2bis}
\end{equation}
where $K=\frac{\rho_s \hbar^2}{m^2}$ is the superfluid stiffness expressed in terms of the $^4$He atomic mass, $m$, and the superfluid mass density, $\rho_s$.
The potential $U_G({\bf u}_i)$
is obtained from solving a covariant Poisson equation with the Gaussian curvature, $G(\mathbf{u_i})$,
acting as a source
\begin{equation}
\nabla^2 U_G(\mathbf{u_i}) = G(\mathbf{u_i}).
\label{eq:geompdeq}
\end{equation}
Vortices (and anti-vortices) are attracted (repelled) to regions of negative (positive) Gaussian curvature.
These geometric interactions, while more exotic, are similar in origin to boundary-vortex interactions and
can be suitably treated by the method of conformal mapping.
Similar ideas naturally arise in a variety of soft-matter systems which have been confined
in a thin layer wetting a curved substrate. The specific form of the resulting geometric interactions
depends on the symmetry of the order parameter as well as on the shape of the substrate. Examples that have been studied both theoretically and experimentally include
colloidal crystals on curved interfaces \cite{bowickB,VitelliLucks,Bausch03}, columnar phases of block co-polymers \cite{Santangelo07, Alex-thesis} as well as thin layers of nematic liquid crystals \cite{park,geomgenerate,Vitelli06,Fernandez-Nieves07}. Fueled by the drive towards technological applications based on the notion of self-assembly directed by geometry \cite{Dinsmore02,Nelson02,DeVries07}, the study of these $frustrated$ materials aims at predicting how the non-uniform distribution of curvature of the underlying substrates induces an inhomogeneous phase in the curved monolayer. An understanding of the resulting macroscopic properties can be built from a mesoscopic description cast in terms of the energetics of the topological defects which often exist even in the ground state and play a crucial role in determining how the material melts or ruptures. The advantage of this approach stems from the huge reduction in degrees of freedom achieved upon re-expressing the energy stored in the elastic field in terms of a few topological defects, rather than keeping track of the state of all the microscopic components, e.g. individual particle positions or molecular orientations. This step allows one to carry out efficient computational studies \cite{Hexemer07,bowickB} and provides a suitable starting point for analytical work in the form of effective free energies derived from continuum elastic theory.
Many of the mathematical techniques employed in this article to study vortices in curved superfluid films find application in the soft matter domain, in particular in those contexts where bond-orientational order is important \cite{swiss}. In flat space both superfluid and liquid crystal films can be described, as a first approximation, by an XY model\cite{Nels-Kost}.
Both liquid crystal disclinations and vortices are modeled as a Coulomb gas of charged particles interacting logarithmically.
However, the quantum nature of the problem considered in the present work introduces a fundamental difference between these two classes of systems that is best illustrated by contrasting the angle of the liquid crystal director with the phase of the superfluid's collective wave function. The former represents the orientation of a vector (with both ends identified in the case of a nematic) that lives in the tangent space of the surface while the latter is a quantum mechanical object that transforms like a scalar since it is defined in an internal space.
This subtle difference resurfaces upon considering the distinct curved-space generalizations of the XY model that apply to each of these two systems.
The free energy functional ${\cal
F}_{v}$ to be minimized for the case of orientational order on a surface with points labeled by the coordinates ${\bf u}=(u_1,u_2)$ reads
\cite{Davidreview}:
\begin{equation}
{\cal F}_{v} = \frac{K}{2}\int d^{2}u\sqrt{g} g^{\alpha\beta}
(\partial_{\alpha}\theta({\bf u}) - \Omega_{\alpha}({\bf
u}))(\partial_{\beta}\theta({\bf u}) - \Omega_{\beta}({\bf
u}))\!\!\!\! \quad , \label{eq:patic-ener}
\end{equation}
where $g_{\alpha\beta}$ and $g$ indicate the metric tensor and its determinant while $\Omega_{\alpha}({\bf u})$ is a connection that compensates for
the rotation of the 2D basis vectors ${\bf E}_{\alpha}({\bf u})$ (with respect to which $\theta({\bf u})$ is measured)
in the direction of $u_{\alpha}$ \cite{Kami02}. Since the curl of the field
$\Omega_{\alpha}({\bf u})$ is equal to the Gaussian curvature $G({\bf u})$ \cite{Davidreview},
the integrand in Eq. (\ref{eq:patic-ener}) never vanishes
because $\Omega_{\alpha}({\bf u}) \ne \partial_{\alpha}\theta$ on a surface with $G({\bf u})\ne0$.
As the
substrate becomes more curved, the resulting energy cost of geometric
frustration can be lowered by generating disclination-dipoles in the ground
state even in the absence of topological constraints.
The connection $\Omega_{\alpha}({\bf u})$ is a geometric gauge field akin to the electromagnetic
vector potential, with the Gaussian curvature playing the role of a magnetic field. If topological defects are present,
they appear as monopoles in the singular part of $\partial_{\alpha}\theta({\bf u})$. In analogy with electromagnetic theory, their interaction with the Gaussian curvature arises mathematically from the cross-products between $\partial_{\alpha}\theta({\bf u})$ and the geometry
induced vector potential $\Omega_{\alpha}({\bf u})$, see Eq. (\ref{eq:patic-ener}). As a result of this interaction, disclinations in a liquid crystal
are attracted to regions of the substrate whose curvature
has the same sign as the defect's topological charge \cite{park}, whereas
vortices in a superfluid favor negatively curved regions independently of their
sense of circulation. The anomalous coupling between vortices and Gaussian curvature introduced in Equations (\ref{eq:ecurv-s2bis}) and (\ref{eq:geompdeq})
originates from the distortion of the flow pattern by the protrusions and wrinkles
of the surface.
For a disclination with topological index $n_{i}$ (defined by the amount $\theta$ increases along a
path enclosing the defect's core) the geometric potential $E_{v}({\bf u}_{i})$ reads \cite{swiss}
\begin{equation}
E_{v}({\bf u}_{i})= 2 \pi K \!\!\!\!\! \quad n_{i}
\left(1-\frac{n_{i}}{2 } \right) \!\!\!\!\! \quad U_G({\bf
u}_{i}) \!\!\!\! \quad , \label{eq:ecurv-v}
\end{equation}
where $K$ is the elastic stiffness and $U_G$ is the same potential defined in Eq. (\ref{eq:geompdeq}).
Note that the anomalous coupling also contributes to determine the energetics of liquid crystal disclinations but, in this case,
the gauge coupling, which is linear in $n$, is stronger for small $n$.
To understand the physical and mathematical origin of these distinct coupling mechanisms, note
that in the ground state of a $^4$He film, the phase $\theta({\bf u})$ can be
constant throughout the surface so that the corresponding energy
vanishes. In a system with geometric frustration, the gauge coupling between defects and the underlying curvature is mediated by the deformed ground state texture that exists in the liquid crystal layer prior to the introduction of the defects simply as a result of geometrical constraints. Once a defect is introduced it interacts with these preexisting elastic deformations. Unlike the case of orientational order considered previously, no geometric frustration exists in the superfluid film.
The superfluid free energy ${\cal F}_{s}$ to be minimized is a simple
scalar generalization of the familiar flat space counterpart
\begin{equation} {\cal F}_{s} = \frac{K}{2}\int
d^{2}u \!\!\!\!\!\! \quad \sqrt{g} \!\!\!\!\! \quad
g^{\alpha\beta}
\partial_{\alpha}\theta({\bf u}) \!\!\!\!\! \quad \partial_{\beta}\theta({\bf u}) \!\!\!\! \quad .
\label{eq:supfl-I}
\end{equation}
The crucial point is that no connection $\Omega_{\alpha}({\bf u})$ is necessary to write down the covariant derivative for this simpler case of a scalar order parameter. Therefore the ground state is given by $\theta(\mathbf{x})$ equal to a constant. There is no preexisting texture for a vortex to
interact with, and so another mechanism is required to explain the coupling of vortices to geometry.
In the following sections, we will employ the method of conformal mapping to demonstrate that when an isolated vortex is placed on a curved surface it feels
a force as if there were a smeared out topological ``image charge,"
jointly proportional to the vortex's own circulation and
the Gaussian curvature across the substrate. Such an imaginary topological
charge distribution produces a \emph{real} force analogous to the
force on an electrostatic charge due to its mirror image
in a conducting surface.
The method of conformal mapping may seem, prima facie, a surprising route to derive a coupling between vortices and geometry, because the free energy of Eq. (\ref{eq:supfl-I}) is invariant under conformal transformations that introduce a non-uniform compression of the surface while keeping local angles unchanged.
This
invariance property at first seems to rule out the possibility of a geometrical
interaction! This apparent contradiction
can be seen by choosing a special set of (isothermal) coordinates that can always bring the two dimensional metric tensor in the diagonal form $g_{\alpha \beta}({\bf u})= e^{2\omega({\bf u})}\delta_{\alpha \beta}({\bf u})$ \cite{Davidreview}. The result of this step is to eliminate
the geometry dependence from the free energy of Eq. (\ref{eq:supfl-I}) since the product $g^{\alpha \beta}({\bf u}) \sqrt{g}=\delta_{\alpha \beta}({\bf u})$ and ${\cal F}_{s}$ reduces to its counterpart for a planar surface, where there
is of course no geometry dependence.
An interaction between vortices and geometry
violates this conformal symmetry
of the free energy from which it emerges, but in fact the conformal symmetry
is not an exact symmetry when vortices are present.
Analogous subtleties frequently arise in the study of fields that fluctuate
thermally or quantum mechanically, due to the occurrence
of a cut-off length scale below which fluctuations cannot
occur. A conformal mapping is a strange type of symmetry that stretches
lengths
and thus does not preserve the microscopic structure of a system. At finite
temperatures, the discreteness of a system, such as a thermally
fluctuating membrane \cite{wends} which is actually made up of a network
of molecules, can have an important effect because the fluctuations
excite modes with microscopic wavelengths. This produces violations
of the conformal symmetry at every point of the surface.
In a superfluid at zero temperature, however, short wavelengths not describable
by the continuum free energy ${\cal F}_{s}$ are excited only in
the cores of vortices. Obtaining a finite value for
the energy necessitates the removal of
vortex cores of a certain fixed radius in the local tangent plane,
so a conformal mapping
is not a symmetry in the neighborhood of a vortex. However the amount by
which this symmetry fails can be calculated in a simple form (intriguingly
independent of the microscopic model of the cores) in terms
of the rescaling function $\omega(\mathbf{u})$
evaluated at the locations of the vortices,
where the symmetry fails.
Rather than ruling out the possiblity of
a geometric interaction, a realistic
treatment of conformal mapping becomes
a powerful mathematical tool for deriving these interactions, a technique
which is relevant especially to other branches of theoretical physics such as the
study of scattering amplitudes in string theory \cite{threedee}.
While the free energy, ${\cal F}_{s}$, of the curved superfluid layer in Eq. (\ref{eq:supfl-I}) does not exhibit a
\emph{geometric} gauge field, rotating the sample at a constant angular
velocity leads
to an energy of the same form as Eq. (\ref{eq:patic-ener}).
The resulting forces exerted on the vortices compete with the geometric interactions to determine the
equilibrium configurations of an arrangement of topological defects. This simple idea is behind some of the experimental suggestions
put forward in this article to map out the geometric potential by progressively increasing the rotational speed while monitoring the equilibrium position
of say a single vortex on a helium coated surface shaped like the bottom of a wine bottle \cite{zieve}. Since the position dependence of the force induced by the rotation is easily calculated, one can read off the geometric interaction by simply assuming force balance.
The theory of curved helium films also helps build intuition
for the more
general case of vortex lines confined in a bounded three dimensional region such as
the cavity shown in cross-section in Fig. \ref{fig:slickslopes}A.
The vortex, drawn as a bold black line, can be pinned by the constriction of
the container. The classic problem of understanding
the interaction of the vortex
with itself and with the bump as the superfluid flows past \cite{vortexhill}, is of crucial importance in
elucidating how vortices can be produced when a superfluid starts
rotating despite the absence of any friction. A possible mechanism, known as the ``vortex mill" , assumes that vortex rings break
off a pinned vortex line, while the pinned vortex remains in place
\cite{vortexmill,workingmill}.
The common route to studying vortex dynamics in three-dimensional geometries is
the ``local induction approximation" which
\begin{figure}
\includegraphics[width=0.45\textwidth]{slickslopes3}
\caption{\label{fig:slickslopes}
Cross sections of two regions in which one can study vortex energetics.
A) A region with nonparallel boundaries. The vortex is pushed
to the right. This tendency
can be interpreted either in terms of a drive toward
a shorter length or as the local induction force due to the curvature in
the vortex enforced by the boundaries. B) A cross-section
of a constant thickness layer of helium bounded above by air and below by
the substrate. The vortices keep a constant length and remain straight
while moving around. Hence there is no local induction
force/thickness-variation
force to overwhelm the geometrical forces that we focus on.}
\end{figure}
assumes that each element of a vortex
experiences a force determined only by its local radius of curvature. This simplifying assumption
omits any long range forces experienced by the vortices as they interact with the boundaries (or among themselves).
In the opposite limit of films with uniform thickness (which can be
thought of as special types of bulk superfluid regions with two parallel boundaries,
as in Fig. \ref{fig:slickslopes}B), $all$ the forces exerted on the vortices are long-range. This is the regime of interest to our
investigation.
This article is organized in two tracks. The first, comprising sections \ref{subsec:effective}$-$\ref{sec:experimental},
is phenomenological in nature and emphasizes intuitive analogies between the
(non-linear) geometric forces and conventional electrostatics, simple illustrations of the main results and
experimental ideas. The second track, sections \ref{sec:complex}$-$\ref{sec:geomineq} is more technical and presents
a unified derivation of the geometric potential
by the method of conformal mapping and its application to the study of complex surface morphologies.
The first track starts with a review of superfluid dynamics
that can be used to relate
the anomalous coupling to hydrodynamic lift.
In Sec. \ref{subsec:anomalous}, the geometrical force
is evaluated, using a mapping between the geometric potential studied here and the familiar Newton's theorem that allows an efficient calculation of the gravitational field for a spherically symmetric mass distribution. An intriguing consequence of this analogy is that vortices on saddle
surfaces can be trapped in regions of negative curvature leading to geometrically
confined persistent currents as discussed in section \ref{BS}. Section
\ref{earnshaw} relates this observation
to Earnshaw's theorem from electrostatics.
Upon heating and subsequently cooling a curved superfluid film, some of the
thermally generated defects can remain trapped in metastable states located at the saddles of the substrate. The existence of such geometry
induced vortex hysteresis is conjectured in section \ref{subsec:cande}.
In section \ref{subsec:Rotation}, we derive the forces experienced by vortices when the vessel containing the superfluid layer is rotated
around the axis of symmetry of a curved surface shaped as a Gaussian bump. The dependence of single and multiple defect-configurations on different angular speeds and aspect ratios of the bump is studied in sections \ref{subsec:single} and \ref{subsec:multiple}. The Abrikosov lattice of vortices on a curved surface is discussed in section \ref{subsec:abr}. In realistic experimental situations the thickness of the superfluid layer will not be uniform and additional forces will drive vortices towards thinner regions of the sample. The strength of these forces is assessed in section \ref{sec:experimental} and related to spatial variations of the film thickness due to gravity and surface tension.
A short discussion of choice of parameters for the proposed
rotation experiments follows in section \ref{subsec:nucleation}. The relevance of our discussion to experiments performed in bounded three dimensional samples is addressed in section
\ref{sec:unevenexptl}.
The second track starts with a general derivation of the geometric potential
by the method of conformal mapping in section \ref{sec:map}. The computational efficiency of this approach is illustrated in section \ref{sec:bubble} where
the geometric potential of a vortex is evaluated on an Enneper disk, a strongly deformed minimal surface.
We show that
changing the geometry of the substrate has interesting effects not only on the one-body geometric potential but also on the two-body interaction between vortices. In section
\ref{sec:bumps}
we use conformal methods to show how a periodic lattice of bumps can cause
the vortex interaction to become anisotropic.
In section \ref{sec:zucchini}, we demonstrate that
the quantization of circulation
leads to an extremely long-range force on an elongated surface with the topology of a sphere. The interaction energy is no longer logarithmic, but now grows linearly with the distance
between the two vortices. As we demonstrate,
the whole notion of splitting the energy in a one body geometric potential and a vortex-vortex interaction is subject to ambiguities on deformed spheres. Section \ref{sec:noodles} provides some guidance on how to perform calculations in this context by choosing a convenient Green's function among the several available. Finally,
in section \ref{sec:geomineq}, we present a discussion of some general upper bounds which constrain the strength of geometric forces.
The conclusion serves as a concise summary and contains a table designed to locate at a glance our main results throughout the article including the more technical points relegated
to appendices but useful to perform calculations.
\section{\label{subsec:effective}Fluid Dynamics and Vortex-Curvature Interactions}
We start by writing down the collective wave function of the superfluid as
\begin{equation}
\Psi(\mathbf{ u})=\sqrt{\frac{\rho_{s}(\mathbf{u})}{m_4}} \!\!\!
\quad e^{i\theta (\mathbf{u})} \!\!\!\! \quad,
\label{wave-function}
\end{equation}
where $\mathbf{u}=\{u_{1},u_{2}\}$ is a set of curvilinear coordinates for the
surface, $m$ is the mass of a $^{4}$He atom and $\rho_{s}$ is
the superfluid mass density that we shall assume to be constant in what
follows. To obtain the superfluid current we can use the standard
expression
$
j_{\alpha}(\mathbf{u})=\frac{i \hbar}{2 m_{4}} \left(\Psi
\partial_{\alpha}\Psi^{\ast}-\Psi^{\ast}\partial_{\alpha} \Psi \right) \!\!\!\!
\quad $
showing that the superfluid velocity is given by
\begin{equation}
v_{\alpha}(\mathbf{
u})=\frac{\hbar}{m_{4}} \!\!\!\! \quad
\partial_{\alpha} \theta(\mathbf{u}).
\label{eq:velocity}
\end{equation}
The circulation along a path $C$ enclosing a vortex is given
by
\begin{equation}
\oint_{C}\! d u^{\alpha} \!\!\!\!\!\! \quad v_{\alpha} = n \kappa
\!\!\!\! \quad , \label{quantization}
\end{equation}
where the quantum of circulation, $\kappa=\frac{h}{m_4}$, is equal
to 9.98 10$^{-8}$ $m^2$ $s^{-1}$ and the integer $n$ is the topological index
of the vortex. The free energy can be cast in the form
\begin{equation}
F = \frac{1}{2} \rho_s\frac{\hbar^2}{m_4^2}
\int_{S} d^{2}u \sqrt{g} \!\!\!\!\! \quad
g^{\alpha\beta}\partial_{\alpha}\theta\partial_{\beta}\theta \!\!\!\! \quad ,
\label{eq:free-energy}
\end{equation}
where $g^{\alpha\beta}$ is the (inverse) metric tensor describing the
surface on which the superfluid layer lies and $g$ is its
determinant. We will often use the superfluid stiffness
\begin{equation}
K=\frac{\rho_s\hbar^2}{m^2}.\label{eq:stiffness}
\end{equation}
This expression for the free-energy can be parameterized in terms of the vortex positions once the seemingly divergent
kinetic energy near a vortex core is correctly accounted for.
As is well known, the radius-independence of the circulation about a vortex
implies that the velocity in its proximity
is given by $\frac{\hbar}{m_4r}$, which leads to a logarithmic
divergence in
(\ref{eq:free-energy}). The energy stored in an
annulus of internal radius $r_{\rm{in}}$
and outer radius $r_{\rm{out}}$ reads
\begin{equation}
E_{\rm{near}}=\pi K\ln\frac{r_{\rm{out}}}{r_{\rm{in}}}.
\label{eq:log-div}
\end{equation}
which diverges as $r_{\rm{in}}\rightarrow 0$. A physical trait of superfluid
helium prevents this from happening: it cannot sustain speeds which are greater
than $v_c$, the critical velocity. Thus the superfluidity is destroyed
below a core radius of $a\sim\frac{\hbar}{m_4v_c}$. This breakdown may be modeled
by excising a disk of radius $a$ around each vortex and by adding a constant core energy
$\epsilon_c$ to account for the energy associated with
the disruption of the superfluidity in the core.
Starting on the flat plane,
the interaction of two vortices can now be determined.
Superimposing the fields of the two vortices and integrating the cross-term
in the kinetic energy of Eq. (\ref{eq:free-energy})
leads to a Coulomb-like interaction,
$V_{ij}=-2\pi Kn_in_j\ln\frac{|\mathbf{u}_i-\mathbf{u}_j|}{a}$ in addition to vortex
self-energies. In deducing
the force between the vortices from this expression, it is useful to assume that
$a$ does not vary significantly with position. The justification for this simplification is that the
background flow due to other vortices only gives a fractionally small correction
to the $\frac{1}{r}$ flow near each vortex, and therefore barely affects
where the critical velocity is attained.
For the more complicated case of a \emph{curved} surface, with a very distant
boundary (see \cite{geomgenerate}
for the discussion of effects due to a boundary
at a finite distance), we found in Ref. \cite{swiss}
that the energy including both
single-particle and two-particle
interactions is,
\begin{equation}
\frac{E(\{q_i,\mathbf{u}_i\})}{K}=\sum_{i<j} 4\pi^2
n_in_j V_{ij}({\bf u}_i,{\bf u}_j)
+\sum_i \left(-\pi n_i^2U_G({\bf u}_i)\right),
\label{eq:particlemodel}
\end{equation}
apart from a position-independent term (given for a distant circular
boundary of radius $R$ by
$\pi(\sum_i n_i)^2\ln\frac{R}{a}+N\frac{\epsilon_c}{K}$, with
$N$ the total number of vortices and $\epsilon_c$ the
core energy of one of them).
The pair potential $V_{ij}=\Gamma(\mathbf{u_i},\mathbf{u_j})$
is expressed in terms of $\Gamma$,
the Green's function of the covariant Laplacian defined by:
\begin{equation}
\nabla_u^2 \Gamma(\mathbf{u},\mathbf{v})=-\delta_c(\mathbf{u},\mathbf{v})
\label{eq:BasicGreen}
\end{equation}
Note that the covariant delta
function $\delta_c$
includes a factor of $\frac{1}{\sqrt{g}}$ so that its integral with
respect to the ``proper area" $\sqrt{g}du_1du_2$ is normalized. Equation (\ref{eq:BasicGreen})
determines the Green's function up to a constant provided that
we assume additionally that the Green's function is symmetric
between its two arguments. The constant is fixed by assuming
that at large separations, the Green's function approaches the Green's function
of an undeformed plane. This expression shows that vortices behave like
electrostatic particles, with charges given by $2\pi n_i$ and coupling constant
$K$.
The single-particle potential $U_G(\mathbf{u})$ is the ``geometric potential"
defined in Eq. (\ref{eq:geompdeq}).
This potential entails a repulsion $\nabla\pi K U_G$ of vortices of either sign from positive curvature and an attraction to negative curvature. The
following analogy with boundary interactions and image charges is useful.
The flow-field
is modified by having to conform to the curvature, leading to an image
charge of the vortex which is spread continuously over the surface, with
density $-\frac{n}{2}G(\mathbf{u})$. Just like the image of a vortex in a circular boundary has an equal and opposite circulation, the continuous image of the
vortex in the curvature has a charge density proportional to the number of
quanta $n$ in the vortex.
This point of view may be connected to fluid mechanics by analyzing the streamlines on the bump and in the presence of a circular boundary as illustrated in Fig. \ref{fig:volcano}. Streamlines are tangent to the direction of flow \emph{and} their density is
proportional to the local speed.
Only an incompressible velocity field ($\mathrm{div}\ \mathbf{v}=0$)
may be described by
streamlines, since incompressibility ensures that
any closed curve has an equal number of streamlines
entering and exiting. This condition is satisfied for superfluids (far below the critical speed) since
minimizing (\ref{eq:free-energy}) leads
to $\mathbf{\nabla}\cdot{\bf \nabla}\theta=0$, or $\mathrm{div}\ \mathbf{v}=0$
according to Eq. (\ref{eq:velocity}). Thus the flow is both irrotational (the
circulation around any curve, not enclosing a vortex, is $0$) and
incompressible:
\begin{eqnarray}
\mathrm{div}\ {\bf v}&=&0\nonumber\\
\mathrm{curl}\ {\bf v}&=&0.
\label{eq:veleqns}
\end{eqnarray}
The former relation implies that we may write
\begin{equation}
{\bf v}=\mathrm{curl}\ \chi \hat{\mathbf{n}};
\label{eq:chidef}
\end{equation}
so that the streamlines are equally spaced level curves of $\chi$. (For example,
around a vortex, the radii of the successive streamlines forms a geometric sequence,
$r(1-\epsilon)^l$ where $\epsilon$ sets the ratio
between flowline density and speed.)
Now consider, as an illustration, the flow field on the slope of the bump represented in
Fig. \ref{fig:volcano}.
The curves must spread out to go over
the bump, leading to a lower velocity above the vortex. By Bernoulli's
principle (true for irrotational flows),
this creates a high pressure that pushes the vortex away from
the bump. Note, however, that the actual motion of a vortex is
more subtle: Although the gradient of the energy points away from the bump, a vortex (disregarding friction)
always moves at right angles to the gradient of the energy. Thus a
vortex in the absence of drag forces
actually circles around the bump, in the same direction as the fluid
flows around the vortex. Dissipation converts the motion into
an outward spiral\cite{nelambhalsig}. We will not study the dynamics.
\begin{figure*}
\includegraphics[width=\textwidth]{volcano3}
\caption{\label{fig:volcano}
Left, the flow around a vortex situated on the side of a Gaussian bump,
calculated with the methods described in this paper. The low density of flow
lines above the vortex indicates a lower velocity and thus a higher pressure,
leading to the repulsion represented in Eq. (\ref{eq:particlemodel}),
$-\bf{\nabla} (-\pi K n^2U_G)$. Right, the analogous flow around a vortex in
a disk with a solid boundary. The attraction to the boundary is also seen
to result from high speeds, since the flow lines are compressed in between
the vortex and the boundary.}
\end{figure*}
A convenient mathematical formulation of the problem of determining the flow
pattern of a collection of vortices is obtained by introducing
the scalar function $\chi(\mathbf{u})$ which satisfies
\begin{equation}
\nabla^2\chi(\mathbf{u})=-\sum_i 2\pi n_i \delta_c(\mathbf{u},\mathbf{u_i})
\equiv -\sigma(\mathbf{u}).
\label{eq:flowdeq}
\end{equation}
The sum can be described as a singular distribution of surface charge.
This relation follows from the circulation condition,
$2\pi n_i=\oint \nabla \theta\cdot\mathbf{dl}$, which can be rewritten
as the integral of the flux of $\nabla\chi$ through the boundary,
$\oint \nabla\chi\cdot\mathbf{\hat{n}}dl$,
by
using Eq. (\ref{eq:chidef}).
In analogy with Gauss's law, there must
therefore be delta-function sources for $\chi$ at
the locations of the vortices, as described by Eq. (\ref{eq:flowdeq}).
Solving Eq. (\ref{eq:flowdeq}) in terms of the Green's function gives:
\begin{equation}
\chi(\mathbf{u})=\sum_i\frac{hn_i}{m}\Gamma(\mathbf{u},\mathbf{u}_i).
\label{eq:green-stream1}
\end{equation}
The flow due to a given vortex is proportional to its ``charge" $2\pi n_i$.
The energy as a function of the
positions of the vortices, Eq. (\ref{eq:particlemodel}),
can now be derived by integrating the kinetic energy in the flow determined
by Eq. (\ref{eq:flowdeq}) for each placement of the vortices.
We will begin by discussing applications of Eq. (\ref{eq:particlemodel}),
saving its derivation until later.
Interestingly,
the \emph{energy} of the vortices can
be described by a differential equation analogous to
Eq. (\ref{eq:flowdeq}) for the \emph{flow}.
We choose one vortex
and fix the positions of all the others. We take the Laplacian of Eq.
(\ref{eq:particlemodel}) with respect to the position of the chosen vortex
and use Eq. (\ref{eq:geompdeq}) and Eq. (\ref{eq:BasicGreen}).
The energy as a function of the chosen vortex $E(\mathbf{u_i})$ satisfies:
\begin{equation}
\nabla^2 \frac{E(\mathbf{u_i})}{2\pi K n_i}=-\sigma_i(\mathbf{u_i})-
\frac{n_i}{2}G(\mathbf{u_i}).
\label{eq:endeq}
\end{equation}
The notation $\sigma_i$ stands for the delta function charge
distributions of all the vortices with the exception of the $i^{\mathrm{th}}$,
so
that
$\sigma_i(u)=\sum_{j,j\neq i}{2\pi n_j\delta(\mathbf{u}-\mathbf{x}_j)}$.
The self-charge term which we have had to remove (so that $\mathbf{u}_i$ can
be substituted in place of $\mathbf{u}$ as in Eq. (\ref{eq:endeq}))
is replaced here
by a spread-out charge proportional to the removed term.
\subsection{\label{subsec:anomalous}
Anomalous force on rotationally symmetric surfaces}
On an azimuthally symmetric surface the force on a defect can be found
by exploiting Gauss's law. This is analogous to the familiar
``Newton's Shell" theorem that predicts
the gravitational field at the surface of a sphere surrounding a
spherically symmetric mass
distribution
by concentrating all the enclosed mass at the center.
Points on an azimuthally symmetric two dimensional
surface embedded in three dimensional Euclidean space are
specified by a three dimensional vector ${\bf R}(r,\phi)$ given by
\begin{equation}{\bf R}(r,\phi)=\left(\begin{array}{c} r\cos\phi \\ r\sin\phi \\h(r)\end{array}\right)
\!\!\!\! \quad, \label{eq-coord-polar}
\end{equation}
where $r$ and $\phi$ are plane polar coordinates in the $x$$y$
plane of Fig. \ref{fig:bump}, and $h(r)$ is the height as a function of radius;
e.g. $h(r)=h_0e^{-\frac{r^2}{2r_0^2}}$ for the Gaussian bump with height $h_0$ and spatial
extent $\sim r_0$.
\begin{figure}
\psfrag{phi}{$\phi$}
\includegraphics[width=0.45\textwidth]{driedfig3}
\caption{\label{fig:bump}(a) A bumpy surface shaped as a Gaussian.
(b) Top view of (a) showing a schematic representation of the
positive and negative intrinsic curvature as a non-uniform
background ``charge" distribution that switches sign at $r=r_{0}$.
The varying density of + and - signs tries to mimic the changing
curvature of the bump.}
\end{figure}
It is useful to characterize the deviation of the bump from a
plane in terms of a dimensionless aspect ratio
\begin{equation}
\alpha \equiv \frac{h_0}{r_{0}} \!\!\!\! \quad. \label{aspect ratio}
\end{equation}
The metric tensor, $g_{\alpha\beta}$, is diagonal for this choice
of coordinates. In general, $g_{\phi\phi}=r^2$, $g_{rr}=1+h'(r)^2$,
and for the Gaussian bump we have
\begin{equation}
g_{\alpha\beta} = \begin{pmatrix} 1+ \alpha^2\frac{r^2}{r_0^2}\exp
\left(-\frac{r^2}{r_{0}^2}\right) & 0 \\
0 & r^2
\end{pmatrix} \!\!\!\! \quad .
\label{metric-mat}
\end{equation}
Note that the $g_{\phi\phi}$ entry is equal to the flat space
result $r^{2}$ in polar coordinates while $g_{rr}$ is modified in
a way that depends on $\alpha$ but tends to the plane result
$g_{rr}=1$ for both small and large $r$.
The Gaussian curvature for the bump is readily found from the
eigenvalues of the second fundamental form \cite{Dubrovinbook}; e.g., for
the Gaussian bump,
\begin{equation}
G_{\alpha}(r)=\frac{\alpha^2 e^{-\frac{r^2}{r_{0}^2}}}{r_{0}^2
\!\!\!\! \quad \left(1+ \frac{{\alpha}^2 r^2}{r_{0}^2} \exp
\left(-\frac{r^2}{r_{0}^2}\right)\right)^2}
\left(1-\frac{r^2}{r_{0}^2}\right) \!\!\!\! \quad .
\label{Gaussian Gaussian curvature}
\end{equation}
Note that $\alpha$ controls the overall magnitude of $G(r)$ and
that $G(r)$ changes sign at $r=r_{0}$ (see Fig. \ref{fig:bump}b).
The integrated Gaussian curvature inside a cup of radius $r$
centered on the bump vanishes as $r \rightarrow \infty$. The
positive Gaussian curvature enclosed within the radius $r_{0}$
(see Fig. \ref{fig:bump}) approaches $2\pi$ for $\alpha\gg1$, half
the integrated Gaussian curvature of a sphere. We shall show below
that there is always more positive than negative curvature within any
given radius, for an azimuthally symmetric surface. It will follow that
the force on a vortex is
repulsive at any distance.
In general an individual vortex of index $n_{i}$ confined on a curved surface
at position ${\bf u}_{i}$ feels a geometric interaction described by the energy
\begin{equation}
E({\bf u}_i)=- \pi K n_i^2 U_G({\bf u}_i).
\label{eq:ecurv-s2}
\end{equation}
For an azimuthally symmetric surface such as the bump represented
in Fig. \ref{fig:bump}, we can derive Newton's theorem as follows.
Define $\mathbf{E}=-\nabla U_G$ so that the covariant radial
component of $\mathbf{E}$
is
$E_r=-\partial_{r}U_G$. Then
$-\nabla^2 U_G=\mathrm{div\ }
\mathbf{E}=\frac{1}{\sqrt{g}}\partial_r \sqrt{g}g^{rr}
E_r$,
and if we integrate both sides of Eq. (\ref{eq:geompdeq}) out to $r$,
\begin{equation}
2\pi\sqrt{g}g^{rr}E_r=-\iint {\sqrt{g}drd\phi G(r)}
\label{eq:covariant}
\end{equation}
so that $E_r$ has a simple expression in terms of the net Gaussian curvature
at a radius less than $r$. Now $E_r$ is the ``covariant component" of the
geometrical ``electric field", not the actual field, which would be obtained by
differentiating with respect to arclength rather than the projected
coordinate $r$. Therefore
the magnitude of $\mathbf{E}$ is $\frac{E_r}{\sqrt{g_{rr}}}$, which,
rephrasing Eq. (\ref{eq:covariant}), obeys
this generalized version of Newton's theorem:
\noindent\emph{The magnitude of}
$\mathbf{E}$ \emph {is}
$-\frac{1}{2\pi r}$
\emph{times the integrated Gaussian curvature}.
\noindent(Recall that $g^{rr}=\frac{1}{g_{rr}}$ and
$g=\mathrm{det}\ g_{\alpha\beta}=rg_{rr}$.)
Note that the force on the
vortex is $\mathbf{F}=-\nabla E=-\pi K n_i^2 \mathbf{E}$
according to Eq. (\ref{eq:ecurv-s2}), so
that the geometrical force is proportional
to the integrated Gaussian curvature divided by the distance of the vortex from
the axis of symmetry of the surface;
the expression for the force on the vortex
obtained in the next section by integrating
the Gaussian curvature is
\begin{equation}
F_{geom}=\frac{K\pi}{r}(1-\frac{1}{\sqrt{1+h'^2}}),
\label{eq:aziforce}
\end{equation}
if $n_i=\pm 1$. Note that this force is always \emph{repulsive} since
the integrated curvature is positive.
The geometric potential can now be expressed explicitly by integrating
$\mathbf{E}_r=-\partial_r U_G$ with the aid of Eq. (\ref{eq:covariant}):
\begin{equation}
U_G(r) = - \int_{r}^{\infty} \! dr' \frac{\sqrt{1+ \frac{{\alpha}^2
r^2}{r_{0}^2} \exp \left(-\frac{r^2}{r_{0}^2}\right)}-1}{r'}
\!\!\!\! \quad . \label{potential4-bis}
\end{equation}
The resulting potential $U_{G}(r)$ vanishes at infinity. Its
range and strength are given respectively by the linear size of the
bump and its aspect ratio squared (see Fig. \ref{fig:geompot1})
\begin{figure}
\psfrag{X}{$E(r)$}
\includegraphics[width=0.45\textwidth]{geompot1}
\caption{Plot of the interaction energy $E(r)=-\pi K U_G(r)$ between a singly
quantized vortex and a Gaussian bump with $\alpha=1$. The energy is measured
in units of $K$ and the radius is measured in units of $r_0$.
Note that the force points away from the bump
and has its maximum strength near $r_0$.} \label{fig:geompot1}
\end{figure}
We now summarize
an intuitive argument that explains why the energy of a vortex on top
of the bump is greater than the energy of a vortex that is far away
\cite{swiss,Halperin-private}.
Fig. \ref{fig:intuition},
reproduced from \cite{swiss},
focuses on a rotationally symmetric bump
coated by a helium film (of a constant thickness).
We can estimate the energy of a vortex on top of the bump by comparing
the situation to a vortex on a plane, illustrated vertically below
the bump. Rotational symmetry implies that the superfluid phase is
given by $\theta=\phi$, the azimuthal angle. The velocity
depends on the rate of change on the phase according
to Eq. (\ref{eq:velocity}), so since an infinitesimal
arc of the circle
concentric with the top of the bump has size $rd\phi$, the velocity
is $\frac{\hbar}{mr}$. Here $r$ is the
radius of the circle measured horizontally to the axis of the bump.
This calculation shows that the velocity, and thus the energy \emph{density},
are the same at any point on the bump and its projection into the plane.
However, the \emph{energy} contained in the tilted annulus on
the bump stretching from $r$ to $r+dr$ is greater than the energy
in the annulus directly below it because, though the energy density is
the same, the annulus's area is greater. Hence a vortex
on a bump has a greater energy than a vortex in a plane, whether
it is the vortex at $P'$ in the projection plane, or the vortex
at $Q$ which is very far from the bump. This reasoning indicates
a repulsive force, since
the vortex lowers its energy by moving away from the bump.
The intuitive argument suggests that the Gaussian curvature should appear
in the force law, as in Eq. (\ref{eq:geompdeq}); in fact, it is a widely
known fact that the sign of the Gaussian curvature of a surface
determines how fast
the circumference of a circle on the surface increases, relative
to the circumference of a circle on the plane, as a function of the radius.
Of course,
for less symmetric surfaces, comparing the
energy on a curved surface to that on a flat reference plane directly below it
will be more complicated, since symmetry and the constant circulation
constraint do not force the energy densities to be equal. Thus, a simple
vertical projection will not set up a monotonic relation between energies.
The conformal mapping technique which we use in Sec. \ref{sec:complex}
is a variation on the idea of comparing a ``target" substrate to a simple
``reference" surface
which is in principle applicable to arbitrary surfaces, and furthermore not
only allows one to compare energies, but also to calculate them quantitatively.
The technique can also be used to give a concise derivation of
Eq. (\ref{eq:geompdeq}).
\begin{figure}
\includegraphics[width=0.48\textwidth]{figintuition}
\caption{\label{fig:intuition} An azimuthally symmetric substrate and its
downward projection on a flat plane. The shaded strip surrounding
$P$ is more stretched than the one surrounding $Q$ despite their
projections onto the plane having the same area. As discussed in the
text it follows that the energy stored
in the field will be lower if the center of the vortex is located at
$Q$ rather than $P$.}
\end{figure}
Such an intuitive argument
applies only for azimuthally symmetric surfaces.
For less symmetric surfaces, comparing the
energy on a curved surface to that on a flat reference plane directly below it
will be more complicated, since symmetry and the constant circulation
constraint do not force energy densities to be equal at corresponding postions.
Thus, a simple
vertical projection will not set up a monotonic relation between energies.
The conformal mapping technique which we use in Section \ref{sec:complex}
is a variation on the idea of comparing a ``target" substrate to a simple
``reference" surface
which is in principle applicable to arbitrary surfaces, and furthermore not
only allows one to compare energies, but also to calculate them quantitatively.
The technique can also be used to give a concise derivation of
Eq. (\ref{eq:geompdeq}).
\subsection{\label{BS}Vortex-trapping surfaces}
In order to illustrate the consequences of the curvature-induced
interaction for different surface morphologies, we study a ``Gaussian saddle"
surface (suggested to us by Stuart Trugman)
for which the geometric potential has its absolute minimum at the origin.
\begin{figure}
\includegraphics[width=0.45\textwidth]{trap}
\caption{Plot of vortex trapping surface. \label{fig:antibump}}
\end{figure}
First, we show that an alternative design for a vortex trap geometry fails
because of the long-range nature of the curvature-induced interaction. Fig. \ref{fig:bump} shows
that bumps have negative curvature on their flanks; it might seem possible
that a well-chosen bump
would have enough negative curvature to hold
a vortex. However, a vortex cannot be held by nearby negative curvature alone;
it also feels the positive curvature at the center of symmetry because, according
to Gauss's law applied to the azimuthally symmetric region, the force is
due to the \emph{net} curvature contained in any circle
concentric with the top of the bump.
This curvature is given generally by
\begin{equation}
G=-\frac{1}{r\sqrt{1+h'(r)^2}}\partial_{r}\frac{1}{\sqrt{1+h'(r)^2}},
\end{equation}
\label{eq:tower} and the net curvature within radius $r_v$ is thus
\begin{eqnarray}
\int_0^{r_v}\sqrt{g}\ dr\ d\phi\
G(r)&=&2\pi(1-\frac{1}{\sqrt{1+h'(r)^2}})\nonumber\\
&=&2\pi(1-\cos\theta[r_v])
\label{eq:stuckring}
\end{eqnarray}
where $\theta$ is the angle between the surface at the location of the vortex
and the horizontal plane.
This formula also describes the cone angle of a cone tangent to
the surface at radius $r_v$; it can also be derived from
the Gauss-Bonnet theorem which implies that the net curvature of a
curved region depends only on the boundary of the region and how
it is embedded in a small strip containing it; thus the net curvature
of the cone (concentrated at the sharp point of the cone)
is the same as the net curvature
of the bump which it is tangent to.
Since this curvature is always positive, the defect is always
repelled from the top of an azimuthally symmetric bump\footnote{More specifically, all bumps with azimuthally symmetric \emph{embeddings} repel vortices from
their tops. The negative curvature
cones discussed in
Sec. \ref{sec:geomineq} have an internal azimuthal symmetry but their three dimensional
embeddings are not symmetric.}.
To find a way to trap a vortex,
one must therefore investigate some non-symmetric surfaces.
The curvature-defect interaction energy on a generic surface
is mediated by the Green's function
of the surface, Eq. (\ref{eq:BasicGreen}), as can be seen by solving Eq.
(\ref{eq:geompdeq}):
\begin{equation}
E(\mathbf{u}) = K\pi\int\ \!\!\! d^2\mathbf{u} \!\!\!\!\quad \Gamma(\mathbf{u},\mathbf{v})\!\!\!\!\! \quad G(\mathbf{v}),
\label{eq:curvature-defect}
\end{equation}
for a singly quantized vortex. One such
surface, which we will treat perturbatively in the amount of
deformation from flatness
(Section \ref{sec:bubble} treats a different confining
surface exactly)
is the Gaussian saddle
represented in Fig. \ref{fig:antibump} and described by
the height function
\begin{equation}
h_{\lambda}(x,y)=\frac{\alpha}{r_0} (x^2-\lambda y^2)
\!\!\!\!\quad e^{-\frac{x^2+y^2}{2 r_0^2}} \label{eq:saddlemesa}
\end{equation}
where the exponential factor was included to make the surface flat
away from the saddle \cite{trugman}. Here,
$\lambda$ is a parameter which we later
vary to illustrate the
non-locality of the interaction.
The leading order contribution to the curvature-defect interaction
(for $\alpha<<1$) is of the same order
$\alpha^2$ as the curvature corrections to the defect-defect interaction
(calculated in Appendix \ref{app:calmseas}). In fact $\Gamma$ is multiplied
in Eq. (\ref{eq:curvature-defect})
by the Gaussian curvature $G(\mathbf{u}')$, of order $\alpha^2$.
Thus, for a single defect, it is
sufficient to use the flat space Green's function $\Gamma_{flat}$
\begin{equation}
\Gamma _{flat} (x-x',y-y') =
-\frac{1}{2\pi}\log\sqrt{(x-x')^2+(y-y')^2})
\end{equation}
for calculating the defect-curvature interaction.
Furthermore, the Gaussian
curvature that acts as the source of the geometric potential can be
calculated from the usual second-order approximation:
\begin{equation}
G_{\lambda}(x,y)\approx\frac{\partial^2 h_{\lambda}}{\partial x^2}
\frac{\partial^2
h_{\lambda}}{\partial y^2}-
\left(\frac{\partial^2 h_\lambda}{\partial x\partial y}\right)^2
\end{equation}
This function is plotted in Fig. \ref{fig:antibumpcurv}, and its
sign is represented in
the middle frame of Figure \ref{fig:occult}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{antibumpcurv2}
\caption{Plot of the curvature of the $\lambda=1$ vortex trapping surface.}
\label{fig:antibumpcurv}
\end{figure}
The graph of the vortex-curvature interaction energy for this surface,
Fig. \ref{fig:peacefulvalley},
shows that a vortex is indeed
confined at the center of the saddle; the energy graphed in this figure
is given by
\begin{eqnarray}
E_{\lambda}(x,y) \approx K\pi\int\ \!\!\! dx'\ dy' \!\!\!\!\quad \Gamma _{flat}
(x-x',y-y') \!\!\!\!\! \quad G_{\lambda}(x',y'),\nonumber\\
\label{eq:smallasp-saddle}
\end{eqnarray}
for $\lambda=1$.
For realistic film thicknesses and $\alpha$ of order unity
(see Sec. \ref{sec:experimental}),
the depth of the well is about $50$ Kelvin! We have
found that the energy associated with a vortex at the origin
is less than for any other position. Of course, the configuration with
a vortex at the origin cannot beat the configuration with no vortices
at all! The latter has zero kinetic energy; when the vortex is at the
origin, the energy is positive provided that Eq. (\ref{eq:particlemodel})
is supplemented by the position-independent contribution $\pi K \ln\frac{R}{a}$.
This
term is always necessary for comparing configurations with different numbers of
defects, as when one studies the formation of a vortex lattice at increasing
rotational frequencies\cite{campbell}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{potentialantibump2}
\caption{Plot of the geometric potential for the $\lambda=1$ vortex trapping
surface.} \label{fig:peacefulvalley}
\end{figure}
\subsection{\label{earnshaw}Negative curvature which does not trap}
In this section, we shall discuss what happens when the parameter $\lambda$
of the saddle surface is increased; Fig. \ref{fig:brokentrap}
illustrates such a
surface corresponding to $\lambda=17$. To give a hint of what causes the
equilibrium to change its character,
Fig. \ref{fig:occult}
shows the sign of the curvature
for the Gaussian bump, the saddle with $\lambda=1$, and the saddle with
$\lambda=17$.
\begin{figure}
\includegraphics[width=.47\textwidth]{brokentrap}
\caption{\label{fig:brokentrap}
A saddle surface with $\lambda=17$; this parameter value is just
large enough to destabilize a vortex at the center.}
\end{figure}
\begin{figure*}
\includegraphics[width=1.0\textwidth]{occult}
\caption{\label{fig:occult}
Plots of the sign of the curvature, with white for positive curvature. (A) is for the Gaussian bump, and (B)
and (C) are
for the saddle surfaces with $\lambda=1$ and $17$ respecitvely.
Because of the lack of symmetry
in the third figure, the center point becomes a saddle point of the energy; the vortex
is pushed away by the strong positive curvature
in the ellipsoidal regions at positive and negative $y$.}\end{figure*}
In the graph of the defect-curvature interaction
energy with $\lambda=17$, one notices that the
origin is an unstable equilibrium position for the vortex. We will derive
the exact value of $\lambda$ where this instability first occurs below.
However, symmetry considerations alone show
that the origin \emph{is} a stable
equilibrium point when $\lambda=1$, as Fig. \ref{fig:peacefulvalley}
shows.
One might be tempted to argue from Newton's theorem
that a vortex at a small enough
radius $r$ is always attracted to the origin by the negative curvature
at radii smaller than $r$. However, the
asymmetry of the saddle surfaces invalidates Newton's theorem
and
positive curvature more distant from the origin than the vortex might be
able to push the vortex toward infinity.
This does not occur for the saddle surface with $\lambda=1$; although
the rotational symmetry needed
for Newton's theorem is absent, the surface does have order four symmetry,
under a $90$ degree rotation combined with the isometry $z \rightarrow -z$.
Upon expanding the defect-curvature interaction energy about the origin,
we obtain
\begin{equation}
E=E_0+ax+by+cx^2+2dxy+ey^2+\cdots
\end{equation}
This energy must be invariant under the symmetries of the surface (without
a sign change). Order two symmetry implies that the linear terms vanish,
so the center point is an equilibrium. The order four symmetry
(apparent in Fig. \ref{fig:occult}B) implies that it is either a maximum or a
minimum (a quadratic function with a saddle point has only
$180$ degree symmetry).
In more detail, $90^{\circ}$ rotational symmetry, given by
$x\rightarrow y,\ y\rightarrow -x$,
implies that $c=e,d=0$.
Since the Laplacian of $E$ at the origin is proportional to
minus the \emph{local} curvature, $2c=2e=c+e$
is positive, so the origin is a local minimum. Without the order four
symmetry the negative
curvature only ensures that $c+e>0$.
Earnshaw's theorem of electrostatics\cite{scott,earnshaw,jeans}
states that an electric charge
cannot have a stable equilibrium at a point where the charge density is zero or has the same sign as the
charge. The charge cannot be confined by electric fields produced
by electrostatic charge distributions in a surrounding apparatus.
(This theorem
motivated the design of magnetic and electrodynamic traps for trapping
charged particles in plasma physics
and atomic physics.)
The argument provided here can be
generalized to give the following
converse rule based on discrete symmetries (whereas Newton's theorem applies
only for continuous azimuthal symmetry):
If $P$ is a symmetry point of a charge distribution with rotation
angle $\frac{2\pi}{m}$, and $m\geq 3$, and the charge density at $P$ is positive, then $P$ is a point of stable equilibrium for particles of negative charge.
This is the formulation for electrical charges in two dimensions; for vortices,
the sense of the rotation of the vortex does not matter of course, since the
vortex interacts with its own image charge distribution. Hence if the
curvature at $P$ is negative, then a vortex
will be trapped there.
Similar reasoning can be used to show that a generalization of
Eq. (\ref{eq:saddlemesa}), the ``Gaussian Monkey Saddle"
given by $h(x,y)=\frac{\alpha}{r_0^2}\Re(x-iy)^3e^{-\frac{x^2+y^2}{2r_0^2}}$,
traps vortices in an energy well of the form
$E=E_0+\frac{9\pi K}{4}\frac{r^4}{r_0^4}+(cnst.+cnst. \cos 6\theta)r^6+\dots$.
The reasoning needs to be modified because the curvature at the
origin of the monkey-saddle is zero and the trapping is due to the negative
curvature near the origin.
At a point of low symmetry
(such as the origin in Eq. (\ref{eq:saddlemesa}) when $\lambda\neq 1$),
the character of an extremum
depends on the charge distribution elsewhere, since the previous argument
only implies that $c+e>0$. Fig. \ref{fig:occult}C suggests that a vortex
at the origin is destabilized by its repulsion from the positive
curvature above and below the origin, which is not balanced by enough
positive curvature to the left and right.
In fact, more detailed calculations show that
the range of $\lambda$ for which
the origin is an energy minimum is $\sqrt{65}-8<\lambda<\sqrt{65}+8$; the
origin is a saddle point outside this range, as is just barely visible
for the case of $\lambda=17$
in Fig. \ref{fig:dangerous}. (Likewise, for negative values of
$\lambda$, the origin is a maximum when $\sqrt{65}-8<-\lambda<\sqrt{65}+8$,
but a saddle point outside this range.)
\begin{figure}
\includegraphics[width=.47\textwidth]{dangerousland}
\caption{\label{fig:dangerous}
The geometry-defect interaction energy of a vortex
on the saddle surface with $\lambda=17$. One notices
a slight instability in the $x$ direction.}
\end{figure}
These results follow by changing the integration variables to
$\xi=x-x'$,$\eta=y-y'$ in Eq.(\ref{eq:smallasp-saddle})
and then expanding to second order about the origin $(x,y)=(0,0)$. The integral expressions for second derivatives of the energy can be evaluated explicitly,
\begin{multline}
E_{\lambda}(x,y)=K\pi[\alpha^2\frac{1+\lambda^2-6\lambda}{16}+
\\+\frac{x^2}{4}(\alpha^2\frac{\lambda^2-1}{4r_0^2}-G_0)
+\frac{y^2}{4}(\alpha^2\frac{1-\lambda^2}{4r_0^2}-G_0)]
\end{multline}
where $G_0=-4\lambda\frac{\alpha^2}{r_0^2}$ is the curvature at the origin.
In Appendix
\ref{app:multipole}, we
determine the geometric potential for arbitrary $x$ and $y$
in (unwieldy) closed form.
\subsection{\label{subsec:cande}Hysteresis of vortices and trapping strength}
The geometrical interaction has its maximum strength when the Gaussian
curvature is the strongest. However, the geometric charge (i.e.,
integrated Gaussian curvature) of any
particular feature on a surface has a strength roughly equivalent at most to
the charge of one or two vortices. Eq. \ref{eq:aziforce} therefore
suggests that the force on a vortex due to a feature of the
surface is less than the force due to a couple vortices at the same distance.
Precise limits on the
strength of the geometric interaction will be stated and proven in
Section \ref{sec:geomineq}, for arbitrary geometries.
As a consequence
\emph{the geometric interaction has its most significant effects
when the number of vortices is comparable to
the number of bumps and saddles on the surface}, so that the geometrical force is not
obscured by interactions with the other vortices. This is a recurring
(melancholy) theme of our calculations, to be illustrated in Section
\ref{subsec:abr} for arrangements of vortices in a rotating film. The current
section illustrates the point by discussing hysteresis on a surface with
multiple saddle points (i.e., traps). If a vortex-free superfluid film
is heated, many
vortices form in pairs of opposite signs. When it is cooled again, positive and
negative vortices can remain trapped in metastable states in the saddles,
but even with the strongest curvature possible, the argument
above suggests that not more than one vortex can be trapped per saddle.
\begin{comment}
We first note that the geometric potential is scale independent.
Suppose a feature has a length-scale of $L$. Dimensional analysis of
Eq. (\ref{eq:geompdeq}) show that the geometric potential is
independent of $L$. The energy
of a single vortex can be expressed in the form $K f(\mathbf{u}/L)$ where
the function $f$ only depends on the rescaled position of the vortex. The
force on a vortex therefore has the form $\frac{K}{L} \nabla_{\mathbf{u}/L}
f(\mathbf{u}/L)$
(where the derivatives are taken with respect to the dimensionless variables)
and therefore becomes stronger as $L$ is decreased, provided
the vortex is at the same relative position
To illustrate this point, note from the previous section
that near a vortex trap with $90^{\circ}$
symmetry the energy as a function of distance from the saddle point $r$ is
approximately parabolic
\begin{equation}
E(r)\approx-\frac{\pi}{4} K|G(0)|r^2.
\label{eq:springpot}
\end{equation}
The force at the radius
$tr_0$ (where $t$ is a dimensionless measure of the displacement of the vortex)
is therefore
\begin{equation}
F(r)\approx-\frac{\pi}{2}Kt\frac{r_0}{r_{\mathrm{curv}}^2}
\label{eq:spring}
\end{equation}
where the principle
curvatures at the saddle point are $\pm r_{\mathrm{curv}}$. Thus the
force grows inversely with $\frac{1}{r_0}$ as the trap is miniaturized,
because $r_\mathrm{curv}=\frac{r_0}{2\alpha}$ where the
dimensionless aspect ratio $\alpha$ is held fixed.
Now let us estimate how many vortices can be trapped by the heating
and cooling process, as a function of the aspect ratio $\alpha$ of the surface,
assumed to be small.
Each one of the vortex traps has
a certain integrated curvature $I$ (the net curvature in the central
region of the trap, of size comparable to $r_0$). $I$ will usually be
less than $2\pi$. But this does not
mean that vortices sitting on two of the traps, whose charge $2\pi$ is greater
than $I$, would be able to pull each other out of the traps and annihilate.
The deciding feature is the relation between the distance between the vortices,
$D$, and the distance from the vortices to their traps. Each vortex is pulled
slightly off center from its trap by the other vortex, so that the restoring
force increases according to Hooke's law. The maximum force
(found by applying Gauss's law
and using the approximate rotational symmetry of the trap) is rougly
$K\frac{\pi}{2}\frac{r_0}{\rho^2}$,
at a distance ($\sim r_0$)
where the integrated
curvature is maximum. (This assumes $\frac{1}{r_{\rm{curv}}}\gtrsim
\frac{1}{r_0}$, or equivalently $\alpha\lesssim1$; a stronger curvature
cannot exist out to the radius $r_0$ without making the surface extremely
distorted and possibly self-intersecting. If one tries to obtain
a stronger curvature by using the same function form as in Eq.
(\ref{eq:saddlemesa}) with a large value of $\alpha$, one finds that
all the curvature is concentrated close to the origin.)
This restoring force can overcome the attraction
between the vortices $K\frac{2\pi}{D}$ provided
$\frac{r_0^2}{\rho^2}>\frac{4r_0}{D}$.
For a lattice of saddles (width $r_0$) which make a bumpy texture like an
egg carton or a
chicken skin, the
fraction of saddles which hold vortices is
$\frac{r_0^2}{D^2}$, and the maximum value of this is $\frac{r_0^4}{16r_{\mathrm{curv}}^4}$,
or $\propto\alpha^4$.
An appreciable fraction of the traps could hold vortices
provided that the vortices have both signs so that only the nearby vortices have an effect, and so that the distance vortices cannot act collectively
to produce a large force.
Of course, a configuration with this many vortices will
only be metastable if $T$ is less than the Kosterlitz-Thouless superfluid
phase transition temperature.
For large values of $\alpha$, the linear approximation used here
suggests that the geometrical force can be arbitrarily large.
The results in Section \ref{sec:geomineq} show that this is not the
case, limiting the strength of the geometric force even for strong curvatures.
\end{comment}
The effectiveness of the defect trapping by geometry
is determined mainly by the ratio of the saddle density to
vortex density. As shown in the previous section, the geometric
energy near the center of a vortex trap with $90^{\circ}$ symmetry is given by
\begin{equation}
E(r)\approx \frac{\pi}{4}K|G_0|r^2 .
\end{equation}
The force on the vortex found by differentiating
the energy
reads
\begin{equation}
F(r)\approx -\frac{\pi}{2}K|G_0|r\label{eq:earthcore}.
\end{equation}
Eq. (\ref{eq:earthcore}) shows
that the trap pulls the vortex more and more strongly as the vortex is pulled away from the center,
like a spring, until the vortex reaches the end of the trap at
a distance of the order of $r_0$ where the force starts decreasing.
Since $G_0\sim\frac{\alpha^2}{r_0^2}$ (which is valid for a small aspect ratio $\alpha$), ``the spring breaks down" when the vortex is pulled with a force greater than
\begin{equation}
F_{max}\sim F(r_0)\sim \frac{K\alpha^2}{r_0}.
\label{eq:escapeforce}
\end{equation}
Let us consider a pair of saddles separated by distance $d$.
It is possible
that one vortex can be trapped in each saddle even for a small $\alpha$ provided that
$d$ is large enough. Remote vortices
do not interact strongly enough to push one another out
of their traps. The Coulomb attraction or repulsion of the vortices must be weaker than
the breakdown force of the trap $F_{max}$, i.e., $\frac{K\alpha^2}{r_0}\gtrsim \frac{K}{d}$.
The minimum distance between the
two saddles is therefore
\begin{equation}
d_{min}\sim\frac{r_0}{\alpha^2}.
\label{eq:dmin}
\end{equation}
Let us find the maximum density of trapped vortices that can remain
when the helium film is cooled through the Kosterlitz-Thouless temperature.
Let us suppose there is a lattice of saddles forming a bumpy
texture like a chicken skin. Suppose bumps
cover the whole surface, so that the
spacing between the saddles is of order $r_0$. Then not every
saddle can trap a vortex; the largest density of saddles which trap
vortices is of the order of $1/d_{min}^2$, so the fraction
of saddles which ultimately contain vortices is at most
$\frac{r_0^2}{d_{min}^2}\propto\alpha^4$. Note that not as many
vortices can be trapped if they all have the same sign, since the interactions from
distant vortices add up producing a very large net force.
On the other hand, producing vortices of both signs by heating and then cooling the helium film
results in screened vortex interactions which are weaker and
hence less likely to push the defects out of the metastable states in which they are trapped.
\section{\label{sec:Rotation}Rotating Superfluid Films on a Corrugated Substrate}
\subsection{\label{subsec:Rotation}The effect of rotation}
Suppose that the vessel containing the superfluid layer is rotated
around the axis of symmetry of the Gaussian bump with angular velocity
${\bf \Omega} = \Omega \!\!\!\! \quad \mathbf{\hat{z}}$, as might occur at
the bottom of a spinning wine bottle.
The container can rotate independently of the superfluid
in it because there is no friction between
the two. However, a state with vanishing superfluid angular momentum is not
the ground state. To see this, note
that the energy, $E_{rot}$, in a frame rotating at angular
velocity ${\bf \Omega}$ is given by:
\begin{equation}
E_{rot}=E - {\bf L} \cdot {\bf \Omega} \!\!\!\! \quad .
\label{eq:Erot}
\end{equation}
where $E$ is the energy in the laboratory frame and
$\mathbf{L}$ is the angular momentum. Hence $E_{rot}$ is
lowered when ${\bf L} \cdot {\bf \Omega} > 0$, that is, when the
circulation in the superfluid is non-vanishing. This is achieved
by introducing quantized vortices in the system (see
Eq.(\ref{quantization})), whose microscopic core radius (of the
order of a few \AA) is made of normal rather than superfluid
component. The energy of rotation, ${\bf L} \cdot {\bf \Omega}$,
corresponding to a vortex at position ${x,y}$ on the bump can be
evaluated from
\begin{equation}
L_{z}= \rho_{s}\int_{S} dx dy \sqrt{g(x,y)} \left(x v_{y} - y
v_{x} \right) \!\!\!\!\! \quad . \label{eq:Lz}
\end{equation}
Upon casting the integral in Eq.(\ref{eq:Lz}) in polar coordinates
${r,\phi}$ and using the identity
\begin{equation}
\left(x v_{y} - y v_{x} \right) = r \mathbf{\hat{\phi}} \cdot \mathbf{v}
\!\!\!\!\! \quad \!\!\!\! \quad, \label{eq:identity}
\end{equation}
we obtain
\begin{equation}
L_{z}= \rho_{s}\int_{0}^{R} dr \sqrt{g(r)} \oint_{C}\! d
u^{\alpha} \!\!\!\!\!\! \quad v_{\alpha} \!\!\!\!\! \quad .
\label{eq:Lz-b}
\end{equation}
where $R$ is the size of the system. The line integral in
Eq.(\ref{eq:Lz-b}) of radius is evaluated over circular contours of radius
$r$ centered at
the origin of the bump. The circulation vanishes if the vortex of strength
$n$ at
distance $r_v$ is not enclosed by the contour of radius $r$:
\begin{equation}
\oint_{C_r}\! d u^{\alpha} \!\!\!\!\!\! \quad v_{\alpha} = n
\kappa \theta(r-r_{v}) \!\!\!\! \quad . \label{eq:count-int}
\end{equation}
Upon substituting in Eq.(\ref{eq:Lz-b}), we obtain
\begin{eqnarray}
L_{z}&=& n \rho_{s} \kappa \int_{r_v}^{R} dr \sqrt{g(r)} \nonumber\\
&=& \frac{n \rho_{s} \kappa}{2 \pi} \left(A(R)-A(r_{v})\right)
\!\!\!\!\! \quad , \label{eq:Lz-c}
\end{eqnarray}
where $A(R)$ is the total area spanned by the bump and $A(r_{v})$ is the
area of the cup of the bump bounded by the position of the vortex.
Thus, after suppressing a constant,
the rotation generates an approximately parabolic potential
energy $E_{\Omega}(r)$ (see Fig.(\ref{fig:area})) that confines a vortex of
positive index $n$ close to the axis of rotation as in flat space:
\begin{equation}
E_{\Omega}(r_v)= n \frac{\hbar \Omega \rho_{s}}{m_{4}} A(r_v) \!\!\!\!\!
\quad, \label{eq:area}
\end{equation}
where a constant has been neglected.
One recovers the flat space result\cite{Vine} by setting
$\alpha$ equal to zero.
Eq.(\ref{eq:Lz-c}) has an appealing intuitive interpretation
as the total number of superfluid atoms beyond the vortex,
$\frac{\rho_{s}}{m} (A(R)-A(r))$,
times a quantum of angular momentum $\hbar$ carried by each of
them. The closer the vortex is to the axis, the more atoms there
are rotating
with the container.
Above a critical frequency $\Omega_1$, the restoring force due to the
rotation (the gradient of Eq. (\ref{eq:area})) is greater than
the attraction to the boundary. The energy of attraction to the boundary is
approximately
$\pi K\ln(1-\frac{r^2}{R^2})$,
where we assume the aspect ratio of the bump is small
so that the flat space result is recovered. Upon expanding this boundary
potential
harmonically about the origin and comparing to Eq. (\ref{eq:area}),
one sees that
\begin{equation}
\Omega_1\sim\frac{\hbar}{mR^2}.
\end{equation}
Above $\Omega_1$, the origin is a local minimum in the
energy function for a single vortex, though higher frequencies are
necessary to produce the vortex in the first place. What determines
the critical frequency for producing a vortex is unclear.
There is a higher frequency $\Omega_{1}'\sim \frac{\hbar}{mR^2}\ln\frac{R}{a}$,
at which the single vortex actually has a lower energy (according to Eq. (\ref{eq:Lz-c}))
than no vortex at all, but critical speeds are rarely in agreement with the
measured values \cite{criticalvinen}. In the context of thin
layers, it is likely that a third, much larger critical speed
$\Omega_{\mathrm{crit}}\sim\frac{\hbar}{mRD_0}$,
is necessary before vortices form spontaneously, where $D_0$ is the thickness
of the film (see Sec. \ref{subsec:nucleation}).
\subsection{\label{subsec:single}Single defect ground state}
The equilibrium position of an isolated vortex far from the
boundary is determined from the competition between the confining
potential caused by the rotation and the geometric interaction
that pushes the vortex away from the top of the bump. The energy
of the vortex, $E(r)$, as a function of its radial distance from
the center of the bump is given up to a constant by the sum of the geometric
potential and the potential due to rotation,
\begin{equation}
\frac{E(r)}{K}= -\pi U_G(r)+
\frac{A(r)}{\lambda ^2} \!\!\!\! \quad , \label{eq:E-tot}
\end{equation}
where we have ignored the
effects of the distant boundary, boundary effects are discussed
in the next section. The ``rotational length"
$\lambda$ is defined as
\begin{equation}
\lambda \equiv \sqrt{\frac{\hbar}{m \Omega}} \!\!\!\! \quad .
\label{eq:bohr}
\end{equation}
A helium atom at radius $\lambda$ from the origin rotating with
the frequency of the substrate has a single quantum of angular momentum.
The geometric contribution to $E(r)$
( see Fig. \ref{fig:geompot}) varies strongly as the shape
of the substrate is changed. The rotation contribution to $E(r)$ confinement
(see Fig. \ref{fig:area}) varies predominantly as the frequency is changed;
near the center of rotation, where the substrate is parallel to
the horizontal plane, the rotational contribution barely changes as $\alpha$
is increased.
\begin{figure}
\psfrag{X}{$-U_G(r)$}
\includegraphics[width=0.45\textwidth]{geompotnew}
\caption{Plot of minus the geometric potential $-U_G(r)$ for $\alpha=0.5,
1, 1.5, 2$. The arrow indicates increasing $\alpha$. The radial
coordinate $r$ is measured in units of $\lambda$ and
$r_{0}=\lambda$.} \label{fig:geompot}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{rotpot}
\caption{Plot of the area of a cup of radius $r$ for $\alpha=0.5,
1, 1.5, 2$. The arrow indicates increasing $\alpha$. The radial
coordinate $r$ is measured in units of $\lambda$ and
$r_{0}=\lambda$.} \label{fig:area}
\end{figure}
As one varies $\alpha$ (fixing $r_0$ and $\Omega$)
there is a transition to an asymmetric minimum.
In fact, Fig. \ref{fig:totalpot} reveals that for $\alpha$ greater than a critical
value $\alpha_c$ the total energy $E(r)$ assumes a
Mexican hat shape whose minimum is offset from the top
of the bump. The position of this minimum is found by
taking a derivative of Eq. (\ref{eq:E-tot})
with
respect to $r$:
\begin{equation}
\pi\frac{dU_G}{dr}=\frac{1}{\lambda^2}\frac{dA}{dr}.
\label{eq:*}
\end{equation}
Now $\frac{dA}{dr}$ can be shown to equal $2\pi r\sqrt{1+h'^2}$ by
differentiating Eq. \ref{eq:Lz-c} and $\frac{dU_G}{dr}$, which
is the same as $F_G\sqrt{1+h'^2}$ can be evaluated by substituting
for $F_G$ from Eq. \ref{eq:aziforce}.
This
leads to an implicit equation for the position of
the minimum, $r_{m}$, namely
\begin{equation}
\frac{r_{m}}{\lambda} = \sin(\frac{\theta[r_{m}]}{2}) \!\!\!\! \quad .
\label{eq:constr}
\end{equation}
Here $\theta(r)$, defined in Sec. \ref{BS},
is the angle that the tangent at $r$ to the bump
forms with a horizontal plane. A simple
construction allows one to solve Eq.(\ref{eq:constr}) graphically by
finding the intercept(s) of the curve on the right-hand side with the
straight line of slope $\frac{1}{\lambda}$ on the left-hand side (see Fig.
\ref{fig:constr}). A brief calculation based on this construction
shows that for $\alpha>\alpha_c=\frac{2r_0}{\lambda}$, there are two
intercepts: one at $r=0$ (the maximum) and one at $r=r_{m}$, the
minimum; whereas for $\alpha < \alpha_{c}$ only a minimum at $r=0$
exists exactly like in flat space.
\begin{figure}
\includegraphics[width=0.45\textwidth]{totalpot}
\caption{Plot of $E(r)$ measured in units of $K=\frac{\hbar ^2
\rho_s}{m^2}$ as $\alpha$ is varied.
In these units, the thermal energy $k_{B} T$ is less
than 0.1 below the Kosterlitz-Thouless temperature, for $200$\AA$\ $films.
The radial coordinate $r$ is measured in units of
$\lambda$ and $r_{0}=\frac{\lambda}{2}$. Note that this plot is a
2D slice of a 3D potential. For $\alpha < \alpha _c$, $E(r)$ is
approximately a paraboloid while, for $\alpha > \alpha _{c}$, we
have a Mexican hat potential.} \label{fig:totalpot}
\end{figure}
It is possible to go through this second order transition by
changing other parameters such as the rotational frequency.
See Figs. \ref{fig:totalpot} and \ref{fig:vario} for
illustrations of how the transition occurs when the shape of
the substrate is varied.
More details
on the choice of substrate parameters are given in Sec.
\ref{sec:experimental}. Once these parameters have been chosen, changing
the frequency would likely be more convenient;
Fig. \ref{fig:constr} shows how the equilibrium position of the vortex varies.
If the vortex position $r_m$ can be measured precisely as a function of $\Omega$
and if there is not too much pinning, then
the geometrical potential can
even be reconstructed by integrating
$U_G=-\int_{\Omega}^{\Omega_c} {2\frac{m\Omega'}{\hbar}r_m(\Omega')\sqrt{1+h'(r_m(\Omega'))^2}\frac{dr_m}{d\Omega}\big(\Omega'\big)d\Omega'}
+cnst.$ which follows from Eq. \ref{eq:*}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{vario}
\caption{Plot of $E(r)$ in units of $\frac{\hbar^2\rho_s}{m^2}$ versus $r$
as $r_0$ is varied.
The aspect ratio is kept fixed at
$\alpha=2$ while the range of the geometric potential
(corresponding to the width of the bump) is varied so that $r_{0}=
0.2, 0.4, 0.6, 0.8, 1$ in units of $\lambda$.
As $r_{0}$ decreases, the geometric force becomes stronger, so
the system goes
through a transition analogous to the one displayed in Fig.
\ref{fig:totalpot}.} \label{fig:vario}
\end{figure}
\begin{figure}
\includegraphics[width=.45\textwidth]{potentialtracewithlabels}
\caption{\label{fig:constr} Graphical method for determining equilibrium positions
of one vortex. The equilibrium position is at the intersection of
$\sin\frac{\theta(r)}{2}$ and $\frac{r}{\lambda}$. If we fix
$r_0$ and set $\alpha=1$, the rotational frequency
will control the position of the vortex. The four lines
correspond to $\Omega=\frac{\hbar}{mr_0^2}, \frac{\hbar}{4mr_0^2}$ (which is the critical frequency $\Omega_c$), $\frac{\hbar}{25mr_0^2}
,\ \frac{\hbar}{100mr_0^2}$.}
\end{figure}
\subsection{\label{subsec:multiple} Multiple defect configurations}
As the angular speed is raised, a cascade of
transitions characterized by an increasing number of vortices
occurs just as in flat space.
In order to facilitate the mathematical analysis we
introduce a conformal set of coordinates $\{\mathcal{R}(r),\phi\}$ (see
\cite{geomgenerate} for details). The function $\mathcal{R}(r)$
corresponds to a nonlinear stretch of the radial coordinate that
``flattens" the bump, leaving the points at the origin and infinity
unchanged:
\begin{equation} \mathcal{R}(r)= r \!\!\!\!\!
\quad e^{U_G(r)} \!\!\!\! \quad , \label{solution2-bis}
\end{equation}
Note the unwonted appearance of the geometric potential $U_G(r)$ playing
the role of the conformal scale factor; this surprise is the starting
point for our derivation of the geometric interaction in Section \ref{sec:map}.
The free energy of $N_{v}$ vortices on a bump bounded by a
circular wall at distance $R$ from its center is given by
\begin{eqnarray}
\frac{E}{4\pi^2 K} &=& \frac{1}{2}\sum_{j \neq
i}^{N_{d}} n_{i} n_{j} \!\!\!\!\! \quad \Gamma^{D}(x_{i};x_{j}) +
\sum_{i=1}^{N_{d}}
\frac{{n_{i}}^{2}}{4 \pi} \ln \left[1-x_{i}^{2}\right] \nonumber\\
&-& \sum_{i=1}^{N_{d}}\frac{{n_{i}}^{2}}{4 \pi}U_G(r_{i}) +
\sum_{i=1}^{N_{d}} \frac{{n_{i}}^{2}}{4 \pi} \ln
\left[\frac{\mathcal{R}(R)}{a}\right] \!\!\!\! \quad . \label{eq:longD1}
\end{eqnarray}
The Green's function expressed in scaled coordinates reads
\begin{eqnarray}
\Gamma^{D}(t_{i};t_{j})=\frac{1}{4 \pi} \ln\left(\frac{1 +t_{i}^{2}
t_{j}^{2} - 2 t_{i} t_{j} \cos \left(\phi_{i}- \phi_{j}\right)
}{t_{i}^{2}+t_{j}^{2}-2t_{i} t_{j} \cos \left( \phi_{i}-
\phi_{j}\right) }\right) \!\!\!\! \quad . \nonumber\\
\label{eq:green-norm1}
\end{eqnarray}
where $\phi_i$ is the usul polar angle and
the dimensionless vortex position $t_{i}$ is defined by
\begin{eqnarray}
t_{i} \equiv \frac{\mathcal{R}(r_{i})}{\mathcal{R}(R)} \!\!\!\! \quad .
\label{eq:scaled-coord1}
\end{eqnarray}
Eq.(\ref{eq:longD1}) is now cast in a form equivalent to the flat
space expression apart from the third term which results from the
curvature of the underlying substrate and vanishes when
$\alpha=0$. However, we emphasize that the Green's function
$\Gamma^{D}$ also is modified by the curvature of the surface and
thus depends on $\alpha$.
The contributions from the second term and the numerator of the Green's
functions in the
first term account for the interaction of each vortex with its own
image and with the images of the other vortices present on the
bump (see \cite{geomgenerate}). If $R\gg r_0$, and all the vortices
are near the top of the bump (i.e., $r_i\sim r_0$) then these
boundary effects may all be omitted when determining equilibrium positions,
as the forces which they imply are on
the order of $K\frac{r_0}{R^2}$, small compared to the intervortex forces
and geometric forces, which have a typical value of $\frac{K}{r_0}$.
Let us imagine rotating the superfluid, so that each vortex is confined
by a potential of the form Eq.(\ref{eq:area}).
In flat space, the locally stable configurations usually involve concentric
rings of vortices\cite{campbell}. In particular, there are two
stable configurations of six vortices. The lower energy
configuration has one vortex in the center and five in a pentagon surrounding
it. The other configuration, six vortices in a hexagon, has a slightly higher
energy, and Ref. \cite{yarmchuk} saw the configuration
fluctuating randomly between the two,
probably due to mechanical vibrations since thermal oscillations
would not be strong enough to move the vortices.
(The experiment
used a $D_0=2$ cm high column of superfluid; if one
regards the problem as two dimensional by considering flows that
are homogeneous in the $z$ direction, $\rho_s=D_0\rho_3$ is so large
that $K=\frac{\hbar^2}{m^2}\rho_s$ is on the order of millions of degrees
Kelvin.) There are no other stable configurations. However, on
the curved surface of a bump, there are several more configurations which
can be found by numerically minimizing Eq. (\ref{eq:longD1});
the progression of patterns as $\alpha$ increases depends
on how tightly confined the vortices are compared to
the size of the bump, as illustrated in Fig. \ref{fig:colonies}. If the
vortices are tightly confined, the interactions of the vortices (which are different in curved space) stabilize the new vortex arrangements.
If the vortices are spaced
far apart, the geometric interaction between the bump and the central
vortex causes a transition akin to the decentering transition in the previous
section.
For example, if $\Omega=9\frac{\hbar^2}{mr_0^2}$,
then at $\alpha=0$, the five off-center vortices start out
in a ring of radius $.6 r_0$.
This pentagonal arrangement (see Fig. \ref{fig:colonies}A)
is locally stable for $\alpha<\alpha_1=2.7$.
However, for
$\alpha>\alpha_2=2.1$, another arrangement with less symmetry is also stable
(see
frame B of Fig. \ref{fig:colonies}), and above $\alpha_1$ it takes over
from the pentagon. For $\alpha_2<\alpha<\alpha_1$, both arrangements are
locally stable, with the asymmetric shape becoming energetically favored at some
intermediate aspect ratio. (There is also a third arrangement which coexists
with the less symmetric arrangement
for the larger aspect ratios, seen in the frame C of Fig. \ref{fig:colonies}.)
In the plane, the configuration labelled B, for example, is unstable,
because
the outer rectangle of vortices can rotate through angle $\epsilon$, decreasing its
interaction energy with the two interior vortices while keeping the rotational
confinement energy constant.
(That the interaction energy decreases can be demonstrated
by expanding it in powers of $\epsilon$.)
Because the Green's functions are different on the
curved surface (they do not depend solely on the distance between the
vortices in the projected view shown), figures B and C are stabilized.
At lower rotational frequencies, the equilibria which occur are
even less symmetric. For $\Omega=\frac{\hbar^2}{mr_0^2}$, the
vortices form a pentagon of radius $r_0$ when the surface is flat.
This pentagon is far enough away that it has a minor influence
on the central vortex, which undergoes a transition similar to
the one discussed in Sec. \ref{subsec:single}. At
$\alpha_1'=1.4$, the central vortex moves off-axis (the transition
is continuous), causing only a slight deformation of the pentagon (see
frame D of Fig. \ref{fig:colonies}).
As for the single vortex on a rotating bump, the
geometric potential has pushed the central vortex away from the maximum, and
the other vortices are far enough away that they are not influenced much.
At higher aspect ratios, the figure distorts further, taking a shape
similar to the one which occurs for $\Omega=9\frac{\hbar^2}{mr_0^2}$,
but offset due to the geometric interaction. For these two rotational
frequencies the
hexagonal configuration is less stable than the pentagon; it will not
take the place of the pentagon once the pentagon is destabilized.
The hexagon is of course metastable for nearly flat surfaces.
\begin{figure}
\psfrag{a}{A}
\psfrag{b}{B}
\psfrag{c}{C}
\psfrag{d}{D}
\psfrag{e}{E}
\psfrag{f}{F}
\includegraphics[width=.47\textwidth]{colonies2}
\caption{\label{fig:colonies} Arrangements of $6$ vortices
that can occur on a curved surface. A circle of radius $r_0$
is drawn to give a sense that the confinement is tighter
in the top row ($\Omega=9\frac{\hbar^2}{mr_0^2}$) than in
the bottom row ($\Omega=\frac{\hbar^2}{mr_0^2}$).
The upper row shows the patterns
which occur at large angular frequencies
($\Omega=9\frac{\hbar^2}{mr_0^2}$). The transition from
the pentagon to the rectangle with two interior points is
discontinuous, and there is a range of aspect ratios $2.1<\alpha<2.7$
where both configurations are metastable. The third configuration
is nearly degenerate with the second configuration. The lower
row shows the configurations which occur for $\Omega=\frac{\hbar^2}{mr_0^2}$
as $\alpha$ increases. The first transition is continuous and caused
by the central vortex's being repelled from the top
by the geometric interaction. The third configuration is similar to the second
large $\Omega$ configuration
but the effect of the geometric repulsion is seen in its asymmetry.}
\end{figure}
\subsection{\label{subsec:abr}Abrikosov lattice on a curved surface}
As in Section \ref{subsec:cande}
the geometric potential will have significant consequences only
when the number of vortices near each
geometrical feature such as a bump is of order unity.
As an example, consider the triangular vortex lattice that forms
at higher rotational frequencies ($\Omega\gg \frac{\hbar}{mr_0^2}$
is the criterion for a large number of vortices to reside on top of the Gaussian
bump). In flat space,
the vortex number density is approximately constant and equal to \cite{tilleybook}
\begin{equation} \nu({\bf u})= \frac{4\pi m\Omega}{\hbar}=\frac{2\Omega}{\kappa} \!\!\!\! \quad .
\label{eq:dens}
\end{equation}
At equilibrium, the force exerted
on an arbitrary vortex as a result of the rotation exactly
balances the force resulting from the interaction with the other
vortices in the lattice and from the anomalous coupling to the
Gaussian curvature. We can determine the distribution
on a curved substrate by making the continuum approximation to Eq.
(\ref{eq:endeq}).
The sum of delta-functions $\sigma$ gets replaced by $2\pi\nu(r)$ and the
self-charge subtraction can be neglected in the continuum approximation. The Gaussian curvature can be neglected because it is small compared to the large density of vortex charge.
Upon
applying Gauss's theorem to the vortex charge distribution
in an analogous way to Section
\ref{subsec:anomalous}, we
find that the force on a vortex at radius $r$ is given by
\begin{equation}
F_v=\frac{1}{r}\int_0^r 4\pi^2K\nu(r')r'\sqrt{1+h'^2}dr'
\label{eq:field1}
\end{equation}
while the rotational confinement force, obtained by differentiating
Eq. (\ref{eq:Lz-c}), is
\begin{equation}
F_{\Omega}=-\rho_s\frac{2\pi\hbar\Omega}{m} r.
\label{eq:Fomega}
\end{equation}
Balancing the two forces leads to an areal density of vortices,
\begin{equation}
\nu(r)=\frac{ m\Omega}{\pi\hbar\sqrt{1+h'^2}}.
\label{eq:n}
\end{equation}
Eq.(\ref{eq:n}) has a succinct geometric interpretation: the
vortex density $\nu(r)$ arises from distributing the vortices
on the bump so that the projection of this density on
the $xy$ plane is uniform and equal to the flat space result.
The superfluid tries to mimic a rigidly rotating curved body
as much
as possible given that the flow must be
irrotational outside of vortex cores as for the case of a rotating
cylinder of helium \cite{tilleybook}. To check this, first notice
that the approximate
rigid rotation entails a flow speed of $\Omega r$
at points whose projected distance from the rotation axis is $r$. Hence,
the circulation increases according to the quadratic law
$\oint {\mathbf{v\cdot dl}}=2\pi\Omega r^2$. Since this quantity is proportional
to the \emph{projected} area of the surface out to radius $r$,
the discretized version
of such a distribution would consist of
vortices, each with circulation $\kappa=\frac{2\pi\hbar}{m}$,
with a constant \emph{projected}
density $\frac{2 \Omega}{\kappa}$ as in flat space.
This result can be generalized with some effort
to any
surface rotated at a constant rate, whether the surface is
symmetric or not.
\begin{comment}
We omit the derivation here, however. The derivation
is more difficult on an asymmetric surface because the tangential
flow is not incompressible (the first of Eqs. \ref{eq:veleqns}). To see this,
note that an asymmetric rotating surface pushes the helium around, leading
to the boundary condition
$\bm{v_{\perp}}=\bm{\hat{n}}\cdot(\bm{\Omega}\times\bm{r})$
where $\bm{\hat{n}}$ is the normal to the surface.
This condition implies that
$\nabla_{\parallel}\cdot\bm{v_{\parallel}}=
-\partial_{\perp}\bm{v_{\perp}}\neq 0$, so that a modified definition of the
stream function for the tangential flow, Eq. \ref{eq:chidef},
is necessary.
\end{comment}
\begin{comment}
To derive this for a \emph{non-rotationally} symmetric substrate,
we use the similarity between
the flow equation, Eq. (\ref{eq:flowdeq}), and the energy equation,
Eq. (\ref{eq:endeq}):
the flow due to a vortex at a certain point is proportional
to the force produced by this vortex on a vortex at the same point.
We aim to show that the force on a single vortex
derived from
the gradient of the $-\mathbf{L}\cdot\Omega$ term in Eq. (\ref{eq:Erot})
is balanced
by repulsion from the other vortices, provided these vortices have a
density distribution proportional to $\frac{1}{\sqrt{1+h_x^2+h_y^2}}$
as in Eq. (\ref{eq:n}).
To see this we note that the velocity field is the curl
of $\chi(\mathbf{u})=\sum_{i=1}^N\chi_i(\mathbf{u})$, where the stream
function $\chi(\mathbf{u})$ has been
decomposed into the fields due to the individual vortices.
Since the angular momentum is a linear function of the velocity, the rotational
force on a single vortex is the same whether the vortices are present or not.
Thus we focus on the angular momentum contained in the flow field of a
particular vortex. We balance the gradient of this angular momentum against
the interactions of the vortex with all the other vortices.
Upon integrating the angular momentum density (see
App. \ref{app:Lz}) we find that $E_{\Omega}=-\mathbf{L}\cdot\Omega
=-2\Omega\iint{\chi_i(x,y)dxdy}$
(using the coordinates $x,y$ of the point projected into the $xy$-plane).
Now $\chi_i=\kappa_0\rho_s n_i \Gamma_D(\mathbf{u},\mathbf{u}_i)$
($\Gamma_D(\mathbf{u},\mathbf{u}_i)$ is
the Green's function
with Dirichlet boundary conditions), so we have
\begin{equation}
E_{\Omega}(\mathbf{u})
=-4\pi n_i\frac{\hbar\Omega}{m}\iint{\Gamma_D(\mathbf{u},\mathbf{u}_i) dxdy}.
\label{eq:lopsided}
\end{equation}
This
expression is the same as the continuum approximation to
minus the interaction energy of vortex i with all the other vortices, namely
$4\pi^2Kn_i\iint{\nu(\mathbf{x})\Gamma_D(\mathbf{x},\mathbf{x}_i) \sqrt{g}dxdy}$,
provided that $\nu=\frac{2\Omega}{\kappa\sqrt{1+h_x^2+h_y^2}}$. Therefore
the confinement and intervortex repulsion forces arising from these two energies
cancel one another. The energy expression omits the geometric force and
the interaction of the vortex with its own image in the boundary,
but neither of these is
proportional to the density of vortices, so they can be neglected when $\Omega$
is large.
This very general result arises because both
the flow field and the vortex-vortex interaction are given by the same
Green's function.
\end{comment}
The geometric force has to compete with the interactions among the
many vortices expected
at high angular frequencies. More precisely, the maximum force at radius
$r_0$ according to Eq. (\ref{eq:aziforce}) is of order $\frac{K\pi}{r_0}$ while
the force due to all the vortices Eq. (\ref{eq:field1})
is of order $K\frac{(2\pi)(\pi r_0^2)(2\pi\nu(0))}
{2\pi r_0}=2\pi^2 K r_0 \nu(0)$. The last expression greatly exceeds
$\frac{K\pi}{r_0}$ in the limit of high angular velocity.
The geometrical repulsion
leads to a small depletion of the vortex density of the order of one vortex in
an area of order $\pi r_0^2$.
The vortex arrangements produced by rotation are reminiscent of Abrikosov lattices in a superconductor \cite{Vine}.
In fact an analogy exists between a \emph{thin film}
of superconductor in a magnetic field and a rotating film of superfluid.
A major difference between bulk superfluids and bulk superconductors
is that the vortices in a bulk superconductor have
an exponentially decaying interaction rather than a logarithmic one because of the magnetic field (produced by the vortex current) which screens
the supercurrent. The analogy is more appropriate
in a thin superconducting film,
where the supercurrents (being confined to the film)
produce a much weaker magnetic field. In fact, Abrikosov vortices in a
superconducting film exhibit helium-like unscreened
logarithmic interactions out to length scales of order $\lambda'=\frac{\lambda^2}{D}$ where $\lambda$ is the bulk London penetration depth and $D$ is the film
thickness (see \cite{pearl} and, for a review, section 6.2.5 of
\cite{nelsonbook}). Our results on helium superfluids
without rotation therefore apply
also to vortices in a curved superconducting layer in the absence of an
\emph{external} magnetic field. Curved superconducting layers in external
magnetic fields can be understood as well by replacing the magnetic field by
rotation
of the superfluid.
Let us review the analogy between a container of superfluid helium
rotating at angular speed ${\bf \Omega}$ and a superconductor in a
magnetic field ${\bf H}$ \cite{Vine}. Note that in Eq. \ref{eq:Erot},
$E$ is given by $\frac{1}{2}\rho_s\iint d^2\mathbf{u}\frac{\hbar^2}{m^2}
\left(\nabla\theta\right)^2$ and $\hbar\nabla\theta$ is the \emph{momentum}
in the rest frame, $\bm{p}$, although we are working in the rotating frame
(the frame in which a vortex lattice would be at rest).
For helium, the momentum in
the rest frame ${\bf p}$ is related to the momentum in the
rotating frame ${\bf p'}$ by the ``gauge" transformation
\begin{eqnarray}
{\bf p} \rightarrow {\bf p'} + m \!\!\!\!\! \quad {\bf r} \times
{\bf \Omega} \!\!\!\! \quad . \label{eq:gauge1}
\end{eqnarray}
Similarly, in the case of a superconductor the momentum ${\bf p}$
in the absence of a magnetic field is related to the momentum
${\bf p'}$ in the presence of the field by the familiar relation
\cite{Tinkhambook}
\begin{eqnarray}
{\bf p} \rightarrow {\bf p'} + \left(\frac{e}{c}\right){\bf A}
\!\!\!\! \quad , \label{eq:gauge2}
\end{eqnarray}
where ${\bf A}$ is the vector potential. Comparison of
Eq. (\ref{eq:gauge1}) and Eq. (\ref{eq:gauge2}) suggests a
formal analogy between the two problems,
\begin{eqnarray}
{\bf A} \leftrightarrow \left(\frac{m c}{e}\right){\bf r} \times
{\bf \Omega} \!\!\!\! \quad . \label{eq:gauge3}
\end{eqnarray}
Eq. (\ref{eq:gauge2}) establishes a correspondence between the
angular velocity ${\bf \Omega}$ and the magnetic field ${\bf H}$
that allows to convert most of the relations we derived for helium
to the problem of a superconducting layer, with the identification
\begin{eqnarray}
{\bf \Omega} \leftrightarrow \left(\frac{e}{2 m c}\right) {\bf H}
\!\!\!\! \quad . \label{eq:gauge4}
\end{eqnarray}
Of course, one should keep the external magnetic field small so that
a dense Abrikosov lattice does not form, since (as for superfluids) when there
are too many vortices, the curvature interaction is overcome by the vortex
interactions.
\section{\label{sec:experimental}Experimental Considerations}
Vortices in bulk fluids are extended objects such as curves connecting opposite boundaries, rings or knots. A vortex interacts with itself and with its image generated by the boundary of the fluid. However,
if the vortex is curved, such forces (the three-dimensional generalization
of the geometric force) are usually dominated by a force which
depends on the curvature of the vortex called the
``local induction force." This force
has a strength per unit length \cite{saffman} of
\begin{equation}
f_{LIA}=\pi\frac{\hbar^2}{m^2}\rho_3 \kappa \ln\frac{1}{\kappa a},
\label{eq:LIA}
\end{equation}
where $\rho_3$ is the bulk superfluid density and $\kappa$ is the curvature
of the vortex at the point where this force acts.
This force is in danger of dominating the long range
forces because of the core size appearing in the logarithm.
``Two-dimensional" regions are a special case of three-dimensional
regions in which two of the boundaries are parallel and at a distance $D_0$ much less than the radius of curvature of the boundaries. The two dimensional superfluid density is given by $\rho_s=\rho_3 D_0$,
and the interactions of the vortices should be captured by the
two dimensional theory described in this paper once this substitution is made.
A discrepancy will occur, however, if the boundaries of the film
are not exactly parallel because the vortices are
forced to curve in order to meet both boundaries at right angles. In this case, there is a force which is
a relic of the local induction force (see Sec. \ref{sec:uneven}),
\begin{equation}
F_{th}=-\frac{\pi\hbar^2}{m^2}\rho_s \frac{\nabla D}{D_0}\ln\frac{r_0}{a},
\label{eq:relic}
\end{equation}
where $r_0$ is the relevant curvature scale. According to this formula,
vortices are attracted to the thinnest portions of the film.
We will need to ensure that the thickness of the film is uniform
enough so that this force does not dominate over the geometric interactions we are interested
in.
There is a maximum film thickness for which the geometric
force is relevant. The most stringent
requirement arises from demanding that
the van der Waals force causes wetting
of the surface with a sufficiently uniform film.
Van der Waals forces compete against gravity, which thickens
the superfluid at lower portions of the substrate, and surface tension, which thickens the superfluid where the mean curvature of the substrate is
negative. Both gravity and surface tension
thin the film on hills and thicken it in valleys, but if the
film is thin enough, the van der Waals force can keep the nonuniformity very
small.
Section \ref{subsec:nucleation} discusses the critical speeds for the nucleation of
vortices in thin films, which are typically higher
than those required in long thin rotating cylinders \cite{yarmchuk}. We assume
that vorticity is not created from scratch, but from
pinned vortices present even before the rotation has begun \cite{tilleybook}.
Finally in Secs. \ref{sec:uneven} and \ref{sec:unevenexptl} a comparison is made between forces on
vortex lines in three-dimensional geometries and on point vortices in
two dimensions.
\subsection{\label{subsec:thickness}The Van der Waals force and thickness
variation}
We start by providing an estimate of the
variation in the relative thickness
\begin{equation}
\epsilon \equiv \frac{D_t-D_0}{D_{0}}. \label{eq:epsilon}
\end{equation}
for a liquid layer which wets a bump and apply it to thin helium films.
$D_t$ denotes the thickness on top of the bump
and $D_0$ is the thickness far away.
The wetting properties of
very thin films ($\sim$ 100 \AA) of dodecane on polymeric fibers
of approximately cylindrical shape have been thoroughly
investigated in \cite{Quer89}. We start by reviewing a
theoretical treatment of the statics of wetting on rough surfaces
by \cite{Ande88}. A film on a solid substrate that
is curved has a mean curvature determined by the shape of the substrate,
unlike in the case of a large drop of water on a non-wetting surface. By
choosing an appropriate shape, the drop can adjust its mean curvature
(and thereby balance surface tension against gravity). The shape
is therefore described by a differential equation.
A thin film on a solid substrate, in contrast,
has approximately the same curvature as
the substrate that it outlines.
Consider a film that completely wets a solid surface. The surface
itself is described by
its height function $h({\bf x})$, where {\bf x} denotes a pair of
Cartesian coordinates in the horizontal plane below the surface (see Fig. \ref{fig:paralgau}).
\begin{figure}
\psfrag{hL}{$h_L$}
\psfrag{h}{$h$}
\psfrag{D}{$D$}
\psfrag{(x,y)}{$(x,y)$}
\includegraphics[width=.45\textwidth]{paralgau3}
\caption{\label{fig:paralgau} Definition plot for
a laminating film. $h(\mathbf{x})$ is the height of the
substrate
above the horizontal surface at a point $\mathbf{x}=(x,y)$,
and $h_L(\mathbf{x})$ is the
height of the upper surface of the film. $D(\mathbf{x})$ is the
thickness of the film which (if the film has a slowly varying
thickness) is given by $(h_L(\mathbf{x})-h(\mathbf{x}))\cos\theta(\mathbf{x})$
where $\theta(\mathbf{x})$ is the local inclination angle of the substrate.}
\end{figure}
The height function for the liquid-vapor interface $h_{L}(\bf x)$ can
be determined by minimizing the free energy $F$,
\begin{alignat}{1}
F = \iint d^2{\bf x} &[\gamma \sqrt{1+|\nabla h_{L}({\bf
x})|^2}
\frac{\rho_3 g}{2}(h_L(\mathbf{x})^2-h(\mathbf{x})^2) \nonumber\\&-
\mu \!\!\!\!\! \quad
(h_{L}({\bf x})-h({\bf x})) ]\nonumber\\+
\iint d^2\mathbf{x}&\int_{h_L(\mathbf{x})}^{\infty}
dz \iint d^2\mathbf{x'}\int_{-\infty}^{h(\mathbf{x'})}dz'\nonumber\\
&\ \ w(\sqrt{(\mathbf{x}-\mathbf{x'})^2+(z-z')^2}) \!\!\!\!\!\! \quad ,
\label{eq:wet-1}
\end{alignat}
where $\gamma$, $\rho_3$, and $\mu$ are respectively the liquid-vapor surface
tension, the total mass density, and
the chemical potential (per unit volume). (Note that $\nabla$ here is
not the covariant gradient for the surface; it is the gradient in the $xy$
plane.) The second term describes the gravitational potential energy
integrated through the thickness of the film.
The second and fourth terms model the force between the helium atoms and the substrate assuming for simplicity a non-retarded van der Waals interaction. The last term involves
an integral over interactions between pairs of points, one
above the helium film and one in the substrate, but with no points
in the liquid helium itself. This is equivalent to
including interactions between all pairs of atoms contained in all
combinations of the vapor, liquid and solid regions, as long as
$w(r)=- \!\!\!\!\! \quad \alpha \!\!\!\!\! \quad
r^{-\!\!\!\!\! \quad 6}$ where $\alpha$
is the appropriate combination of parameters for
these phases \cite{Ande88}.
Minimization of
Eq. (\ref{eq:wet-1}) leads to a differential equation for
$h_{L}({\bf x})$ that is a suitable starting point for evaluating
the profile of the liquid-vapor interface numerically \cite{Ande88}.
In what follows, we will instead work within an
approximation valid when $D_0\ll r_0,h_0$; in this case, the curvature of
the film is fixed. The $local$ film thickness is described by
$D({\bf x})=(h_L({\bf x})-h({\bf x}))/\sqrt{1+|
\nabla h({\bf x})^2)|}$, see Fig.(\ref{fig:paralgau}).
We need to determine
how each contribution to the free energy
per unit area at a point $\mathbf{u}$
is changed by an increase in thickness $\delta D({\bf x})$.
First let us
consider the variation of the van der Waals energy in order to understand
how this attraction sets the thickness of the film.
When the film thickens by $\delta D$ over a small area $A$ of the film (centered
at $\bm{x},z$),
the change in the van der Waals energy
is given by $-A\delta D\Pi(\mathbf{x})$ where the disjoining pressure
is
\begin{equation}
\Pi(\mathbf{x})=
\iint d^2\mathbf{x'}\int_{-\infty}^{h(\mathbf{x'})}dz'w(\sqrt{(\mathbf{x}-\mathbf{x'})^2+(z-z')^2}).
\label{eq:wet-4}
\end{equation}
For a film on a horizontal surface at $h=0$,
the surface area and gravitational potential energy
do not increase when the film is thickened.
The equilibrium thickness is determined by balancing the variation of
the chemical potential contribution, $-\mu A\delta D$, against the
disjoining pressure, giving $\mu=-\Pi(D_0)$.
The disjoining pressure obtained by integrating Eq. (\ref{eq:wet-4})
for a flat surface is
\begin{equation}
\Pi (D) = - \frac{A_{H}}{6\pi D_0^3}
\!\!\!\! \quad . \label{eq:wet-3}
\end{equation}
$A_{H}=\pi^2\alpha$ is the Hamaker constant for the solid and the vapor
interacting across a liquid layer of thickness $D_0$
\cite{Israelachvili-book}. One sees that a negative value of $A_H=\pi^2\alpha$
is necessary for wetting.
The equilibrium thickness is
\begin{equation}
D_{0}=\sqrt[3]{\frac{A_H}{6 \pi \mu}} \!\!\!\! \quad.
\label{eq:wet-5}
\end{equation}
(For example, liquid $^4$He on a CaF$_2$ surface has $A_H\approx -10^{-21}$ J,
and has a liquid-vapor surface tension of $3\times 10^{-4}$ J/m$^2$.)
When there is a bump on the surface, Eq. (\ref{eq:wet-5}) gives
the equilibrium thickness far from
the bump. Note that both $A_H$ and $\mu$ are negative in this
expression. Increasing $\mu$ therefore increases the thickness of the film
as expected.
Now let us continue by considering the effects of gravity and surface
tension for a curved substrate.
The increase in gravitational potential energy is
$\rho_3 g h\delta D$, just because there is an additional mass per unit area of
the fluid
$\rho_3 \delta D$ at height $h$. (The additional elevation from adding the fluid at
the top of the fluid that was already present can be ignored if the layer
is very thin.)
The variation of the surface tension energy can
be related to the mean curvature \cite{Kami02}
using the fact that
the area of a small patch of the liquid vapor interface $A({\bf
x})$ (at a distance $D$ from the substrate) is related to the
corresponding area of the solid surface $A_{0}({\bf x})$ by the
relation \cite{Hide-book}
\begin{equation}
A({\bf x}) = A_{0}({\bf x}) \left[1 + 2 H({\bf x}) D({\bf x})+
G({\bf x}) D^2({\bf x}) \right] \!\!\!\! \quad . \label{eq:wet-6}
\end{equation}
The second term is proportional to the mean curvature $H=\frac{1}{2}(\kappa_1+\kappa_2)$
of the surface, where we use the convention that the principal curvatures
$\kappa_1,\ \kappa_2$
are positive when the surface curves away from the outward-pointing
normal. The last term, proportional to the Gaussian curvature, can be
ignored relative to the previous term since it is smaller by a factor of
$\frac{D_0}{r_0}$. The mean curvature of the upper surface of the fluid is
nearly the same as for the substrate,
so the energy required to increase the thickness
of the film is $2\gamma H({\bf x})\delta D$. For example, at the top of
the bump, an increased thickness leads to an increased area, so surface
tension prefers a smaller thickness there. Gravity also
thins the film at the top of the bump so that vortices are attracted to the top.
\begin{comment}
The mean curvature $H(r)$ of a Gaussian bump
as a function of the scaled radial distance
$x\equiv \frac{r}{r_{0}}$ from the top is
\begin{equation}
H=\frac{\alpha}{2 r_{0}} \!\!\!\! \quad \frac{\exp \left[
\frac{-x^2}{2}\right] \!\!\!\!\! \quad \left(2- x^{2} + \alpha
^{2} x^{2} \exp[-x^{2}] \right)}{\left(1+ \alpha ^{2} x^{2} \exp
[-x^{2}]\right)^{3/2}}
\!\!\!\!
\quad . \label{eq:wet-5bis}
\end{equation}
\end{comment}
Now we must balance these forces against the disjoining pressure. The flat space form
of the disjoining pressure is not significantly altered by the
curvature of the substrate for very thin films.
According to Eq. (\ref{eq:wet-4}), the disjoining pressure
is the sum of all the van der Waals interaction energies between the points of
the substrate and a fixed point at the surface of the film. The integral
(evaluated in Appendix \ref{app:vdW})
for a point at a distance
$D$ away from the substrate shows
\begin{equation}
\Pi[D({\bf x})] \approx \frac{-A_{H}}{6 \pi D({\bf x})^3} \left(1
- \frac{3}{2} \!\!\!\! \quad H({\bf x}) D({\bf x}) \right)
\!\!\!\! \quad . \label{eq:wet-7}
\end{equation}
The curvature correction in the second term of Eq.(\ref{eq:wet-7})
arises (when $H>0$ as at the top of the bump)
because the surface bends away from the vapor
molecules which interact only with the very nearest atoms of
the solid substrate. This effect is small if $D_0<<r_0$ and
will be neglected in our estimates.
We can now collect the various contributions to set up a pressure
balance equation that allows us to estimate the relative change in
layer thickness $\epsilon$ defined in Eq. (\ref{eq:epsilon}). This
equation reads:
\begin{equation}
\frac{A_{H}}{6 \pi D({\bf x})^3} + 2 \!\!\!\!\! \quad \gamma
\!\!\!\!\! \quad H({\bf x}) + \rho_{3} \!\!\!\!\! \quad g
\!\!\!\!\! \quad h({\bf x}) - \mu = 0 \!\!\!\! \quad .
\label{eq:wet-8}
\end{equation}
Apart from the lengths $r_{0}$, $h_{0}$ and $D_0$ inherited
from the geometry of the system, it is convenient to define three
characteristic length scales $\delta$, $\varrho$ and $l_c$,
obtained by pairwise balancing of the first three terms of
Eq. (\ref{eq:wet-8}):
\begin{eqnarray}
\delta &\equiv& \sqrt{\frac{-A_{H}}{6 \pi \gamma}} \nonumber\\
\varrho &\equiv& \sqrt[4]{\frac{-A_{H}}{6 \pi \rho _{3} g }} \nonumber\\
l_{c} &\equiv& \sqrt{\frac{\gamma}{\rho_{3} g }} \!\!\!\! \quad .
\label{eq:wet-9}
\end{eqnarray}
The last relation in Eq.(\ref{eq:wet-9}) defines the familiar
capillary length \cite{Guyon-book} below which surface tension dominates
over gravity while the first and the second
give the length scales involving the disjoining pressure.
For $^4$He on CaF$_{2}$, $\delta \simeq 10$ \AA (for most liquids it
is one order of magnitude less), $\varrho \simeq 0.7 \mu m$ and
$l_c \simeq 0.4 mm$.
Upon substituting Eq. (\ref{eq:wet-5}) in Eq. (\ref{eq:wet-8}), we obtain
an approximate relation between $D({\bf x})$ and $D_{0}$,
\begin{eqnarray}
\frac{D({\bf x})}{D_0} \approx 1- \frac{D_{0}^3}{3} \left(\frac{2
H({\bf x})}{\delta ^2} + \frac{h({\bf x})}{\varrho ^4} \right)
\!\!\!\! \quad . \label{eq:wet-10}
\end{eqnarray}
This relation leads to an estimate of the relative change in layer thickness, $\epsilon$, valid for thin
films (if we take $\alpha\sim 1$), namely
\begin{eqnarray}
\epsilon \sim \frac{D_{0}^3}{r_{0} \delta ^2} \left(1 +
(\frac{r_{0}}{l_{c}})^2 \right)\!\!\!\! \quad , \label{eq:wet-11}
\end{eqnarray}
Now the thickness-variation force (Eq. (\ref{eq:relic}))
is small compared to the coupling to the geometry only if
\begin{equation}
|\epsilon|<\frac{\alpha^2}{2\ln\frac{r_0}{a}}.
\label{eq:epsilonmanners}
\end{equation}
where we estimated the maximum of the geometrical force to be $\frac{K\pi\alpha^2}{2r_0}$, see Eq. (\ref{eq:aziforce}).
This limit on $\epsilon$ leads in turn to an upper bound for the film thickness $D_0$ for each choice
of $r_0$. Assuming $\alpha\sim 1$ and
splitting into cases according to the size of the bump gives the limits:
\begin{eqnarray}
D_0\lesssim \frac{r_0^{\frac{1}{3}}\delta^{\frac{2}{3}}}
{(\ln\frac{r_0}{a})^{\frac{1}{3}}}\ \ \mathrm{for}\ r_0<l_c
\label{eq:badsurften}\\
D_0\lesssim \frac{\rho^{\frac{4}{3}}}
{(r_0\ln\frac{r_0}{a})^{\frac{1}{3}}}\ \ \mathrm{for}\ r_0>l_c.
\label{eq:badgravity}
\end{eqnarray}
For smaller bumps, surface tension plays the main role in creating
thickness variation and for larger bumps, gravity has the largest
effects. In order for the vortices to be easily observable,
$D_0$ should be as large as possible. Eqs. (\ref{eq:badsurften}) and
(\ref{eq:badgravity})
show that the film
can be made thickest (while retaining its approximate uniformity) when
gravity and surface tension have comparable effects, $r_0\simeq l_c$.
The optimal thickness
(calculated with $\alpha=1$ and for the Gaussian bump; similar numbers
are optimal for the saddle surface) is about $150$ \AA\ at
$r_0=.5$ mm.\footnote{Interestingly, even
without rotating the bump, there are two radii
where the vortex could rest for a film
thickness of $200$ \AA. Then the confinement due to the
varying film thickness and
the geometric repulsion are comparable in magnitude, producing
an equilibrium off-center position for the vortex.}
A smaller
value of $r_0$ might be required if there is a lower
limit $\Omega_{min}$ on the rotational frequency as discussed in the next
section, requiring a slightly thinner film, about $100$ \AA.
The restriction on the film thickness is the most serious obtacle to studying
two-dimensional superflows experimentally. The method of observing vortices
described in \cite{yarmchuk} requires the vortices to be long enough to
be able to trap an observable number of electrons. If
the method of \cite{yarmchuk} turns out to be unsuitable for thin films and
an alternative method cannot be found, one might also study a saturated
superfluid layer confined
between two solid surfaces. In
this case, none of the considerations on wetting are relevant, and the
``film" could have a large thickness.
The two solid surfaces would have to be parallel to one another, and hence
not congruent. (Congruent surfaces displaced
by a fixed distance in the \emph{vertical} direction lead to a
thickness varying
as $\cos\theta(\mathbf{x})$ where $\theta$ is again the inclination angle.) The two surfaces would have to be very accurately
shaped in order to make the film uniformly thick.
Another concern is that a vortex may be pinned to an irregularity on
the substrate strongly enough that it will not
move to the location favored by the geometrical force.
On the other hand, the geometrical force is much stronger than
random forces due to thermal energy.
Even with a film as thin as hundreds of Angstroms, the geometric force is very
strong. With $\rho_s=\rho_3 D_0=.2\rm{g}/\rm{cc} D_0$,
which assumes that the superfluid density
is not depleted too much by thermal effects or by the thinness of the film,
the value of $K=\frac{\rho_s\hbar^2}{m^2}$ is
about $40$ Kelvin. Since the potential wells which trap the vortices
on the saddle surface or on the rotating Gaussian bump
have depths on the order of $K\alpha^2$ the geometric force
will be strong enough
to prevent the vortex from wandering out of the trap due to thermal Brownian
motion except very close to the Kosterlitz-Thouless transition where
$\rho_s$ is depleted.
\begin{comment}
It is interesting to discuss some of the peculiarities of wetting
on curved surfaces that arise in
the opposite limit to the one studied above, namely the case of
vanishing external pressure $\mu$. If a surface shaped as a bump
(with $h_{0}$ and $r_{0}$ much less than $l_c$ and $\alpha \sim
1$) is located at a height $\Delta h$ above a liquid reservoir,
the chemical potential (per unit volume) is given by $\mu = \rho_s
g \Delta h $. As a thought experiment consider how the thickness
of the liquid layer $D({\bf x})$ varies in the limit $\Delta h
\rightarrow 0$. In the region close to the top of the bump,
surface tension drastically suppresses the equilibrium thickness
of the layer $D_t$ to lengths of the order of $r_{0}
^{\frac{1}{3}} \delta ^{\frac{2}{3}}$. However, as one moves away
from the cup of the bump, the mean curvature decreases
exponentially with the radial distance from the top and the
equilibrium thickness set by the vanishing capillary pressure
becomes unstable. For radial distances greater than $r_{0}$, the
disjoining pressure is unbalanced and the thickness of the layer
grows up to a length scale where gravity sets in fixing the new
equilibrium $D_0$ to be of the order of $\varrho$. As a result of
the separation of length scales in which gravity and capillary
effect act, one has $D_{t} \gg D_{0}$. In order to study how the
liquid profile interpolates between these very different lengths
one needs to trade the constant $\mu$ in the left hand side of
Eq.(\ref{eq:wet-3}) with a thickness dependent term $\rho g
(h_{L}({\bf x})+h({\bf x}))$ and solve the integro-differential
equation numerically.
This illustrative example highlights the difficulties inherent in
the task of uniformly coating surfaces of varying mean curvature
such as the bottom of an egg-carton for which capillary effects
are crucial. Unless a stabilizing external pressure is applied,
the liquid will form thin layers of varying thickness on top of
the hills and thick puddles in the valleys where the capillary
pressure switches sign.
\end{comment}
\subsection{\label{subsec:nucleation}Parameters for the rotation experiment}
A practical consideration can limit the thickness of the film further.
In the experiments of \cite{yarmchuk}, uniform rotations slower than
$\Omega_{min}=1$ rad/s or so were hard to attain. The maximum
value of $\Omega$ for which the geometrical force can displace
the vortex as in Sec. \ref{subsec:single} can be determined by taking the limit
$r_m\rightarrow 0$ in Eq. (\ref{eq:constr}); this shows
that for a given $r_0$,
the vortex is displaced from the top of the bump only if
\begin{equation}
\Omega<\frac{\hbar\alpha^2}{4mr_0^2},
\label{eq:rotnvsgeom}
\end{equation}
Hence, if details of the rotational apparatus require
that $\Omega> 1 $ rad/s, then the size of the bump should
be less than $.1$ mm (for $\alpha=1$). Consequently Eq. (\ref{eq:badsurften})
requires even thinner films than for the case discussed in
the previous section where the bump size is determined purely
by the natural forces of gravity and surface tension
and is equal to the capillary length.
An additional issue is whether it is possible to create just one
vortex reliably.
In practice, the number of vortices in a rotating film is a property
of a history-dependent metastable state whose dynamics are not
completely understood (see Chapter 6 of Ref. \cite{tilleybook}).
We can discuss this in the context
of a flat rotating disk. The critical frequency
for local stability of a single vortex at the
center of the disk is $\Omega_{stab1}=\frac{\hbar}{mR^2}$.
There is a barrier to creating such a
vortex, so the frequency initially would have to be raised up to a higher
frequency in order for a vortex to form at the boundary and then move in to the center
of the rotating helium. The critical frequency for creating
vorticity at the boundary is
$\Omega_{\rm{crit}}$.
It is not clear what this critical speed
is. Perhaps
$\Omega_{\rm{crit}}$ for a thin disk of helium is determined
by the height $D$ of the disk;
the critical \emph{linear} speed\footnote{At
the critical speed, vortices are believed to form by breaking away from
pinned vortices. This implies that the critical velocity is determined
by the shortest dimension $H$ of the chamber which is perpendicular to
the flow of the normal fluid. The critical velocity is estimated in the remanent vorticity
theory\cite{vortexmill}
by assuming that ring vortices break off of pinned vortices stretching across this
shortest dimension and
then expand until they hit the surfaces of the superfluid and break into
two line vortices with opposite circulations. The critical speed is the speed
at which a ring vortex of size $H$
would expand due to the motion of the fluid rather than contract.
The critical velocity is determined by balancing the Magnus force
of the fluid moving past the vortex (which expands the vortex) against the attraction of the vortex
for itself, and it is $v_c=\frac{\hbar}{mH}\ln{\frac{H}{2a}}$.
Thus the critical speed for a narrow cylinder is determined
by setting $H$ equal to the radius $R$ of the cylinder so
$\omega_c=v_c/R=\frac{\hbar}{mR^2}\ln\frac{R}{a}$, as seen experimentally
in \cite{yarmchuk}.
For a thin film on a disk, the shortest
dimension is the height, so $H=D$.}
would
then be $R\Omega_{\rm{crit}}\sim \frac{\hbar\ln\frac{D}{a}}{mD}$
\cite{criticalvinen}.
Now because $\Omega_{\rm{crit}}$ is so much greater
than $\Omega_{stab1}$, a lattice of many vortices will form. According
to Eq. (\ref{eq:n}), there will be about $\frac{m\Omega R^2}{\hbar}\sim\frac{R}{D}\ln\frac{D}{a}$ of them. By slowing down to just above
$\Omega_{stab1}$, where the rotational confinement is not
strong enough to confine more than one vortex (even metastably), one would
hope to retain just a single vortex.
\subsection{\label{sec:uneven}Films of varying thickness from
the three-dimensional point of view
In a chamber of arbitrary shape the length of a vortex line
crossing through the chamber changes as the
vortex moves, and therefore knowledge of the core energy per unit length
is crucial for determining the forces experienced by the vortex. The energy
of a length $D$ isolated vortex
in a cylinder of radius $R$ including the core energy is given by
\begin{eqnarray}
E_{line}&=&\rho_3 D\frac{\pi\hbar^2}{m^2}\ln\frac{R}{a}+\epsilon_{c3}D\nonumber\\
&=&\rho_3D\frac{\pi\hbar^2}{m^2}\ln\frac{R}{\tilde{a}}
\label{eq:linetension}
\end{eqnarray}
where $\tilde{a}=ae^{-\frac{m^2\epsilon_{c3}}{\rho_3\pi\hbar^2}}$ is the
core radius rescaled to account for the effects of the core energy \emph{per unit length},
$\epsilon_{c3}$.
A vortex line connecting two approximately parallel parts of the boundary feels
a force as a consequence of the variation of $E_{line}$.
Let the two nearly parallel boundaries be
$\mathcal{S}$ and $\mathcal{S'}$ as illustrated
in Fig. \ref{fig:railroad}.
Let us define $D(\mathbf{x})$
to be the length of the line which is perpendicular to $\mathcal{S}$ at
$\mathbf{x}$ and which extends to $\mathbf{x'}$ on $\mathcal{S'}$ (see Fig. \ref{fig:railroad}).
If a vortex is attached to $\mathcal{S}$ at $\mathbf{x}$
then
since
the fluid flow is fastest at the vortex, the energy depends primarily
on the thickness of the helium film where the vortex is located. This
thickness
is approximately given by $D(\mathbf{x})$ no matter how the vortex
connects the two surfaces (unless it is extremely wiggily).
\begin{figure}
\psfrag{d}{$D$}
\psfrag{X'}{$X'$}
\psfrag{X}{$X$}
\psfrag{S}{$\mathcal{S}$}
\psfrag{S1}{$\mathcal{S'}$}
\psfrag{beta}{$\beta$}
\includegraphics[width=.47\textwidth]{railroad}
\caption{\label{fig:railroad}
Illustration of the definition of $D(\mathbf{x})$, for nonparallel surfaces. The
distance between $\mathcal{S'}$ and $\mathcal{S}$ is measured along the segment which is
perpendicular to $\mathcal{S}$ at $\mathbf{x}$. The segment meets $\mathcal{S'}$ (obliquely) at
a point which we call $\mathbf{x'}$. A segment displaced to the right is also drawn, illustrating
the derivation of Eq. (\ref{eq:ezekiel}).}
\end{figure}
A useful fact is that,
if the two surfaces are at a constant separation, then the line between
$\mathbf{x}$ and $\mathbf{x'}$ is perpendicular to both surfaces.
This can be derived by differentiating,
$D(\mathbf{x})^2=|\mathbf{x}-\mathbf{x'}|^2$. Upon displacing
$\mathbf{x}$ slightly, $2 D\delta D=2(\mathbf{x}-\mathbf{x'})\cdot(\delta\mathbf{x}-\delta\mathbf{x'})=2(\mathbf{x'}-\mathbf{x})\cdot\delta\mathbf{x'}$, since the line connecting $\mathbf{x}$
to $\mathbf{x'}$ was chosen to be
perpendicular to $\mathcal{S}$. Expressing the inner product in terms of the
angle $\beta$ indicated in Fig. \ref{fig:railroad}, this reads,
\begin{equation}
\delta D=|\delta\mathbf{x'}|\cos\beta\approx|\delta\mathbf{x}|\cos\beta.
\label{eq:ezekiel}
\end{equation}
In order to calculate the kinetic energy of a vortex in a curved film of
varying thickness, we will derive a more detailed form of
Eq. (\ref{eq:linetension}) for a curved film.
First, the core
energy due to the local disruption of superfluidity has the same form,
$\epsilon_{c3} D(\mathbf{x})$ where $D(\mathbf{x})$ is the thickness
of the film at the location $\mathbf{x}$ of the vortex. The major
component of the energy is given by $\frac{\hbar^2}{2m^2}
\rho_3\iiint{d\zeta rdrd\phi\frac{1}{r^2}}$ where it is assumed that the flow is
parallel to the boundaries and roughly independent of $\zeta$,
the coordinate normal to $\mathcal{S}$ and approximately
normal to $\mathcal{S'}$.
We integrate over $\zeta$ and divide the remaining two-dimensional
integration into two parts:
\begin{alignat}{1}
E=\epsilon_c D(\mathbf{x})+&\frac{\hbar^2}{2m^2}
\big(\rho_3\iint_{r<L_{th}} D(r,\phi)\frac{drd\phi}{r}
\nonumber\\&+\rho_3
\iint_{r>L_{th}} D(r,\phi)\frac{drd\phi}{r}\big),
\label{eq:epsilonc}
\end{alignat}
where we are using polar coordinates centered on the location
of the vortex.
Here, $L_{th}$ is the distance over which $D$ varies appreciably
so that $L_{th}\sim\frac{D}{\nabla D}$. We regard the thickness
as a constant in the second term because the integral starts far enough away
from the vortex that it does not depend on the specific thickness of the
film at the location of the vortex.
The first term may
be approximated by replacing $D(r,\phi)$ by $D(\mathbf{x})$
(the thickness of the film
at the location of the vortex, where the energy is very big).
Therefore,
\begin{alignat}{1}
E\approx\epsilon_{c3} D(\mathbf{x})
+\frac{\pi\hbar^2}{m^2}&\rho_3 D(\mathbf{x})\ln\frac{D}{a|\nabla D|}\nonumber\\
&+\mathrm{
energy\ of\ the\ distant\ flow}.
\label{eq:youtalktoomuch}
\end{alignat}
The force obtained by taking the gradient of the kinetic energy
reads
\begin{equation}
\mathbf{F}=-\epsilon_c\nabla D-\frac{\pi\hbar^2}{m^2}\rho_3\nabla
D\ln\frac{D}{a\nabla D}+\mathbf{F_G}
\label{eq:shrinkforce}
\end{equation}
where $\mathbf{F_G}$ represents the variation in energy of the long range
portion of the flow, which includes the geometric forces.
We now show that the thickness variation force can be interpreted as the Biot-Savart self-interraction
of the curved vortex. The net Biot-Savart force points towards the center of curvature of the vortex,
and so favors shrinkage of its length. We will use the ``local induction approximation" to the Biot-Savart self-interaction, given in Eq. (\ref{eq:LIA}). This expression results from integrating the interaction of a particular element of the vortex line with nearby elements; the peculiar dependence on the core radius $a$ arises from the need to place a cut-off in the diverging interaction for nearby elements.
\begin{comment}
This expression
is the
only appearance of
the core radius and core energy in a \emph{force} in this article
since for uniformly thick films, they are both constant contributions
to the energy.
Our thickness-variation-force is a more detailed
form of Eq. (\ref{eq:rollingball}),
which assumed that $L_{th}\sim r_0$; in any case this length occurs within
a logarithm, so that the estimated energy is nearly independent of its value.
Interactions between vortices in three dimensional can be described
using
an analogue of the Biot-Savart force law that
gives the force on an element of vortex-line as an integral of
its interactions with all the other elements of vortex-line. When the
helium has boundaries, a distribution of ``image vorticity" beyond
the boundaries can be supplied to simulate the effects of the boundary
conditions\cite{saffman}, and then the Biot-Savart integral would have
to include interactions with this imaginary vorticity as well.
For a stubby vortex connecting the upper and lower surfaces of
a layer of helium of uniform thickness,
the interaction with the image vorticity
is equivalent to the geometric interaction (the final term in Eq.
(\ref{eq:shrinkforce}))
which is the subject
of this paper. The integral of the Coulomb interaction over
the distribution of surface curvature (see Eq. (\ref{eq:curvature-defect}))
is reminiscent
of the long-range integral of the Biot-Savart interaction with the
distribution of current on the surface.
Now the second term in Eq. (\ref{eq:shrinkforce})
does not seem likely to be described by any sort of Biot-Savart interaction
since it does not have any long-range geometrical contributions.
In fact, the second term of
Eq. (\ref{eq:shrinkforce}) is the interaction of the vortex line with itself.
For films of constant thickness, the surfaces $\mathcal{S},\mathcal{S'}$
can be connected by straight vortices. According
to the Biot-Savart law a straight vortex has zero interaction with itself.
On the other hand a
curved vortex acts on itself with a divergent force (made finite by
cutting off the integral at a distance on the order of the core radius).
This force can
be described by the ``local induction approximation"\cite{saffman},
Eq. (\ref{eq:LIA}).
This force points
toward the center of curvature of the vortex, therefore encouraging
it to shrink. To see that this is the same as the force
(Eq. \ref{eq:shrinkforce}) which
drives the vortex towards thinner regions where the kinetic energy
is less,
\end{comment}
To see this, note that
if the opposing boundaries $\mathcal{S},\mathcal{S'}$ are not at a constant
distance, the vortex must
curve in order to connect them, because it meets both boundaries at right
angles. The resulting curvature of the vortex, $\kappa$, is equal to
$\frac{\nabla D}{D}$ (see Fig. \ref{fig:crevice}), so that the force per
\begin{figure}
\psfrag{S}{$\mathcal{S}$}
\psfrag{S1}{$\mathcal{S'}$}
\psfrag{beta}{$\beta$}\includegraphics[width=.47\textwidth]{trigdiag}
\caption{\label{fig:crevice}A figure for reference in determining the
curvature of a vortex. If the upper and lower surface $\mathcal{S}$ and
$\mathcal{S'}$ are nearly but not quite parallel, the curvature of a vortex
connecting them can be estimated simply. Extend the tangents at
the end-points $X,Y$ of the vortex until they intersect at the center $O$
of curvature of the vortex (which happens because the vortex
is perpendicular to $\mathcal{S,S'}$). The radius of curvature, $OX$, can then be
found by using Eq. (\ref{eq:ezekiel}) and noticing that $OX=XX'\tan\beta$ and $\sin\beta\approx 1$.
}
\end{figure}
unit length given by
Eq. (\ref{eq:LIA}), multiplied by $D$, agrees with the force derived by the energy method, Eq.
(\ref{eq:shrinkforce}).
On the other hand, for films of constant thickness, the surfaces $\mathcal{S},\mathcal{S'}$
can be connected by straight vortices. According to the Biot-Savart law a straight vortex has zero
interaction with itself. In addition, there is still an interaction energy with its image, which
is where the geometric force, the third term in Eq. \ref{eq:shrinkforce}, comes from.
When the helium has boundaries, a distribution of ``image vorticity" beyond the boundaries
can be supplied to simulate the effects of the boundary conditions \cite{saffman}. The integral of the
Coulomb interaction with the distribution of surface curvature (Eq. (\ref{eq:curvature-defect}) is reminiscent of the
long-range integral of the Biot-Savart interaction with the distribution of vorticity beyond the surface.
\begin{comment}
The mathematical principle connecting these different points of view is the following (see Fig. \ref{fig:railroad}): The constancy of $d$ is equivalent to the possibility of drawing straight lines
from any point on $\mathcal{S}$ to a point $\mathcal{S'}$, which are perpendicular to \emph{both}
surfaces. The equivalence is derived by differentiating
$d(\mathbf{x})^2=|\mathbf{x}-\mathbf{x'}|^2$; if the two surfaces can be connected
by uncurved vortices then $2 d\delta d=2(\mathbf{x}-\mathbf{x'})\cdot(\delta\mathbf{x}-\delta\mathbf{x'})=2(\mathbf{x'}-\mathbf{x})\cdot\delta\mathbf{x'}$, since the line connecting $\mathbf{x}$
to $\mathbf{x'}$ is perpendicular to $\mathcal{S}$. In other words,
\begin{equation}
\delta d=d\cos\beta
\label{eq:ezekiel}
\end{equation}
where $\beta$ is illustrated in Fig. \ref{fig:railroad}.
Hence the local induction force is equivalent to a vortex energy
per unit length
which is roughly equal to $\frac{\rho_3\hbar^2}{m^2}\ln\frac{L_{th}}{a}+\epsilon_{c3}$.
Hence if the vortex connecting the points$\mathbf{x}$ and $\mathbf{x'}$ is a straight line normal to both surfaces (so that there is no
local induction force), then $d$ is constant (so that there is no variation in
the logarithmically
divergent energy around the core to compete with the geometrical force).
\end{comment}
\subsection{\label{sec:unevenexptl}Vortex Depinning}
Because of the divergence of the force described by Eq. (\ref{eq:relic}) in the limit
$a\rightarrow 0$, it will be hard to see effects of the geometric force without
films of very nearly uniform thickness.
In \cite{zieve} an experiment is described in which a vortex extending
along
a wire in a helium-filled tube leaves the wire
more easily
when the wire
is connected to a bump on the bottom of the tube
than when the bottom is flat. In this case, any geometric
repulsion from the bump should be negligible in comparison to the extra
energy associated with the stretching of the vortex.
(See Fig. \ref{fig:last}.)
Geometric energies should be of order $R\rho_3\frac{\hbar^2}{m^2}$
where $R$ is the radius of the tube and
approximately the height of the bump,
but the vortex has to stretch an additional
length on the order of $R$ in order to leave the bump. Therefore,
the extra kinetic energy from Eq. (\ref{eq:linetension}),
$R\rho_3\frac{\hbar^2}{m^2}\ln\frac{R}{a}$, probably overwhelms the geometric
effects. Thus, the energy barrier for detachment of the vortex
line should increase when the wire is attached to a bump.
\begin{figure}
\psfrag{A}{$A$}
\psfrag{B}{$B$}
\psfrag{C}{$C$}
\psfrag{D}{$D$}
\includegraphics[width=.2\textwidth]{last}
\caption{\label{fig:last}
Sketch of the experimental geometry from Ref. \cite{zieve}. This
experiment may give some indication about the difference in energy
between a vortex stretching from $A$ to $B$ and one
stretching from $C$ to $D$. A wire is placed along $AB$ and
a vortex is formed around it. Ref. \cite{zieve} found
that this vortex depins more easily than a vortex in a tube with a flat bottom.
This suggests that the energy of a vortex (when there is no wire) would
decrease by moving from $AB$ to $CD$, but estimates seem to rule out this
explanation.}
\end{figure}
This suggests
that depinning is not simply caused by thermally activated
barrier crossings (which would occur at the rate proportional
to $e^{-\Delta E/T}$). In fact,
for the length scale of the bump in the experiment (a few millimeters),
the additional energy
barrier due to the stretching of the vortex line
is millions of degrees kelvin at the temperature where the vortex
depins! The depinning may instead depend on
``remanent vorticity" \cite{tilleybook} in the form of extra pinned
vortices stretching from the top of the bump to the cylinder's walls.
This would decrease the energy barrier because the vortex could leave
the wire by attaching itself to some of the vortices that already exist.
In a more nearly two-dimensional geometry,
Eq. \ref{eq:shrinkforce} shows that
observing the equilibrium positions of vortices in a film of
varying thickness could shed light on the value of the core
size and core-energy and whether they are fixed functions of temperature
as most models imply.
The structure of vortex cores is still
not well understood, and there are several alternative models
\cite{tilleybook}.
Experiments on the geometric force in contexts where the two contributions
to Eq. (\ref{eq:shrinkforce}) are comparable could give information
about the effective core radius $\tilde{a}$. By allowing
the film thickness to vary
by about $\epsilon = \frac{\Delta D}{D}\sim\frac{1}{\ln\frac{r_0}{a}}$, see Eq.(\ref{eq:epsilonmanners}), one ensures
that the vortex core size has a decisive effect on the
equilibrium positions of the vortices (comparable to the effects of
the long range forces).
\begin{comment}
Whereas the local induction approximation
does not tell the whole story for vortices in bulk because
the long-range portions of the Biot-Savart interaction
depend on the complicated tangling of the vortex line, the self interaction
of a vortex connecting nearly parallel surfaces is independent of
the configuration of the vortex as long as it is reasonably straight.
The force
on a short vortex is accurately
described by just the local induction aproximation
together with the geometrical potential, which describes all the long-range
forces. If these two forces are comparable
(i.e., $\epsilon\sim\frac{1}{\ln\frac{r_0}{a}}$), then the competition
of these forces determines the vortex-equilibrium positions. If pinning
is not important and the vortices can be observed, this would help
to measure the quantity
$\tilde{a}$ the only quantity
which can be determined based on macroscopic properties of vortices.
\end{comment}
\section{\label{sec:complex}Complex Surface Morphologies}
Up to this point, our discussion has been confined to rotationally symmetric
surfaces and slightly deformed
surfaces for which the electrostatic analogy and perturbation theory can be successfully employed to determine
the geometric potential. To investigate geometric effects that arise for
strong deformations and for surfaces with the topology of a sphere, we adopt a more
versatile geometric approach based on the method of conformal mapping, often employed in the study of complicated boundary problems in electromagnetism and fluid mechanics.
This approach also sheds light on the physical origin
of the geometric potential. A concrete goal is to solve for the energetics (and the associated flows)
of topological defects on a complicated substrate $T$, the target surface, whose metric tensor we denote by $g_{Tab}$.
This is accomplished by means of a conformal
map $C$ that transforms the target surface into a reference surface $R$, with metric tensor $g_{Rab}$.
The computational advantages result from choosing the conformal map so that $R$ is a simple surface (e.g. an infinite flat plane, a flat disk, or a regular
sphere) that preserves the topology of the target surface. Figure \ref{fig:splat} represents a complicated planar domain denoted by $T$ which can be mapped conformally onto a simple annulus labeled by $R$. We will introduce all the basic concepts in the context of this simple planar problem before turning to the conformal mapping between target and reference surfaces which is represented schematically in Fig. \ref{fig:emental}. Such mappings
can always be found in principle \cite{Davidreview}.
The conformal transformation will map the original positions of the defects on $T$, denoted by $\mathbf{u}$, onto a new set of coordinates on $R$ denoted by $\mathbf{\mathcal{U}}=C(\mathbf{u})$. In what follows,
capital calligraphic fonts always indicate coordinates on
the reference surface. For sufficiently small objects near a point $\mathbf{u}$ the map will act as a similarity transformation; that is, an infinitesimal length, $ds_T =\sqrt{ g_{Tab} du^{a} du^{b}}$, will be rescaled by a scale factor $e^{\omega(\mathbf{u})}$ which is independent of the orientation of
the length on $T$:
\begin{equation}
ds_R=e^{\omega(\mathbf{u})} ds_T ,
\label{eq:confdef}
\end{equation}
where $ds_R=\sqrt{g_{RAB}d\mathcal{U}^Ad\mathcal{U}^B}$. This result
in turn implies a simple relation between the metric tensors of the two
surfaces:
\begin{equation}
g_{RAB}=e^{2 \omega(\mathbf{u})} g_{TAB},
\label{eq:confdef2}
\end{equation}
where we have assumed for simplicity that the coordinates used on
the target surface are chosen so that corresponding points on the two
surfaces have the same coordinates $\mathcal{U}^A$.
We will demonstrate that, once the geometric quantity $\omega(\mathbf{u})$ is calculated, the geometric potential of an
isolated vortex interacting with the curvature is automatically determined. For multiple vortices,
the energy consists of single-vortex terms and vortex-vortex interactions. On a deformed sphere or plane,
the geometric potential reads
\begin{equation}
E_1(\mathbf{u_i})=-\pi n_i^2 K\omega(\mathbf{u}_i),
\label{eq:conformalself-energy}
\end{equation}
where $K$ is the stiffness parameter defined in Eq. (\ref{eq:stiffness}).
For a deformed disk, there are boundary
interactions not included in Eq. (\ref{eq:conformalself-energy}).
(We will not consider multiply connected surfaces here, but
the single particle energy on a multiply connected surface has
additional
contributions which cannot be described by a local Poisson equation.)
The interaction energy is
\begin{equation}
E_2(\mathbf{u}_i,\mathbf{u}_j)=-2\pi n_i n_j K\ln\frac{\mathcal{D}_{ij}}{a},
\label{eq:conformalpair-energy}
\end{equation}
where $\mathcal{D}_{ij}$ is the distance between the two \emph{image} points on the
reference surface. When the reference
surface is an undeformed sphere (the other possibilities are a plane or disk),
$\mathcal{D}_{ij}$ is the
distance between
the points along a chord rather than a great circle\cite{lube92}.
We will show below that on a deformed plane $\omega$ is equal to $U_G$,
but from now on we will use $\omega$ instead; the two functions
are conceptually different, and are not even equal on a deformed sphere.
Equation (\ref{eq:conformalself-energy}) is derived
by the method of conformal mapping in section \ref{sec:map} and its computational efficiency is illustrated in section \ref{sec:bubble} where
the geometric potential of a vortex is evaluated on an Enneper disk, a minimal surface
that naturally arises in the context of soap films, but whose geometry is
distorted enough compared to flat space
that it cannot be analyzed with perturbation theory. Changing the geometry of the substrate has interesting effects not only on the one-body geometric potential but also on the two-body interaction between vortices. In section
\ref{sec:bumps}
we use conformal methods to show how a periodic lattice of bumps can cause
the vortex interaction to become anisotropic.
In section \ref{sec:zucchini}, we demonstrate that
the quantization of circulation
leads to an extremely long-range force on an elongated surface with the topology of a sphere. The interaction energy is no longer logarithmic as in Eq.(\ref{eq:conformalpair-energy}), but now grows linearly with the distance
between the two vortices. Indeed the charge neutrality constraint imposed by the compact topology of a sphere, blurs the distinction between geometric potential and vortex-interaction drawn in Equations (\ref{eq:conformalself-energy}) and (\ref{eq:conformalpair-energy}). This is most easily seen by bypassing
vortex energetics on the reference surface (which is after all an auxiliary concept) and opting for a more
direct restatement of the problem in terms of Green's functions on the actual
target surface coated
by the helium layer. The interaction energy now reads
\begin{equation}
E_2'(\mathbf{u}_i,\mathbf{u}_j)=
4\pi^2 K n_i n_j \Gamma(\mathbf{u}_i,\mathbf{u}_j),
\label{eq:greenpair-energy}
\end{equation}
and the single-particle energy takes the form of a self-energy
\begin{equation}
E_1'(\mathbf{u}_i)=-\pi n_i^2 KU_G(\mathbf{u}_i)=\pi n_i^2 K
\iint G(\mathbf{u'})\Gamma(\mathbf{u_i},\mathbf{u'})
d^2\mathbf{u'},
\label{eq:greenself-energy}
\end{equation}
where $\Gamma$ is a Green's function for the surface that generalizes the logarithmic potential familiar from two dimensional electrostatics. For a deformed
plane the two descriptions of the interaction energy are equivalent
since the Green's
function on a deformed plane can be obtained by conformal mapping,
\begin{equation}
\Gamma(\mathbf{u}_i,\mathbf{u}_j)=-\frac{1}{2\pi}\ln\frac{\mathcal{D}_{ij}}{a}.
\label{eq:greentransf}
\end{equation}
We will see that the expressions for the single-particle
energies are also equivalent. In contrast, for a deformed sphere, we show
in section \ref{sec:noodles} and appendix \ref{app:CG}
that
the two formulations do not agree term by term
( $E_1'\neq E_1$ and
$E_2'\neq E_2$), although the \emph{combined} effect of one-particle and
interaction
terms is the same (up to an additive constant). Both self-energies and interaction energies
include effects of the geometry and explicit formulas are provided on an azimuthally symmetric deformed sphere. Finally,
in section \ref{sec:geomineq}, we present a discussion of some general upper bounds to which the strength of geometric forces is subjected (even in the regime of strong deformations) which are useful in experimental
estimates and which illustrate a major difference between electrostatic and geometric forces: The former can always be increased by piling-up
physical charges but the latter are generated by the Gaussian curvature that can grow only at the price of ``warping" the
underlying geometry of space.
Too much warping either leads to self-intersection of the surface
or a dilution of the long-range force.
\subsection{\label{sec:map}Using conformal mapping}
We start by proving a simple relationship between the total
energies (including self- and interaction parts)
$E_T$ and $E_R$ of two corresponding vortex-configurations on the target and reference surfaces respectively:
\begin{equation}
E_T=E_R-\pi K\sum_{i=1}^N\omega(\mathbf{u}_i).
\label{eq:anom}
\end{equation}
The right-hand side of Eq. (\ref{eq:anom}) can be calculated for
the reference surface and then subsequently decomposed into single-vortex and vortex-vortex
interactions; several examples are worked out in detail in Sections \ref{sec:bubble} and \ref{sec:bumps}.
\begin{figure}
\psfrag{Reference}{Reference}
\psfrag{Target}{Target}
\includegraphics[width=0.45\textwidth]{splatmap2}
\caption{\label{fig:splat}
The flow in a wiggily annular region (the target substrate $T$),
obtained by conformally
mapping the flow from a circular annulus (the reference substrate $R$).
Since every pair of radial spokes in the
first picture comprises the same energy, this is true in the conformal image
as well. The small region in the constriction on the right manages this
by compensating for its small area by having a high flow speed.}
\end{figure}
The general approach is best illustrated by considering the planar flow in the complicated annular container
shown in Fig. \ref{fig:splat}. which can be tackled by conformally mapping it
to a simpler circular annulus.
The flow in the reference annulus is clearly circular, and it has
the same $\frac{1}{r}$ dependence as for a vortex.
A crucial property
of conformal transformation allows us to transplant this understanding of the reference flow
to the target annulus. This property concerns
the following mapping of the stream function $\chi$ (see Eq. (\ref{eq:chidef}))
from the reference surface to the target surface:
\begin{equation}
\chi_T(\mathbf{u})=\chi_R(\mathbf{\mathcal{U}})
\label{eq:mapping}
\end{equation}
Once the coordinate change $\mathbf{\mathcal{U}}=C(\mathbf{u})$ is found, the prescription to determine $\chi_T$ provided in Eq.(\ref{eq:mapping}) guarantees that the
corresponding flow will be irrotational and incompressible, as required. Visually, all we are doing is taking the streamlines on
the reference surface, which are level curves
of $\chi_R$, and mapping them by $C^{-1}$ to the target surface.
We can informally state this first property of conformal maps as follows:
\indent{{\it 1) The conformal image of a physical flow pattern is still a physical pattern.}}
Note that any multiple of the mapped stream function,
$\alpha\chi_R(C(\mathbf{u}))$, corresponds to an irrotational
and incompressible flow but in this case the rates of flow
in the target and reference substrate are different. Only the choice $\alpha=1$ ensures that
both flows have the same number $n$ of
circulation quanta around the hole (or around each
vortex if some are present.) This
follows from another basic property of conformal maps:
\indent{{\it 2) In flow patterns related by a conformal map, according to
Eq. (\ref{eq:mapping}), circulation integrals around corresponding
curves are the same.}}
This property follows from the fact that the contribution to the circulation from each element of the contour, $\mathbf{v\cdot dl}$, is conformally invariant. The
infinitesimal length $\mathbf{dl}$ and the velocity $\mathbf{v}$ scale inversely to each other under conformal transformations and the angle between them is preserved by the map.
To understand why, note that
flow lines are compressed together (stretched apart) when they are
mapped to the target space if the conformal parameter
$\omega$ is greater (less) than zero.
As a result, the velocity increases (decreases) by the same factor as distances
are decreased (increased). This heuristic argument is confirmed
by noting that the velocity is given by the $covariant$ curl of the stream function,
see Equations (\ref{eq:chidef}) and (\ref{eq:mapping}). By definition, the covariant curl carries a multiplicative factor of $g^{-1/2}=e^{\omega}$ \cite{Dubrovinbook}, hence
\begin{equation}
v_T=e^{\omega}v_R.
\label{eq:mapvelocity}
\end{equation}
On the other hand, $|\mathbf{dl}_T|= e^{-\omega} |\mathbf{dl}_R|$ rescales in the opposite way, as indicated in Eq. (\ref{eq:confdef}).
The problem of
finding the \emph{energy}
in the deformed annulus can now be reduced to a simple-rotationally
symmetric problem by appealing to a third property of the conformal mapping:
\indent{{\it 3) The kinetic energies in corresponding regions are equal, provided
the regions do not contain vortices.}}
The proof of this statement relies on the previous discussion:
the kinetic energy contained in an element of the area of the surface $dA$
is $\frac{1}{2}\rho_s \mathbf{v}^2dA$. By Eq. (\ref{eq:mapvelocity}), $\mathbf{v}^2$ scales as $e^{2\omega}$ whereas $dA$ scales by $e^{-2\omega}$ making the energy conformally invariant.
Figure \ref{fig:splat} illustrates pictorially that the energy
density in the original flow on the target surface
is smoothed out and simplified,
its variations being replaced by variations in the conformal scale factor.
We now return to the energetics of flows containing point vortices.
The starting point of our analysis follows from the defining property of a conformal map, namely that
a conformal image of a small figure has the same shape as the original
figure, while a larger shape becomes distorted (consider Greenland, which
has an elongated shape, but appears to round out at the top in a Mercator
projection, which is itself a conformal map).
To quantify the size limits, note that if a shape has size $l$, $\omega$ changes
by about $l\nabla\omega$ across the shape. Thus,
as long as
\begin{equation}
l<<\frac{1}{|\nabla\omega|},
\label{eq:columbus}
\end{equation}
the mapping rescales the shape uniformly. The right-hand side
is ordinarily of order $L$, the curvature scale of the surface. As a result we can conclude
that
\indent{{\it 4) The circular shape of the streamlines near a microscopic vortex core on a substrate of slowly varying curvature is preserved.}}
On a deformed substrate with a flow induced by vortices,
the flow speeds will increase
or decrease not just depending on the distance to the vortex, but also
depending on the shape of the surface. For example, the vortex on top of
the bump in the example of Sec. \ref{subsec:anomalous} has a flow that decays
more slowly with distance than in flat space. Also, for a vortex
well off to the side of a bump, if the bump's height $h$
is larger than its width $2r_0$, it turns out that
the flow pattern penetrates only
up to an elevation of about $r_0$ up the side of the bump.
\begin{figure*}
\includegraphics[width=\textwidth]{emental}
\caption{\label{fig:emental} Comparing the energy
of $T$ and $R$ by splitting $T$ up into portions $I$ and $O$ and
using different maps to map them to $R$.
Left, a target surface, which has
the topology of an infinite plane and is distorted by a three dimensional
lump. Right, the reference plane (shown in the plane of the page). $I$
consists of the interior of radius $l$
disks in $T$, and is mapped rigidly to the heavily demarcated disks in $R$.
$O$
consists of the gray portion of $T$, and is mapped conformally to the
gray portion of $R$.
$I$ and $O$ contain the same kinetic energy as their images in $R$, but
the images do not fit together perfectly. Thus the difference in energy
between the flows in $T$ and in $R$ is determined by calculating how much
energy is contained in the annuli which are either mismatched or covered
twice.}
\end{figure*}
The method of conformal mapping elucidates these geometrical rearrangements of the flow pattern. To find the flow pattern around the vortices at positions
$\mathbf{u}_i$, we find
the flow pattern around vortices at the corresponding positions
$\mathbf{\mathcal{U}}_i$ on the reference surface and then map these
streamlines onto the target surface
by Eq. (\ref{eq:mapping}). The energies are \emph{not}
equal in this case, in spite of property 3.
Property
3 does not apply to a region containing vortex cores, because
we would have to
suppose the area of the cores on the reference surface
to be greater by $e^{2\omega}$
and the energy in the cores to be smaller by a factor of $e^{2\omega}$, in order for the conformal relation Eq. $\frac{1}{2}\rho_s v_T^2dA_T=
\frac{1}{2}\rho_s v_R^2dA_R$ to continue to hold.
In contrast, the core radius is fixed by the short-distance correlations of the
helium atoms and the core energy is related to the interaction energy of
the atoms.
The vortex cores are not significantly affected by
the curvature of the substrate; moreover, the whole flow pattern
in the vicinity of the core is nearly independent of the location of a vortex.
We observe that
each vortex has a ``dominion," a region where the flows are forced by
the presence of the vortex to be
\begin{equation}
v=n\frac{\hbar}{mr}+\delta v
\label{eq:alsouseful}
\end{equation}
The leading term has the same form as one expects for
a vortex in a rotationally symmetric situation, and the effects
of geometry are accounted for by $\delta v$ ; by dimensional
analysis, this error is of the order of $\frac{\hbar}{mL}$ where
$L$ is the radius of curvature of the substrate (or possibly the
distance to another vortex or to the boundary, whichever is shortest).
Therefore we can introduce any length $l<<L$ and note
that $l$ is then a distance below which the
effects of curvature do not have a significant effect (compared
to the diverging velocity field).
The geometry correction gives a contribution
to the energy within this radius that is also small, as seen
by integrating the kinetic energy over the annulus
between $a$ and $l$ (using Eq. (\ref{eq:alsouseful})):
\begin{equation}
\pi K\ln\frac{l}{a}+ \epsilon_c + O(K)\frac{l^2}{L^2}
\label{eq:dominionenergy}
\end{equation}
where $a$ is the core radius and $\epsilon_c$ the core energy.
The error term is \emph{quadratic}
in $\frac{l}{L}$ because the integral over the cross-term from
squaring Eq. (\ref{eq:alsouseful}) cancels.
When we wish to find the kinetic energy of the superflow, these near-vortex
regions are thus the simplest to account for, as their energy is nearly
independent of their positions relative features such as bumps on the substrate
as long as
\begin{equation}
l<<L.
\label{eq:useful}
\end{equation} Since the target and reference configurations have
the same number $N$ of vortices, the energies contained within
radius $l$ of the vortices are the same:
\begin{equation}
E^T_{<l}=E^R_{<l}=\pi NK\ln\frac{l}{\tilde{a}}
\label{eq:inert}
\end{equation}
In order to find the forces on a set of vortices, we need to account
for all the energy of the vortices in regions away from the
vortices where the flow \emph{has}
been affected by the curvature. Let us imagine cutting the target surface
up into an inner region $I$ (the union of the radius $l$ disks around each vortex) and an outer region $O$ (consisting of everything else) as illustrated
in Fig. \ref{fig:emental}. We can
map $I$ to the reference surface by simply translating each of the disks
so that they surround the vortices on $R$.
The modifications to the flow are all in $O$;
for example, streamlines there are deformed from their circular shape.
These irregularities can be removed (or at least simplified) by using
the
conformal mapping to map $O$ to $R$, just as in the case of the annulus
illustrated above. The inhomogeneity of the conformal
map compensates for the irregularity of the flow.
Some portions of $O$ are expanded and some are contracted,
but its circular boundaries are small enough (see Eqs. (\ref{eq:useful}) ,
(\ref{eq:columbus}))
that they
are simply rescaled into circles of different radii (property 4):
\begin{equation}
l_i=e^{\omega(\mathbf{u}_i)}l
\label{eq:scaleregion}
\end{equation}
Now we have mapped $I$ and $O$ from the target surface to regions
on the reference surface which contain the same energy.
But these
images of the regions on $R$ do not fit together.
The conformal map on $O$ stretches or contracts each hole in it, to
circles of radii $e^{\omega(\mathbf{u}_i)}l$. These stretched
edges (corresponding to
the solid black circle in Fig. \ref{fig:emental}) do not fit together with
the images of $O$, which have been moved rigidly from the target surface.
The energies are related by
\begin{equation}
E_T=E^T_{<l}+E^T_{>l}=E^R_{<l}+E^R_{>l_i}
\label{eq:mismatch}
\end{equation}
We must correct for the gaps and overlaps between the two image regions
on $R$ in order to relate the last expression to $E_R$.
If $\omega(\mathbf{u}_i)>0$, there is an annular gap near vortex $i$;
using Eq. (\ref{eq:alsouseful}) (since this gap is part of the flow
controlled by this vortex):
\begin{eqnarray}
\Delta E_i&=&
-\frac{1}{2}\rho_s\int_{l}^{l_i}2\pi rdr\frac{n_i^2\hbar^2}{m^2r^2}\\
&=&-\pi Kn_i^2\omega(\mathbf{u}_i)
\end{eqnarray}
Summing all these contributions gives our desired result, Eq. (\ref{eq:anom}).
We emphasize
the energy difference is not produced within the cores, or anywhere
near the vortices. In fact, the fact that the energies on $T$ and $R$
differ is a result of assuming that there is \emph{no change} in the flow
within a macroscopic distance $l$ of the vortex. The scale
$l\ll L$ only has to be small compared to the geometry of
the system and has no relation to the atomic structure of the core.
On the other hand, taking $l$ as small as possible has an elegant
consequence: Eq. (\ref{eq:anom}) actually has an error which is
$O\left(K\left(\frac{a}{L}\right)^2\right)$, smaller than
$O\left(K\left(\frac{l}{L}\right)^2\right)$ as predicted at first.
Taking smaller values of $l$ gives a more accurate result, since
the conformal mapping (which does not suffer from the error in Eq.
(\ref{eq:dominionenergy}))
is used to calculate the energy of a larger portion of the flow pattern.
Now Eq. (\ref{eq:anom}) shows that the position-dependent scale
factor, $\omega(\mathbf{u})$, plays
the role of a single-particle
energy. Additionally, this single particle energy can be
regarded as the ``geometric potential," since it turns out
to be related to the curvature in
a way analogous to how the electrostatic potential is related
to the charge. The function $\omega$ depends on the
shape of the boundaries and on the curvature of the surface.
A varying scale factor is necessary to map between
surfaces with different distributions of curvature (such as planes
with
and without bumps). (A constant scale
factor only rescales the curvature.) The curvature therefore depends on the
\emph{variation} of $\omega$. In fact it can be shown that
\begin{alignat}{1}
G_T(u,v)=e^{2\omega(u,v)}&G_R(U(u,v),V(u,v))\nonumber\\
&+\frac{1}{\sqrt{det(g_{T,cd})}}
\partial_{a}\sqrt{det(g_{T,cd})}g_T^{ab}\partial_{b}\omega.
\label{eq:Greenland}
\end{alignat}
The second term is the Laplacian of $\omega$ as a function on the
target surface\footnote{As a check of this
identity, imagine reversing the roles
of the reference and target surfaces. Then $\omega$ should
be regarded as a function on the reference
surface. This changes the Laplacian
by a factor of $e^{2\omega}$ (because $g_R$ is replaced by $g_T$). Also,
the sign of $\omega$ should be reversed. Rearranging the equation now
brings it back into the original form with $T$ and $R$ switched.}. The
correspondence with electrostatic potentials, with $G_T(u,v)$ as a source,
is clear if the reference surface is a plane
or disk and $G_R=0$. Then Eq. (\ref{eq:Greenland}) reduces to
Eq. (\ref{eq:geompdeq}).
For a deformed sphere or
plane, single particle potentials come entirely from the second term in
Eq. (\ref{eq:anom}).
The reference energy, corresponding to vortices on a sphere or plane cannot favor
one position over another because of the homogeneity of these reference surfaces. The first term, $E_R$, leads to
the vortex-vortex interactions which depend
only on the separation of the vortices \emph{on the reference surface},
again by symmetry; this energy is
\begin{equation}
E_R=4\pi^2 K\sum_{i<j}n_in_j\Gamma(\mathbf{\mathcal{U}_i},\mathbf{\mathcal{U}_j})
\label{eq:refenergy}
\end{equation}
where $\Gamma$ depends on the reference surface:
\begin{equation}
\Gamma_{plane}(\mathbf{\mathcal{X}},\mathbf{\mathcal{Y}})=
-\frac{1}{2\pi} \ln\frac{|\mathbf{\mathcal{X}}-\mathbf{\mathcal{Y}}|}{a}
\end{equation}
and
\begin{eqnarray}
\Gamma_{sphere}(\mathbf{\mathcal{X}},\mathbf{\mathcal{Y}})&=&
-\frac{1}{2\pi} \ln\frac{2R\sin\frac{\gamma}{2}}{a}\\
&=&-\frac{1}{2\pi}\ln\frac{|\mathbf{\mathcal{X}}-\mathbf{\mathcal{Y}}|}{a}
\label{eq:sphereen}
\end{eqnarray}
where $\gamma$ is the angle between the two points. ($2R\sin\frac{\gamma}{2}$
is the \emph{chordal} distance between the points, not the geodesic distance
along the surface, as one might have guessed for the natural generalization
of a Green's function to curved space.)
The detailed derivation of the second formulation
of the vortex interaction energies in terms of the Green's functions
on the deformed surface (see Eqs.
(\ref{eq:greenpair-energy}) and (\ref{eq:greenself-energy}))
is contained in Appendix
\ref{app:CG}.
\subsection{\label{sec:bubble}Vortices on a ``Soap Film" Surface}
There are experimental and theoretical motivations for
studying substrates shaped as minimal surfaces.
An example of a minimal surface is easy to make
by dipping a loop of wire in soap;
the spanning soap film tries to minimize its area. Vortices can
be studied on a helium film coating a \emph{solid} substrate whose
surface has
the shape of such a film.
Such surfaces are characterized by a vanishing mean curvature, $H({\bf x})$,
so
the contribution of the surface tension $2\gamma H(\mathbf{x})$
to the thickness variation equation,
Eq. (\ref{eq:wet-8})
is drastically reduced.
From the mathematical point of view, there is a
widely known parametrization due to Weierstrass \cite{Hide-book} which readily leads to an
\emph{exact} expression for the geometric potential of a vortex on such
a surface.
Weierstrass's formulae, which give a
minimal surface for each choice of an analytic function $R(\zeta)$, read:
\begin{eqnarray}
x(\zeta)&=&Re\int_0^\zeta {R(\zeta')(\zeta'^2-1)d\zeta'}\nonumber\\
y(\zeta)&=&Im\int_0^\zeta {R(\zeta')(\zeta'^2+1)d\zeta'}\nonumber\\
z(\zeta)&=&Re\int_0^\zeta {R(\zeta')2\zeta'd\zeta'}
\label{eq:weierstrass}
\end{eqnarray}
The correspondence between this parametric surface
and the complex variable $\zeta=\mathcal{X}+i\mathcal{Y}$ is a
conformal map, and
the conformal factor can be expressed in terms of $R(\zeta)$.
Therefore, the analysis of
vortices on such a surface is not difficult at all when the
$\mathcal{X},\mathcal{Y}$-plane is used as the reference surface.
As an example, let $R(\zeta)$ be equal to $L\zeta$ where $L$ controls the
size of the target surface.
Then the surface produced is given in parametric form by
\begin{eqnarray}
x&=&L\left[\frac{\mathcal{X}^3}{3}-\mathcal{X}\mathcal{Y}^2
-\mathcal{X}\right]\nonumber\\
y&=&L\left[\frac{-\mathcal{Y}^3}{3}+\mathcal{Y}\mathcal{X}^2+\mathcal{Y}\right]
\nonumber\\
z&=&L(\mathcal{X}^2-\mathcal{Y}^2)
\label{eq:enneper}
\end{eqnarray}
\begin{figure}
\psfrag{x}{$x$}
\psfrag{y}{$y$}
\psfrag{z}{$z$}
\includegraphics[width=.47\textwidth]{ennepescape2}
\caption{\label{fig:escaping?}Vortex and its streamlines on an ``Enneper Disk".}
\end{figure}
We consider a superfluid film coating only a circle of radius $A$ about the origin of the
$\mathcal{X,Y}$ plane because the complete
surface has self-intersections.
This surface can be called the Enneper disk and is illustrated in Fig.
\ref{fig:escaping?}. The figure illustrates that the left and right hand sides
of the saddle fold over it and would pass through each other if allowed to
extend further while
the front and the back would eventually
intersect each other underneath the saddle.
The former pair of intersection
curves correspond to the two branches
of the hyperbola $\mathcal{X}^2=3(\mathcal{Y}^2+1)$. When the
reference surface is curved into the Enneper surface, the $X$ axis
bends upward so that the branches map to the same
intersection curve in the $yz$ plane.
(The other intersection curves are obtained
by exchanging $\mathcal{X}$ and $\mathcal{Y}$.)
Since the points where these
hyperbolae are closest to the origin are $(\pm\sqrt{3},0),(0,\pm\sqrt{3})$,
a non-self intersecting portion of the Enneper surface results as long
as $A<\sqrt{3}$.
Now we explicitly calculate how a single vortex interacts with the curvature of such a surface
by using Eq. (\ref{eq:anom}). (We will use conformal
mapping instead of Eq. (\ref{eq:curvature-defect}) since
the latter equation does not hold on a surface with the topology
of a disk, as $\omega$ does not satisfy the
Dirichlet boundary conditions which are implied by such an expression.)
The metric obtained from (\ref{eq:enneper}) is given by
\begin{equation}
dx^2+dy^2+dz^2=L^2
(d\mathcal{X}^2+d\mathcal{Y}^2)(1+\mathcal{X}^2+\mathcal{Y}^2)^2.
\end{equation}
(Surprisingly, this \emph{metric} is rotationally symmetric. This implies
that the surface may be slid along itself without stretching, but with changing
amounts of bending.)
Hence
\begin{equation}
\omega_{Enneper}=-\ln L(1+\mathcal{R}^2),
\end{equation}
where $\mathcal{R}^2=\mathcal{X}^2+\mathcal{Y}^2$. According to Eq.
(\ref{eq:anom}),
this indicates that the vortex should be attracted to the middle
of the surface,
but of course this force competes with
the boundary interaction $K\pi \ln\frac{A^2-\mathcal{R}^2}{aA}$
which tries to pull the vortex to the edge. This expression
for the boundary interaction is obtained from the familiar formula for the energy of a vortex interacting with its image
in a flat
reference disk\cite{geomgenerate}.
The total energy is then
\begin{equation}
E=K[\pi\ln\frac{L}{aA}+ \pi\ln (A^2-\mathcal{R}^2)(1+\mathcal{R}^2)].
\end{equation}
As long as $A>1$,
the central point of the saddle is a local minimum and this condition is compatible
with the requirement $A<\sqrt{3}$ for non-self-intersecting
disks. Fig. \ref{fig:escaping?}
shows the flow lines of
a vortex forced by the geometric interactions towards the center of an Enneper surface with $A=1.5$.
In general, conformal mapping
allows us to express the energy of a single vortex on a deformed surface
with a boundary
in the form:
\begin{equation}
E=\pi K[\ln\frac{A^2-\mathcal{R}(\mathbf{u})^2}{aA}-\omega(\mathbf{u})],
\label{eq:netenergy}
\end{equation}
where $\mathcal{R}(\mathbf{u})$ refers to the image of a defect at $\mathbf{u}$
under a conformal map to a flat, circular disk of radius $A$.
The Green's function method cannot be used to determine the energy
of defects
on a surface with a boundary. Although
the conformal factor $\omega(\mathbf{u})$
satisfies the Poisson equation, Eq. (\ref{eq:geompdeq}),
it cannot be expressed
as the integral of the curvature times the Green's function (as
in Eq. (\ref{eq:greenself-energy})),
since $\omega$ does not satisfy simple boundary conditions.
In any case, the first term in Eq. (\ref{eq:netenergy})
has no general expression in terms of $\omega$ either.
Interestingly, the \emph{total}
single-particle
energy satisfies a \emph{nonlinear} differential equation (the Liouville
equation):
\begin{equation}
\nabla_{\mathbf{u}}^2E(\mathbf{u})=-\pi K G(\mathbf{u})-
\frac{4\pi K}{a^2}e^{-\frac{2E(\mathbf{u})}{\pi K}}.
\label{eq:liouvilledisk}
\end{equation}
This result can be derived by using Eq. (\ref{eq:Greenland})
to calculate the Laplacian of the first term and using
$\nabla^2=e^{2\omega(\mathbf{u})}\frac{1}{\mathcal{R}}
\frac{\partial}{\partial\mathcal{R}}\mathcal{R}
\frac{\partial}{\partial\mathcal{R}}$
to calculate the Laplacian of the second term.
$E(\mathbf{u})$ also satisfies an asymptotic boundary condition:
\begin{equation}
e^{\frac{E(\mathbf{u})}{\pi K}}\rightarrow \frac{2d}{a}
\end{equation}
as $d$, the distance from $\mathbf{u}$ to the boundary, approaches 0.
Together the differential equation and the boundary condition
should determine the total geometrical
and boundary energy of a single vortex, although the nonlinear Eq.
(\ref{eq:liouvilledisk})
is difficult to solve.
\subsection{\label{sec:bumps}Periodic surfaces}
In this section, we illustrate how a periodically curved substrate distorts the flat space
interaction energies between vortices,
besides generating the single-particle geometric potential.
This effect is shown to be a consequence of the action of a conformal map
which generally will map the target surface into
a periodic reference
substrate with different lattice vectors from the vectors of the
target substrate.
According to the general relation Eq. (\ref{eq:greentransf}),
the long-distance behavior of
the Green's function is given by the logarithm of a distorted distance.
Consider a surface with a periodic height function
$z=h(x,y)$, i.e., say $h$ satisfies
\begin{equation}
h(x+\lambda_i,y+\mu_i)=h(x,y) \quad \ \mathrm{for\ }i=\{1,2\} ,
\label{eq:goingtoseed}
\end{equation}
where $i$ labels the two basis vectors,
which are not assumed to be orthogonal.
Figure (\ref{tulipbulbs}) shows the corresponding periods $(\lambda_i,\mu_i)$. A conformal mapping can be chosen to preserve the fact that the substrate
is periodic but not the actual values of the periods, which are therefore
given on the reference substrate by two new pairs denoted by $(\Lambda_i,M_i)$.
In other words, we suppose that a tesselation of
the target substrate by congruent unit cells is
mapped to a set of congruent unit cells on the reference
substrate. Then
the map transforming the original coordinates $(x,y)$ into the target coordinates $(\mathcal{X},\mathcal{Y})$ satisfies
\begin{eqnarray}
\mathcal{X}(x+\lambda_i , y+\mu_i)=\mathcal{X}(x, y)+\Lambda_i \nonumber\\
\mathcal{Y}(x+\lambda_i , y+\mu_i)=\mathcal{Y}(x, y)+M_i .
\label{eq:periods}
\end{eqnarray}
There is no simple formula for
the new set of primitive lattice vectors $(\Lambda_i,M_i)$ for the reference space. In some cases, though, precise information can be derived
from the fact that
the $(\Lambda_i,M_i)$
share the symmetry of the topography of the original substrate.
For example, if the lattice is composed of bumps which have
a $90^{\circ}$ rotational point symmetry, then the reference lattice
will be square. On the other hand, the topology of the periodic surface with a
$square$ lattice shown in the contour plot of Fig. (\ref{fig:squaredaway}) does not posses a $90^{\circ}$ rotational symmetry and hence its
conformal image will have a $rectangular$ lattice.
\begin{figure}
\psfrag{l1}{$\lambda_1$}
\psfrag{m1}{$\mu_1$}
\psfrag{o1}{}
\psfrag{l2}{$\lambda_2$}
\psfrag{m2}{$\mu_2$}
\psfrag{o2}{}
\psfrag{x}{$x$}
\psfrag{y}{$y$}
\includegraphics[width=.45\textwidth]{tulipbulbs3}
\caption{\label{tulipbulbs} An illustration of the period
lattice of the target surface, projected into the $xy$-plane.
The coordinates $\lambda_1,\mu_1$ and $\lambda_2,\mu_2$
describe the basis vectors,
which we do not assume to be orthogonal.
}
\end{figure}
To get an idea how the conformal mapping behaves
macroscopically, we try to decompose it into a linear
transformation $\mathcal{L}$
with matrix coefficients $\{A, B; C, D\}$ and a periodic modulation
captured by the functions $\xi(x,y)$ and $\eta(x,y)$ that
distort the $\mathcal{X}$ and $\mathcal{Y}$ axes respectively:
\begin{eqnarray}
\mathcal{X}(x,y)=Ax+By+\xi(x,y) \nonumber\\
\mathcal{Y}(x,y)=Cx+Dy+\eta(x,y) .
\label{eq:shiftperiods}
\end{eqnarray}
(This decomposition is justified by the self-consistency of the
following calculations.)
The matrix coefficients of the linear transformation can be determined by
requiring
consistency with Eq. (\ref{eq:periods}). Start by evaluating the
left hand sides of the two Equations (\ref{eq:shiftperiods}) at the positions $\{x+\lambda_i,y+ \mu_i \}$ which are
shifted by the two pairs of periods $\{\lambda_i,\mu_i\}$, so that the right hand sides become $\mathcal{X}(x,y)+\Lambda_i$ and $\mathcal{Y}(x,y)+M_{i}$, according to Eq. (\ref{eq:periods}). Then subtract the resulting equations from the corresponding unshifted Equations (\ref{eq:shiftperiods}), for each value of $i$. We then obtain two pairs of equations
\begin{eqnarray}
A \lambda_i + B \mu_i&=&\Lambda_i \nonumber\\
C \lambda_i + D \mu_i&=&M_i \quad \ \mathrm{for\ }i=\{1,2\} ,
\label{eq:int1}
\end{eqnarray}
where we have used the fact that the periodic functions $\xi(x,y)$ and $\eta(x,y)$ are unchanged when shifted by the periods. We can now solve the four
equations of Eq. (\ref{eq:int1})
simultaneously for $A,\ B,\ C,\ \mathrm{and\ }D$
to see that the linear transformation matrix $\mathcal{L}$ reads
\begin{equation}
\mathcal{L}=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)=
\left(\begin{array}{cc}\Lambda_1&\Lambda_2\\M_1&M_2\end{array}\right)
\left(\begin{array}{cc}\lambda_1&\lambda_2\\\mu_1&\mu_2\end{array}\right)^{-1}.
\label{eq:affine}
\end{equation}
(Now we can justify the original decomposition, Eq. (\ref{eq:shiftperiods}),
by \emph{defining}
$\mathcal{L}$ by Eq. (\ref{eq:affine}) and \emph{defining} $\xi(x,y)$ and
$\eta(x,y)$ as the discrepancy between the conformal
map $\mathcal{X}(x,y),\mathcal{Y}(x,y)$ and the linear map $\mathcal{L}(x,y)$
as in Eq. (\ref{eq:shiftperiods}). We can then check that $\xi(x,y)$
and $\eta(x,y)$ \emph{are} periodic functions of the coordinates.)
The linear transformation can be used to calculate approximately the long distance behavior of the Green's function
\begin{eqnarray}
\Gamma(x,y;x',y')&=&-\frac{1}{4\pi}
\ln[(\Delta\mathcal{X})^2+(\Delta\mathcal{Y})^2]\nonumber\\
&\approx&-\frac{1}{4\pi}\ln[(A\Delta x+B\Delta y)^2\nonumber\\
&&\ \ \ \ \ \ +(C\Delta x+D\Delta y)^2] ,
\label{eq:ellipsie}
\end{eqnarray}
where we used the fact that the periodic functions $\xi(x,y)$ and $\eta(x,y)$ are bounded and hence negligible
in comparison to $\Delta x$ and $\Delta y$ at long distances. This
expression illustrates the fact that
the matrix $\mathcal{L}$ captures the long distance lattice distortions
induced by the conformal mapping,
apart from the additional waviness described by $\xi(x,y)$ and $\eta(x,y)$. The linear transformation determined by $\mathcal{L}$
is by itself typically \emph{not conformal}, meaning
that it generates an anisotropic deformation of the target lattice which does not preserve the angle between the original
lattice vectors.
\begin{figure}
\includegraphics[width=.47\textwidth]{squared}
\caption{\label{fig:squaredaway}A periodic surface with a square lattice
but not square symmetry.
This surface is illustrated by its contour plot.
It has the two reflectional symmetries but
not the $90^{\circ}$ symmetry of a square lattice; hence,
generically, the conformal image will have a \emph{rectangular} lattice.}
\end{figure}
The deformation of the lattice is controlled by the curvature of the substrate. To spell out this connection and allow an explicit evaluation of the long-distance
Green's function in Eq. (\ref{eq:ellipsie}), we explicitly
evaluate the matrix elements $L_{ij}$ in terms
of the height function $h(x,y)$ of
a gently curved (or low-aspect-ratio) surface, one for which
$h(x,y)\ll \{\lambda_i,\mu_i\}$, $\{\xi(x,y),\eta(x,y)\}\ll1$
and $\{A-1,D-1,B, C\}\ll1$.
The new set of (isothermal) coordinates $\mathcal{X}$ and $\mathcal{Y}$,
used to implement the conformal transformation, are found by solving the Cauchy-Riemann Equations
(\ref{eq:tack})
\begin{eqnarray*}
&&\partial_x\mathcal{X}
=\sqrt{g}g^{yy}\partial_y\mathcal{Y}+\sqrt{g}g^{xy}\partial_x\mathcal{Y}
\nonumber\\
&&\partial_y\mathcal{X}
=-\sqrt{g}g^{xx}\partial_x\mathcal{Y}-\sqrt{g}g^{xy}\partial_y\mathcal{Y},
\end{eqnarray*}
which, upon substituting from Eq. (\ref{eq:shiftperiods}) and
making the small aspect ratio approximation discussed in Appendix
\ref{app:calmseas}, reduce to
\begin{eqnarray}
&&A+\partial_x\xi=D+\partial_y\eta+\frac{1}{2}(h_x^2-h_y^2)\\
&&B+\partial_y\xi=-C-\partial_x\eta+h_xh_y.
\label{eq:alphabet}
\end{eqnarray}
We now proceed to show that these equations do not have solutions unless the lattice is distorted, that is to say
the matrix $\mathcal{L}$ cannot be the identity for a generic
periodic function $h(x,y)$. Note that
the periodicity of $\xi(x,y)$ and $\eta(x,y)$ implies that the integral
of either one
over any unit cell, e.g.,
\begin{equation}
\iint_{\mathrm{cell}}dxdy\ \xi(a+x,b+y)
\end{equation}
is independent of the quantities $(a,b)$ by which the unit cell is shifted. Upon differentiating the integral with respect to $a$, one obtains
\begin{equation}
\iint_{\mathrm{cell}}dxdy\ \xi_x(x,y)=0.
\label{eq:zero}
\end{equation}
Similarly, the averages of $\eta_x,\eta_y,$ and $\xi_y$ are also equal to zero.
Hence, upon averaging Eq. (\ref{eq:alphabet}) over a unit cell we obtain the key relations
\begin{eqnarray}
A-D&=&\frac{1}{2}<h_x^2-h_y^2>\nonumber\\
B+C&=&<h_xh_y>,
\label{eq:approxperiods}
\end{eqnarray}
which prove our assertion that $L$ cannot be a simple dilation or rotation. Some shear is naturally introduced by the non trivial metric (or height function) of the
underlying surface.
The $\mathcal{L}$ matrix is undetermined
up to a dilation and a rotation but this
is of no consequence to the determination of the Green's function. In fact,
to
find the Green's function, note that Eq. (\ref{eq:approxperiods}) allows
us to
write the
matrix coefficients in terms of two undetermined constants $\epsilon_1$ and $\epsilon_2$ that will drop out of the final answer:
\begin{eqnarray}
&&A=1+\epsilon_1+\frac{1}{4}<h_x^2-h_y^2>\\
&&D=1+\epsilon_1+\frac{1}{4}<h_y^2-h_x^2>\\
&&B=\epsilon_2+\frac{1}{2}<h_xh_y>\\
&&C=-\epsilon_2+\frac{1}{2}<h_xh_y> ,
\end{eqnarray}
so that consistency with Equations (\ref{eq:approxperiods}) is guaranteed.
(The variables $\epsilon_1$ and $\epsilon_2$ parameterize an overall
infinitesimal
scaling (by $1+\epsilon_1$) and a rotation (by angle $\epsilon_2$)
respectively.)
Substitution of these equations into Eq. (\ref{eq:ellipsie}) gives the desired long-distance behavior
of the Green's function purely in terms of derivatives of the height function, which we assume to be known:
\begin{eqnarray}
\Gamma(x,y;x',y')&\approx& -\frac{1}{4\pi}\ln [\Delta x^2+\Delta y^2\nonumber\\
&&\ \ \ \
+\frac{1}{2}<h_x^2-h_y^2>((\Delta x)^2-(\Delta y)^2)\nonumber\\
&&\ \ \ \ \ +2<h_xh_y>\Delta x\Delta y] .
\end{eqnarray}
This is the central result of this section; it can also be applied
to interactions between disclinations in liquid crystals \cite{geomgenerate}
and dislocations in crystals \cite{VitelliLucks}.
The anisotropic correction to the Green's function, captured by the second and third term, suggests that a distorted version of the triangular
lattice of vortices expected on a flat substrate may form
when the helium-coated surface is rotated slowly enough that there
is only one vortex to several unit cells. However, the
actual ground state is likely to be difficult to observe, as the geometric potential will try to trap the vortices near saddles as discussed in Section
\ref{BS}.
\subsection{Band-flows on elongated surfaces\label{sec:zucchini}}
In this section, we show that
the quantization of circulation can induce
an extremely long-range force on a stretched-out sphere (such as a
surface with the shape of
a zucchini or a very prolate spheroid). We
first demonstrate the main result in the context of a simple
example before presenting a general formula for the forces experienced
by vortices on azimuthally
symmetric surfaces. Details are presented in
Appendix \ref{app:deimos}.
Consider
a cylinder
of length $2H$ and radius $R<<H$ with hemispherical caps of radius $R$
at the ends,
depicted in Fig. \ref{fig:goodnplenty}, and imagine a symmetric
arrangement of a vortex ($n$=1) and an anti-vortex ($n$=-1) at the
north and south poles respectively.
\begin{figure}
\includegraphics[width=.2\textwidth]{goodnplenty}
\caption{\label{fig:goodnplenty}A capped cylinder; a cylinder of length
$2H$ is closed off by hemispheres at the north and south poles
of radius $R$. The circulation around
every lattitude is the same.}
\end{figure}
Extrapolating our intuition from flat space
suggests that the energy of the vortex and
anti-vortex
is $2\pi K\ln\frac{D}{a}$, where $D$ is the distance between the vortices.
However, more careful reasoning shows that the energy
grows \emph{linearly} rather than logarithmically with $D$.
The reason is that, unlike in flat space, the velocity field
does not fall off like the inverse of the distance from each vortex.
Note that the azimuthal symmetry of the arrangement of the
vortices implies that the flow
is parallel to the lines of latitude of the surface.
Since the circulation around each latitude must be $\frac{h}{m}$,
the flow speed on the cylindrical part reads
\begin{equation}
v=\frac{h}{2\pi m R}.
\label{eq:farina}
\end{equation}
The
kinetic energy of this part of the flow is
$[4\pi RH]\frac{1}{2}\rho_s v^2=2\pi\frac{KH}{R}$. Since this
cylindrical part
of the flow forms the main contribution to the kinetic energy when $H>>R$
we find that the energy of a vortex-antivortex pair situated
at opposite poles is linear,
\begin{equation}
E_{poles}\approx 2\pi\frac{KH}{R}.
\label{eq:lateral}
\end{equation}
(The exact expression also includes a near-vortex energy of approximately
$2\pi K\ln\frac{R}{a}$.)
In contrast, when
the vortices forming the neutral pair
are across from each other on the same latitude, the aforementioned long-range
persistence of the velocity field is absent because the vorticity is
screened within a distance of order $R$. The resulting
kinetic energy follows the familiar logarithmic growth
\begin{equation}
E_{equator}\approx 2\pi K\ln\frac{2R}{a}.
\label{eq:opposite}
\end{equation}
More generally, consider an azimuthally symmetric surface described by the
radial distance, $r(z)$, as a function of height, $z$, as indicated in Fig.
\ref{fig:kitestring} A. If the north and south poles of
the surface are at $z_s$ and $z_n$,
then $r(z_s)=r(z_n)=0$ since the surface closes at the top and bottom.
A point
on such a surface can be identified by the coordinates
$(\phi,\sigma)$ where $\phi$ is the azimuthal angle and $\sigma$ is the
distance to the point from the north pole along one of the longitudes such
as the one shown
in Fig. \ref{fig:kitestring}A.
In Appendix \ref{app:deimos}, we develop an approximation scheme which
rests on the observation that the flow pattern becomes mostly
azimuthally symmetric
if $\frac{dr}{dz}<<1$, even if the vortices break the azimuthal symmetry
of the surface because they are not at the poles.
If a pair of $n=\pm 1$ vortices are present at different heights $z_{1,2}$,
then
the fluid in the band between them
flows almost horizontally and at a nearly $\phi$-independent speed
(except for irregularities near the vortices) while the fluid beyond
them is approximately stagnant (see Fig. \ref{fig:kitestring}).
\begin{figure}
\includegraphics[width=.45\textwidth]{kitestring3}
\caption{\label{fig:kitestring}The flow on an azimuthally symmetric
surface, described by the coordinates $(z,r)$ and an azimuthal
angle $\phi$ (not shown).
A) The surface is defined as the surface of
revolution of the curve $r(z)$ in the $r-z$ plane.
The other two images show the flow on an ellipsoid and
compare
the flow pattern predicted by the band model (B)
and the exact solution determined by conformal mapping (C).
The flow lines in the band between the two vortices
become close to horizontal and are approximately azimuthally
symmetric.
Beyond the vortices, they are spaced far apart indicating a
vanishingly small speed for a greatly elongated surface.
}
\end{figure}
Along any latitude inside the band the circulation is \emph{exactly}
$\frac{h}{m}$, while it is zero above and below it.
These properties approximately determine the flow away from
the vortices since the asymmetric irregularities near the
vortices decay
exponentially, giving a speed of
\begin{equation}
\frac{h}{2\pi m r(z)}
\label{eq:simplebandspeed}
\end{equation}
in between $z_1$ and $z_2$, the locations of the vortices,
and zero elsewhere. We describe this approximation as the ``band model".
This expression shows that constrictions in the surface cause the speed
to increase. For the cylinder with spherical caps
and arbitrarily placed vortices,
Eq. (\ref{eq:simplebandspeed}) shows that the speed
is approximately constant within the band. (It increases within a distance
on the order of $R$ from the vortices, which are on the edges of the band.)
The kinetic energy can be determined approximately by noting that
the energy in a thin ring on the surface between the vortices
(extending from the longitudinal arclength $\sigma$ to $\sigma+d\sigma$)
is
\begin{equation}
[2\pi r(z)d\sigma][\frac{1}{2}\rho_s (\frac{\hbar}{m r(z)})^2]=\pi K
\frac{d\sigma}{r(z)}
\nonumber
\end{equation}
where the first factor represents the area of the ring since
$\sigma$ is the geodesic distance along the surface and
the second factor represents the included kinetic energy.
The flow is zero
past the two vortices, so the total energy is
\begin{equation}
E=K\pi \int_{\sigma_1}^{\sigma_{2}}\frac{d\sigma}{r}.
\label{eq:bandenergy1}
\end{equation}
For the capped cylinder with a constant $r(z)$ the integral is
$\pi\frac{KD}{R}$, where $D$ is the vertical distance between the vortices.
This expression generalizes the result for vortices at the poles of the surface,
Eq. (\ref{eq:lateral}). However, when the vortices are on opposite ends of
an equator, they are too close for the asymmetries to be neglected. The
energy in this case is calculated in Appendix \ref{app:deimos}.
The force $F_{1,band}$, experienced by
vortex $1$ can be determined by taking the
gradient of Eq. (\ref{eq:bandenergy1}). If vortex 1 is moved downward, the
band
of moving fluid shrinks so the energy drops.
The force on the vortex therefore reads
\begin{equation}
F_{1,band}=\frac{\pi K}{r(\sigma_1)}
\bm{\hat{\sigma}}.
\label{eq:bandforce1}
\end{equation}
On the capped cylinder, this force
is independent of the positions of the vortices. Even on
an arbitrary elongated surface, a noteworthy feature is
that the force on vortex $1$ does
not depend on the position of vortex $2$!
This force can be explained with the
familiar phenomenon of lift: the vortex is on the boundary between
stationary and moving fluid, so there is a pressure difference
due to the Bernoulli principle.
Approximating the flow pattern generated by multiple vortices in
a similar fashion
requires only minor modifications of the previous argument.
In a low-resolution snapshot of the flow, the point-vortices
would appear as circles of discontinuity in the velocity
field that go all the way around the axis (the analogue for a layer
of superfluid
of a two
dimensional
vortex sheet).
If the vortices are
labeled in order of decreasing $z$ a loop just below
the $l^{\mathrm{th}}$ vortex contains
\begin{equation}
N_l=\sum_{i=1}^l n_i
\label{eq:balancesheet}
\end{equation}
units of circulation above it.
Approximate azimuthal symmetry of the flow then implies that,
\begin{equation}
\mathbf{v}(z,\phi)_{band}=N_l\frac{\hbar}{m r(z)}\mathbf{\hat{\phi}}\ \ \
(\mathrm{for\ }z_l<z<z_{l+1}),
\label{eq:bandspeed}
\end{equation}
a natural generalization of Eq. (\ref{eq:simplebandspeed}) that is proved
in Appendix \ref{app:deimos}.
Conformal mapping can be employed to justify (without detailed calculations)
the
decay of the nonazimuthally symmetric parts of the flow that are not
determined by the quantization condition.
We sketch the basic reasoning here by focusing
(for simplicity) on the flow pattern \emph{near the
equator} of the surface, at a distance
$\sigma_{eq}$ from the north pole. The conformal
transformation that maps the elongated sphere onto a regular
reference sphere with coordinates $\Theta,\Phi$ reads
(see Appendix \ref{app:deimos})
\begin{eqnarray}
&&\sin\Theta=\mathrm{sech}\int_{\sigma}^{\sigma_{eq}}
{\frac{d\sigma'}{r(\sigma')}}\nonumber\\
&&\Phi=\phi.
\label{eq:deimosmap}
\end{eqnarray}
The upper and lower
halves of the elongated sphere can be
mapped to the upper and lower hemispheres
by choosing appropriately between the two values of $\Theta$
that correspond to a given value of $\sin \Theta$.
Near the equator the integral can be approximated
by $\frac{\sigma-\sigma_{eq}}{r_{eq}}$ since $r$ varies slowly.
Suppose the vortices are far from the equator,
at a distance greater than $kr_{eq}$ for a large $k$. Then the vortices
above the equator are mapped
exponentially close (at a distance less than $e^{-k}$)
to the sphere's north pole. Likewise vortices
on the southern half of the surface map exponentially close
to the sphere's south pole. We have thus reduced the task of
finding the flow due to a complicated arrangement of
vortices to a symmetric case. In fact, after mapping the
flow on the long, thin surface to the reference sphere, nothing can
be resolved beyond a pair of
multiply quantized vortices at the north and south poles
containing $N$ and $-N$
units of circulation respectively, where $N$ is the
total circulation number of
all the vortices above the equator.
Since the image vortices are very close to the poles,
their flow pattern on the reference surface is approximately azimuthally
symmetric near the equator. When mapped back to
the elongated surface, the flow retains its approximate azimuthal symmetry
in the region around the equator, completing our argument.
A similar argument proves the approximate azimuthal symmetry
of the flow
near lines of latitude other than the equator;
one simply adjusts the conformal map
in Eq. (\ref{eq:deimosmap})
so that another latitude of the target surface
is mapped to the equator of the reference sphere.
Now the geometrical force derived in Eq. (\ref{eq:bandforce1})
can compete with physical forces such as those induced by rotating an ellipsoid about its long axis
with angular velocity $\bm{\Omega} = \Omega \!\!\!\! \quad \bm{\hat{z}}$.
Let us extend the
treatment of rotational forces on curved substrates, introduced in section \ref{sec:Rotation}, to the case of an
ellipsoid described by the radial function
\begin{equation}
r(z)=R\sqrt{1-\frac{z^2}{H^2}}.
\label{eq:prolate}
\end{equation}
Let us
use the aspect ratio $\alpha=\frac{H}{R}$ to describe how elongated this
ellipsoid is, and determine
how a vortex-anti-vortex pair
is torn apart by the rotation as
the angular frequency is increased. As in Section \ref{sec:Rotation}, metastable
vortex configurations can often be found, so we will consider
transitions between different \emph{local} minima of the vortex-energy function.
Recall that the effect of the rotation on the superfluid energy is expressed in terms of the angular momentum $L_z$ by an extra term $-\Omega L_z$ in Eq. (\ref{eq:Erot})
which must be evaluated for (at least)
a pair of opposite-signed
vortices to satisfy the topological constraints imposed by the spherical
topology of the
surface. We find, in analogy to Eq. (\ref{eq:Lz-c}), that the
extra contribution to the vortex energetics is additive and reads
\begin{equation}
-\Omega L_z=n_1\hbar \Omega \frac{\rho_s}{m} [A(\sigma_1)-A(\sigma_2)],
\label{eq:Lzvortex}
\end{equation}
where $A(\sigma)$ represents the area of the ellipsoid out to
a distance $\sigma$ from the north pole while $\sigma_1$ and $\sigma_2$ represent the positions of the two defects ($\sigma_1<\sigma_2$).
If $n_1=1$, the energy in
Eq. (\ref{eq:Erot}) is
decreased by moving the positive vortex closer to the north pole and the negative one closer to the south pole. To understand this,
note that the sense of rotation
of the superfluid around a vortex is defined by an observer facing
the surface. Hence, a negative vortex on the southern half of the surface
rotates in the same direction as a positive one on the northern half
(relative to the positive $z$-axis), and both agree with the sense of rotation
of the substrate.
The rotational force on vortex 1
derived from Eq. (\ref{eq:Erot}) is
\begin{equation}
\mathbf{F_{\Omega}}=-\frac{n_1\hbar\rho_s}{m}2\pi r(\sigma)\mathbf{\hat{\sigma}} .
\label{eq:Frotsequel}
\end{equation}
As $\Omega$ increases, \emph{pairs} of positive and negative vortices will
appear in this geometry.
As each vortex pair is created, the positive vortex will move to the top side
of the surface, and the negative one to the bottom. There is a critical frequency,
$\Omega_b$, at which a single pair of vortices, once created, can
exist metastably in a configuration symmetric about the $xy$-plane.
As the angular
frequency is increased, the vortices are gradually pulled apart
until at the frequency $\Omega_a$, they reach the poles. (The latter
transition is
analogous to the center/off-center transition for a single vortex described
in Section \ref{subsec:single}.)
When $\Omega_{a}>\Omega>\Omega_b$, the equilibrium condition, obtained by balancing the forces in
Equations (\ref{eq:bandforce1}) and (\ref{eq:Frotsequel}), reads
\begin{equation}
\frac{\pi K}{r_1}=2\pi\frac{\hbar\rho_s}{m}\Omega r_1
\label{eq:heartgrowsfonder}
\end{equation}
Since $r$ and $z$ are connected by the equation of the ellipsoid,
the vortices are located at heights
\begin{equation}
\pm z\approx \pm\alpha\sqrt{R^2-\frac{\hbar}{2m\Omega}}.
\label{eq:prize}
\end{equation}
The vortices first become metastable when force balance is achieved with both vortices close to the equator. Upon substituting the equatorial value
$r_1=R$
into Eq. (\ref{eq:heartgrowsfonder}) an estimate of $\Omega_b$ is obtained
\begin{equation}
\Omega_b\approx\frac{\hbar}{2mR^2}.
\label{eq:newadventure}
\end{equation}
When the pair first appears, there will actually
be a non-zero defect separation,
although substituting Eq. (\ref{eq:newadventure}) into Eq. (\ref{eq:prize})
suggests otherwise. Imagine slowing the
rotation speed through $\Omega_b$.
The vortices will approach each other gradually; within the
large vortex-separation approximation
of Eq.
(\ref{eq:bandforce1}), the attraction between them will \emph{decrease}
as they become closer because $r(z)$ increases. However, when the
vortices become close enough, the attraction between them
starts increasing and the vortices are suddenly pulled together.
The minimum $z$-coordinate for metastable vortices is
derived along these lines
in Appendix \ref{app:deimos} (which also discusses what
happens at $\Omega=\Omega_a$) and reads
\begin{equation}
z_1=-z_2=z_b\approx R\ln\alpha .
\label{eq:logs}
\end{equation}
The transition through $\Omega_b$ is illustrated pictorially
in Fig. \ref{fig:spndl} which shows
how the local minimum in the energy function disappears as
the frequency decreases.
\begin{figure}
\psfrag{Energy/K}{Energy/$K$}
\psfrag{sigma/R}{$\sigma/R$}
\includegraphics[width=.45\textwidth]{spndl2}
\caption{\label{fig:spndl}The rotational and fluid energy
(units of $K$) as a function of $\sigma_1=\sigma_{tot}-\sigma_2=\sigma$
(units of $R$) for $H=3.5 R$ and $\frac{m\omega}{\hbar}=.49,.61,.74 R^{-2}$.
The middle curve, roughly at $\omega=\omega_b$,
shows the last position where the vortex is stable as $\omega$ is decreased.}
\end{figure}
\subsection{\label{sec:noodles}Interactions on a closed surface}
To understand interactions between vortices on an arbitrary
deformed sphere one must come to
terms with the neutrality constraint on the total
circulation of a flow. On any compact surface,
\begin{equation}
\sum_i n_i=0.
\label{eq:neutrality}
\end{equation}
This constraint on the sum of the circulation indices $\{n_i\}$ always holds:
if the surface is divided into two pieces by a curve, the sum
of the quantum numbers
on the top and bottom half must be equal and opposite (because they
are both equal to the circulation
around the dividing curve).
As we shall see, this relation implies that there are multiple ways of splitting up the energy into single-particle energies and two-particle interaction
energies, despite the fact that the total
energy is well-defined.
The behavior of the one-particle
and interaction terms depends on how the splitting is carried out.
To illustrate this ambiguity, multiply Eq. (\ref{eq:neutrality}) by
$4\pi^2 K n_1 f(\bm{u}_1)$ and separating out the $i=1$ term, to obtain
\begin{equation}
4\pi^2 n_1^2 f(\mathbf{u}_1)=-\sum_{i\neq 1} 4\pi^2 n_1 n_i f(\mathbf{u}_1).
\end{equation}
Hence, a portion $4\pi^2 K f(\mathbf{u}_1)$ of the ``geometrical
energy" of vortex 1 can be reattributed to this vortex's interaction with all
the other vortices.
This can be seen explicitly by checking
that the \emph{net}
energy according to Eqs. (\ref{eq:greenself-energy}) and
(\ref{eq:greenpair-energy}),
\begin{alignat}{1}
E(\{n_i,\bm{u}_i\})=\sum_{i<j}&4\pi^2 Kn_in_j\Gamma(\bm{u}_i,\bm{u}_j)\\
&-\sum_i\pi n_i^2 KU_G(\bm{u}_i)
\label{eq:redouble}
\end{alignat}
is not changed by the following transformation:
\begin{eqnarray}
&&\Gamma'(\mathbf{u}_1,\mathbf{u}_2)=\Gamma(\mathbf{u}_1,\mathbf{u}_2)
-f(\mathbf{u}_1)-f(\mathbf{u}_2)\\
&&U_G'(\mathbf{u})=U_G(\mathbf{u})+4\pi f(\mathbf{u})
\label{eq:ces}
\end{eqnarray}
This flexibility
is reflected in the possibility of choosing different Green's functions
$\Gamma'(\mathbf{u}_1,\mathbf{u}_2)$ for the covariant Laplacian on a deformed
compact surface. A detailed discussion of Green's functions
is given in Appendix \ref{app:CG}.
Here we higlight this ambiguity by performing explicit calculations using two distinct choices of Green's functions on a model surface formed
from a unit sphere. First cut the sphere in halves along a great circle.
Choose one of the hemispheres and bring opposite sides of the great
circle bounding it together and glue them. The result is
a pointed sphere resembling a conchigliette noodle (i.e., a shell noodle)
sealed shut (see Fig. \ref{fig:noodlemachine}A). The surface closes up
smoothly since opposite sides of the seal have matching curvatures;
the surface also turns out to be rotationally symmetric. (A similar
shape forms when a pollen grain with a weak sector
(such as the pollen of a lily) dries out \cite{microscopic}.)
\begin{figure}
\psfrag{u1}{$\mathbf{u}_1$}
\psfrag{u2}{$\mathbf{u}_2$}
\psfrag{P2}{$Q_2$}
\psfrag{P1}{$Q_1$}
\psfrag{P1*}{$Q_1^*$}
\psfrag{P2*}{$Q_2^*$}
\includegraphics[width=.5\textwidth]{noodleseal4}
\caption{\label{fig:noodlemachine} A) The process of folding a hemisphere
into a pointed sphere, bounded at the north and south poles by two $180^{\circ}$
disclinations.
The north and south poles move outward along the axis, while
the latitudes stay horizontal. The Gaussian curvature is
invariant because the decreasing curvature of the lines of longitude
is compensated by the tighter curvature around the lines of lattitude.
B) A top view of vortices during the furling-up of one hemisphere.
The first stage shows both the $(0<\alpha<\pi)$
hemisphere that is furled up and the other
hemisphere together with two defects
and their images. The hemisphere is furled up into the pointed sphere
so that the left and right halves of the cut (which appears as a horizontal
diameter in this top view) are brought together
to form the seam on the pointed sphere; simultaneously, the defects $Q_1,Q_2$
move to positions $\mathbf{u}_1$ and $\mathbf{u}_2$ on the pointed sphere.
The furling process leads to a continuous flow pattern
on the pointed sphere. For example, the two points marked with an x on the cut
in the original sphere are sealed together. Both points feel
a strong flow, the one on the right because it is close to the
vortex at $Q_1$ and the one on the left
because it is close to this
vortex's image.
}
\end{figure}
The geometrical interaction on this surface can be given an appealing
interpretation in terms of the method of images from electrostatics.
When one uses the method of images to study charges in a
half-space bounded by a conducting plane, one completes the space,
with the other half-space. Then one introduces charges into this
fictitious region to ensure that the right boundary conditions (orthogonality
of the field lines to the original boundary plane) are satisfied.
For the pointed sphere, we first complete the surface by opening it
up again and
adding back the second hemisphere. We can describe points on the pointed
sphere by $(\phi,\sigma)$ (where $\sigma$ is the distance from
the north point and $\phi$ is the azimuthal angle, as
for the general case described in Sec. \ref{sec:zucchini}).
Points on the completed sphere are naturally labeled by
the standard spherical
coordinates $(\alpha,\theta)$ that represent
the azimuthal and polar angles respectively. The mapping from the
pointed sphere that returns each point to its position
on the original sphere is
given by
\begin{eqnarray}
\alpha=\frac{\phi}{2}\nonumber\\
\theta=\sigma.
\label{eq:half-rations}
\end{eqnarray}
As $\phi$
ranges from $0$ to $2\pi$, $\alpha$ goes from $0$ to $\pi$, across
the hemisphere used to make the pointed sphere.
Note that $\theta=\sigma$ since
the pointed sphere is formed by isometrically bending the hemisphere
(the angle $\theta$ is equal to the geodesic distance to the north pole
of the hemisphere).
Now for each vortex at $\mathbf{u}=(\phi,\sigma)$
on the pointed sphere, we introduce two vortices on the sphere,
one at $Q$ with $(\alpha,\theta)=(\frac{\phi}{2},\sigma)$ and one
at
$Q^*=(\frac{\phi}{2}+\pi,\sigma)$. The latter is the image vortex
of the former, obtained by rotating $Q$ by $180^{\circ}$ around
the $z$-axis.
The energy of a defect configuration on the pointed sphere is derived by halving
the energy of the flow pattern produced by the doubled set of vortices
on the full sphere. In analogy with the electrostatic problem,
the purpose of situating
the image defects in the way just described
is to preserve the continuity of flows across the seam. Imagine
drawing the flow pattern of all the vortices on the
sphere. Focus on the hemisphere $0<\alpha<\pi$.
Because the vortices are placed
symmetrically about the sphere's axis, the flow near
$\alpha=0$ will match the flow near $\alpha=\pi$ when the
surface is sealed. (See Fig. \ref{fig:noodlemachine}B.)
The flow pattern on the pointed sphere results from rolling up
half of the flow pattern on the sphere.
Once the positions of the image defects are chosen, the flow pattern
is found by deriving it from the stream function $\chi(\bm{u})$
introduced in Sec. \ref{subsec:effective}. The stream function
at a point $\bm{u}$ on the pointed sphere can
be expressed in terms of the Green's function of the sphere, according
to Eq. (\ref{eq:green-stream1}):
\begin{equation}
\chi(\mathbf{u})
=\sum_i{\frac{n_ih}{m}[\Gamma^{sphere}(Q,Q_i)+\Gamma^{sphere}(Q,Q_i^*)]}
\label{eq:darkandlight}
\end{equation}
where $Q$ is the point on the sphere corresponding to $\mathbf{u}$
on the pointed sphere and
$Q_i$ and $Q_i^*$ are the locations of the $i^{\mathrm{th}}$ pair of vortices
on the sphere.
The energy of the vortices on the sphere
takes up the familiar electrostatic form of a
sum of the interactions between all pairs of
defects and/or their images. The energy stored in the
flow pattern on the pointed sphere (which is half as large as
on the complete sphere) reads
\begin{align}
\frac{E_N}{K}&=\frac{1}{4}\sum_{i\neq j} 4\pi^2n_in_j[\Gamma^{sphere}(Q_i,Q_j)
+\Gamma^{sphere}(Q_i,Q_j^*)\nonumber\\
&\ \ \ \ \ \ \ \ \ +\Gamma^{sphere}(Q_i^*,Q_j)
+\Gamma^{sphere}(Q_i^*,Q_j^*)]\nonumber\\
&\ \ \ +\frac{1}{2}\sum 4\pi^2n_i^2\Gamma^{sphere}(Q_i,Q_i^*)\nonumber\\
&=\frac{1}{2}\sum_{i\neq j} 4\pi^2 n_in_j[\Gamma^{sphere}(Q_i,Q_j)
+\Gamma^{sphere}(Q_i,Q_j^*)]\nonumber\\
&\ \ \ +\sum 2\pi^2n_i^2\Gamma^{sphere}(Q_i,Q_i^*)
\label{eq:shadows}
\end{align}
In the second expression, we note that the terms in the first line
are equal in pairs, so that a factor of $\frac{1}{2}$ cancels.
This energy is given the same form as Eq. (\ref{eq:redouble})
by separating out
the part which depends on the positions
of \emph{two} vortices, proportional to
\begin{equation}
\Gamma_s(\mathbf{u}_1,\mathbf{u}_2)
=\Gamma^{sphere}(Q_1,Q_2)+\Gamma^{sphere}(Q_1,Q_2^*)
\label{eq:standnoodleg}
\end{equation}
and the part which depends on one vortex,
\begin{equation}
U_s(\mathbf{u})=-2\pi\Gamma^{sphere}(Q,Q^*).
\label{eq:standnoodlev}
\end{equation}
The function in Eq. (\ref{eq:standnoodleg}) is
a Green's function for the pointed sphere, as
the placement of the images guarantees that this function
is well-defined on the \emph{pointed sphere}. It appears
in the stream function, Eq. (\ref{eq:darkandlight}), as well as in
the energetics, as expected for a Green's function.
The potential which describes the single-particle
energy of a vortex
becomes singular as the
vortex approaches the apex of the cone at the north or south pole,
since then the vortex $Q$ approaches its image $Q^*$.
This is in accord with the result, Eq. (\ref{eq:greenself-energy}),
that the Gaussian curvature is the source of the single-particle energy since
the pointed sphere has delta-function concentrations of curvature
at its north and south poles:
\begin{equation}
G(\mathbf{u})=1+\pi\delta_N(\mathbf{u})+\pi\delta_S(\mathbf{u}).
\label{eq:ouch}
\end{equation}
where $\delta_N(\mathbf{u})$ and $\delta_S(\mathbf{u})$ are the appropriate
delta funnctions.
The geometric repulsion from the positive curvature
points arises from the repulsion between vortices
and their images!
We can check step-by-step that $U_s$ is sourced
by the Gaussian curvature,
\begin{equation}
U_s(\mathbf{u})=-\iint \Gamma_s(\mathbf{u},\mathbf{u'}) G(\mathbf{u'}) d^2\mathbf{u'}.
\label{eq:selfconsistent}
\end{equation}
We substitute for $G(\mathbf{u'})$ from Eq. (\ref{eq:ouch}) and
for $\Gamma_s$ from Eq. (\ref{eq:standnoodleg}) which can be written
in the form,
\begin{align}
&\Gamma_{s}(\sigma_1,\phi_1;\sigma_2,\phi_2)=
\Gamma^{sphere}(\sigma_1,\frac{\phi_1}{2};\sigma_2,\frac{\phi_2}{2})\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\Gamma^{sphere}
(\sigma_1,\frac{\phi_1}{2};\sigma_2,\frac{\phi_2}{2}+\pi)\nonumber\\
&=-\frac{1}{4\pi}\ln
4[(1-\cos\sigma_1\cos\sigma_2)^2
\nonumber\\
&\ \ \ \ \ \ \ \ \ \ -\sin^2\sigma_1\sin^2\sigma_2\cos^2\frac{\phi_1-\phi_2}{2}].
\label{eq:orbitrig}
\end{align}
In the last line, we have evaluated
the Green's function for the sphere
by writing the chordal distance between, e.g., $Q_1$
and $Q_2$ in terms of the spherical
coordinates $(\alpha_{1,2},\theta_{1,2})$,
$D^2=2[1-\cos\theta_1\cos\theta_2-
\sin\theta_1\sin\theta_2\cos(\alpha_1-\alpha_2)]$ (see \cite{lube92}),
and then combining the two terms together.
To evaluate the integral in Eq. (\ref{eq:selfconsistent}), we have
to note that
the area element of this integral is $d^2\mathbf{u}=\frac{1}{2}
\sin\sigma d\sigma d\phi$. The area of a region
on the pointed sphere is the \emph{same} as the area
$\sin\theta d\theta d\alpha$ of the
corresponding region
on the original sphere, and the factor of $\frac{1}{2}$ results from
how the angles are related, $\alpha=\frac{\phi}{2}$, see
Eq. (\ref{eq:half-rations}).
Now the integral on the right-hand side of Eq. (\ref{eq:selfconsistent}) can
be shown to be equal to the left-hand side using the identities
\begin{eqnarray}
\int_0^{2\pi}\ln|A+B\cos t|dt
&=&2\pi\ln\frac{A+\sqrt{A^2-B^2}}{2}\ \mathrm{if}\ B<A\nonumber\\
&=&2\pi\ln\frac{B}{2}\ \mathrm{if}\ B>A\nonumber.
\end{eqnarray}
We have now derived one formulation of the energetics in terms of $\Gamma_s$
and $U_s$, the corresponding geometric potential.
Let us contrast this isometric mapping method with
the conformal mapping method in order to illustrate how different
approaches can naturally lead to different delineations
between vortex-vortex and
vortex-curvature interactions. (The net result is of course the
same from either point of view.)
As a result of the isometric mapping
each point is doubled, whereas the distance-distorting conformal mapping
transforms each point on the
pointed sphere to \emph{one} point on the reference sphere.
We first use Eq. (\ref{eq:deimosmap}) to find that
the conformal map is given by
\begin{equation}
\tan\frac{\Theta}{2}=\tan^2\frac{\sigma}{2}.
\end{equation}
Comparing the conformal mapping results, Eqs. (\ref{eq:conformalself-energy})
and
(\ref{eq:conformalpair-energy}) to the Green's function formulation,
Eqs. (\ref{eq:greenself-energy}) and (\ref{eq:greenpair-energy})
suggests the following identification of
the interaction potential (or Green's function)
and single-particle potential:
\begin{eqnarray}
&&\Gamma_c(\mathbf{u}_1,\mathbf{u}_2)
=\Gamma^{sphere}(\Theta(\sigma_1),\phi_1;\Theta(\sigma_2),\phi_2)\nonumber\\
&&U_c(\mathbf{u})=\omega=\ln\frac{2\sin\sigma}{1+\cos^2\sigma}.
\label{eq:confnoodle}
\end{eqnarray}
These expressions differ from Equations (\ref{eq:standnoodleg}) and
(\ref{eq:standnoodlev}).
Nevertheless, as promised,
the net energy is the same whether the pairs $(\Gamma_s,U_s)$
or $(\Gamma_c,U_c)$ are used in place of $\Gamma$ and $U_G$.
In fact,
\begin{eqnarray}
&&\Gamma_c(\mathbf{u}_1,\mathbf{u}_2)=\Gamma_s(\mathbf{u}_1,\mathbf{u}_2)
-f(\mathbf{u}_1)-f(\mathbf{u}_2)\nonumber\\
&&U_c(\mathbf{u})=U_s(\mathbf{u})+4\pi f(\mathbf{u})
\label{eq:cesnoodle}
\end{eqnarray}
where $f(\mathbf{u})=-\frac{1}{4\pi}\ln(1+\cos^2\sigma)$. This transforms
the energy from the single-particle to the interaction terms consistently
as described at the beginning of the section. Appendix \ref{app:CG}
shows that the Green's function formulation is generally
equivalent to the conformal mapping result derived in Section \ref{sec:map},
even when there is no method of images
that can be used to determine the Green's function explicitly in general.
\section{\label{sec:geomineq}
Limits on the Strength and Range of Geometrical Forces}
Geometrical forces are limited in strength due to the nonlinear relation
between the curvature and the geometric potential. Curvature
affects both the \emph{source} of the geometrical force and the \emph{force law},
as illustrated in the examples of Secs. \ref{sec:bumps} and \ref{sec:zucchini}.
As a consequence,
even on a wildly distorted surface (with planar topology), there is
a precise
limit on the strength of the force on
a single vortex. This result has the character of a geometrical
optimization problem, like maximizing the capacitance of a solid
when the surface area is given.
Consider a vortex located at the center
of a geodesic disk of radius $R$. Assume that the
Gaussian curvature is zero within the disk, but may be different
from zero elsewhere.
Then the force $\mathbf{F}$ due to
the curvature satisfies
\begin{equation}
|\mathbf{F}|\leq\frac{4\pi Kn_1^2}{R}.
\label{eq:limit1}
\end{equation}
where $n_1$ is the number of circulation quanta in the vortex. This relation
between $R$ and $\mathbf{F}$ is proven in Appendix \ref{app:geomineq}.
If
one warps a surface in a vain attempt to overcome the limit, but
the force gets diluted
because the distortion of the region around the curvature pulls the
force-lines apart,
as we can understand
from the simple example of vortices on
cones.
A cone of \emph{cone-angle} $\theta$
is obtained by taking a segment of paper with an angle $\theta$ and gluing
the opposite edges of the angle together. This is most familiar when
$\theta<2\pi$.
If $\theta=2\pi m+\beta$, such a cone
can be produced by adding $m$ extra sheets of paper,
as illustrated in Fig. \ref{fig:firstcontortion}.
We slit the $m$ sheets of
paper and put them together with an angle of size
$\beta$ cut out of an additional sheet. By gluing the edges of the
slits
together cyclically, a cone of arbitrary angle $\theta$ is made.
\begin{figure}
\psfrag{1}{$1$}
\psfrag{2}{$2$}
\psfrag{beta}{$\beta$}
\includegraphics[width=.47\textwidth]{torment3}
\caption{\label{fig:firstcontortion}How to form cones of
negative curvature. One complete sheet of paper is slit and
an angle is cut out of an additional sheet of paper.
The edges labeled 1 are taped together, and then the
edges labeled 2 are taped. The circular arcs join together
to form an extra-large circle. The cone angle is $\theta=2\pi+\beta$,
and cones with even larger cone angles can be formed by using additional
sheets.}
\end{figure}
A cone has a delta function of curvature at its apex, but
no Gaussian curvature elsewhere
because the surface can be formed from
a flat piece of paper without stretching.
The weight $2 \pi - \theta$ of the delta function
is expressed, according to the Gauss-Bonnet theorem, as an integral of the Gaussian curvature in any region
containing the apex\cite{Kami02}
\begin{equation}
\iint G(\mathbf{u})d^2\mathbf{u}=2\pi-\oint \kappa ds
\label{eq:paralleltrans}
\end{equation}
where $\kappa$ is
the geodesic curvature along the boundary of the region and $s$ its arc length.
Apply this formula to the circle of radius $D$
centered at the apex of the cone. Imagine the circle as it would appear on the original sheets
of paper, as in Fig. \ref{fig:firstcontortion}. Its measure
in radians is $\beta+2\pi m=\theta$ since it consists of $m$
complete circles together with an additional arc. The length is therefore $S=D\theta$.
The \emph{geodesic} curvature of the circle does not change when the
cone is unfolded, so it is equal to $\frac{1}{D}$.
Upon substituting in Eq. (\ref{eq:paralleltrans}), we obtain
\begin{equation}
\iint G(\mathbf{u})d^2\mathbf{u}=2\pi-S\frac{1}{D}=2\pi-\theta.
\label{eq:conepoint}\end{equation}
When $\theta>2\pi$ the curvature is negative.
Now imagine a vortex (with $n_1=\pm 1$, say) at a distance $D$ from
the cone point, on the circle of circumference $S$ just considered.
The arbitrarily large negative curvature which is possible by making $m$
large seems
to defy the general upper bound on the geometric force. According
to Newton's theorem, applied to the radius $D$ circle centered
at the cone's apex and passing through the vortex,
the force on the vortex is $F=\frac{\pi K\iint G d^2\mathbf{u}}{S}$. Since
the circumference $S=D\theta$ is larger than it would be in the plane, the force
is diluted; substituting the integrated curvature from Eq. (\ref{eq:conepoint}),
we find that it is given by
\begin{equation}
F=\pi \frac{K}{D} \frac{2\pi-\theta}{\theta}.
\label{eq:cone}
\end{equation}
This satisfies Eq. (\ref{eq:limit1}) for all negatively curved cones ($\theta>2\pi$); even
when $\theta\rightarrow\infty$
the magnitude of the force is less than $4\pi\frac{K}{D}$
because the large circumference in the denominator of the Newton's theorem expression cancels
the large integrated curvature in the numerator.
In the opposite limit $\theta\rightarrow 0$, the theorem described by Eq. (\ref{eq:limit1})
is still correct of course. One has to be careful about applying it, however. The force on
a vortex at radius $D$ (given by Eq. (\ref{eq:cone})) is \emph{not} bounded
by $\frac{4\pi K}{R}$ with $R$ set equal to $D$ when $\theta$ is small enough (in fact, for an
extremely pointed cone, $\theta\ll 1$, the force given by Eq. (\ref{eq:cone}) diverges),
but this does
not contradict the inequality because the circle of radius $D$ centered at the vortex
is pathological: although it does not contain any curvature, the circle wraps around the cone and intersects itself. Taking $R$ to be the radius of the largest circle centered at the
defect which does not intersect itself, one finds that the inequality \emph{is} satisfied,
with room to spare, for all values of the cone angle $\theta$ (see Appendix \ref{app:geomineq}).
One can describe
a more awkwardly shaped surface such that the force on a singly-quantized
vortex is arbitrarily close to the upper
bound $\frac{4\pi K}{R}$ (see Appendix \ref{app:geomineq}).
One can also provide limits to the strength of the geometric
force from a localized source of curvature.
Rotationally symmetric surfaces such as the Gaussian bump
have force fields that do not extend beyond the bump, since
the net Gaussian curvature is zero, and Newton's theorem says that only
the $integrated$ Gaussian
curvature can have a long range effect for a rotationally
symmetric surface. To
get a longer-range force, one must focus on non-symmetric surfaces, like
the saddle surface of Sec. \ref{BS}.
The integration methods of Appendix \ref{app:calmseas} can
be used to show that this surface's potential has a quadrupole form at
long distance. Let us consider, more generally, a plane which is flat
except for a non-rotationally symmetric deformation
confined within radius $R$ of the origin. (The result
will not apply directly to the saddle
surface since its curvature extends
out to infinity.) In this case, the total integrated Gaussian curvature is zero,
implying that the long-range force law cannot have any monopole component.
A dipole component is not ruled out by this simple reasoning,
but Appendix \ref{app:geomineq} shows that the
limiting form of the potential is at least a \emph{quadrupole}
(or a faster
decaying field),
\begin{equation}
E(\mathbf{r})\sim n_1^2\frac{\mu_2\cos(2\phi-\gamma_2)}{r^2},
\label{eq:quadrupole}
\end{equation}
where $r$ and $\phi$ are the polar coordinates of the vortex relative
to the origin, and
$\mu_2$ and $\gamma_2$ are constants that
depend on the shape of the deformation in the vicinity of the origin.
As in the previous case, there is an upper limit
on the quadrupole moment $\mu_2$, no matter how strong the curvature of
the deformation is:
\begin{equation}
\mu_2\leq \pi K R^2.
\label{eq:limitq}
\end{equation}
For electrostatics in the plane, the
maximum quadrupole moment of $N$ particles with charge $2\pi$
and $N$ with charge $-2\pi$
in a region of radius $R$ is at most of the order of $KNR^2$,
which has the same form as the bound in Eq. (\ref{eq:limitq}), except
for the factor of $N$.
Because of the
nonlinearity of the
geometrical force and
restrictions on how much positive and negative curvature can be separated
from each other, the quadrupole moment is bounded no matter how drastically
curved the surface is.
These results describe key physical \emph{differences} (resulting from the
fact that the curvature cannot be adjusted without changing the surface)
between the geometrical forces discussed in this work and their electrostatic counterparts despite the close resemblance from a formal viewpoint.
\section{Conclusion}
\begin{comment}
\begin{table*}
\begin{tabular}{l|c|c|c}
&Deformed Plane&Deformed Disk&Deformed Sphere\\ \hline
Interaction &
$K n_i n_
[4\pi^2\Gamma(\mathbf{u}_i,\mathbf{u}_j)$& $4\pi^2 K n_in_j\Gamma_D(\mathbf{u}_i,\mathbf{u}_j)
& $4\pi^2 K n_in_j\Gamma(\mathbf{u}_i,\mathbf{u}_j)$\\
$E_2(\mathbf{u}_i,\mathbf{u}_j)$ &$-2\pi \ln\frac{R}{a}]$
&&\\
\hline
Green's &
$-\frac{1}{2\pi}\ln\frac{|\bm{\mathcal{U}}_i-\bm{\mathcal{U}}_j|}{a}$
&$\frac{1}{4 \pi} \ln\left(\frac{1+
\mathcal{R}_{i}^{2} \mathcal{R}_{j}^{2} -
2 \mathcal{R}_{i} \mathcal{R}_{j} \cos \left(\Phi_{i}-
\Phi_{j}\right) }{\mathcal{R}_{i}^{2}+\mathcal{R}_{j}^{2}-2\mathcal{R}_{i}
\mathcal{R}_{j} \cos \left(
\Phi_{i}-
\Phi_{j}\right) }\right) $
& $-\frac{1}{2\pi}\ln\frac{\mathcal{D}_{ij}}{a}+\frac{1}{4\pi}[\omega(\mathbf{u}_i)+\omega(\mathbf{u}_j)]$ \\
Function &&&\\ \hline
Geom.-Potl. & $K\pi[\iint d^2\mathbf{u}\Gamma(\mathbf{u}_i,\mathbf{u})G(\mathbf{u})$& $\pi K\ln \frac{1-\mathcal{R}_i^2}{a}
-\pi K\omega(\mathbf{u})$& 0 \\
$E_1(\mathbf{u}_i)$&
$+\ln\frac{R}{a}]$
&& \\ \hline
Diff'l Eq.
&
$\nabla ^2 E_1(\mathbf{u}_i)$& $\nabla^2 E_1(\mathbf{u}_i)$\\
for $E_1(\mathbf{u}_i)$ &$=-\pi K G(\mathbf{u}_i) $
&
$=-\pi K G(\mathbf{u}_i)-\frac{4\pi K}{a^2}e^{-\frac{2E_1(\mathbf{u}_i)}{\pi K}}$
&\\
\hline
Applications& Gaussian bumps and saddles: &Interactions with images:
& Band-Flows: \\
&Secs. \ref{subsec:anomalous}-\ref{earnshaw}; &
Sec. \ref{subsec:multiple};
&Sec. \ref{sec:zucchini}; \\
&Periodic surfaces:&Bounded Enneper surface &Multiple Green's functions:\\
&Sec. \ref{sec:bumps};&Sec. \ref{sec:bubble}&App. \ref{app:CG}\\
&Nearly flat geometries:&&\\
&App. \ref{app:calmseas}&&\\
\hline
\end{tabular}
\caption{\label{table:chair}The conformal map of a deformed plane to a flat
plane is chosen
so that it approaches the identity at infinity. The conformal
map of the deformed disk into the flat plane
is chosen so that the boundary maps to a \emph{unit} circle.
$\mathcal{D}_{ij}$ is the chordal distance between the images of
the $i^{\mathrm{th}}$ and $j^{\mathrm{th}}$ defects.}
\centering\end{table*}
\end{comment}
\begin{table*}
\begin{tabular}{c}
\includegraphics[width=\textwidth]{chair}
\end{tabular}
\caption{\label{table:chair}An outline of vortex interactions on curved surfaces.
The net energy of a set of vortices on a surface with the topology of
a plane, disk, or sphere is given by $\sum_i n_i^2E_1(\mathbf{u}_i)+\sum_{i<j}E_2(\mathbf{u}_i,\mathbf{u}_j)$, where simple expressions for
the single-particle (or geometric) potential
and two-particle potentials are given in the table. A conformal mapping
is necessary for evaluating some of these expressions. For example,
$\bm{\mathcal{U}}_i$ (in the expression for the Green's function on a deformed
plane)
is the Cartesian coordinates of the conformal image of vortex $i$.
}
\end{table*}
In this article, we have laid out a mathematical formalism based on
the method of conformal mapping that allows one to calculate the energetics
of topological defects on arbitrary deformed substrates with a focus
on applications to superfluid helium films. The starting point of
our approach is the
observation that upon a change of coordinate the metric tensor of a complicated
surface can be brought in the diagonal form $g_{ab}=e^{2 \omega(\mathbf{u})}
\delta_{ab}$. This corresponds to the metric of a \emph{flat}
plane which is locally
stretched or compressed by the conformal factor $e^{2 \omega(\mathbf{u})}$.
Many of the geometric interactions experienced by topological defects on curved
surfaces are simply determined once the function $\omega(\mathbf{u})$ is known.
Vortices in thin helium layers wetting a curved surface are a natural arena to
explore this interplay between geometry and physics but our
approach is of broader applicability.
The curved geometry results in a modified law for defect interaction as well as
in a
one body geometric potential. On a deformed plane, the latter is obtained by
solving a covariant Poisson equation with the Gaussian curvature as a source.
Table \ref{table:chair} presents a summary of the general form that the defect
interaction (first row) and the
geometric potential (third row) take up in curved spaces with the topology of a
deformed plane (first column), disk (second column) and
sphere (third column). These results can be
derived starting from the differential
equations that the geometric potential
satisfies or the appropriate Green's functions that we list in the second and
fourth row respectively for each of the three
surface topologies. The fifth row of Table \ref{table:chair} directs the
reader towards the relevant sections and appendices
of the paper where he will be able to find
some concrete applications
of the formalism and technical derivations.
For example, the geometric potential of an Enneper disk (a minimal surface with
negative curvature described in Sec. \ref{sec:bubble})
is given by the conformal factor
$\omega({\bf u})$ evaluated at the point $P=\{u_1,u_2\}$ where the vortex is
located combined with an ``electrostatic-like" interaction
with an image defect located at the inverse of $P$ with respect to the circular
boundary. The geometric potential
satisfies the Liouville (non-linear differential) equation that reduces to the
Poisson equation derived for the plane in the limit of
an infinitely large disk. In the case of deformed spheres, we showed in
Appendix \ref{app:CG}
that one can make a convenient choice of
Green's function so that \emph{all} the geometric effects are included in the
defect-defect interactions without introducing
a one-body geometric potential explicitly. An interesting application
naturally arises on vesicles deformed into an elongated shape, like
a zucchini.
The range of
the defect interaction becomes much longer and its functional form different
from the logarithmic dependence expected in flat two dimensional spaces.
We hope that the discussion of the geometric effects presented in this work may
pave the way for their observation in thin superfluid or liquid crystal layers
on a curved substrate. A useful starting point could be the design of
experiments to detect the geometric potential by balancing it with forces
exerted on the defects by external fields or rotation of the sample as
discussed in Section \ref{sec:Rotation}. Such experiments should focus
on single vortices, or on situations where the separation between vortices
is comparable to the length scale of the geometry.
Signatures of the geometric interactions described
here may also survive in defect pinning experiments carried out in some bounded
three dimensional geometries \cite{zieve}.
\begin{comment}
An infinite plane is not really infinite; it is just a surface where
the boundary is very far away. A distant boundary adds only
constants to the energy, and these are given in the table
for the case where the boundary
is a radius $R$ circle.
Conformal mapping is the only way to find the energy of
defects on a deformed disk.
The conformal factor $\omega(\mathbf{u})$ is not
given by the integral of the curvature times the Green's function,
since $\omega$ does not satisfy Dirichlet boundary conditions.
Besides this, the interaction of a defect with its image
has to be included, and images have to be defined in the
reference plane. Interestingly, the \emph{total}
single-particle
energy satisfies a \emph{nonlinear} differential equation (the Liouville
equation), which is given in the table.
We have used the Green's function which eliminates
self-energies on the sphere (see Appendix \ref{app:CG} for other formulations).
\end{comment}
\section{Acknowledgments}
We thank B. Halperin, R. D. Kamien, S. Trugman and
R. Zieve for helpful suggestions.
AMT, VV and DRN acknowledge financial support from the National Science Foundation,
through Grant DMR-0654191, and through
the Harvard Materials Research Science and Engineering Center through
Grant DMR-0213805. VV acknowledges financial support from NSF Grant DMR05-47230. It is a pleasure
to acknowledge the Aspen Center for Physics for providing an interactive research environment where this article was completed. |
0807.0761 | \section{Introduction}
An optical lattice is produced by pairs of counter propagating laser beams, which introduce standing waves of lattice constant of half wave length, $a=\lambda/2$ \cite{Zoller}. The laser beams have a given wave length, intensity, and polarization, and where they are off resonance to the atomic internal transitions. A boson gas of ultracold atoms loaded into the optical lattice can be described by Bose-Hubbard model, and a quantum phase transition from the superfluid into the Mott insulator phase occurs by changing the laser intensity \cite{Jaksch,Bloch}. Here we consider the Mott insulator phase with one atom per lattice site, and we concentrate only in an resonant excitation between two internal atomic levels. The atoms experience optical lattice potentials corresponding to the polarizability of each internal atomic state. The optical lattice potentials are in general different, but here we assume ground and excited state optical lattice potentials with minima at the same positions, which can be obtained for suitable choice of laser wave length \cite{Katori,Barber}. For a deep lattice, around the minimum of the optical lattice at each site the potential can be approximated by a harmonic potential with discrete levels \cite{Zoller}. Throughout the paper we assume the atoms to be localized at the ground state of these harmonic states, and we neglect excitations to higher levels. Note that already in previous works we studied collective electronic excitations (excitons) in such a system, and within a cavity we introduced polaritons \cite{ZoubiA,ZoubiB,ZoubiC,ZoubiD}.
The internal atomic transition between the ground state $|g\rangle$, of energy $\hbar\omega_g$, and the excited state $|e\rangle$, of energy $\hbar\omega_e$, has a transition energy of $\hbar\omega_A=\hbar\omega_e-\hbar\omega_g$. The expectation value of the transition dipole operator $\bar{\mu}$ is $\vec{\mu}=\langle e|\bar{\mu}|g\rangle$. The transition dipoles have a given orientation, with a direction fixed mainly in applying external static fields, e.g. electric or magnetic fields, and by the polarization of the optical lattice laser beams. The fact that the transition dipole has a fixed direction makes the system strongly anisotropic. A system of an optical lattice in the Mott insulator phase with one atom per site and a fixed transition dipole direction at each site, is similar to an artificial anisotropic crystal of Uniaxial type \cite{Born}. Identical considerations hold for molecular optical lattice of one molecule per site, e.g., for diatomic molecules the excited state has an oriented transition dipole, where due to the optical lattice polarization and in applying external static fields all the dipoles of different sites are organized in the same direction \cite{Rempe}.
In the present paper we study optical lattices with an anisotropy defined in the following, and through investigating optical spectra we achieve different physical properties of the system. To achieve our goals in an easily controllable system, the 2D anisotropic optical lattice is taken to be localized between two planar cavity mirrors \cite{ZoubiA}. For each cavity mode exists two possible orthogonal degenerate polarizations, TE and TM polarizations, of transverse electric and transverse magnetic fields, respectively \cite{Haroche}. We consider only a single perpendicular cavity mode, the one which is close to resonance to the above internal atomic transition. As the atomic transition dipole has a fixed direction, which is in general different from the direction of the cavity electric field, the coupling of the internal atomic excitations and the photons is a function of the angle between the transition dipoles and the photon polarizations. In the strong coupling regime the electronic excitations and the cavity photons of both polarizations are coherently mixed to form the new system diagonal eigenmodes, which are called cavity polaritons \cite{ZoubiE,Litinskaya}. Such photon polarization mixing can be observed via linear optical spectra \cite{ZoubiA,ZoubiE}. For an incident field which is, e.g., TE polarized, we expect, due to the anisotropic optical lattice, to observe transmitted and reflected fields of TE and TM polarizations. Photon polarization mixing can be also observed via the phase shift of the transmitted and reflected fields relative to an incident filed of a fixed polarization.
The paper is organized as follows. In section 1 we present anisotropic optical lattice within a cavity. The polarization mixing in the formation of cavity polaritons appears in section 2. Linear optical spectra is derived in section 3 with the optical light shift, for a given incident field. A summary is given in section 4.
\section{An anisotropic optical lattice within a cavity}
We consider a simple model of ultracold two-level atoms in an optical lattice in the Mott insulator phase with one atom per site. The atoms are prepared in a state, where the transition dipole moment matrix element has a fixed direction at each site, e.g., as in the case of figure (1). In fact we have multilevel atoms, but the previous case can be achieved in preparing all the ultracold atoms in a state of a fixed angular momentum, and to load them on an optical lattice with a fixed polarization accompanied with external static fields.
The atomic excitation Hamiltonian is given by
\begin{equation}
H_A=\sum_i\hbar\omega_A\ B_i^{\dagger}B_i,
\end{equation}
where $B_i^{\dagger}$ and $B_i$ are the creation and annihilation operators of electronic excitation at lattice site $i$, respectively. Here we neglect electrostatic interactions between atoms at different sites, as for small wave vectors they result only in an energy shift which can be absorbed in $\hbar\omega_A$, (their role in the formation of excitons was discussed widely by us in \cite{ZoubiA,ZoubiB}). In using the lattice symmetry, we can formally transform the excitation Hamiltonian into the momentum space in applying the transformation
\begin{equation}
B_i=\frac{1}{\sqrt{N}}\sum_{\bf q}e^{i{\bf q}\cdot{\bf n}_i}B_{\bf q},
\end{equation}
where $N=M\times M$ is the number of sites in the optical lattice, ${\bf n}_i$ is the position of site $i$, and ${\bf q}$ is the in-plane wave vector, which takes the values ${\bf q}=(q_x,q_y)=\frac{2\pi}{\sqrt{S}}(n_x,n_y)$, where $n_x,n_y=0,\pm 1,\pm 2,\cdots,\pm \frac{M}{2}$, with $S=Na^2$. The Hamiltonian is now written as
\begin{equation}
H_A=\sum_{\bf q}\hbar\omega_A\ B_{\bf q}^{\dagger}B_{\bf q},
\end{equation}
\begin{figure}
\centerline{\epsfxsize=6.0cm \epsfbox{Fig1.eps}}
\caption{The optical lattice plane with oriented transition dipoles.}
\end{figure}
The 2D optical lattice is taken to be localized in the middle and parallel to planner cavity mirrors, as presented in figure (2). First we consider the case of perfect cavity mirrors, (later in order to get the linear optical spectra we consider the case of non-perfect mirrors). Here the electromagnetic field is free in the cavity plane with in-plane wave-vector ${\bf k}$, and is quantized in the perpendicular, $z$, direction with wave numbers $k_z=\frac{m\pi}{L}$, where $L$ is the distance between the cavity mirrors, and $m$ takes integer numbers, ($m=0,1,2,3,\cdots$). For each cavity mode $({\bf k}, m)$ we have two possible polarizations TE and TM, which are denoted by $(s)$ and $(p)$, respectively. The two cavity photon polarizations, $(\nu=s,p)$, are degenerate, where $\omega_{{\bf k}ms}=\omega_{{\bf k}mp}=\omega_{{\bf k}m}$, with the cavity photon dispersion
\begin{equation}
\omega_{{\bf k}m}=c\ \sqrt{k^2+\left(\frac{m\pi}{L}\right)^2},
\end{equation}
where $k=|{\bf k}|$. The cavity photon Hamiltonian is
\begin{equation}
H_C=\sum_{{\bf k},m,\nu}\hbar\omega_{{\bf k}m}\ a_{{\bf k}m\nu}^{\dagger}a_{{\bf k}m\nu},
\end{equation}
where $a_{{\bf k}m\nu}^{\dagger}$ and $a_{{\bf k}m\nu}$ are the creation and annihilation operators of a cavity photon in the mode $({{\bf k},m,\nu})$, respectively.
The electric field operator is defined by
\begin{eqnarray}
{\bf E}({\bf r},z)&=&-i\sum_{{\bf k},m,\nu}\sqrt{\frac{\hbar\omega_{{\bf k}m}}{LS\epsilon_0}}\ \left\{{\bf u}^m_{\nu}({\bf k},z)e^{i{\bf k}\cdot{\bf r}}\ a_{{\bf k}m\nu}\right. \nonumber \\
&-&\left.{\bf u}^{m\ast}_{\nu}({\bf k},z)e^{-i{\bf k}\cdot{\bf r}}\ a_{{\bf k}m\nu}^{\dagger}\right\},
\end{eqnarray}
where $S$ is the cavity mirror quantization area, ${\bf r}$ is the in-plane position, and $z$ is the perpendicular position. The cavity mirrors are taken to be localized at the positions $z=\pm L/2$, see figure (2). The field vector functions are defined by \cite{Haroche}
\begin{equation}
{\bf u}^m_{s}({\bf k},z)=\sin\left[\frac{m\pi}{L}\left(z+\frac{L}{2}\right)\right]\ \hat{n}_{\bf k},
\end{equation}
where $m=0,1,2,\cdots$, and
\begin{eqnarray}
{\bf u}^m_{p}({\bf k},z)&=&-\frac{cm\pi}{L\omega_{{\bf k}m}}\left\{i\ \sin\left[\frac{m\pi}{L}\left(z+\frac{L}{2}\right)\right]\ \hat{e}_{\bf k}\right. \nonumber \\
&-&\left.\frac{kL}{m\pi}\ \cos\left[\frac{m\pi}{L}\left(z+\frac{L}{2}\right)\right]\ \hat{e}_{z}\right\},
\end{eqnarray}
where $m=1,2,3,\cdots$, and for $m=0$ we multiply ${\bf u}^0_{p}({\bf k},z)$ by the factor $1/\sqrt{2}$. The unit vectors are: $\hat{e}_{z}$ is along the $z$ axis, $\hat{e}_{\bf k}$ is along ${\bf k}$, that is $\hat{e}_{\bf k}={\bf k}/k$, and $\hat{n}_{\bf k}=\hat{e}_{\bf k}\times\hat{e}_{z}$, as illustrated in figure (3).
\begin{figure}
\centerline{\epsfxsize=6.0cm \epsfbox{Fig2.eps}}
\caption{An optical lattice within a cavity.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=4.0cm \epsfbox{Fig3.eps}}
\caption{The transition dipole, and the unit vectors.}
\end{figure}
The optical lattice is located at the middle and parallel to the cavity mirrors at $z=0$. We assume only cavity modes with $m=1$ which are close to resonance to the atomic transition. We neglect all the other perpendicular modes. For multilevel atoms the cavity photons, of TE or TM linear polarizations, are close to resonance to the appropriate electronic transition with a fixed angular momentum. For the case of ground and/or excited states with degenerate multi-levels, where several allowed transitions close to resonance to the cavity photons of TE or TM polarizations, in applying external static fields the degenerate states split, and we can stay with a single electronic transition which is close to resonance to the cavity photons.
The coupling between the atomic transition and the cavity modes in the electric dipole approximation is given by the Hamiltonian $H_{AC}=-\bar{\mu}\cdot {\bf E}$, where the material electric dipole operator is defines as $\bar{\mu}=\sum_i\left(\vec{\mu}\ B_i^{\dagger}+\vec{\mu}^{\ast}\ B_i\right)$, and the electric field operator is evaluated at the atom positions. In the rotating wave approximation, the interaction is given by
\begin{eqnarray}
H_{AC}&=&i\sum_{{\bf k},\nu,i}\sqrt{\frac{\hbar\omega_{\bf k}}{LS\epsilon_0}}\left\{\left(\vec{\mu}\cdot{\bf u}_{\nu}({\bf k})\right)e^{i{\bf k}\cdot{\bf n}_i}\ B_i^{\dagger}a_{{\bf k}\nu}\right. \nonumber \\
&-&\left.\left(\vec{\mu}\cdot{\bf u}_{\nu}({\bf k})\right)^{\ast}e^{-i{\bf k}\cdot{\bf n}_i}\ a_{{\bf k}\nu}^{\dagger}B_i\right\}.
\end{eqnarray}
In transforming the electronic excitation operators into the momentum space we get
\begin{eqnarray}
H_{AC}&=&i\sum_{{\bf k},{\bf q},\nu,i}\sqrt{\frac{\hbar\omega_{\bf k}}{NLS\epsilon_0}}\left\{\left(\vec{\mu}\cdot{\bf u}_{\nu}({\bf k})\right)e^{i({\bf k-q})\cdot{\bf n}_i}\ B_{\bf q}^{\dagger}a_{{\bf k}\nu}\right. \nonumber \\
&-&\left.\left(\vec{\mu}\cdot{\bf u}_{\nu}({\bf k})\right)^{\ast}e^{-i({\bf k-q})\cdot{\bf n}_i}\ a_{{\bf k}\nu}^{\dagger}B_{\bf q}\right\}.
\end{eqnarray}
In using the property $\frac{1}{N}\sum_i e^{i({\bf k-q})\cdot{\bf n}_i}=\delta_{\bf k,q}$, we have
\begin{equation}
H_{AC}=\hbar \sum_{{\bf k},\nu}\left\{f_{\bf k}^{\nu}\ B_{\bf k}^{\dagger}a_{{\bf k}\nu}+f_{\bf k}^{\nu\ast}\ a_{{\bf k}\nu}^{\dagger}B_{\bf k}\right\},
\end{equation}
where the coupling parameter is
\begin{equation}
\hbar f_{\bf k}^{\nu}=i\sqrt{\frac{\hbar\omega_{\bf k}}{La^2\epsilon_0}}\ \left(\vec{\mu}\cdot{\bf u}_{\nu}({\bf k})\right).
\end{equation}
Due to the lattice translational symmetry, the coupling is between cavity photons and excitations with the same in-plane wave vector.
Explicitly, we have
\begin{eqnarray}
\hbar f_{\bf k}^{s}&=&i\sqrt{\frac{\hbar\omega_{\bf k}}{La^2\epsilon_0}}\ \left(\vec{\mu}\cdot\hat{n}_{\bf k}\right), \nonumber \\
\hbar f_{\bf k}^{p}&=&\sqrt{\frac{\hbar\omega_{\bf k}}{La^2\epsilon_0}}\left(\frac{\omega_0}{\omega_{\bf k}}\right)\ \left(\vec{\mu}\cdot\hat{e}_{\bf k}\right),
\end{eqnarray}
where $\omega_0=c\pi/L$. We interest in the case of small wave vectors ${\bf k}\approx {\bf 0}$, hence we neglected the contribution of the $z$ direction, (or we can assume $\mu_z\approx 0$).
We take $\vec{\mu}$ to be real, e.g. for $\vec{\mu}=\mu\hat{x}$, (see figure (3)), and also we have
\begin{equation}
\hat{e}_{\bf k}=\cos\theta\ \hat{x}+\sin\theta\ \hat{y}\ ,\ \hat{n}_{\bf k}=\sin\theta\ \hat{x}-cos\theta\ \hat{y},
\end{equation}
to get
\begin{equation}
\hbar f_{\bf k}^{s}=iC_{\bf k}\ \sin\theta\ ,\ \hbar f_{\bf k}^{p}=C_{\bf k}\left(\frac{\omega_0}{\omega_{\bf k}}\right)\ \cos\theta,
\end{equation}
where $C_{\bf k}=\sqrt{\frac{\hbar\omega_{\bf k}\mu^2}{La^2\epsilon_0}}$. For example, if $\theta=\pi/4$, we get $\hbar f_{\bf k}^{s}=iC_{\bf k}/\sqrt{2}$, and $\hbar f_{\bf k}^{p}=C_{\bf k}\left(\frac{\omega_0}{\omega_{\bf k}\sqrt{2}}\right)$. For $\theta=0$, we get $\hbar f_{\bf k}^{s}=0$, and $\hbar f_{\bf k}^{p}=C_{\bf k}\left(\frac{\omega_0}{\omega_{\bf k}}\right)$. For $\theta=\pi/2$, we get $\hbar f_{\bf k}^{s}=iC_{\bf k}$, and $\hbar f_{\bf k}^{p}=0$.
In the case of cavity photons of standing waves without propagations, we need only to substitute ${\bf k}=0$ in the above results. The cavity modes are of transverse electric and magnetic fields, that is TEM modes, with two orthogonal polarizations, which are denoted here also by $(s)$ and $(p)$. For $(m=1)$ the coupling parameters are $\hbar f_0^{s}=iC_0\sin\theta$ and $\hbar f_0^{p}=C_0\cos\theta$, where $C_0=\sqrt{\frac{\hbar\omega_0\mu^2}{La^2\epsilon_0}}$.
\section{Polarization mixing in the strong coupling regime}
The total Hamiltonian of the coupled electronic excitations and cavity photons is given by
\begin{eqnarray}
H&=&\hbar\sum_{\bf k}\left\{\omega_A\ B_{\bf k}^{\dagger}B_{\bf k}+\sum_{\nu}\omega_{\bf k}\ a_{{\bf k}\nu}^{\dagger}a_{{\bf k}\nu}\right. \nonumber \\
&+&\left.\sum_{\nu}\left(f_{\bf k}^{\nu}\ B_{\bf k}^{\dagger}a_{{\bf k}\nu}+f_{\bf k}^{\nu\ast}\ a_{{\bf k}\nu}^{\dagger}B_{\bf k}\right)\right\}.
\end{eqnarray}
In the strong coupling regime, where the coupling is larger than the atomic excitation and the cavity photon line-width, the real system eigenmodes are cavity polaritons \cite{ZoubiA,ZoubiE}, which are obtained in diagonalizing the above Hamiltonian, to get
\begin{equation}
H=\sum_{{\bf k},r}\hbar\Omega_{{\bf k}r}\ A_{{\bf k}r}^{\dagger}A_{{\bf k}r},
\end{equation}
where we obtain three polariton branches, with the eigenfrequencies
\begin{equation}
\Omega_{{\bf k}\pm}=\frac{\omega_{\bf k}+\omega_A}{2}\pm\Delta_{\bf k}\ ,\ \Omega_{{\bf k}0}=\omega_{\bf k},
\end{equation}
where
\begin{equation}
\Delta_{\bf k}=\sqrt{\delta_{\bf k}^2+|f_{\bf k}|^2}\ , \ |f_{\bf k}|^2=\sum_{\nu}|f_{\bf k}^{\nu}|^2,
\end{equation}
with the excitation-photon detuning
\begin{equation}
\delta_{\bf k}=\frac{\omega_{\bf k}-\omega_A}{2}.
\end{equation}
For the case of atomic transition frequency of $\omega_A/2\pi=2.5\times10^{14}\ Hz$, and for the first cavity mode $m=1$ with a distance between the cavity mirrors of $L=c\pi/\omega_0\approx 3.77\ \mu m$, where we get zero detuning between the atomic transition and the cavity photon at $k=0$, we plot in figure (4) the three polariton frequency dispersions, $\Omega_{{\bf k}r}/2\pi$, as a function of $k$, at the angle $\theta=\pi/4$, and for transition dipole of $\mu=2\ e\AA$, and lattice constant of $a=2000\ \AA=0.2\ \mu m$.
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig4.eps}}
\caption{The three polariton frequency dispersions $\Omega_{r}/2\pi$ vs. $k$, at $\theta=\pi/4$.}
\end{figure}
In the limit of large detuning, where $\delta_{\bf k}\gg|f_{\bf k}|$, we get $\Omega_{{\bf k}+}\approx\omega_{\bf k}+\frac{|f_{\bf k}|^2}{2\delta_{\bf k}}$, $\Omega_{{\bf k}-}\approx\omega_A-\frac{|f_{\bf k}|^2}{2\delta_{\bf k}}$, and $\Omega_{{\bf k}0}=\omega_{\bf k}$. The upper branch is a photon with a shift due to the coupling to the excitation, and the lower is an excitation with a light shift due to the coupling to the cavity photon. Here the middle branch still a photon as before. In this limit we get birefringence, as we have two refracted cavity fields, the ordinary field of the $(0)$ branch, and the extraordinary field of the $(+)$ branch.
The polariton operators are in general a coherent superposition of atomic excitations and cavity photons of both polarizations, namely we have
\begin{equation}
A_{{\bf k}r}=X_{{\bf k}r}\ B_{\bf k}+\sum_{\nu}Y_{{\bf k}r}^{\nu}\ a_{{\bf k}\nu},
\end{equation}
with the relation $|X_{{\bf k}r}|^2+\sum_{\nu}|Y_{{\bf k}r}^{\nu}|^2=1$, where the excitation and photon amplitudes of the upper $(+)$ and the lower $(-)$ polariton branches are
\begin{equation}
X_{{\bf k}\pm}=\pm\sqrt{\frac{\Delta_{\bf k}\mp\delta_{\bf k}}{2\Delta_{\bf k}}}\ ,\ Y_{{\bf k}\pm}^{\nu}=\frac{f_{\bf k}^{\nu}}{\sqrt{2\Delta_{\bf k}(\Delta_{\bf k}\mp\delta_{\bf k})}},
\end{equation}
and the excitation and photon amplitudes of the middle $(0)$ polariton branch are
\begin{equation}
X_{{\bf k}0}=0\ ,\ Y_{{\bf k}0}^{\nu}=\frac{f_{\bf k}^{\nu}}{|f_{\bf k}|}.
\end{equation}
Note that the middle polariton branch is pure photonic. Exist a direction, for each ${\bf k}$, where the polarization direction of a photon, which is a superposition of cavity photons of both polarizations, is orthogonal to the transition dipole, and hence no interaction between this photon and the material is obtained. In other words, as seen in figure (3), the superposition of the two photon polarizations gives two components. The longitudinal component along the transition dipole, which forms in the strong coupling regime with the excitation the two upper and lower polariton branches. The transverse component which is orthogonal to the transition dipole and decouple to the excitation, to form the photonic middle branch.
At the intersection point between the transition frequency and the cavity photon dispersion, where $\delta_{\bf k}=0$, the polaritons are half excitation and half photon, namely $|X_{{\bf k}r}|^2=1/2$ and $\sum_{\nu}|Y_{{\bf k}r}^{\nu}|^2=1/2$. At large wave vectors the upper branch became photonic, with $|X_{{\bf k}r}|^2\approx 0$ and $\sum_{\nu}|Y_{{\bf k}r}^{\nu}|^2\approx 1$, and the lower branch became excitation, with $|X_{{\bf k}r}|^2\approx 1$ and $\sum_{\nu}|Y_{{\bf k}r}^{\nu}|^2\approx 0$.
In figures (5-9) we plot the excitation and photon weights in the three polariton branches, $|X_{{\bf k}r}|^2$ and $|Y_{{\bf k}r}^{\nu}|^2$, as a function of $k$, for the angle $\theta=\pi/4$. It is seen that for small $k$ the upper and lower polariton branches, in figures (5,8), are a coherent mix of the excitation and the cavity photon of both polarizations. The upper branch, for large $k$, becomes photonic, and for larger $k$ it becomes photon of $(s)$ polarization, as seen in figure (5) and more clearly in figures (6-7). While the lower branch becomes excitation for large $k$, as appears in figure (8). The middle branch is pure photonic, as seen in figure (9), and for large $k$ becomes $(s)$ polarized photon. In figures (10-12) we plot the excitation and photon weights as a function of the angle $\theta$ at $k=5\times10^{-7}\ \AA^{-1}$. The plots indicate the significant role of the angle controllability.
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig5.eps}}
\caption{The excitation and photon of both polarization weights in the upper branch, $|X_{{\bf k}+}|^2$, $|Y_{{\bf k}+}^s|^2$, and $|Y_{{\bf k}+}^p|^2$ vs. $k$, at $\theta=\pi/4$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig6.eps}}
\caption{The excitation and photon of both polarization weights in the upper branch, $|X_{{\bf k}+}|^2$, $|Y_{{\bf k}}^s|^2$, and $|Y_{{\bf k}+}^p|^2$ vs. $k$, for small $k$ and at $\theta=\pi/4$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig7.eps}}
\caption{The excitation and photon of both polarization weights in the upper branch, $|X_{{\bf k}+}|^2$, $|Y_{{\bf k}+}^s|^2$, and $|Y_{{\bf k}+}^p|^2$ vs. $k$, for larger $k$ and at $\theta=\pi/4$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig8.eps}}
\caption{The excitation and photon of both polarization weights in the lower branch, $|X_{{\bf k}-}|^2$, $|Y_{{\bf k}-}^s|^2$, and $|Y_{{\bf k}-}^p|^2$ vs. $k$, at $\theta=\pi/4$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig9.eps}}
\caption{The excitation and photon of both polarization weights in the middle branch, $|X_{{\bf k}0}|^2$, $|Y_{{\bf k}0}^s|^2$, and $|Y_{{\bf k}0}^p|^2$ vs. $k$, at $\theta=\pi/4$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig10.eps}}
\caption{The excitation and photon of both polarization weights in the upper branch, $|X_{{\bf k}+}|^2$, $|Y_{{\bf k}+}^s|^2$, and $|Y_{{\bf k}+}^p|^2$ vs. $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig11.eps}}
\caption{The excitation and photon of both polarization weights in the lower branch, $|X_{{\bf k}-}|^2$, $|Y_{{\bf k}-}^s|^2$, and $|Y_{{\bf k}-}^p|^2$ vs. $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig12.eps}}
\caption{The excitation and photon of both polarization weights in the middle branch, $|X_{{\bf k}0}|^2$, $|Y_{{\bf k}0}^s|^2$, and $|Y_{{\bf k}0}^p|^2$ vs. $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$.}
\end{figure}
\section{Anisotropic Linear Optical Spectra}
To observe the system eigenmodes we need to couple the internal system to the external world. Our observation tool is Linear Optical Spectra \cite{ZoubiA,ZoubiE}. In assuming non-perfect mirrors, the internal system get coupled to their environment. For an incident external field with a fixed in-plane wave vector and polarization we calculate the Transmission, Reflection, and Absorption Spectra, as seen in figure (2). Now the electromagnetic field is divided into two parts, the cavity field and the environment field, and they coupled through the non-perfect cavity mirrors. The life time of the excited state is included phenomenologically.
The external electromagnetic field is given by the following Hamiltonians, for fields at the two sides of the cavity,
\begin{equation}
H_U=\sum_{{\bf k},\nu}\int d\omega_{\bf k}\ \hbar\omega_{\bf k}\ b_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})b_{{\bf k}\nu}(\omega_{\bf k}),
\end{equation}
and
\begin{equation}
H_L=\sum_{{\bf k},\nu}\int d\omega_{\bf k}\ \hbar\omega_{\bf k}\ c_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})c_{{\bf k}\nu}(\omega_{\bf k}),
\end{equation}
where $b_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})$ and $b_{{\bf k}\nu}(\omega_{\bf k})$ are the creation and annihilation operators of an external photon in the upper side of the cavity, and $c_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})$ and $c_{{\bf k}\nu}(\omega_{\bf k})$ are the creation and annihilation operators of an external photon in the lower side of the cavity. The external photon energy is $\hbar\omega_{\bf k}$. The coupling of the cavity photons to the external fields is given by
\begin{equation}
V_U=\sum_{{\bf k},\nu}\int d\omega_{\bf k}\ i\hbar u(\omega_{\bf k})\ \left\{b_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})a_{{\bf k}\nu}-a_{{\bf k}\nu}^{\dagger}b_{{\bf k}\nu}(\omega_{\bf k})\right\},
\end{equation}
and
\begin{equation}
V_L=\sum_{{\bf k},\nu}\int d\omega_{\bf k}\ i\hbar v(\omega_{\bf k})\ \left\{c_{{\bf k}\nu}^{\dagger}(\omega_{\bf k})a_{{\bf k}\nu}-a_{{\bf k}\nu}^{\dagger}c_{{\bf k}\nu}(\omega_{\bf k})\right\},
\end{equation}
where $u(\omega_{\bf k})$ is the coupling parameter through the upper mirror, and $v(\omega_{\bf k})$ through the lower mirror. The coupling is between cavity photons and external photons with the same in-plane wave vector and polarization. No polarization mixing or wave vector scattering can be obtained due to the cavity mirrors.
The total Hamiltonian of the coupled internal system and external field is separable for each in-plane wave vector ${\bf k}$, hence we can treat the whole system for each ${\bf k}$ separately. The Hamiltonian for a fixed ${\bf k}$, where we can drop ${\bf k}$ in the following, is given by
\begin{eqnarray}
H&=&\sum_{r}\hbar\Omega_{r}\ A_{r}^{\dagger}A_{r}+\sum_{\nu}\int d\omega\ \hbar\omega\ b_{\nu}^{\dagger}(\omega)b_{\nu}(\omega) \nonumber \\
&+&\sum_{\nu}\int d\omega\ \hbar\omega\ c_{\nu}^{\dagger}(\omega)c_{\nu}(\omega) \nonumber \\
&+&\sum_{r,\nu}\int d\omega\ i\hbar u(\omega)\ \left\{Y_{\nu}^{r\ast}\ b_{\nu}^{\dagger}(\omega)A_{r}-Y_{\nu}^{r}\ A_{r}^{\dagger}b_{\nu}(\omega)\right\} \nonumber \\
&+&\sum_{r,\nu}\int d\omega\ i\hbar v(\omega)\ \left\{Y_{\nu}^{r\ast}\ c_{\nu}^{\dagger}(\omega)A_{r}-Y_{\nu}^{r}\ A_{r}^{\dagger}c_{\nu}(\omega)\right\}, \nonumber \\
\end{eqnarray}
where we used the cavity photon operator in terms of polariton operators, in using the inverse transformation $a_{\nu}=\sum_r Y_{\nu}^{r\ast}\ A_{r}$.
The equations of motion for the external field operators are
\begin{eqnarray}
\frac{d}{dt}b_{\nu}(\omega)&=&-i\omega\ b_{\nu}(\omega)+u(\omega)\ \sum_rY_{\nu}^{r\ast}\ A_r, \nonumber \\
\frac{d}{dt}c_{\nu}(\omega)&=&-i\omega\ c_{\nu}(\omega)+v(\omega)\ \sum_rY_{\nu}^{r\ast}\ A_r,
\end{eqnarray}
and for the polariton operator we get
\begin{eqnarray}
&&\frac{d}{dt}A_r=-i\Omega_{r}\ A_r \nonumber \\
&-&\sum_{\nu}\int d\omega\ Y_{\nu}^{r}\ \left\{u(\omega)\ b_{\nu}(\omega)+v(\omega)\ c_{\nu}(\omega)\right\}.
\end{eqnarray}
Using the input-output formalisms \cite{GardinerA}, the equations for the external field operators are solved formally for the initial and final conditions and substituted back in the polariton equation, and after applying the Markov approximation \cite{GardinerB}, we get the two equations
\begin{eqnarray}
\frac{d}{dt}A_r&=&-i\Omega_{r}\ A_r-\gamma\sum_{\nu}\ Y_{\nu}^{r}\ a_{\nu} \nonumber \\
&+&\sum_{\nu}\ Y_{\nu}^{r}\ \left\{\sqrt{\gamma_U}\ b^{\nu}_{in}+\sqrt{\gamma_L}\ c^{\nu}_{in}\right\},
\end{eqnarray}
and
\begin{eqnarray}
\frac{d}{dt}A_r&=&-i\Omega_{r}\ A_r+\gamma\sum_{\nu}\ Y_{\nu}^{r}\ a_{\nu} \nonumber \\
&-&\sum_{\nu}\ Y_{\nu}^{r}\ \left\{\sqrt{\gamma_U}\ b^{\nu}_{out}+\sqrt{\gamma_L}\ c^{\nu}_{out}\right\},
\end{eqnarray}
where we defined the damping rates at the two mirrors to be constants, and are given by $\gamma_U=2\pi\ u^2(\omega)$, and $\gamma_L=2\pi\ v^2(\omega)$, and we defined also $\gamma=\frac{\gamma_U+\gamma_L}{2}$. Furthermore, we defined the input and output fields at the upper mirror by $b^{\nu}_{in}$ and $b^{\nu}_{out}$, and at the lower mirror by $c^{\nu}_{in}$ and $c^{\nu}_{out}$, respectively.
The boundary condition between the cavity and the external fields at the upper and lower mirrors are given by
\begin{equation}
\sqrt{\gamma_U}\ a_{\nu}=b^{\nu}_{in}+b^{\nu}_{out}\ ,\ \sqrt{\gamma_L}\ a_{\nu}=c^{\nu}_{in}+c^{\nu}_{out}.
\end{equation}
The electronic excitation damping rate is $\Gamma_{ex}$. Phenomenologically, the $r$ polariton branch damping rate is $\Gamma_r=\Gamma_{ex}\ |X^r|^2$, where $|X^r|^2$ is the excitation weight of the $(r)$ polariton. We define the polariton complex frequency by $\bar{\Omega}_r=\Omega_r-i\Gamma_r$.
In applying the Fourier transform, from time $t$ into frequency $\omega$ space, we get the system of equations
\begin{eqnarray}
i(\bar{\Omega}_r-\omega)\ \tilde{A}_r&=&\gamma\sum_{\nu}\ Y_{\nu}^{r}\ \tilde{a}_{\nu} \nonumber \\
&-&\sum_{\nu}\ Y_{\nu}^{r}\ \left\{\sqrt{\gamma_U}\ \tilde{b}^{\nu}_{out}+\sqrt{\gamma_L}\ \tilde{c}^{\nu}_{out}\right\}, \nonumber \\
i(\bar{\Omega}_r-\omega)\ \tilde{A}_r&=&-\gamma\sum_{\nu}\ Y_{\nu}^{r}\ \tilde{a}_{\nu} \nonumber \\
&+&\sum_{\nu}\ Y_{\nu}^{r}\ \left\{\sqrt{\gamma_U}\ \tilde{b}^{\nu}_{in}+\sqrt{\gamma_L}\ \tilde{c}^{\nu}_{in}\right\},
\end{eqnarray}
and
\begin{equation}
\sqrt{\gamma_U}\ \tilde{a}_{\nu}=\tilde{b}^{\nu}_{in}+\tilde{b}^{\nu}_{out}\ ,\ \sqrt{\gamma_L}\ \tilde{a}_{\nu}=\tilde{c}^{\nu}_{in}+\tilde{c}^{\nu}_{out}.
\end{equation}
Two assumptions will be done here, in order to simplify the system of equations. The first is to assume an input external field from only the upper mirror, namely we have $\tilde{c}^{\nu}_{in}=0$. Second, we assume the upper and lower mirrors to be identical, namely we have $\gamma_U=\gamma_L=\gamma$. Hence, in term of cavity and external photon operators, we get the system of equations
\begin{eqnarray}
\tilde{a}_{\alpha}&=&\sum_{\beta}\Lambda_{\alpha\beta}\ \left\{\gamma\ \tilde{a}_{\beta}-\sqrt{\gamma}\ \left(\tilde{b}^{\beta}_{out}-\tilde{c}^{\beta}_{out}\right)\right\}, \nonumber \\
\tilde{a}_{\alpha}&=&\sum_{\beta}\Lambda_{\alpha\beta}\ \left\{-\gamma\ \tilde{a}_{\beta}+\sqrt{\gamma}\ \tilde{b}^{\beta}_{in}\right\},
\end{eqnarray}
and
\begin{equation}
\sqrt{\gamma}\ \tilde{a}_{\alpha}=\tilde{b}^{\alpha}_{in}+\tilde{b}^{\alpha}_{out}\ ,\ \sqrt{\gamma}\ \tilde{a}_{\alpha}=\tilde{c}^{\alpha}_{out},
\end{equation}
where we defined the matrix
\begin{equation}
\Lambda_{\alpha\beta}=i\sum_r\frac{Y_{\alpha}^{r\ast}Y_{\beta}^{r}}{\omega-\bar{\Omega}_r}.
\end{equation}
Here we will solve the above system of equation for the following case.
For incident field of only $(s)$ polarization, where $\tilde{b}^{p}_{in}=0$, the solution is
\begin{eqnarray} \label{SPOL}
\frac{\tilde{c}^{s}_{out}}{\tilde{b}^{s}_{in}}&=&\frac{\gamma\ \Lambda_{ss}(1+\gamma\ \Lambda_{pp})-\gamma^2\ \Lambda_{sp}\Lambda_{ps}}{D}=t^{(s)}_s\ e^{i\Delta\phi^{(s)t}_s}, \nonumber \\
\frac{\tilde{c}^{p}_{out}}{\tilde{b}^{s}_{in}}&=&\frac{\tilde{b}^{p}_{out}}{\tilde{b}^{s}_{in}}=\frac{\gamma\ \Lambda_{ps}}{D}=t^{(s)}_p\ e^{i\Delta\phi^{(s)t}_p}=r^{(s)}_p\ e^{i\Delta\phi^{(s)r}_p}, \nonumber \\
\frac{\tilde{b}^{s}_{out}}{\tilde{b}^{s}_{in}}&=&\frac{-(1+\gamma\ \Lambda_{pp})}{D}=r^{(s)}_s\ e^{i\Delta\phi^{(s)r}_s},
\end{eqnarray}
where
\begin{equation}
D=(1+\gamma\ \Lambda_{ss})(1+\gamma\ \Lambda_{pp})-\gamma^2\ \Lambda_{sp}\Lambda_{ps}.
\end{equation}
The transmission and reflection of $(s)$ polarized fields are
\begin{eqnarray}
T_s^{(s)}&=&\left(t^{(s)}_s\right)^2=\frac{\gamma^2\ |\Lambda_{ss}(1+\gamma\ \Lambda_{pp})-\gamma\ \Lambda_{sp}\Lambda_{ps}|^2}{|D|^2}, \nonumber \\
R_s^{(s)}&=&\left(r^{(s)}_s\right)^2=\frac{|1+\gamma\ \Lambda_{pp}|^2}{|D|^2}.
\end{eqnarray}
The transmission and reflection of $(p)$ polarized fields are
\begin{equation}
T_p^{(s)}=R_p^{(s)}=\left(t^{(s)}_p\right)^2=\left(r^{(s)}_p\right)^2=\frac{\gamma^2\ |\Lambda_{ps}|^2}{|D|^2}.
\end{equation}
Even though the incident field is $(s)$ polarized, due to the anisotropy in the optical lattice we get transmitted and reflected fields which are $(p)$ polarized. Moreover, due to the assumption of identical mirrors, the $(p)$ polarized transmission and reflection fields are equal.
The absorption, $A$, in the cavity medium is calculated from the identity relation
\begin{equation}
T_s^{(s)}+T_p^{(s)}+R_s^{(s)}+R_p^{(s)}+A^{(s)}=1.
\end{equation}
Also from the system of equations (\ref{SPOL}) we deduce the phase shift in the $(s)$ and $(p)$ polarized transmitted fields, $\Delta\phi^{(s)t}_s$ and $\Delta\phi^{(s)t}_p$, and reflected fields, $\Delta\phi^{(s)r}_s$ and $\Delta\phi^{(s)r}_p$, relative to the incident field, respectively.
In figures (13-14) we plot the transmission and reflection spectra of the $(s)$ polarized fields, $T_s^{(s)}$ and $R_s^{(s)}$, as a function of frequency, $\omega\rightarrow\omega/2\pi$, respectively, and for different angles $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$. The three peaks and three dips correspond to the three polariton branches. At $\theta=0$ we get zero transmission and complete reflection. The largest transmission peaks and the deepest reflection dips are obtained at $\theta=\pi/2$. In figure (15) we plot the transmission and reflection spectra of the $(p)$ polarized fields, $T_p^{(s)}$ and $R_p^{(s)}$, which are equal for identical cavity mirrors. Even though the incident field is $(s)$ polarized we get transmission and reflection of $(p)$ polarized fields, with maximum at $\theta=\pi/4$, and zeros at $\theta=0$ and $\theta=\pi/2$. In figure (16) we plot the absorption spectra $A^{(s)}$. Only two peaks are obtained correspond to the upper and lower branches, as the middle branch is pure photonic and no absorption take place, where the absorption is only for the polariton excitation part. Also here zero absorption at $\theta=0$, and maximum absorption at $\theta=\pi/2$, are obtained. Here, for the excitation and photon damping rates we used $\gamma/2\pi= 10^9\ Hz$ and $\Gamma_{ex}/2\pi= 10^8\ Hz$. The line width is wider for peaks and dips which are more photonic than excitation, where the line width for peaks and dips which are more photonic is dominated by the cavity line width, while for excitation dips and peaks the line width approaches the excitation line width.
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig13.eps}}
\caption{The $(s)$ polarized field transmission spectra, $T_s^{(s)}$, for different angles, $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$ of $(s)$ polarized incident field.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig14.eps}}
\caption{The $(s)$ polarized field reflection spectra, $R_s^{(s)}$, for different angles, $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$ of $(s)$ polarized incident field.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig15.eps}}
\caption{The $(p)$ polarized field transmission and reflection spectra, $T_p^{(s)}$ and $R_p^{(s)}$, for different angles, $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$ of $(s)$ polarized incident field.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig16.eps}}
\caption{The absorption spectra, $A^{(s)}$, for different angles, $\theta$, at $k=5\times10^{-7}\ \AA^{-1}$ of $(s)$ polarized incident field.}
\end{figure}
The phase shift in the $(s)$ polarized transmitted field relative to the incident field, $\Delta\phi^{(s)t}_s$, as a function of frequency, at $k=5\times10^{-7}\ \AA^{-1}$ and for $\theta=\pi/4$, is plotted in figure (17). And the phase shift in the $(s)$ polarized reflected field, $\Delta\phi^{(s)r}_s$, is plotted in figure (18). In figure (19) we plot the phase shift in the transmitted and reflected $(p)$ polarized fields, $\Delta\phi^{(s)t}_p$ and $\Delta\phi^{(s)r}_p$. As the coupling through the mirrors is taken to be a real constant, $\gamma$, we neglect here phase shifts due to the cavity mirrors. In the general case they can be easily included.
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig17.eps}}
\caption{The phase shift in the $(s)$ polarized transmitted field, $\Delta\phi^{(s)t}_s$, at $k=5\times10^{-7}\ \AA^{-1}$ and for $\theta=\pi/4$ of $(s)$ polarized incident field.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig18.eps}}
\caption{The phase shift in the $(s)$ polarized reflected field, $\Delta\phi^{(s)r}_s$, at $k=5\times10^{-7}\ \AA^{-1}$ and for $\theta=\pi/4$ of $(s)$ polarized incident field.}
\end{figure}
\begin{figure}
\centerline{\epsfxsize=7.0cm \epsfbox{Fig19.eps}}
\caption{The phase shift in the $(p)$ polarized transmitted and reflected fields, $\Delta\phi^{(s)t}_p$ and $\Delta\phi^{(s)r}_p$, at $k=5\times10^{-7}\ \AA^{-1}$ and for $\theta=\pi/4$ of $(s)$ polarized incident field.}
\end{figure}
The $(s)$ polarized cavity mean photon number relative to the incident $(s)$ polarized photon number, in using $\sqrt{\gamma}\ \tilde{a}_{s}=\tilde{c}^{s}_{out}$, is given by
\begin{equation}
I^{(s)}_s=\frac{\langle\tilde{a}_{s}^{\dagger}\tilde{a}_{s}\rangle}{\langle\tilde{b}_{in}^{s\dagger}\tilde{b}^{s}_{in}\rangle}=\frac{\gamma\ |\Lambda_{ss}(1+\gamma\ \Lambda_{pp})-\gamma\ \Lambda_{sp}\Lambda_{ps}|^2}{|D|^2},
\end{equation}
and the $(p)$ polarized cavity mean photon number relative to the incident $(s)$ polarized photon number, in using $\sqrt{\gamma}\ \tilde{a}_{p}=\tilde{c}^{p}_{out}$, is given by
\begin{equation}
I^{(s)}_p=\frac{\langle\tilde{a}_{p}^{\dagger}\tilde{a}_{p}\rangle}{\langle\tilde{b}_{in}^{s\dagger}\tilde{b}^{s}_{in}\rangle}=\frac{\gamma\ |\Lambda_{ps}|^2}{|D|^2}.
\end{equation}
Note that $I^{(s)}_s=T^{(s)}_s/\gamma$, and $I^{(s)}_p=T^{(s)}_p/\gamma=R^{(s)}_p/\gamma$. Up to the division by $\gamma$, the plot of $I^{(s)}_s$ is as that of $T^{(s)}_s$ in figure (13), and the plot of $I^{(s)}_p$ is as that of $T^{(s)}_p$ and $R^{(s)}_p$ in figure (15). Hence the cavity mean photon number of both polarizations can be observed directly through the optical linear spectra.
Identical results are obtained for the case of $(p)$ polarized incident field only in exchanging $(s)$ and $(p)$ in all the previous equations. The incident field can be also in a superposition of both polarizations.
\section{Summary}
We investigated an anisotropic optical lattice, for the case of the Mott insulator phase with one atom per site. The anisotropy is induced by the atomic transition dipole of a fixed direction, which is fixed by the combination of the optical lattice laser polarization and external static fields. In the strong coupling regime, the cavity photon polarizations, of TE and TM modes, are coherently mixed with the electronic excitations to form two cavity polariton branches. As the superposition of the cavity photons of both polarizations has a component which is orthogonal to the atomic transition dipole, a third photonic branch is obtained that decouple to the electronic transitions.
The system eigenmodes are observed by linear optical spectra, which also proved the polarization mixing. For an incident field which is TE polarized, we get TE and TM transmission and reflection spectra, where the TM spectra is induced by the anisotropic optical lattice. The absorption spectrum of the cavity medium is calculated in including the atom excited state life time phenomenologically. Furthermore, the eigenmodes and the polarization mixing can be observed through the phase shift of the transmitted and reflected fields relative to the incident field.
Polarization mixing is of big importance for electro-optics devices and quantum information, and can be used as an observation tool of many properties of anisotropic optical lattices. The system can serve as a linear optical switch, as the transmitted and reflected field intensities and phase shifts are a function of the angle between the incident field and the transition dipole, they can be controlled in changing the angle. Moreover, the results allow us to fix the mean photon number in the cavity, for both polarizations, in fixing the intensity of the incident field.
\begin{acknowledgments}
The work was supported by the Austrian Science Fund (FWF), through the Lise-Meitner Program (M977).
\end{acknowledgments} |
0807.1200 | \section{Introduction}
The mysteries of the nature of dark matter and dark energy are perhaps the
most important ones of contemporary cosmology. Dark matter, which accounts for
the observed discrepancy between the dynamical and luminous masses of bounded
astrophysical systems, is usually formulated within the so-called
\textit{particle dark matter} approach, in which the dark matter consists of
unknown non-baryonic particles, e.g. neutralinos as predicted by
super-symmetric extensions of the standard model of particle physics
(see~\cite{Be.al.05} for a review). Furthermore, the dark matter triggers the
formation of large-scale structures by gravitational collapse and explains the
distribution of baryonic matter from galaxy cluster scales up to cosmological
scales by the non-linear growth of initial perturbations. Simulations suggest
some universal dark matter density profile around distributions of ordinary
baryonic matter~\cite{Na.al.97}. An important characteristic of dark matter,
required by the necessity of clustering matter on small scales, is that it
should be cold (or non-relativistic) at the epoch of galaxy
formation. Together with the observational evidence of dark energy (presumably
a cosmological constant $\Lambda$) measured from the Hubble diagram of
supernovas, the particle dark matter hypothesis has yielded the successful
concordance model of cosmology called $\Lambda$-CDM, which reproduces
extremely well the observed cosmic microwave background spectrum~\cite{HuD02}.
However, despite these successes at cosmological scales, the particle dark
matter approach has some difficulties~\cite{McBl.98} at explaining in a
natural way the flat rotation curves of galaxies, one of the most persuasive
evidence for the existence of dark matter, and the Tully-Fisher empirical
relation between the observed luminosity and the asymptotic rotation velocity
of spiral galaxies. In order to deal with these difficulties, all linked with
the properties of dark matter at galactic scales, an alternative paradigm has
emerged in the name of the modified Newtonian dynamics
(MOND)~\cite{Mi1.83,Mi2.83,Mi3.83}. Although MOND in its original formulation
cannot be considered as a viable physical model, it is now generally admitted
that it does capture in a very simple and powerful ``phenomenological recipe''
a large number of observational facts, that any pertinent model of dark matter
should explain.
It is frustrating that the two alternatives $\Lambda$-CDM and MOND, which are
successful in complementary domains of validity (say the cosmological scale
for $\Lambda$-CDM and the galactic scale for MOND), seem to be fundamentally
incompatible. In the present paper, we shall propose a different approach,
together with a new interpretation of the phenomenology of MOND, which has the
potential of bringing together $\Lambda$-CDM and MOND into a single unifying
relativistic model for dark matter and dark energy. This relativistic model
will be shown to benefit from both the successes of $\Lambda$-CDM at
cosmological scales, and MOND at galactic scales.
\section{The modified Newtonian dynamics (MOND)}
The original idea behind MOND~\cite{Mi1.83,Mi2.83,Mi3.83} is that there is no
dark matter, and we witness a violation of the fundamental law of gravity (or
of inertia). MOND is designed to account for the basic features of galactic
dark matter halos. It states that the ``true'' gravitational field $\bm{g}$
experienced by ordinary matter, say a test particle whose acceleration would
thus be $\bm{a}=\bm{g}$, is not the Newtonian gravitational field
$\bm{g}_\mathrm{N}$, but is actually related to it by
\begin{equation}\label{recipe1}
\mu\!\left(\frac{g}{a_0}\right)\bm{g} = \bm{g}_\mathrm{N} \, .
\end{equation}
Here $\mu$ is a function of the dimensionless ratio $g/a_0$ between the norm
of the gravitational field $g=\vert\bm{g}\vert$, and the constant MOND
acceleration scale $a_0 \simeq 1.2 \times 10^{-10}~\text{m} / \text{s}^2$,
whose numerical value is chosen to fit the data. The specific MOND regime
corresponds to the weak gravity limit, much weaker than the scale $a_0$. In
this regime (where formally $g \rightarrow 0$) we
have~\cite{Mi1.83,Mi2.83,Mi3.83}
\begin{equation}\label{recipe2}
\mu\!\left(\frac{g}{a_0}\right) = \frac{g}{a_0} + \mathcal{O}(g^2) \, .
\end{equation}
On the other hand, when $g$ is much larger than $a_0$ (formally $g \rightarrow
+\infty$) the usual Newtonian law is recovered, i.e. $\mu \rightarrow
1$. Various functions $\mu$ interpolating between the MOND regime and the
Newtonian limit are possible, but most of them appear to be rather \textit{ad
hoc}. Taken for granted, the MOND ``recipe''~\eqref{recipe1}--\eqref{recipe2}
beautifully predicts a Tully-Fisher relation and is very successful at fitting
the detailed shape of rotation curves of galaxies from the observed
distribution of stars and gas (see~\cite{Mi.02,SaMc.02} for reviews). So MOND
appears to be more than a simple recipe and may well be related to some new
fundamental physics. In any case the agreement
of~\eqref{recipe1}--\eqref{recipe2} with a large number of observations calls
for a clear physical explanation.
Taking the divergence of both sides of~\eqref{recipe1}, and using the usual
Poisson equation for the Newtonian field $\bm{g}_\mathrm{N}$, we obtain a
local formulation of MOND in the form of the modified Poisson equation
\cite{BeMi.84}
\begin{equation}\label{MONDeq}
\bm{\nabla} \! \cdot \! \left( \mu \, \bm{g} \right) = -4 \pi G \,
\rho_\text{b} \, ,
\end{equation}
where $\rho_\text{b}$ is the density of baryonic matter. This equation can be
derived from a Lagrangian, and that Lagrangian has been the starting point for
constructing relativistic extensions of MOND. Such extensions postulate the
existence of extra (supposedly fundamental) fields associated with gravity
besides the spin-$2$ field of general relativity. Promoting the Newtonian
potential $U$ to a scalar field $\phi$, scalar-tensor theories for MOND have
been constructed~\cite{BekS94} but shown to be non viable; essentially because
light signals do not feel the presence of the scalar field, since the physical
(Jordan-frame) metric is conformally related to the Einstein-frame metric, and
the Maxwell equations are conformally invariant. This is contrary to
observations: huge amounts of dark matter are indeed observed by gravitational
(weak and strong) lensing.
Relativistic extensions of MOND that pass the problem of light deflection by
galaxy clusters have been shown to require the existence of a time-like vector
field. The prototype of such theories is the tensor-vector-scalar (TeVeS)
theory~\cite{Sa.97,Be.04,Sa.05}, whose non-relativistic limit reproduces MOND,
and which has been extensively investigated in cosmology and at the
intermediate scale of galaxy clusters~\cite{An.al.06}. Modified gravity
theories such as TeVeS have evolved recently toward Einstein-{\ae}ther like
theories~\cite{JaMa.01,Zl.al.07,Ha.al.08}. Still, recovering the level of
agreement of the $\Lambda$-CDM scenario with observations at cosmological
scales remains an issue for such theories. In the present paper we shall
follow a completely different route from that of modified gravity and/or
Einstein-{\ae}ther theories. We shall propose an alternative to these theories
in the form of a specific \textit{modified matter theory} based on an
elementary interpretation of the MOND equation~\eqref{MONDeq}.
\section{Interpretation of the phenomenology of MOND}
The physical motivation behind our approach is the striking (and presumably
deep) analogy between MOND and the electrostatics of dielectric
media~\cite{Bl1.07}. From electromagnetism in dielectric media we know that
the Maxwell-Gauss equation can be written in the two equivalent forms
\begin{equation}
\bm{\nabla} \! \cdot \! \bm{E} = \frac{1}{\varepsilon_0} \, \bigl(
\sigma_\text{free} + \sigma_\text{pol} \bigr) \quad
\Longleftrightarrow \quad \bm{\nabla} \! \cdot \! \left(
\varepsilon_\text{r} \, \bm{E} \right) = \frac{1}{\varepsilon_0} \,
\sigma_\text{free} \, ,
\end{equation}
where $\bm{E}$ is the electric field, $\sigma_\text{free}$ and
$\sigma_\text{pol}$ are the densities of free and polarized (electric) charges
respectively, and $\varepsilon_\text{r} = 1 + \chi_\text{e}$ is the relative
permittivity of the dielectric medium. Such an equivalence is only possible
because the density of polarized charges reads $\sigma_\text{pol} = -
\bm{\nabla}\!\cdot\!\bm{P}$, where the polarization field $\bm{P}$ is
aligned with the electric field according to $\bm{P} = \varepsilon_0
\chi_\text{e} \, \bm{E}$, the proportionality coefficient $\chi_\text{e}(E)$
being known as the electric susceptibility.
By full analogy, one can write the MOND equation \eqref{MONDeq} in the form of
the usual Poisson equation but sourced by some additional distribution of
``polarized gravitational masses'' $\rho_\text{pol}$ (to be interpreted as
dark matter, or a component of dark matter), namely
\begin{equation}\label{Poisson_MOND}
\bm{\nabla} \! \cdot \bm{g} = -4 \pi G \, \bigl( \rho_\text{b} +
\rho_\text{pol} \bigr) \quad \Longleftrightarrow \quad
\bm{\nabla} \! \cdot \! \left( \mu \, \bm{g} \right) = -4 \pi G \,
\rho_\text{b} \, .
\end{equation}
This rewriting stands as long as the mass density of polarized masses appears
as the divergence of a vector fied, namely takes the \textit{dipolar} form
\begin{equation}\label{rhodm}
\rho_\text{pol} = - \bm{\nabla}\!\cdot\!\bm{\Pi}\, ,
\end{equation}
where $\bm{\Pi}$ denotes the (gravitational analogue of the) polarization
field. It is aligned with the gravitational field $\bm{g}$ (i.e. the dipolar
medium is polarized) according to
\begin{equation}\label{Pig}
\bm{\Pi} = -\frac{\chi}{4 \pi G} \, \bm{g} \, .
\end{equation}
Here the coefficient $\chi$, which depends on the norm of the gravitational
field $g=\vert\bm{g}\vert$ in complete analogy with the electrostatics of
dielectric media, is related to the MOND function by
\begin{equation}\label{muchi}
\mu=1+\chi\, .
\end{equation}
Obviously $\chi$ can be interpreted as a ``gravitational susceptibility''
coefficient, while $\mu$ itself can rightly be called a ``digravitational''
coefficient. It was shown in~\cite{Bl1.07} that in the gravitational case the
sign of $\chi$ should be negative, in perfect agreement with what MOND
predicts; indeed, we have $\mu<1$ in a straightforward interpolation between
the MOND and Newtonian regimes, hence $\chi<0$. This finding is in contrast to
electromagnetism where $\chi_\text{e}$ is positive. It can be viewed as some
``anti-screening'' of gravitational (baryonic) masses by polarization masses
--- the opposite effect of the usual screening of electric (free) charges by
polarization charges. Such anti-screening mechanism results in an enhancement
of the gravitational field \textit{\`a la} MOND, and offers a very nice
interpretation of the MOND phenomenology. Furthermore, it was pointed out that
the stability of the dipolar dark matter medium requires the existence of some
internal force, which turned out to be simply (in a crude quasi-Newtonian
model~\cite{Bl1.07}) that of an harmonic oscillator. This force could then be
interpreted as the restoring force in the gravitational analogue of a plasma
oscillating at its natural plasma frequency. Finally, it seems from this
discussion that the dielectric interpretation of MOND is deeper than a mere
formal analogy. However the model~\cite{Bl1.07} is clearly non-viable because
it is non-relativistic, and it involves negative gravitational-type masses and
therefore a violation of the equivalence principle at a fundamental level.
\section{Relativistic model for the dipolar dark fluid}\label{sec3}
Here we shall take seriously the physical intuition that MOND has something to
do with a mechanism of gravitational polarization. We shall build a fully
relativistic model based on a matter action in standard general
relativity. Note that this means we are changing the point of view of the
original MOND proposal. Instead of requiring a modification of the laws of
gravity in the absence of dark matter, we advocate that the phenomenology of
MOND results from a physically well-motivated mechanism acting on a new type
of dark matter, very exotic compared to standard particle dark matter. Thus,
we are proposing a modification of the \textit{dark matter} sector rather than
a modification of gravity as in TeVeS like theories.
\subsection{Action and equations of motion}
From the previous discussion, the necessity of endowing dark matter with a new
vector field to build the polarization field is clear. However this vector
field will not be expected to be fundamental as in TeVeS like
theories. Extending previous work~\cite{Bl2.07}, our model will be based on a
\textit{matter} action (in Eulerian fluid formalism) in general relativity of
the form
\begin{equation}\label{S}
S = \int \ud^4 x \, \sqrt{-g} \, L \bigl[ J^\mu, \xi^\mu,
\dot{\xi}^\mu, g_{\mu \nu} \bigr] \, .
\end{equation}
This action is to be added to the Einstein-Hilbert action for gravity, and to
the actions of all the other matter fields. It contains \textit{two} dynamical
variables: (i) a conserved current $J^\mu = \sigma u^\mu$ satisfying
$\nabla_\mu J^\mu=0$, where $u^\mu$ is the normalized four-velocity and
$\sigma = (- J_\nu J^\nu)^{1/2}$ is the rest mass energy density (we pose
$c=1$ throughout); (ii) the vector field $\xi^\mu$ representing the dipole
four-vector moment carried by the fluid particles. This extra field being
dynamical, the Lagrangian will also depend on its covariant derivative
$\nabla_\nu \xi^\mu$; but in our model this dependence will occur only through
the covariant time derivative $\dot{\xi}^\mu \equiv u^\nu \nabla_\nu
\xi^\mu$. The Lagrangian explicitly reads (see~\cite{BlLe.08} for details)
\begin{equation}\label{L}
L = \sigma \left[ -1 - \Xi + \frac{1}{2} \dot{\xi}_\mu \dot{\xi}^\mu
\right] - \calW (\Pi_\perp) \, ,
\end{equation}
where $\Xi \equiv \bigl\{ \bigl( u_\mu - \dot{\xi}_\mu \bigr) \bigl( u^\mu -
\dot{\xi}^\mu \bigr) \bigr\}^{1/2}$. The first term is a mass term in the
ordinary sense (i.e. the Lagrangian of a pressureless perfect fluid), the
second one is inspired by the action of spinning particles in general
relativity~\cite{BaIs.80}, and the third term is a kinetic term for the dipole
moment. Finally, the last term represents a potential $\calW$ describing some
internal interaction, function of the polarization (scalar) field $\Pi_\perp =
(\perp_{\mu\nu}\!\Pi^\mu\Pi^\nu)^{1/2}$, where $\Pi^\mu = \sigma \xi^\mu$ is
the polarization four-vector, and $\perp_{\mu\nu} \, = g_{\mu\nu} + u_\mu
u_\nu$ is the projector orthogonal to the four-velocity.
By varying the action \eqref{S}--\eqref{L} with respect to both the current
$J^\mu$ and the dipole moment $\xi^\mu$, we get two dynamical equations: an
equation of motion for the dipolar fluid, and an evolution equation for the
dipole moment $\xi^\mu$. From these equations it can be shown~\cite{BlLe.08}
that we can impose the constraint $\Xi = 1$ as a particular way of selecting a
physically interesting solution, such that the final equations depend only on
the \textit{space-like} projection $\xi_\perp^\mu=\,\perp_{\mu\nu}\!\xi^\nu$
of the dipole moment $\xi^\mu$. The final equations we obtain are
\begin{align}
\dot{u}^\mu &= - \mathcal{F}^\mu \equiv - \hat{\xi}_\perp^\mu \, \calW' \, ,
\label{motion} \\ \dot{\Omega}^\mu &= \frac{1}{\sigma} \nabla^\mu \left( \calW
- \Pi_\perp \calW' \right) - \xi_\perp^\nu R^\mu_{\phantom{\mu} \rho \nu
\lambda} u^\rho u^\lambda \, , \label{evolution}
\end{align}
where we denote $\Omega^\mu \equiv u^\mu \bigl( 1 + \xi_\perp \calW' \bigr) \,
+ \! \perp^\mu_\nu \dot{\xi}_\perp^\nu$, and employ the notations
$\hat{\xi}_\perp^\mu \equiv \xi_\perp^\mu / \xi_\perp = \Pi_\perp^\mu /
\Pi_\perp$ and $\calW' \equiv \ud \calW / \ud \Pi_\perp$. The motion of the
dipolar fluid as given by \eqref{motion} is non-geodesic, and driven by the
internal force $\mathcal{F}^\mu$ derived from the potential $\calW$. Observe
the coupling to the Riemann curvature tensor in the equation of evolution
\eqref{evolution} of the dipole moment. By varying the action with respect to
the metric $g_{\mu \nu}$ we obtain the stress-energy tensor $T^{\mu
\nu}$. Using the canonical decomposition $T^{\mu \nu} = r \, u^\mu u^\nu +
\mathcal{P} \! \perp^{\mu \nu} + \, 2 \, Q^{(\mu} u^{\nu)} + \Sigma^{\mu
\nu}$, we find the energy density $r$, pressure $\mathcal{P}$, heat flux
$Q^\mu$ (such that $u_\mu Q^\mu=0$) and anisotropic stresses $\Sigma^{\mu
\nu}$ ($u_\nu \Sigma^{\mu\nu}=0$ and $\Sigma^\nu_\nu=0$) as
\begin{subequations}\label{rPQSigma}\begin{align}
r &= \calW - \Pi_\perp \calW' \! + \rho \, , \label{r} \\
\mathcal{P} &= - \calW + \frac{2}{3} \, \Pi_\perp \calW' \, ,
\label{P} \\ Q^\mu &= \sigma \, \dot{\xi}_\perp^\mu +
\Pi_\perp \calW' u^\mu - \Pi_\perp^\lambda \nabla_\lambda
u^\mu \, , \label{Q} \\ \Sigma^{\mu \nu} &= \biggl(
\frac{1}{3} \! \perp^{\mu \nu} \! - \, \hat{\xi}_\perp^{\mu}
\hat{\xi}_\perp^{\nu}\biggr) \Pi_\perp \calW'
\label{Sigmamunu} \, .
\end{align}\end{subequations}
Here the contribution $\rho$ to the energy density involves a monopolar term
$\sigma$ and a dipolar term $- \nabla_\lambda \Pi_\perp^\lambda$ which clearly
appears as a relativistic generalisation of \eqref{rhodm}, and will play the
crucial role when recovering MOND:
\begin{equation}\label{rho}
\rho = \sigma - \nabla_\lambda \Pi_\perp^\lambda\, .
\end{equation}
\subsection{Weak field expansion of the internal potential}
The dipolar fluid dynamics in a given background metric, and its influence on
spacetime are now known; in the following we shall apply this model to
large-scale cosmology and to galactic halos. For both applications we shall
need to consider the model in a regime of \textit{weak gravity}, which will be
either first-order perturbations around a Friedman-Lema\^itre-Robertson-Walker
(FLRW) background in cosmology, or the non-relativistic limit for galaxies. A
crucial assumption we make is that the potential function $\calW$ admits a
minimum when the polarization $\Pi_\perp$ is zero, and can be Taylor-expanded
around that minimum, with coefficients being entirely specified (modulo an
overall factor $G$) by the single surface density scale built from the MOND
acceleration $a_0$,
\begin{equation}\label{Sigma}
\Sigma \equiv \frac{a_0}{2 \pi G}\, .
\end{equation}
These coefficients in the expansion of $\calW$ when $\Pi_\perp\rightarrow 0$
will be \textit{fine-tuned} in order to recover the relevant physics at
cosmological and galactic scales. Physically, this expansion corresponds to
$\Pi_\perp \ll \Sigma$ and is valid in the weak gravity limit $g\ll
a_0$. Clearly, the minimum of $\calW$ is nothing but a cosmological constant
$\Lambda$, and we find
\begin{equation}\label{W}
\calW(\Pi_\perp) = \frac{\Lambda}{8 \pi G} + 2 \pi G \, \Pi_\perp^2 +
\frac{8 \pi G}{3 \Sigma} \, \Pi_\perp^3 + \mathcal{O}(\Pi_\perp^4) \,
.
\end{equation}
The expansion is thereby determined up to third order inclusively.
Our assumption that the function $\calW$ involves the single fundamental scale
$\Sigma$ implies in particular that the cosmological constant $\Lambda$ should
scale with $G^2\Sigma^2\sim a_0^2$. We thus introduce a dimensionless
parameter $\alpha$ through
\begin{equation}\label{Lambda_a0}
\alpha \, a_0 = \frac{1}{2\pi}\sqrt{\frac{\Lambda}{3}} \equiv
a_\Lambda \, .
\end{equation}
This parameter represents a conversion factor between $a_0$ and the natural
acceleration scale $a_\Lambda$ associated with the cosmological constant
$\Lambda$. Posing $x \equiv \Pi_\perp / \Sigma$, we find that \eqref{W} can be
recast in the form $\calW(\Pi_\perp) = 6 \pi G \, \Sigma^2 \, w(x)$, where
\begin{equation}\label{W2}
w(x) = \alpha^2 \pi^2 + \frac{1}{3} x^2 + \frac{4}{9} x^3 +
\mathcal{O}(x^4) \, .
\end{equation}
The present model should be considered only as ``effective'' or
phenomenological, in the sense that the weak-gravity expansion \eqref{W} or
\eqref{W2} should come from a more fundamental (presumably quantum) underlying
theory. Therefore, the coefficients in \eqref{W2} should not be given by
exceedingly large or small numbers (like $10^{10}$ or $10^{-10}$), but rather
be numerically of the order of one, up to a factor of, say, ten. Hence, we
expect that $\alpha$ itself should be around one, and we see by
\eqref{Lambda_a0} that the cosmological constant $\Lambda$ should naturally be
numerically of the order of $a_0^2$. Notice though that our model does not
provide a way to compute the exact value of $\alpha$. However, one can say
that the ``cosmic coincidence'' (see e.g.~\cite{Mi.02}) between the values of
$a_0$ and $a_\Lambda$ --- with the measured conversion factor being $\alpha
\simeq 0.8$ --- finds a natural explanation if dark matter is made of a fluid
of polarizable gravitational dipole moments.
\section{Recovering the $\Lambda$-CDM scenario at cosmological scales}\label{sec4}
Consider a small perturbation of a background FLRW metric valid between, say,
the end of the inflationary era and the recombination. The dipolar fluid is
described by its four-velocity $u^\mu = \overline{u}^\mu + \delta u^\mu$,
where $\delta u^\mu$ is a perturbation of the background comoving
four-velocity $\overline{u}^\mu=(1,\bm{0})$, and by its rest mass density
$\sigma = \overline{\sigma} + \delta \sigma$, where $\delta \sigma$ is a
perturbation of the mean cosmological value $\overline{\sigma}$. The crucial
point in our analysis of the dipolar fluid in cosmology is that the background
value of the dipole moment field $\xi_\perp^\mu$ (which is orthogonal to the
four-velocity and therefore is \textit{space-like}) must vanish in order to
preserve the spatial isotropy of the FLRW background. We shall thus write
$\xi_\perp^\mu = \delta \xi_\perp^\mu$, and similarly $\Pi_\perp^\mu = \delta
\Pi_\perp^\mu$ for the polarization.
At first perturbation order, the stress-energy tensor with explicit components
\eqref{rPQSigma} can be naturally recast, using also \eqref{W}, in the form
$T^{\mu \nu} = T_\text{de}^{\mu \nu} + T_\text{dm}^{\mu \nu}$, where the
explicit expressions of the dark energy and dipolar dark matter components are
\begin{align}
T_\text{de}^{\mu \nu} &= - \frac{\Lambda}{8 \pi G} \, g^{\mu
\nu} \, , \\ T_\text{dm}^{\mu \nu} &= \rho \, u^\mu u^\nu + 2
\, Q^{(\mu} u^{\nu)} \label{exprTdm}\, .
\end{align}
At that order, we find that the dipolar dark matter density reduces to $\rho$
given by \eqref{rho}. The heat flux in \eqref{exprTdm} reads as $Q^\mu =
\sigma \, \dot{\xi}_\perp^\mu - \Pi_\perp^\lambda \nabla_\lambda u^\mu$; it is
non zero, however it is perturbative because so are both $\xi_\perp^\mu$ and
$\Pi_\perp^\mu$. The point for our purpose is that at linear perturbation
order $Q^\mu$ can be absorbed into a redefinition of the perturbed
four-velocity of the dipolar fluid. Posing $\delta \widetilde{u}^\mu = \delta
u^\mu + Q^\mu / \overline{\sigma}$ and introducing the effective four-velocity
$\widetilde{u}^\mu = \overline{u}^\mu + \delta \widetilde{u}^\mu$, we find
that at first order the dipolar dark matter fluid is described by the
stress-energy tensor
\begin{equation}
T_\text{dm}^{\mu \nu} = \rho \, \widetilde{u}^\mu \widetilde{u}^\nu \, ,
\end{equation}
which is that of a perfect fluid with four-velocity $\widetilde{u}^\mu$,
vanishing pressure and energy density \eqref{rho}. In the linear cosmological
regime, the dipolar fluid therefore behaves as standard cold dark matter (a
pressureless fluid) plus standard dark energy (a cosmological
constant). Adjusting the background value $\overline{\sigma}$ so that
$\Omega_\mathrm{dm}\simeq 23\%$, the model is thus consistent with the
standard $\Lambda$-CDM scenario and the cosmological observations of the CMB
fluctuations (see~\cite{BlLe.08} for more details).
\section{Recovering the MOND phenomenology at galactic scales}
Next, we turn to the study of the dipolar dark fluid in a typical galaxy at
low redshift. We have to consider the non-relativistic limit ($c \rightarrow
+\infty$) of the model; we consistently neglect all relativistic terms
$\mathcal{O}(c^{-2})$. It is straightforward to check that in the
non-relativistic limit the equation of motion \eqref{motion} of the dipolar
fluid reduces to
\begin{equation}\label{motion_NR}
\frac{\ud \bm{v}}{\ud t} = \bm{g} - \hat{\bm{\Pi}}_\perp \calW' \, ,
\end{equation}
where $\bm{g}$ is the \textit{local} gravitational field generated by both the
baryonic matter in the galaxy and the dipolar dark matter. Applying next the
standard minimal coupling to gravity in general relativity, we find that the
gravitational field obeys the Poisson equation
\begin{equation}\label{Poisson}
\bm{\nabla}\!\cdot \bm{g} = -4 \pi G \, \bigl( \rho_\text{b} + \rho
\bigr) \, ,
\end{equation}
where $\rho_\text{b}$ and $\rho$ are respectively the baryonic and dipolar
dark matter mass densities. From \eqref{rho} the dipolar dark matter density
reduces in the non-relativistic limit to
\begin{equation}\label{rhosigma}
\rho = \sigma - \bm{\nabla} \! \cdot \! \bm{\Pi}_\perp \, .
\end{equation}
The first term is a usual monopolar contribution: the rest mass density
$\sigma$ of the fluid, while the second term is the dipolar contribution, and
can be interpreted as coming from the fluid's internal energy. Here
$\bm{\Pi}_\perp$ denotes the spatial components of the polarization.
In order to recover MOND, we need two things: (i) to find a mechanism for the
alignment of the polarization field $\bm{\Pi}_\perp$ with the gravitational
field $\bm{g}$, so that a relation similar to \eqref{Pig} will apply; (ii) to
justify that the rest mass density $\sigma$ of dipole moments in
\eqref{rhosigma} is small with respect to the baryonic density
$\rho_\text{b}$, hence the galaxy will mostly appear as baryonic in MOND fits
of rotation curves. We have proposed in~\cite{BlLe.08} a single mechanism able
to answer positively these two points. We call it the hypothesis of
\textit{weak clustering} of dipolar dark matter; it is an hypothesis because
it has been conjectured but not proved, and should be checked using numerical
simulations. The weak clustering hypothesis is motivated by a solution of the
full set of equations describing the non-relativistic motion of dipolar dark
matter in a typical baryonic galaxy whose mass distribution $\rho_\text{b}$ is
spherically symmetric (see the Appendix in~\cite{BlLe.08}). This particular
solution corresponds to an equilibrium configuration in spherical symmetry,
for which $\bm{v} = \bm{0}$ and $\sigma = \sigma_0(r)$. The dipole moments
remain at rest because the gravitational field $\bm{g}$ is balanced by the
internal force $\bm{\mathcal{F}} = \hat{\bm{\Pi}}_\perp \calW'$; for that
solution the right-hand-side of \eqref{motion_NR} vanishes.
During the cosmological evolution we expect that the dipolar medium will not
cluster much because the internal force may balance part of the local
gravitational field generated by an overdensity. The dipolar dark matter
density contrast in a typical galaxy at low redshift should thus be small, at
least smaller than in the standard $\Lambda$-CDM scenario. Hence $\sigma \ll
\rho_\text{b}$, and we could even envisage that $\sigma$ stays around its mean
cosmological value, $\sigma \sim \overline{\sigma} \ll \rho_\text{b}$. Now,
because of its size and typical time-scale of evolution, a galaxy is almost
unaffected by the cosmological expansion of the Universe. The cosmological
mass density $\overline{\sigma}$ of the dipolar dark matter is not only
homogeneous, but also almost constant in this galaxy. The continuity equation
reduces to $\bm{\nabla} \! \cdot \! \left( \overline{\sigma} \, \bm{v} \right)
\simeq \bm{0}$, and the most simple solution corresponds to a static fluid
verifying $\bm{v} \simeq \bm{0}$. By \eqref{motion_NR} we see that the
polarization field $\bm{\Pi}_\perp$ is then aligned with the gravitational
field $\bm{g}$, namely
\begin{equation}\label{geq}
\bm{g} \simeq \hat{\bm{\Pi}}_\perp \calW' \, .
\end{equation}
On the other hand, because $\sigma \ll \rho_\text{b}$ by the same mechanism,
we observe that the gravitational field equation \eqref{Poisson} with
\eqref{rhosigma} becomes
\begin{equation}\label{Poisson2}
\bm{\nabla} \! \cdot \! \left( \bm{g} - 4 \pi G \, \bm{\Pi}_\perp
\right) \simeq - 4 \pi G \, \rho_\text{b} \, ,
\end{equation}
which according to \eqref{Poisson_MOND} is found to be rigorously equivalent
to the MOND equation. Finally, by inserting the expression of the potential
\eqref{W} into \eqref{geq} and comparing with the defining equation
\eqref{Pig}, we readily find the MOND behaviour of the gravitational
susceptibility coefficient as
\begin{equation}\label{chiMOND}
\chi(g) = - 1 + \frac{g}{a_0} + \mathcal{O}(g^2) \, ,
\end{equation}
in complete agreement with \eqref{recipe2}. We can thus state that the dipolar
fluid described by the action \eqref{S}--\eqref{L} explains the phenomenology
of MOND in a typical galaxy. Note also that in this model there is no problem
with the light deflection by galaxy clusters. Indeed the standard
general relativistic coupling to gravity implies the usual formula for the
bending of light.
To conclude, the present model reconciles in some sense the observations of
dark matter on cosmological scales, where the evidence is for cold dark
matter, and on galactic scales, which is the realm of MOND. In addition, it
offers a nice unification between the dark energy in the form of $\Lambda$ and
the dark matter in galactic halos. More work should be done to test the model,
either by studying second-order perturbations in cosmology, or by computing
numerically the non-linear growth of perturbations and comparing with
large-scale structures, or by studying the intermediate scale of clusters of
galaxies.
\section*{References} |
2105.09242 | \section{Introduction}
The inflationary scenario offers the most attractive mechanism for
the generation of the primordial perturbations (for the original discussions,
see Refs.~\cite{Hawking:1982cz,Guth:1982ec,Starobinsky:1982ee,Bardeen:1983qw};
for reviews, see, for example, Refs.~\cite{Mukhanov:1990me,Martin:2003bt,
Martin:2004um,Bassett:2005xm,Sriramkumar:2009kg,Baumann:2008bn,Baumann:2009ds,
Sriramkumar:2012mik,Linde:2014nna,Martin:2015dha}).
The existence of primordial gravitational waves (GWs) is one of the profound
predictions of inflationary dynamics (for the initial discussions, see, for
example, Refs.~\cite{Grishchuk:1974ny,Starobinsky:1979ty}; for recent reviews
on the generation of primary and secondary GWs, see, for instance,
Refs.~\cite{Guzzetti:2016mkm,Caprini:2018mtu}).
If the primordial GWs or their imprints are detected, it will not only prove
the quantum origin of the perturbations, it can also, in principle, provide
us with insights into the fundamental nature of gravitation.
The primordial GWs provide a unique window to probe the dynamics of our
universe during its very early stages, which seems difficult to observe
by any other known means.
However, the extremely weak nature of the gravitational force makes the
detection of GWs rather challenging.
Decades of effort towards detecting GWs have finally achieved success only
in the last few years with the observations of GWs from merging binary black
holes and neutron stars by LIGO~\cite{TheLIGOScientific:2016agk,TheLIGOScientific:2016qqj,TheLIGOScientific:2016wfe,Abbott:2016blz,Abbott:2016nmj,Abbott:2017vtc,Abbott:2017gyy,Abbott:2017oio,TheLIGOScientific:2017qsa,LIGOScientific:2020stg,Abbott:2020uma,Abbott:2020khf}.
These observations have led to a surge of experimental proposals across the
globe to observe GWs over a wide range of frequencies.
The proposed GW observatories include
advanced LIGO ($10$--$10^3\, \mathrm{Hz}$)~\cite{TheLIGOScientific:2016dpb},
ET ($1$--$10^4\, \mathrm{Hz}$)~\cite{Punturo:2010zz,Sathyaprakash:2012jk},
BBO ($10^{-3}$--$10\,
\mathrm{Hz}$)~\cite{Crowder:2005nr,Corbin:2005ny,Baker:2019pnp},
DECIGO ($10^{-3}$--$1\, \mathrm{Hz}$)~\cite{Seto:2001qf,Kawamura:2011zz,Sato:2017dkf,Kawamura:2019jqt},
eLISA ($10^{-5}$--$1\,\mathrm{Hz}$)~\cite{AmaroSeoane:2012km,Audley:2017drz,
Barausse:2020rsu}, and
SKA ($10^{-9}$--$10^{-6}\, \mathrm{Hz}$)~\cite{Janssen:2014dka}.
Apart from various astrophysical mechanisms that can generate GWs, as
we mentioned, inflation provides an exclusive mechanism to produce GWs
of quantum mechanical origin (for discussions on the generation of
primary and secondary GWs, see, for instance,
Refs.~\cite{Easther:2006gt,Ananda:2006af,Baumann:2007zm,
Saito:2008jc,Saito:2009jt,Kuroyanagi:2018csn,Kohri:2018awv,
Inomata:2019zqy,Inomata:2019ivs,Braglia:2020eai,Ragavendra:2020sop,
Ragavendra:2020vud,Bhattacharya:2020lhc} and references therein).
The tensor perturbations generated from the quantum vacuum are amplified
during inflation, which subsequently evolve through the various phases of
the universe until they reach the GW detectors today.
Therefore, the spectrum of primordial GWs today is a convolution of their
origin as well as dynamics.
On the one hand, they contain the signatures of the mechanism that generates
them, viz. the specific model that drives inflation as well as the initial
conditions from which the perturbations emerge.
On the other, they also carry the imprints of the dynamics of the subsequent
cosmological phases as the GWs propagate through them.
As is well known, immediately after inflation, the universe is expected to be
reheated through the decay of the inflaton into radiation, which eventually
leads to the epoch of radiation domination.
In this work, we shall examine the evolution of primordial GWs with special
emphasis on the effects due to the epoch of reheating.
Specifically, our aim is to decode the mechanism of reheating from the
spectrum of GWs today.
Over the years, major cosmological observations have considerably
improved the theoretical understanding of the various epochs of our
universe~\cite{Akrami:2018odb,Aubourg:2014yra,Vazquez:2018qdg}.
However, due to the lack of direct observational signatures, the phase of
reheating remains poorly understood.
The effects of reheating on the dynamics of GWs have already been
examined in the standard cosmological scenario (see, for instance,
Refs.~\cite{Turner:1993vb,Boyle:2005se,Nakayama:2008wy,Nakayama:2008ip,
Kuroyanagi:2011fy,Kuroyanagi:2014nba})
as well as in certain non-standard scenarios (see, for example,
Refs.~\cite{Assadullahi:2009nf,Nakayama:2009ce,Durrer:2011bi,Alabidi:2013lya,
DEramo:2019tit,Ricciardone:2016lym,Koh:2018qcy,Fujita:2018ehq,Bernal:2019lpc,
Bernal:2020ywq,Mishra:2021wkm}).
Moreover, the imprints of specific microscopic physical effects
on the spectrum of GWs~---~such as decoupling
neutrinos~\cite{Weinberg:2003ur,Mangilli:2008bw}
or the variation in the number of relativistic degrees of freedom in the early
universe~\cite{Watanabe:2006qe,Kuroyanagi:2008ye,Caldwell:2018giq}~---~have
also been explored.
In this work, we shall study the effects of reheating on the spectrum of primordial GWs over a wide range of scales and illustrate the manner in
which the spectrum captures specific aspects of the different phases.
We shall consider two of the simplest reheating mechanisms.
We shall first consider a scenario wherein the reheating phase is described
by an averaged equation of state (EoS) parameter, with reheating ending
instantaneously~\cite{Martin:2010kz,Dai:2014jja,Cook:2015vqa}.
We shall then consider a scenario wherein there is a gradual change in
the EoS parameter from its initial value during the phase of coherent
oscillations to its eventual value during radiation domination achieved
through the perturbative decay of the
inflaton~\cite{Chung:1998rq,Giudice:2000ex,Maity:2018dgy,Haque:2019prw,Haque:2020zco}.
We shall explicitly illustrate the effects of the reheating dynamics on
the spectrum of GWs.
It seems worthwhile to highlight here that the following aspects of the
reheating dynamics can, in principle, be decoded from the spectrum of
primordial GWs: (i)~the shape of the inflaton potential near its minimum
which is responsible for the end of inflation and the dynamics during reheating,
(ii)~the decay width of the inflaton, which governs the entire process of
reheating and therefore determines the reheating temperature, and
(iii)~the thermalization time scale over which the EoS parameter during
the period of coherent oscillations of the inflaton say, $w_{\phi}$,
is modified to the EoS parameter corresponding to radiation.
Further, it is expected that determining the inflaton decay width and the
thermalization time scale would permit us not only to arrive at the form of
the coupling between the inflaton and radiation, but also help us understand
the nature of the coupling amongst all the relativistic degrees of freedom.
We shall briefly discuss these possibilities in this work, and we shall return
to examine the issue in greater detail in a future effort.
Finally, we shall also consider the implications of our analysis on the
recent observations involving the pulsar-timing data by the North American
Nanohertz Observatory for Gravitational Waves (NANOGrav), which has been
attributed to stochastic GWs~\cite{Arzoumanian:2020vkk,Pol:2020igl}.
A variety of mechanisms that can possibly occur in the early universe
have been explored in the literature to explain this interesting
observation~\cite{Kuroyanagi:2020sfw,Inomata:2020xad,Tahara:2020fmn,
Kitajima:2020rpm,Bhattacharya:2020lhc,Domenech:2020ers,Li:2020cjj,
Bian:2020bps,Blasi:2020mfx,DeLuca:2020agl,Vaskonen:2020lbd,
Ellis:2020ena,Buchmuller:2020lbh,Kohri:2020qqd,Vagnozzi:2020gtf,
Bhattacharya:2020lhc}.
We find that introducing a secondary phase of reheating~---~apart
from the original, inflaton driven, primary reheating phase~---~can
account for the NANOGrav observations.
We introduce an exotic, non-canonical scalar field to drive such a phase
and show that a suitable EoS for the non-canonical field can
lead to primordial GWs of strength as observed by NANOGrav.
This paper is structured as follows.
In Sec.~\ref{sec:pgw}, we shall briefly sketch the arguments leading
to the standard scale-invariant spectrum of GWs generated in de
Sitter inflation.
We shall also discuss the typical inflationary model that we shall have
in mind when we later discuss the effects due to reheating.
Moreover, we shall introduce the dimensionless density parameter~$\Omega_{_{\mathrm{GW}}}$
characterizing the spectrum of GWs.
In Sec.~\ref{sec:egw-drh}, we shall discuss the evolution of the tensor
perturbations during the epoch of reheating.
We shall first consider the scenario wherein the epoch of reheating is
described by an averaged EoS parameter associated with the inflaton.
Such a description allows us to arrive at analytic solutions for the
tensor perturbations during the epoch.
We shall also consider the scenario of perturbative reheating wherein
we take into account the continuous decay of the inflaton into radiation.
As it proves to be involved in constructing analytical solutions for the
background as well as the tensor perturbations in such a case, we shall
resort to numerics.
In Sec.~\ref{sec:egw-drd}, we shall briefly discuss the evolution of
the tensor perturbations during the epoch of radiation domination and
arrive at the spectrum of GWs today by comparing the behavior of the
the energy density of GWs in the sub-Hubble domain with that of radiation.
In Secs.~\ref{sec:averaged} and~\ref{sec:actual}, we shall evaluate the
dimensionless energy density of GWs today that arise in the two types
of reheating scenarios mentioned above.
We shall focus on the spectrum of primordial GWs over wave numbers (or,
equivalently, frequencies) that correspond to small scales which reenter
the Hubble radius either during the epochs of reheating or radiation
domination.
In Sec.~\ref{sec:actualinflation}, we shall numerically evaluate the
inflationary tensor power spectrum and discuss the behavior of the
spectrum of GWs today close to the scale that leaves the Hubble
radius at the end of inflation.
In Sec.~\ref{sec:probe}, we shall outline the manner in which we should
be able to decode various information concerning the epochs of inflation
and reheating from the observations of the anisotropies in the cosmic
microwave background (CMB) and the spectrum of GWs today.
In Sec.~\ref{sec:NANOGrav}, we shall evaluate the spectrum of GWs in
a scenario involving late time production of entropy and discuss the
implications for the recent observations by NANOGrav.
Lastly, in Sec.~\ref{sec:c}, we shall conclude with a summary of the
main results.
Before we proceed further, a few clarifications concerning the conventions
and notations that we shall adopt are in order.
We shall work with natural units such that $\hbar=c=1$, and set the reduced
Planck mass to be $M_{_{\mathrm{Pl}}}=\left(8\,\pi\, G\right)^{-1/2}$.
We shall adopt the signature of the metric to be~$(-,+,+,+)$.
Note that Latin indices shall represent the spatial coordinates, except
for~$k$, which shall be reserved for denoting the wavenumber of the tensor
perturbations, i.e.~GWs.
We shall assume the background to be the spatially flat
Friedmann-Lema\^itre-Robertson-Walker (FLRW) line element described by the
scale factor~$a$ and the Hubble parameter~$H$.
Also, an overprime shall denote differentiation with respect to the
conformal time~$\eta$.
We should mention that the frequency, say, $f$, is related to the wave
number~$k$ of the tensor perturbations through the relation
\begin{equation}
f = \frac{k}{2\,\pi}
= 1.55\times10^{-15}\,\left(\frac{k}{1\, \mathrm{Mpc}^{-1}}\right)\,\mathrm{Hz}
\label{eq:f}
\end{equation}
and, as convenient, we shall refer to the spectrum of GWs either in
terms of wave numbers or frequencies.
\section{Spectrum of GWs generated during inflation}\label{sec:pgw}
In this section, we shall briefly recall the equation governing the tensor
perturbations and arrive at the spectrum of GWs generated in de Sitter
inflation.
We shall also introduce the dimensionless energy density of GWs, which is
the primary observational quantity of interest in this work.
\subsection{Generation of GWs during inflation}
Let $h_{ij}$ denote the tensor perturbations characterizing the GWs in a
FLRW universe.
When these tensor perturbations are taken into account, the line-element
describing the spatially flat, FLRW universe can be expressed
as~\cite{Maggiore:1999vm}
\begin{equation}
{\rm d} s^2
= a^2(\eta)\,\biggl\{-{\rm d}\eta^2+\left[\delta_{ij}+h_{ij}(\eta,{\bm x})\right]\,
{\rm d} x^i {\rm d} x^j\biggr\}.
\end{equation}
Since the tensor perturbations are transverse and traceless, they satisfy
the conditions $\partial^i\,h_{ij}=0$ and $h^i_i=0$.
We shall assume that no anisotropic stresses are present.
In such a case, the first order Einstein's equations then lead to the following
equation of motion for the tensor perturbations~$h_{ij}$:
\begin{equation}
h_{ij}'' + 2\,\frac{a'}{a}\,h_{ij}'
-{\bm \nabla^2}\,h_{ij}=0.
\end{equation}
On quantization, the tensor perturbations can be decomposed in terms of the
Fourier modes~$h_k$ as follows~\cite{Mukhanov:1990me,Martin:2003bt,
Martin:2004um,Bassett:2005xm,Sriramkumar:2009kg,Baumann:2008bn,Baumann:2009ds,
Sriramkumar:2012mik,Linde:2014nna,Martin:2015dha}:
\begin{eqnarray}
\hat{h}_{ij}(\eta, {\bm x})
&=& \int \frac{{\rm d}^{3}{\bm k}}{\left(2\,\pi\right)^{3/2}}\,
\hat{h}_{ij}^{\bm k}(\eta)\, \mathrm{e}^{i\,{\bm k}\cdot{\bm x}}\nn\\
&=& \sum_{\lambda={+,\times}}\int \frac{{\rm d}^{3}{\bm k}}{(2\,\pi)^{3/2}}\,
\left[\hat{a}^{\lambda}_{\bm k}\, \varepsilon^{\lambda}_{ij}({\bm k})\,
h_{k}(\eta)\, \mathrm{e}^{i\,{\bm k}\cdot{\bm x}}
+\hat{a}^{\lambda\dag}_{\bf k}\,\varepsilon^{\lambda\ast}_{ij}({\bm k})\,
h^{\ast}_{k}(\eta)\,\mathrm{e}^{-i\,{\bm k}\cdot{\bm x}}\right],\label{eq:tp-m-dc}
\end{eqnarray}
where the quantity $\varepsilon^{\lambda}_{ij}({\bm k})$ represents the
polarization tensor, with the index~$\lambda$ denoting the
polarization~$+$ or~$\times$ of the GWs.
The polarization tensor obeys the relations
$\delta^{ij}\,\varepsilon^{\lambda}_{ij}({\bm k})
=k^{i}\,\varepsilon_{ij}^\lambda({\bm k})=0$,
and we shall work with the normalization such
that $\varepsilon^{ij\,\lambda}({\bm k})\,
\varepsilon_{ij}^{\lambda'\ast}({\bm k})=2\,\delta^{\lambda\lambda'}$.
In the above decomposition, the operators $(\hat{a}_{\bm k}^{\lambda},
\hat{a}^{\lambda\dagger}_{\bm k})$ denote the annihilation and creation
operators corresponding to the tensor modes associated with the wave
vector~${\bm k}$.
They obey the following commutation
relations:~$[\hat{a}^{\lambda}_{\bm k},\hat{a}_{\bm k'}^{\lambda'}]
=[\hat{a}^{\lambda\dag}_{\bm k},\hat{a}_{\bm k'}^{\lambda'\dagger}]=0$ and
$[\hat{a}^{\lambda}_{\bm k},\hat{a}_{\bm k'}^{\lambda'\dagger}]
=\delta^{(3)}({\bm k}-{\bm k}')\, \delta^{\lambda\lambda'}$.
In the absence of sources with anisotropic stresses, the Fourier mode $h_k$
satisfies the differential equation
\begin{equation}
h_k''+2\,\frac{a'}{a}\,h_k'+k^2\,h_k=0.\label{eq:em-hk}
\end{equation}
The tensor power spectrum $\mathcal{P}_{_{\mathrm{T}}}(k)$ is defined through the relation
\begin{equation}
\langle 0 \vert {\hat h}^{ij}_{\bm k}(\eta)\,
{\hat h}_{ij}^{\bm k'}(\eta)\vert 0\rangle
=\frac{(2\,\pi)^2}{2\, k^3}\, \mathcal{P}_{_{\mathrm{T}}}(k)\;
\delta^{(3)}({\bm k}+{\bm k'}),\label{eq:tps-d}
\end{equation}
where the vacuum state $\vert 0\rangle$ is defined as
${\hat a}_{\bm k}^{\lambda}\vert 0\rangle=0$ for all ${\bm k}$ and $\lambda$.
On utilizing the above decomposition in terms of the Fourier modes, we
obtain that
\begin{equation}
\mathcal{P}_{_{\mathrm{T}}}(k)=4\,\frac{k^3}{2\, \pi^2}\, \vert h_k\vert^2,\label{eq:tps}
\end{equation}
and it is often assumed that the spectrum is evaluated on super-Hubble
scales during inflation.
Motivated by form of the second order action governing the tensor
perturbation~$h_{ij}$, the Fourier mode $h_k$ is usually written in
terms of the Mukhanov-Sasaki variable $u_k$ as $h_k=(\sqrt{2}/M_{_{\mathrm{Pl}}})\, (u_k/a)$.
The Mukhanov-Sasaki variable $u_k$ satisfies the differential
equation~\cite{Mukhanov:1990me,Martin:2003bt,Martin:2004um,
Bassett:2005xm,Sriramkumar:2009kg,Baumann:2008bn,Baumann:2009ds,
Sriramkumar:2012mik,Linde:2014nna,Martin:2015dha}
\begin{equation}
u_k''+ \left(k^2 - \frac{a''}{a}\right)\, u_k = 0.\label{inflation eom1}
\end{equation}
We shall focus on the simple case of slow roll inflation and work in the de Sitter
approximation wherein the scale factor is given by $a(\eta)=(1-H_{_{\mathrm I}}\,\eta)^{-1}$,
with $H_{_{\mathrm I}}$ denoting the constant Hubble scale during inflation.
In such a case, $a''/a=2\,H_{_{\mathrm I}}^2/\left(1-H_{_{\mathrm I}}\,\eta\right)^2$, and the
solution to Eq.~\eqref{inflation eom1} corresponding to the
Bunch-Davies initial condition is given by
\begin{equation}
u_k(\eta)
= \frac{1}{\sqrt{2\,k}}\,
\left[1 + \frac{i\,H_{_{\mathrm I}}\, a(\eta)}{k}\right]\,
\mathrm{e}^{-i\,k\,\eta}.\label{inflation solution1}
\end{equation}
Or, equivalently, we can write that
\begin{eqnarray}
h_k(\eta) = h_k(a)
= \frac{\sqrt{2}}{M_{_{\mathrm{Pl}}}}\,\frac{i\,H_{_{\mathrm I}}}{\sqrt{2\,k^3}}\,
\left[1-\frac{i\,k}{H_{_{\mathrm I}}\,a(\eta)}\right]\,\mathrm{e}^{-i\,k/H_{_{\mathrm I}}}\;
\mathrm{e}^{i\,k/[H_{_{\mathrm I}}\,a(\eta)]}\label{inflation solution2}
\end{eqnarray}
with $a(\eta)$ being given by the de Sitter form mentioned above.
Let us assume that inflation ends at the conformal time~$\eta_{\mathrm{f}}$ such
that $0< \eta_{\mathrm{f}} <H_{_{\mathrm I}}^{-1}$, and let $a_{\mathrm{f}}=a(\eta_{\mathrm{f}})$.
Under these conditions, upon using the above solution, the tensor
power spectrum at~$a_{\mathrm{f}}$ can be obtained to be
\begin{equation}
\mathcal{P}_{_{\mathrm{T}}}(k) = \frac{2\,H_{_{\mathrm I}}^2}{\pi^2\,M_{_{\mathrm{Pl}}}^2}\,\left(1+\frac{k^2}{k_{\mathrm{f}}^2}\right),
\end{equation}
where $k_{\mathrm{f}} =a_{\mathrm{f}}\,H_{_{\mathrm I}}$ is the mode that leaves the Hubble radius at
the end of inflation.
For $k \ll k_{\mathrm{f}}$, the above spectrum reduces to
\begin{equation}
\mathcal{P}_{_{\mathrm{T}}}(k) \simeq \frac{2\,H_{_{\mathrm I}}^2}{\pi^2\,M_{_{\mathrm{Pl}}}^2}\,\label{eq:pt-i},
\end{equation}
which is the well known scale invariant spectrum often discussed in the context
of de Sitter inflation~\cite{Mukhanov:1990me,Martin:2003bt,
Martin:2004um,Bassett:2005xm,Sriramkumar:2009kg,Baumann:2008bn,Baumann:2009ds,
Sriramkumar:2012mik,Linde:2014nna,Martin:2015dha}.
Actually, in slow roll inflation, the tensor power spectrum will contain a
small spectral tilt, which we shall choose to ignore in our discussion.
We should emphasize the point that the above scale invariant spectrum is valid
only for $k \ll k_{\mathrm{f}}$ since the de Sitter form for the scale factor would not
hold true close to the end of inflation.
Therefore, in our discussion below, we shall mostly restrict ourselves to wave
numbers such that $k < 10^{-2}\,k_{\mathrm{f}}$.
In Sec.~\ref{sec:actualinflation}, we shall evaluate tensor power spectrum
numerically in the inflationary model of interest to arrive at the present
day spectrum of GWs near~$k_{\mathrm{f}}$.
\subsection{A typical inflationary model of interest}
In order to illustrate the results later, we shall focus on a specific
inflationary model that permits slow roll inflation.
We shall consider the so-called $\alpha$-attractor model, which unifies
a large number of inflationary potentials (in this context, see
Refs.~\cite{Kallosh:2013hoa,Kallosh:2013yoa}).
If $\phi$ is the canonical scalar field driving inflation, the model is
described by the potential~$V(\phi)$ of the form
\begin{equation}
V(\phi) = \Lambda^4\,
\left[1 - \mathrm{exp}\left(-\sqrt{\frac{2}{3\,\alpha}}\,\frac{\phi}{M_{_{\mathrm{Pl}}}}\right)\right]^{2\,n},
\label{eq:aap}
\end{equation}
where, as we shall soon discuss, the scale $\Lambda$ can be determined
using the constraints from the observations of the anisotropies in the
CMB.
It is worth pointing out here that, for $n = 1$ and $\alpha = 1$,
the above potential reduces to the Higgs-Starobinsky
model~\cite{Starobinsky:1980te,Bezrukov:2007ep}.
We should also mention that the potential~\eqref{eq:aap} contains a
plateau at suitably large values of the field, which is favored by the
CMB data~\cite{Akrami:2018odb}.
Let~$N_k$ denote the e-fold at which the wave number $k$ leaves the Hubble
radius, \textit{when counted from the end of inflation}.\/
For the potential~\eqref{eq:aap}, the quantity $N_k$ can be easily determined
in the slow roll approximation to be (see, for instance, Ref.~\cite{Drewes:2017fmn})
\begin{equation}
N_k \simeq \frac{3\,\alpha}{4\,n}\,
\left[\mathrm{exp}\left({\sqrt{\frac{2}{3\,\alpha}}}\,\frac{\phi_k}{M_{_{\mathrm{Pl}}}}\right)
- \mathrm{exp}\left({\sqrt{\frac{2}{3\,\alpha}}}\,\frac{\phi_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}}\right)
- \sqrt{\frac{2}{3\,\alpha}}\,\left(\frac{\phi_k}{M_{_{\mathrm{Pl}}}} - \frac{\phi_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}}\right)\right],
\label{e-folding-alpha-attractor}
\end{equation}
where $\phi_k$ and $\phi_{\mathrm{f}}$ denote the values of the field when the mode with wave
number~$k$ crosses the Hubble radius and at the end of inflation, respectively.
It is easy to show that $\phi_{\mathrm{f}}$ can be expressed in terms of the parameters~$\alpha$
and~$n$ as
\begin{eqnarray}
\frac{\phi_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}}
\simeq \sqrt{\frac{3\,\alpha}{2}}\,\mathrm{ln}\left(\frac{2\,n}{\sqrt{3\,\alpha}} + 1\right).
\label{end of phi}
\end{eqnarray}
As we mentioned, the parameter~$\Lambda$ can be determined by utilizing the
constraints from the CMB.
One finds that the parameter can be expressed in terms of the scalar
amplitude~$\mathcal{A}_{_{\mathrm{S}}}$, the scalar spectral index~$n_{_{\mathrm{S}}}$ and the tensor-to-scalar
ratio~$r$ as follows:
\begin{equation}
\Lambda = M_{_{\mathrm{Pl}}}\, \left(\frac{3\,\pi^2\, r\,\mathcal{A}_{_{\mathrm{S}}}}{2}\right)^{1/4}\,
\left[\frac{2\,n\,(1+2\,n)
+ \sqrt{4\,n^2
+ 6\,\alpha\,(1+n)\,\left(1-n_{_{\mathrm{S}}}\right)}}{4\,n\,(1+n)}\right]^{n/2}.\label{lambda}
\end{equation}
Given the best-fit values for the inflationary parameters~$\mathcal{A}_{_{\mathrm{S}}}$ and $n_{_{\mathrm{S}}}$ as
well as the upper bound on~$r$ from Planck~\cite{Akrami:2018odb}, evidently,
using the above relation, we can choose a set of values for the parameters~$\Lambda$,
$\alpha$ and~$n$ describing the potential~\eqref{eq:aap} that are compatible
with the CMB data.
As we shall see, to evolve the background beyond inflation, we shall require
the value of the energy density of the scalar field at the end of inflation,
say, $\rho_\mathrm{f}$.
One finds that the quantity $\rho_\mathrm{f}$ can be expressed in terms of
the corresponding value of the potential, say, $V_\mathrm{f}$, as
\begin{equation}
\rho_\mathrm{f}= \frac{3}{2}\,V_\mathrm{f}.\label{eq:rhof}
\end{equation}
It is useful to note here that the value of the potential~\eqref{eq:aap} at
the end of inflation is given by
\begin{eqnarray}
V_\mathrm{f} = V(\phi_{\mathrm{f}})
=\Lambda^4\,\left(\frac{2\,n}{2\,n
+ \sqrt{3\,\alpha}}\right)^{2\,n}.\label{end of potential}
\end{eqnarray}
We should mention that, hereafter, we shall set the parameter $\alpha$ to be
unity.
\subsection{The dimensionless energy density of GWs}
Let us turn to now discuss the observable quantity of our interest, viz.
the dimensionless energy density of GWs.
We shall be interested in evaluating the quantity over the domain of wave
numbers which reenter the Hubble radius during the epochs of reheating and
radiation domination.
The energy density of GWs at any given time, say, $\rho_{_{\mathrm{GW}}}(\eta)$, is given
by (see, for example, Refs.~\cite{Boyle:2005se,Maggiore:1999vm})
\begin{equation}
\rho_{_{\mathrm{GW}}}(\eta)=\frac{M_{_{\mathrm{Pl}}}^2}{4\, a^2}\, \left(\frac{1}{2}\,\langle {\hat h}_{ij}^{\prime 2}\rangle
+\frac{1}{2}\,\langle \vert{\bm \nabla} {\hat h}_{ij}\vert^2\rangle\right),
\end{equation}
where the expectation values are to be evaluated in the initial Bunch-Davies
vacuum imposed in the sub-Hubble regime during inflation.
The energy density per logarithimic interval, say, $\rho_{_{\mathrm{GW}}}(k,\eta)$, is
defined through the relation
\begin{equation}
\rho_{_{\mathrm{GW}}}(\eta)=\int_{0}^{\infty}{\rm d}\, \mathrm{ln}\, k\; \rho_{_{\mathrm{GW}}}(k,\eta).
\end{equation}
Upon using the mode decomposition~\eqref{eq:tp-m-dc} and the above expressions
for $\rho_{_{\mathrm{GW}}}(\eta)$, we obtain the quantity~$\rho_{_{\mathrm{GW}}}(k,\eta)$ to be
\begin{equation}
\rho_{_{\mathrm{GW}}}(k,\eta)=\frac{M_{_{\mathrm{Pl}}}^2}{a^2}\, \frac{k^3}{2\,\pi^2}\,
\left(\frac{1}{2}\,\vert h_k'(\eta)\vert^2+\frac{k^2}{2}\,\vert h_k(\eta)\vert^2\right).
\label{eq:rho-gw-at}
\end{equation}
The corresponding dimensionless energy density $\Omega_{_{\mathrm{GW}}}(k,\eta)$ is defined as
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k,\eta)=\frac{\rho_{_{\mathrm{GW}}}(k,\eta)}{\rho_\mathrm{c}(\eta)}
=\frac{\rho_{_{\mathrm{GW}}}(k,\eta)}{3\,H^2\,M_{_{\mathrm{Pl}}}^2},
\end{equation}
where $\rho_\mathrm{c}=3\,H^2\,M_{_{\mathrm{Pl}}}^2$ is the critical density at time~$\eta$.
The observable quantity of interest is the dimensionless energy
density~$\Omega_{_{\mathrm{GW}}}(k,\eta)$ evaluated \textit{today},\/ which we shall
denote as $\Omega_{_{\mathrm{GW}}}(k)$.
We shall often refer to $\Omega_{_{\mathrm{GW}}}(k)$ or, equivalently, $\Omega_{_{\mathrm{GW}}}(f)$ [recall
that $f$ is the frequency associated with the wave number $k$, cf.
Eq.~\eqref{eq:f}], as the spectrum of GWs today.
In the following sections, we shall evolve the tensor perturbations
through the epochs of reheating and radiation domination.
As we shall see, at late times during radiation domination, once all
the wave numbers of interest are well inside the Hubble radius, the
energy density of GWs behave in a manner similar to that of the
energy density of radiation.
We shall utilize this behavior to arrive at the spectrum of GWs today.
\section{Evolution of GWs during reheating}\label{sec:egw-drh}
In order to follow the evolution of the tensor perturbations post
inflation, it proves to be convenient to write (in this context,
see, for instance, Ref.~\cite{Boyle:2005se})
\begin{equation}
h_k(\eta) = h_k^{^{_{\mathrm{P}}}}\,\chi_k(\eta),\label{transfer function}
\end{equation}
where $h_k^{^{_{\mathrm{P}}}}$ denotes the primordial value evaluated
at the end of inflation, and is given by [cf. Eq.~\eqref{inflation solution2}]
\begin{equation}
h_k^{^{_{\mathrm{P}}}}
=h_k(a_{\mathrm{f}})
=\frac{\sqrt{2}}{M_{_{\mathrm{Pl}}}}\,\frac{i\,H_{_{\mathrm I}}}{\sqrt{2\,k^3}}\,
\left(1-\frac{i\,k}{k_{\mathrm{f}}}\right)\,\mathrm{e}^{-i\,k/H_{_{\mathrm I}}}\;
\mathrm{e}^{i\,k/k_{\mathrm{f}}}.\label{eq:hkp}
\end{equation}
The quantity $\chi_k$ is often referred to as the tensor transfer function,
which obeys same equation of motion as $h_k$, viz. Eq.~\eqref{eq:em-hk}.
Clearly, the strength of the primordial GWs observed today will not only
depend on the amplitude of the tensor perturbations generated during
inflation, but also on their evolution during the subsequent epochs.
In this section, we shall discuss the evolution of the transfer function $\chi_k$
during the epoch of reheating.
As we mentioned earlier, we shall consider two types of scenarios for reheating.
We shall first consider the case wherein the period of reheating is described by
the constant EoS parameter, say, $w_\phi$, often associated with the coherent
oscillations of the scalar field around the minimum of the inflationary potential.
We shall assume that the transition to radiation domination occurs instantaneously
after a certain duration of time.
In such a case, as we shall see, the transfer function $\chi_k$ can be arrived
at analytically.
We shall then consider the scenario of perturbative reheating wherein there is a
gradual transfer of energy from the inflaton to radiation.
It seems difficult to treat such a situation analytically and, hence, we shall
examine the problem numerically.
To follow the evolution of GWs in the post-inflationary regime, we shall
choose to work with rescaled scale factor as the independent variable rather
than the conformal time coordinate.
If we define $A=a/a_{\mathrm{f}}$, where $a_{\mathrm{f}}=a(\eta_{\mathrm{f}})$ is the scale factor at the end
of inflation, then we find that the equation governing the transfer function
$\chi_k$ is given by
\begin{equation}
\frac{{\rm d}^2\chi_k}{{\rm d} A^2} + \left(\frac{4}{A}+\frac{1}{H}\,\frac{{\rm d} H}{{\rm d} A}\right)\,
\frac{{\rm d}\chi_k}{{\rm d} A}
+ \frac{(k/k_{\mathrm{f}})^2}{(H/H_{_{\mathrm I}})^2\,A^4}\,\chi_k=0,\label{eq:de-tf}
\end{equation}
where, recall that, $k_{\mathrm{f}}=a_{\mathrm{f}}\,H_{_{\mathrm I}}$ denotes the wave number that leaves the
Hubble radius at the end of inflation.
Our aim now is to solve the above equation for the transfer function during
the epoch of reheating.
As is evident from the equation, the evolution of GWs is dictated by the
behavior of the Hubble parameter.
Therefore, if we can first determine the behavior of the Hubble parameter during
reheating, we can solve the above equation to understand the evolution of
transfer function~$\chi_k$ during the epoch.
We shall also require the initial conditions for the transfer function~$\chi_k$
and its derivative ${\rm d} \chi_k/{\rm d} A$ at the end of inflation, i.e. when $a=a_{\mathrm{f}}$
or, equivalently, when $A=1$.
Since we have introduced the transfer function through
the relation~\eqref{transfer function}, clearly,
\begin{equation}
\chi_k^{_\mathrm{I}}(A=1)=1.\label{boundary condition inflation1}
\end{equation}
Also, on using the solution~\eqref{inflation solution2} and the
expression~\eqref{eq:hkp} for $h_k^{^{_{\mathrm{P}}}}$, we find that
\begin{equation}
\frac{{\rm d} \chi_k^{_\mathrm{I}}(A=1)}{{\rm d} A} =-\frac{(k/k_{\mathrm{f}})^2}{1-i\,(\,k/k_{\mathrm{f}})}
\simeq 0,\label{boundary condition inflation2}
\end{equation}
where the final equality is applicable when $k \ll k_{\mathrm{f}}$.
We shall make use of these initial conditions to determine the transfer
function during the epoch of reheating.
In the following two sub-sections, we shall discuss the solutions in the two
scenarios involving instantaneous and gradual transfer of energy from the
inflaton to radiation.
\subsection{Reheating described by an averaged EoS parameter}\label{subsec:averaged}
Let us first consider the scenario wherein the epoch of reheating is dominated
by the dynamics of the scalar field as it oscillates at the bottom of an
inflationary potential, such as the $\alpha$-attractor model~\eqref{eq:aap}
we had introduced earlier.
In such a case, the evolution of the scalar field can be described by an
averaged EoS parameter, say,~$w_\phi$~\cite{Martin:2010kz,Dai:2014jja,Cook:2015vqa}.
The conservation of energy implies that the energy density of the inflaton
behaves as $\rho_\phi \propto a^{-3\,(1+w_\phi)}$, which, in turn,
implies that the Hubble parameter behaves as
$H^2 =H_{_{\mathrm I}}^2\, A^{-3\,(1 + w_\phi)}$.
Note that, $H=H_{_{\mathrm I}}$ when $A=1$, as required.
As a result, during such a phase, equation~\eqref{eq:de-tf}
governing the tensor transfer function reduces to the form
\begin{equation}
\frac{{\rm d}^2\chi_k}{{\rm d} A^2}
+ (5-3\,w_\phi)\,\frac{1}{2\,A}\,\frac{{\rm d}\chi_k}{{\rm d} A}
+ \frac{(k/k_{\mathrm{f}})^2}{A^{1-3\,w_\phi}}\,\chi_k = 0.
\label{transfer equation reheating case1}
\end{equation}
The general solution to this differential equation can be expressed as
\begin{equation}
\chi_k^{_\mathrm{RH}}(A)
=A^{-\nu}\, \left[C_k\, J_{-\nu/\gamma}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\,A^{\gamma}\right)
+ D_k\,J_{\nu/\gamma}\left(\frac{k}{\gamma\,k_{\mathrm{f}}}\,A^{\gamma}\right)\right],
\label{transfer solution reheating case1}
\end{equation}
where $J_{\alpha}(z)$ denote Bessel functions of order $\alpha$, while
the quantities $\nu$ and $\gamma$ are given by
\begin{equation}
\nu=\frac{3}{4}\,(1-w_\phi),\quad
\gamma=\frac{1}{2}\,\left(1+3\,w_\phi\right).\label{eq:n-g}
\end{equation}
The coefficients $C_k$ and $D_k$ can be arrived at by using the
conditions~\eqref{boundary condition inflation1}
and~\eqref{boundary condition inflation2} for the transfer function
$\chi_k$ and its derivative ${\rm d} \chi_k/{\rm d} A$ at end of inflation, i.e. when $A=1$.
We find that the coefficients $C_k$ and $D_k$ can be expressed as follows:
\begin{subequations}\label{eq:Ck-Dk}
\begin{eqnarray}
C_k &=&\frac{\pi\,k}{2\,\gamma\,k_{\mathrm{f}}}\, \left[\frac{1}{1-i\,(k/k_{\mathrm{f}})}\right]\,
\left[\frac{k}{k_{\mathrm{f}}}\;J_{\nu/\gamma}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\right)
- \left(1-\frac{i\, k}{k_{\mathrm{f}}}\right)\,J_{(\nu/\gamma)+1}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\right) \right]\,
\csc\left(\frac{\pi\,\nu}{\gamma}\right),\\
D_k &=& -\frac{\pi\,k}{2\,\gamma\,k_{\mathrm{f}}}\, \left[\frac{1}{1-i\,(k/k_{\mathrm{f}})}\right]\,
\left[\frac{k}{k_{\mathrm{f}}}\;J_{-\nu/\gamma}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\right)
+ \left(1-\frac{i\,k}{k_{\mathrm{f}}}\right)\,J_{-(\nu/\gamma)-1}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\right) \right]\,
\csc\left(\frac{\pi\,\nu}{\gamma}\right).
\end{eqnarray}
\end{subequations}
We shall later make use of these coefficients and the
solution~\eqref{transfer solution reheating case1} to eventually arrive at
the spectrum of GWs today.
It is useful to note here that the quantity ${\rm d} \chi_k^{_\mathrm{RH}}/{\rm d} A$
is given by
\begin{equation}
\frac{{\rm d} \chi_k^{_\mathrm{RH}}}{{\rm d} A}
=\frac{k}{k_{\mathrm{f}}}\, A^{-1+\gamma-\nu}\,
\left[C_k\, J_{-(\nu/\gamma)-1}\left(\frac{k}{\gamma\, k_{\mathrm{f}}}\,A^{\gamma}\right)
- D_k\,J_{(\nu/\gamma)+1}\left(\frac{k}{\gamma\,k_{\mathrm{f}}}\,A^{\gamma}\right)\right].
\label{eq:dchi-dA}
\end{equation}
\color{black}
We should mention here that the duration of the reheating phase characterized by
the number of e-folds~$N_{\mathrm{re}}$ and the reheating temperature~$T_{\mathrm{re}}$ can be expressed
in terms of the equation of state parameter~$w_\phi$ and the inflationary
parameters as follows (in this context, see, for example,
Refs.~\cite{Dai:2014jja,Cook:2015vqa}):
\begin{subequations}
\begin{eqnarray}
N_{\mathrm{re}} &=& \frac{4}{(3\,w_\phi-1)}
\left[N_\ast
+\frac{1}{4}\,\ln \left(\frac{30}{\pi^2\,g_\mathrm{r, re}}\right)
+ \frac{1}{3}\ln \left(\frac{11\,g_{s,\mathrm{re}}}{43}\right) + \ln \left(\frac{k_\ast}{a_0\,T_0}\right)
+ \ln \left(\frac{\rho_\mathrm{f}^{1/4}}{H_{_{\mathrm I}}}\right)\right],\label{reheating e-folding}\\
T_{\mathrm{re}} &=& \left(\frac{43}{11\,g_{s,\mathrm{re}}}\right)^{1/3}
\left(\frac{a_0\,H_{_{\mathrm I}}}{k_\ast}\right)\,
\mathrm{e}^{-(N_\ast + N_{\mathrm{re}})}\;
T_0,\label{eq:Tre}
\end{eqnarray}
\end{subequations}
where $T_0 = 2.725\,\mathrm{K}$ is the present temperature of the CMB and
$H_0$~denotes the current value of the Hubble parameter.
Moreover, note that, $k_\ast/a_0$ represents the CMB pivot scale, with $a_0$
denoting the scale factor today.
We shall assume that $k_\ast/a_0 \simeq 0.05\,\mathrm{Mpc}^{-1}$.
We should also point out that $N_\ast$ denotes the number of e-folds prior to
the end of inflation when the pivot scale~$k_\ast$ leaves the Hubble radius.
\subsection{The case of perturbative reheating}\label{subsec:actual}
As a second possibility, we shall consider the perturbative
reheating scenario (for recent discussions, see, for instance,
Refs.~\cite{Maity:2018dgy,Haque:2020zco}).
In such a case, after inflation, the inflaton energy density, say,
$\rho_\phi$, gradually decays into the radiation energy density, say,
$\rho_{_{\mathrm{R}}}$, with the decay process being governed by the Boltzmann equations.
As a result, the effective EoS parameter during the reheating phase becomes
time dependent.
In our analysis below, for simplicity, we shall assume that the EoS
parameter~$w_\phi$ of the scalar field during the reheating phase
remains constant.
Such an assumption is valid as far as the oscillation time scale of the
inflaton is much smaller than the Hubble time scale.
This turns out to be generally true when the field is oscillating immediately
after the end of inflation, and we should point out that such a behavior has
also been seen in lattice
simulations~\cite{Podolsky:2005bw,Figueroa:2016wxr,Maity:2018qhi}.
Hence, for a wide class of inflationary potentials which behave as $V(\phi)
\propto \phi^{2\,n}$ near the minimum, the time averaged EoS parameter of the
inflaton can be expressed as $w_\phi = (n-1)/(n+1)$~\cite{Mukhanov:2005sc}.
We should emphasize that this scenario is different from the one considered
in the previous section wherein the explicit decay of the inflaton field was
not taken into account.
Specifically, in the earlier case, the energy density of the inflaton was
supposed to be converted instantaneously into the energy density of radiation
at a given time, leading to the end of the phase of reheating.
In due course, we shall demonstrate the manner in which the detailed mechanism
of reheating leaves specific imprints on the spectrum of primordial GWs observed
today.
Let us define the following dimensionless variables to describe the comoving
energy densities of the scalar field and radiation
\begin{equation}\label{rescale}
\Phi(A) = \frac{\rho_\phi}{m_\phi^4}\, A^{3\,(1 + w_\phi)}, \quad
R(A) = \frac{\rho_{_{\mathrm{R}}}}{m_\phi^4}\, A^4,
\end{equation}
where $m_\phi$ denotes the mass of the inflaton.
Also, let $\Gamma_\phi$ represent the decay rate of the
inflaton to radiation.
In such a case, the Boltzmann equations governing the evolution of the energy
densities $\rho_\phi$ and $\rho_{_{\mathrm{R}}}$ can be
expressed as~\cite{Maity:2018dgy,Haque:2020zco,Haque:2020bip}
\begin{subequations}\label{eq:be}
\begin{eqnarray}
\frac{{\rm d}\Phi}{{\rm d} A} + \frac{\sqrt{3}\, M_{_{\mathrm{Pl}}}\, \Gamma_\phi}{m_\phi^2}\, (1 + w_\phi)\;
\frac{A^{1/2}\,\Phi}{(\Phi/A^{3\, w_\phi}) + (R/A)} &=& 0,\label{Boltzmann equation1}\\
\frac{{\rm d} R}{{\rm d} A} - \frac{\sqrt{3}\, M_{_{\mathrm{Pl}}}\, \Gamma_\phi}{m_\phi^2}\, (1 + w_\phi)\;
\frac{A^{3\, (1 - 2\,w_\phi)/2}\,\Phi}{(\Phi/A^{3\, w_\phi})
+ (R/A)} &=& 0.\label{Boltzmann equation2}
\end{eqnarray}
\end{subequations}
Note that the tensor transfer function $\chi_k$ is essentially governed by
the behavior of the Hubble parameter~$H$ [cf. Eq.~\eqref{eq:de-tf}].
It proves to be difficult to solve the above set of equations analytically.
Therefore, to arrive at the Hubble parameter during the epoch of reheating, we
shall solve the Boltzmann equations~\eqref{eq:be} numerically with the following
conditions imposed at the end of inflation:
\begin{equation}
\rho_\phi(A=1) = \rho_\mathrm{f}=\frac{3}{2}\,V_\mathrm{f},\quad
\rho_{_{\mathrm{R}}}(A=1)= 0,\label{boundary Boltzmann equation1}
\end{equation}
where, recall that, $V_\mathrm{f}$ is the value of the inflationary potential
at the end of inflation.
We should point out that, for the inflationary potential~\eqref{eq:aap}
of our interest, $V_\mathrm{f}$ is given by Eq.~\eqref{end of potential} .
We should also clarify a couple of points in this regard.
In the case of perturbative reheating, the phase of reheating is assumed to
be complete when $H = \Gamma_\mathrm{\phi}$.
In other words, we require
\begin{equation} \label{reheating 1}
H^2(A_{\mathrm{re}})
= \frac{1}{3\,M_{_{\mathrm{Pl}}}^2}\,
\left[\rho_\mathrm{\phi}(A_{\mathrm{re}},n_{_{\mathrm{S}}},\Gamma_\mathrm{\phi})
+ \rho_{_{\mathrm{R}}}(A_{\mathrm{re}},n_{_{\mathrm{S}}},\Gamma_\phi)\right]= \Gamma_\phi^2,
\end{equation}
where $A_{\mathrm{re}}=a_\mathrm{re}/a_{\mathrm{f}}$, with $a_\mathrm{re}$ denoting the scale factor
when reheating has been achieved.
In fact, this corresponds to the point in time when the rate of transfer of
the energy from the inflaton to radiation is the maximum.
The associated reheating temperature can then be determined from the energy
density of radiation~$\rho_{_{\mathrm{R}}}$ through the relation
\begin{equation} \label{reheating 2}
T_{\mathrm{re}}= \left(\frac{30}{\pi^2\, g_{r, \mathrm{re}}}\right)^{1/4}\,
\rho_{_{\mathrm{R}}}^{1/4}(A_{\mathrm{re}},n_{_{\mathrm{S}}},\Gamma_\phi).
\end{equation}
In fact, later, to arrive at the results on the spectrum of GWs today, along
with specific values for the parameters describing the inflationary potential
(that are consistent with the CMB data), we shall also choose a value of~$T_{\mathrm{re}}$.
Having fixed the value of $T_{\mathrm{re}}$, we shall make use of Eq.~\eqref{eq:Tre} to
arrive at $N_{\mathrm{re}}$ and thereby determine the value $\Gamma_\phi$ using the
condition~$H(A_{\mathrm{re}})=\Gamma_\phi$.
With the solutions to the coupled background equations~\eqref{eq:be} at hand, we
shall proceed to solve the differential equation~\eqref{eq:de-tf} for the transfer
function~$\chi_k$ during reheating.
To illustrate the nature of the solutions, we have plotted the behavior of the Hubble
parameter~$H$ and the transfer function~$\chi_k$ in Fig.~\ref{plot_solution_reheating}
during the period of perturbative reheating.
We have chosen specific values for the EoS parameter~$w_\phi$, the reheating
temperature~$T_{\mathrm{re}}$, and wave number~$k$ in plotting the figures.
\begin{figure}[t!]
\centering
\includegraphics[width=8.25cm]{H-reheating.pdf}
\hskip 5pt
\includegraphics[width=8.00cm]{tr-reheating.pdf}
\caption{The evolution of the Hubble parameter~$H$ (in blue, on the left)
and the amplitude of the tensor transfer function~$\chi_k$ for a given wave
number (in blue, on the right), obtained numerically in the case of the
perturbative reheating scenario, have been plotted as a function of $A=a/a_{\mathrm{f}}$
over the domain $1 \leq A \leq A_{\mathrm{re}}$, which corresponds to the period of
reheating.
We have also indicated the lapse in time in terms of e-folds (counted from the
end of inflation) on the top of the two figures.
We have assumed that $w_{\phi} = 0$ and have set $T_{\mathrm{re}} = 10^{3}\,\mathrm{GeV}$
in plotting the quantities.
We have chosen the wave number to be $k\simeq 2\times 10^{12}\,\mathrm{Mpc}^{-1}$,
which corresponds to the frequency of $f\simeq 3\times10^{-3}\,\mathrm{Hz}$ [cf.
Eq.~\eqref{eq:f}].
The wave number has been chosen so that it renters the Hubble radius during the
epoch of reheating.
The slope of the straight line describing $H(A)$ (on the left) is $-3/2$, which is
consistent with $w_\phi=0$.
We find that the slope changes as $A$ approaches $A_{\mathrm{re}}$ indicating the beginning of
the transition to the radiation dominated epoch.
The vertical line (in red, on the right) indicates the time when the mode reenters
the Hubble radius.
As expected, the transfer function proves to be constant on super-Hubble scales and
it oscillates once the mode is inside the Hubble radius.}
\label{plot_solution_reheating}
\end{figure}
We have considered a wave number so that the mode reenters the Hubble radius
during the period of reheating.
As one would have expected, while the amplitude of the transfer function is a
constant on super-Hubble scales, it decreases as the mode reenters the Hubble
radius and begins to oscillate once inside.
\section{Evolution during radiation domination and the spectrum
of GWs today}\label{sec:egw-drd}
The Hubble parameter during the radiation dominated epoch evolves as
\begin{eqnarray}
H^2=H_{\mathrm{re}}^2\, \frac{A_\mathrm{re}^4}{A^4}\label{connection1}
\end{eqnarray}
with $H_{\mathrm{re}}$ and $A_{\mathrm{re}}$ denoting the Hubble parameter and the rescaled
scale factor at the end of reheating, respectively.
During radiation domination, the transfer function is governed by the
equation
\begin{equation}
\frac{{\rm d}^2 \chi_k}{{\rm d} A^2}
+ \frac{2}{A}\,\frac{{\rm d}\chi_k}{{\rm d} A}
+ \frac{(k/k_{\mathrm{re}})^2}{A_{\mathrm{re}}^2}\,\chi_k = 0,\label{transfer equation radiation}
\end{equation}
where $k_{\mathrm{re}} = a_\mathrm{re}\,H_{\mathrm{re}}$ is the mode which reenters the Hubble
radius at the end of the reheating era.
The above differential equation can be immediately solved to arrive at
the following general solution:
\begin{equation}
\chi_k^{_\mathrm{RD}}(A)
= \frac{1}{A}\,\biggl\{E_k\, \mathrm{e}^{-i\,(k/k_{\mathrm{re}})\,\left[(A/A_{\mathrm{re}})-1\right]}
+ F_k\,\mathrm{e}^{i\,(k/k_{\mathrm{re}})\,\left[(A/A_{\mathrm{re}})-1\right]}\biggr\}.
\label{transfer solution radiation 1}
\end{equation}
The coefficients $E_k$ and $F_k$ need to be determined by matching this solution
and its derivative with the solution~\eqref{transfer solution reheating case1}
during the epoch of reheating and its derivative at~$A=A_{\mathrm{re}}$.
Upon carrying out the matching at the junction between the eras of reheating
and the radiation domination, we find that the coefficients $E_k$ and $F_k$
can be expressed as
\begin{subequations}\label{eq:Ek-Fk}
\begin{eqnarray}
E_k &=&\frac{A_{\mathrm{re}}}{2}\, \left[\left(1+ \frac{i\,k_{\mathrm{re}}}{k}\right)\,\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
+ \frac{i\,k_{\mathrm{re}}}{k}\,A_{\mathrm{re}}\,\frac{{\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})}{{\rm d} A}\right]
=i\,\frac{A_{\mathrm{re}}}{2}\ \frac{k_{\mathrm{re}}}{k}\,\mathcal{E}_k,\\
F_k &=& \frac{A_{\mathrm{re}}}{2}\, \left[\left(1- \frac{i\,k_{\mathrm{re}}}{k}\right)\,\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
- \frac{i\,k_{\mathrm{re}}}{k}\,A_{\mathrm{re}}\,\frac{{\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})}{{\rm d} A}\right]
=-i\,\frac{A_{\mathrm{re}}}{2}\ \frac{k_{\mathrm{re}}}{k}\,\mathcal{F}_k,
\end{eqnarray}
\end{subequations}
where we have also introduced the quantities $\mathcal{E}_k$ and
$\mathcal{F}_k$ which we shall make use of later.
At this stage, we should mention that the quantities $\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$
and ${\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})/{\rm d} A$ and hence the coefficients $E_k$ and~$F_k$
will depend on the details of the reheating mechanism.
In particular, they will be different for two the types of reheating mechanisms under
consideration.
Since we have set $h_k=h_k^{^{_{\mathrm{P}}}}\,\chi_k$, where $h_k^{^{_{\mathrm{P}}}}$
denotes the primordial amplitude of the tensor perturbations
[cf. Eq.~\eqref{eq:rho-gw-at}], the energy density of GWs $\rho_{_{\mathrm{GW}}}(k,\eta)$ is
given by [cf. Eq.~\eqref{transfer function}]
\begin{equation}
\rho_{_{\mathrm{GW}}}(k,\eta)=\frac{M_{_{\mathrm{Pl}}}^2}{a^2}\,
\frac{k^3}{2\,\pi^2}\, \vert h_k^{^{_{\mathrm{P}}}}\vert^2\,
\left(\frac{1}{2}\,\vert {\chi_k^{_\mathrm{RD}}}'(\eta)\vert^2
+\frac{k^2}{2}\,\vert \chi_k^{_\mathrm{RD}}(\eta)\vert^2\right).
\end{equation}
On using the definition~\eqref{eq:tps} of the inflationary tensor power
spectrum~$\mathcal{P}_{_{\mathrm{T}}}(k)$, this energy density of GWs can be expressed as
\begin{equation}
\rho_{_{\mathrm{GW}}}(k,\eta)=\frac{M_{_{\mathrm{Pl}}}^2}{4\,a^2}\, \mathcal{P}_{_{\mathrm{T}}}(k)\,
\left(\frac{1}{2}\,\vert {\chi_k^{_\mathrm{RD}}}'(\eta)\vert^2
+\frac{k^2}{2}\,\vert \chi_k^{_\mathrm{RD}}(\eta)\vert^2\right).
\end{equation}
Upon substituting the solution~\eqref{transfer solution radiation 1} in the
above expression, we obtain $\rho_{_{\mathrm{GW}}}(k,\eta)$ to be
\begin{eqnarray}
\rho_{_{\mathrm{GW}}}(k,\eta)
&=& \frac{M_{_{\mathrm{Pl}}}^2\,k^2}{8\,a_{\mathrm{f}}^2\,A^4}\,\mathcal{P}_{_{\mathrm{T}}}(k)\,
\biggl\{\left(\vert E_k\vert^2 +\vert F_k \vert^2\right)\,
\left[2+\left(\frac{k_{\mathrm{re}}\,A_{\mathrm{re}}}{k\,A}\right)^2\right]\nn\\
& &+\,E_k\,F_k^\ast\,\left(\frac{k_{\mathrm{re}}\,A_{\mathrm{re}}}{k\,A}\right)^2\,
\left(1+\frac{2\,i\,k\,A}{k_{\mathrm{re}}\,A_{\mathrm{re}}}\right)\,
\mathrm{e}^{-2\,i\,(k/k_{\mathrm{re}})\,\left[(A/A_{\mathrm{re}})-1\right]}\nn\\
& &+\,E_k^\ast\,F_k\, \left(\frac{k_{\mathrm{re}}\,A_{\mathrm{re}}}{k\,A}\right)^2\,
\left(1-\frac{2\,i\,k\,A}{k_{\mathrm{re}}\,A_{\mathrm{re}}}\right)\,
\mathrm{e}^{2\,i\,(k/k_{\mathrm{re}})\,\left[(A/A_{\mathrm{re}})-1\right]}\biggr\}.
\end{eqnarray}
We had mentioned earlier that we shall be interested in the range of wave numbers
which reenter the Hubble radius during the epochs of reheating and radiation
domination.
At late times during radiation domination such that $A/A_{\mathrm{re}}\gg 1$, all the modes
of our interest would be well inside the Hubble radius, i.e. $k\, A \gg 1$.
In such a case, we find that, the above expression for $\rho_{_{\mathrm{GW}}}(k,\eta)$
simplifies to be
\begin{equation}
\rho_{_{\mathrm{GW}}}(k,\eta)
= \frac{M_{_{\mathrm{Pl}}}^2\,k^2}{4\,a_{\mathrm{f}}^2\,A^4}\,\mathcal{P}_{_{\mathrm{T}}}(k)\,
\left(\vert E_k\vert^2 +\vert F_k \vert^2\right).\label{eq:rho-gw-rd}
\end{equation}
Hence, the corresponding dimensionless parameter describing the energy
density of GWs, viz. $\Omega_{_{\mathrm{GW}}}(k,\eta)$, is given by
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k,\eta)
= \frac{k^2}{12\,a_{\mathrm{f}}^2\,H^2\,A^4}\,\mathcal{P}_{_{\mathrm{T}}}(k)\,
\left(\vert E_k\vert^2 +\vert F_k \vert^2\right)
= \frac{k_{\mathrm{re}}^2\,A_{\mathrm{re}}^2}{48\,a_{\mathrm{f}}^2\,H^2\,A^4}\,\mathcal{P}_{_{\mathrm{T}}}(k)\,
\left(\vert \mathcal{E}_k\vert^2
+\vert \mathcal{F}_k \vert^2\right),\label{PGW_spectrum_main}
\end{equation}
where we have made use of the expressions~\eqref{eq:Ek-Fk} relating
the coefficients $E_k$ and $F_k$ to the quantities $\mathcal{E}_k$
and $\mathcal{F}_k$.
During radiation domination, $H^2\,A^4=H_{\mathrm{re}}^2\,A_{\mathrm{re}}^4$.
Also, recall that, $k_{\mathrm{re}}=a_{\mathrm{re}}\, H_{\mathrm{re}}$.
On using these relations, at late times during radiation domination,
we obtain that
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k,\eta)
= \frac{\mathcal{P}_{_{\mathrm{T}}}(k)}{48}\, \left(\vert \mathcal{E}_k\vert^2
+\vert \mathcal{F}_k \vert^2\right).\label{PGW_spectrum_main2}
\end{equation}
The task that remains is to explicitly determine the quantities
$\mathcal{E}_k$ and $\mathcal{F}_k$.
In the special case of instantaneous reheating, $A_{\mathrm{re}}=1$ and $k_{\mathrm{re}}=k_{\mathrm{f}}$.
Therefore, on using the conditions~\eqref{boundary condition inflation1}
and~\eqref{boundary condition inflation2}, one can readily show that
\begin{equation}
\mathcal{E}_k
=\frac{1-2\,i\,(k/k_{\mathrm{f}})-2\,(k^2/k_{\mathrm{f}}^2)}{1-i\,(k/k_{\mathrm{f}})},\quad
\mathcal{F}_k=\frac{1}{1-i\,(k/k_{\mathrm{f}})}.
\end{equation}
For $k\ll k_{\mathrm{f}}$, we find that, $\mathcal{E}_k \simeq F_k \simeq 1$,
which lead to
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k,\eta)= \frac{\mathcal{P}_{_{\mathrm{T}}}(k)}{24}=\frac{H_{_{\mathrm I}}^2}{12\,\pi^2\,M_{_{\mathrm{Pl}}}^2},
\end{equation}
where, in arriving at the final expression, we have made use of the scale invariant
inflationary tensor power spectrum~\eqref{eq:pt-i}.
In other words, the dimensionless density parameter $\Omega_{_{\mathrm{GW}}}(k,\eta)$ is strictly scale
invariant over all wave numbers in the instantaneous reheating scenario.
Evidently, such a behavior can be expected to be hold true even when we have an
epoch of reheating with $w_\phi=1/3$, a result we shall encounter in due course.
Note that the energy density of GWs behaves as $a^{-4}$
[cf. Eq.~\eqref{eq:rho-gw-rd}], exactly as the energy density of radiation does.
Such a behavior should not come as a surprise and it arises due to the fact that
the modes of interest are well inside the Hubble radius at late times (say,
close to the epoch of radiation-matter equality) during radiation domination.
On utilizing this property, the dimensionless energy density
parameter~$\Omega_{_{\mathrm{GW}}}(k)$ \textit{today}\/ can be expressed in terms
of $\Omega_{_{\mathrm{GW}}}(k,\eta)$ as follows:
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k)\,h^2
= \left(\frac{g_{r,\mathrm{eq}}}{g_{r,0}}\right)\,
\left(\frac{g_{s,0}}{g_{s,\mathrm{eq}}}\right)^{4/3}\,
\Omega_{_{\mathrm{R}}}\, h^2\; \Omega_{_{\mathrm{GW}}}(k,\eta)
\simeq \left(\frac{g_{r,0}}{g_{r,\mathrm{eq}}}\right)^{1/3}\,
\Omega_{_{\mathrm{R}}}\, h^2\; \Omega_{_{\mathrm{GW}}}(k,\eta),\label{presentrelic}
\end{equation}
where $\Omega_{_\mathrm{R}}$ denotes the present day dimensionless
energy density of radiation.
We should mention that, while $g_{r,\mathrm{eq}}$ and $g_{r,0}$ represent
the number of relativistic degrees of freedom at equality and today,
respectively, $g_{s,\mathrm{eq}}$ and $g_{s,0}$ represent the number of
such degrees of freedom that contribute to the entropy at these epochs.
Further, the Hubble parameter today, as usual, has been expressed as
$H_0=100\,h\,\mathrm{km}\,\mathrm{sec}^{-1}\,\mathrm{Mpc}^{-1}$.
The spectrum of primordial GWs in the reheating scenario with an averaged EoS
parameter can be expected to be different when compared to the one arising in
the perturbative reheating scenario,
In the following two sections, we shall derive the spectrum of primordial GWs
at the present epoch in the two cases.
\section{Spectrum of GWs in reheating described by an averaged EoS
parameter}\label{sec:averaged}
Before we go on to discuss the results, we should mention that the
spectrum of GWs arising in the scenario wherein the epoch of reheating
is described by an averaged EoS parameter and the transition to radiation
domination is assumed to occur instantaneously at a given time has been
evaluated earlier in the literature (see, for instance,
Refs.~\cite{Turner:1993vb,Nakayama:2008wy,Nakayama:2009ce}; for a recent
discussion, see Ref.~\cite{Mishra:2021wkm}).
However, we find that, in the earlier investigations, the initial conditions
that determine the dynamics during reheating have not always been chosen to
be consistent with the dynamics during inflation.
In this work, we shall consider a specific model of inflation and we
shall show that model dependent initial conditions play a primary role
in determining the range of frequencies which reenter the Hubble radius
during reheating.
Therefore, in this section, we shall reanalyze the effect of the averaged
EoS parameter during reheating (with appropriate initial conditions) on
the spectrum of GWs.
We shall briefly discuss the derivation of the spectrum of GWs and arrive
at the shape of the spectrum in the domains $k < k_{\mathrm{re}}$ and $k > k_{\mathrm{re}}$.
In the next section, we shall compare the results with those that arise
in the case of perturbative reheating.
It should be clear from the expression~\eqref{PGW_spectrum_main2} that
we shall require the quantities~$\mathcal{E}_k$ and $\mathcal{F}_k$ to
arrive at the spectrum of GWs.
Using Eqs.~\eqref{eq:Ek-Fk}, we can express~$\mathcal{E}_k$
and $\mathcal{F}_k$ as
\begin{subequations}\label{eq:cal-Ek-Fk}
\begin{eqnarray}
\mathcal{E}_k
&=&\left(1-\frac{i\,k}{k_{\mathrm{re}}}\right)\,\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
+ A_{\mathrm{re}}\,\frac{{\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})}{{\rm d} A},\\
\mathcal{F}_k
&=& \left(1+\frac{i\,k}{k_{\mathrm{re}}}\right)\,\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
+ A_{\mathrm{re}}\,\frac{{\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})}{{\rm d} A}.
\end{eqnarray}
\end{subequations}
Also, recall that the transfer function at the end of the epoch
of reheating~$\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$ is given by
Eq.~\eqref{transfer solution reheating case1}, with the coefficients
$C_k$ and $D_k$ being described by Eqs.~\eqref{eq:Ck-Dk}.
During radiation domination, we have $H^2\,A^4=H_{\mathrm{re}}^2\,A_{\mathrm{re}}^4$.
Since $k_{\mathrm{f}}=a_{\mathrm{f}}\,H_{_{\mathrm I}}$ and $k_{\mathrm{re}}=a_{\mathrm{re}}\,H_{\mathrm{re}}$, we find that we can
write $A_{\mathrm{re}}=(k_{\mathrm{f}}/k_{\mathrm{re}})^{1/\gamma}$.
As a result, the Bessel functions in the
expression~\eqref{transfer solution reheating case1}
for~$\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$ depend on the
ratio $(k/k_{\mathrm{re}})$.
Note that, in contrast, the coefficients $C_k$ and $D_k$ depend
only on the ratio~$k/k_{\mathrm{f}}$.
As we mentioned earlier, we have been interested in arriving
at the spectrum over wave numbers such that $k< 10^{-2}\,k_{\mathrm{f}}$.
For small $z$, the Bessel function $J_\alpha(z)$ behaves
as (see, for instance, Ref.~\cite{gradshteyn2007})
\begin{equation}
\lim_{z \ll 1} J_\alpha(z)
\simeq \frac{1}{\Gamma(1+\alpha)}\,\left(\frac{z}{2}\right)^\alpha,
\label{eq:J-sz}
\end{equation}
where $\Gamma(z)$ denotes the Gamma function.
Clearly, in such a limit, the Bessel functions involving the largest
negative value for the index~$\alpha$ can be expected to dominate.
Since $0 \leq w_\phi \leq 1$, the quantities $\nu$ and
$\gamma$ are always positive [cf. Eq.~\eqref{eq:n-g}].
Hence, in the limit $k \ll k_{\mathrm{f}}$, we find that it is the term involving
$D_k$ in Eq.~\eqref{transfer solution reheating case1} that will dominate.
For the above reasons, the quantity~$\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$
can be approximated as follows:
\begin{equation}
\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
\simeq A_{\mathrm{re}}^{-\nu}\, D_k\, J_{\nu/\gamma}\left(\frac{k}{\gamma\,k_{\mathrm{re}}}\right),
\label{eq:chi-RH-Are-af}
\end{equation}
with the coefficient $D_k$ being given by
\begin{equation}
D_k \simeq -\frac{\pi}{\Gamma(-\nu/\gamma)}\,
\csc\left(\frac{\pi\,\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\, k_{\mathrm{f}}}\right)^{-\nu/\gamma}.\label{eq:Dk-sk}
\end{equation}
Let us first arrive at the shape of the spectrum in the domain $k \ll k_{\mathrm{re}}$.
In such a domain, we can use the form~\eqref{eq:J-sz} for the
Bessel function $J_{\nu/\gamma}[k/(\gamma\,k_{\mathrm{re}})]$ that
appears in the expression~\eqref{eq:chi-RH-Are-af} above
for $\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$.
On doing so and utilizing the identity $\Gamma(z)\,\Gamma(1-z)
=\pi/\mathrm{sin}\,(\pi\,z)$~\cite{gradshteyn2007}, we find
that, in the domain $k \ll k_{\mathrm{re}}$, the quantity
$\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$ reduces to unity.
Under the same conditions, we find that the quantity
${\rm d} \chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})/{\rm d} A$ vanishes.
Therefore, it should be evident from the expressions~\eqref{eq:cal-Ek-Fk}
that, in the limit $k \ll k_{\mathrm{re}}$, $\mathcal{E}_k\simeq \mathcal {F}_k
\simeq 1$.
In other words, the spectrum of GWs today is scale invariant over this domain
and its present day amplitude is given by
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k)\,h^2
\simeq \left(\frac{g_{r,0}}{g_{r,\mathrm{eq}}}\right)^{1/3}\,
\Omega_{_{\mathrm{R}}}\, h^2\; \frac{\mathcal{P}_{_{\mathrm{T}}}(k)}{24}
\simeq \Omega_{_{\mathrm{R}}}\, h^2\;
\frac{H_{_{\mathrm I}}^2}{12\,\pi^2M_{_{\mathrm{Pl}}}^2}.\label{eq:ogw-sk}
\end{equation}
In arriving at the final expression, we have assumed that
$g_{r,0}\simeq g_{r,\mathrm{eq}}$ and have made use of the tensor
power spectrum~\eqref{eq:pt-i} arising in de Sitter inflation.
We should mention that this result is the same as in the case of
instantaneous reheating.
This result should come as a surprise since these large scale modes
are on super-Hubble scales during the epoch of reheating and hence
are not influenced by it.
Let us now turn to the domain $k\gg k_{\mathrm{re}}$.
Since the limit $k\ll k_{\mathrm{f}}$ continues to be valid, the term involving
$D_k$ in Eq.~\eqref{transfer solution reheating case1} remains the
dominant term.
Therefore, $\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$ is again described by
Eq.~\eqref{eq:chi-RH-Are-af}, with $D_k$ given by Eq.~\eqref{eq:Dk-sk}.
However, the argument of the Bessel function in Eq.~\eqref{eq:chi-RH-Are-af}
is now large.
For large $z$, the Bessel function $J_\alpha(z)$ behaves as (see,
for instance, Ref.~\cite{gradshteyn2007})
\begin{equation}
\lim_{z \gg 1} J_\alpha(z)
\simeq \sqrt{\frac{2}{\pi\,z}}\,
\mathrm{cos}\left[z-\pi\,\alpha-(\pi/4)\right].\label{eq:J-lz}
\end{equation}
Therefore, in the domain $k\gg k_{\mathrm{re}}$, we find that the quantity
$\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})$ and its derivative
${\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})/{\rm d} A$ behave as
\begin{subequations}
\begin{eqnarray}
\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})
&\simeq& -\frac{1}{\sqrt{\pi}}\;\Gamma\left(1+\frac{\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}\right)^{-(\nu/\gamma)-(1/2)}\,
\mathrm{cos} \left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}-\frac{\pi\,\nu}{\gamma}-\frac{\pi}{4}\right),\\
A_{\mathrm{re}}\,\frac{{\rm d}\chi_k^{_\mathrm{RH}}(A_{\mathrm{re}})}{{\rm d} A}
&\simeq& -\frac{2\,\gamma}{\sqrt{\pi}}\;\Gamma\left(1+\frac{\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}\right)^{-(\nu/\gamma)+(1/2)}\,
\mathrm{sin} \left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}-\frac{\pi\,\nu}{\gamma}-\frac{\pi}{4}\right),
\end{eqnarray}
\end{subequations}
with $\nu$ and $\gamma$ being given by Eq.~\eqref{eq:n-g}.
On substituting these expressions in Eqs.~\eqref{eq:cal-Ek-Fk}, we
obtain the corresponding~$\mathcal{E}_k$ and~$\mathcal{F}_k$ to be
\begin{equation}
\mathcal{E}_k
\simeq \mathcal{F}_k^\ast
\simeq -\frac{2\,i\,\gamma}{\sqrt{\pi}}\;\Gamma\left(1+\frac{\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}\right)^{-(\nu/\gamma)+(1/2)}\,
\mathrm{exp}\; i\,\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}-\frac{\pi\,\nu}{\gamma}
-\frac{\pi}{4}\right)
\end{equation}
so that
\begin{equation}
\vert \mathcal{E}_k\vert^2 = \vert \mathcal{F}_k\vert^2
= \frac{4\,\gamma^2}{\pi}\, \Gamma^2\left(1+\frac{\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}\right)^{n_{_\mathrm{GW}}},
\end{equation}
where we have defined $n_{_\mathrm{GW}}$ to be
\begin{equation}
n_{_\mathrm{GW}}=1-\frac{2\,\nu}{\gamma}=-\frac{2\,(1-3\,w_\phi)}{1+3\,w_\phi}.
\end{equation}
If we substitute these results in the expression~\eqref{presentrelic},
we obtain the spectrum of GWs today in the domain $k\gg k_{\mathrm{re}}$ to be
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k)\,h^2
\simeq \left(\frac{g_{r,0}}{g_{r,\mathrm{eq}}}\right)^{1/3}\,
\Omega_{_{\mathrm{R}}}\, h^2\; \frac{\mathcal{P}_{_{\mathrm{T}}}(k)}{24}\,\vert \mathcal{E}_k\vert^2
\simeq \Omega_{_{\mathrm{R}}}\, h^2\;
\frac{H_{_{\mathrm I}}^2}{12\,\pi^2M_{_{\mathrm{Pl}}}^2}\,
\frac{4\,\gamma^2}{\pi}\, \Gamma^2\left(1+\frac{\nu}{\gamma}\right)\,
\left(\frac{k}{2\,\gamma\,k_{\mathrm{re}}}\right)^{n_{_\mathrm{GW}}}.\label{GW spectrum- second domain}
\end{equation}
In other words, for wave numbers such that $k\gg k_{\mathrm{re}}$, the spectrum of
GWs today has the index~$n_{_\mathrm{GW}}$.
Notably, the index vanishes when $w_\phi=1/3$.
Also, while the spectrum is blue for $w_\phi>1/3$, it is red for~$w_\phi< 1/3$.
Moreover, in the extreme cases wherein $w_\phi$ vanishes or is unity,
we have $n_{_\mathrm{GW}}=-2$ and $n_{_\mathrm{GW}}=1$, respectively.
On utilizing the expression~\eqref{transfer solution reheating case1} for
the transfer function during reheating and the expressions~\eqref{eq:Ek-Fk}
to determine the quantities~$\mathcal{E}_k$ and~$\mathcal{F}_k$, we can
arrive at the complete spectrum of GWs by substituting the expressions in
Eq.~\eqref{PGW_spectrum_main2}.
In Fig.~\ref{plot_analytic1}, we have plotted the spectrum of GWs today that
arise in the case of the $\alpha$-attractor model~\eqref{eq:aap} for a set of
values of the EoS parameter $w_\phi$.
\begin{figure}[t!]
\centering
\includegraphics[width=12.50cm]{differentomega.pdf}
\caption{The behavior of the dimensionless energy density of primordial GWs
observed today, viz. $\Omega_{_{\mathrm{GW}}}(f)$, has been plotted over a wide range of frequencies.
The spectrum has been obtained analytically and it corresponds to the case wherein
the post-inflationary phase is described by the EoS parameter~$w_\phi$
and reheating is expected to occur instantaneously at a given time.
We have considered the scenario wherein the inflationary potential is described
by the $\alpha$-attractor model~\eqref{eq:aap}.
We have illustrated the spectra for the cases wherein $n=(1,2,3,5)$ (in black,
brown, green and magenta), which correspond to $w_\phi=(0,1/3,1/2,2/3)$.
We should mention that we have chosen the parameters such that $T_{\mathrm{re}} = 3\,
\mathrm{GeV}$ in all the cases.
In the figure, we have also included the sensitivity curves of the different
ongoing and forthcoming GW observatories (in varied colors, on top).
Note that, as expected, the spectrum is strictly scale invariant for frequencies
such that $f < f_\mathrm{re}=k_{\mathrm{re}}/(2\,\pi)$.
However, for larger frequencies such that $f>f_\mathrm{re}$, while the spectrum
has a red tilt for $w_\phi < 1/3$, it has a blue tilt for $w_\phi> 1/3$.
Interestingly, we find that, for a suitably large value of $w_\phi$,
the spectrum of GWs already intersect the sensitivity curves of some of
the observatories over a certain range of frequencies.
Moreover, we find that, for a high value of $w_\phi$, the spectra cross the
BBN bound of $\Omega_{_{\mathrm{GW}}}\,h^2<10^{-6}$ at suitably large
frequencies.}\label{plot_analytic1}
\end{figure}
In plotting the spectra, we have chosen the other parameters in such a fashion
that the reheating temperature is $T_{\mathrm{re}} = 3\, \mathrm{GeV}$ in all the cases.
The figure clearly illustrates the qualitative features we discussed above:
(i)~the spectrum is strictly scale invariant for $k<k_{\mathrm{re}}$, and (ii)~the spectrum
has the index $n_{_\mathrm{GW}}$ for $k> k_{\mathrm{re}}$.
We have plotted the spectra for $w_{\phi} = (0,1/3,1/2,2/3)$, which correspond
to the values $n=(1,2,3,5)$ for the index in the potential~\eqref{eq:aap}.
We should mention that these cases lead to the indices $n_{_\mathrm{GW}} = (-2,0,2/5,2/3)$,
as expected.
In the figure, we have also included the sensitivity curves of the some of the
current and forthcoming GW observatories (for a discussion on the sensitivity
curves, see Ref.~\cite{Moore:2014lga} and the associated web-page).
Interestingly, we find that, for a set of inflationary and reheating parameters,
the spectra already intersect the sensitivity curves.
Moreover, we find that the BBN bound, viz. $\Omega_{_{\mathrm{GW}}}(k_{\mathrm{f}})\,h^2
\leq 10^{-6}$ (in this context, see, for instance, Ref.~\cite{Pagano:2015hma}
and the reviews~\cite{Caprini:2018mtu,Guzzetti:2016mkm}),
can be violated for $w_\phi>1/3$, which leads to constraints
on the EoS parameter~$w_\phi$ for a given~$k_{\mathrm{f}}$ and vice-versa.
As we have emphasized, Fig.~\ref{plot_analytic1} depicts the interesting
dependence of the value of~$k_{\mathrm{f}}$ on the inflationary model parameter~$n$
due to different initial conditions at the beginning of reheating.
This interdependence of $k_{\mathrm{f}}$ and the EoS parameter~$w_\phi$ can be translated
into the constraints on the reheating temperature~$T_{\mathrm{re}}$ and scalar spectral
index~$n_{_{\mathrm{S}}}$ through the aforementioned BBN bound.
For instance, in the figure, the spectrum corresponding to $w_\phi=2/3$
clearly crosses the BBN bound at large frequencies.
These clearly suggest that observations of the spectrum of GWs today can lead
to interesting constraints on the primordial physics.
\section{Spectrum of GWs in the case of perturbative reheating}\label{sec:actual}
In the perturbative reheating scenario, the inflaton continuously transfers
its energy to radiation after the end of the inflationary epoch.
As a result, the effective EoS parameter during the reheating era, say,
$w_\mathrm{eff}$, becomes time dependent.
It can be expressed as
\begin{eqnarray}
w_\mathrm{eff} = \frac{3\,w_{\phi}\,\rho_{\phi}
+ \rho_{_{\mathrm{R}}}}{3\,\left(\rho_{\phi} + \rho_{_{\mathrm{R}}}\right)},\label{effective EoS}
\end{eqnarray}
where, recall that, the evolution of the energy densities of the inflaton and
radiation, viz. $\rho_{\phi}$ and $\rho_{_{\mathrm{R}}}$, are governed by the Boltzmann
equations~\eqref{eq:be}, while $w_{\phi}$ is the EoS parameter describing
the inflaton.
Such a time dependence of the effective EoS parameter has been explicitly
demonstrated earlier (see, for example, Refs.~\cite{Maity:2018dgy,Haque:2020zco}).
It has been illustrated that, while immediately after the termination of
inflation, $w_\mathrm{eff}$ is approximately equal to $w_{\phi}$, after
a certain time, the effective EoS parameter smoothly transits from
$w_{\phi}$ to $1/3$, which indicates the onset of the epoch of radiation
domination.
We should emphasize again here that such a reheating scenario is different
from the case considered in the previous section where the inflaton energy
density is assumed to be converted instantaneously into radiation after a
certain period of time.
Specifically, in the previous reheating scenario, $w_\mathrm{eff}$
remains equal to $w_{\phi}$ during the whole of reheating era and,
at a particular time, $w_\mathrm{eff}$ sharply changes to $1/3$.
These differences in the dynamics of the reheating scenarios should be
reflected in spectrum of GWs today.
The corresponding features in the GW spectrum can, in principle, help us
probe the microscopic mechanisms operating during the era of reheating.
We shall now proceed to compute the spectrum of GWs at the present time,
i.e. $\Omega_{_{\mathrm{GW}}}(k)$ or, equivalently, $\Omega_{_{\mathrm{GW}}}(f)$, in the case of the perturbative
reheating scenario.
As we had discussed, we shall analyze this case numerically.
With the solution to the Hubble parameter~$H(A)$ at hand, we proceed to
solve for the transfer function~$\chi_k(A)$ during the epoch of reheating,
as we had outlined in section~\ref{sec:egw-drh}.
The numerical solutions are determined using the initial
conditions~\eqref{boundary condition inflation1}
and~\eqref{boundary condition inflation2}.
With the solutions at hand, we arrive at the spectrum of GWs at the
current epoch for different sets of reheating temperature and the EoS
parameter $w_\phi$ describing the inflaton.
The results we have obtained are illustrated in Figs.~\ref{plot_perturbative1}
and~\ref{plot_perturbative2} for the cases of $w_{\phi} < 1/3$
and~$w_{\phi} > 1/3$, respectively.
Let us now highlight a few points concerning the results plotted in the two
figures.
Let us first broadly understand the spectra in Fig.~\ref{plot_perturbative1}
wherein we have plotted the results for~$w_{\phi} = 0$.
\begin{figure}[!t]
\includegraphics[height=5.25cm,width=8.50cm]{variationofomegawithfco.pdf}
\hskip 5pt
\includegraphics[height=5.25cm,width=8.40cm]{higgsrelicplotdifferenttemperatureco.pdf}
\caption{The behavior of the dimensionless energy density of primordial GWs
today, viz. $\Omega_{_{\mathrm{GW}}}(f)$, has been plotted over a small (in black, on the left)
as well as a wide range of frequencies (in red, green, brown and black, on
the right).
We have considered the scenario wherein the inflationary potential is described
by the $\alpha$-attractor model~\eqref{eq:aap} with $n=1$,
which corresponds to $w_\phi = 0$.
We have plotted the spectrum of GWs for the following values of the reheating
temperature~$T_{\mathrm{re}}$:~$10^{10}\, \mathrm{GeV}$ (in red, on the right), $10^{6}\,
\mathrm{GeV}$ (in black on the left and green on the right), $2\, \mathrm{TeV}$
and $2\, \mathrm{GeV}$ (in brown and black on the right).
Note that, we have also illustrated the behavior of the effective EoS
parameter~$w_\mathrm{eff}$ (in blue, in the figure on the left)
as a function of the frequency~$f$, which has been determined using
the relation $f = a\,H/(2\,\pi)$.
In other words, $w_\mathrm{eff}(f)$ (marked on the $y$-axis on the right
hand side of the figure on the left) represents the effective EoS parameter at
the instant when the mode with the frequency~$f$ reenters the Hubble radius.
We have also indicated the frequencies associated with the wave numbers~$k_{\mathrm{re}}$
and $k_{\mathrm{f}}$ (as vertical red and green lines, on the left).
Moreover, we have demarcated the regime (in pink) wherein the transition from
$w_\mathrm{eff}=0$ to $w_\mathrm{eff}=1/3$ occurs.
We should point out that the spectrum of GWs exhibit oscillations in the region
of the transition.
Further, we have included the sensitivity curves of different ongoing and
forthcoming GW observatories (in the figure on the right).}\label{plot_perturbative1}
\end{figure}
In the figure, we have illustrated the dimensionless energy density of GWs today
as a function of the frequency~$f$.
We have considered the case wherein the inflationary potential is described by
the $\alpha$-attractor model~\eqref{eq:aap},
and have plotted the results for~$n=1$ (which corresponds to $w_\phi=0$)
and a set of values of~$T_{\mathrm{re}}$.
We have also included behavior of the effective EoS parameter $w_\mathrm{eff}$,
which we have plotted as a function of frequency using the relation $f=a\,H/(2\pi)$.
The plot indicates the evolution of the parameter $w_\mathrm{eff}$ as the modes
with different frequencies~$f$ reenter the Hubble radius (in this context, also
see the earlier efforts~\cite{Maity:2018dgy,Haque:2020zco}).
Note that larger wave numbers or, equivalently, larger frequencies reenter the
Hubble radius earlier than the smaller ones.
The plot clearly highlights the transition from the inflaton dominated universe to
the epoch of radiation domination, achieved through the mechanism of perturbative
reheating.
The transition is clearly reflected in the behavior of the effective EoS parameter
which changes smoothly from $w_\mathrm{eff}=0$ at early times (i.e. at large
frequencies) to $w_\mathrm{eff}=1/3$ at late times (i.e. at small frequencies).
In the figure, we have indicated the frequencies associated with the wave numbers
$k_{\mathrm{f}}$ and $k_{\mathrm{re}}$ and have also marked the domain of the transition to highlight
these points.
Let us now point out a few more aspects of the results presented in
Fig.~\ref{plot_perturbative1}.
In the figure, we have also plotted the spectrum of GWs for a few different values of
the reheating temperature.
Further, we have included the sensitivity curves of different current and forthcoming
GW observatories.
Note that the plots suggest that the spectra of GWs remain scale invariant over
wave numbers~$k < k_{\mathrm{re}}$ which reenter the Hubble radius during the radiation
dominated epoch.
This result should not come as surprise.
As we have pointed our earlier, these modes are on super-Hubble scales during the
period of reheating and hence are unaffected by the process.
Therefore, they carry the scale invariant nature of the spectrum of GWs generated
during inflation.
However, modes with wave numbers $k_{\mathrm{re}} < k < k_{\mathrm{f}}$ reenter the Hubble radius
during the epoch of reheating and hence they carry the signatures of the
mechanism of reheating.
For the value of $w_\phi=0$ we have worked with in Fig.~\ref{plot_perturbative1},
we find that the spectrum exhibits a strong red tilt for $k > k_{\mathrm{re}}$.
In fact, we find that the red tilt occurs over this range of wave numbers whenever
$w_\phi <1/3$.
Moreover, for lower values of the reheating temperature, the red tilt begins to
occur at smaller wave numbers.
Such a behavior can be attributed to the fact that, when the reheating temperature
is lower, the epoch of reheating lasts longer.
Since reheating is delayed, the mode with wave number $k_{\mathrm{re}}$ reenters the Hubble
radius at a later time or, equivalently, leaves the Hubble radius during inflation
at an earlier time thereby suggesting that it will have a smaller wave number.
We should mention here that these features in the spectrum of GWs are similar to
the behavior in the simpler reheating scenario we had discussed in the last section.
Interestingly, we find that the perturbative reheating scenario leaves tell tale
imprints on the spectrum of GWs which can possibly help us decipher finer details
of the mechanism of reheating.
We find that the spectrum exhibits a burst of oscillations near~$k_{\mathrm{re}}$.
It should be clear from Fig.~\ref{plot_perturbative1} that the oscillations occur
over modes which leave the Hubble radius during the period of the transition when
$w_\mathrm{eff}$ changes from its initial value of $w_\phi$ to the final
value of~$1/3$.
Recall that, in the perturbative reheating scenario, the reheating temperature
is identified as the temperature associated with the energy density of radiation
at the instance when $H(A_{\mathrm{re}}) = \Gamma_{\phi}$.
Consequently, it is at this point of time that the change in the effective EoS
parameter with respect to the scale factor is the maximum.
This aspect is reflected in the peak that arises in the spectrum of GWs exactly
at the wave number $k_{\mathrm{re}}$ which re-enters the Hubble radius when $H(A_{\mathrm{re}}) =
\Gamma_{\phi}$.
We should also mention that these features in the spectrum of GWs spectrum are
not limited only to the case of $w_{\phi} = 0$ and $T_{\mathrm{re}} = 10^{6}\, \mathrm{GeV}$,
but also arise for all possible sets of values of~$(w_{\phi} < 1/3, T_{\mathrm{re}})$.
In order to highlight this point, in Fig.~\ref{plot_perturbative1}, we have
illustrated the spectrum $\Omega_{_{\mathrm{GW}}}(f)$ for different values of reheating temperature
$T_{\mathrm{re}}$.
However, note that, the frequency around which the spectrum begins to exhibit
a red tilt increases as the reheating temperature increases.
This is expected for the reason we discussed above, viz. that the period of
reheating is shorter for a higher reheating temperature as a result of which
the wave number~$k_{\mathrm{re}}$ of the mode which re-enters at the end of reheating
turns out to be larger.
Lastly, we should mention that, for $w_{\phi} < 1/3$, the spectrum of GWs
is indeed compatible with the BBN constraints.
To illustrate the dependence of the spectrum of GWs on $w_\phi$, in
Fig.~\ref{plot_perturbative2}, we have plotted the spectrum for~$w_{\phi}
= 1/2$ [i.e. for $n=3$ in the potential~\eqref{eq:aap}]
and a set of values of the reheating temperature, just as in
Fig.~\ref{plot_perturbative1}.
\begin{figure}[t!]
\includegraphics[height=5.25cm,width=8.50cm]{variationofomegawithfn3co.pdf}
\hskip 5pt
\includegraphics[height=5.25cm,width=8.50cm]{relicplotdifferenttemperaturen3co.pdf}
\caption{The spectrum of GWs today have been illustrated in the same manner as
in the last figure.
But, in contrast to the previous figure wherein we had considered the case
$w_\phi=0$ [or, equivalently, $n=1$ in the potential~\eqref{eq:aap}], we
have set $w_\phi=0.5$ (i.e. $n=3$) in arriving at the plots above.
Note that, as in the case of $w_\phi=0$, the spectrum is scale invariant
over frequencies corresponding to $k < k_{\mathrm{re}}$.
However, the spectrum spectrum has a strong blue tilt at higher frequencies.
Importantly, for some values of the reheating temperature, the spectra intersect
the sensitivity curves of the various GW observatories which immediately
translate to constraints on the parameters $w_\phi$ and $T_{\mathrm{re}}$ that
characterize the epoch of reheating.
In the figure, we have also included the BBN constraint (as the horizontal black
line, on the right), which corresponds to~$\Omega_{_{\mathrm{GW}}}\,h^2< 10^{-6}$.}
\label{plot_perturbative2}
\end{figure}
Clearly, the spectrum of GWs is scale invariant over frequencies corresponding
to $k< k_{\mathrm{re}}$.
But, in contrast to the $w_{\phi} = 0$ case, the spectrum has a strong blue
tilt at higher frequencies.
Also, as expected, the higher the reheating temperature, the larger is $k_{\mathrm{re}}$,
for reasons we have discussed earlier.
Moreover, we find that the spectrum exhibits a burst of oscillation around $k_{\mathrm{re}}$,
exactly as in the $w_{\phi} = 0$ case.
Further, the maximum of the oscillation occurs at the instance when $k_{\mathrm{re}}$ reenters
the Hubble radius and the width of the oscillation coincides with the period of
transition from $w_\mathrm{eff} = w_\phi=0.5$ to $w_\mathrm{eff}=1/3$.
Finally, we should mention that the spectra exhibit a blue tilt for $k > k_{\mathrm{re}}$
whenever $w_\phi>1/3$.
The above arguments clearly indicate that the details of epoch of reheating
significantly affects the spectrum of GWs.
Therefore, the characteristic features of $\Omega_{_{\mathrm{GW}}}(f)$ can considerably aid us
in garnering adequate amount of information regarding the reheating phase.
Upon comparing Figs.~\ref{plot_analytic1} and~\ref{plot_perturbative1}
(or~\ref{plot_perturbative2}), it is clear that the perturbative reheating
mechanism, wherein the transfer of energy from the inflaton to radiation
occurs smoothly, leads to oscillations in the spectrum of GWs in contrast
to the simpler model wherein the transition to radiation domination
occurs instantaneously.
We believe that such quantitative differences can provide us with stronger
constraints on the mechanism of reheating.
Specifically,
\begin{itemize}
\item
The presence of the oscillating feature in the spectrum of GWs, in particular,
the width of the oscillation can provide us information concerning the time
scale over which $w_\mathrm{eff}$ makes the transition from $w_{\phi}$
to $1/3$.
\item
As we have discussed above, the peak of the oscillation occurs at $k = k_{\mathrm{re}}$.
Thus, identifying the location of the peak of the oscillation in the spectrum
can help us determine the Hubble scale at the end of reheating or, equivalently,
the decay rate $\Gamma_\phi$ of the inflaton to radiation.
In other words, the observation of the peak can indicate the strength of the
coupling between the inflaton and radiation in given a decay channel.
\end{itemize}
\section{Spectrum of GWs near the end of the inflation}\label{sec:actualinflation}
Until now, while discussing the tensor power spectrum generated during inflation,
for simplicity, we had assumed that inflation was of the de Sitter form.
This had led to a scale invariant power spectrum for scales such that $k \ll k_{\mathrm{f}}$
[cf. Eq.~\eqref{eq:pt-i}], where $k_{\mathrm{f}}$ denotes the wave number that leaves the
Hubble radius at the end of inflation.
However, potentials such as the $\alpha$-attractor model~\eqref{eq:aap} of our
interest actually lead to slow roll inflation and, as we had mentioned earlier,
in such cases, there will arise a small tensor spectral tilt.
Moreover, even the slow roll approximation will cease to be valid towards the
end of inflation.
Therefore, to understand the nature of the inflationary tensor power spectrum
close to the wave number~$k_{\mathrm{f}}$, the easiest method seems to evaluate the
spectrum numerically.
There exists a standard procedure to evaluate the spectrum of perturbations generated
during inflation (in this context, see, for instance, Ref.~\cite{Hazra:2012yn}).
The modes are typically evolved from the Bunch-Davies initial conditions when they
are well inside the Hubble radius and the spectra are evaluated in the super-Hubble
domain when the amplitude of the perturbations have frozen.
Such an approach works well for the large scale modes.
But, since we are interested in the tensor power spectrum over small scales, in
particular, with wave numbers close to $k_{\mathrm{f}}$, these modes would not be able to
spend adequate amount of time in the super-Hubble regime.
Hence, in these situations, the best approach would be evaluate the spectrum at
the end of inflation.
In Fig.~\ref{numericalsolution}, we have plotted the inflationary tensor power
spectrum computed numerically in the $\alpha$-attractor model of our interest.
\begin{figure}[t!]
\includegraphics[width=8.50cm]{ptf1.pdf}
\hskip 5pt
\includegraphics[width=8.10cm]{numericalsolution1.pdf}
\caption{The inflationary tensor power spectrum arising in the
$\alpha$-attractor model of interest (on the left) as well as
the corresponding spectrum of GWs today (on the right) have
been plotted for frequencies close to the wave number~$k_{\mathrm{f}}$.
The inflationary spectra have been computed numerically and
we have made use of the analytical solutions for the tensor
transfer function $\chi_k$ during the post-inflationary epochs
to arrive at the spectrum of GWs today.
We have plotted the spectrum of GWs today, viz. $\Omega_{_{\mathrm{GW}}}(f)$, for a
few different values of the reheating EoS parameter~$w_\phi$
and a specific reheating temperature.
We find that, while the inflationary power spectrum begins to
behave as $k^2$ near~$k_{\mathrm{f}}$, the corresponding spectra of GWs
today behave as~$k^4$.}
\label{numericalsolution}
\end{figure}
Actually, in the figure, we have also plotted the spectra of GWs today~$\Omega_{_{\mathrm{GW}}}(f)$
for a few sets of values of the EoS parameter $w_\phi$ and a specific value
of the reheating temperature.
Having computed the inflationary spectra numerically, we have used the analytical
forms for the tensor transfer function post-inflation to arrive at the~$\Omega_{_{\mathrm{GW}}}(f)$.
Note that the inflationary spectral shape begins to change for wave numbers close
to $k_{\mathrm{f}}$.
In fact, we find that, the inflationary tensor power spectrum $\mathcal{P}_{_{\mathrm{T}}}(k)$ behaves
as~$k^2$ for wave numbers close to and beyond~$k_{\mathrm{f}}$.
This is not surprising and occurs due to the fact that these modes have either
hardly left or remain inside the Hubble radius at the end of inflation.
Therefore, the modes are essentially of the Minkowskian form leading to
the~$k^2$ behavior of the power spectrum.
From the structure of energy density of GWs [cf. Eq.~\eqref{eq:rho-gw-at}],
it is easy to establish that the corresponding $\Omega_{_{\mathrm{GW}}}(k)$ would behave as~$k^4$
over this domain of wave numbers.
It is easy to see from Fig.~\ref{numericalsolution} that $\Omega_{_{\mathrm{GW}}}(f)$ indeed
behaves as expected around and beyond $k_{\mathrm{f}}$.
Further, from the figure, we can see that $k_{\mathrm{f}}$ is crucially dependent on the
structure of the inflationary potential.
For reheating dynamics described by an effective EoS parameter~$w_\phi$, we can
write $k_{\mathrm{f}}$ in terms of the potential parameter~$n$, the reheating
parameters~$(T_{\mathrm{re}}\,N_{\mathrm{re}}$) and the inflationary parameter~$N_\ast$ as follows:
\begin{equation}
k_{\mathrm{f}}=a_{\mathrm{f}}\, H_\mathrm{f}
=k_\ast\,\frac{H_\mathrm{f}}{H_{_{\mathrm I}}}\,\mathrm{e}^{N_\ast}
=k_\ast\,\left(\frac{V_\mathrm{f}}{2\,H_{_{\mathrm I}}^2\,M_{_{\mathrm{Pl}}}^2}\right)^{1/2}\,\mathrm{e}^{N_\ast}
=k_\ast\,\left(\frac{\pi^2\,g_{r, \mathrm{re}}}{90}\right)^{1/2}\,
\frac{T_{\mathrm{re}}^2}{H_{_{\mathrm I}}\,M_{_{\mathrm{Pl}}}}\,\mathrm{e}^{N_\ast+[3\,n/(n+1)]\,N_{\mathrm{re}}},
\end{equation}
where we have been careful to distinguish between the value of $H_{k_\ast}\simeq
H_{_{\mathrm I}}$ and $H_\mathrm{f}$, i.e. the Hubble parameters evaluated at the moment
when the pivot scale $k_\ast$ crosses the Hubble radius and at the end of
the inflation, respectively.
\section{CMB, spectrum of GWs, and the microscopic reheating
parameters}\label{sec:probe}
As is well known, the observations of the anisotropies in the CMB by
missions such as Planck can be explained in a simple and successful
manner by invoking an early phase of
inflation~\cite{Mukhanov:1990me,Martin:2003bt,Martin:2004um,
Bassett:2005xm,Sriramkumar:2009kg,Baumann:2008bn,Baumann:2009ds,
Sriramkumar:2012mik,Linde:2014nna,Martin:2015dha}).
Nevertheless, the characterization of the inflaton is far from complete
because of the lack of adequate observational constraints, particularly
over scales smaller than the CMB scales.
The spectrum of GWs is possibly the only probe which can provide us direct
access to the physics operating during the epochs of inflation and reheating.
In this section, we shall discuss the manner in which we can extract the
properties of the inflaton by the combining the observations of the CMB
and GWs.
As we have already mentioned, the spectrum of GWs carries signatures which
reflect some details of the mechanism of reheating.
Recall that, the primary aspect of the reheating phase is the time evolution
of the EoS parameter from the initial value of $w_{\phi}$ associated with
the inflaton to the final value of~$1/3$ corresponding to radiation.
The phase can be generically divided into three stages based on the underlying
physical processes that operate.
In what follows, we shall discuss these stages and the corresponding
imprints on~$\Omega_{_{\mathrm{GW}}}(f)$.
To facilitate the discussion, let us introduce the spectral index $n_{_\mathrm{GW}}
={\rm d}\,\mathrm{ln}\,\Omega_{_{\mathrm{GW}}}/{\rm d}\, \mathrm{ln}\, k$ associated with the spectrum
of GWs.
Interestingly, we find that all the three stages leave distinct imprints on
the spectral index~$n_{_\mathrm{GW}}$, and we have illustrated the behavior of $n_{_\mathrm{GW}}(f)$
for the $\alpha$-attractor model~\eqref{eq:aap} in Fig.~\ref{plotnomega}.
\begin{figure}[t!]
\includegraphics[height=5.25cm,width=8.50cm]{nomegaplotco.pdf}
\hskip 10pt
\includegraphics[height=5.25cm,width=8.50cm]{frensplotc.pdf}
\caption{\textit{Left:}\/~The variation of the index $n_{_\mathrm{GW}}$ associated the
spectrum of primordial GWs observed today in the case of the perturbative
reheating scenario has been plotted as a function of frequency~$f$ for two
different values of the inflaton EoS parameter $w_\phi=(0,1/2)$ (as
solid and dashed lines) for the $\alpha$-attractor potential~\eqref{eq:aap}.
We have fixed value of the reheating temperature to be $T_{\mathrm{re}}=10^6\,
\mathrm{GeV}$ in arriving at the plot.
In the figure (on the left), we have also explicitly highlighted the
behavior of $n_{_\mathrm{GW}}$ for modes that re-enter the Hubble radius during
the following regimes: (i)~reheating phase dominated by the inflaton
(as solid and dashed lines, in magenta), (ii)~the period of rapid
transition of the EoS parameter from $w_\phi$ to $1/3$ (as solid
and dashed lines in blue), and (iii)~the radiation dominated epoch
(in green).
We have also demarcated the domain in frequency associated with the
thermalization time scales in the figure (as shaded regions in dark
and light blue).
These quantities have been denoted as $\Delta f_\mathrm{th}^1$ and
$\Delta f_\mathrm{th}^3$ for $w_\phi=0$ and $1/2$,
respectively.
\textit{Right:}\/~We have illustrated the variation of the frequency
$f_\mathrm{re}=k_{\mathrm{re}}/(2\,\pi)$ as a function of the scalar spectral
index $n_{_{\mathrm{S}}}$ for three different values of $n=(1,3,5)$ (in blue,
black and pink) for the $\alpha$-attractor model.
We have also indicated the $1$-$\sigma$ and $2$-$\sigma$ confidence
regions (as light and dark bands in red) associated with the constraint
on scalar spectral index~$n_{_{\mathrm{S}}}$ from Planck~\cite{Akrami:2018odb}.}
\label{plotnomega}
\end{figure}
Note that the first and longest stage is when $H \ll \Gamma_{\phi}$, i.e. when
the inflaton is decaying very slowly and hence the background dynamics is
dominated by the EoS parameter~$w_{\phi}$ governing the inflaton.
The spectral index $n_{_\mathrm{GW}}$ associated with modes which reenter the Hubble
radius during the stage is given by $n_{_\mathrm{GW}} = 2\, (3\, w_{\phi} -1)/(3\,
w_{\phi}+ 1)$.
In Fig.~\ref{plotnomega}, we have indicated the $n_{_\mathrm{GW}}(f)$ associated with
$w_\phi=0$ and $1/2$.
In the subsequent stage, as the Hubble parameter approaches $\Gamma_{\phi}$,
the decay of the inflaton becomes increasingly efficient, and the effective
EoS parameter begins to change rapidly.
The corresponding effects are reflected in the variation of $n_{_\mathrm{GW}}$ over modes
which reenter the Hubble radius during the transition, as highlighted in
Fig.~\ref{plotnomega}.
However, this intermediate stage is the shortest among the three stages and
it ends when $H_{\mathrm{re}} = \Gamma_{\phi}$, i.e. when the rate of decay of the
inflaton to radiation is at its maximum.
The most important of the three stages is the final stage of thermalization
which is characterized by the time scale~$\Delta t_\mathrm{th}$.
It is the time scale over which the decay products of the inflaton thermalize
among themselves.
In fact, it is this mechanism that determines the actual initial temperature
of the radiation dominated phase contrary to the conventional definition of
reheating temperature~$T_{\mathrm{re}}$ defined when $H = \Gamma_{\phi}$.
The thermalization process and the associated time
scale~$\Delta t_\mathrm{th}$ would crucially depend upon nature of all the
decay products of the inflaton as well as the detailed dynamics of the decay
process.
These details will determine the manner in which the EoS parameter changes during
this stage and its variation will be imprinted in the behavior of $n_{_\mathrm{GW}}(f)$ as we
have illustrated in Fig.~\ref{plotnomega}.
Note that there arises a frequency range, say, $\Delta f_\mathrm{th}$, associated
with the time scale $\Delta t_\mathrm{th}$, and the variation of the spectral
index $n_{_\mathrm{GW}}$ over this domain can provide us with clues to the physics operating
during the stage.
Therefore, from the CMB observations and the spectrum of GWs, we can, in
principle, extract the following essential information regarding the nature
of the inflaton and the underlying physical process taking place during
reheating.
\vskip 4pt\noindent
(i) {\it Effective inflaton EoS parameter~$w_{\phi}$:}\/~The nature of inflaton
potential near its minimum or, equivalently, the effective EoS parameter $w_\phi$
associated with the decay of the inflaton can be determined from the spectral index
of GWs through the relation
\begin{equation}
w_{\phi} = \frac{1}{3}\, \left(\frac{2 + n_{_\mathrm{GW}}}{2 - n_{_\mathrm{GW}}}\right).
\end{equation}
\vskip 4pt\noindent
(ii)~{\it Inflaton decay width $\Gamma_{\phi}$:}~Once we have arrived at the EoS
parameter describing the inflaton, the effective inflaton decay constant~$\Gamma_\phi$
can be determined from the CMB and the spectrum of GWs in the following fashion.
As we have already mentioned, for modes with wave numbers
$k < k_{\mathrm{re}}$, the amplitude of the tensor perturbations will remain approximately
constant from the time the modes leave the Hubble radius during inflation till
they renter the Hubble radius during the epoch of radiation domination.
Due to this reason, over these range of modes, the spectrum of GWs at late times
retains the same shape as the spectrum of tensor perturbations generated during
inflation.
Therefore, using Eq.~\eqref{eq:ogw-sk}, the scale invariant amplitude of the
spectrum of GWs can be utilized to estimate the approximate energy scale near
the end of inflation.
In the limit of high reheating temperature, we have
\begin{equation}
\Omega_{_{\mathrm{GW}}}\, h^2 \simeq \Omega_{_{\mathrm{R}}}\,h^2\,\frac{H_{_{\mathrm I}}^2}{6\,\pi^2\,M_{_{\mathrm{Pl}}}^2}
\label{infscale}
\end{equation}
and, as a result,
\begin{equation}
H_{_{\mathrm I}} \simeq \left(\frac{6\,\pi^2\,M_{_{\mathrm{Pl}}}^2\,\Omega_{_{\mathrm{GW}}}\, h^2}{\Omega_{_{\mathrm{R}}}\,
h^2}\right)^{1/2}
\end{equation}
which, in turn, allows us to express the energy density at the end of inflation as
follows:
\begin{equation}
\rho_\mathrm{f}
\simeq \frac{18\,\pi^2\, M_{_{\mathrm{Pl}}}^4\, \Omega_{_{\mathrm{GW}}}\, h^2}{\Omega_{_{\mathrm{R}}}\, h^2}.
\end{equation}
From the observed spectrum, we can, in principle, determine wave numbers $k_{\mathrm{re}}$
and $k_{\mathrm{f}}$, which are the wave numbers that reenters at the end of the epoch of
reheating and leaves the Hubble radius at the end of inflation, respectively.
In Fig.~\ref{plotnomega}, we have illustrated the dependence of $k_{\mathrm{re}}$ on
the scalar spectral~$n_{_{\mathrm{S}}}$ for different values of the parameter~$n$ of the
$\alpha$-attractor model.
In a manner similar to the existence of a maximum possible reheating temperature,
we notice that there arises a maximum possible value for the frequency associated
with the wave number~$k_{\mathrm{re}}$.
We find that, generically, $f_\mathrm{re}^\mathrm{max} \simeq 10^7\, \mathrm{Hz}$,
irrespective of the values of the other parameters involved.
It would be interesting to study further implications of this point.
Nonetheless, the aforementioned wave numbers satisfy the following relations
\begin{equation}
k_{\mathrm{f}} = a_\mathrm{f}\, H_\mathrm{f} \simeq a_\mathrm{f}\, H_{_{\mathrm I}},\quad
k_{\mathrm{re}}=a_\mathrm{re}\, H_{\mathrm{re}} = A_{\mathrm{re}}\, a_\mathrm{f}\, \Gamma_\phi.
\end{equation}
Considering perturbative reheating, one can obtain an approximate analytical
expression for the normalized scale factor~$A_{\mathrm{re}}$ at the end of the reheating
to be (in this context, see Ref.~\cite{Haque:2020zco})
\begin{equation}
A_{\mathrm{re}}=\left(\frac{4\, \rho_\mathrm{f}\,(1+w_\phi)^2}{\mathcal{G}^4\,\beta\,
(5-3\,w_\phi)^2}\right)^{-1/(1-3\,w_\phi)},\quad
\beta=\frac{\pi^2\,g_{r, \mathrm{re}}}{30},\quad
\mathcal{G}=\left(\frac{43}{11\,g_\mathrm{re}}\right)^{1/3}\,
\left(\frac{a_0\,HI}{k_\ast}\right)\,T_0\, \mathrm{e}^{-N_\ast}.
\end{equation}
The primary assumption in arriving at the above expressions is that the energy
scale does not change significantly throughout the entire period of inflation.
With all the above expressions at hand, we find that the inflaton decay
constant $\Gamma_\phi$ can be written in terms of the observable
quantities as
\begin{equation}
\Gamma_\phi
= \frac{k_{\mathrm{re}}\, H_{_{\mathrm I}}}{A_{\mathrm{re}}\, k_{\mathrm{f}}}
=\frac{k_{\mathrm{re}}}{k_{\mathrm{f}}}\,
\left(\frac{6\,\pi^2\,M_{_{\mathrm{Pl}}}^2\,\Omega_{_{\mathrm{GW}}}\, h^2}{\Omega_{_{\mathrm{R}}}\, h^2}\right)^{1/2}
\left(\frac{72\, \pi^2\,M_{_{\mathrm{Pl}}}^4\, \Omega_{_{\mathrm{GW}}}\, h^2\,(1+w_\phi)^2}{\Omega_{_{\mathrm{R}}}\,
h^2\,\mathcal{G}^4\,\beta\,(5-3\,w_\phi)^2}\right)^{1/(1-3\omega_{\phi})}.
\label{gammaphi}
\end{equation}
One can then immediately obtain the following analytic expression for the reheating temperature:
\begin{equation}
T_{\mathrm{re}} = \frac{\mathcal{G}}{A_{\mathrm{re}}}
= \left(\frac{43}{11\,g_\mathrm{re}}\right)^{1/3}\,
\frac{k_{\mathrm{f}}\,H_{_{\mathrm I}}}{k_\ast\,H_0}\,
\left(\frac{72\,\pi^2\, M_{_{\mathrm{Pl}}}^4\, \Omega_{_{\mathrm{GW}}}\, h^2\, (1+w_\phi)^2}{\Omega_{_{\mathrm{R}}}\,
h^2 \mathcal{G}^4\,\beta\,(5-3\,w_\phi)^2}\right)^{1/(1-3\,w_{\phi})}\, T_0.
\end{equation}
So far, we have expressed the inflation decay constant and the reheating
temperature in terms of the observables associated with the CMB and the
spectrum of GWs today.
To understand the exact nature of the coupling, the subsequent thermalization
processes right after reheating (i.e. when $\Gamma_{\phi} = H_{\mathrm{re}}$) becomes
important.
Therefore, let us now compute the thermalization time scale.
\vskip 4pt\noindent
(iii)~{\it Thermalization time scale $\Delta t_\mathrm{th}$:}\/~Thermalization
is an important non-equilibrium phenomenon that is ubiquitous in nature.
At the end of the phase of reheating, when the rate at which the inflaton
decays into radiation has attained a maximum, the subsequent thermalization
phase leads to the epoch of radiation domination.
In this process, thermalization time scale $\Delta t_\mathrm{th}$ is an
important observable which crucially depends on the nature of the initial
state as well as the interactions among the internal degrees of freedom.
Also, it is the initial state which encodes the information about the
coupling between the inflaton and the other fields to which the energy is
being transferred.
Hence, if we can arrive at $\Delta t_\mathrm{th}$ from the spectrum of
GWs, valuable information regarding the fundamental nature of the coupling
parameters between the inflation and other fields at very high energies can,
in principle, be extracted.
The thermalization time scale is defined as
\begin{equation}
\label{ther}
\Delta t_\mathrm{th}=t_\mathrm{th}-t_\mathrm{re},
\end{equation}
where $t_\mathrm{re}$ is the time corresponding to the end of reheating
and $t_\mathrm{th}$ denotes the time at the end of the thermalization
process, which leads to the beginning of the actual radiation dominated
epoch.
In order to obtain an approximate analytic expression, during this regime,
we shall assume that the scale factor behaves as $a\propto t^{2/3\,(1+w)}$.
This can be justified since we can express the effective EoS parameter
during the thermalization phase using the following perturbative
expansion:
\begin{equation}
w= \frac{1}{t_\mathrm{th} - t_\mathrm{re}}\,
\int_{t_\mathrm{re}}^{t_\mathrm{th}} {\rm d} t\,
\left[w_{_\mathrm{R}} + \left(w_{\phi} - w_{_{\mathrm{R}}}\right)\,
\frac{\rho_{\phi}}{\rho_{_\mathrm{R}}} + \cdots\right]
\simeq \frac{1}{3} + \left(w_{\phi} -\frac{1}{3}\right)\,x
+ {\cal O}(x^2),\quad x= \frac{1}{t_\mathrm{th} - t_\mathrm{re}}\,
\int_{t_\mathrm{re}}^{t_\mathrm{th}}{\rm d} t\,\frac{\rho_{\phi}}{\rho_{_\mathrm{R}}},
\end{equation}
where $w_{_\mathrm{R}} = 1/3$ is the EoS parameter describing radiation.
Note that, during the thermalization phase $\rho_{\phi} \ll \rho_{_\mathrm{R}}$
and, hence, $x \ll 1$.
This particular fact enables us to obtain the leading order expression for the
thermalization time scale in terms of the observable quantities that we discussed.
On utilizing the above form for the EoS parameter, we find that the leading order
behavior of the scale factor at $\Gamma_{\phi} = H_{\mathrm{re}}$ can be written as
\begin{equation}\label{are}
a_\mathrm{re}
\simeq \left(\frac{t_\mathrm{re}}{t_1}\right)^{2/[3\,(1+\omega_{_\mathrm{R}})]}\,
\left[1 - \frac{2\,x\,(w_{\phi} - w{_\mathrm{R}})}{3\, (1+w_{_\mathrm{R}})^2}\;
\mathrm{ln}\, \left(\frac{t_\mathrm{re}}{t_1}\right) + \cdots \right],
\end{equation}
where $t_1$ is a constant we have introduced for purposes of normalization.
Upon using the relations $k_{\mathrm{re}} = a_\mathrm{re}\, H_{\mathrm{re}}$, $k_\mathrm{th}
= a_\mathrm{th}\, H_\mathrm{th}$ and $H_{\mathrm{re}} = \Gamma_{\phi}$, we can
express the reheating time $t_\mathrm{re}$ and the thermalization time
$t_\mathrm{th}$ in terms of the wave numbers $k_{\mathrm{re}}$ and $k_\mathrm{th}$ as
\begin{equation}
t_\mathrm{re}
= \left(\frac{k_\mathrm{th}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}\, t_1^{p_{_\mathrm{R}}/(p_\mathrm{R}-1)}
+ \mathcal{O}(x),\quad
t_\mathrm{th} = \left(\frac{k_\mathrm{re}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}
t_1^{p_{_\mathrm{R}}/(p_{_\mathrm{R}}-1)} + \mathcal{O} (x).
\end{equation}
We can also express $\Gamma_{\phi}$ in terms of normalized time $t_1$ as
\begin{equation}
\label{gamma}
\Gamma_\phi \sim \left(\frac{p_{_\mathrm{R}}}{t_1}\right)^{p_{_\mathrm{R}}/(p_{_\mathrm{R}}-1)}\;
k_{\mathrm{re}}^{-1/(p_{_\mathrm{R}}-1)}+ {\cal O}(x),
\end{equation}
where we have introduced the quantity $p_{_\mathrm{R}}
=2/[3\,(1+\omega_{_\mathrm{R}})]$.
Utilizing all the above equations, we finally obtain the final expression
for the thermalization time scale to the leading order in $x$ to be
\begin{equation}
\renewcommand*{\arraystretch}{2.5}
\Delta t_\mathrm{th}
\sim \left\{\begin{array}{ll}
\left[\left(\frac{k_\mathrm{th}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}
- \left(\frac{k_{\mathrm{re}}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}\right]\, p_{_\mathrm{R}}^{p_{_\mathrm{R}}/(p_{_\mathrm{R}}-1)}\;
\left(\frac{k_{\mathrm{re}}^{-1/(p_{_\mathrm{R}}-1)}}{\Gamma_\phi}\right)
+ \mathcal{O}(x), &\mbox{in terms of $\Gamma_{\phi}$},\\
\left[\left(\frac{k_\mathrm{th}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}
- \left(\frac{k_{\mathrm{re}}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}\right]\,
p_{_\mathrm{R}}^{p_{_\mathrm{R}}/(p_{_\mathrm{R}}-1)}\;
\left(\frac{G\, k_{\mathrm{re}}^{-p_{_\mathrm{R}}/(p_{_\mathrm{R}}-1)}}{H_{_{\mathrm I}}\, T_{\mathrm{re}}}\right)
+ {\cal O}(x), &\mbox{in terms of $T_{\mathrm{re}}$}.
\end{array}\right.\label{thermalization}
\end{equation}
In the above expressions, an important point one should remember is that the
value of the wave numbers $k_{\mathrm{re}}$ and $k_\mathrm{th}$ are, in general,
dependent on the decay width of the inflaton.
Therefore, the overall thermalization time scale is a non-trivial function
of $\Gamma_{\phi}$.
The thermalization time scale crucially depends on the initial number density
of the decayed particles compared with the thermalized ones.
The initial number density at the instant when $\Gamma_{\phi} = H_{\mathrm{re}}$ can be
approximately estimated to be $n_\mathrm{i} \simeq M_{_{\mathrm{Pl}}}^2\, H_{\mathrm{re}}^2/m_{\phi}
= M_{_{\mathrm{Pl}}}^2\, \Gamma^2_{\phi}/m_{\phi}$, and, in arriving at this expression, it
has been assumed that the momentum of the decay products is as large as the
mass $m_\phi$ of the inflaton~\cite{Kurkela:2011ti,Harigaya:2013vwa}.
If we consider the particles to have thermalized at the reheating temperature
$T_{\mathrm{re}}$, then the number density can again be approximately determined to be
$n_\mathrm{th} \simeq T_{\mathrm{re}}^3 \simeq \Gamma_{\phi}^{3/2} M_{_{\mathrm{Pl}}}^{3/2}$.
Hence, the ratio of the particle number densities $n_\mathrm{th}$ and
$n_\mathrm{i}$ turns out to be
\begin{equation}
\frac{n_\mathrm{th}}{n_\mathrm{i}}
\simeq \frac{m_{\phi}}{\sqrt{\Gamma_\phi\,M_{_{\mathrm{Pl}}}}}
= \left(\frac{m_{\phi}^2\, A_{\mathrm{re}}\, k_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}\,k_{\mathrm{re}}\,H_{_{\mathrm I}}}\right)^{1/2}.\label{ninth}
\end{equation}
This is one of the crucial parameters in the context of the thermalizing
plasma which dictates the kind of physical processes that occur during
thermalization~\cite{Ellis:1987rw,McDonald:1999hd,Allahverdi:2000ss,
Kurkela:2011ti,Harigaya:2013vwa}.
At this stage, we are unable to extract the inflaton mass $m_{\phi}$ from
the observations in a model independent manner.
However, given the mass of the inflaton, along with the CMB observations,
Eq.~\eqref{ninth} will have two generic possibilities, viz.
\begin{itemize}
\item
${n_\mathrm{i} < n_\mathrm{th}}$:\/~The particle number density is smaller
than the thermalized ones, which means that, at the end of reheating, the
universe is under occupied.
For example if one considers marginal inflaton-scalar ($\xi$) coupling such
$\beta\, \phi\, \xi^3$, inflaton-Fermion ($\psi$) Yukawa coupling $\beta\,
\phi\, \bar{\psi}\, \psi$, the inflaton decay width behaves as $\Gamma_{\phi}
\sim \beta^2\, m_{\phi}$ which implies that
$n_\mathrm{i}/n_\mathrm{th} \sim \sqrt{m_\phi/M_{_{\mathrm{Pl}}}} < 1$.
\item
$n_\mathrm{i} > n_\mathrm{th}$:\/~The particle number density is larger than
the thermalized ones, i.e. at the end of reheating, the universe is over
occupied.
For example, if one considers any Planck suppressed operator containing a
coupling between the inflaton and the reheating field, the decay width behaves
as $\Gamma_{\phi} \sim m_{\phi}^3/M_{_{\mathrm{Pl}}}^2$, which implies that $n_\mathrm{i}/
n_\mathrm{th} \sim \sqrt{M_{_{\mathrm{Pl}}}}/m_{\phi} > 1$.
\end{itemize}
\vskip 4pt\noindent
(iv) {\it Determination of microscopic interactions:}\/~From our discussion
above for the two cases, it is clear that if we can determine the value of
$n_\mathrm{th}/n_\mathrm{i}$ from the combined observations of the CMB and
GWs, the fundamental nature of the inflaton-reheating field coupling such
as `$\beta$' can be extracted.
Furthermore, interestingly, it has been pointed out that, depending on the
aforementioned two conditions for the initial, non-thermal states generated by
the end of reheating, the behavior of the thermalization time scale in terms
of the microscopic variables will be very different, and will behave as (in
this context, see Ref.~\cite{Kurkela:2011ti})
\begin{equation}
\renewcommand*{\arraystretch}{1.5}
\Delta t_\mathrm{th}
\sim \left\{\begin{array}{ll}
\alpha^{-2}\, m_{\phi}^{1/2}\, T_\mathrm{th}^{-3/2},
& \mbox{for under-occupied initial states such
that $n_\mathrm{i} < n_\mathrm{th}$},\\
\alpha^{-2}\, T_\mathrm{th}^{-1},
& \mbox{for the over-occupied initial state scuh that
$n_\mathrm{i} > n_\mathrm{th}$},\\
\end{array}\right.\label{alpha}
\end{equation}
where~$\alpha$ denotes the gauge interaction strength among the decayed particles,
and $T_\mathrm{th}$ is the final thermalization temperature.
Hence, it is extremely important to recognize that, once we know the inflaton
mass $m_{\phi}$ and the final thermalization temperature $T_\mathrm{th}$
[the latter can be computed once the reheating dynamics is fixed, by using Eqs.~\eqref{ninth} and~\eqref{alpha}], the gauge interaction strength can,
in principle, be computed in terms of the observable quantities through the
relations
\begin{equation}
\renewcommand*{\arraystretch}{2.5}
\alpha \sim
\left\{\begin{array}{ll}
T_\mathrm{th}^{-1/2 }\,
\left[\left(\frac{k_\mathrm{th}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}
- \left(\frac{k_{\mathrm{re}}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}\right]^{-1/2}\;
p_{_\mathrm{R}}^{-p_{_\mathrm{R}}/[2\,(p_{_\mathrm{R}}-1)]}\;
\left(\frac{k_{\mathrm{re}}^{-1/(p_{_\mathrm{R}}-1)}}{\Gamma_\phi}\right)^{-1/2},
& \mbox{for $H_I > \frac{m_{\phi}^2\, A_{\mathrm{re}}\, k_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}\, k_{\mathrm{re}}}$},\\
\left(\frac{m_{\phi}}{T_\mathrm{th}^3}\right)^{1/2}\;
\left[\left(\frac{k_{th}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}
- \left(\frac{k_{re}}{p_{_\mathrm{R}}}\right)^{1/(p_{_\mathrm{R}}-1)}\right]^{-1/2}\; p_{_\mathrm{R}}^{p_{_\mathrm{R}}/[-2\,(p_{_\mathrm{R}}-1)]}\;
\left(\frac{k_{\mathrm{re}}^{-1/(p_{_\mathrm{R}}-1)}}{\Gamma_\phi}\right)^{-1/2},
& \mbox{for $H_I < \frac{m_{\phi}^2\, A_{\mathrm{re}}\, k_{\mathrm{f}}}{M_{_{\mathrm{Pl}}}\, k_{\mathrm{re}}}$}.
\end{array}\right. \label{alpha2}
\end{equation}
We expect to carry out a detailed study on these important issues in a
future publication.
Having examined the effects of the epoch of reheating on the spectrum of
GWs today, let us now turn to discuss the effects that arise due to a
secondary phase of reheating.
As we shall see, such a phase can have an important implication for
the recent observational results reported by NANOGrav~\cite{Arzoumanian:2020vkk,Pol:2020igl}.
\section{Spectrum of GWs with late time entropy production and implications
for the recent NANOGrav observations}\label{sec:NANOGrav}
In Secs.~\ref{sec:averaged} and~\ref{sec:actual}, while arriving at the
spectrum of GWs $\Omega_{_{\mathrm{GW}}}(f)$ today, we had assumed that the perturbations were
generated during inflation and had evolved through the epochs of reheating
and radiation domination.
In such a scenario, the entropy of the universe is conserved from the end
of reheating until today.
In fact, we had earlier utilized the conservation of entropy to relate the
temperature~$T_{\mathrm{re}}$ at the end of reheating to the temperature~$T_0$ today.
Recall that, for simplicity, we had assumed that the spectrum of tensor
perturbations generated during inflation was strictly scale invariant
[cf. Eq.~\eqref{eq:pt-i}].
We had also found that, for wave numbers $k < k_{\mathrm{re}}$, the evolution of the
tensor perturbations through the standard epochs of reheating and radiation
domination does not alter the shape of the spectrum of GWs observed today,
i.e. $\Omega_{_{\mathrm{GW}}}(f)$ remains scale invariant for $f < f_{\mathrm{re}}=k_{\mathrm{re}}/(2\,\pi)$.
Over the last decade or so, there has been an interest in examining
scenarios wherein there arises a short, secondary phase of reheating
some time after the original phase of reheating which immediately
follows the inflationary epoch (see, for example,
Ref.~\cite{Kawasaki:2004rx}).
It has been shown that such a modified scenario can also be consistent
with the the various observations~\cite{Nakayama:2009ce,Kuroyanagi:2013ns,
Hattori:2015xla}.
A secondary phase of entropy production can occur due to the decay of an
additional scalar field (which we shall denote as~$\sigma$) that can be
present, such as the non-canonical scalar fields often considered in
high energy physics or the moduli fields encountered in string
theory\footnote{It is for this reason that the secondary phase is
sometimes referred to as the moduli dominated epoch.
The scalar field could have emerged from an extra-dimensional modulus field
or due to some higher curvature effects \cite{Banerjee:2017lxi,Elizalde:2018rmz}}.
In this section, we shall discuss the effects of such a secondary phase of
reheating (which occurs apart from the primary phase of reheating considered
earlier) on the spectrum of GWs observed today.
As we shall see, the secondary phase of entropy production leads to unique
imprints on the spectrum of GWs which has interesting implications for the
recent observations by NANOGrav~\cite{Arzoumanian:2020vkk,Pol:2020igl}.
Let us first calculate the reheating temperature associated with the secondary
phase of reheating.
We can expect the entropy to be conserved during the radiation dominated epoch
sandwiched between the two phases of reheating.
On following the chronology of evolution mentioned above and, upon demanding
the conservation of entropy, we can arrive at the relation between the
temperature $T_{\mathrm{re}}$ at the end of the first phase of reheating and the temperature
at the beginning of the second phase of reheating, say, $T_{\sigma R}$.
We find that they can be related as follows:
\begin{equation}
g_{s,\mathrm{re}}\, a_{\mathrm{re}}^3\, T_{\mathrm{re}}^3
= g_{s,\sigma R}\, a_{\sigma R}^3\, T_{\sigma R}^3,\label{entropy conservation1}
\end{equation}
where $(g_{s,\mathrm{re}},g_{s,\sigma R})$ and $(a_{\mathrm{re}},a_{\sigma R})$ denote
the relativistic degrees of contributing to the entropy and the scale factor
at the end of primary reheating phase and at the start of the second phase of
reheating (or, equivalently, at the end of the first epoch of radiation
domination), respectively.
Using the above relation, we can express the original reheating
temperature~$T_{\mathrm{re}}$ in terms of the temperature~$T_{\sigma R}$
at the beginning of the secondary phase of reheating as
\begin{equation}
T_{\mathrm{re}}= \left(\frac{g_{s,\sigma R}}{g_{s,\mathrm{re}}}\right)^{1/3}\,
\mathrm{e}^{N_{_\mathrm{RD}}^{(1)}}\;T_{\sigma R},\label{temperature1}
\end{equation}
where $N_{_\mathrm{RD}}^{(1)} = \mathrm{ln}\, (a_{\sigma R}/a_{\mathrm{re}})$ denotes
the duration of first epoch of radiation domination in terms of the number
of e-folds.
Similarly, we can relate the temperature at the end of the secondary phase
of reheating, say, $T_\sigma$, to the temperature $T_0$ today by demanding
the conservation of entropy after the onset of the second epoch of radiation
domination.
On doing so, we obtain that
\begin{equation}
T_{\sigma} = \left(\frac{43}{11\, g_{s,\sigma}}\right)^{1/3}\,
\left(\frac{a_0}{a_{\sigma}}\right)\,T_0,\label{temperature2}
\end{equation}
where $g_{s,\sigma}$ and $a_{\sigma}$ represent the degrees of freedom
contributing to the entropy and the scale factor at the end of the
secondary phase of reheating.
If $a_\mathrm{eq}$ denotes the scale factor at the epoch of matter-radiation
equality, then the above expression for $T_{\sigma}$ can be written as
\begin{equation}
T_{\sigma} = \left(\frac{43}{11\, g_{s,\sigma}}\right)^{1/3}\,
\left(\frac{a_0}{a_\mathrm{eq}}\right)\,
\mathrm{e}^{N_{_\mathrm{RD}}^{(2)}}\,T_0.\label{modified temperature2}
\end{equation}
The factor $a_0/a_\mathrm{eq}$ can be expressed in terms of the quantity
$a_0/a_k$ through the relation
\begin{equation}
\frac{a_0}{a_\mathrm{eq}}
= \left(\frac{a_0}{a_k}\right)\,\mathrm{e}^{-\left[N_k+N_{\mathrm{re}}+N_{_\mathrm{RD}}^{(1)}\,
+N_{\mathrm{sre}}+N_{_\mathrm{RD}}^{(2)}\right]},\label{connection}
\end{equation}
where $a_k$ denotes the scale factor when the mode with the wave
number~$k$ crosses the Hubble radius during inflation, while~$N_k$
represents the number of e-folds from the time corresponding
to $a_k$ to the end of inflation.
Moreover, recall that, $N_{\mathrm{re}}$ denotes the duration of the first phase
of reheating.
It should be evident that the quantities $N_\mathrm{sre}$ and
$N_{_\mathrm{RD}}^{(2)}$ represent the duration (in terms of
e-folds) of the secondary phase of (say, moduli dominated) reheating
and the second epoch of radiation domination, respectively.
With $k$ set to be the pivot scale $k_\ast$, on substituting the above
expression for $a_0/a_\mathrm{eq}$ in Eq.~\eqref{modified temperature2},
we obtain that
\begin{equation}
T_{\sigma} = \left(\frac{43}{11\, g_{s,\sigma}}\right)^{1/3}\,
\left(\frac{a_0\,H_{_{\mathrm I}}}{k_\ast}\right)\,
\mathrm{e}^{-\left[N_\ast+N_{\mathrm{re}}+N_{_\mathrm{RD}}^{(1)}
+N_{\mathrm{sre}}\right]}\;T_0,\label{modified temperature3}
\end{equation}
which is the temperature at the end of the secondary phase of reheating.
Note that the above expression for $T_\sigma$ can be inverted to
write $N_{_\mathrm{RD}}^{(1)}$ as
\begin{equation}
\mathrm{e}^{N_{_{\mathrm{RD}}}^{(1)}}
= \left(\frac{43}{11\, g_{s,\sigma}}\right)^{1/3}\,
\left(\frac{a_0\,H_{_{\mathrm I}}}{k_ast}\right)\,
\mathrm{e}^{-\left[N_\ast+N_{\mathrm{re}}+N_{\mathrm{sre}}\right]}\,\left(\frac{T_0}{T_{\sigma}}\right).
\end{equation}
This relation, along with Eq.~(\ref{temperature1}), immediately leads to
the following expression for the original reheating temperature~$T_{\mathrm{re}}$ in
terms of the parameters associated with the late time entropy production:
\begin{equation}
T_{\mathrm{re}} = \left(\frac{43}{11\, g_{s,re}}\right)^{1/3}\,
\left(\frac{a_0\,H_{_{\mathrm I}}}{k_\ast}\right)\, F^{-1/3}\,
\mathrm{e}^{-(N_\ast+N_{\mathrm{re}})}\,T_0.\label{final reheating temperature late entropy}
\end{equation}
In this relation, the factor $F$ represents the ratio of the entropy at the
end and at the beginning of the secondary phase of reheating, and it is given by
\begin{equation}
F = \frac{s(T_{\sigma})\,a_{\sigma}^3}{s(T_{\sigma R})~a_{\sigma R}^3},\label{F1}
\end{equation}
where $s(T)$ denotes the entropy at the temperature~$T$.
If we now assume that the secondary phase of reheating is described
by the EoS parameter $w_\sigma$, then we can arrive at the
following useful relations between the Hubble parameter and the
temperature at the end and at the beginning of the secondary phase
of reheating:
\begin{equation}
H_\sigma=\left(\frac{\gamma_1\,T_\sigma}{\gamma_2\,F^{1/3}\,
T_{\sigma R}}\right)^{3\,(1+w_\sigma)/2}\,H_{\sigma R},\quad
T_\sigma=\left(\frac{\gamma_ 1}{\gamma_2\,F^{1/3}}\right)^{3\,(1
+w_\sigma)/(1-3\,w_\sigma)}\,
\left(\frac{g_{r,\sigma R}}{g_{r,\sigma}}\right)^{1/(1-3\,w_\sigma)}\,
T_{\sigma R},
\end{equation}
where the quantities $\gamma_1$ and $\gamma_2$ are defined as
\begin{equation}
\gamma_1=\left(\frac{g_{r,\mathrm{re}}}{g_{r,\sigma R}}\right)^{1/4},\quad
\gamma_2=\left(\frac{g_{s,\mathrm{re}}}{g_{s,\sigma}}\right)^{1/3}.
\end{equation}
Clearly, the factor~$F$ controls the extent of entropy produced at late times.
And, in the absence of such entropy production, the factor~$F$ reduces to unity.
Also, in such a case, the expression~\eqref{final reheating temperature late entropy}
for the reheating temperature~$T_{\mathrm{re}}$ reduces to the earlier expression~\eqref{eq:Tre},
as required.
Let us now turn to discuss the spectrum of GWs that arises in such a
modified scenario.
As we mentioned above, for simplicity, we shall assume that the secondary
phase of reheating is described by the EoS parameter~$w_\sigma$.
In order to arrive at $\Omega_{_{\mathrm{GW}}}(k)$ in the new scenario, we shall follow the
calculations described in Sec.~\ref{sec:averaged} wherein we have evaluated
the spectrum analytically.
To highlight all the relevant scales involved and also to aid our
discussion below, in Fig.~\ref{diagram-scales}, we have illustrated
the evolution of the comoving Hubble radius in the modified scenario.
\begin{figure}[t!]
\centering
\includegraphics[width=12.50cm]{gwdiagramc.pdf}
\caption{A schematic diagram illustrating the evolution of the
comoving Hubble radius $(a\,H)^{-1}$ plotted (in green) against
the number of e-folds~$N=\mathrm{ln}\,a$.
In the diagram, we have also delineated the various epochs that
are relevant for our discussion.
In the above plot, we have assumed that the EoS
parameter~$w_\phi$ describing the primary
reheating phase is less than~$1/3$.
However, we have assumed that the EoS parameter, say,
$w_\sigma$, associated with a string modulus or
a non-canonical scalar field driving the secondary
phase of reheating is greater than~$1/3$.
In the figure, apart from the wave numbers $k_\ast$, $k_{\mathrm{re}}$
and~$k_{\mathrm{f}}$ we had encountered earlier (indicated as red,
brown and yellow, dashed lines), we have indicated the
scales $k_{\sigma R}$ and $k_{\sigma}$ (as dashed lines
in purple and black) which correspond to wave numbers
that reenter the Hubble radius at the beginning and at
the end of the second phase of reheating, respectively.}
\label{diagram-scales}
\end{figure}
Specifically, in the figure, we have indicated the new scales $k_{\sigma R}$
and $k_\sigma$ which are the wave numbers that reenter the Hubble radius at
the start and at the end of the secondary phase of reheating.
Before we go on to illustrate the results, let us try to understand the
shape of $\Omega_{_{\mathrm{GW}}}(k)$ that we can expect in the modified scenario involving
late time production of entropy.
\begin{itemize}
\item
{\it For wave numbers $k < k_{\sigma}$}:\/~As we mentioned above, $k_{\sigma}$
represents the wave number of the mode that reenters the Hubble radius at
the onset of the second epoch of radiation domination or, equivalently, at
the end of the secondary phase of reheating.
Therefore, the range of wave numbers $k < k_{\sigma}$ (but with wave numbers
larger than those corresponding to the CMB scales) reenter the the Hubble
radius during the second epoch of radiation domination.
Since they are on super-Hubble scales prior to their reentry, they are not
influenced by the background dynamics during the earlier epochs.
Hence, the spectrum of GWs for these range of modes can be expected to be
scale invariant, which implies that the corresponding spectral index~$n_{_\mathrm{GW}}$
vanishes identically.
\item
{\it For wave numbers $k_{\sigma} < k < k_{\sigma R}$}:\/~Recall that,
$k_{\sigma R}$ denotes the wave number that reenters the Hubble radius
at the beginning of the secondary phase of reheating.
As in the case of modes that reenter the primary phase of reheating, we
can expect the spectrum of GWs over this range of wave numbers to exhibit
a spectral tilt which depends on the EoS parameter~$w_\sigma$.
We find that, over this range of wave numbers, $\Omega_{_{\mathrm{GW}}}(k)$ behaves as
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k) \sim k^{2\,(3\,w_\sigma - 1)/(3\,w_\sigma + 1)}\label{2}.
\end{equation}
Consequently, the spectral index of the primordial GWs over this range of
wave numbers turns out to be $n_{_\mathrm{GW}} = 2\,(3\,w_\sigma - 1)/(3\,w_\sigma + 1)$.
In others words, over the domain $k_{\sigma} < k < k_{\sigma R}$, the spectrum
has a blue tilt for $w_\sigma > 1/3$ and a red tilt for $w_\sigma < 1/3$.
\item
{\it For wave numbers $k_{\sigma R} < k < k_{\mathrm{re}}$}:\/~These range of wave numbers
reenter the Hubble radius during the first epoch of radiation domination.
Hence, we can expect the spectrum of GWs to be scale invariant over this range.
It is important to recognize that the amplitude of the spectrum $\Omega_{_{\mathrm{GW}}}(k)$ over
this range will be greater or lesser than the amplitude over $k < k_\sigma$ (i.e.
over wave numbers which reenter the Hubble radius during the second epoch of
radiation domination) depending on whether the EoS parameter~$w_\sigma$
(characterizing the second phase of reheating) is greater than or less than~$1/3$.
\item
{\it For wave numbers $k_{\mathrm{re}} < k < k_{\mathrm{f}}$}:\/~These correspond to wave numbers
that reenter the Hubble radius during the first phase of reheating and, as
we have discussed before, the spectrum of GWs over this range of wave numbers
is expected to behave as
\begin{equation}
\Omega_{_{\mathrm{GW}}}(k) \sim k^{2\,(3\,w_\phi - 1)/(3\,w_\phi + 1)}.\label{4}
\end{equation}
In other words, the corresponding spectral index is given by $n_{_\mathrm{GW}} = 2\,(3\,w_\phi
- 1)/(3\,w_\phi + 1)$, which has a blue or red tilt depending on whether $w_\phi$
is greater than or less than $1/3$.
\end{itemize}
The behavior we have highlighted above can be clearly seen in
Fig.~\ref{plot_latetimeentropy} wherein we have plotted the
quantity $\Omega_{_{\mathrm{GW}}}(f)$ in scenarios involving the second phase of
reheating.
\begin{figure}[t!]
\includegraphics[height=5.25cm,width=8.50cm]{gwlatetimeentropydifferentc.pdf}
\includegraphics[height=5.25cm,width=8.50cm]{gwlatetimeentropydifferentnsc.pdf}
\caption{The spectrum of GWs observed today $\Omega_{_{\mathrm{GW}}}(f)$ has been plotted in
the modified scenario with late time production of entropy.
We have illustrated the results for the cases wherein $n_{_{\mathrm{S}}}$ is fixed and the
quantity $F$ is varied (on the left) as well as for the cases wherein $F$ is
fixed and $n_{_{\mathrm{S}}}$ is varied (on the right).
We have set $w_\phi = 0$ and $T_{\sigma R} = 1\,\mathrm{GeV}$ in arriving
at the above plots.
Also, we have considered the extreme values for $w_\sigma$ to demonstrate
the maximum levels of impact that the generation of entropy at late times
can have on the spectrum of GWs.
Interestingly, we find that for a set of values of the parameters associated
the secondary phase of reheating, the spectrum $\Omega_{_{\mathrm{GW}}}$ can have amplitudes as
suggested by the recent observations
by NANOGrav~\cite{Arzoumanian:2020vkk,Pol:2020igl}.}\label{plot_latetimeentropy}
\end{figure}
In arriving at the plots in the figure, we have set the inflaton the
EoS parameter to be $w_{\phi}=0$ and have assumed that $T_{\sigma R}
= 1\,\mathrm{GeV}$.
Note that, for a given inflaton EoS parameter~$w_{\phi}$, we have two
parameters, viz. $n_{_{\mathrm{S}}}$ and $F$, which control the global shape of the
spectrum of GWs.
We have plotted the spectrum for different values of~$F$ with a fixed
value of~$n_{_{\mathrm{S}}}$ as well as for different values of $n_{_{\mathrm{S}}}$ with a fixed~$F$.
In order to highlight the effects due to the additional generation of
entropy, we have plotted the results for the limiting values of zero
and unity for the EoS parameter~$w_\sigma$ governing the second phase
of reheating.
We should point that, if a canonical scalar field also dominates the
secondary phase of reheating, then for $V(\sigma)\propto \sigma^{2\,n}$,
we have $w_\sigma = (n-1)/(n+1)$ so that for the above mentioned
limiting values can be achieved for $n=1$ and $n \to \infty$,
respectively.
One can also consider a more exotic, non-canonical, scalar field with a
Lagrangian density of the form ${\mathcal L} \sim (\partial \sigma)^\mu
- \sigma^{2\,n}$, where $\mu$ is a rational number, to drive the second
phase of reheating.
In such a case, it can be shown that the EoS parameter is given by (in
this context, see, for example, Ref.~\cite{Unnikrishnan:2012zu})
\begin{equation}
w_{\sigma} = \frac{n-\mu}{n\,(2\,\mu -1) + \mu},
\label{noncanEoS}
\end{equation}
with the expression reducing to the canonical result for $\mu=1$,
as required.
Such a model can lead to the extreme values of $w_\sigma = 0$ (for
$n = \mu$) and $w_\sigma \simeq 1$ for [$n \simeq \mu/
(1-\mu)$] that we have considered, without unnaturally large values
for a dimensionless number, as it occurs in the canonical case.
It seems worthwhile to explore such models in some detail as they could
have interesting phenomenological implications.
We find that the effects on $\Omega_{_{\mathrm{GW}}}(f)$ over the range $f < f_{\mathrm{re}}$ due to
the late time creation of entropy have important implications for the
recent observations by the NANOGrav mission.
Recall that, recent observations by the NANOGrav mission suggest a stochastic
GW background with an amplitude of $\Omega_{_{\mathrm{GW}}}\, h^2\simeq 10^{-11}$ around the
frequency of~$10^{-8}\,\mathrm{Hz}$~\cite{Arzoumanian:2020vkk,Pol:2020igl}.
Clearly, the frequency lies in the domain $f < f_{\mathrm{re}}$.
In the absence of a second phase of reheating, evidently, the amplitude
of $\Omega_{_{\mathrm{GW}}}$ in the nano-Hertz range of frequencies is rather small, much
below the sensitivity of the NANOGrav mission, as we had seen in
Secs.~\ref{sec:averaged} and~\ref{sec:actual}.
However, as we have discussed, the late time decay of an additional scalar
field such as the moduli field leads to a spectrum with a blue tilt for
$w_\sigma > 1/3$ over the frequency range $f_\sigma < f< f_{\sigma R}$,
where $f_\sigma$ and $f_{\sigma R}$ are the frequencies associated with
the wave numbers $k_\sigma$ and $k_{\sigma R}$.
Therefore, in such a modified scenario, it is possible to construct situations
that result in $\Omega_{_{\mathrm{GW}}}$ of the strength indicated by NANOGrav, albeit with rather
large values for $\omega_\sigma$ and relatively low values of the reheating
temperature of $10 < T_{\mathrm{re}} < 10^3\,\mathrm{GeV}$, as illustrated in
Fig.~\ref{plot_latetimeentropy}.
To motivate high values for the EoS parameter, as we mentioned, it seems
interesting to consider a non-canonical model of a scalar field that leads
to an EoS parameter as in Eq.~\eqref{noncanEoS}.
We should clarify that, to avoid pathological behaviour, in the model,
we can consider parameters lying within the domains $n>0$ and $\mu> 0$.
In such a case, one can obtain that $w_{\sigma} \sim 1$ for $\mu$ in the
range $0 < \mu < 1$.
Moreover, note that, in the modified scenario with late time entropy production,
to be compatible with the NANOGrav results, the reheating temperature $T_{\mathrm{re}}$ has
to be less than $10^{3}\,\mathrm{GeV}$, which implies a low decay width for the
inflaton.
Further, from Fig.~\ref{plot_latetimeentropy}, it can be easily seen that
the modified scenario can be strongly constrained by many of the forthcoming
GW observatories such as SKA, BBO, LISA and DECIGO.
In the pulsar-timing data considered by NANOGrav, the spectrum of the
characteristic strain $h_\mathrm{c}(f)$ induced by the GWs is assumed
to be a power law of the form~\cite{Arzoumanian:2020vkk,Pol:2020igl}
\begin{equation}
h_\mathrm{c}(f) = A_{_\mathrm{CP}}\,
\left(\frac{f}{f_\mathrm{yr}}\right)^{(3-\gamma_{_\mathrm{CP}})/2},
\label{eq:cs}
\end{equation}
where $A_{_\mathrm{CP}}$ refers to the amplitude at the reference frequency
$f_\mathrm{yr}=1\,\mathrm{yr}^{-1} =3.17\times10^{-8}\, \mathrm{Hz}$, and
$\gamma_{_\mathrm{CP}}$ is the timing-residual cross-power spectral density.
The dimensionless energy density of GWs today~$\Omega_{_{\mathrm{GW}}}(f)$ is related to
characteristic strain~$h_\mathrm{c}(f)$ induced by the GWs through the
relation (in this context, see the recent review~\cite{Yokoyama:2021hsa})
\begin{equation} \label{omegahc}
\Omega_{_{\mathrm{GW}}}(f)=\frac{2\,\pi^2\,f^2}{3\,H_0^2}\,h_\mathrm{c}^2(f).
\end{equation}
Upon utilizing the form~\eqref{eq:cs} for the characteristic strain,
the energy density of GWs today can be expressed in terms of the
amplitude $A_{_\mathrm{CP}}$ and the index $\gamma_{_{\mathrm{CP}}}$
as follows:
\begin{equation}
\Omega_{_{\mathrm{GW}}}(f)=\frac{2\,\pi^2\,f_\mathrm{yr}^2}{3\,H_0^2}\,
A_{_\mathrm{CP}}^2\,\left(\frac{f}{f_\mathrm{yr}}\right)^{5-\gamma_{_\mathrm{CP}}}.
\label{eq:omegaACP}
\end{equation}
As we have discussed above, in the scenario with late time entropy production,
it is the power associated with the modes that reenter the Hubble radius during
the secondary phase of reheating that are consistent with the NANOGrav results
[cf. Fig.~\ref{plot_latetimeentropy}].
Over this domain of wave numbers, since the index of the spectrum of GWs is
given by $n_{_\mathrm{GW}}=2\,(3\,w_\sigma - 1)/(3\,w_\sigma + 1)$, where $w_\sigma$
is the EoS parameter describing the secondary phase of reheating, clearly,
we can set $\gamma_{_\mathrm{CP}}=5-n_{_\mathrm{GW}}$.
We can utilize the constraints from the NANOGRav results on the parameters
$A_{_\mathrm{CP}}$ and $\gamma_{_{\mathrm{CP}}}$ to arrive at the corresponding
constraints on, say, the EoS parameter $w_\sigma$ and the reheating
temperature~$T_{\mathrm{re}}$ associated with the primary phase.
We have illustrated these constraints in Fig.~\ref{fig:gammacp}.
\begin{figure}[t!]
\centering
\includegraphics[width=8.00cm]{Acpgammacpomega.pdf}
\hskip 5pt
\includegraphics[width=8.75cm]{gammacp0.99.pdf}
\caption{The constraints from NANOGrav on the parameters $A_{_\mathrm{CP}}$
and $\gamma_{\mathrm{CP}}$ have been utilized to illustrate the corresponding
constraints on the EoS parameter~$w_\sigma$ describing the secondary phase
of reheating and the reheating temperature~$T_{\mathrm{re}}$ associated with the primary
phase.
In the figures, we have included the 1-$\sigma$ and 2-$\sigma$ contours (in
black) arrived at by the NANOGrav analysis based on the five-frequency
power-law fit~\cite{Arzoumanian:2020vkk}.
We have projected the results into the $\gamma_{_\mathrm{CP}}$-$A_{_\mathrm{CP}}$
plane for the following two possibilities: (i)~a scenario wherein~$T_{\mathrm{re}}$ is
fixed and the parameter~$w_\sigma$ is varied (on the left), and (ii)~a scenario
wherein~$w_\sigma$ is fixed and~$T_{\mathrm{re}}$ is varied (on the right).
Note that we have assumed that $w_\phi = 0$ and $T_{\sigma R} = 0.1\,\mathrm{GeV}$
in arriving at the above plots.}\label{fig:gammacp}
\end{figure}
We have illustrated the constraints for the two possibilities, viz. with the
reheating temperature~$T_{\mathrm{re}}$ associated with the primary phase fixed and the
EoS parameter~$w_\sigma$ describing the secondary phase of reheating
varied as well as with~$w_\sigma$ fixed and~$T_{\mathrm{re}}$ varied.
These constraints suggest that only low reheating temperatures (say,
$T_{\mathrm{re}}< 10\, \mathrm{GeV}$) and very high values of the EoS
parameter~$w_\sigma$ (with $w_\sigma \simeq 1$) are compatible with
the NANOGrav data.
\section{Conclusions}\label{sec:c}
In this work, we have attempted to understand the effects of reheating on
the spectrum of primordial GWs observed today.
As a specific example, we had considered the so-called $\alpha$-attractor
model of inflation and had evolved the tensor perturbations from the end
of inflation through the epochs of reheating and radiation domination to
eventually arrive at the spectrum of GWs today.
Moreover, we had considered two different scenarios to achieve reheating.
In the first scenario, the epoch is described by a constant EoS
parameter~$w_\phi$ and the transition to radiation domination
is expected to occur instantaneously~\cite{Dai:2014jja}.
Such a simpler modeling of the reheating mechanism had allowed us to study
the evolution of the GWs analytically.
In the second and more realistic scenario of perturbative reheating wherein
the evolution of the energy densities are governed by the Boltzmann equations,
the inflaton decays gradually and the transition to the epoch of radiation
domination occurs smoothly.
The effective EoS parameter in such a scenario changes continuously from its initial
value of $w_\mathrm{eff} = w_\phi$ to the final value of
$w_\mathrm{eff} = 1/3$.
As it proves to be difficult to obtain analytical solutions in the perturbative
reheating scenario, we had examined the problems at hand numerically.
Note that we are interested in the spectrum of GWs today over scales that are
considerably smaller than the CMB scales.
These scales reenter the Hubble radius either during the phase of reheating or
during the epoch of radiation domination.
In both the models of reheating we have considered, the spectrum of GWs today
$\Omega_{_{\mathrm{GW}}}(f)$ is scale invariant over wave numbers that reenter the Hubble radius
during the epoch of radiation domination (i.e. for $k <k_{\mathrm{re}}$).
The scale invariant amplitude over this domain can help us extract the
inflationary energy scale in terms of the present radiation abundance
since $H_{_{\mathrm I}}^2/M_{_{\mathrm{Pl}}}^2 \sim 6\,\pi^2\,\Omega_{_{\mathrm{GW}}}/ \Omega_{_\mathrm{R}}$
[cf. Eq.~\eqref{infscale}].
The spectrum of GWs today has a tilt over wave numbers $k_{\mathrm{re}} <k <k_{\mathrm{f}}$ which
reenter the Hubble radius during the phase of reheating.
The spectral tilt $n_{_\mathrm{GW}} = 2\,(3\,w_{\phi} - 1)/(3\,w_\phi + 1)$ is red
or blue depending on whether $w_\phi < 1/3$ or $w_\phi > 1/3$.
Clearly, the observation of the tilt will allow us to determine the reheating
parameters such as the EoS parameter~$w_\phi$ and the reheating temperature~$T_{\mathrm{re}}$.
These will allow us to determine the behavior of the inflaton near the minimum
of the potential.
Moreover, the constraint on the reheating parameters, in turn, can help us
constrain the inflationary parameters such as the scalar spectral index~$n_{_{\mathrm{S}}}$.
Further, in the realistic perturbative reheating scenario, there arises an important
feature in the spectrum of GWs around wave numbers that reenter the Hubble radius
towards the end of the phase of reheating.
The spectrum exhibits distinct oscillations near the frequency $f_{\mathrm{re}}$ and we find
that the width of the oscillation reflects the time scale over which the EoS parameter
changes from the inflaton dominated value of $w_\phi$ to that of $1/3$ in the
radiation dominated epoch (cf. Figs.~\ref{plot_perturbative1} and
\ref{plot_perturbative2}).
In fact, the peak of the oscillation occurs at $f_{\mathrm{re}}$, which can be associated with
the end of perturbative reheating (i.e. when $H =\Gamma_{\phi}$).
We find that, at this instant, the rate of change of the effective
EoS parameter exhibits a maximum.
This can help us further in determining the inflaton decay rate $\Gamma_{\phi}$
in terms of the observed quantities, as expressed in Eq.~\eqref{gammaphi}.
The end of reheating is to be followed by the most important stage of
thermalization.
From the width of the oscillation in the spectrum, one can extract information
about thermalization time scale $\Delta t_\mathrm{th}$
[cf. Eq.~\eqref{thermalization}] as well as the nature of the thermalization
process depending upon over-occupied or under occupied initial state set by
the end of reheating [cf. Eq.~\eqref{alpha}].
It turns out that the ratio of the thermalized particle density to the initial decaying
particle density (i.e. $n_\mathrm{th}/n_\mathrm{i}$) depends on the inflaton mass,
inflationary energy scale and the wave numbers $k_{\mathrm{re}}$ and $k_{\mathrm{f}}$.
Therefore, given the inflaton mass, the spectrum of primordial GWs at the present
epoch can be utilized to determine the ratio $n_\mathrm{th}/n_\mathrm{i}$, which
is one of the important parameters describing the thermalizing plasma.
Once the value of $n_\mathrm{th}/n_\mathrm{i}$ is extracted, the strength of the
gauge interaction, say, $\alpha$, among the reheating decay products can be obtained
[cf. Eq.~\eqref{alpha2}].
We should mention that the oscillatory feature in the spectrum of GWs is encountered
for all possible values of $w_{\phi}$ and $T_{\mathrm{re}}$.
We find that the range of frequencies around which the oscillations arise shifts
towards higher frequencies as the reheating temperature increases.
This can be attributed to the fact that, for a higher value of $T_{\mathrm{re}}$, the duration of
reheating proves to be shorter and, as a result, the wave number~$k_{\mathrm{re}}$ which reenters
at the end of reheating becomes larger.
Apart from the effects due to the standard phase of reheating, we had also
considered the signatures on the spectrum of GWs due to a secondary phase
of reheating.
Such a secondary phase can arise due to the decay of additional scalar fields
such as the moduli fields, which can lead to the production of entropy at late
times.
The moduli dominated phase, which can be described by a constant EoS
parameter~$w_\sigma$ (as the primary phase of reheating), leads
to a tilt in the spectrum over the range of wave numbers that reenter
the Hubble radius during the epoch.
Remarkably, we have shown that, for $w_\sigma > 1/3$ and certain values of
the reheating temperature, the production of entropy at late times can lead to
$\Omega_{_{\mathrm{GW}}}(k)$ that correspond to the strengths of the stochastic GW background
suggested by the recent NANOGrav observations~\cite{Arzoumanian:2020vkk,Pol:2020igl}.
We should clarify that such a possibility cannot arise in the conventional
reheating scenario wherein the entropy remains conserved from the end of
reheating until the present epoch.
In fact, the assumption of the secondary phase of reheating and the NANOGrav
observations indicate a low reheating temperature of $T_{\mathrm{re}} \lesssim 10^{3}\,
\mathrm{GeV}$. We are currently examining different models to understand the wider implications
of such interesting possibilities.
\acknowledgments
MRH would like to thank the Ministry of Human Resource Development,
Government of India (GoI), for financial assistance.
DM and LS wish to acknowledge support from the Science and Engineering
Research Board~(SERB), Department of Science and Technology~(DST), GoI,
through the Core Research Grant CRG/2020/003664.
LS also wishes to acknowledge support from SERB, DST, GoI, through the
Core Research Grant CRG/2018/002200.
\bibliographystyle{apsrev4-1} |
1711.10660 | \section{Introduction and Motivation}
Realization of the fact that black holes behave as thermodynamical objects \cite{Bekenstein:1974ax,Bardeen:1973gs,Gibbons:1977mu,Hawking:1976de, Bekenstein:1973ur,Bekenstein:1972tm,Hawking:1974sw,Jacobson:1995ab,Padmanabhan:2009vy,Padmanabhan:2013nxa} has, since long, been fuelling the hope of reconciling the quantum theory with the framework of gravity in a successful and consistent manner. In particular, the thermodynamic nature of black holes suggest a strong connection between macroscopic geometrical constructs and microscopic quantum behaviour. Thus demystifying and understanding the black hole geometry has been one of the pioneering goals of modern physics, which in all likelihood, will provide an useful insight to the quest of arriving at a successful quantum theory of gravity \cite{Maldacena:1996ky,Rovelli:1996dv,Dowker:2005tz,Ashtekar:1997yu}. Different approaches towards quantizing gravity \cite{Maldacena:1996ky,Rovelli:1996dv,Dowker:2005tz,Ashtekar:1997yu}, emerging out since the discovery of black hole thermodynamics, have been accordingly oriented, keeping the hope of unifying gravity and quantum theory in sight. Unfortunately, despite decades of efforts along numerous directions, which have resulted into many promising candidates for quantum gravity, we have not been able to come up with a theory that can actually provide a consistent description to unify quantum theory and gravity. Each of these candidates has offered its own different pathway of studying and understanding different constructions of gravity, more particularly the black holes. However with no black holes available to experiment with, and a plethora of quantum gravity theories with their own predictions, the black hole mysteries \cite{Hawking:1976ra,Visser:2014ypa,Mathur:1997wb,Mathur:2009hf,Hooft:2015jea,Chakraborty:2017pmn,Hawking:2016msc,Hawking:2016sgy,Modak:2014qja} have led this search to a cross road.
The detections of gravitational waves \cite{Abbott:2016blz,Abbott:2016nmj} from merger of two black holes have recently ignited the debates regarding the inner workings of black holes \cite{Hawking:1976ra,Visser:2014ypa,Mathur:1997wb,Mathur:2009hf,Hooft:2015jea,Chakraborty:2017pmn,Hawking:2016msc,Hawking:2016sgy,Modak:2014qja,Kraus:2015zda,Hawking:2014tga,Frolov:2014wja,Vaz:2014era} and its possible implications for quantum gravity theories. There have been studies \cite{Abedi,Foit:2016uxn} to explore the imprints of some non-intuitive quantum features of the black holes, which appear in many frameworks of quantum gravity. Therefore, the dynamical evolution of black holes can really act as a lead for testing of our theoretical models claiming to quantize gravity \cite{Cardoso:2017cqb,Maselli:2017kic,Cardoso:2017cfl}.
Apart from the classical gravitational wave emissions, there is also a quantum channel by which a black hole can emit, known as the famous {\it Hawking radiation} \cite{Hawking:1974sw}. There is a renewed interest in the quantum emission profile of black holes after the realization gained ground that quantum emission of black holes may well be hiding its intrinsic quantum face \cite{Alonso-Serrano:2015trn, Lochan:2015oba,Lochan:2016nbs,Chakraborty:2016fye}. Studies of such quantum emissions hold their own ground as the tussle between non-compatibility of the quantum theory with principles of gravity become manifest here, thanks to the thermal nature of the radiation \cite{Chakraborty:2017pmn,Saini:2015dea,Chen:2014jwq}.
In this work, we will assume that the notion of black hole entropy as perceived in the context of thermodynamics of macroscopic black holes holds true through the microcanonical counting, in the quantum domain (see, for example \cite{Bekenstein:1995ju,Bekenstein1974,Bekenstein:1997bt}). Given the above assumption on the nature of black hole entropy at a microscopic level we would like to address the question, across these class of theories, is there any signature of the underlying quantum nature of black holes which gets fossilized in its emission? In previous works \cite{Lochan:2015bha,Chakraborty:2016fye}, we have explicitly demonstrated that if black holes are indeed a macroscopic realization of a microscopic quantum geometry and the thermodynamic relation between entropy and area still holds in the microscopic level having holographic properties, then it is highly unlikely that the emission from them will have the suggested thermal behaviour. If the geometry (of black hole spacetime) gets discretized {\it in any fashion}, the emission lines from the macroscopic black hole become hugely discrete in the low frequency regime. Similar conclusions were reached in many other contexts and their far reaching implications are still being investigated \cite{Gray:2015pma,Hod:2015wva,Hod:2016xbu,Hod:2016rmg}.
However the scenario depicted in earlier works was for a Schwarzschild black hole, with mass as the only hair \cite{Bekenstein:1995ju,Hod:1998vk,Hod:2000it,Barreira:1996dt,Lochan:2015bha,Hod:2015wva,Hod:2016xbu,Hod:2016rmg,Chakraborty:2016fye}, while astrophysical black holes are most likely to inherit non-zero angular momentum as well. Thus in the process of decay, the black hole loses both mass and angular momentum and as we will show in this work, the inclusion of angular momentum modifies the quantum spectrum of a black hole drastically. In particular, as we will depict, there will be lot of interesting physics happening near the extremal limit, which was non-existent for Schwarzschild black hole such as occurrence of a dense emission spectrum. Further, since black holes are supposed to be thermodynamical objects in classical theory, the emission lines must be consistent with both thermodynamics and the underlying quantum structure. As we will see, these two demands are strong enough that result into a discrete quantum spectrum for black holes \cite{Bekenstein:1995ju,Hod:1998vk,Hod:2000it,Lochan:2015bha,Chakraborty:2016fye}. This fact has been aptly demonstrated in the so called \emph{Bekenstein-Mukhanov} effect, showing the existence of a minimum frequency and hence the largest wavelength that a black hole can emit. Note that the above result is based on the assumption that the entropy-area relation, obtained from the thermodynamics of macroscopic black holes, comes through in {\it exact form} from the underlying quantum counting of this macroscopic realization of the quantum discrete geometry. However, typically there can be sub-leading corrections to such a relation, which will come through the exact (say) von-Neumann counting. In \ref{AppB}, we have addressed a class of such subleading correction which appear across many theories of quantum gravity \cite{Lochan:2012in,Lochan:2012sp,Sen:2012dw,Ghosh:2004rq}. Interestingly, for this class of corrections too, the results obtained in this work remain valid. Once we have minimum frequency of emission, we can track the dynamics of the black hole evaporation, counting the {\it maximum possible} number of emissions as well as the time taken by the black hole to turn Planckian in size where the laws of emission are supposed to get compromised.
The paper is organized as follows: In \ref{GeometricOp} we analyze the relation of the geometric aspects of a Kerr-Newman black hole in order to determine the mass gap between the nearest macroscopic configuration emerging out from a quantum operator which is discretized in integer steps along with the universal relation connecting various length scales. Subsequently, in \ref{Lifetime}, we estimate the maximum number of quanta that a Kerr-Newman black hole can emit and hence will provide a rough estimate of the lifetime of a black hole. Some discussions regarding extremal limit in this discretization scheme has been presented in \ref{ExtremalLimit}. We finally argue the robustness of our results for higher dimensional rotating as well as charged black holes in \ref{HighDim}, before summarizing our main findings in \ref{Conclusion}.
\section{Holography and quantum spectrum of black holes} \label{GeometricOp}
We will work with a premise that a quantum theory of gravity discretizes geometry in any (hitherto unknown) fashion. There are many candidate theories with different geometric operators getting quantized as well as with different spectrum of quantization \cite{Rovelli:1994ge,Vaz:2007td,Vaz:2009jj}. The results obtained in this paper remain fairly insensitive to the details of the geometry quantization as they care only about the {\it course gained} quantities at the macroscopic levels (by which we mean fairly away from being Planck sized). Throughout this work we will exclusively concentrate on the Kerr-Newman family of black holes, which is constructed at the macroscopic level from any of the geometric quantization theories, characterized by the mass $M$, angular momentum $J$ and charge $Q$ with the horizon located at,
\begin{align}
r_{\rm h}=M\left[1+\sqrt{1-\left\{\left(\frac{a}{M}\right)^{2}+\left(\frac{Q}{M}\right)^{2}\right\}}\right]~,
\end{align}
where $a=J/M$ is the rotation parameter associated with the Kerr-Newman black hole. Since we are interested in black hole spacetime, in what follows we will assume that $(a^{2}+Q^{2})<M^{2}$. Note that in the limit $\{(a/M)^{2}+(Q/M)^{2}\}\rightarrow1$ the term inside square root vanishes and is known as the extremal limit.
In this context of Kerr-Newman black hole we will assume that certain geometric feature, (e.g., area) gets discretized, without referring to any particular scheme of quantum gravity. In this approach we will consider any classical black hole to be a macroscopic realization of the underlying quantum theory, which still awaits discovery. Remarkably enough, our results will be independent on the nitty gritties of such a quantum theory, except for the fact that some geometrical object has been discretized. Let such a geometrical construct, denoted by $\mathcal{Z}$ is assumed to have a discrete spectral profile characterized by $f(j)$, where $j$ symbolizes the quantum level, which, motivated by the compactness of the horizon we consider to be discrete. The particular form for $f(j)$ will depend on the details of the discretization scheme, which we will not be bothered about and shall keep it arbitrary.
The classical geometry, which in the quantum theory of gravity will be given by some state $\hat{\rho}$ and is understood to emerge in the macroscopic limit, when the expectation value of the geometric quantity $\mathcal{Z}$ becomes
\begin{align}\label{Arb_Spec}
{\cal Z}= \text{Tr}[\hat{\rho}\hat{\cal Z}] =\alpha \sum_jn_j f(j)~,
\end{align}
where the fluctuations are understood to be heavily suppressed for macroscopic size system.
As mentioned earlier the discretization spectrum $f(j)$ as of now will be arbitrary, which will be known only when the correct quantum theory of black holes becomes available. The parameter $\alpha$ is a new scale which is brought along by the underlying quantum theory and is possibly related to the Planck scale. Further, the presence of a minimal Planck length in any theory of quantum gravity tells us that ${\cal Z}$ must be bounded from below.
In the macroscopic limit let the area of the event horizon get related to this characterizing geometric parameter as
\begin{align}\label{A-Z Rel}
A=\kappa{\cal Z}^{\gamma}=\kappa\alpha^{\gamma}\Big\{\sum_{j}n_{j} f(j)\Big\}^{\gamma}~,
\end{align}
where $\gamma$ is assumed to be some real positive number\footnote{We can generalize the relation of the area to the discretized variable in a polynomial fashion. Appendix A discusses the generalization.}. For example, when $\mathcal{Z}$ corresponds to square root of black hole area, then $\kappa=1$ and $\gamma=2$. If, we further accept that a macroscopic description emerging from the basic quantum theory should respect the Bekenstein's entropy-area relation, then there must also be a way to connect entropy with the quantum description, such that,
\begin{align}\label{Holography}
S=\frac{A}{4}=\frac{\bar{\alpha}^{\gamma}}{4}\Big\{\sum_jn_j f(j)\Big\}^{\gamma}
=\ln g(n)~,
\end{align}
where $\bar{\alpha}^{\gamma}=\kappa\alpha^{\gamma}$ and $g(n)$ is effective number of micro states giving rise to the macroscopic black hole configuration, as we consider the black hole as a system in a micro-canonical set up. Many quantum gravity theories attempt to visualize this number of available microstates through their counting schemes and have been successful in doing so \cite{Sen2014,Rovelli:1996dv,Kaul:2012pf}. Therefore, recovery of this relation from microscopic counting does not really serve as a very strong discriminator between the available quantum gravity models. However, we will see (in \ref{MassGap}) that characterizing the nearest allowed configuration may well be such a tool. Furthermore one can argue that this feature remains true in different ensemble pictures of analysis as well. From \ref{Holography} it is simple to compute the number of micro states, leading to,
\begin{align}\label{IntegerConstraint}
g(n)=\exp{S}=\exp\left({\frac{A}{4}}\right)=\exp\left[{\frac{\bar{\alpha}^{\gamma}}{4} F^{\gamma}}\right];\qquad F\equiv \Big\{\sum_jn_j f(j)\Big\}~.
\end{align}
Clearly, as the number of microstates in the micro-canonical picture has to be an integer, \ref{IntegerConstraint} demands $\mathcal{B}F^{\gamma}/4=\ln K$ for some variable integer $K$, corresponding to different area values of the horizon. This can happen in two possible ways --- (a) $\mathcal{B}/4 =\ln \mathcal{I}_{0}$ and $F^{\gamma}=\mathcal{I}$; or --- (b) $\mathcal{B}/4 =\mathcal{I}_0$ for some fixed integer $\mathcal{I}_0$, and $F^{\gamma} =\ln \mathcal{I}$ for variable integer $\mathcal{I}$, for all possible $\{n_j\}$ (at least) for which $\sum_j n_j \gg 1$, i.e., in the macroscopic limit. The choice (a) essentially presents a integer shift \cite{Bekenstein:1995ju,Bekenstein:1997bt} in the macroscopic expectation of the geometric variable, while choice (b) corresponds to a logarithmically discretized geometric operator \cite{Visser:1992ck}. We will discuss the scenario depicted by case (a) alone, i.e., in which $A=\mathcal{B} \mathcal{I}$. Here $\mathcal{B}$ carries the imprint of the underlying quantum structure and the integer $\mathcal{I}$ marks the micro states $\{n_{j}\}$ in a collective manner. Following the footsteps, the case (b) can also be taken up with equal ease and the results remain qualitatively similar to case (a), which we would not report here. Such an integerly discretized scheme may be advocated on various grounds including quantum gravity as well \cite{Barreira:1996dt,Rovelli:1996dv,Rovelli:2017mzl,Ashtekar:1997yu,Chakraborty:2017s}. As the black hole makes a transition from one macroscopic configuration characterized by $\{n_{j}\}$ to another macroscopic state denoted by $\{n_{j}'\}$, the integer changes from $\mathcal{I}$ to $\mathcal{I}'$.
Before concluding this section we would like to emphasize that the quantization scheme presented here is completely general, since we have \emph{not} assumed any particular form for the function $F(j)$ determining the quantization levels of black holes. As and when a viable theory of quantum gravity becomes available, it will certainly produce some $F(j)$ and the results derived in this work being independent of $F(j)$, will be readily applicable. Hence all the results we will derive in this work will have applicability for any theory of quantum gravity, as long as it predicts quantization of certain geometrical quantity. It is \emph{not at all dependent} on the details of the quantization procedure.
\subsection{General Analysis}\label{MassGap}
If the macroscopic black hole of the quantum theory, closely resembles the classical black hole configuration, the area and the mass are related through,
\begin{align}\label{I_R}
\frac{\mathcal{B} \mathcal{I}}{4\pi}=r_{\rm h}^{2}+a^{2}
=2M^{2}\left[1+\sqrt{1-\left\{\left(\frac{a}{M}\right)^{2}+\left(\frac{Q}{M}\right)^{2}\right\}}\right]-Q^{2}~.
\end{align}
It is possible to invert the above relation in order to write down the mass in terms of the discretized area as
\begin{align}
2M = \frac{\left(\mathcal{B} \mathcal{I}/4\pi\right)+Q^{2}}{\sqrt{\left(\mathcal{B} \mathcal{I}/4\pi\right)-a^{2}}}
= \sqrt{\left(\mathcal{B} \mathcal{I}/4\pi\right)-a^{2}}+\frac{\left(a^{2}+Q^{2}\right)}{\sqrt{\left(\mathcal{B} \mathcal{I}/4\pi\right)-a^{2}}}
\end{align}
We are interested in obtaining the smallest mass difference between two (mass-wise) nearest black holes. In principle, the nearest neighbour of a black hole of mass M (and hence integer $\mathcal{I}$) can only be larger than the mass identified by the integer value $\mathcal{I}+1$. In general, as both the mass and the rotation parameter of black hole are changed, the area of the black hole can either increase or decrease. For example, if the mass does not change, but only the rotation parameter decreases, the area will increase. However, for emission of a particle with certain energy, the mass parameter must be modified. Thus in general there will be a competition between the change in mass and the change in rotation parameter. For black holes which is near extremal it is difficult to argue, which one among these two will dominate. But for black holes having $a\ll M$, of course the change in mass will dominate and one can safely ignore the effects of rotation. A similar consideration applies to the charge parameter associated with the black hole as well. Since we are interested in black holes which are far from being extremal to start with, the area will decrease as it emits quanta of radiation.
Following the above discussion we will be considering \emph{only} those transitions for which both the area and rotation parameter \emph{decrease}, since these are the ones which are relevant for a black hole away from extremality. Furthermore, we will be excluding transitions among black hole micro-states having identical area. Thus it follows that the mass change will be minimum when both $Q$ and $a$ are kept the same. Otherwise, the change in $Q$ or $a$ will add to the area change and hence the shift in black hole mass will be bigger. Thus the mass difference between two nearest
micro-canonical configurations turns out to be
\begin{align}\label{MGap}
\Delta M _{\rm min}=\frac{\mathcal{B}}{16\pi} \frac{\left(\mathcal{B} \mathcal{I}/4\pi\right)-2a^{2}-Q^{2}}{\Big\{
\left(\mathcal{B} \mathcal{I}/4\pi\right)-a^{2}\Big\}^{3/2}}~.
\end{align}
In order to arrive at the above expression we have assumed the black hole to be macroscopic, which justifies expansion in inverse powers of $\mathcal{I}$. This is because, large value of $\mathcal{I}$ implies that most of the micro-states are already occupied, resulting into a macroscopic description for black holes. However, the above expression is not very illuminating. To cast the above relation to a more useful form we have to provide an expression for $(\mathcal{B} \mathcal{I}/4\pi)$ in terms of the hairs of the black hole. Such a relation is being supplied by \ref{I_R}, yielding,
\begin{align}
\frac{\mathcal{B} \mathcal{I}}{4\pi}-a^{2}=\left[M+\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}\right]^{2}~,\label{I_R2}\\
\frac{\mathcal{B} \mathcal{I}}{4\pi}-2a^{2}-Q^{2}=r_{\rm h}^{2}-\left(a^{2}+Q^{2}\right)
=2r_{\rm h}\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}, \label{I_R3}
\end{align}
which along with \ref{MGap} leads us to the relation,
\begin{align} \label{MGap2}
\Delta M _{\rm min}&=\frac{\mathcal{B}}{8\pi}\frac{\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}}{\left[M+\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}\right]^{2}}
=\frac{\mathcal{B}}{2}\frac{T_H(\eta)}{1+\sqrt{1-\eta ^{2}}}~.
\end{align}
Here $T_H(\eta)$ corresponds to the Hawking temperature associated with the Kerr-Newman black hole and $\eta ^{2}=(a^{2}+Q^{2})/M^{2}$, a dimensionless parameter approaching unity as the black hole approaches the extremal limit. The above expression clearly demonstrates that the minimum mass gap identically vanishes in the extremal limit. This is because, the term in the denominator takes a value $2$ in the $\eta \rightarrow 0$ limit, while it becomes $1$ in the extremal limit. On the other hand, the black hole temperature $T_H(\eta)$ is directly proportional to $1-\eta ^{2}$ and hence identically vanishes in the extremal limit. Therefore the minimum mass gap also vanishes, implying continuous black hole spectrum in the extremal limit as shown in \ref{fig_01}. One can further verify that this minimum mass gap exactly coincides with the one derived from thermodynamical consideration, a crucial test for consistency of the scenario depicted here.
\begin{figure*}
\begin{center}
\includegraphics[height=2in, width=3in]{Continuous_Spectrum_Final.pdf}~~
\includegraphics[height=2in, width=3in]{Continuous_Spectrum_03.pdf}\\
\caption{The figure on the left depicts how the minimum mass gap, expressed for convenience as $M\Delta M_{\rm min}$, changes as $(\sqrt{a^{2}+Q^{2}}/M)$ approaches unity. As the figure clearly demonstrates, with the black hole approaching extremal limit, the mass gap also tends to vanish. On the other hand, the right hand figure schematically illustrates how the quantum spectrum gradually becomes continuous as the black hole approaches the extremal limit. The blue, dot-dashed curve at the top illustrates the case of Schwarzschild black hole (i.e., with $a=Q=0$), from which it is clear that the spectrum will be largely discrete, as the minimum mass gap appears around the maxima of the distribution. While the red, dashed curve depicts the corresponding situation for $a=Q=0.35~M$ and the lowermost green curve illustrates the case with $a=Q=0.6~M$ respectively. As evident from the red and green curves, the region spanned by the dense spectrum gradually increases as the $(a/M)$ and $(Q/M)$ increases. Ultimately, when the black hole becomes near-extremal the quantum spectrum will become continuous.}\label{fig_01}
\end{center}
\end{figure*}
We see that the (in principle) smallest mass gap gets related to the thermodynamic temperature of the black hole. If this mass gap gives rise to the emission of a quanta in the form of loss of energy, the frequency of the emitted quanta (ignoring the $\mathcal{B}/2$ factor), will always be less compared to the thermal frequency associated with black hole temperature. This is because the term $1+\sqrt{1-\eta ^{2}}$ is always greater than unity. Notably as, $a=Q=0$, then $\Delta M\sim 1/M$, as it should for a macroscopic Schwarzschild black hole \cite{Bekenstein:1995ju,Lochan:2015bha,Chakraborty:2016fye}. We would also like to point out that as the black hole approaches the classical extremal limit, i.e., $M\rightarrow (a^{2}+Q^{2})$, $\Delta M \sim 0$, remarkably the mass gap vanishes, leading to continuous quantum spectrum of emission, though at extremely small temperature. It turns out that continuous emission from rotating black hole in the extremal limit can also be derived from calculation of characteristic time scale of the emission of a quanta from black hole, see \cite{Hod:2015wva}.
The last property, presented above, is very unique for Kerr-Newman class of black holes and is absent in Schwarzschild black holes. The result clearly shows that near the extremal limit the quantum spectrum of a geometry discretized hole is almost continuous, unlike the Schwarzschild scenario. Thus the initial discreteness in the quantum spectrum of a Kerr-Newman black hole gradually makes its way to a continuous spectrum as the black hole approaches extremality (see also \cite{Hod:2015wva}).
Before concluding this discussion, let us briefly mention what happens to the mass gap in a more general scenario, in which both the mass and rotation parameter $a$ change. This is because we normally expect a black hole to emit its multi-pole moments which apart from carrying energy will carry away some angular momentum as well. In this situation both area and rotation parameter change once the black hole makes a jump to the nearest allowed configuration. That immediately leads to the following most general expression for the mass gap,
\begin{align}
\Delta M=\frac{\mathcal{B}}{8\pi}\frac{\sqrt{M^{2}-a^{2}}}{\left[M+\sqrt{M^{2}-a^{2}}\right]^{2}}
+\frac{Ma \Delta a}{\left[M+\sqrt{M^{2}-a^{2}}\right]^{2}}~.
\end{align}
As evident from the above relation, by setting $\Delta a=0$ we get back the original relation for the minimum mass gap derived in \ref{MGap2} (with $Q=0$). This explicitly shows that the mass gap due to quantum nature of underlying geometry does not betray black hole thermodynamics, since it directly follows from the first law of black hole thermodynamics. Hence our initial assumption that black holes are macroscopic, behaving as a thermodynamic object holds good even when the geometry is discretized (see also \cite{Bekenstein:1997bt,Bekenstein:1995ju,Hod:1998vk}). Also note that for nonzero $\Delta a$ the mass gap will be higher from the minimum value. In what follows, we will concentrate on this characteristic minimum energy that a quantized (yet, macroscopic) black hole may emit and whether using this distinctive signature originating from the interface of quantum theory and gravity one can make any concrete prediction which can be falsified.
\subsection{Connecting micro and macro length scales}\label{Universal}
In order to understand the behaviour of the mass gap, let us now introduce two more length scales in the problem --- (a) the thermal de Broglie wavelength (for massless particles) $\lambda _{\rm T}\sim (1/T)$, signifying the scale set by the thermality of the black hole \cite{Barcelo:2010pj,Barcelo:2007yk,Visser:2001kq} and (b) the Compton wavelength $\lambda _{\rm c}\sim r_{\rm h}$ \cite{Bekenstein1974,Bekenstein:1997bt,Kotwal:2002ch,Hod:1998vk}, where $r_{\rm h}$ is the location of the event horizon, marking the {\it size} of the black hole. The smallest mass difference $\Delta M_{\rm min}$ corresponds to a minimum emission frequency and hence to a maximum emission wavelength $\lambda_{\rm max}$. Thus given \ref{MGap2}, the maximum wavelength takes the following form,
\begin{align}
\lambda _{\rm max}=\left(\frac{8\pi}{\mathcal{B}}\right)\frac{\left(M+\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}\right)^{2}}{\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}}~.
\end{align}
On the other hand, the thermal wavelength $\lambda _{\rm T}$ and the Compton wavelength $\lambda _{\rm c}$ for Kerr-Newman black hole take the following form,
\begin{align}
\lambda _{\rm T}&=\frac{4\pi M\left(M+\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}\right)}{\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}}~,
\label{thermal_W}
\\
\lambda _{\rm c}&=M+\sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}~.
\label{comp_W}
\end{align}
It is possible to express the largest emission wavelength $\lambda _{\rm max}$ in terms of both $\lambda _{T}$ and $\lambda _{c}$ individually, i.e.,
\begin{align}
\lambda _{\rm max}&=\left(\frac{8\pi}{\mathcal{B}}\right)\left(\frac{\lambda _{\rm T}}{4\pi}\right)\left[1+\frac{1}{\left(\frac{\lambda _{\rm T}}{4\pi}\right)\frac{1}{M}-1}\right]~,
\end{align}
and
\begin{align}
\frac{1}{\lambda _{\rm max}}&=\frac{\mathcal{B}}{8\pi}\left(\frac{1}{\lambda _{\rm c}}-\frac{M}{\lambda _{\rm c}^{2}}\right).
\end{align}
Hence, we readily obtain that in the near-extremal limit, i.e., when $\left(a^{2}+Q^{2}\right)\rightarrow M^{2}$ and hence $\lambda _{\rm max}\sim \lambda _{\rm T}$, but \emph{not} as $\lambda _{\rm c}$. Thus for nearly extremal black holes the maximum wavelength (or, minimum frequency) scales with the thermal wavelength but \emph{not} the Compton wavelength. Since the minimum emission frequency is intimately connected to the quasi-normal mode frequency of a black hole \cite{Dreyer:2002vy,Berti:2009kk,Chakraborty:2017qve}, the above observation suggests that in the near extremal limit, the minimum quasi-normal mode frequency of a black hole, has nothing to do with its size (which is $\sim M$) but with the thermal scale (which scales as $\sim \sqrt{M^{2}-\left(a^{2}+Q^{2}\right)}$)! Moreover, interestingly, it is possible to combine all the
three length scales in a convenient manner to obtain,
\begin{align}\label{ScaleRel1}
\lambda _{\rm max}=\left(\frac{8\pi}{\mathcal{B}}\right)\frac{\lambda _{\rm c}\lambda _{\rm T}}{4\pi M}~.
\end{align}
This expression relates the three distinct length scales of a black hole through its mass. Note that in the limit of vanishing charge and rotation, the thermal and the Compton wavelength both scale proportional to the black hole mass, such that the maximum wavelength for Schwarzschild black hole becomes $\lambda _{\rm max,sch}=(8\pi/\mathcal{B})4M$. Therefore, using the maximum wavelength for Schwarzschild, one readily arrives at the following dimensionless ratio,
\begin{align}\label{Scale_Rel2}
\frac{\lambda _{\rm max}}{\lambda _{\rm max,sch}}=\frac{\lambda _{\rm c}\lambda _{\rm T}}{16\pi M^{2}}~.
\end{align}
Since the maximum wavelength $\lambda _{\rm max}$ dictates whether the quantum spectrum of a black hole is dense or not, the above relation is a measure of the denseness of the emission spectral profile of a Kerr-Newman black hole viz-\'{a}-viz its Schwarzschild counterpart. One can actually write down a constant, independent of the nature of black hole hairs except its mass, out of the above length scales. This constant may be dubbed as universal among all the black holes having identical mass, i.e., it is independent of the charge and/or rotation parameter of the black hole. This can be achieved by noting that $\lambda _{\rm c,sch}=2M$ and $\lambda _{\rm T,sch}=8\pi M$, thus we obtain
\begin{align}
\frac{\lambda _{\rm max}}{\lambda _{\rm c}\lambda_{\rm T}}=\frac{\lambda _{\rm max,sch}}{\lambda _{\rm c,sch}\lambda_{\rm T,sch}}~.
\end{align}
The universality of the ratio $(\lambda _{\rm max}/\lambda _{\rm T}\lambda_{\rm c})$ stems from the fact that it {\it is independent of whether the black hole is Kerr-Newman or Schwarzschild} and speaks about the fundamental length scale of quantization of geometry (see \ref{ScaleRel1}). Further, this tri-ratio of infrared physics cooks up an ultra-violet invariant of the theory of the black holes. Since this gives the parameter strength of the underlying quantum theory of black holes, it is an observationally realizable infrared test on the models of quantum gravity. When compared to using sub-leading corrections to the area-entropy relation \cite{Sen:2012dw,Chakraborty:2016dwb,Bhattacharya:2017bpl}, this tri-ratio is much more favourable to decide in favour of a theory observationally. Measurement of $\lambda_{c}$ for a black hole will be feasible in near future as the Einstein telescope starts operating \cite{Sathyaprakash:2012jk} as it can determine the photon radius of the black hole to sufficient accuracy. While measurement of $\lambda _{\rm T}$ and $\lambda _{\rm max}$ is not going to happen for a real black hole in foreseeable future, but it is indeed possible for analogue systems. There one can simply check whether such a relation exists and whether it can directly tell us something useful about the underlying quantum structure without having to probe ``quantum gravity''.
\section{How long can a astrophysical black hole live?} \label{Lifetime}
We have obtained the expression for the smallest frequency to be emitted from a black hole as long as the mass of the black hole is large compared to the Planck mass. Therefore, in principle, we can follow the footprints of black hole emission obtained in earlier sections, till the time it turns into a Planck mass object, where the approximation of large mass, would no longer remain applicable. Therefore, through these calculations, we could in principle evaluate how many quanta a black hole could radiate before it settles into such a Planck mass system. Since we are interested in astrophysical black holes, the presence of the electric charge can be safely neglected and hence we will only concentrate on Kerr black hole in what follows.
\subsection{Number of emitted quanta}
In order to obtain an upper bound on the number of emitted quanta, we will assume that throughout its evaporation the mass change brought about in the hole is only through the minimum mass change at each step. Thus difference between the initial $\mathcal{I}_{i}$ (which signifies the starting hole mass) and the final $\mathcal{I}_{f}$ (signifying the Planck mass hole) should yield the number of quanta emitted. Let the initial mass be $M_{i}$ with the rotation parameter being $a_{i}$. Similarly, for the final state we have $a_{f}$ representing the final rotation parameter and $M_{f}$ being the final mass. Let $\eta ^{2} =a^{2}/M^{2}$, then for initial and final state we immediately obtain,
\begin{align}
\frac{\mathcal{B}}{4\pi}\mathcal{I}_{i}&=2M_{i}^{2}\left[1+\sqrt{1-\eta _{i}^{2}}\right]~,
\label{eq01}
\\
\frac{\mathcal{B}}{4\pi}\mathcal{I}_{f}&=2M_{f}^{2}\left[1+\sqrt{1-\eta _{f}^{2}}\right]~.
\label{eq02}
\end{align}
Introducing the final black hole mass a fraction of the initial mass $M_{f}=\mu M_{i}$, we obtain the (largest) number of emitted quanta for the process $M_i \rightarrow M_f$, for the integer discretized black hole as,
\begin{align}\label{Num_Quant}
N_{\rm if}=\frac{8\pi}{\mathcal{B}}M_{i}^{2}\left[\left(1-\mu ^{2}\right)+\left(\sqrt{1-\eta _{i}^{2}}-\mu ^{2}\sqrt{1-\eta _{f}^{2}}\right)
\right]~.
\end{align}
In general if a macroscopic black hole emits thermal radiation as prescribed by semiclassical physics \cite{Hawking:1974sw,Chakraborty:2015nwa,Singh:2014paa}, then the quanta emission for a Kerr black hole will go on till it either becomes extremal or Planck sized. At the extremal limit, the temperature of the black hole becomes vanishingly small and hence the minimum energy will also go to zero (see \ref{fig_01}). In such a scenario, if the final state of the hole turns out to be extremal, then, $\eta _{f}=1$, hence the number of quanta emitted would be given by,
\begin{align}
N_{\rm extremal}=\frac{8\pi}{\mathcal{B}}M_{i}^{2}\left[\left(1-\mu ^{2}\right)+\sqrt{1-\eta _{i}^{2}}\right]~.
\end{align}
With the end state being an extremal black hole, the maximum number of emitted quanta would correspond to the situation when besides being extremal the final black hole also turns out to be Planck sized, such that $\mu \rightarrow 0$, in which case the maximum number of quanta would be,
\begin{align}
N^{\rm max}_{\rm extremal}\simeq \frac{8\pi}{\mathcal{B}}M_{i}^{2}\left[1+\sqrt{1-\eta _{i}^{2}}\right]=\mathcal{I}_i~.
\end{align}
While the minimum number of emitted quanta for the extremal case would correspond to the case $\mu \sim 1$, i.e., the black hole becomes extremal very quickly much before reaching the Planck size and hence,
\begin{align}
N_{\rm extremal}^{\rm min}=\frac{8\pi}{\mathcal{B}}M_{i}^{2}\sqrt{1-\eta _{i}^{2}}~.
\end{align}
Further it will be useful to ensure that, the black hole, by emission of these quanta, is indeed going towards extremality. This can be done by imposing the condition that the rotation parameter and the black hole mass $M$ are changing in such a manner that $\eta _{f}>\eta _{i}$. For example, if $a_{f}^{2}>a_{i}^{2}$, while $M_{f}<M_{i}$ then the above criteria will be trivially satisfied. Also, note that for $\eta _{f}>\eta _{i}$, it follows that $(1-\eta _{f}^{2})<(1-\eta _{i}^{2})$ and hence, we arrive at, the inequality, $(\mathcal{I}_{f}/\mathcal{I}_{i})<\mu ^{2}$. Since the black hole is supposed to decay to lower and lower $\mathcal{I}$ values, the above condition ensures that it will certainly be driven to extremality.
However, as the black hole tends to reach an extremal configuration, i.e., the mass and the rotation parameter are comparable with one another, there are many factors that comes into play. First of all, in this case even for a finite decrease of mass and rotation parameter, the area can increase. Further, the determination of minimum mass gap becomes non-trivial and hence cannot be performed in a general context. This complicates the scenario significantly. In the semi-classical context it is certainly possible to perform a numerical analysis of the perturbation equations for various spin fields and hence determine the final state of the black hole. This has been worked out in \cite{Page:1976df,Page:1976ki} using numerical techniques, whose results we briefly recall. First of all for generic situations, with all possible spin fields into account, it is expected that the rotating black hole will end up as a Schwarzschild one with spin-1/2 particles carrying away maximum energy. On the other hand, if all the emissions are through scalar fields alone, it follows that the end product will be a rotating black hole with $a/M\sim 0.555$ \cite{Chambers:1997ai}. In our case as well, we are interested in change in black hole area such that mass changes by the minimum amount $\Delta M_{\rm min}$. This is certainly the case for scalar particles, as they do not take away any angular momentum. Further, the assumption that the black hole is always away from extremality is also borne out by the fact that the end product of a rotating black hole will have $a/M\sim 0.555<1$ \cite{Chambers:1997ai}. Hence, semi-classical physics would suggest to substitute $\eta _{\rm f}=0.555$ in \ref{Num_Quant}. Thus the analysis presented above naturally adopts to the semiclassical emission rates as well.
Furthermore, note that in our analysis we have considered quantum emission from black holes, rather than emission of quanta through semiclassical processes. Factors like grey body will make the emission process less efficient, leading to enhancement in lifetimes. However, we, in this work are more interested in finding out the quantum steps allowed for a macroscopic non extremal black hole. Hence in this case rather than the grey body factors, the distribution and probability of emission of these quanta is more important. These are the factors, which have been used in \cite{Bekenstein:1995ju} to determine the time scale of a black hole as we will discuss in the next section. Moreover, the analysis presented above provides an order of magnitude estimate and is by no means exact. There can be several additional factors including some form of the grey body factors, which will modify the numerical estimations. However, the key result, namely the number of quanta scaling as $M^{2}$ will not be affected.
Thus given the initial mass and the final mass, along with the charge and rotation parameters it is possible to obtain the maximum number of quanta a black hole could emit. Depending upon the spin ${\cal S}$ of the emitted quanta, we can associate the dimensionality $(2{\cal S}+1)^N$ to the emitted radiation's Hilbert space quantum theoretically. Since the characteristic emission time for a quanta is related to the frequency it is emitted at, we can calculate, dynamically how long a black hole will last before it emits the quanta to turn into a Planck star or a extremal remnant. We can visualize the black hole emissions as jumps within macroscopically distinguishable configurations which must be microscopically orthogonal too. In that case we can also find out the fastest time in which a black hole can vanish \cite{MARGOLUS1998188}. This will also give the fastest route for a black hole to emit all its information. This exercise we leave for a future computation.
\subsection{Lifetime of a black hole}
Given the number of maximum possible quanta of emission we can estimate {\it the maximum possible lifetime} of a black hole. To this end, we will assume that all the emitted quanta have the minimum energy $\omega _{\rm min}$, so that one can introduce a characteristic time scale $\tau$, identical for all the emitted quanta. A natural way to determine the time scale $\tau$ is through the following equation for the mass loss rate \cite{Bekenstein:1995ju}
\begin{align}\label{Ch_time}
\frac{dM}{dt}=-\frac{2\omega _{\rm min}}{\tau}~.
\end{align}
At this stage one assumes some form for the mass loss rate and hence determine $\tau$. Subsequently this time scale is coupled to the number of emitted quanta, leading to an estimation of the lifetime of a black hole. Since we are considering only the minimum energy quanta, it immediately follows that the time period computed here will provide an upper bound to the lifetime of a black hole.
To proceed further it is natural to assume that the above mass loss is due to thermal emission. From which it is possible to determine the mass loss rate in terms of the area and the temperature of a astrophysical Kerr black hole as
\begin{align}\label{Eq_mass_loss}
\frac{dM}{dt}&=-\left(\frac{2\pi ^{2}}{120}\right)\left(\textrm{Area}\right)~T^{4}
\nonumber
\\
&=\left(-\frac{1}{1920 \pi}\right)\frac{\left[M^{2}-a^{2}\right]^{2}}{M^{3}\left(M+\sqrt{M^{2}-a^{2}}\right)^{3}}~,
\end{align}
where we have used expressions for the horizon area $A$ and the black hole temperature $T$ in term of the black hole parameters $(M,a)$. Note that the minus sign in front of \ref{Eq_mass_loss} ensures that the mass decreases with time. Finally equating \ref{Eq_mass_loss} with the right hand side of \ref{Ch_time}, we obtain the characteristic time scale $\tau$ of emission of a minimum energy quanta to be,
\begin{align}\label{timescale}
\tau =480\mathcal{B} M^{3} ~\frac{\left(M+\sqrt{M^{2}-a^{2}}\right)}{\Big\{M^{2}-a^{2}\Big\}^{3/2}}~.
\end{align}
Note that in the case of Schwarzschild black hole (i.e., with $a=0$) the time scale becomes proportional to $M$ \cite{Bekenstein:1995ju}, while in the extremal limit, obtained by taking $a^{2}\rightarrow M^{2}$, $\tau$ diverges. This suggests that in the near extremal region it takes a large time to emit the minimum energy quanta. This is expected as in the extremal limit the temperature of the black hole vanishes. The total lifetime of a black hole can be obtained by multiplying the average time scale $\tau$ with the total number of emitted quanta. In principle it should be obtained by integrating over the total time period of the black hole and since $\tau$ diverges, it takes an infinite time to reach the extremal limit. On the other hand, if we follow the analysis of \cite{Chambers:1997ai} with a large number of scalar fields being present whose quanta are emitted from the black hole, then for the final state $a/M\sim 0.555$. Thus one can simply substitute the same in \ref{timescale} to obtain $\tau \sim 1528 \mathcal{B}M$. This is three times larger compared to the corresponding Schwarzschild scenario. Thus the time scale of emission of a single quanta for a rotating black hole is larger compared to a static black hole. While, if the black hole is far from being extremal, as a crude estimate we can take the number of emitted quanta to be $\sim M^{2}$ and time scale of emission of a individual quanta as $\sim M$, then the total lifetime of a black hole will be $\sim M^{3}$, which is certainly large. The feasibility of this scheme for an isolated primordial black hole survivability is an interesting point to dwell upon, which we defer for future.
\section{Extremal limit of a black hole} \label{ExtremalLimit}
In this section we would like to show that a few results derived in the context of extremal limit of a macroscopic black hole also hold even when the black hole is assumed to have discrete structure. In the context of macroscopic physics it has been demonstrated that, if one throws a charged particle inside a Kerr (or, Kerr-Newman) black hole then one can gradually arrive at an extremal black hole. We would like to show that the same is true for our case as well, i.e., if we consider the black hole characterized by mass $M$ and angular momentum $J\equiv Ma$ to transform into another black hole with parameters $(\bar{M},\bar{J})$ by jumping down the discrete geometry spectrum, it actually evolves to an extremal configuration. We further assume that the rotating black hole emits an quanta with the minimum energy $\omega _{\rm min}$ with respect to the asymptotic observers, such that its mass change of the black hole corresponds to $\Delta M=\omega _{\rm min}$ and the change in angular momentum being $\Delta J=(
J/M)\Delta M$, such that $\delta a\equiv \delta (J/M)=0$. Then one can construct the following dimensionless quantity,
\begin{align}
\Theta (M,a)=\frac{T_{\rm new}-T_{\rm old}}{T_{\rm old}}~,
\end{align}
where, $T_{\rm old}$ corresponds to a rotating black hole with parameters $(M,J)$, while $T_{\rm new}$ relates to the black hole with $(M-\Delta M,J-\Delta J)$. Assuming $\Delta M \ll M$, we have the dimensionless ratio $\Theta$ taking the following form,
\begin{align}
\Theta (M,a)=\frac{M\Delta M}{M^{2}-a^{2}}\left[\sqrt{1-\eta ^{2}}-\eta ^{2}\right];\qquad \eta =\frac{a}{M}~.
\end{align}
Thus note that for $\eta \geq 0.79$ (which is a solution of the algebraic equation $\eta^{4}+\eta^{2}-1=0$), the dimensionless quantity $\Theta<0$ \cite{Hod:2017dck}. Hence as the black hole emits the minimum quanta the $\eta$ gradually increases as $a$ is fixed, while mass $M$ decreases, thus one will eventually end up with a higher negative value of the $\Theta$ parameter. This suggests that one arrives at the extremal limit as the black hole loses mass gradually by jumping into lower quantum states. An interesting fact with the above expression is that for $\eta<0.79$, the temperature associated with the new black hole is actually greater than the temperature of the old black hole, which is the signal of domination of Schwarzschild character again.
Let us now briefly mention how to connect this up with the physical process of lowering a charged particle gradually into a Kerr black hole. In particular we will consider a situation where a particle of mass $\mu$ and charge $q$ is being dropped slowly into a Kerr black hole. We will assume that this will make the black hole to absorb $n$ quanta, such that it jumps from initial configuration $\mathcal{I} \rightarrow \mathcal{I}+n$ while the angular momentum $J$ is unaltered. Thus the original black hole with mass $M$ and rotation parameter $a=J/M$ becomes another black hole with mass $M+\mathcal{E}$ and rotation parameter $J/(M+\mathcal{E})$. Here $\mathcal{E}$ corresponds to the energy contributed by the charged object. Then from \ref{I_R} we obtain,
\begin{align}
\frac{\mathcal{B} n}{4\pi}&=2(M+\mathcal{E})\left[(M+\mathcal{E})+\sqrt{(M+\mathcal{E})^{2}-\frac{J^{2}}{(M+\mathcal{E})^{2}}-q^{2}}\right]-q^{2}
-2M\left[M+\sqrt{M^{2}-\frac{J^{2}}{M^{2}}}\right]
\nonumber
\\
&=-\frac{Mq^{2}}{\sqrt{M^{2}-a^{2}}}-q^{2}+2\mathcal{E}\left[M+\sqrt{M^{2}-a^{2}}\right]
\left[1+\frac{M}{M+\sqrt{M^{2}-a^{2}}}+\frac{M^{2}+a^{2}}{\sqrt{M^{2}-a^{2}}\left[M+\sqrt{M^{2}-a^{2}}\right]} \right]~.
\end{align}
The above expression must remain finite in the extremal limit as well, since we would like to impose the condition that if the initial black hole is extremal then the final black hole should also remain extremal at most (i.e., does not become a naked singularity). Thus the requirement of divergent terms in the above expression in the extremal limit forces the energy associated with the charged particle to be
\begin{align}
\mathcal{E}=\frac{q^{2}}{4M}~.
\end{align}
Incidentally, this exactly coincides with the universal minimum energy that can be supplied by a charged body and is also the energy supplied in extremal situation by a charged particle \cite{Hod:2017dck}. This result was derived earlier using the classical black hole picture, while here we have merely computed the changed in the energy levels due to lowering of this charged particle in the black hole. Note that if the particle had no charge, i.e., if $q=0$ then the above expression will require $\mathcal{E}$ to identically vanish. Further using the above expression for $\mathcal{E}$, in the extremal limit we obtain,
\begin{align}\label{Quant_charge}
\frac{\mathcal{B} n}{4\pi}=\frac{\Delta A}{4\pi}=-q^{2}+4\mathcal{E}M=0~.
\end{align}
Thus remarkably, the area does not change when the energy carried by the charged body coincides with $q^{2}/4M$, the universal minimum energy that a charged object can supply. Hence as the black hole approaches to the extremal limit, the classical universal features associated with them are borne out by our quantum schemes as well.
\section{Generalization to Higher Dimensional Black Holes} \label{HighDim}
So far, our discussion was concentrated on Kerr-Newman solution in four spacetime dimensions alone. In this setting we have observed that in the extremal limit the black hole emission spectra becomes almost continuous. We can similarly ask if such characteristic features of a rotating black hole are retained in higher dimensions.
Let us start by considering Kerr black hole in higher dimensions, characterized with a single rotation parameter $a$ \cite{Emparan:2008eg,Myers:1986un,Kanti:2004nr,Frolov:2007nt,gravitation}. In $D$ spacetime dimensions the horizon of such a rotating black hole is located at $r=r_{h}$, where $r_{h}$ is a solution of the following algebraic equation,
\begin{align}
r^{2}+a^{2}-Mr^{5-D}=0~.
\end{align}
In general obtaining a solution for $r_{h}$ for arbitrary $D$ is difficult, so we will consider only the case $D=5$. For a five dimensional spacetime the above algebraic equation can be readily solved resulting into the following horizon location: $r_{h}=\sqrt{M-a^{2}}$. Note that in this case the extremal limit is defined as $M\rightarrow a^{2}$. The area of the event horizon for the rotating black hole in five dimensions can be easily computed, resulting into $A= 2\pi^{2} r_{h}(r_{h}^{2}+a^{2})$. Thus if the black hole is discretized in integer steps (see \ref{GeometricOp}) then the following relation can be obtained,
\begin{align}
\frac{\mathcal{B} \mathcal{I}}{2\pi ^{2}}=M\sqrt{M-a^{2}}~.
\end{align}
We can compute the minimum mass change for the higher dimensional solution following exactly an identical procedure as in \ref{MassGap}. In particular we keep the rotation parameter $a$ fixed and then consider the mass change as the integer $\mathcal{I}$ drops to $(\mathcal{I}-1)$. This results into the following expression for minimum mass change,
\begin{align}
\Delta M_{\rm min}=\frac{\mathcal{B}}{\pi ^{2}}\frac{\sqrt{M-a^{2}}}{3M-2a^{2}}~.
\end{align}
Thus again the mass change vanishes in the extremal limit, which corresponds to $M=a^{2}$. Thus our conclusion remains unchanged as we consider a five dimensional rotating solution.
For completeness let us also consider a charged black hole in five dimensions. One can immediately compute the location of the event horizon by setting $g^{rr}=0$. This in turn implies that the horizon location can be obtained by solving the following algebraic equation, $r^{4}-2Mr^{2}+Q^{2}=0$. The corresponding solution for the event horizon becomes
\begin{align}
r_{h}^{2}=M+\sqrt{M^{2}-Q^{2}}~.
\end{align}
Here $M\rightarrow Q$ corresponds to the extremal limit. In this context the horizon area is being given by $A=2\pi ^{2}r_{h}^{3}$, such that for integer discretization scheme, the above area-mass relation reduces to,
\begin{align}
\mathcal{B} ^{2/3}\mathcal{I}^{2/3}=M+\sqrt{M^{2}-Q^{2}}~,
\end{align}
One can in principle invert the above relation and obtain the mass in terms of the integer $\mathcal{I}$. Then again considering the transition of the black hole from $\mathcal{I}$ to $\mathcal{I}-1$, the minimum mass separation can be immediately obtained as,
\begin{align}
\Delta M=-\frac{\mathcal{B} Q^{2}}{3}\left[M+\sqrt{M^{2}-Q^{2}}\right]^{-5/2}+\frac{\mathcal{B}}{3}\left[M+\sqrt{M^{2}-Q^{2}}\right]^{-1/2}~.
\end{align}
In this case as well, note that in the extremal limit, i.e., as $M\rightarrow Q$, one essentially obtains a vanishing contribution to $\Delta M$. Thus in the case of a higher dimensional charged black hole, in the extremal limit, the quantum spectrum becomes dense again. Thus one can expect that in higher dimensions too, in the extremal limit, all the black holes in the Kerr-Newman class do produce continuous quantum spectrum, visually contrasting them with the Schwarzschild case.
\section{Conclusion} \label{Conclusion}
In this work, we have studied the emission pattern of a classical black hole which is a macroscopic realization of an underlying quantum gravity theory, where some geometric feature of spacetime geometry gets discretized. We have obtained the spacing between distinguishable macroscopic configurations such a hole is allowed to have, which has revealed many interesting features when applied to the Kerr-Newman class of black holes. The smallest frequency of emission gets related to the temperature of the hole through the de-Broglie thermal wavelength associated with the emission profile. This is unlike a normal blackbody where the minimum frequency of emission is determined by the size of the blackbody. The computations presented here have many significant implications involving black hole information release, life time, scrambling time etc. Below we provide the key findings of this work:
\begin{itemize}
\item {\it Generality of the Approach:} The approach of area quantization presented here is completely general and certainly encompasses the more familiar integer area quantization scheme. Note that unlike previous works in this direction, the geometric operator which is being quantized is taken to be arbitrary, it could be area but need not be so. Then, the scheme of quantization is also arbitrary and determined through the function $f(j)$. Interestingly, all the results presented in this work goes through for any choices of $f(j)$ and even in this general setting the quantization of area can be established. Hence, as and when any theory of quantum gravity provides us a quantization scheme for geometric operators, the results presented here will remain directly applicable.
\item {\it Smallest frequency of emission:} The emission profile of the black hole admits a smallest mass (or frequency) gap. This comes just from the micro canonical counting of the entropy and is independent of any quantum modelling whatsoever. If the area changes by integer steps, the minimum frequency a black hole can emit, while jumping to nearest allowed configuration, scales inversely proportional to the hole's mass for the Schwarzschild case. Therefore, the collection of black hole radiation looks vastly discrete given the fact that the semi-classical radiation is expected to be thermal with the temperature inversely related to the mass. Similarly, for the Kerr-Newman black holes too, the minimum frequency gap between the nearest allowed configuration is set by the temperature the black hole is at.
\item {\it Dense spectrum and extremal limit:} For Kerr-Newman black hole it is possible to consider the extremal limit. In this case the minimum mass gap becomes vanishing. Thus unlike the Schwarzschild scenario in this case the quantum spectrum can become dense, as the rotation parameter and/or charge of the black hole increases.
\item {\it Connecting micro-physics to macro-physics:} For a macroscopic black hole, much of the information about its inherent quantum nature gets washed out. For example, its entropy, mass, maximum number of quanta it could emit would not change. However there are also quantities which know about the effect the quantum theory brought into the macroscopic parameters of the hole. We identified three such quantities: mass gap, time scale and a relation among the length scales. The last one among these is of significant interest. It relates three length scales associated with a black hole in a universal fashion. Therefore observation of these associated quantities presumably will reveal the parameter (e.g. Immirizi parameter \cite{Ghosh:2004rq}, string tension \cite{Callan:1988hs}, non commutativity parameter of space-time \cite{Nicolini:2005vd} etc.) associated with the black holes' discretization scheme.
\item {\it A road to Planck scale remnant:} Based upon our estimates of the minimum frequency of emission, we can calculate {\it maximum possible} number of quanta a black hole can emit before it turns extremal and/or becomes Planck sized. One can determine the number of quanta starting from the initial configuration, upto the time where the classical/semiclassical approximation breaks down. Therefore, one can also estimate the maximum possible information content, available with the radiation, once the evaporation has ended. With such emission profile one can also obtain a characteristic time scale of the black hole, which tells us about the time phase corresponding to a particular configuration a black hole adopts in course of emitting radiation. These studies may potentially provide insights regarding at least two crucial issues --- (a) a qualitative estimate of the amount of information a black hole is entitled to emit with its correspondence with information loss paradox and (b) an estimate for the lifetime of a black hole, i.e., survivability of a black hole.
\end{itemize}
The results obtained here open up many frontiers of analysis, some of which are already discussed. We will like to pursue these aspects elsewhere. As the most useful insight obtained from this work, we would like to stress here is the infrared region of the black hole emission may be much more richer in terms of revealing the ultraviolet physics a microscopic theory is built up of. Further research in this direction is indeed warranted.
\section*{Acknowledgements}
Research of SC is funded by the INSPIRE Faculty Fellowship (Reg. No. DST/INSPIRE/04/2018/000893) from Department of Science and Technology, Government of India. Research of KL is supported by INSPIRE Faculty Fellowship grant by Department of Science and Technology, Government of India. |
1610.04499 | \section{Introduction}
Bootstrap percolation, also known as the {\em irreversible $r$-threshold process} \cite{DreyerRoberts, Roberts} or the \emph{target set selection} is a deterministic cellular automaton first introduced by Chalupa, Leath, and Reich \cite{CLR}. Vertices of a graph are in one of two states, ``dormant'' or ``active.'' Given an integer $r$, a dormant vertex becomes active only if it is adjacent to at least $r$ active vertices. Once a vertex is activated, it remains in that state for the remainder of the process.
More formally, consider a graph $G$ and let $A$ denote the initial set of active vertices. For a fixed $r \in \mathbb{N}$, the {\em $r$-neighbor bootstrap percolation} process on $G$ occurs recursively by setting $A = A_0$ and for each time step $t\ge 0$, \[ A_t = A_{t-1} \cup \{ v \in V(G) : | A_{t-1} \cap N(v) | \ge r\},\] where $N(v)$ denotes the neighborhood of the vertex $v$. If all of the vertices of $G$ eventually become active, regardless of order, then we say that $A$ is {\em $r$-contagious} or that $G$ {\em $r$-percolates from $A$}. Given $G$ and $r$, let $m(G,r)$ denote the minimum size of an $r$-contagious set in $G$. (Observe that $m(G, r) \geq \min\{r, |V(G)|\}$.)
Originally, bootstrap percolation was studied on lattices by statistical physicists as a model of ferromagnetism \cite{CLR}, and it can also be viewed as a model of discrete epidemiology, wherein a virus or other contagion is being transmitted across a network (cf.~\cite{BaloghPete,Roberts}). (In the latter context, each vertex is either ``infected'' or ``uninfected''.) Further applications include the spread of influence in social networks \cite{Chen, KKT} and market stability in finance \cite{ACM}.
Much attention has been devoted to examining percolation in a probabilistic setting, referred to in~\cite{BaloghPete} as the \textit{random disease problem}. In this setting, the initial activated set $A$ is selected according to some probability distribution. The parameter of interest is then the probability that $G$ $r$-percolates from $A$, and in particular determining the threshold probability $p$ for which $G$ almost surely does (or does not) $r$-percolate when vertices are placed in $A$ independently with probability~$p$. Results have been obtained in this setting for a number of families of graphs, including random regular graphs \cite{BaloghPittel}, the Erd\H{o}s--R\'enyi random graph~$G_{n,p}$ \cite{FKR,JLTV}, hypercubes \cite{BB:hypercube}, trees \cite{BaloghPeresPete}, and grids \cite{AL,BBDCM,BBM:high}.
In addition, there has recently been interest in extremal problems concerning bootstrap percolation in various families of graphs~\cite{BenevidesPrzykucki15,Przykucki,Riedl:tree}.
The problem has also been studied from the point of view of computational complexity.
For $r \geq 3$, determining $m(G, r)$ is NP-complete~\cite{DreyerRoberts}, and determining $m(G, 2)$ is NP-complete even for graphs with maximum degree $4$~\cite{Centeno2011,irrbernard}.
Furthermore, it is computationally difficult to approximate $m(G, r)$~\cite{Chen,charikar_et_al:LIPIcs:2016:6627}.
Notice that $m(G,1)$ is always equal to the number of connected components of $G$.
\subsection{Degree-Based Results}
In this paper, we are interested in degree-based density conditions that ensure that a graph $G$ will percolate from a small set of initially activated vertices. Freund, Poloczek, and Reichman~\cite{FPR} showed that for each $r \geq 2$, if $G$ has order~$n$ and $\delta(G) \geq \frac{r-1}{r}n$, then $m(G, r) = r$. Note that when $r = 2$, this is the same as Dirac's condition for hamiltonicity~\cite{D}. Recently, Gunderson~\cite{Gunderson} showed that if $n \geq 30$ and $\delta(G) \geq \lfloor n/2 \rfloor + 1$, then $m(G, 3) = 3$, and that for each $r \geq 4$, if $n$ is sufficiently large and $\delta(G) \geq \lfloor n/2 \rfloor + r - 3$, then $m(G, r) = r$. Moreover, both bounds are sharp.
Let $\sigma_2(G)$ denote the minimum degree sum of a pair of nonadjacent vertices in a graph $G$. Ore~\cite{O} proved that every graph $G$ of order~$n\ge 3$ that satisfies $\sigma_2(G)\ge n$ is hamiltonian. Freund, Poloczek, and Reichman \cite{FPR} also showed that Ore's condition is sufficient to ensure that a graph 2-percolates from the smallest possible initially activated set.
\begin{theorem}[\cite{FPR}]\label{th:FPROre}
Let $n \geq 2$. If $G$ is a graph of order~$n$ and $\sigma_2(G) \geq n$, then $m(G, 2) = 2$.
\end{theorem}
Note that hamiltonicity alone is not sufficient to conclude that a graph~$G$ satisfies $m(G, 2) = 2$, as $m(C_n,2) = \lceil n/2 \rceil$, which tends to infinity with $n$. Rather, Theorem \ref{th:FPROre} is part of a diverse collection of results that demonstrate that many sufficient density conditions for hamiltonicity imply a much richer structure that allows for stronger conclusions (cf.~\cite{Bondy:metaconjecture,BCFGL}).
In this paper, we improve Theorem~\ref{th:FPROre} in several ways. First, we characterize graphs of order $n$ with $\sigma_2\ge n-2$ and $m(G,2)>2$.
These will consist of four infinite families of graphs $\mathcal{G}_0$, $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$ and a finite set of graphs~$\mathcal{X}$.
The graphs in $\mathcal{X}$ are depicted in Figure \ref{fig:extremal}.
The class $\mathcal{G}_0$ consists of all graphs which are unions of two disjoint non-empty cliques $X$,~$Y$.
Note that $X$ and $Y$ can be of different sizes.
Graphs in $\mathcal{G}_1$, $\mathcal{G}_2$ and $\mathcal{G}_3$ are formed from $\mathcal{G}_0$ by selecting $\{x,x'\}\subseteq X$, $\{y,y'\}\subseteq Y$, adding the edges $xy$, $x'y'$, and deleting the edges $xx'$ and $yy'$ if they exist (see Figure~\ref{fig:Gi}). For simplicity, we have distinguished the cases where $x=x'$ and $y=y'$ ($\mathcal{G}_1$), $x\ne x'$ and $y\ne y'$ ($\mathcal{G}_2$) and $x=x'$ and $y\ne y'$ ($\mathcal{G}_3$). It is easy to see that any graph $G\in \mathcal{G}_0\cup\mathcal{G}_1\cup\mathcal{G}_2\cup\mathcal{G}_3$ containing at least one vertex in each of $X$ and $Y$ that is not adjacent to any vertex in the other set has $\sigma_2(G)=|V(G)|-2$ and $m(G,2)>2$.
\begin{figure}
\begin{center}
\[ \begin{array}{cccccccc}
\begin{tikzpicture}[scale=1]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (\a*4,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5] at (A.corner \i) {};
}
\end{tikzpicture}
&
\begin{tikzpicture}[scale=1]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\node[circle,fill=black,scale=.5] (a) at (0, 0) {};
\draw (y3) -- (a) -- (y4);
\end{tikzpicture}
&
\begin{tikzpicture}[scale=1.2]
\node[circle,fill=black,scale=.5] (a) at (0,0){};
\node[circle,fill=black,scale=.5] (b) at (1,0) {};
\node[circle,fill=black,scale=.5] (c) at (1,.75) {};
\node[circle,fill=black,scale=.5] (d) at (1, 1.5) {};
\node[circle,fill=black,scale=.5] (e) at (0 , 1.5) {};
\node[circle,fill=black,scale=.5] (f) at (0,.75) {};
\draw (a) -- (b);
\draw (b) -- (c);
\draw(c) -- (d);
\draw(d) -- (f);
\draw(e) -- (c);
\draw(e) -- (f);
\draw(f) -- (a);
\end{tikzpicture}
&
\begin{tikzpicture}[scale=1.2]
\node[circle,fill=black,scale=.5] (a) at (0,0){};
\node[circle,fill=black,scale=.5] (b) at (1,0) {};
\node[circle,fill=black,scale=.5] (c) at (1,.75) {};
\node[circle,fill=black,scale=.5] (d) at (1, 1.5) {};
\node[circle,fill=black,scale=.5] (e) at (0 , 1.5) {};
\node[circle,fill=black,scale=.5] (f) at (0,.75) {};
\draw (a) -- (b);
\draw (b) -- (c);
\draw(c) -- (d);
\draw(d) -- (e);
\draw(e) -- (f);
\draw(f) -- (c);
\draw(f) -- (a);
\end{tikzpicture}
\\[10pt]
\begin{tikzpicture}[xscale=0.6,yscale=0.7]
\node[circle,fill=black,scale=.5](x) at (-2,0) {};
\node[circle,fill=black,scale=.5](y) at (2,0) {};
\foreach \a in {0,1,2,3}{
\node[circle,fill=black,scale=.5](z\a) at (0,-1.5+\a) {};
}
\node[circle,fill=black,scale=.5](zy) at (1,0) {};
\node[circle,fill=black,scale=.5](zx) at (-1,0) {};
\draw (x)--(zx)--(zy)--(y)--(z0)--(x)--(z3)--(y) (z0)--(z1)--(z2)--(z3) (zx)--(z2)--(zy);
\end{tikzpicture}
&
\begin{tikzpicture}[scale=.9]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\node[circle,fill=black,scale=.5] (a) at (270+72:0.4) {};
\node[circle,fill=black,scale=.5] (b) at (270-72:0.4) {};
\node[circle,fill=black,scale=.5] (c) at (90:0.4) {};
\draw
(a)--(c)--(b)
(y1)--(c)
(y2)--(b)--(y3)
(y4)--(a)--(y5)
;
\end{tikzpicture}
&
\begin{tikzpicture}[scale=.8]
\foreach \a in {4}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\foreach \a in {4}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (.75,.75) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw
(x1)--(y2)
(x2)--(y1)
(x3)--(y3)
(x4)--(y4)
;
\end{tikzpicture}
&
\begin{tikzpicture}[scale=.8]
\foreach \a in {4}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\foreach \a in {4}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (.75,.75) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw
(x1)--(y1)
(x2)--(y2)
(x3)--(y3)
(x4)--(y4)
;
\end{tikzpicture}
\end{array}\]
\caption{The family $\mathcal{X}$: Small exceptional graphs for Theorem \ref{th:bestOre}.}
\label{fig:extremal}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\tikzset{vtx/.style={inner sep=1.7pt, outer sep=0pt, circle,fill},}
\[ \begin{array}{cccccccc}
\begin{tikzpicture}[scale=.9]
\foreach \a in {4}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\draw (x1)--(x3)--(x2)--(x4)--(x1);
\foreach \a in {6}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (3,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw (y1)--(y3)--(y5)--(y2)--(y4)--(y1) (y1)--(y6)--(y2) (y3)--(y6)--(y4) (y5)--(y1);
\end{tikzpicture}
&
\begin{tikzpicture}[scale=.9]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\draw (x1)--(x3)--(x5)--(x2)--(x4)--(x1);
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (3,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw (y1)--(y3)--(y5)--(y2)--(y4)--(y1);
\draw (x5) -- (y2);
\end{tikzpicture} \\
\mathcal{G}_0 & \mathcal{G}_1 \\
\begin{tikzpicture}[scale=.9]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\draw (x1)--(x3)--(x5)--(x2)--(x4)--(x1);
\draw (x1)--(x2)--(x3)--(x4) (x5)--(x1);
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm] at (3,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw (y1)--(y3)--(y5)--(y2)--(y4)--(y1);
\draw (y1)--(y2) (y3)--(y4)--(y5)--(y1);
\draw (x5) -- (y2) (x4)--(y3);
\begin{scope}[node distance=3pt]
\node [above=of x5] {$x$};
\node [below=of y3] {$v$};
\end{scope}
\end{tikzpicture}
&
\begin{tikzpicture}[scale=.9]
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm, draw] at (0,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](x\i) at (A.corner \i) {};
}
\draw (x1)--(x3)--(x5)--(x2)--(x4)--(x1);
\foreach \a in {5}{
\node[regular polygon, regular polygon sides=\a, minimum size=2cm] at (3,0) (A) {};
\foreach \i in {1,...,\a}
\node[circle,fill=black,scale=.5](y\i) at (A.corner \i) {};
}
\draw (y1)--(y3)--(y5)--(y2)--(y4)--(y1);
\draw (y1)--(y2) (y3)--(y4)--(y5)--(y1);
\draw (x5) -- (y2) (x5)--(y3);
\begin{scope}[node distance=3pt]
\node [above=of x5] {$x$};
\node [below=of y3] {$v$};
\end{scope}
\end{tikzpicture}
\\
\mathcal{G}_2 & \mathcal{G}_3
\end{array}\]
\end{center}
\caption{Examples of graphs in $\mathcal{G}_0$, $\mathcal{G}_1$, $\mathcal{G}_2$, and $\mathcal{G}_3$ for $n=10$. The labeled vertices in the third and fourth graphs refer to the proof of Theorem~\ref{th:bestOre}.}
\label{fig:Gi}
\end{figure}
\begin{theorem}\label{th:bestOre}
Let $G$ be a graph of order~$n \geq 2$ such that $G$ is not in $\mathcal{G}_0$, $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$ or $\mathcal{X}$. If $\sigma_2(G) \geq n - 2$, then $m(G, 2) = 2$.
\end{theorem}
In particular, Theorem \ref{th:bestOre} implies that $C_5$ is the only graph~$G$ with $\sigma_2(G) = |V(G)| - 1$ and $m(G, 2) > 2$.
Second, we prove a degree sequence condition for $m(G,2) = 2$.
Let $G$ be a graph with degree sequence $d_1 \leq \cdots \leq d_n$.
We say that $G$ satisfies \emph{Chv\'atal's condition} if
\begin{align}
d_i \geq i+1 \quad\text{ or }\quad d_{n-i} \geq n-i, \quad \forall i, 1 \leq i < \tfrac{n}{2}.\label{eqch}
\end{align}
In \cite{Chv:hamiltonian}, Chv\'{a}tal proved that a graph $G$ of order $n\ge 3$ that satisfies Chv\'{a}tal's condition is hamiltonian.
Here, we show that, with only a few exceptions, a slightly weaker Chv\'{a}tal-type condition implies that $m(G,2)=2$. We say that a graph $G$ satisfies the \emph{weak Chv\'{a}tal condition} if
\begin{align}
d_i \geq i+1\quad\text{ or }\quad d_{n-i} \geq n-i-1, \quad \forall i, 1 \leq i < \tfrac{n}{2},\label{eqwch}
\end{align}
and prove the following. We denote the path on $k$ vertices by $P_k$ and the cycle on $k$ vertices by $C_k$.
\begin{theorem}\label{thm:chvatalsharp}
If $G$ is a graph with degree sequence $d_1 \leq \cdots \leq d_n$ that satisfies the weak Chv\'{a}tal condition \eqref{eqwch}, then either $m(G,2)=2$ or one of the following holds:
\begin{itemize}
\item $G$ is disconnected,
\item $G$ contains exactly two vertices of degree one and $G \not\in\{P_2,P_3\}$, or
\item $G$ is $C_5$.
\end{itemize}
\end{theorem}
Note that the ordinary Chv\'atal condition~\eqref{eqch} rules out the last three cases in Theorem~\ref{thm:chvatalsharp}.
\begin{corollary}\label{cor:chvatalcorollary}
If $G$ is a graph with degree sequence $d_1 \leq \cdots \leq d_n$ that satisfies Chv\'{a}tal's condition~\eqref{eqch}, then $m(G,2)=2$.
\end{corollary}
Much as Chv\'{a}tal's Theorem implies Ore's Theorem for hamiltonicity, each of Theorems \ref{th:bestOre} and~\ref{thm:chvatalsharp} and Corollary~\ref{cor:chvatalcorollary} implies Theorem~\ref{th:FPROre}.
\subsection{Notation}
Let $G$ be a graph, let $U \subseteq V(G)$ and let $v \in V(G)$.
We denote by $G[U]$ the subgraph of $G$ induced by $U$.
The notation $\Delta(G)$ means the maximum degree of $G$ and $N(v)$ is theset of neighbors of~$v$.
We denote by $N_U(v)$ the set of neighbors in $U$, that is $N(v) \cap U$.
The notation $d(v)$ means the degree of $v$
and $d_U(v)$ is $|N_U(v)|$.
\section{Proof of Theorem~\ref{th:bestOre}}
\begin{proof}[Proof of Theorem~\ref{th:bestOre}]
Let $G$ be a graph of order $n$ with $\sigma_2(G)\ge n-2$ that is not in one of the exceptional classes $\mathcal{X}$, $\mathcal{G}_0$, $\mathcal{G}_1$, $\mathcal{G}_2$, or $\mathcal{G}_3$. Throughout the proof, amongst all subsets of $V(G)$ that can be activated from a starting set of two vertices, let $I$ (for ``infected'') have maximum size, and let $U = V(G) \setminus I$ denote the set of vertices that remain dormant from this starting set.
We repeatedly use the following observation that follows from the maximality of $I$.
\begin{observation}\label{oneneighbor}
Each vertex in $U$ has at most~one neighbor in $I$.
\end{observation}
Notice that our assumption on $\sigma_2(G)$ implies that $\Delta(G) \geq (n - 2)/2$.
For $n\le 11$, Nauty \cite{McKay2014} was utilized to generate all graphs with $m(G,2)>2$, which is precisely the set $\mathcal{X}$.
The program we used is available at \url{https://arxiv.org/abs/1610.04499}.
Thus, we may assume that $n\ge 12$ as we proceed. Further, if $G$ is disconnected, the degree sum condition guarantees that $G$ has exactly two complete components and so $G \in \mathcal{G}_0$, a contradiction. We may therefore assume that $G$ is connected. The following sequence of claims establishes important facts about the size and structure of $I$ and $U$.
\begin{claim}\label{claim:bigI}
$|I|>\frac{n}{2}$ and $G[U]$ is complete.
\end{claim}
\begin{proof}
First we show that $|I| \geq 4$.
Suppose for a contradiction that $|I| < 4$. If $G$ contains a triangle~$T$, then since $G$ is connected some vertex $y\in V(T)$ has a neighbor $y'$ in $G-T$. So, if we initially activate $y'$ and some $x\in V(T) \setminus \{y\}$, then at least 4 vertices are activated.
If $G$ contains a $C_4$, we can activate the entire cycle starting with either pair of nonadjacent vertices. Hence we may suppose that $G$ contains neither a triangle nor $C_4$.
Let $w$ be a vertex with $d(w)=\Delta(G)\ge (n-2)/2\ge 4$. As $G$ is triangle-free, $N(w)$ is independent. Let $x$ and $y$
be distinct neighbors of $w$.
As $G$ contains no $C_4$, we have $N(x) \cap N(y) = \{w\}$, which means that $d(x)+d(y) \leq n - 3$,
a contradiction. So, we may assume that $|I|\ge 4$.
Next we establish that $|I|\ge |U|$.
It suffices to show that $m(G[U],2)=2$, which would imply that $|U|\le |I|$ since $|I|$ is maximum among all sets activated by two vertices.
To that end, let $u$ and $v$ be nonadjacent vertices in $U$ and recall that every vertex in $U$ has at most~one neighbor in $I$. Consequently, as $|I|\ge 4$,
\[
d_U(u)+d_U(v) \ge d(u)-1 + d(v)-1 \ge n-4\ge |U|.
\]
Thus, $m(G[U],2) = 2$ by Theorem~\ref{th:FPROre}, so $|U|\le |I|$.
Therefore, the first condition $|I| > \frac{n}{2}$ holds unless $|U|=|I|=\frac{n}{2}$.
We will deal with this case after establishing that $G[U]$ is complete.
Suppose $G[U]$ is not a complete graph, and let $u$ and $v$ be nonadjacent vertices in $U$.
Then, as $|U|\le\frac{n}{2}$,
\[
d(u)+d(v)\le 2(|U|-1) \le n-2,
\]
which is a contradiction unless equality holds. If equality holds, then $U$ induces a complete graph on exactly $\frac{n}2$ vertices minus a matching $M$, where every vertex of~$M$ has a neighbor in $I$. Notice that in this case, $G[U]$ percolates from any choice of two vertices in $U$ (since $\frac{n}{2} \geq 5$). Activating two neighbors of a vertex of~$M$ -- one in $I$ and one in $U$ -- results in at least $|U|+1$ activated vertices, a contradiction to $|I|=\frac{n}{2}$ being maximum. Therefore, $G[U]$ must be a complete graph.
We finally return to the case where $|U|=|I|=\frac{n}{2}$.
Let $v \in I$ have a neighbor $u$ in $U$. For any $z$ in $U\setminus \{u\}$,
initially activating $\{v,z\}$ leads to (at least) the activation of $U \cup \{v\}$, contradicting the maximality of $|I|$ and establishing Claim~\ref{claim:bigI}.
\end{proof}
Partition $I$ into sets $I_0$ and $I_1$, where $I_1$ is the set of vertices of~$I$ with at least~one neighbor in $U$, so that vertices in $I_0$ have no neighbors in $U$.
Since $|I|>|U|$, and no vertex in $U$ has more than one neighbor in $I$, there exists a vertex $w \in I_0$. Let $u\in U$ and observe that
\begin{equation}\label{I0degreebound}
n-2\le d(w)+d(u)\le |I|-1+|U|=n-1.
\end{equation}
The bound on $d(w)$ in~\eqref{I0degreebound} has the following useful consequences.
\begin{observation}\label{I0}
Each vertex in $I_0$ has at most~one non-neighbor in $I$. Furthermore, if any three vertices in $I$ are activated, then all of $I_0$ will be activated in the following step of the percolation.
\end{observation}
\begin{claim}\label{allinU}
Every vertex in $U$ has exactly one neighbor in $I$.
\end{claim}
\begin{proof}
By Observation~\ref{oneneighbor}, it suffices to show that each vertex in $U$ has at least one neighbor in $I$.
Suppose otherwise, so that there exists $z \in U$ with no neighbors in $I$, and therefore there are at least two~vertices $w_1$ and $w_2$ in $I_0$.
Then $$n-2 \leq d(w_i) + d(z) \leq |I|-1+|U|-1 = n-2$$ for $i \in \{1,2\}$. Hence $d(w_1)=d(w_2)=|I|-1$
and $w_1,w_2$ are adjacent to all vertices of $I$.
Let $v \in I$ and $u \in U$ be adjacent vertices. If we initially activate $\{w_1,u\}$, this in turn would activate at least~$I\cup\{u\}$, contradicting the maximality of $|I|$. Consequently, every vertex in $U$ has a neighbor in $I$, establishing Claim~\ref{allinU}.
\end{proof}
\begin{claim}\label{bigI0}
$|I_0| \geq 2$.
\end{claim}
\begin{proof}
As observed above, $I_0$ is non-empty, so suppose for a contradiction that $|I_0| = 1$. It follows from Claims \ref{claim:bigI} and~\ref{allinU} that
\[
|I| > |U| \geq |I_1| = |I| - 1,
\]
which is a contradiction unless $|U| = |I| - 1$.
Because $|U| = |I_1|$, Claim~\ref{allinU} implies that there is a perfect matching between $U$ and $I_1$. Also, $I_1$ cannot be an independent set, or else for all $a$,~$b \in I_1$, $d(a) + d(b) \leq 4 < n - 2$, a contradiction. So, let $x_1$ and $x_2$ be adjacent vertices of~$I_1$ and let $u_1$ and $u_2$ be their respective neighbors in $U$. If we initially activate $\{u_1, x_2\}$, then $x_1$ will also become active. Furthermore, by Claim~\ref{claim:bigI}, $U$ is a clique, so all of $U$ will become active, for a total of at least~$|U| + 2 =|I| + 1$ active vertices. This contradiction completes the proof of Claim~\ref{bigI0}.
\end{proof}
\begin{claim}\label{cl:three}
Let $v \in I_1$ have at least two neighbors in $U$ and let $u$ be one such neighbor. Also, let $D$ be a subset of $I$ containing at least three vertices, including $v$, and let $x\in I_1\setminus \{v\}$. The following hold:
\begin{itemize}
\item[(1)] There is no set of size 2 that activates $U \cup D$;
\item[(2)] $N_I(v)$ is an independent set;
\item[(3)] If there is a vertex $y$ in $N_I(v) \cap I_1$, then $y$ is the only neighbor of $v$ in $I$;
\item[(4)] $v$ and $x$ have no common neighbor;
\item[(5)] $|N_I(v)|=1$, $I_1 = \{v,x\}$, $x$ has exactly one neighbor in $U$ and $v$ is adjacent to every other vertex of~$U$.
\end{itemize}
\end{claim}
\begin{figure}
\begin{center}
\tikzset{vtx/.style={inner sep=1.7pt, outer sep=0pt, circle,fill},}
\def1.6{1.6}
\def\hskip 0.8em{\hskip 0.8em}
\begin{tikzpicture}
\draw
(0,0) ellipse (0.5 and 1.5)
(0,0.8) node[vtx,label=above:$v$](v){}
(1.6,0) ellipse (0.5 and 1.5)
(1.6,0.8) node[vtx,label=above:$u$](u){}--(v)
(1.6,0.5) node[vtx,label=above:$ $](up){}--(v)
(u)--(up)
(1.6,-1.9) node{$U$}
(0,-1.9) node{$I$}
(0,0.3) node[vtx]{}
(0,0.55) node[vtx]{} ellipse(0.2 and 0.4)
(0,-0.1) node{$D$}
(0.5*1.6,-2.5) node{(1)};
;
\end{tikzpicture}
\hskip 0.8em
\begin{tikzpicture}
\draw
(0,0) ellipse (0.5 and 1.5)
(0,0.8) node[vtx,label=above:$v$](v){}
(1.6,0) ellipse (0.5 and 1.5)
(1.6,0.8) node[vtx,label=above:$u$](u){}--(v)
(1.6,0.5) node[vtx,label=above:$ $](up){}--(v)
(u)--(up)
(1.6,-1.9) node{$U$}
(0,-1.9) node{$I$}
(v)--(-0.22,0.3) node[vtx,label=below:$w_1$]{}--
(0.22,0.3) node[vtx,label=below:$w_2$]{}--(v)
;
\draw (0.5*1.6,-2.5) node{(2)};
\end{tikzpicture}
\hskip 0.8em
\begin{tikzpicture}
\draw
(0,0) ellipse (0.5 and 1.5)
(0,0.8) node[vtx,label=above:$v$](v){}
(1.6,0) ellipse (0.5 and 1.5)
(1.6,0.8) node[vtx,label=above:$u$](u){}--(v)
(1.6,0.5) node[vtx,label=above:$ $](up){}--(v)
(u)--(up)
(1.6,-1.9) node{$U$}
(0,-1.9) node{$I$}
(v)--(-0.22,0.3) node[vtx,label=below:$w$]{}
(0.22,0.3) node[vtx,label=below:$y$](y){}--(v)
(y) -- (1.6,0) node[vtx](z){}
(u)--(z) (u) to[bend left=50] (z)
;
\draw (0.5*1.6,-2.5) node{(3)};
\end{tikzpicture}
\hskip 0.8em
\begin{tikzpicture}
\draw
(0,0) ellipse (0.5 and 1.5)
(0,0.8) node[vtx,label=above:$v$](v){}
(1.6,0) ellipse (0.5 and 1.5)
(1.6,0.8) node[vtx,label=above:$u$](u){}--(v)
(1.6,0.5) node[vtx,label=above:$ $](up){}--(v)
(u)--(up)
(0,-0.7) node[vtx,label=below:$x$](x){}
(1.6,-0.7) node[vtx,label=above:$ $](z){}--(x)
(1.6,-1.9) node{$U$}
(0,-1.9) node{$I$}
(v)--(0.15,0) node[vtx,label=left:$w$]{}--(x)
(u)--(z) (u) to[bend left=40] (z)
;
\draw (0.5*1.6,-2.5) node{(4)};
\end{tikzpicture}
\hskip 0.8em
\begin{tikzpicture}
\draw
(0,0) ellipse (0.5 and 1.5)
(0,0.8) node[vtx,label=above:$v$](v){}
(1.6,0) ellipse (0.5 and 1.5)
(1.6,0.8) node[vtx,label=above:$u$](u){}--(v)
(1.6,0.5) node[vtx,label=above:$ $](up){}--(v)
(u)--(up)
(0,-0.7) node[vtx,label=below:$x$](x){}
(1.6,-0.7) node[vtx,label=above:$ $](z){}--(x)
(1.6,-1.9) node{$U$}
(0,-1.9) node{$I$}
(v)--(-0.22,0.3) node[vtx,label=below:$w_1$]{}
(0.22,0.3) node[vtx,label=below:$w_2$]{}--(v)
(u)--(z) (u) to[bend left=40] (z)
;
\draw (0.5*1.6,-2.5) node{(5)};
\end{tikzpicture}
\end{center}
\caption{The situation in Claim~\ref{cl:three}.}\label{figlemma}
\end{figure}
\begin{proof}
Before we begin, it is useful to note that if $u$ and $v$ are activated, then so too will be all of~$U$, as $G[U]$ is complete. Also, the proofs of {\it(1)--(5)} are illustrated in Figure~\ref{figlemma}.\\
\noindent{\it (1):} Suppose that $U \cup D$ can be activated starting from two vertices $a$ and $b$.
By Observation~\ref{I0}, all of~$I_0$ is activated.
Thus, the set of vertices activated starting with $\{a,b\}$ contains $U \cup I_0 \cup \{v\}$. Since $|U| \geq |I_1|$ by Claim~\ref{allinU}, we obtain a contradiction to the maximality of~$|I| = |I_0 \cup I_1|$.\\
\noindent{\it (2):} Suppose otherwise, and let $w_1$ and $w_2$ be adjacent vertices in $N_I(v)$. Initially activate $\{w_1,u\}$. In the first three steps, all of $\{u,v,w_1,w_2\}\cup U$ is activated, contradicting {\it(1)} with $D=\{v,w_1,w_2\}$.\\
\noindent{\it (3):} Assume otherwise, that $v$ has two neighbors $y$ and $w$ in $I$, where $y$ has a neighbor in $U$. Initially activating $\{w,u\}$ activates $v$ in the first step, and $U$ in the step that follows. Consequently, $y$ is activated, contradicting {\it(1)} with $D=\{v,w,y\}$.\\
\noindent{\it (4):} Let $x$ be in $I_1\setminus \{v\}$, as given, and assume that $w$ is a common neighbor of $v$~and~$x$. Initially activating $\{w,u\}$ then activates $v$, $x$ and the entirety of $U$ in four iterations, again contradicting {\it(1)}.\\
\noindent{\it (5):} Suppose first that $w_1$ and $w_2$ are distinct neighbors of $v$ in $I$.
By {\it(2)}, {\it(3)} and~{\it(4)}, they are nonadjacent, they have no neighbors in $U$, and neither is adjacent to $x$. Furthermore, because $w_1$,~$w_2 \notin I_1$, both vertices are distinct from $x$. Then $d(w_1) + d(u) \leq |I| - 3 + |U| = n-3$, a contradiction.
Hence $|N_I(v)| \leq 1$.
By Claim~\ref{bigI0}, $|I_0| \geq 2$, so there exists $w \in I_0 \setminus N(v)$.
By applying the assumption $\sigma_2(G)\ge n-2$ to the nonadjacent vertices $v$ and $w$,
we obtain
\[
|I|-2+d(v)\ge d(w)+d(v)\ge n-2 = |U|+|I|-2.
\]
Hence $|U| \leq d(v)$. On the other hand $d(v) \leq |U \setminus N(x)|+1$, so $d(v)=|U|$. It follows that $v$ is adjacent to all but one vertex of $U$, so that $I_1 = \{v,x\}$ and $x$ has exactly one neighbor in $U$. This completes the proof of Claim~\ref{cl:three}.
\end{proof}
Let $v$ and $x$ be as in the statement of Claim~\ref{cl:three}. By Claims \ref{claim:bigI} and~\ref{cl:three}{\it(5)}, $U$ is a clique, $x$ has exactly~one neighbor in $U$, and $v$ is adjacent to every other vertex of~$U$.
If $v$ and $x$ are not adjacent, let $z$ be the (only) neighbor of~$v$ in $I$. By Claim~\ref{cl:three}{\it(5)}, $z \in I_0$, and by Claim~\ref{cl:three}{\it(4)}, $z$ is not adjacent to~$x$. It follows from Observation~\ref{I0} that all other pairs of vertices in $I_0 \cup \{x\}$ are adjacent (cf.~Figure~\ref{fig:Gi}). Thus, $G \in \mathcal{G}_2$.
If $v$ and $x$ are adjacent, then Claim~\ref{cl:three}{\it(5)} and Observation~\ref{I0} imply that $I_0 \cup \{x\}$ is a clique. It follows that $G \in \mathcal{G}_3$.
In either case, we have a contradiction, the final one needed to complete the proof of Theorem~\ref{th:bestOre}.\end{proof}
\section{Proof of Theorem~\ref{thm:chvatalsharp}}
\begin{proof}[Proof of Theorem~\ref{thm:chvatalsharp}]
For $n \leq 12$, Theorem~\ref{thm:chvatalsharp} was verified using Nauty \cite{McKay2014}, so throughout the proof, we may assume that $n \ge 13$.
The program we used is available at \url{https://arxiv.org/abs/1610.04499}.
Suppose then that $G$ is a graph of order $n$ that satisfies the weak Chv\'atal condition~\eqref{eqwch}. Further, by way of contradiction, suppose that $m(G,2)> 2$ and that $G$ is connected and has at most one vertex of degree~$1$. Also, let $V(G)=\{v_1,v_2, \ldots, v_n\}$, where $d(v_i)=d_i$ and $d_i\le d_j$ whenever $i\le j$, and let $L$ be the set of vertices of degree at least~$\frac{n-1}{2}$.
If $G$ satisfies the full Chv\'{a}tal condition~\eqref{eqch}, one of the following must hold:
\begin{itemize}
\item
$d(v)\ge \frac{n}{2}$ for all $v \in L$ and $|L| > \frac{n}{2}$ or
\item
$d(v)>\frac{n}{2}$ for all $v \in L$ and $|L| \ge \frac{n}{2}$.
\end{itemize}
As one would expect, the weak Chv\'atal condition~\eqref{eqwch} results in slightly weaker conclusions. However, \eqref{eqwch} still implies that $|L| \geq n/2$.
Also, notice that if $n$ is even, then $d(v) \ge \frac{n}{2}$ for all $v \in L$, and if $n$ is odd, then $|L| \geq \frac{n+1}{2}$.
Let $I$ have maximum size among sets that can be actived by starting from two active vertices and that satisfy $I\cap L\ne \emptyset$.
Furthermore, let $U=V\setminus I$, $L_I=L\cap I$ and $L_U=L\cap U$. Notice that Observation~\ref{oneneighbor}, which says that each vertex in $U$ has at most~one neighbor in $I$, continues to hold.
\begin{claim}\label{claim:smallishI}
$|I| \le \frac{n+1}{2}$.
\end{claim}
\begin{proof}
Suppose, towards a contradiction, that $|I| > \frac{n+1}{2}$, so that $|U| < \frac{n-1}{2}$. Let $u \in U$ and note that since $u$ has at most one neighbor in $I$, $d(u) \leq |U|$, which implies that $d(v_{|U|}) \leq |U|$. The weak Chv\'{a}tal condition~\eqref{eqwch} then implies that $d(v_{n-|U|}) \geq n-|U|-1$.
Let $X = \{v_{n-|U|},\ldots,v_n\}$, so that $d(v) \geq n-|U|-1=|I|-1> |U|$ for all $v \in X$. Note that $|X|=|U|+1$.
As $|U| \leq |I| - 2$, we have $n-|U|-1 > |U|$, which means that
$X\subseteq I$.
Suppose that there are vertices $v$ and $w$ in $X$ with no neighbors in $U$, implying that they have degree equal to $|I|-1$ in $I$.
Because $G$ is connected, there exists $u \in U$ with a neighbor in $I$. Initially activating $\{v,u\}$ then activates $w$
in the second round and consequently activates $I \cup \{u\}$, which contradicts the maximality of~$|I|$.
Thus there is at most one vertex $v \in X$ with no neighbors in $U$. Indeed, since $|X| = |U|+1$, $v$ is the unique member of $X$ that has no neighbors in $U$
and, by Observation~\ref{oneneighbor}, every $y \in X\setminus\{v\}$ has exactly one neighbor in $U$. Hence, there is a perfect matching between $U$ and $X\setminus\{v\}$.
Let $w \in X \setminus\{v\}$ and let $u \in U$ be adjacent to $w$. Initially activating $\{v,u\}$ activates $w$ in the first round, followed by all vertices in $I \cap N(w)$, since they are also adjacent to $v$. By the maximality of $|I|$ and the fact that $|N(w) \cap I| \geq |I|-2$,
there is exactly one vertex~$z \in I \setminus \{w\}$ that is not adjacent to $w$.
Moreover, $z$ must be adjacent only to $v$ in $I$, otherwise $I\cup\{u\}$ would be activated.
If $z$ had a neighbor in $U$, then $z \in X$.
Hence, $2 \geq d(z) \geq |I| - 1$, which implies that $|I| \leq 3$ and thus $n \leq 5$, a contradiction. Hence, $z$ is a vertex of degree one. Also, because $z \notin X$, all of~$X$ becomes active. If $u$ has a neighbor $u'$ in $U$, then $u'$ has a neighbor in $X$, and hence also becomes active since all of $X$ is activated. This would contradict the maximality of $|I|$. Hence $u$ is also a vertex of degree one, a contradiction to the assumption that $G$ has at most one vertex of degree one. This concludes the proof of Claim~\ref{claim:smallishI}.
\end{proof}
\begin{claim}\label{claim:strictodd}
$|I| < \frac{n+1}{2}$.
\end{claim}
\begin{proof}
Assume otherwise. By Claim~\ref{claim:smallishI}, $n$ is odd, $|I| = \frac{n+1}{2}$ and $|U| = \frac{n-1}{2}$.
First we show that $G[I]$ does not contain two universal vertices.
Suppose for a contradiction that $x$ and $y$ are two vertices of $I$, each with $|I|-1$ neighbors in $I$.
By the connectivity of $G$, there exists an edge $zu$, where $z \in I$ and $u \in U$.
By symmetry, assume $x \neq z$.
Initially activating $\{x,u\}$ activates $z$ in the first step. After the second step, $y$ is activated.
This activates $I \cup \{u\}$, contradicting the maximality of $|I|$.
For $i = (n-3)/2$, the weak Chv\'{a}tal condition \eqref{eqwch} states that either
$d_i \geq i+1 = (n-3)/2+1 = (n-1)/2$ or
$d_{n-i} \geq n-i-1 = n - (n-3)/2 - 1 = (n+1)/2$.
This implies there exists $L'$ such that
\begin{itemize}
\item[(A)]
$d(v)\ge \frac{n-1}{2}$ for all $v \in L'$ and $|L'| \ge \frac{n-1}{2}+3$ or
\item[(B)]
$d(v)\ge \frac{n+1}{2}$ for all $v \in L'$ and $|L'| \ge \frac{n-1}{2}$.
\end{itemize}
If (B) holds, Observation~\ref{oneneighbor} implies that $L' \subset I$.
Moreover, every vertex in $L'$ must have at least one neighbor in $U$.
Since $|U|=|L'|$, every vertex in $|L'|$ has exactly one neighbor in $U$ and thus every vertex in $L'$ is universal in $G[I]$.
Since $|L'| \geq 2$, this contradicts that $G[I]$ has at most~one universal vertex.
Now we assume that (A) holds.
Since $|I| = \frac{n+1}{2}$ and $|L'| \geq \frac{n+1}{2}+2$, there are at least two vertices, call them $u$ and $v$, in $L' \cap U$.
Denote the neighbors of $u$ and $v$ in $I$ by $u_I$ and $v_I$, respectively; we first show that every vertex in $I$ has at most one neighbor in $U$. Suppose otherwise, so that there exists $x \in I$ with at least two distinct neighbors $a$ and $b$ in $U$. If $u_I \neq x$, then initially activating $\{u_I,v\}$ eventually activates $U \cup \{u_I,x\}$, contradicting the maximality of $|I|$. Hence we may assume that $u_I = v_I = x$. If $x$ has a neighbor $y \in I$, then initially activating $\{y,v\}$ eventually activates $U \cup \{y,x\}$, again contradicting the maximality of $|I|$. Consequently, we may then assume that $x$ has no neighbors in $I$; this implies, however, that $x$ was in the initially activated set and therefore $|I|=2$ and thus $n\le 3$, a contradiction.
Consider then initially activating $\{u_I,v\}$, which activates $U \cup \{u_I\}$.
As $|U \cup \{u_I\}| = |I|$, $U \cup \{u_I\}$ has the same properties as $I$.
In particular, this implies that $u_I$ has at most one neighbor in $I$.
Therefore, $d(u_I) = 2$, and by symmetry, $d(v_I) = 2$.
Since every vertex in $I$ has at most one neighbor in $U$, $\Delta(G) \leq \frac{n+1}{2}$.
The weak Chv\'atal condition \eqref{eqwch} therefore implies that the initial part of the degree sequence of $G$ term wise dominates $2,3,4,\ldots$. In particular, there is at most one vertex of degree at most 2 in $G$, contradicting the existence of $u_I$ and $v_I$.
\end{proof}
\begin{claim}\label{claim:strict}
$|I| < \frac{n}{2}$.
\end{claim}
\begin{proof}
Assume otherwise. By Claim~\ref{claim:strictodd}, $|I| = |U|=\frac{n}{2}$, which implies that $n$ is even. If $u \in L_U$, then $d(u)\geq \frac{n}{2}$.
So, by Observation~\ref{oneneighbor}, $u$ is adjacent to all other vertices in $U$ and has exactly one neighbor in $I$. If $|L_U| \geq 2$, let $u$,~$x \in L_U$, let $v \in N_I(u)$ and initially activate $\{v,x\}$. It follows that all of~$U$ is activated by the end of the second round. Because this activates $U\cup\{v\}$, it contradicts the maximality of $I$.
Suppose then, that $|L_U| \leq 1$, so that $|L_I| \geq \frac{n}{2}-1$. As $n$ is even, every vertex in $L$ has degree at least $\frac{n}{2}$.
Hence every vertex in $L_I$ has at least one neighbor in $U$.
Since the number of edges between $I$ and $U$ is at most $\frac{n}{2}$, there is at most one vertex in $L_I$ with more than
one neighbor in $U$ and all the other vertices of $L_I$ are complete to $I$.
Because
$|L_I| - 1 \geq \frac{n}{2} - 2 \geq 2$,
there are vertices $v$ and $w$ in $I$ such that each vertex is adjacent to all of~$I$ except for itself and such that $v$ has a neighbor $u$ in $U$. Activating $u$ and $w$ results in the activation of $I \cup \{u\}$, contradicting the maximality of $I$.
This concludes the proof of Claim~\ref{claim:strict}.
\end{proof}
Let
\begin{equation}\label{pdef}
p= \frac{n}{2}-|I|\ge \frac{1}{2}.
\end{equation}
Notice that $p$ is an integer if $n$ is even.
\begin{claim}\label{claim:bigLU}
$|L_U|\ge 3$.
\end{claim}
\begin{proof}
If $p \geq 3$, the claim follows from $|L_I| \leq |I| \leq \frac{n}{2}-3$ and $|L_I|+|L_U| \geq \frac{n}{2}$. So, we let $p \leq 2$
and assume for a contradiction that $|L_U| \leq 2$.
We distinguish the following three cases based on the parity of $n$ and the value of~$p$.\\
\textbf{Case 1:} $n$ is even.
Since $n$ is even, \eqref{pdef} implies that $p \in \{1,2\}$. Also, if $v\in L_I$, then $d(v) \geq n/2$. It follows from~\eqref{pdef} that $d_U(v) \geq \frac{n}{2} - (|I| - 1) = p+1$. Furthermore, every vertex in $U$ has at most one neighbor in $I$, so $|U|\ge |L_I|(p+1)$.
Since $|L_U| \leq 2$, we get $|L_I| \geq |L| - 2 \geq \frac{n}{2}-2$.
Therefore,
\[
\frac{n}{2}-2 \le |L_I|\le \frac{|U|}{p+1}=\frac{n/2+p}{p+1}.
\]
However, as $p \in \{1,2\}$ and thus $n \le 10$, this is a contradiction.\\
\textbf{Case 2:} $n$ is odd and $p>\frac{1}{2}$.
Since $n$ is odd and $p > \frac12$, for every $v\in L_I$, we have $d_U(v)\ge p+\frac12$. Every vertex in $U$ has at most one neighbor in $I$, so $|U|\ge |L_I|(p+1/2)$.
Since $|L| \geq \frac{n+1}{2}$, we obtain $|L_I| \geq \frac{n-3}{2}$.
Therefore,
\[
\frac{n-3}{2} \le |L_I|\le \frac{|U|}{p+1/2}=\frac{n/2+p}{p+1/2}.
\]
However, as $p \in \{\frac{3}{2},\frac{5}{2}\}$ and $n > 9$, this inequality fails, a contradiction.\\
\textbf{Case 3:} $n$ is odd and $p = \frac{1}{2}$.
In this case, $|I| = \frac{n-1}{2}$ and, by assumption, $|L_I| \geq |L| - 2 \geq \frac{n-1}{2} - 1$.
Every vertex in $L_I$ has degree at least $\frac{n-1}{2}$,
and therefore must have a neighbor in $U$.
Since every vertex in $U$ has at most one neighbor in $I$, there
are at most $\frac{n+1}{2}$ edges between $L_I$ and $U$.
Hence there are at most two vertices in $L_I$ with more than one neighbor in $U$.
If there are two such vertices in $L_I$, then both have exactly two neighbors in $U$.
If there is exactly one such vertex in $L_I$, then it has at most three neighbors in $U$.
Consequently, $I$ is a complete graph of order at least $5$ except for either a single edge or two incident edges.
As each vertex in $L_I$ has at least one neighbor in $U$, it is straightforward to select two vertices that, when initially activated, activate all of $I$ and at least~one vertex in $U$, a contradiction.
This concludes the proof of Claim~\ref{claim:bigLU}.
\end{proof}
\begin{claim}\label{luclique}
$L_U$ is a clique.
\end{claim}
\begin{proof}
Assume otherwise, and let $u$ and $v$ be nonadjacent vertices in $L_U$.
We claim that initially activating $u$ and $v$ generates a contradiction to the maximality of $I$.
As every vertex in $U$ has at most one neighbor in $I$, both $u$ and $v$ have at least
$\frac{n-1}{2}-1$ neighbors among the other $\frac{n}{2}+p-2$ vertices of~$U$.
Let $r$ be the number of vertices in $U$ that are common neighbors of $u$~and~$v$.
By counting edges from $u$ and $v$ to the remainder of~$U$, we obtain
\[
2\left( \frac{n-1}{2}-1\right) \leq d_U(u)+d_U(v) \leq 2r + \bigl(|U \setminus\{u,v\} | - r\bigr) \leq 2r + \left(\frac{n}{2}+p-2 - r\right),
\]
which implies that
\[
\frac{n}{2} - p - 1 \leq r.
\]
Together with $\{u,v\}$, a total of $r+2\ge\frac{n}{2} - p +1 = |I| + 1$ vertices become activated by the second round, contradicting the maximality of $|I|$ and proving Claim~\ref{luclique}.
\end{proof}
\begin{claim}\label{cl:almostdone}
Vertices in $L_U$ have no neighbors in $I$.
\end{claim}
\begin{proof}
Suppose for a contradiction that $v \in I$ and $u \in L_U$ are adjacent. Let $w$ and $z$ be two vertices
in $L_U$ aside from $u$, which exist by Claim \ref{claim:bigLU}. Initially activate the set~$\{v,w\}$. In the first round, $u$ is activated, and in the second round, the remainder of $L_U$, including $z$, is activated.
By counting edges from $u$, $w$, and $z$ to the remainder of $U$, and letting $r$ denote the number of vertices adjacent to at least two of $u$, $w$ or $z$, we get
\[
3\left( \frac{n-1}{2}-3\right) \leq d_U(u)-2+d_U(z)-2+d_U(w)-2 \leq 3r + \bigl(|U \setminus\{u,v,w\} | - r\bigr) \leq 3r + \left(\frac{n}{2}+p-3 - r\right),
\]
which implies that
\[
\frac{n}{2} - \frac{p}{2} - \frac{15}{4} \leq r.
\]
When we include $u$, $v$, $w$, and~$z$, we see that there are at least $\frac{n}{2} - \frac{p}{2} + \frac{1}{4} \geq \frac{n}{2}$ activated vertices.
This contradicts the maximality of $|I| = \frac{n}{2} - p$
and concludes the proof of Claim~\ref{cl:almostdone}.
\end{proof}
To finish the proof, let $u$, $w$, $z \in L_U$ and initially activate the set $\{z,w\}$, so that $u$ is activated in the first round.
Now we count edges from $\{u,w,z\}$ to $U \setminus \{u,w,z\}$, again letting $r$ denote the number of vertices in $U \setminus \{u,w,z\}$ that are adjacent to at least two vertices in $\{u,w,z\}$. By Claim~\ref{cl:almostdone},
\[
3\left( \frac{n-1}{2}-2\right) \leq d_U(u)-1+d_U(z)-1+d_U(w)-1 \leq 3r + \bigl(|U \setminus\{u,v,w\} | - r\bigr)\leq 3r + \left(\frac{n}{2}+p-3 - r\right).
\]
Hence,
\[
\frac{n}{2} - \frac{p}{2} - \frac{11}{4} \leq r.
\]
Together with $u$, $w$, and $z$, we get at least $\frac{n}{2} - \frac{p}{2} + \frac{1}{4} \geq \frac{n}{2} > |I|$ activated vertices, the final contradiction to the maximality of $|I|$ necessary to complete the proof of Theorem~\ref{thm:chvatalsharp}.
\end{proof}
\section*{Conclusion}
Theorem \ref{thm:chvatalsharp} gives a sharp degree condition that ensures that a graph $G$ satisfies $m(G,2)=2$, in that it provides a class of graphs that demonstrates the sharpness of the weak Chv\'{a}tal condition for this property. However, Chv\'{a}tal-type conditions are often shown to be best possible in a different manner, which gives rise to a perhaps challenging open problem related to the work in this paper.
Let $S=(d_1,\dots, d_n)$ and $S'=(d_1',\dots,d_n')$ be sequences of real numbers. We say that $S$ \textit{majorizes} $S'$, and write $S\succeq S'$, if $d_i\ge d_i'$ for every $i$. A sufficient degree condition ${\mathcal C}$ for a graph property ${\mathcal P}$ is \textit{monotone best possible} if whenever ${\mathcal C}$ does not imply that every realization of a degree sequence $\pi$ has property ${\mathcal P}$, there is some graphic sequence $\pi'\succeq \pi$ such $\pi'$ has at least one realization without property ${\mathcal P}$. Note that it is possible that $\pi'=\pi$.
The Chv\'{a}tal condition is a monotone best possible degree condition for hamiltonicity~\cite{Chv:hamiltonian}, and~\cite{BK} is a thorough survey of monotone best possible degree criteria for a number of graph properties. However, it is easy to show that it is not the case that either the Chv\'{a}tal condition~\eqref{eqch} or the weak Chv\'{a}tal condition~\eqref{eqwch} is monotone best possible for the property $m(G,2)= 2$.
To see that the Chv\'{a}tal condition is not monotone best possible, consider the graphic sequence $$\pi=(i^i, (n-i-1)^{n-2i},(n-1)^i),$$ where $2\le i<\frac{n}{2}$ and the exponents represent the multiplicities of the terms of $\pi$. The unique realization of $\pi$ is $K_i\vee(\overline{K_i}\cup K_{n-2i})$. However, any sequence $\pi'$ such that $\pi'\succeq\pi$ must have at least two vertices of degree $n-1$, implying every realization $G$ of $\pi'$ has $m(G,2)=2.$ If $n$ is even, the sequence $$\pi=(i^i, (n-i-2)^{n-2i},(n-1)^i),$$ suffices to show that the
weak Chv\'{a}tal condition is also not monotone best possible for the property $m(G,2)=2$.
This gives rise to the following problem:
\begin{problem}
Determine a monotone best possible degree condition for the property ``$m(G,2)=2$''.
\end{problem}
\textbf{Acknowledgement:} The work in this paper was primarily completed during the 2016 Rocky Mountain - Great Plains Graduate Research Workshop in Combinatorics, held at the University of Wyoming in July 2016. The authors would like to thank Gavin King and Tyrrell McAllister for many fruitful discussions during the workshop, and the University of Wyoming for its hospitality.
\bibliographystyle{amsplain} |
2010.09989 | \section{Introduction}
Single particle cryo-electron microscopy (cryo-EM) is a powerful method for reconstructing the 3D structure of individual proteins and other macromolecules \citep{Frank2006,Cheng2018}.
In a cryo-EM experiment, a sample of the molecule of interest is rapidly frozen, forming a thin sheet of vitreous ice, and then imaged using a transmission electron microscope.
This results in a set of images that contain noisy tomographic projections of the electrostatic potential of the molecule at random orientations.
The images then undergo a series of processing steps, including:
\begin{itemize}
\item Motion correction.
\item Estimation of the contrast transfer function (CTF) or point-spread function.
\item Particle picking, where the position of the molecules in the ice sheet is identified.
\item 2D class averaging, where similar particle images are clustered together and averaged.
\item Filtering of bad images (typically, entire clusters from the previous step).
\item Reconstruction of a 3D molecular model (or models).
\end{itemize}
All of these steps are done using special-purpose software, for example: \citet{TangEtal2007,Scheres2012b,LyumkisEtal2013,Xmipp2013,RohouGrigorieff2015,PunjaniEtal2017,HeimowitzAndenSinger2018,GrantRohouGrigorieff2018}.
Cryo-EM has been gaining popularity and its importance has been recognized in the 2017 Nobel prize in Chemistry \citep{CryoEMNobel2017}. An examination of the Protein Data Bank \citep{BermanEtal2000} shows that in 2020, 16\% of entries had structures that were determined by cryo-EM, compared to 12\% in 2019, and 7\% in 2018.
For more on cryo-EM and its current challenges, see the reviews of \cite{VinothkumarHenderson2016,Lyumkis2019,SingerSigworth2020}.
\subsection{Image formation in cryo-EM}
Given a 3D molecule with an electrostatic potential function $\rho: \mathbb{R}^3 \rightarrow \mathbb{R}$ (henceforth ``particle''), the images captured by the electron microscope are modeled as noisy tomographic projections along different orientations.
A tomographic projection of the particle onto the x-y plane is defined as $\left(\mathcal{T}\rho\right)(x,y) = \int_\mathbb{R} \rho(x,y,z)dz$.
More general tomographic projections are given by a rotation $R$ of the potential function $\rho$ followed by projection onto the x-y plane,
\begin{equation}
\mathcal{T}_R \rho = \int_\mathbb{R} \left( R \rho \right)(x,y,z)dz
\end{equation}
where $R \in \text{SO}(3)$, and we define $(R\rho)(x,y,z) = \rho(R^T(x,y,z))$.
Particle images in cryo-EM are typically modeled as follows,
\begin{align}
\text{Image} = h * \mathcal{T}_{R}\rho + {\boldsymbol \epsilon},
\end{align}
where $h$ is a point spread function that is convolved with the projection and ${\boldsymbol \epsilon}$ is Gaussian noise.
See Figure \ref{fig:experiments-3D-ribosome} for an example of a tomographic projection. We define the viewing angle of our particle as a unit vector $\v$ that represents orientation of our particle $R \in \mathrm{SO}(3)$ modulo in-plane rotations (which can be represented as members of $\mathrm{SO}(2)$).
\subsection{Clustering tomographic projections}
The imaging process in cryo-EM involves high-levels of noise.
To improve the signal-to-noise ratio (SNR), it is common to cluster the noisy projection images with ones that are similar up to an in-plane rotations.
This clustering task is called ``2D classification and averaging'' in the cryo-EM literature and the clusters are called classes.
The results of this stage have multiple uses, including:
\begin{itemize}
\item Template selection from 2D class averages for particle picking \citep{FrankWagenkrecht1983,ChenGrigorief2007,Scheres2014}.
\item Particle sorting \citep{ZhouMoscovichBendoryBartesaghi2020}, discarding images that belong to bad clusters.
\item Visual assessment and symmetry detection.
\item Ab-initio modeling, where an initial 3D model (or set of models) is constructed to be refined in later stages \citep{GreenbergShkolnisky2017,PunjaniEtal2017}.
\end{itemize}
Existing methods for 2D clustering include the reference-free alignment algorithm that tries to find a global alignment of all images \citep{PenczekRadermacherFrank1992}, clustering based on invariant functions and multivariate statistical analysis \citep{SchatzVanheel1990}, rotationally invariant k-means \citep{PenczekZhuFrank1996} and the Expectation-Maximization (EM) based approach of \cite{ScheresValleCarazo2005}.
All of these methods use the $L_2$ distance metric on rotated images.
Let us consider a simple centroid+noise model, where each image in a cluster is generated by adding Gaussian noise to a particular clean view ${\boldsymbol \mu}$ that is the (oracle) centroid,
\begin{align}
\text{Image}_i \sim \mathcal{N}({\boldsymbol \mu}, \sigma^2 I).
\end{align}
In that case the $L_2$ metric is (up to constants) nothing but the log-likelihood of the image patch, conditioned on $\boldsymbol \mu$,
\begin{align}
\log \mathcal L(\text{Image}_i | {\boldsymbol \mu}) = -\|\text{Image}_i - {\boldsymbol \mu}\|^2/2 \sigma^2 + C_\sigma.
\end{align}
Under the centroid+noise model, the commonly-used $L_2$ distance metric seems perfectly suitable.
However, real particle images have many other sources of variability, including angular differences, in-plane shifts and various forms of molecular heterogeneity.
For these sources of variability, the $L_2$ distance metric is ill-suited.
See Section \ref{sec:theory} for more on this.
In this work we propose to use the Wasserstein-1 metric $W_1$, also known as the Earthmover's distance, as an alternative to the $L_2$ distance for comparing particle images.
In Section \ref{sec:methods} we describe a variant of the rotationally invariant k-means that is based on a fast linear-time approximation of $W_1$ and in Section \ref{sec:results} we demonstrate superior performance on a synthetic dataset of Ribosome projections.
In Section \ref{sec:theory} we analyze the behavior of the $W_1$ and $L_2$ metrics theoretically with respect to angular differences of tomographic projections.
In particular we show that the rotationally-invariant $W_1$ metric has the nice property that it is bounded from above by the angular difference of the projections.
On the other hand, the $L_2$ metric shows no such relation.
\section{Methods} \label{sec:methods}
In cryo-EM, in-plane rotations of the molecular projections are assumed to be uniformly distributed, hence is is desirable for the distance metric to be invariant to in-plane rotations.
A natural candidate is the rotationally-invariant Euclidean distance,
\begin{equation}
L_2^R(I_1, I_2) := \min_{R \in \mathrm{SO}(2)} ||I_1 - R I_2||_2.
\label{eq:rotationally_invariant_L2}
\end{equation}
A drawback of this metric is that visually similar projection images that have a small viewing angle difference can have an arbitrarily large $L_2$ distance.
See discussion in Section \ref{sec:theory}.
To address this, we define a metric based on the Wasserstein-$p$ Metric between two probability distributions \citep{Villani2003}. This metric measures the ``work'' it takes to transport the mass of one probability distribution to the other, where work is defined as the amount of mass times the distance (on the underlying metric space) between the origin and destination of the mass. More formally, the Wasserstein-$p$ distance between two normalized greyscale $N \times N$ images is defined as
\begin{equation}
W_p(I_1, I_2) = \inf_{\pi \in \Pi(I_1, I_2)} \sum_{u \in [N]^2}\sum_{v \in [N]^2}||u-v||^p\pi(u, v),
\end{equation}
where $\Pi(I_1, I_2)$ is the set of joint distributions on $[N]^2$ with marginals $I_1, I_2$ respectively.
In cryo-EM, the Wasserstein-1 metric has been used to understand the conformation space of 3D volumes of molecules \citep{ZeleskoMoscovichKileelSinger2020}. For clustering tomographic projections, we construct a rotationally-invariant $W_p$ distance to be our clustering metric analogously to Eq. \eqref{eq:rotationally_invariant_L2}
\begin{equation} \label{def:rotinv_wasserstein}
W_p^R(I_1, I_2) := \min_{R \in \mathrm{SO}(2)} W_p(I_1, R I_2).
\end{equation}
\subsection{Clustering with rotationally-invariant metrics}
Algorithm \ref{algo:k-means} describes a generic rotationally-invariant k-means algorithm based on an arbitrary image patch distance metric $d$.
The choice $d = L_2$ gives the rotationally-invariant k-means of \cite{PenczekZhuFrank1996}.
By supplying $d = W_1$ we get a new rotationally-invariant k-means algorithm based on the $W_1$ distance. We choose $p=1$ for the $W_p$ distance as it admits a fast linear-time wavelet approximation as we will see in the next section.
For both choices of the metric, we initialize the centroids using a rotationally-invariant k-means++ initialization which we describe in Algorithm \ref{algo:k++}. When we denote $r \in \mathrm{rot}$ we mean that $r$ performs an in-plane rotation of an image. We approximate the space of all in-plane rotation angles by a discrete set of angles.
\begin{algorithm}[ht]
\SetAlgoLined
\DontPrintSemicolon
\KwParams{$k, n, d$}
\;\vspace{-10pt}
\KwOutput{$\{C_j\}_{j \in [k]}$}
\;\vspace{-11pt}
\KwData{$\{I_i\}_{i \in [n]}$}
\;\vspace{-5pt}
$\{C_j\}_{j \in [k]} := \mathrm{InitializeCenters}(\{I_i\}_{i \in [n]})$\\
\;\vspace{-9pt}
\While{\textnormal{loss decreases}}{
$\mathrm{loss} = 0$\\
\For{$i \in [n]$} {
$(j_i, r_i) := \argmin_{(j,r):j \in [k], r \in \mathrm{rot}} d(I_i, r(C_j))$\\
$\mathrm{loss} = \mathrm{loss} + d(I_i, r_i(C_{j_i}))^2$\\
}
\For{$j \in [k]$}{
$C_j := \mathrm{mean}(\{r_i^{-1}(I_i) : j_i = j\})$\\
}
}
\;\vspace{-5pt}
\KwRet$\{C_j\}_{j \in [k]}$
\caption{Rotationally invariant k-means}
\label{algo:k-means}
\end{algorithm}
\begin{algorithm}[ht]
\SetAlgoLined
\SetAlgoLined
\DontPrintSemicolon
\KwParams{$k, n, d$}
\;\vspace{-10pt}
\KwOutput{$\{C_j\}_{j \in [k]}$}
\;\vspace{-11pt}
\KwData{$\{I_i\}_{i \in [n]}$}
\;\vspace{-5pt}
$C_1 = \mathrm{RandomSelect}(\{I_i\}_{i \in [n]}$)\\[1pt]
\For{$j \in \{2, ..., k\}$}{
\For{$i \in [n]$}{
$p_{i} := \left( \min_{m \in [j-1], r \in \mathrm{rot}} d(I_i, r(C_m)) \right)^2$
}
$j = \mathrm{DrawFrom}({\bf p})$
\tcp{Draw index $j$ with probability proportional to $p_j$}
$C_j = I_j$
}
\;\vspace{-5pt}
\KwRet $\{C_j\}_{j \in [k]}$
\caption{Rotationally invariant k-means++ initialization}
\label{algo:k++}
\end{algorithm}
\subsection{Computing the Earthmover's distance}
Computing the distance $W_1(I_1, I_2)$ (also known as the Earthmover's distance) between two $N \times N$ pixels can be formulated as a linear program in $O(N^2)$ variables and $O(N)$ constraints. Given $n$ images, $k$ centers, and $m$ rotations for each image and $t$ iterations of k-means, we have to compute $O(n\times m \times k \times t)$ distances. Computing the $W_1$ distance between two $100 \times 100$ size images using a standard LP solver takes on average of 5 seconds to compute using the Python Optimal Transport Library of \cite{FlameryCourty2017} on a single core of a 7th generation 2.8 GHz Intel Core i7 processor. Computing the exact $W_1$ distance over all the rotations of all the images over all the iterations is prohibitively slow.
Fortunately, the $W_1$ distance admits a wavelet-based approximation with a runtime that is linear in the number of pixels \citep{ShirdhonkarJacobs2008}. Let $\mathcal{W}I_i$ be the 2D wavelet transform of $I_i$. We can approximate the $W_1$ distance between two images $I_1, I_2$ by a weighted $\ell_1$ distance between their wavelet coefficients,
\begin{align} \label{eq:approximate_emd}
\widetilde{W_1}(I_1, I_2) := \sum_{\lambda} 2^{-2 \lambda_s}|(\mathcal{W}I_1)(\lambda) - (\mathcal{W}I_2)(\lambda)|,
\end{align}
where $\lambda$ goes over all the wavelet coefficients and $\lambda_s$ is the scale of the coefficient.
This metric is strongly equivalent to $W_1$. i.e. there exist constants $0 < c < C$ such that for all $I_1, I_2 \in \mathbb{R}^{N^2}$
\begin{equation}
c \cdot \widetilde{W_1}(I_1, I_2) \leq W_1(I_1, I_2) \leq C \cdot \widetilde{W_1}(I_1, I_2).
\end{equation}
Different choices of the wavelet basis give different ratios $C/c$. We chose the symmetric Daubechies wavelets of order 5 due to the quality of their approximation \citep{ShirdhonkarJacobs2008}.
The sum in Eq. \eqref{eq:approximate_emd} was computed with scale up to $6$ using the PyWavelets package \citep{LeeEtal2019}.
\section{Experimental results} \label{sec:results}
\subsection{Dataset generation}
We built a synthetic dataset of 10,000 tomographic projections of the Plasmodium falciparum 80S ribosome bound to the anti-protozoan drug emetine, EMD-2660 \citep{WongEtal2014}, shown in Figure~\ref{fig:experiments-3D-ribosome}.
To generate each image, we randomly rotated the ribosome around its center of mass using a uniform draw of $\mathrm{SO}(3)$, projected it to 2D by summing over the $z$ axis and resized the resulting image to $128 \times 128$.
Finally we added i.i.d. $\mathcal{N}(0,\sigma^2)$ noise to the image at different signal-to-noise (SNR) levels. Given a dataset of images $S \in \mathbb{R}^{N \times N \times n} $, the SNR is defined as
\begin{equation}
\text{SNR} := \frac{ \|D\|^2}{nN^2\sigma^2}.
\end{equation}
We produce three datasets to run experiments on by adding noise at SNR values $\{1/8, 1/12, 1/16\}$.
\begin{figure}[h]
\centering
\includegraphics[width=4cm]{figures/experiments-ribosome-image}
\includegraphics[width=9cm]{figures/example_projection}
\caption{(left) Surface plot of the 3D electrostatic potential of the 80S ribosome; (middle) Example tomographic projection; (right) The same projection with Gaussian noise at SNR=1/16.}
\label{fig:experiments-3D-ribosome}
\end{figure}
\subsection{Simulation results} \label{sec:results}
We performed rotationally-invariant 2D clustering on our three datasets using rotationally-invariant k-means++ (Algorithms \ref{algo:k-means}, \ref{algo:k++}) with the $W_1$ and $L_2$ distance using $k=150$ clusters.
All in-plane rotations at increments of $\pi/100$ radians were tested in the Algorithm \ref{algo:k-means}.
To quantify the difference in performance, we computed the distribution of within-cluster viewing angle differences for all pairs of images assigned to the same cluster. For a molecule like the Ribosome that has no symmetries, we expect these angular differences to be concentrated around zero, since large angular differences typically give large differences in the projection images.
For all SNR levels, we see that $W_1$ gives better angular coherence than $L_2$.
Note that some of these distributions have a small peak towards 180 degrees which physically represents our algorithms confusing
antipodal viewing angles.
\begin{figure}
\includegraphics[width=0.33\textwidth]{figures/snr--angle_dist-8.pdf}
\includegraphics[width=0.33\textwidth]{figures/snr--angle_dist-12.pdf}
\includegraphics[width=0.33\textwidth]{figures/snr--angle_dist-16.pdf}
\caption{Distribution of within-cluster pairwise angular differences (narrower is better) at SNR values $1/8,\ 1/12,\ 1/16$ from left to right.}
\label{fig:angulardistribution}
\end{figure}
In Figure \ref{fig:ribosome_centroids} we show the cluster means of the 8 largest clusters at SNR $1/16$. Visually, we can see that the algorithm based on $W_1$ produces sharper mean images.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/snr--top_centers.pdf}
\caption{Means of the largest 8 clusters for the SNR=$1/16$ dataset, sorted by cluster size from left to right.
(top) $L_2$ distance; (bottom) $W_1$ metric. We can see that the $W_1$ metric preserves more details than the $L_2$ distance. The portion of the image that does not contain our particle appears less noisy in the $L_2$ averages, however this is just an artifact of the cluster size. By averaging a large number of images, the noise decreases but the signal also deteriorates.}
\label{fig:ribosome_centroids}
\end{figure}
Finally, we examine the occupancy of the clusters. The $W_1$ algorithm provides more evenly sized clusters, whereas for the $L_2$ algorithm we see a few very populated centers and a large dropoff in occupancy in the other clusters.
When clustering images with Gaussian noise, the averages of larger clusters will tend to be less noisy, since the noise variance is inversely proportional to the cluster size. Due to the lower noise levels, more of the images will be assigned to the larger clusters, making them even larger.
This ``rich get richer'' phenomenon has been observed in cryo-EM \citep{SorzanoEtal2010}.
It can explain the large differences in occupancy visible in the top panels of Figure \ref{fig:occupancy}, despite the fact that the angles were drawn uniformly.
The Wasserstein distance is more resilient to i.i.d. noise and this may explain the uniformity in the resulting cluster sizes seen in the bottom panels of Figure \ref{fig:occupancy}.
Finally, clustering with the $W_1$ distance does not increase the runtime of the clustering algorithm by much. We include timing results per iteration of k-means with each metric, along with the number of iterations it took for the algorithm to converge in Table \ref{table:k-means-timing}.
\begin{figure}
\includegraphics[width=0.33\textwidth]{figures/snr--occupancy_dist-8.pdf}
\includegraphics[width=0.33\textwidth]{figures/snr--occupancy_dist-12.pdf}
\includegraphics[width=0.33\textwidth]{figures/snr--occupancy_dist-16.pdf}
\caption{Cluster sizes for the datasets with signal-to-noise $1/8,\, 1/12,\, 1/16$ from left to right.}
\label{fig:occupancy}
\end{figure}
\begin{table}[h]
\caption{Seconds per iteration averaged (over two runs) for the $L_2$ and $W_1$ metrics and the number of iterations before
convergence at different SNRs (for one run). The programs were run using 32 Intel core i7 CPU cores which allow us to compute the distances from images in our dataset to all the cluster averages in a parallelized fashion across the averages.}
\label{table:k-means-timing}
\centering
\begin{tabular}{rcccc}
Metric & k-means (sec/iteration) & $n_{\text{iter}}$ (SNR $=1/8$) & $n_{\text{iter}}$ (SNR $=1/12$) & $n_{\text{iter}}$ (SNR $=1/16$)\\
\toprule
$L_2$ & 1204 s & 13 & 31 & 22\\
$W_1$ & 1676 s & 23 & 27 & 26\\
\end{tabular}
\end{table}
\subsection{Sensitivity to noise}
To examine the effect of noise on the $W_1^R$ and $L_2^R$ distances, we plot the $W_1^R$ and $L_2^R$ distances against the viewing angle difference between projections of the ribosome. In Figure \ref{fig:noise} we can see that $W_1^R$ continues to give meaningful distances under noise compared to the $L_2^R$ distance.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{figures/l216noise.pdf}
\includegraphics[width=0.4\textwidth]{figures/emd16noise.pdf}
\caption{$W_1^R$ and $L_2^R$ distances versus the viewing angle difference between a fixed random clean projection and $1000$ random noisy projections (SNR=1/16)}
\label{fig:noise}
\end{figure}
\section{Theory} \label{sec:theory}
For a given particle, two tomographic projections from similar viewing angles typically produce images that are similar.
Hence, it is desirable that a metric for comparing tomographic projections will a assign small distance to projections that have a small difference in viewing angle.
We show that the rotationally-invariant Wasserstein metric satisfies this property.
\begin{proposition} \label{prop:w1UpperBound}
Let $\rho: \mathbb{R}^3 \rightarrow \mathbb{R}_{\geq 0}$ be a probability distribution supported on the 3D unit ball and
let $I_1$ and $I_2$ be its tomographic projections along the vectors $\u$ and $\v$ respectively.
Denote by $\measuredangle(\u,\v) \in [0,\pi]$ the angle between the vectors, then
\begin{align}
W_p^R(I_1, I_2)^p \leq [2\sin(\measuredangle(\u,\v)/2)]^p \leq \measuredangle(\u, \v)^p
\end{align}
where $W_p^R$ is the rotationally-invariant Wasserstein metric defined in Eq. \eqref{def:rotinv_wasserstein}.
\end{proposition}
See Appendix \ref{sec:proofs} for the proof.
A similar upper-bound for the rotationally-invariant $L_2$ distance cannot hold for all densities $\rho$.
To see why, consider an off-center point mass. Any two projections at slightly different viewing angles will have a large $L_2^R$ distance no matter how small their angular difference is.
However, for densities with bounded gradients it is possible to produce upper bounds.
\begin{proposition} \label{prop:L2UpperBound}
Let $B = \sup_{\bf x}|\nabla\rho({\bf x})|$ be an upper bound on the absolute gradient of the density.
Under the same conditions of Proposition \ref{prop:w1UpperBound} we have
\begin{align}
L_2^R(I_1, I_2) \leq 2\sqrt{\pi}B\measuredangle(\u, \v).
\end{align}
\label{eq:l2UpperBound}
\end{proposition}
The proof is in Appendix \ref{sec:proofs}.
This bound suggests that $L_2^R$ is a reasonable metric to use for very smooth signals. For non-smooth signals, or signals with very large $B$, this means that there is no guarantee that the $L_2^R$ distance will assign a small distance between projections with a small viewing angle.
\section{Discussion}
From our numerical experiments, we see that the rotationally-invariant Wasserstein-1 k-means clustering algorithm produces clusters that have more angular coherence and sharper cluster means than the rotationally-invariant $L_2$ clustering algorithm.
Thus, we believe it is a promising alternative to commonly used rotationally-invariant clustering algorithms based on the
$L_2$ distance.
Recently, there has been an explosion of interest in the analysis of molecular samples with continuous variability \citep{NakaneEtal2018,SorzanoEtal2019,MoscovichHaleviAndenSinger2020,LedermanAndenSinger2020,ZhongBeplerDavisBerger2020,DashtiEtal2020}.
In the presence of continuous variability, it is much more challenging to employ 2D class averaging and related techniques.
This is due to the fact that the clusters need to account not only for the entire range of viewing directions but also
for all the possible 3D conformations of the molecule of interest.
In future work we would like to test the performance of Wasserstein-based k-means clustering for datasets with continuous variability.
Wasserstein metrics seem like a natural fit since, by definition, motion of a part of the molecule incurs a distance no greater than the mass of the moving part multiplied by its displacement.
There are many other directions for future work, including incorporating Wasserstein metrics into EM-ML style algorithms \citep{Scheres2014},
Wasserstein barycenters for estimating the cluster centers \citep{CuturiDoucet2014},
rotationally-invariant denoising procedures \citep{ZhaoSinger2014},
incorporating translations, experiments on more realistic datasets with a contrast transfer function and analysis of real datasets.
\section{Broader impact}
Our work demonstrates that the use of Wasserstein metrics can improve the quality of clustering tomographic projections in the context of cryo-electron microscopy. This can reduce the number of images required to produce high quality cluster averages. Improved data efficiency in the cryo-EM pipeline will accelerate discovery in structural biology (with broad implications to medicine) and lower the barriers of entry to cryo-EM for resource-constrained labs around the world. |
1701.01074 | \section{Local degree and defect}\label{SecLocDeg}
We will use the following criterion to measure defect, which is Proposition 3.4 \cite{C12}. This result is implicit in \cite{CP} with the assumptions of Proposition \ref{Prop1}.
\begin{Proposition}\label{Prop15}
Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$. Further suppose that $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field
$K^*$ such that $S$ dominates $R$.
Suppose that $\nu^*$ is a valuation of $K^*$ such that
$\nu^*$ dominates $S$, the residue field $V_{\nu^*}/m_{\nu^*}$ of $V_{\nu^*}$ is algebraic over $S/m_S$ and
the value group $\Phi_{\nu^*}$ of $\nu^*$ has rational rank 1.
Let $\nu$ be the restriction of $\nu^*$ to $K$. There
exists a local ring $R'$ of $K$ which is essentially of finite type over $R$, is dominated by
$\nu$ and dominates $R$ such that if we have a commutative diagram
\begin{equation}\label{eq31}
\begin{array}{lll}
V_{\nu}&\rightarrow&V_{\nu^*}\\
\uparrow&&\uparrow\\
R_1&\rightarrow&S_1\\
\uparrow&&\\
R'&&\uparrow\\
\uparrow&&\\
R&\rightarrow&S
\end{array}
\end{equation}
where
$R_1$ is a regular local ring of $K$ which is essentially of finite type over $R$ and dominates $R$, $S_1$ is a regular local ring of $K^*$ which is essentially of finite type over $S$ and dominates $S$,
$R_1$ has a regular system of parameters $u,v$ and $S_1$ has a regular system of parameters $x,y$ such that there is an expression
$$
u=\gamma x^a, v=x^bf
$$
where $a>0$, $b\ge 0$, $\gamma$ is a unit in $S$, $x\not\,\mid f$ in $S_1$ and $f$ is not a unit in $S_1$, then
\begin{equation}\label{eq37}
ad[S_1/m_{S_1}:R_1/m_{R_1}]=e(\nu^*/\nu)f(\nu^*/\nu)p^{\delta(\nu^*/\nu)}
\end{equation}
where $d=\overline\nu(f\mbox{ mod }x)$ with $\overline \nu$ being the natural valuation of the DVR $S/xS$.
\end{Proposition}
\vskip .2truein
We now prove Lemma \ref{LemmaN66} from the introduction. Let $\nu_1=\nu^*,\nu_2,\ldots,\nu_r$ be the extensions of $\nu$ to $K^*$. Let $T$ be the integral closure of $V_{\nu}$ in $K^*$. Then $T=V_{\nu_1}\cap\cdots\cap V_{\nu_r}$ is the integral closure of $V_{\nu^*}$ in $K^*$ (by Propositions 2.36 and 2.38 \cite{RTM}). Let $m_i=m_{\nu_i}\cap T$ be the maximal ideals of $T$. By the Chinese remainder theorem, there exists $u\in T$ such that $u\in m_1$ and $u\not\in m_i$ for $2\le i\le r$. Let
$$
u^n+a_1u^{n-1}+\cdots+a_n=0
$$
be an equation of integral dependence of $u$ over $V_{\nu}$. Let $A$ be the integral closure of $R[a_1,\ldots,a_n]$ in $K$ and let $R'=A_{A\cap m_{\nu}}$. Let $T'$ be the integral closure of $R'$ in $K^*$. We have that $u\in T'\cap m_i$ if and only if $i=1$. Let $S'=T'_{T'\cap m_1}$. Then $\nu$ does not split in $S'$ and $R'$ has the property of the conclusions of the lemma.
\section{Generating Sequences}\label{SecGen}
Given an additive group $G$ with $\lambda_0,\ldots, \lambda_r\in G$, $G(\lambda_0,\ldots,\lambda_r)$ will denote the subgroup generated by $\lambda_0,\ldots,\lambda_r$. The semigroup generated by $\lambda_0,\ldots, \lambda_r$ will be denoted by $S(\lambda_0,\ldots,\lambda_r)$.
In this section, we will suppose that $R$ is a regular local ring of dimension two, with maximal ideal $m_R$ and residue field $R/ m_R$.
For $f\in R$, let $\overline f$ or $[f]$ denote the residue of $f$ in $R/ m_R$.
The following theorem is Theorem 4.2 of \cite{CV1}, as interpreted by Remark 4.3 \cite{CV1}.
\begin{Theorem}\label{Theorem1*} Suppose that $\nu$ is a valuation
of the quotient field of $R$ dominating $R$. Let $L=V_{\nu}/m_{\nu}$ be the residue field of the valuation ring $V_{\nu}$ of $\nu$. For $f \in V_{\nu}$, let $[f]$ denote the class of $f$ in $L$. Suppose that $x,y$ are regular parameters in $R$.
Then there exist $\Omega\in{\NZQ Z}_+\cup\{\infty\}$ and
$P_i(\nu,R)\in m_R$ for $i\in{\NZQ Z}_+$ with $i<\min\{\Omega+1,\infty\}$
such that $P_0(\nu,R)=x$, $P_1(\nu,R)=y$ and for $1\le i<\Omega$, there is an expression
\begin{equation}\label{eq11*}
P_{i+1}(\nu,R) = P_i(\nu,R)^{n_i(\nu,R)}+\sum_{k=1}^{\lambda_i} c_kP_0(\nu,R)^{\sigma_{i,0}(k)}P_1(\nu,R)^{\sigma_{i,1}(k)}\cdots P_{i}(\nu,R)^{\sigma_{i,i}(k)}
\end{equation}
with $n_i(\nu,R)\ge 1$, $\lambda_i\ge 1$,
\begin{equation}\label{eq12*}
0\ne c_k\mbox{ units in }R
\end{equation}
for $1\le k\le \lambda_i$,
$\sigma_{i,s}(k)\in{\NZQ N}$ for all $s,k$, $0\le \sigma_{i,s}(k)<n_s(\nu,R)$ for $s\ge 1$.
Further,
$$
n_i(\nu,R)\nu(P_i(\nu,R))=\nu(P_0(\nu,R)^{\sigma_{i,0}(k)}P_1(\nu,R)^{\sigma_{i,1}(k)}\cdots P_{i}(\nu,R)^{\sigma_{i,i}(k)})
$$
for all $k$.
For all $i\in{\NZQ Z}_+$ with $i<\Omega$, the following are true:
\begin{enumerate}
\item[1)] $\nu(P_{i+1}(\nu,R))>n_i(\nu,R)\nu(P_i(\nu,R))$.
\item[2)] Suppose that $r\in{\NZQ N}$, $m\in {\NZQ Z}_+$, $j_k(l)\in{\NZQ N}$ for $1\le l\le m$ and $0\le j_k(l)<n_k(\nu,R)$ for $1\le k\le r$ are such that $(j_0(l),j_1(l),\ldots,j_r(l))$ are distinct for $1\le l\le m$, and
$$
\nu(P_0(\nu,R)^{j_0(l)}P_1(\nu,R)^{j_1(l)}\cdots P_r(\nu,R)^{j_r(l)})=\nu(P_0(\nu,R)^{j_0(1)}\cdots P_r(\nu,R)^{j_r(1)})
$$
for $1\le l\le m$.
Then
$$
1,\left[\frac{P_0(\nu,R)^{j_0(2)}P_1(\nu,R)^{j_1(2)}\cdots P_r(\nu,R)^{j_r(2)}}{P_0(\nu,R)^{j_0(1)}P_1(\nu,R)^{j_1(1)}\cdots P_r(\nu,R)^{j_r(1)}}\right],
\ldots,
\left[\frac{P_0(\nu,R)^{j_0(m)}P_1(\nu,R)^{j_1(m)}\cdots P_r(\nu,R)^{j_r(m)}}{P_0(\nu,R)^{j_0(1)}P_1(\nu,R)^{j_1(1)}\cdots P_r(\nu,R)^{j_r(1)}}\right]
$$
are linearly independent over $R/ m_R$.
\item[3)] Let
$$
\overline n_i(\nu,R)=[G(\nu(P_0(\nu,R)),\ldots, \nu(P_(\nu,R)i)):G(\nu(P_0(\nu,R)),\ldots, \nu(P_{i-1}(\nu,R)))].
$$
Then $\overline n_i(\nu,R)$ divides $\sigma_{i,i}(k)$ for all $k$ in (\ref{eq11*}). In particular, $n_i(\nu,R)=\overline n_i(\nu,R)d_i(\nu,R)$ with $d_i(\nu,R)\in {\NZQ Z}_+$
\item[4)] There exists $U_i(\nu,R)=P_0(\nu,R)^{w_0(i)}P_1(\nu,R)^{w_1(i)}\cdots P_{i-1}(\nu,R)^{w_{i-1}(i)}$ for $i\ge 1$ with $w_0(i),\ldots, w_{i-1}(i)\in{\NZQ N}$
and $0\le w_j(i)<n_j(\nu,R)$ for $1\le j\le i-1$ such that
$\nu(P_i(\nu,R)^{\overline n_i})=\nu(U_i(\nu,R))$ and setting
$$
\alpha_i(\nu,R)=\left[\frac{P_i(\nu,R)^{\overline n_i(\nu,R)}}{U_i(\nu,R)}\right]
$$
then
$$
\begin{array}{lll}
b_{i,t}&=&\left[\sum_{\sigma_{i,i}(k)=t\overline n_i(\nu,R)}c_k\frac{P_0(\nu,R)^{\sigma_{i,0}(k)}P_1(\nu,R)^{\sigma_{i,1}(k)}\cdots P_{i-1}(\nu,R)^{\sigma_{i,i-1}(k)}} {U_i(\nu,R)^{(d_i(\nu,R)-t)}}\right]\\
&&\in R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{i-1}(\nu,R))
\end{array}
$$
for $0\le t\le d_i(\nu,R)-1$ and
$$
f_i(u)=u^{d_i(\nu,R)}+b_{i,d_i(\nu,R)-1}u^{d_i(\nu,R)-1}+\cdots+b_{i,0}
$$
is the minimal polynomial of $\alpha_i(\nu,R)$ over $R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{i-1}(\nu,R))$.
\end{enumerate}
The algorithm terminates with $\Omega<\infty$ if and only if either
\begin{equation}\label{eqL15}
\overline n_{\Omega}(\nu,R)=[G(\nu(P_0(\nu,R)),\ldots, \nu(P_{\Omega}(\nu,R))):G(\nu(P_0(\nu,R)),\ldots, \nu(P_{\Omega-1}(\nu,R)))]=\infty
\end{equation}
or
\begin{equation}\label{eqL10}
\begin{array}{l}
\mbox{$\overline n_{\Omega}(\nu,R)<\infty$ (so that $\alpha_{\Omega}(\nu,R)$ is defined as in 4)) and}\\
\mbox{$d_{\Omega}(\nu,R)=[R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{\Omega}(\nu,R)):R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{\Omega-1}(\nu,R))]=\infty$.}
\end{array}
\end{equation}
If $\overline n_{\Omega}(\nu,R)=\infty$, set $\alpha_{\Omega}(\nu,R)=1$.
\end{Theorem}
Let notation be as in Theorem \ref{Theorem1*}.
The following formula is formula $B(i)$ on page 10 of \cite{CV1}.
\vskip .2truein
\begin{equation}\label{eqZ51}
\begin{array}{l}
\mbox{ Suppose that $M$ is a Laurent monomial in $P_0(\nu,R),P_1(\nu,R),\ldots, P_i(\nu,R)$}\\
\mbox{ and $\nu(M)=0$. Then there exist $s_i\in{\NZQ Z}$ such that }\\
\,\,\,\,\,\,\,\,\,\, M=\prod_{j=1}^i\left[\frac{P_j(\nu,R)^{\overline n_j}}{U_j(\nu,R)}\right]^{s_j},\\
\mbox{ so that}\\
\,\,\,\,\,\,\,\,\,\, [M]\in R/m_R[\alpha_1(\nu,R),\ldots,\alpha_i(\nu,R)].
\end{array}
\end{equation}
\vskip .2truein
Define $\beta_i(\nu,R)=\nu(P_i(\nu,R))$ for $0\le i$.
Since $\nu$ is a valuation of the quotient field of $R$, we have that
\begin{equation}\label{eq50}
\Phi_{\nu}=\cup_{i=1}^{\infty}G(\beta_0(\nu,R),\beta_1,\ldots,\beta_i(\nu,R))
\end{equation}
and
\begin{equation}\label{eq51}
V_{\nu}/m_{\nu}=\cup_{i=1}^{\infty} R/m_R[\alpha_1(\nu,R),\ldots,\alpha_i(\nu,R)]
\end{equation}
The following is Theorem 4.10 \cite{CV1}.
\begin{Theorem}\label{TheoremG2}
Suppose that $\nu$ is a valuation dominating $R$.
Let
$$
P_0(\nu,R)=x, P_1(\nu,R)=y, P_2(\nu,R),\ldots
$$
be the sequence of elements of $R$ constructed by Theorem \ref{Theorem1*}. Suppose that $f\in R$ and there exists $n\in{\NZQ Z}_+$ such that $\nu(f)<n\nu(m_R)$.
Then there exists an expansion
$$
f=\sum_{I}a_IP_0(\nu,R)^{i_0}P_1(\nu,R)^{i_1}\cdots P_r(\nu,R)^{i_r}+\sum_J\phi_JP_0(\nu,R)^{j_0}\cdots P_r(\nu,R)^{j_r}+h
$$
where $r\in{\NZQ N}$, $a_{I}$ are units in $R$, $I,J\in {\NZQ N}^{r+1}$, $\nu(P_0(\nu,R)^{i_0}P_1(\nu,R)^{i_1}\cdots P_r(\nu,R)^{i_r})=\nu(f)$ for all $I$ in the first sum, $0\le i_k<n_k(\nu,R)$ for $1\le k\le r$, $\nu(P_0(\nu,R)^{j_0}\cdots P_r(\nu,R)^{j_r})>\nu(f)$ for all terms in the second sum, $\phi_J\in R$ and $h\in m_R^n$.
The terms in the first sum are uniquely determined, up to the choice of units $a_i$, whose residues in $R/m_R$ are uniquely determined.
\end{Theorem}
Let $\sigma_0(\nu,R)=0$ and inductively define
\begin{equation}\label{eq3}
\sigma_{i+1}(\nu,R)=\min\{j>\sigma_i(\nu,R)\mid n_j(\nu,R)>1\}.
\end{equation}
In Theorem \ref{TheoremG2}, we see that all of the monomials in the expansion of $f$ are in terms of the $P_{\sigma_i}$.
We have that
$$
S(\beta_0(\nu,R),\beta_1(\nu,R),\ldots,\beta_{\sigma_j(\nu,R)})=S(\beta_{\sigma_0}(\nu,R),\beta_{\sigma_1(\nu,R)},\ldots,\beta_{\sigma_j(\nu,R)})
$$
for all $j\ge 0$ and
$$
\begin{array}{l}
R/m_R[\alpha_1(\nu,R),\alpha_2(\nu,R),\ldots,\alpha_{\sigma_j(\nu,R)}(\nu,R)]\\
=R/m_R[\alpha_{\sigma_1(\nu,R)}(\nu,R),\alpha_{\sigma_2(\nu,R)}(\nu,R),\ldots,\alpha_{\sigma_j(\nu,R)}(\nu,R)]
\end{array}
$$
for all $j\ge 1$.
Suppose that $R$ is a regular local ring of dimension two which is dominated by a valuation $\nu$. The quadratic transform $T_1$ of $R$ along $\nu$ is defined as follows. Let $u,v$ be a system of regular parameters in $R$,
Then $R[\frac{v}{u}]\subset V_{\nu}$ if $\nu(u)\le \nu(v)$ and $R[\frac{u}{v}]\subset V_{\nu}$ if $\nu(u)\ge \nu(v)$. Let
$$
T_1=R\left[\frac{v}{u}\right]_{R[\frac{v}{u}]\cap m_{\nu}}\mbox{ or }T_1=R\left[\frac{u}{v}\right]_{R[\frac{u}{v}]\cap m_{\nu}},
$$
depending on if $\nu(u)\le\nu(v)$ or $\nu(u)>\nu(v)$.
$T_1$ is a two dimensional regular local ring which is dominated by $\nu$.
Let
\begin{equation}\label{eqX3}
R\rightarrow T_1\rightarrow T_2\rightarrow \cdots
\end{equation}
be the infinite sequence of quadratic transforms along $\nu$, so that $V_{\nu}=\cup_{i\ge 1} T_i$ (Lemma 4.5 \cite{RTM}) and $L=V_{\nu}/m_{\nu}=\cup_{i\ge 1} T_i/m_{T_i}$.
For $f\in R$ and $R\rightarrow R^*$ a sequence of quadratic transforms along $\nu$, we define a strict transform of $f$ in $R^*$ to be $f_1$ if $f_1\in R^*$ is a local equation of the strict transform in $R^*$ of the subscheme $f=0$ of $R$. In this way, a strict transform is only defined up to multiplication by a unit in $R^*$. This ambiguity will not be a difficulty in our proof. We will denote a strict transform of $f$ in $R^*$ by $\mbox{st}_{R^*}(f)$.
We use the notation of Theorem \ref{Theorem1*} and its proof for $R$ and the $\{P_i(\nu,R)\}$. Recall that $U_1=U^{w_0(1)}$. Let $w=w_0(1)$. Since $\overline n_1(\nu,R)$ and $w$ are relatively prime, there exist $a,b\in{\NZQ N}$ such that
$$
\epsilon:=\overline n_1(\nu,R)b-wa=\pm 1.
$$
Define
elements of the quotient field of $R$ by
\begin{equation}\label{eqX50}
x_1=(x^by^{-a})^{\epsilon}, y_1=(x^{-w}y^{\overline n_1(\nu,R)})^{\epsilon}.
\end{equation}
We have that
\begin{equation}\label{eqZ1}
x=x_1^{\overline n_1(\nu,R)}y_1^a, y=x_1^wy_1^b.
\end{equation}
Since $\overline n_1(\nu,R)\nu(y)=w\nu(x)$, it follows that
$$
\overline n_1(\nu,R)\nu(x_1)=\nu(x)>0\mbox{ and }\nu(y_1)=0.
$$
We further have that
\begin{equation}\label{eqZ3}
\alpha_1(\nu,R)=[y_1]^{\epsilon}\in V_{\nu}/m_{\nu}.
\end{equation}
Let $A=R[x_1,y_1]\subset V_{\nu}$ and $ m_A= m_{\nu}\cap A$.
Let $R_1=A_{ m_A}$. We have that $R_1$ is a regular local ring and the divisor of $xy$ in $R_1$
has only one component ($x_1=0$). In particular, $R\rightarrow R_1$ is ``free'' (Definition 7.5 \cite{CP}).
$R\rightarrow R_1$ factors (uniquely) as a product
of quadratic transforms and the divisor of $xy$ in $R_1$ has two distinct irreducible factors in all intermediate rings.
The following is Theorem 7.1 \cite{CV1}.
\begin{Theorem}\label{birat}
Let $R$ be a two dimensional regular local ring with regular parameters $x,y$. Suppose that $R$ is dominated by a valuation $\nu$. Let $P_0(\nu,R)=x$, $P_1(\nu,R)=y$ and $\{P_i(\nu,R)\}$ be the sequence of elements of $R$ constructed in Theorem \ref{Theorem1*}. Suppose that $\Omega\ge 2$. Then there exists some smallest value $i$ in the sequence (\ref{eqX3}) such that
the divisor of $xy$ in $\mbox{Spec}(T_i)$ has only one component. Let $R_1=T_i$.
Then $R_1/m_{R_1}\cong R/m_R(\alpha_1(\nu,R))$, and there exists $x_1\in R_1$ and $w\in{\NZQ Z}_+$ such that
$x_1=0$ is a local equation of the exceptional divisor of $\mbox{Spec}(R_1)\rightarrow \mbox{Spec}(R)$, and $Q_0=x_1$, $Q_1=\frac{P_2}{x_1^{wn_1}}$ are regular parameters in $R_1$. We have that
$$
P_i(\nu,R_1)=\frac{P_{i+1}(\nu,R)}{P_0(\nu,R_1)^{w n_1(\nu,R)\cdots n_i(\nu,R)}}
$$
for $1\le i< \max\{\Omega,\infty\}$
satisfy the conclusions of Theorem \ref{Theorem1*} for the ring $R_1$.
\end{Theorem}
We have that
$$
G(\beta_0(\nu,R_1),\ldots,\beta_i(\nu, R_1))=G(\beta_0(\nu,R),\ldots,\beta_{i+1}(\nu,R))
$$
for $i\ge 1$ so that
$$
\overline n_i(\nu,R_1)=\overline n_{i+1}(\nu,R)\mbox{ for }i\ge 1
$$
and
$$
R_1/m_{R_1}[\alpha_1(\nu,R_1),\ldots,\alpha_i(\nu,R_1)]=R/m_R[\alpha_1(\nu,R),\ldots,\alpha_{i+1}(\nu,R)]\mbox{ for }i\ge 1
$$
so that
$$
d_i(\nu,R_1)=d_{i+1}(\nu,R)\mbox{ and }n_{i}(\nu,R_1)=n_{i+1}(\nu,R)\mbox{ for }i\ge 1.
$$
Let $\sigma_0(\nu,R_1)=0$ and inductively define
$$
\sigma_{i+1}(\nu,R_1)=\min\{j>\sigma_i(1)\mid n_j(\nu,R_1)>1\}.
$$
We then have that
$\sigma_0(\nu,R_1)=0$ and for $i\ge 1$, $\sigma_i(\nu,R_1)=\sigma_{i+1}(\nu,R)-1$ if $n_1(\nu,R)>1$ and $\sigma_i(\nu,R_1)=\sigma_i(\nu,R)-1$ if $n_1(\nu,R)=1$, and for all $j\ge 0$,
$$
S(\beta_0(\nu,R_1),\beta_1(\nu,R_1),\ldots,\beta_{\sigma_{j+1}(\nu,R_1)}(\nu,R_1))=S(\beta_{\sigma_0(1)}(\nu,R_1),\beta_{\sigma_1(\nu,R_1)},\ldots,\beta_{\sigma_j(\nu,R_1)}(\nu,R_1))
$$
Iterating this construction, we produce a sequence of sequences of quadratic transforms along $\nu$,
$$
R\rightarrow R_1\rightarrow \cdots\rightarrow R_{\sigma_1(\nu,R)}.
$$
Now $x, \overline y=P_{\sigma_1(\nu,R)}$ are regular parameters in $R$. By (\ref{eqX50}) (with $y$ replaced with $\overline y$) we have that $R_{\sigma_1(\nu,R)}$ has regular parameters
\begin{equation}\label{eq8}
x_1=(x^b\overline y^{-a})^{\epsilon},\,\,\,
y_1=(x^{-\omega}\overline y^{\overline n_{\sigma_1(\nu,R)}(\nu,R)})^{\epsilon}
\end{equation}
where $\omega,a,b\in {\NZQ N}$ satisfy
$\epsilon=\overline n_{\sigma_1(\nu,R)}(\nu,R)b-\omega a=\pm 1$.
Further, $R_{\sigma_1(\nu,R_1)}$ has regular parameters $x_{\sigma_1(\nu,R)}, y_{\sigma_1(\nu,R)}$ where $x=\delta x_{\sigma_1(\nu,R_1)}^{\overline n_{\sigma_1(\nu,R)}(\nu,R_1)}$ and $y_{\sigma_1(\nu,R_1)}={\rm st}_{R_{\sigma_1}(\nu,R_1)}P_{\sigma_1(\nu,R)}(\nu,R)$ with $\delta\in R_{\sigma_1(\nu,R)}$ a unit.
For the remainder of this section, we will suppose that $R$ is a two dimensional regular local ring and $\nu$ is a non discrete rational rank 1 valuation of the quotient field of $R$ with valuation ring $V_{\nu}$, so that $V_{\nu}/m_{\nu}$ is algebraic over $R/m_R$. Suppose that $f\in R$ and $\nu(f)=\gamma$. We will denote the class of $f$ in $\mathcal P_{\gamma}(R)/\mathcal P^+_{\gamma}(R)\subset {\rm gr}_{\nu}(R)$ by ${\rm in}_{\nu}(f)$.
By Theorem \ref{TheoremG2}, we have that ${\rm gr}_{\nu}(R)$ is generated by the initial forms of the $P_i(\nu,R)$ as an $R/m_R$-algebra. That is,
$$
\begin{array}{l}
{\rm gr}_{\nu}(R)=R/m_R[{\rm in}_{\nu}(P_0(\nu,R)),{\rm in}_{\nu}(P_1(\nu,R)),\ldots]\\
=R/m_R[{\rm in}_{\nu}(P_{\sigma_0(\nu,R)}(\nu,R)),{\rm in}_{\nu}(P_{\sigma_1(\nu,R)}(\nu,R)),\ldots].
\end{array}
$$
Thus the semigroup
$S^R(\nu)=\{\nu(f)\mid f\in R\}$ is equal to
$$
S^R(\nu)=S(\beta_0(\nu,R),\beta_1(\nu,R),\ldots)=S(\beta_{\sigma_0(\nu,R)}(\nu,R),\beta_{\sigma_1(\nu,R)}(\nu,R),\ldots)
$$
and the value group
$$
\Phi_{\nu}=G(\beta_0(\nu,R),\beta_1(\nu,R)\ldots)
$$
and the residue field of the valuation ring
$$
V_{\nu}/m_{\nu}=R/m_R[\alpha_1(\nu,R),\alpha_2(\nu,R),\ldots]=R/m_R[\alpha_{\sigma_1}(\nu,R),\alpha_{\sigma_2}(\nu,R),\ldots]
$$
By 1) of Theorem \ref{Theorem1*}, every element $\beta\in S^{R}(\nu)$ has a unique expression
$$
\beta=\sum_{i=0}^ra_i\beta_i(\nu,R)
$$
for some $r$ with $a_i\in {\NZQ N}$ for all $i$ and $0\le a_i<n_i(\nu,R)$ for $1\le i$. In particular, if $a_i\ne 0$ in the expansion then $\beta_i(\nu,R)=\beta_{\sigma_j(\nu,R)}(\nu,R)$ for some $j$.
\begin{Lemma}\label{Lemma1} Let
$$
\sigma_i=\sigma_i(\nu,R), \beta_i=\beta_i(\nu,R), P_{i}=P_i(\nu,R), n_i=n_i(\nu,R), \overline n_i=\overline n_i(\nu,R),
$$
$$
\sigma_i(1)=\sigma_i(\nu,R_{\sigma_1}), \beta_i=\beta_i(\nu,R_{\sigma_1}), P_i(1)=P_i(\nu,R_{\sigma_1}), n_i(1)=n_i(\nu,R_{\sigma_1}), \overline n_i(1)=\overline n_i(\nu,R_{\sigma_1}).
$$
Suppose $i\in {\NZQ N}$, $r\in {\NZQ N}$ and $a_j\in {\NZQ N}$ for $j=0,\ldots, r$ with $0\le a_j<n_{\sigma_j}$ for $j\ge 1$ are such that
$$
\nu(P_{\sigma_0}^{a_0}\cdots P_{\sigma_r}^{a_r})> \nu(P_{\sigma_i})
$$
or $r<i$ and
$$
\nu(P_{\sigma_0}^{a_0}\cdots P_{\sigma_r}^{a_r})= \nu(P_{\sigma_i}).
$$
By (\ref{eqZ1}) and Theorem \ref{birat}, we have expressions in
$$
R_{\sigma_1}=R[x_1,y_1]_{m_{\nu}\cap R[x_1,y_1]}
$$
where $x_1,y_1$ are defined by (\ref{eq8})
$$
P_{\sigma_0}^{a_0}\cdots P_{\sigma_r}^{a_r}=y_1^{aa_0+ba_1}P_{\sigma_1(1)}(1)^{a_2}\cdots P_{
\sigma_{r-1}(1)}(1)^{a_r}P_{\sigma_0(1)}(1)^t
$$
where $t=\overline n_{\sigma_1}a_0+\omega a_1+\omega n_{\sigma_1}a_2+\cdots+\omega n_{\sigma_1}\cdots n_{\sigma_{r-1}}a_r$ and
$$
P_{\sigma_i}=
\left\{
\begin{array}{ll}
y_1^aP_{\sigma_0(1)}(1)^{\overline n_{\sigma_1}}&\mbox{ if }i=0\\
y_1^bP_{\sigma_0(1)}(1)^{\omega}&\mbox{ if }i=1\\
P_{\sigma_{i-1}(1)}(1)P_{\sigma_0(1)}(1)^{\omega n_{\sigma_1}\cdots n_{\sigma_{i-1}}}&\mbox{ if }i\ge 2.
\end{array}\right.
$$
Let
$$
\lambda=\left\{\begin{array}{ll}
\overline n_{\sigma_1}&\mbox{ if }i=0\\
\omega&\mbox{ if }i=1\\
\omega n_{\sigma_1}\cdots n_{\sigma_{i-1}}&\mbox{ if }i\ge 2.
\end{array}\right.
$$
Then
$$
t>\lambda,
$$
except in the case where $i=1$, $P_{\sigma_0}^{a_0}\cdots P_{\sigma_r}^{a_r}=P_{\sigma_0}$, and $\overline n_{\sigma_1}=\omega=1$.
In this case $\lambda=t$.
\end{Lemma}
\begin{proof} First suppose that $i\ge 2$ and $r\ge i$. Then
$$
t-\lambda=(\overline n_{\sigma_1}a_0+\omega a_1+\omega n_{\sigma_1}a_2+\cdots +\omega n_{\sigma_1}\cdots n_{\sigma_{r-1}}a_r)-\omega n_{\sigma_1}\cdots n_{\sigma_{i-1}}>0.
$$
Now suppose that $i\ge 2$ and $r<i$.
We have that
$$
\begin{array}{l}
(\overline n_1a_0+\omega a_1+\ldots+\omega n_1\cdots n_{r-1}a_r-\omega n_1(\nu,R)\cdots n_{i-1})\beta_{\sigma_0(1)}(1)\\
\ge \beta_{\sigma_{i-1}(1)}(1)-a_2\beta_{\sigma_1(1)}(1)-\ldots-a_r\beta_{\sigma_{r-1}(1)}(1)>0
\end{array}
$$
since $n_{\sigma_j(1)}(1)=n_{\sigma_{j+1}}$ for all $j$, and so $n_{\sigma_{j+1}}\beta_{\sigma_j(1)}(1)<\beta_{\sigma_{j+1}(1)}(1)$ for all $j$.
Now suppose that $i=1$. As in the proof for the case $i\ge 2$ we have that $t-\lambda>0$ if $r\ge 1$, so suppose that $i=1$ and $r=0$. Then $\overline n_{\sigma_1}\beta_{\sigma_1}=\omega \beta_{\sigma_0}$.
From our assumption $a_0\nu(P_0)\ge \nu(P_1)$ we obtain $t-\lambda=\overline n_{\sigma_1}a_0-\omega\ge 0$ with equality if and only if $a_0=\omega=\overline n_{\sigma_1}=1$ since ${\rm gcd}(\omega,\overline n_{\sigma_1})=1$.
Now suppose $i=0$. As in the previous cases, we have $t-\lambda>0$ if $r>1$ and $t-\lambda>0$ if $r=1$ except possibly if $P_0^{a_0}\cdots P_r^{a_r}=P_1^{a_1}$. We then have that $\nu(P_{\sigma_1}^{a_1})>\nu(P_{\sigma_0})$, and so
$$
a_1\frac{\beta_{\sigma_1}}{\beta_{\sigma_0}}>1.
$$
Since
$$
\frac{\beta_{\sigma_1}}{\beta_{\sigma_0}}=\frac{\omega}{\overline n_{\sigma_1}},
$$
we have that $t-\lambda=\omega a_1-\overline n_{\sigma_1}>0$.
\end{proof}
\begin{Lemma}\label{Lemma2} Let notation be as in Lemma \ref{Lemma1}. Suppose that $f\in R$, with $\nu(f)=\nu(P_{\sigma_i})$ for some $i\ge 0$, and that $f$ has an expression of the form of Theorem \ref{TheoremG2},
$$
f=cP_{\sigma_i}+\sum_{j=1}^sc_iP_{\sigma_0}^{a_0(j)}P_{\sigma_1}^{a_1(j)}\cdots P_{\sigma_r}^{a_r(j)}+h
$$
where $s,r\in{\NZQ N}$, $c, c_j$ are units in $R$, with $0\le a_k(j)<n_k$ for $1\le k\le r$ for $1\le j\le s$,
$$
\nu(f)=\nu(P_{\sigma_i})\le \nu(P_{\sigma_0}^{a_0(j)}P_{\sigma_1}^{a_1(j)}\cdots P_{\sigma_r}^{a_r(j)})
$$
for $1\le j\le s$, $a_k(j)=0$ for $k\ge i$ if $\nu(f)=\nu(P_{\sigma_0}^{a_p(j)}\cdots P_{\sigma_r}^{a_r(j)})$
and $h\in m_R^n$ with $n>\nu(f)$.
Then
${\rm st}_{R_{\sigma_1}}(f)$ is a unit in $R_{\sigma_1}$ if $i=0$ or 1 and if $i>1$, there exists a unit $\overline c$ in $R_{\sigma_1}$ and $\Omega\in R_{\sigma_1}$ such that
$$
{\rm st}_{R_{\sigma_1}}(f)=\overline cP_{\sigma_{i-1}(1)}(1)+x_1\Omega
$$
with $\nu({\rm st}_{R_{\sigma_1}}(f))=\nu(P_{\sigma_{i-1}(1)}(1))$ and $\nu(P_{\sigma_{i-1}(1)}(1))\le\nu(x_1\Omega)$.
\end{Lemma}
\begin{proof} Let
$$
\lambda=\left\{\begin{array}{ll}
\overline n_1&\mbox{ if }i=0\\
\omega&\mbox{ if }i=1\\
\omega n_{\sigma_1}\cdots n_{\sigma_{r-1}}&\mbox{ if }i\ge 2
\end{array}\right.
$$
Then
$$
f=cH_i+\sum_{j=1}^sc_j(y_1)^{aa_0(j)+ba_1(j)}P_{\sigma_0(1)}(1)^{t_j}P_{\sigma_1}(1)^{a_2(j)}\cdots P_{\sigma_{r-1}(1)}(1)^{a_r(j)}
+P_{\sigma_0(1)}(1)^th'
$$
with
$$
H_i=\left\{
\begin{array}{ll}
(y_1)^aP_{\sigma_0(1)}(1)^{\overline n_1}&\mbox{ if }i=0\\
(y_1)^bP_{\sigma_0(1)}(1)^{\omega}&\mbox{ if }i=1\\
P_{\sigma_0(1)}(1)^{\omega n_1\cdots n_{i-1}}P_{\sigma_{i-1}(1)}(1)&\mbox{ if }i\ge 2
\end{array}\right.
$$
and
$$
t_j=\overline n_1a_0(j)+\omega a_1(j)+\omega n_{\sigma_1}a_2+\cdots+\omega n_{\sigma_1}\cdots n_{\sigma_{r-1}}a_r(j)
$$
for $1\le j\le s$, $t>\lambda$ and $h'\in R_{\sigma_1}$.
By Lemma \ref{Lemma1}, if $i\ge 2$ or $i=0$, we have that $t_j>\lambda$ for all $j$. Thus $f=P_{\sigma_0(1)}(1)^{\lambda}\overline f$
where
$$
\overline f=cG_i+\sum_{j=1}^sc_jP_{\sigma_0(1)}(1)^{t_j-\lambda}P_{\sigma_1(1)}(1)^{a_2(j)}\cdots P_{\sigma_{r-1}(1)}(1)^{a_r(j)}
+P_{\sigma_0(1)}(1)^{t-\lambda}h'
$$
with
$$
G_i=\left\{\begin{array}{ll}
(y_1)^a&\mbox{ if }i=0\\
(y_1)^b&\mbox{ if }i=1\\
P_{\sigma_{i-1}(1)}(1)&\mbox{ if }i\ge 2
\end{array}\right.
$$
is a strict transform $\overline f={\rm st}_{R_1}(f)$ of $f$ in $R_1$.
If $i=1$, then by Lemma \ref{Lemma1}, $t_j>\lambda$ for all $j$, except possibly for a single term (that we can assume is $t_1$) which is $P_{\sigma_0}$, and we have that $\omega=\overline n_{\sigma_1}=1$. In this case $t_1=\lambda$. Then
$$
\left[\frac{P_{\sigma_1}}{P_{\sigma_0}}\right]=\alpha_{\sigma_1}(\nu,R)\in V_{\nu}/m_{\nu}
$$
which has degree $d_{\sigma_1}(\nu,R)=n_{\sigma_1}>1$ over $R/m_R$. By (\ref{eqZ1}), $x=x_1$, $y=x_1y_1$ and
$$
f=x_1[c+c_1y_1+x_1\Omega]
$$
with $\Omega\in R_{\sigma_1}$. We have that $c+c_1y_1$ is a unit in $R_{\sigma_1}$ since
$$
[y_1]=\left[\frac{P_{\sigma_0}}{P_{\sigma_1}}\right]\not\in R/m_R.
$$
\end{proof}
\section{Finite generation implies no defect}
Suppose that $R$ is a two dimensional regular local ring of $K$ and $S$ is a two dimensional regular local ring such that $S$ dominates $R$
Let $K$ be the quotient field of $R$ and $K^*$ be the quotient field of $S$. Suppose that $K\rightarrow K^*$ is a finite separable field extension. Suppose that $\nu^*$ is a non discrete rational rank 1 valuation of $K^*$
such that $V_{\nu^*}/m_{\nu^*}$ is algebraic over $S/m_S$ and that $\nu^*$ dominates $S$. Then we have a natural graded inclusion ${\rm gr}_{\nu}(R)\rightarrow {\rm gr}_{\nu^*}(S)$, so that for $f\in R$, we have that ${\rm in}_{\nu}(f)={\rm in}_{\nu^*}(f)$. Let $\nu=\nu^*|K$. Let $L=V_{\nu^*}/m_{\nu^*}$. Suppose that
${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra.
Let $x,y$ be regular parameters in $R$, with associated generating sequence to $\nu$, $P_0=P_0(\nu,R)=x,P_1=P_1(\nu,R)=y,P_2=P_2(\nu,R),\ldots$ in $R$ as constructed in Theorem \ref{Theorem1*}, with $U_i=U_i(\nu,R)$, $\beta_i=\beta_i(\nu,R)=\nu(P_i)$, $\gamma_i=\alpha_i(\nu,R)$, $m_i=m_i(\nu,R)$, $\overline m_i=\overline m_i(\nu,R)$, $d_i=d_i(\nu,R)$ and $\sigma_i=\sigma_i(\nu,R)$ defined as in Section \ref{SecGen}.
Let $u,v$ be regular parameters in $S$, with associated generating sequence to $\nu^*$, $Q_0=P_0(\nu^*,S)=u,Q_1=P_1(\nu^*,S)=v,Q_2=P_2(\nu^*,S),\ldots$ in $S$ as constructed in Theorem \ref{Theorem1*}, with $V_i=U_i(\nu^*,S)$, $\gamma_i=\beta_i(\nu^*,S)=\nu^*(Q_i)$, $\delta_i=\alpha_i(\nu^*,S)$, $n_i=n_i(\nu^*,S)$, $\overline n_i=\overline n_i(\nu^*,S)$, $e_i=\alpha_i(\nu^*,S)$ and $\tau_i=\sigma_i(\nu^*,S)$ defined as in Section \ref{SecGen}.
With our assumption that ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, we have that for all sufficiently large $l$,
\begin{equation}\label{eq6}
{\rm gr}_{\nu^*}(S)={\rm gr}_{\nu}(R)[{\rm in}_{\nu^*}Q_{\tau_0},\ldots,{\rm in}_{\nu^*}Q_{\tau_l}].
\end{equation}
\begin{Proposition}\label{Prop3} With our assumption that ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, there exist integers $s>1$ and $r>1$ such that for all $j\ge 0$,
$$
\beta_{\sigma_{r+j}}=\gamma_{\tau_{s+j}},
\overline m_{\sigma_{r+j}}=\overline n_{\tau_{s+j}},
d_{\sigma_{r+j}}=e_{\tau_{s+j}},
m_{\sigma_{r+j}}=n_{\tau_{s+j}},
$$
$$
G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+j}})\subset G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+j}}),
$$
$$
[G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+j}}):G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+j}})]=e(\nu^*/\nu),
$$
$$
R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_{r+j}}]\subset S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_{s+j}}]
$$
and
$$
[S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_{s+j}}]:R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_{r+j}}]]=f(\nu^*/\nu).
$$
\end{Proposition}
\begin{proof} Let $l$ be as in (\ref{eq6}). For $s\ge l$, define the sub algebra $A_{\tau_s}$ of ${\rm gr}_{\nu^*}(S)$ by
$$
A_{\tau_s}=S/m_S[{\rm in}_{\nu^*}Q_{\tau_0},\ldots,{\rm in}_{\nu^*}Q_{\tau_s}].
$$
For $s\ge l$, let
$$
r_s=\max\{j\mid {\rm in}_{\nu^*}P_{\sigma_j}\in A_{\tau_s}\},
$$
$$
\lambda_s=[G(\gamma_{\tau_0},\ldots,\gamma_{\tau_s}):G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r_s}})],
$$
and
$$
\chi_s=[S/m_S[\epsilon_{\tau_0},\ldots,\epsilon_{\tau_s}]:R/m_R[\delta_{\sigma_0},\ldots,\delta_{\sigma_{r_s}}]].
$$
To simplify notation, we will write $r=r_s$.
We will now show that $\beta_{\sigma_{r+1}}=\gamma_{\tau_{s+1}}$.
Suppose that $\beta_{\sigma_{r+1}}>\gamma_{\tau_{s+1}}$. We have that
$$
{\rm in}_{\nu^*}Q_{\tau_{s+1}}\in {\rm gr}_{\nu}(R)[{\rm in}_{\nu^*}Q_{\tau_0},\ldots,{\rm in}_{\nu^*}Q_{\tau_s}].
$$
Since
$$
\beta_{\sigma_{r+1}}<\beta_{\sigma_{r+2}}<\cdots
$$
we then have that
${\rm in}_{\nu^*}Q_{\tau_{s+1}}\in A_{\tau_s}$ which is impossible. Thus $\beta_{\sigma_{r+1}}\le \gamma_{\tau_{s+1}}$.
If $\beta_{\sigma_{r+1}}<\gamma_{\tau_{s+1}}$, then since
$$
\gamma_{\tau_{s+1}}<\gamma_{\tau_{s+2}}<\cdots
$$
and ${\rm in}_{\nu^*}P_{\sigma_{r+1}}\in {\rm gr}_{\nu^*}(S)$, we have that ${\rm in}_{\nu^*}P_{\sigma_{r+1}}\in A_{\tau_s}$, which is impossible. Thus
$\beta_{\sigma_{r+1}}=\gamma_{\tau_{s+1}}$.
We will now establish that either we have a reduction $\lambda_{s+1}<\lambda_s$ or
\begin{equation}\label{eq22}
\lambda_{s+1}=\lambda_s,\,\, \beta_{\sigma_{r+1}}=\gamma_{\tau_{s+1}}\mbox{ and } \overline m_{\sigma_{r+1}}=\overline n_{\tau_{s+1}}.
\end{equation}
Let $\omega$ be a generator of the group $G(\gamma_{\tau_1},\ldots,\gamma_{\tau_s})$, so that $G(\gamma_{\tau_1},\ldots,\gamma_{\tau_s})={\NZQ Z}\omega$. We have that
$$
G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+1}})=\frac{1}{\overline n_{\tau_{s+1}}}{\NZQ Z}\omega
$$
and
$$
G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+1}})=\frac{1}{\overline m_{\sigma_{r+1}}}{\NZQ Z}(\lambda_s\omega).
$$
There exists a positive integer $f$ with $\mbox{gcd}(f,\overline n_{\tau_{s+1}})=1$ such that
$$
\gamma_{\tau_{s+1}}=\frac{f}{\overline n_{\tau_{s+1}}}\omega
$$
There exists a positive integer $g$ with $\mbox{gcd}(g,\overline m_{\sigma_{r+1}})=1$ such that
$$
\beta_{\sigma_{r+1}}=\frac{g}{\overline m_{\sigma_{r+1}}}\lambda_s\omega.
$$
Since $\beta_{\sigma_{r+1}}=\gamma_{\tau_{s+1}}$, we have
$$
g\lambda_s \overline n_{\tau_{s+1}}=f \overline m_{\sigma_{r+1}}.
$$
Thus $\overline n_{\tau_{s+1}}$ divides $\overline m_{\sigma_{r+1}}$ and $\overline m_{\sigma_{r+1}}$ divides $\lambda_s \overline n_{\tau_{s+1}}$, so that
$$
a=\frac{\overline m_{\sigma_{r+1}}}{\overline n_{\tau_{s+1}}}
$$
is a positive integer and defining
$$
\overline\lambda=\frac{\lambda_s}{a},
$$
we have that $\overline\lambda$ is a positive integer with
$$
\frac{\lambda_s}{\overline m_{\sigma_{r+1}}}=\frac{\overline \lambda}{\overline n_{\tau_{s+1}}}
$$
and
$$
\overline\lambda =[G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+1}}):G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+1}})].
$$
Since $\lambda_{s+1}\le \overline\lambda$, either $\lambda_{s+1}<\lambda_s$ or $\lambda_{s+1}=\lambda_s$ and $\overline m_{\sigma_{r+1}}=\overline n_{\tau_{s+1}}$.
We will now suppose that $s$ is sufficiently large that (\ref{eq22}) holds. Since
$$
{\rm in}_{\nu^*}Q_{\tau_{s+1}}\in {\rm gr}_{\nu^*}(S)={\rm gr}_{\nu}(R)[{\rm in}_{\nu^*}Q_{\tau_0},\ldots,{\rm in}_{\nu^*}Q_{\tau_s}],
$$
if $\overline n_{\tau_{s+1}}>1$ we have an expression
\begin{equation}\label{eqN2}
{\rm in}_{\nu^*}P_{\sigma_{r+1}}={\rm in}_{\nu^*}(\alpha){\rm in}_{\nu^*}Q_{\tau_{s+1}}
\end{equation}
in $\mathcal P_{\gamma_{\tau_{s+1}}}(S)/\mathcal P^+_{\gamma_{\tau_{s+1}}}(S)$ with $\alpha$ a unit in $S$ and if $\overline n_{\tau_{s+1}}=1$,
since ${\rm in}_{\nu^*}P_{\sigma_{r+1}}\not\in A_{\tau_s}$, we have an expression
\begin{equation}\label{eqN3}
{\rm in}_{\nu^*}P_{\sigma_{r+1}}={\rm in}_{\nu^*}(\alpha){\rm in}_{\nu^*}Q_{\tau_{s+1}}
+\sum {\rm in}_{\nu^*}(\alpha_J)({\rm in}_{\nu^*}Q_{\tau_0})^{j_0}\cdots ({\rm in}_{\nu^*}Q_{\tau_s})^{j_s}
\end{equation}
in $\mathcal P_{\gamma_{\tau_{s+1}}}(S)/\mathcal P^+_{\gamma_{\tau_{s+1}}}(S)$ with $\alpha$ a unit in $S$ and the sum is over certain $J=(j_i,\ldots,j_s)\in {\NZQ N}^s$ such that the $\alpha_J$ are units in $S$, and the terms
${\rm in}_{\nu^*}Q_{\tau_{s+1}}$ and the $({\rm in}_{\nu^*}Q_{\tau_0})^{j_0}\cdots ({\rm in}_{\nu^*}Q_{\tau_s})^{j_s} $ are linearly independent over $S/m_S$.
The monomial $U_{\sigma_{r+1}}$ in $P_{\sigma_0},\ldots, P_{\sigma_r}$ and the monomial $V_{\tau_{s+1}}$ in $Q_{\tau_0},\ldots, Q_{\tau_s}$ both have the value
$\overline n_{\tau_{s+1}}\gamma_{\tau_{s+1}}=\overline m_{\sigma_{r+1}}\beta_{\sigma_{r+1}}$, and satisfy
$$
\epsilon_{\tau_{s+1}}=\left[\frac{Q_{\tau_{s+1}}^{\overline n_{\tau_{s+1}}}}{V_{\tau_{s+1}}}\right]
$$
and
$$
\delta_{\sigma_{r+1}}=\left[\frac{P_{\sigma_{r+1}}^{\overline n_{\tau_{s+1}}}}{U_{\sigma_{r+1}}}\right].
$$
Since $U_{\sigma_{r+1}}, V_{\tau_{s+1}}\in A_{\tau_s}$ and by (\ref{eqZ51}) and 2) of Theorem \ref{Theorem1*},
we have that
$$
\left[\frac{V_{\tau_{s+1}}}{U_{\sigma_{r+1}}}\right]\in S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}].
$$
If $\overline n_{\tau_{s+1}}>1$, then by (\ref{eqN2}), we have
$$
\left[\frac{P_{\sigma_{r+1}}^{\overline n_{\tau_{s+1}}}}{U_{\sigma_{r+1}}}\right]
=\left[\frac{V_{\tau_{s+1}}}{U_{\sigma_{r+1}}}\right]
\left(\left[\alpha\right]^{\overline n_{\tau_{s+1}}}\left[\frac{Q_{\tau_{s+1}}^{\overline n_{\tau_{s+1}}}}{V_{\tau_{s+1}}}\right]\right)
$$
in $L=V_{\nu^*}/m_{\nu^*}$, and if $\overline n_{\tau_{s+1}}=1$, then by (\ref{eqN3}), we have
$$
\left[\frac{P_{\sigma_{r+1}}}{U_{\sigma_{r+1}}}\right]
=\left[\frac{V_{\tau_{s+1}}}{U_{\sigma_{r+1}}}\right]
\left(\left[\alpha\right]\left[\frac{Q_{\tau_{s+1}}}{V_{\tau_{s+1}}}\right]
+\sum\left[\alpha_J\right]\left[\frac{Q_{\tau_0}^{j_0}\cdots Q_{\tau_s}^{j_s}}{V_{\tau_{s+1}}}
\right] \right) .
$$
Thus by equation (\ref{eqZ51}),
\begin{equation}\label{eqN4}
S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}][\epsilon_{\tau_{s+1}}]=S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}][\delta_{\sigma_{r+1}}].
\end{equation}
We have a commutative diagram
$$
\begin{array}{ccccc}
S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}]&\rightarrow& S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s},\epsilon_{\tau_{s+1}}]&=& S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}][\delta_{\sigma_{r+1}}]\\
\uparrow&&\uparrow\\
R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_r}]&\rightarrow&R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_r}][\delta_{\sigma_{r+1}}].
\end{array}
$$
Let
$$
\overline\chi=[S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s},\epsilon_{\tau_{s+1}}]:R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_r},\delta_{\sigma_{r+1}}]].
$$
Since
$$
S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s},\epsilon_{\tau_{s+1}}]= S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}][\delta_{\sigma_{r+1}}],
$$
we have that $e_{\tau_{s+1}}|d_{\sigma_{r+1}}$. Further,
$$
\frac{d_{\sigma_{r+1}}}{e_{\tau_{s+1}}}\overline\chi=\chi_s,
$$
whence $\overline\chi\le\chi_s$. Thus $\chi_{s+1}\le\chi_s$ and if $\chi_{s+1}=\chi_s$, then $d_{\sigma_{r+1}}=e_{\tau_{s+1}}$ and $r_{s+1}=r_s+1$ since $P_{\sigma_{r+2}}\in A_{\tau_{s+1}}$ implies $\lambda_{s+1}<\lambda_s$ or $\chi_{s+1}<\chi_s$.
We may thus choose $s$ sufficiently large that there exists an integer $r>1$ such that for all $j\ge 0$,
$$
\beta_{\sigma_{r+j}}=\gamma_{\tau_{s+j}},
\overline m_{\sigma_{r+j}}=\overline n_{\tau_{s+j}},
d_{\sigma_{r+j}}=e_{\tau_{s+j}},
m_{\sigma_{r+j}}=n_{\tau_{s+j}},
$$
$$
G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+j}})\subset G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+j}}),
$$
there is a constant $\lambda$ (which does not depend on $j$) such that
$$
[G(\gamma_{\tau_0},\ldots,\gamma_{\tau_{s+j}}):G(\beta_{\sigma_0},\ldots,\beta_{\sigma_{r+j}})]=\lambda
$$
$$
R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_{r+j}}]\subset S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_{s+j}}]
$$
and there is a constant $\chi$ (which does not depend on $j$) such that
$$
[S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_{s+j}}]:R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_{r+j}}]]=\chi.
$$
Then
$$
\Phi_{\nu^*}=\cup_{j\ge 1}\frac{1}{\overline n_{\tau_{s+1}}\cdots\overline n_{\tau_{s+j}}}{\NZQ Z}\omega
$$
where $G(\gamma_{\tau_0},\ldots,\gamma_{\tau_s})={\NZQ Z}\omega$,
and
$$
\Phi_{\nu}=\cup_{j\ge 1}\frac{1}{\overline m_{\sigma_{r+1}}\cdots\overline m_{\sigma_{r+j}}}\lambda{\NZQ Z}\omega= \cup_{j\ge 1}\frac{1}{\overline n_{\tau_{s+1}}\cdots\overline n_{\tau_{s+j}}}\lambda{\NZQ Z}\omega
$$
so that
$$
\lambda=[\Phi_{\nu^*}:\Phi_{\nu}]=e(\nu^*/\nu).
$$
For $i\ge 0$, let $K_i=R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_{r+i}}]$ and $M_i=S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_{s+i}}]$.
We have that $M_{i+1}=M_i[\delta_{\sigma_{r+i+1}}]$ for $i\ge 0$ and $\chi=[M_i:K_i]$ for all $i$. Further,
$$
\cup_{i=0}^{\infty}M_i=V_{\nu^*}/m_{\nu^*}\mbox{ and }\cup_{i=0}^{\infty}K_i=V_{\nu}/m_{\nu}.
$$
Thus if $g_1,\ldots,g_{\lambda}\in M_0$ form a basis of $M_0$ as a $K_0$-vector space, then $g_1,\ldots,g_{\lambda}$ form a basis of $M_i$ as a $K_i$-vector space for all $i\ge 0$. Thus
$$
\chi=[V_{\nu^*}/m_{\nu^*}:V_{\nu}/m_{\nu}]=f(\nu^*/\nu).
$$
\end{proof}
Let $r$ and $s$ be as in the conclusions of Proposition \ref{Prop3}.
There exists $\tau_t$ with $t\ge s$ such that we have a commutative diagram of inclusions of regular local rings (with the notation introduced in Section \ref{SecGen})
$$
\begin{array}{ccc}
R_{\sigma_r}&\rightarrow &S_{\tau_t}\\
\uparrow&&\uparrow\\
R&\rightarrow&S.
\end{array}
$$
After possibly increasing $s$ and $r$, we may assume that $R'\subset R_{\sigma_r}$, where $R'$ is the local ring of the conclusions of Proposition \ref{Prop15}.
Recall that $R$ has regular parameters $x=P_0$, $y=P_1$ and $S$ has regular parameters $u=Q_0$, $v=Q_1$, $R_{\sigma_r}$ has regular parameters $x_{\sigma_r}$, $y_{\sigma_r}$ such that
$$
x=\delta x_{\sigma_r}^{\overline m_{\sigma_1}\cdots \overline m_{\sigma_r}},\,\, y_{\sigma_r}={\rm st}_{R_{\sigma_r}}P_{\sigma_{r+1}}
$$
where $\delta$ is a unit in $R_{\sigma_r}$ and
$S_{\tau_t}$ has regular parameters $u_{\tau_t}$, $v_{\tau_t}$ such that
$$
u=\epsilon u_{\tau_t}^{\overline n_{\tau_1}\cdots \overline n_{\tau_t}},\,\,v_{\tau_t}={\rm st}_{S_{\tau_t}}Q_{\tau_{t+1}}
$$
where $\epsilon$ is a unit in $S_{\tau_t}$. We may choose $t\gg 0$ so that we we have an expression
\begin{equation}\label{eq7}
x_{\sigma_r}=\phi u_{\tau_t}^{\lambda}
\end{equation}
for some positive integer $\lambda$ where $\phi$ is a unit in $S_{\tau_t}$, since $\cup_{t=0}^{\infty}S_{\tau_t}=V_{\nu^*}$.
We have expressions $P_i=\psi_i x_{\sigma_r}^{c_i}$ in $R_{\sigma_r}$ where $\psi_i$ are units in $R_{\sigma_r}$ for $i\le \sigma_r$ so that
$P_i=\psi_i^* u_{\tau_t}^{c_i\lambda}$ in $S_{\tau_t}$ where $\psi_i^*$ are units in $S_{\tau_t}$ for $i\le \sigma_r$ by (\ref{eq7}).
\begin{Lemma}\label{Lemma3} For $j\ge 1$ we have
$$
{\rm st}_{R_{\sigma_r}}(P_{\sigma_{r+j}})=u_{\tau_t}^{\lambda_j}{\rm st}_{S_{\tau_t}}(P_{\sigma_{r+j}})
$$
for some $\lambda_j\in {\NZQ N}$, where we regard $P_{\sigma_{r+j}}$ as an element of $R$ on the left hand side of the equation and regard $P_{\sigma_{r+j}}$ as an element of $S$ on the right hand side.
\end{Lemma}
\begin{proof} Using (\ref{eq7}), we have
$$
P_{\sigma_{r+j}}={\rm st}_{R_{\sigma_r}}(P_{\sigma_{r+j}})x_{\sigma_r}^{f_j}
={\rm st}_{R_{\sigma_r}}(P_{\sigma_{r+j}})u_{\tau_t}^{\lambda f_j}\phi^{f_j}
$$
where $f_j\in {\NZQ N}$.
Viewing $P_{\sigma_{r+j}}$ as an element of $S$, we have that
$$
P_{\sigma_{r+j}}={\rm st}_{S_{\tau_t}}(P_{\sigma_{r+j}})u_{\tau_t}^{g_j}
$$
for some $g_j\in {\NZQ N}$. Since $u_{\tau_t}\not\,\mid {\rm st}_{S_{\tau_t}}(P_{\sigma_{r+j}})$, we have that $f_j\lambda \le g_j$ and so $\lambda_j=g_j-f_j\lambda\ge 0$.
\end{proof}
By induction in the sequence of quadratic transforms above $R$ and $S$ in Lemma \ref{Lemma2}, and since $\nu^*(P_{\sigma_{r+j}})=\beta_{\sigma_{r+j}}=\gamma_{\tau_{s+j}}$ by Proposition \ref{Prop3}, we have by (\ref{eqN2}) and (\ref{eqN3}) an expression
\begin{equation}\label{eqN60}
{\rm st}_{S_{\tau_t}}(P_{\sigma_{r+j}})=c{\rm st}_{S_{\tau_t}}(Q_{\tau_{s+j}})+u_{\tau_t}\Omega
\end{equation}
with $c\in S_{\tau_t}$ a unit, $\Omega\in S_{\tau_t}$ and $\nu^*(u_{\tau_t}\Omega)\ge\nu^*({\rm st}_{S_{\tau_t}}(Q_{\tau_{s+j}}))$ if $s+j>t$ and
\begin{equation}\label{eqN61}
S_{\tau_t}(P_{\sigma_{r+j}})\mbox{ is a unit in $S_{\tau_t}$}
\end{equation}
if $s+j\le t$.
Thus
$P_{\sigma_{r+j}}=u_{\tau_t}^{d_j}\overline\phi_j$ in $S_{\tau_t}$ where $d_j$ is a positive integer and $\overline\phi_j$ is a unit in $S_{\tau_t}$ if $s+j\le t$.
Suppose $s<t$. Then
$$
y_{\sigma_r}={\rm st}_{R_{\sigma_r}}(P_{\sigma_{r+1}})=\tilde\phi u_{\tau_t}^{h}
$$
where $\tilde\phi$ is a unit in $S_{\tau_{t}}$ and $h$ is a positive integer.
As shown in equation (\ref{eq8}) of Section \ref{SecGen},
$$
R_{\sigma_{r+1}}=R_{\sigma_r}[\overline x_1,\overline y_1]_{m_{\nu}\cap R_{\sigma_r}[\overline x_1,\overline y_1]}
$$
where
$$
\overline x_1=(x_{\sigma_r}^{b}y_{\sigma_r}^{-a})^{\epsilon},\,\,\, \overline y_1=(x_{\sigma_r}^{-\omega}y_{\sigma_r}^{m_{\sigma_r}})^{\epsilon}
$$
with $\epsilon=m_{\sigma_r}b-\omega a=\pm 1$,
$\nu(\overline x_1)>0$ and $\nu(\overline y_1)=0$.
Substituting
$$
x_{\sigma_r}=\phi u_{\tau_t}^{\lambda}\mbox{ and }y_{\sigma_1}=\tilde\phi u_{\tau_t}^h
$$
we see that
$R_{\sigma_{r+1}}$ is dominated by $S_{\tau_t}$.
We thus have a factorization
$$
R_{\sigma_r}\rightarrow R_{\sigma_{r+1}}\rightarrow S_{\tau_t}
$$
with $x_{\sigma_{r+1}}=\overline x_1=\hat\phi u_{\tau_t}^{\lambda'}$ where $\hat\phi$ is a unit in $S_{\tau_t}$ and $\lambda'$ is a positive integer. We may thus replace $s$ with $s+1$, $r$ with $r+1$ and $R_{\sigma_r}$ with $R_{\sigma_{r+1}}$.
Iterating this argument, we may assume that $s=t$ (with $r=r_s$) so that by Lemma \ref{Lemma3}, (\ref{eqN61}) and (\ref{eqN60}),
$$
y_{\sigma_r}={\rm st}_{R_{\sigma_r}}(P_{\sigma_{r+1}})=u_{\tau_s}^{\mu}{\rm st}_{S_{\tau_s}}(P_{\sigma_{r+1}})
$$
where
$$
{\rm st}_{S_{\tau_s}}(P_{\sigma_{r+1}})=c\,{\rm st}_{S_{\tau_s}}(Q_{\tau_{s+1}})+u_{\tau_s}\Omega
$$
with $c$ a unit in $S_{\tau_s}$ and $\Omega\in S_{\tau_s}$. Thus by (\ref{eq7}), we have an expression
$$
x_{\sigma_r}= \phi u_{\tau_s}^{\lambda},\,\, y_{\sigma_r}=\overline\epsilon u_{\tau_s}^{\alpha}(v_{\tau_s}+u_{\tau_s}\Omega)
$$
where $\lambda$ is a positive integer, $\alpha\in {\NZQ N}$, $\phi$ and $\overline\epsilon$ are units in $S_{\tau_s}$ and $\Omega\in S_{\tau_s}$.
We have that $\nu^*(x_{\sigma_r})=\lambda\nu^*(u_{\tau_s})$,
$$
\begin{array}{lll}
\nu(x_{\sigma_r}){\NZQ Z}&=&G(\nu(x_{\sigma_r}))=G(\beta_{\sigma_0},\ldots,\beta_{\sigma_r})\mbox{ and }\\
\nu^*(u_{\tau_s}){\NZQ Z}&=&G(\nu^*(u_{\tau_s}))=G(\gamma_{\tau_0},\ldots,\gamma_{\tau_s}).
\end{array}
$$
Thus
$$
\lambda=[G(\gamma_{\tau_0},\ldots,\gamma_{\tau_s}):G(\beta_{\sigma_0},\ldots,\beta_{\sigma_r})]=e(\nu^*/\nu)
$$
by Proposition \ref{Prop3}.
By Theorem \ref{birat}, we have that
$$
R_{\sigma_r}/m_{R_{\sigma_r}}=R/m_R[\delta_{\sigma_1},\ldots,\delta_{\sigma_r}]\mbox{ and }S_{\tau_s}/m_{S_{\tau_s}}=S/m_S[\epsilon_{\tau_1},\ldots,\epsilon_{\tau_s}].
$$
Thus
$$
[S_{\tau_s}/m_{S_{\tau_s}}:R_{\sigma_r}/m_{R_{\sigma_r}}]=f(\nu^*/\nu)
$$
by Proposition \ref{Prop3}.
Since the ring $R'$ of Proposition \ref{Prop15} is contained in $R_{\sigma_r}$ by our construction, we have by Proposition \ref{Prop15} that $(K,\nu)\rightarrow (K^*,\nu^*)$ is without defect, completing the proofs of Proposition \ref{Prop1} and Theorem \ref{Theorem4}.
\section{non splitting and finite generation}
In this section, we will have the following assumptions. Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$. Further suppose that $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field
$K^*$ such that $S$ dominates $R$.
Suppose that $\nu^*$ is a valuation of $K^*$ such that
$\nu^*$ dominates $S$. Let $\nu$ be the restriction of $\nu^*$ to $K$.
Suppose that $\nu^*$ has rational rank 1 and $\nu^*$ is not discrete. Then $V_{\nu^*}/m_{\nu^*}$ is algebraic over $S/m_S$, by Abhyankar's inequality, Proposition 2 \cite{Ab1}.
\begin{Lemma} \label{LemmaN1} Let assumptions be as above. Then the associated graded ring ${\rm gr}_{\nu^*}(S)$ is an integral extension of ${\rm gr}_{\nu}(R)$.
\end{Lemma}
\begin{proof} It suffices to show that ${\rm in}_{\nu^*}(f)$ is integral over ${\rm gr}_{\nu}(R)$ whenever $f\in S$. Suppose that
$f\in S$. There exists $n_1>0$ such that $n_1\nu^*(f)\in \Phi_{\nu}$. Let $x\in m_R$ and $\omega=\nu(x)$. Then there exists a positive integer $b$ and natural number $a$ such that $bn_1\nu^*(f)=a\omega$, so
$$
\nu^*\left(\frac{f^{bn_1}}{x^a}\right)=0.
$$
Let
$$
\xi=\left[\frac{f^{bn_1}}{x^a}\right]\in V_{\nu^*}/m_{\nu^*},
$$
and let $g(t)=t^r+\overline a_{r-1}t^{r-1}+\cdots+\overline a_0$ with $\overline a_i\in R/m_R$ be the minimal polynomial of $\xi$ over $R/m_R$. Let $a_i$ be lifts of the $\overline a_i$ to $R$.
Then
$$
\nu^*(f^{b_1n_1r}+a_{r-1}x^af^{bn_1(r-1)}+\cdots+a_0x^{ar})>\nu^*(f^{bn_1r})=\nu^*(a_{r-1}x^af^{bn_1(r-1)})=\cdots=\nu^*(a_0x^{ar}).
$$
Thus
$$
{\rm in}_{\nu^*}(f)^{b_1n_1r}+{\rm in}_{\nu}(a_{r-1}x^a){\rm in }_{\nu^*}(f)^{bn_1(r-1)}+\cdots+{\rm in}_{\nu}(a_0x^{ar})=0
$$
in ${\rm gr}_{\nu^*}(S)$. Thus ${\rm in}_{\nu^*}(f)$ is integral over ${\rm gr}_{\nu^*}(R)$.
\end{proof}
We now establish Theorem \ref{ThmN2}. Recall (as defined after Proposition \ref{Prop1}) that $\nu^*$ does not split in $S$ if $\nu^*$ is the unique extension of $\nu$ to $K^*$ which dominates $S$.
\begin{Theorem} Let assumptions be as above and suppose that $R$ and $S$ are regular local rings. Suppose that ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra. Then $S$ is a localization of the integral closure of $R$ in $K^*$, the defect $\delta(\nu^*/\nu)=0$ and $\nu^*$ does not split in $S$. \end{Theorem}
\begin{proof} Let $s$ and $r$ be as in the conclusions of Proposition \ref{Prop3}. We will first show that $P_{\sigma_{r+j}}$ is irreducible in $\hat S$ for all $j>0$. There exists a unique extension of $\nu^*$ to the quotient field of $\hat S$ which dominates $\hat S$ (\cite{Sp}, \cite{CV1}, \cite{GAST}). The extension is immediate since $\nu^*$ is not discrete; that is, there is no increase in value group or residue field for the extended valuation. It has the property that if $f\in \hat S$ and $\{f_i\}$ is a a Cauchy sequence in $\hat S$ which converges to $f$, then $\nu^*(f)=\nu^*(f_i)$ for all $i\gg 0$.
Suppose that $P_{\sigma_{r+j}}$ is not irreducible in $\hat S$ for some $j>0$. We will derive a contradiction. With this assumption, $P_{\sigma_{r+j}}=fg$ with $f,g\in m_{\hat S}$. Let $\{f_i\}$ be a Cauchy sequence in $S$ which converges to $f$ and let $\{g_i\}$ be a Cauchy sequence in $S$ which converges to $g$. For $i$ sufficiently large, $f-f_i,g-g_i\in m_{\hat S}^n$ where $n$ is so large that $n\nu^*(m_{\hat S})=n\nu^*(m_S)>\nu(P_{\sigma_{r+j}})$.
Thus $P_{\sigma_{r+j}}=f_ig_i+h$ with $h\in m_{\hat S}^n\cap S=m_S^n$, and so ${\rm in}_{\nu^*}(P_{\sigma_{r+j}})={\rm in}_{\nu^*}(f_i){\rm in}_{\nu^*}(g_i)$. Now
$$
\nu^*(f_i),\nu^*(g_i)<\nu(P_{\sigma_{r+j}})=\beta_{\sigma_{r+j}}=\gamma_{\tau_{s+j}}=\nu^*(Q_{\tau_{s+j}})
$$
so that
$$
{\rm in}_{\nu^*}(f_i),{\rm in}_{\nu^*}(g_i)\in S/m_S[{\rm in }_{\nu^*}(Q_{\tau_0}),\ldots, {\rm in}_{\nu^*}(Q_{\tau_{s+j-1}})]
$$
which implies
$$
{\rm in}_{\nu^*}(P_{\sigma_{r+j}})\in S/m_S[{\rm in }_{\nu^*}(Q_{\tau_0}),\ldots, {\rm in}_{\nu^*}(Q_{\tau_{s+j-1}})].
$$
But then (\ref{eqN3}) implies
$$
{\rm in}_{\nu^*}(Q_{\tau_{s+j}})\in S/m_S[{\rm in }_{\nu^*}(Q_{\tau_0}),\ldots, {\rm in}_{\nu^*}(Q_{\tau_{s+j-1}})]
$$
which is impossible. Thus $P_{\sigma_{r+j}}$ is irreducible in $\hat S$ for all $j>0$.
If $S$ is not a localization of the integral closure of $R$ in $K^*$, then by Zariski's Main Theorem (Theorem 1 of Chapter 4 \cite{R}), $m_RS=fN$ where $f\in m_S$ and $N$ is an $m_S$-primary ideal. Thus $f$ divides $P_i$ in $S$ for all $i$, which is impossible since we have shown that $P_{\sigma_{r+j}}$ is analytically irreducible in $S$ for all $j>0$; we cannot have $P_{\sigma_{r+j}}=a_jf$ where $a_j$ is a unit in $S$ for $j>0$ since
$\nu(P_{\sigma_{r+j}})=\nu^*(Q_{\tau_{s+j}})$ by Proposition \ref{Prop3}.
Now suppose that $\nu^*$ is not the unique extension of $\nu$ to $K^*$ which dominates $S$. Recall that $V_{\nu}$ is the union of all quadratic transforms above $R$ along $\nu$ and $V_{\nu^*}$ is the union of all quadratic transforms above $S$ along $\nu^*$ (Lemma 4.5 \cite{RTM}).
Then for all $i\gg 0$, we have a commutative diagram
$$
\begin{array}{lll}
R_{\sigma_i}&\rightarrow& T_i\\
\uparrow&&\uparrow\\
R&\rightarrow &T
\end{array}
$$
where $T$ is the integral closure of $R$ in $K^*$, $T_i$ is the integral closure of $R_{\sigma_i}$ in $K^*$, $S=T_{\mathfrak p}$ for some maximal ideal $\mathfrak p$ in $T$ which lies over $m_R$,
and there exist $r\ge 2$ prime ideals $\mathfrak p_1(i),\ldots,\mathfrak p_r(i)$ in $T_i$ which lie over $m_{R_{\sigma_i}}$ and whose intersection with $T$ is $\mathfrak p$. We may assume that $\mathfrak p_1(i)$ is the center of $\nu^*$.
There exists an $m_R$-primary ideal $I_i$ in $R$ such that the blow up of $I_i$ is $\gamma:X_{\sigma_i}\rightarrow \mbox{Spec}(R)$ where $X_{\sigma_i}$ is regular and $R_{\sigma_i}$ is a local ring of $X_{\sigma_i}$. Let $Z_{\sigma_i}$ be the integral closure of $X_{\sigma_i}$ in $K^*$. Let $Y_{\sigma_i}=Z_{\sigma_i}\times_{\mbox{Spec}(T)}\mbox{Spec}(S)$.
We have a commutative diagram of morphisms
$$
\begin{array}{lll}
Y_{\sigma_i}&\stackrel{\beta}{\rightarrow}&X_{\sigma_i}\\
\delta\downarrow&&\gamma\downarrow\\
\mbox{Spec}(S)&\stackrel{\alpha}{\rightarrow}&\mbox{Spec}(R)
\end{array}
$$
The morphism $\delta$ is projective (by Proposition II.5.5.5 \cite{EGAII} and Corollary II.6.1.11 \cite{EGAII} and it is birational, so since $Y_{\sigma_i}$ and $\mbox{Spec}(S)$ are integral, it is a blow up of an ideal $J_i$ in $S$ (Proposition III.2.3.5 \cite{EGAIII}), which we can take to be $m_S$-primary since $S$ is a regular local ring and hence factorial. Define curves $C=\mbox{Spec}(R/(P_{\sigma_i}))$ and $C'=\alpha^{-1}(C)=\mbox{Spec}(S/(P_{\sigma_i}))$. Denote the Zariski closure of a set $W$ by $\overline W$. The strict transform $C^*$ of $C'$ in $Y_{\sigma_i}$ is the Zariski closure
\begin{equation}\label{eqN21}
\begin{array}{lll}
C^*&=&\overline{\delta^{-1}(C'\setminus m_S)}=\overline{\delta^{-1}\alpha^{-1}(C\setminus m_R)}
=\overline{\beta^{-1}\gamma^{-1}(C\setminus m_R)}\\
&=&\beta^{-1}(\overline{\gamma^{-1}(C\setminus m_R)})\mbox{ since $\beta$ is quasi finite}\\
&=& \beta^{-1}(\tilde C)
\end{array}
\end{equation}
where $\tilde C$ is the strict transform of $C$ in $X_{\sigma_i}$. We have that $Z_{\sigma_i}\times_{X_{\sigma_i}}\mbox{Spec}(R_{\sigma_i})\cong\mbox{Spec}(T_i)$, so
$$
Y_{\sigma_i}\times_{X_{\sigma_i}}\mbox{Spec}(R_{\sigma_i})\cong \mbox{Spec}(T_i\otimes_TS).
$$
Let $x_{\sigma_i}$ be a local equation in $R_{\sigma_i}$ of the exceptional divisor of $\mbox{Spec}(R_{\sigma_i})\rightarrow \mbox{Spec}(R)$ and let $y_{\sigma_i}=\mbox{st}_{R_{\sigma_i}}(P_{\sigma_i})$. Then
$x_{\sigma_i},y_{\sigma_i}$ are regular parameters in $R_{\sigma_i}$. We have that
$$
\sqrt{m_{R_{\sigma_i}}(T_i\otimes_TS)}=\cap_{j=1}^r\mathfrak p_j(i)(T_i\otimes_TS).
$$
The blow up of $J_i(S/(P_{\sigma_i}))$ in $C'$ is $\overline\delta:C^*\rightarrow C'$, where $\overline\delta$ is the restriction of $\delta$ to $C^*$ Corollary II.7.15 \cite{H}). Since $y_{\sigma_i}$ is a local equation of $\tilde C$ in $R_{\sigma_i}$, we have by (\ref{eqN21}) that
$$
\mathfrak p_1(i),\ldots,\mathfrak p_r(i)\in \overline\delta^{-1}(m_S)\subset C^*.
$$
Since $\overline\delta$ is proper and $C'$ is a curve, $C^*=\mbox{Spec}(A)$ for some excellent one dimensional domain $A$ such that the inclusion $S/(P_{\sigma_i})\rightarrow A$ is finite (Corollary I.1.10 \cite{Mi}).
Let $B=A\otimes_{S/(P_{\sigma_i})}\hat S/(P_{\sigma_i})$. Then
$$
C^*\times_{\mbox{Spec}(S/(P_{\sigma_i}))}\mbox{Spec}(\hat S/(P_{\sigma_i}))=\mbox{Spec}(B)\rightarrow \mbox{Spec}(\hat S/(P_{\sigma_i}))
$$
is the blow up of $J_i(\hat S/(P_{\sigma_i}))$ in $\hat S/(P_{\sigma_i})$. The extension $\hat S/(P_{\sigma_i})\rightarrow B$ is finite since $S/(P_{\sigma_i})\rightarrow A$ is finite.
Now assume that $S/(P_{\sigma_i})$ is analytically irreducible. Then $B$ has only one minimal prime since the blow up $\mbox{Spec}(B)\rightarrow \mbox{Spec}(\hat S/(P_{\sigma_i}))$ is birational.
Since a complete local ring is Henselian, $B$ is a local ring (Theorem I.4.2 on page 32 of \cite{Mi}), a contradiction to our assumption that $r>1$.
\end{proof}
As a consequence of the above theorem (Theorem \ref{ThmN2}), we now obtain Corollary \ref{CorN32}.
\begin{Corollary} Let assumptions be as above and suppose that $R$ is a regular local ring. Suppose that $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $\nu$. Then
$\mbox{gr}_{\nu}(R')$ is not a finitely generated $\mbox{gr}_{\nu}(R)$-algebra.
\end{Corollary}
The conclusions of Theorem \ref{ThmN2} do not hold if we remove the assumption that $\nu^*$ is not discrete, when $V_{\nu}/m_{\nu}$ is finite over $R/m_R$. We give a simple example. Let $k$ be an algebraically closed field of characteristic not equal to 2 and let $p(u)$ be a transcendental series in the power series ring $k[[u]]$ such that $p(0)=1$. Then $f=v-up(u)$ is irreducible in the power series ring $k[[u,v]]$ and $k[[u,v]]/(f)$ is a discrete valuation ring with regular parameter $u$. Let $\nu$ be the natural valuation of this ring.
Let $R=k[u,v]_{(u,v)}$ and $S=k[x,y]_{(x,y)}$. Define a $k$-algebra homomorphism $R\rightarrow S$ by $u\mapsto x^2$ and $v\mapsto y^2$. The series
$f(x^2,y^2)$ factors as $f=(y-x\sqrt{p(x^2)})(y+x\sqrt{p(x^2)})$ in $k[[x,y]]$. Let $f_1=y-x\sqrt{p(x^2)}$ and $f_2=y+x\sqrt{p(x^2)}$. The rings $k[[x,y]]/(f_i)$ are discrete valuation rings with regular parameter $x$.
Let $\nu_1$ and $\nu_2$ be the natural valuations of these ring.
Let $\nu$ be the valuation of the quotient field of $R$ which dominates $R$ defined by the natural inclusion $R\rightarrow k[[u,v]]/(f)$ and let $\nu_i$ for $i=1,2$ be the valuations of the quotient field of $S$ which dominate $S$ and are defined by the respective natural inclusions $S\rightarrow k[[x,y]]/(f_i)$ . Then $\nu_1$ and $\nu_2$ are distinct extensions of $\nu$ to the quotient field of $S$ which dominate $S$. However,
we have that
$ {\rm gr}_{\nu}(R)=k[{\rm in}_{\nu}(u)]$ and ${\rm gr}_{\nu_i}(S)=k[{\rm in}_{\nu^*}(x)]$ with ${\rm in}_{\nu^*}(x)^2={\rm in}_{\nu}(u)$. Thus ${\rm gr}_{\nu_i}(S)$ is a finite $ {\rm gr}_{\nu}(R)$-algebra.
We now give an example where $\nu^*$ has rational rank 2 and $\nu$ splits in $S$ but ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra. Suppose that $k$ is an algebraically closed field of characteristic not equal to 2.
Let $R=k[x,y]_{(x,y)}$ and $S=k[u,v]_{(u,v)}$. The substitutions $u=x^2$ and $v=y^2$ make $S$ into a finite separable extension of $R$.
Define a valuation $\nu_1$ of the quotient field $K^*$ of $S$ by $\nu_1(x)=1$ and $\nu_1(y-x)=\pi+1$ and define a valuation $\nu_2$ of the quotient field $K^*$ by $\nu_2(x)=1$ and $\nu_2(y+x)=\pi+1$.
Since $u=x^2$ and $v-u=(y-x)(y+x)$, we have that $\nu_1(u)=\nu_2(u)=2$ and $\nu_1(v-u)=\nu_2(v-u)=\pi+2$. Let $\nu$ be the common restriction of $\nu_1$ and $\nu_2$ to the quotient field $K$ of $R$.
Then $\nu$ splits in $S$. However, ${\rm gr}_{\nu_1}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra since
${\rm gr}_{\nu_1}(S)=k[{\rm in}_{\nu_1}(x),{\rm in }_{\nu_1}(y-x)]$ is a finitely generated $k$-algebra. Note that ${\rm gr}_{\nu}(R)=k[{\rm in}_{\nu}(u),{\rm in }_{\nu}(v-u)]$ with
${\rm in}_{\nu_1}(x)^2={\rm in}_{\nu}(u)$ and ${\rm in}_{\nu}(v-u)=2{\rm in}_{\nu_1}(y-x){\rm in}_{\nu_1}(x)$. |
2102.13377 | \section{Introduction}
The galaxy population in the local Universe is observed to be bimodal. This bimodality manifests in multiple properties such as colour, morphology, metallicity, light profile shape and environment (e.g. \citealt{Kauffmann03}; \citealt{Baldry04}; \citealt{Brinchmann04}). This bimodality is also found to extend to earlier epochs (see e.g., \citealt{Strateva01}; \citealt{Hogg02}; \citealt{Bell04}; \citealt{Driver06}; \citealt{Taylor09}; \citealt{Brammer09}; \citealt{Williams09}; \citealt{Brammer11}) across many measurable parameters, for example in colour (e.g. \citealt{Cirasuolo07}; \citealt{Cassata08}; \citealt{Taylor15}), morphological type (e.g. \citealt{Kelvin14}; \citealt{Whitaker15}; \citealt{Moffett16a}; \citealt{Krywult17}), size (e.g. \citealt{Lange15}), and specific star formation rate (e.g. \citealt{Whitaker14}; \citealt{Renzini15} and the references therein). The inference from these observations is that there are likely two evolutionary pathways regulated by mass and environment giving rise to this bimodality (\citealt{Driver06}; \citealt{Scarlata07b}; \citealt{DeLucia07}; \citealt{Peng10}; \citealt{Trayford16}). However, it is unclear as to whether studying the global properties of galaxies or their individual morphological components can better explain the origin of this bimodality \citep{Driver13} or whether more complex astrophysics is required.
In this study, we explore the origin of this bimodality by using the global properties of galaxies to study the evolution of their stellar mass function (SMF). The SMF, a statistical tool for measuring and constraining the evolution of the galaxy population, is defined as the number density of galaxies per logarithmic mass interval (\citealt{Schechter76}). The SMF of galaxies in the local Universe is now very well studied and found to be described by a two-component \cite{Schechter76} function with a characteristic cut-off mass of between $10^{10.6}$-$10^{11}\mathrm{M}_\odot$ and a steepening to lower masses (for example see: \citealt{Baldry08}; \citealt{Peng10}; \citealt{Baldry12}; \citealt{Kelvin14}; \citealt{Weigel16}; \citealt{Moffett16a}; \citealt{Wright17} ). Several studies have also investigated the evolution of the SMF at higher redshifts (\citealt{Pozzetti10}; \citealt{Muzzin13}; \citealt{Whitaker14}; \citealt{Leja15}; \citealt{Mortlock15}; \citealt{Wright18}; \citealt{Kawinwanichakij20}). A fingerprint of this bimodality is also observed in the double-component Schechter function required to fit the local SMF (e.g. \citealt{Baldry12}; \citealt{Wright18}). At least two distinct galaxy populations corresponding to \textit{star-forming} and \textit{passive} systems are thought to be the origin of this bimodal shape, with \textit{star-forming} systems dominating the low-mass tail and \textit{passive} galaxies dominating the high-mass ``hump'' of the SMF (\citealt{Baldry12}; \citealt{Muzzin13}; \citealt{Wright18}).
Many previous studies have separated galaxies into two main populations of star forming and passive (or a proxy thereof), and measured their individual stellar mass assemblies (e.g. \citealt{Pozzetti10}; \citealt{Tomczak14}; \citealt{Leja15}; \citealt{Davidzon17}). For example, by separating their sample into early- and late-type galaxies based on colour, \cite{Vergani08} studied the SMF of the samples and confirmed that $\sim 50\%$ of the red sequence galaxies were already formed by $z \sim 1$.
\cite{Pannella06} used a sample of $\sim 1600$ galaxies split into early- intermediate- and late-type galaxies classified according to their S\'ersic index. Similar work was carried out by \cite{Bundy05} and \cite{Ilbert10}. In the local Universe, some studies have classified small samples of galaxies and investigated the SMF of different morphological types (\citealt{Fukugita07} and \citealt{Bernardi10} using The Sloan Digital Sky Survey, SDSS). For example \cite{Bernardi10} reported that elliptical galaxies contain 25\% of the total stellar mass density and 20\% of the luminosity density of the local Universe.
Splitting galaxies into only two broad populations of star-forming and passive (or early- and late-type) galaxies might be inadequate to comprehensively study all aspects of galaxy evolution (e.g., \citealt{Siudek18}). Most previous studies at higher redshifts have separated galaxies into early- and late-type according to their colour distribution mainly due to the lack of spatial resolution as at high redshifts galaxies become unresolved in ground-based imaging. This has restricted true morphological comparisons in most studies to the local Universe (see e.g., \citealt{Fontana04}; \citealt{Baldry04}). A disadvantage of a simple colour-based classification is at higher redshifts, where for example regular disks occupied by old stellar populations are likely to be classified as early-type systems (see e.g., \citealt{Pannella06}). This is however unlikely to have an impact on the visual morphological classification, particularly when using high resolution imaging such as the Hubble Space Telescope (HST) data (\citealt{Huertas-Company15}). As such, we need better ways of quantifying the structure of galaxies throughout the history of the Universe.
Consequently, some studies have started to probe visually classified morphological types (e.g., \citealt{Sandage05}; \citealt{Fukugita07}; \citealt{Nair10}; \citealt{Lintott11} and the references therein). In effect, there are two morphologies one might track: the end-point morphology at $z \sim 0$ or the instantaneous morphology at the observed epoch. Observing galaxies at higher redshifts we find a different distribution of morphological types. For example, in earlier works using HST, \cite{vandenBergh96} and \cite{Abraham96} presented a morphological catalogue of galaxies in the Hubble Deep Field and found significantly more interacting, merger and asymmetric galaxies than in the nearby Universe.
By studying the SMF of various morphological types in the local Universe ($z < 0.06$), visually classified in the Galaxy and Mass Assembly survey (GAMA, \citealt{Driver11}), \cite{Kelvin14} and \cite{Moffett16a} found that the local stellar mass density is dominated by spheroidal galaxies, defined as E/S0/Sa. They reported the contribution of spheroid- (E/S0ab) and disk-dominated (Sbcd/Irr) galaxies of approximately 70\% and 30\%, respectively, towards the total stellar mass budget of the local Universe. These studies, however, are limited to the very local Universe. For a better understanding of the galaxy formation and evolution processes at play in the evolution of different morphological types, and their rates of action, we need to extend similar analyses to higher redshifts. This will allow us to explore the contribution of different morphological types to the stellar mass budget as a function of time and elucidate the galaxy evolution process.
In the present work, we perform a visual morphological classification of galaxies in the DEVILS-D10/COSMOS field (\citealt{Scoville07}; \citealt{Davies18}). We make use of the high resolution HST imaging and use a sample of galaxies within $0 \le z \le 1$ and $\mathrm{M}_* > 10^{9.5}$, selected from the DEVILS sample and analysis, for which classification is reliable. This intermediate redshift range is a key phase in the evolution of the Universe where a large fraction ($\sim 50\%$) of the present-day stellar mass is formed and large structures such as groups, clusters and filaments undergo a significant evolution (\citealt{Davies18}). During this period the sizes and masses of galaxies also appear to undergo significant evolution (e.g. \citealt{vanderWel12}).
By investigating the evolution of the SMF for different morphological types across cosmic time we can probe the various systems within which stars are located, and how they evolve. In this paper, we investigate the evolution of the stellar mass function of different morphological types and explore the contribution of each morphology to the global stellar mass build-up of the Universe. Shedding light on the redistribution of stellar mass in the Universe and also the transformation and redistribution of the stellar mass between different morphologies likely explains the origin of the bimodality in galaxy populations observed in the local Universe.
The ultimate goal of this study and its companion papers is to not only probe the evolution of the morphological types but also study the formation and evolution of the galaxy structures, including bulges and disks. This will be presented in a companion paper (Hashemizadeh et al. in prep.), while in this paper we explore the visual morphological evolution of galaxies and conduct an assessment into the possibility of the bulge formation scenarios by constructing the global stellar mass distributions and densities of various morphological types.
The data that we use in this work are presented in Section \ref{sec:AvailData}. In Section \ref{subsec:SampleSel}, we define our sample and sample selection. Section \ref{sec:MorphClass} presents different methods that we explore for the morphological classification. In Section \ref{sec:FitSMF}, we describe the parameterisation of the SMF. We then show the \textit{total} SMF at low-$z$ in Section \ref{subsec:MfunZ0}. Section \ref{subsec:LSS_cor} describes the effects of the cosmic large scale structure due to the limiting size of the DEVILS/COSMOS field, and the technique we use to correct for this. The evolution of the SMF and the SMD and their subdivision by morphological type are discussed in Sections \ref{subsec:MfunEvol}-\ref{sec:rho}. We finally discuss and summarize our results in Sections \ref{sec:discussion}-\ref{sec:summary}.
Throughout this paper, we use a flat standard $\Lambda$CDM cosmology of $\Omega_{\mathrm{M}} = 0.3$, $\Omega_\Lambda = 0.7$ with $H_0 = 70 \mathrm{km}\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$. Magnitudes are given in the AB system.
\subsection{DEVILS: Deep Extragalactic VIsible Legacy Survey}
\label{sec:DEVILS}
The Deep Extragalactic VIsible Legacy Survey (DEVILS) \citep{Davies18}, is an ongoing magnitude-limited spectroscopic and multi-wavelength survey. Spectroscopic observations are currently being undertaken at the Anglo-Australian Telescope (AAT), providing spectroscopic redshift completeness of $> 95\%$ to Y-mag $< 21.2$ mag.
This spectroscopic sample is supplemented with robust photometric redshifts, newly derived photometric catalogues (Davies et al. in prep.) and derived physical properties (\citealt{Thorne20}, and the work here) all undertaken as part of the DEVILS project. These catalogues extend to Y$\sim25$ mag in the D10 region.
The objective of the DEVILS campaign is to obtain a sample with high spectroscopic completeness extending over intermediate redshifts ($0.3 < z < 1.0$). At present, there is a lack of high completeness spectroscopic data in this redshift range (e.g., \citealt{Davies18}; \citealt{Scodeggio18}). DEVILS will fill this gap and allow for the construction of group, pair and filamentary catalogues to fold in environmental metrics. The DEVILS campaign covers three well-known fields: the XMM-Newton Large-Scale Structure field (D02: XMM-LSS), the Extended Chandra Deep Field-South (D03: ECDFS), and the Cosmological Evolution Survey field (D10: COSMOS). In this work, we only explore the D10 region as this overlaps with the HST COSMOS imaging. The spectroscopic redshifts used in this paper are from the DEVILS combined spectroscopic catalogue which includes all available redshifts in the COSMOS region, including zCOSMOS (\citealt{Lilly07}, \citealt{Davies15a}), hCOSMOS \citep{Damjanov18} and DEVILS \citep{Davies18}. As the DEVILS survey is ongoing the spectroscopic observations for our full sample are still incomplete (the completeness of the DEVILS combined data is currently $\sim 90$ per cent to Y-mag $= 20$). Those objects without spectroscopic redshifts are assigned photometric redshifts in the DEVILS master redshift catalogue (\texttt{DEVILS\_D10MasterRedshiftCat\_v0.2} catalogue), described in detail in \cite{Thorne20}. In this work, we also use the stellar mass measurements for the D10 region (\texttt{DEVILS\_D10ProSpectCat\_v0.3} catalogue) reported by \cite{Thorne20}.
Briefly, to estimate stellar masses they used the {\sc ProSpect} SED fitting code \citep{Robotham20} adopting \cite{BC03} stellar libraries, the \cite{Chabrier03} IMF together with \cite{Charlot00} to model dust attenuation and \cite{Dale14} to model dust emission. This study makes use of the new multiwavelength photometry catalogue in the D10 field (\texttt{DEVILS\_PhotomCat\_v0.4}; Davies et al. in prep.) and finds stellar masses $\sim 0.2$ dex higher than in COSMOS2015 \citep{Laigle16} due to differences in modelling an evolving gas phase metallicity. See \cite{Thorne20} for more details.
\begin{figure}
\centering
\includegraphics[width = 0.53 \textwidth]{Pics/F814W_DEVILS_sample_z1_3.png}
\caption{ Background shows the ACS/F814W mosaic image of the COSMOS field. The cyan rectangle represents the D10 region in the DEVILS survey. Gold points are the D10/ACS sources used in this work consisting of $\sim 36$k galaxies. See section \ref{subsec:SampleSel} for more detail.
}
\label{fig:F814W_DEVILS_smp_z1}
\end{figure}
\subsection{COSMOS ACS/WFC imaging data}
\label{subsec:ACS/HST}
The Cosmic Evolution Survey (COSMOS) is one of the most comprehensive deep-field surveys to date, covering almost 2 contiguous square degrees of sky, designed to explore large scale structures and the evolution of galaxies, AGN and dark matter \citep{Scoville07}. The high resolution Hubble Space Telescope (HST) F814W imaging in COSMOS allows for the study of galaxy morphology and structure out to the detection limits. In total COSMOS detects $\sim 2$ million galaxies at a resolution of $< 100$ pc \citep{Scoville07}.
The COSMOS region is centred at RA = $150.121$ ($10:00:28.600$) and DEC = $+2.21$ ($+02:12:21.00$) (J2000) and is supplemented by 1.7 square degrees of imaging with the Advanced Camera for Surveys (ACS\footnote{ACS Hand Book: \href{http://www.stsci.edu/hst/acs/documents/handbooks/current/c05\_imaging7.html\#357803}{www.stsci.edu/hst/acs/documents/}}) on HST. This 1.7 square degree region was observed during 590 orbits in the F814W (I-band) filter and also, 0.03 square degrees with F475W (g-band). In this study we exclusively use the F814W filter, not only providing coverage but also suitable rest-frame wavelength for the study of optical morphology of galaxies out to $z\sim1$ \citep{Koekemoer07}. The original ACS pixel scale is 0.05 arcsec and consists of a series of overlapping pointings. These have been drizzled and re-sampled to 0.03 arcsec resolution using the MultiDrizzle code (\citealt{Koekemoer03}), which is the imaging data we use in this work. The frames were downloaded from the public NASA/IPAC Infrared Science Archive (IRSA) webpage\footnote{\href{https://irsa.ipac.caltech.edu/data/COSMOS/images/acs\_2.0/I/}{irsa.ipac.caltech.edu/data/COSMOS/images/acs\_2.0/I/} } as fits images.
\begin{figure}
\centering
\includegraphics[width = 0.48\textwidth]{Pics/102850238.png}
\caption{ A random galaxy in our sample at redshift $z \sim 0.47$ showing an example of the postage stamps we generate to perform our visual inspection. Main image is the HST ACS/F814W and the inset is SUBARU $gri$ colour image. The image shows a cutout of $5\times \mathrm{R90}$ on each side, where $\mathrm{R90} = 3.23$ arcsec measured from UltraVISTA Y-band \citep{Davies18}. }
\label{fig:smplPostageStamp}
\end{figure}
\subsection{Sample selection: D10/ACS Sample}
\label{subsec:SampleSel}
As part of the DEVILS survey we have selected an HST imaging sample with which to perform various morphological and structural science projects.
Figure \ref{fig:F814W_DEVILS_smp_z1}, shows the ACS mosaic imaging of the COSMOS field with the full D10 region overlaid as a cyan rectangle (defined from the Ultra VISTA imaging). Our final sample, D10/ACS, is the common subset of sources from ACS and D10. The position of the D10/ACS sample on the plane of RA and DEC is overplotted on the same figure as yellow dots. Note that as shown in Figure \ref{fig:F814W_DEVILS_smp_z1}, we exclude objects in the jagged area of the ACS imaging leading to a rectangular effective area of $1.3467$ square degrees.
Although our HST imaging data is high resolution (see Figure \ref{fig:smplPostageStamp} for a random galaxy at redshift $z \sim 0.47$), we are still unable to explore galaxy morphologies beyond a certain redshift and stellar mass in our analysis as galaxies become too small in angular scale or too dim in surface brightness to identify morphological substructures.
We first try to select a complete galaxy sample from the DEVILS-D10/COSMOS data to define the redshift and stellar mass range for which we can perform robust morphological classification and structural decomposition (Hashemizadeh et al. in prep.). Note that we make use of a combination of photometric and available spectroscopic redshifts as well as stellar masses as described in Section \ref{sec:DEVILS}. In total, at the time of writing this paper, $23,264$ spectroscopic redshifts are available in the D10 region (excluding the jagged edges) out of which $2,903$ redshifts are observed by DEVILS. See the DEVILS website\footnote{\href{https://devilsurvey.org}{https://devilsurvey.org}} for a full description of these data.
We select 284 random galaxies drawn from across the entire redshift and stellar mass distribution, and visually inspect them. These 284 galaxies are shown as circles in Figure \ref{fig:mzsmplsel}. From our visual inspection, we identify the boundaries within which we believe the majority of galaxies are sufficiently resolved that morphological classifications should be possible (i.e., not too small or faint). Our visual assessments are indicated by colour in Figure \ref{fig:mzsmplsel} showing two-component (grey), single component (blue), and problematic cases (red; merger, disrupted, and low S/N). Unsurprisingly, in agreement with other studies (e.g. \citealt{Conselice05}), we find the fraction of problematic galaxies increases drastically at high redshifts ($z > 1.4$). Beyond this redshift an increasing number of galaxies ($\sim 50\%$) become complex and no longer adhere to a simplistic picture of a central bulge plus disk system. A large fraction of galaxies appear interacting, clumpy, very faint and/or extremely \textit{compact}, hence the notion of galaxies as predominantly bulge plus disk systems becomes untenable. Note that the vast majority of galaxies in this redshift range are disturbed interacting systems, however our observation of a fraction of the clumpy galaxies could be due to the fact that the bluer rest-frame emission is more sensitive to star forming regions dominating the flux. The clumps and potential bulges at this epoch, are mostly comparable or smaller than the HST PSF, hence conventional 2D bulge+disk fits are also unlikely to be credible even at HST resolution.
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width=1.3\textheight]{Pics/SmplSel.pdf}
\caption{ (a) The relation between stellar mass and lookback time (redshift) of our sample. All galaxies are shown in cyan in background. Circles represent 284 galaxies that we randomly sample for initial visual inspection. Grey and blue circles show double and single component galaxies, respectively. Red symbols are complicated galaxies which consist of merging systems, perturbed galaxies, low S/N or high redshift clumpy galaxies. The dotted rectangle corresponds to our initial sample region ($z < 1.4$ and log$(\mathrm{M}_*/\mathrm{M}_{\odot}) > 9$). The solid rectangle shows our final sample region which covers galaxies up to $z < 1.0$ and log$(\mathrm{M}_*/\mathrm{M}_{\odot}) > 9.5$. The solid black line represents the spline fit to the modal value of the stellar mass in bins of lookback time indicating the stellar mass completeness. Black dashed line shows \protect\cite{Thorne20} completeness limit. See text for more details. Panels (b) and (c) display the distribution of stellar mass and redshift of our final sample (i.e. within the solid rectangle). Note that the PDFs are smoothed by a kernel with standard deviation of 0.02. }
\label{fig:mzsmplsel}
\end{figure}
\end{landscape}
\begin{figure*}
\centering
\includegraphics[width = \textwidth]{Pics/Fraction_CI.png}
\caption{ The number fraction of each morphological type (BD: \textit{bulge+disk}, D: \textit{pure-disk}, E: \textit{elliptical}, C: \textit{compact}, H: \textit{hard}) in bins of lookback time (redshift) and total stellar mass, left and right panels, respectively. Each colour represents a morphology as indicated in the inset legend. Dashed lines are the $\sim4000$ random sample that we visually classified ($z < 1.4$) while solid lines are our full visual inspection of the final sample. Shaded stripes around lines display 95\% confidence intervals from beta distribution method as calculated using {\fontfamily{qcr}\selectfont prop.test} in R package {\fontfamily{qcr}\selectfont stats}.}
\label{fig:frac_z1}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth, height=11.5cm]{Pics/GZH_decision_tree2.jpeg}
\caption{ This flowchart shows the decision tree that we adopt to translate Galaxy Zoo: Hubble (GZH) tasks and outputs into the desired morphologies to be used in this study. The weight of final arrows are proportional to the number of galaxies following those paths. In addition, the fraction of galaxies falling into each morphological category is annotated.}
\label{fig:GZH}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth, height=11.5cm]{Pics/Mass_z_agreement.png}
\caption{Stellar mass versus redshift, colour coded by the level of agreement in our final classifications (by three co-authors), i.e. the percentage of objects in each cell consistently classified by at least two of our classifiers. Note that we increase the resolution of the colour map of agreement $\gtrsim 0.92$ to highlight the variation of agreement in this level. See the text for more details.}
\label{fig:agreent}
\end{figure*}
Note that as can be seen in Figure \ref{fig:mzsmplsel}, we also start to suffer significant mass incompleteness at low redshifts due to the COSMOS F814W sensitivity limit. The sample completeness limit is shown here as a smooth spline fitted to the modal values of stellar mass in bins of lookback time (shown as a solid line), i.e., peaks of the stellar mass histograms in narrow redshift slices. We also show the completeness limit reported by \cite{Thorne20} and shown as a dashed line in Figure \ref{fig:mzsmplsel}, which is based on a $g-i$ colour analysis and is formulated as $\mathrm{log(M_*)} = 0.25t + 7.25$, where $t$ is lookback time in Gyr. From this initial inspection we define our provisional window of 105,185 galaxies at $z < 1.4$ and log$(\mathrm{M}_*/\mathrm{M}_{\odot}) > 9$, shown as a dotted rectangle in Figure \ref{fig:mzsmplsel}.
To further tune this selection, we then generated postage stamps of $4,000$ random galaxies (now initial selection) using the HST F814W imaging and colour $gri$ insets from the Subaru Suprime-Cam data \citep{Taniguchi07}. Figure \ref{fig:smplPostageStamp} shows an example of the cutouts we generated for our visual inspection. The postage stamps are generated with $5 \times \mathrm{R90}$ on each side, where $\mathrm{R90}$ is the radius enclosing 90\% of the total flux in the UltraVISTA Y-band (soon to be presented in DEVILS photometry catalogue, Davies et al. in prep.).
These stamps were independently reviewed by five authors (AH, SPD, ASGR, LJMD, SB) and classified into single component, double component and complicated systems (hereafter: \textit{hard}). The single component systems were later subdivided into disk or elliptical systems. Note that the \textit{hard} class consists of asymmetric, merging, clumpy, extremely compact and low-S/N systems, for which 2D structural decomposition would be unlikely to yield meaningful output. Objects with three or more votes in one category were adopted and more disparate outcomes discussed and debated until an agreement was obtained. In this way, we established a ``gold calibration sample" of 4k galaxies to justify our final redshift and stellar mass range, and for later use as a training sample in our automated-classification process, see Section \ref{sec:MorphClass} for more details.
Figure \ref{fig:frac_z1} shows the fraction of each of the above classifications versus redshift and total stellar mass (dashed lines). As the left panel shows, the fraction of \textit{hard} galaxies (gray dashed line) drastically increases at $z > 1$. At the highest redshift of our sample, $z \sim 1.4$, 40 percent of the galaxies are deemed unfittable, or at least inconsistent with the notion of a classical/pseudo-bulge plus disk systems (this is consistent with \citealt{Abraham96b} and \citealt{Conselice05}). Also see the review by \cite{Abraham01}. We therefore further restrict our redshift range to $z \leq 1$ in our full-sample analysis. Additionally, we increase our stellar mass limit to $\mathrm{log(M}_*/\mathrm{M}_{\odot}) = 9.5$ to reduce the effects of incompleteness (see Figure \ref{fig:mzsmplsel}) and to restrict our sample to a manageable number for morphological classifications.
In Figure \ref{fig:mzsmplsel}, our final sample selection region is now shown as a solid rectangle. The distribution of the stellar mass and redshift of our final sample is also displayed in the right panels (b and c) of the same figure.
Overall, our analysis of galaxies in different regions of the $M_*$-$z$ parameter space leads us to a final sample of galaxies for which we can confidently study their morphology and structure. We conclude that we can study the structure of galaxies up to $z \sim 1$ and down to log$(\mathrm{M}_{*}/\mathrm{M}_{\odot}) \geq 9.5$. Within this selection, our sample consists of $35,803$ galaxies with $14,036$ available spectroscopic redshifts.
\begin{figure}
\centering
\centering
\includegraphics[width=0.48\textwidth]{Pics/sSFR_stMass.png}
\label{subfig:stMass_PDF}
\caption{Top panel: the PDF of the specific star formation rate (sSFR = SFR/M$_*$) of the same morphologies. Bottom panel: the stellar mass probability density function (PDF) of three morphologies in our sample (all redshifts), i.e. \textit{pure-disk} (cyan), \textit{double-component} (green) and \textit{elliptical} (magenta) galaxies. Note that for clarity, the PDFs are slightly smoothed by a kernel with standard deviation of 0.02. }
\label{fig:stMass_sSFR_PDF}
\end{figure}
\section{Final Morphological Classifications}
\label{sec:MorphClass}
In this section, we initially aim to develop a semi-automatic method for morphological classification. Our ultimate goal would be to reach a fully automatic algorithm for classifying galaxies into various morphological classes. While this ultimately proves unsuitable we discuss it here to explain why we finally visually inspect all systems.
These classes are as mentioned above \textit{bulge+disk} (BD; double-component), \textit{pure-disk} (D), \textit{elliptical} (E) and \textit{hard} (H) systems. In order to overcome this problem we test various methods including: cross-matching with Galaxy Zoo, Hubble catalogue (GZH) and Zurich Estimator of Structural Type (ZEST) catalogues. In the end, none of the methods proved to be robust and we resort to full visual inspection.
\subsection{Using Galaxy Zoo: Hubble}
\label{subsec:GZH}
A large fraction of COSMOS galaxies with I$_{F814W} < 23.5$ have been classified in the Galaxy Zoo: Hubble (GZH) project \citep{Willett17}. Like other Galaxy Zoo projects, this study classifies galaxies using crowdsourced visual inspections. They make use of images taken by the ACS in various wavelengths including 84,954 COSMOS galaxies in the F814W filter. Our sample has more than 80\% overlap with GZH, so we can cross-match and use their classifications to partially improve our final sample. We translate the GZH classifications into our desired morphological classes by using the decision tree shown in figure 4 of \cite{Willett17}, as well as the suggested thresholds for morphological selection presented in Table 11 of the same paper. Our final decision tree is displayed in the flowchart shown in Figure \ref{fig:GZH}. We refer to \cite{Willett17} for a detailed description of each task. The thresholds are shown as P values in the flowchart. In addition to using different combinations and permutations of tasks as shown in Figure \ref{fig:GZH}, the only part that we add to the GZH tasks is where an object is voted to have a smooth light profile. In this case, the object is likely to be an E or S0 or a smooth \textit{pure-disk} galaxy. To distinguish between them, we make use of the single S\'ersic index (n). As shown in the flowchart, $n > 2.5$ and $n \leq 2.5$ are classified as spheroid and \textit{pure-disk}, respectively. The S\'ersic indices are taken from our structural analysis which will be described in Hashemizadeh et al. (in prep.). In order to capture the S0 galaxies or \textit{double-component} systems with smooth disk profiles we add an extra condition as to whether there are at least 2 Galaxy Zoo votes for a prominent bulge. If so then it is likely that the galaxy is a \textit{double-component} system.
An advantage of using the GZH decision tree is that it can identify a vast majority of galaxies with odd features such as merger-induced asymmetry etc.
Table \ref{tab:GZH_cofmat} shows a confusion matrix comparing the GZH predictions with the visual inspection of our 4k gold sample as the ground truth. Double component galaxies are predicted by the GZH with maximum accuracy 81\%. Single component (i.e. \textit{pure-disk} and \textit{elliptical} galaxies) and \textit{hard} galaxies are predicted at a significantly lower accuracy with high misclassification rates. We further visually inspected misclassified galaxies and do not find good agreement between our classification and those of GZH. As such, we do not trust the GZH classification for our sample.
\begin{figure*}
\centering
\includegraphics[width = \textwidth, height = 11.7cm]{Pics/Random_of_all3_lo.png}
\caption{ Random postage stamps of various morphologies that we generate to perform our visual inspection. Background images are HST ACS/F814W while insets are combined SUBARU $gri$ colour image. Rows represent different morphologies. Galaxies are randomly selected within different redshift bins shown in columns. Redshifts annotated in the first row are the mean of the associated redshift bins. Cutouts indicate $5\times \mathrm{R90}$ on each side, where $\mathrm{R90}$ is measured from UltraVISTA Y-band in the DEVILS photometry catalogue (Davies et al. in prep.). }
\label{fig:PostageStamp1}
\end{figure*}
\subsection{Using Zurich Estimator of Structural Type (ZEST)}
\label{subsec:ZEST}
Another available morphological catalogue for COSMOS galaxies is the Zurich Estimator of Structural Type (ZEST) \citep{Scarlata07a}. In ZEST, \cite{Scarlata07a} use their single-S\'ersic index (n) as well as five other diagnostics: asymmetry, concentration, Gini coefficient, second-order moment of the brightest 20\% of galaxy flux, and ellipticity. ZEST includes a sample of $\sim 56,000$ galaxies with I$_{F814W}\leq 24$. More than 90\% of our sample is cross-matched with ZEST which we use as a complementary morphological classification. We use the ZEST TYPE flag with four values of 1,2,3,9 representing early type galaxy, disk galaxy, irregular galaxy and no classification. For Type 2 (i.e. disk galaxy) we make use of the additional flag BULG, which indicates the level of bulge prominence. BULG is flagged by five integers as follows: 0 = bulge dominated, 1,2 = intermediate-bulge, 3 = \textit{pure-disk} and 9 = no classification.
We present the accuracy of ZEST predictions in a confusion matrix in Table \ref{tab:ZEST_cofmat} which shows we do not find an accurate morphological prediction from ZEST in comparison to our gold calibration sample. While \textit{double-component} systems are classified with an accuracy of 74\%, other classes are classified poorly with high error ratios. We confirm this by visual inspection of the misclassified objects where we still favour our visual classification.
\\[2\baselineskip]
Overall, by analysing both of the above catalogues we do not find them to be sufficiently accurate for predicting the proper morphologies of galaxies when we compare their estimates with our 4k gold calibration visual classification.
\begin{table}
\centering
\caption{The confusion matrix comparing the morphological predictions of the GZH with our visual inspection of 4k gold calibration sample as the ground truth. For example, 0.81 means 81\% of double component galaxies (in our visual classification) are also correctly classified by the GZH. }
\begin{tabular}{c|ccc}
\hline \hline
\backslashbox[35mm]{GZH Pred}{Ground Truth}
& Double & Hard & Single \\ \hline \\
Double & \textbf{0.81} & 0.28 & 0.32 \\
Hard & 0.0038 & \textbf{0.18} & 0.011 \\
Single & 0.18 & 0.53 & \textbf{0.67} \\ \\
\lasthline
\end{tabular}
\label{tab:GZH_cofmat}
\end{table}
\begin{table}
\centering
\caption{The confusion matrix comparing the morphological predictions of the ZEST catalogue with our visual inspection of 4k gold calibration sample as the ground truth. }
\begin{tabular}{c|ccc}
\hline \hline
\backslashbox[35mm]{ZEST Pred}{Ground Truth}
& Double & Hard & Single \\ \hline \\
Double & \textbf{0.74} & 0.34 & 0.42 \\
Hard & 0.047 & \textbf{0.37} & 0.036 \\
Single & 0.21 & 0.28 & \textbf{0.55} \\ \\
\lasthline
\end{tabular}
\label{tab:ZEST_cofmat}
\end{table}
\subsection{Visual inspection of full D10/ACS sample}
\label{subsec:VisInsp_full}
As no automatic classification robustly matches our gold calibration sample we opt to visually inspect all galaxies in our full sample.
However, we can use the predictions from GZH and ZEST as a pre-classification decision and put galaxies in master directories according to their prediction. The \textit{hard} class is adopted from the GZH prediction as it is shown to perform well in detecting galaxies with odd features. In addition, from analysing the distribution of the half-light radius (R50) of our 4k gold calibration sample, extracted from our DEVILS photometric analysis, we know that resolving the structure of galaxies with a spatial size of R50 $\leq 0.15$ arcsec is nearly impossible. R50 is measured by using the {\sc ProFound} package \citep{Robotham18}, a tool written in the {\sc R} language for source finding and photometric analysis. So, we put these galaxies into a separate \textit{compact} class (C).
Having pre-classified galaxies, we now assign each class to one of our team members (AH, SPD, ASGR, LJMD, SB) so each of the authors is independently a specialist in, and responsible for, only one morphological class. Initially, we inspect galaxies and relocate incorrectly classified galaxies from our master directories to transition folders for further inspections by the associated responsible person. In the second iteration, the incorrectly classified systems will be reclassified and moved back into master directories. The left-over sources in the transition directories are therefore ambiguous. For these galaxies, all classifiers voted and we selected the most voted class as the final morphology. As a final assessment and quality check three of our authors (SPD, LJMD, SB) independently reviewed the entire classifications ($\sim 15k$ each) and identified $\sim 7k$ that were still felt to be questionable. These three classifiers then independently classified these 7k objects. The final classification was either the majority decision or in the case of a 3-way divergence the classification of SPD.
Figure \ref{fig:agreent} shows the stellar mass versus redshift plane colour coded according to the level of agreement between our 3 classifiers. Colour indicates the percentage of objects in each cell consistently classified by at least two classifiers, obtained as follows:
\begin{equation}
\mathrm{\scriptstyle Agreement} = \frac{\mathrm{\scriptstyle Number~of~objects~with~two~or~more~agreement~in~the~cell}}{\mathrm{\scriptstyle Total~number~of~objects~in~the~cell}},
\label{eq:agree}
\end{equation}
This figure implies that we have the highest agreement ($\sim 100\%$) for high stellar mass objects at low redshifts. The agreement in our classification decreases to $\sim 90\%-95\%$ towards lower stellar mass regime at high redshift. This figure indicates that, on average, our visual classifications performed independently by different team members agree by $\sim95\%$ over the complete sample.
We report the number of objects in each morphological class in Table \ref{tab:FinalClassStats}. $44.4\%$ of our sample (15,931) are classified as \textit{double-component} (BD) systems. We find 13,212 \textit{pure-disk} (D) galaxies ($37\%$) while only 3,890 ($11\%$) \textit{elliptical} (E) galaxies. The \textit{Compact} (C) and \textit{hard} (H) systems, in total, occupy $7.6\%$ of our D10/ACS sample.
The visual morphological classification is released in the team-internal DEVILS data release one (DR1) in D10VisualMoprhologyCat data management unit (DMU).
\subsection{Possible effects from the ``Morphological K-correction'' on our galaxy
classifications}
\label{subsec:K-corr}
The morphology of a subset of nearby galaxy classes shows some significant dependence on rest-frame wavelengths, especially below the Balmer break and towards the UV (e.g. \citealt{Hibbard97}; \citealt{Windhorst02}; \citealt{Papovich03}; \citealt{Taylor05}; \citealt{Taylor-Mager07}; \citealt{Huertas-Company09}; \citealt{Rawat09}; \citealt{Holwerda12}; \citealt{Mager18}). This ``Morphological K-correction'' can be quite significant, even between the B- and near-IR bands (e.g., \citealt{Knapen96}), and must be quantified in order to distinguish genuine evolutionary effects from simple bandpass shifting. Hence, the results of faint galaxy classifications may, to some extent, depend on the rest-frame wavelength sampled.
Our D10/ACS classifications are done in the F814W filter (I-band), and the largest redshift in our sample, $z \sim$ 1, samples $\sim$412 nm (B-band), so for our particular case, the main question is, to what extent galaxy rest-frame morphology changes significantly from 412--823 nm across the BVRI filters. Here we briefly summarize how any such effects may affect our classifications.
\cite{Windhorst02} imaged 37 nearby galaxies of all types with the Hubble Space Telescope (HST), gathering available data mostly at 150, 255, 300, 336, 450, 555, 680, and/or 814 nm, including some ground based images to complement filters missing with HST. These nearby galaxies are all large enough that a ground-based V-band image yields the same classification as an HST F555W
or F606W image. These authors conclude that the change in rest-frame morphology going from the red to the mid--far UV is more pronounced for early type galaxies (as defined at the traditional optical wavelengths or V-band), but not necessarily negligible for all mid-type spirals or star-forming galaxies. Late-type galaxies generally look more similar in morphology from the mid-UV to the red due to their
more uniform and pervasive SF that shows similar structure for young--old stars in all filters from the mid-UV through the red. \cite{Windhorst02} conclude {\it qualitatively} that in the rest-frame mid-UV, early- to mid-type galaxies are more likely to be misclassified as later types than late-type galaxies are likely to be misclassified as earlier types.
To {\it quantify} these earlier qualitative findings regarding the morphological K-correction, much larger sample of 199 nearby galaxies across all Hubble types (as defined in V-band) was observed by \cite{Taylor05} and \cite{Taylor-Mager07}, and 2071 nearby galaxies were similarly analyzed by \cite{Mager18}. They determined their SB-profiles, radial light-profiles, color gradients, and CAS parameters (Concentration index, Asymmetry, and Clumpiness; e.g. \citealt{Conselice04}) as a function of rest-frame wavelength from 150-823 nm.
\cite{Taylor-Mager07} and \cite{Mager18} conclude that early-type galaxies (E--S0) have CAS parameters that appear, within their errors, to be similar at all wavelengths longwards of the Balmer break, but that in the far-UV, E--S0 galaxies have concentrations and asymmetries that more closely resemble those of spiral and peculiar/merging galaxies in the optical. This may be explained by extended disks containing recent star formation. The CAS parameters for galaxy types later than S0 show a mild but significant wavelength dependence, even over the wavelength range 436-823 nm, and a little more significantly so for the earlier spiral galaxy types (Sa--Sc). These galaxies generally become less concentrated and more asymmetric and somewhat more clumpy towards shorter wavelengths. The same is true for mergers when they progress from pre-merger via major-merger to merger-remnant stages.
While these trends are mostly small and within the statistical error bars for most galaxy types from 436--823 nm, this is not the case for the Concentration index and Asymmetry of Sa--Sc galaxies. For these galaxies, the Concentration
index decreases and the Asymmetry increases going from 823 nm to 436 nm (Fig. 17 of \citealt{Taylor-Mager07} and Fig. 5 of \citealt{Mager18}). Hence, to the extent that our visual classifications of apparent Sa--Sc galaxies depended on their Concentration index and Asymmetry, it is possible that some of these objects may have been misclassified. E-S0 galaxies show no such trend in Concentration with wavelength for 436--823 nm, and have Concentration indices much higher than Sa--Sc galaxies and Asymmetry and Clumpiness parameters generally much lower than Sa--Sc galaxies. Hence, it is not likely that a significant fraction of E--S0 galaxies are misclassified as Sa--Sc galaxies at $z \lesssim 0.8$ in the 436--823 nm filters due to the Morphological K-correction.
Sd-Im galaxies show milder trends in their CAS parameters and in the same direction as the Sa--Sc galaxies. It is thus possible that a small fraction of Sa-Sc galaxies --- if visually relying heavily on Concentration and Asymmetry index --- may have been visually misclassified as Sd--Im galaxies, while a smaller fraction of Sd--Im galaxies may have been misclassified as Sa--Sc galaxies. The available data on the rest-frame wavelength dependence of the morphology of nearby galaxies does not currently permit us to make more precise statements. In any case, these CAS trends with restframe wavelengths as observed at 436--823 nm for redshifts $z\simeq$ 0.8--0 in the F814W filter are shallow enough that the fraction of misclassifications between Sa--Sc and Sd--Im galaxies, and vice versa, is likely small.
\subsection{Review}
\label{subsec:VisInsp_verify}
In Figure \ref{fig:frac_z1}, we show the associated fractions of our final visual inspection as solid lines. The global trends are in good agreement with our initial 4k gold calibration sample.
We find that, although our inspection procedure may have slightly changed from the 4k gold calibration sample to the full sample, the outcome classifications are consistent in the three primary classes making up more than $92\%$ of our sample.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Pics/Mfunc_morph_compare_lowZ.png}
\caption{Top panel: Single and double Schechter functions fitted to a low-$z$ sample of D10/ACS ($0.0 < z < 0.25$). The fits are performed by using {\fontfamily{qcr}\selectfont dftools} down to the stellar mass limit of the sample, i.e. $10^{9.5} M_\odot$.
Transparent region shows the error range calculated by 1000 times sampling of the full posterior probability distribution of the single Schechter fit parameters. Black data points show the binned galaxy counts for which the stellar mass range from minimum to maximum ($9.5 \le$ log$(\mathrm{M}_{*}/\mathrm{M}_{\odot}) \le 12$) is binned into 25 equal-sized bins. Note that the Schechter functions are not fitted directly to the galaxy counts. Bottom panel: Comparison of our SMF at $0.0 < z < 0.25$ with \protect\cite{Muzzin13} and \protect\cite{Davidzon17}. For completeness, the SMF of the GAMA local galaxies from several studies are shown in different colours. Solid and dashed black lines represent the double and single Schechter function fits to our data, respectively (eq. \ref{eq:SingleSchechter} \& \ref{eq:DoubleSchechter}). Note that, the fits to other studies (colour solid lines) are double Schechter fits.}
\label{fig:Mfunc_z0}
\end{figure*}
\begin{table}
\centering
\caption{Final number of objects in each of our morphological classes. }
\begin{tabular}{ccc}
\firsthline \firsthline
Morphology & Object Number & Percentage \\ \hline \\
Double & 15,931 & 44.4\% \\
Pure Disk & 13,212 & 37\% \\
Elliptical & 3,890 & 11\% \\
Compact & 1,124 & 3.1\% \\
Hard & 1,600 & 4.5\% \\ \\
\lasthline
\end{tabular}
\label{tab:FinalClassStats}
\end{table}
Figure \ref{fig:stMass_sSFR_PDF} shows the probability density function (PDF) of the sSFR (upper panel) and total stellar mass (lower panel) for galaxies classified as D, BD and E. We use the SFR from the {\sc ProSpect} SED fits described in \cite{Thorne20}. These figures indicate that, as one would expect, D galaxies dominate lower stellar mass and higher sSFR regime, opposite to Es which occupy the high stellar mass end and low sSFR. BD galaxies are systems located in-between these two classes in terms of both stellar mass and sSFR. These results are as expected and provide some confidence that our classifications are sensible.
Figure \ref{fig:PostageStamp1} displays a set of random galaxies in each of the morphological classes within different redshift intervals. In addition, we present 50 random galaxies of each morphology in Figures \ref{fig:contact_sheets_double}-\ref{fig:contact_sheets_comp}.
Having finalised our morphological classification, we now investigate the stellar mass functions for different morphologies and their evolution from $z = 1$.
\section{Fitting galaxy stellar mass function}
\label{sec:FitSMF}
For parameterizing the SMF, we assume a fitting function that can describe the galaxy number density, $\Phi\left(\mathrm{M}\right)$. The typical function adopted is that described by \cite{Schechter76} as:
\begin{equation}
\Phi(M)\mathrm{d}M=\Phi^{*} e^{-M / M^{*}}\left(\frac{M}{M^{*}}\right)^{\alpha} \mathrm{d}M,
\label{eq:SingleSchechter}
\end{equation}
\noindent where the three key parameters of the function are, $\alpha$, the power low slope or the slope of the low-mass end, $\Phi^{*}$, the normalization, and, $M^{*}$, the characteristic mass (also known as the break mass or the knee of the Schechter function).
At very low redshifts a number of studies have argued that the shape of the SMF is better described by a double Schechter function (\citealt{Baldry08}; \citealt{Pozzetti10}; \citealt{Baldry12}), i.e. a combination of two single Schechter functions, parameterized by a single break mass ($M^{*}$), and given by:
\begin{equation}
\Phi(M) \mathrm{d} M=e^{-M / M^{*}}\left[\Phi_{1}^{*}\left(\frac{M}{M^{*}}\right)^{\alpha_{1}}+\Phi_{2}^{*}\left(\frac{M}{M^{*}}\right)^{\alpha_{2}}\right] \frac{\mathrm{d} M}{M^{*}},
\label{eq:DoubleSchechter}
\end{equation}
\noindent where $\alpha_2 < \alpha_1$, indicating that the second term predominantly drives the lower stellar mass range.
To fit our SMFs, we use a modified maximum likelihood (MML) method (i.e., not $1/V_{\mathrm{max}}$) as implemented in {\fontfamily{qcr}\selectfont dftools}\footnote{\href{https://github.com/obreschkow/dftools}{https://github.com/obreschkow/dftools}} \citep{Obreschkow18}. This technique has multiple advantages including: it is free of binning, accounts for small statistics and Eddington bias. {\fontfamily{qcr}\selectfont dftools} recovers the mass function (MF) while simultaneously handling any complex selection function, Eddington bias and the cosmic large-scale structure (LSS). Eddington bias tends to change the distribution of galaxies, particularly in the low-mass regime which is more sensitive to the survey depth and S/N, as well as high-mass regime due to the exponential cut-off which is sensitive to the scatters by noise (e.g. \citealt{Ilbert13}; \citealt{Caputi15}). Eddington bias occurs because of the observational/photometric errors and often can dominate over the shot noise \citep{Obreschkow18}. Photometric uncertainties, which are introduced in the redshift estimation, as well as the stellar mass measurements are fundamentally the source of this bias (\citealt{Davidzon17}). We account for Eddington bias in {\fontfamily{qcr}\selectfont dftools} by providing the errors on the stellar masses from the {\sc ProSpect} analysis by \cite{Thorne20}.
The R language implementation and MML method make {\fontfamily{qcr}\selectfont dftools} very fast. In the fitting procedures described in this work, we use the inbuilt {\fontfamily{qcr}\selectfont optim} function with the default optimization algorithm of \cite{Nelder65} for maximizing the likelihood function. In order to account for the volume corrections, we use the effective unmasked area of the D10/ACS region which we calculate to be $1.3467$ square degrees. This is exclusive of the masked areas from bright stars and the non-uniform edges of the ACS mosaic (see Figure \ref{fig:F814W_DEVILS_smp_z1}) and this is calculated using the process outlined in Davies et al. (in prep.). Our selection function, required for a proper volume corrected distribution function, is essentially a volume limited sample, i.e. a constant volume across adopted mass range. We refer the reader to \cite{Obreschkow18} for full details regarding {\fontfamily{qcr}\selectfont dftools} and its methodology. In this paper, we fit both single and double Schechter functions to the SMF. We examine both functions and will discuss further in Section \ref{subsec:MfunZ0}.
\subsection{Verification of our SMF fitting process at low redshift}
\label{subsec:MfunZ0}
We first validate our \textit{total} SMF fitting process at low-$z$ and compare with the known literature, prior to splitting by redshift and morphology.
To achieve this, we compare the SMF of D10/ACS galaxies in our lowest-$z$ bin, i.e. $0 < z < 0.25$, with literature studies. We primarily choose this redshift range to compare with the SMF \cite{Muzzin13} and \protect\cite{Davidzon17} at $0.2 < z < 0.5$ in the COSMOS/UltraVISTA field and local GAMA galaxies at $z < 0.06$ (\citealt{Baldry12}; \citealt{Kelvin14}; \citealt{Moffett16a}; \citealt{Wright17}). We fit the SMF within this redshift range using both single and double Schechter functions (Equations \ref{eq:SingleSchechter} and \ref{eq:DoubleSchechter}, respectively). The upper panel of Figure \ref{fig:Mfunc_z0}, shows our single and double Schechter fit to this low-$z$ sample. As annotated in the figure, we report the best fit single Schechter parameters of log$(\mathrm{M}^{*}/\mathrm{M}_{\odot}) = 11.01\pm0.05$, $\alpha = -1.15\pm0.05$, log$(\Phi^*) = -2.69\pm0.06$ and log$(\mathrm{M}^{*}/\mathrm{M}_{\odot}) = 10.96\pm0.06$ $\alpha_1 = -1.07\pm0.08$, $\alpha_2 = -4.91\pm3$, log$(\Phi^*_1) = -2.62\pm0.07$ and log$(\Phi^*_2) = -8.6\pm5$ in our double Schechter fit. Therefore, the double Schechter fit involves a broad range of uncertainty, in particular at the low-mass range.
Larger uncertainty is likely due to the fact that the stellar mass range of the D10/ACS sample does not extend to log$(M_*/M_\odot) < 9.5$, where the pronounced upturn in the SMF occurs.
The error ranges shown in the upper panel of Figure \ref{fig:Mfunc_z0} as transparent curves are obtained from 1000 samples drawn from the full posterior probability distribution of all the single Schechter parameters.
Despite larger errors on the double Schechter parameters, we elect to use this function for our future analysis at all redshifts as it has been shown that a double Schechter can better describe the stellar mass distribution even at higher-$z$ (e.g., \citealt{Wright18}).
In the lower panel of Figure \ref{fig:Mfunc_z0}, we show that our SMF is in good agreement with \cite{Muzzin13} and \protect\cite{Davidzon17} within the quoted errors.
Overall, we see a good agreement with the mass function of galaxies at low-$z$. We observe no significant flattening of the SMF at the intermediate stellar masses ($10^{9.5}-10^{10.5}$) as reported in the local Universe by e.g., \cite{Moffett16a}. We are unsurprised that we do not see this flattening as the D10/ACS contains only 3 galaxies in the redshift range comparable to GAMA ($z < 0.06$).
One might expect an evolution of the SMF within $0 < z < 0.25$ that would explain the slightly higher number density in the intermediate mass range. Our fitted Schechter functions however also deviate from the GAMA SMFs in the high stellar mass end. We find that this is systematically due to the Schechter function fitting process. Unlike D10/ACS, as shown in Figure \ref{fig:Mfunc_z0}, the local literature data extend to lower stellar mass regimes (log$(\mathrm{M}_{*}/\mathrm{M}_{\odot}) = 8$ and $7.5$ in the case of \citealt{Wright17}), influencing the bright end fit and the position of $\mathrm{M}^*$. In this regime the upturn of the stellar mass distribution is remarkably more pronounced. This strong upturn drives the optimization fitting and impacts the high-mass end. This highlights the difficulty in directly comparing fitted Schechter values if fitted over different mass ranges.
\subsection{Cosmic Large Scale Structure Correction}
\label{subsec:LSS_cor}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Pics/zdens_withSHARK.png}
\caption{ The $\mathrm{N}(z)$ distribution of the D10/ACS sample (solid line) compared with the SHARK semi-analytic model prediction (dashed line). SHARK data represent a light-cone covering 107.889 square degrees with Y-mag $< 23.5$. Colour bands represent the redshift bins that we consider in this work. Note that the PDFs are smoothed by a kernel with standard deviation of 0.3 so are non-zero beyond the nominal limits.}
\label{fig:zdens}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 0.49\textwidth, angle = 0]{Pics/LSS_cor.png}
\caption{ Top panel: our measurement of the evolution of the \textit{total} SMD (per comoving Mpc$^{3}$) compared with a compilation of GAMA, COSMOS and 3D-HST by \protect\cite{Driver18}. The black curve represents a spline fit to the \protect\cite{Driver18} data. Middle panel: the large scale structure correction factor applied to our SMD in order to meet the predictions of the spline fit. Bottom panel: the residual of the SMDs before and after the LSS correction indicating the correction coefficient we apply to each redshift bin. }
\label{fig:LSS_cor}
\end{figure}
All galaxy surveys are to some extent influenced by the cosmic large scale structure (LSS, \citealt{Obreschkow18}). Generally, the LSS produces local over- and under-densities of the galaxies at particular redshifts in comparison to the mean density of the Universe at that epoch. For example, GAMA regions are $\sim 10\%$ underdense compared with SDSS (\citealt{Driver11}).
We observe this phenomenon in the nonuniform D10/ACS redshift distribution, $\mathrm{N}(z)$.
To highlight this, Figure \ref{fig:zdens} compares the $\mathrm{N}(z)$ distribution of the D10/ACS sample with the prediction of the SHARK semi-analytic model (\citealt{Lagos18b}; \citealt{Lagos19}). SHARK data in this figure represent a light-cone covering 107.889 square degrees with Y-mag $< 23.5$, and because of the much larger simulated volume, is less susceptible to LSS. In this figure the redshift bins we shall use later in our analysis are shown as background colour bands indicating redshift intervals of (0, 0.25, 0.45, 0.6, 0.7, 0.8, 0.9 and 1.0). SHARK predicts a nearly uniform galaxy distribution with no significant over- and under-density regions, while the empirical D10/ACS sample shows a nonuniform $\mathrm{N}(z)$ with overdensities and underdensities.
These density fluctuations can introduce systematic errors in the construction of the SMF by, for example, overestimating the number density of very low-mass galaxies which are only detectable at lower redshifts \citep{Obreschkow18}. In other words, the LSS can artificially change the shape/normalization of the SMF.
Using the distance distribution of galaxies, {\fontfamily{qcr}\selectfont dftools} (\citealt{Obreschkow18}) internally accounts for the LSS by modeling the relative density in the survey volume, $g(r)$, i.e. it measures the mean galaxy density of the survey at the comoving distance $r$, relative to the mean density of the Universe. Incorporating this modification into the effective volume, the MML formalism works well for a sensitivity-limited sample (see \citealt{Obreschkow18} for details). However, for our volume limited sample, this method is unable to thoroughly model the density fluctuations. Therefore, to take this non-uniformity into account, we perform a manual correction as follows.
In a comprehensive study of the cosmic star formation history, \cite{Driver18} compiled GAMA, COSMOS and 3D-HST data to estimate the total stellar mass density from high redshifts to the local Universe ($0 < z < 5$). Figure \ref{fig:LSS_cor} (upper panel) shows these data. As we know the total stellar mass density must grow smoothly and hence we assume that perturbations around a smooth fit represent the underlying LSS.
In Figure \ref{fig:LSS_cor} (middle panel), we fit a smooth spline with the degree of freedom $3.5$ to the \cite{Driver18} data to determine a uniform evolution of the total $\rho_*$. We then introduce a set of correction factors ($\beta$) to our empirical measurements of the \textit{total} SMDs to match the prediction of the spline fit (middle panel of Figure \ref{fig:LSS_cor}). We then apply these correction factors (bottom panel of Figure \ref{fig:LSS_cor}) to our estimations of the morphological SMD values.
This ignores any coupling between morphology and LSS which we consider a second order effect.
We report the $\beta$ correction factors in Table \ref{tab:rho} and show them in the bottom panel of Figure \ref{fig:LSS_cor}.
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width = 1.3\textwidth, angle = 0]{Pics/SMF_morph_newZbin_LSScor_FINAL.png}
\caption{ The \textit{total} and morphological SMFs in eight redshift bins. Top row highlighted by yellow colour represents the GAMA SMFs ($0 \le z \le 0.08$). Data points are galaxy counts in each of equal-size stellar mass bins. Width of stellar mass bins are shown as horizontal bars on data points. Vertical bars are poisson errors. Shaded regions around the best fit curves are 68 per cent confidence regions. Black solid and dashed curves demonstrate double and single Schechter functions of all galaxies, respectively, while dotted curves over-plotted on higher-$z$ bins are the GAMA $z=0$ SMFs to highlight the evolution of the SMF.}
\label{fig:Mfunc_6z}
\end{figure}
\end{landscape}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Pics/Mfunc_morph_Par_evol_LSScor_WO_C.png}
\caption{ Left: Evolution of the single Schechter best fit parameters, $\Phi^*$, $M^*$ and $\alpha$. Error bars are the standard deviations on each of the most likely parameters. Black line is the \textit{total} SMF while colour coded data represent the best fit parameters of the Schechter function of different morphologies. Note that for simplicity we remove \textit{compacts} as their trends are very noisy and washes out the trends of other morphological types. Right: Zoomed-in plot showing the evolution of our \textit{total} Schechter function parameters compared with a compilation of other studies. Highlighted region shows the epoch covered by the GAMA data.}
\label{fig:Mfunc_6z_par_evol}
\end{figure*}
\subsection{Evolution of the SMF since \lowercase{$z$} $= 1$}
\label{subsec:MfunEvol}
With the LSS correction in place, we now split the sample into 7 bins of redshift (0-0.25, 0.25-0.45, 0.45-0.6, 0.6-0.7, 0.7-0.8, 0.8-0.9 and 0.9-1.0) so that we have enough numbers of objects in each bin and are not too wide to incorporate a significant evolution (these redshift ranges are shown by the colour bands in Figure \ref{fig:zdens}). Each bin contain 1,108, 5,041, 4,216, 4,867, 5,277, 7,090, 8,207 galaxies, respectively.
Figure \ref{fig:Mfunc_6z} shows the SMF in each redshift bin for the full sample, as well as for different morphological types of: \textit{double-component} (BD), \textit{pure-disk} (D), \textit{elliptical} (E), \textit{compact} (C) and \textit{hard} (H). These SMF measurements include the correction for LSS as discussed in Section \ref{subsec:LSS_cor}. Note that the first row of Figure \ref{fig:Mfunc_6z} highlighted by yellow shade shows GAMA $z = 0$ SMFs (Driver et al., in prep.).
As shown in Figure \ref{fig:Mfunc_6z}, we fit the \textit{total} SMF within all 8 redshift bins (including GAMA) by both single and double Schechter functions (displayed as dashed and solid black curves, respectively), while the morphological SMFs are well fit by single Schechter function at all epochs. The difference between double and single Schechter fits is insignificant, compared to the error on individual points. However, as noted in Section \ref{subsec:MfunZ0}, for calculating the stellar mass density we will use our double Schechter fits.
As we will see later, the effect of this choice on the measurement of our total stellar mass density is negligible. In Figure \ref{fig:Mfunc_6z}, we over-plot the GAMA $z = 0$ SMFs (Driver et al., in prep.) on higher-$z$ to highlight the evolution (dotted curves).
In the \textit{total} SMF we see a growth at the low-mass end and a relative stability at high-mass end, particularly when comparing high-$z$ with the D10/ACS low-$z$ ($0 < z < 0.25$).
At face value, this suggests that since $z = 1$ in-situ star formation and minor mergers (low-mass end) play an important role in forming or transforming mass within galaxies \citep{Robotham14}.
Looking at the SMF of the morphological sub-classes we find that the BD systems show no significant growth with time at their high-mass end, while at the low-mass end we see a noticeable increase in number density.
In D galaxies, at both low- and high-mass ends we find a variation in the number density with the low-mass end increasing and extreme steepening at the lower-$z$ and high-mass end decreasing with time. Note that we do not rule out the possible effects of some degree of incompleteness in this mass regime on the evolution of the low-mass end.
For E galaxies, however, we report a modest growth in their high-mass end (again when comparing with D10/ACS low-$z$) and a significant growth in intermediate- and low-mass regimes from $z = 1$ to $z = 0$.
Finally, H and C systems become less prominent with declining redshift as fewer galaxies occupy these classes. The physical implications of these trends will be discussed in Section \ref{sec:discussion}.
Figure \ref{fig:Mfunc_6z_par_evol} further summarizes the trends in Figure \ref{fig:Mfunc_6z}, showing the evolution of our best fit single Schechter parameters $\Phi^*$, $M^*$ and $\alpha$. We report the best Schechter fit parameters in Table \ref{tab:Morph_MF_par}. For comparison, we also show a compilation of literature values in the right panel. The literature data show only single Schechter parameters of the \textit{total} SMF. Comparing our \textit{total} SMF with other studies, including \cite{Muzzin13}, we find a good agreement within the quoted errors.
Note that the three parameters are, of course, highly correlated and the mass limits to which they are fitted vary.
The $\Phi^*$ of BD systems has the largest value and almost mimics the trend of the \textit{total} $\Phi^*$ which is largely consistent with no evolution. This is expected as BD galaxies dominate the sample at almost all redshifts. We report a slight decrease in the $\Phi^*$ value of D galaxies, while almost no evolution in E systems. Note that as mentioned earlier, our lowest redshift bin ($0.0 < z < 0.25$) contains only $1,108$ galaxies of which only $102$ are D systems incorporating an uncertainty in our fitting process. This can be seen in the deviation of the second data point of D galaxies (cyan lines) from other redshifts in Figure \ref{fig:Mfunc_6z_par_evol}.
H systems also contribute more at higher redshifts (higher $\Phi^*$ values) owing to the fact that the abundance of mergers, clumpy and disrupted systems dramatically increases at higher redshifts (despite lower resolution). We note that due to our LSS corrections (Section \ref{subsec:LSS_cor}) the $\Phi^*$ reported here is not the directly measured parameter but modified according to our LSS correction coefficients (reported in Table \ref{tab:rho}). This normalization smooths out the evolutionary trends, otherwise fluctuating due to the large scale structures, but does not impact their global trends.
The characteristic mass, $M^*$, of the BD systems presents a stable trend since $z~=~1$. We interpret this behaviour to be a result of the lack of significant evolution of the massive/intermediate-mass regime of the SMF of this morphological class. D galaxies also show slight evidence of evolution with some fluctuations in $M^*$ value. E galaxies, likewise, evolve only a small amount. Overall, we observe a behaviour consistent with no evolution in $M^*$ for all morphological types. Note that large variation of the $M^*$ in H and C types, at low-$z$ in particular, is not physical. This is massively driven by the lack of H and C galaxies at low-$z$ resulting in an unconstrained turnover and as can be seen in Figure \ref{fig:Mfunc_6z_par_evol} large uncertainties.
The low-mass end slope, $\alpha$, of different morphological classes also shows some evolution. Similar to D systems, BD galaxies show a marked steepening increase in their slope at later times, indicating that the SMF steepens at lower redshifts. The steepest mass function at almost all times is for the D galaxies. E galaxies occupy the lowest steepening but constantly growing from $\alpha = 0.34$ to $-0.75$ at $z \sim 0.2$.
\section{The Evolution of the Stellar Mass Density Since \lowercase{$z$} $= 1$}
\label{sec:rho}
In this section, we investigate the evolution of the Stellar Mass Density (SMD) as a function of morphological types. To determine the SMD, we integrate under the best fit Schechter functions over all stellar masses from $10^{9.5}$ to $\infty$. This integral can be expressed as a gamma function:
\begin{equation}
\rho_* = \int_{M=10^{9.5}}^{\infty} M^\prime \Phi(M^\prime) dM^\prime = \Phi^* M^* \Gamma(\alpha+2,10^{9.5}/M^*),
\label{eq:massDens}
\end{equation}
\noindent where $\Phi^*$, $M^*$ and $\alpha$ are the best regressed Schechter parameters. Figure \ref{fig:Mdens_6z} shows the evolution of the distribution of the SMDs, term $M^\prime \Phi(M^\prime)$, for different morphologies. We also illustrate the fitting errors by sampling 1000 times the full posterior probability distribution of the fit parameters. These are shown in Figure \ref{fig:Mdens_6z} as transparent shaded regions around the best fit curves. The standard deviations of the integrated SMD calculated from each of these functions are reported as the fit error on $\rho_*$ in Table \ref{tab:rho}. As can be seen in Figure \ref{fig:Mdens_6z}, the distribution of the stellar mass density of all individual morphological types in almost all redshift bins is bounded, implying that integrating under these curves will capture the majority of the stellar mass for each class.
Figure \ref{fig:MassBuildUp_LSS} then shows the evolution of the integrated SMD, $\rho_*$, in the Universe between $z = 0-1$.
This includes the LSS correction by forcing our \textit{total} SMD values to match the smooth spline fit to the \cite{Driver18} data, as described in Section \ref{subsec:LSS_cor}. The uncorrected evolutionary path of $\rho_*$ is shown in the upper panel of Figure \ref{fig:LSS_cor}.
We report the empirical LSS corrected $\rho_*$ values in Table \ref{tab:rho}. This table also provides the LSS correction factor, $\beta$, so one might obtain the original values by $\rho_*^{Orig} = \rho_*^{corr}/\beta$.
For completeness, we show the evolution of the SMDs before we apply the LSS corrections in Figure \ref{fig:SMD_evol_noLSS} of Appendix \ref{sec:SMD_evol_noLSS}. In this Figure, the trends are not as smooth as one would expect without the LSS correction being applied, given the evident structure in the $N(z)$ distribution from Figure \ref{fig:zdens}. However, even without the LSS corrections, the main trends are still present, albeit not as strong. This highlights that the LSS correction is an important aspect and also the need for much wider deep coverage than that currently provided by HST. This will become possible in the coming Euclid and Roman era.
Before we analyse the evolution of the SMDs in Figure \ref{fig:MassBuildUp_LSS}, below we investigate the errors that are involved in this calculation.
\begin{landscape}
\begin{figure}
\centering
\includegraphics[width = 1.3\textwidth, angle = 0]{Pics/SMD_morph_LSScor_newZbin_FINAL.png}
\caption{The distribution of \textit{total} and morphological stellar mass density in different redshifts. Redshift bins are the same as Figure \ref{fig:Mfunc_6z}. Points and lines indicate $M^\prime \Phi(M^\prime)$ in Equation \ref{eq:massDens}, where $\Phi(M^\prime)$ is our Schechter function fit. The shaded transparent regions represent the error range calculated by 1000 times sampling of the full posterior probability distribution of the fit parameters. The distribution of the stellar mass density of all individual morphological types in almost all redshift bins is bounded. Top row highlighted by yellow colour represents the GAMA data ($0 \le z \le 0.08$).}
\label{fig:Mdens_6z}
\end{figure}
\end{landscape}
\begin{figure*}
\centering
\includegraphics[width = \textwidth, angle = 0]{Pics/rho_Morph_newZbin_EC.png}
\caption{The evolution of the \textit{total} and morphological stellar mass density, $\rho_{*}$, in the last 8 Gyr of the cosmic age. $\rho_{*}$ expresses our measurements of the analytical integration under the best Schechter function fits in 8 redshift bins, i.e. Equation \ref{eq:massDens}. Highlighted region shows the epoch covered by the GAMA data ($0 \le z \le 0.08$). This figure includes the correction for the large scale structure, LSS, that we apply by fitting a smooth spline to the empirical data of GAMA/G10COSMOS/3D-HST by \protect\cite{Driver18}. See the text for more details on our method for this correction. The prediction of the SHARK semi-analytic model is also overlaid as dashed line. Colour codes are similar to Figure \ref{fig:Mfunc_6z}. Vertical bars on the points show the total error budget on each data point including: cosmic variance within the associated redshift bin taken from \protect\cite{Driver10}, classification error, fit error and Poisson error. See Section \ref{sebsec:morph_err_budget} for more details about our error analysis. Horizontal bars indicate the redshift ranges while the data points are plotted at the mean redshift.}
\label{fig:MassBuildUp_LSS}
\end{figure*}
\subsection{Analysing the Error Budget on the SMD}
\label{sebsec:morph_err_budget}
The error budget on our analysis of the SMD includes \textit{cosmic variance } (CV), \textit{fit error}, \textit{classification error} and \textit{Poisson error}. Cosmic variance and classification are the dominant sources of error.
We make use of \cite{Driver10} equation 3 to calculate the \textit{cosmic variance} in the volume encompassed within each redshift bin. This calculation is implemented in the R package: {\fontfamily{qcr}\selectfont celestial}\footnote{ The online version of this cosmology calculator is available at: \href{http://cosmocalc.icrar.org/}{http://cosmocalc.icrar.org/} }. Note that for our low-$z$ GAMA data, instead of using \cite{Driver10} equation that estimates a $\sim 22.8\%$ cosmic variance, we empirically calculate the CV using the variation of the SMD between 3 different GAMA regions of G09, G12, and G15 with a total effective area of 169.3 square degrees (G09:54.93; G12: 57.44; G15: 56.93, \citealt{Bellstedt20a}) and find CV to be $\sim 16\%$.
We measure the \textit{fit error} by 1000 times sampling of the full posterior probability distribution of the Schechter parameters (shown as shaded error regions in Figure \ref{fig:Mdens_6z}) and calculating the associated $\rho_*$ in each iteration.
We calculate the \textit{classification error} by measuring the stellar mass density associated with each of our 3 independent morphological classifications. The range of variation of the SMD between classifiers gives the error of our classification.
The \textit{Poisson error} is calculated by using the number of objects in each morphology per redshift bin.
The combination of all the above error sources will provide us with the total error that is reported in Table \ref{tab:rho}.
\\[2\baselineskip]
As can be seen in Figure \ref{fig:MassBuildUp_LSS}, the extrapolation of our D10/ACS $\rho_*$ to $z = 0$ agrees well with the our local GAMA estimations. Note slight difference in D, but consistent in errors.
The total change in the stellar mass is consistent with observed SFR evolution (e.g., \citealt{Madau14}; \citealt{Driver18}) as we will discuss more in Section \ref{sec:discussion}. Analysing the evolution of the SMD (Figure \ref{fig:MassBuildUp_LSS}), we find that in total (black symbols), 68\% of the current stellar mass in galaxies was in place $\sim 8$ Gyr ago ($z \sim 1.0$). The top panel of Figure \ref{fig:MassDensVar_fs} illustrates the variation of the $\rho_*$ ($\rho_{z}^{*} / \rho_{z=0}^{*}$), where $\rho_{z}^{*}$ is the SMD at redshift $z$ while $\rho_{z=0}^{*}$ represents the final SMD at $z~=~0$.
According to our visual inspections C types are closer to Es than other subcategories. We, therefore, combine C types with E galaxies that shows a smooth growth with time of a factor of $\sim 2.5$ up to $z = 0.25$ and flattens out since then ($0.0 < z < 0.25$). Note that as reported in Table \ref{tab:rho} the amount of mass in the C class is very little. This demonstrates a significant mass build-up in E galaxies over this epoch ($\sim 150\%$). We also note that the large error bars on some of E+C data points are dominantly driven by the large errors in C SMDs.
Having analytically integrated the SMD, we now measure the fraction of baryons in the form of stars ($f_s = \Omega_*/\Omega_b$) locked in each of our morphological types. We adopt $\Omega_b = 0.0493$ as estimated from Planck by \cite{Planck20} and the critical density of the Universe at the median redshift of GAMA, i.e. $z = 0.06$ to be $\rho_c = 1.21 \times 10^{11} \mathrm{M}_{\odot} \mathrm{Mpc}^{-3}$ in a 737 cosmology.
As shown in the bottom panel of Figure \ref{fig:MassDensVar_fs}, at our D10/ACS lowest redshift bin $\overline{z} \sim 0.18$ we find the fraction of baryons in stars $f_s \sim 0.033\pm0.009$. Including our GAMA measurements at $\overline{z} \sim 0.06$ we find this fraction to be $f_s \sim 0.035\pm0.006$. As shown in Figure \ref{fig:MassDensVar_fs}, this result is consistent with other studies within the quoted errors for example: \cite{Baldry06}, \cite{Baldry12} and \cite{Moffett16a}.
The evolution of the fraction of baryons locked in stars, $f_s$, shows that it has increased from $(2.4\pm0.5)$\% at $z \sim 1$ to $(3.5\pm0.6)$\% at $z \sim 0$ indicating an increase of a factor of $\sim 1.5$ during last 8 Gyr.
Figure \ref{fig:MassDensVar_fs} also shows the breakdown of the total $f_s$ to each of our morphological subcategories highlighting that as expected BD systems contribute the most to the stellar baryon fraction increasing from $f_s = 0.011\pm0.006$ to $f_s = 0.022\pm0.005$ at $0 < z < 1$.
E+C systems take less percentage of the total $f_s$ but increase their contribution from $0.005\pm0.001$ to $0.012\pm0.002$ while D galaxies decrease from $f_s = 0.006\pm0.002$ to $f_s = 0.001\pm0.005$ between $0 < z < 1$.
We report our full $f_s$ values for all morphological types at all redshifts in Table. \ref{tab:fs}.
In summary, over the last 8 Gyr, double component galaxies clearly dominate the overall stellar mass density of the Universe at all epochs. The second dominant system is E galaxies. However the extrapolation of the trends to higher redshifts in Figures \ref{fig:MassBuildUp_LSS} and \ref{fig:MassDensVar_fs} indicates that D systems are likely to dominate over Es in the very high-$z$ regime ($z > 1$), which is reasonable according to the rise of the cosmic star formation history and the association of star-formation with disks.
\begin{table*}
\centering
\caption{The \textit{total} and morphological stellar mass density in different redshift ranges displayed in Figure \ref{fig:MassBuildUp_LSS} (left panel). $\rho_*$ values are calculated from the integration under the best Schechter function fits. Note that the $\rho_*$ are presented after applying the LSS correction. Columns represent: $z-$bin: redshift bins, $\overline{z}$: mean redshift, LBT: lookback time, T (S-Schechter): \textit{total with single Schechter}, T (D-Schechter): \textit{total with double Schechter}, BD: \textit{bulge+disk}, pD: \textit{pure-disk}, E: \textit{elliptical}, C: \textit{compact}, H: \textit{hard}, $\beta$: the large scale structure correction factor. Errors incorporate all error sources including, cosmic variance, fit error, Poisson error and classification error (see Section \ref{sebsec:morph_err_budget} for details). }
\begin{adjustbox}{angle = 0, scale = 0.8}
\begin{tabular}{ccccccccccccc}
\firsthline \firsthline \\
& & & \multicolumn{6}{c}{log10$(\rho_*/M_\odot) $} \\ \\
\cline{4-9} \\
$z-$bin & $\overline{z}$ & LBT (Gyr) & T (S-Schechter) & T (D-Schechter) & BD & pD & E & C & H & $\beta$ \\ \\ \hline \\
$0.00 \le z < 0.08$ & $0.06$ & $0.82$ & $8.315\pm0.08$ & $8.316\pm0.11$ & $8.109\pm0.08$ & $6.977\pm0.26$ & $7.835\pm0.10$ & $-$ & $5.818\pm1.25$ & $1.40$ \\
$0.00 \le z < 0.25$ & $0.18$ & $2.27$ & $8.289\pm0.14$ & $8.289\pm0.14$ & $8.063\pm0.14$ & $6.735\pm0.53$ & $7.863\pm0.20$ & $5.332\pm0.83$ & $6.054\pm0.38$ & $0.89$ \\
$0.25 \le z < 0.45$ & $0.36$ & $3.95$ & $8.258\pm0.10$ & $8.257\pm0.10$ & $8.054\pm0.10$ & $7.025\pm0.17$ & $7.740\pm0.12$ & $5.513\pm1.23$ & $6.350\pm0.43$ & $0.74$ \\
$0.45 \le z < 0.60$ & $0.53$ & $5.23$ & $8.231\pm0.10$ & $8.230\pm0.16$ & $8.008\pm0.10$ & $7.376\pm0.12$ & $7.594\pm0.13$ & $5.936\pm1.01$ & $6.635\pm0.21$ & $1.21$ \\
$0.60 \le z < 0.70$ & $0.65$ & $6.05$ & $8.210\pm0.12$ & $8.210\pm0.21$ & $7.962\pm0.12$ & $7.395\pm0.13$ & $7.603\pm0.13$ & $5.922\pm0.60$ & $6.688\pm0.25$ & $0.74$ \\
$0.70 \le z < 0.80$ & $0.74$ & $6.55$ & $8.195\pm0.11$ & $8.194\pm0.12$ & $7.936\pm0.12$ & $7.468\pm0.12$ & $7.540\pm0.13$ & $6.032\pm0.95$ & $6.692\pm0.17$ & $0.81$ \\
$0.80 \le z < 0.90$ & $0.85$ & $7.07$ & $8.171\pm0.11$ & $8.170\pm0.13$ & $7.877\pm0.11$ & $7.497\pm0.12$ & $7.508\pm0.12$ & $6.258\pm0.55$ & $6.863\pm0.15$ & $0.57$ \\
$0.90 \le z \le 1.00$ & $0.95$ & $7.51$ & $8.150\pm0.11$ & $8.149\pm0.13$ & $7.795\pm0.11$ & $7.574\pm0.12$ & $7.473\pm0.12$ & $6.430\pm0.43$ & $6.946\pm0.14$ & $0.55$ \\
\\
\lasthline
\end{tabular}
\end{adjustbox}
\label{tab:rho}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Pics/MassDensVar_fsEvol_z006.png}
\caption{Top panel: the variation of the stellar mass density showing the fraction of final stellar mass density assembled or lost by each redshift, i.e. $\rho_{*z}/\rho_{*z=0}$. $\rho_{*z=0}$ is taken from GAMA estimations of the local Universe. Bottom panel: the evolution of the baryon fraction in form of stars ($f_s$). For simplicity, in this figure we only show the main morphological types and remove the \textit{hard} and \textit{compact} sub-classes. Highlighted region shows the epoch covered by the GAMA data.}
\label{fig:MassDensVar_fs}
\end{figure}
\section{Discussion}
\label{sec:discussion}
Making use of our morphological classifications, we explore the evolution of the stellar mass function at $0 \le z \le 1$, to assess the physical processes that are likely affecting the individual morphological SMF. In particular, major mergers are thought to primarily occur between comparable mass companions (1:3) resulting in the fast growth of the SMF \citep{Robotham14}. Conversely, secular activities, minor mergers and tidal interactions will primarily alter the number density at the low-stellar mass end of the SMF \citep{Robotham14}. Investigating the \textit{total} SMF (Figure \ref{fig:Mfunc_6z}) shows that unlike high-mass end the low-mass end grows significantly. This suggests that since $z~\sim~1$, the galaxy population goes through mainly in-situ/secular processes at the low-mass end.
An important caveat, in what follows, is that although we have undertaken multiple tests of our visual classifications, possible uncertainties due to human classification error or inconsistencies will be present. Nevertheless some clear trends, which we believe are resilient to the classifications uncertainties are evident. Furthermore, we do not rule out the effects of dust in distinguishing bulges, particularly at high-$z$ leading to overestimating the number of \textit{pure-disk} systems at high redshifts. Although, high level of agreement in our visual classifications (see Figure \ref{fig:agreent}) gives us some confidence that our evolutionary trends are unlikely dominated by incorrect classifications, it is possible that all classifiers agree on the incorrect classification.
Firstly, double component galaxies display a modest decrease with redshift in the high-stellar mass end of their SMF (Figure \ref{fig:Mfunc_6z}), whilst their low-stellar mass end steepens significantly. This could be interpreted as most of the stellar mass in double component systems evolving via lower mass weighted star formation or secular activity, rather than major merging, i.e., BD systems are not merging together to form higher-mass BD systems.
Secondly, \textit{pure-disk} systems show a strong increase at the low-mass end and a {\it decrease} at their high-mass end. The low-mass end evolution suggests in-situ star formation of the disk and/or the formation of new disks. However, the decrease at the high-mass end is to some extent unphysical, unless these systems are undergoing a transformation from the D class to another class. The most likely prospect is the secular formation of a central bulge component, resulting in a morphological transformation into the BD class. Hence, as time progresses and the second component forms, galaxies exit the D class leading to a mass-deficit at the high-mass end.
The new component forming through such an in-situ process is most likely a pseudo-bulge (pB), resulting in mass loss from the high-mass D SMF and a corresponding mass gain at comparable mass in the BD class.
Finally, \textit{elliptical} galaxies, generally thought to be inert, show little growth in their high-mass end but a significant growth of low- and intermediate mass end presumably due to mergers.
The evolution of the global and morphological SMDs indicates that BD galaxies dominate ($\sim 60\%$ on average) the stellar mass density of the Universe since at least at $z < 1$. This morphological class also shows a constant mass growth, compared to other morphologies. As mentioned, D systems slightly decrease their stellar mass density with time. Presumably, the mass transfer out of the D class out-weighs the mass gain due to in-situ star-formation for this class.
We remark that the \textit{rate} of mass growth in the BD systems also decreases with time. This is reflected in the decrease of their SMD slope although it is still steeper than the \textit{total} SMD evolution and consistent with the general decline in the cosmic star-formation history. On the other hand, the E galaxies experience an initial growth until $z\sim0.2$ and a recent flattening in their stellar-mass growth from $z=1-0$ (see Figure \ref{fig:MassBuildUp_LSS} and the top panel of Figure \ref{fig:MassDensVar_fs}).
\begin{table*}
\centering
\caption{\textit{Total} and morphological stellar baryon fraction ($f_s$) in different times. See the text for details.}
\begin{adjustbox}{angle = 0, scale = 0.8}
\begin{tabular}{lcccccccc}
\firsthline \firsthline \\
& \multicolumn{7}{c}{Redshift} \\ \\
\cline{2-8} \\
Morphology Type & $0.00 \le z < 0.08$ & $0.00 \le z < 0.25$ & $0.25 \le z < 0.45$ & $0.45 \le z < 0.60$ & $0.60 \le z < 0.70$ & $0.70 \le z < 0.80$ & $0.80 \le z < 0.90$ & $0.90 \le z < 1.0$ \\ \hline \\
Total & $0.035\pm0.006$ & $0.033\pm0.009$ & $0.030\pm0.006$ & $0.029\pm0.006$ & $0.027\pm0.006$ & $0.026\pm0.006$ & $0.025\pm0.005$ & $0.024\pm0.005$ \\
Double & $0.022\pm0.004$ & $0.019\pm0.005$ & $0.019\pm0.004$ & $0.017\pm0.004$ & $0.015\pm0.004$ & $0.015\pm0.003$ & $0.013\pm0.003$ & $0.010\pm0.002$ \\
Pure-Disk & $0.002\pm0.001$ & $0.001\pm0.001$ & $0.002\pm0.001$ & $0.004\pm0.001$ & $0.004\pm0.001$ & $0.005\pm0.001$ & $0.005\pm0.001$ & $0.006\pm0.001$ \\
Elliptical & $0.012\pm0.002$ & $0.012\pm0.004$ & $0.009\pm0.002$ & $0.007\pm0.002$ & $0.007\pm0.002$ & $0.006\pm0.002$ & $0.005\pm0.001$ & $0.005\pm0.001$ \\
Compact & $-$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ \\
E+C & $0.012\pm0.002$ & $0.012\pm0.004$ & $0.009\pm0.002$ & $0.007\pm0.002$ & $0.007\pm0.002$ & $0.006\pm0.002$ & $0.006\pm0.001$ & $0.005\pm0.001$ \\
Hard & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.000\pm0.000$ & $0.001\pm0.000$ & $0.001\pm0.000$ & $0.001\pm0.000$ & $0.001\pm0.000$ & $0.001\pm0.000$ \\
\lasthline
\end{tabular}
\end{adjustbox}
\label{tab:fs}
\end{table*}
\section{Summary and Conclusions}
\label{sec:summary}
We have presented a visual morphological classification of a sample of galaxies in the DEVILS/COSMOS survey with HST imaging, from the D10/ACS sample. The quality of the imaging data (HST/ACS) provides arguably the best current insight into galaxy structures and therefore the best pathway with which to explore morphological evolution.
We summarize our results as below:
\begin{description}
\item[$\bullet$] By visually inspecting galaxies out to $z \sim 1.5$, we find that morphological classification becomes far more challenging at $z > 1.0$ as many galaxies ($> 40\%$) no longer adhere to the classical notion of spheroids, bulge+disk or disk systems (\citealt{Abraham01}). We see a dramatic increase in strongly disrupted systems, presumably due to interactions and gas clumps. Nevertheless, at all redshifts we find that more mass is involved in star-formation than in merging and most likely it is the high star-formation rates that are driving the irregular morphologies.
\item[$\bullet$] The SMF of the D10/ACS sample in our lowest redshift bin ($z<0.25$) is consistent with previous measurements from the local Universe.
\item[$\bullet$] The evolution of the global SMF shows enhancement in both low- and slightly in high-mass ends. We interpret this as suggesting that at least two evolutionary pathways are in play, and that both are significantly impacting the SMF.
\item[$\bullet$] Despite a slight decrease in their high-mass end, BD systems demonstrate a non-negligible growth in the low-mass end of their SMF. In the D type, we witness a significant variation in both high- and low-mass ends of the SMF. We interpret the high-mass end decrease in D systems, which is at first sight unphysical, as an indication of significant secular mass transfer through the formation of pseudo-bulges and hence an apparent mass loss as galaxies transit to the BD category.
\item[$\bullet$] E galaxies experience a modest growth in their high-mass end as well as an enhancement in their low/intermediate-mass end which we interpret as a consequence of major mergers resulting in the relentless stellar mass growth of this class.
\item[$\bullet$] Despite the above shuffling of mass we find that the best regressed Schechter function parameters in the \textit{total} SMF are observed to be relatively stable from $z = 1$. This is consistent with previous studies (\citealt{Muzzin13}; \citealt{Tomczak14}; \citealt{Wright18}). Conversely, the component SMFs show significant evolution. This implication is that while stellar-mass growth is slowing, mass-transformation processes via merging and in-situ evolution are shuffling mass between types behind the scenes.
\item[$\bullet$] We measured the integrated total stellar mass density and its evolution since $z = 1$ and find that approximately $32\%$ of the current stellar mass in the galaxy population was formed during the last 8 Gyr.
\item[$\bullet$] As shown in Figure \ref{fig:MassBuildUp_LSS}, we find that the BD population dominates the SMD of the Universe within $0 \le z \le 1$ and has constantly grown by a factor of $\sim 2$ over this timeframe. On the other hand, the SMD of Ds declines slowly and eventually loses $\sim 85\%$ of it's original value until $z \sim 0.2$. On the contrary, the E population experiences constant growth of a factor of $\sim 2.5$ since $z = 1$. We observe that the extrapolation of the trends of all of our morphological SMD estimations meets GAMA measurements at $z = 0$ (see Figure \ref{fig:MassBuildUp_LSS}) except for the \textit{pure-disk} systems which is likely due to unbound distribution of their SMD (see Figure \ref{fig:Mdens_6z}).
\end{description}
One clear outcome of our analysis is that the late Universe ($z<1$) appears to be a time of profound spheroid and bulge growth/emergence. To move forward and explore this further we conclude that to move forward it is key to decompose the double component morphological type, which significantly dominates the stellar mass density, into disks, classical bulges, and pseudo-bulges. To do this, requires robust structural decomposition which we will describe in Hashemizadeh et al. (in prep.) using our morphological classifications to guide the decomposition process.
\section{Acknowledgements}
DEVILS is an Australian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The DEVILS input catalogue is generated from data taken as part of the ESO VISTA-VIDEO \citep{Jarvis13} and UltraVISTA \citep{McCracken12} surveys. DEVILS is part funded via Discovery Programs by the Australian Research Council and the participating institutions. The DEVILS website is \href{https://devilsurvey.org}{https://devilsurvey.org}. The DEVILS data is hosted and provided by AAO Data Central (\href{https://datacentral.org.au}{https://datacentral.org.au}). This work was supported by resources provided by The Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. We also gratefully acknowledge the contribution of the entire COSMOS collaboration consisting of more than 200 scientists. The HST COSMOS Treasury program was supported through NASA grant HST-GO-09822. SB and SPD acknowledge support by the Australian Research Council's funding scheme DP180103740. MS has been supported by the European Union's Horizon 2020 research and innovation programme under the Maria Skłodowska-Curie (grant agreement No 754510), the National Science Centre of Poland (grant UMO-2016/23/N/ST9/02963) and by the Spanish Ministry of Science and Innovation through Juan de la Cierva-formacion program (reference FJC2018-038792-I). ASGR and LJMD acknowledge support from the Australian Research Council's Future Fellowship scheme (FT200100375 and FT200100055, respectively).
This work was made possible by the free and open R software (\citealt{R-Core-Team}).
A number of figures in this paper were generated using the R \texttt{magicaxis} package (\citealt{Robotham16b}). This work also makes use of the \texttt{celestial} package (\citealt{Robotham16a}) and \texttt{dftools} (\citealt{Obreschkow18}).
\section{Data Availability}
\label{sec:AvailData}
In this work, we draw upon two datasets; the established HST imaging of the COSMOS region (\citealt{Scoville07}, \citealt{Koekemoer07}), and multiple data products produced as part of the DEVILS survey \citep{Davies18}, consisting of a spectroscopic campaign currently being conducted on the Anglo-Australian Telescope, photometry (Davies et al. in prep.), and deep redshift catalogues, and stellar mass measurements from \cite{Thorne20}.
These datasets are briefly described below.
\bibliographystyle{mnras} |
1902.06196 | \section{Introduction}
\label{sec:intro}
\subsection{Digital fingerprinting}
Fingerprinting techniques for digital content provide a way for copyright holders to uniquely mark each copy of their content, to prevent unauthorized redistribution of this content: if a digital ``pirate'' nevertheless decides to publicly share his (fingerprinted) content with others, the owner of the content can obtain this copy, extract the fingerprint, link it to the responsible user, and take appropriate steps. Digital pirates may try to prevent being caught by collaborating, and forming a mixed copy of the content from their individual copies, thus mixing up the embedded fingerprint as well. To guarantee that collusions of pirates cannot get away with this, \textit{collusion-resistant fingerprinting schemes} are needed.
Mathematically speaking, collusion-resistant fingerprinting can be modeled as follows. First, the content owner generates code words $\vc{x}_j \in \{0,1\}^{\ell}$ for $j = 1, \dots, n$, corresponding to fingerprints for the $n$ users, where each of the $\ell$ columns defines one segment of the content. Then, a collusion $\mathcal{C}$ of $c$ colluders applies a mixing strategy to their code words $\{\vc{x}_j\}_{j \in \mathcal{C}}$ to form a new pirate copy $\vc{y}$. Here the critical condition we impose on this mixing procedure is the \textit{marking assumption}, stating that if $x_{j,i} = b$ for all $j \in \mathcal{C}$, then also $y_i = b$, for $b \in \{0,1\}$. Finally, the owner of the content obtains $\vc{y}$, applies a decoding algorithm to $\vc{y}$ and all the code words $\{\vc{x}_j\}_{j=1}^n$, and outputs a subset of user indices $j$. This method is successful if, with high probability over the randomness in the code generation and mixing strategy, the decoding algorithm outputs (a subset of) the colluders, without incriminating any innocent, legitimate users.
\subsection{Related work}
In the late 1990s, Boneh--Shaw~\cite{boneh98} were the first to design a somewhat practical, combinatorial solution for collusion-resistant fingerprinting. Their construction based on error-correcting codes achieved high success probabilities with a code length of $\ell = O(c^4 \log n)$, i.e.\ scaling logarithmically in the often large number of users $n$, and quarticly in the number of colluders $c$. The milestone work of Tardos~\cite{tardos03} later improved upon this with a code length $\ell = O(c^2 \log n)$, and he proved that this quadratic scaling in $c$ is optimal. Later work focused on bringing down the leading constants in Tardos' scheme~\cite{skoric08b, blayer08, furon08, furon09c, meerwald12, oosterwijk13, furon14, skoric15}, leading to an optimal code length of $\ell \propto \frac{1}{2} \pi^2 c^2 \ln n$ for the original (symmetric) Tardos score function~\cite{skoric08, laarhoven14dcc, laarhoven13ihmmsec}, and an optimal overall code length of $\ell \propto 2 c^2 \ln n$ when using further improved decoders~\cite{oosterwijk13, oosterwijk13b, desoubeaux13, laarhoven14ihmmsec, furon14}.
The main focus of most literature on collusion-resistant fingerprinting has been on decreasing $\ell$ -- the shorter the fingerprints, the faster the colluders can be traced. Other aspects of these fingerprinting schemes, however, have received considerably less attention in the literature, and in particular the decoding procedure is often neglected. Indeed, with $n$ users and a code length of $\ell = O(c^2 \log n)$, the decoding time is commonly $O(\ell \cdot n) = O(c^2 n \log n)$ for linear decoders, and up to $O(c^2 n^k \log n)$ for \textit{joint decoders}, attempting to decode to groups of $k \leq c$ colluders simultaneously. In particular, past work has shown that joint decoders achieve superior performance to simple decoders~\cite{amiri09, charpentier09, furon09b, meerwald12, furon12, berchtold12, laarhoven14ihmmsec, laarhoven16phd}, but are often considered infeasible due to their high decoding complexity. Moreover, in \textit{dynamic} settings~\cite{laarhoven13tit, laarhoven15ihmmsec} where decisions about the accusation of users need to be made swiftly, an efficient decoding method is even more critical. Techniques that can speed up the decoding procedure may therefore be useful for further improving these schemes in practice.
\subsection{Contributions}
In this work, we study how the decoding method in the score-based fingerprinting framework can be improved, leading to faster decoding times. In particular, we show that we can typically bring down the decoding costs of simple decoders from $O(\ell n)$ to $O(\ell n^{\rho})$ where $\rho \leq 1$ is determined by $c$ and the instantiation of the score-based framework. This usually comes at a higher space requirement for indexing the code words in a query-efficient data structure, although arbitrary trade-offs between the space and query time can be obtained by tweaking the parameters.
To obtain these improved results, we show that we can model the decoding procedure of Tardos-like fingerprinting as a high-dimensional \textit{nearest neighbor search} problem. This field of research studies methods of storing data in more refined data structures, such that highly similar vectors to any given query point can be found faster than with a linear search through the data. Applying practical, state-of-the-art techniques from this area, such as the locality-sensitive hashing mechanisms of~\cite{charikar02, andoni15cp} and the more recent locality-sensitive filtering of \cite{becker16lsf, andoni17}, this allows us to obtain faster, sublinear decoding times, both in theory and in practice. We give a recipe how to apply these techniques to any score-based fingerprinting framework, and give an explicit, detailed analysis of what happens when we use these techniques in combination with the symmetric score function of \v{S}kori\'{c}--Katzenbeisser--Celik~\cite{skoric08}.
To illustrate these results for the symmetric score function, let us provide some explicit decoding costs for the case of $c = 2$. We obtain a decoding time of $O(\ell n^{3/4})$ without using additional memory; a decoding cost of $O(\ell n^{1/3})$ when using $O(\ell n^{4/3})$ memory; and a subpolynomial decoding cost $\ell n^{o(1)}$ when using up to $O(\ell n^4)$ storage. For larger $c$ the improvement becomes smaller, and in the limit of large $c$ nearest neighbor techniques offer only minor improvements over existing decoding schemes -- the main benefit lies in defending against small or moderate collusion sizes.
\subsection{Outline.}
The remainder of this paper is organized as follows. In Section~\ref{sec:pre} we first introduce notation, we describe the score-based fingerprinting framework, and we cover basics on nearest neighbor searching. Section~\ref{sec:main} describes how to apply nearest neighbor techniques to fingerprinting, and analyzes the theoretical impact of this application. Section~\ref{sec:exp} covers basic experiments, to illustrate the potential effects of these techniques in practice, and Section~\ref{sec:disc} finally discusses other aspects of our proposed improvement.
\newpage
\section{Preliminaries}
\label{sec:pre}
\begin{table}
\caption{Notation used throughout the paper. The first five rows indicate how concepts in fingerprinting translate to concepts in nearest neighbor searching.}
\label{tab:not}
\begin{tabular}{clccl}
\toprule
& FP terminology & & & NNS terminology \\
\midrule
$n$ & Number of users & $\sim$ & $n$ & Number of points \\
$\ell$ & Code length & $\sim$ & $d$ & Dimension of data \\
$\vc{x}_j$ & Code word & $\sim$ & $\vc{v}_j$ & Data point \\
$\vc{y}$ & Pirate copy & $\sim$ & $\vc{q}$ & Query point \\
$s_j$ & User score & $\sim$ & $\langle \vc{v}_j, \vc{q} \rangle$ & Dot product \\
\midrule
$c$ & Colluders & & $\alpha$ & Approximation factor \\
$\vc{p}$ & Probability vector & & $\rho_q$ & Query time exponent \\
$g$ & Score function & & $\rho_s$ & Space exponent \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Score-based fingerprinting schemes}
We first recall the score-based fingerprinting framework, introduced by Tardos~\cite{tardos03}, and describe the model considered in this paper. The fingerprinting game consists of three phases: (1) \textit{encoding}, generating the fingerprints and embedding them in the content; (2) the \textit{collusion attack}, constructing the mixed fingerprint; and (3) \textit{decoding}, mapping the mixed fingerprint to a set of accused users.
\subsubsection{Encoding.} First, the copyright holder generates code words $\vc{x}_j \in \{0,1\}^{\ell}$ for each of the users $j = 1, \dots, n$. To do this, he first generates a probability vector $\vc{p} \in [0,1]^{\ell}$, where each coordinate $p_i$ is drawn independently from some fixed probability distribution $F$. In the original Tardos scheme, and many of its variants, a truncated version of the arcsine distribution is used, which has cumulative density function $F$ given below.
\begin{align}
F(p) := \frac{2}{\pi} \arcsin \sqrt{p}. \qquad (0 \leq p \leq 1) \label{eq:F}
\end{align}
For small $c$ and the symmetric Tardos score function, certain discrete distributions are known to be optimal~\cite{nuida09}, leading to a better performance and shorter code lengths. For large $c$, these optimal discrete distributions converge to the arcsine distribution~\eqref{eq:F}.
After generating each entry $p_i$ from the chosen bias distribution, the code words for the users are generated as follows: for each user $j$, the $i$th entry of their codeword $x_{j,i}$ is set to $1$ with probability $p_i$, and $0$ with probability $1 - p_i$. This assignment is done independently for each $i$ and $j$. These fingerprints are then embedded in the content and sent to the users. (Note that it is crucial that the bias vector $\vc{p}$ remains secret, and is not known to the colluders during the attack.)
\subsubsection{Collusion attack.} Given a collusion $\mathcal{C}$ of some size $c$, the pirates employ a strategy $\theta$, mapping their code words $\{\vc{x}_j\}_{j \in \mathcal{C}}$ to a mixed copy $\vc{y} \in \{0,1\}^{\ell}$. Often the attack is modeled by a vector $\vc{\theta} = (\theta_0, \dots, \theta_c)$, where $\theta_k := \Pr(y_i = 1 | \sum_{j \in \mathcal{C}} x_{j,i} = k)$. By the marking assumption, we assume $\theta_0 = 0$ and $\theta_c = 1$.
\subsubsection{Decoding.} Given $\vc{y}$, the content holder attempts to deduce who were responsible for creating this pirate copy. For this, he computes scores $s_{j,i} := g(x_{j,i}, y_i, p_i)$ for some score function $g$, and then computes the total user scores as $s_j := \sum_{i=1}^n s_{j,i}$. For well-chosen parameters, the cumulative scores $s_j$ are significantly higher for colluders than for innocent users. The actual decision whom to accuse is then made by e.g.\ setting a threshold $z$ and accusing users $j$ with $s_j > z$, or by accusing the user with the highest cumulative score. As an example, the symmetric score function of \v{S}kori\'{c}--Katzenbeisser--Celik~\cite{skoric08} is given below.
\begin{align}
g(x, y, p) := \begin{cases}
+\sqrt{p/(1-p)}, & \text{if } x=0 \text{ and } y=0; \\
-\sqrt{(1-p)/p}, & \text{if } x=1 \text{ and } y=0; \\
-\sqrt{p/(1-p)}, & \text{if } x=0 \text{ and } y=1; \\
+\sqrt{(1-p)/p}, & \text{if } x=1 \text{ and } y=1.
\end{cases}
\end{align}
\subsubsection{Equivalent decoding.} Since the decoding step remains equally valid after linear transformations (i.e.\ scaling all user scores by a common positive factor, or shifting all user scores by the same amount), the following scoring function is equivalent to the symmetric score function described above:
\begin{align}
\hat{g}(x, y, p) := \begin{cases}
+1/\sqrt{p(1-p)}, & \text{if } x = y; \\
-1/\sqrt{p(1-p)}, & \text{if } x \neq y.
\end{cases}
\end{align}
To see why, note that the contribution of a segment for the symmetric score function, in terms of how far the scores for a match and a difference are apart, is independent of $y$:
\begin{align}
g(1,1,p) - g(0,1,p) = g(0,0,p) - g(1,0,p) = \frac{1}{\sqrt{p(1-p)}}.
\end{align}
In other words, as long as the difference between a match and a difference in segment $i$ is proportional to $1/\sqrt{p_i(1-p_i)}$ (with a positive contribution for a match, and a negative contribution for a difference), this only constitutes a scaling/transformation of the scores. By scaling the scores by a factor $2$, and centering the scores at $0$, we obtain the score function $\hat{g}$. (Note that the threshold $z$ will have to be scaled and translated by the same amounts to guarantee equivalent error probabilities.)
\subsection{(Approximate) nearest neighbor searching}
Next, let us recall some definitions, techniques, and results from nearest neighbor searching (NNS). Given a data set $\{\vc{v}_1, \dots, \vc{v}_n\} \subset \mathbb{R}^d$, this problem asks to index these points in a data structure such that, when later given a query vector $\vc{q} \in \mathbb{R}^d$, one can quickly identify the nearest vector to $\vc{q}$ in the data set. To measure the performance of an NNS method, we consider the space complexity $S = O(n^{1 + \rho_s})$ and the query time complexity $T = O(n^{\rho_q})$ to process a query $\vc{q}$. Note that a naive linear search, without any indexing of the data, achieves $T = S = O(n)$ or $\rho_s = 0$ and $\rho_q = 1$. Ideally a good NNS method should achieve $\rho_q < 1$, perhaps with $\rho_s > 0$.
In this paper we restrict our attention to the NNS problem on the \textit{unit sphere}, under the $\ell_2$-norm: we assume that $\|\vc{v}_j\| = 1$ for all $j$, and $\|\vc{q}\| = 1$. The following lemma states that, if the entire data set has a small dot product $\langle \vc{v}_j, \vc{q} \rangle := \sum_{j=1}^d v_{j,d} q_j$ with $\vc{q}$, except for one near neighbor $\vc{v}_{j^*}$, which has a large dot product with $\vc{q}$, then finding this unique near neighbor can be done efficiently in sublinear time. The parameter $\alpha \geq 1$ below is commonly referred to as the \textit{approximation factor} (denoted $c$ in e.g.\ \cite{andoni17}) -- one obtains a sublinear time complexity for NNS only when either an approximate solution suffices, or there is a guarantee that the data set contains unique nearest neighbors which are a factor $\alpha$ closer (under the $\ell_2$-norm) than all other vectors in the data set.
\begin{lemma}[NNS complexities~\cite{andoni17}] \label{lem:nns}
Suppose that the data points $\vc{v}_i$ and query $\vc{q}$ have norm $1$, and we are given two guarantees:
\begin{itemize}
\item For the nearest neighbor $\vc{v}_{j^*}$, we have $\langle \vc{v}_{j^*}, \vc{q} \rangle \geq d_1$;
\item For all other vectors $\vc{v}_j \neq \vc{v}_{j^*}$, we have $\langle \vc{v}_j, \vc{q} \rangle \leq d_0$.
\end{itemize}
Let $\alpha = \sqrt{1 - d_0} / \sqrt{1 - d_1} \geq 1$, and let $\rho_q, \rho_s \geq 0$ satisfy:
\begin{align}
\alpha^2 \sqrt{\rho_q} + (\alpha^2 - 1) \sqrt{\rho_s} \geq \sqrt{2\alpha^2 - 1}. \label{eq:tradeoff}
\end{align}
Then we can construct a data structure with $\tilde{O}(n^{1 + \rho_s})$ space and preprocessing time, allowing to answer any query $\vc{q}$ correctly (with high probability) with query time complexity $\tilde{O}(n^{\rho_q})$.
\end{lemma}
To achieve the above complexities, at a high level the data structure looks as follows. Given the normalized data points $\vc{v}_j$, all lying on the unit sphere, we first sample many random vectors $\vc{r}_k$ on the sphere, and for each of these vectors we store which vectors are close to $\vc{r}_k$ in a bucket $B_k$. The key property we use here is (approximate) \textit{transitivity} of closeness on the sphere: if $\vc{x}$ and $\vc{y}$ are close, and $\vc{y}$ and $\vc{z}$ are close, then also $\vc{x}$ and $\vc{z}$ are more likely to be close than usual. In other words, if $\vc{v}_j$ is close to $\vc{q}$, and $\vc{v}_j$ is close to $\vc{r}_k$ (and contained in bucket $B_k$), then likely $\vc{q}$ will also be close to $\vc{r}_k$. Therefore, if we create and index many of these buckets $B_k$ and, given $\vc{q}$, we compute the dot products of $\vc{q}$ with the vectors $\vc{r}_k$ and only check those buckets $B_k$ for potential near neighbors for which $\langle \vc{q}, \vc{r}_k \rangle$ is large, then we may only check a small fraction of the entire data set for potential nearest neighbors, while still finding all near neighbors. (The actual state-of-the-art techniques from~\cite{andoni17} are slightly more sophisticated. For further details, we refer the reader to~\cite{andoni15cp, andoni17, becker16lsf}.)
Note that Lemma~\ref{lem:nns} only states the \textit{scaling} behavior of the time and space complexities, and does not state how large the real overhead is in practical scenarios. For actual applications of NNS techniques, we refer the reader to e.g.\ the benchmarks of~\cite{aumueller17}, comparing implementations of various NNS techniques for their practicality on real-world data sets, including data sets on the sphere.
\section{Nearest neighbor decoding}
\label{sec:main}
\subsection{Score-based decoding as an NNS problem}
To apply NNS techniques to score-based fingerprinting, let us first show how we can phrase the decoding step of score-based fingerprinting as an NNS problem on the sphere. First, we map the $n$ code words $\vc{x}_j \in \{0,1\}^{\ell}$ to $n$ data points $\vc{v}_j \in \{-1,1\}^{\ell}$ by the linear operation $\vc{v}_j = 2 \vc{x}_j - \vc{1}$: a $1$ in $\vc{x}_j$ is mapped to a $1$ in $\vc{v}_j$, and a $0$ in $\vc{x}_j$ to a $-1$ in $\vc{v}_j$. Next, given $\vc{y} \in \{0,1\}^{\ell}$, we map it to a query vector $\vc{q}$ as $q_i = (2 y_i - 1) / \sqrt{p_i (1 - p_i)}$: the entries of $\vc{q}$ are $\pm 1 / \sqrt{p_i (1 - p_i)}$, depending on the value of $y_i$. Note that the Euclidean norms of the data and query vectors are given by:
\begin{align}
\|\vc{v}_j\| &= \sqrt{\ell}, \qquad %
\|\vc{q}\| = \sqrt{\sum_{i=1}^{\ell} \frac{1}{p_i (1 - p_i)}}. \label{eq:norms}
\end{align}
To guarantee that all vectors are normalized, we will later have to scale everything down by $\|\vc{v}_j\|$ and $\|\vc{q}\|$ accordingly. Observe that with the modified symmetric score function $\hat{g}$, the user score $s_j$ can now be equivalently expressed in terms of dot products as follows:
\begin{align}
s_j = \sum_{i=1}^{\ell} \hat{g}(x_{j,i},y_i,p_i) = \sum_{i=1}^{\ell} \frac{(2 x_{j,i} - 1)(2 y_i - 1)}{\sqrt{p_i (1-p_i)}} = \langle \vc{v}_j, \vc{q} \rangle.
\end{align}
Therefore, a user score $s_j$ is large iff the dot product between $\vc{v}_j$ and $\vc{q}$ is large, and $\vc{v}_j$ and $\vc{q}$ are near neighbors in space.
With the above translation in mind, we can now apply the aforementioned NNS techniques. To apply Lemma~\ref{lem:nns}, after normalization we need to provide two guarantees:
\begin{itemize}
\item A nearest neighbor $\vc{v}_{j^*}$ (i.e.\ a code word $\vc{x}_{j^*}$ of a colluder $j^* \in \mathcal{C}$) must have a large dot product with the query vector $\vc{q}$ (i.e.\ must have a high score $s_j$);
\item Other neighbors $\vc{v}_j$ (i.e.\ innocent users $j$) must have a small dot product with $\vc{q}$ (i.e.\ must have a low score $s_j$).
\end{itemize}
For this, we could derive similar proven bounds on $\Pr(s_j > z)$ for innocent and guilty users, as previously done in e.g.\ \cite{tardos03, skoric08, skoric08b, blayer08, laarhoven14dcc}, taking into account that the scores have been transformed. Instead let us give a slightly informal, high-level description of what these results may be.
Let $H_0$ be the hypothesis that user $j$ is innocent, and $H_1$ the hypothesis that user $j$ is a colluder. Let $\mu_b = \mathbb{E}_{\vc{p}, \vc{x}_j, \vc{y}}(s_j | H_b)$ for $b \in \{0,1\}$. By the central limit theorem, cumulative user scores are distributed approximately normally for large $\ell$, and if both variances $\sigma_b^2 = \mathbb{E}_{\vc{p}, \vc{x}_j, \vc{y}}(s_j^2 | H_b) - \mu_b^2$ are small, we may conclude that with high probability these scores are closely concentrated around their means. Then we can estimate the parameters $d_0$ and $d_1$ for Lemma~\ref{lem:nns}, after normalization, as follows:
\begin{align}
d_0 &= \frac{\mu_0}{\|\vc{v}_j\| \cdot \|\vc{q}\|}, \qquad %
d_1 = \frac{\mu_1}{\|\vc{v}_j\| \cdot \|\vc{q}\|}.
\end{align}
Here the expressions for $\|\vc{v}_j\|$ and $\|\vc{q}\|$ follow from~\eqref{eq:norms}. Note however that both $\mu_0$ and $\mu_1$ are likely dependent on the collusion attack, and may not be known in advance, before the decoding stage.
\subsection{Two colluders}
Let us first investigate the simplest case of $c = 2$, i.e.\ having two colluders. Under the assumption that the colluders work symmetrically, there is no collusion strategy to consider: if they have the same symbol, they output this symbol, and if they receive both a $0$ and a $1$ they can choose either with equal probability (i.e.\ $\vc{\theta} = (0, \frac{1}{2}, 1)$). For $c = 2$ the best choice is to fix $p_i = \frac{1}{2}$ for all $i$, leading to uniformly random codes. This implies $\|\vc{q}\| = 2 \sqrt{\ell}$, $\mu_0 = 0$ and $\mu_1 = \ell$, resulting in $d_0 = 0$ and $d_1 = \frac{1}{2}$, as in Table~\ref{tab:num}. In Lemma~\ref{lem:nns} this leads to an approximation factor $\alpha = \sqrt{2}$, and a trade-off between the time and space exponents $\rho_q$ and $\rho_s$ of:
\begin{align}
2 \sqrt{\rho_q} + \sqrt{\rho_s} \geq \sqrt{3}.
\end{align}
Without increasing the memory (i.e.\ for $\rho_s = 0$), we obtain $\rho_q \geq \frac{3}{4}$, i.e.\ an asymptotic query time complexity for decoding of $O(n^{3/4} \ell)$. Setting $\rho_q = \rho_s$ we obtain $\rho_q \geq \frac{1}{3}$, i.e.\ with $O(n^{4/3} \ell)$ memory, we can obtain a query complexity of $O(n^{1/3} \ell)$. With a large amount of memory and preprocessing time, we can further get a subpolynomial query time $n^{o(1)} \ell$ at the cost of $O(n^4 \ell)$ memory.
\begin{table}
\caption{Numerical data for the interleaving attack, using the optimal discrete distributions of~\cite{nuida09}. The last three columns correspond to three extreme time--space trade-offs: (I) $\rho_q$ for $\rho_s = 0$; (II) $\rho_q$ for $\rho_q = \rho_s$; and (III) $\rho_s$ for $\rho_q = 0$.}
\label{tab:num}
\begin{tabular}{c|cccccc|ccc}
\toprule
$c$ & $\frac{\|\vc{q}\|}{\sqrt{\ell}}$ & $\frac{\mu_0}{\ell}$ & $\frac{\mu_1}{\ell}$ & $d_0$ & $d_1$ & $\alpha$ & I & II & III \\
\midrule
$1$ & $2.00$ & $0.00$ & $2.00$ & $0.00$ & $1.00$ & $\infty$ & $0.00$ & $0.00$ & $0.00$ \\
$2$ & $2.00$ & $0.00$ & $1.00$ & $0.00$ & $0.50$ & $1.41$ & $0.75$ & $0.33$ & $5.00$ \\
$3$ & $2.45$ & $0.82$ & $1.36$ & $0.33$ & $0.56$ & $1.22$ & $0.89$ & $0.50$ & $8.00$ \\
$4$ & $2.45$ & $0.82$ & $1.22$ & $0.33$ & $0.50$ & $1.15$ & $0.94$ & $0.60$ & $15.0$ \\
$5$ & $2.83$ & $1.26$ & $1.56$ & $0.45$ & $0.55$ & $1.11$ & $0.96$ & $0.68$ & $25.8$ \\
$6$ & $2.83$ & $1.26$ & $1.51$ & $0.45$ & $0.54$ & $1.09$ & $0.97$ & $0.72$ & $37.6$ \\
$7$ & $3.16$ & $1.57$ & $1.78$ & $0.50$ & $0.56$ & $1.07$ & $0.98$ & $0.77$ & $57.3$ \\
$8$ & $3.16$ & $1.57$ & $1.75$ & $0.50$ & $0.56$ & $1.06$ & $0.99$ & $0.79$ & $75.2$ \\
\midrule
$\infty$ & $\infty$ & $\infty$ & $\infty$ & $0$ & $0$ & $1$ & $1$ & $1$ & $\infty$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{More colluders}
For $c \geq 3$, the collusion strategy affects $d_0$ and $d_1$, and the resulting space and time exponents for the decoding phase. For simplicity, let us focus on the strongest and most natural attack, the \textit{interleaving attack}, where given $k$ ones and $c - k$ zeros in segment $i$, the collusion sets $y_i = 1$ with probability $k/c$ (i.e.\ $\theta_k = \frac{k}{c}$). Equivalently, for each segment the colluders random choose one of their members, and output his content. In that case $\mathbb{E}_{p_i}(y_i) = p_i$ and we can further simplify the expressions for $\mu_0$ and $\mu_1$:
\begin{align}
\mu_0 &= \mathbb{E}_{\vc{p}, \vc{x}_j, \vc{y}}(s_j | H_0) = \ell \cdot \mathbb{E}_p\left(\frac{p^2 + (1-p)^2 - 2 p (1-p)}{\sqrt{p(1-p)}}\right), \\
\mu_1 &= \mathbb{E}_{\vc{p}, \vc{x}_j, \vc{y}}(s_j | H_1) = \left(1 - \frac{1}{c}\right) \cdot \mu_0 + \frac{\ell}{c} \cdot \mathbb{E}_p\left(\frac{1}{\sqrt{p(1-p)}}\right).
\end{align}
Using the optimal discrete distributions of~\cite{nuida09} for small $c$, optimized for the symmetric score function, and computing the resulting parameters, we obtain Table~\ref{tab:num}. Although the entire asymptotic trade-off spectrum is defined by Equation~\ref{eq:tradeoff} and $\alpha$, we explicitly instantiate these trade-offs in the last three columns, for the near-linear space regime ($\rho_s = 0$), the balanced regime ($\rho_q = \rho_s$), and the subpolynomial query time regime ($\rho_q = 0$).
\subsection{Many colluders}
As one can see in the table, as $c$ increases the time and space exponents for the decoding phase quickly increase. For instance, for $c = 6$, we can obtain a query time complexity scaling as $n^{0.72}$, with space scaling as $n^{1.72}$, or if we insist on using only quasi-linear memory in $n$, the best query time complexity scales as $n^{0.97}$.
For asymptotically large $c$, using the arcsine distribution with a cut-off $\delta > 0$ (optimally scaling as $\delta \propto c^{-4/3}$~\cite{laarhoven14dcc}), both the innocent and guilty scores scale such that, after normalization, we get $d_{0,1} \propto \delta^{1/4} \to 0$. The fact that $d_0 \approx d_1$ for large $c$ logically follows from the fact that colluders are able to blend in with the crowd better and better as $c$ increases, requiring large code length. Since $d_1/d_0 \to 1$, NNS techniques do not give any improvement in the limit of large $c$, and the main benefits are obtained when more memory is available to index the code words, and when $c$ is small.
\newpage
\section{Experiments}
\label{sec:exp}
To give an example of the potential speed-up in practice, we performed experiments for the interleaving attack with $c = 3$ colluders. In total we simulated $n = 10^5$ innocent users in each of $1000$ trials, where we used a code length of $\ell = 5000$. For the bias distribution we used the optimal $p$-values of~\cite{nuida09} of $p = 0.5 \pm 0.289$ with equal weights for both possibilities.
\paragraph{NNS data structure.} For the NNS data structure and decoder, we implemented the asymptotically suboptimal but often more practical hyperplane locality-sensitive hashing method of Charikar~\cite{charikar02}. For the hyperplane LSH data structure, we chose the number of hash tables as $t = 100$, and we used a hash length of $k = 16$ (see~\cite{charikar02} for more details). For each of the $t$ hash tables, each of the $n$ data vectors is stored in one of the $2^k = 65536$ hash buckets. Given a query vector $\vc{q}$, for each of the $t$ hash tables we (1) compute $k$ inner products with random (sparse~\cite{achlioptas01}) unit vectors, with a total cost of $1600$ sparse dot products, and (2) do look-ups in these hash buckets for potential near neighbors (colluders), by computing their scores.
\paragraph{Results.} Figure~\ref{fig:exp} illustrates (running averages of) how many user scores are commonly computed, i.e.\ how much work is done in the decoding stage, depending on how high the user scores are; the higher the score $s_j$, the larger the dot product $\langle \vc{v}_j, \vc{q} \rangle$, and the more likely it is we will find user $j$ (vector $\vc{v}_j$) colliding with $\vc{q}$ in one of the hash tables. From $1000$ simulations of the collusion process, approximately $4.2\%$ of all innocent users were considered as potential colluders (i.e.\ on average $4200$ of $10^5$ user scores were computed), and over $31\%$ of all colluders were found through collisions in the hash tables (i.e.\ on average approximately $1$ of the $3$ colluders was found). On average, the decoding consists of computing $1600$ dot products for the hash table look-ups, and $4200$ score computations of innocent users, for a total of around $5800$ dot products of length $\ell$. Compared to a naive linear search, which requires computing all $10^5$ user scores, the decoding is a factor $17$ faster. This comes at the cost of requiring $100$ hash tables, which each store pointers to all $n$ vectors in memory; since pointers are much smaller than the actual vectors, in practice the NNS data structure only required a factor $2$ more memory compared to no indexing.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{fig-exp}
\caption{An illustration of (running averages of) how many user scores are computed, as a function of the user scores. \label{fig:exp}}
\end{figure}
\paragraph{Theory vs. Practice.} Theoretically, with $t = 100$ hash tables we are using of the order $t \cdot n = n^{1.40}$ memory, i.e.\ setting $\rho_s = 0.40$, (although in practice the memory only increases by a small amount). With $c = 3$, according to Table~\ref{tab:num} we have $\alpha \approx 1.22$ for the optimal asymptotic trade-offs, which according to~\eqref{eq:tradeoff} would thus result in $\rho_q \approx 0.58$. In reality the average query cost is computing $5800$ dot products, corresponding to a query exponent $\rho_q = \log(5800)/\log(10^5) \approx 0.75$. In practice, one may indeed notice that $\rho_q$ is slightly higher than the theoretical values suggest, but the memory increase is commonly much less than expected.
Note that although in most runs at least one colluder is successfully found in the hash tables (and has a large score), sometimes none of the colluders are found, and in some applications finding all colluders may be required. To get a higher success rate of finding colluders, one could for instance use multiprobing~\cite{andoni15cp} to still get a significantly lower decoding cost for finding all colluders compared to a linear search, without further increasing the memory.
\section{Discussion}
\label{sec:disc}
Besides the main analysis on the costs of the decoding method, and the associated effect on the memory complexity, let us finally discuss a few more properties and aspects of the techniques outlined in this paper, which may affect how practical this method truly is.
\paragraph{Score function.} Although we only explicitly analyzed the application to the symmetric score function~\cite{skoric08}, the same techniques can be applied to any score-based scheme. For other score functions, the normalization factor $\vc{q}$ may depend on the attack strategy however, making an accurate instantiation of the NNS data structure harder unless the attack is known in advance.
\paragraph{Effects on encoding and embedding.} Even if the decoding method is more efficient, deploying this method in practice may not be cost-effective if the method for generating fingerprints and embedding these in the data needs to be modified. This is fortunately not a concern here, as the only thing that needs to be modified is how the owner of the content stores the code words $\vc{x}_j$ for decoding purposes: the \textit{exact same} encoding and embedding techniques can still be used.
\paragraph{Decoding accuracy.} One of the main reasons NNS techniques are fast, is that they allow for a small margin of error in the decoding procedure. In the application to fingerprinting, this means that the decoder may not always identify colluders from straightforward look-ups in the hash tables. This problem can be mitigated with multiprobing techniques~\cite{andoni15cp, panigrahy06}, or one could use NNS techniques without false negatives, such as~\cite{pagh16}.
\paragraph{Overhead of NNS techniques.} With NNS techniques, we reduce the asymptotic decoding time from $O(\ell n)$ to $O(\ell n^{\rho})$ with $\rho \leq 1$. Depending on how the hidden constants change, the effects may not immediately be visible for small $n$. Note however that the setting considered here, of solving NNS for data on the sphere, can be handled effectively with practical NNS techniques such as~\cite{charikar02, andoni15cp}, which have previously been proven to be much faster than linear searches on various benchmarks~\cite{aumueller17}, and our preliminary experiments confirm the improvement in practice.
\paragraph{Dynamic settings.} For streaming applications~\cite{fiat99, tassa05, laarhoven13tit, laarhoven15ihmmsec}, decisions on whether to accuse users or not need to be made in real-time as well. As the NNS techniques considered here commonly rely on static data, it is not directly obvious whether the same speed-ups can be obtained when the data arrives in a streaming fashion. Interested readers may consider~\cite{law05, liu03} for further reading on NNS techniques that may be relevant for streaming data.
\paragraph{Joint decoding.} While simple decoders with a decoding cost linear in $n$ might be considered reasonably efficient, in the joint decoding setting, the decoding cost of $O(\ell n^k)$ for $k \geq 2$ is a real problem. Similar techniques can be applied there, by slightly changing how the decoding problem is modeled as a near neighbor problem. In that case, the decoding time becomes $O(\ell n^{k \rho})$, and the improvement may be even more noticeable than for simple decoders.
\paragraph{Group testing.} As discussed in e.g.\ \cite{meerwald11b, laarhoven13allerton, skoric15b, laarhoven16phd, laarhoven14ihmmsec}, the group testing problem of detecting infected individuals among a large population using simultaneous testing~\cite{dorfman43}, is equivalent to the fingerprinting problem where the collusion strategy is fixed to the all-$1$ attack: whenever allowed by the marking assumption, the colluders output the symbol $1$. Similar techniques as described above can be applied there to reduce the decoding time complexity both for simple and joint group testing methods.
\begin{acks}
The author thanks Peter Roelse for discussions on the potential relevance of these techniques. The author is supported by an NWO Veni Grant under project number 016.Veni.192.005.
\end{acks}
\bibliographystyle{ACM-Reference-Format} |
1605.01123 | \section{Introduction}
Neutrinos are very special among the currently known elementary particles \cite{Agashe:2014kda}. They have tiny masses compared to the rest of the elementary fermions in all known processes. So far they are produced only at relativistic energies.
Their interactions are very weak, so that an overwhelming majority of them escape direct detection. Their flavor mixings turn out to be much larger that those of quarks. If there are right handed chiral components, they must be sterile under the known interactions. The masses of sterile components, if they exist, could be Dirac or Majorana and could be as large as GUT size. Large scale masses could explain the smallness of the known neutrino components by the so called
seesaw mechanisms (in several versions) \cite{see-saw}; there could be neutrino components that comprise the Dark Matter of the Universe \cite{DarkMatter}; and the CP violation in the lepton sector could provide explanation for the baryon asymmetry of the universe \cite{BA-Lep}.
In this work, we address a simple way to discriminate the Dirac vs. Majorana character of sterile neutrinos, provided they exist with masses near and below the $W$ boson mass. Currently, the main experiments that are sensitive to the Majorana character of neutrinos are those that search for neutrinoless double beta decays ($0\nu\beta\beta$) \cite{0nuBB}. These experiments are sensitive to Majorana neutrinos, in principle of any masses, including the known light neutrinos. However, the extraction of parameters from $0\nu\beta\beta$ experiments will not be an easy task, due to at least two reasons: firstly, there are large theoretical uncertainties in the estimation of the nuclear matrix elements involved; secondly, there could be cancellations of interfering amplitudes which will not be possible to disentangle without further inputs from other experiments. Therefore, it can be important to have additional ways, such as BaBar \cite{BABAR:2012aa, Lees:2013gdj}, BELLE \cite{Seon:2011ni, Petric:2011zz} and LHCb \cite{Adeva:2013csa, Khanji:2014dca} studies on rare lepton flavor and lepton number violating decays of heavy mesons \cite{Cvetic:2010rw, Helo:2010cw}, to discriminate between the Dirac vs. Majorana character of neutrinos.
One such additional way is the search for equal sign dileptons in high energy colliders, which are suitable for large neutrino masses. Indeed, for neutrino masses above $\sim 100$ GeV the LHC experiments can use the mode $pp \to \ell^\pm \ell^\pm j j$ \cite{Aad:2015xaa, Khachatryan:2015gha,Helo:2013esa, Keung:1983uu}. On the other hand, for neutrino masses below $M_W$ the produced jets in the final state $\ell^\pm \ell^\pm j j$ may not pass the cuts required to reduce backgrounds, so that purely leptonic modes such as $pp \to \ell^\pm \ell^\pm \ell^{\prime\mp}\nu$ can be more favorable \cite{Izaguirre:2015pga}.
In our previous work \cite{Dib:2015oka} we studied the signal $W^\pm \to e^\pm e^\pm \mu^\mp \nu$, which will appear resonantly enhanced provided there exist neutrinos with masses below $M_W$, through the subprocess $W^\pm \to e^\pm N$ followed by
$N\to e^\pm\mu^\mp\nu$. The choice of having no opposite-sign same-flavor (no-OSSF) lepton pairs in the final state helps eliminate a serious standard model radiative background $\gamma^*/Z \to \ell^+\ell^-$ \cite{p2mu}.
Now, a heavy neutrino $N$, if it is Majorana, will induce the lepton number conserving (LNC) $W^+ \to e^+ e^+ \mu^- \nu_e$ as well as the lepton number violating (LNV) $W^+ \to e^+ e^+ \mu^- \bar\nu_\mu$ process, while if it is of Dirac type, it will induce only the LNC process (for simplicity of notation we considered the $W^+$ decays, but the same is valid for the $W^-$).
Consequently, one could in principle try to observe a difference in these processes that could test whether the neutrino $N$ is Majorana or Dirac.
However, since the final neutrino escapes detection, the observed final state is just $e^\pm e^\pm \mu^\mp$ or $\mu^\pm \mu^\pm e^\mp$ { plus missing energy. Hence, apparently, it is not a simple task} to distinguish between the LNC and the LNV processes.
Therefore, although this tri-lepton channel is appropriate for searching sterile neutrinos with masses below $M_W^{}$, the determination of the Dirac or Majorana character of the sterile neutrino is quite challenging.
In our previous work \cite{Dib:2015oka} we found that one could still discriminate the Dirac vs. Majorana nature of the heavy neutrino by studying the energy distribution of the lepton in the final state with charge opposite to the decaying $W$ boson
(e.g. $\mu^-$ in the process $W^+ \to e^+ e^+ \mu^- \nu$).
In this work, we present a simpler method to distinguish between Majorana and Dirac $N$ by examining the integrated decay rates, not the spectra, for all the channels $e^\pm e^\pm \mu^\mp$ and $\mu^\pm \mu^\pm e ^\mp$, because the discrimination through the spectra could be a highly challenging task for experiments. This method, on the other hand, is useful only in the case that the mixing parameters $U_{Ne}$ and $U_{N\mu}$ are considerably different from each other.
This method works due to the fact that, for Dirac sterile neutrinos, no matter whether the mixing angles with electrons and muons are equal or not, the decays into the channels $e^\pm e^\pm \mu^\mp$ and $\mu^\pm \mu^\pm e ^\mp$ are all LNC processes and are all the same, while for Majorana sterile neutrinos, with different mixing parameters there is always one LNV process which dominates over the LNC processes. Consequently, if the true nature of sterile neutrinos is of Majorana type and their mixing angles with electrons and muons are different enough, the observed number of events in the $e^\pm e^\pm \mu^\mp$ and $\mu^\pm \mu^\pm e^\mp$ channels will be different, unlike the case for Dirac sterile neutrinos. Details of the method are described in the next section.
It should be noted that such a finding is only valid at the theoretical level. In an actual collider search, because of the detector effects, the observed numbers of events in the $e^\pm_{} e^\pm_{} \mu^\mp_{}$ and $\mu^\pm_{} \mu^\pm_{} e^\mp_{}$ channels could be different even for the Dirac sterile neutrino case.
Therefore, it is worthwhile to perform a careful collider simulation and carry out a detailed statistical study for this scenario, which we present in detail in Section \ref{sec:simulation}.
Now, in the less fortunate scenario where $N$ has equal --or nearly equal-- mixing parameters with electrons and muons, the above method no longer applies, as both Dirac or Majorana neutrino will give the same number of events in the above two channels, in which case one can only resort to the spectral distributions of final state leptons, as shown in the Ref. \cite{Dib:2015oka}. A further investigation along this line has also been pursued by us recently, and the obtained results will be presented elsewhere \cite{DKWZ}.
The rest of the paper is organized as follows. In Section \ref{sec:rate}, we review the expressions for the decay branching ratios of $W$ to tri-leptons via the sterile neutrino $N$. The detailed collider simulation for the scenario with disparate mixing is carried out in Section \ref{sec:simulation}, followed by a statistical analysis on excluding the Dirac sterile neutrino case in Section \ref{sec:stat}. We conclude in Section \ref{sec:summary}.
\section{Leptonic Decay of $W$ via Sterile Neutrino $N$}
\label{sec:rate}
As mentioned in the Introduction section, we will study decays into final states without dileptons of the same flavor and opposite sign in order to avoid large SM backgrounds from radiative pair production, namely we consider $W^+ \to e^+ e^+ \mu^{-}
{\nu}$ and $W^+ \to \mu^+ \mu^+ e^{-}
{\nu}$ (or their charge conjugates) but neither
$W^+ \to e^+ e^{-} \mu^+
{\nu} $ nor $W^+ \to \mu^+ \mu^{-} e^+
{\nu}$. For illustration, in Fig.~\ref{fig1} we show the corresponding Feymann diagrams for the LNC and LNV processes in the $e^+_{} e^+_{} \mu^-_{}$ channel.
\begin{figure}[h]
\centering
\begin{minipage}[b]{.35\linewidth}
\includegraphics[width=\linewidth]{LNV.png}
\end{minipage}
\hspace{18pt}
\begin{minipage}[b]{.35\linewidth}
\vspace{0pt}
\includegraphics[width=\linewidth]{LNC.png}
\end{minipage}
\vspace{-0.2cm} \caption{ The lepton number violating (LNV) process $W^+\to e^+ e^+ \mu^-\bar\nu_\mu$, mediated by a heavy sterile neutrino of Majorana type (left) and the lepton number conserving (LNC) process $W^+\to e^+ e^+ \mu^- \nu_e$, mediated by a heavy sterile neutrino of Majorana or Dirac type (right).} \label{fig1}
\end{figure}
For the LNV processes $W^+ \to e^+ e^+ \mu^{-} \bar\nu_\mu$ and $W^+ \to \mu^+ \mu^+ e^{-} \bar\nu_e$, the rates are:
\begin{eqnarray}
\Gamma (W^+ \to e^+ e^+\mu^-\bar\nu_\mu) &=&
|U_{N e}|^4 \times \hat\Gamma \ \ \propto \ \ |U_{N e}|^4, \nonumber \\
\Gamma (W^+ \to \mu^+ \mu^+ e^{-} \bar\nu_e) &=&
|U_{N \mu}|^4 \times \hat\Gamma \ \ \propto\ \ |U_{N \mu}|^4.
\label{LNV_rate}
\end{eqnarray}
where the factor $\hat\Gamma$ is
\begin{equation}
\hat\Gamma = \frac{G_F^3 M_W^3}{12\times 96 \sqrt{2} \ \pi^4} \frac{ m_N^{5}}{\Gamma_N}
\left( 1-\frac{m_N^2}{M_W^2} \right)^2 \left(1 + \frac{m_N^2}{2 M_W^2} \right).
\end{equation}
$G_F^{}$ is the Fermi's constant and $M_W^{}$ is the $W$-boson mass. Here we have assumed that only a single sterile neutrino $N$ participates in the decay of $W$, with its total decay width denoted as $\Gamma_N^{}$, its mass as $m_N^{}$, and its mixing with a charged lepton $\ell$ denoted as $U_{N \ell}$. Notice that, although the processes in Eq.~\ref{LNV_rate} appear to be quartic in the mixings $U_{N\ell}$, they are effectively quadratic when $N$ goes on its mass shell, because $\Gamma_N$ is quadratic in $U_{N\ell}$.
On the other hand, the corresponding rates for the LNC processes $W^+ \to e^+ e^+ \mu^{-} \nu_e$ and $W^+ \to \mu^+ \mu^+ e^{-} \nu_\mu$ are given by the similar expressions:
\begin{eqnarray}
\Gamma (W^+\to e^+ e^+\mu^-\nu_e) &=&
|U_{N e} U_{N\mu}|^2 \times \hat\Gamma \ \ \propto\ \ |U_{N e} U_{N\mu}|^2 \nonumber, \\
\Gamma (W^+\to \mu^+ \mu^+ e^{-} \nu_\mu) &=&
|U_{N e} U_{N\mu}|^2\times \hat\Gamma \ \ \propto\ \ |U_{N e} U_{N\mu}|^2.
\label{LNC_Rate}
\end{eqnarray}
The expressions differ only in the lepton mixing factors: $|U_{N e}|^4$ for the LNV process
$W^+ \to e^+ e^+ \mu^{-} \bar\nu_\mu$, $|U_{N \mu}|^4$ for the LNV process $W^+ \to \mu^+ \mu^+ e^{-} \bar\nu_e$,
and $|U_{N e} U_{N \mu}|^2$ for the two LNC processes $W^+ \to e^+ e^+ \mu^{-} \nu_e$ and $W^+ \to \mu^+ \mu^+ e^{-} \nu_\mu$. So, clearly both LNC processes have always equal rates, while the LNV processes are equal only if $|U_{Ne}|^2 =|U_{N\mu}|^2$. With the fact that a Dirac sterile neutrino will produce only the LNC processes while a Majorana neutrino will produce both the LNC and LNV processes, one can compare the production of $e^+ e^+ \mu^-$ and $\mu^+ \mu^+ e^- $ (or their charge conjugates) induced by a Dirac or a Majorana sterile neutrino.
\begin{table}[h]
\centering
\begin{tabular}{c | c | c}
\hline
\hline
& Dirac & Majorana \\
\hline
$e^+ e^+ \mu^-$ & $1$ & $1+r$ \\
$\mu^+ \mu^+ e^-$ & $1$ & $1+1/r$ \\
\hline
\hline
\end{tabular}
\caption{Relative factors in the branching ratios of $W^+ \rightarrow e^+ e^+ \mu^- {\nu}$ and $W^+ \rightarrow \mu^+ \mu^+ e^- {\nu}$ for both Dirac and Majorana sterile neutrino scenarios, where ${\nu}$ represents a standard neutrino or anti-neutrino. The same applies for the respective charge conjugate modes. Here $r$ is defined as $r \equiv |U_{Ne}^{}|^2/|U_{N\mu}^{}|^2$.}
\label{tb:comparison}
\end{table}
In Table \ref{tb:comparison}, we present such a comparison, where the rate of $W^+ \rightarrow e^+ e^+ \mu^- \nu$ in the Dirac case is chosen as the reference value, by which the rates of other cases are normalized. It is now apparent that in the case of a Dirac sterile neutrino the production rates of $e^+_{} e^+_{} \mu^-_{}$ and $\mu^+_{} \mu^+_{} e^-_{}$ should be equal, while for the Majorana case they will differ, depending on the \emph{disparity factor}, $r$, defined as:
\begin{equation}
r \equiv \frac{ |U_{Ne}^{}|^2}{|U_{N\mu}^{}|^2}.
\label{disparity}
\end{equation}
For $r> 1$, the number of $e^+_{} e^+_{} \mu^-_{}$ events should be
larger than the $\mu^+_{} \mu^+_{} e^-_{}$ events, and viceversa, if $r < 1$ the $\mu^+_{} \mu^+_{} e^-_{}$ events will be more abundant than $e^+_{} e^+_{} \mu^-_{}$. Similar comparison also exists in the corresponding charge-conjugated processes.
The essence of the above feature can be attributed to the requirement of having no lepton pairs with opposite sign and same flavor (no-OSSF) in the final state. With such requirement, the diagrams of the LNV and LNC processes have different topological structures. As shown in Fig.~\ref{fig1}, in the LNC process the fermion line containing the sterile neutrino $N$ must be attached to final leptons of opposite charge, and consequently these must have different flavors,
hence the mixing factor $|U_{Ne} U_{N\mu}|$.
On the other hand, in the LNV process, the sterile neutrino line is attached to two leptons of same sign and same flavor, hence the factor $|U_{N e}|^4$. Actually, since this difference is valid irrespectively of the mass of $N$, one may generalize our current study to the case where $m_N^{} > M_W^{}$, although for larger masses the dilepton-dijet processes $\ell\ell j j$ are favored as they tend to give larger rates even after cuts to reduce the hadronic background.
Consequently, in the following section we will restrict our study to the case where $m_N^{} < M_W^{}$,
for different values of the $r$ parameter (see Table \ref{tb:comparison}).
We will perform the statistical analysis with collider simulations for both background and signals, and determine the statistical level at which one can distinguish a Majorana vs. a Dirac sterile neutrino case at the LHC.
\section{Collider Simulation}
\label{sec:simulation}
In our simulation, we first build a Universal FeynRules Output \cite{Degrande:2011ua} model file using $\texttt{FeynRules}$ \cite{Christensen:2008py} which extends the SM model with additional sterile neutrino interactions. Both signal and background events are generated within the framework of $\texttt{MadGraph 5}$ \cite{Alwall:2014hca}, where the parton showering and detector simulation are carried out by $\texttt{PYTHIA 6}$ \cite{Sjostrand:2006za} and $\texttt{DELPHES 3}$ \cite{deFavereau:2013fsa}, respectively. At the parton level, we include up to two extra partons for both signal and background processes, and perform the jet matching using the MLM-based shower-$k_\bot^{}$ scheme \cite{Alwall:2008qv}. Lastly, to maintain consistency across the processes we are considering, we present the results using the cross sections from the $\texttt{MadGraph 5}$ output.
Although in this trilepton search we demand no OSSF lepton pairs in the final state, there still exists non-negligible background from various processes. We divide them into two categories. In the first category, we have the pair production of of $WZ$ with $W$ decaying leptonically and $ Z \rightarrow \tau^+_{} \tau^-_{}$. The subsequent decay of the $\tau$'s can lead to trilepton events with no OSSF lepton pairs. We estimate this background process via the Monte Carlo simulation.
The second category of background consists in ``fake" leptons which mainly originate from heavy-flavor meson decays. Although in general leptons from such a heavy-flavor meson decay are not well isolated, there are still rare occasions when they can pass the lepton isolation criteria \cite{Sullivan:2006hb,Sullivan:2008ki,Sullivan:2010jk}. Dominant background processes of this kind are $\gamma^*_{}/Z$+jets and $t\bar{t}$, where an event with no OSSF lepton pairs
arises from $\gamma^*_{}/Z \rightarrow \tau^+_{} \tau^-_{}$ or the prompt decay of $t$ and $\bar{t}$, and a third lepton is faked from jets containing heavy-flavor mesons. Because these processes have large cross sections and small fake probabilities, it is very challenging to obtain enough statistics for background study in the pure MC simulation. Moreover, simulating such processes requires a detailed modelling of the jet fragmentations, and current level of MC simulation may not be accurate enough. For these reasons, data-driven methods are used by the ATLAS and CMS collaborations to estimate the fake lepton contributions \cite{Khachatryan:2015gha,Chatrchyan:2014aea,ATLAS:2014kca}.
In this work, we adopt a phenomenological Fake Lepton (FL) simulation method originally introduced in Ref.~\cite{Curtin:2013zua} and later also implemented in
Ref.~\cite{Izaguirre:2015pga}. In this FL simulation, one employs the fact that FL's originate from jets, and therefore they inherit parts of the kinematics of the original jets. Two modelling functions are introduced: one is called ``mistag efficiency'', $\epsilon_{j \rightarrow \ell}^{}$, which represents the probability of a particular jet faked into a lepton, and the other is called ``transfer function", $\mathcal{T}_{j\rightarrow \ell}^{}$, which is a probability distribution that determines how much the jet momentum is transferred into the faked lepton. These two functions consist of a few modelling parameters, which can be pinned down by validating simulated results against actual experimental ones. We revisit the validation performed in Ref.~\cite{Izaguirre:2015pga}, and find that the modelling parameters they obtained can be consistent with the experimental results. Thus, the same set of parameters are used here, and we also assume the same fake efficiency for electrons and muons. Details of this FL simulation method and the validation can be found in Refs.~\cite{Izaguirre:2015pga,Curtin:2013zua}. Our validation results are given in Appendix \ref{sec:FL}.
We now discuss the various kinematic cuts applied in this simulation study.
At first, we impose the basic cuts for leptons and jets: for leptons, $p_T^\ell \geq 10~\text{GeV}$ and rapidity $|\eta_\ell^{}| \leq 2.5$; and for jets, $p_T^j \geq 20~\text{GeV}$ and rapidity $|\eta_j^{}| \leq 5.0$.
We then require the transverse mass $M_T^{}(\text{leps}, \cancel{E}_T^{}) < 90~\mathrm{GeV}$, with $M_T^{}(\text{leps}, \cancel{E}_T^{})$ defined as
\begin{eqnarray}
M_T^{2}(\text{leps}, \cancel{E}_T^{}) = m_{\text{leps}}^{2} + 2 \left( \sqrt{\left(m_{\text{leps}}^{2} +\vec{p}_{\text{leps},T}^{2} \right) ~ \vec{\cancel{p}}_T^{2} } - \vec{\cancel{p}}_T^{} \cdot \vec{p}_{\text{leps},T}^{} \right),
\end{eqnarray}
where $m_{\text{leps}}^{}$ and $ \vec{p}_{\text{leps},T}^{}$ are the invariant mass and transverse momentum of all three visible leptons in the final state, and $\vec{\cancel{p}}_T^{}$ is the missing transverse momentum. For the signal processes, this $M_T^{}$ is bounded by the W-boson mass, because all the charged leptons and neutrinos in the final state come from the decay of a W-boson. Considering the detector resolution and the W-boson width, we find that $M_T^{}(\text{leps}, \cancel{E}_T^{}) < 90~\mathrm{GeV}$ is an good discriminating condition between the signal and background, especially in the elimination of large background from $t\bar{t}$. For illustration, in Fig.~\ref{fg:MtLepsMET}, we show the distributions of $M_T^{}(\text{leps}, \cancel{E}_T^{})$ after basic cuts for both background and signal processes in the $e^+_{} e^+_{} \mu^-_{}$ channel.
\begin{figure}[h]
\centering
\includegraphics[scale=0.85]{mtMETBeforeCuts_pl.pdf}
\caption{Distributions of $M_T^{}(\text{leps}, \cancel{E}_T^{})$ for both signal and background processes after basic cuts. For signal processes, we choose the LNC case in the $e^+_{} e^+_{} \mu^-_{}$ channel as an example.
}
\label{fg:MtLepsMET}
\end{figure}
To further suppress background, we impose the cuts $\cancel{E}_T^{} < 40~\mathrm{GeV}$, zero b-jets, and $H_T^{} < 50~\mathrm{GeV}$, where $H_T^{}$ is the scalar sum of all jets $p_T^{j}$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.65]{MdLep2sLep_20_pl.pdf}
\includegraphics[scale=0.65]{MdLep2sLep_50_pl.pdf}
\caption{Distributions of $M(\ell_N^{},\ell^\prime)$ for both signal and background processes after applying all cuts in Table \ref{tb:bkgd} except for the last one. For signal processes, we choose the LNC case in the $e^+_{} e^+_{} \mu^-_{}$ channel as an example.}
\label{fg:MdLep2sLep}
\end{figure}
The above cuts are proved to be very effective in reducing the background from $WZ$ and $t \bar t$.
However, the background from $\gamma^*/Z$+jets is still large (see Table \ref{tb:bkgd}). As was also observed in Ref.~\cite{Izaguirre:2015pga}, for the $m_N^{} = 20~{\text{GeV}}$ case one is able to impose a further cut using the fact that the invariant mass of the two leptons originating from the $N$ decay should be bounded from above by the mass of $N$. Such a cut requires a correct identification of the lepton pair, as there will be two final state leptons which are indistinguishable by their signs and flavors. In Appendix \ref{sec:SL_order} we provide a method for identifying the correct lepton pair. The three leptons in the final state are labeled as $\ell_
W^\pm \ell_N^\pm \ell^{\prime \mp}$, with $\ell_W^{}$ and $\ell_N^{}$ denoting the leptons from the prompt decays of $W$ and $N$, respectively. Note that this additional cut on the invariant mass of $M(\ell_N^{}, \ell_{}^{\prime})$ is effective in the $m_N^{} = 20~{\text{GeV}}$ case while not as much in the $m_N^{} = 50~{\text{GeV}}$ case. This can be seen from Fig.~\ref{fg:MdLep2sLep}, where the distributions of $M(\ell_N^{},\ell^\prime)$ for both signal and background processes are shown. As one can see, for the $m_N^{} = 20~{\text{GeV}}$ case one can further reduce the background by requiring $M(\ell_N^{}, \ell_{}^{\prime}) < 20~\text{GeV}$, however, a similar cut cannot be applied to the $m_N^{} = 50~{\text{GeV}}$ case, as the signal process exhibits almost the same distribution as the background ones.
\begin{table}[h]
\centering
\begin{tabular}{c | c | c | c | c}
\hline
\hline
\multirow{2}{*}{Cuts} & \multicolumn{2}{c|}{$WZ$} & $\gamma^* / Z$+jets & $t\bar{t}$ \\
\cline{2-5}
& $\ell^+_{} \ell^+_{} \ell^{\prime -}_{}$ & $\ell^-_{} \ell^-_{} \ell^{\prime +}_{}$ & $\ell^\pm_{} \ell^\pm_{} \ell^{\prime \mp}_{}$ & $\ell^\pm_{} \ell^\pm_{} \ell^{\prime \mp}_{}$ \\
\hline
Basic cuts & 779 & 550 & 1055 & 17147 \\
\hline
$M_T^{}(\text{leps}, \cancel{E}_T^{}) < 90~\mathrm{GeV}$ & 52 & 34 & 374 & 160 \\
\hline
$\cancel{E}_T^{} < 40~\mathrm{GeV}$ & 46 & 28 & 356 & 113 \\
\hline
$N(\text{b-jets}) = 0, H_T^{} < 50~\mathrm{GeV}$ & 39 & 23 & 323 & 15 \\
\hline
$M(\ell_N^{}, \ell^\prime) <20~\mathrm{GeV}$ & 7.4 & 4.4 & 62 & 2.7 \\
\hline
\hline
\end{tabular}
\caption{Cut flow for background processes. Numbers of events correspond to an integrated luminosity of $3000~\mathrm{fb}^{-1}$ at the $14~\mathrm{TeV}$ LHC; $\ell$ includes both $e$ and $\mu$.
}
\label{tb:bkgd}
\end{table}
In Tables \ref{tb:bkgd}, \ref{tb:mN20} and \ref{tb:mN50} we show the cut flow tables for all the background processes, and two benchmark signal scenarios with $m_N^{} = 20~\text{GeV}$ and $50~\text{GeV}$, respectively. The numbers of events are calculated for an integrated luminosity of $3000~\text{fb}^{-1}_{}$ at the $14~\text{TeV}$ LHC. For $\gamma^*/Z$+jets and $t\bar{t}$, although the FL simulation method is adopted, it is still difficult to obtain enough statistics to resolve the small difference between the modes $e^\pm e^\pm \mu^\mp$ and $\mu^\pm \mu^\pm e^\mp$. Hence we choose to combine them. The symbol $\ell$ in Table \ref{tb:bkgd} includes both $e$ and $\mu$. The same treatment is also made for $WZ$, where $\ell^+_{} \ell^+_{} \ell^{\prime -}_{}$ and $\ell^-_{} \ell^-_{} \ell^{\prime +}_{}$ are separated due to the different production rates of $W^+_{}Z$ and $W^-_{}Z$. For signal processes we distinguish not only different tri-lepton modes but also the LNC and LNV processes of interest here.
For illustration purposes, we take $|U_{N e}|^2=|U_{N \mu}|^2 = 1\times 10^{-6}$ and $U_{\tau N}^{} = 0$. For other values of the mixings, the number of events can be scaled accordingly. It is worth noting that from Table \ref{tb:mN20} and \ref{tb:mN50}, the number of events for LNV process is always larger compared with LNC process after applying basic cuts. This is due to the fact that kinematical distrubtions of final state leptons are different between LNV and LNC processes. The leptons in the LNV process are more efficient to pass the basic selection cuts.
\begin{table}[h]
\centering
\begin{tabular}{c | c | c | c | c | c | c | c | c}
\hline
\hline
\multirow{2}{*}{Cuts} & \multicolumn{2}{c |}{$e^+ e^+ \mu^-$} & \multicolumn{2}{c |}{$\mu^+ \mu^+ e^-$} & \multicolumn{2}{c |}{$e^- e^- \mu^+$} & \multicolumn{2}{c }{$\mu^- \mu^- e^+$}\\
\cline{2-9}
& LNC & LNV & LNC & LNV & LNC & LNV & LNC & LNV \\
\hline
Basic cuts & 13.6 & 19.5 & 15.0 & 22.0 & 12.1 & 18.2 & 13.3 & 19.5 \\
\hline
$M_t(\text{leps, MET}) < 90~\mathrm{GeV}$ & 12.7 & 18.3 & 13.9 & 20.3 & 11.3 & 17.0 & 12.3 & 18.3 \\
\hline
$\text{MET} < 40~\mathrm{GeV}$ & 12.5 & 18.3 & 13.8 & 20.3 & 11.2 & 17.0 & 12.3 & 18.3 \\
\hline
$N(\text{b-jets}) = 0, H_t < 50~\mathrm{GeV}$ & 11.1 & 16.6 & 12.2 & 18.5 & 10.0 & 15.6 & 11.0 & 16.6\\
\hline
$M(\ell_N^{}, \ell^\prime) <20~\mathrm{GeV}$ & 10.8 & 16.3 & 11.8 & 17.8 & 9.8 & 15.1 & 10.7 & 16.1 \\
\hline
\hline
\end{tabular}
\caption{Cut flow for signal processes with $m_N^{} = 20~\mathrm{GeV}$, $|U_{N e}|^2=|U_{N \mu}|^2 = 1\times 10^{-6}$ and $U_{\tau N}^{} = 0$. Numbers of events correspond to an integrated luminosity of $3000~\mathrm{fb}^{-1}$ at the $14~\mathrm{TeV}$ LHC.
}
\label{tb:mN20}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{c | c | c | c | c | c | c | c | c}
\hline
\hline
\multirow{2}{*}{Cuts} & \multicolumn{2}{c |}{$e^+ e^+ \mu^-$} & \multicolumn{2}{c |}{$\mu^+ \mu^+ e^-$} & \multicolumn{2}{c |}{$e^- e^- \mu^+$} & \multicolumn{2}{c }{$\mu^- \mu^- e^+$}\\
\cline{2-9}
& LNC & LNV & LNC & LNV & LNC & LNV & LNC & LNV \\
\hline
Basic cuts & 27.7 & 30.7 & 30.7 & 33.3 & 23.7 & 26.6 & 26.3 & 29.8 \\
\hline
$M_t(\text{leps, MET}) < 90~\mathrm{GeV}$ & 26.4 & 29.0 & 29.2 & 31.7 & 22.5 & 25.1 & 25.0 & 28.1 \\
\hline
$\text{MET} < 40~\mathrm{GeV}$ & 26.1 & 28.7 & 28.9 & 31.4 & 22.3 & 25.1 & 24.8 & 28.1 \\
\hline
$N(\text{b-jets}) = 0, H_t < 50~\mathrm{GeV}$ & 23.7 & 26.0 & 26.2 & 28.4 & 20.1 & 22.8 & 22.4 & 25.5\\
\hline
\hline
\end{tabular}
\caption{Cut flow for signal processes with $m_N^{} = 50~\mathrm{GeV}$, $|U_{N e}|^2=|U_{N \mu}|^2 = 1\times 10^{-6}$ and $U_{\tau N}^{} = 0$. Numbers of events correspond to an integrated luminosity of $3000~\mathrm{fb}^{-1}$ at the $14~\mathrm{TeV}$ LHC.}
\label{tb:mN50}
\end{table}
\section{Statistical Analysis}
\label{sec:stat}
We now turn to a detailed statistical analysis of our scenario of interest, where the sterile neutrino mixings $U_{Ne}^{}$ and $U_{N\mu}^{}$ are different. In the analysis, $N$ is assumed to be of Majorana type, and the question is whether it can be distinguished from a Dirac neutrino. As mentioned in Section \ref{sec:rate}, in this scenario one is able to distinguish Dirac from Majorana sterile neutrinos by comparing the numbers of $e^\pm e^\pm \mu^\mp$ events with $\mu^\pm \mu^\pm e^\mp$ events. We now study such a discrimination quantitatively based on the previous simulation results.
The essence of this statistical analysis is to perform a hypothesis test on two competing hypotheses, namely, the nature of sterile neutrino can be either Dirac or Majorana.
We follow here the frequentist approach and consider two benchmark scenarios: $m_N^{} = 20~\text{GeV}$ and $50~\text{GeV}$. The observables are the numbers of events in the various final states $\ell^\pm_{} \ell^\pm_{} \ell^{\prime \mp}_{}$, which are treated as the data set. In Tables \ref{tb:mN20} and \ref{tb:mN50}, we have listed the numbers of \emph{expected} events for the case of the mixing angles $|U_{eN}|^2=|U_{\mu N}|^2 = 1\times 10^{-6}$ and $U_{\tau N}^{} = 0$. Apparently, one can easily obtain the expected numbers of events for other mixing angles by simple rescaling. Moreover, by further applying statistical fluctuations, one can generate realistic pseudo-data (simulated data) samples. For convenience, in addition to the \emph{disparity factor} $r = |U_{Ne}/U_{N\mu}|^2$, we introduce another parameter, $s$, a \emph{normalization factor} defined as:
\begin{equation}
s \equiv 2\cdot10^6\, \times \frac{|U_{Ne}^{}U_{N\mu}^{}|^2}{ |U_{Ne}|^2+ |U_{N\mu}|^2 },
\label{normalization_factor}
\end{equation}
i.e. $s$ is a measure of the smallest of the two mixings. Thus, for the case given in Tables \ref{tb:mN20} and \ref{tb:mN50}, we have $s = r = 1$.
Having obtained the realistic pseudo-data samples,
we next fit them with the two competing hypotheses. A $\chi^2_{}$ function is built so as to characterize how well the pseudo-data sets are described within a given hypothesis,
\begin{eqnarray}
\chi^2_{H} = -2 \underset{s,r \subset H}{\text{min}} \left \{ \text{ln} \left( \prod_i ~ \text{Poiss} \left [ N_i^{\text{expc}}(s,r;H), N_i^{\text{obs}} \right ] \right) \right \},
\end{eqnarray}
where $H$ stands for the Dirac or Majorana hypothesis of sterile neutrinos, $i$ denotes a particular trilepton final state, and $\text{Poiss}(N^{\text{expc}}_{}, N^{\text{obs}}_{})$ is the probability of observing $N^{\text{obs}}_{}$ events in Poisson statistics when the number of expected events is $N^{\text{expc}}_{}$. The above definition also involves a minimization procedure that is taken upon the free parameters i.e., $s$ and $r$ in the hypothesis.
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{test_statistic.pdf}
\caption{Distributions of test statistic $T$ for the case with $s = 1$ and $ r= 5$, given the true nature of sterile neutrinos is Dirac (dashed) or Majorana (solid).}
\label{fg:test_statistic}
\end{figure}
To quantify which hypothesis is more favorable, we then define a test statistic $T$ as
\begin{eqnarray}
T = \chi^2_{\text{Dirac}} - \chi^2_{\text{Maj}}.
\end{eqnarray}
Because of statistical fluctuations, the obtained values of $T$ for different pseudo-data samples, which correspond to the same expected numbers of events (or the same set of input parameters $s$ and $r$), can be different. In Figure \ref{fg:test_statistic} we show the probability distributions of the test statistic $T$ for 1000 pseudo-data samples that all have $s = 1$ and $r = 5$, assuming either the true Dirac (dashed) or Majorana (solid) nature of sterile neutrinos. As expected, when sterile neutrinos are truly Dirac particles, we obtain almost equally well fitting for both hypotheses so that $T$ is centered around 0. Namely, in this case no discrimination power can be obtained. However, when the true nature of sterile neutrinos is Majorana, the Majorana hypothesis has a better fit (a smaller $\chi^2_{}$ value), resulting in a positive value of $T$. Statistical fluctuation causes the spread of two distributions, and the level of their overlap determines the confidence level of discriminating these two hypotheses.
\begin{figure}[h]
\centering
\includegraphics[scale=0.65]{scenario_1_mN_20.pdf}~ \includegraphics[scale=0.65]{scenario_1_mN_50.pdf}
\caption{Confidence levels of excluding Dirac type given the true nature of sterile neutrinos is of Majorana type, for two benchmark scenarios of $m_N^{} = 20~\text{GeV}$ (left) and $50~\text{GeV}$ (right). Horizontal and vertical axes respectively denote the true values of the mixing angle ratio $r$ and the normalization factor $s$ in logarithmic scales. Solid, dashed and dot-dashed curves correspond to the $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels of excluding the Dirac type, respectively.
}
\label{fg:scenario_1}
\end{figure}
To simplify the complexities caused by the statistical fluctuations, we consider a ``median" discrimination. Namely, for the true Dirac case, where the distribution of $T$ is sharply peaked at zero, we therefore choose $T = 0$ as the median possible value of $T$.
Then, given the true Majorana nature of sterile neutrinos, the confidence level of excluding the Dirac hypothesis can be quantified as $1-\alpha$, where $\alpha $ is the probability of explaining the true Majorana nature with the wrong Dirac one. In terms of Figure \ref{fg:test_statistic}, this $\alpha$ is the area under the blue curve for $T < 0$.
Finally, in Figure \ref{fg:scenario_1}, we present our main point: the numerical results on the discrimination of Majorana vs. Dirac neutrinos based on the disparity of the mixings $|U_{Ne}|^2$ and $|U_{N\mu}|^2$. We do this for two benchmark scenarios $m_N^{} = 20~\text{GeV}$ (left) and $50~\text{GeV}$ (right). The main question is to determine how far from unity the disparity factor $r$ has to be in order to tell a Majorana character apart from Dirac. Horizontal and vertical axes respectively denote the true values of the disparity factor $r$ and the normalization factor $s$, which are used in generating the pseudo-data samples. Solid, dashed and dot-dashed curves correspond to excluding the Dirac hypothesis given the true Majorana nature at $1\sigma$, $2\sigma$ and $3\sigma$ levels, respectively. As one can see, for both benchmark scenarios, at least a $3\sigma$ level exclusion can be reached for disparities as mild as e.g. $r \lesssim 0.7$ (or $1/r \lesssim 0.7$), provided $s \gtrsim 5$. For smaller $s$ (smaller mixings), which means fewer events, one clearly requires larger values of $r$ to reach the same level of discrimination; in the same way, as $r$ approaches 1, larger values of $s$ are required as it becomes more and more difficult to exclude the Dirac case.
Further discriminating power will require additional information from the spectral distributions of the produced leptons, an issue that we will discuss in a later work.
\section{Conclusions}
\label{sec:summary}
In this work we focus on the question of determining the nature of sterile neutrino with mass below $M_W^{}$ at the LHC. Because of such a low mass for the neutrino, the conventional same sign dilepton plus jet search for Majorana sterile neutrinos at the LHC suffers from the issue of insufficient phase space for final state leptons and jets passing the necessary detector cuts. Therefore, we choose to study the alternative tri-lepton search channel, and to reduce SM backgrounds we further require no opposite-sign same-flavor lepton pairs in the final state. Although this tri-lepton search is ideal for such a low mass sterile neutrino search, it turns out to be a non-trivial task of pinning down the underlying nature of sterile neutrinos, as the final state neutrinos or anti-neutrinos, which carry valuable information about lepton number, are not detected at current colliders.
A simple scenario is identified in this tri-lepton search, so that a discrimination on the nature of sterile neutrinos can be possible. This fortunate scenario could arise if the underlying nature of sterile neutrinos are of Majorana type, and their mixing angles with electrons and muons are different enough. We find that in this fortunate scenario, one is able to exclude the Dirac sterile neutrino case by simply counting and then comparing the numbers of events in the $e^+_{} e^+_{} \mu^-_{}$ and $\mu^+_{} \mu^+_{} e^-_{}$ channels (or the corresponding charge-conjugated ones). We perform a careful collider simulation for this scenario. According to our statistical analysis, at the $14~\text{TeV}$ LHC with an integrated luminosity of $3000~\text{fb}^{-1}$, at least a $3\sigma$ level of exclusion on the Dirac case can be achieved, depending on the size and disparity of the two relevant mixing parameters. For example, such is the case for $s \gtrsim 5$ and $r \lesssim 0.7$ (or $1/r \lesssim 0.7$), where $s = 2\cdot 10^6 \times
|U_{Ne}^{}U_{N\mu}^{}|^2/ \left(
|U_{N e}|^2 +|U_{N\mu}|^2
\right) $, and $r = |U_{Ne}/U_{N\mu}|^2$. For other values of mixings, see Fig.~\ref{fg:scenario_1}.
Therefore, in the current collider search for sterile neutrinos with masses below $M_W^{}$, a quick check of this scenario via the above method can be very rewarding, as if sterile neutrinos were indeed in this mass range and their mixing to electron and muon flavor were different enough, we might know the nature of those sterile neutrinos quite shortly.
\section*{Acknowledgements}
J.Z. thanks Prof. Shun Zhou for helpful discussions, encouragement and support, Brian Shuve for sharing details on fake lepton simulation, and Yan-dong Liu for help on using $\texttt{DELPHES 3}$; K.W. thanks Prof. Cai-Dian L$\rm \ddot{u}$ for his help; and C.D. thanks Juan C. Helo for helpful discussions.
J.Z. was supported in part by the Innovation Program of the Institute of High Energy Physics under Grant No. Y4515570U1; K.W. by the International Postdoctoral Exchange Fellowship Program (No.90 Document of OCPC, 2015); C.S.K. by the NRF grant funded by the Korean government of the MEST (No. 2011-0017430) and (No. 2011-0020333); and C.D. by Chile grants Fondecyt No.~1130617, Conicyt FB0821 and ACT1406. |
1108.5875 | \section{Introduction}
The suppression of quarkonium has been hypothesized 25 years ago \cite{Matsui:1986dk} to represent a signature of the formation of a deconfined medium and has been ever since intensely investigated, both theoretically and experimentally. Here we address the problem, central to these studies, of the behaviour of a quarkonium bound state in a deconfined thermal medium. To this end we shall illustrate the Effective Field Theory (EFT) framework that has been recently constructed in \cite{Brambilla:2008cx,Brambilla:2010vq,Brambilla:2011mk} (see also \cite{Escobedo:2008sy,Escobedo:2010tu,Escobedo:2011ie} for an analogous EFT of QED) by generalizing the successful zero-temperature framework of Non-Relativistic (NR) EFTs for heavy quarkonia to finite temperatures. These NR EFTs exploit the hierarchy $m\gg mv\gg mv^2$ that characterizes any NR binary bound state, $m$ being in this case the heavy quark mass and $v$ the relative velocity. $mv$ is then the typical momentum transfer or inverse radius and $mv^2\sim E$ the binding energy. The low-lying quarkonium states, especially the bottomonium ground states $\Upsilon(1S)$ and $\eta_b$, are believed to be approximately Coulombic. That corresponds to having $mv\sim m\alpha_{\mathrm{s}}\gg\Lambda_{\mathrm{QCD}}$ and $mv^2\sim m\alpha_{\mathrm{s}}^2{\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower6pt\vbox{\hbox{$\sim$}}}}\ } \Lambda_{\mathrm{QCD}}$.\\
In a weakly-coupled plasma, which we consider in our study, the temperature $T$ and the chromoelectric screening mass $m_D$ are larger than $\Lambda_{\mathrm{QCD}}$ and, since $m^2_D\sim g^2 T^2$, $T\gg m_D$. Under these conditions one can then calculate observables relevant for the phenomenology of low-lying states to a large extent analytically in perturbation theory, which makes them extremely interesting, also in the light of the recent CMS measurements of the suppression of the $\Upsilon$ family \cite{Chatrchyan:2011pe}.\\
In \cite{Laine:2006ns} the perturbative $Q\overline{Q}$ static potential was computed in QCD for distances $r$ such that $T\gg1/r{\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower6pt\vbox{\hbox{$\sim$}}}}\ } m_D$. The resulting potential surprisingly shows an imaginary part which is larger than the screened real part for $1/r\sim m_D$. This imaginary part can be traced back to the imaginary part of the gluon self-energy and is due to the Landau-damping phenomenon; it eventually leads to a thermal width for the bound state, which is in turn responsible for its dissociation, representing a change from the previous colour-screening paradigm. This change was further reinforced by the introduction of a \emph{dissociation temperature} in \cite{Escobedo:2008sy,Laine:2008cf}, defined as the temperature for which the imaginary part of the potentials becomes of the same size of its real part; parametrically it is of order $m\alpha_{\mathrm{s}}^{2/3}$ and hence smaller than the temperature at which screening sets in. A quantitative calculation of the dissociation temperature for the $\Upsilon(1S)$ can be found in \cite{Escobedo:2010tu} and a phenomenological analysis, based on these imaginary parts, of $b\overline{b}$ bound states at LHC energies can be found in \cite{Strickland:2011mw}.\\
In \cite{Brambilla:2008cx} the static $Q\overline{Q}$ was first studied in an EFT framework, systematically exploring the hierarchy of different energy scales in the problem. Many possibilities were considered, from temperature smaller than $E$ to temperatures much larger than $1/r$, where the results of \cite{Laine:2006ns} were recovered in a rigorous EFT derivation. Furthermore a new dissociation mechanism, the colour-singlet to octet decay, was identified; it is the leading one when $E\gg m_D$. In \cite{Brambilla:2010xn} the relation between the proper real-time quarkonium potential and the correlator of two Polyakov loops, a quantity often measured on the lattice and used as input for potential models, was investigated. The breaking of Lorentz invariance induced by the preferred reference frame introduced by the medium was instead analyzed in \cite{Brambilla:2011mk} in the spin-orbit sector of the EFT. In the following we will report about the findings of \cite{Brambilla:2010vq}, where in a specific range of temperatures the spectrum and width of quarkonia have been computed up to order $m\alpha_{\mathrm{s}}^5$. To this end, the specific global hierarchy that the NR and thermodynamical scales fulfill in the assumed range of parameters has been exploited by constructing a corresponding tower of EFTs.
\section{Energy scales, the EFT formalism and the results}
The aforementioned global hierarchy we assume is $m\gg m\alpha_{\mathrm{s}}\gg \pi T\gg m\alpha_{\mathrm{s}}^2\gg m_D$,
which implies a temperature below the dissociation temperature. We also remark that for the $\Upsilon(1S)$ at the LHC it may hold that $m_b\approx 5\;\text{GeV}>m_b\alpha_{\mathrm{s}}\approx1.5\;\text{GeV}>\pi T\approx 1\;\text{GeV}>m\alpha_{\mathrm{s}}^2\approx 0.5\;\text{GeV}{\ \lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower6pt\vbox{\hbox{$\sim$}}}}\ } m_D$.\\
Given this hierarchy, we now proceed to integrate out each scale in sequence. The integration of the mass scale $m$ yields non-relativistic QCD (NRQCD) \cite{Caswell:1985ui} and the further integration of the scale $m\alpha_{\mathrm{s}}$ from NRQCD gives potential non-relativistic QCD (pNRQCD) \cite{Pineda:1997bj}. Since the temperature is much smaller than these two scales, it may be set to zero in the matching and the Lagrangians of NRQCD and pNRQCD are the same as at zero temperature.\\
Integrating out $T$ from pNRQCD modifies it into its Hard Thermal Loop (HTL) version, pNRQCD${}_{\mathrm{HTL}}$ \cite{Brambilla:2008cx,Vairo:2009ih}, where the light degrees of freedom are described by the HTL effective Lagrangian \cite{Braaten:1991gm} and the pNRQCD potentials receive a thermal part. Finally, within this EFT we can compute contribution to the spectrum and width from the scales $E$ and $m_D$. Diagrams contributing to the calculation are shown in Fig.~\ref{figure}.\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.4]{figure.pdf}
\end{center}
\caption{The diagrams contributing to our calculation. Single lines are colour-singlet $Q\overline{Q}$ states, double lines are colour octets, curly lines are gluons, vertices are chromoelectric dipoles and the blob is the gluon self-energy. The imaginary part of the first diagram yields the singlet-to-octet decay mechanism, whereas the second one gives the Landau damping contribution to the width.}
\label{figure}
\end{figure}\\
Let us now show the final results for the thermal contribution to the spectrum and to the width up to order $m\alpha_{\mathrm{s}}^5$. We recall that for a Coulombic bound state the spectrum is at LO given by the Bohr levels $E_n=-mC_F^2\alpha_{\mathrm{s}}^2/(4n^2)$ and the Bohr radius is $a_0=2/(mC_F\alpha_{\mathrm{s}})$. The vacuum contribution to the spectrum up to order $m\alpha_{\mathrm{s}}^5$ can be read from \cite{Brambilla:1999xj}.\\
The thermal contribution to the spectrum then reads
\begin{eqnarray}
\nonumber
\delta E_{n,l}^{(\mathrm{thermal})}&=&
\frac{\pi}{9}N_c C_F \,\alpha_{\mathrm{s}}^2 \,T^2 \frac{a_0}{2}\left[3n^2-l(l+1)\right] +\frac{\pi}{3}C_F^2\, \alpha_{\mathrm{s}}^2\, T^2\,a_0
\\
\nonumber&&
+\frac{E_n\alpha_{\mathrm{s}}^3}{3\pi}\left[\log\left(\frac{2\pi T}{E_1}\right)^2-2\gamma_E\right]
\left\{\frac{4 C_F^3\delta_{l0} }{n}+\frac{2N_c^2C_F}{n(2l+1)} +\frac{N_c^3}{4}
\right.
\\
\nonumber
&& \hspace{3.2cm}
\left. +N_c C_F^2\left[\frac{8}{n (2l+1)}-\frac{1}{n^2}
- \frac{2\delta_{l0}}{ n } \right]\right\}
+\frac{2E_nC_F^3\alpha_{\mathrm{s}}^3}{3\pi}L_{n,l}
\\
\nonumber
&&
+ \frac{a_0^2n^2}{2}\left[5n^2+1-3l(l+1)\right]
\left\{- \left[\frac{3}{2\pi} \zeta(3)+\frac{\pi}{3}\right] C_F \, \alpha_{\mathrm{s}} \, T \,m_D^2
\right.
\\
&& \hspace{6.7cm}
+ \left. \frac{2}{3} \zeta(3)\, N_c C_F \, \alpha_{\mathrm{s}}^2 \, T^3\right\},
\label{finalspectrum}
\end{eqnarray}
where $L_{n,l}$ is the QCD Bethe log \cite{Brambilla:1999xj}. The terms on the first line are the leading ones the power counting of the EFT and, being positive, lead to an increase in the mass of the bound state quadratic with the temperature.\\
For what concerns the thermal width, we have
\begin{eqnarray}
\nonumber
\Gamma_{n,l}^{(\mathrm{thermal})}&=&
\frac{1}{3}N_c^2C_F\alpha_{\mathrm{s}}^3T+\frac{4}{3}\frac{C_F^2\alpha_{\mathrm{s}}^3 T}{n^2}(C_F+N)
\\
\nonumber&&
+\frac{2E_n\alpha_{\mathrm{s}}^3}{3}\left\{\frac{4 C_F^3\delta_{l0} }{n}+N_c C_F^2
\left[\frac{8}{n (2l+1)}-\frac{1}{n^2} - \frac{2\delta_{l0}}{ n } \right]
+\frac{2N_c^2C_F}{n(2l+1)}+\frac{N_c^3}{4}\right\}
\\
\nonumber&&
-a_0^2n^2\left[5n^2+1-3l(l+1)\right]\left[
\left(\ln\frac{E_1^2}{T^2}+ 2\gamma_E -3 -\log 4- 2 \frac{\zeta^\prime(2)}{\zeta(2)} \right)\right.\\
&& \hspace{2cm}\left.\times\frac{C_F}{6} \alpha_{\mathrm{s}} T m_D^2+\frac{4\pi}{9} \ln 2 \; N_c C_F \, \alpha_{\mathrm{s}}^2\, T^3 \right]
+\frac{8}{3}C_F\alpha_{\mathrm{s}}\, Tm_D^2\,a_0^2n^4
\,I_{n,l}\;,
\label{finalwidth}
\end{eqnarray}
where $I_{n,l}$ is a new Bethe logarithm \cite{Brambilla:2010vq}. The terms in the first two lines are the leading ones and are caused by singlet-to-octet decay, whereas those on the last two lines are due to Landau damping. The width is at leading order linear in the temperature and much smaller than the binding energy. This small width is certainly not in contradiction with the recent CMS results \cite{Chatrchyan:2011pe} that point to a substantial survival of the $\Upsilon(1S)$ at the LHC. |
1804.02682 | \section{Phase space with respect to the two-photon formalism}
\label{app:twophoton_gaussian}
From the two-photon operators
\begin{align}
\op{a}_1 &= \frac{ \op{a}_{ \omega + \Omega } + \op{a}_{ \omega - \Omega }^{\dagger} }{ \sqrt{2} },
&
\op{a}_2 &= \frac{ \op{a}_{ \omega + \Omega } - \op{a}_{ \omega - \Omega }^{\dagger} }{ i\sqrt{2} },
\end{align}
one can construct hermitian position and momentum operators from \( \op{a}_1 \) and \( \op{a}_2 \) as
\begin{align}
\op{x}_1 &= \frac{ \op{a}_1 + \op{a}_1^{\dagger} }{ \sqrt{2} } = \frac{ \op{a}_{+} + \op{a}_{-}^{\dagger} + \op{a}_{+}^{\dagger} + \op{a}_{-} }{ 2 } = \frac{ \op{x}_{+} + \op{x}_{-} }{ \sqrt{2} }, \\
\op{x}_2 &= \frac{ \op{a}_2 + \op{a}_2^{\dagger} }{ \sqrt{2} } = \frac{ \op{a}_{+} - \op{a}_{-}^{\dagger} - \op{a}_{+}^{\dagger} + \op{a}_{-} }{ 2i } = \frac{ \op{p}_{+} + \op{p}_{-} }{ \sqrt{2} }, \\
\op{p}_1 &= \frac{ \op{a}_1 - \op{a}_1^{\dagger} }{ i\sqrt{2} } = \frac{ \op{a}_{+} + \op{a}_{-}^{\dagger} - \op{a}_{+}^{\dagger} - \op{a}_{-} }{ 2i } = \frac{ \op{p}_{+} - \op{p}_{-} }{ \sqrt{2} }, \\
\op{p}_2 &= \frac{ \op{a}_2 - \op{a}_2^{\dagger} }{ i\sqrt{2} } = \frac{ - \op{a}_{+} + \op{a}_{-}^{\dagger} - \op{a}_{+}^{\dagger} + \op{a}_{-} }{ 2 } = \frac{ - \op{x}_{+} + \op{x}_{-} }{ \sqrt{2} },
\end{align}
where we adopt \( \op{a}_{\pm} \) as shorthand for \( \op{a}_{ \omega \pm \Omega } \), \( \op{x}_{\pm} \) for \( \op{x}_{ \omega \pm \Omega } \), and \( \op{p}_{\pm} \) for \( \op{p}_{ \omega \pm \Omega } \).
We can recognise from this that the position and momentum operators of the two-photon creation/annihilation operators correspond to a rotation of the sideband frequency creation/annihilation operators.
Thus we shall express states as a function of the \( \{ \op{x}_1, \op{x}_2, \op{p}_1, \op{p}_2 \} \) operators which we see is equivalent to the use of the \( \{ \op{x}_+, \op{p}_+, \op{x}_-, \op{p}_- \} \) operators through a simple change of basis.
Here the non-zero commutators,
\(
\commutator{ \op{x}_+(\Omega_A) }{ \op{p}_+(\Omega_B) } = \commutator{ \op{x}_-(\Omega_A) }{ \op{p}_-(\Omega_B) } = i\delta(\Omega_A-\Omega_B)
\)
in terms of the frequency quadratures, have become
\(
\commutator{ \op{x}_1(\Omega_A) }{ \op{x}_2(\Omega_B) } = \commutator{ \op{p}_1(\Omega_A) }{ \op{p}_2(\Omega_B) } = i\delta(\Omega_A-\Omega_B)
\).
\section{Input-output relations for a gravitational-wave interferometer}
Optical interferometers typically require mirrors to redirect light back to a common point in order to generate interference.
When the motion of the mirrors are disturbed by the reflected light a squeezing of the optical fields is produced through the interaction between optical and mechanical modes.
This optomechanical effect is particularly apparent in systems used to resolve the displacement of the mechanical system such as laser-interferometric \ac{GW} detectors.
For the tuned interferometer optical fields evolve through the interferometer as~\cite{miao_quantum_2012}
\begin{align}
\op{b}_1^{(j)}(t) &= \op{a}_1^{(j)}(t-2\tau),\label{eq:b1time}\\
\op{b}_2^{(j)}(t) &= \op{a}_2^{(j)}(t-2\tau) + \sqrt{\frac{2I_j}{\hbar\omega_j}}\frac{\omega_j}{c} \op{x}_d(t-\tau),\label{eq:b2time}
\end{align}
where \( \op{x}_d \) is the differential motion of the two mirrors and common to all optical modes.
The signal to sense \( h(t) \) acts on the mechanical part of the optomechanical sensor as a force causing the interferometer arm lengths to vary.
In the tuned configuration differential motion of the mirrors is thus
\begin{equation}
m \ddot{\op{x}}_d (t) + m \Omega_p \op{x}_d (t) = 4 \sum\limits_{i} \sqrt{\frac{2\hbar\omega_iI_i}{c^2}} \op{a}_1^{(i)}(t-\tau) + mL\ddot{h}(t).
\label{eq:xd_time}
\end{equation}
Translating the optical field evolution to the frequency domain Eqs.~\eqref{eq:b1time} and~\eqref{eq:b2time} become
\begin{align}
\op{b}_1^{(j)}(\Omega) &= e^{2i\Omega\tau_j} \op{a}_1^{(j)}(\Omega), \\
\op{b}_2^{(j)}(\Omega) &= e^{2i\Omega\tau_j} \op{a}_2^{(j)}(\Omega) - \frac{e^{i\Omega\tau_j}}{m(\Omega^2 - \Omega_p^2)}\sqrt{\frac{2I_j}{\hbar\omega_j}}\frac{\omega_j}{c} \left[ \sum\limits_i 4e^{i\Omega\tau_i} \sqrt{\frac{2\hbar\omega_iI_i}{c^2}} \op{a}_1^{(i)}(\Omega) - m L \Omega^2 h(\Omega) \right],
\label{eq:app:b2freq}
\end{align}
where the \( 1/\left( \Omega^2 - \Omega_p^2 \right) \) term produces a resonance at \( \Omega = \Omega_p \).
From this we derive the expressions for multi-mode input-output relations (Eq.~(2) in the main text)
\begin{equation}
\begin{aligned}
\mathcal{M}_{jk} &=
e^{i(\beta_j+\beta_k)}
\begin{pmatrix}
\delta_{jk} & 0 \\
- \chi\sqrt{ \kappa_j \kappa_k } & \delta_{jk}
\end{pmatrix},
&
\mathcal{V}_{j} &=
\chi
\frac{ e^{i\beta_j} }{ h_{\text{SQL}} }
\begin{pmatrix}
0 \\
\sqrt{ 2 \kappa_j }
\end{pmatrix}.
\end{aligned}
\label{eq:app:multiMV_inout}
\end{equation}
where
\begin{equation}
\begin{aligned}
\chi &= \sign \left( \Omega^2 - \Omega_p^2 \right) &
\kappa_i &= \left| \frac{2\sqrt{2} I_i \omega_i}{ m c^2 \left( \Omega^2 - \Omega_p^2 \right) } \right| &
\beta_i &= \Omega \tau_i &
h_{\text{SQL}} &= \sqrt{\frac{4\hbar}{m L^2 \Omega^2}}
\end{aligned}
\end{equation}
give the input-output relations for a simplified interferometer, accounting for the cavity modes retrieves the form given in Eq.~(3) in the main text~\cite{miao_quantum_2012}.
In the tuned configuration the pendulum response can typically be found around \( \SI{\sim1}{\Hz} \)~\cite{corbitt_squeezed-state_2006,the_ligo_scientific_collaboration_advanced_2015} which lies below the principal frequency range of advanced LIGO~\cite{the_ligo_scientific_collaboration_advanced_2015} and so it is sufficient to take \( \Omega_p \ll \Omega \) and consider \( \chi=1 \) and drop the \( \Omega_p \) from the definition of \( \kappa_i \).
Through the signal-recycling mirror~\cite{buonanno_quantum_2001,buonanno_signal_2002} the mechanical response is modified to exhibit a spring-like reaction at some higher frequency \( \Theta \) which does appear at larger frequencies.
While the input-output relations are generally more complicated than those of Eq.~\eqref{eq:app:multiMV_inout} in the low-frequency regime---where radiation pressure dominates---the input-output relations can be reduced to follow the form of Eq.~\eqref{eq:app:multiMV_inout}~\cite{corbitt_squeezed-state_2006}.
In the low-frequency domain, with mirror motion well below the cavity bandwidth (\( \Omega \ll \gamma_i \)) the signal-recycling mirror configuration reduces to the same form of input-output relations as Eq.~\eqref{eq:app:multiMV_inout} with the optomechanical couplings \( \kappa_i \) being proportional to \( 1/\left( \Omega^2 - \Theta^2 \right) \) and \( \chi = \sign (\Omega^2 - \Theta^2) \).
The two cases \( \chi = 1 \) and \( \chi = -1 \) can be related through the phase shift \( D = \bigoplus\limits_{i=1}^d \begin{pmatrix} 1 & 0 \\ 0 & \chi \end{pmatrix} \) and so the \( \chi = 1 \) response is equivalent to an interferometer with negative response which undergoes a \( \pi \) phase shift acting on \( \opvec{x}_2 \) and \( \opvec{p}_2 \) before the initial state is input to the sensor and after the state is output from the sensor.
Specifically for a squeezed vacuum input such as in Eq.~\eqref{eq:app:squeezed_input}
\[
\sigma = \bigoplus_{i=1}^2 \left[ \bigoplus_{j=1}^d \begin{pmatrix} \cosh 2r_j + \sinh 2r_j \cos 2\phi_j & \sinh 2r_j \sin 2\phi_j \\ \sinh 2r_j \sin 2\phi_j & \cosh 2r_j - \sinh 2r_j \cos 2\phi_j \end{pmatrix} \right],
\]
if then followed by a phase shift \( D \) which acts as \( (D \oplus D) \sigma (D \oplus D) \) leaving the squeezing maginitudes unchanged and negates the squeezing angles
\[
\begin{aligned} (D \oplus D) & \sigma(r_1,\phi_1,\cdots,r_d,\phi_d) (D \oplus D) \\
&= (D \oplus D) \left\{ \bigoplus_{i=1}^2 \left[ \bigoplus_{j=1}^d \begin{pmatrix} \cosh 2r_j + \sinh 2r_j \cos 2\phi_j & -\sinh 2r_j \sin 2\phi_j \\ -\sinh 2r_j \sin 2\phi_j & \cosh 2r_j - \sinh 2r_j \cos 2\phi_j \end{pmatrix} \right] \right\} (D \oplus D) \\
&= \sigma(r_1,-\phi_1,\cdots,r_d,-\phi_d),
\end{aligned}
\]
the squeezing angles \( \{ \phi_i \} \) when \( \chi = -1 \) give the same sensitivity as the squeezing angles \( \{ -\phi_i \} \) in the absence of the phase shift.
Similarly performing a rotation \( \theta_i \) between \( \op{x}_1^{(i)} \) and \( \op{x}_2^{(i)} \) as done to model homodyne detection in Sec.~\ref{app:hom_squeezeloss} after the phase operation \( D \) is equivalent to performing the rotation \( -\theta_i \) without the phase shift.
This allows us to recover sensitivities for the \( \chi = -1 \) response from the expressions for the \( \chi = 1 \) response and apply our conclusions to either sign of the response.
\section{Quantum Cramér-Rao bound for estimating a displacement from input-output relations}
\label{app:inout_qcrb}
For a Gaussian input state whose covariance matrix is block-diagonal, namely \( \sigma_{\text{dark}} = \sigma_0 \oplus \sigma_0 \), and has an input-output relation of the form
\begin{equation}
\opvec{b}(\Omega) = B M B \opvec{a}(\Omega) + h(\Omega) B \vec{V}(\Omega),
\end{equation}
where \( M \in \mathbb{R}^{2d{\times}2d} \) and \( \vec{V} \in \mathbb{R}^{2d} \), and
\(
B = \diag ( e^{i\beta_1}, e^{i\beta_1}, e^{i\beta_2}, e^{i\beta_2}, \cdots, e^{i\beta_d}, e^{i\beta_d} )
\).
The evolved state is given by
\begin{equation}
\begin{aligned}
\vec{d} &= \sqrt{2} \mathcal{S}_B \begin{pmatrix} \Re[h] \vec{V} \\ \Im[h] \vec{V} \end{pmatrix},
&
\sigma &=
\mathcal{S}_B
\begin{pmatrix}
M \left( \Re[B] \sigma_0 \Re[B] + \Im[B] \sigma_0 \Im[B] \right) M^T & 0 \\
0 & M \left( \Re[B] \sigma_0 \Re[B] + \Im[B] \sigma_0 \Im[B] \right) M^T
\end{pmatrix}
\mathcal{S}_B^T.
\end{aligned}
\label{eq:app:ideal_moments}
\end{equation}
The \ac{QFI} for the magnitude of any signal \( |h| = \sqrt{( \Re h )^2 + ( \Im h )^2}\) is given by
\begin{equation}
H(|h|) = 2 ( \partial_{|h|} \vec{d} )^T \sigma^{-1} ( \partial_{|h|} \vec{d} ),
\end{equation}
which for \( \vec{d} \) and \( \sigma \) of form
\begin{equation}
\begin{aligned}
\vec{d} &= \mathcal{S}\begin{pmatrix} \Re[h] \vec{W} \\ \Im[h] \vec{W} \end{pmatrix}, &
\sigma &= \mathcal{S}\begin{pmatrix} \sigma_N & 0 \\ 0 & \sigma_N \end{pmatrix}\mathcal{S}^T,
\end{aligned}
\label{eq:app:abstract_moments}
\end{equation}
is
\begin{equation}
H(|h|) = 2\left[ ( \partial_{|h|} \Re h )^2 + ( \partial_{|h|} \Im h )^2 \right] \vec{W}^T \sigma_N^{-1} \vec{W},
\end{equation}
as \( ( \partial_{|h|} \Re h )^2 + ( \partial_{|h|} \Im h )^2 = 1 \) the \ac{QFI} can thus be reduced to
\begin{equation}
H(|h|) = 2\vec{W}^T \sigma_N^{-1} \vec{W},
\end{equation}
which is equivalent to the \ac{QFI} obtained by taking the signal to be real.
The ideal state in Eq.~\eqref{eq:app:ideal_moments} is of the form of Eq.~\eqref{eq:app:abstract_moments} and remains such under mixture with any Gaussian state of the same form, including thermal states which have a diagonal covariance matrix, producing a state \( \eta \left( \Re[B] \sigma_0 \Re[B] + \Im[B] \sigma_0 \Im[B] \right) + (1-\eta)\sigma_1 \).
\begin{equation}
H(|h|) = 4 \vec{V}^T
\left[
\eta M \left( \Re[B] \sigma_0 \Re[B] + \Im[B] \sigma_0 \Im[B] \right) M^T + (1-\eta) \sigma_1
\right]^{-1}
\vec{V},
\label{eq:app:qfi_magnitude}
\end{equation}
Thus taking \( h \in \mathbb{R} \) allows us to consider only the \( \opvec{x} \) modes as the state has a covariance matrix with block form \( \sigma_0 \oplus \sigma_0 \) it is separable between the \( \opvec{x} \) and \( \opvec{p} \) modes; with the latter modes contain no parameter dependence and being uncorrelated with any of the modes which have a parameter-dependence the \( \opvec{p} \) modes can be discarded from our analysis.
If the input covariance matrix \( \sigma_0 \) and the covariance matrix the state is mixed with at the output \( \sigma_1 \) are block-diagonal with block size \( n \) and \( B \) is of the form \( \bigoplus_j e^{i\beta_j} \identity_{n {\times} n } \), then this simplifies through \( \Re[B] \sigma_0 \Re[B] + \Im[B] \sigma_0 \Im[B] = \sigma_0 \) and \( \Re[B] \sigma_1 \Re[B] + \Im[B] \sigma_1 \Im[B] = \sigma_1 \).
\begin{equation}
H(|h|) = 4 \vec{V}^T \left( \eta M \sigma_0 M^T + (1-\eta) \sigma_1 \right)^{-1} \vec{V}.
\end{equation}
This is the case for systems considered in Secs.~\ref{app:qcrb_squeezeloss} and~\ref{app:hom_squeezeloss} where the externally input squeezing is localised to a carrier and so \( \sigma_0 \) is block-diagonal with \( 2 {\times} 2 \) blocks.
\section{Relation between the Cramér-Rao bound and spectral density for a signal with white Gaussian noise}
\label{app:crb_spectraldensities}
For estimating a signal \( h(t) \) from a measured signal \( y(t) = h(t) + w(t) \), which is a stationary process, the sensitivity can be measured by the single-sided spectral density~\cite{braginsky_quantum_1992,thorne_modern_2017}
\begin{equation}
S(\Omega) = 2 \int\limits_{-\infty}^{\infty} \intd \tau \, C_w(\tau) \cos(\Omega\tau),
\end{equation}
where \( C_w (\tau) \) is the autocorrelation function
\begin{equation}
C_w (\tau) = \lim\limits_{T\to\infty} \frac{1}{2T} \int\limits_{-T}^{T} \intd t \, [w (t) - \bar{w}] [w (t+\tau) - \bar{w}],
\end{equation}
where \( \bar{w} \) is the time-average of \( w \).
For a white Gaussian noise process \( C_w(\tau) = \upsilon \delta (\tau) \), this is simply
\begin{equation}
S(\Omega) = 2 \upsilon.
\label{eq:app:spectral_gen}
\end{equation}
For the same signal \( y(t) = h(t) + w(t) \) the precision of any estimator of a parameter \( g \) of the signal is~\cite{kay_fundamentals_1998}
\begin{equation}
(\Delta h)^2 \geq
\frac{\upsilon}{\mathbb{E} \left[ \left( \frac{ \partial h(t) }{ \partial g } \right)^2 \right]}.
\end{equation}
The equivalent case of interest to the spectral noise density is the amplitude of the frequency modes \( h(\Omega) \), for which the relevant derivative is
\begin{equation}
\begin{aligned}
\frac{\partial h(t)}{\partial |h(\Omega)|} &=
\frac{\partial}{\partial |h(\Omega)|}
\int\limits_{-\infty}^{\infty} \intd \Omega' \, h(\Omega') e^{i\Omega' t} \\
&=
\int\limits_{-\infty}^{\infty} \intd \Omega' \, \left[ \delta(\Omega-\Omega') e^{i\arg (h(\Omega'))} + \delta(\Omega+\Omega') e^{i\arg (h(\Omega'))} \right] e^{i\Omega' t}\\
&=
2\cos \left[ \Omega t + \arg(h(\Omega)) \right],
\end{aligned}
\end{equation}
where the derivative yields two terms due to the Fourier transform property \( h(-\Omega) = h(\Omega)^{\dagger} \).
The expectation of the square of \( \partial_{|h(\Omega)|} h(t) \) is then simply \( 2 \) giving a \ac{CRB} of
\begin{equation}
(\Delta |h(\Omega)|)^2 \geq \frac{\upsilon}{2},
\label{eq:app:crb_gen}
\end{equation}
showing a proportionality constant of \( 4 \) relating these two methods of calculating sensitivities given by Eqs.~\eqref{eq:app:spectral_gen} and~\eqref{eq:app:crb_gen}.
\section{Quantum Cramér-Rao bound for a lossy interferometer with squeezed vacuum input}
\label{app:qcrb_squeezeloss}
Parallel squeezing corresponds to the covariance matrix of the \( \opvec{x} \) operators on the input dark port being
\begin{equation}
\sigma_{\text{dark}}
=
\bigoplus_{i=1}^d
\begin{pmatrix}
\cosh 2r_i + \sinh 2r_i \cos 2\phi_i &
\sinh 2r_i \sin 2\phi_i \\
\sinh 2r_i \sin 2\phi_i &
\cosh 2r_i - \sinh 2r_i \cos 2\phi_i
\end{pmatrix},
\label{eq:app:squeezed_input}
\end{equation}
which evolves through the interferometer to
\begin{equation}
\sigma_{ij}
=
\begin{multlined}
\left(
\begin{array}{c}
\delta_{ij} (\cosh 2r_i + \sinh 2r_i \cos 2\phi_i) \\
\delta_{ij} \sinh 2r_i \sin 2\phi_i - \sqrt{\kappa_i \kappa_j}(\cosh 2r_j + \sinh 2r_j \cos 2\phi_j)
\end{array}
\right.
\\
\shoveright{
\left.
\begin{array}{c}
\delta_{ij} \sinh 2r_i \sin 2\phi_i - \sqrt{\kappa_i \kappa_j}(\cosh 2r_i + \sinh 2r_i \cos 2\phi_i) \\
\delta_{ij} (\cosh 2r_i - \sinh 2r_i \cos 2\phi_i) - \sqrt{\kappa_i \kappa_j} \left( \sinh 2r_i \sin 2\phi_i + \sinh 2r_j \sin 2\phi_j - K_{\text{Tot}} \right)
\end{array}
\right)
}
\end{multlined}
\end{equation}
where each mode has a squeezing \( \xi_k = r_k e^{i\phi_k} \), and
\[
K_{\text{Tot}} = \sum\limits_{k=1}^d \kappa_k \left( \cosh 2r_k + \sinh 2r_k \cos 2\phi_k \right),
\]
represents a squeezed version of \( \kappa_{\text{Tot}} \).
The effect of loss on \( \sigma \) is to mix the matrix with the state \( \sigma_{\text{Loss}} \), which we will take to be the vacuum state, under
\[
\sigma \to \eta \sigma + (1-\eta) \sigma_{\text{Loss}},
\]
produces the Gaussian state on the \( \opvec{x} \) modes
\begin{equation}
\sigma_{ij}
=
\begin{multlined}[t]
\delta_{ij}
\begin{pmatrix}
\eta\left( \cosh 2r_i + \sinh 2r_i \cos 2\phi_i \right) + (1-\eta) &
\eta\sinh 2r_i \sin 2\phi_i \\
\eta\sinh 2r_i \sin 2\phi_i &
\eta\left( \cosh 2r_i - \sinh 2r_i \cos 2\phi_i \right) + (1-\eta)
\end{pmatrix}\\
-\eta\sqrt{\kappa_i \kappa_j}
\begin{pmatrix}
0 &
\cosh 2r_i + \sinh 2r_i \cos 2\phi_i \\
\cosh 2r_j + \sinh 2r_j \cos 2\phi_j &
\sinh 2r_i \sin 2\phi_i + \sinh 2r_j \sin 2\phi_j - K_{\text{Tot}}
\end{pmatrix}.
\end{multlined}
\label{eq:app:cov_squeezeloss}
\end{equation}
This covariance matrix has no obvious inverse, however we can rearrange \( \sigma \) in block form such that the top left quarter is the covariances of the \( \opvec{x}_1^{(\omega_i)} \) operators and observe that this can be rewritten in terms of the matrices \( Q \), \( R \), \( S \), and \( L \) where
\begin{equation}
\begin{aligned}
Q_{ij} &= \delta_{ij} \left( \cosh 2r_i + \sinh 2r_i \cos 2\phi_i \right),\\
R_{ij} &= \delta_{ij} \left( \cosh 2r_i - \sinh 2r_i \cos 2\phi_i \right),\\
S_{ij} &= \delta_{ij} \sinh 2r_i \sin 2\phi_i, \\
L_{ij} &= \vec{k}\vec{k}^T = \sqrt{\kappa_i \kappa_j},
\end{aligned}
\end{equation}
such that
\begin{equation}
\sigma
=
\begin{pmatrix}
(1-\eta) \identity + \eta Q &
\eta \left( S - Q L \right) \\
\eta \left( S - L Q \right) &
(1-\eta) \identity + \eta \left( R - S L - L S + K_{\text{Tot}} L \right)
\end{pmatrix},
\label{eq:app:cov_squeezeloss_rearranged}
\end{equation}
the inverse can then be found, with all blocks being invertible with the Woodbury matrix identity
\begin{equation}
\left( A + U C V \right)^{-1}
=
A^{-1} - A^{-1} U \left( C^{-1} + V A^{-1} U \right)^{-1} V A^{-1},
\label{eq:woodbury}
\end{equation}
The parameter information is encoded in the \( \{ \op{x}_2^{(\omega_i)} \} \) modes and so calculation of the \ac{QCRB} requires the lower right quarter of the inverse.
For a block matrix, the inverse is
\[
\begin{pmatrix}
A & B \\ C & D
\end{pmatrix}^{-1}
=
\begin{pmatrix}
\left( A - B D^{-1}C \right)^{-1} &
- \left( A - B D^{-1}C \right)^{-1} B D^{-1} \\
- D^{-1} C \left( A - B D^{-1}C \right)^{-1} &
D^{-1} + D^{-1} C \left( A - B D^{-1}C \right)^{-1} B D^{-1}
\end{pmatrix}.
\]
The relevant inverses are given---using Eq.~\eqref{eq:woodbury}---by
\begin{equation}
D^{-1} =
T^{-1}
-
\frac{\eta}{\alpha}
T^{-1}
\begin{pmatrix} \vec{k} & S\vec{k} \end{pmatrix}
\begin{pmatrix}
K_{\text{Tot}} - \eta\expect{ST^{-1}S} &
-1 + \eta\expect{T^{-1}S} \\
-1 + \eta\expect{T^{-1}S} &
-\eta\expect{T^{-1}}
\end{pmatrix}
\begin{pmatrix} \vec{k}^T \\ \vec{k}^T S \end{pmatrix}
T^{-1},
\end{equation}
where we introduce the definitions \( \expect{Z} = \trace{L Z} \), and can thus rewrite \( K_{\text{Tot}} \) as \( \expect{Q} \), and define \( T = (1-\eta) \identity + \eta R \) and \( \alpha = 1 + \eta\left[ K_{\text{Tot}}\expect{T^{-1}} - 2 \expect{S T^{-1} S}+ \eta \left( \expect{S T^{-1}}^2 - \expect{T^{-1}}\expect{S T^{-1} S} \right) \right] \).
As well as
\begin{equation}
\left( A - B D^{-1} C \right)^{-1}
=
W^{-1}
-
W^{-1}
\begin{pmatrix} S T^{-1} \vec{k} & U \vec{k} \end{pmatrix}
X^{-1}
\begin{pmatrix} \vec{k}^T T^{-1} S \\ \vec{k}^T U \end{pmatrix}
W^{-1},
\end{equation}
where \( U = \left( Q - \eta S T^{-1} S \right) \), \( W = (1-\eta)\identity + \eta Q - \eta^2 S T^{-1} S \), and
\[
X
=
\begin{pmatrix}
\frac{\expect{T^{-1}}}{\eta^2} + \expect{T^{-1} S W^{-1} S T^{-1}} &
\frac{1 - \eta \expect{S T^{-1}}}{\eta^2} + \expect{T^{-1} S W^{-1} U} \\
\frac{1 - \eta \expect{S T^{-1}}}{\eta^2} + \expect{T^{-1} S W^{-1} U} &
\frac{-\expect{Q} + \eta \expect{S T^{-1} S}}{\eta} + \expect{U W^{-1} U}
\end{pmatrix}.
\]
From this the \ac{QFI} can be evaluated with
\[
H(h) =
\frac{8\eta}{h_{\text{SQL}}^2}
\sum\limits_{i,j=1}^d
\sqrt{\kappa_i \kappa_j} \left( \sigma^{-1} \right)_{d+i,d+j}.
\]
From these expressions we can (after much simplification) construct the \ac{QFI}
\begin{equation}
H(h)
=
\frac{8\eta}{h_{\text{SQL}}^2}
\frac{(1-\eta)\expect{\Gamma} + \eta \expect{Q \Gamma}}
{
1 -
(1-\eta)\eta \left\{
\expect{S \Gamma} \left[ 2 - (1-\eta)\eta\expect{S \Gamma} \right]
- \left[ \eta \expect{\Gamma} + (1-\eta) \expect{Q \Gamma} \right] \left[ (1-\eta) \expect{\Gamma} + \eta \expect{Q \Gamma} \right] \right\}
},
\end{equation}
where
\[
\Gamma = T^{-1} W^{-1} = \left\{\left[(1-\eta)^2 + \eta^2\right]\identity + \eta(1-\eta)\left( Q + R \right)\right\}^{-1},
\]
from which the \ac{QCRB} is given by
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8}
\Bigg\{
\frac{1}{\eta(1-\eta)\expect{\Gamma} + \eta^2 \expect{Q \Gamma}}
- (1-\eta) \Bigg[
2\frac{\expect{S \Gamma}}{(1-\eta)\expect{\Gamma} + \eta \expect{Q \Gamma}}
- \eta(1-\eta) \frac{\expect{S \Gamma}^2}{(1-\eta)\expect{\Gamma} + \eta \expect{Q \Gamma}} - \left(\eta \expect{\Gamma} + (1-\eta) \expect{Q \Gamma} \right)
\Bigg]\Bigg\}.
\label{eq:app:qcrb_loss_squeeze}
\end{equation}
If we assume an equal squeezing mode and angle in each mode this \ac{QCRB} reduces to
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8}
\left\{
\frac{
1+2\eta(1-\eta)\left( \cosh 2r - 1 \right) -2\eta(1-\eta)\kappa_{\text{Tot}}\sinh 2r \sin 2\phi + \eta(1-\eta)\kappa_{\text{Tot}}K_{\text{Tot}}
}{\eta\left[ (1-\eta)\kappa_{\text{Tot}} + \eta K_{\text{Tot}} \right]}
\right\}.
\label{eq:app:qcrb_loss_equal_squeeze}
\end{equation}
\section{Cramér-Rao bounds for a lossy interferometer with squeezed vacuum input under homodyne detection}
\label{app:hom_squeezeloss}
The results of homodyne measurement are given by the marginal distribution of the Wigner function consisting of a set of commuting quadratures~\cite{adesso_continuous_2014}.
Homodyne measurement local to each carrier-mode would consist of performing homodyne read-out with some angle \( \theta_i \) between the two \( \{ \op{x}_1^{(\omega_i)} , \op{x}_2^{(\omega_i)} \} \) modes, where we omit the \( \opvec{p} \) modes by takin \( h \) to be real and thus leaving the statistics of these modes independent of \( h \).
Providing this angle is equivalent to performing a rotation \( \theta_i \) between the \( \op{x}_1^{(i)} \) and \( \op{x}_2^{(i)} \) modes
\begin{equation}
\mathcal{S}_{\text{Hom.}}(\vec{\theta})
=
\begin{pmatrix}
\cos \theta_1 & -\sin \theta_1 & \cdots & 0 & 0 \\
\sin \theta_1 & \cos \theta_1 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & \cos \theta_d & -\sin \theta_d \\
0 & 0 & \cdots & \sin \theta_d & \cos \theta_d
\end{pmatrix}
\end{equation}
and then measuring the \( \op{x}_2^{(i)} \) quadratures.
The resultant probability distributions are Gaussian with first-order moments
\begin{equation}
\vec{w} = \frac{2h}{h_{\text{SQL}}}
\begin{pmatrix} \sqrt{\kappa_1} \cos \theta_1 \\ \sqrt{\kappa_2} \cos \theta_2 \\ \vdots \\ \sqrt{\kappa_d} \cos \theta_d \end{pmatrix},
\end{equation}
and the second-order moments are given by
\begin{equation}
\Sigma =
(1-\eta) \identity + \eta \left( FQF + FSG + GSF + GRG - \left( GS + FQ \right) L G - G L \left( SG + QF \right) + K_{\text{Tot}} GLG \right),
\end{equation}
where
\(
G_{ij} = \delta_{ij} \cos \theta_i,
\)
and
\(
F_{ij} = \delta_{ij} \sin \theta_i.
\)
The \ac{CFI} of a Gaussian probability distribution can similarly be evaluated in terms of its moments as~\cite{kay_fundamentals_1998}
\begin{equation}
\mathcal{I}(h)
=
2\frac{ \partial \vec{w}^T }{ \partial h } \Sigma^{-1} \frac{ \partial \vec{w} }{ \partial h }
+
\frac{1}{2}\trace{ \left( \frac{ \partial \Sigma }{ \partial h } \Sigma^{-1} \right)^2 },
\label{eq:cfi_gaussian}
\end{equation}
where our definition of the covariance matrix \( \sigma \) differs from that of Ref.~\cite{kay_fundamentals_1998} by a factor of \( 2 \), leading to the extra factor in the first term of Eq.~\eqref{eq:cfi_gaussian}.
The \ac{CFI} is then given by
\[
\mathcal{I}(h)=
\frac{8\eta}{h_{\text{SQL}}^2}
\sum\limits_i
\cos \theta_i \cos \theta_j \sqrt{\kappa_i \kappa_j}
\left( \Sigma^{-1} \right)_{ij}
=
\frac{8\eta}{h_{\text{SQL}}^2}
\expect{G^2\Sigma^{-1}},
\]
where \( \Sigma^{-1} \) can be calculated with Eq.~\eqref{eq:woodbury} to be
\begin{equation}
\begin{aligned}
\Sigma^{-1}
&=
Y^{-1}
\\
&\mkern32mu -
Y^{-1}
\begin{pmatrix}
G \vec{k} &
\left( G S + F Q \right) \vec{k}
\end{pmatrix}
\begin{pmatrix}
\expect{G^2 Y^{-1}} & -\frac{1}{\eta} + \expect{G Y^{-1} \left( G S + F Q \right)} \\
-\frac{1}{\eta} + \expect{G Y^{-1} \left( G S + F Q \right)} & -\frac{(1-\eta)}{\eta}\expect{Q Y^{-1}} - \expect{G^2 Y^{-1}}
\end{pmatrix}^{-1}
\begin{pmatrix}
\vec{k}^T G \\
\vec{k}^T \left( S G + Q F \right)
\end{pmatrix}
Y^{-1},
\end{aligned}
\end{equation}
where \( Y = (1-\eta)\identity + \eta \left( F^2 Q + 2 F G S + G^2 R \right) \).
Thus \(\mathcal{I} \) is given by
\begin{equation}
\begin{aligned}
\mathcal{I}(h)
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\expect{G^2 \Sigma^{-1}}
\\
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\Bigg[\expect{G^2 Y^{-1}}
-
\begin{pmatrix}
\expect{G^2 Y^{-1}} & \expect{G Y^{-1} \left( G S + F Q \right)}
\end{pmatrix}
\\
&\mkern128mu
\begin{pmatrix}
\expect{G^2 Y^{-1}} & -\frac{1}{\eta} + \expect{G Y^{-1} \left( G S + F Q \right)} \\
-\frac{1}{\eta} + \expect{G Y^{-1} \left( G S + F Q \right)} & -\frac{(1-\eta)}{\eta}\expect{Q Y^{-1}} - \expect{G^2 Y^{-1}}
\end{pmatrix}^{-1}
\begin{pmatrix}
\expect{G^2 Y^{-1}} \\
\expect{G Y^{-1} \left( G S + F Q \right)}
\end{pmatrix}
\Bigg]
\\
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\frac{\expect{G^2 Y^{-1}}}{\left[ 1 - \eta\expect{G Y^{-1} (G S + F Q)} \right]^2 + \eta \expect{G^2 Y^{-1}}\left[ (1-\eta)\expect{Q Y^{-1}} + \eta \expect{G^2 Y^{-1}}\right]}.
\end{aligned}
\label{eq:app:squeezed_hom_general}
\end{equation}
which gives a \ac{CRB} of
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8}
\left\{
\frac{\left[ 1-\eta\left( \expect{G^2 Y^{-1} S} + \expect{F G Y^{-1} Q} \right) \right]^2}{\eta\expect{G^2 Y^{-1}}} + (1-\eta) \expect{Q Y^{-1}} + \eta \expect{G^2 Y^{-1}}
\right\}.
\label{eq:app:hom_crb}
\end{equation}
\subsection{Cramér-Rao bounds for measurement along the signal quadratures}
For homodyne measurements along the signal quadrature Eq.~\eqref{eq:app:squeezed_hom_general} reduces to
\begin{equation}
\begin{aligned}
\mathcal{I}
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\expect{\Sigma^{-1}}
\\
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\left[\expect{T^{-1}}
-
\begin{pmatrix}
\expect{T^{-1}} & \expect{T^{-1} S}
\end{pmatrix}
\begin{pmatrix}
\expect{T^{-1}} & -\frac{1}{\eta} + \expect{T^{-1} S} \\
-\frac{1}{\eta} + \expect{T^{-1} S} & -\frac{K_{\text{Tot}}}{\eta} + \expect{S T^{-1} S}
\end{pmatrix}^{-1}
\begin{pmatrix}
\expect{T^{-1}} \\
\expect{T^{-1} S}
\end{pmatrix}
\right]
\\
&=
\frac{8\eta}{h_{\text{SQL}}^2}
\frac{\expect{T^{-1}}}{\left( 1 - \eta\expect{T^{-1}S} \right)^2 + \eta \expect{T^{-1}}\left( \expect{Q} - \eta \expect{S T^{-1} S} \right)},
\end{aligned}
\end{equation}
where the limit is given by \( \theta_i = 0, \forall i\), which implies \( F = 0 \), \( G = \identity \), and \( Y = T \).
With \ac{CRB}
\begin{equation}
\left( \Delta h \right)^2 \geq
\frac{h_{\text{SQL}}^2}{8}
\left[ \frac{ ( 1 - \eta \expect{T^{-1}S} )^2}{\eta\expect{T^{-1}}} + \expect{Q} - \eta \expect{ST^{-1}S} \right]
\label{eq:app:signalhom_crb}
\end{equation}
In the lossless regime the bound becomes
\begin{equation}
(\Delta h)^2 \geq
\frac{h_{\text{SQL}}^2}{8}
\left[
\frac{\left( 1 - \expect{R^{-1}S} \right)^2}{\expect{R^{-1}}} + \expect{R^{-1}}
\right].
\label{eq:app:lossless_signalhom_crb}
\end{equation}
\subsection{Cramér-Rao bounds for measurement along the optimal quadrature}
\label{app:hom_optimal}
In the limit \( r_i = r, \forall i \), \( \phi_i = \phi, \forall i \), and \( \theta_i = \theta, \forall i \); Eq.~\eqref{eq:app:hom_crb} reduces to
\begin{equation}
\begin{aligned}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8\eta}
\Bigg[&
\frac{(1-\eta)\sec^2 \theta + \eta \left( \cosh 2r - \sinh 2r \cos 2\phi + 2\sinh 2r \sin 2\phi \tan \theta + \left( \cosh 2r + \sinh 2r \cos 2\phi \right) \tan^2 \theta \right)}{\kappa_{\text{Tot}}}
\\
&\mkern32mu
+ \eta \left( K_{\text{Tot}}
- 2\left( \sinh 2r \sin 2\phi+ \left( \cosh 2r + \sinh 2r \cos 2\phi \right) \tan \theta \right)
\right)
\Bigg].
\end{aligned}
\label{eq:app:hom_signal_crb}
\end{equation}
The optimal homodyne angle for this case is given by
\begin{equation}
\theta =
\arctan \left(
\eta
\frac{K_{\text{Tot}} - \sinh 2r \sin 2\phi}{ 1-\eta + \eta ( \cosh 2r + \sinh 2r \cos 2\phi )}
\right),
\end{equation}
at which homodyne angle the \ac{CRB} becomes
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8\eta}
\frac{
\eta^2
+
(1-\eta)
\left[
\left( 1-\eta \right) + 2\eta \cosh 2r
-2\kappa_{\text{Tot}} \sinh 2r\sin 2\phi
+\eta \kappa_{\text{Tot}} K_{\text{Tot}}
\right]
}{(1-\eta) \kappa_{\text{Tot}} + \eta K_{\text{Tot}}},
\end{equation}
attaining the \ac{QCRB} given in Eq.~\eqref{eq:app:qcrb_loss_equal_squeeze}.
In the lossless limit \( \eta \to 1 \) (with squeezings unconstrained) the \ac{CRB} becomes
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8}
\left\{
\frac{\left[ 1-\left( \expect{G^2 Z^{-1} S} + \expect{F G Z^{-1} Q} \right) \right]^2}{\expect{G^2 Z^{-1}}} + \expect{G^2 Z^{-1}}
\right\},
\label{eq:app:lossless_hom_crb}
\end{equation}
where \( Z = F^2 Q + 2 F G S + G^2 R \).
The homodyne angle
\begin{equation}
\theta_i = \arctan
\left( \frac{K_{\text{Tot}} - \sinh 2r_i \sin 2\phi_i}{\cosh 2r_i + \sinh 2r_i \cos 2\phi_i} \right),
\label{eq:app:hom_angle_lossless}
\end{equation}
then attains the \ac{QCRB} of
\begin{equation}
\left( \Delta h \right)^2
\geq
\frac{h_{\text{SQL}}^2}{8}
\frac{1}{K_{\text{Tot}}}.
\end{equation}
\section{Multi-carrier optimum sensitivity}
The Cramér-Rao bounds Eqs.~\eqref{eq:app:qcrb_loss_squeeze},~\eqref{eq:app:hom_crb}, and~\eqref{eq:app:signalhom_crb} all have form
\begin{equation}
\frac{8}{h_{\text{SQL}}^2}\left( \Delta h \right)^2 \geq B(\kappa_1,\cdots,\kappa_d) = \frac{\Big( 1 - \sum\limits_i c_1^{(i)} \kappa_i \Big)^2}{\sum\limits_i c_2^{(i)} \kappa_i} + \sum\limits_i c_3^{(i)} \kappa_i,
\label{eq:app:multi_abstract}
\end{equation}
with \( c_2^{(i)} \geq 0 \) and \( c_3^{(i)} \geq 0 \) with equality only holding only (but not necessarily) if \( \eta = 1 \).
The optimum sensitivity is achieved by minimising \( B \) which---unlike typical interferometry cases---generally does not find maximum at \( \kappa_i \to \infty \) as \( c_3^{(i)} > 0 \) means \( B \) diverges when any \( \kappa_i \to \infty \) and similarly at \( \kappa_i = 0, \forall i \) \( B \) diverges.
The derivatives and Hessian of Eq.~\eqref{eq:app:multi_abstract} are
\begin{gather}
\frac{\partial B}{\partial \kappa_j} = -c_2^{(j)} \left( \frac{ 1 - \sum\limits_i c_1^{(i)} \kappa_i }{\sum\limits_i c_2^{(i)} \kappa_i} \right)^2 - 2c_1^{(j)} \left( \frac{ 1 - \sum\limits_i c_1^{(i)} \kappa_i }{\sum\limits_i c_2^{(i)} \kappa_i} \right) + c_3^{(j)},
\label{eq:app:B_grad} \\
\frac{\partial^2 B}{\partial \kappa_j \partial \kappa_k} =
\frac{2}{\sum\limits_i c_2^{(i)} \kappa_i}
\left[ c_1^{(j)} + c_2^{(j)} \left( \frac{ 1 - \sum\limits_i c_1^{(i)} \kappa_i }{\sum\limits_i c_2^{(i)} \kappa_i} \right) \right]
\left[ c_1^{(k)} + c_2^{(k)} \left( \frac{ 1 - \sum\limits_i c_1^{(i)} \kappa_i }{\sum\limits_i c_2^{(i)} \kappa_i} \right) \right],
\label{eq:app:B_hessian}
\end{gather}
which being positive semi-definite for \( \sum\limits_i c_2^{(i)} \kappa_i > 0 \) identifies \( B \) as convex.
Solving for roots of Eq.~\eqref{eq:app:B_grad} we find
\begin{equation}
\frac{ 1 - \sum\limits_i c_1^{(i)} \kappa_i }{ \sum\limits_i c_2^{(i)} \kappa_i }
=
- \frac{c_1^{(j)}}{c_2^{(j)}} \pm \sqrt{\left(\frac{c_1^{(j)}}{c_2^{(j)}}\right)^2 + \frac{c_3^{(j)}}{c_2^{(j)}} },
\label{eq:app:B_grad_soln}
\end{equation}
where the \( + \) and \( - \) solutions are valid for \( \sum\limits_i c_1^{(i)} \kappa_i < 1 \) and \( \sum\limits_i c_1^{(i)} \kappa_i > 1 \) respectively; and are always positive and negative respectively.
This indicates that a solution to \( \partial_{\kappa_1} B = \cdots = \partial_{\kappa_d} B = 0 \) cannot be found unless
\begin{equation}
- \frac{c_1^{(j)}}{c_2^{(j)}} + \sqrt{\left(\frac{c_1^{(j)}}{c_2^{(j)}}\right)^2 + \frac{c_3^{(j)}}{c_2^{(j)}} }
=
- \frac{c_1^{(k)}}{c_2^{(k)}} + \sqrt{\left(\frac{c_1^{(k)}}{c_2^{(k)}}\right)^2 + \frac{c_3^{(k)}}{c_2^{(k)}} },
\label{eq:app:multiroot_plus}
\end{equation}
for all \( j \) and \( k \); or
\begin{equation}
- \frac{c_1^{(j)}}{c_2^{(j)}} - \sqrt{\left(\frac{c_1^{(j)}}{c_2^{(j)}}\right)^2 + \frac{c_3^{(j)}}{c_2^{(j)}} }
=
- \frac{c_1^{(k)}}{c_2^{(k)}} - \sqrt{\left(\frac{c_1^{(k)}}{c_2^{(k)}}\right)^2 + \frac{c_3^{(k)}}{c_2^{(k)}} },
\label{eq:app:multiroot_minus}
\end{equation}
for all \( j \) and \( k \).
Considering first the case when neither Eq.~\eqref{eq:app:multiroot_plus} nor Eq.~\eqref{eq:app:multiroot_minus} hold for any \( j \) or \( k \).
As \( \kappa_i = 0, \forall i \) and \( \exists i | \kappa_i = \infty \) lead \( B \) to diverge, \( B \) must be extremised for \( \kappa_i \in [0,\infty) \) with \( \sum\limits_i \kappa_i \in (0,\infty) \).
If \( \partial_{\kappa_i} B = 0\) and \( \partial_{\kappa_j} B = 0 \) cannot be simultaneously satisfied then we find instead a set of possible minima \( \{ \kappa_i | \kappa_j = 0, \forall j\neq k\} \) where \( \kappa_k \) is such that \( \partial_{\kappa_k} B = 0 \).
These solutions are exactly the single-carrier configurations for the interferometer, which have optimal precision at
\begin{equation}
\kappa_i = \pm \frac{1}{\sqrt{(c_1^{(i)})^2 + c_2^{(i)} c_3^{(i)}}},
\end{equation}
where the \( - \) solution can be discarded as unphysical.
The overall optimum sensitivity is thus
\begin{equation}
\min\limits_i \left\{ -2\frac{c_1^{(i)}}{c_2^{(i)}} + 2\sqrt{\left(\frac{c_1^{(i)}}{c_2^{(i)}}\right)^2 + \frac{c_3^{(i)}}{c_2^{(i)}} } \right\},
\end{equation}
which is obtained using the carrier \( l \) where
\begin{equation}
l = \argmin\limits_i \left\{ -2\frac{c_1^{(i)}}{c_2^{(i)}} + 2\sqrt{\left(\frac{c_1^{(i)}}{c_2^{(i)}}\right)^2 + \frac{c_3^{(i)}}{c_2^{(i)}} } \right\}.
\end{equation}
When the solutions to \( \partial_{\kappa_i} B = 0 \) and \( \partial_{\kappa_j} B = 0 \) are compatible with one another then a family of potential solutions exist satisfying
\begin{equation}
\frac{1 - \sum\limits_{i \in \mathcal{A}_j^{\pm}} c_1^{(i)} \kappa_i}{\sum\limits_{i \in \mathcal{A}_j^{\pm}} c_2^{(i)} \kappa_i}
=
-\frac{c_1^{(i)}}{c_2^{(i)}} \pm \sqrt{\left(\frac{c_1^{(i)}}{c_2^{(i)}}\right)^2 + \frac{c_3^{(i)}}{c_2^{(i)}} },
\end{equation}
which reduces to
\begin{equation}
\sum\limits_{i \in \mathcal{A_j^{\pm}}} \sqrt{(c_1^{(j)})^2 + c_2^{(j)}c_3^{(j)}} \kappa_j = \pm 1,
\label{eq:app:degenerate_set}
\end{equation}
where
\begin{equation}
\mathcal{A}_j^{\pm} =
\left\{ i \middle|
- \frac{c_1^{(j)}}{c_2^{(j)}} \pm \sqrt{\left(\frac{c_1^{(j)}}{c_2^{(j)}}\right)^2 + \frac{c_3^{(j)}}{c_2^{(j)}} }
=
- \frac{c_1^{(i)}}{c_2^{(i)}} \pm \sqrt{\left(\frac{c_1^{(i)}}{c_2^{(i)}}\right)^2 + \frac{c_3^{(i)}}{c_2^{(i)}} }
\right\}.
\end{equation}
The \( - \) solutions again require some \( \kappa_i < 0 \) which is unphysical and so we can consider only the positive solutions.
When solutions \( \{ \kappa_i \} \) satisfy Eq.~\eqref{eq:app:degenerate_set} the precision attained is
\begin{equation}
-2\frac{c_1^{(i)}}{c_2^{(i)}} + 2\sqrt{\left(\frac{c_1^{(i)}}{c_2^{(i)}}\right)^2 + \frac{c_3^{(i)}}{c_2^{(i)}} },
\end{equation}
which is identical for all \( i \in \mathcal{A}_j^+ \).
For a set of configurations of which the minimum and maximum \( \kappa_{\text{Tot}} \) cases are single-carrier configurations.
\section{Applications to LIGO}
\footnotetext[1]{Values used are \( I_{\text{Tot}} = \SI{840}{\kW} \), \( L = \SI{4}{\km} \), \( M = \SI{40}{\kg} \), \( \gamma_i \approx \gamma = \SI[product-units=single]{2\pi x 500}{\per\s} \), \( \omega_i \approx \omega = \SI[product-units=single]{2\pi x 2.82e14}{\per\s} \)}
In the absence of external squeezing the fundamental limit (Eq.~(20) in the main text) is minimised by \( \kappa_{\text{Tot}} = \left[ (1-\eta)\eta \right]^{-\frac{1}{2}} \) while optimising the precision of measurement along the signal quadrature (Eq.~(21) in the main text) requires \( \kappa_{\text{Tot}} = \eta^{-\frac{1}{2}} \).
The latter limit is already achieved by the current LIGO interferometer around \( \Omega = \SI[product-units=single]{2\pi x 90}{\per\s} \)~\cite{note1}, while the introduction of frequency-dependent homodyne detection would increase the required intensity by \( 1/\sqrt{1-\eta} \) which for \( \eta \leq 0.99 \) is no more than an order of magnitude increase in the circulating power.
\begin{figure}[htb]
\includegraphics{sensitivity_plot}
\footnotetext[1]{Plots are for the given equation in the \( \eta = 1 \) limit}
\caption{
Plots of precision attainable unsqueezed and lossless, squeezed and lossless, and unsqueezed and lossy interferometers.
Values based on LIGO setup~\cite{danilishin_quantum_2012,note1} in the \( \kappa_{\text{Tot}} \approx g I_{\text{Tot}} \) regime, detector loss of \num{0.05} (\( \eta = 0.95 \)) and an equal squeezing amplitude \( e^{-2r} = 0.1 \) in each mode is used.
Bounds on \( 2\Delta h \) are plotted to give equivalent values to the spectral noise density.
Equation numbers refer to the main text.
}
\label{fig:plots}
\end{figure}
For the tuned gravitational-wave detector these sensitivity plots are given for identical squeezing in Fig.~\ref{fig:plots}, the case of zero external squeezing, zero loss, and zero loss and zero squeezing simultaneously with the optimal squeezing angle.
Values used are \( I_{\text{Tot}} = \SI{840}{\kW} \), \( L = \SI{4}{\km} \), \( M = \SI{40}{\kg} \), \( \gamma_i \approx \gamma = \SI[product-units=single]{2\pi x 500}{\per\s} \), \( \omega_i \approx \omega = \SI[product-units=single]{2\pi x 2.82e14}{\per\s} \) |
1906.00428 | \section{Introduction}
\label{introduction}
An (integer) partition of $n$ is a non-increasing sequence of positive integers $\lambda_1 \geq \lambda_2 \cdots \geq \lambda_r \geq 1$ that sum to $n$. Let $p(n)$ be the number of partitions of $n$. By convention, we take $p(0)=1$ and $p(n)=0$ for negative $n$.
This function has been extensively studied in the last century. In the 1920's
Ramanujan discovered amazing congruence properties for $p(n)$.
\begin{theorem*}
For all positive integers $j$, we have,
\begin{align*}
p(5^jn+\delta_{5,j}) & \equiv0\pmod{5^j}, \\
p(7^jn+\delta_{7,j}) & \equiv0\pmod{7^{[\frac{j+2}{2}]}}, \\
p(11^jn+\delta_{11,j}) & \equiv0\pmod{11^j},
\end{align*}
where $24\delta_{\ell,j}\equiv1\pmod{\ell^j}$ for $\ell \in \{5,7,11\}$.
\end{theorem*}
Ramanujan in \cite{RS1} proved the first two congruences for the case of $j=1$ by using the Jacobi triple product and later in \cite{RS2} using the theory of modular forms on $SL_2(\mathbb{Z})$. For arbitrary $j\geq 1$, Watson in \cite{W} gave a proof using modular equations for prime $5$ and $7$. Ramanujan in \cite{RS3} stated that he found a proof for the third congruence for $j=1,2$, but did not include the proof. In $1967$, Atkin in \cite{A2} gave a proof for the third congruence.
These fascinating congruence properties not only hold for the partition function itself, but also for the restricted partitions. To study a large class of restricted partitions, we study the partition function $p_{[1^c\ell^d]}(n)$. This partition function also well studied in recent years, for example see Chan and Toh \cite{CHT}, and Liuquan Wang \cite{WL}.
The partition function $p_{[1^c\ell^d]}(n)$ is defined using the generating function in the following way.
\[\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}=\sum_{n=0}^{\infty}p_{[1^c\ell^d]}(n)q^n.\]
To illustrate the importance of studying this partition function, let's look at the following four examples .
\begin{itemize}
\item $d=0$ and $c>0$: $c-$color partitions.\\
This partition function generates partitions of $n$ in to $c$ colors. See Gordon \cite{GB} and Atkin \cite{A1} for interesting congruence relations for primes less than or equal to 13.\\
\item $c=1, d=-1$: $\ell$-regular partitions.\\
This partition function generates partitions of $n$ with a restriction that no parts divisible by $\ell$. This partitions also well studied in recent years. See Wang \cite{WL2} and \cite{WL3} for divisibility properties of $5$- regular and $7$-regular partitions.\\
\item $c=1, d=-\ell$: $\ell$-core partitions.\\
This partition function generates partition of $n$ with a restriction that no hook numbers are divisible by $\ell$. See Wang \cite{WL} to see the divisibility properties of this partition function.\\
\item $c=1, d=1$ and $\ell=2$: The cubic partition function.\\
This partition function has a deep connection to the Ramanujan's cubic continued fraction, in \cite{CH} and \cite{CH2} Chan used this connection to obtain interesting congruences.
\end{itemize}
\vspace{3mm}
In 2016, in \cite{WL}, Wang proved the following congruences.\\
\begin{theorem*}[Wang, 2016]
For any integers $n\geq 0$ and $k\geq 1$,
\begin{align*}
p_{[1^111^{-11}]}\left(11^kn+11^k-5\right) & \equiv 0\pmod{11^k},\\
p_{[1^111^{-1}]}\left(11^{2k-1}n+\dfrac{7\cdot11^{2k-1}-5}{12}\right) & \equiv 0\pmod{11^k},\\
p_{[1^111^{1}]}\left(11^kn+\dfrac{11^k+1}{2}\right) & \equiv 0\pmod{ 11^k}.
\end{align*}
\end{theorem*}
Furthermore in \cite{WL}, he stated that it possible to obtain congruences for each value $c,d\in \mathbb{Z}$ separately.The primary goal of this paper is to find a unified way to prove congruences for the partition function $p_{[1^c{11}^d]}(n)$ for any $c,d\in \mathbf{Z}$.
\begin{theorem}\label{T:1.1}
For any integers $c$, $d$ and for any positive integer $r$,
\begin{equation}
p_{[1^c{11}^d]}({11}^rm+n_r)\equiv 0\pmod{{11}^{A_r}}
\end{equation}
where $24n_r\equiv(c+{11}d)\pmod{{11}^r}$.
\end{theorem}
Here $A_r$ only depends on the integers $c,d$ and it can be calculated explicitly.\\
Moreover, we can obtain the following corollary, this is similar to the Gordon's Theorem 1.2 in \cite{GB}.
\begin{corollary}\label{C:1.2}
For any positive integer $r$,
\[
p_{[1^c11^d]}(11^rm+n_r)\equiv 0\pmod{11^{\frac{1}{2}\alpha r +\epsilon}}
\]
where $24n_r\equiv(c+11d)\pmod{11^r}$, $\epsilon=\epsilon(c,d)=O(\log|c+11d|)$ and when $c+11d\geq 0$ , $\alpha$ depends on the residue of $c+11d\pmod{120}$ which is shown in Table \ref{T:1}.
\begin{table}[htp]
\centering
\footnotesize\setlength{\tabcolsep}{2.3pt}
\begin{tabular}{l@{\hspace{6pt}} *{25}{c}}
\cmidrule(l){2-25}
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
\midrule
\bfseries 0
& 2 & 1 & 2 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0\\
\bfseries 24
& 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0\\
\bfseries 48
& 1 & 1 & 2 & 2 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0\\
\bfseries 72
& 2 & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 1 & 1 & 0\\
\bfseries 96
& 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0\\
\bottomrule
\addlinespace
\end{tabular}
\caption{Values of $\alpha$}\label{T:1}
\end{table}
Here the entry is $\alpha(24i+j)$ where row labelled $24i$ and column labeled $j$. When $c+11d<0$, the last column must be changed to $2, 2, 2, 0, 2$.
\end{corollary}
\begin{remark}
This is the same shape as Gordon's result for $k-$colored partitions, with k replaced by $c+11d$.
\end{remark}
The remainder of the paper is as follows. Section \ref{prem} reviews properties of modular forms and operators on their coefficients. In Section \ref{proofs} the main results are proven. The paper closes with several examples in Section \ref{ex}.
\section{Preliminaries}\label{prem}
For a Laurent series $f(\tau)=\sum_{n\geq N} a(n)q^n$, we define the $U_p$ operator by,
\begin{equation}\label{E:2.1}
U_p\left(f(\tau)\right)=\sum_{pn\geq N} a(pn)q^n.
\end{equation}
\vspace{3mm}
Let $g(\tau)=\sum_{n\geq N} b(n)q^{n}$ be another Laurent series. The following simple property will play a key role in our proof.
\begin{equation}\label{E:2.2}
U_p\left(f(\tau)g(p\tau)\right)=g(\tau)U_p\left(f(\tau)\right).
\end{equation}
\begin{proposition}[\cite{AL}, lemma 7]\label{P:Gamma0N}
If $f(\tau)$ is a modular function for $\Gamma_0(N)$, if $p^2|N $, then $U_p\Big{(}f(\tau)\Big{)}$ is a modular function for $\Gamma_0(N/p)$.
\end{proposition}
\vspace{2mm}
Let $V$ be the vector space of modular functions on $\Gamma_0(11)$, which are holomorphic everywhere except possibly at 0 and $\infty$.\\
Atkin constructed a basis for $V$, Gordon in [2] slightly modified these basis elements and defined $J_{\nu}(\tau)$. For detailed information about the construction of the basis elements see \cite{A2}.
\begin{theorem}[Gordon \cite{GB}]
For all $\nu\in \mathbf{Z}$, we have:\\
\begin{enumerate}
\item $J_\nu(\tau)=J_{\nu-5}(\tau)J_5(\tau)$,
\item $\{J_\nu(\tau)|-\infty <\nu<\infty \}$ is a basis of $V$,
\item ${\rm ord}_{\infty}J_\nu(\tau)=\nu$,
\item $${\rm ord}_0J_\nu(\tau)=
\begin{cases}
-\nu \qquad & \text{if } \nu \equiv0\pmod{5}, \\
-\nu-1 & \text{if } \nu\equiv 1,2 \text{ or } 3\pmod{5}, \\
-\nu-2 & \text{if } \nu\equiv 4\pmod{5}.
\end{cases}
$$
\item The Fourier series of $J_\nu(\tau)$ has integer coefficients, and is of the form $J_\nu(\tau)=q^{\nu}+\dots$.
\end{enumerate}
\end{theorem}
Now let
\[\phi(\tau):=q^5\prod_{n=1}^{\infty}\left(\frac{1-q^{121n}}{1-q^n}\right)=\frac{\eta(121\tau)}{\eta(\tau)}.\]
This is a modular function on $\Gamma_0(121)$ by proposition \ref{P:Gamma0N}, hence $V$ is mapped to itself by the linear transformation,
\[T_{\lambda}:f(\tau)\rightarrow U_{11}\left(\phi(\tau)^{\lambda}f(\tau)\right)\]
where $\lambda$ is an integer. Let $(C_{\mu,v}^{\lambda})_{\mu,\nu}$ be the matrix of the linear transfomation $T_\lambda$ with respect to the basis elements $J_{\nu}$.
\begin{equation}\label{E:2.3}
U_{11}\left(\phi(\tau)^{\lambda}J_{\mu}(\tau)\right)
=\sum_{\nu}C_{\mu,\nu}^{\lambda}J_{\nu}(\tau)
\end{equation}
Gordon in \cite{GB}, proved an inequality (equation (17)) about the $11$-adic orders of the matrix elements (denoted by $\pi(C_{\mu,\nu})$).
\begin{equation}\label{E:2.4}
\pi(C_{\mu,\nu}^{\lambda})\geq \left[\dfrac{11{\nu}-\mu-5\lambda+\delta}{10}\right].
\end{equation}
Here $[x]$ means the floor function of the real number $x$ and $\delta=\delta(\mu,\nu)$ depends on the residues of $\mu$ and $\nu \pmod{5}$ according to the Table \ref{T:2}.\\
\begin{center}
\begin{table}[htp]
\begin{tabular}{|l@{\hspace{16pt}}|*{6}{c|}}
\hline
& \multicolumn{5}{c|}{$\nu$} \\
\cline{1-6}
\hline
$\mu$& 0 & 1 & 2 & 3 & 4 \\
\hline\hline
\cline{1-6}
0 & -1 & 8 & 7 & 6 & 15\\
1 & 0 & 9 & 8 & 2 & 11 \\
2 & 1 & 10 & 4 & 13 & 12\\
3 & 2 & 6 & 5 & 4 & 13 \\
4 & 3 & 7 & 6 & 5 & 9\\ \hline
\end{tabular}\\
\vspace{2mm}
\caption{Values of $\delta(\mu,\nu)$.}\label{T:2}
\end{table}
\end{center}
\vspace{-5mm}
We can clearly see from the table that $\delta \geq -1$, so we can rewrite \ref{E:2.4} as,
\begin{equation}\label{E:2.5}
\pi(C_{\mu,\nu}^{\lambda})\geq \left[\dfrac{11\nu-\mu-5\lambda-1}{10}\right].
\end{equation}
Now by Lemma $2.1(v)$, the Fourier series of $T_\lambda(J_{\mu})$ has all coefficients divisible by 11 if and only if,
$$C_{\mu,\nu}^{\lambda}\equiv0\pmod{11} \ \mbox{for all $\nu$}.$$
Now we define $\theta(\lambda,\mu)=1$ if all the coefficients of $U(\phi^{\lambda}J_{\mu})$ divisible by $11$. Otherwise we put $\theta(\lambda,\mu)=0$.\\
Now from the recurrences obtained in \cite{GB} page 119, we have
\begin{equation}\label{E:2.6}
\theta(\lambda-11,\mu)=
\theta(\lambda+12,\mu-5)=\theta(\lambda,\mu).
\end{equation}
Therefore the values of $\theta(\lambda,\mu)$ is completely determined by its values in the range $0\leq \lambda \leq 10$ and $0\leq \mu \leq 4$. Those values are listed in Table \ref{T:3}.
\begin{center}
\begin{table}[htp]
\begin{tabular}{|*{12}{c|}}
\hline
& \multicolumn{11}{c|}{$\lambda$} \\
\cline{1-12}
\hline
$\mu$ &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
\hline\hline
\cline{1-12}
0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0\\
1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
3 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\
4 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0\\ \hline
\end{tabular}\\
\vspace{2mm}
\caption{Values of $\theta(\lambda, \mu)$.}\label{T:3}
\end{table}
\end{center}
Now we define,
\begin{equation}\label{E:2.7}
A_r(c,d):=\sum_{i=1}^{r-1}\theta(\lambda_{i},\mu_i),
\end{equation}
for any positive integer $r$ and integers $c,d$. We put $A_0:=0$.\\
\vspace{2mm}
We construct a sequence of modular functions that are the generating functions for the $p_{[1^c{11}^d]}(n)$ restricted to certain arithmetic progressions. This generalizes Gordon's construction for "$k-$color" partitions. Here we use \eqref{E:2.2} repeatedly.
$L_0 := 1$, and
\begin{align*}
L_1(\tau):&=U_{11}\left(\phi(\tau)^c\prod_{n=1}^{\infty}\frac{(1-q^{11n})^d}{(1-q^{11 n})^d}\right),\\
&=U_{11}\left(q^{5c}\prod_{n=1}^{\infty}\frac{(1-q^{121n})^c(1-q^{11n})^d}{(1-q^n)^c(1-q^{11 n})^d}\right),\\
&=\prod_{n=1}^{\infty}(1-q^{11 n})^c(1-q^n)^d\sum_{m\geq \lceil\frac{5c}{11}\rceil}^{\infty}p_{[1^c11^d]}(11m-5c)q^m.
\end{align*}
Similarly we can define,
\begin{align*}
L_2(\tau) :&=U_{11}\left(\phi^d(\tau)L_1(\tau)\right),\\
L_2(\tau)& =\prod_{n=1}^{\infty}(1-q^{11 n})^d(1-q^n)^c\sum_{m\geq \left\lceil\frac{5\cdot d+\lceil\frac{5c}{11}\rceil }{11}\right\rceil}^{\infty}p_{[1^c11^d]}(11^2m-5\cdot 11\cdot d -5\cdot c)q^m.
\end{align*}
Now, to get an equation for higher powers, We define,
\begin{equation}\label{E:2.8}
L_r:=U_{11}\left(\phi^{\lambda_{r-1}}(\tau)L_{r-1}\right),
\end{equation}
where
\[ \quad \lambda_{r}=
\left\{
\begin{array}{ll}
c & \mbox{if $r$ is even }, \\
d & \mbox{if $r$ is odd}.
\end{array}
\right. \]\\
Then by a short calculation using \eqref{E:2.2} gives,
\begin{align}
\label{E:L_r}
L_{2r}(\tau)& =\prod_{n=1}^{\infty}(1-q^n)^c(1-q^{11 n})^d\sum_{m\geq \mu_{2r}}p_{[1^c11^d]}(11^{2r}m+n_{2r})q^m, \\
\notag
L_{2r-1}(\tau)& =\prod_{n=1}^{\infty}(1-q^{11 n})^c(1-q^n)^d\sum_{m\geq \mu_{2r-1}}p_{[1^c11^d]}(11^{2r-1}m+n_{2r-1})q^m.
\end{align}
\vspace{3mm}
From \eqref{E:2.5}, \eqref{E:2.6} and \eqref{E:2.8} we can see that,
\[n_{2r}=-5\cdot d\cdot 11^{2r-1}+n_{2r-1},\]
\[n_{2r-1}=-5\cdot c\cdot 11^{2r-2}+n_{2r-2}.\]
Since $n_0=0$, using above recurrence relations we have,
\vspace{2mm}
\begin{itemize}[label={}]
\item $n_1=-5\cdot c$
\item $n_2=-5\cdot 11\cdot d -5\cdot c$
\end{itemize}
Using induction,
\begin{align}
\label{E:n_r}
n_{2r-1}&=-c\left(\dfrac{11^{2r}-1}{24}\right)-11\cdot d\left(\dfrac{11^{2r-2}-1}{24}\right).\\
\notag
n_{2r}&=-c\left(\dfrac{11^{2r}-1}{24}\right)-11\cdot d\left(\dfrac{11^{2r}-1}{24}\right) .
\end{align}
From this we have that,\\
$24\cdot n_{2r-1}\equiv (c+11\cdot d) \mod 11^{2r-1}$ and
$24\cdot n_{2r}\equiv (c+11\cdot d) \mod 11^{2r}$ .\\
Therefore, for each $n_r$ are integers such that,
\[24n_{r}\equiv (c+11\cdot d) \mod 11^r.\]
Now, we need to find $\mu_r$ in terms of integers $c,d$. Notice that $\mu_r$ is the least integer $m$ such that $11^r m+n_r\geq 0$, which implies that,
\begin{align}
\label{E:mur}
\mu_{2r-1}& =\left\lceil\frac{11\cdot c+d}{24}-\dfrac{c+11\cdot d}{24\cdot11^{2r-1}}\right\rceil,\\
\notag
\mu_{2r}& =\left\lceil\frac{c+11\cdot d}{24}-\dfrac{c+11\cdot d}{24\cdot11^{2r}}\right\rceil.
\end{align}
Following Gordon, we represent these formulas in the following form.
\begin{align}
\label{E:2.11}
\mu_{2r-1}& =\left\lceil{\frac{11\cdot c+d}{24}}\right\rceil+\omega(c,d) \quad \mbox{if $|c+11\cdot d|<11^{2r-1}$}.\\
\notag
\mu_{2r}& =\left\lceil{\frac{c+11\cdot d}{24}}\right\rceil+\omega(c,d) \quad \mbox{if $|c+11\cdot d|< {11}^{2r}$ }.
\end{align}
\[\omega(c,d)=
\left\{
\begin{array}{ll}
1 & \mbox{if $c+ 11\cdot d<0$ and $24|(c+11\cdot d)$}, \\
0 & \mbox{Otherwise}. \\
\end{array} \right. \]
\vspace{2mm}
\section{The proofs}\label{proofs}
\vspace{2mm}
If $$f(\tau):=\sum_{n\geq n_0}a(n)q^n,$$ we define,
\begin{equation}\label{E:3.1}
\pi\left(f(\tau)\right):=\mbox{min}_{n\geq n_0}\left\{\pi(a(n)\right\}.
\end{equation}
Here we follow Gordon's argument to prove $\pi(L_r)\geq A_r(c,d)$.
\begin{proof}[Proof of Theorem \ref{T:1.1}]
We see that using Proposition \ref{P:Gamma0N}, $L_r\in V$ for all $r$. So we have,
\begin{equation}\label{E:3.2}
L_r=\sum_{\nu}a_{r,\nu}J_{\nu}.
\end{equation}
Now by \eqref{E:2.1}, \eqref{E:2.8} and \eqref{E:3.2},
\begin{equation}\label{E:3.3}
a_{r,\nu}=\sum_{\mu\geq\mu_{r-1}}a_{r-1,\mu}C_{\mu,\nu}^{\lambda_{r-1}}.
\end{equation}
\vspace{2mm}
Now we prove by induction,
\begin{equation}\label{E:3.4}
\pi(L_r)\geq A_r+\left[\dfrac{\nu-\mu_r}{2}\right] \quad \mbox{for $\nu\geq \mu_r$}.
\end{equation}
\begin{equation*}
\pi(a_{r,\nu}) \geq A_r + \left[\dfrac{\nu-\mu_r}{2}\right] \quad \text{for } \nu \geq \mu_r.
\end{equation*}
Since $A_0=0$, the result holds for $r=0$.
Now assume the result is true for $r-1$. Using \eqref{E:3.4} we have,
\begin{equation}\label{E:3.5}
\pi(a_{r-1,\nu})\geq A_{r-1}+\left[\dfrac{\nu-\mu_{r-1}}{2}\right] \quad \text{for $\nu\geq \mu_{r-1}$}.
\end{equation}
Now from equation \eqref{E:3.3} we have,
\begin{equation}\label{E:3.6}
\pi(a_{r,\nu})\geq \min_{\mu\geq\mu_{r-1}}\left(\pi(a_{r-1,\mu})+\pi(C_{\mu,\nu}^{\lambda_{r-1}})\right).
\end{equation}
From \eqref{E:2.5} and \eqref{E:3.5} the right hand side of \eqref{E:3.6} is at least equal to,
\begin{equation}
A_{r-1}+\left[\dfrac{\mu-\mu_{r-1}}{2}\right]+\left[\dfrac{11\nu-\mu-5\lambda_{r-1}-1}{10}\right].
\end{equation}
This expression cannot decrease if $\mu$ is increased by $2$, so its minimum occurs when $\mu=\mu_{r-1}+1$, therefore at,
\begin{equation}\label{E:3.8}
A_{r-1}+\left[\dfrac{11\nu-\mu_{r-1}-5\lambda_{r-1}-2}{10}\right].
\end{equation}
Now from \eqref{E:2.8} we have,
\begin{equation}\label{E:3.9}
\mu_r=\left\lceil\dfrac{5\lambda_{r-1}+\mu_{r-1}}{11}\right\rceil,
\end{equation}
therefore $\mu_r\geq\left(\dfrac{5\lambda_{r-1}+\mu_{r-1}}{11}\right).$\\
Plugging it in \eqref{E:3.8}, the right hand side of \eqref{E:3.5} is at least equal to
\begin{align*}
A_{r-1}+\left[\dfrac{11\nu-11\mu_{r}-2}{10}\right] &=A_{r-1}+1+\left[\dfrac{11(\nu-\mu_r)-12}{10}\right]\\
&\geq A_r+\left[\dfrac{\nu-\mu_r}{2}\right] \mbox{ for all $\nu\geq\mu_r+2$,}
\end{align*}
since $A_{r-1}+1\geq A_r$.\\
Now consider $\nu=\mu_r$ or $\nu=\mu_r+1$.
\vspace{3mm}
If $\mu=\mu_{r-1},$
\[\pi(a_{r-1},\mu_{r-1})+\pi\left(C_{\mu_{r-1},\nu}^{\lambda_{r-1}}\right)\geq A_{r-1}+\theta(\mu_{r-1},\lambda_{r-1})=A_r.\]
This also works when $\mu\geq \mu_{r-1}+2$, since by induction hypothesis,
\[\pi(a_{r-1},\mu)\geq A_{r-1}+\left[\dfrac{\mu-\mu_{r-1}}{2}\right]\geq A_r.\]
Now consider $\mu=\mu_{r-1}+1$,
\vspace{3mm}
Now we need to show,
\[\pi(a_{r-1},\mu_{r-1}+1)+\pi(C_{\mu_{r-1}+1,\nu}^{\lambda_{r-1}})\geq A_{r-1}+\left[\dfrac{11\nu-(\mu_{r-1}+1)-5\lambda_{r-1}+\delta(\mu_{r-1}+1,\nu)}{10}\right].\]
\vspace{3mm}
Since $\nu=\mu_r$ or $\mu_r+1$, it suffices to show that when $\theta(\lambda_{r-1},\mu_{r-1})=1$,
\[\left[\dfrac{11\mu_r-\mu_{r-1}-1-5\lambda_{r-1}+\delta(\mu_{r-1}+1,\mu_r)}{10}\right]\geq 1,\]
and
\[ \left[\dfrac{11(\mu_r+1)-\mu_{r-1}-1-5\lambda_{r-1}+\delta(\mu_{r-1}+1,\mu_r+1)}{10}\right]\geq 1.\]
\vspace{3mm}
Now from \eqref{E:2.6}, Table \ref{T:2} and Table \ref{T:3} we see that the above claims hold.\\
\end{proof}
\begin{proof}[Proof of Corollary \ref{C:1.2}].\\
Recall \eqref{E:2.7},\\
\[A_r(c,d)=\sum_{i=0}^{r-1}\theta(\lambda_i,\mu_i).\]
Here we follow Gordon's argument from Section 4 of \cite{GB}, again with $k$ replaced by $c + 11d$. Note that Gordon's calculations had terms involving $11k$, which will be written in a more symmetric shape here using the fact that $11(c+11d) \equiv 11c + d \pmod{24}$.
\begin{align*}
A_r(c,d) & =\sum_{i=0}^{\log_{11}(c+11d)}\theta(\lambda_i,\mu_i)+\sum_{i=\log_{11}(c+11d)}^{r-1}\theta(\lambda_i,\mu_i) \\
&=\sum_{i=0}^{\log_{11}(c+11d)}\theta(\lambda_i,\mu_i)+N_1\cdot\theta\left(d,\ceil[\Big]{\dfrac{11c+d}{24}}+\omega(c,d)\right) \\
& \qquad \qquad \qquad \qquad +N_2\cdot\theta\left(c,\ceil[\Big]{\dfrac{c+11d}{24}}+\omega(c,d)\right).
\end{align*}
Here $N_1$ is the number of odd integers and $N_2$ is the number of even integers in the interval $\left[\log_{11}|c+11d|,r-1\right]$ respectively.\\
\begin{equation}
\mbox{Set} \quad \alpha:=\alpha(c,d)=\theta\left(d,\ceil[\Big]{\dfrac{11c+d}{24}}+\omega(c,d)\right)+\theta\left(c,\ceil[\Big]{\dfrac{c+11d}{24}}+\omega(c,d)\right)
.\end{equation}
Now if $r\leq \log_{11}|c+11d|+1$ then $N_1=N_2=0$,
\[A_r\leq \log_{11}|c+11d|.\]
If $r>\log_{11}|c+11d|+1$,
\[\left|N_1-\frac{1}{2}(r-1-\log_{11}(c+11d))\right|+\left|N_2-\frac{1}{2}(r-1-\log_{11}(c+11d))\right|<1.\]
\vspace{2mm}
Now consider,
\[\left|A_r-\frac{1}{2}\alpha(r-1-\log_{11}(c+11d))\right|<2+\log_{11}|c+11d|,\]
\[\left|A_r-\frac{\alpha r}{2} \right|<2+\frac{\alpha}{2}+(1+\frac{\alpha}{2})\log_{11}|c+11d|.\]\\
So we have $A_r=\frac{1}{2}\alpha r+\mathcal{O}\left(\log{}|c+11d|\right)$.
\vspace{2mm}\\
Now we prove the condition for $\alpha$. As in [2], the proof is complete once we show that first $\alpha$ only depends $c+11d$ with the period $120$. Periodicity follows by the fact that $\alpha(c+11d)$ is invariant under the each maps,
\[c\rightarrow{c+120-11k} \quad\mbox{and}\quad d\rightarrow{d+k}\quad \mbox{for each integer k}.\]
\end{proof}.
\section{Examples}\label{ex}
Here we give three examples to demonstrate our method. The first two examples are Wang's results and the final one is a new example.\\
\begin{example}[$c=1$ and $d=1$]
\[A_r(1,1)=\sum_{i=0}^{r-1}\theta(\lambda_{i},\mu_{i}).\]
In this case, $\lambda_i=1$, $\mu_i=1$ for all $i$ and $\theta(1,1)=1$. So $A_r=r$ and $n_r=-\frac{11^r-1}{2}$.
thus we have,
\[p_{[1^1{11}^1]}\left(11^rm+\frac{11^r-1}{2}\right)\equiv 0\pmod{11^{r}}.\]
In [1], Wang obtained $n_r=-\frac{11^r+1}{2}$, which is slightly different from the $n_r$ given here, though one can obtain Wang's $n_r$
by setting $m \mapsto m-1$, thus both congruences are equivalent.\\
Now, we illustrate corollary \ref{C:1.2} using this example. Since $c+11d=12$ from table \ref{T:1}, we have $\alpha=2$ so $A_r$ is asymptotically equals to $r$.
\end{example}
\begin{example}[$c=1$ and $d=-1$]
In this case, $\lambda_i$ is 1 if $i$ even or is -1 if $i$ is odd, we also have using \eqref{E:3.9}, $\mu_i$ is 1 if $i$ is odd and it is 0 if $i$ is even. By \eqref{E:n_r} and table \ref{T:3}, we have $n_{2r}=5\cdot \frac{11^{2r}-1}{12}$ and $A_{2r}=r$.
\[p_{[1^111^{-1}]}\left(11^{2r}m+5\cdot \frac{11^{2r}-1}{12}\right)\equiv 0\pmod{11^{r}}.\]
In view of corollary \ref{C:1.2}, we have $\alpha=1$ from table \ref{T:1} since $c+11d=-10$. Notice here we used the fact that $\alpha$ is a periodic function of $c+11d$ with period 120. Now we have $A_{2r}$ asymptotically equals to $r$ with the error ‘$\epsilon’=0$.
\end{example}
\begin{example}[$c=2$ and $d=7$]
Then we have $\lambda_i$ is 2 if $i$ is even or 7 if $i$ is odd, we also have $n_{2r}=\frac{7\cdot11^{2r}-79}{24}$ and $A_{2r}=2r-1$. In this case and the most cases $\mu_r$ are not immediately periodic. Here $\mu_0=0$ and $\mu_1=1$ afterwords, $\mu_i$ is $4$ if $i$ is odd or 2 if $i$ is even.
\[p_{[1^211^{7}]}\left(11^{2r}m-\frac{7\cdot11^{2r}-79}{24}\right)\equiv 0\pmod{11^{2r-1}}.\]
In this case $c+11d=79$, now from the table \ref{T:1}, $\alpha=2$ so $A_{2r}$ asymptotically $2r$ with the error $\epsilon=-1$.
\end{example}
\section*{Acknowledgements}
The author thanks thesis advisor Karl Mahlburg for suggesting this problem and for his guidance during this project. |
2106.10447 | \section{Introduction}
\subsection{Framework}
When dealing with PDEs coming from the Euler--Lagrange equations of some energy functional, existence and multiplicity results of weak solutions are usually achieved via the so-called Variational Method.
In the recent years, this approach has been employed by many authors in order to deal with a large variety of interesting PDEs on graphs, see~\cites{CGY97,CGY00,CCP19,CG98,G18,G20,GHJ18,GJ18-1,GJ18-2,GJ19,Gri01,Gri18,GLY16-K,GLY16-Y,GLY17,HSZ20,LW17-2,LY20,Z17,ZZ18,ZC18,ZL18,ZL19} and the references therein.
Of particular interest for the scopes of the present paper is the work~\cite{GLY16-Y}, where the authors proved existence of weak solutions for a Yamabe-type equation on locally finite weighted graphs via the celebrated Mountain Pass Theorem due to Ambrosetti and Rabinowitz~\cite{AR73}.
The main aim of this note is twofold.
On one hand, by exploiting some ideas developed in~\cites{FMBR16,MBR17} in the context of Carnot groups we prove the existence of weak solutions for a Yamabe-type equation on locally finite weighted graphs.
Our result is similar to the one of~\cite{GLY16-Y} but holds under a different set of assumptions.
On the other hand, we adapt the strategy of~\cite{BMP07} developed in the Euclidean setting to establish a uniqueness result for the weak solutions of Yamabe-type equations on locally finite weighted graphs in the spirit of the celebrated Brezis--Strauss Theorem~\cite{BS73}.
\subsection{Main notation}
Before stating our main results, we need to recall some notation, see \cref{dfe} for the precise definitions.
Given $G=(V,E)$ a locally finite non-oriented graph, the \emph{vertex boundary} $\partial\Omega$
and the \emph{vertex interior} $\Omega^\circ$ of a connected subgraph $\Omega\subset V$ are defined as
\begin{align*}
\partial\Omega=\set*{x\in\Omega : \exists y\notin\Omega\ \text{such that}\ xy\in E},
\qquad
\Omega^\circ=\Omega\setminus\partial\Omega.
\end{align*}
We say that $\Omega$ is \emph{bounded} if it is a bounded subset of~$V$ with respect to the usual \emph{vertex distance} $\d\colon V\times V\to[0,+\infty)$.
Once a symmetric \emph{weight function} $w\colon V\times V\to [0,\infty)$ is given, we can define the \emph{Laplacian} of a function $u\colon V\to\mathbb{R}$ as
\begin{equation}\label{eq:laplacian}
\Delta u(x)=\frac{1}{\mathfrak{m}(x)}\sum_{y\in V}w_{xy}(u(y)-u(x))
\quad
\text{for}\ x\in V,
\end{equation}
where $\mathfrak{m}\colon V\to[0,+\infty)$ is the measure function
\begin{equation}\label{defW}
\mathfrak{m}(x)=\sum_{y\in V}w_{xy}
\quad
\text{for all}\ x\in V.
\end{equation}
The \emph{gradient form} associated to the Laplacian operator is the bilinear symmetric form
\begin{equation*}
\Gamma(u,v)(x)
=
\frac{1}{2\mathfrak{m}(x)}\sum_{y\in V}w_{xy}(u(y)-u(x))(v(y)-v(x)),
\quad
x\in V,
\end{equation*}
defined for any couple of functions $u,v\colon V\to\mathbb{R}$.
As a consequence, the \emph{slope} of the function $u\colon V\to\mathbb{R}$ is given by
\begin{equation*}
|\nabla u|(x)
=
\sqrt{\Gamma(u,u)(x)}
=
\left(\frac{1}{2\mathfrak{m}(x)}\sum_{y\in V}w_{xy}(u(y)-u(x))^2\right)^\frac{1}{2}
\quad
\text{for}\ x\in V.
\end{equation*}
Note that
$|\Gamma(u,v)|\le |\nabla u|\,|\nabla v|$
for any couple of functions $u,v\colon V\to\mathbb{R}$.
In analogy with the Euclidean framework, for any $m\in\mathbb{N}$ we recursively define the $m$-\emph{slope} of the function~$u$ as
\begin{equation*}
|\nabla^m u|=
\begin{cases}
|\nabla(\Delta^\frac{m-1}{2} u)| & \text{if $m$ is odd},\\[3mm]
|\Delta^\frac{m}{2} u| & \text{if $m$ is even},
\end{cases}
\end{equation*}
where $|\Delta^\frac{m}{2} u|$ denotes the usual absolute value of the function $\Delta^\frac{m}{2} u$.
The natural operator associated to the Sobolev spaces $(W^{m,p}_0(\Omega),\|\cdot\|_{W^{m,p}_0(\Omega)})$ (see~\eqref{eq:sobolev_norm} below for the precise definition) is the \emph{$(m,p)$-Laplacian operator}
\begin{equation*}
\mathcal{L}_{m,p}\colon W^{m,p}_0(\Omega)\to L^p(\Omega)
\end{equation*}
defined in the distributional sense for all $u\in W^{m,p}_0(\Omega)$ as
\begin{equation}
\label{eq:def_mp_Lap}
\int_\Omega \mathcal{L}_{m,p}u\,\phi\di\mathfrak{m}=
\begin{cases}
\displaystyle\int_\Omega |\nabla^m u|^{p-2}\,\Gamma(\Delta^{\frac{m-1}{2}}u,\Delta^{\frac{m-1}{2}}\phi)\di\mathfrak{m}
& \text{if $m$ is odd},\\[5mm]
\displaystyle\int_\Omega |\nabla^m u|^{p-2}\,\Delta^{\frac{m}{2}}u\,\Delta^{\frac{m}{2}}\phi\di\mathfrak{m}
& \text{if $m$ is even},
\end{cases}
\end{equation}
whenever $\phi\in W^{m,p}_0(\Omega)$.
The $(m,p)$-Laplacian $\mathcal{L}_{m,p} u$ can be explicitly computed at any point of $\Omega$.
In particular, $\mathcal{L}_{1,p}$ is the \emph{$p$-Laplacian operator}, given by
\begin{equation}
\label{eq:p-laplacian}
\Delta_p u(x)
=
\frac{1}{\mathfrak{m}(x)}\sum_{y\in\Omega}\left(|\nabla u|^{p-2}(y)+|\nabla u|^{p-2}(x)\right)w_{xy}(u(y)-u(x)),
\quad
x\in\Omega,
\end{equation}
for all $u\in W^{1,p}_0(\Omega)$.
When $p=2$, we recover the usual Laplacian operator defined in~\eqref{eq:laplacian}.
\subsection{Main results}
We are now ready to state our main results.
Our first main theorem is the following existence result for a Yamabe-type equation for the $(m,p)$-Laplacian operator on locally finite weighted graphs.
\begin{theorem}\label{th:main}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $m\in\mathbb{N}$, $p\in(1,+\infty)$ and $q\in[p-1,+\infty)$.
Let $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function such that
\begin{equation}\label{eq:growth}
|f(x,t)|\le a(x)+b(x)\,|t|^q
\quad
\text{for every }
(x,t)\in\Omega\times\mathbb{R}
\end{equation}
for some non-negative $a,b\in L^1(\Omega)$ with $\|a\|_{L^1(\Omega)},\|b\|_{L^1(\Omega)}>0$.
There exists
\begin{equation}\label{eq:Lambda}
\Lambda
=
\Lambda(m,p,q,\|a\|_{L^1(\Omega)},\|b\|_{L^1(\Omega)})
>0
\end{equation}
such that the Yamabe-type problem
\begin{equation}\label{eq:yamabe}
\begin{cases}
\mathcal{L}_{m,p} u=\lambda f(x,u) & \text{in } \Omega^\circ\\[2mm]
|\nabla^j u|=0 & \text{on } \partial\Omega, \quad 0\le j\le m-1,
\end{cases}
\end{equation}
admits at least one non-trivial solution $u_\lambda\in W^{m,p}_0(\Omega)$ for every $0<\lambda<\Lambda$.
\end{theorem}
We observe that the growth condition of the function $f$ assumed in~\eqref{eq:growth} of \cref{th:main} is different from the one assumed in~\cite{GLY16-Y}*{Theorem~3}.
In particular, we do not assume that $f(x,0)=0$ for all $x\in\Omega$.
We also underline that the existence threshold~\eqref{eq:Lambda} depends uniquely on the growth of the function~$f$ and not on the first eigenvalue of the $(m,p)$-Laplacian, as instead it happens in~\cite{GLY16-Y}*{Theorem~3}.
Our second main result is the following uniqueness theorem for a Yamabe-type equation for the $p$-Laplacian operator on locally finite weighted graphs in the spirit of the famous Brezis--Strauss Theorem, see~\cites{BS73,BMP07}.
\begin{theorem}\label{res:main_uniqueness}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $p\in[1,+\infty)$ and let $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a function such that $g(x,0)=0$ and $t\mapsto g(x,t)$ is non-decreasing for all $x\in\Omega$.
If $f_1,f_2\in L^1(\Omega)$, $h\in L^1(\partial\Omega)$ and $u_1,u_2\in W^{1,p}(\Omega)$ solve the problems
\begin{equation*}
\begin{cases}
-\Delta_p u_i + g(x,u_i) = f_i & \text{in}\ \Omega^\circ\\[2mm]
u_i = h & \text{on}\ \partial\Omega
\end{cases}
\qquad
\text{for}\ i=1,2,
\end{equation*}
then
\begin{equation}
\label{eq:oscillation}
\int_\Omega |g(x,u_1)-g(x,u_2)|\di\mathfrak{m}
\le
\int_\Omega |f_1-f_2|\di\mathfrak{m}.
\end{equation}
As a consequence, for every $f\in L^1(\Omega)$ and $h\in L^1(\partial\Omega)$ the problem
\begin{equation}\label{eq:uniqueness_main}
\begin{cases}
-\Delta_p u + g(x,u) = f & \text{in}\ \Omega^\circ\\[2mm]
u = h & \text{on}\ \partial\Omega
\end{cases}
\end{equation}
admits at most one solution $u\in W^{1,p}(\Omega)$.
\end{theorem}
By combining \cref{th:main} and \cref{res:main_uniqueness}, we get the following well-posedness result for a Yamabe-type problem for the $p$-Laplacian on locally finite weighted graphs.
\begin{proposition}\label{propcomb}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $p\in(1,+\infty)$, $q\in[p-1,+\infty)$ and $a,b\in L^1(\Omega)$ with $\inf_\Omega b\ge0$.
There exists
\begin{equation*}
\Lambda
=
\Lambda(m,p,q,\|a\|_{L^1(\Omega)},\|b\|_{L^1(\Omega)})
>0
\end{equation*}
such that the Yamabe-type problem
\begin{equation}\label{eq:yamabe_cor}
\begin{cases}
-\Delta_p u + b|u|^{q-1}u = a
& \text{in}\ \Omega^\circ\\[2mm]
u=0
& \text{on}\ \partial\Omega
\end{cases}
\end{equation}
has a unique solution $u\in W^{1,p}_0(\Omega)$.
\end{proposition}
\subsection{Organization of the paper}
The structure of the paper is the following.
In \cref{dfe} we recall the preliminary definitions and notions needed in the paper.
Sections~\ref{proof1} and~\ref{proof2} are devoted to the proofs of Theorems~\ref{th:main} and~\ref{res:main_uniqueness} respectively.
Finally, in \cref{proof3} we provide some applications of our main results, along with the proof of \cref{propcomb}.
\section{Preliminaries}\label{dfe}
In this section, we introduce the main notation and some preliminary results we will need in the sequel of the paper.
\subsection{Non-oriented graphs}
Let $V$ be a non-empty set and let $E\subset V\times V$.
We write
\begin{equation*}
x\sim y
\iff
xy=(x,y)\in E.
\end{equation*}
We will always assume that
\begin{equation*}
xy\in E \iff yx\in E.
\end{equation*}
We say that the couple $G=(V,E)$ is a \emph{non-oriented graph} with \emph{vertices} $V$ and \emph{edges} $E$.
The non-oriented graph $G$ is \emph{locally finite} if
\begin{equation*}
\#\set*{y\in V : xy\in E}<+\infty
\quad
\text{for all}\
x\in V,
\end{equation*}
that is, each vertex in $V$ belongs to a finite number of edges in $E$.
Given $n\in\mathbb{N}$, a \emph{path} on $G$ is any finite sequence of vertices $\set{x_k}_{k=1,\dots,n}\subset V$ such that
\begin{equation*}
x_kx_{k+1}\in E
\quad
\text{for all}\ k=1,\dots,n-1.
\end{equation*}
The \emph{length} of a path on $G$ is the number of edges in the path.
We say that $G$ is \emph{connected} if, for any two vertices $x,y\in V$, there is a path connecting $x$ and $y$.
If $G$ is connected, then the function $\mathsf{d}\colon V\times V\to[0,+\infty)$ given by
\begin{equation*}
\mathsf{d}(x,y)=\min\set*{n\in\mathbb{N}_0 :\text{$x$ and $y$ can be connected by a path of length $n$}},
\end{equation*}
for $x,y\in V$, is a distance on $V$.
As a consequence, any connected locally finite non-oriented graph has at most countable many vertices.
Let $G=(V,E)$ be a locally finite non-oriented graph.
A \emph{weight} on $G$ is a function $w\colon V\times V\to[0,+\infty)$, $w(x,y)=w_{xy}$ for $x,y\in V$, such that
\begin{equation*}
w_{xy}=w_{yx}
\quad
\text{and}
\quad
w_{xy}>0
\iff
xy\in E
\end{equation*}
for all $x,y\in V$.
We conclude this section by pointing out that the function $\mathfrak{m}\colon V\to[0,+\infty)$ defined in \eqref{defW}
can be interpreted as a measure on the graph by simply setting
\begin{equation*}
\int_V u\di\mathfrak{m}
=
\int_V u(x)\di\mathfrak{m}(x)
=
\sum_{y\in V} u(x)\,\mathfrak{m}(x)\in[0,+\infty]
\end{equation*}
for any function $u\colon V\to[0,+\infty)$.
\subsection{Sobolev spaces on bounded domains}
Let $G=(V,E)$ be a weighted locally finite graph and let $\Omega\subset V$ be a bounded domain.
Note that the integral
\begin{equation*}
\int_\Omega u\di\mathfrak{m}
=
\int_\Omega u(x)\di\mathfrak{m}(x)
=
\sum_{x\in\Omega}u(x)\,\mathfrak{m}(x)
\end{equation*}
of a function $u\colon\Omega\to\mathbb{R}$ is well defined, since $\Omega$ is a finite set.
Let $p\in[1,+\infty]$ and $m\in\mathbb{N}_0$.
The \emph{Sobolev space $W^{m,p}(\Omega)$} is the set of all functions $u\colon\Omega\to\mathbb{R}$ such that
\begin{equation}\label{eq:sobolev_norm}
\|u\|_{W^{m,p}(\Omega)}
=
\sum_{k=0}^m\|\nabla^k u\|_{L^p(\Omega)}<+\infty.
\end{equation}
When $m=0$, this space is simply the \emph{Lebesgue space} $L^p(\Omega)$.
Since $\Omega$ is a finite set, the Banach space $(W^{m,p}(\Omega),\|\cdot\|_{W^{m,p}(\Omega)})$ is finite dimensional and, actually, coincides with the set of all real-valued functions on~$\Omega$.
For $m\in\mathbb{N}$, we define
\begin{equation}
\label{eq:def_C_m0}
C^m_0(\Omega):=\set*{u\colon\Omega\to\mathbb{R} : |\nabla^k u|=0\ \text{on}\ \partial\Omega\ \text{for all}\ 0\le k\le m-1}
\end{equation}
and we let $W^{m,p}_0(\Omega)$ be the completion of $C^m_0(\Omega)$ with respect to the Sobolev norm~\eqref{eq:sobolev_norm}.
The following result is proved in~\cite{GLY16-Y}*{Theorem~7}.
\begin{theorem}[Sobolev embedding]\label{th:sobolev}
Let $G=(V,E)$ be a locally finite graph and let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $m\in\mathbb{N}$ and $p\in[1,+\infty)$.
The space $W^{m,p}_0(\Omega)$ is continuously embedded in $L^q(\Omega)$ for all $q\in[1,+\infty]$, i.e.\ there exists a constant $C_{m,p}>0$, depending only on $m$, $p$ and $\Omega$, such that
\begin{equation}\label{eq:sobolev}
\|u\|_{L^q(\Omega)}
\le
C_{m,p}
\|\nabla^m u\|_{L^p(\Omega)}
\end{equation}
for all $q\in[1,+\infty]$ and $u\in W^{m,p}_0(\Omega)$.
\end{theorem}
By \cref{th:sobolev}, the space $(W^{m,p}_0(\Omega),\|\cdot\|_{W^{m,p}_0(\Omega)})$ is a finite dimensional Banach space, where
\begin{equation}\label{eq:sobolev_0_norm}
\|u\|_{W^{m,p}_0(\Omega)}
=
\|\nabla^m u\|_{L^p(\Omega)}
\end{equation}
is a norm on $W^{m,p}_0(\Omega)$ equivalent to the norm~\eqref{eq:sobolev_norm}.
Since $\Omega$ is a finite set, the Banach space $(W^{m,p}_0(\Omega),\|\cdot\|_{W^{m,p}_0(\Omega)})$ is finite dimensional and, actually, coincides with the set~$C^m_0(\Omega)$ defined in~\eqref{eq:def_C_m0}.
\section{Proof of \texorpdfstring{\cref{th:main}}{Theorem 1.1}}\label{proof1}
In this section, we prove our first main result following the strategy outlined in~\cite{FMBR16}. Given $\lambda>0$, we define
\begin{equation}\label{eq:def_Phi_Psi}
\Phi(u)
=
\|u\|_{W^{m,p}_0(\Omega)},
\qquad
\Psi_\lambda(u)
=
\lambda\int_\Omega F(x,u)\di\mathfrak{m},
\end{equation}
for all $u\in W^{m,p}_0(\Omega)$, where
\begin{equation}\label{eq:def_F}
F(x,t)
=
\int_0^t f(x,\tau)\di\tau
\quad
\text{for all}\ t\in\mathbb{R}.
\end{equation}
Note that, thanks to the assumption in~\eqref{eq:growth}, the functional $\Psi_\lambda$ is well defined and (strongly) continuous on $W^{m,p}_0(\Omega)$.
Indeed, we can estimate
\begin{align*}
|F(x,t)|
\le
\int_0^{|t|}|f(x,\tau)|\di\tau
\le
\int_0^{|t|}a(x)+b(x)\,|\tau|^q\di\tau
=
a(x)\,|t|+b(x)\,\frac{|t|^{1+q}}{1+q}
\end{align*}
for all $(x,t)\in\Omega\times\mathbb{R}$, so that
\begin{equation*}
|\Psi_\lambda(u)|
\le
\lambda\left(
\|a\|_{L^1(\Omega)}\|u\|_{L^\infty(\Omega)}
+
\|b\|_{L^1(\Omega)}\,\frac{\|u\|_{L^\infty(\Omega)}^{1+q}}{1+q}
\right)
\end{equation*}
which is finite for all $u\in W^{m,p}_0(\Omega)$ by \cref{th:sobolev}.
In addition, if $(u_n)_{n\in\mathbb{N}}\subset W^{m,p}_0(\Omega)$ is converging to some $u\in W^{m,p}_0(\Omega)$, then $u_n\to u$ in $L^\infty(\Omega)$ as $n\to+\infty$ and thus
\begin{align*}
\lim_{n\to+\infty}
\Psi_\lambda(u_n)
=
\lim_{n\to+\infty}
\sum_{x\in\Omega} F(x,u_n(x))\,\mathfrak{m}(x)
=
\sum_{x\in\Omega} F(x,u(x))\,\mathfrak{m}(x)
=
\Psi_\lambda(u)
\end{align*}
by the continuity of the function $t\mapsto F(x,t)$ for $x\in\Omega$ fixed.
The following two results are proved in~\cite{FMBR16}*{Lemma~3.2 and Lemma~3.3} respectively for the case $p=2$.
Here we reproduce the proofs in our setting in the more general case $p\in(1,+\infty)$ for the reader's ease.
\begin{lemma}\label{lemma:tic}
Let $p\in(1,+\infty)$ and $\lambda>0$.
If
\begin{equation}
\label{eq:tic_ipotesi}
\limsup_{\varepsilon\to0+}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\varepsilon}
<
\rho^{p-1}
\end{equation}
for some $\rho>0$, then
\begin{equation}\label{eq:inf_rho_sigma}
\inf_{\sigma<\rho}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\sigma])}\Psi_\lambda(u)}{\rho^p-\sigma^p}
<
\frac{1}{p}.
\end{equation}
\end{lemma}
\begin{proof}
Let $\varepsilon\in(0,\rho)$ and note that
\begin{align*}
\lim_{\varepsilon\to0^+}
\frac{\varepsilon}{\rho^p-(\rho-\varepsilon)^p}
=
\frac{1}{p\rho^{p-1}}.
\end{align*}
Therefore, in virtue of~\eqref{eq:tic_ipotesi}, we get that
\begin{align*}
\limsup_{\varepsilon\to0^+}\,
&
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\rho^p-(\rho-\varepsilon)^p}
\\
&=
\limsup_{\varepsilon\to0^+}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\varepsilon}
\cdot
\frac{\varepsilon}{\rho^p-(\rho-\varepsilon)^p}
\\
&=
\frac{1}{p\rho^{p-1}}\,
\limsup_{\varepsilon\to0^+}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\varepsilon}
<
\frac1p.
\end{align*}
Thus we can find $\bar\varepsilon\in(0,\rho)$ such that
\begin{align*}
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\bar\varepsilon])}\Psi_\lambda(u)}{\rho^p-(\rho-\bar\varepsilon)^p}
<
\frac1p
\end{align*}
and so $\bar\sigma=\rho-\bar\varepsilon<\rho$ gives
\begin{align*}
\inf_{\sigma<\rho}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\sigma])}\Psi_\lambda(u)}{\rho^p-\sigma^p}
<
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\bar\sigma])}\Psi_\lambda(u)}{\rho^p-\bar\sigma^p}
<
\frac1p
\end{align*}
proving~\eqref{eq:inf_rho_sigma}.
The proof is complete.
\end{proof}
\begin{lemma}\label{lemma:tac}
Let $p\in(1,+\infty)$ and $\lambda>0$. If~\eqref{eq:inf_rho_sigma} holds for some $\rho>0$, then
\begin{equation}
\label{eq:tac_inf}
\inf_{u\in\Phi^{-1}([0,\rho))}
\frac{\sup\limits_{v\in\Phi^{-1}([0,\rho])}\Psi_\lambda(v)-\Psi_\lambda(u)}{\rho^p-\|u\|_{W^{m,p}_0(\Omega)}^p}
<
\frac{1}{p}.
\end{equation}
\end{lemma}
\begin{proof}
In virtue of~\eqref{eq:inf_rho_sigma}, we can find $\bar\sigma\in(0,\rho)$ such that
\begin{align*}
\sup\limits_{u\in\Phi^{-1}([0,\bar\sigma])}\Psi_\lambda(u)
>
\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)
-
\frac{1}{p}(\rho^p-\bar\sigma^p).
\end{align*}
Since the functional $\Psi_\lambda$ is continuous on $W^{m,p}_0(\Omega)$, we can find $\bar u\in W^{m,p}_0(\Omega)$ with $\|\bar u\|_{W^{m.p}_0(\Omega)}=\bar\sigma$ such that
\begin{align*}
\sup\limits_{u\in\Phi^{-1}([0,\bar\sigma])}\Psi_\lambda(u)
=
\sup\limits_{\|u\|_{W^{m,p}_0(\Omega)}=\,\bar\sigma}\Psi_\lambda(u)
=
\Psi_\lambda(\bar u)
\end{align*}
and so
\begin{align*}
\Psi_\lambda(\bar u)
>
\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)
-
\frac{1}{p}(\rho^p-\bar\sigma^p).
\end{align*}
We thus conclude that
\begin{align*}
\inf_{u\in\Phi^{-1}([0,\rho))}
\frac{\sup\limits_{v\in\Phi^{-1}([0,\rho])}\Psi_\lambda(v)-\Psi_\lambda(u)}{\rho^p-\|u\|_{W^{m,p}_0(\Omega)}^p}
<
\frac{\sup\limits_{v\in\Phi^{-1}([0,\rho])}\Psi_\lambda(v)-\Psi_\lambda(\bar u)}{\rho^p-\|\bar u\|_{W^{m,p}_0(\Omega)}^p}
<
\frac1p
\end{align*}
proving~\eqref{eq:tac_inf}.
The proof is complete.
\end{proof}
We are now ready to prove our first main result, in analogy with~\cite{FMBR16}*{Theorem~3.1}.
\begin{proof}[Proof of \cref{th:main}]
Let $\lambda>0$ and consider the energy functional $\mathcal{E}_\lambda\colon W^{m,p}_0(\Omega)\to\mathbb{R}$ defined as
\begin{align*}
\mathcal{E}_\lambda(u)
=
\frac{\Phi(u)^p}{p}-\Psi_\lambda(u)
\quad
\text{for all}\ u\in W^{m,p}_0(\Omega),
\end{align*}
where $\Phi$ and $\Psi_\lambda$ are as in~\eqref{eq:def_Phi_Psi}.
By the growth condition~\eqref{eq:growth} and \cref{th:sobolev}, we have that $\mathcal{E}_\lambda\in C^1(W^{m,p}_0(\Omega);\mathbb{R})$, with derivative at $u\in W^{m,p}_0(\Omega)$ given by
\begin{align*}
\mathcal{E}'_\lambda(u)[\phi]
=
\begin{cases}
\displaystyle\int_\Omega |\nabla^m u|^{p-2}\,\Gamma(\Delta^{\frac{m-1}{2}}u,\Delta^{\frac{m-1}{2}}\phi)\di\mathfrak{m}
-
\lambda\int_\Omega f(x,u)\,\phi\di\mathfrak{m}
&
\text{if $m$ is odd},\\[5mm]
\displaystyle\int_\Omega |\nabla^m u|^{p-2}\,\Delta^{\frac{m}{2}}u\,\Delta^{\frac{m}{2}}\phi\di\mathfrak{m}
-
\lambda\int_\Omega f(x,u)\,\phi\di\mathfrak{m}
&
\text{if $m$ is even},
\end{cases}
\end{align*}
for any $\phi\in W^{m,p}_0(\Omega)$.
In particular, the solutions of the problem~\eqref{eq:yamabe} are exactly the critical points of the functional $\mathcal{E}_\lambda$.
Now let $\rho>0$ to be fixed later.
Since $\mathcal{E}_\lambda$ is a continuous functional on~$W^{m,p}_0(\Omega)$, there exists $u_{\lambda,\rho}\in\Phi^{-1}([0,\rho])$ such that
\begin{equation}\label{eq:minimality_u}
\mathcal{E}_\lambda(u_{\lambda,\rho})
=
\inf_{u\in\Phi^{-1}([0,\rho])}\mathcal{E}_\lambda(u).
\end{equation}
To conclude the proof, we just need to show that $\|u_{\lambda,\rho}\|_{W^{m,p}_0(\Omega)}<\rho$.
To this aim, for $\varepsilon\in(0,\rho)$ we consider
\begin{equation*}
\Lambda(\rho,\varepsilon)
=
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\varepsilon}.
\end{equation*}
Recalling the definition of $\Psi_\lambda$ in~\eqref{eq:def_Phi_Psi}, we have
\begin{align*}
\Lambda(\rho,\varepsilon)
&=
\frac{1}{\varepsilon}\,
\bigg(\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)\bigg)\\
&\le
\frac{\lambda}{\varepsilon}\,
\sup_{u\in\Phi^{-1}([0,1])}\,
\int_\Omega\,\abs*{\int_{(\rho-\varepsilon)u(x)}^{\rho u(x)}|f(x,t)|\di t\,}\di\mathfrak{m}(x).
\end{align*}
Thanks to the growth condition~\eqref{eq:growth}, we can estimate
\begin{align*}
\frac{\lambda}{\varepsilon}\int_\Omega\bigg|\int_{(\rho-\varepsilon)u(x)}^{\rho u(x)}&|f(x,t)|\di t\,\bigg|\di\mathfrak{m}(x)\\
&\le
\frac{\lambda}{\varepsilon}\int_\Omega \varepsilon\,a(x)|u(x)|
+
b(x)\left(\frac{\rho^{q+1}-(\rho-\varepsilon)^{q+1}}{q+1}\right)|u(x)|^{q+1}\di\mathfrak{m}(x)\\
&\le
\lambda\|u\|_{L^\infty(\Omega)}\|a\|_{L^1(\Omega)}+\frac{\lambda\|u\|_{L^\infty(\Omega)}^{q+1}\|b\|_{L^1(\Omega)}}{q+1}\left(\frac{\rho^{q+1}-(\rho-\varepsilon)^{q+1}}{\varepsilon}\right)
\end{align*}
for all $u\in W^{m,p}_0(\Omega)$.
Thus, by the embedding inequality~\eqref{eq:sobolev}, we get
\begin{equation*}
\Lambda(\rho,\varepsilon)
\le
\lambda\,C_{m,p}\|a\|_{L^1(\Omega)}
+
\frac{\lambda\, C_{m,p}^{q+1}\|b\|_{L^1(\Omega)}}{q+1}\left(\frac{\rho^{q+1}-(\rho-\varepsilon)^{q+1}}{\varepsilon}\right)
\end{equation*}
and so
\begin{equation*}
\limsup_{\varepsilon\to0+}
\Lambda(\rho,\varepsilon)
\le
\lambda
\left(
C_{m,p}\|a\|_{L^1(\Omega)}
+
C_{m,p}^{q+1}\|b\|_{L^1(\Omega)}\,\rho^q
\right).
\end{equation*}
We now define
\begin{equation}\label{eq:precise_Lambda}
\lambda_\rho
=
\frac{\rho^{p-1}}{C_{m,p}\|a\|_{L^1(\Omega)}+C_{m,p}^{q+1}\|b\|_{L^1(\Omega)}\,\rho^q}
\in
(0,+\infty)
\end{equation}
and, consequently,
\begin{equation*}
\Lambda
=
\sup_{\rho>0}\lambda_\rho\in(0,+\infty)
\end{equation*}
(note that $\Lambda<+\infty$ is ensured by the fact that $q\ge p-1$).
Now fix $\lambda<\Lambda$ and choose the parameter $\rho>0$ in such a way that $\lambda<\lambda_\rho<\Lambda$.
This choice implies that
\begin{align*}
\limsup_{\varepsilon\to0+}
\Lambda(\rho,\varepsilon)
&\le
\lambda
\left(
C_{m,p}\|a\|_{L^1(\Omega)}
+
C_{m,p}^{q+1}\|b\|_{L^1(\Omega)}\,\rho^q
\right)
\\
&<
\lambda_\rho
\left(
C_{m,p}\|a\|_{L^1(\Omega)}
+
C_{m,p}^{q+1}\|b\|_{L^1(\Omega)}\,\rho^q
\right)
<\rho^{p-1},
\end{align*}
so that
\begin{align*}
\limsup_{\varepsilon\to0+}\,
\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\rho-\varepsilon])}\Psi_\lambda(u)}{\varepsilon}
<
\rho^{p-1}.
\end{align*}
We can now apply \cref{lemma:tic} to get that
\begin{equation*}
\inf_{\sigma<\rho}\frac{\sup\limits_{u\in\Phi^{-1}([0,\rho])}\Psi_\lambda(u)-\sup\limits_{u\in\Phi^{-1}([0,\sigma])}\Psi_\lambda(u)}{\rho^p-\sigma^p}<\frac{1}{p}
\end{equation*}
and so, by \cref{lemma:tac}, we infer that
\begin{equation*}
\inf_{u\in\Phi^{-1}([0,\rho))}\frac{\sup\limits_{v\in\Phi^{-1}([0,\rho])}\Psi_\lambda(v)-\Psi_\lambda(u)}{\rho^p-\|u\|_{W^{m,p}_0(\Omega)}^p}<\frac{1}{p}.
\end{equation*}
The above inequality implies that there exists $w_{\lambda,\rho}\in\Phi^{-1}([0,\rho))$ such that
\begin{equation*}
\sup_{v\in\Phi^{-1}([0,\rho])}\Psi_\lambda(v)
<
\Psi_\lambda(w_{\lambda,\rho})
+
\frac{\rho^p-\|w_{\lambda,\rho}\|_{W^{m,p}_0(\Omega)}^p}{p}.
\end{equation*}
Now, if by contradiction we assume that $\|u_{\lambda,\rho}\|_{W^{m,p}_0(\Omega)}=\rho$, then the previous inequality implies that
\begin{equation*}
\Psi_\lambda(u_{\lambda,\rho})
<
\Psi_\lambda(w_{\lambda,\rho})
+
\frac{\rho^p-\|w_{\lambda,\rho}\|_{W^{m,p}_0(\Omega)}^p}{p}
\end{equation*}
which is equivalent to $\mathcal{E}_\lambda(u_{\lambda,\rho})>\mathcal{E}_\lambda(w_{\lambda,\rho})$, contradicting~\eqref{eq:minimality_u}.
The proof is complete.
\end{proof}
\begin{remark}[The precise value of $\Lambda$ in \cref{th:main}]\label{rem:Lambda}
Note that the above proof allows to give a precise value to the existence threshold $\Lambda>0$ in \cref{th:main}.
Indeed, one just need to find the maximal value of the function defined in~\eqref{eq:precise_Lambda}, which is explicitly computable in term of $p$, $q$, $\|a\|_{L^1(\Omega)}$, $\|b\|_{L^1(\Omega)}$ and $C_{m,p}$.
In particular, in the limiting case $q=p-1$, one has
\begin{align*}
\Lambda
=
\lim_{\rho\to+\infty}
\frac{\rho^{p-1}}{C_{m,p}\|a\|_{L^1(\Omega)}+C_{m,p}^{p}\|b\|_{L^1(\Omega)}\,\rho^{p-1}}
=
\frac{1}{C_{m,p}^{p}\|b\|_{L^1(\Omega)}},
\end{align*}
which does not depend on $\|a\|_{L^1(\Omega)}$.
\end{remark}
\section{Proof of \texorpdfstring{\cref{res:main_uniqueness}}{Theorem 1.2}}\label{proof2}
In this section we prove \cref{res:main_uniqueness}.
The overall strategy is to adapt the line developed in~\cite{BMP07}*{Appendix~B} for the Euclidean setting to the present framework.
Note that~\cite{BMP07} is focused on the case $p=2$ only.
Nonetheless, exploiting the explicit expression~\eqref{eq:p-laplacian} of the $p$-Laplacian, we are able to extend the approach of~\cite{BMP07} also to the case $p\ne2$.
We begin with the following result, analogous to~\cite{BMP07}*{Lemma~B.1}.
\begin{lemma}\label{res:H_function}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $p\in[1,+\infty)$ and $f\in L^1(\Omega)$.
If $u\in W^{1,p}_0(\Omega)$ is a solution of the problem
\begin{equation}
\label{eq:p_Lap_H_function}
\begin{cases}
-\Delta_p u = f & \text{in}\ \Omega^\circ\\[1mm]
u = 0 & \text{on}\ \partial\Omega,
\end{cases}
\end{equation}
then
\begin{equation*}
\int_\Omega f\, H(u)\di\mathfrak{m} \ge0
\end{equation*}
for every non-decreasing locally Lipschitz function $H\colon\mathbb{R}\to\mathbb{R}$ such that $H(0)=0$.
\end{lemma}
\begin{proof}
We start by observing that $H(u)\in W^{1,p}_0(\Omega)$.
Indeed, $H(v)\in C^0_0(\Omega)$ for all $v\in C^0_0(\Omega)$ with $|\nabla H(v)|\le L|\nabla v|$ on~$\Omega$, where $L=\mathrm{Lip}(H,[-c,c])$, $c=\|v\|_{L^\infty(\Omega)}$.
Using $H(u)$ as a test function in~\eqref{eq:p_Lap_H_function}, we get
\begin{align*}
\int_\Omega f\,H(u)\di\mathfrak{m}
=
-\int_\Omega\Delta_p u\, H(u)\di\mathfrak{m}
=
\int_\Omega|\nabla u|^{p-2}\,\Gamma(u,H(u))\di\mathfrak{m}\ge0,
\end{align*}
because
\begin{align*}
\Gamma(u,H(u))(x)
=
\frac{1}{2\mathfrak{m}(x)}\sum_{y\in\Omega}w_{xy}(u(y)-u(x))(H(u(y))-H(u(x)))\ge0
\end{align*}
since $H$ is non-decreasing.
The proof is complete.
\end{proof}
As a consequence, and in analogy with~\cite{BMP07}*{Proposition~B.2}, from \cref{res:H_function} we deduce the following result.
\begin{corollary}\label{res:sgn}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $p\in[1,+\infty)$, $M>0$ and $f\in L^1(\Omega)$.
If $u\in W^{1,p}_0(\Omega)$ is a solution of the problem
\begin{equation*}
\begin{cases}
-\Delta_p u = f & \text{in}\ \Omega^\circ\\[1mm]
u = 0 & \text{on}\ \partial\Omega,
\end{cases}
\end{equation*}
then
\begin{equation*}
\int_{\Omega\cap\set*{u\ge M}} f\di\mathfrak{m} \ge0,
\quad
\int_{\Omega\cap\set*{u\le -M}} f\di\mathfrak{m} \ge0.
\end{equation*}
In particular,
\begin{equation*}
\int_{\Omega\cap\set*{|u|\ge M}} f\,\sgn(u)\di\mathfrak{m} \ge0,
\end{equation*}
where $\sgn\colon\mathbb{R}\to\mathbb{R}$ is the \emph{sign function} defined by $\sgn(t)=\frac{t}{|t|}$ for $t\ne0$ and $\sgn(0)=0$.
\end{corollary}
\begin{proof}
For every $n\in\mathbb{N}$ such that $n>\frac1M$, we let $H_n\colon\mathbb{R}\to\mathbb{R}$ be the function
\begin{equation*}
H_n(t)
=
\begin{cases}
0 & \text{for}\ t\le M-\frac1n\\[2mm]
nt-nM+1 & \text{for}\ M-\frac1n<t<M\\[2mm]
1 & \text{for}\ t\ge M.
\end{cases}
\end{equation*}
Since $H_n$ is Lipschitz, non-decreasing and such that $H_n(0)=0$, by \cref{res:H_function} we get that
\begin{equation*}
\int_\Omega f\,H_n(u)\di\mathfrak{m}\ge0.
\end{equation*}
Passing to the limit as $n\to+\infty$, we find that
\begin{equation*}
\int_{\Omega\cap\set*{u\ge M}} f\di\mathfrak{m}\ge0,
\end{equation*}
as desired.
The conclusion thus follows by linearity.
\end{proof}
We are now ready to prove our second main result, in analogy with~\cite{BMP07}*{Corollary~B.1}.
\begin{proof}[Proof of \cref{res:main_uniqueness}]
The function $v=u_1-u_2\in W^{1,p}_0(\Omega)$ solves the problem
\begin{equation}\label{eq:v_problem}
\begin{cases}
-\Delta_p v = F & \text{in}\ \Omega^\circ\\[2mm]
v = 0 & \text{on}\ \partial\Omega
\end{cases}
\end{equation}
with $F=f_1-f_2-g(x,u_1)+g(x,u_2)\in L^1(\Omega)$.
By \cref{res:sgn}, we have that
\begin{equation*}
\int_\Omega F\sgn(v)\di\mathfrak{m}\ge0,
\end{equation*}
which is equivalent to
\begin{align*}
\int_\Omega(g(x,u_1)-g(x,u_2))\,\sgn(u_1-u_2)\di\mathfrak{m}
\le
\int_\Omega (f_1-f_2)\,\sgn(u_1-u_2)\di\mathfrak{m}
\end{align*}
and~\eqref{eq:oscillation} immediately follows.
As a consequence, if $f_1=f_2$ then also $g(x,u_1)=g(x,u_2)$ and thus $F=0$ in~\eqref{eq:v_problem}.
Therefore $\Delta_p v=0$ in $\Omega$ and thus
\begin{align*}
\sup_\Omega |v|
\le
C_{1,p}
\int_\Omega|\nabla v|^p\di\mathfrak{m}
=
-C_{1,p}
\int_\Omega v\,\Delta_p v\di\mathfrak{m}
=0
\end{align*}
by \cref{th:sobolev} and~\eqref{eq:def_mp_Lap}, so that $u_1=u_2$.
The proof is complete.
\end{proof}
\section{Applications}\label{proof3}
In this last section we briefly discuss some applications of our main results.
We begin by stating the following result, which shows that the Dirichlet problem in $W^{1,2}_0(\Omega)$ for the Laplcian operator with sufficiently well-behaved non-linearity admits a unique solution.
\begin{corollary}\label{res:bambi}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a function such that $t\mapsto g(x,t)$ is $C^1$ and non-decreasing with $g(x,0)=\partial_t g(x,0)=0$ for all $x\in\Omega$.
Let us set $\bar f(x)=g(x,0)$ for all $x\in\Omega$.
There exists $\delta>0$ with the following property: if $f\in L^2(\Omega)$ with $\|f-\bar f\|_{L^2(\Omega)}<\delta$, then the problem
\begin{equation*}
\begin{cases}
-\Delta u+g(x,u)=f
& \text{in}\ \Omega^\circ\\[2mm]
u=0
& \text{on}\ \partial\Omega
\end{cases}
\end{equation*}
admits a unique solution $u\in W^{1,2}_0(\Omega)$.
\end{corollary}
The uniqueness part in \cref{res:bambi} is clearly immediately achieved by \cref{res:main_uniqueness}, while the existence part follows from the following result, which is inspired by the work~\cite{BC98}.
\begin{lemma}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a function such that $t\mapsto g(x,t)$ is of class $C^1$ with $\partial_t g(x,0)=0$ for all $x\in\Omega$.
Let us set $\bar f(x)=g(x,0)$ for all $x\in\Omega$.
There exist $\delta,\varepsilon>0$ with the following property: if $f\in L^2(\Omega)$ with $\|f-\bar f\|_{L^2(\Omega)}<\delta$, then the problem
\begin{equation*}
\begin{cases}
-\Delta u+g(x,u)=f
& \text{in}\ \Omega^\circ\\[2mm]
u=0
& \text{on}\ \partial\Omega
\end{cases}
\end{equation*}
admits a unique solution $u\in W^{1,2}_0(\Omega)$ with $\|u\|_{W^{1,2}_0(\Omega)}<\varepsilon$.
\end{lemma}
\begin{proof}
Let us consider the function $\mathcal F\colon W^{1,2}_0(\Omega)\to L^2(\Omega)$ defined by
\begin{equation*}
\mathcal F(u)
=
-\Delta u+g(x,u)
\end{equation*}
for all $u\in W^{1,2}_0(\Omega)$.
Note that the map $\mathcal F$ is well defined, since $W^{1,2}_0(\Omega)\subset L^\infty(\Omega)$ with continuous embedding by \cref{th:sobolev} and thus also $x\mapsto g(x,u(x))\in L^\infty(\Omega)$ by the continuity property of~$g$ and by the fact that the number of vertices in~$\Omega$ is finite.
We additionally note that $\mathcal F\in C^1(W^{1,2}_0(\Omega),L^2(\Omega))$.
Indeed, the Laplacian $\Delta$ is linear and the map $u\mapsto g(x,u)$ is of class $C^1$ thanks to the continuity properties of~$g$.
Finally, we observe that the map $\mathcal F'(0)\colon W^{1,2}_0(\Omega)\to L^2(\Omega)$ is invertible, since $\mathcal F'(0)=-\Delta$ by the assumption that $\partial_t g(x,0)=0$ for all $x\in\Omega$.
Since $\mathcal F(0)=\bar f$, the conclusion follows by the Inverse Function Theorem and the proof is complete.
\end{proof}
Our two main results Theorems~\ref{th:main} and~\ref{res:main_uniqueness} can be combined in order to achieve the well-posedness of a Yamabe-type problem on bounded domains, namely \cref{propcomb}.
\begin{proof}[Proof of \cref{propcomb}]
The function $g(x,t)=b(x)|t|^{q-1}t$, defined for $(x,t)\in\Omega\times\mathbb{R}$, satisfies the assumptions of \cref{res:main_uniqueness}, so that problem~\eqref{eq:yamabe_cor} admits at most one solution and we just need to deal with the existence issue.
If $\|a\|_{L^1(\Omega)}=0$, then clearly \mbox{$a=0$} and thus the null function $u=0$ is the unique solution of problem~\eqref{eq:yamabe_cor}.
If $\|a\|_{L^1(\Omega)}>0$ instead, then we apply \cref{th:main}.
Indeed, the function $f(x,t)=a(x)-b(x)|t|^{q-1}t$, defined for $(x,t)\in\Omega\times\mathbb{R}$, satisfies the assumptions of \cref{th:main}, and the conclusion thus follows in virtue of \cref{rem:Lambda}.
\end{proof}
We conclude our paper with the following uniqueness result for a Kazdan--Warner-type problem on bounded domains.
Its proof is a simple application of \cref{res:main_uniqueness} and is thus left to the reader.
\begin{corollary}
Let $G=(V,E)$ be a weighted locally finite graph.
Let $\Omega\subset V$ be a bounded domain such that $\Omega^\circ\ne\varnothing$ and $\partial\Omega\ne\varnothing$.
Let $p\in[1,+\infty)$ and let $\alpha,\beta\in L^1(\Omega)$ be two non-negative functions.
For every $f\in L^1(\Omega)$ and $h\in L^1(\partial\Omega)$, the Kazdan--Warner-type problem
\begin{equation}\label{eq:kazdan-warner}
\begin{cases}
-\Delta_p u + \alpha\,e^{\beta u} = f & \text{in}\ \Omega^\circ\\[2mm]
u = h & \text{on}\ \partial\Omega
\end{cases}
\end{equation}
admits at most one solution $u\in W^{1,p}(\Omega)$.
\end{corollary}
\begin{bibdiv}
\begin{biblist}
\bib{AR73}{article}{
author={Ambrosetti, Antonio},
author={Rabinowitz, Paul H.},
title={Dual variational methods in critical point theory and
applications},
journal={J. Functional Analysis},
volume={14},
date={1973},
pages={349--381},
}
\bib{BCG01}{article}{
author={Barlow, Martin},
author={Coulhon, Thierry},
author={Grigor'yan, Alexander},
title={Manifolds and graphs with slow heat kernel decay},
journal={Invent. Math.},
volume={144},
date={2001},
number={3},
pages={609--649},
}
\bib{BC98}{article}{
author={Brezis, Ha\"{\i}m},
author={Cabr\'{e}, Xavier},
title={Some simple nonlinear PDE's without solutions},
journal={Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)},
volume={1},
date={1998},
number={2},
pages={223--262},
}
\bib{BMP07}{article}{
author={Brezis, Ha\"{\i}m},
author={Marcus, M.},
author={Ponce, A. C.},
title={Nonlinear elliptic equations with measures revisited},
conference={
title={Mathematical aspects of nonlinear dispersive equations},
},
book={
series={Ann. of Math. Stud.},
volume={163},
publisher={Princeton Univ. Press, Princeton, NJ},
},
date={2007},
pages={55--109},
}
\bib{BS73}{article}{
author={Brezis, Ha\"{\i}m},
author={Strauss, Walter A.},
title={Semi-linear second-order elliptic equations in $L^{1}$},
journal={J. Math. Soc. Japan},
volume={25},
date={1973},
pages={565--590},
}
\bib{CGY97}{article}{
author={Chung, Fan},
author={Grigor'yan, Alexander},
author={Yau, Shing-Tung},
title={Eigenvalues and diameters for manifolds and graphs},
conference={
title={Tsing Hua lectures on geometry \& analysis},
address={Hsinchu},
date={1990--1991},
},
book={
publisher={Int. Press, Cambridge, MA},
},
date={1997},
pages={79--105},
}
\bib{CGY00}{article}{
author={Chung, Fan},
author={Grigor'yan, Alexander},
author={Yau, Shing-Tung},
title={Higher eigenvalues and isoperimetric inequalities on Riemannian
manifolds and graphs},
journal={Comm. Anal. Geom.},
volume={8},
date={2000},
number={5},
pages={969--1026},
}
\bib{CCP19}{article}{
author={Chung, Soon-Yeong},
author={Choi, Min-Jun},
author={Park, Jea-Hyun},
title={On the critical set for Fujita type blow-up of solutions to the
discrete Laplacian parabolic equations with nonlinear source on networks},
journal={Comput. Math. Appl.},
volume={78},
date={2019},
number={6},
pages={1838--1850},
}
\bib{CG98}{article}{
author={Coulhon, T.},
author={Grigoryan, A.},
title={Random walks on graphs with regular volume growth},
journal={Geom. Funct. Anal.},
volume={8},
date={1998},
number={4},
pages={656--701},
}
\bib{FMBR16}{article}{
author={Ferrara, Massimiliano},
author={Molica Bisci, Giovanni},
author={Repov\v s, Du\v san},
title={Nonlinear elliptic equations on Carnot groups},
journal={Rev. R. Acad. Cienc. Exactas F\'{i}s. Nat. Ser. A Math.},
date={2016},
pages={1--12},
}
\bib{G18}{article}{
author={Ge, Huabin},
title={A $p$-th Yamabe equation on graph},
journal={Proc. Amer. Math. Soc.},
volume={146},
date={2018},
number={5},
pages={2219--2224},
}
\bib{G20}{article}{
author={Ge, Huabin},
title={The $p$th Kazdan-Warner equation on graphs},
journal={Commun. Contemp. Math.},
volume={22},
date={2020},
number={6},
pages={1950052, 17},
}
\bib{GHJ18}{article}{
author={Ge, Huabin},
author={Hua, Bobo},
author={Jiang, Wenfeng},
title={A note on Liouville type equations on graphs},
journal={Proc. Amer. Math. Soc.},
volume={146},
date={2018},
number={11},
pages={4837--4842},
}
\bib{GJ18-1}{article}{
author={Ge, Huabin},
author={Jiang, Wenfeng},
title={Kazdan-Warner equation on infinite graphs},
journal={J. Korean Math. Soc.},
volume={55},
date={2018},
number={5},
pages={1091--1101},
}
\bib{GJ18-2}{article}{
author={Ge, Huabin},
author={Jiang, Wenfeng},
title={Yamabe equations on infinite graphs},
journal={J. Math. Anal. Appl.},
volume={460},
date={2018},
number={2},
pages={885--890},
}
\bib{GJ19}{article}{
author={Ge, Huabin},
author={Jiang, Wenfeng},
title={The 1-Yamabe equation on graphs},
journal={Commun. Contemp. Math.},
volume={21},
date={2019},
number={8},
pages={1850040, 10},
}
\bib{Gri01}{article}{
author={Grigor'yan, Alexander},
title={Heat kernels on manifolds, graphs and fractals},
conference={
title={European Congress of Mathematics, Vol. I},
address={Barcelona},
date={2000},
},
book={
series={Progr. Math.},
volume={201},
publisher={Birkh\"{a}user, Basel},
},
date={2001},
pages={393--406},
}
\bib{Gri18}{book}{
author={Grigor'yan, Alexander},
title={Introduction to analysis on graphs},
series={University Lecture Series},
volume={71},
publisher={American Mathematical Society, Providence, RI},
date={2018},
pages={viii+150},
}
\bib{GLY16-K}{article}{
author={Grigor'yan, Alexander},
author={Lin, Yong},
author={Yang, Yunyan},
title={Kazdan-Warner equation on graph},
journal={Calc. Var. Partial Differential Equations},
volume={55},
date={2016},
number={4},
pages={Art. 92, 13},
}
\bib{GLY16-Y}{article}{
author={Grigor'yan, Alexander},
author={Lin, Yong},
author={Yang, Yunyan},
title={Yamabe type equations on graphs},
journal={J. Differential Equations},
volume={261},
date={2016},
number={9},
pages={4924--4943},
}
\bib{GLY17}{article}{
author={Grigor'yan, Alexander},
author={Lin, Yong},
author={Yang, Yunyan},
title={Existence of positive solutions to some nonlinear equations on
locally finite graphs},
journal={Sci. China Math.},
volume={60},
date={2017},
number={7},
pages={1311--1324},
}
\bib{HSZ20}{article}{
author={Han, Xiaoli},
author={Shao, Mengqiu},
author={Zhao, Liang},
title={Existence and convergence of solutions for nonlinear biharmonic equations on graphs},
journal={J. Differential Equations},
volume={268},
date={2020},
number={7},
pages={3936--3961},
}
\bib{LW17-2}{article}{
author={Lin, Yong},
author={Wu, Yiting},
title={The existence and nonexistence of global solutions for a
semilinear heat equation on graphs},
journal={Calc. Var. Partial Differential Equations},
volume={56},
date={2017},
number={4},
pages={Paper No. 102, 22},
}
\bib{LY20}{article}{
author={Liu, Shuang},
author={Yang, Yunyan},
title={Multiple solutions of Kazdan-Warner equation on graphs in the
negative case},
journal={Calc. Var. Partial Differential Equations},
volume={59},
date={2020},
number={5},
pages={Paper No. 164, 15},
}
\bib{MBR17}{article}{
author={Molica Bisci, Giovanni},
author={Repov\v s, Du\v san},
title={Yamabe-type equations on Carnot groups},
journal={Potential Anal.},
volume={46},
date={2017},
number={2},
pages={369--383},
}
\bib{Z17}{article}{
author={Zhang, Dongshuang},
title={Semi-linear elliptic equations on graphs},
journal={J. Partial Differ. Equ.},
volume={30},
date={2017},
number={3},
pages={221--231},
}
\bib{ZZ18}{article}{
author={Zhang, Ning},
author={Zhao, Liang},
title={Convergence of ground state solutions for nonlinear Schr\"{o}dinger
equations on graphs},
journal={Sci. China Math.},
volume={61},
date={2018},
number={8},
pages={1481--1494},
}
\bib{ZC18}{article}{
author={Zhang, Xiaoxiao},
author={Chang, Yanxun},
title={$p$-th Kazdan-Warner equation on graph in the negative case},
journal={J. Math. Anal. Appl.},
volume={466},
date={2018},
number={1},
pages={400--407},
}
\bib{ZL18}{article}{
author={Zhang, Xiaoxiao},
author={Lin, Aijin},
title={Positive solutions of $p$-th Yamabe type equations on graphs},
journal={Front. Math. China},
volume={13},
date={2018},
number={6},
pages={1501--1514},
}
\bib{ZL19}{article}{
author={Zhang, Xiaoxiao},
author={Lin, Aijin},
title={Positive solutions of $p$-th Yamabe type equations on infinite
graphs},
journal={Proc. Amer. Math. Soc.},
volume={147},
date={2019},
number={4},
pages={1421--1427},
}
\end{biblist}
\end{bibdiv}
\end{document} |
0905.3496 | \section{Introduction}
This is the third paper of a series devoted to the analysis of the ultraviolet
(UV) morphology of intermediate and late type stars and evolved stellar
populations. In the first paper \citep[hereafter Paper~I]{lino05}, we
presented the UVBLUE library of synthetic stellar
spectra at high resolution and provided its potential usefulness and
limitations, when applied to complement empirical data in the analysis of
stellar spectra. In a second paper \citep[hereafter Paper~II]{chavez07}, we
applied UVBLUE to the detailed analysis of a set of 17 mid-UV spectroscopic
indices in terms of the main atmospheric parameters: effective temperature,
surface gravity, and metallicity. This study also included the comparison of
synthetic indices with those measured in {\it IUE} spectra of a sample of main
sequence stars. The global result was that, with UVBLUE properly degraded to
match the resolution of {\it IUE} (6~\AA), eleven synthetic indices
(absorption or continuum) either well reproduced the observations or could be
easily transformed to the {\it IUE} system. In this work, we further extend the
investigation to the simplest aged stellar aggregates: the globular clusters.
Globular clusters represent the classical test bench for any population
synthesis approach of the analysis of larger stellar aggregates in the local
as well as the distant universe \citep{schiavon07}. They are commonly
considered the prototypes of a coeval, chemically homogeneous stellar
population, and are used to build up the properties of elliptical galaxies.
Although the integrated ultraviolet light of globulars has been extensively
analyzed, the study of their UV morphology has not yet been fully exploited
spectroscopically through the analysis of their absorption indices
\citep{fanelli90,fanelli92}. This analysis provides
numerous advantages over the full UV spectral energy distribution (SED), since
indices are easily calculated and generally defined in narrow bands,
diminishing substantially the effects of interstellar reddening and flux
calibration. Chemical species can be examined separatedly, at
least on those indices dominated by a single atom (i.e., no
blends).
Over the past decade Fanelli's et al.\ absorption indices have been
incorporated in a variety of analyses. \citet{ponder98}, for instance,
analyzed a sample of four globular clusters in M31 and six elliptical galaxies
observed with the Faint Object Spectrograph on board the {\it Hubble Space
Telescope}. They compare these data, in the form of colors and line indices,
with a sample of Galactic stars, Milky Way globular clusters, and the
integrated UV spectrum of M32. They found that Milky Way systems, clusters in
M31, and elliptical galaxies conform three distinct stellar populations. Almost
contemporarily, \citet{rd99} carried out a detailed comparative analysis of
M32 and NGC~104 (47~Tuc) aimed at constraining the stellar content of M32. In view of
the chemical similarities (both nearly solar) they modelled the UV spectral
indices of M32 on the basis of the better studied Galactic stellar system, in
terms of its leading parameters (age and metallicity). In spite of their
compatible chemical composition, they found that, while spectroscopic indices of
NGC~104 can be interpreted in terms of a dominant main sequence (turn-off) and
various contributions of other stellar types, in particular red objects in the
horizontal branch (HB), M32 requires a significantly more
metal rich turn-off (TO) and a more important contribution
of stars of type A, likely consistent with the observed
paucity of post-asymptotic giant branch stars \citep{bertola95,brown08}.
More recently, \citet{lotz00}
conducted an analysis of the effects of age and chemical composition on the
integrated spectra of simple populations modelled with population synthesis
techniques. They tested and used as ingredients the library of theoretical
fluxes by \citet{Kurucz93} and a modified set of the isochrones of
\citet{bertelli94}; they found that, within the limitations of Kurucz's low
resolution SEDs (see Paper II), which appeared to faithfully reproduce the
stellar indices BL~2538 and S2850, simple population models adequately match
the mid-UV morphology of mean globular groups \citep{bba95}. However, they
failed to match the mid-UV indices of galaxies, ascribing the discrepancies,
among other possible agents, to the presence of a hot stellar component [e.g.,
blue straggler stars (BS)] and a mixture of metal-poor objects (and a
metal-rich young population, as in the case of M32).
It is fair to mention that the above cited papers provide some of the main
drivers of the analysis presented in this paper: a)- Galactic globular
clusters (GGCs) has represented, with a few exceptions, the best reliable set of
single generation populations, therefore, any stellar population modelling
should be tested against the observed integrated properties of globulars; b)-
current empirical stellar libraries still present a marked paucity in the
parameter space, particularly in chemical composition. Such a deficiency has
been managed with the use of either grids of theoretical spectra
\citep{lotz00} or by extending the stellar database with a set of stellar
spectra of metal-rich stars with fiducial chemical compositions
\citep{rd99}. Nonetheless, it was until recently that theoretical databases at
the appropiate resolution to handle, for instance, {\it IUE} spectroscopic data became
available. Additionally, it has been suggested that the mid-UV properties of
Milky Way systems might not be applicable to the study of extragalactic
evolved populations \citep{ponder98}. It is therefore imperative that we test
theoretical predictions with local
globular clusters and establish the necessary corrections, before we can rely on synthetic populations
for studying non-local or more complex stellar systems.
Preliminary results on the application of synthesis techniques to the study
of the UV morphology of old populations have been already presented in
\citet{lino04}, \citet{bertone07}, and \citet{chavez08}. More recently
\citet{maraston08} have made use of our previous results to investigate
the far-UV and mid-UV indices in young populations with the goal of providing
the tools for the study of distant post-starburst galaxies
(age $<$ 1~Gyr). Being based on the same theoretical and empirical framework, it
turns mandatory to at least briefly explain the ``added value" of the present
analysis. First, this paper extends their analyses to older populations,
including the age interval (1--3~Gyr), which incorporates the so far determined
(but apparently still rather inconclusive) ages of red galaxies at redshifts
around $z=1.5$ \citep[see, for instance,][]{Spinrad97}. Second, we provide a
comparison with Galactic globular clusters as a tool for quantifying the
required corrective factors for the theoretical indices. Third, eventhough we
concentrate in ``canonical" evolution, we provide, for the first time, hints
of the effects of non-solar partitions on the integrated ultraviolet energy
distributions of evolved populations. Fourth, we include more indices, some of
which appear to more clearly segregate the effects of age and metallicity in
old populations.
Aimed at complementing the necessary tools for the study of old populations at UV wavelengths, we present in this paper the
calculation of the integrated SEDs of SSPs in the UV spectral interval. We first theoretically
explore, in \S~\ref{sec:syn_pop}, the properties of integrated spectral features (in the form of spectroscopic and continuum indices) in terms of the main population parameters: metallicity and age. We then look, in \S~\ref{sec:obs}, into the empirical correlations between indices and chemical composition in a sample of Galactic globular clusters observed by {\it IUE}. In this section we also show the comparison between synthetic and empirical indices. A brief discussion on the agents that could explain
discrepancies (and their corrections) is provided in \S~\ref{problems}. In \S~\ref{alpha} we demonstrate the importance of adopting the
suitable element partition in particular in low resolution theoretical fluxes. A summary and conclusions are given in \S~\ref{summary}.
\section{Synthetic Single Stellar Populations at UV Wavelengths}
\label{sec:syn_pop}
We have computed new integrated spectra of SSPs following the prescriptions
outlined in \citet{bcf94} and the later revision by \citet{bgs98}.
The main differences with respect to \citet{bgs98} SSPs are summarized below.
In the present paper we incorporate the new Padova
isochrones \citep{bertelli08}.
These isochrones allow the calculation of SSPs with
arbitrary chemical compositions, metallicities that range
from $Z=$0.0001 to $Z=$0.07
in the assumption of a solar partition for the heavy elements,
and He abundance within reasonable ranges for each metallicity.
The asymptotic giant branch (AGB) phase is treated according to \citet{marigo08}.
The new isochrones do not include the post AGB phase and,
since this phase is relevant for the UV properties of SSPs,
we have added it following \citet{bertelli94}.
In our calculations we have considered ages from 100~Myr to 16~Gyr and the
following chemical compositions: Z=0.05, Y=0.28; Z=0.02, Y=0.28; Z=0.008,
Y=0.25; Z=0.004, Y=0.25; Z=0.0004, Y=0.25; Z=0.0001, Y=0.25. The red giant
branch (RGB) mass-loss parameter is very relevant for the present paper,
because it fixes the stellar masses on the horizontal branch (HB) and, in
turn, its hot blue tail. It has been set to $\eta_{RGB}$=0.50
independently from the metallicity as suggested by
\citet{carraro96} and, more recently, by \citet{vanloon08}.
The same RGB mass-loss parameter
has been used to reproduce the mid infrared colours
of 47~Tuc and of selected old metal rich ellipticals in
the Coma cluster \citep{clemens09}.
The youngest populations will not be of any use
in the analysis of globular clusters presented in what follows.
Nevertheless, their calculation will
allow to probe the behavior of metallic features since their very onset, and will certainly
be an important ingredient in the planned analysis of composite systems, such as M32, in
which a young stellar generation (including A-type stars) might be present.
In computing the integrated SEDs\footnote{The full sample of SEDs are
available upon request from the author.} we have made use of the UVBLUE
high-resolution synthetic stellar
library\footnote{http://www.inaoep.mx/$\sim$modelos/uvblue.html} (Paper~I). We
have linearly interpolated within the spectral library in order to match the
parameters considered for the mass bins along the isochrones.
Four sequences of integrated SEDs of SSPs are shown in panels of Fig.~\ref{fig:seds_ages}. In the upper panels, the sequences illustrate the effects of age in two synthetic populations of [M/H]=$-1.7$ and 0.0~dex, whose ages range from 0.1 to 16~Gyr as indicated by the labels on the right.
In the lower panels, we portray the effects of chemical composition for two representative old populations of 4 and 12~Gyr. A simple examination of the figure allows us to easily identify
changes in many spectral features including faint ones, variations that would
have been hidden if we had used a low resolution spectral grid (at, say,
10~\AA\ resolution, as the commonly used Kurucz low resolution database of stellar fluxes). For instance,
most of the spectral features (including the Mg~\textsc{ii} doublet at 2800~\AA) at the metal-poor regime display a slight increase over
the age interval 2--14~Gyr, with an apparent turnover at the oldest populations. In contrast, at solar metallicity we clearly
see that some features sharply increase up to relatively young ages and then
show a shallow decline with age, disappearing at about 12~Gyr. An example of
this behaviour is the series of features in the interval 2400--2500~\AA\ (mainly due to Fe~\textsc{i},
Fe~\textsc{ii}, Ni~\textsc{i}, Co~\textsc{i}, and Mg~\textsc{i}) which are evident all the way to 16~Gyr at low metallicities, but absent in the more metal-rich counterparts.
The quantitative measurement of these variations in terms of the leading
population parameters (age and chemical composition)
is given in the following section.
\begin{figure*}[!t]
\begin{center}
\resizebox{!}{!}{\includegraphics{mchavezfig1.eps}}
\caption{A sequence of synthetic spectral energy distributions for a set of metal-poor and solar metallicity populations is presented in the upper panels. The composite spectra include populations ranging from 100~Myr to 16~Gyr. The labels on the right indicate the age (in Gyr) of the
population. Aimed at illustrating the effects of chemical composition we plot in the lower panels the sequence of integrated
fluxes for two representative ages, 4 and 12~Gyr, and six [Fe/H] values varying from $-2.30$ to $+0.40$.
\label{fig:seds_ages}}
\end{center}
\end{figure*}
\subsection{The Absorption Spectroscopic Indices} \label{sec:indices_ssp}
With the set of synthetic SEDs, we have calculated 17 spectroscopic indices,
whose definition is given in Paper II; most of them were introduced by
\citet{fanelli90}. The indices measure the absorption of the most prominent
features in the mid-UV spectra of intermediate and late-type stars. Briefly,
an index is defined through three wavelength bands, with the two side bands
defining a pseudo-continuum, which is compared to the flux in the central band
pass, very much like the popular Lick indices defined in the optical
\citep[e.g.,][]{Worthey94}. For the calculation, we have
broadened the synthetic spectra with a Gaussian kernel of FWHM=6~\AA\ to
simulate the resolution of {\it IUE} in low resolution mode, which has been
(an still is) the work horse for comparison with empirical data. This step is
important, since we have demostrated that resolution might have non-negligible
effects on indices defined with the narrower bands (Paper~II).
Another important aspect is that we have, in the purely theoretical approach
presented in this section, included also some of the bluer indices. Unlike
previous investigations, we opted to include them foreseeing a potential use
in the analysis of higher quality data collected by either large optical
telescopes (for high redshift objects) or planned spaceborn instruments
\citep{ana08}.
On what follows we present the effects of age and metallicity on each of the 17 indices. Partial results have been already presented in
\citet{lino04} and \citet{bertone07}.
\subsubsection{The Effects of Metallicity and Age}
\label{glob_age_met}
Amongst the most important properties of theoretical stellar libraries, such
as UVBLUE, is that synthetic stellar SEDs can be computed for just about any
combination of stellar parameters, allowing, in turn, the calculation of SSPs
for a wide variety of combinations of age and chemical
composition. Empirically, previous analyses have coped with incompleteness in
the metallicity space, particularly at the metal-rich end, by complementing
available data sets with archival UV data and a homogeneous set of chemical
compositions \citep{rd99}. Another possibility, which however will also
require a metal-rich stellar library, is to assemble an empirical sequence of
more metallic populations. Perhaps, a way to do this task is to extend the
chemical composition with the integrated spectra of old open clusters, but,
hosting a significantly smaller number of stars, the results would be subject
to stocastic processes.
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=16.0cm,angle=90]{mchavezfig2.eps}
\caption{Full set of mid-UV SSP synthetic indices as a function of age.
The different line types stand for different chemical compositions and helium contents:
Z=0.05,Y=0.28; Z=0.02,Y=0.28; Z=0.008,Y=0.25;
Z=0.004,Y=0.25; Z=0.0004,Y=0.25;
Z=0.0001,Y=0.25. Indices are in magnitudes.
\label{fig:seds_indexteffage}}
\end{center}
\end{figure*}
The combined effects of age and chemical composition are displayed in
Fig.~\ref{fig:seds_indexteffage}.
Taking into account that at ages less than 1~Gyr all the
absorption features are diluted by the strong continuum,
we may summarize the following behavior:
i) There are indices that sharply increase to their maximum
value within the first two Gyr and, thereafter,
remain almost constant up to 10~Gyr.
ii) At ages above about 11~Gyr some of the indices in the
most metal-rich populations show a more or less pronounced decline.
This is particularly evident in the case of $Z=0.004$
and likely reflects a combination of age, metallicity,
helium content, and mass-loss parameter that favour the
rapid development of a hot horizontal branch.
In some cases (e.g., Mg~\textsc{ii}~2800, S2850) the index value is
comparable to that of very low metallicity populations.
iii) At solar and super-solar metallicities, the two bluer indices,
Fe~\textsc{ii}~2332 and Fe~\textsc{ii}~2402, behave quite differently with
respect to the other indices: after a rapid rise (1~Gyr), there is a
relatively fast decline, which is steeper for the highest $Z$.
The distinctive behavior of these two indices can be understood by examining
their behavior in stellar data displayed in Fig.~3 of Paper~II (see also
Fig.~2 of that paper). The stellar patterns are directly reflected in the
composite spectra of metal-rich SSPs, whose mid-UV emission is dominated by
turn-off stars. In stars, Fe~\textsc{ii}~2332 and Fe~\textsc{ii}~2402 are the
mid-UV indices that reach the maximum at the highest effective
temperature. Since they peak at about 7000~K at super-solar metallicity, in a
stellar populations the maxima of these
indices are expected to take place in systems where turn-off stars are less
than 2~Gyr old. A similar reasoning is applicable to the solar chemical
composition case. The index peak $T_{\rm eff}$ is decreasing with
decreasing metal abundance, therefore in the less metallic populations the
maxima are reached at older ages.
We have furthermore analyzed the effect of the dominant species in the band
passes of the three indices Fe~\textsc{ii}~2332, Fe~\textsc{ii}~2402, and
Fe~\textsc{ii}~2609: we found that, whilst in fact ionized Fe has a dominant
role in their central band passes, at the same time it also significantly depresses
the blue band of Fe~\textsc{ii}~2609, making this index to be
less sensitive than the bluest two to Fe~\textsc{ii} abundance and to
effective temperature.
These indices mark the presence of a young metal-rich population, teherfore,
might be relevant for the study of high-redshift galaxies and/or to trace
rejuvenation episodes in the local old systems.
iv) Finally, another remarkable feature is that, with the
exception of the indices mentioned in (iii), the metallicity, not the
age, appears to be the main agent regulating the strength of most of
the indices, at least in the age interval 2--10~Gyr.
\section{UV Indices of Galactic Globular Clusters}
\label{sec:obs}
In this section, we present the
comparison of synthetic indices with those measured in a sample of globular
clusters observed by the {\it IUE}. The goal of the comparison is two-fold. On
the one hand, we want to explore the sensitivity of the indices to metallicity
from an empirical point of view, complementing the results of \citet{rd99}. On
the other hand, we want to test and discuss the capabilities of synthetic SSPs
to reproduce what is actually measured in observational data and, therefore,
identify fiducial features that can be modelled for the analysis of more
complex systems, which, due to the limitations of observed data sets, cannot
be studied from a purely empirical perspective.
We have extracted from the {\it IUE}-Newly Extracted Spectra (INES) database
the images of 27 globular clusters observed with the longwave prime and
redundant cameras, large aperture and in low dispersion mode. The sample
represents the full set of GGCs observed with {\it IUE},
available in the INES data base. The working sample does not consider the
cluster NGC~5139 (Omega Cen) and includes four more clusters than in the
work of \citet{rd99}. The analysis of NGC~5139\footnote{There are other GGC in
our sample for which a composite main sequence has been identified
(e.g. NGC~2808), nevertheless the abundance spread of iron peak elements
turns out to be less pronounced that in Omega Cen.} will be
deferred to an on-going mid-UV study of the sample of early-type galaxies
observed by {\it IUE} being developed by our group. The unusual (as compared
to other globulars) composite nature in age and metallicity \citep[see,
e.g.,][]{sollima05,sdn06} of NGC~5139 will in fact serve as template for
testing combinations of SSPs. The four additional
clusters have spectra with a much lower quality than the rest, however, they
are still useful in some spectral intervals.
The sample is described in Table~\ref{glob_data}, where we list the cluster
identification, the iron abundance [Fe/H] from \citet{harris96}, the age
mainly from \citet{sw02}, the horizontal branch ratio\footnote{This quantity
is defined as HBR = (B-R)/(B+V+R), where B, V, and R are the numbers of blue
HB, RR-Lyrae, and red HB stars \citep{lee90}.} compiled
by \citet{harris96},
the horizontal branch type according to \citet{dickens72}, and, in the last
column, the additional references for the age not included in
\citet{sw02}. For two clusters, NGC~6388 and NGC~6441, which have been of
particular interest, because they represent a typical second parameter case,
we were not able to find in the literature age determinations. Based on the
evident similarities between NGC~6388 and NGC~104, \citet{catelan06} point out
that these clusters should be of similar age. Similarly, \citet{cd07} assumed
a reference age of 11~Gyr for NGC~6441.
It is worth to mention that sixteen objects have {\it IUE} images for which
the quality code (ECC) is four, five or six in the first digit. This, in
principle, would indicate {\it bonafide} data. This quality designation does
not guarantee, however, that the full wavelength range is useful. We have to
keep in mind that most globulars are instrinsically faint at the shortest
wavelengths and that the sensitivity curves for both the LWP and LWR cameras
peak at about 2750~\AA\ and significantly degrade in the bluest region as well
as in the red, although in this latter region (say at 3000~\AA) the instrinsic
flux is much larger. These facts have imposed some limitations, which have
prevented in most analyses the use of the complete set of indices. For these
reasons, we have restricted our study to a subsample of ten
indices. The criteria for such a selection is somewhat complicated due to the
mixed instrumental effects and stellar theory limitations. However, within the
goals of this pioneering comparison between synthetic and empirical
integrated indices, we decided to start with indices that accomplish the
following criterion: we include narrow line indices, whose full bands are
defined in the interval 2400--2950~\AA, and broader line and continuum indices
in the range 2400--3130~\AA, hence excluding the indices Fe~\textsc{ii}~2402,
Fe~\textsc{i}~3000, BL~3096, and 2110/2570. Additionally, we have also excluded the
two indices BL~2720 and BL~2740 for which in Paper~II we found, respectively, that indices are largely overestimated by UVBLUE-based measurements and that the saturation of empirical indices is not reproduced by UVBLUE. Regarding this latter point, it should be noted that there are other indices that also get saturated (those displaying a hook-like behavior in Fig.~6 of Paper~II), nevertheless they differ from BL~2740 in an important aspect: the deviation from the linear correlation in this index takes place at values that would correspond to much younger ages than those expected for GGCs; therefore, age effects will blur
any existing correlation with chemical composition and any potential
compatibility with synthetic SSPs based on UVBLUE.
Note that we have also left the index Fe~\textsc{ii}~2332 in spite of not fulfilling the selection criteria.
Though unavoidably affected by the {\it IUE} sensitivity constraints,
we considered this feature important essentially because the synthetic analysis has shown that
the index displays remarkable different behavior with respect to the other
indices, turning it into potential tool for breaking the age-metallicity
degeneracy \citep{buzzoni08}. This index was measured in a restricted sub-sample of objects
(15 out of 27), whose images are labelled with high quality codes.
In Table~\ref{glob_index}, we report the ten selected indices that have been
measured after correcting globular spectra for extinction, using the \citet{ccm89}
extinction curve, and for radial velocity. The error figures for five indices; namely 2600-3000, 2609/2660, 2828/2923, Mg~\textsc{ii}~2800, and Mg~\textsc{i}~2852 have been given by \citet{rd99}, based upon several images of individual clusters. We have independently estimated through an iterative process the typical
error associated with each index. The process consists of randomly adding artificial noise to each original spectrum and computing the index for each iteration. The standard deviation of the resulting (Gaussian) distribution of indices was taken as the error. Our estimates are in agreement with the typical values of \citet{rd99} of 0.03 (and 0.06 for Mg~\textsc{ii}~2800), except for Mg~Wide and Fe~\textsc{ii}~2332 for which we derived 0.01 and 0.05 mag, respectively. Empty entries in Table~\ref{glob_index} are
index values (mainly negative ones) that have been dropped after a detailed
visual evaluation of the quality of spectra in the bands defining each index.
We would like to point out an additional comment on the interstellar
extiction. Eventhough we have mentioned that spectroscopic indices do not
significantly depend on reddening, some of the indices cover up to 660~\AA\
and, bearing in mind that interstellar extinction dramatically increases at
mid-UV and shorter wavelengths, we consider it necessary to correct {\it IUE}
spectra for the effects of extinction. In fact, we conducted a test in which
we measured the set of indices, whose bands are defined at $\lambda >
2300$~\AA\ in the coadded spectrum (extracted from four {\it IUE} images) of
the slightly reddened cluster NGC~104. After correcting the spectrum, assuming
color excesses in the interval 0.0 to 0.8~mag (at a step of 0.1~mag), which
roughly correspond to the values comprised in our cluster sample, we found
that effects of interstellar reddening are far from negligible for the indices
covering the widest wavelength intervals, in particular Mg~Wide. Index
variations turned out to be of the order of 10\% for an E(B-V) difference of
0.1~mag.
\subsection{Globular Cluster Indices vs.\ Chemical Composition}
\label{sec:index_metal}
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=16.0cm]{mchavezfig3.eps}
\caption{Observed GGC selected indices
compared to synthetic indices of SSP of 10 (solid line), 12 (dashed line), and 13~Gyr
(dot-dashed line).
Models are the same of Fig.~\ref{fig:seds_indexteffage}.
The solid circle shows the position of NGC~6388,
while the solid square represents
NGC~6441. The vertical arrows at the extrema of the 10Gyr synthetic indices show the
dilution of an index due to the presence of hot stars in the HB (see section 4.2
for details).
\label{ggcindex_vs_met}}
\end{center}
\end{figure*}
In Fig.~\ref{ggcindex_vs_met}, we show the trends of the ten selected indices
vs.\ metallicity for our sample of 27 GGCs. Among the main features are:
a)- All indices steadily increase with increasing chemical composition and
display remarkable good correlations, most of them increasing a factor of 5--6
with [Fe/H] increasing from about $-2.2$ to $-0.5$ dex. In the work of
\citet{rd99} the correlation was explored aimed at assessing the adequacy of
metal-rich clusters for studying the populations of M32. In addition to their
five indices, we found that five more also display a monotonic behavior up to
the metal-rich edge of our sample. From the analysis of synthetic populations,
we found that this monotonic tendency is, general, continued at super-solar regimes. An
exception to this trend is, for instance, the Mg~\textsc{ii}~2800), for which the oldest
populations present an inflexion and indices decrease towards high metallicities.
b)- It is interesting to note that, while the spread in age to some
extent increases the vertical dispersion of the points, age is not the main
agent, and the dispersion should be abscribed to other reasons, in particular
the UV bright population (stars at the turn-off, hot horizontal branch stars, and
blue stragglers). In this line, from an analysis of the Mg~\textsc{ii}~2800 index,
\citet{rd99} concluded that the observed metallicity-index correlation can be
explained by the dependence of the temperature of the TO on metallicity. They
also examined the effects of the HB morphology and found no correlation
between the dispersion and HB morphology. Such results are confirmed by
scrutiny of the panels in Fig.~\ref{ggcindex_vs_met} only for low metallicity systems, where no clear tendency
is evident. As we shall see in what follows, the presence of extended HBs severely affects some of the indices.
As a supplemental piece of the puzzle, it is noteworthy to mention that there
is evidence that enhancement of elements formed by capture of $\alpha$
particles is present in globular clusters, reflecting the overall chemical evolution of the Milky Way \citep{lw05,mendel07}. In principle, such effects should be taken into account in both the available stellar data sets that include spectra with $\alpha$-enhancement \citep[e.g.,][]{munari05} and in the opacities used to construct evolutionary tracks. Whilst the detailed study of the imprints of $\alpha$-element enrichment in mid-UV indices is beyond the scope of this paper, we consider it convenient to present in \S~\ref{alpha} a brief analysis.
\subsection{Synthetic vs.\ Empirical Indices}\label{glob_vs_synt}
In order to quantitatively compare empirical indices with results from
population synthesis techniques, we have initially considered to apply, whenever
possible, the transformation coefficients needed to match theoretical and
empirical indices of stars as described in Paper II. However, in composite
energy distributions, the discussed limitations of the theoretical atmospheric
data are hauled together with additional potential drawbacks of their own. Instead,
we have conducted
a direct comparison of results of the synthesis codes after matching
the {\it IUE} nominal resolution in low dispersion mode
of 6~\AA.
In Fig.~\ref{ggcindex_vs_met}, we have superimposed to the data of globular
clusters the synthetic integrated indices for the ages of 10~Gyr (solid line), 12~Gyr (dashed line), and 13~Gyr
(dot-dashed line). Only indices for Z=0.0001, Z=0.0004, and
Z=0.004 are plotted here, because they span the range of the
observed values. It is worth stressing that models
are plotted in this diagram with the [Fe/H] values corresponding
to the metallicity of the isochrones, derived from the
abundances of the atmosphere models.
Finally, we remind that the solid circle represents the NGC~6388
and the solid square indicates NGC~6441.
These two clusters display quite extended horizontal branches as
compared to the other three metal rich clusters (characterized by the presence
of a red clump), indicating the possibility of a higher
than normal helium content \citep[e.g.,][]{bcf94}.
Synthetic indices fairly well reproduce the trends of the empirical
data. However, in some cases, especially for the lower metallicity clusters,
the models tend to overestimate the data. This is particularly evident in the
case of BL~2538, Fe~\textsc{ii}~2609, 2609/2660, and 2828/2921. We also notice
that age differences, at fixed metallicity, are visible only for the highest
metallicity bin.
In what follows, we give some remarks on groups of indices, while a
more global discussion is deferred to \S~\ref{problems}.
a)- Fe~\textsc{ii}~2332: The overall trend is faithfully reproduced by the
synthetic indices. This index is the only case where there is a slight
theoretical underestimation of about 0.05~mag.
b)- BL~2538, Fe~\textsc{ii}~2609, and 2609/2660: These indices show general
offsets, which are about 0.2~mag at the lower metallicities. However, at the
highest metallicity, almost all the data are well bracketed by the 10~Gyr and
13.5~Gyr models. In these indices, NGC~6388 is always consistent with a
large age, while NGC~6441 is compatible with different ages
depending on the adopted index.
This fact likely reflects the different stellar
mixture, in particular the presence of a more or less prominent
(with respect to the metallicity) ensemble of hot He-burning stars.
According to the correlations seen in stars, the differences between
theoretical and {\it IUE} indices (obtained
from the least square fit presented in Paper II or estimated by
inspection of Fig.~6 in that paper), for the largest measured {\it IUE}
indices, are approximately 0.34, 0.5, and 0.5~mag. Therefore the overestimation
of these three indices can be amply explained by stellar discrepancies.
c)- Mg~\textsc{ii}~2800: The models match the data fairly well.
All clusters follow a line of constant age around 10--12~Gyr.
Interestingly, the clusters NGC~6388 and NGC~6441 are both well
separated from the other three metal-rich clusters,
reflecting the presence of the hot HB population.
d)- Mg~\textsc{i}~2852: It also shows a quite fair match, apart from the
lowest metallicity case, where the models sligthly overestimate the empirical
data. At high metallicity, the age differences are more pronounced than
in the case of Mg~\textsc{ii}~2800. This index indicates ages around 12~Gyr
for the high metallicity clusters, apart from the two outliers
already discussed above.
e)- Mg~Wide and 2828/2921: These are amongst the best reproduced indices, as
was the case for the stellar indices. It is important to
note that Mg~Wide displays the least dispersion, most probably reflecting the
dilution of discrepancies due to its broader bands. Paradoxically, the features
that are expected to modulate this index are not very well reproduced
individually.Note that for these two indices the clusters NGC~6388 and NGC~6441 are close to
those of the other clusters of similar metallicity, and
consistent with an age of about 12 Gyr.
f)- 2600-3000 and S2850L: These two continuum indices are formed by widely
separated bands. The match with the data is fairly good
and, similarly to indices mentioned in the previous point, metal-rich
clusters display consistent indices, with NGC~6441 attaining the highest
values.
The behavior depicted by the indices in the last two points above is interesting and may be
due to a certain independence of the indices from the extension
of the HB. This will be investigated in a forthcoming paper
(see also section 4.2).
\section{Theoretical Stellar Data Bases: a Need for an Update} \label{problems}
\subsection{Shortcomings of the theoretical spectra} \label{bad_theory}
We have extensively discussed the caveats associated with the UVBLUE spectral
library in Paper~I and their implications in the synthetic stellar indices in
Paper~II. In order to decrease the observed discrepancies in stars, it would be
important to re-calculate the grid following a two step process
(in addition to the calculation of the grid itself). We first
need to incorporate up-dated solar abundances, since the parent model
atmospheres of UVBLUE spectra adopted the chemical composition of
\citet{and_grev89}, while new Kurucz model atmospheres are now available with
the \citet{grev_sauv98} abundances. The differences in these two databases
result in a change of the global metallicity $Z$ of the order of 0.002. We
actually conducted a series of tests by comparing entries in UVBLUE with those
of \citet{munari05}, which incorporates \citet{grev_sauv98} abundances, and
the effects on the overall energy distribution are negligible. More recently,
however, 3-D hydrodynamic calculations by \citet{gas07} have shown that a
major update of solar composition might be required. These latest results
indicate a solar metallicity about one half ($Z=0.012$) of the value reported
in the papers cited above. It is therefore of fundamental importance that
these new values are tested and included in the synthetic atmospheres and
interiors. Once the reference abundances have been anchored, the second step,
perhaps as important as the first one, consists on the evaluation of the
atomic line parameters. The fact that theoretical spectra predict too deep
absorption lines \citep{bell94,bertone08} can be, in some cases, solely
ascribed to the uncertain line parameters (mainly oscillator strengths and Van
der Waals damping constants). A comparison with high-quality and
high-resolution observational data, as in \citet{peterson05}, will certainly
bring the solution to this problem.
\subsection{Theoretical Incompleteness of the Hot Evolutionary Stages}
We have mentioned that hot stars play an important role in modulating
(decreasing) the index strength. Evolutionary tracks should, in principle,
include evolutionary prescriptions that allow the formation of objects that
populate extensions of the main sequence and the horizontal branch in the
color-magnitude diagram of stellar clusters (such as BSs,
blue HB stars, and components of the so-called blue tail and
blue hook of the HB). As far as BSs are concerned, it has been demostrated that
their presence in old open clusters severely affects the integrated
energy distributions, particularly at ultraviolet wavelengths
\citep{Xin07}. More pronounced effects are expected from He-burning objects
that attain high luminosities and temperatures (higher than those expected
from metallicity effects alone) as those present in some GGCs
\citep[see][for the cases of NGC~6441 and NGC~6388]{busso07}. It is clear that a
more appropiate way to carry out the comparisons presented in
\S~\ref{glob_vs_synt} would preferably have to include in the synthesis code
the observed star counts and distributions (extracted from CM diagrams) of
these hot components for each cluster. However, even if we are able to include
the effects of these hot stars, such effects should be modelled on the basis
of an integrated property of the population in order to be applicable to the
analysis of non-resolved stellar aggregates, which is the main scope of this
paper. Perhaps an interesting suggestion has emerged from the magnesium
indices depicted in Fig.~\ref{ggcindex_vs_met}, which appear to segregate
clusters with blue HBs.
\begin{figure}[!t]
\begin{center}
\includegraphics[height=6.cm,width=7.0cm]{mchavezfig4.eps}
\caption{Dilution of the Mg~\textsc{ii}~2800 and 2600-3000 indices as a function of the bolometric luminosity
fraction of artificially added hot stars with respect to the total luminosity of the original population: $L_{\rm
HB}/L_{\rm SSP}$. The thin vertical arrow stands for the index decrease that would correspond to the difference between
the averaged empirical values of the clusters NGC~6388 and NGC~6441 and that of an 10~Gyr population, indicating that
a luminosity ratio of $log(L_{\rm
HB}/L_{\rm SSP})$ = -1.46 can explain that difference. By fixing this ratio we obtained similar {\it dilution} vectors
for the rest of the indices. In the x-axis at the top we include the scale for the integrated flux ratio in the mid-UV.
\label{hot_stars}}
\end{center}
\end{figure}
In Fig.~\ref{hot_stars}, we show the results of a simple but illustrative
exercise in which we analyze the effects of the presence of hot stars on two
of the indices included in Fig.~\ref{ggcindex_vs_met}. For this purpose, we
have artificially added a hot stellar component to a set of synthetic
populations of 10~Gyr and $Z$=0.004. The stellar model flux that we have
added in different amounts, corresponds to the parameters of the hottest HB
star present in the stellar isochrones of Fig.~\ref{ggcindex_vs_met} (Age/$T_{\rm
eff}$/$\log{g}$/[Fe/H])=(13~Gyr/10490/3.72/$-0.68$). At any rate, the qualitative behavior of the results
does not depend of the exact value of the
stellar parameters. In the figure, we plot the index
strength as a function of the bolometric luminosity fraction of the added hot
stars with respect to the total luminosity of the original population: $L_{\rm
HB}/L_{\rm SSP}$. The figure shows that indices preserve their starting value
(that of the SSP without any additional hot star) almost constant until the HB
luminosity accounts for approximately 1\% of that of the parent population. At
about $L_{\rm HB}=L_{\rm SSP}$ the trends asymptotically approach the values
of the HB stars alone. It is important to remark that we have actually
computed the HB effects for the remaining eight indices and they all display a
quite similar behavior. This analysis indicates, as anticipated, that the
presence of hot objects in a stellar population is reflected in a reduction of their
mid-UV indices, and, consequently, could provide an explanation for
the low values of the line indices, in particular for the cluster NGC~6441.
In order to quantitatively establish a {\it dilution} vector for each index,
we have taken the index Mg~\textsc{ii}~2800 as a reference. We assumed that the
low values of this index in NGC~6388 and NGC~6441 are solely due to hot HB stars.
Then, we have calculated the difference between
the averaged empirical values of the clusters NGC~6388 and NGC~6441 and that of
an 10~Gyr population (0.42mag, indicated with an vertical arrow in
Fig.~\ref{hot_stars}) and searched for the
luminosity fraction that accounts for this difference; $log (L_{\rm HB}/L_{SSP}) = -1.46$.
With this luminosity ratio we obtained the vectors for the rest of the indices and
are indicated with arrows in Fig.~\ref{ggcindex_vs_met} on the right of each panel. Note that
these {\it dilution} vectors allow also an explanation for the low index values
for BL~2538, Fe~\textsc{ii}~2609, Mg~\textsc{i}~2852, and 2609/2660 of the cluster
NGC~6388 and for Mg~\textsc{i}~2852 of NGC~6441. To within the uncertainties
associated with the empirical indices Mg~Wide and 2828/2921 and considering
the their rather small vectors, these last two indices are also
appropriately reproduced for both clusters.
The above analysis, however, demands an interpretation of the inconsistent
values of NGC~6441 for the indices Fe~\textsc{ii}~2609, 2609/2660 and
S2850L. We have double checked the only available IUE image of NGC~6441 and found
that the flux errors associated with the blue bands defining these indices
translate into significantly larger uncertainties on the indices of
NGC~6441 than those of NGC~6388.
Somewhat intriguing is the behavior of the 2600--3000 slope index. The two
metal-rich clusters with hot HBs are about in the same loci as the red HB
clusters, eventhough the above analysis indicates that the vector for this
index is large (0.42~mag). One can speculate on a number of reasons that potentially
affect the overall mid-UV energy distribution (such as color excess), nevertheless,
from an empirical point of view it appears that this index is, to some extent, insensitive to the
dilution by hot stars. We want to stress that a more profound examination is
required. This has to include, as mentioned earlier, the appropiate numbers
and distributions of the hot stellar component. Nevertheless, it is clear that
the exclusion of hot objects could lead to wrong predictions for metal rich
systems.
\section{Effects of Alpha Enhancement on the Mid-UV Morphology}\label{alpha}
Closely related to the brief discussion in \S~\ref{bad_theory} is the evidence that the elemental abundances in globular clusters show significant variations with respect to solar partition values. It has been a common practice to incorporate, within population synthesis techniques, adjustments to optical indices (in both theoretical and empirical) that account, for example, for the enhancement of the $\alpha$-elements (O, Na, Ti, Ca, Mg, etc.) \citep{mendel07}. In the mid-UV, the effects of $\alpha$-enhancement have never been explored and in this section we present some preliminary results.
In what follows, we made use of the theoretical database of stellar fluxes
available at Fiorella Castelli's
website\footnote{http://wwwuser.oat.ts.astro.it/castelli/grids/}
\citep{castelli03}. In view of the low resolution of these grids and within
the exploratory nature of this investigation, we decided to concentrate on
showing the effects of $\alpha$-enhancement on the theoretical stellar and
population SEDs, defering the detailed analysis of the impact of these effects
on continuum and line indices for a future investigation in which we also partially cover the points mentioned in the previous section. An important note is that the grid of Castelli's stellar fluxes has been calculated by adopting the \citet{grev_sauv98} solar abundances and consider a set of updated opacity distribution functions (NEWODFs).
In Fig.~\ref{fig:star_alpha}, we show the residuals for a series of
theoretical fluxes for three chemical compositions ($Z$ = 0.0004, 0.02, 0.05),
five different temperatures, and a surface gravity of $\log{g}$=4.0~dex. In the
lower left corner in each panel on the left we indicate the effective
temperature for that row. The vertical axes are the flux residuals $\Delta
f/f$, where $\Delta f=f_{\alpha}-f$ is the difference between the flux with an
$\alpha$-enhancement of [$\alpha$/Fe]=+0.4~dex and those considering
[$\alpha$/Fe]=+0.0~dex. The residuals for temperatures $T_{\rm eff} > 5000$~K have been scaled with offsets of 1, 2, 3, and 4 with respect to that for 5000~K. The thick lines illustrate the relative residuals for the cases in which $Z$ has been fixed. A significant
$\alpha$-enhancement, keeping the total metallicity $Z$ fixed,
implies that the abundances of the non-enhanced
metals (in particular Fe)
have to be rescaled downwards \citep[see, for example,][for the case of Lick indices]
{annibali07,mendel07}. Since oxygen is the most abundant metal,
the decrease on the non-$\alpha$ metals turns out to be significant.
Particularly important for the stellar atmospheres is the decrease of Fe
because of its large contribution to line blanketing.
As an example, in the case of $Z=0.004$, the atmosphere models have
[Fe/H]=$-0.58$ for non-enhanced abundances and [Fe/H]=$-0.89$ if
[$\alpha$/Fe]=+0.4~dex.
To perform this rescaling we have simply interpolated
the needed stellar atmospheres.
Note that the residuals heavily depend on temperature and chemical
composition, with the $\alpha$-enhanced fluxes being on average larger in the
mid-UV and hence delivering positive residuals. The residuals can reach values
in excess of 60\%, particularly at the solar and super solar regimes.
The thin lines in Fig.~\ref{fig:star_alpha} show the residuals when we compare
theoretical fluxes with the same nominal parameters ($T_{\rm
eff}$/$\log{g}$/[M/H]) of the parent model, i.e.\ $Z$ is not fixed. In this case the
residuals are generally negative, reflecting the overestimation of the blanketing.
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=16.cm,width=16.0cm]{mchavezfig5.eps}
\caption{Effects of $\alpha$-enhancement in stellar theoretical fluxes for a
fixed surface gravity of $\log{g}$=4.0~dex and different metallicities and
effective temperatures. The y-axis depicts the relative flux ratio in the
form of $\Delta f/f$ where $\Delta f=f_{\alpha}-f$ is the difference between
the flux with an $\alpha$-enhancement of [$\alpha$/Fe]=+0.4~dex and those
considering [$\alpha$/Fe]=+0.0~dex. For the sake of clarity the flux ratios
have been scaled, from bottom to top, with a unity offset.
The thick solid line shows the case where the $\alpha$-enhancement is
accompanied by a rescaling of all metals to preserve the global metallicity,
while the thin solid line refers to the case where this rescaling is not
performed (see text for more details).
\label{fig:star_alpha}}
\end{center}
\end{figure*}
For the calculation of the composite $\alpha$-enhanced spectra of SSPs, we
remind that the new Padova models do not account yet for
variations of $\alpha$ elements partition.
Thus, in the calculation of present SSP spectra,
the $\alpha$-enhancement is only taken into account for
the effect it has in the stellar atmospheres. The stellar fluxes used in
the synthesis code were those with a renormalized heavy element abundance to
a fixed metallicity ($Z$). The effects on the stellar evolutionary tracks
should not be significant for the low metallicities of GGCs.
In Fig.~\ref{fig:ssp_alpha}, we show the relative differences between
integrated spectra of SSP with and without accounting for $\alpha$-enhancement.
The panels refer to the combinations of three ages (rows) and four chemical
compositions (columns). An examination of the figure allows to point out some
remarkable behaviors. First, the effects of $\alpha$-enhancement heavily
depend on $Z$, with the most pronounced differences at the metal-rich end,
even in the relatively young populations of 1~Gyr. For the most metal-poor
SSPs ($Z= 0.0001$), the flux differences never exceed 10\%. Second, concerning
the age effects, the notable differences seen in the mid-UV residual spectrum
of the 1~Gyr metal-rich population appear gradually shifted to longer
wavelengths. Globally, $\alpha$-enhancement is notoriously more important in
the mid-UV than in the near-UV, at ages and abundances compatible with those
of GGCs, and certainly far more important than in the optical, where it has
been extensively studied. This UV sensitivity suggests, among many other
possible options, to conduct an analysis of mid-UV indices of intermediate and
late-type stars in the context of the chemical evolution scenarios for the
Milky Way.
\begin{figure*}[!t]
\begin{center}
\includegraphics[height=14.cm,width=16.0cm]{mchavezfig6.eps}
\caption{Effects of $\alpha$-enhancement of SSP models for different ages (rows)
and chemical compositions (columns). The y-axis illustrates the flux ratio as in Fig.~\ref{fig:star_alpha}. In this comparison global metallicity of
the $\alpha$-enhanced stellar fluxes have been properly rescaled to preserve the same $Z$.
\label{fig:ssp_alpha}}
\end{center}
\end{figure*}
\section{Summary}\label{summary}
In this work we have studied the ultraviolet properties of evolved stellar populations,
with the purpose of analyzing the behavior of mid-UV indices in terms of
the leading population parameters (chemical composition and age) and checking
the current status of the theoretical models. With these results we trace the
next more important advances to perform.
The results can be summarized as follows:
\begin{itemize}
\item This is, to our knowledge, the first theoretical analysis of integrated
mid-UV indices of old populations. They display a variety of behaviors,
with the main one being the low index insensitivity to age in a wide age interval (age $>$ 2~Gyr). Two indices, Fe~\textsc{ii}~2332 and Fe~\textsc{ii}~2402, show, however, a remarkable distinct behavior with respect to other indices at the metal rich regime. This result is of particular importance, since our work is aimed at providing predictive tools for the analysis of elliptical galaxies of high metallicity.
\item We also present a test of the ability of SSP fluxes to reproduce 10
spectroscopic
indices measured in the {\it IUE} spectra of a sample of 27 GGCs. Theoretical indices well
reproduce the overall trends but show slight discrepancies in which synthetic
indices are sistematically higher. In this respect, we conclude that SSPs
well represent the observational prototypes of simple populations, after the
appropiate scaling of SSPs synthetic indices to the {\it IUE}
system. Therefore, one can be confident that
the approach presented here serves as the basis for studies of metal-rich
populations, which cannot be modelled with existing empirical data bases, due
to their poor coverage of the metallicity space.
\item We provide a brief diagnostics on how the theoretical ingredients should
be improved. We suggest the following three main roads: a)- improvement of the solar reference abundances in the calculation of
theoretical atmospheres and evolutionary tracks; b)- semi-empirically
up-dating of the atomic line parameters to reduce the too strong absorption
lines in synthetic spectra; and c)- check with the new isochrones the
importance of hot evolved evolutionary phases.
\item Together with the above points, it is important to incorporate
the effects of $\alpha$-enhancement in adequate resolution stellar atmosphere
models. In fact, the tests we have illustrated here, though performed with
low resolution models, have shown that an enhancement of +0.4 dex
may result in dramatic differences for the integrated spectra
of the same metallicity and age. In parallel, $\alpha$-enhancement should also be considered
for the computation of the evolutionary tracks, though this effect is expected
to be important only at high metallicity.
\end{itemize}
All the above issues will be the subjects of our future
investigations.
\acknowledgements M.C. and E.B. are pleased to thank financial support from
Mexican CONACyT, via grants 49231-E and SEP-2004-C01-47904. A.B. acknowledges
contract ASI-INAF COFIS I/016/07/0. |
0905.3439 | \section{Introduction}
\label{sect:intro}
In studying the equation of state (EOS) of ordinary quark matter,
the cruial point is to treat quark confinement in a proper way.
Except the conventional bag mechanism (where quarks are
asymptotically free within a large bag), an alternative way to
obtain confinement is based on the density dependence of quark
masses, then the proper variation of quark masses with density would
mimic the strong interaction between quarks, which is the basic idea
of the quark mass-density-dependent model.
Originally, the interaction part of the quark masses was assumed to
be inversely proportional to the density~(Fowler et al. 1981;
Chakrabarty 1991; Chakrabarty 1993; Chakrabarty 1996), and this
linear scaling has been extensively applied to study the properties
of strange quark matter (SQM). However, this class of scaling is
often criticized for its absence of a convincing derivation~(Peng
2000). Then a cubic scaling was derived based on the in-medium
chiral condensates and linear confinement~(Peng 2000). and has been
widely used afterwards~(Lugones $\&$ Horvath 2003; Zheng et al.
2004; Peng et al. 2006; Wen et al. 2007; Peng et al. 2008). But this
deriving procedure is still not well justified since it took only
the first order approximation of the chiral condensates in medium.
Incorporating of higher orders of the approximation would
nontrivially complicate the quark mass formulas~(Peng 2009). In
fact, there are also other mass scalings in the literatures~(Dey et
al. 1998; Wang 2000; Zhang et al. 2001; Zhang $\&$ Su 2002; Zhang
$\&$ Su 2003).
Despite the big uncertainty of the quark mass formulas, this model,
after all, is no doubt only a crude approximation to QCD. For
example, the model may not account for quark system where realistic
quark vector interaction is non-ignorable. However, we can not get a
general idea of how the strong interaction acts from the fundamental
theory of strong interactions in hand, i.e. QCD. Until this
stimulating controversy is solved, we feel safe to take the
pragmatic point of view of using the model. This work does not claim
to answer how Nature works. However, it may shed some light on what
may happen in interesting physical situations. In this respect, the
quark mass-density-dependent model has been, and still is, an
interesting laboratory.
The aim of the present paper then, is to study in what extent this
scaling model is allowed to study the properties of SQM. To this
end, we treat the quark mass scaling as a free parameter, to
investigate the stability of SQM and the variation of the predicted
properties of the corresponding strange stars (SSs) within a wide
scaling range. Furthermore, we try to demonstrate the general
features of SSs related to astrophysics observations, whatever the
value of the free parameters.
The paper is organized as follows. In Section 2 we describe the
formalism applied in calculating the EOS of the SQM in the quark
mass-density-dependent model. In Section 3 we present the structure
of the stars made of this matter, including mass-radius relation,
spin frequency, electric properties of the quark surface. Finally in
Section 4 we address our main conclusions.
\section{{\bf The} Model}
\label{sect:Mod}
As usually done, we consider SQM as a mixture of interacting $u$,
$d$, $s$ quarks, and electrons, where the mass of the quarks $m_{q}$
($q = u, d, s$) is parametrized with the baryon number density
$n_{\mathrm{b}}$ as follows:
\begin{equation}
m_q \equiv m_{q0}+ m_{\mathrm{I}}=m_{q0}+\frac{C}{n_{\mathrm{b}}^x},
\label{mqT0}
\end{equation}
where $C$ is a parameter to be determined by stability arguments.
The density-dependent mass $m_{q}$ includes two parts: one is the
original mass or current mass $m_{q0}$, the other is the interacting
part $m_{\mathrm{I}}$. The exponent of density $x$, i.e. the quark
mass scaling, is treated as a free parameter in this paper.
Denoting the Fermi momentum in the phase space by $\nu_i$ ($i=u, d,
s,e^-$), the particle number densities can then be expressed as
\begin{equation} \label{nimod}
n_i =g_i\int \frac{\mathrm{d}^3{\bf p}}{(2\pi\hbar)^3}
=\frac{g_i}{2\pi^2} \int_0^{\nu_i}\, p^2\,\mbox{d}p
=\frac{g_i\nu_i^3}{6\pi^2}{\bf ,}
\end{equation}
and the corresponding energy density as
\begin{equation} \label{Emod}
\varepsilon
=\sum_i\frac{g_i}{2\pi^2}\int_0^{\nu_i}\sqrt{p^2+m_i^2}\,p^2\,\mbox{d}p{\bf
.}
\end{equation}
The relevant chemical potentials $\mu_u$, $\mu_d$, $\mu_s$, and
$\mu_e$ satisfy the weak-equilibrium condition (we assume that
neutrinos leave the system freely):
\begin{eqnarray}
\mu_u+\mu_e=\mu_d, ~~~\mu_d=\mu_s{\bf .} \label{weak}
\end{eqnarray}
For the quark flavor $i$ we have
\begin{eqnarray}
\mu_i &=& \frac{\mathrm{d} \varepsilon}{\mathrm{d} n_i}
|_{\{n_{k\neq i}\}}= \frac{\partial \varepsilon_i}{\partial \nu_i}
\frac{\mathrm{d}\nu_i}{\mathrm{d} n_i}
+\sum_j \frac{\partial \varepsilon}{\partial m_j}\frac{\partial m_j}{\partial n_i}
\nonumber \\
&=&
\sqrt{\nu_i^2+m_i^2}
+\sum_j n_j\frac{\partial m_j}{\partial n_i}
f\!\left(\frac{\nu_j}{m_j}\right),
\label{mui}
\end{eqnarray}
where
\begin{equation}
f(a) \equiv \frac{3}{2a^3} \left[
a\sqrt{1+a^2}-\ln\left(a+\sqrt{1+a^2}\right)
\right].
\end{equation}
We see clearly from Equ.~(\ref{mui}) that since the quark masses are
density dependent, the derivatives generate an additional term with
respect to the free Fermi gas model.
For electrons, we have
\begin{equation} \label{muevsne}
\mu_e=\sqrt{\left(3\pi^2n_e\right)^{2/3}+m_e^2}{\bf .}
\end{equation}
The pressure is then given by
\begin{eqnarray}
P&=& -\varepsilon + \sum_i \mu_i n_i
\nonumber \\
&=&
-\Omega_0
+\sum_{ij} n_i n_j\frac{\partial m_j}{\partial n_i}
f\left(\frac{\nu_j}{m_j}\right)
\nonumber \\
&=& -\Omega_0
+n_{\mathrm{b}}\frac{\mathrm{d}m_{\mathrm{I}}}{\mathrm{d}n_{\mathrm{b}}}
\sum_{j=u,d,s} n_j\ f\!\left(\frac{\nu_j}{m_j}\right){\bf ,}
\label{pressure}
\end{eqnarray}
with $\Omega_0$\ being the free-particle contribution:
\begin{eqnarray}
\Omega_0 &=& -\sum_i\frac{g_i}{48\pi^2} \left[
\nu_i\sqrt{\nu_i^2+m_i^2}\left(2\nu_i^2-3m_i^2\right)
\right.
\nonumber\\
&& \phantom{-\sum_i\frac{g_i}{48\pi^2}[}
\left.
+3m_i^4\,\mbox{arcsinh}\left(\frac{\nu_i}{m_i}\right)
\right].
\end{eqnarray}
The baryon number density and the charge density can be given as:
\begin{equation} \label{qmeq3}
n_{\mathrm{b}}=\frac{1}{3}(n_u+n_d+n_s){\bf ,}
\end{equation}
\begin{equation} \label{qmeq4}
Q_{\mathrm{q}}=\frac{2}{3}n_u-\frac{1}{3}n_d-\frac{1}{3}n_s-n_e.
\end{equation}
The charge-neutrality condition requires $Q_{\mathrm{q}}=0$.
Solving Equs. (\ref{weak}), (\ref{qmeq3}), (\ref{qmeq4}), we can
determine $n_u$, $n_d$, $n_s$, and $n_e$ for a given total baryon
number density $n_{\mathrm{b}}$. The other quantities are obtained
straightforwardly.
In the present model, the parameters are: the electron mass
$m_e=0.511$ MeV, the quark current masses $m_{u0}$, $m_{d0}$,
$m_{s0}$, the confinement parameter $C$ and the quark mass scaling
$x$. Although the light-quark masses are not without controversy and
remain under active investigations, they are anyway very small, and
so we simply take $m_{u0}=5$ MeV, $m_{d0}=10$ MeV. The current mass
of strange quarks is $95\pm 25$ MeV according to the latest version
of the Particle Data Group~\cite{Yao06}
\begin{figure}
\includegraphics[width=8.0cm]{fig1.eps}
\caption{The stability window of the SQM at zero pressure with the
quark mass scaling parameter $x = 1/10, 1/5, 1/3, 1, 2$. The
stability region (shadow region), is where the energy per particle
is lower than 930 MeV and two-flavor quark matter is unstable. }
\label{fig1}
\end{figure}
We now need to establish the conditions under which the SQM is the
true strong interaction ground state. That is, we must require, at
$P=0, E/A\leq M(^{56}{\rm Fe})c^2/56=930$ MeV for the SQM and
$E/A>930$ MeV for two-flavor quark matter (where $M(^{56}{\rm Fe})$
is the mass of $^{56}{\rm Fe}$) in order not to contradict standard
nuclear physics. The EOS will describe stable SQM only for a set of
values of ($C,m_{s0}$) satisfying these two conditions, which is
given in Fig.~1 as the ``stability window''. Only if the
$(C,m_{s0})$ pair is in the shadow region, SQM can be absolutely
stable, therefore the range of $C$ values is very narrow for a
chosen $m_{s0}$ value. As shown in Fig.~1, the allowed region
decreases for decreasing value of $x$. When $x = 1/10$ it approaches
to a very narrow area around $C$ = 199.1 MeV fm$^{{\rm -3}x}$.
We then illustrate in Fig.~2 the density dependence of
$m_{\mathrm{I}}$ with the quark mass scaling $x = 1/10, 1/3, 1, 3$.
The calculation is done with $m_{s0}=95$ MeV and $C$ values
corresponding to the upper boundaries defined in Fig. 1 (the same
hereafter), that is, the system always lies in the same binding
state (for each $x$), i.e, E/A = 930 MeV. We presented those $C$
values in the last row of the Table.~1. Clearly the quark mass
varies in a very large range from very high density region
(asymptotic freedom regime) to lower densities, where confinement
(hadrons formation) takes place. It is compared with Dey et al.'s
scaling (dash-dotted)~\cite{Dey98}.
\begin{figure}
\includegraphics[width=8.0cm]{fig2.eps}
\caption{The density dependence of $m_{\mathrm{I}}$ with the quark
mass scaling parameter $x = 1/10, 1/3, 1, 3$. The calculation is
done with $m_{s0}=95$ MeV and $C$ values presented in the last row
of the Table.~1 (see text for details). It is compared with Dey et
al.'s scaling (dash-dotted line)~(Dey et al. 1998). } \label{fig2}
\end{figure}
\section{Results and Discussion}
\label{sect:Res}
The resulting EOSs of SQM are shown in Fig.~3 for all considered
models. Because the sound velocity $v = \mid dP/d\rho\mid^{~1/2~}$
should be smaller than $c$ (velocity of light), unphysical region
excluded by this condition has been displayed with scattered dots.
For the $x$ values chosen here, they have quite different behavior
at low density, basically falling into two sequences. At small
scalings ($x = 1/10, 1/5, 1/3$) the pressure increases rather slowly
with density; while the curve turns to rapidly increase with density
at relatively large $x$ values ($x = 1, 2, 3$). They cross at
$\varepsilon \sim $ 800 MeV fm $^{-3}$, then tend to be
asymptotically linear relations at higher densities, and a larger
$x$ value leads to a stiffer EOS. Meanwhile we check also the
stability of such a quark matter, since some EOSs in Fig 3 ($x = 1,
2, 3$) are rather stiff for small pressures. We present in Fig.~4
the total pressure as a function of neutron chemical potential in
quark matter for all considered models, and comparison with that of
typical nuclear matter (obtained from the Brueckner-Hartree-Fock
approach~\cite{Li06}). We see clearly from the figure that the quark
matter tends to be more stable than nuclear matter for all
considered models.
The behavior of EOSs would be mirrored at the prediction of
mass-radius relations of the corresponding SSs, as is shown in the
Fig.~5. For the first sequence, the maximum mass occurs at a low
central density (as shown in Table.~\ref{Table1}), so a higher
maximum mass is obtained due to a stiffer EOS, and with the increase
of $x$ value, the maximum mass is reduced from 1.78$M_{\odot}$ at $x
= 1/10$ down to 1.61 $M_{\odot}$ at $x = 1/3$; While we observe a
slight increase of the maximum mass with $x$ value for the second
sequence: from 1.56$M_{\odot}$ at $x = 1$ up to 1.62 $M_{\odot}$ at
$x = 3$. Anyway the resulting maximum mass lies between
1.5$M_{\odot}$ and 1.8$M_{\odot}$ for a rather wide range of $x$
value chosen here (0.1 -- 3), which may be a pleasing feature of
this model: well-controlled. To see the region of stellar parameters
allowed by this model, we plot in Fig.~5 also the M(R) curves for
lower boundaries defined in Fig.~1 for $x = 1/5, 1/3, 1$ with grey
lines.
The radii, on the other hand, decrease invariably with $x$ value.
Employing the empirical formula connecting the maximum rotation
frequency with the maximum mass and radius of the static
configuration~\cite{Gou99}, we get also the maximum rotational
angular frequency $\Omega_{\rm max}$ as $7730{\Large \left(
\frac{M_{\odot }^{\rm stat}}{M_{\odot }} \right)^{\frac{1}{2}}\left(
\frac{R^{\rm stat}_{M_{\odot }}}{10 {\rm km}}
\right)^{-\frac{3}{2}}}$rad s$^{-1}$. As a result, a larger $x$
value results in a larger maximum spin frequency, SSs with $x = 3$
can rotate at a frequency of $f_{\rm max}$ = 2194 Hz. More detailed
results can be found in Table.~\ref{Table1}.
\begin{table}
\begin{center}
\begin{minipage}[]{85mm}
\caption{\small Calculated results for the gravitational masses,
radii, central baryon densities (normalized to the saturation
density of nuclear matter, $n_0$ = 0.17 fm$^{-3}$), and the maximum
rotational frequencies for the maximum-mass stars of each strange
star sequence. The calculation is done with $m_{s0}=95$ MeV and
$C$ values presented in the last row of this table. } \label{Table1}
\end{minipage}
\end{center}
\begin{center}
\begin{tabular}{cccccccccc}
\hline
$x$ & $1/10$ & $1/5$ & $1/3$ & $1$ & $2$ & $3$\\
\hline
$M/M_{\odot}$ & 1.78 & 1.66 & 1.61 & 1.56 & 1.61 & 1.62\\
$R/{\rm km}$ & 13.2 & 10.5 & 9.38 & 8.10 & 7.97 & 7.89\\
$n_c/n_0$ & 4.35 & 6.47 & 7.88 & 10.1 & 10.2 & 10.3\\
$f_{\rm max}/{\rm Hz}$ & 1066 & 1446 & 1691 & 2072 & 2159 & 2194\\
\hline
$C/{\rm MeV~fm}^{-3x}$& 199.1 & 157.2 & 126.8 & 69.5 & 41.7 & 28.8 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\includegraphics[width=8.0cm]{fig3.eps}
\caption{The EOSs of SQM for all considered models. Unphysical
region excluded by \textbf{this} condition has been displayed with
scattered dots (see text for details). } \label{fig3}
\end{figure}
\begin{figure}
\includegraphics[width=8.0cm]{fig4.eps}
\caption{The total pressure as a function of neutron chemical
potential in SQM for all considered models, and comparison with that
of typical nuclear matter (see text for details).} \label{fig4}
\end{figure}
\begin{figure}
\includegraphics[width=8.0cm]{fig5.eps}
\caption{The mass-radius relations of SSs for all considered models.
M(R) curves for lower boundaries defined in Fig.~1 with the quark
mass scaling parameter $x = 1/5, 1/3, 1$ are presented with grey
lines. Contours of the maximum rotation frequencies are given by the
light grey curves~(Gourgoulhon et al. 1999). } \label{fig5}
\end{figure}
In addition, the surface electric field could be very strong near
the bare quark surface of a strange star because of the mass
difference of the strange quark and the up (or down) quark, which
could play an important role in producing the thermal emission of
bare strange stars by the Usov mechanism~(Usov 1998; Usov 2001). The
strong electric field is also very crucial in forming a possible
crust around a strange star, which has been investigated extensively
by many authors~(for a recent development, see Zdunik et al. 2001).
Furthermore, it should be noted that this electric field may have
some important implications on pulsar radio emission mechanisms~(Xu
et al. 2001). Therefore it is very worthwhile to explore how the
mass scaling influences the surface electric field of the stars, and
possible related astronomical observations in turn may drop a hint
on what the proper mass scaling would be.
Adopting a simple Thomas-Fermi model, one gets the Poisson's
equation~(Alcock et al. 1986):
\begin{equation}
{d^2 V\over dz^2} = \left\{ \begin{array}{ll}
{4\alpha\over 3\pi}(V^3-V_{\rm q}^3) & z\leq 0,\\
{4\alpha\over 3\pi} V^3 & z > 0,
\end{array} \right.
\end{equation}
where $z$ is the height above the quark surface, $\alpha$ is the
fine-structure constant, and $V_{\rm q}^3/(3\pi^2 \hbar^3 c^3)$ is
the quark charge density inside the quark surface.
Together with the physical boundary conditions $\{ z \rightarrow
-\infty: V \rightarrow V_{\rm q}, dV/dz \rightarrow 0;~~ z
\rightarrow +\infty: V \rightarrow 0, dV/dz \rightarrow 0 \}$, and
the continuity of $V$ at $z=0$ requires $V(z=0) = 3V_{\rm q}/4$, the
solution for $z > 0$ finally leads to
\begin{equation}
V={3V_{\rm q}\over \sqrt{6\alpha\over\pi}V_{\rm q}z+4}~~ ({\rm
for}~z > 0).
\end{equation}
The electron charge density can be calculated as $ V^3/(3\pi^2
\hbar^3 c^3) $, therefore the number density of the electrons is
\begin{equation}
n_{\rm e} = {9V_{\rm q}^3\over \pi^2 (\sqrt{6\alpha\over\pi}V_{\rm
q}z+4)^3} \label{ne}
\end{equation}
and the electric field above the quark surface is finally
\begin{equation}
E = \sqrt{2\alpha\over 3\pi} \cdot {9 V_{\rm q}^2 \over
(\sqrt{6\alpha\over \pi} V_{\rm q} \cdot z + 4)^2} \label{E}
\end{equation}
which is directed outward.
We see from Fig.~5 (take $x = 1/3$ for example) that although the
electric field near the surface is about $ 10^{18}$ V cm$^{-1}$, the
outward electric field decreases very rapidly above the quark
surface, and at $z\sim 10^{-8}$ cm, the field gets down to $\sim
5\times 10^{11}$ V cm$^{-1}$, which is of the order of the
rotation-induced electric field for a typical Goldreich-Julian
magnetosphere. To change the mass scaling mainly has two effects:
First, it affects a lot the surface electric field, and a small
scaling parameter leads to an enhanced electric field. The weakening
of electric field would be almost a order of magnitude large (from
$10^{17}$ V cm$^{-1}$ to $10^{18}$ V cm$^{-1}$), which may have some
effect on astronomical observations. Second, a larger scaling would
slow the decrease of the electric field above the quark surface.
\begin{figure}
\includegraphics[width=8.0cm]{fig6.eps}
\caption{The electric field above the quark surface with the quark
mass scaling parameter $x = 1/3, 1, 2, 3$.} \label{fig6}
\end{figure}
\section{Conclusions}
\label{sect:conclusion}
In this paper, we investigate the stability of SQM within a wide
scaling range, i.e. from 0.1 to 3. We study also the properties of
the SSs made of the matter. The calculation shows that the resulting
maximum mass always lies between 1.5$M_{\odot}$ and 1.8$M_{\odot}$
for all the mass scalings chosen here. Strange star sequences with a
linear scaling would support less gravitational mass, a change
(increase or decrease) of the scaling parameter around the linear
scaling would result in a higher maximum mass. Radii invariably
decrease with the mass scaling. Then the larger the scaling, the
faster the star rotates. In addition, the variation of the scaling
may cause an order of magnitude change of the surface electric
field, which may have some effect on astronomical observations.
\section*{Acknowledgments}
We would like to thank an anonymous referee for valuable comments
and suggestions, and acknowledge Dr. Guang-Xiong Peng for beneficial
discussions. This work was supported by the National Basic Research
Program of China under grant 2009CB824800, the National Natural
Science Foundation of China under grants 10778611, 10833002,
10973002 and the Youth Innovation Foundation of Fujian Province
under grant 2009J05013. |
0905.3494 | \section{\@startsection {section}{1}{\z@}%
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\begin{document}
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\begin{flushright}
\parbox[t]{1.8in}{MIT-CTP-4039\\
CAS-KITPC/ITP-111\\
CERN-PH-TH/2009-064\\ MAD-TH-09-04}
\end{flushright}
\vspace*{0.3in}
\begin{center}
{\large \bf Large Primordial Trispectra in General Single Field
Inflation}
\vspace*{0.5in} {Xingang Chen$^1$, Bin Hu$^2$, Min-xin Huang$^3$,
Gary Shiu$^{4,5}$, and Yi Wang$^2$}
\\[.3in]
{\em
$^1$ Center for Theoretical Physics\\
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
\\[.1in]
$^2$ Kavli Institute for Theoretical Physics China, \\Key Laboratory
of Frontiers in Theoretical Physics, \\Institute of Theoretical
Physics,
Chinese Academy of Sciences, \\Beijing 100190, P.R.China \\[.1in]
$^3$ Theory Division, Department of Physics, CERN,\\ CH-1211 Geneva,
Switzerland\\[.1in]
$^4$ Department of Physics, University of Wisconsin, \\Madison, WI
53706, USA \\[.1in]
$^5$School of Natural Sciences, Institute for Advanced Study, \\
Princeton, NJ 08540, USA
\\[0.3in]}
\end{center}
\begin{center}
{\bf Abstract}
\end{center}
\noindent We compute the large scalar four-point correlation functions
in general single field inflation models, where the inflaton
Lagrangian is an
arbitrary function of the inflaton and its first derivative.
We find that the
leading order trispectra have four different shapes determined by
three parameters. We study features in these shapes that can be used to
distinguish among themselves, and between them and the
trispectra of the local form. For the
purpose of data analyses, we give two simple representative forms
for these ``equilateral trispectra''. We
also study the effects on the trispectra if the initial state of inflation
deviates from the standard Bunch-Davies
vacuum.
\vfill
\newpage
\setcounter{page}{1}
\tableofcontents
\newpage
\section{Introduction}
\setcounter{equation}{0}
Primordial non-Gaussianity is potentially one of the most promising
probes of the inflationary universe \cite{Komatsu:2009kd}. Like the role
colliders play in particle physics, measurements of primordial
non-Gaussian features
provide microscopic information on the
interactions of the inflatons and/or the curvatons. Constraining and detecting
primordial non-Gaussianities has become one of the major efforts in
modern cosmology. Theoretical predictions of non-Gaussianities, especially their
explicit forms, play an important role in this program. On the one
hand, they are needed as inputs of data analyses
\cite{Komatsu:2003iq,Creminelli:2006gc,Smith:2006ud,Fergusson:2008ra}
which eventually constrain the parameters defining the non-Gaussian features;
on the other hand, different forms of non-Gaussianities are
associated with different inflaton or curvaton interactions, and so if
detected can help us understand the nature of inflation.
A variety of potentially detectable forms of non-Gaussian features from inflation
models have been
proposed and classified, in terms of their
shapes and running. The scalar three-point functions,
i.e.~the scalar bispectra,
are by far the most well-studied. For single field
inflation, a brief summary of the status is as follows. Minimal slow-roll
inflation gives undetectable amount of primordial non-Gaussianities
\cite{Maldacena:2002vr,Acquaviva:2002ud,Seery:2005wm};
non-canonical kinetic terms can generate large
bispectra of the equilateral shapes
\cite{Chen:2006nt,Cheung:2007st};
non-Bunch-Davies vacuum can boost
the folded shape \cite{Chen:2006nt,Holman:2007na,Meerburg:2009ys}; and
features in the Lagrangian (sharp or periodic)
can give rise to large bispectra with oscillatory
running \cite{Chen:2006xjb,Chen:2008wn}. Multifield inflation models
provide many other possibilities due to various kinds of
isocurvature modes, such as curvatons \cite{Lyth:2002my}, turning
\cite{Vernizzi:2006ve,Huang:2007hh,Gao:2008dt,Langlois:2008qf,Arroja:2008yy,Byrnes:2008zy}
or bifurcating \cite{Naruko:2008sq,Li:2009sp} trajectories, thermal
effects \cite{Moss:2007cv,Chen:2007gd} and etc. These models give
many additional forms of large bispectra, notably ones with a large
local shape.
We will be getting much more data in the near future from new
generations of experiments, ranging from cosmic microwave
background, large scale structure and possibly even 21-cm hydrogen
line. Compared with the current WMAP, these experiments will be
measuring signals from shorter scales and/or in three dimensions.
Therefore a significant larger number of modes will become
available. This makes the study of four- or higher point functions
interesting, as they provide information on new interaction terms
and refined distinctions among models. In this paper we extend the
work of Ref.~\cite{Chen:2006nt} and classify the forms of large
scalar trispectra (i.e.~the scalar four-point function) in general
single field inflation models. There have been some preliminary
works in this direction \cite{Huang:2006eh,Arroja:2008ga},
calculating contributions from the contact interaction diagram
(Fig.~\ref{Fdiagrams} (A)). For models with a large trispectrum,
there is yet another set of diagrams involving the exchange of a
scalar (Fig.~\ref{Fdiagrams} (B)) that contributes at the same order
of magnitude.\footnote{Note that for
slow-roll inflation, the contribution from the scalar-exchange
diagram Fig.~\ref{Fdiagrams} (B)
is subleading, while the graviton-exchange contribution belongs to
the leading order \cite{Seery:2008ax}.} In this paper, we
complete this program and classify all possible shapes arising in this
framework.
For the bispectra in general single field inflation,
the leading large non-Gaussianities have two different
shapes controlled by two parameters \cite{Chen:2006nt}.
As we will see here, for
trispectra, we have four different shapes controlled by three
parameters. Some of them have complicated momentum-dependence.
For the purpose of data analyses, we give simple
representative shapes that can capture the main features of these
functions. We point out
the features in the shapes that can be used to distinguish among
themselves, as well as to distinguish them
from the trispectra of the local form. We also study the effects of a
non-Bunch-Davies initial state of inflation on these trispectra.
This paper is organized as follows. In Section 2, we review the
basic formalisms and main results for the power spectrum and
bispectra in general single field inflation. In Section 3, we
calculate the leading order trispectra, and summarize the final
results. At leading order, the trispectra can be classified into
four shapes, controlled by three parameters. In Section 4, we
investigate the shapes of the trispectra, including consistency
relations, figures in various limits, and also give two simple
representative forms of these equilateral trispectra to facilitate future data
analyses. In Section 5, we discuss DBI and K-inflation as two
examples to illustrate our results. In Section 6, we study the trispectra
when the initial state of inflation is in a non-Bunch-Davies vacuum. We conclude in Section 7.
\begin{figure}
\begin{center}
\epsfig{file=Fdiagrams.eps, width=10cm}
\end{center}
\medskip
\caption{Two diagrams that contribute to the large trispectra.}
\label{Fdiagrams}
\end{figure}
\section{Formalism and review}
\label{SecReview}
\setcounter{equation}{0}
In this section, we review the formalisms and main results of
Ref.~\cite{Chen:2006nt}.
As in Ref.~\cite{Garriga:1999vw}, we consider the following
general Lagrangian
for the inflaton field
$\phi$,
\begin{eqnarray}
S= \frac{1}{2} \int d^4x \sqrt{-g} \left[ M_{\rm Pl}^2 R + 2 P(X,\phi) \right] ~,
\end{eqnarray}
where $X \equiv - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu
\phi$ and
the signature of the metric is $(-1,1,1,1)$.
Irrespective of the specific mechanism that is responsible for the
inflation, once it is achieved we require the following set of
slow-variation parameters to be small, \begin{eqnarray} \epsilon = -\frac{\dot
H}{H^2} ~, ~~~~\eta = \frac{\dot \epsilon}{\epsilon H} ~, ~~~~s =
\frac{\dot c_s}{c_s H} ~, \end{eqnarray} where $H$ is Hubble parameter and
\begin{eqnarray} c_s^2 \equiv \frac{P_{,X}}{P_{,X} + 2X P_{,XX}} \end{eqnarray} is the
sound speed. The slow-variation parameters can be large temporarily
or quickly oscillating \cite{Chen:2006xjb,Chen:2008wn,Bean:2008na},
but we do not consider such cases here.
The power spectrum $P_\zeta$ is defined from the two-point function
of the curvature perturbation $\zeta$, \begin{eqnarray} \langle \zeta({\bf k}_1)
\zeta({\bf k}_2) \rangle = (2\pi)^5 \delta^3 ({\bf k}_1 +{\bf k}_2)
\frac{1}{2k_1^3} P_\zeta ~. \end{eqnarray} For the class of inflation models
that we consider, \begin{eqnarray} P_\zeta =
\frac{1}{8\pi^2M_{\rm Pl}^2}\frac{H^2}{c_s \epsilon} ~. \end{eqnarray}
In order to parametrize the three-point function, we need to define
a parameter $\lambda/\Sigma$ related to the third derivative of
the inflaton Lagrangian $P$ with respect to $X$,
\begin{eqnarray}
\lambda &=& X^2 P_{,XX} + \frac{2}{3} X^3 P_{,XXX} ~,
\\
\Sigma &=& X P_{,X} + 2 X^2 P_{,XX} = \frac{H^2\epsilon}{c_s^2} ~.
\end{eqnarray}
The bispectrum form factor ${\cal A}(k_1,k_2,k_3)$ is defined as
\begin{eqnarray}
\langle \zeta({\bf k}_1) \zeta({\bf k}_2) \zeta({\bf k}_3) \rangle
= (2\pi)^7 \delta^3 ({\bf k}_1 +{\bf k}_2 +{\bf k}_3)
P_\zeta^2 \prod_{i=1}^{3} \frac{1}{k_i^3} ~{\cal A} ~.
\end{eqnarray}
Up to ${\cal O}(\epsilon)$, this bispectrum is determined by five
parameters, $c_s$, $\lambda/\Sigma$, $\epsilon$, $\eta$ and $s$.
For the most interesting cases $c_s \ll 1$ or $\lambda/\Sigma \gg 1$
where the non-Gaussianities are large, the leading bispectrum is given
by
\begin{eqnarray}
{\cal A} &=& \left( \frac{1}{c_s^2} - 1 - \frac{2\lambda}{\Sigma} \right)
\frac{3k_1^2k_2^2k_3^2}{2K^3}
\cr
&+& \left( \frac{1}{c_s^2} - 1 \right)
\left( - \frac{1}{K} \sum_{i>j} k_i^2 k_j^2 + \frac{1}{2K^2}
\sum_{i\ne j} k_i^2 k_j^3 + \frac{1}{8} \sum_i k_i^3 \right) ~.
\label{Aform}
\end{eqnarray}
So we have two different forms determined by two parameters.
In such cases, the effect of the non-canonical kinetic terms of
the inflaton has to
become large enough so that the inflationary mechanism is no longer
slow-roll (in slow-roll the canonical kinetic term dominates over
the non-canonical terms).
Since
inflation gives approximately scale-invariant spectrum, ignoring the
mild running of the non-Gaussianity \cite{Chen:2005fe}, the bispectrum
is approximately a function of two variables in terms of the momentum
ratios, $k_2/k_1$ and $k_3/k_1$ \cite{Babich:2004gb}.
The two forms in (\ref{Aform}) have
very similar shapes and
they are usually referred to as the equilateral shapes. Because the two
shapes do have a small difference, for fine-tuned model parameters
$c_s$ and $\lambda/\Sigma$, they can cancel each other to a
large extent
and leave an approximately orthogonal component. One can use this
component and the one of the originals to form a new bases of the
shapes.\footnote{We would like to thank Eiichiro Komatsu for
discussions on this point \cite{SenatoreUnpub}.}
\section{Large trispectra}
\label{SecTri}
\setcounter{equation}{0}
As in the bispectrum case, we are most interested in cases where the
trispectra are large. In general single field inflationary models,
this is achieved by the non-canonical kinetic terms.
The origin of large non-Gaussianities
come from terms with derivatives of the
inflaton
Lagrangian $P$ with respect to $X$.
The contribution from the gravity sector is negligibly
small. The derivative of $P$ with respective to $\phi$ is also
small due to the approximate shift symmetry associated with the
inflaton. Another equivalent way to see this is to work in the
comoving gauge \cite{Maldacena:2002vr}
where the scalar perturbation $\zeta$ only appears in the metric.
So $P_{,\phi}$ explicitly does not appear in the
expansion. Using the leading order relation
\begin{eqnarray}
\zeta \approx - \frac{H}{\dot \phi} \alpha
\label{zetaalpha}
\end{eqnarray}
to convert $\zeta$ into
$\alpha \equiv \delta \phi$, again we see that $P_{,\phi}$ does not
appear.
Therefore for our purpose, it is convenient to choose the inflaton
gauge where the scalar perturbation only appears in the inflaton
\cite{Maldacena:2002vr},
\begin{eqnarray}
\phi = \phi_0(t) + \alpha(t,{\bf x}) ~;
\end{eqnarray}
and when we expand the inflaton Lagrangian $P$,
we only concentrate on terms that have derivatives with respect to
$X$. Such a method has also been used in
Ref.~\cite{Creminelli:2003iq,Gruzinov:2004jx}.
\subsection{Scalar-exchange diagram}
\label{SecSE}
In this subsection, we compute the scalar-exchange diagram,
Fig.~\ref{Fdiagrams} (B).
Using the inflaton gauge, we get the cubic terms of the Lagrangian
in the small $c_s$ or large $\lambda/\Sigma$ limit,
\begin{eqnarray}
{\cal L}_3 = \left(\frac{1}{2} P_{,XX} \dot \phi + \frac{1}{6} P_{,XXX} \dot
\phi^3 \right) a^3 \dot \alpha^3
-\frac{1}{2} P_{,XX} \dot \phi ~a \dot \alpha (\nabla \alpha)^2 ~.
\end{eqnarray}
Written in terms of $\zeta$ using (\ref{zetaalpha}), we get
\begin{eqnarray}
{\cal L}_3 =
-2a^3 \frac{\lambda}{H^3} \dot \zeta^3 + a \frac{\Sigma}{H^3}
(1-c_s^2) \dot
\zeta (\partial_i \zeta)^2 ~.
\label{CL3}
\end{eqnarray}
Despite of its
different appearance from the three leading cubic terms in
\cite{Chen:2006nt}, one can show, using the linear equation of
motion and integration by part, that the difference is a total
derivative.
In terms of the interaction Hamiltonian, ${\cal H}^I_3 = - {\cal L}_3$,
we denote the two terms in (\ref{CL3}) as
\begin{eqnarray} H^I_3 = - \int d^3x {\cal L}_3 = H_a + H_b ~, \end{eqnarray}
where
\begin{eqnarray} H_a(t) &=& 2a^3 \frac{\lambda}{H^3}
\int \prod_{i=1}^3
\frac{d^3{\bf p}_i}{(2\pi)^3} \dot\zeta_I({\bf p}_1,t) \dot\zeta_I({\bf p}_2,t)
\dot\zeta_I({\bf p}_3,t)
(2\pi)^3
\delta^3(\sum_{i=1}^3 {\bf p}_i) ~,
\label{H_a}
\\
H_b(t) &=& a \frac{\Sigma}{H^3} (1-c_s^2)
\int \prod_{i=1}^3
\frac{d^3{\bf p}_i}{(2\pi)^3} ({\bf p}_2 \cdot {\bf p}_3)
\dot\zeta_I({\bf p}_1,t) \zeta_I({\bf p}_2,t)
\zeta_I({\bf p}_3,t)
(2\pi)^3 \delta^3(\sum_{i=1}^3 {\bf p}_i) ~.
\label{H_b}
\end{eqnarray}
The $\zeta_I$ is in the interaction picture and satisfies the equation
of motion followed from the kinematic Hamiltonian.
The scalar trispectrum is the expectation value of the curvature
perturbation $\zeta_I^4$ in the interaction vacuum. According to the
in-in formalism \cite{Weinberg:2005vy}, there are three terms
contributing to the diagram Fig.~\ref{Fdiagrams} (B),\footnote{The
in-in
formalism is often used in the literature in terms of a
commutator form which is equivalent to the form presented here.
However, for a subset of terms,
the algebra in the
commutator form is more complicated than the one we use here. We
discuss this equivalence in Appendix \ref{AppCom}.}
\begin{eqnarray}
\langle \zeta^4
\rangle &=& \langle 0| \left[{\bar T} e^{i\int_{t_0}^t dt'
H_I(t')} \right]
\zeta_I({\bf k}_1,t) \zeta_I({\bf k}_2,t) \zeta_I({\bf k}_3,t)
\zeta_I({\bf k}_4,t)
\left[ T e^{-i\int_{t_0}^t dt' H_I(t')} \right] |0 \rangle
\label{Def}
\\
&\supset& \int_{t_0}^t dt' \int_{t_0}^t dt'' ~\langle 0| ~H_I(t')
~\zeta_I^4 ~H_I(t'') ~|0 \rangle
\nonumber \\
&-& \int_{t_0}^t dt' \int_{t_0}^{t'} dt'' ~\langle 0| ~H_I(t'')
~H_I(t') ~\zeta_I^4 ~|0\rangle
\nonumber \\
&-& \int_{t_0}^t dt' \int_{t_0}^{t'} dt'' ~\langle 0| ~\zeta_I^4
~H_I(t') ~H_I(t'') ~|0\rangle ~.
\label{4pt3terms}
\end{eqnarray}
Here $t$ is
a time several efolds after the modes exit the horizon and $t_0$ is
a time when modes are all well within the horizon. In terms of the
conformal time $\tau$, $dt=a(\tau) d\tau$,
we take $\tau=0$ and $\tau_0=-\infty$.
We evaluate (\ref{4pt3terms}) using the standard technique of normal
ordering. We decompose (omitting the subscript ``I'' for $\zeta$ in
the following)
\begin{eqnarray} \zeta ({\bf k},\tau) = \zeta^+ + \zeta^- = u({\bf k},\tau) a_{\bf k} +
u^*(-{\bf k},\tau) a^\dagger_{-{\bf k}} ~, \end{eqnarray} where \begin{eqnarray} u({\bf k},\tau) =
\frac{H}{\sqrt{4\epsilon c_s k^3}} (1+i k c_s \tau)e^{-i k c_s\tau}
~, \end{eqnarray}
and
\begin{eqnarray} [a_{\bf k}, a^\dagger_{\bf p}] = (2\pi)^3 \delta^3({\bf k}-{\bf p})
~. \end{eqnarray}
After normal ordering, the only terms that are non-vanishing
are those with all terms contracted. A contraction between the two
terms, $\zeta({\bf k},\tau')$ (on the left) and $\zeta({\bf p},\tau'')$ (on
the right), gives
\begin{eqnarray}
[\zeta^+({\bf k},\tau'), \zeta^-({\bf p},\tau'')] = u({\bf k},\tau')
u^*(-{\bf p},\tau'') (2\pi)^3 \delta^3({\bf k}+{\bf p}) ~.
\end{eqnarray}
We sum over all possible
contractions that represent the Feynman
diagram Fig.~\ref{Fdiagrams} (B), where the four external
legs are connected to $\zeta({\bf k}_i,t)$'s.
To give an example,
we look at the 1st term of (\ref{4pt3terms}) with the
component (\ref{H_a}). One example of such contractions is
\begin{eqnarray}
&&
\contraction{\dot\zeta(} {{\bf p}_1 }{ ,t') \dot\zeta({\bf p}_2,t')
\dot\zeta({\bf p}_3,t') \zeta( } { {\bf k}_1 }
\contraction[2ex]{\dot\zeta({\bf p}_1,t') \dot\zeta(} {{\bf p}_2} {,t')
\dot\zeta({\bf p}_3,t') \zeta({\bf k}_1,t) \zeta(} {{\bf k}_2}
\contraction[1ex]{\dot\zeta({\bf p}_1,t') \dot\zeta({\bf p}_2,t')
\dot\zeta({\bf p}_3,t') \zeta({\bf k}_1,t) \zeta({\bf k}_2,t) \zeta(} {{\bf k}_3}
{,t) \zeta({\bf k}_4,t) \dot\zeta(} {{\bf q}_1}
\contraction[2ex]{\dot\zeta({\bf p}_1,t') \dot\zeta({\bf p}_2,t')
\dot\zeta({\bf p}_3,t') \zeta({\bf k}_1,t) \zeta({\bf k}_2,t) \zeta({\bf k}_3,t)
\zeta(} {{\bf k}_4} {,t) \dot\zeta({\bf q}_1,t'') \dot\zeta(} {{\bf q}_2}
\bcontraction[1ex]{\dot\zeta({\bf p}_1,t') \dot\zeta({\bf p}_2,t')
\dot\zeta(} {{\bf p}_3} {,t') \zeta({\bf k}_1,t) \zeta({\bf k}_2,t) \zeta({\bf k}_3,t)
\zeta({\bf k}_4,t) \dot\zeta({\bf q}_1,t'') \dot\zeta({\bf q}_2,t'')
\dot\zeta(} {{\bf q}_3}
\dot\zeta({\bf p}_1,t') \dot\zeta({\bf p}_2,t')
\dot\zeta({\bf p}_3,t') \zeta({\bf k}_1,t) \zeta({\bf k}_2,t) \zeta({\bf k}_3,t)
\zeta({\bf k}_4,t) \dot\zeta({\bf q}_1,t'') \dot\zeta({\bf q}_2,t'')
\dot\zeta({\bf q}_3,t'')
\nonumber\\
&=&
[\dot\zeta^+({\bf p}_1,t'), \zeta^-({\bf k}_1,t)]
[\dot\zeta^+({\bf p}_2,t'), \zeta^-({\bf k}_2,t)]
[\zeta^+({\bf k}_3,t), \dot\zeta^-({\bf q}_1,t'')]
[\zeta^+({\bf k}_4,t), \dot\zeta^-({\bf q}_2,t'')]
\nonumber \\
&&[\dot\zeta^+({\bf p}_3,t'), \dot\zeta^-({\bf q}_3,t'')] ~.
\end{eqnarray}
There are three ways of picking two of the three ${\bf p}_i$'s (${\bf q}_i$'s),
so we have a symmetry factor 9. Also, there are 24 permutations of the
${\bf k}_i$'s.
The overall contribution
to the correlation function is
\footnote{The integrations
are conveniently done in terms of the conformal time $\tau$.
Integrals such as
$\int_{-\infty}^0 dx~ x^2 e^{\pm ix} = \pm 2i$ are constantly used
in the evaluation in this paper. As in \cite{Maldacena:2002vr}, the
convergence at $x\to -\infty$ is achieved by $x\to x(1\mp
i\varepsilon)$.}
\begin{eqnarray}
&& 9\cdot 4\frac{\lambda^2}{H^6} ~u_{k_1}^*(t)
u_{k_2}^*(t) u_{k_3}(t) u_{k_4}(t)
\nonumber \\
&\times& \left[ \int_{t_0}^t dt' \int_{t_0}^t dt'' a^3(t') a^3(t'')
\dot u_{k_1}(t') \dot u_{k_2}(t') \dot u_{k_3}^*(t'') \dot
u_{k_4}^*(t'') \dot u_{k_{12}}(t') \dot u_{k_{12}}^*(t'') \right]
\nonumber \\
&\times& (2\pi)^3 \delta^3(\sum_{i=1}^4 {\bf k}_i) + {\rm 23~ perm.}
\nonumber \\
&=& \frac{9}{8} \left( \frac{\lambda}{\Sigma} \right)^2
\frac{k_{12}}{k_1k_2k_3k_4} \frac{1}{(k_1+k_2+k_{12})^3}
\frac{1}{(k_3+k_4+k_{12})^3}
\nonumber \\
&\times& (2\pi)^9 P_\zeta^3 \delta^3(\sum_{i=1}^4 {\bf k}_i) + {\rm 23~
perm.} ~,
\label{aa1}
\end{eqnarray}
where
\begin{eqnarray}
{\bf k}_{12} = {\bf k}_1 + {\bf k}_2 ~.
\end{eqnarray}
The 2nd and 3rd term in (\ref{4pt3terms}) has a time-ordered double
integration, and so is more complicated. Their integrands are complex
conjugate to each other, and we get
\begin{eqnarray}
&& 2\cdot \frac{9}{8}
\left( \frac{\lambda}{\Sigma} \right)^2 \frac{k_{12}}{k_1k_2k_3k_4}
\frac{1}{(k_1+k_2+k_{12})^3}
\nonumber \\
&\times& \left[ 6 \frac{(k_1+k_2+k_{12})^2}{K^5} +
3\frac{k_1+k_2+k_{12}}{K^4} +
\frac{1}{K^3} \right]
\nonumber \\
&\times& (2\pi)^9 P_\zeta^3 \delta^3(\sum_{i=1}^4 {\bf k}_i) + {\rm 23~
perm.} ~,
\label{aa2}
\end{eqnarray}
where
\begin{eqnarray} K=k_1+k_2+k_3+k_4 ~. \end{eqnarray}
The other terms are similarly computed. We leave the details to
Appendix \ref{AppSEDetail}.
\subsection{Contact-interaction diagram}
\label{SecCI}
In this subsection, we compute the contact-interaction diagram,
Fig.~\ref{Fdiagrams} (A).
We define
\begin{eqnarray}
\mu \equiv \frac{1}{2} X^2 P_{,XX} + 2X^3 P_{,XXX} + \frac{2}{3} X^4
P_{,XXXX} ~.
\end{eqnarray}
The fourth order expansion is \cite{Huang:2006eh}
\begin{eqnarray}
{\cal L}_4 = a^3 \frac{\mu}{H^4} \dot \zeta^4 - \frac{a}{H^4}
\left(3\lambda-\Sigma(1-c_s^2)\right) (\partial \zeta)^2 \dot
\zeta^2
+ \frac{1}{4aH^4} \Sigma (1-c_s^2) (\partial \zeta)^4 ~.
\label{CL4}
\end{eqnarray}
Generally speaking, the Lagrangian of the form
\begin{eqnarray}
{\cal L}_2 &=& f_0 \dot \zeta^2 + j_2 ~,
\\
{\cal L}_3 &=& g_0 \dot \zeta^3 + g_1 \dot \zeta^2 + g_2 \dot \zeta + j_3
~,
\\
{\cal L}_4 &=& h_0 \dot \zeta^4 + h_1 \dot \zeta^3 + h_2 \dot \zeta^2 + h_3
\dot \zeta + j_4 ~
\end{eqnarray}
gives the following interaction Hamiltonian at the fourth order in
$\dot \zeta_I$ \cite{Huang:2006eh},
\begin{eqnarray}
{\cal H}_4^{I} &=& \left( \frac{9g_0^2}{4f_0} - h_0 \right) \dot\zeta_I^4
+ \left( \frac{3g_0g_1}{f_0} - h_1 \right) \dot \zeta_I^3
\cr
&+& \left( \frac{3g_0g_2}{2f_0} + \frac{g_1^2}{f_0} -
h_2 \right) \dot\zeta_I^2
+ \left(\frac{g_1g_2}{f_0} - h_3 \right) \dot \zeta_I
+ \frac{g_2^2}{4f_0} - j_4 ~,
\end{eqnarray}
where $f$, $g$, $h$ and $j$'s are functions of $\zeta$, $\partial_i
\zeta$ and $t$, and the subscripts denote the orders of $\zeta$.
So for (\ref{CL4}) we have
\begin{eqnarray}
{\cal H}_4^{I} &=& \frac{a^3}{H^4} (-\mu + 9\frac{\lambda^2}{\Sigma})
\dot\zeta_I^4 + \frac{a}{H^4} \left( 3\lambda c_s^2 - \Sigma (1-c_s^2)
\right) (\partial \zeta_I)^2 \dot\zeta_I^2
\cr
&+& \frac{1}{4aH^4} \Sigma (-c_s^2 + c_s^4) (\partial \zeta_I)^4 ~.
\label{CH4}
\end{eqnarray}
Note that in the second term the order $\lambda$ term
cancelled, in the third term the order $\Sigma$ term
cancelled.
The following are the contributions to the form factor ${\cal T}$ defined
as
\begin{eqnarray}
\langle \zeta^4 \rangle =
(2\pi)^9 P_\zeta^3 \delta^3 (\sum_{i=1}^4 {\bf k}_i)
~\prod_{i=1}^4 \frac{1}{k_i^3} ~{\cal T} ~.
\end{eqnarray}
The contribution from the first term in (\ref{CH4}) is
\begin{eqnarray}
36 \left( \frac{\mu}{\Sigma} - \frac{9 \lambda^2}{\Sigma^2}
\right)
\frac{\prod_{i=1}^4 k_i^2}{K^5} ~;
\label{CT_c1}
\end{eqnarray}
from the second term,
\begin{eqnarray}
-\frac{1}{8} \left(\frac{3\lambda}{\Sigma} -
\frac{1}{c_s^2} +1 \right)
\frac{k_1^2k_2^2 ({\bf k}_3 \cdot {\bf k}_4)}{K^3}
\left[ 1+ \frac{3(k_3+k_4)}{K} + \frac{12k_3k_4}{K^2} \right]
+ {\rm 23~perm.} ~;
\end{eqnarray}
from the third term,
\begin{eqnarray}
&&\frac{1}{32} \left( \frac{1}{c_s^2} -1 \right)
\frac{({\bf k}_1 \cdot {\bf k}_2)({\bf k}_3 \cdot {\bf k}_4)}{K}
\left[ 1+ \frac{\sum_{i<j} k_i k_j}{K^2}
+ \frac{3k_1k_2k_3k_4}{K^3} (\sum_{i=1}^4 \frac{1}{k_i} )
+ 12 \frac{k_1k_2k_3k_4}{K^4} \right]
\cr
&&+ ~{\rm 23 ~ perm.} ~.
\label{CT_c3}
\end{eqnarray}
\subsection{Summary of final results}
Here we summarize the final results from Sec.~\ref{SecSE}, \ref{SecCI}
and Appendix \ref{AppSEDetail}.
For the general single field inflation
${\cal L}({\phi,X})$, we define
\begin{eqnarray}
c_s^2 &\equiv& \frac{P_{,X}}{P_{,X}+2X P_{,XX}} ~,
\cr
\Sigma &\equiv& X P_{,X} + 2X^2 P_{,XX} ~,
\cr
\lambda &\equiv& X^2 P_{,XX} + \frac{2}{3} X^3 P_{,XXX} ~,
\cr
\mu &\equiv& \frac{1}{2} X^2 P_{,XX} + 2X^3 P_{,XXX} + \frac{2}{3} X^4
P_{,XXXX} ~.
\end{eqnarray}
If any of $\mu/\Sigma$, $\lambda^2/\Sigma^2$,
$1/c_s^4 \gtrsim 1$, the single field inflation generates
a large primordial trispectrum, whose
leading terms are given by
\begin{eqnarray}
\langle \zeta^4 \rangle = (2\pi)^9
P_\zeta^3 \delta^3(\sum_{i=1}^4 {\bf k}_i)
\prod_{i=1}^4 \frac{1}{k_i^3} ~ {\cal T}(k_1,k_2,k_3,k_4,k_{12},k_{14}) ~,
\end{eqnarray}
where ${\cal T}$ has the following six components:
\begin{eqnarray}
{\cal T} &=&
\left( \frac{\lambda}{\Sigma} \right)^2 T_{s1}
+ \frac{\lambda}{\Sigma} \left( \frac{1}{c_s^2}-1 \right) T_{s2}
+ \left( \frac{1}{c_s^2}-1 \right)^2 T_{s3}
+ \left( \frac{\mu}{\Sigma} - \frac{9\lambda^2}{\Sigma^2} \right)
T_{c1}
\cr
&+&
\left(\frac{3\lambda}{\Sigma} - \frac{1}{c_s^2} +1 \right) T_{c2}
+ \left( \frac{1}{c_s^2} -1 \right) T_{c3} ~.
\label{shapesum}
\end{eqnarray}
The $T_{s1,s2,s3}$ are contributions from the scalar-exchange
diagrams and are given in Appendix~\ref{AppSEDetail}, $T_{c1,c2,c3}$ are
contributions from the contact-interaction diagram and are given by
(\ref{CT_c1})-(\ref{CT_c3}).
For the most interesting cases, where any of
$\mu/\Sigma$, $\lambda^2/\Sigma^2$,
$1/c_s^4 \gg 1$, the first four terms in (\ref{shapesum})
are the leading contributions. So we have four shapes determined by
three parameters,\footnote{More generally,
we have six shapes controlled by three parameters. However,
the second line of (\ref{shapesum}) are negligible unless
$\mu/\Sigma$, $\lambda^2/\Sigma^2$,
$1/c_s^4$ are all $\sim 1$, in which case the trispectra is only
marginally large, $\sim {\cal O}(1)$. Also note that, for $\mu/\Sigma$,
$\lambda^2/\Sigma^2$,
$1/c_s^4 \gg 1$, the second line of (\ref{shapesum}) does not capture all
the subleading contributions.} $\lambda/\Sigma$,
$1/c_s^2$ and $\mu/\Sigma$.
A large bispectrum necessarily implies a
large
trispectrum, because either $1/c_s^4$ or $(\lambda/\Sigma)^2$ is
large. But the reverse is not necessarily true. One can in principle
have a large $\mu/\Sigma$ but small $1/c_s^4$ and
$(\lambda/\Sigma)^2$.
To quantify the size (i.e., magnitude) of the non-Gaussianity for
each shape, we define the following estimator $t_{NL}$ for each
shape component,
\begin{eqnarray}
\langle \zeta^4 \rangle_{\rm component}
\xrightarrow[\rm limit]{\rm RT} (2\pi)^9 P_\zeta^3 \delta^3 (\sum_i
{\bf k}_i) \frac{1}{k^9} ~ t_{NL} ~, \label{tNLdef}
\end{eqnarray}
where the RT
limit stands for the regular tetrahedron limit
($k_1=k_2=k_3=k_4=k_{12}=k_{14}\equiv k$). The parameter $t_{NL}$ is
analogous to the $f_{NL}$ parameter for bispectra.
This definition applies to
both the cases of interest here, and the non-Gaussianities of the
local form that we will discuss shortly. Unlike the convention in
the bispectrum case where the normalization of $f_{NL}$ is chosen
according to the local form non-Gaussianity, here we conveniently
choose the normalization of $t_{NL}$ according to (\ref{tNLdef}).
This is because, for the trispectra, even the local form has two
different shapes.
The size of non-Gaussianity for
each shape in (\ref{shapesum}) is then given by
\begin{eqnarray}
t_{NL}^{s1} = 0.250 \left(\frac{\lambda}{\Sigma} \right)^2 ,
~~
t_{NL}^{s2} = 0.420 \frac{\lambda}{\Sigma}
\left(\frac{1}{c_2^2}-1\right) ,
~~
t_{NL}^{s3} = 0.305 \left( \frac{1}{c_s^2}-1 \right)^2 ,
\cr
t_{NL}^{c1} = 0.0352 \left(
\frac{\mu}{\Sigma}-\frac{9\lambda^2}{\Sigma^2} \right) ,
~~
t_{NL}^{c2} = 0.0508 \left(
\frac{3\lambda}{\Sigma}-\frac{1}{c_s^2}+1\right) ,
~~
t_{NL}^{c3} = 0.0503 \left(\frac{1}{c_s^2}-1\right) ~.
\end{eqnarray}
For comparison, let us also look at the trispectrum of the local
form. This is obtained from the ansatz in real space
\cite{Okamoto:2002ik,Kogo:2006kh},
\begin{eqnarray}
\zeta ({\bf x})
= \zeta_g + \frac{3}{5} f_{NL} \left( \zeta_g^2-\langle \zeta_g^2
\rangle \right) + \frac{9}{25} g_{NL} \left( \zeta_g^3 -3\langle
\zeta_g^2 \rangle \zeta_g \right) ~,
\end{eqnarray}
where $\zeta_g$ is Gaussian and the shifts in the 2nd and 3rd
terms are introduced to cancel the disconnected diagrams.
Such a form constantly arises in multi-field models, where
the large non-Gaussianities are converted from
isocurvature modes at super-horizon scales.
The resulting trispectrum is
\begin{eqnarray}
{\cal T} = f_{NL}^2 T_{loc1} + g_{NL} T_{loc2} ~.
\end{eqnarray}
The two shapes are
\begin{eqnarray}
T_{loc1} &=& \frac{9}{50} \left( \frac{k_1^3 k_2^3}{k_{13}^3} + {\rm
11~perm.} \right) ~,
\label{Tloc1}
\\
T_{loc2} &=&
\frac{27}{100} \sum_{i=1}^4 k_i^3 ~,
\label{Tloc2}
\end{eqnarray}
where the 11 permutations includes $k_{13} \to k_{14}$ and 6 choices of
picking two momenta such as $k_1$ and $k_2$.
The size of the trispectrum for each shape is
\begin{eqnarray}
t_{NL}^{loc1} = 2.16 f_{NL}^2 ~, ~~~
t_{NL}^{loc2} = 1.08 g_{NL} ~.
\end{eqnarray}
So again a large bispectrum implies a large trispectrum, but not
reversely.
\section{Shapes of trispectra}
\setcounter{equation}{0}
\label{shapesection}
In this section, we investigate the shape of the trispectra. We take
various limits of the shape functions $T_{s1}$, $T_{s2}$, $T_{s3}$ and
$T_{c1}$, and then compare among themselves, and with the local
shapes $T_{loc1}$ and $T_{loc2}$. We will summarize the main results
at the end of this section.
\begin{figure}
\center
\includegraphics[width=0.6\textwidth]{reseps/t.eps}
\caption{\label{tFig} This figure illustrates the tetrahedron we
consider. }
\end{figure}
Before the discussion of the shape functions, we note that the
arguments of the shape functions are six momenta
$k_1,k_2,k_3,k_4,k_{12},k_{14}$. In order for these momenta to form a
tetrahedron (as in Fig. \ref{tFig}), the following two conditions are
required:
Firstly, we define three angles at one vertex:
\begin{align}
\cos(\alpha) &= \frac{k_1^2+k_{14}^2-k_4^2}{2k_1k_{14}}~, \nonumber \\
\cos(\beta) &= \frac{k_2^2+k_{14}^2-k_3^2}{2k_2k_{14}}~, \nonumber \\
\cos(\gamma) &= \frac{k_1^2+k_{2}^2-k_{12}^2}{2k_1k_{2}}~.
\end{align}
These three angles should satisfy $\cos(\alpha-\beta)\geq \cos(\gamma)
\geq \cos(\alpha+\beta)$. This inequality is equivalent to
\begin{align}\label{cosineq}
1-\cos^2(\alpha)-\cos^2(\beta)-\cos^2(\gamma)+2\cos(\alpha)\cos(\beta)\cos(\gamma) \geq 0~.
\end{align}
Secondly, the four momenta should satisfy all the triangle
inequalities. We need
\begin{align}
\label{ineq}
k_1+k_4>k_{14}~,\quad k_1+k_2>k_{12}~,\quad
k_2+k_3>k_{14}~,\nonumber\\
k_1+k_{14}>k_4~,\quad k_1+k_{12}>k_2~,\quad
k_2+k_{14}>k_3~,\nonumber\\
k_4+k_{14}>k_1~,\quad k_2+k_{12}>k_1~,\quad
k_3+k_{14}>k_2~.
\end{align}
The last triangle inequality involving $(k_3,k_4,k_{12})$ is always
satisfied given Eq. \eqref{cosineq} and Eqs. \eqref{ineq}.
We also would like to mention a symmetry in our trispectrum.
As $k_1,
k_2,k_3,k_4$ are symmetric in our model,
we have
\begin{align}\label{symeq}
{\cal T}(k_1,k_2,k_3,k_4,k_{12},k_{14})= {\cal
T}(k_1,k_2,k_4,k_3,k_{12},k_{13}) = {\cal
T}(k_1,k_3,k_2,k_4,k_{13},k_{14}) ~,
\end{align}
and etc, where
\begin{eqnarray}
k_{13}\equiv |{\bf k}_1+{\bf
k}_3|=\sqrt{k_1^2+k_2^2+k_3^2+k_4^2-k_{12}^2-k_{14}^2} ~.
\label{k13Ex}
\end{eqnarray}
The first set of limits we would like to take is those involved in the
consistency relations. There are two known consistency relations for
the trispectra to satisfy.
Firstly, we discuss the consistency relation in the squeezed limit.
When one external momentum, say, $k_4$ goes to zero,
this mode can be treated as a classical background for the other
modes, and the
trispectrum should reduce to the product of a
power spectrum and a running of bispectrum
\cite{Seery:2006vu,Huang:2006eh,Li:2008gg}:
\begin{align}\label{consistency1}
\langle \zeta_{{\bf k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf
k}_3}\zeta_{{\bf k}_4} \rangle
\sim -
{\mathbb P}(k_4)
\frac{d}{d\ln a}\langle \zeta_{{\bf k}_1}\zeta_{{\bf
k}_2}\zeta_{{\bf k}_3} \rangle~,
\end{align}
where ${\mathbb P}(k)\equiv \frac{2\pi^2}{k^3} P_\zeta(k)$,
and $P_\zeta(k)$ is
the dimensionless power spectrum.
In our case, the leading order contribution to the trispectra scales
as $c_s^{-4}$, or $c_s^{-2}\lambda/\Sigma$, or $(\lambda/\Sigma)^2$,
or $\mu/\Sigma$.
However, RHS scales as $c_s^{-3}$, or $c_s^{-1}\lambda/\Sigma$. In
order that Eq. \eqref{consistency1} holds, $k_4^3\langle \zeta_{{\bf
k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf k}_3}\zeta_{{\bf k}_4} \rangle$
must vanish at the leading order in the $k_4\rightarrow 0$
limit. One can check that our result indeed vanish in this limit,
$T_{s1,2,3},T_{c1} \to {\cal O}(k_4^2)$.
Secondly, we check the folded limit, say $k_{12} \to 0$.
For the s-channel (in which the exchanged scalar carries the momentum
${\bf k}_{12}$),
the four-point function can be regarded as
a pair of two-point functions modulated by the same classical
background generated by
the long wave mode ${\bf k}_{12}$, and we have \cite{Seery:2008ax}
\begin{align}\label{consistency2}
\langle \zeta_{{\bf k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf
k}_3}\zeta_{{\bf k}_4} \rangle
\sim (n_s-1)^2 {\mathbb P}(k_1){\mathbb P}(k_3)\langle \zeta_{-{\bf
k}_{12}} \zeta_{{\bf k}_{12}}\rangle~.
\end{align}
Note that the RHS takes the same shape as (\ref{Tloc1}).
Again, RHS scales as $c_s^{-3}$ in our case.
So in the $k_{12}\rightarrow 0$
limit, we expect $k_{12}^3\langle \zeta_{{\bf
k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf k}_3}\zeta_{{\bf k}_4} \rangle$
to vanish for the s-channel.
For the t-, u- and the contact interaction channels, there are
neither
propagators that give rise to the pole behavior $1/k_{12}^3$ nor
inverse Laplacians in our Lagrangian. Therefore in our case
$k_{12}^3\langle \zeta_{{\bf
k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf k}_3}\zeta_{{\bf k}_4} \rangle
\to 0$
trivially for these channels.
One can check that our results indeed satisfy the condition.
In fact, we have
$\langle \zeta_{{\bf
k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf k}_3}\zeta_{{\bf k}_4} \rangle
\to {\cal O}(k_{12})$ for the s-channel, so the pole behavior at $k_{12}=0$
is cancelled more than enough to satisfy the condition. (Note that
summing over all channels gives $\langle \zeta^4 \rangle \to {\rm
constant}$.)
After checking the consistency relations, now we shall plot the
shape functions. To do so, we shall take various limits to reduce
the number of variables. We set the shape function to zero when the
momenta do not form a tetrahedron. We consider the following
cases:
\begin{enumerate}
\item Equilateral limit: $k_1=k_2=k_3=k_4$. In Fig. \ref{equilateral}, we
plot $T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$, $T_{loc1}$ and $T_{loc2}$ as
functions of $k_{12}/k_1$ and $k_{14}/k_1$. (We would like to
remind the reader that unlike the first four
shape functions, $T_{loc1}$ and $T_{loc2}$ are not obtained in our
model. We plot them for the purpose of comparison.)
One observes that
$T_{loc1}$ blows up at all boundaries. This feature can distinguish
our shape functions
from the local shape $T_{loc1}$ originated from the local
$f_{NL}$.
\item Folded limit: $k_{12}= 0$. (This limit is also related
to the parallelogram limit, ${\bf k}_1={\bf k}_3$, by the symmetry
(\ref{symeq}).)
In this limit, $k_1=k_2$
and $k_3=k_4$. We plot
$T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$ and $T_{loc2}$
as functions of $k_{4}/k_1$ and $k_{14}/k_1$ in
Fig. \ref{folded}.
(Note that $T_{loc1}$ blows up in this limit). We assumed $k_4<k_1$
without losing
generality. Note that $T_{loc2}$ does not vanish in the
$k_4\rightarrow 0$ limit. This can be used to distinguish our
shape functions from the local shape originated from $g_{NL}$.
\item Specialized planar limit: We take $k_1=k_3=k_{14}$, and
additionally the
tetrahedron to be a planar quadrangle. In this limit,
one can solve for $k_{12}$ from \eqref{cosineq}:
\begin{align}
k_{12}=\left[
k_1^2+\frac{k_2 k_4}{2 k_1^2}\left( k_2 k_4 \pm
\sqrt{(4k_1^2-k_2^2)(4k_1^2-k_4^2)} \right) \right]^{1/2}~.
\end{align}
The minus sign solution can be related to another plus sign solution
in the $k_1=k_2=k_{14}$ limit through a symmetry discussed in
Appendix~\ref{planarappendix}. We will only consider the plus sign
solution in our following discussion.
We plot the shape functions as functions
of $k_2/k_1$ and $k_4/k_1$ in Fig. \ref{specplanar}. These figures
illustrate two important distinctions between our shape functions
and the local form shape functions. At the $k_2 \rightarrow k_4$
limit, we have $k_{13} \to 0$, so $T_{loc1}$ blows up, while the
others are all finite. At the $k_2\rightarrow 0$ and $k_4\rightarrow
0$ boundaries, our shapes functions vanish as ${\cal O}(k_2^2)$ and
${\cal O}(k_4^2)$ respectively,
while $T_{loc1}$ and $T_{loc2}$ are non-vanishing.
\item Near the double-squeezed limit: we consider the case where
${k}_3={k}_4=k_{12}$ and the tetrahedron
is a planar quadrangle. We are interested in the behavior of the
shape functions as $k_3=k_4=k_{12} \to 0$, i.e.~as the planar
quadrangle is doubly squeezed.
In this case, Eq. \eqref{cosineq} takes the
equal sign. One can solve for $k_{2}$
from \eqref{cosineq}. The solution is presented in Eq.
\eqref{planark2}.
We plot $T_{s1}/(\prod_{i=1}^4 k_i)$, $T_{s2}/(\prod k_i)$,
$T_{s3}/(\prod k_i)$, $T_{c1}/(\prod k_i)$, $T_{loc1}/(\prod k_i)$
and $T_{loc2}/(\prod k_i)$ as
functions of $k_{12}/k_1$ and $k_{14}/k_1$ in
Fig. \ref{doublesqueeze}. To reduce the
range of the plot, we only show the figures partially with
$k_4<k_1$. Note that in this figure, we divided the
shape functions by $\prod k_i$ in order to have better distinction
between contact-interaction and scalar-exchange contributions.
Fig. \ref{doublesqueeze} shows simultaneously the three
differences among the four shapes $T_{s1}$ ($\sim T_{s2,3}$),
$T_{c1}$, $T_{loc1}$ and $T_{loc2}$. 1) In the double-squeezed
limit, $k_3=k_4\rightarrow
0$, the scalar-exchange contributions $T_{s1}/(\prod k_i)$,
$T_{s2}/(\prod k_i)$, $T_{s3}/(\prod
k_i)$ are nonzero and finite, and the contact-interaction
$T_{c1}/(\prod k_i)$ vanishes.
As a comparison, the local form terms $T_{loc1}/(\prod k_i)$ and
$T_{loc2}/(\prod k_i)$ blow up.
2) In the folded limits, at the $(k_4/k_1=1,k_{14}/k_1=0)$ corner where
$k_{14}\to 0$, and close to the $(k_4/k_1=1,k_{14}/k_1=2)$ area where
$k_{13} \to 0$, $T_{loc1}/(\prod k_i)$ blows up. 3) In the squeezed
limit, at
$(k_4/k_1=1,k_{14}/k_1=1)$ where $k_2\to 0$, the $T_{loc1}/(\prod
k_i)$ and
$T_{loc2}/(\prod k_i)$ blow up. The last two behaviors have also
appeared in the previous figures.
\end{enumerate}
\begin{figure}
\center
\includegraphics[width=0.43\textwidth]{reseps/EqCof1.eps}
\hspace{0.02\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/EqCof2.eps}
\includegraphics[width=0.4\textwidth]{reseps/EqCof3.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/EqCof4.eps}
\includegraphics[width=0.4\textwidth]{reseps/EqLoc1.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/EqLoc2.eps}
\caption{\label{equilateral} In this group of figures, we consider
the equilateral limit $k_1=k_2=k_3=k_4$, and plot $T_{s1}$,
$T_{s2}$, $T_{s3}$, $T_{c1}$, $T_{loc1}$ and $T_{loc2}$,
respectively, as
functions of $k_{12}/k_1$ and
$k_{14}/k_1$. Note that
$T_{loc1}$ blows up when $k_{12}\ll k_1$ and $k_{14} \ll
k_1$. $T_{loc1}$ also blows up in the other boundary, because this
boundary corresponds to $k_{13}\ll k_1$. So $T_{loc1}$ is
distinguishable from all other shapes in this limit. We also note
that $T_{c1}$ and $T_{loc2}$ are both independent of $k_{12}$ and $k_{14}$.}
\end{figure}
\begin{figure}
\center
\includegraphics[width=0.43\textwidth]{reseps/k12eq0Cof1.eps}
\hspace{0.02\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k12eq0Cof2.eps}
\includegraphics[width=0.4\textwidth]{reseps/k12eq0Cof3.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k12eq0Cof4.eps}
\includegraphics[width=0.4\textwidth]{reseps/k12eq0Loc2.eps}
\caption{\label{folded} In this group of figures, we consider
the folded limit $k_{12}=0$, and plot
$T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$ and $T_{loc2}$,
respectively, as
functions of $k_{14}/k_1$ and
$k_{4}/k_1$. $T_{loc1}$
blows up in this limit. Note that when $k_4\rightarrow 0$, all
shape functions except $T_{loc1}$ and $T_{loc2}$ vanish.}
\end{figure}
\begin{figure}
\center
\includegraphics[width=0.41\textwidth]{reseps/k1eqk4eqk13Cof1.eps}
\hspace{0.04\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13Cof2.eps}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13Cof3.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13Cof4.eps}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13Loc1.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13Loc2.eps}
\caption{\label{specplanar} In this group of figures, we consider
the specialized planar limit with $k_1=k_3=k_{14}$, and plot
$T_{s1}$, $T_{s2}$, $T_{s3}$, $T_{c1}$, $T_{loc1}$ and
$T_{loc2}$, respectively, as
functions of $k_{2}/k_1$ and
$k_{4}/k_1$. Again, in
the $k_2\rightarrow 0$ or $k_4\rightarrow 0$ limit, our shape
functions vanish as ${\cal O}(k_2^2)$ and ${\cal O}(k_4^2)$ respectively.
This is different from that of the local
shape.
$T_{loc1}$ blows up when $k_2 \rightarrow k_4$. This is because in
this limit, $k_{13}\rightarrow 0$. }
\end{figure}
\begin{figure}
\center
\includegraphics[width=0.48\textwidth]{reseps/dsqs1.eps}
\hspace{0.\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/dsqs2.eps}
\includegraphics[width=0.4\textwidth]{reseps/dsqs3.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/dsqc1.eps}
\includegraphics[width=0.4\textwidth]{reseps/dsqloc1.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/dsqloc2.eps}
\caption{\label{doublesqueeze} In this group of figures, we look
at the shapes near
the double squeezed limit: we consider the case where
${k}_3={k}_4=k_{12}$ and
the tetrahedron
is a planar quadrangle. We plot
$T_{s1}/(\prod_{i=1}^4 k_i)$,
$T_{s2}/(\prod k_i)$, $T_{s3}/(\prod k_i)$, $T_{c1}/(\prod k_i)$, $T_{loc1}/(\prod k_i)$ and $T_{loc2}/(\prod k_i)$,
respectively, as
functions of $k_{12}/k_1$ and
$k_{14}/k_1$. Note that, taking the double-squeezed limit $k_4\rightarrow
0$, the scalar-exchange contributions $T_{s1}/(\prod k_i)$, $T_{s2}/(\prod k_i)$, $T_{s3}/(\prod
k_i)$ are nonzero and finite, and the contact-interaction
$T_{c1}/(\prod k_i)$ vanishes.
As a comparison, the local form terms $T_{loc1}/(\prod k_i)$ and $T_{loc2}/(\prod
k_i)$ blow up. The different behaviors in the folded and squeezed
limit can also been seen from this figure (see the main text for details).}
\end{figure}
In the second, third and fourth limits, the tetrahedron reduce to a
planar quadrangle. We collectively denote this group of limits as the
planar limit. This planar limit is
of special importance, because one of the most important ways to
probe trispectrum is the small (angular) scale CMB experiments.
These experiments directly measure signals contributed mainly from the planar quadrangles. The
more general plot for the planar limit is presented in
Appendix \ref{planarappendix}.
We can see that while very different from the two local shapes, the
three shapes $T_{s1}$, $T_{s2}$ and $T_{s3}$ are overall similar. Of
course like in the bispectrum case, we can tune the parameters to
subtract out the similarities and form new bases for the shapes.
We end this section by emphasizing a couple of important points:
\begin{itemize}
\item {\em The equilateral
trispectra forms}:
The scalar-exchange contributions $T_{s1,2,3}$
and the contact-interaction contribution $T_{c1}$
are similar at most regions, but
can be distinguished in the double-squeezed limit (e.g.~$k_3=k_4\to
0$), where the two
kinds of forms approach
zero at different speeds, $T_{s1,2,3} \to {\cal O}(k_3^2)$,
$T_{c1}\to{\cal O}(k_3^4)$.
Within the scalar-exchange contributions, the three shapes $T_{s1}$,
$T_{s2}$, $T_{s3}$ are very similar overall, having only small
differences.\footnote{For example, in Fig.~\ref{equilateral} or
\ref{folded}, if
we look at the double folded limit, $k_{12} \to 0$ and $k_{14} \to
0$, $T_{s2}$ and $T_{s3}$ go from positive to negative ($T_{s2} \to
-0.066$ and $T_{s3} \to -0.030$), while $T_{s1}$ remains positive
($T_{s1} \to 0.092$).}
For the purpose of data analyses, one can then use the following two
representative forms for the ``equilateral
trispectra''. One is $T_{c1}$, given in (\ref{CT_c1}). This ansatz
can be used to represent all four leading shapes at most
regions. For a refined data analysis, for example to distinguish
shapes in the double-squeezed limit, one can add another form
$T_{s1}$, given in (\ref{aa1_1}) and (\ref{aa23_1}). This ansatz
represents very well the three scalar-exchange contributions
$T_{s1,2,3}$.
The first ansatz is factorizable (in terms of the six variables
$k_{1,2,3,4},
k_{12}, k_{14}$) by introducing an integral $1/K^n=
(1/\Gamma(n)) \int_0^{\infty} t^{n-1} e^{-Kt}$
\cite{Smith:2006ud}; while the second ansatz cannot be easily
factorized due to the presence of $k_{13}$ given by (\ref{k13Ex}).
\item {\em Distinguishing between the equilateral and local forms}:
In the following limits, the equilateral and local forms behave very
differently.
At the folded limit (e.g.~$k_{12}\to 0$), $T_{loc1}$ generically
blows up, while
the four equilateral shapes and $T_{loc2}$
approach constants. At the squeezed limit (e.g.~$k_4 \to 0$), the
four equilateral shapes all vanish as ${\cal O}(k_4^2)$,
while the local forms $T_{loc1}$
and $T_{loc2}$ do not.
\end{itemize}
\section{Examples}
\setcounter{equation}{0}
For DBI inflation
\cite{Silverstein:2003hf,Alishahiha:2004eh,Chen:2004gc,Chen:2005ad,Kecskemeti:2006cg,Shandera:2006ax},
$P = -f(\phi)^{-1} \sqrt{1-2X f(\phi)} + f(\phi)^{-1} -V(\phi)$,
\begin{eqnarray} c_s \ll 1 ~, ~~~~ \frac{\lambda}{\Sigma}=\frac{1}{2} \left(
\frac{1}{c_s^2}-1 \right) ~, ~~~~ \frac{\mu}{\Sigma}= \frac{1}{4}
\left( \frac{5}{c_2^2} -4 \right) \left( \frac{1}{c_s^2} -1 \right)
~. \end{eqnarray}
The dominant contribution come from the scalar-exchange terms
${\cal T}_{s1,s2,s3}$ and one contact-interaction term ${\cal T}_{c1}$, which
are of order $1/c_s^4$. The ${\cal T}_{c2}$ and ${\cal T}_{c3}$ are of order
$1/c_s^2$, so belong to subleading contributions.
Therefore the shape function defined in \eqref{shapesum} takes the
form
\begin{align}
{\cal T}^{\rm DBI}\approx &\left( \frac{T_{s1}}{4}+\frac{T_{s2}}{2}+T_{s3}-T_{c1}
\right)\frac{1}{c_s^4} ~.
\end{align}
According to the definition (\ref{tNLdef}), $t_{NL}^{\rm DBI} \approx
0.542/c_s^4$.
For k-inflation
\cite{Armendariz-Picon:1999rj,Li:2008qc,Engel:2008fu}, we look at
the example $P \sim (-X+X^2)/\phi^2$, \begin{eqnarray} c_s \ll 1 ~, ~~~~
\frac{\lambda}{\Sigma} = \frac{2X}{-1+6X} =\frac{1-c_s^2}{2} ~, ~~~~
\frac{\mu}{\Sigma} = \frac{X}{-1+6X} = \frac{1-c_s^2}{4} ~. \end{eqnarray}
The only dominant term is
one of the scalar-exchanging terms ${\cal T}_{s3}$, the others all belong
to subleading contributions.
So
\begin{align}
{\cal T}^{\rm K}\approx&\frac{T_{s3}}{c_s^4} ~,
\end{align}
and $t_{NL}^{\rm K} \approx 0.305/c_s^4$.
As mentioned in the previous section, there are some differences
among the shapes $T_{s1}$, $T_{s2}$ and
$T_{s3}$, and especially between them and $T_{c1}$. These differences
may be used to distinguish some special models within this class. But
unfortunately, for the above two examples,
after summing over all contributions for DBI inflation,
the trispectrum difference
between the DBI inflation and this specific k-inflation example
becomes smaller, and we do not find any
features that can very sharply distinguish them. This is because
the four leading shapes $T_{si}$ and $T_{c1}$ are similar in most regions and in the discriminating double-squeezed limit, the trispectra in
both examples take the form $T_{si}$ (which are similar among
themselves) since $T_{c1}$ vanishes faster.
\section{Non-Bunch-Davies vacuum}
\setcounter{equation}{0}
We now study the shape of trispectrum if the initial state of inflation deviates from the standard
Bunch-Davies vacuum of
de Sitter space. This is an interesting question
because
short distance physics may give rise to
such deviation \cite{EGKS,Danielsson}, and one might argue whether its effects on the power spectrum of the CMB are observable \cite{Kaloper}\footnote{The choice of initial state is often discussed in the context of
trans-Planckian effects \cite{Martin:2000bv} though the issue has more general applicability.
See
e.g.~\cite{Greene:2005aj}
for a review and references, and \cite{BoundaryEFT} for a discussion of how to capture the initial state effects in terms of a boundary effective field theory.}.
The effects of the non-Bunch-Davies vacuum on the bispectrum have been
studied in
\cite{Chen:2006nt,Holman:2007na,Meerburg:2009ys}, where it was found
that the non-Gaussianities are boosted in the folded triangle limit
(e.g.~$k_1+k_2-k_3 \sim 0$).
A general vacuum state for the fluctuation of the inflaton during inflation can be written as
\begin{eqnarray}
u_k = u(\textbf{k},\tau) = \frac{H}{\sqrt{4\epsilon c_s k^3}}
(C_{+}(1+i k c_s\tau )e^{-i k c_s\tau}+ C_{-} (1-i k c_s\tau) e^{i k
c_s \tau}) ~.
\label{nonBDuk}
\end{eqnarray}
Here a small and non-zero $C_{-}$ parametrizes a deviation from the
standard Bunch-Davies vacuum which has $C_{+}=1, C_{-}=0$. We
consider the corrections of a non-zero $C_{-}$ to the leading shapes
assuming $C_{-}$ is small, we keep only terms up to linear order in $C_{-}$.
Similar to the case of the bispectrum \cite{Chen:2006nt},
the first sub-leading corrections
come from replacing one of the $u(\tau,\textbf{k})$'s with their
$C_{-}$ components, and since the correction from $u(0,\textbf{k})$
only has the same shape as that of the Bunch-Davies vacuum, we only
need to consider the case that $u(\tau,\textbf{k})$ comes from the
interacting Hamiltonian, where $\tau$ is not zero. In the following
we discuss the contact-interaction diagram and the scalar-exchange
diagram respectively.
For the contact-interaction diagram, the corrections consist of four
terms from replacing $k_i$ with $-k_i$ in the shape for Bunch-Davies
vacuum. For the leading shape $T_{c1}$ we denote the
correction as $\tilde{T}_{c1}$ and we find
\begin{eqnarray} \label{nonBDc1}
\tilde{T}_{c1} &=& 36 ~{\rm Re}(C_{-}) \prod_{i=1}^4 k_i^2 \left[\frac{1}{(k_1+k_2+k_3-k_4)^5}+\frac{1}{(k_1+k_2-k_3+k_4)^5} \right.\nonumber \\
&& +\left.
\frac{1}{(k_1-k_2+k_3+k_4)^5}+\frac{1}{(-k_1+k_2+k_3+k_4)^5}\right]~.
\end{eqnarray}
Then let us consider scalar-exchange diagram. For illustration we
only consider the $T_{s1}$ term. The calculations for $T_{s2}$ and
$T_{s3}$ are similar but more complicated. Now we have six $u_k$
modes from the interaction Hamiltonian. Replacing each mode with
its $C_{-}$ component gives rise to six terms in the corrections.
They correspond to replacing $k_i$ with $-k_i$, or one of the two
$k_{12}$'s with $-k_{12}$ in the calculations for the Bunch-Davies
vacuum. The corrections to Eq.~(\ref{aa1_1}) are
\begin{eqnarray} \label{nonBDs11}
&& \frac{9~{\rm Re}(C_{-})}{8}
k_1^2k_2^2k_3^2k_4^2k_{12}\left\{\left[\sum _{k_i\rightarrow -k_i,
i=1}^4
\frac{1}{(k_3+k_4+k_{12})^3}\frac{1}{(k_1+k_2+k_{12})^3}\right]\right.
\nonumber \\
&&\left. + \frac{1}{(k_1+k_2-k_{12})^3}\frac{1}{(k_3+k_4+k_{12})^3}
+\frac{1}{(k_1+k_2+k_{12})^3}\frac{1}{(k_3+k_4-k_{12})^3} \right\}
\nonumber \\&& + \textrm{23 perm.}
\end{eqnarray}
In Eq.~(\ref{aa23_1}), the two $k_{12}$
cancelled in the final expression for the Bunch-Davies vacuum, so we
have to recover them in the calculations. Denoting
$M=k_3+k_4+k_{12}$ and $K=k_1+k_2+k_3+k_4$, we find the corrections
\begin{eqnarray} \label{nonBDs12}
&& \frac{9~{\rm Re}(C_{-})}{4} k_1^2k_2^2k_3^2k_4^2 k_{12}
\left\{\left[\sum _{k_i\rightarrow -k_i, i=1}^4
\frac{1}{M^3}\left(\frac{6M^2}{K^5}+\frac{3M}{K^4}+\frac{1}{K^3}\right)\right]\right.
\nonumber \\
&& +\frac{1}{(k_3+k_4-k_{12})^3}\left(\frac{6(k_3+k_4-k_{12})^2}{(K-2k_{12})^5}+\frac{3(k_3+k_4-k_{12})}{(K-2k_{12})^4}+\frac{1}{(K-2k_{12})^3}\right) \nonumber \\
&& \left.
+\frac{1}{M^3}\left(\frac{6M^2}{(K+2k_{12})^5}+\frac{3M}{(K+2k_{12})^4}+\frac{1}{(K+2k_{12})^3}\right)\right\}
+ \textrm{23 perm.}
\end{eqnarray}
To summarize, the correction $\tilde T_{s1}$ to
$T_{s1}$ is the summation of Eqs. \eqref{nonBDs11}
and \eqref{nonBDs12}.
\begin{figure}
\center
\includegraphics[width=0.4\textwidth]{reseps/EqNBDc.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/EqNBD1.eps}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13NBDc.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/k1eqk4eqk13NBD1.eps}
\includegraphics[width=0.4\textwidth]{reseps/dsqNBDc.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth]{reseps/dsqNBD1.eps}
\caption{\label{NBDplot} In this group of figures, we plot
the $\tilde T_{c1}/{\rm
Re}(C_-)$ (the left column) and $\tilde T_{s1}/{\rm Re}(C_-)$ (the
right column) in the equilateral limit, specialized planar limit,
and near double squeezed limit (we plot $\tilde T_{c1}/[{\rm
Re}(C_-)\Pi_i k_i]$ and $\tilde T_{s1}/[{\rm Re}(C_-)\Pi_i k_i]$ in
near double squeezed limit) respectively. Note that, in order to show
clearly the locations of the divergence, in some figures we have taken the
cutoffs of the z-axes to be extremely large.}
\end{figure}
We can look for the analogue of the folded triangle limit discovered
in the study of bispectrum where the corrections due to
deviation from the Bunch-Davies vacuum diverge. Here we see when any one
triangle, e.g., $(k_1,k_2,k_{12})$, in the momentum tetrahedron
becomes folded, the corrections to $T_{s1}$ (see (\ref{nonBDs11}) and (\ref{nonBDs12}))
become divergent. Furthermore, when $k_1+k_2+k_3-k_4=0$, the two
triangles $(k_1,k_2,k_{12})$ and $(k_3,k_4,k_{12})$ become folded
simultaneously, and the correction (\ref{nonBDc1}) to $T_{c1}$
also diverges. We will refer to such configurations as the folded
sub-triangle configurations.\footnote{Due to permutations, the three
momenta do not have to be next to each other in terms of
Fig.~\ref{tFig}, for example, $k_1,k_3,k_{13}$.}
As discussed in \cite{Chen:2006nt}, these divergences are
artificial, and do not correspond to real infinities in observables.
Rather, the divergences appear because it is not realistic to assume
a non-standard vacuum to exist in the infinite past. A cutoff on
momenta should be imposed at the same time when a non-Bunch-Davies
vacuum is considered.
We would like to point out two interesting aspects of the effects of
the non-Bunch-Davies vacuum
on trispectra, and more generally on higher point
functions.
Let us first look at the regions away from the folded sub-triangle
configurations.
In the regular tetrahedron limit, in terms of \eqref{tNLdef}, the
corresponding $t_{NL}$ for the non-Bunch-Davies contribution are
\begin{equation}
\tilde t_{NL}^{c1}=4.50 ~ {\rm
Re}(C_-)\left(\frac{\mu}{\Sigma}-\frac{9\lambda^2}{\Sigma^2}\right)~,\qquad
\tilde t_{NL}^{s1}=401 ~ {\rm
Re}(C_-)\left(\frac{\lambda}{\Sigma}\right)^2~.
\end{equation}
Note that $\tilde t_{NL}^{c1}/{\rm Re}(C_-)$ is 128 times larger
than $t_{NL}^{c1}$; $\tilde t_{NL}^{s1}/{\rm Re}(C_-)$ is about 1600
times larger than $t_{NL}^{s1}$. These large numbers arise because
some plus signs become minus signs in the denominators, and there are
also more terms to consider in the non-Bunch-Davies case. So in the
context of general single field inflation, even if ${\rm Re}(C_-)$
is as small as one part in one thousand, it becomes important
phenomenologically. Generalize this to higher point functions, we see
that, no matter how small the $C_-$ is,
there always exists a high point function beyond
which the contribution from the $C_-$ component becomes comparable to
that from the $C_+$ component. For such functions,
we should use the whole
wavefunction (\ref{nonBDuk}) to get the correct shapes instead of
treating the $C_-$ component as a correction, even away from the
folded sub-triangle limit.
Therefore generally speaking, higher
point function is a more sensitive probe to the
non-Bunch-Davies component. However, on the other hand,
higher point functions
contribute less to the total non-Gaussianities and will be more
difficult to measure experimentally.
We next look at the region near the folded sub-triangle limits.
Because the denominators here have larger
powers than those in 3pt, the corrections grow faster as we
approach the folded sub-triangle limit. This also indicates that
the trispectrum, and more generally higher point functions, is a
nice probe of the non-Bunch-Davies vacuum.
But, on the other hand, because the higher point function has a
larger momentum phase space (three more dimensions here in trispectra)
than the
3pt, the phase space for the folded limits becomes relatively
smaller.
It will be interesting to see how the two factors in each of the
above two
aspects play out in the data analyses.
In Fig. \ref{NBDplot}, we plot $\tilde T_{c1}/{\rm Re}(C_-)$ and
$\tilde T_{s1}/{\rm Re}(C_-)$ in the equilateral limit, specialized
planar limit, and near double squeezed limit ($\tilde T_{c1}/[{\rm
Re}(C_-)\Pi_i k_i]$ and $\tilde T_{s1}/[{\rm Re}(C_-)\Pi_i k_i]$ in
this case) respectively.
\section{Conclusion}
\setcounter{equation}{0}
To conclude, we have calculated the leading order trispectra for
general single field inflation. As in the case of bispectra, the
trispectra turns out to be of ``equilateral shape'' in general
single field inflation. Compared with the local shape trispectra,
the equilateral shape trispectra has not been extensively
investigated in the literature. It is clear that there are a lot of
work worthy to be done on this topic in the future. Directions for
future work include:
\begin{itemize}
\item It is useful to extend our calculation to more
general cases. We have focused here on the leading order
contribution and single field inflation. It is interesting to
generalize this calculation to next-to-leading order which may be
also potentially observable, or
multifield inflation \cite{XGao}. It is also useful to perform a unified
analysis for general single field inflation and slow roll
inflation.
\item The shape of the trispectra is much more complicated
compared with that of the bispectra. In our paper, we have
obtained a lot of features of the shape functions by taking various
limits. However, it is still a challenge to find improved
representations to understand the shape functions. For example,
one can find new bases for the shapes by tuning parameters to subtract out the
similarities. Also, we focus on the planar limit in plotting
the figures (except for Fig. \ref{equilateral}), because the planar
limit is of special importance for the CMB data analysis. It is
interesting to investigate the non-planar parameter region in more
details for the large scale structure and 21-cm line surveys.
\item In the discussion of non-Bunch-Davies vacuum, we have
calculated contributions from the two representative shapes.
However, as we have seen in Section 6, trispectra is a powerful
probe for non-Bunch-Davies vacuum. Even one part in $10^3$
deviation from the Bunch-Davies vacuum could lead to an order
one correction to the trispectra. So it is valuable to perform the
full calculation for the non-Bunch-Davies vacuum, and to study the
effects of the cutoff.
\item Most importantly, one would like to apply these shape
functions to data analyses and see how they are constrained.
\end{itemize}
\noindent {\it Note added:} On the day this work appeared on the
arXiv, the paper \cite{Arroja:2009pd} was also submitted to the
arXiv, which overlaps with our Sec.3 and 5.
\medskip
\section*{Acknowledgments}
We thank Bin Chen for his participation in the early stage of this
work. We thank Eiichiro Komatsu, Miao Li, Eugene Lim for helpful
discussions. XC and MH would like to thank the hospitality of the
organizers of the program ``Connecting fundamental physics with
observations'' and the KITPC, where this work was initiated.
GS would like to thank Kazuya Koyama for independently
pointing out to him the
importance of the scalar-exchange term.
XC was
supported by the US Department of Energy under cooperative research
agreement DEFG02-05ER41360. BH was supported in part by the Chinese
Academy of Sciences with Grant No. KJCX3-SYW-N2 and the NSFC with
Grant No. 10821504 and No. 10525060.
GS was supported in part by NSF CAREER Award No. PHY-0348093, DOE grant DE-FG-02-95ER40896, a Research Innovation Award and a Cottrell Scholar Award from Research
Corporation, a Vilas Associate Award from the University of Wisconsin, and a John Simon Guggenheim Memorial Foundation Fellowship. GS would also like to acknowledge support from the Ambrose Monell Foundation during his stay at the Institute for Advanced Study.
YW was supported in part by a NSFC grant No. 10535060/A050207, a NSFC group grant No. 10821504, and a 973 project grant No. 2007CB815401. |
2003.05021 | \section{Introduction}
In quantum theory,
partition functions or expectation values of observables are central objects.
For a Lagrangian field theory,
the path-integral provides these objects,
though how to integrate is obscure except for free theories.
The perturbative path-integral is a standard technique that enables us to treat interacting fields in terms of free theories.
In this paper,
we show explicitly that the perturbative path-integral can be regarded as a morphism of the (quantum) $A_{\infty }$ structure intrinsic to each quantum field theory.
Such a perspective provides simple explanations of some algebraic properties of the quantities based on the perturbative path-integral,
which will be useful for
calculating the scattering amplitudes,
deriving effective theories,
fixing gauge degrees,
studying flows of exact renormalization group and so on.
\vspace{2mm}
Homotopy algebras,
such as quantum $A_{\infty }$ or $L_{\infty }$,
arise naturally in the context of the ordinary Lagrangian description of quantum field theory.
As is known among experts,
they describe not only the gauge invariance of Lagrangian but also the Feynman graph expansion.
Thus,
theoretical physicists already know some of these structures,
albeit implicitly,
even for the theory without gauge degrees.\footnote{String field theories will be typical examples revealing these explicitly. }
The Batalin-Vilkovisky (BV) formalism makes these structures visible and provides the translation between Lagrangian field theories and homotopy algebras \cite{Zwiebach:1992ie}.
The BV formalism is one of the most powerful and general frameworks for quantization of gauge theories,
which is based on the homological perturbation \cite{Batalin:1981jr, Henneaux}.
For a given Lagrangian field theory,
we can defines a complex with an appropriate BV differential by solving the BV master equation,
which is one equivalent description of a quantum $A_{\infty }$ algebra \cite{Barannikov:2010np, Doubek:2013dpa}.
This $A_{\infty }$ algebra reduces to an $L_{\infty }$ algebra whenever multiplications of space-time fields are graded commutative.
Since the BV formalism assigns a homotopy algebra to each quantum field theory,
we can extract the intrinsic $A_{\infty }$ structure explicitly by casting the BV master action into the homotopy Maurer-Cartan form \cite{Schwarz:1992nx, Alexandrov:1995kv, Jurco:2018sby, Macrelli:2019afx, Jurco:2019yfd}.\footnote{The original action is recovered by the homotopy Maurer-Cartan action by setting antifields to zero,
whose $A_{\infty }$ structure is just a piece of the full (quantum) $A_{\infty }$ structure of the BV master action. }
\vspace{2mm}
As is well-known,
in the BV formalism,
any effective action also solves the BV master equation.
This fact implies that the path-integral can be understood as a morphism of the BV differential.
Since any solution of the BV master equation is in one-to-one correspondence with a quantum $A_{\infty }$ structure,
the path-integral preserves this intrinsic $A_{\infty }$ structure of quantum field theory.
Although these properties may valid for non-perturbative path-integral,
in this paper,
we consider the perturbative path-integral.
We first show that the perturbative path-integral can be performed as a result of the homological perturbation for the intrinsic $A_{\infty }$ structure and thus it gives a morphism of this (quantum) $A_{\infty }$ structure in any BV-quantizable quantum field theory.
Then,
we apply these ideas to string field theory and consider some quantities based on the perturbative path-integral of string fields.
As a result of the homological perturbation,
we derive effective theory with finite $\alpha '$ \cite{Sen:2016qap },
the Light-cone reduction \cite{Matsunaga:2019fnc, EM},
and string S-matrix in a simple way.
In addition to these (re-)derivations,
we explain that this approach may enable us to use unconventional pieces of perturbative calculus.
We discuss the open string $S$-matrix based on unconventional propagators whose $4$-point amplitude reproduces the gauge invariant quantity given by \cite{Masuda:2019rgv} directly.
\vspace{2mm}
This paper is organized as follows.
In section 2,
after giving a brief review of the BV formalism and its relation to the quantum $A_{\infty }$ structure,
we show explicitly that the homological perturbation indeed performs the perturbative path-integral.
This result would be known among experts except for incidental details.
The quantum $A_{\infty }$ structure of effective theory and the classical limit are also discussed.
In section 3,
we translate the results based on the BV formalism into corresponding results based on the (quantum) $A_{\infty }$ structure.
We show that when the original BV master action includes source terms,
its effective theory must have a twisted $A_{\infty }$ structure.
In section 4,
we apply these results to string field theory.
We (re-)derive effective theories with finite $\alpha '$,
reduction of gauge degrees,
string amplitudes,
and gauge invariant quantities discussed in \cite{Masuda:2019rgv} in a simple way.
In section 5,
we conclude with summary and mentioning earlier works.
In appendix,
we explain two natural gradings of our $A_{\infty }$ structure and corresponding bases of the symplectic pairing in the BV formalism.
In the rest of this section,
we summarize basic facts of the perturbative path-integral and the relation between $A_{\infty }$ and $L_{\infty }$.
\subsection{Perturbative path-integral}
A classical action $S_{\mathrm{cl}} [\psi _{\mathrm{cl}} ] = S_{\mathrm{cl \, free}} [\psi _{\mathrm{cl}} ] + S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl}} ] $ is a functional of classical fields $\psi _{\mathrm{cl}} $.
The action consists of the kinetic term $S_{\mathrm{cl \, free}} [\psi _{\mathrm{cl}} ]$ and the interacting terms $S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl}} ]$,
which we write
\begin{align}
\label{classical physical action}
S_{\mathrm{cl \, free}} [\psi _{\mathrm{cl}} ]
\equiv - \frac{1}{2} \int dx \,
\psi _{\mathrm{cl}} \, \mu ^{\mathrm{cl}}_{1} ( \psi _{\mathrm{cl}} )
\, , \hspace{5mm}
S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl}} ]
\equiv - \sum_{n} \frac{1}{n+1} \int dx \,
\psi _{\mathrm{cl}} \, \mu ^{\mathrm{cl}}_{n} ( \underbrace{\psi _{\mathrm{cl}} , ... , \psi _{\mathrm{cl}} }_{n} ) \, .
\end{align}
In this section,
we assume that (\ref{classical physical action}) consists of the physical degrees only for simplicity.
In a Lagrangian field theory,
the expectation value of observables $\langle ... \rangle _{J}$ is described by using the path-integral of fields $\psi _{\mathrm{cl}} $ as follows
\begin{align}
\label{non-perturbative}
Z_{J}^{-1}
\int \mathcal{D} [\psi _{\mathrm{cl}} ] \, \big{(} \, ... \, \big{)} \, e^{S[\psi _{\mathrm{cl}} ] + J\psi _{\mathrm{cl}} } \, ,
\hspace{5mm}
Z_{J} = \int \cD [\psi _{\mathrm{cl}} ] \, e^{S [ \psi _{\mathrm{cl}} ] + J \psi _{\mathrm{cl}} } \, ,
\end{align}
where $Z_{J}$ denotes the partition function.\footnote{We set $\hbar = 1$ for convenience.
If necessary,
we write $\hbar $ explicitly,
such as $\exp ( \hbar ^{-1} S )$. }
Although the non-perturbative path-integral of interacting fields is a deep question,
we can perform it for free theories since free actions are at most quadratic.
When the free theory $S_{\mathrm{cl \, free} } [\psi _{\mathrm{cl}} ]$ is solved and the value of $\sqrt{\mathrm{det}\, (\mu ^{\mathrm{cl}}_{1})^{-1} }$ is given,
the path-integral integral of free fields is performed as a Gaussian integral and is normalized as
\begin{align}
\label{Gaussian}
1 = \int \cD [\psi _{\mathrm{cl}} ] \, e^{S_{\mathrm{cl \, free}} [ \psi _{\mathrm{cl}} ] } \, .
\end{align}
The perturbative expansion enables us to perform the path-integral of interacting fields formally,
which is a standard procedure in a Lagrangian field theory.
In terms of the free theory,
which should be well solved,
we can represent (\ref{non-perturbative}) as the following expectation values
\begin{align}
\label{free rep}
\big{\langle } \, ... \,
e^{S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl} }] } \, ... \, \big{\rangle } _{\mathrm{free}, \, J}
= Z_{J}^{-1}
\int \mathcal{D} [\psi _{\mathrm{cl}} ] \, \big{(} \, ... \,
e^{S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl} } ]} \, ... \, \big{)} \, e^{S_{\mathrm{cl \, free}} [\psi _{\mathrm{cl}} ] + J\psi _{\mathrm{cl}} }
\, .
\end{align}
The partition function $Z_{J}$ can be represented as $Z_{J} = \langle \, e^{S_{\mathrm{cl \, int}} [\psi _{\mathrm{cl}} ] } \, \rangle _{\mathrm{free}, \, J}$.
We then consider to \textit{replace} a given functional of $\psi _{\mathrm{cl}} $ by using a formal power series of $\psi _{\mathrm{cl} }$,
for which we write $F [\psi _{\mathrm{cl}} ] $,
and \textit{replace} the expectation value of a given functional by $\langle \, F[\psi _{\mathrm{cl} } ] \, \rangle _{\mathrm{free}, \, J}$ formally.
This type of integral reduces to the Gaussian integral (\ref{Gaussian}) because of $F [\psi ] \, e^{J \psi } = F[\partial _{J} ] \, e^{J \psi }$.
Hence,
whenever the free theory is well solved,
we can perform the \textit{perturbative} path-integral of (\ref{free rep}) as follows
\begin{align}
\label{J rep}
\big{\langle } \, ... \, F' [ \psi _{\mathrm{cl}} ] \, ...\, \big{\rangle } _{\mathrm{free} , \, J}
& \equiv \Big{(} \, ...\, F' [ \, \partial _{J} ] \, ... \,
\Big{)} \,
e^{\frac{1}{2} J \, \mu _{1}^{-1} \, J } \, ,
\end{align}
where $F' [\psi _{\mathrm{cl}} ] $ is a formal power series of $\psi _{\mathrm{cl} }$.
The expectation value $\langle ... \rangle _{J} \equiv \langle ... e^{S_{\mathrm{int} } [\psi _{\mathrm{cl} } ] } ... \rangle _{\mathrm{free}, \, J}$ is always defined by the perturbative path-integral (\ref{J rep}) in the rest of this paper.
The Feynman graph expansion of $F [\psi _{\mathrm{cl} }]$ is an alternative representation of (\ref{J rep}) with $F' [\psi _{\mathrm{cl}} ] = F[\psi _{\mathrm{cl} }] \, e^{S_{\mathrm{int} } [\psi _{\mathrm{cl} } ] }$.
As is well-known,
by adding the source term $e^{J \, \psi _{\mathrm{ev} }}$,
(\ref{J rep}) can be cast as
\begin{align}
\label{Feynman rep}
\big{\langle } \, ... \, F[ \psi _{\mathrm{cl}} ] \, ...\, \big{\rangle } _{J}
\equiv e^{ \frac{1}{2} \partial _{\psi _{\mathrm{ev} } } \mu _{1}^{-1} \partial _{\psi _{\mathrm{ev} }} }
\Big{[} ( \, ... \, F [\psi _{\mathrm{ev} } ] \, ... \, e^{S_{\mathrm{cl \, int} } [\psi _{\mathrm{ev} } ] } \, ) \,
e^{ J \psi _{\mathrm{ev} } } \, \Big{]} _{\psi _{\mathrm{ev} } = 0 } \, .
\end{align}
In this paper,
the words ``perturbative path-integral'' always mean (\ref{J rep}) or (\ref{Feynman rep}).
Note that fields $\psi _{\mathrm{cl} }$ are integrated by using (\ref{Gaussian}) in both representations (\ref{J rep}) and (\ref{Feynman rep}).
\subsection{(Quantum) $A_{\infty }$ reduces to (quantum) $L_{\infty }$}
Let $\mathcal{\widehat{H}}$ be the state space of fields and $\mathcal{T} (\mathcal{\widehat{H}} )$ be the tensor algebra.
In this paper,
we regard products or multiplications of space-time fields $\psi _{1}, ... , \psi _{n}$ as values of multilinear maps $\mu _{n}$ acting on the tensor product $\psi _{1} \otimes \cdots \otimes \psi _{n}$,
which may be non-commutative in general.
We write
\begin{align}
\label{multilinear mu}
\mu _{n} ( \psi _{1} , ... , \psi _{n} )
\equiv \mu _{n} \, ( \psi _{1} \otimes \cdots \otimes \psi _{n} ) \, .
\end{align}
For instance,
the vertices appearing in the action (\ref{classical physical action}) can be specified by these multilinear maps.
In most cases,
these multilinear maps acting on fields associate with algebraic relations,
such as coupling constant,
delta functions of momentum conservation,
contractions of indices,
cut-off functions,
space-time differentials or structure constants of Lie algebras.
The $A_{\infty }$ or $L_{\infty }$ structure we consider in this paper is a special combination of such algebraic relations,\footnote{As we see, the BV formalism uniquely specifies such a set of algebraic relations for a give field theory. }
$\mu = \{ \mu _{n} \}_{n}$,
that can be identified with properties of multilinear maps $\{ \mu _{n} \}_{n}$ acting on the tensor algebra.
\vspace{2mm}
When we consider ordinary quantum field theory,\footnote{A quantum field theory of particles that is not based on the non-commutative geometry. }
because of the graded commutativity of fields $\psi _{1} \cdot \psi _{2} = (-)^{\psi _{1} \psi _{2}} \psi _{2} \cdot \psi _{1}$,
it is economical to consider the symmetric tensor algebra $\mathcal{S} (\mathcal{\widehat{H}} )$ instead of $\mathcal{T} (\mathcal{\widehat{H}})$.
The symmetrized tensor product,
\begin{align}
\label{symmetrized tensor product}
\psi _{1} \wedge \cdots \wedge \psi _{n} \equiv \sum_{\sigma \in \mathbb{S} } (-)^{\sigma (\psi ) }
\psi _{\sigma (1) } \otimes \cdots \otimes \psi _{\sigma (n) } \, ,
\end{align}
is a natural product of the symmetric tensor algebra $\mathcal{S} (\widehat{\mathcal{H}} )$.
Then,
instead of (\ref{multilinear mu}),
it is reasonable to consider the values of multilinear maps $\{ \mu _{n} \}_{n}$ acting on the symmetrized tensor product
\begin{align}
\mu ^{\mathsf{sym}}_{n} (\psi _{1} , ... , \psi _{n} ) \equiv \mu _{n} (\psi _{1} \wedge \cdots \wedge \psi _{n} ) \, .
\end{align}
This kind of $\mu ^{\mathsf{sym} }(\psi , ... , \psi )$ gives natural algebraic structures appearing in \textit{commutative} quantum field theory.
Note that $\mu$ and $\mu ^{\mathsf{sym}}$ are distinguished by just input states:
the commutativity of space-time fields can be identified with a property of inputs.
When we consider \textit{commutative} quantum field theory,
we can obtain $\mu ^{\mathrm{sym}} (\psi , ... , \psi )$ from a given $\mu (\psi , ... , \psi )$ as follows
\begin{align}
\label{A to L}
\mu _{n} (\psi _{1} , ... , \psi _{n} )
= \frac{1}{n!} \, \mu ^{\mathsf{sym}}_{n} (\psi _{1} , ... , \psi _{n} ) \, .
\end{align}
The factor $n !$ comes from the symmetrization of the tensor product.
In this paper,
we thus consider properties of algebraic structures $\mu (\psi , ... , \psi )$ that do not depend on the graded commutativity of space-time fields.
As (\ref{A to L}),
our $\mu (\psi, ... , \psi )$ reduces to $\mu ^{\mathsf{sym} }(\psi , ... , \psi )$ automatically whenever we consider ordinary quantum field theory.
Actually,
the relation of $\mu $ and $\mu ^{\mathsf{sym} }$ is nothing but that of (quantum) $A_{\infty }$ and $L_{\infty }$\,.
The quantum $A_{\infty }$ structure appearing in this paper can be always switched to the quantum $L_{\infty }$ structure for ordinary quantum field theory.\footnote{Note that it does not imply that the $A_{\infty }$ products can be written in terms of the $L_{\infty }$ products for ordinary field theory:
the $A_{\infty }$ or $L_{\infty }$ products we consider are not elementary objects of a given theory but special combinations of them.
Even for closed string field theory,
the $L_{\infty }$ products or string vertices are constructed by specifying conformal mappings or differential forms on the (puncture-symmetrized) moduli space of Riemann surfaces. }
\vspace{2mm}
As we see later,
physical gradings,
such as the space-time ghost number or Grassmann parity of fields,
do not give the $A_{\infty }$ degree directly.
In addition,
by using appropriate (de-)suspension maps,
the change of the grading of $A_{\infty }$ algebras does not change the physics.
Hence,
it is useful to set all $A_{\infty }$ products to have degree $1$,
which we call a natural $A_{\infty }$ degree.
\subsubsection*{Quantum $A_{\infty }$ structure}
An $A_{\infty }$ structure $\boldsymbol{\mu } = \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{2} + \cdots$ is a (co-)derivation acting on $\mathcal{T}(\mathcal{\widehat{H}} )$ such that $( \boldsymbol{\mu } )^{2}= 0$.
For a given $\psi _{1} \otimes \cdot \cdot \cdot \otimes \psi _{n} \in \mathcal{T}(\mathcal{\widehat{H}} ) $ with fixed $n \geq 1$,
the $A_{\infty }$ relations $( \boldsymbol{\mu } )^{2} =0$ can be represented as
\begin{align}
\label{def of A-relations}
\sum _{k+l=n} \sum_{m=0}^{k} (-)^{\epsilon (\psi ) }
\boldsymbol{\mu }_{k+1}
\big{(} \, \underbrace{\psi _{1} , ... , \psi _{m} }_{m} , \,
\boldsymbol{\mu }_{l} (\psi _{m+1} , ... , \psi _{m+l} ) , \, \underbrace{\psi _{m+l+1} , ... , \psi _{n}}_{k-m} \, \big{)} = 0 \, ,
\end{align}
where $\epsilon (\psi )$ denotes the sign factor arising from $\boldsymbol{\mu }_{l}$ passing $\psi _{1} \otimes \cdot \cdot \cdot \otimes \psi _{m}$\,.
Let $\omega $ be a graded symplectic structure of degree $-1$ and $\{ \hat{e}_{-s} , \hat{e}_{1+s} \} _{s \geq 0}$ be a set of complete basis such that $\omega ( \hat{e}_{-s} , \hat{e}_{1+s'} ) = (-)^{s} \delta _{s,s'}$.
A cyclic $A_{\infty }$ structure is an $A_{\infty }$ structure $\boldsymbol{\mu }$ satisfying $\omega (\boldsymbol{\mu } \otimes 1 + 1 \otimes \boldsymbol{\mu } )=0$,
which is the classical limit of a quantum $A_{\infty }$ structure.
\vspace{2mm}
A quantum $A_{\infty }$ structure $\boldsymbol{\mu } + \hbar \, \mathfrak{L}$ is a linear map acting on $\mathcal{T} (\mathcal{\widehat{H}} )$ such that $(\boldsymbol{\mu } + \hbar \, \mathfrak{L} )^{2} = 0$
where $\boldsymbol{\mu } = \sum_{n} \boldsymbol{\mu }_{n\, [0] } + \sum _{n,g } \hbar ^{g} \, \boldsymbol{\mu }_{n, [g]}$ is a (co-)derivation and $\mathfrak{L}$ is a second order (co-)derivation.
For fixed $n\geq 1$ and $g \geq 0$,
the quantum $A_{\infty }$ relations $(\boldsymbol{\mu } + \hbar \, \mathfrak{L} )^{2} = 0$ can be represented as
\begin{align}
\label{def of quantum A-relations}
& \hspace{7mm}
\sum _{\substack{k+l=n \\ g_{1}+g_{2} =g } } \sum_{m=0}^{k} (-)^{\epsilon (\psi ) }
\boldsymbol{\mu }_{k+1 ,\, [g_{1}] } \big{(} \underbrace{\psi _{1} , ... , \psi _{m} }_{m} , \,
\boldsymbol{\mu }_{l ,\, [g_{2} ] } (\psi _{m+1} , ... , \psi _{m+l} ) , \, \underbrace{\psi _{m+l+1} , ... , \psi _{n}}_{k-m} \big{)}
\nonumber\\ & +
\sum _{s \in \mathbb{Z} } \sum_{i = 0}^{n} \sum_{j=0}^{n-i} (-)^{\epsilon (s,i,j)}
\boldsymbol{\mu }_{n+2 , [g-1] } \big{(} \underbrace{\psi _{1} , ... , \psi _{i} }_{i} , \hat{e}_{-s} , \underbrace{\psi _{i+1} , ... , \psi _{i+j} }_{j} , \hat{e}_{1+s} , \psi _{i+j+1} , ... , \psi _{n} \, \big{)}
= 0 \, ,
\end{align}
where the sign factor $\epsilon (s,i,j)$ arises from $\hat{e}_{1+s}$ passing $\psi _{1} \otimes \! \cdot \! \cdot \! \cdot \! \otimes \psi _{i+j}$ and $\hat{e}_{-s}$ passing $\psi _{1} \otimes \! \cdot \! \cdot \! \cdot \! \otimes \psi _{i}$\,.
\subsubsection*{Quantum $L_{\infty }$ structure}
An $L_{\infty }$ structure $\boldsymbol{\mu }^{\mathsf{sym }}= \boldsymbol{\mu }^{\mathsf{sym} }_{1} + \boldsymbol{\mu }^{\mathsf{sym} }_{2} + \cdots$ is a (co-)derivation acting on $\mathcal{S}(\mathcal{\widehat{H}} )$ such that $( \boldsymbol{\mu }^{\mathsf{sym} } )^{2}= 0$.
For fixed $n \geq 1$,
the $L_{\infty }$ relations $( \boldsymbol{\mu }^{\mathsf{sym }} )^{2} =0$ can be represented as follows
\begin{align}
\label{def of L-relations}
\sum _{k+l=n} \sum_{\sigma \in \mathsf{S}_{l,k} } (-)^{\sigma (\psi ) }
\boldsymbol{\mu }^{\mathsf{sym}}_{k+1} \big{(} \, \boldsymbol{\mu }^{\mathsf{sym}}_{l} (\psi _{\sigma (1) } , ... , \psi _{\sigma (l) } ) , \psi _{\sigma (l+1)} , ... , \psi _{\sigma (n) } \, \big{)} = 0 \, ,
\end{align}
where $\sigma (\psi )$ denotes the sign factor arising from the $(l,k)$-unshuffle of $\psi _{\sigma (1)} \wedge \cdot \cdot \cdot \wedge \psi _{\sigma (n)} \in \mathcal{S}(\mathcal{\widehat{H}} )$.
A cyclic $L_{\infty }$ structure is an $L_{\infty }$ structure $\boldsymbol{\mu }^{\mathsf{sym} }$ satisfying $\omega (\boldsymbol{\mu }^{\mathsf{sym} } \otimes 1 + 1 \otimes \boldsymbol{\mu }^{\mathsf{sym} } ) =0$,
which is the classical limit of a quantum $L_{\infty }$ structure.\footnote{When we consider other gradings,
such as $2-n$ for $\mu _{n}$,
the same relations hold as (\ref{def of A-relations}),
(\ref{def of quantum A-relations}),
(\ref{def of L-relations}) or (\ref{def of quantum L-relations}) except for the sign factors:
(de-)suspension maps relate them.
See also \cite{Jurco:2018sby, Markl:1997bj, Herbst:2006kt, Munster:2011ij}. }
\vspace{2mm}
A quantum $L_{\infty }$ structure $\boldsymbol{\mu }^{\mathsf{sym}} + \hbar \, \mathfrak{L}$ is a linear map acting on $\mathcal{S} (\mathcal{\widehat{H}})$ such that $(\boldsymbol{\mu }^{\mathsf{sym}} + \hbar \, \mathfrak{L})^{2} = 0$
where $\boldsymbol{\mu }^{\mathsf{sym} } = \sum_{n} \boldsymbol{\mu }_{n\, [0] }^{\mathsf{sym} } + \sum _{n,g } \hbar ^{g} \, \boldsymbol{\mu }_{n, [g]}^{\mathsf{sym} }$ is a (co-)derivation and $\mathfrak{L}$ is a second order (co-)derivation.
For fixed $n > 0$ and $g \geq 0$,
the quantum $L_{\infty }$ relations $(\boldsymbol{\mu }^{\mathsf{sym}} + \hbar \, \mathfrak{L})^{2} = 0$ can be represented as
\begin{align}
\label{def of quantum L-relations}
&
\sum_{\substack{k+l = n \\ g_{1} + g_{2} = g }} \sum_{\sigma \in \mathsf{S}_{l,k} } (-)^{\sigma (\psi ) }
\boldsymbol{\mu }^{\mathsf{sym}}_{k+1, \,[g_{1} ] } \big{(} \, \boldsymbol{\mu }^{\mathsf{sym}}_{l , \, [g_{2} ] } (\psi _{\sigma (1) } , ... , \psi _{\sigma (l) } ) , \psi _{\sigma (l+1)} , ... , \psi _{\sigma (n) } \, \big{)}
\nonumber\\ & \hspace{25mm}
+ \frac{1}{2} \sum_{s \in \mathbb{Z} } \boldsymbol{\mu }^{\mathsf{sym}}_{n+2, \,[g-1] } \big{(} \, \hat{e}_{-s} , \hat{e}_{1+s} , \psi _{1} , ... , \psi _{n} \, \big{)} = 0 \, .
\end{align}
As we see in section 2,
a quantum $A_{\infty }$ structure can be assigned to \textit{every} quantum field theory that solves the BV master equation.
Most of our results will be presented in terms of $A_{\infty }$ since the results based on $A_{\infty }$ can be always switched to those based on $L_{\infty }$ for ordinary quantum field theory.
As long as we consider an $A_{\infty }$ structure $\mu $ that can be represented by the form of (\ref{multilinear mu}),
the (quantum) $A_{\infty }$ structure $\mu $ of \textit{commutative} quantum field theory reduces the (quantum) $L_{\infty }$ structure $\mu ^{\mathsf{sym }}$ automatically just as (\ref{A to L}).
We end this section by giving two examples.
\subsubsection*{$4$ point amplitude}
The amplitudes of Lagrangian field theory have a quantum $A_{\infty }$ structure,
which we will explain in more detail in section 4.
Let us consider the cubic action,
which is (\ref{classical physical action}) with $\mu _{n >2} = 0$.
It can be a non-commutative field theory.
We write $\mu _{1}^{-1}$ for a propagator of this theory.
The $4$ point amplitude $\mathcal{A}_{4}$ is given by
\begin{align}
\label{ST rep}
\mathcal{A}_{4} \sim \big{\langle } \psi _{0} , \,
\mu _{2} ( \mu_{1}^{-1} \mu _{2} ( \psi _{1} , \psi _{2} ) , \psi _{3} ) \big{\rangle }
+
\big{\langle } \psi _{0} , \, \mu _{2} ( \psi _{1} , \mu_{1}^{-1} \mu _{2} ( \psi _{2} , \psi _{3} ) ) \big{\rangle } \, .
\end{align}
It consists of the $S$-channel and $T$-channel.
When multiplications of space-time fields are commutative,
$\mu $ reduces to $\mu ^{\mathsf{sym}}$ as (\ref{A to L}).
Then,
the expression (\ref{ST rep}) reduces to
\begin{align}
\mathcal{A}_{4} & \sim
\big{\langle } \psi _{0} , \, \mu ^{\mathsf{sym} }_{2} ( \mu_{1}^{-1} \mu ^{\mathsf{sym}}_{2} ( \psi _{1} , \psi _{2} ) , \psi _{3} ) \big{\rangle }
+\big{\langle } \psi _{0} , \mu ^{\mathsf{sym} }_{2} ( \mu_{1} ^{-1} \mu ^{\mathsf{sym}}_{2} ( \psi _{2} , \psi _{3} ) , \psi _{1} ) \big{\rangle }
\nonumber\\ & \hspace{25mm}
+ \big{\langle } \psi _{0} , \, \mu ^{\mathsf{sym} }_{2} ( \mu_{1}^{-1} \mu ^{\mathsf{sym}}_{2} ( \psi _{3} , \psi _{1} ) , \psi _{2} ) \big{\rangle } \, .
\end{align}
It consists of the $S$-channel,
the $T$-channel and the $U$-channel.
As is known,
this is a $4$ point amplitude of commutative Lagrangian field theory.
\subsubsection*{Yang-Mills theory}
Let us consider the $A_{\infty }$ structure of the ordinary Yang-Mills action $S [A] = - \frac{1}{2} \int \, \langle F , \star \, F \rangle $\,,
which is a commutative Lagrangian field theory.
Yang-Mills fields $A$ are Lie-algebra-value $1$-forms.
The first $A_{\infty }$ structure is given by the kinetic operator
\begin{align}
\mu _{1} ( A_{1} ) & = d \, \star \, d \, A_{1} \, ,
\end{align}
where $d$ denotes the exterior differential and $\star $ denotes the Hodge dual operation.
By casting the Yang-Mills action as the form of (\ref{classical physical action}),
vertices provides higher $A_{\infty }$ products
\begin{align}
\mu _{2} ( A_{1} , A_{2} ) & = d \, \star \big{(} A_{1} \wedge A_{2} \big{)}
- \big{(} \star \, d \, A_{1} \big{)} \wedge A_{2}
+ A_{1} \wedge \big{(} \star \, d \, A_{2} \big{)} \, ,
\\
\mu _{3} (A_{1} , A_{2} , A_{3} ) & = A_{1} \wedge \big{(} \star ( A_{2} \wedge A_{3} ) \big{)}
- \big{(} \star ( A_{1} \wedge A_{2} ) \big{)} \wedge A_{3} \, ,
\end{align}
where $\wedge $ denotes the exterior product of forms.
Note that this $\wedge $ is different from the symmetrized tensor product of (\ref{symmetrized tensor product}).
As a symmetrization of exterior products,
we can consider the graded commutator of exterior products,
$[A_{1} , A_{2} ]_{\wedge } \equiv A_{1} \wedge A_{2} - (-)^{A_{1} A_{2} } A_{2} \wedge A_{1}$.
We find
\begin{align}
\mu ^{\mathsf{sym} }_{2} ( A_{1} , A_{2} ) & = d \, \star \big{[} \, A_{1} \, , \, A_{2} \, \big{]}_{\wedge }
- \big{[} \star d \, A_{1} \, , \, A_{2} \, \big{]}_{\wedge }
+ \big{[} \, A_{1} \, , \star \, d \, A_{2} \, \big{]}_{\wedge } \, ,
\\
\mu ^{\mathsf{sym} }_{3} (A_{1} , A_{2} , A_{3} ) &
= \big{[} \, A_{1} \, , \star [ A_{2} , A_{3} ]_{\wedge } \, \big{]}_{\wedge }
+ \big{[} \, A_{2} \, , \star [ A_{3} , A_{1} ]_{\wedge } \, \big{]}_{\wedge }
+ \big{[} \, A_{3} \, , \star [ A_{1} , A_{2} ]_{\wedge } \, \big{]}_{\wedge } \, .
\end{align}
These are the $L_{\infty }$ structure of the Yang-Mills theory.
These $A_{\infty }$ and $L_{\infty }$ structures are related to each other by (\ref{A to L}).
As is known,
the above \textit{incomplete} $A_{\infty }$ (or $L_{\infty }$) structure is just a piece of the complete $A_{\infty }$ (or $L_{\infty }$) structure of the BV master action for the Yang-Mills theory.
When we consider non-commutative quantum field theory,
it has only an $A_{\infty }$ structure obtained by replacing the exterior product with the non-commutative product.
\section{Path-integral as a morphism of BV}
On the basis of the BV formalism,
we show a statement that the homological perturbation performs the perturbative path-integral and discuss several properties effective theories have as a consequence of it.
Most of them would be known among experts:
except for incidental details,
this section is a review.
We first explain that solving the BV master equation is equivalent to extracting the quantum $A_{\infty }$ structure intrinsic to each Lagrangian field theory.
Next,
we give a brief review of basic facts of the BV formalism which are related to properties of the path-integral.
Then,
we show the statement and properties of the effective $A_{\infty }$ structure.
Note that
quantum field theories \textit{without} gauge degrees can be also treated within the BV formalism.
Although it trivially solves the BV master equation,
it provides non-trivial results.
\vspace{2mm}
The BV formalism is one of the most general and systematic prescription to quantize gauge theories,
which enable us to treat open or redundant gauge algebras \cite{Batalin:1981jr, Henneaux}.
As is known,
a gauge-fixing is necessary to perform the path-integral for a given gauge theory,
to which we can apply the BV formalism even if ordinary methods such as fixing-by-hand,
deriving the Dirac bracket,
brute-force computations and the BRST procedure do not work.
To carry out (\ref{non-perturbative}) or (\ref{Gaussian}),
the action must provides a regular Hessian.
In addition,
symmetries proportional to the equations of motion are redundant and must be taken into account.
We thus introduce antifields $\psi ^{\ast }_{\mathrm{cl}}$,
ghost fields $c$,
antifields for ghosts $c^{\ast }$ and pairs of higher fields-antifields as much as needed,
\begin{align}
\label{example of BV fields}
S_{\mathrm{cl} }[\psi _{\mathrm{cl}} ] \,\, \longrightarrow \,\,
S [\psi ] = S_{\mathrm{cl} } [\psi _{\mathrm{cl}} ]
+ \underbrace{ \psi ^{\ast }_{\mathrm{cl }} ( S , \psi ) }_{\mathrm{ghost\,\,terms}}
+ \underbrace{ c^{\ast } \, ( S , c ) + \cdots }_{\mathrm{higher\,\,ghost\,\, terms}} \, .
\end{align}
We write $\psi $ for the sum of all fields and antifields.
This extended action $S[\psi ]$ is called a BV master action,
which can provide a regular Hessian,
and enables us to perform the path-integral of gauge theory.
The BV master action $S[\psi ]$ is a solution of the BV master equation
\begin{align}
\label{quantum BV master equation}
\hbar \, \Delta \, e^{S[\psi ] }
=
\Big[ \,
\hbar \, \Delta S [\psi ]
+ \frac{1}{2} \big{(} \, S [\psi ] , \, S[\psi ] \, \big{)}
\, \Big] \, e^{S[\psi ] }
= 0 \, .
\end{align}
The BV master equation guarantees that the theory is independent of gauge-fixing conditions and has no gauge anomaly arising from the measure factor of the path-integral.
A gauge-fixing is carried out by choosing appropriate gauge-fixing fermions,
which determines a Lagrangian submanifold.
\vspace{2mm}
We write $\psi _{g}$ and $\psi ^{\ast }_{g}$ for fields and antifields having space-time ghost number $g$ and $-g-1$ respectively:
$\psi _{0} \equiv \psi _{\mathrm{cl}}$,
$\psi ^{\ast }_{0} \equiv \psi ^{\ast }_{\mathrm{cl}}$,
$\psi _{1} \equiv c$ and $\psi ^{\ast }_{1} \equiv c^{\ast }$ in (\ref{example of BV fields}) for example.
We also use $\psi _{-1-g} = \psi _{g}^{\ast } $ for brevity.
The BV Laplacian $\Delta $ is a second-order odd derivative,
which is defined by
\begin{align}
\Delta \equiv \sum_{g} (-)^{g} \frac{\partial }{\partial \psi _{g} } \frac{\partial }{\partial \psi _{g}^{\ast }}
= \frac{\partial }{\partial \psi _{\mathrm{cl} } } \frac{\partial }{\partial \psi ^{\ast }_{\mathrm{cl} } }
- \frac{\partial }{\partial c } \, \frac{\partial }{\partial c^{\ast } }
+ \cdots \, .
\end{align}
It is a fundamental object in the BV formalism and has geometrical meaning \cite{Schwarz:1992nx, Alexandrov:1995kv}.
The BV bracket is defined by $ (-)^{F} ( F , G ) \equiv \Delta (FG) - (\Delta F) G - (-)^{F} F (\Delta G )$,
where $F$ and $G$ are any functionals of fields and antifields.
The BV bracket can be cast as
\begin{align}
\label{antibracket}
\big{(} \, F \, , \, G \, \big{)}
= \sum_{g} \bigg{[} \,
\frac{\partial _{r} F}{\partial \psi _{g} } \, \frac{\partial G}{\partial \psi ^{\ast }_{g} }
- \frac{\partial _{r} F}{\partial \psi ^{\ast }_{g} } \, \frac{\partial G}{\partial \psi _{g} } \,
\bigg{]} \, .
\end{align}
Note that $\partial _{r}$ denotes the right derivative and it satisfies $\frac{\partial }{\partial \psi _{g} } F = (-)^{g ( F +1 )} \frac{\partial _{r} }{\partial \psi _{g} } F$\,.
\subsection{$A_{\infty }$ structure of the BV master equation}
Suppose that for a given Lagrangian field theory,
its BV master action $S[\psi ]$ was obtained by solving the BV master equation.
When the theory consists of physical degrees only,
the BV master action is the classical action itself.
We start with a given $S [\psi ]$\,:
see appendix A for notation.
\vspace{2mm}
We first consider the simplest case.
Suppose that a solution $S$ of the classical master equation $(S,S)=0$ also solves the quantum master equation $\hbar \, \Delta S + \frac{1}{2} (S , S) = 0$ \textit{without any modification}.
The cyclic $A_{\infty }$ structure $\mu $ can be read from the derivative $(S , \psi )$ as follows
\begin{align}
\label{(S,psi) in sec2}
\big{(} \, S \, , \, \psi \, \big{)}
= \sum_{g} (-)^{g} \bigg{[} \, \frac{\partial S}{\partial \psi _{g}}
+ \frac{\partial S}{\partial \psi ^{\ast }_{g}} \, \bigg{]}
= \sum_{n} (-)^{|\mu _{n}(\psi) |} \mu _{n} ( \psi , ... , \psi ) \, ,
\end{align}
where $\psi $ is the sum of all fields and antifields $\psi = \sum \psi _{g} + \sum \psi ^{\ast }_{g}$ and $|\mu _{n} (\psi )|$ denotes the total ghost number of the inputs of $\mu _{n}$.
Let $\mathcal{H}$ be the state space of fields and antifields in the BV formalism:
the tensor algebra $\mathcal{T}(\mathcal{H})$ on which $\mu $ acts consists of only one kind of \textit{field} $\psi $.
The BV master action $S[\psi ]$ has neutral ghost number and the BV derivation $(S , \,\,\, )$ has ghost number one,
although $\psi = \sum \psi _{g} + \sum \psi ^{\ast }_{g}$ includes fields having different ghost numbers.
We write $\mu _{n} (\psi , .. , \psi ) |_{-g}$ for the restriction onto the ghost number $-g$ sector.
The $A_{\infty }$ relations can be read from
\begin{align}
\label{A in sec2}
0 = \big{(} \, S \, , ( S , \psi ) \big{)}
= \sum_{n , g} \sum_{k+l=n} \sum _{m=0}^{k}
(-)^{\epsilon (\psi ) } \mu_{k+1} ( \underbrace{\psi , ... , \psi }_{m}, \mu _{l} (\psi , ... , \psi ) , \underbrace{\psi , ... ,\psi }_{\epsilon (\psi )}) \Big{|}_{g} \, ,
\end{align}
where $\epsilon (\psi )$ denotes the sum of $\psi $'s ghost numbers.
See appendix for adjusting the sign factors.
In terms of these $A_{\infty }$ products $\{ \mu _{n} \} _{n}$,
the BV master action $S[\psi ]$ can be always cast into the following form of homotopy Maurer-Cartan action,\footnote{
To extract the (quantum) $L_{\infty }$ structure,
instead of (\ref{cl bv}) and (\ref{(S,psi) in sec2}),
one should start with the $L_{\infty }$ homotopy Maurer-Cartan form $S [\psi ] = \sum_{n=1}^{\infty } \frac{1}{(n+1)!} \langle \psi , \, \mu ^{\mathsf{sym} }_{n} ( \psi , \cdots , \psi ) \rangle $ and $(-)^{\psi }(S , \psi ) = - \sum_{n} \frac{1}{n!} \mu ^{\mathsf{sym}}_{n} (\psi , ... , \psi )$.}
\begin{align}
\label{cl bv}
S [\psi ] = \frac{1}{2} \big{\langle } \, \psi \, , \, \mu _{1} \, \psi \, \big{\rangle }
+ \sum_{n=2}^{\infty } \frac{1}{n+1} \big{\langle } \, \psi \, , \, \mu _{n} ( \psi , \cdots , \psi ) \, \big{\rangle } \, ,
\end{align}
where $\langle \,\,\, , \,\,\, \rangle$ denotes a symplectic form in the BV formalism,
which is explained in appendix.
Note that the space-time ghost number is not a natural grading of the $A_{\infty }$ structure $\mu$\,.
Since $\mu $ consists of kinetic operators and interacting vertices,
$\mu $ has neutral ghost number.
\vspace{2mm}
Next,
we consider a generic case.
Suppose that a solution $S_{[0]}$ of the classical master equation $(S_{[0]} ,S_{[0]} ) = 0$ does not solve the quantum master equation,
such as $\hbar \Delta S_{[0]} \not=0$\,.
Then,
we need to construct correcting terms $\hbar S_{[1]} + \hbar ^{2} S_{[2]} + \cdots$ such that $S \equiv S_{[0]} + \hbar S_{[1]} + \hbar ^{2} S_{[2]} + \cdots$ satisfies the quantum master equation $\hbar \Delta S + \frac{1}{2} (S , S) = 0$.
In this case,
the quantum BV master action $S$ induces the quantum $A_{\infty }$ structure $\mu _{n, [l]}$ as follows
\begin{align}
(-)^{\psi +1} \big{(} \, S \, , \, \psi \, \big{)}
= \sum_{g} \bigg{[}
\frac{\partial S_{[0]}}{\partial \psi _{g}}
+ \sum_{l >0} \hbar ^{l} \frac{\partial S_{[l]} }{\partial \psi _{g} }
\bigg{]}
= \sum_{n,g} \bigg{[}
\mu_{n, [0]} (\psi , ... , \psi ) +
\sum_{l} \hbar ^{l} \mu _{n,[l]} (\psi , ... , \psi )
\bigg{]}_{-g}
\, .
\end{align}
The quantum BV master action $S$ provides a natural nilpotent operation $\Delta _{S}$ defined by
\begin{align}
\label{BV diff}
\hbar \, \Delta _{S} \equiv \hbar \, \Delta + ( \, S \, , \hspace{3mm} ) \, .
\end{align}
The quantum $A_{\infty }$ relation is encoded in (\ref{BV diff}) as follows
\begin{align}
\label{quantum A}
(\hbar \, \Delta _{S} )^{2} \psi ^{\ast }_{g} =
& \sum_{n,l} \bigg{[} \,
\hbar \, \sum_{s \in \mathbb{Z} } \sum_{i=0}^{n} \sum_{j=0}^{n-i} (-)^{\epsilon (s,i,j) }
\mu _{n+2 , [l-1]} \big{(} \underbrace{\psi , ... , \psi }_{i} , e_{-s} , \underbrace{\psi , ... , \psi }_{j} , e_{1+s} , \psi , ... , \psi \big{)}
\nonumber\\ & \hspace{3mm}
+
\sum_{\substack{n_{1}+n_{2}=n \\ l_{1}+l_{2} = l}}
\sum_{m=0}^{n_{1} } (-)^{\epsilon (\psi ) }
\mu_{n_{1}+1, [l_{1}]} \big{(} \underbrace{ \psi , ... , \psi }_{m } , \,
\mu _{n_{2} , [l_{2}] } ( \psi , ... , \psi ) , \,
\underbrace{\psi , ... , \psi }_{n_{1} - m} \big{)}
\bigg{]}_{1-g} \, ,
\end{align}
where the sign factor $\epsilon (s,i,j)$ arises from $e_{1+s}$ passing $\psi ^{\otimes (i+j)}$ and $e_{-s}$ passing $\psi ^{\otimes i}$.
These complicated sign factors can be simplified when we use a degree $-1$ symplectic form $\omega $ and assign a basis carrying unphysical grading to each field or antifield:
see appendix A.
For $s \geq 0$,
these $e_{-s}$ and $e_{1+s}$ are defined by $e_{-s} \equiv \frac{\partial }{\partial \psi _{s}} \psi$ and $e_{1+s} \equiv (-)^{s} \frac{\partial }{\partial \psi ^{\ast }_{s}} \psi $ respectively.
They enable us to get the following useful representation
\begin{align}
\Delta \, \mu _{n,[l]}(...) = \sum_{s \in \mathbb{Z}} (-)^{\epsilon (s)} \mu _{n , [l]} ( ... , e_{-s} , ... , e_{1+s} , ... ) \,.
\end{align}
Note that the condition $(\hbar \, \Delta _{S} )^{2} = 0$ is equivalent to the BV master equation (\ref{quantum BV master equation}).
Hence,
a solution of the BV master equation assigns a quantum $A_{\infty }$ structure to each Lagrangian field theory \cite{Barannikov:2010np}:
see \cite{Kajiura:2003ax} for the case of cyclic $A_{\infty}$.
In terms of these quantum $A_{\infty }$ products,
the quantum BV master action $S[\psi ]$ can be cast into the form of homotopy Maurer-Cartan action
\begin{align}
\label{master action}
S [\psi ] = S_{[0]} [\psi ]
+ \sum_{n,l} \frac{\hbar ^{l}}{n+1} \big{\langle } \psi , \, \mu _{n,[l]} ( \psi , \cdots , \psi ) \big{\rangle } \, ,
\end{align}
where the classical master action $S_{[0]} [\psi ]$ takes the same form as (\ref{cl bv}) with $\mu _{n,[0]} \equiv \mu _{n}$\,.
\subsection{On the BV differential}
In the classical theory,
a solution of the equations of motion determines physical states up to gauge degrees.
These information are encoded into the classical BV differential
\begin{align}
\label{classical BV diff}
Q_{S} \equiv ( \, S \, , \hspace{3mm} ) \,
\end{align}
acting on the space $\mathcal{F} (\mathcal{H} )$ spanned by functionals
of fields and antifields $\psi \in \mathcal{H}$.
Note that $\mathcal{F} (\mathcal{H} )$ includes the space of polynomials of space-time-field multiplications,
which is often identified with the tensor algebra $\mathcal{T} (\mathcal{H} )$ in this paper.
For a given master action (\ref{master action}),
the equation of motion for the field $\psi _{g}$ can be represented by using the BV differential and its antifield $\psi ^{\ast }_{g}$ as follows
\begin{align}
\label{stationary pt}
0 = (-)^{g} \frac{\partial S }{\partial \psi _{g}} =\sum_{n} (-)^{g} \mu _{n} (\psi , \dots , \psi ) \Big{|}_{-g}
= Q_{S} \, \psi ^{\ast }_{g} \, .
\end{align}
It implies that the on-shell states are $Q_{S}$-closed.
Likewise,
the gauge transformation of the master action implies that its gauge degrees are $Q_{S}$-exact.
The space $\mathcal{H}_{\mathrm{phys}}$ of the physical states are described by the $Q_{S}$-cohomology.
The classical observables $F$ are functionals of physical states,
which we write $F \in \mathcal{F} (\mathcal{H}_{\mathrm{phys} } )$.
Solving the classical theory is equivalent to finding the cohomology of complex with the classical BV differential (\ref{classical BV diff}),
which we often represent by
\begin{align}
\label{cl BV cohomology}
\big{(} \, \mathcal{F} (\mathcal{H} ) , \, Q_{S} \, \big{)}
\hspace{3mm}
\overset{p }{\underset{i}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} (\mathcal{H}_{\mathrm{phys} } ) , \, 0 \, \big{)}
\, ,
\end{align}
where $p$ denotes a restriction to on-shell and $i$ denotes an embedding to off-shell.
\vspace{2mm}
In the quantum theory,
the stationary point of (\ref{stationary pt}) does not completely determine the physical states.
In addition to solve (\ref{stationary pt}),
we need to replace functionals $F$ of physical states by their expectation values $\langle F \rangle$,
which is given by the path-integral
\begin{align}
\label{expectation value}
F \,\, \overset{P}{\longrightarrow } \,\,
\big{\langle } F \big{\rangle } \equiv \int \mathcal{D}[ \psi ] \, F \, e^{S [\psi ]} \, .
\end{align}
Solving the BV master equation was necessary to define a regular Hessian for $S[\psi ]$\,.
As the case of $F = 1$,
the integrand $Fe^{S}$ must be $\Delta$-closed in order to obtain the gauge independent path-integral.
Hence,
for a given theory $S[\psi ]$,
its observables $F = F [\psi ]$ satisfy
\begin{align}
\hbar \, \Delta _{S} \, F [\psi ] = 0 \, .
\end{align}
The equation of motions can be also cast as $\hbar \, \Delta _{S} \psi = 0$.
Note however that the $\Delta _{S}$-exact transformation,
such as $\delta \psi = \hbar \, \Delta _{S} \, \epsilon $,
is not the invariance of the action.
It is the invariance of the partition function or expectation values defined by the above path-integral:
for example,
the $\Delta _{S}$-exact deformation $F \mapsto F + \Delta _{S} \Lambda $ does not change $\langle F \rangle $ because of $\int \mathcal{D}[\psi ] \Delta (...)=0$\,.
In this sense,
the physics of BV-quantizable field theory is described by the $\Delta _{S}$-cohomology,
\begin{align}
\label{BV cohomology}
\big{(} \, \mathcal{F} ( \mathcal{H} ) , \, \hbar \, \Delta + Q_{S} \, \big{)}
\hspace{3mm}
\overset{P}{\underset{I}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}_{\mathrm{q\mathchar`-phys} } ) , \, 0 \, \big{)}
\hspace{3mm}
\overset{\mathrm{i}}{\underset{\mathrm{ev} }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}_{\mathrm{phys} } ) , \, 0 \, \big{)}
\, ,
\end{align}
where $\mathcal{H}_{\mathrm{q\mathchar`-phys}}$ denotes the space of physical state in the quantum theory.
While $p$ and $i$ of (\ref{cl BV cohomology}) denote restriction and embedding to on-shell and off-shell respectively,
$\mathrm{ev}$ and $\mathrm{i}$ of (\ref{BV cohomology}) denote substituting the expectation values and returning values to variables respectively.
The path-integral $P$ should be identified with $P = \mathrm{ev} \circ p$,
the composition of $p$ and $\mathrm{ev}$,
because it not only gives the expectation value but also condenses field configurations onto the stationary point.
\vspace{2mm}
In this paper,
we consider the perturbative path-integral (\ref{Feynman rep}),
which is written in terms of the free theory (\ref{Gaussian}).
For the perturbative path-integral,
there would be two options for realizing $\mathrm{ev}$ of (\ref{BV cohomology}).
The first option is to identify $\mathrm{ev} : \mathcal{F} (\mathcal{H}_{\mathrm{phys} } ) \rightarrow \mathcal{F} ( \mathcal{H}_{\mathrm{q\mathchar`- phys}} )$ with imposing
\begin{align}
\label{imposing free BV}
\Delta _{S_{\mathrm{free}} } F [\psi ] = 0
\end{align}
for any $F [\psi ] \in \mathcal{F} ( \mathcal{H}_{\mathrm{phys}} )$.
The second option is to identify $\mathrm{ev}$ with performing the Gaussian integral (\ref{Gaussian}).
While $p$ gives $p ( \psi ) = \psi _{\mathrm{phys}}$ for $\psi = \psi _{\mathrm{phys }} + \psi _{\mathrm{gauge} } + \psi _{\mathrm{unphys}}$,
the map $\mathrm{ev}$ evaluates the expectation value of a free physical field $\psi _{\mathrm{phys }} \in \mathcal{H} _{\mathrm{phys} }$,
namely,
\begin{align}
\label{ev}
\mathrm{ev} \, (\psi _{\mathrm{phys} } ) = 0 \, ,
\end{align}
which comes from $\langle \psi _{\mathrm{phys }} \rangle _{\mathrm{free}} =0$.
We may write $\psi _{\mathrm{ev}} = \mathrm{i} (\langle \psi _{\mathrm{phys }} \rangle )$ and $\mathrm{ev} (\psi _{\mathrm{phys }} ) = \langle \psi _{\mathrm{phys }} \rangle $ with $\mathrm{ev} \circ \mathrm{i} = 1$,
although both of them are $0$ for the perturbative path-integral in most cases.
As we will see,
both of (\ref{imposing free BV}) and (\ref{ev}) provide the perturbative path-integral map $P$.
The equivalence of these two options comes from the fact that the Gaussian integral (\ref{Gaussian}) can be understood as a result of the homological perturbation for the BV differential of the free theory.
\vspace{2mm}
As we see later,
we do not need to require (\ref{imposing free BV}) or (\ref{ev}) explicitly when we consider the path-integral to obtain $S$-matrix or to remove gauge and unphysical degrees.
The condition (\ref{imposing free BV}) or (\ref{ev}) should be explicitly imposed when we consider the path-integral of fields whose momentum are higher than some cut-off scale or the path-integral of all massive fields for example.
\subsection{Path-integral preserves the BV master equation}
For a given BV master action $S[\psi ]$,
we split fields $\psi $ into two components $\psi '$ and $\psi ''$,
\begin{align}
\psi = \psi ' + \psi '' \, .
\end{align}
By performing the path-integral of the fields $\psi ''$,
we obtain the BV effective action $A[\psi ' ]$ from the original BV action $S[\psi '+ \psi '' ]$.
The effective action can be written as follows
\begin{align}
\label{effective BV master action}
A [\psi ' ] \equiv \ln \int \mathcal{D} [\psi '' ] \, e^{S[\psi ' + \psi '' ] } \, .
\end{align}
Although it is independent of $\psi ''$ on the mass shell,
the form of $A[\psi ']$ depends on the Lagrangian submanifold of $\psi ''$:
the path-integral of $\psi ''$ imposes nonlinear constraints arising from $\frac{\partial S }{\partial \psi ''} = 0$ on the remaining fields $\psi '$ when two fields $\psi '$ and $\psi ''$ interact.
It is well-known that the BV effective action $A[\psi ']$ also solves the BV master equation
\begin{align}
\label{effective BV master equation}
\hbar \, \Delta ' A [\psi ' ]
+ \frac{1}{2} \big{(} \, A [\psi ' ] \, , \, A [\psi '] \, \big{)}' = 0 \, .
\end{align}
The effective BV Laplacians $\Delta '$ and $\Delta ''$ of $\Delta = \Delta ' + \Delta ''$ are defined by
\begin{align}
\Delta' \equiv \sum (-)^{g} \frac{\partial }{\partial \psi '_{g} } \frac{\partial }{\partial \psi _{g}^{\prime \, \ast }} \, ,
\hspace{5mm}
\Delta '' \equiv \sum (-)^{g} \frac{\partial }{\partial \psi ''_{g} } \frac{\partial }{\partial \psi _{g}^{\prime \prime \, \ast }} \, .
\end{align}
As $\Delta $ provides the BV bracket (\ref{antibracket}),
the effective BV Laplacian $\Delta '$ also provides the effective BV bracket $(-)^{A}(A ,B)' \equiv \Delta ' (AB) - (\Delta ' A)B - (-)^{A} A (\Delta ' B)$.
Because of the effective BV master equation (\ref{effective BV master equation}),
the operator $\hbar \, \Delta '_{A} \equiv \hbar \, \Delta ' + ( A , \,\, )'$ satisfies $(\hbar \, \Delta _{A}' )^{2} = 0$.
Hence,
the effective action also has a quantum $A_{\infty }$ structure $\mu '$ and takes the homotopy Maurer-Cartan form
\begin{align}
\label{effective BV action}
A [\psi ' ] =
\sum_{n} \frac{1}{n+1} \big{\langle } \psi ' , \, \mu '_{n,[0]} ( \psi ' , ... , \psi ' ) \big{\rangle }
+ \sum_{n,l} \frac{\hbar ^{l}}{n+1} \big{\langle } \psi ' , \, \mu '_{n,[l]} ( \psi ' , ... , \psi ' ) \big{\rangle } \, .
\end{align}
\vspace{2mm}
This fact implies that the path-integral of fields $\psi ''$ gives a morphism $P$ preserving the BV master equation such that
\begin{align}
P \, \Delta _{S} = \Delta '_{A } \, P \, .
\end{align}
As long as the original action $S[\psi ] $ satisfies the BV master equation,
these operators $\Delta _{S}$ and $\Delta '_{A}$ are nilpotent and the morphism $P$ preserves the cohomology.
Because of $\mu(\psi , ..., \psi ) |_{-g} = \Delta _{S} ( \psi ^{\ast }_{g})$ and $\mu ' (\psi ' , ... , \psi ' ) |_{-g} = \Delta '_{A} ( \psi ^{\prime \, \ast }_{g})$,
as we will explain in section 3,
the path-integral $P$ induces a morphism $\textsf{p}$ between these $A_{\infty }$ structures $\mu $ and $\mu '$ such that
\begin{align}
\textsf{p} \, ( \mu _{1} + \mu _{2} + \cdots )
= (\mu '_{1} + \mu '_{2} + \cdots ) \, \textsf{p} \, .
\end{align}
In the rest of this section,
we show that the path-integral can be understood as a morphism $P$ preserving the cohomology of the BV differentials,
\begin{align}
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}'' ) , \, \Delta _{S} \, \big{)}
\hspace{3mm}
\overset{P}{\underset{I}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}''_{\mathrm{phys} } ) , \, \Delta '_{A} \, \big{)}
\, ,
\end{align}
where $\mathcal{H}'$ and $\mathcal{H}''$ denotes the state spaces of $\psi ' $ and $\psi ''$ respectively,
$\mathcal{H}''_{\mathrm{phys} } $ denotes the physical space of $\psi ''$.
On the basis of the homological perturbation,
we can construct this morphism $P$ explicitly and show that $P$ gives
\begin{align}
\label{morphism P}
P ( F [\psi ] ) = \big{\langle } \, F [ \psi ''+ \psi '' ] \, \big{\rangle } ''
\equiv Z_{\psi '}^{-1} \int \mathcal{D} [\psi '' ] \, F[\psi ] \, e^{S[\psi ' + \psi '' ] } \, ,
\end{align}
where $F [\psi ]$ is any functional of fields $\psi$ and $Z_{\psi '}$ is defined by
\begin{align}
Z_{\psi '} \equiv \int \mathcal{D} [\psi '' ] \, e^{S[\psi ' + \psi '' ] } \, .
\end{align}
\subsection{Homological perturbation performs the path-integral I}
In this section,
we construct a morphism $\widehat{P}$ performing the path-integral without normalization.
The perturbative path-integral $P$ is constructed by using this $\widehat{P}$ in the next section.
We split the action $S = S_{\mathrm{free}}+ S_{\mathrm{int}} $ into the kinetic part $S_{\mathrm{free}} $ and interacting pert $S_{\mathrm{int} }$.
Since the perturbative path-integral is based on the free theory,
we construct a map $\widehat{P}$ such that
\begin{align}
\label{pre-statement}
\widehat{P} ( e^{S_{\mathrm{int} }[\psi '' ] } )
= \big{\langle } \, e^{S_{\mathrm{int}} [ \psi '' ] } \, \big{\rangle } _{\mathrm{free} } ''
\equiv \int \mathcal{D} [\psi ''] \, e^{S[\psi '' ] } \, ,
\end{align}
where we set $\psi ' = 0$ for simplicity.
Clearly,
such $\widehat{P}$ satisfies $\widehat{P} (1) = 1$ as (\ref{Gaussian}) and describes the perturbative path-integral based on the free field theory.
\vspace{2mm}
We assume that the kinetic terms of $\psi '$ and $\psi ''$ have no cross term $S_{\mathrm{free} } [\psi ' + \psi ''] = S_{\mathrm{free} } [\psi '] + S_{\mathrm{free}} [\psi '']$.
We also assume that the free theory of $\psi ''$ is solved and takes
\begin{align}
S_{\mathrm{free}} [ \psi '' ]
= \frac{1}{2} \big{\langle } \, \psi '' , \, \mu _{1}'' \, \psi '' \, \big{\rangle }
= \frac{1}{2} \big{\langle } \, \psi ''_{0} , \, K_{0} \, \psi ''_{0} \, \big{\rangle }
+ \sum_{g} \big{\langle } \, \psi ^{\prime \prime \, \ast }_{g-1} , \, K_{g} \, \psi ''_{g} \, \big{\rangle }
\, ,
\end{align}
where $\psi ''_{g}$ is the $g$-th ghost field of $\psi '' = \sum_{g} \psi ''_{g} + \sum _{g} \psi ^{\prime \prime \, \ast }_{g}$ and $K_{g}$ is its kinetic operator.
We write $K^{-1}_{g}$ for a propagator of the kinetic operator $K_{g}$.
In order to derive the propagators,
we need to add an appropriate gauge-fixing fermion into the action with trivial pairs.
Note that we now consider the path-integral over a corresponding Lagrangian submanifold
and thus $\psi ^{\prime \prime \, \ast }$ should be understood as functionals of fields and trivial pairs determined by the gauge-fixing fermion.\footnote{Recall that propagators are singular in the gauge invariant basis.
If we want propagators not to be singular,
we need to switch to appropriate gauge-fixed basis via canonical transformations of fields-antifields.}
\vspace{2mm}
Let us consider a projection $\pi : \mathcal{H}'' \rightarrow \mathcal{H}''_{\mathrm{phys}}$ onto the physical space of the $\psi ''$ fields,
in which the free equations of motion $\mu ''_{1} \, \psi = 0$ holds.
We may represent $\iota \, \pi = e^{-\infty |\mu ''_{1} |}$ by using a natural embedding $\iota : \mathcal{H}''_{\mathrm{phys}} \rightarrow \mathcal{H}''$ satisfying $\mu ''_{1} \, \iota = 0$\,.
We write $K^{-1} = \sum_{g} K^{-1}_{g}$ and $\mu _{1}'' =\sum_{g} K_{g}$ for brevity.
Once $K^{-1}$ is given,
we get the abstract Hodge decomposition
\begin{align}
\label{dec in qft}
\mu ''_{1} \, K^{-1} + K^{-1} \, \mu ''_{1} = 1 - \iota \, \pi \, .
\end{align}
We often impose the conditions $\pi \, K^{-1}= 0$,
$K^{-1} \, \iota = 0$ and $(K^{-1} )^{2}=0$,
which is always possible without additional assumptions \cite{Crainic}.
As we see later,
these conditions are naturally realized under the $i \epsilon$-trick.
The decomposition (\ref{dec in qft}) induces a homotopy contracting operator $\mathsf{k}^{-1}_{\psi ''}$ for $Q_{S_{\mathrm{free}} [\psi '' ] } = ( S_{\mathrm{free}} [\psi '' ] , \hspace{2mm} )$ and it provides an alternative expression of (\ref{dec in qft})
\begin{align}
\label{Hodge relation}
Q_{S_{\mathrm{free}} [\psi '' ] } \, \mathsf{k}^{-1}_{\psi ''} + \mathsf{k}^{-1}_{\psi ''} \, Q_{S_{\mathrm{free}} [\psi '' ] } = 1 - \iota \, \pi \, .
\end{align}
Note that $\mathsf{k}^{-1}_{\psi ''}$ decreases space-time ghost number $1$ since $Q_{S_{\mathrm{free}} [\psi '' ] } $ increases $1$.
In terms of the kinetic operators $K_{g}$ and their propagators $K_{g}^{-1}$,
these BV operations can be represented as
\begin{align}
Q_{S_{\mathrm{free}} [\psi '' ]} & = - K_{0} \, \psi ''_{0} \frac{\partial }{\partial \psi _{0}^{\prime \prime \, \ast } }
- \sum_{g>0}
K_{g} \, \Big{[}
\psi _{g-1}^{\prime \prime \, \ast } \frac{\partial }{\partial \psi _{g}^{\prime \prime \, \ast }}
+ \psi ''_{g} \frac{\partial }{\partial \psi ''_{g-1}} \Big{]} \, ,
\\
\mathsf{k}^{-1}_{\psi ''} & = - \frac{K^{-1}_{0} }{n_{0} } \, \psi _{0}^{\prime \prime \, \ast } \, \frac{\partial }{\partial \psi ''_{0} }
- \sum_{g>0}
\frac{K^{-1}_{g} }{n_{g} } \, \Big{[}
\psi _{g}^{\prime \prime \, \ast } \, \frac{\partial }{\partial \psi _{g-1}^{\prime \prime \, \ast } }
+ \psi ''_{g-1} \, \frac{\partial }{\partial \psi ''_{g} } \Big{]} \, ,
\end{align}
where $n_{0}$ and $n_{g}$ are determined by the relation (\ref{Hodge relation}).
In the above normalization,
the operator $n_{g}$ counts the $\psi ''_{g}$-$\psi ''_{g-1}$ polynomial degree as $n_{g} ( \psi ''_{g} )^{\otimes m} (\psi ''_{g-1})^{\otimes n} = (m+n) (\psi ''_{g} )^{\otimes m} (\psi ''_{g-1})^{\otimes n}$\,.
Likewise,
by identifying $\psi ''_{-g-1} = \psi ^{\prime \prime \, \ast }_{g}$ for $g\geq 0$,
we find $n_{0} ( \psi ''_{0} )^{\otimes m} (\psi ^{\prime \prime \, \ast }_{0})^{\otimes n} = (m+n) (\psi ''_{0} )^{\otimes m} (\psi ^{\prime \prime \, \ast }_{0})^{\otimes n}$ and $n_{g} ( \psi _{g-1} ^{\prime \prime \, \ast })^{\otimes m} (\psi ^{\prime \prime \, \ast }_{g})^{\otimes n} = (m+n) (\psi _{g-1} ^{\prime \prime \, \ast } )^{\otimes m} (\psi ^{\prime \prime \, \ast }_{g})^{\otimes n}$\,.
We thus obtain $n_{g} (\psi '' )^{\otimes n} = n \, (\psi '')^{\otimes n}$.
\vspace{2mm}
Now,
we have the following homological data of the classical theory of free fields
\begin{align}
\label{old dr}
\mathsf{k}^{-1}_{\psi ''} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}'' ) , \, Q_{S_{\mathrm{free} } [\psi ']}
+ Q_{S_{\mathrm{free} } [ \psi '' ] } \, \big{)}
\hspace{3mm}
\overset{\pi }{\underset{\iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} (\mathcal{H}' \oplus \mathcal{H}''_{\mathrm{phys}} ), \, Q_{S_{\mathrm{free} } [\psi '] } \, \big{)}
\, ,
\end{align}
which is called a deformation retract.
Note that we must solve the equations of motion of $\psi ''$ to specify $\pi $ or $\iota $.
In order to define a propagator $\mathsf{k}^{-1}_{\psi ''}$,
we have to specify the off-shell and carry out its gauge-fixing if $\psi ''$ has any gauge or unphysical degree.
Therefore,
we must know how to solve the theory to obtain this homological data.
\vspace{2mm}
We expect that the perturbative path-integral (\ref{pre-statement}) can be found by transferring the relation (\ref{old dr}) into its quantum version (\ref{BV cohomology}) \textit{without} interactions since (\ref{pre-statement}) is an expectation value of the free theory.
The homological perturbation lemma enables us to perform such a transfer of homological data.
Clearly,
we can take a perturbation $\hbar \, \Delta $ since $\hbar \, \Delta _{S_{\mathrm{free} } [\psi '+ \psi '' ] } = \hbar \, \Delta + Q_{S_{\mathrm{free} } [\psi '+ \psi '' ] } $ is nilpotent.
Aa a result of the homological perturbation,
we obtain a new deformation retract
\begin{align}
\label{new dr}
\widehat{\mathsf{K}}^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}'' ) , \, \hbar \, \Delta _{S_{\mathrm{free} } [\psi '+ \psi '' ] } \, \big{)}
\hspace{3mm}
\overset{\widehat{P} }{\underset{\widehat{I} }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}''_{\mathrm{phys }} ) , \, \hbar \, \Delta '_{S_{\mathrm{free} } [\psi '] } \, \big{)}
\, ,
\end{align}
where morphisms $\iota $, $\pi $ and a contracting homotopy $\mathsf{k}^{-1}_{\psi ''}$ of the initial data (\ref{old dr}) are replaced by perturbed ones
\begin{align}
\widehat{I}
= \big{(} 1 + \mathsf{k}^{-1}_{\psi ''} \hbar \,\Delta \big{)}^{-1} \, \iota
\, ,
\hspace{5mm}
\widehat{P}
= \pi \, \big{(} 1+ \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} \big{)}^{-1} \, ,
\hspace{5mm}
\widehat{\mathsf{K}}^{-1} & = \mathsf{k}^{-1}_{\psi '' } \, \big{(} 1 + \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} \big{)}^{-1} \, .
\end{align}
As (\ref{Hodge relation}),
these operators satisfy the abstract Hodge decomposition with $\hbar \, \Delta _{S_{\mathrm{free} [\psi '+ \psi '' ] } }$ on the left side of (\ref{new dr}).
Note that $\widehat{I} = \iota $ follows from $\mathsf{k}^{-1}_{\psi ''} \Delta ' + \Delta ' \mathsf{k}^{-1}_{\psi '' } = 0$,
$(\Delta )^{2} = (\Delta ')^{2} = 0$ and $\mathsf{k}^{-1}_{\psi ''} ( \Delta '') \, \iota = 0$.
On the right side of (\ref{new dr}),
a new differential operator is given by
\begin{align}
\hbar \, \Delta '_{S_{\mathrm{free} }[\psi ']}
& \equiv Q_{S_{\mathrm{free} } [\psi ' ] }
+ \pi \, \hbar \, \Delta \, \widehat{I}
= Q_{S_{\mathrm{free} } [\psi ' ] }
+ \hbar \, \pi \, \Delta \, \iota \, .
\end{align}
Note that the differential $\pi \, \Delta '' \iota $ must vanish on $\mathcal{F} (\mathcal{H}''_{\mathrm{phys }} )$ to obtain $\Delta '' = \pi \, \Delta \, \iota $,
which is automatic when we consider the path-integral of off-shell states or gauge-and-unphysical degrees.\footnote{For the Hodge decomposition $\psi = \psi _{p} + \psi _{g} + \psi _{u}$,
the BV Laplacian $\Delta $ takes the form $\frac{\partial }{\partial \psi } \frac{\partial }{\partial \psi ^{\ast } } = \frac{\partial }{\partial \psi _{p} } \frac{\partial }{\partial \psi ^{\ast }_{p} } + \frac{\partial }{\partial \psi _{g} } \frac{\partial }{\partial \psi ^{\ast }_{u} } + \frac{\partial }{\partial \psi _{u} } \frac{\partial }{\partial \psi ^{\ast }_{g} }$.
We find $\pi \, \frac{\partial }{\partial \psi } \frac{\partial }{\partial \psi ^{\ast } } \, \iota = \pi \, \frac{\partial }{\partial \psi _{p} } \frac{\partial }{\partial \psi ^{\ast }_{p} } \, \iota $.
}
In order to consider the path-integral of physical fields,
we assume (\ref{imposing free BV}) or (\ref{ev}) explicitly.
While (\ref{imposing free BV}) gives $\pi \, \hbar \, \Delta '' \, \iota = 0$ directly,
(\ref{ev}) replaces $\pi $ by the composition $\mathrm{ev} \circ \pi $ that provides $\mathrm{ev} \circ \pi (\psi ) = 0$ for all $\psi \in \mathcal{H}$ and removes physical degrees as a result of (\ref{Gaussian}).
\vspace{2mm}
We show that the above $\widehat{P}$ obtained as a result of the homological perturbation indeed realizes the perturbative path-integral (\ref{pre-statement}).
Note that when we impose $\pi \, K^{-1} = 0$ in (\ref{dec in qft}),
the operator $\mathsf{k}^{-1}_{\psi ''}$ commutes with $\Delta '$ and vanishes on $\mathcal{H}''_{\mathrm{phys} }$.
We thus consider off-shell fields $\pi \, \psi '' = 0$,
where $\psi '' = \sum _{g \in \mathbb{Z}} \psi ''_{g}$ with $\psi ''_{-g} \equiv \psi ^{\prime \prime \, \ast }_{g-1}$ having ghost number $-g$.
We find
\begin{align}
\widehat{P} (\psi ''{}^{\otimes 2n})
& = \pi \, ( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} )^{n} (\psi '' {}^{\otimes 2n})
= \pi \, \frac{1}{n!} \Big{(} \frac{\hbar }{2} \, \sum_{g} K^{-1}_{g} \frac{\partial }{\partial \psi ''_{g } } \, \frac{\partial }{\partial \psi ''_{-g} } \Big{)}^{n} (\psi '' {}^{\otimes 2n})
\end{align}
because of $\pi \, ( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} )^{m} (\psi '' {}^{\otimes 2n}) = 0$ for $m \not=n$ and obtain
\begin{align}
\pi \, ( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} )^{n} (\psi '' {}^{\otimes 2n})
& = \pi \, ( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} )^{n-1}
\underbrace{ \Big{(} \frac{\hbar }{2n} \, \sum_{g} K^{-1}_{g} \frac{\partial }{\partial \psi ''_{g } } \, \frac{\partial }{\partial \psi ''_{-g} } \Big{)} (\psi '' {}^{\otimes 2n}) }_{\psi '' {}^{\otimes 2(n-1)} }
\nonumber\\
& = \pi \, ( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi ''} )^{n-2}
\frac{1}{n (n-1)} \underbrace{ \Big{(} \frac{\hbar }{2} \, \sum_{g} K^{-1}_{g} \frac{\partial }{\partial \psi ''_{g } } \, \frac{\partial }{\partial \psi ''_{-g} } \Big{)}^{2} (\psi ''{}^{\otimes 2n}) }_{\psi '' {}^{\otimes 2(n-2)} } \, .
\end{align}
It leads the Feynman graph expansion (\ref{Feynman rep}) thanks to $\pi \, K^{-1}=0$.
The condition $\pi \, K^{-1} = 0$ can be regarded as a result of the $i \epsilon $-trick:\footnote{More precisely,
to choose a pair $(\pi , \iota , K^{-1} )$ satisfying $\pi K^{-1} = 0$ and $K^{-1} \iota = 0$ implies that we use a propagator without propagation of on-shell states or we modify the mass shell itself to avoid poles,
which can be imposed for generic propagators by hand.
What is special about the $i \epsilon $-trick is that it automatically sets this condition. }
when we apply the $i \epsilon $-trick to the propagator in order for choosing a contour avoiding the on-shell poles,
each operators of the Hodge decomposition (\ref{Hodge relation}) are $i \epsilon$-modified.
Then we can impose $\pi \, \psi '' = 0$ on the original physical space $\mathcal{H}''_{\mathrm{phys} }$ because the poles of the mass shell are $i \epsilon$-shifted.
We thus obtain
\begin{align}
\label{integrated psi''}
\widehat{P} \big{(} e^{S_{\mathrm{int} } [\psi '+ \psi '' ] } \big{)}
& = \pi \, \sum_{n=0}^{\infty }( \hbar \, \Delta \, \mathsf{k}^{-1}_{\psi '' } )^{n} \big{(} e^{S_{\mathrm{int} } [\psi ' + \psi '' ] } \big{)}
\nonumber\\
& = \pi \exp{ \bigg{[} \frac{\hbar }{2} \, \sum_{g} K^{-1}_{g} \frac{\partial }{\partial \psi ''_{g } } \, \frac{\partial }{\partial \psi ''_{-g} } \bigg{]} } \big{(} e^{S_{\mathrm{int} } [\psi ' + \psi '' ] } \big{)} \, ,
\end{align}
which gives a functional of on-shell states $\pi \, \psi '' \in \mathcal{H}_{\mathrm{phys} } $.
In addition,
by introducing a source term $e^{J \psi ''}$ as (\ref{Feynman rep}),
we can represent (\ref{integrated psi''}) by the formula
\begin{align}
\label{realizing widehat P}
\widehat{P} \big{(} e^{S_{\mathrm{int} } [\psi ' + \psi '' ] + J \psi '' } \big{)}
= \exp{ \bigg{[} \frac{\hbar }{2} \, \sum_{g} K^{-1}_{g} \frac{\partial }{\partial \psi ''_{\mathrm{ev}\, g } } \, \frac{\partial }{\partial \psi ''_{\mathrm{ev} \, -g} } \bigg{]} } \big{(} e^{S_{\mathrm{int} } [\psi ' + \psi ''_{\mathrm{ev}} ] + J \psi ''_{\mathrm{ev}} } \big{)} \bigg{|}_{\psi ''_{\mathrm{ev}}
= \pi \, \psi '' }
\, .
\end{align}
These (\ref{integrated psi''}) and (\ref{realizing widehat P}) are nothing but the Feynman graph expansion (\ref{Feynman rep}) in the perturbative quantum field theory.
Note that the term $\frac{\hbar }{2} K^{-1}_{0} (\partial _{\psi _{\mathrm{cl} } })^{2} $ consists of classical fields and their propagators
and thus we obtain non-zero value after removing all antifields (and also ghosts) from (\ref{realizing widehat P}).
Hence,
quantum field theory \textit{without} gauge degrees can be treated within this framework and does not provide trivial results after the homological perturbation,
although its BV master action is the same as the classical action and the BV master equation looks trivial.
\vspace{2mm}
Note that the condition (\ref{imposing free BV}) or (\ref{ev}) is not necessary to derive (\ref{integrated psi''}) or (\ref{realizing widehat P}).
The formula (\ref{realizing widehat P}) takes the completely same form as (\ref{Feynman rep}) under (\ref{ev}).
More conventional calculations of the $S$-matrix may be provided by using (\ref{ev}),
rather than (\ref{imposing free BV}).
Actually,
when we calculate $S$-matrix with (\ref{ev}) providing $(\mathrm{ev} \circ \pi ) \, \psi '' = 0$,\footnote{
The equation $(\mathrm{ev} \circ \pi ) \, \psi '' = 0$ should be understood as a result of solving the $\psi ''$ equations of motion,
substituting it into $S[\psi ' + \psi '' ]$,
and evaluating the expectation value (\ref{ev}). }
we need to use $e^{\psi _{as} \mu _{1} \psi _{\mathrm{phys} } }$ as usual.
When we calculate $S$-matrix with (\ref{imposing free BV}),
$\pi \, \psi ''$ of (\ref{realizing widehat P}) should be understood as $\psi _{as}$ as we see in section 4.
\subsection{Homological perturbation performs the path-integral II}
In this section,
we construct a morphism $P$ performing the perturbative path-integral such that
\begin{align}
\label{statement}
P ( ... ) \equiv Z_{\psi '}^{-1} \, \widehat{P} \big{(} (...)e^{S [\psi ' + \psi '' ] - S_{\mathrm{free} } [\psi ''] } \big{)}
\overset{(\ref{pre-statement})}{=} Z_{\psi '}^{-1} \int \mathcal{D} [\psi ''] \, (...) \, e^{S[\psi ' + \psi '' ] } \, .
\end{align}
We expect that it can be found by transferring the homological data of (\ref{old dr}) into its fully quantum version including interactions.
Again,
the homological perturbation enables us to perform such a transfer.
We can take $\hbar \, \Delta _{S_{\mathrm{int}}[ \psi ] } = \hbar \, \Delta + Q_{S_{\mathrm{int} [\psi ] }}$ as a perturbation since $\hbar \, \Delta _{S [\psi ] } = \hbar \, \Delta + Q_{S [\psi ] } $ is nilpotent.
After the perturbation,
we obtain a new homological data as follows
\begin{align}
\label{quantum dr}
\mathsf{K}^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}'' ) , \, \hbar \, \Delta _{S[\psi ' + \psi '' ] } \,
\big{)}
\hspace{3mm}
\overset{P}{\underset{I}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}''_{\mathrm{phys }} ) , \, \hbar \, \Delta '_{A[\psi ' ]} \,
\big{)}
\, .
\end{align}
The perturbation lemma tells us how to construct morphisms $I$ and $P$ explicitly,
\begin{align}
I = \big{(} 1 + \mathsf{k}^{-1}_{\psi ''} \hbar \,\Delta _{S_{\mathrm{int}}[ \psi ] } \big{)}^{-1} \, \iota \, ,
\hspace{5mm}
P = \pi \, \big{(} 1+ \hbar \, \Delta _{S_{\mathrm{int}}[ \psi ] } \, \mathsf{k}^{-1}_{\psi ''} \big{)}^{-1} \, .
\end{align}
Likewise,
a contracting homotopy for $\hbar \, \Delta _{S[\psi ] }$ and the induced differential $\hbar \, \Delta '_{A[\psi ' ] }$ are given by
\begin{align}
\mathsf{K}^{-1} = \mathsf{k}^{-1}_{\psi ''} \, \big{(} 1 + \hbar \, \Delta _{S_{\mathrm{int}}[ \psi ] } \, \mathsf{k}^{-1}_{\psi ''} \big{)}^{-1}
\, \hspace{5mm}
\hbar \, \Delta '_{A[\psi ']} =
P \, \hbar \, \Delta _{S[\psi ] } \, \iota
= \pi \, \hbar \, \Delta _{S [\psi ] } \, I \, .
\end{align}
The statement (\ref{statement}) can be proved by tedious but direct calculations as section 2.4.
We however takes another pedagogical approach given by \cite{Doubek:2017naz}.
See also \cite{Albert} and \cite{Plumann}.
\vspace{2mm}
As is known,
the homological perturbation transfers a given deformation retract to a new deformation retract.
It therefore enables us to obtain the new Hodge decomposition
\begin{align}
\label{Hodge dec of F}
\big{(} \, 1 - I \, P \, \big{)} \, F = \Big{[} ( \hbar \, \Delta _{S[\psi ] } ) \, \mathsf{K}^{-1} + \mathsf{K}^{-1} \, ( \hbar \, \Delta _{S[\psi ] } ) \Big{]} \, F
\end{align}
for any $F [ \psi ] \in \mathcal{H}' \oplus \mathcal{H}''$\,.
Note that since $[ \, \mathsf{k}^{-1}_{\psi ''} , \Delta \, ] = \sum_{g} n_{g}^{-1} K^{-1}_{g} \partial _{\psi ''_{-g}} \partial _{\psi ''_{g}} $ acts on the off-shell states satisfying $K \, \psi '' \not= 0$,
it does not act on $\iota (...)$\,.
Because of $\widehat{P} \, I \, (...) = \widehat{P} \, \iota (...)$ with $\pi \, \mathsf{k}^{-1}_{\psi ''} =0$\,,
we find the following property of $I$ and $\widehat{P}$,
\begin{align}
\widehat{P} \, \Big{(} \, I \, (...) \, e^{S_{\mathrm{int} } [\psi ' + \psi '' ] } \Big{)}
& = \pi \, \Big{[} \,
\big{(} ( 1 + \hbar \, \Delta '' \, \mathsf{k}^{-1}_{\psi ''} )^{-1} \, e^{S_{\mathrm{int}} [\psi ' + \psi '' ] } \big{)} \,
\iota \, (...) \Big{]}
\nonumber\\ &
= \widehat{P} \, \Big{(} e^{S_{\mathrm{int}} [\psi ' + \psi '' ] } \Big{)} \,
\pi \, \iota \, (...) \, .
\end{align}
In other words,
since $\pi \, \iota =1$ on $\mathcal{H}''_{\mathrm{phys} }$,
we proved that $I(...)$ passes the $\psi ''$ integral as follows
\begin{align}
Z_{\psi '}^{-1} \int \mathcal{D} [\psi ''] \, I \, ( P F ) \, e^{S [\psi ' + \psi '' ] }
= Z_{\psi '}^{-1} \int \mathcal{D} [\psi ''] \, \iota \, ( P F ) \, e^{S [\psi ' + \psi '' ] }
= P F \, .
\end{align}
\vspace{2mm}
The abstract Hodge decomposition (\ref{Hodge dec of F}) elucidates that our morphism $P$,
a result of homological perturbation,
indeed performs the perturbative path-integral as follows
\begin{align}
P (F)
= Z_{\psi '}^{-1} \int \mathrm{D} [\psi ''] \, F \, e^{S [\psi ' + \psi '' ] }
- Z_{\psi '}^{-1} (\mathrm{extra}) \, .
\end{align}
We show that the extra term vanishes
\begin{align}
(\mathrm{extra}) \equiv
\hbar \, \widehat{P} \, \Big{(}
\big{(} \, \Delta _{S[\psi ] } \, \mathsf{K}^{-1} F + \mathsf{K}^{-1} \, \Delta _{S[\psi ] } \, F \, \big{)} \, e^{S [\psi ' + \psi '' ] - S_{\mathrm{free} } [\psi '' ] } \Big{)} = 0 \, .
\end{align}
The second term is trivially zero when we use $\mathsf{k}^{-1}_{\psi ''}$ satisfying the subsidiary condition $(\mathsf{k}^{-1}_{\psi ''})^{2} =0$ and $\pi \, \mathsf{k}^{-1}_{\psi ''} = 0$, which can be always imposed by dressing the old $\mathsf{k}^{-1}_{\psi ''}$ without any additional condition \cite{Crainic}.
Note that $(\mathsf{k}^{-1}_{\psi ''})^{2} =0$ gives $\widehat{P} \, \mathsf{K}^{-1} = \pi \, \mathsf{K}^{-1}$.
Thus,
$\pi \, \mathsf{k}^{-1}_{\psi ''} = 0$ provides
\begin{align}
\widehat{P} \, \Big{(} \, \mathsf{K}^{-1}(...) \, e^{S_{\mathrm{int} } [\psi ' + \psi '' ] } \, \Big{)}
= \pi \, \Big{[} \,
\mathsf{k}^{-1}_{\psi ''} \, (1 + \hbar \, \Delta _{S_{\mathrm{int} } [\psi ] } \mathsf{k}^{-1}_{\psi '' } )^{-1} \, (...)
e^{S_{\mathrm{int}} [\psi ' + \psi '' ] } \,
\Big{]}
= 0 \, .
\end{align}
This fact implies that after the path-integral,
as expected,
the $\mathsf{K}^{-1}$-exact quantities vanish
\begin{align}
\int \mathcal{D} [\psi '' ] \, \mathsf{K}^{-1} (...) \, e^{S[\psi ' + \psi '' ]} = 0 \, .
\end{align}
Actually,
the first term vanishes for similar reasons.
The morphism $\widehat{P}$ satisfies $\widehat{P} \, \Delta _{S_{\mathrm{free} }[\psi '']} = \Delta ' \, \widehat{P} $ because of its defining properties $\widehat{P} \, \Delta _{S_{\mathrm{free} } [\psi ] } = \Delta '_{S_{\mathrm{free} } [ \psi ' ] } \, \widehat{P}$ and $\widehat{P} \, e^{S_{\mathrm{free} } [\psi ' ] } = e^{S_{\mathrm{free} } [\psi ' ] } \, \widehat{P}$\,.
We find
\begin{align}
\widehat{P} \, \Big{(} \big{[} \Delta _{S [\psi ] } (...) \big{]} \, e^{S [\psi ] - S_{\mathrm{free} }[\psi '' ] } \, \Big{)}
= \widehat{P} \, \Big{(} \Delta _{S_{\mathrm{free} } [\psi '' ] } \big{[} (...) e^{S [\psi ] - S_{\mathrm{free} }[\psi '' ] } \big{]} \, \Big{)}
= \Delta ' \widehat{P} \, \Big{(} \big{[} (...) e^{S [\psi ] - S_{\mathrm{free} } [\psi '' ] } \big{]} \, \Big{)}
\, .
\end{align}
It implies that the $\psi ''$ integral maps the $\Delta _{S[\psi ]}$-exacts into $\Delta '$-exact quantities,
\begin{align}
\label{PD=D'P}
\int \mathcal{D} [\psi '' ] \, \Delta _{S [\psi ] } (...) \, e^{S[\psi ' + \psi '' ]}
= \Delta ' \, \bigg{[} \int \mathcal{D} [\psi '' ] \, (...) \, e^{S[\psi ' + \psi '' ]} \bigg{]} \, .
\end{align}
After applying this property,
the integrand of the first term becomes $\mathsf{K}^{-1}$-exact and gives zero.
Hence,
the statement (\ref{statement}) is proved.
Note also that
because of $Z_{\psi '} \, P(1) = \widehat{P} (1) = e^{A[\psi '] }$,
the relation (\ref{PD=D'P}) is nothing but the condition of morphism
\begin{align}
P \, \Delta _{S[\psi ]} = Z_{\psi '}^{-1} \, \Delta ' \, Z_{\psi '} \, P = \Delta '_{A[\psi ']} \, P \, .
\end{align}
\subsection{$A_{\infty }$ structure of the effective theory}
In the rest of this section,
we explain several properties that effective theories have as a result of the homological perturbation.
We consider the (quantum) $A_{\infty }$ structure of the effective theory,
\begin{align}
\mu ' ( \psi ', ... , \psi ' ) = \mu '_{1} (\psi ' ) + \mu '_{\mathrm{int}} ( \psi ' , ... , \psi ' ) \, ,
\end{align}
which is given by $\mu ' (\psi ' , ... , \psi ') \equiv \hbar \, \Delta '_{A[\psi ' ]} \psi ' $ for $\psi ' = \sum_{g} [ \psi '_{g} + \psi ' {}^{\ast }_{g} ]$.
The $A_{\infty }$ structure of the effective theory can be obtained by calculating the perturbed BV differential $\hbar \, \Delta '_{A[\psi ' ] } $.
Since $\mathsf{k}^{-1}_{\psi ''}$ commutes with $\Delta '$,
we find that it takes
\begin{align}
\hbar \, \Delta '_{A[\psi ' ] }
= Q_{S_{\mathrm{free} } [\psi ' ] }
+ \pi \, \sum_{n} ( \hbar \, \Delta ''_{S_{\mathrm{int}} [\psi ' +\psi ''] } \, \mathsf{k}^{-1}_{\psi ''} )^{n} \,
\hbar \, \Delta ' _{ S_{\mathrm{int } } [ \psi ' + \psi '' ] }
\, \iota \, .
\end{align}
Note that the commutator of the full perturbation $\hbar \, \Delta _{S_{\mathrm{int}} [\psi ] }$ and the propagator $\mathsf{k}^{-1}_{\psi ''}$,
\begin{align}
\big{[} \, \hbar \,\Delta _{S_{\mathrm{int}}[ \psi ] } , \, \mathsf{k}^{-1}_{\psi ''} \, \big{]}
= \hbar \, \sum_{g} \frac{K^{-1}_{g}}{n_{g} } \frac{\partial }{\partial \psi ''_{-g} } \frac{\partial }{\partial \psi ''_{g} }
+ \sum_{g} \frac{K^{-1}_{g}}{n_{g} } \, \mu _{\mathrm{int} } ( \psi , ... \psi ) \big{|}_{g}
\frac{\partial }{\partial \psi ''_{g} } \, ,
\end{align}
naturally includes the loop operator $L_{\check{\psi }'' \check{\psi }'' }$ and the tree grafting operator $T_{\check{\psi }''}$ defined by
\begin{align}
\hbar \, L_{\check{\psi }'' \check{\psi }'' } \equiv \hbar \, \sum_{g} \frac{K^{-1}_{g}}{n_{g} } \frac{\partial }{\partial \psi ''_{-g} } \frac{\partial }{\partial \psi ''_{g} } \, ,
\hspace{5mm}
T_{\check{\psi }'' } \equiv \sum_{g} \frac{K^{-1}_{g}}{n_{g} } \, \mu _{\mathrm{int} } ( \psi '+ \psi '' , ... , \psi ' + \psi '') \big{|}_{g} \frac{\partial }{\partial \psi ''_{g} } \, .
\end{align}
These provide basic manipulations of the $\psi ''$ Feynman graphs as follows
\begin{align}
L_{\check{\psi }'' \check{\psi }'' } \, \mu _{n+2} (
\psi , ... , \psi
) \big{|}_{\psi '' = 0}
& = \frac{1}{2} \sum_{s \in \mathbb{Z} } \sum_{i,j} \mu _{n+2} \big{(} \underbrace{
\psi ' , ... , \psi '}_{i} , \,
K^{-1}_{s} e_{s} , \, \underbrace{\psi ' , ... , \psi ' }_{n-i-j} , \,
e_{-s} , \, \underbrace{\psi ' , ... ,\psi '
}_{j} \big{)} \, ,
\\
T_{\check{\psi }'' } \, \mu _{n+1} ( \psi, ... , \psi ) \big{|}_{\psi '' = 0}
& = \sum_{k} \mu _{n+1} \big{(}
\underbrace{\psi ' ,
... , \psi ' }_{k} , \,
\mu_{\mathrm{int} } ( \psi ' , ... , \psi ' ) , \,
\underbrace{\psi ' , ... , \psi '
}_{n-k}
\big{)} \, .
\end{align}
Note that
since $n_{g} \, \psi ''{}^{\otimes m} = m \, \psi ''{}^{\otimes m}$,
each graph has appropriate coefficient,
such as
\begin{align}
\label{mention}
T_{\check{\psi } '' } \, T_{\check{\psi }'' } \, \,
\mu _{\mathrm{int}} \big{|}_{\psi ''=0}
= & \sum \mu _{\mathrm{int} }
\big{(} ... , \, K^{-1} \mu_{\mathrm{int} } ( ... , \, K^{-1} \mu_{\mathrm{int} } \, , \, ... ) , \, ... \big{)}
\nonumber\\ &\hspace{5mm}
+ 2 \sum \, \frac{1}{2} \,
\mu_{\mathrm{int} } ( ... , K^{-1} \mu_{\mathrm{int} } \, , ... , K^{-1} \mu_{\mathrm{int} } \, , ... ) \, .
\end{align}
We write $\pi (\psi ' + \psi '' ) = \varphi $ and $\iota (\varphi ) = \psi ' $ for clarity,
although $\varphi = \psi '$ for our case.
From $\hbar \, \Delta '_{A[\varphi ]}$ acting on $\varphi $,
we obtain the quantum $A_{\infty }$ structure of the effective theory as follows
\begin{align}
\label{effective A}
\mu ' ( \varphi , ... , \varphi )
= \mu _{1} (\varphi ) +
\sum_{n=0}^{\infty } \Big{[} \,
\hbar \, L_{\check{\psi }''_{\mathrm{ev} } \check{\psi }''_{\mathrm{ev}} } + T_{\check{\psi }''_{\mathrm{ev} } } \,
\Big{]}^{n}
\mu _{\mathrm{int} } ( \varphi + \psi ''_{\mathrm{ev} } , ... , \varphi + \psi ''_{\mathrm{ev} } ) \bigg{|}_{\psi ''_{\mathrm{ev} }= 0} \, .
\end{align}
Note that the effective vertices $\mu '_{\mathrm{int}} = \mu '_{2} + \mu '_{3} + \cdots $ have the $\hbar $ dependent parts,
\begin{align}
\label{expansion of mu'}
\mu '_{n} (\varphi , ... \varphi ) =
\mu '_{n, [0]} (\varphi , ... \varphi )
+ \hbar \, \underbrace{ \mu ' _{n,[1]} (\varphi , ... \varphi )
+ \hbar ^{2} \, \mu '_{n,[2] } (\varphi , ... \varphi )
+ \cdots }_{\mathrm{made \,\, from}\,\, L_{\check{\psi }''_{\mathrm{ev} } \check{\psi }''_{\mathrm{ev} } } } \,.
\end{align}
We consider $\varphi (t)$ such that $\varphi (0) = 0$ and $\varphi (1) = \varphi$ for $t \in \mathbb{R}$.
As a functional of $\varphi $,
by using $\varphi (t)$,
the effective action (\ref{effective BV action}) can be cast as
\begin{align}
\label{effective BV action alt}
A [\varphi ] = \int _{0}^{1} \, dt \, \big{\langle } \, \partial _{t} \, \varphi (t) , \, \mu ' \big{(} \, \varphi (t) , \dots , \varphi (t) \, \big{)} \, \big{\rangle } \, .
\end{align}
\subsection{The classical limit and cyclic $A_{\infty }$}
The classical part of the effective theory has a cyclic $A_{\infty }$ structure.
The effective $A_{\infty }$ structure (\ref{effective A}) has the non-trivial classical limit $\mu '_{\mathrm{tree} } \equiv \lim _{\hbar \rightarrow 0} \mu ' $,
which is obtained by setting $\hbar \rightarrow 0$ in (\ref{expansion of mu'}) as follows,
\begin{align}
\label{tree A}
\mu '_{\mathrm{tree}} ( \varphi , ... , \varphi )
= \sum_{n=0}^{\infty } \bigg{[}
\sum_{g} \frac{K^{-1}_{g}}{n_{g} } \, \mu _{\mathrm{int} } ( \varphi +\psi ''_{\mathrm{ev} } , ... , \varphi +\psi ''_{\mathrm{ev} } ) \Big{|}_{g}
\frac{\partial }{\partial \, \psi ''_{\mathrm{ev} \, g} } \bigg{]}^{n} \mu ( \varphi + \psi ''_{\mathrm{ev} } ) \bigg{|}_{\psi ''_{\mathrm{ev} } = 0} \, .
\end{align}
We write $A_{\mathrm{tree}} [\varphi ]$ for the classical part of the effective action $A[\varphi ]$,
which consists of tree graphs only.
By construction of (\ref{effective A}) and $\mu '_{\mathrm{tree} } (\varphi ) = Q_{A_\mathrm{tree} [\varphi ]} \, \varphi$,
we find
\begin{align}
Q_{A_{\mathrm{tree} } [\varphi ] } = \pi \, Q_{S [\psi ] } \, I_{\mathrm{tree}}
= P_{\mathrm{tree}} \, Q_{S [ \psi ] } \, \iota \, ,
\end{align}
where $I_{\mathrm{tree}}$ and $P_{\mathrm{tree}}$ are the classical limits of $I$ and $P$ respectively,
\begin{align}
I_{\mathrm{tree} } = \big{(} 1 +
\mathsf{k}^{-1}_{\psi ''} \, Q_{S_{\mathrm{int}}[ \psi ] } \, \big{)}^{-1} \, \iota \, ,
\hspace{5mm}
P_{\mathrm{tree} } = \pi \, \big{(} 1 +
Q_{S_{\mathrm{int}}[ \psi ] } \, \mathsf{k}^{-1}_{\psi ''} \, \big{)}^{-1} \, .
\end{align}
We can obtain these classical limits as a result of the perturbation $Q_{S_{\mathrm{int}} [ \psi ]}$ to (\ref{old dr}),
\begin{align}
\label{classical dr}
\mathsf{K}_{\mathrm{tree} }^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}'' ) , \, Q_{S[ \psi ] } \,
\big{)}
\hspace{3mm}
\overset{P_{\mathrm{tree} }}{\underset{I_{\mathrm{tree} }}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{F} ( \mathcal{H}' \oplus \mathcal{H}''_{\mathrm{phys }} ) , \, Q_{A[\varphi ]} \,
\big{)}
\, .
\end{align}
This fact implies that a morphism $P_{\mathrm{tree} }$ performs the classical part of the perturbative path-integral,
or the Feynman graph expansion grafting only trees,
as follows
\begin{align}
\label{tree partition function}
P_{\mathrm{tree} } ( ... ) = (Z^{\mathrm{tree} } _{\varphi } )^{-1}
\lim _{\hbar \rightarrow 0 }
\int \mathcal{D} [\psi '' ] \, (...) \, e^{S[\psi ' + \psi ''] }
\, .
\end{align}
The same result was obtained in \cite{Matsunaga:2019fnc} in terms of cyclic $A_{\infty }$ algebras.
In (\ref{tree partition function}),
we assumed that the perturbative partition function $Z_{\varphi }$ splits into the tree and loop parts,
\begin{align}
Z_{\varphi } = Z_{\varphi }^{\mathrm{tree}} \, \cdot \, Z_{\varphi }^{\mathrm{loop}} \, ,
\hspace{8mm}
Z_{\varphi }^{\mathrm{tree} } \equiv e^{ A_{\mathrm{tree} } [\varphi ] }
\, .
\end{align}
Thus,
if we interested in the tree part only,
the classical perturbation (\ref{classical dr}) is enough.
Actually,
by using these $I_{\mathrm{tree} }$, $P_{\mathrm{tree} }$,
a first few terms of (\ref{tree A}) are also calculated as follows
\begin{align*}
& \hspace{15mm} \mu '_{\mathrm{tree} ,\, 1} (\varphi ) = \mu _{1} ( \varphi ) \, ,
\hspace{5mm}
\mu '_{\mathrm{tree} ,\, 2} (\varphi , \varphi ) = \mu _{2} ( \varphi , \varphi ) \, ,
\\ & \hspace{-5mm}
\mu '_{\mathrm{tree} ,\, 3} (\varphi , \varphi , \varphi ) = \mu _{3} (\varphi , \varphi , \varphi )
+ \mu _{2} ( K^{-1} \mu _{2} ( \varphi , \varphi ) , \varphi )
+ \mu _{2} ( \varphi , K^{-1} \mu _{2} (\varphi , \varphi ) ) \, ,
\\
\mu '_{\mathrm{tree} ,\, 4} (\varphi , ... , \varphi ) & = \mu _{4} (\varphi , ... , \varphi )
+ \sum \mu _{3} ( \varphi , \varphi , K^{-1} \mu _{2} (\varphi , \varphi ) )
+ \mu_{2} ( K^{-1} \mu _{2} (\varphi , \varphi ) , K^{-1} \mu _{2} (\varphi , \varphi ) )
\nonumber\\ & \hspace{5mm}
+ \sum \mu _{2} ( \varphi , K^{-1} \mu _{3} (\varphi , \varphi , \varphi ) )
+ \sum \mu_{2} ( \varphi , K^{-1} \sum \mu _{2} ( \varphi , K^{-1} \mu _{2} (\varphi , \varphi ) ) ) \, ,
\end{align*}
where $\Sigma $ denotes the cyclic sum.
Note that as we mentioned in (\ref{mention}),
the propagator $\mathsf{k}^{-1}_{\psi ''}$ adjusts the coefficients and the restriction $\psi ''_{\mathrm{ev} } = 0$ picks up the appropriate contributions.
\section{Path-integral as a morphism of $A_{\infty }$}
All results obtained in section 2 can be written in terms of the (quantum) $A_{\infty }$ algebra and its morphism directly.
In this section,
we explicitly construct an effective $A_{\infty }$ structure $\mu '$ and a morphism $\textsf{p}$ between two $A_{\infty }$ structures $\mu $ and $\mu '$ such that
\begin{align}
\textsf{p} \, ( \mu _{1} + \mu _{2} + \cdots )
= (\mu '_{1} + \mu '_{2} + \cdots ) \, \textsf{p} \, ,
\end{align}
where $\mu = \mu _{1} + \mu _{2} + \cdots$ and $\mu ' = \mu '_{1} + \mu '_{2} + \cdots $ are (higher order) differentials acting on tensor algebras.
The perturbative path-integral map $P$ induces such $\mathsf{p}$,
which we explain.
\subsection{Tensor trick}
In order to extract $A_{\infty }$ products from (\ref{old dr}),
we first consider the state space $\mathcal{H}$ instead of $\mathcal{F}(\mathcal{H} )$,
on which $Q_{S_{\mathrm{free}} [\psi ] } \psi = \mu _{1} (\psi )$ and $\mathsf{k}^{-1}_{\psi ''} \psi = K^{-1} ( \psi )$ hold.
For brevity,
we write (\ref{old dr}) as
\begin{align}
\label{dr to co-dr}
\kappa ^{-1}_{\psi ''} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H} , \, \mu _{1} \, \big{)}
\hspace{3mm}
\overset{\pi }{\underset{\iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}' , \, \mu _{1}' \, \big{)}
\, .
\end{align}
By applying the tensor trick to each component of (\ref{dr to co-dr}),
we can obtain corresponding deformation retract of tensor algebras.
The identity map $\mathbb{I}$ of $\mathcal{H}$ and morphisms $\pi $ and $\iota $ can be extended to the identity $1 = 1_{\mathcal{T}(\mathcal{H} )}$ of $\mathcal{T}(\mathcal{H})$ and morphisms $\pi $ and $\iota $ of tensor algebras by defining
\begin{align}
1 |_{\mathcal{H}^{\otimes n } } = (\mathbb{I} )^{\otimes n} \, ,
\hspace{5mm}
\pi |_{\mathcal{H}^{\otimes n} } \equiv (\pi )^{\otimes n} \, ,
\hspace{5mm}
\iota |_{\mathcal{H ' }^{\otimes n} } \equiv (\iota )^{\otimes n} \, .
\end{align}
These are morphisms of tensor algebra preserving the cohomology
\begin{align}
\label{morph prop}
\pi \, (1 \otimes 1 ) = \pi \otimes \pi \, ,
\hspace{5mm}
\iota \, (1 \otimes 1 ) = \iota \otimes \iota \, ,
\end{align}
where $\otimes $ is the product $\otimes : \mathcal{T}(\mathcal{H} ) \otimes \mathcal{T}(\mathcal{H} ) \rightarrow \mathcal{T}(\mathcal{H} )$ of the tensor algebra.
The tensor algebra $\mathcal{T}(\mathcal{H} )$ can be regarded as a coalgebra.
Note that these $\pi $ and $\iota $ are also coalgebra morphisms
\begin{align}
\label{comor prop}
\widehat{\Delta } \, \pi = ( \pi \otimes \pi ) \, \widehat{\Delta } \, ,
\hspace{5mm}
\widehat{\Delta }\, \iota = ( \iota \otimes \iota ) \, \widehat{\Delta } \, ,
\end{align}
where $\widehat{\Delta }$ denotes the coproduct $\widehat{\Delta } : \mathcal{T}(\mathcal{H} ) \rightarrow \mathcal{T}(\mathcal{H} ) \otimes \mathcal{T}(\mathcal{H} )$ of coalgebra.
The $k$-linear map $\mu _{k} : \mathcal{H}^{\otimes k} \rightarrow \mathcal{H}$ can be extended to a linear map $\mu _{k}$ acting on the tensor algebra,
which becomes a (co-)derivation on $\mathcal{T} (\mathcal{H} )$,
and the contracting homotopy $\mathsf{k}^{-1}_{\psi ''}$ between $\mathbb{I}$ and $\iota \, \pi $ becomes a homotopy $\kappa ^{-1}$ between two morphisms $1$ and $\iota \, \pi $ of $\mathcal{T}(\mathcal{H} )$ by defining
\begin{align}
\label{deriv}
\mu _{k} |_{\mathcal{H}^{\otimes n} } \equiv \sum_{l} \mathbb{I}^{\otimes n-l } \otimes \mu _{k} \otimes \mathbb{I}^{\otimes l-k } \, ,
\hspace{5mm}
\kappa ^{-1} |_{\mathcal{H}^{\otimes n}} \equiv \sum_{l} \mathbb{I}^{\otimes n-l-1} \otimes \kappa ^{-1}_{\psi ''} \otimes ( \iota \, \pi )^{\otimes l} \, .
\end{align}
Namely,
$\mu _{k}$ is a derivation on the tensor products and $\kappa ^{-1}$ is a $(1, \iota \pi )$-derivation
\begin{align}
\label{deriv prop}
\mu _{k} \, (1 \otimes 1 ) = \mu _{k} \otimes 1 + 1 \otimes \mu _{k} \, ,
\hspace{5mm}
\kappa ^{-1} \, ( 1 \otimes 1 ) = \kappa ^{-1} \otimes \iota \pi + 1 \otimes \kappa ^{-1} \, .
\end{align}
Note that $\mu _{k}$ is a coderivation acting on $\mathcal{T} (\mathcal{H})$ and $\kappa ^{-1}$ is also a $(1, \iota \pi )$-coderivation
\begin{align}
\label{coder prop}
\widehat{\Delta } \, \mu _{k} \, (1 \otimes 1 ) = ( \mu _{k} \otimes 1 + 1 \otimes \mu _{k} ) \,\widehat{\Delta } \, ,
\hspace{5mm}
\widehat{\Delta } \, \kappa ^{-1} = ( \kappa ^{-1} \otimes \iota \pi + 1 \otimes \kappa ^{-1} ) \, \widehat{\Delta } \, ,
\end{align}
and thus $\kappa ^{-1}$ satisfies the characteristic property with the coproduct $\widehat{\Delta }$ as follows
\begin{align}
(\kappa ^{-1} \otimes 1 - 1 \otimes \kappa ^{-1} ) \, \widehat{\Delta } \, \kappa ^{-1} = ( \kappa ^{-1} \otimes \kappa ^{-1} ) \, \widehat{\Delta } \, .
\end{align}
We obtain the abstract Hodge decomposition on $\mathcal{T}(\mathcal{H} )$
\begin{align}
1 - \iota \, \pi = \mu _{1} \, \kappa ^{-1} + \kappa ^{-1} \, \mu _{1} \, ,
\end{align}
and thus we can consider a deformation retract of tensor algebras,
induced from (\ref{dr to co-dr}),
\begin{align}
\label{free co-dr}
\kappa ^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} (\mathcal{H} ) , \, \mu _{1} \, \big{)}
\hspace{3mm}
\overset{ \pi }{\underset{\iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T}(\mathcal{H}' ) , \, \mu '_{1} \, \big{)}
\, ,
\end{align}
which can be also regarded as a deformation retract of coalgebras.
The similar construction can be applied to (\ref{new dr}) or (\ref{classical dr}).
Note that $\pi $ and $\iota $ are $A_{\infty }$ morphisms such that
\begin{align}
\label{pi-iota}
\pi \, \mu _{1} = \mu '_{1} \, \pi \, ,
\hspace{5mm}
\mu _{1} \, \iota = \iota \, \mu '_{1} \, .
\end{align}
\vspace{2mm}
In the rest of this paper,
for simplicity,
we use the degree $0$ fields $\Psi \in \mathcal{\widehat{H}}$,
the degree $1$ linear maps $\{ \boldsymbol{\mu }_{n} \}_{n}$ acting on $\mathcal{T} ( \hat{\mathcal{H}} )$ and the degree $-1$ symplectic $\omega $ by using a set of bases $\{ \hat{e}_{a} \}_{a} $ carrying unphysical gradings such that $\Psi = \sum _{g} e_{-g} \otimes \psi _{g} + \sum _{g} e_{1+g} \otimes \psi ^{\ast }_{g}$,
which is explained in the appendix.
For a given derivation $\boldsymbol{\mu }_{1}$ acting on $\mathcal{T} ( \hat{\mathcal{H}} )$,
which can be defined as (\ref{adjusted A}) for a given $\mu _{1}$,
there is a contracting homotopy $\boldsymbol{\kappa }^{-1}$ satisfying
\begin{align}
\boldsymbol{\kappa }^{-1} \, (\Psi _{-g} ) \equiv - \mathsf{k}^{-1}_{\psi } \Psi _{g}
= (-)^{g+1} \hat{e}_{1+g} \otimes K^{-1} \, (\psi _{-g}) \, .
\end{align}
Likewise,
morphisms $\pi $ and $\iota $ of (\ref{free co-dr}) are extended in a natural way
\begin{align}
\boldsymbol{\pi } (\Psi _{g} ) \equiv \hat{e}_{-g} \otimes \pi (\psi _{g} ) \, ,
\hspace{5mm}
\boldsymbol{\iota } (\Psi _{g} ) \equiv \hat{e}_{-g} \otimes \iota (\psi _{g} ) \, .
\end{align}
Notice that these degree-adjusted operators can be defined for given $\mu _{1}$,
$\kappa ^{-1}$,
$\pi$ and $\iota$ uniquely once a set of symplectic bases is fixed.
They satisfy the abstract Hodge decomposition
\begin{align}
\label{Hodge ad}
1 - \boldsymbol{\iota } \, \boldsymbol{\pi } = \boldsymbol{\mu }_{1} \, \boldsymbol{\kappa }^{-1} + \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{1} \, ,
\end{align}
where $1$ denotes the unit of the tensor algebra $\mathcal{T} (\hat{\mathcal{H} } )$.
Therefore,
instead of (\ref{free co-dr}),
we can consider a deformation retract of the degree-adjusted $A_{\infty }$ algebra
\begin{align}
\label{dr free}
\boldsymbol{\kappa }^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} (\hat{\mathcal{H}} ) , \, \boldsymbol{\mu }_{1} \, \big{)}
\hspace{3mm}
\overset{ \boldsymbol{\pi } }{\underset{ \boldsymbol{\iota } }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T}( \hat{\mathcal{H}}' ) , \, \boldsymbol{\mu }'_{1} \, \big{)}
\, ,
\end{align}
which has the same algebraic or coalgebraic properties as (\ref{free co-dr}).
\subsection{Perturbing $A_{\infty }$ structure}
We consider the degree-adjusted $A_{\infty }$ structure (\ref{adjusted A}) and the perturbation of (\ref{dr free}).
We would like to extract information of the perturbation of $A_{\infty }$ structure from (\ref{classical dr}) or (\ref{quantum dr}).
Let us consider a derivation $Q_{S_{\mathrm{int} } [\Psi ]}$ acting on the tensor algebra $\mathcal{T} (\hat{\mathcal{H} } )$,
via the tensor trick.
We find
\begin{align}
Q_{S_{\mathrm{int} } [\Psi ] } \, \Psi ^{\otimes n} =
\sum_{k=1}^{n}
\Psi ^{\otimes k-1} \otimes \sum_{m \geq 2} \boldsymbol{\mu }_{m} ( \Psi ^{\otimes m} ) \otimes \Psi ^{\otimes n - k}
= \sum_{m \geq 2} \boldsymbol{\mu }_{m} \, \Psi ^{\otimes n+m-1} \, .
\end{align}
These derivations give the same results on the tensor algebra $\mathcal{T} (\hat{\mathcal{H}} ) $,
which becomes explicit if we take the sum of $\Psi ^{\otimes n}$.
We thus consider the group-like element of the tensor algebra $\mathcal{T} ( \hat{\mathcal{H} } )$,
\begin{align}
\frac{1}{1- \Psi } \equiv \mathbb{I} + \Psi + \Psi \otimes \Psi + \Psi \otimes \Psi \otimes \Psi + \cdots + \Psi ^{\otimes n} + \cdots \, .
\end{align}
By using this element $(1-\Psi )^{-1} \in \mathcal{T} ( \hat{\mathcal{H}} )$,
we find the equality of (co-)derivations
\begin{align}
\label{Q=mu}
\Big{[} \,
Q_{S_{\mathrm{int} } [\Psi ] } - \boldsymbol{\mu }_{\mathrm{int} } \, \Big{]} \, \frac{1}{1- \Psi } = 0 \, .
\end{align}
Hence,
as long as we consider the operators acting on the vector space $\mathcal{T} ( \hat{\mathcal{H} } )$,
we obtain the same results as the previous section even if we replace $Q_{S_{\mathrm{int}} }$ by $\boldsymbol{\mu }_{\mathrm{int} }$ in the homological perturbation.
\vspace{2mm}
We can extend the BV Laplacian $\Delta $ to a linear map acting on the tensor algebra $\mathcal{T} (\hat{\mathcal{H} } )$,
a second order derivation of the tensor algebra $\mathcal{T} ( \hat{\mathcal{H} } )$,
which provides
\begin{align}
\hbar \, \Delta \, \Psi ^{\otimes n }
= \hbar \, \sum_{k,l} \sum_{s \in \mathbb{Z}}
\Psi ^{\otimes k-1} \otimes \hat{e}_{-s} \otimes \Psi ^{\otimes n -k - l} \otimes \hat{e}_{1+s} \otimes \Psi ^{l-1}
= \hbar \,
\mathfrak{L} \, \Psi ^{\otimes n-2} \, .
\end{align}
The higher order coderivation $\mathfrak{L}$ is defined as follows.
For a given base $\hat{e}_{-s} (= \mathbb{I} \otimes \hat{e}_{-s}) \in \hat{\mathcal{H}}$\,,
we consider a derivation $\hat{e}_{-s}$ acting on $\mathcal{T}( \hat{\mathcal{H}})$ by defining $\hat{e}_{-s} |_{\hat{\mathcal{H}}^{\otimes n} } : \hat{\mathcal{H}}^{n} \rightarrow \hat{\mathcal{H}}^{n+1}$ as follows
\begin{align}
\hat{e}_{-s} |_{\hat{\mathcal{H}}^{\otimes n} } = \sum_{l}
\mathbb{I}^{\otimes l} \otimes \hat{e}_{-s} \otimes \mathbb{I}^{\otimes n-l}
\, ,
\end{align}
which is also a coderivation $\widehat{\Delta } \, \hat{e}_{-s} = ( \hat{e}_{-s} \otimes 1 + 1 \otimes \hat{e}_{-s} ) \, \widehat{\Delta }$\,.
Then,
a higher order coderivation $\mathfrak{L}$ is defined by
\begin{align}
\mathfrak{L} |_{\mathcal{H}^{\otimes n} }
= \sum_{l,m} \sum_{s \in \mathbb{Z} }
\mathbb{I}^{\otimes l} \otimes \hat{e}_{-s} \otimes \mathbb{I}^{\otimes m} \otimes \hat{e}_{1+s} \otimes \mathbb{I}^{\otimes n-l-m} \, .
\end{align}
This $\mathfrak{L}$ does not satisfy (\ref{coder prop}) as $\widehat{\Delta } \, \mathfrak{L} = ( \mathfrak{L} \otimes 1 + \sum_{s} (-)^{s} \hat{e}_{-s} \otimes \hat{e}_{1+s} + 1 \otimes \mathfrak{L} ) \, \widehat{\Delta }$\,.
Instead,
it satisfies the relation of order 2 coderivation as follows
\begin{align}
( \widehat{\Delta } \otimes 1 ) \widehat{\Delta } \, \mathfrak{L}
- ( \widehat{\Delta } \mathfrak{L} \otimes 1 + 1 \otimes \widehat{\Delta } \mathfrak{L} ) \, \widehat{\Delta }
+ ( \mathfrak{L} \otimes 1^{\otimes 2} + 1 \otimes \mathfrak{L} \otimes 1 + 1^{ \otimes 2} \otimes \mathfrak{L} ) \, ( \widehat{\Delta } \otimes 1) \widehat{\Delta } = 0 \, .
\end{align}
The equivalence of $\hbar \, \Delta$ and $\hbar \, \mathfrak{L}$ becomes manifest if we take the sum of $\Psi ^{\otimes n}$,
\begin{align}
\label{Delta=L}
\Big{[} \,
\hbar \, \Delta - \hbar \, \mathfrak{L} \,
\Big{]} \, \frac{1}{1- \Psi } = 0 \, .
\end{align}
Hence,
we can obtain the perturbed (quantum) $A_{\infty }$ structure directly by replacing $\hbar \, \Delta _{S_{\mathrm{int}} }$ with $\hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} }$ in the homological perturbation.
\subsection{Morphism of the cyclic $A_{\infty }$ structure}
Recall that the perturbed BV differential $h \Delta '_{A}$ provides the perturbed $A_{\infty }$ structure of (\ref{effective A}),
which is a result of the homological perturbation (\ref{new dr}).
We can derive the perturbed (quantum) $A_{\infty }$ structure $\boldsymbol{\mu }'$ directly by applying homological perturbation to this coalgebraic homological data (\ref{dr free}).
We first consider the classical part with the equality (\ref{Q=mu}).
We can take
\begin{align}
\boldsymbol{\mu }_{\mathrm{int} } = \boldsymbol{\mu }_{2} + \boldsymbol{\mu }_{3} + \boldsymbol{\mu }_{4} + \cdots
\end{align}
acting on $\mathcal{T} ( \mathcal{H} )$ as a perturbation for (\ref{dr free}) because of the $A_{\infty }$ relations of $\boldsymbol{\mu }= \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int}}$\,.
Note that this $\boldsymbol{\mu }_{\mathrm{int}}$ is a coderivation
$\widehat{\Delta } \, \boldsymbol{\mu }_{\mathrm{int}} = ( \boldsymbol{\mu }_{\mathrm{int}} \otimes 1 + 1 \otimes \boldsymbol{\mu }_{\mathrm{int}} ) \, \widehat{\Delta }$ and the coderivation $ \boldsymbol{\mu } = \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int} } $ is nilpotent $(\boldsymbol{\mu })^{2} = 0$\,.
We obtain the deformation retract of tensor algebras or coalgebras
\begin{align}
\mathsf{k}^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} (\hat{\mathcal{H}} ) , \, \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int}} \, \big{)}
\hspace{3mm}
\overset{\mathsf{p} }{\underset{\mathsf{i}}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T}(\hat{\mathcal{H}}' ) , \, \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}} \, \big{)}
\, ,
\end{align}
where $\mathsf{p}$ and $\mathsf{i}$ are morphisms preserving its cohomology and $\mathsf{k}^{-1}$ is a contracting homotopy.
The perturbed data also satisfy the abstract Hodge decomposition
\begin{align}
1 - \mathsf{i} \mathsf{p} = \boldsymbol{\mu } \, \mathsf{k}^{-1} + \mathsf{k}^{-1} \, \boldsymbol{\mu } \, .
\end{align}
The morphisms of tensor algebra $\mathsf{p}$ and $\mathsf{i}$ and the contracting homotopy $\mathsf{k}^{-1}$ of tensor algebra are obtained by solving recursive relations
\begin{align}
\label{defining properties}
\mathsf{p} = \boldsymbol{\pi } - \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\kappa }^{-1} \, ,
\hspace{5mm}
\mathsf{i} = \boldsymbol{\iota } - \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i} \, ,
\hspace{5mm}
\mathsf{k}^{-1} = \boldsymbol{\kappa }^{-1} - \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int}} \, \mathsf{k}^{-1} \, ,
\end{align}
where $\boldsymbol{\pi }$ and $\boldsymbol{\iota }$ are morphisms satisfying (\ref{morph prop}) and (\ref{comor prop}),
$\boldsymbol{\mu }_{\mathrm{int}}$ is a (co-)derivation satisfying (\ref{deriv prop}) and (\ref{coder prop}),
and $\boldsymbol{\kappa }$ is a contracting homotopy of the tensor algebra satisfying (\ref{deriv prop}) and (\ref{coder prop}).
By using this morphism $\mathsf{p}$ or $\mathsf{i}$\,,
the effective $A_{\infty }$ structure $\boldsymbol{\mu }' = \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}}$ is given by
\begin{align}
\label{effective co tree A}
\boldsymbol{\mu }'_{\mathrm{int}}
\equiv \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\iota }
= \boldsymbol{\pi } \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i} \, ,
\end{align}
where the second equality follows form (\ref{defining properties}) quickly
\begin{align}
\mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, ( \mathsf{i} + \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int}} \, \mathsf{i} )
= ( \mathsf{p} + \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} ) \, \boldsymbol{\mu }_{\mathrm{int}} \, \mathsf{i}
\, .
\end{align}
As a result of the perturbation,
the $A_{\infty }$ relations are automatic
\begin{align}
\big{(} \, \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int} } \, \big{)} ^{2} = 0 \, ,
\end{align}
which come from $( \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int}} )^{2} = 0$ and the defining properties (\ref{defining properties}),
\begin{align}
( \boldsymbol{\mu }'_{\mathrm{int}} )^{2} & = \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, ( \boldsymbol{\iota } \, \boldsymbol{\pi } ) \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i}
= \mathsf{p} \, ( \boldsymbol{\mu }_{\mathrm{int}} )^{2} \, \mathsf{i}
+ \underbrace{( - \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\kappa } )}_{\mathsf{p} - \boldsymbol{\pi } } \, \boldsymbol{\mu }_{1} \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i}
+ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\mu }_{1} \,
\underbrace{( - \boldsymbol{\kappa } \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i} ) }_{\mathsf{i} - \boldsymbol{\iota } }
\nonumber\\
& = \mathsf{p} \, \Big{[} ( \boldsymbol{\mu }_{\mathrm{int}} )^{2} + \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{1} \, \boldsymbol{\mu }_{\mathrm{int}} \Big{]} \, \mathsf{i}
- \boldsymbol{\mu }'_{1} \, \underbrace{( \boldsymbol{\pi } \, \boldsymbol{\mu }_{\mathrm{int}} \, \mathsf{i} )}_{ \boldsymbol{\mu }'_{\mathrm{int}} }
- \underbrace{( \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\iota } )}_{ \boldsymbol{\mu }'_{\mathrm{int}} } \, \boldsymbol{\mu }'_{1} \, .
\end{align}
Likewise,
the morphisms $\mathsf{p}$ and $\mathsf{i}$ become $A_{\infty }$ morphisms such that
\begin{align}
\label{p-i}
\mathsf{p} \, \boldsymbol{\mu } = \boldsymbol{\mu }' \, \mathsf{p} \, ,
\hspace{5mm}
\mathsf{i} \, \boldsymbol{\mu }' = \boldsymbol{\mu } \, \mathsf{i} \, ,
\end{align}
as long as the assumptions of the perturbation $\boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} \not= -1$ and $\boldsymbol{\kappa }^{-1} \boldsymbol{\mu }_{\mathrm{int}} \not= -1$ are provided.
Apparently,
when $\sum_{n}(- \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1})^{n}$ converges,
$1 + \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1}$ is invertible and (\ref{p-i}) follows from
\begin{align}
\label{p mu = mu' p}
\mathsf{p} \, \boldsymbol{\mu } - \boldsymbol{\mu }' \, \mathsf{p}
& = ( \boldsymbol{\pi } - \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\kappa }^{-1} ) \, \boldsymbol{\mu }_{1}
+ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, (
\boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{1}
+ \boldsymbol{\iota } \, \boldsymbol{\pi }
+ \boldsymbol{\mu }_{1} \, \boldsymbol{\kappa }^{-1} )
- \boldsymbol{\mu }' \, \mathsf{p}
\nonumber\\
& = ( \boldsymbol{\mu }'_{1} + \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \boldsymbol{\iota } ) \, \boldsymbol{\pi }
+ \mathsf{p} \, ( \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\mu }_{1} ) \, \boldsymbol{\kappa }^{-1}
- \boldsymbol{\mu }' \, \mathsf{p}
\nonumber\\
& = ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}} ) \, ( \boldsymbol{\pi } - \mathsf{p} )
- \mathsf{p} \,( \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int} } ) \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1}
\nonumber\\
& = \big{(} \boldsymbol{ \mu }' \, \mathsf{p}
- \mathsf{p} \, \boldsymbol{\mu } \, \big{)} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} \, .
\end{align}
Likewise,
we find $\boldsymbol{\mu } \, \mathsf{i} - \mathsf{i} \, \boldsymbol{\mu }' = - \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int}} \, \big{(} \boldsymbol{\mu } \, \mathsf{i} - \mathsf{i} \, \boldsymbol{\mu }' \big{)}$ and (\ref{p-i}) follows from it.
\vspace{2mm}
By solving (\ref{defining properties}),
we obtain the following expression of morphisms $\mathsf{p}$ and $\mathsf{i}$\,,
\begin{align}
\label{geo}
\mathsf{p} = \boldsymbol{\pi } \, \frac{1}{1+ \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int} } } \, ,
\hspace{5mm}
\mathsf{i} = \frac{1}{1+ \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int} } } \, \boldsymbol{\iota } \, .
\hspace{5mm}
\end{align}
Note that these expressions enable us to obtain (\ref{p-i}) directly from the second line of (\ref{p mu = mu' p}),
$\mathsf{p} \, ( \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int}} ) = ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}} ) \, \boldsymbol{\pi }
- \mathsf{p} \,( \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int} } ) \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1}$.
By substituting (\ref{geo}) into (\ref{effective co tree A}),
the effective cyclic $A_{\infty }$ structure can be cast as follows
\begin{align}
\boldsymbol{\mu }' = \boldsymbol{\mu }'_{1}
+ \boldsymbol{\pi } \, \boldsymbol{\mu }_{\mathrm{int} } \, \frac{1}{1 + \boldsymbol{\kappa }^{-1} \, \boldsymbol{\mu }_{\mathrm{int} }} \, \boldsymbol{\iota } \, .
\end{align}
In this tree-level case,
because of the coalgebraic properties (\ref{coder prop}) of $ \boldsymbol{\mu }_{\mathrm{int} }$\,,
tensor algebra morphisms $\mathsf{p}$ and $\mathsf{i}$ are also coalgebra morphisms and the derivation $ \boldsymbol{\mu }'$ is also a coderivation
\begin{align}
\label{co p-i}
\widehat{\Delta } \, \mathsf{p} = (\mathsf{p} \otimes \mathsf{p} )\, \widehat{\Delta } \, ,
\hspace{5mm}
\widehat{\Delta } \, \mathsf{i} = ( \mathsf{i} \otimes \mathsf{i} ) \, \widehat{\Delta } \, ,
\hspace{5mm}
\widehat{\Delta } \, \boldsymbol{\mu }' = ( \boldsymbol{\mu }' \otimes 1 + 1 \otimes \boldsymbol{\mu }' ) \, \widehat{\Delta } \, .
\end{align}
The third property quickly follows from the first or second property.
As long as $ \boldsymbol{\mu }_{\mathrm{int} }$ is a well-defined perturbation such that $ \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} \not= -1$ or $ \boldsymbol{\kappa }^{-1} \boldsymbol{\mu }_{\mathrm{int}} \not=-1$,
the first property follows from
\begin{align}
( \mathsf{p} \otimes \mathsf{p} ) \, \widehat{\Delta } \, (1+ \boldsymbol{\mu }_{\mathrm{int} } \boldsymbol{\kappa }^{-1}) =
( \boldsymbol{\pi } \otimes \boldsymbol{\pi } ) \, \widehat{\Delta } = \widehat{\Delta } \, \boldsymbol{ \pi } = \widehat{\Delta } \, \mathsf{p} \,
(1+ \boldsymbol{\mu }_{\mathrm{int} } \, \boldsymbol{\kappa }^{-1} ) \, ,
\end{align}
where the first equality follows from direct computation
\begin{align}
( \mathsf{p} \otimes \mathsf{p} ) \widehat{\Delta } ( \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} )
& = ( \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \otimes \mathsf{p} + \mathsf{p} \otimes \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} ) \widehat{\Delta } ( \boldsymbol{\kappa }^{-1} )
\nonumber\\
& = \Big{[} \,
\mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \otimes \boldsymbol{\pi } + \boldsymbol{\pi } \otimes \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}}
- \underbrace{ ( \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \otimes \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} + \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} \otimes \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} )
\Big{]}
\widehat{\Delta } ( \boldsymbol{\kappa }^{-1} ) }_{ ( \boldsymbol{\kappa }^{-1} \otimes 1 - 1 \otimes \boldsymbol{\kappa }^{-1} ) \widehat{\Delta } ( \boldsymbol{\kappa }^{-1} ) = ( \boldsymbol{\kappa }^{-1} \otimes \boldsymbol{\kappa }^{-1} ) \widehat{\Delta } }
\nonumber\\
& = ( \underbrace{ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} }_{ \boldsymbol{\pi } - \mathsf{p}} \otimes \boldsymbol{\pi }
+ \boldsymbol{\pi } \otimes \underbrace{ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} }_{ \boldsymbol{\pi } - \mathsf{p} } ) \, \widehat{\Delta }
- ( \underbrace{ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} }_{ \boldsymbol{\pi } - \mathsf{p}} \otimes \underbrace{ \mathsf{p} \, \boldsymbol{\mu }_{\mathrm{int}} \, \boldsymbol{\kappa }^{-1} }_{ \boldsymbol{\pi } - \mathsf{p}} ) \, \widehat{\Delta }
\, .
\end{align}
Likewise,
the second property of (\ref{co p-i}) holds.
Again,
(\ref{geo}) can solve (\ref{co p-i}) easily.
\subsection{Cyclicity of the effective $A_{\infty }$ structure}
Before adding the quantum part,
we consider the cyclicity.
Note that the cyclic property of the (perturbed) $A_{\infty }$ structure is manifest from the beginning because we consider the (effective) $A_{\infty }$ structure of the (effective) BV master action.
However,
we would like to show that the homological perturbation itself preserves the cyclic $A_{\infty }$ structure whenever a contracting homotopy satisfies the compatibility condition (\ref{bpz of h}).
In quantum field theory,
it is nothing but a Hermitian property of the propagators.
In string field theory,
it is the BPZ property.
\vspace{2mm}
We may write $\langle \omega | A \otimes B \equiv \omega ( A , B )$ for the symplectic structure on $\hat{\mathcal{H}}$ for simplicity.
The cyclic property of the $A_{\infty }$ structure $\boldsymbol{\mu } = \boldsymbol{\mu }_{1} + \boldsymbol{\mu }_{\mathrm{int} }$ can be cast as follows
\begin{align}
\label{bpz of mu}
\big{\langle } \, \omega \, \big{|} \, \boldsymbol{\mu }_{n} \otimes \mathbb{I} =
- \big{\langle } \, \omega \, \big{|} \, \mathbb{I} \otimes \boldsymbol{\mu }_{n} \, .
\end{align}
The perturbed $A_{\infty }$ structure $\boldsymbol{\mu }'$ provides an effective homotopy Maurer-Cartan action
\begin{align}
\label{hMC A}
\hat{A} [ \Psi ' ] = \sum_{n} \frac{1}{n+1} \omega ' \big{(} \, \Psi ' , \, \boldsymbol{\mu }'_{n} ( \Psi ' , ... , \Psi ' ) \, \big{)} \, ,
\end{align}
where $\Psi ' \in \hat{\mathcal{H} }' \equiv E \otimes \mathcal{H}'$ denotes effective fields.
When $\boldsymbol{\mu }$ is given by (\ref{adjusted A}),
it equals to (\ref{effective BV action}).
The symplectic structure $\omega ' $ on $\hat{\mathcal{H}}'$ is defined by using the inner product on $\mathcal{H}'$,
\begin{align}
\big{\langle } \, A' , \, B' \, \big{\rangle } '
= \big{\langle } \, \iota \, A' , \, \iota \, B' \, \big{\rangle }
\hspace{10mm}
A' , B' \in \mathcal{H}' \, ,
\end{align}
and the symplectic form $\hat{\omega }$ on $E$,
just as (\ref{symplec}).
We write $\langle \omega ' | A' \otimes B' = \omega ' ( A' , B' ) $ for this symplectic structure on $\hat{\mathcal{H}}$ for simplicity,
which provides
\begin{align}
\big{\langle } \, \omega ' \, \big{|} = \big{\langle } \, \omega \, \big{|} \, \boldsymbol{\iota } \otimes \boldsymbol{\iota } \, .
\end{align}
When we take Hermit propagators $K^{-1}$,
we find $\omega ( \boldsymbol{\kappa }^{-1} A , B ) =
(-)^{A} \omega ( A , \boldsymbol{\kappa }^{-1} B )$ quickly.
This compatibility of $\boldsymbol{\kappa }^{-1}$ and $\omega $ can be cast as follows
\begin{align}
\label{bpz of h}
\big{\langle } \, \omega \, \big{|} \, \boldsymbol{\kappa }^{-1} \otimes \mathbb{I}
= \big{\langle } \, \omega \, \big{|} \, \mathbb{I} \otimes \boldsymbol{\kappa }^{-1} \, .
\end{align}
As we see,
this property (\ref{bpz of h}) ensures the cyclicity of the perturbed $A_{\infty } $ structure.
\vspace{2mm}
When we have (\ref{bpz of mu}) and (\ref{bpz of h}),
the abstract Hodge decomposition (\ref{Hodge ad}) implies
\begin{align}
\big{\langle } \, \omega \, \big{|} \, \boldsymbol{\iota } \, \boldsymbol{\pi } \otimes \boldsymbol{\iota }
= \big{\langle } \, \omega \, \big{|} \, \mathbb{I} \otimes \boldsymbol{\iota } \, ,
\hspace{5mm}
\big{\langle } \, \omega \, \big{|} \, \boldsymbol{\iota } \otimes \boldsymbol{\iota } \, \boldsymbol{\pi }
= \big{\langle } \, \omega \, \big{|} \,\boldsymbol{\iota } \otimes \mathbb{I} \, .
\end{align}
Because of (\ref{pi-iota}),
it provides the cyclic property of $\boldsymbol{\mu }'_{1} = \boldsymbol{\pi } \, \boldsymbol{\mu }_{1} \, \boldsymbol{\iota }$ quickly
\begin{align}
\big{\langle } \, \omega ' \, \big{|} \, \boldsymbol{\mu }'_{1} \otimes \mathbb{I}' =
- \big{\langle } \, \omega ' \, \big{|} \, \mathbb{I}' \otimes \boldsymbol{\mu }'_{1} \, ,
\end{align}
where $\langle \omega ' | = \langle \omega | ( \boldsymbol{\iota } \otimes \boldsymbol{\iota } )$ is the symplectic form on $\hat{\mathcal{H} }'$ and $\mathbb{I}' \equiv \boldsymbol{\pi } \, \boldsymbol{\iota }$ denotes the unit of $\hat{\mathcal{H}}'$\,.
Likewise,
(\ref{bpz of mu}) and (\ref{bpz of h}) guarantees the cyclic property of $\boldsymbol{\mu }'_{\mathrm{int}} = \boldsymbol{\pi } \, \boldsymbol{\mu }_{\mathrm{int} } \, \mathsf{i}$ as follows
\begin{align}
\big{\langle } \, \omega \, \big{|} \, \boldsymbol{\mu }_{\mathrm{int} } \mathsf{i} \otimes \underbrace{(\mathsf{i} + \boldsymbol{\kappa }^{-1} \boldsymbol{\mu }_{\mathrm{int} } \mathsf{i} )}_{\boldsymbol{\iota }}
= - \big{\langle } \, \omega \, \big{|} \underbrace{(\mathsf{i} + \boldsymbol{\kappa }^{-1} \boldsymbol{\mu}_{\mathrm{int} } )}_{\boldsymbol{\iota }} \otimes \boldsymbol{\mu }_{\mathrm{int} } \mathsf{i}
\, .
\end{align}
Hence,
as long as we take $ \boldsymbol{\kappa }^{-1}$ satisfying (\ref{bpz of h}),
the cyclic property of $\boldsymbol{\mu }'$ is manifest
\begin{align}
\big{\langle } \, \omega ' \, \big{|}\, ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int} } ) \otimes \mathbb{I}'
= - \big{\langle } \, \omega ' \, \big{|} \, \mathbb{I}' \otimes ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int} } ) \, .
\end{align}
\vspace{2mm}
Note also that in the context of the quantum $A_{\infty }$ algebra,
the cyclic property is a part of the quantum $A_{\infty }$ relations and thus manifest by definition.
If the path-integral or corresponding results of homological perturbation can be understood as a morphism of the quantum $A_{\infty }$ structure,
the above result arises as a natural consequence of its classical limit.
\subsection{Morphism of the quantum $A_{\infty }$ structure}
Finally,
we include the quantum part with (\ref{Delta=L}).
Suppose that a solution $S$ of the classical master equation also solves the quantum one $\Delta S = 0$.
Then,
the cyclic $A_{\infty }$ structure $\mu = \mu _{1} + \mu _{\mathrm{int} }$ induced from $S$ satisfies the quantum $A_{\infty }$ relation
\begin{align}
\label{co quantum A}
\big{(} \, \boldsymbol{\mu } + \hbar \, \mathfrak{L} \, \big{)}^{2} = 0 \, ,
\end{align}
which is the coalgebraic representation of (\ref{quantum A}).
Since $\boldsymbol{\mu } + \hbar \, \mathfrak{L}$ is a nilpotent linear map acting on \textit{the vector space} $\mathcal{T} ( \hat{\mathcal{H}} )$,
we can take the following perturbation for (\ref{dr free}),
\begin{align}
\label{quantum mu}
\boldsymbol{\mu }_{\mathrm{int}} + \hbar \, \mathfrak{L} \, .
\end{align}
As a result of the homological perturbation,
we obtain the deformation retract describing the perturbative path-integral of quantum field theory
\begin{align}
\mathsf{K}^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} ( \hat{\mathcal{H}} ) , \, \boldsymbol{\mu }
+ \hbar \, \mathfrak{L} \, \big{)}
\hspace{3mm}
\overset{\mathsf{P} }{\underset{\mathsf{I}}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T}( \hat{\mathcal{H}}' ) , \, \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{eff}} \, \big{)}
\, .
\end{align}
Morphisms $\mathsf{P}$ and $\mathsf{I}$ and a contracting homotopy $\mathsf{K}^{-1}$ satisfy the abstract Hodge decomposition
\begin{align}
1 - \mathsf{I} \mathsf{P} = ( \boldsymbol{\mu } + \hbar \, \mathfrak{L} ) \, \mathsf{K}^{-1} + \mathsf{K}^{-1} \, (
\boldsymbol{\mu } + \hbar \, \mathfrak{L} ) \, .
\end{align}
These morphisms $\mathsf{P}$ and $\mathsf{I}$ are obtained by solving recursive relations
\begin{align}
\label{def of P-I}
\mathsf{P} = \boldsymbol{\pi } - \mathsf{P} \, ( \, \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } \, ) \, \boldsymbol{\kappa }^{-1} \, ,
\hspace{5mm}
\mathsf{I} = \boldsymbol{\iota } - \boldsymbol{\kappa }^{-1} \, ( \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } \, ) \, \mathsf{I} \, .
\end{align}
Note however that these $\mathsf{P}$ and $\mathsf{I}$ are not coalgebra morphisms and do not satisfy (\ref{comor prop}) because $\mathfrak{L}$ is a higher coderivative and does not satisfy (\ref{coder prop}).
The morphism $\mathsf{P}$ or $\mathsf{I}$ enables us to obtain the effective quantum $A_{\infty }$ structure $ \boldsymbol{\mu }' = \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{eff}}$ with
\begin{align}
\label{effective co quantum A}
\boldsymbol{\mu }'_{\mathrm{eff}} \equiv \boldsymbol{\pi } \, ( \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } ) \, \mathsf{I} = \mathsf{P} \, ( \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } ) \, \boldsymbol{\iota } \, .
\end{align}
Note also that $ \boldsymbol{\mu }'$ is not a coderivation and does not satisfy (\ref{coder prop}),
which may be regarded as a higher order coderivation of $IBL_{\infty }$ (or $IBA_{\infty }$) algebra \cite{Munster:2011ij, Munster:2012gy},
since $\mathfrak{L}$ is a second order.
These maps $\mathsf{P}$ and $\mathsf{I}$ are just morphisms of the vector space $\mathcal{T} ( \hat{\mathcal{H}})$ such that
\begin{align}
\label{P-I}
\mathsf{P} \, ( \boldsymbol{\mu } + \hbar \, \mathfrak{L} ) = ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}} ) \, \mathsf{P} \, ,
\hspace{5mm}
( \boldsymbol{\mu } + \hbar \, \mathfrak{L} ) \, \mathsf{I} = \mathsf{I} \, ( \boldsymbol{\mu }'_{1} + \boldsymbol{\mu }'_{\mathrm{int}} ) \, ,
\end{align}
which we call \textit{a morphism of the (quantum)} $A_{\infty }$ \textit{structure}.
These relations (\ref{P-I}) are proven by the same way as (\ref{p-i}),
which follows from the homological perturbation lemma.
\vspace{2mm}
In general,
when a given solution $S_{[0]}$ of the classical master equation $(S_{[0]} , S_{[0]} ) = 0$ does not solve the quantum master action,
it necessitates quantum corrections $\hbar \, S_{[1]} + \hbar ^{2} \, S_{[2]} + \cdots $ such that $S \equiv S_{[0]} + \hbar \, S_{[1]} + \hbar ^{2} \, S_{[2]} + \cdots $ satisfies $\hbar \, \Delta S + \frac{1}{2} ( S , S) = 0$\,.
Then,
the cyclic $A_{\infty }$ structure $\mu _{[0]}$ induced from $S_{[0]}$ does not satisfy the quantum $A_{\infty }$ relation,
$( \boldsymbol{\mu }_{[0]} + \hbar \, \mathfrak{L} )^{2} \not=0$.
Each correction $S_{[l]}$ gives components of quantum $A_{\infty }$ maps $\mu _{n , \, [l] } : \mathcal{H}^{\otimes n} \rightarrow \mathcal{H}$\,,
which is extended to coderivation of $\mathcal{T} (\hat{\mathcal{H}})$ by defining $ \boldsymbol{\mu }_{n , \, [l] } |_{\hat{\mathcal{H}}^{\otimes m} } : \hat{\mathcal{H}}^{m} \rightarrow \hat{\mathcal{H}}^{\otimes m-n+1}$ for $m \geq n$ otherwise zero as (\ref{deriv}).
For a given $S_{[l]}$\,,
we obtain the coderivation
\begin{align}
\boldsymbol{\mu }_{[l]} = \boldsymbol{\mu }_{0 , [l]} + \boldsymbol{\mu }_{1, [l] } + \boldsymbol{\mu }_{2, [l] } + \cdots \, ,
\end{align}
which are necessary for the quantum $A_{\infty }$ relations $( \boldsymbol{\mu }_{[0]} + \sum_{l} \hbar^{l} \, \boldsymbol{\mu }_{[l]} + \hbar \, \mathfrak{L} )^{2} =0$\,.
Hence,
in this case,
the above $ \boldsymbol{\mu }_{\mathrm{int}}$ of (\ref{quantum mu}) must be replaced by
\begin{align}
\boldsymbol{\mu }_{\mathrm{int}} = \boldsymbol{\mu }_{\mathrm{int} , [0] } + \sum_{l\geq1} \hbar ^{l} \, \boldsymbol{\mu }_{ [l] } \, .
\end{align}
This replacement of (\ref{quantum mu}) enables us to obtain the appropriate perturbed data in the same way.
We can express the solutions of the defining equations (\ref{def of P-I}) as follows
\begin{align}
\mathsf{P} = \boldsymbol{\pi } \, \frac{1}{ 1+ ( \, \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } \, ) \, \boldsymbol{ \kappa }^{-1} } \, ,
\hspace{5mm}
\mathsf{I} = \frac{1}{1+ \boldsymbol{ \kappa }^{-1} \, ( \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } \, ) } \, \boldsymbol{\iota } \, .
\end{align}
The perturbed quantum $A_{\infty }$ structure can be written as
\begin{align}
\label{solution mu'}
\boldsymbol{\mu }'
= \boldsymbol{\mu }'_{1}
+ \boldsymbol{\pi } \, ( \hbar \, \mathfrak{L} + \boldsymbol{\mu }_{\mathrm{int} } \, ) \, \frac{1}{1+ \boldsymbol{\kappa }^{-1} \, ( \hbar \, \mathfrak{L} + \boldsymbol{ \mu }_{\mathrm{int} } \, ) } \, \boldsymbol{\iota } \, ,
\end{align}
which takes the same form as (\ref{effective A}).
Its homotopy Maurer-Cartan action is given by (\ref{hMC A}) with replacing $\boldsymbol{\mu }'$ by (\ref{solution mu'}),
which equals to (\ref{effective BV action}) or (\ref{effective BV action alt}) derived in section 2.
\subsection{Twisted $A_{\infty }$ and source terms}
The BV master action including source terms or external fields gives an effective theory with a twisted $A_{\infty }$ structure.\footnote{It is an $A_{\infty }$ structure including the zeroth product $\mu '_{0}$, which is also called a weak $A_{\infty }$ or curved $A_{\infty }$. }
For a given BV master action,
we can couple $V$ as follows
\begin{align}
S_{V} [ \Psi ] \equiv S [\Psi ] - \omega \big{(} \, \Psi , \, V \, \big{)}
\end{align}
and suppose that this $S_{V} [\Psi ]$ and its free part also satisfy the BV master equation $\hbar \, \Delta \, e^{S_{V} [\Psi ] } = 0$\,.
Then,
source terms $V$ must satisfy the following properties with the $A_{\infty }$ structure of $S[\Psi ]$,
\begin{align}
\mu _{1} (V) = 0 \, ,
\hspace{5mm}
\sum _{n} \sum_{k=0}^{n} \mu _{n+1} ( \underbrace{\Psi , ... , \Psi }_{k} , V , \Psi , ... , \Psi ) = 0 \, ,
\end{align}
which we call gauge invariant source terms.
Then,
the source terms $\mu _{0} \equiv V$ become the zeroth product of a twisted $A_{\infty }$ structure $\mu + V$.
Note that $\mu $ itself is the $A_{\infty }$ structure and thus this $\mu +V$ is stronger than a generic twisted $A_{\infty }$ structure.
If we add such $V$ to our perturbation,
we find that the effective $A_{\infty }$ structure is twisted by $\kappa ^{-1} V$ as follows
\begin{align}
\mu _{V} ' = e^{- \kappa ^{-1} V} \, \mu ' \, e^{\kappa ^{-1} V} + V \, .
\end{align}
It becomes a twisted (quantum) $A_{\infty }$ structure,
whose zeroth element is
\begin{align}
\mu _{V , 0}' = V + \sum_{k=0}^{\infty }
\mu '_{k} \big{(} \, ( \kappa ^{-1} V)^{\otimes k } \, \big{)} \ .
\end{align}
Note that $\mu '$ itself is the $A_{\infty }$ structure and
the twisted $n$-product is given by
\begin{align}
\mu '_{V , n} ( \Psi ^{ \prime \, \otimes n} ) = \sum_{k=0}^{\infty } \sum _{k_{0} + \cdots + k_{n} =k}
\mu '_{n+k} \big{(} \, (\kappa ^{-1} V)^{\otimes k_{0} } , \Psi ' , (\kappa ^{-1} V)^{\otimes k_{1} } ,
... , \Psi ' , ( \kappa ^{-1} V)^{\otimes k_{n}} \, \big{)} \, .
\end{align}
\section{Application to string field theory}
In most string field theories,
fortunately,
we have classical or quantum BV master actions.
Hence,
we can apply the previous results and perform the perturbative path-integral of string fields on the basis of the homological perturbation.
Let us consider the BV master action of string field theory
\begin{align}
S [\Psi ]
= \frac{1}{2} \omega \big{(} \, \Psi , \, \boldsymbol{\mu }_{1} ( \Psi ) \, \big{)}
+ \sum_{n} \frac{1}{n+1} \omega \big{(} \, \Psi , \, \boldsymbol{\mu }_{n} ( \Psi , ... , \Psi ) \, \big{)} \, ,
\end{align}
where $\mu _{1} = Q$ is the BRST operator of strings,
$\{ \boldsymbol{\mu }_{n} \}_{n}$ denotes the set of the string products and $\omega $ is the degree $-1$ symplectic form induced from the BPZ inner product.
A state $\Psi = \sum_{g \in \mathbb{Z}} \hat{e}_{-g} \otimes \psi _{g}$ is a string field that includes all fields and antifields:
classical fields $\psi _{0} (x) $,
ghosts $\{ \psi _{g} (x) \}_{g>0}$ and antifields $\{ \psi _{g} (x) \}_{g<0}$ are associated with the (suspended) complete basis $\{ \hat{e}_{-g} \} _{g \in \mathbb{Z}}$ of conformal field theory,
where $\Psi $ corresponds to (\ref{total state}) and we write $\mathcal{H}$ for the state space spanned by $\Psi$ in this section.
The action and the sum of string products $\boldsymbol{\mu } (\Psi ) = \sum_{n} \boldsymbol{\mu }_{n} (\Psi ^{\otimes n})$ satisfy (\ref{def of A-relations}) or (\ref{def of quantum A-relations}) with $\boldsymbol{\mu } (\Psi ) = \hbar \, \Delta _{S} \Psi $.
By solving the free theory,
we obtain the Hodge decomposition
\begin{align}
\label{solving free theory}
1 - \iota \pi = \mu _{1} \, \kappa ^{-1} + \kappa ^{-1} \, \mu_{1} \, ,
\end{align}
which is the starting point of performing the perturbative path-integral.
\subsection{Effective theories with finite $\alpha '$}
Each effective theory based on the perturbative path-integral,
\begin{align}
\label{effective sft}
A [ \Psi ' ] = \frac{1}{2} \omega ' \big{(} \, \Psi ' , \, \boldsymbol{\mu }'_{1} ( \Psi ' ) \, \big{)}
+ \sum_{n} \frac{1}{n+1} \omega ' \big{(} \, \Psi ' , \, \boldsymbol{\mu }'_{n} ( \Psi ' , ... , \Psi ' ) \, \big{)} \, ,
\end{align}
always has the (quantum) $A_{\infty }$ or $L_{\infty }$ structure trivially,
as a result of the homological perturbation,
as long as the original action $S[\Psi ]$ solves the BV master equation.
In addition,
when the original action includes source terms $\omega ( \Psi ,V )$,
the effective action (\ref{effective sft}) has a weak (quantum) $A_{\infty }$ structure $\mu '_{V} = e^{-\kappa ^{-1} V } \mu ' e^{\kappa ^{-1} V }$ as shown in the previous section.
\vspace{2mm}
We can integrate all massive space-time fields $\Psi _{\mathrm{massive}}$ out from the string field $\Psi = \Psi _{\mathrm{massless}} + \Psi _{\mathrm{massive}}$ and get an effective field theory that consists of massless (plus auxiliary) fields $\Psi _{\mathrm{massless}}$ by using the Hodge decomposition
\begin{align}
1 - (\iota \pi )_{\mathrm{massless} } = \mu _{1} \, \kappa _{\mathrm{massive} }^{-1} + \kappa _{\mathrm{massive} }^{-1} \, \mu_{1} \, ,
\end{align}
where $\kappa _{\mathrm{massive}}^{-1}$ denotes propagators of massive fields and $(\iota \pi )_{M=0}$ denotes a projector onto the massless fields $\Psi _{\mathrm{massless}} = (\iota \pi )_{\mathrm{massless}} \Psi $\,.
We can construct these $\psi _{\mathrm{massless}}$,
$\iota _{\mathrm{massless}} $ and $\kappa _{\mathrm{massive}}^{-1}$ explicitly by solving the free theory,
which gives the effective action (\ref{effective sft}) with $\Psi ' = \pi _{\mathrm{massless}} ( \Psi ) = \pi _{\mathrm{massless}} ( \Psi _{\mathrm{massless}}) \in \mathcal{H}'$\,.
Likewise,
we can integrate space-time fields $\Psi _{\mathrm{UV}}$ having higher momentum $p > \Lambda $ out from the string field $\Psi = \Psi _{\mathrm{IR}} + \Psi _{\mathrm{UV}}$ and construct a Wilsonian effective action with the cut-off scale $\Lambda$ perturbatively.
It is obtained by using the Hodge decomposition
\begin{align}
1 - (\iota \pi )_{p \leq \Lambda } = \mu _{1} \, \kappa _{\mathrm{UV} }^{-1} + \kappa _{\mathrm{UV}}^{-1} \, \mu_{1} \, ,
\end{align}
where $(\iota \pi )_{p \leq \Lambda }$ denotes the restriction onto the lower momentum fields $(\iota \pi )_{p \leq \Lambda } \Psi = \Psi _{\mathrm{IR} }$ and $\kappa _{\mathrm{UV}}^{-1}$ denotes propagators of the higher momentum fields.
It provides (\ref{effective sft}) with $\Psi ' = \pi _{p \leq \Lambda } (\Psi ) \in \mathcal{H} '$\,.
In the same manner,
for any decomposition (\ref{solving free theory}),
we can obtain corresponding effective action.
\vspace{2mm}
We would like to emphasize that the physically important information is in how to construct these projectors and (regular) propagators concretely: to give the Hodge decomposition is equivalent to solving the theory.
We thus started from the free theory and considered perturbations.
\subsection{Light-cone reduction}
In string field theory,
explicit Lorentz covariance is given in return for adding the gauge and unphysical degrees.
Thus,
while the light-cone theory consists of physical degrees,
the covariant theory has the gauge invariance.
We can remove theses extra degrees by using the path-integral and obtain a light-cone string field theory for each covariant string field theory \cite{Matsunaga:2019fnc, EM}.
\vspace{2mm}
We write $Q$ for the BRST operator of the world-sheet theory and $\omega $ for the BPZ inner product of its conformal field theory.
We consider a covariant string field theory,
\begin{align}
S [\Psi ] = \frac{1}{2} \omega \big{(} \Psi , \, Q \, \Psi \big{)}
+ \frac{1}{3} \omega \big{(} \Psi , \, \boldsymbol{m}_{2} ( \Psi , \Psi ) \, \big{)}
+ \cdots \, .
\end{align}
It has an $A_{\infty }$ (or $L_{\infty }$) structure $\boldsymbol{m}$ with $\boldsymbol{m}_{1} = Q$ as long as it satisfies the BV master equation.
There exists a similarity transformation $U$ connecting the BRST operator $Q$ and the kinetic operator $L_{0}^{lc}$ in the light-cone gauge plus the differential $d$ acting on the gauge and unphysical degrees \cite{Aisaka:2004ga},
which diagonalize physical and extra degrees as follows
\begin{align}
Q = U^{-1} \,
\big{(}
c_{0} \, L_{0}^{lc} + d
\, \big{)} \,
U \, .
\end{align}
Note that $(c_{0} L^{lc}_{0} ) ^{2} = 0$ holds in addition to $(c_{0} L_{0}^{lc} + d \, )^{2} = 0$ and these are defined on the critical dimention.\footnote{
For bosonic open strings,
by using $\lambda =1$ Virasoro generators $\tilde{L}_{n} = \tilde{L}_{n}^{\parallel }+ \tilde{L}_{n}^{\perp }$,
these are given by
\begin{align*}
U \equiv \exp \Big{[} - c_{0} \underbrace{\frac{1}{p^{+} } \sum_{n \not= 0} a^{+}_{-n} b_{n} }_{h }
\Big{]}
\exp \Big{[} \frac{1}{p^{+}} \sum _{n \not= 0} \frac{1}{n} a^{+}_{-n} \tilde{L}_{n} \Big{]} \, ,
\hspace{3mm}
d \equiv - p^{+} \sum_{n \not= 0} a^{-}_{n} \, c_{-n} \, ,
\hspace{3mm}
\kappa ^{-1} \equiv h \int_{0}^{1} dt \, e^{- t \tilde{L}_{0}^{\parallel } } \, .
\end{align*}
Note that that $\tilde{L}_{n}$ commutes with $a ^{+}_{n}$ and $L_{0}^{\parallel } = dh+hd$ counts excitations of $\{ a^{\pm }_{n}, b_{n} , c_{n} \}_{n\not= 0}$.
See \cite{Aisaka:2004ga, Matsunaga:2019fnc}. }
The similarity transformation $U$ becomes an isomorphism $\boldsymbol{U} \,\boldsymbol{m} = \boldsymbol{\mu } \, \, \boldsymbol{U}$ and gives the diagonalized $A_{\infty }$ structure $\boldsymbol{\mu }$ which is defined by $\boldsymbol{\mu }_{1} \equiv c_{0} \, L_{0}^{lc} + d$ and
\begin{align}
\boldsymbol{\mu }_{n} \equiv U \, \boldsymbol{m}_{n} \,( U^{-1} \otimes \cdots \otimes U^{-1} ) \, .
\end{align}
It gives a linear transformation of the conformal basis and thus provides a linear string-field redefinition $\Phi \equiv U \, \Psi$\,.
We obtain the diagonalized action with the $A_{\infty }$ structure $\boldsymbol{\mu }$\,,
\begin{align}
\label{cov}
S [\Phi ] = \frac{1}{2} \omega \big{(} \Phi , \, ( \underbrace{ c_{0} \, L_{0}^{lc} + d }_{\mu _{1} } \, ) \Phi \big{)}
+ \sum_{n=2}^{\infty } \frac{1}{n+1} \omega \big{(} \Phi , \, \boldsymbol{\mu }_{n} ( \underbrace{\Phi , \cdots , \Phi }_{n} ) \, \big{)}
\, .
\end{align}
The extra degrees become the BRST quartets and thus $d$ has no cohomology unless there is no quartet excitation.
As is known,
the integration of the BRST quartets is volume $1$ since bosonic and fermionic integrations exactly cancel each other.
We can start with the BRST quartets,
\begin{align}
\label{quartet}
\kappa ^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H} , \, d \, \big{)}
\hspace{3mm}
\overset{\pi }{\underset{\iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}_{\mathrm{lc} } , \, 0\, \big{)}
\, ,
\end{align}
where $\kappa ^{-1} $ denotes the propagator for $d$ and $\mathcal{H}_{lc}$ is the state space of string fields in the light-cone gauge.
We take $\pi : \mathcal{H} \rightarrow \mathcal{H}_{\mathrm{lc}}$ and $\iota : \mathcal{H}_{\mathrm{lc}} \rightarrow \mathcal{H}$ as natural projection and embedding.\footnote{
For the Fock vacuum $| \Omega \rangle \equiv | \mathrm{lc} \rangle \otimes | a^{\pm}, b,c \rangle$,
we define $\pi : |\Omega \rangle \mapsto |\mathrm{lc} \rangle$ and $\iota : |\mathrm{lc} \rangle \mapsto |\Omega \rangle $.
For excitations on these vacua, we define
$\pi \circ ( p^{\mu } , a_{n}^{I} , c_{0} ; a_{n}^{\pm } , c_{n} , b_{n} ) = ( p^{\mu } , a_{n}^{I} , c_{0} ) \circ \pi$ and $\iota \circ ( p^{\mu } , a_{n}^{I} , c_{0} ) = ( p^{\mu } , a_{n}^{I} , c_{0} ; 0 , 0, 0) \circ \iota $ for $n \not=0$. }
We can take $c_{0} L^{lc}$ as a perturbation to (\ref{quartet}) and get
\begin{align}
\label{no-ghost}
\kappa ^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H} , \, c_{0} L_{0}^{lc} + d \, \big{)}
\hspace{3mm}
\overset{\pi }{\underset{\iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}_{\mathrm{lc} } , \, c_{0} \, L_{0}^{lc} \, \big{)}
\, .
\end{align}
It describes the no-ghost theorem of covariant strings \cite{Kato:1982im}.
We can take a further perturbation $\mu _{\mathrm{int} }$ for (\ref{no-ghost}) because of the $A_{\infty }$ structure $(\mu _{1} + \mu _{\mathrm{int} } )^{2} = 0$ and obtain
\begin{align}
\label{lc red}
\mathsf{k}^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} ( \mathcal{H} ) , \, \boldsymbol{\mu _{1} } + \boldsymbol{\mu _{\mathrm{int} }}
\, \big{)}
\hspace{3mm}
\overset{\mathsf{P} }{\underset{\mathsf{I} }{
\scalebox{2.5}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T} ( \mathcal{H}_{\mathrm{lc} } ) , \, \boldsymbol{\nu _{1}^{lc}} +
\boldsymbol{\nu _{\mathrm{int}}^{lc} } \, \big{)}
\, .
\end{align}
While the left side has the $A_{\infty }$ structure $\boldsymbol{\mu }$ of the covariant string field theory (\ref{cov}),
the right side provides the transferred $A_{\infty }$ structure $\boldsymbol{\nu ^{lc}}$ of the light-cone string field theory.
By using the light-cone string field $\varphi \in \mathcal{H}_{\mathrm{lc}}$ and the light-cone vertices $\boldsymbol{\nu _{\mathrm{int} }^{lc} } \equiv \boldsymbol{\pi } \, \boldsymbol{\mu _{\mathrm{int}} } \, \mathsf{I} $,
we obtain the light-cone string field theory $S_{\mathrm{lc} } [\varphi ]$ extracted from the covariant theory (\ref{cov}),
\begin{align}
\label{lc}
S_{\mathrm{lc} } [ \varphi ]
= \frac{1}{2} \omega \big{(} \varphi , \, c_{0} \, L_{0}^{lc} \, \varphi \big{)}
+ \sum_{n=2}^{\infty } \frac{1}{n+1} \omega \big{(} \varphi , \, \boldsymbol{\nu }^{lc}_{n} (\underbrace{ \varphi , \dots , \varphi }_{n} ) \big{)} \, ,
\end{align}
where we used loose notation $\varphi = \iota (\varphi ) $ and $c_{0} \, L^{lc}_{0} = \pi \, ( c_{0} \, L_{0}^{lc} ) \, \iota = \mu _{1}$ for simplicity.
Note that the vertices $\boldsymbol{\nu _{\mathrm{int} }^{lc} }$ consists of the original vertices $\boldsymbol{\mu _{\mathrm{int} } }$ (with projections and embeddings) and \textit{effective} vertices $\mu _{\mathrm{eff} }$ including propagators $\boldsymbol{\kappa ^{-1}}$ as follows
\begin{align*}
\boldsymbol{\nu _{\mathrm{int} }^{lc} }
= \boldsymbol{\pi } \,
\boldsymbol{\mu _{\mathrm{int} } }
\, \boldsymbol{\iota }
+ \boldsymbol{\pi } \,
\underbrace{
\bigg[
\sum_{n=1}^{\infty }
(-)^{n} \boldsymbol{\mu _{\mathrm{int} } }
\big{(} \boldsymbol{\kappa ^{-1} } \, \boldsymbol{\mu _{\mathrm{int} } } \big{)}^{n}
+ \sum_{g} \hbar ^{g} \, (\mathrm{g\mathchar`-loop})
\bigg]
}_{\mu _{\mathrm{eff} } (\varphi , ... , \varphi ) }
\, \boldsymbol{\iota }
\, .
\end{align*}
In this sense,
the light-cone reduction (\ref{lc red}) can be cast as the form which consists of
the light-cone kinetic term,
the original vertices,
and effective vertices.
Hence,
the action (\ref{lc}) has higher interacting terms and takes the different form from the original covariant theory (\ref{cov}) unless all of the effective vertices $\mu _{\mathrm{eff}} (\varphi , ... , \varphi )$ exactly equal to zero.
See \cite{EM} for further discussions.
\subsection{S-matrix and asymptotic string fields}
When a given (quantum) $A_{\infty }$ structure $\boldsymbol{\mu } = \boldsymbol{\mu }_{2} + \cdots $ has no linear part $\boldsymbol{\mu }_{1}$,
it is called minimal.
The $S$-matrix is realized as a minimal model,
which can be obtained by using the homological perturbation.
The uniqueness of the minimal $A_{\infty }$ structure is ensured by the minimal model theorem in mathematics.
In terms of physics,
it implies that the \textit{on-shell} amplitudes are independent of given gauge-fixing conditions or propagators and thus are unique.
\vspace{2mm}
The S-matrix is a set of multi-linear forms $\{ \mathcal{A} _{n} \} _{n \geq 3}$ defined on the tensor algebra $\mathcal{T} (\mathcal{H}_{as} )$ of the state space $\mathcal{H}_{as}$,
whose inputs are asymptotic string fields $\Psi _{as} \in \mathcal{H} _{as }$.
We consider the free action of asymptotic string fields,
\begin{align}
\label{asympt}
S_{as} [ \Psi _{as} ]
= \frac{1}{2} \omega \big{(} \, \Psi _{as} , Q \, \Psi _{as} \, \big{)} \, .
\end{align}
The asymptotic string field $\Psi _{as} \in \mathcal{H} _{as}$ has the linear gauge invariance $\delta \Psi _{as} = Q \, \lambda _{as}$ and the physical states condense on the cohomology of $Q$ acting on $\mathcal{H} _{as}$.
We assume that the cohomology $\mathcal{H} _{as \, \mathrm{phys}}$ of the asymptotic theory is isomorphic to that of the free theory,
$\mathcal{H} _{\mathrm{phys}} \equiv I \, ( \mathcal{H}_{as \, \mathrm{phys}} )$.
\vspace{2mm}
We first solve the free theory and derive a propagator $\kappa ^{-1}$,
which gives the Hodge decomposition (\ref{solving free theory}).
Then,
by defining morphisms $\iota _{as } \equiv \iota \, I$ and $\pi _{as } \equiv I^{-1} \pi $ that satisfy $\pi _{as} \, \kappa ^{-1} = \iota _{as} \, \kappa ^{-1} = 0$,
we can consider
\begin{align}
\label{asymptotic}
\boldsymbol{\kappa ^{-1} } \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} ( \mathcal{H} ) , \, \boldsymbol{\mu _{1} } \, \big{)}
\hspace{3mm}
\overset{\boldsymbol{\pi _{as}} }{\underset{\boldsymbol{\iota _{as}}}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T} ( \mathcal{H} _{as \, \mathrm{phys} } ) , \, \boldsymbol{0} \, \big{)}
\hspace{3mm}
\overset{\boldsymbol{I } }{\underset{\boldsymbol{I^{-1} }}{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{T} ( \mathcal{H}_{\mathrm{phys} } ) , \, \boldsymbol{0} \, \big{)}
\, .
\end{align}
Because of $\iota \, \pi = \iota \, (I \, I^{-1} ) \, \pi = \iota _{as } \, \pi _{as} $,
we quickly find $1 - \iota _{as} \pi _{as} = \mu _{1} \kappa ^{-1} + \kappa ^{-1} \mu _{1}$\,.
The minimal model is obtained by taking interacting terms $\boldsymbol{\mu _{\mathrm{int}} } $ as the perturbation to (\ref{asymptotic}).
The (quantum) $A_{\infty }$ structure of the $S$-matrix is given by the right side of
\begin{align}
\mathsf{K}^{-1}_{as} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{T} ( \mathcal{H} ) , \, \boldsymbol{\mu _{1}} + \boldsymbol{\mu _{\mathrm{int} } } \, \big{)}
\hspace{3mm}
\overset{ \mathsf{P}_{as} }{\underset{\mathsf{I}_{as} }{
\scalebox{4.5}[1]{$\rightleftarrows $} }
}
\hspace{3mm}
\big{(} \, \mathcal{T} ( \mathcal{H} _{as \, \mathrm{phys} } ) , \, \boldsymbol{\mu '_{\mathrm{int} } } \, \big{)}
\, .
\end{align}
This is a minimal model because $\mu '_{1} \equiv Q$ vanishes
and it has no gauge degree.
The morphism $\mathsf{P}_{as}$ determines a nonlinear field relation between interacting and asymptotic theories on-shell.
The $(n+1)$-point amplitude is given by the $\mu '_{n}$ part of the homotopy Maurer-Cartan action
\begin{align}
\label{S-matrix}
\mathcal{A} [ \Psi ' ]
= \sum_{n} \frac{1}{n+1} \omega ' \big{(} \, \Psi '_{as} , \, \boldsymbol{\mu }'_{n} ( \Psi '_{as} , ... , \Psi '_{as} ) \, \big{)} \, .
\end{align}
It defines multi-linear maps acting on the on-shell asymptotic string fields.
As we showed,
it is the same as the Feynman graph expansion and thus gives the amplitudes correctly.
In addition,
as long as it is minimal,
the $A_{\infty }$ relation $(\mu '_{\mathrm{int} } )^{2} = 0$ implies the BRST identities
\begin{align}
\label{BRST id}
\omega ' \big{(} \, Q \, \Psi '_{0} , \, \boldsymbol{\mu }'_{n} ( \Psi '_{1} , ... , \Psi '_{n} ) \, \big{)}
+ \sum_{k=1}^{n} \omega ' \big{(} \, \Psi '_{0} , \, \boldsymbol{\mu }'_{n} ( \Psi '_{1} , ... , Q \, \Psi '_{k} , ... , \Psi '_{n} ) \, \big{)} = 0 \, ,
\end{align}
which corresponds to the Stokes theorem.
Hence,
even if we replace $\mathcal{H}_{as \, \mathrm{phys} }$ by $\mathrm{Ker}[ \, Q \, ]$,
the amplitudes (\ref{S-matrix}) reproduce the same values because of the BRST identities (\ref{BRST id}).
\vspace{2mm}
\subsection*{Open string field theory}
We obtained a generic result (\ref{S-matrix}) which is valid whenever we consider ordinary perturbative calculations,
in which propagators of $S$-matrix and gauge-fixing conditions should be written in terms of the free theory.
When we apply it to open string field theory,
our homological techniques suggest that
we can consider somewhat unconventional situations where each pieces of $S$-matrix are given by using information of interacting terms:
formally,
we may use these \textit{unconventional} gauge-fixing conditions or propagators in the Feynman graph calculations.
\vspace{2mm}
Let us consider Witten's open string field theory,
which satisfies the classical BV master equation.\footnote{If this open string field theory gives a well-defined quantum theory,
it solves the quantum BV master equation without any modification.
Then,
we can extend these results to the loop amplitudes since it guarantees that the theory gives amplitudes independent of the gauge-fixing condition.
}
We can obtain the tree amplitudes on the basis of the classical limit of the homological perturbation \cite{Matsunaga:2019fnc}.
Since it is a cubic theory,
the $A_{\infty }$ structure has no higher product $\boldsymbol{\mu }_{n} = 0$ for $n >2$.
The interacting vertex $\boldsymbol{\mu _{\mathrm{int} }} = \boldsymbol{\mu }_{2}$ is given by the star product
\begin{align}
\boldsymbol{\mu }_{2} ( A , B ) \equiv (-)^{A} \, A \, \ast \, B \, .
\end{align}
We first consider the Siegel gauge and the linear $b$-gauge,
which give a standard perturbative calculus and valid results.
Next,
we consider formal gauges,
the dressed $\mathcal{B}_{0}^{-}$ gauge and $A_{T}$ gauge,
which are singular but reproduce correct on-shell amplitudes under appropriate assumptions.
\subsubsection*{Siegel gauge}
In the Siegel gauge $b_{0} \Psi = 0$,
the propagator $\kappa ^{-1}_{Siegel} \equiv b_{0} \, L_{0}^{-1}$ has poles on the kernel of $L_{0}$.
We can represent the projector onto the physical states as $(\iota \, \pi )_{Siegel} \equiv e^{-\infty L_{0}}$\,.
Note that the Schwinger representation of the inverse of $L_{0}$ naturally includes $e^{- \infty L_{0} }$ as a boundary term \cite{Sen:2019jpm}
\begin{align}
\label{Siegel}
b_{0} \, L_{0}^{-1} \equiv
b_{0} \int_{0}^{\infty } e^{- t \, L_{0} } \, dt
= \frac{b_{0} }{L_{0}} ( 1 - e^{- \infty L_{0} } ) \, .
\end{align}
Since $\mu _{1} \equiv Q$ is the BRST operator of open strings,
we obtain the decomposition
\begin{align}
1 - e^{-\infty L_{0}} = Q \, ( b_{0} \, L_{0}^{-1} ) + ( b_{0} \, L_{0}^{-1} ) \, Q \, .
\end{align}
As is known,
the Siegel gauge is the standard gauge used in perturbative calculations and it provides a conventional propagator.
\subsubsection*{Linear $b$-gauge}
Let us consider a linear combination of the oscillators $b_{n}$,
which we write $\mathcal{B}_{(g)}$,
that can be encoded in a vector field $v (z) = \sum_{n \in \mathbb{Z} } v _{n} z^{n+1}$.
The linear $b$-gauge is given by
\begin{align}
\label{linear b}
\mathcal{B}_{(g)} \, \Psi _{g} = 0
\hspace{5mm}
\mathrm{with}
\hspace{5mm}
\mathcal{B }_{(g)} \equiv \sum_{n \in \mathbb{Z} } v_{n} b_{n} = \oint \frac{d z}{2 \pi i } v (z) b (z)
\, ,
\end{align}
where $g$ denotes the label of the space-time ghost number.
Note that the BPZ properties $\mathcal{B}_{(-g)} = \mathcal{B}_{(g-1)}^{\ast} $ must be satisfied for the consistency.
For each $\mathcal{B}_{(g)}$ or $\mathcal{B}_{(g)}^{\ast }$,
we define a linear combination of the Virasoro generators $\mathcal{L}_{(g)} \equiv Q \, \mathcal{B}_{(g)} + \mathcal{B}_{(g)} \, Q$,
which appears in propagators.
\vspace{2mm}
In general,
the linear $b$-gauge may not be invariant under the BPZ conjugation $\mathcal{B}_{(g) } \not= \mathcal{B}_{(g)}^{\ast }$ and then we cannot impose the same gauge-fixing condition for all space-time ghost numbers,
such as $\mathcal{B}_{(g-1) } = \mathcal{B}_{(-g) }^{\ast } \not= \mathcal{B}_{(-g)}$.
We write $\Psi = \sum \Psi _{g}$,
$\mathcal{B} \equiv \sum _{g} \mathcal{B}_{(g)}$ and $\mathcal{L}_{0} \equiv \sum \mathcal{L}_{(g)}$ for simplicity.
The double Schwinger representation of the propagator
\begin{align}
\label{double Schwinger}
\kappa _{double}^{-1} \equiv
\big{(} \mathcal{B}^{\ast } \mathcal{L}_{0}^{\ast \, -1} \big{)} \, Q \, \big{(} \mathcal{B} \mathcal{L}_{0}^{-1} \big{)}
= \frac{\mathcal{B}^{\ast } }{\mathcal{L}^{\ast }_{0} } \, Q \, \frac{\mathcal{B} }{\mathcal{L}_{0} } \big{(} \, 1-e^{- \infty \mathcal{L}_{0} } \, \big{)} \big{(} \, 1- e^{- \infty \mathcal{L}_{0}^{\ast } } \, \big{)}
\end{align}
provides the decomposition (\ref{solving free theory}) with $1- ( \iota \pi )_{double} \equiv
[1 + Q ( \frac{\mathcal{B} }{\mathcal{L}_{0}} - \frac{\mathcal{B}^{\ast } }{\mathcal{L}_{0}^{\ast } } ) ]
( 1- e^{- \infty \mathcal{L}_{0} } ) (1- e^{- \infty \mathcal{L}_{0}^{\ast } }) $,
where we assume that $e^{- \infty \mathcal{L}_{0}^{\ast } }$ commutes with $\mathcal{B}$, $\mathcal{L}_{0}$ and $e^{- \infty \mathcal{L}_{0} }$.
It gives correct on-shell amplitudes unless the vector field $v(z)$ is singular.\footnote{For singular $v(z)$,
such as a sliver frame,
we can obtain correct on-shell tree amplitudes.
However,
for loops,
\cite{Kiermaier:2008jy} suggests a gauge dependent result.
}
Calculations of homological perturbation suggest us an interesting but slightly unconventional propagator\footnote{In principle,
more unconventional propagator $\frac{1}{2} ( \mathcal{B} (\mathcal{L}_{0})^{-1} + \mathcal{B}^{\ast }(\mathcal{L}_{0}^{\ast } )^{-1})$ may be allowed since $(\iota \pi )$ does not have to be a projector to apply the homological perturbation,
which gives $( \iota \pi ) = \frac{1}{2}( e^{- \infty \mathcal{L}_{0} } + e^{- \infty \mathcal{L}_{0}^{\ast } } )$.
}
\begin{align}
\label{unconventional Schwinger}
\kappa ^{-1}_{mean} \equiv
\frac{1}{2} \Big{(} \mathcal{B} \, ( \mathcal{L}_{0} )^{-1} +\mathcal{B}^{\ast } \, ( \mathcal{L}^{\ast }_{0} )^{-1} \Big{)}
\big{(} \, 1-e^{- \infty \mathcal{L}_{0} } \, \big{)} \big{(} \, 1- e^{- \infty \mathcal{L}_{0}^{\ast } } \, \big{)}
\end{align}
with the gauge-fixing condition $( \mathcal{B} + \mathcal{B}^{\ast } ) \Psi = 0$,
which gives the decomposition (\ref{solving free theory}) with $1 - (\iota \, \pi )_{mean} \equiv (1-e^{- \infty \mathcal{L}_{0} } ) ( 1- e^{- \infty \mathcal{L}_{0}^{\ast } } )$\,.
Both of (\ref{double Schwinger}) and (\ref{unconventional Schwinger}) reduces to the ordinary propagator $( \mathcal{B} + \mathcal{B}^{\ast } ) ( \mathcal{L}_{0} + \mathcal{L}_{0}^{\ast } )^{-1}$ with the gauge-fixing condition $(\mathcal{B} + \mathcal{B}^{\ast }) \Psi = 0$ when $\mathcal{B}^{\ast }_{(g)} = \mathcal{B}_{(g)}$ holds.
\subsubsection*{Dressed $\mathcal{B}_{0}^{-}$ gauge}
We consider formal properties of singular gauge fixing conditions on the basis of the homological perturbation.
Let $z$ be a coordinate of the sliver frame.
We set $\mathcal{B}^{-}_{0} = \mathcal{B}_{0} + \mathcal{B}_{0}^{\ast }$ for $\mathcal{B}_{0}$ defined by $v(z) = z$ of (\ref{linear b}).
Although the $\mathcal{B}_{0}^{-}$ gauge would be understood as a special case of the linear $b$-gauge defined in the sliver frame,
it may have more unconventional or non-perturbative aspects.
We can regard it as a gauge-fixing condition based on the star product multiplications \cite{Erler:2009uj}.
In the sliver frame,
the conformal stress tensor $T(z)$ naturally defines a state
\begin{align}
K \equiv \int_{i \infty }^{-i \infty } dz \, T(z) \, \big{|} \, \mathrm{id} \, \big{\rangle } \, ,
\end{align}
where $| \mathrm{id }\rangle $ denotes the identity state of the star product.
By using any functions $F = F(K) $ and $G = G(K)$ of \textit{the string field} $K$,
where multiplications are given by the star product $\ast $,
we can consider the operator $\mathcal{B}_{F, G}$ defined by
\begin{align}
\label{dressed b}
\mathcal{B}_{F, G} \, \Phi \equiv \frac{1}{2} F (K) \ast \mathcal{B}_{0}^{-} \Big{[} F(K)^{-1} \ast \Phi \ast G(K) \Big{]} \ast G(K) \, .
\end{align}
Since the interactions of open string fields are given by the star product,
(\ref{dressed b}) gives a gauge-fixing condition $\mathcal{B}_{F, G} \, \Phi = 0$ written by using information of interacting terms and would be unconventional in a perturbation from the free theory.
While the linear $b$-gauge is written in terms of the free theory or the world-sheet theory,
the dressed $\mathcal{B}_{0}^{-}$ gauge needs the star product defining the interacting term and deviates from the free theory.
In this sense,
it seems that we cannot use (\ref{dressed b}) within an ordinary perturbation from the free theory.
It however gives a Hodge decomposition of operators acting on the identity state,
which implies that we can apply the homological perturbation.
As long as the gauge-fixing condition $\mathcal{B}_{F, G} \, \Phi = 0$ is valid,
which we just assume,
it gives (\ref{S-matrix}) correctly.
For any state $\Phi \in \mathcal{H}$,
the identity state $|\mathrm{id} \rangle $ satisfies
\begin{align}
\big{|} \, \mathrm{id} \big{\rangle } \, \ast \, \Phi = \Phi = \Phi \, \ast \, \big{|} \, \mathrm{id} \big{\rangle } \, .
\end{align}
Recall that we can represent a given state $\Psi $ as a set of operators $\mathcal{O}_{\Psi }$ acting on the conformal vacuum $|0\rangle $.
Likewise,
we may represent $\Psi $ as a set of operators $\widehat{\Psi }$ acting on the identity state $| \mathrm{id} \rangle $,
\begin{align}
\label{op on id}
\mathcal{O}_{\Psi } \, \big{|} \, 0 \, \big{\rangle }
= \Psi
= (\widehat{\Psi })_{L} \, \big{|} \, \mathrm{id} \big{\rangle }
= (\widehat{\Psi })_{R} \, \big{|} \mathrm{id} \big{\rangle } \, ,
\end{align}
where $(\widehat{\Psi } )_{L} \, \Phi = \Psi \ast \Phi $ and $(\widehat{\Psi } )_{R} \, \Phi = (-)^{\Psi \, \Phi } \Phi \, \ast \, \Psi $ for any state $\Phi \in \mathcal{H}$\,.
The propagator (\ref{dressed b}) gives a decomposition on the identity state and in this sense reproduces (\ref{S-matrix}).
\subsection*{$A_{T}$ gauge}
In open string field theory,
in addition to the perturbative vacuum,
the tachyon vacuum is well studied \cite{Sen:1999xm, Schnabl:2005gv}.
We consider the tachyon vacuum solution $\Psi _{T}$ of the Witten theory,
\begin{align}
Q \, \Psi _{T} + \Psi _{T} \ast \Psi _{T} = 0 \, .
\end{align}
As is known,
the tachyon vacuum has empty cohomology:
there exist a state $A_{T}$ satisfying
\begin{align}
\label{homotopy state}
Q_{T} \, A_{T} = \big{|} \, \mathrm{id} \big{\rangle } \, ,
\end{align}
which is called a homotopy contracting state \cite{Ellwood:2001ig}.
The BRST operator around the tachyon vacuum,
$Q_{T} \equiv Q + (\widehat{\Psi }_{T})_{L} - (\widehat{\Psi }_{T} )_{R}$,
satisfies $Q_{T} \, | \mathrm{id} \rangle =0$,
as $Q \, | \mathrm{id} \rangle = 0$.
We show that this non-perturbative relation (\ref{homotopy state}) gives an interesting Hodge decomposition with helps of a state defined by
\begin{align}
W \equiv \Psi _{T} \, \ast \, A_{T} + A_{T} \, \ast \, \Psi _{T} \, .
\end{align}
As we see,
it provides the tree $S$-matrices based on unconventional propagators,\footnote{For an $A_{\infty }$ type string field theory,
when its tachyon solution $\Psi _{T}$ and an operator $\widehat{A}_{T}$ satisfying (\ref{tachyon sft}) are given,
the same computations can be done by setting $W = \sum_{n} \sum_{\mathrm{cyclic} } \mu _{n+1} ( \Psi _{T} , ... , \Psi _{T} , A_{T} )$. }
whose $4$-point amplitude reproduces that of \cite{Masuda:2019rgv}.
In the rest of this section,
we consider formal but interesting properties suggested by $A_{T} \, \phi = 0$ on the basis of the homological perturbation.
We would like to emphasise that the following discussions are not a proof of the validity of this gauge fixing condition or the correctness of obtained amplitudes.
\vspace{2mm}
Let us consider the free action $S_{T} [\phi _{T} ] = \frac{1}{2} \omega ( \phi _{T} , Q_{T} \, \phi _{T} ) $ for string field theory around the tachyon vacuum.
We write $\mathcal{H}_{\Psi _{T}}$ for the state space of this free theory around $\Psi _{T}$,
namely, $\phi _{T} \in \mathcal{H}_{\Psi _{T}}$.
We assume $A_{T} \ast A_{T} =0$\,.
The relation (\ref{homotopy state}) implies that this theory has no physical state,
which gives the following deformation retract
\begin{align}
\label{tachyon sft}
\widehat{A}_{T} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H}_{\Psi _{T} } , \, Q_{T} \, \big{)}
\hspace{3mm}
\overset{ \pi }{\underset{ \iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, 0 , \, 0 \, \big{)}
\hspace{5mm}
\mathrm{with}
\hspace{3mm}
\widehat{1} = Q_{T} \, \widehat{A}_{T} + \widehat{A}_{T} \, Q_{T}
\, ,
\end{align}
where operators $\widehat{1}$ and $\widehat{A}_{T}$ are defined by $\widehat{1} \, \Phi \equiv | \, \mathrm{id} \rangle \ast \Phi = \Phi \ast | \, \mathrm{id} \rangle $ and $2 \, \widehat{A}_{T} \equiv (\widehat{A})_{L} + (\widehat{A})_{R}$.
It is helpful to introduce the operator $\widehat{W} \equiv \frac{1}{2} (\widehat{W})_{L} + \frac{1}{2} (\widehat{W})_{R}$,
which commutes with $Q$ and $\widehat{A}_{T}$ because of $Q \, W = 0$ and $W \ast A_{T} = A_{T} \ast W $.
The relation (\ref{homotopy state}) or the above Hodge decomposition of (\ref{tachyon sft}) can be cast as
\begin{align}
\label{A-W}
Q \, \widehat{A}_{T} + \widehat{A}_{T} \, Q + \widehat{W} = \widehat{1} \, .
\end{align}
We would like to transfer the Hodge decomposition (\ref{A-W}) obtained by the non-perturbative relation (\ref{homotopy state}) to that of the perturbation theory.
\vspace{2mm}
We consider the free action $S [\phi ] = \frac{1}{2} \omega ( \phi , Q \, \phi )$ for string field theory around the perturbative vacuum and write $\mathcal{H}$ for its state space:
$\phi \in \mathcal{H}$.
This theory has physical states and the $Q$-cohomology $\mathcal{H}_{\mathrm{phys} }$ is not empty.
A gauge fixing gives a decomposition $\mathcal{H} = \mathcal{H}_{\mathrm{phys} } \oplus \mathcal{H}_{\mathrm{unphys} } \oplus \mathcal{H}_{\mathrm{gauge} }$.
We assume $\mathcal{H} \cap \mathcal{H}_{\Psi _{T}} \not= 0$.
When we consider a restriction of the nonperturbative relation (\ref{homotopy state}) or (\ref{A-W}) onto the perturbative state space $\mathcal{H}$,
there would be two possibility:
the relation holds in the same form as (\ref{A-W}) in $\mathcal{H}$,
which we expect for most cases,
or not.
When (\ref{A-W}) holds in $\mathcal{H}$,
for any physical state $\phi _{\mathrm{phys}} \in \mathcal{H}_{\mathrm{phys}}$,
the relation (\ref{A-W}) provides
\begin{align}
\label{value of W}
\widehat{W} \, \phi _{\mathrm{phys} } = \phi _{\mathrm{phys}} + Q \, \Lambda
\end{align}
with the gauge parameter $\Lambda \equiv - \widehat{A}_{T} \, \phi _{\mathrm{phys} }$.\footnote{Note that for any $n \in \mathbb{N}$,
we have $\widehat{1}^{n} \, \phi = 1^{n} \, \phi = \phi $,
where $1 \in \mathbb{R}$ and $\phi \in \mathcal{H}$.
If $\lim_{n \rightarrow \infty } \widehat{1}^{n}$ exists and equals to $1$,
a possible choose of $W = e^{K}$ suggests that we may represent a given $\phi _{\mathrm{phys} }$ as
\begin{align*}
\phi _{\mathrm{phys }} = \Omega _{\infty } \, \phi _{\mathrm{phys} } \, \Omega _{\infty } + Q \, \lambda \, ,
\end{align*}
where $\Omega _{\infty } = \lim_{n \rightarrow \infty } e^{n \, K}$ denotes the sliver state and $\lambda$ is a gauge parameter.
This relation implies that we may use $\Omega _{\infty } \, \phi _{\mathrm{phys} } \, \Omega _{\infty } $ as external lines of the S-matrix,
which gives the same value as the $S$-matrix whose external lines are $ \phi _{\mathrm{phys} }$ with the $Q$-exact shifts.
A state interposed between sliver states belongs to the kernel of $(1- \widehat{W})$. }
Note that the $Q$-exact term of (\ref{value of W}) can be set to zero by imposing $\widehat{A}_{T} \, \phi _{\mathrm{phys}} = 0$
We write $\iota \, \pi$ for a projector onto $\mathrm{Ker}[ 1 - \widehat{W} ]$.
With $\widehat{A}_{T} \, \phi =0$,
the relation (\ref{value of W}) implies that $ \iota \, \pi \, \mathcal{H}$ includes $\mathcal{H}_{\mathrm{phys} }$,
which suggests that we may use
\begin{align}
\label{unconventional propagator}
\widehat{\kappa }^{-1} \equiv \widehat{A}_{T} \, ( \, \widehat{1} - \widehat{W} \, )^{-1} \, ( \, \widehat{1} - \iota \, \pi \, )
\end{align}
as a propagator:
these $\iota \, \pi $,
$\widehat{\kappa }^{-1}$ and $Q$ give a Hodge decomposition of $\mathcal{H}$.
We assume that they satisfy $\widehat{\kappa }^{-1} \, \iota \, \pi = \iota \, \pi \, \widehat{\kappa }^{-1} = 0$.\footnote{When $\widehat{\kappa }^{-1} \, \iota \, \pi = 0$ or $ \iota \, \pi \, \widehat{\kappa }^{-1} = 0$ does not hold,
instead of (\ref{unconventional propagator}),
we use $\widehat{\kappa }^{\prime -1} = \widehat{\kappa }^{-1} ( Q \, \widehat{\kappa }^{-1} + \widehat{\kappa }^{-1} Q )$ or $\widehat{\kappa }^{\prime -1} = ( Q \, \widehat{\kappa }^{-1} + \widehat{\kappa }^{-1} Q ) \, \widehat{\kappa }^{-1} $ as a propagator,
respectively. }
Then,
we can start with the deformation retract
\begin{align}
\label{transfer of most cases}
\widehat{\kappa }^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H}, \, Q \, \big{)}
\hspace{3mm}
\overset{ \pi }{\underset{ \iota }{
\scalebox{2}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}_{\mathrm{phys} }
, \, 0 \, \big{)}
\hspace{5mm}
\mathrm{with}
\hspace{3mm}
\widehat{1} - \iota \, \pi = Q \, \widehat{\kappa }^{-1} + \widehat{\kappa }^{-1} \, Q
\, .
\end{align}
The above $\mathcal{H}_{\mathrm{phys} }$ should be relaxed to $\mathcal{H}_{\mathrm{phys} } \oplus \mathcal{H}_{\mathrm{gauge} }$ when we use the left hand side of (\ref{value of W}) as external lines instead of $\phi _{\mathrm{phys}}$.
As a result of the perturbation $\mu _{2}$ to (\ref{transfer of most cases}),
we obtain an unconventional $S$-matrix whose propagator is (\ref{unconventional propagator}).
A gauge fixing is needed to specify the external lines $\phi _{\mathrm{phys}}$ explicitly.
The relation (\ref{value of W}) suggest that the condition $\widehat{A}_{T} \, \phi = 0$ for $\phi \in \mathcal{H}$ works as a gauge fixing condition,
although it may take a singular expression.
\vspace{2mm}
Next,
we consider the other case that (\ref{A-W}) does not hold in the same form after the restriction onto $\mathcal{H}$.
We assume that such a perturbative breakdown of (\ref{A-W}) occurs in $\mathcal{H}_{\mathrm{phys} }$ only.
In this case,
there exists $\phi _{\mathrm{phys} } \in \mathcal{H}_{\mathrm{phys}}$ such that the equality of the relation (\ref{value of W}) or (\ref{A-W}) does not hold.\footnote{It includes a situation where we cannot define an inner product of $\widehat{W} \phi _{\mathrm{phys}}$ and a given $\psi _{\mathrm{phys}} \in \mathcal{H}_{\mathrm{phys} }$. }
Then,
we need to replace the operator $\widehat{W}$ by a projected one $\widehat{W} (\, \widehat{1} - ( \iota \, \pi )_{\mathrm{phys}} )$ to transfer the relation (\ref{tachyon sft}) to that of $\mathcal{H}$,
where $( \iota \, \pi )_{\mathrm{phys}} : \mathcal{H} \rightarrow \mathcal{H}_{\mathrm{phys} }$ denotes a projector onto the physical space.
We consider
\begin{align}
\label{other case}
\widehat{A}_{T} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H}, \, Q_{T} \, \big{)}
\hspace{3mm}
\overset{ \pi _{\mathrm{phys}} }{\underset{ \iota _{\mathrm{phys}} }{
\scalebox{2.5}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}_{\mathrm{phys} } , \, 0 \, \big{)}
\hspace{5mm}
\mathrm{with}
\hspace{3mm}
\widehat{1} - ( \iota \, \pi )_{\mathrm{phys}} = Q_{T} \, \widehat{A}_{T} + \widehat{A}_{T} \, Q_{T}
\, ,
\end{align}
where we set $\iota _{\mathrm{phys}} =1$ and used $\mathcal{H}_{\mathrm{phys} } = \pi _{\mathrm{phys}} \, \mathcal{H}$ for simplicity.
As a result of the perturbation $Q - Q_{T}$ to (\ref{other case}),
we obtain the transferred relation
\begin{align}
\label{transfer of other case}
\widetilde{\kappa }^{-1} \,
\rotatebox{-70}{$\circlearrowright $} \,
\big{(} \, \mathcal{H}, \, Q \, \big{)}
\hspace{3mm}
\overset{ \pi _{\mathrm{phys}}}{\underset{ \iota _{\mathrm{phys}}}{
\scalebox{2.5}[1]{$\rightleftarrows $}
}}
\hspace{3mm}
\big{(} \, \mathcal{H}_{\mathrm{phys} } , \, 0 \, \big{)}
\hspace{5mm}
\mathrm{with}
\hspace{3mm}
\widehat{1} - (\iota \, \pi )_{\mathrm{phys}} = Q \, \widetilde{\kappa } + \widetilde{\kappa } \, Q
\, ,
\end{align}
where the perturbed homotopy contracting operator $\widetilde{\kappa }^{-1}$ is given by
\begin{align}
\label{formal A_T}
\widetilde{\kappa }^{-1} \equiv \widehat{A}_{T} \, \big{(} \, \widehat{1} - \widehat{W} \, \big{)}^{-1} \big{(} \, \widehat{1} - (\iota \, \pi )_{\mathrm{phys}} \big{)} \, .
\end{align}
It solves the Hodge decomposition (\ref{solving free theory}) on $| \mathrm{id} \big{\rangle } $ and thus provides the on-shell $S$-matrix (\ref{S-matrix}) formally.
The propagator (\ref{formal A_T}) consists of $A_{T}$,
$\Psi _{T}$ and the projector $(\iota \, \pi )_{\mathrm{phys}}$.
The explicit form of $A_{T}$ is determined by specifying the explicit form of $\Psi _{T}$,
which is free from the gauge choice.
Note however that we need to impose a gauge fixing condition to determine an explicit form of the projector $(\iota \, \pi )_{\mathrm{phys}}$ or external lines and to compute the on-shell $S$-matrix.
In this sense,
unless the choice of external lines are taken into account,
the $S$-matrix (without external lines) obtained by using (\ref{formal A_T}) is free from a gauge-fixing.
\vspace{2mm}
The $S$-matrix obtained from (\ref{transfer of most cases}) has the same form as the $S$-matrix obtained from (\ref{transfer of other case}) except for the external lines and projectors.
Actually,
we can check that (\ref{unconventional propagator}) or (\ref{formal A_T}) indeed gives a correct $4$-point amplitude.
For any state $\phi \in \mathcal{H}$,
we find
\begin{align}
\widehat{\kappa }^{-1} \, \widehat{W} \, \phi
= - \big{(} \, A_{T} - \widehat{\kappa }^{-1} \, \big{)} \, \phi \, .
\end{align}
It resembles (\ref{Siegel}) and can be understood as separating the main contribution from the boundary contribution.
By using the cyclic property,
the $4$-point amplitude reduces to
\begin{align}
\label{Vene}
\mathcal{A}_{4} (\phi ' , ... ,\phi ' )
= \frac{1}{2} \Big{\langle } ( \widehat{\kappa }^{-1} -\widehat{A}_{T} ) \widehat{\phi }' \, (\widehat{W} \, \widehat{\phi }' )^{3} \Big{\rangle } _{\mathrm{sliver}} \, ,
\end{align}
where $\langle ... \rangle _{\mathrm{sliver}}$ denotes the correlation function of the conformal field theory on the sliver frame.
As shown by \cite{Masuda:2019rgv},
when external lines $\phi '$ are specified,
the expression (\ref{Vene}) reproduces the on-shell $4$-point amplitudes of the world-sheet theory,
where $\widehat{\kappa }^{-1}$ of (\ref{Vene}) is identified with $A_{\Psi }$ of \cite{Masuda:2019rgv}.
This result also supports the validity of the formal object (\ref{unconventional propagator}) or (\ref{formal A_T}) as a propagator.
\section{Conclusion and Discussions}
Every BV-quantizable field theory has own $A_{\infty }$ structure:
this is because each solution of the BV master equation is in one-to-one correspondence with a quantum $A_{\infty }$ structure.
In this paper,
we showed that the perturbative path-integral can be performed as a morphism of this quantum $A_{\infty }$ structure intrinsic to each quantum field theory,
which is a result of the homological perturbation.
As we checked explicitly,
the homological perturbation for $A_{\infty }$ is an alternative representation of the perturbative path-integral.
Therefore,
when the original BV master action includes source terms,
its effective theory must have a twisted (quantum) $A_{\infty }$ structure.
We also discussed that such a homological approach may enable us to use unconventional propagators for calculating $S$-matrix,
which may provide further applications.
As long as physicists believe that the path-integral condenses configurations of integrated fields onto the on-shell physical ones,
our results seem to be a quite natural (or trivial) because the BV formalism itself is based on the homological perturbation and determines the physical states from it.
\vspace{1mm}
As we explained,
the BV master equation and the intrinsic $A_{\infty }$ structure play central roles in perturbative quantum field theory.
Once we solve the BV master equation,
we can quickly obtain each quantity given by the perturbative path-integral,
such as effective theory or scattering amplitude.
Thus,
it would be important tasks to try to derive BV master actions for some superstring field theories \cite{Berkovits:2012np, Matsunaga:2016zsu, Erler:2017onq, Matsunaga:2018hlh}.
It would be worth mentioning that we checked how the perturbative path-integral preserves the $A_{\infty }$ structure,
although it may be originally a property of the non-perturbative path-integral (\ref{effective BV master action}).
Our results suggest that the non-perturbative corrections would preserve the $A_{\infty }$ structure we consider.
\vspace{1mm}
We would like to emphasize that such algebraic approaches to Lagrangian field theory have been exploited since long-time before:
it is not a new idea.
However,
the link between homotopy algebras and the BV formalism have developed recently and minimal models of quantum homotopy algebras are now available \cite{Braun:2017ikg, Doubek:2017naz}.
We thus believe that it would be worth studying these approaches more explicitly and physicist-friendly in terms of higher algebra.\footnote{Also,
homotopy algebras would be more accessible to mathematicians,
rather than the BV formalism. }
We end this section by mentioning related earlier works.
The earliest and outstanding work would be \cite{Zwiebach:1992ie},
which introduced quantum $L_{\infty }$ algebras and established the link to the BV master equation.
The geometry and meaning of the classical BV formalism were given by various authors in the early days,
for example \cite{Schwarz:1992nx, Alexandrov:1995kv}.
Recently,
a nice review was given by \cite{Jurco:2018sby}.
Also \cite{Albert} is very suggestive.
Application of minimal models of homotopy algebras to field theory was given by \cite{Kajiura:2003ax},
which pointed out that minimal models give S-matrices.
Quantum minimal models is given by \cite{Braun:2017ikg, Doubek:2017naz} recently.
Derivations of S-matrix based on the homological perturbation were given by many authors.
For example,
see \cite{Konopka:2015tta, Matsunaga:2019fnc, Macrelli:2019afx} for the tree level and \cite{Plumann, Doubek:2017naz, Jurco:2019yfd} for the loop.
See also \cite{Arvanitakis:2019ald}.
The work \cite{Nakatsu:2001da} discussed the classical part of effective theory and renormalization group by using the $A_{\infty }$ structure.
The work \cite{Costello:2007ei} presented that the BV formalism is very useful to discuss renormalization group flows.
Also,
the works \cite{Gwilliam:2012jg, JohnsonFreyd:2012ww} derived Wick's theorem and Feynman rules for finite-dimensional integrals by using BV differentials.
The link between solutions of BV master equation and homotopy algebras originates from their operadic relations,
which were studied by \cite{Barannikov:2010np, Doubek:2013dpa, Jurco}.
\vspace{-3mm}
\subsection*{Acknowledgments}
H.M. would like to thank Martin Markl and Jan Plumann for helpful discussions of homological perturbation.
Also,
the authors would like to thank Ted Erler, Carlo Maccaferri and Yuji Okawa for discussions of SFT
at the GGI workshop ``String Theory from the World-Sheet Perspective'',
at the YITP workshop ``Strings and Fields 2019'',
or at Italian or Czech restaurants.
\vspace{1mm}
This work has been supported by GACR Grant EXPRO 19-28268X.
The work of T.M. has been supported by the GACR grant 20-25775X.
The work of H.M. was supported by Praemium Academiae and RVO:\,67985840 within 2019,
in which main part of this work was done. |
2003.05074 | \section{Introduction}
Khovanov homology \cite{Khov1} is a bigraded homology theory which is an invariant of knots and links, categorifying the Jones polynomial. In general, the structure of Khovanov homology and the types of torsion which occur may vary widely \cite{DBN, MPS, Stosic1}. For certain links, there is a partial isomorphism between the extreme gradings of Khovanov homology and chromatic graph homology, a categorification of the chromatic polynomial for graphs \cite{AP,PS}. The isomorphism between these two theories describes a part of Khovanov homology that is supported on two diagonals and has only $\mathbb{Z}_2$ torsion, similar to the Khovanov homology of an alternating link. Moreover, this correspondence allows us to describe ranks of groups in Khovanov homology in terms of combinatorial information from a diagram, or a graph associated to the diagram.
Khovanov homology of alternating knots is determined by the Jones polynomial and the signature of a knot and, similarly chromatic graph homology over the algebra $\mathcal{A}_2 = \mathbb{Z}[x]/(x^2)$ is determined by the chromatic polynomial \cite{LS}. This approach enables us to determine some extremal Khovanov homology groups based on combinatorial results about the chromatic polynomial of a graph which determines its chromatic homology. The following theorem illustrates the type of the results we obtain.
\noindent{\bf Theorem~\ref{rankKhovanovgirth}}\textit{ Let $D$ be a diagram of a link $L$ such that the all-positive graph of $D$ has girth $\ell$ and satisfies the conditions of Theorem \ref{rankHGRgirth}. For $0 < i < \ell$, the ranks of Khovanov homology groups of $L$ are given by:
\begin{equation*}
\textnormal{rk} Kh^{i-c_-(D),N+2i}(L) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{p_1-2+k}{k} \right)-n_{i+1} +(-1)^{i+1} \delta^b
\end{equation*}
where $p_1$ is the cyclomatic number of the graph, $n_{i+1}$ is the number of $(i+1)$-cycles, and $\delta^b$ measures bipartiteness.}
The applicability of our results depends on a quantity defined in Section \ref{Girth1} that we call the girth of a link. We find upper bounds for the value of this invariant based on Khovanov homology and the Jones polynomial. We prove results on the girth of connected sums and of alternating knots, describing another upper bound in terms of crossing number and signature.
Analyzing girth of a link leads to a somewhat surprising characterization of the types of graphs that can be obtained from a homogeneous resolution of diagrams of a given knot (all-positive or all-A state graph)
\noindent{\bf Theorem~\ref{girthPossibleKhovanov}}\textit{ Let $D$ be a diagram of a non-trivial link $L$ such that the all-positive graph of $D$ has girth $\ell.$ Then either the girth of a link equals $\textnormal{gr}(L)=\ell$ or $\ell \in \{1,2\}.$
}
As a consequence we get that if a link has a diagram such that the girth of the corresponding graph is equal to some $\ell>2$, than the girth of the link is equal to $\ell$, see Corollary \ref{girth3more}. In other words, this is saying that if a link $L$ has girth greater than two, all of the corresponding all-A graphs have girth equal to $\textnormal{gr}(L)$, one or two.
\section*{Acknowledgements}
We are grateful to Adam Lowrance for many ideas and useful discussions. RS was partially supported by the Simons Foundation Collaboration Grant 318086 and NSF Grant DMS 1854705.
\section{Background}
\subsection{Jones polynomial}
Let $D$ be a diagram of link $L$. Each crossing of $D$ can be resolved with a positive or negative resolution as shown below. The positive and negative resolutions are sometimes referred to as the A and B resolutions, respectively (see e.g. \cite{DasLin}).
\begin{figure}[h]
\centering
\includegraphics[scale = 0.6]{resolutions2.pdf}
\caption{Positive and negative resolutions at a crossing.}
\end{figure}
The resolution of all crossings in a diagram $D$ produces a collection of disjoint circles known as Kauffman states. From any Kauffman state $s$, we may construct a graph whose vertices correspond to the circles of $s$, and whose edges connect circles whose arcs were obtained by smoothing a single crossing.
The Kauffman state $s_+(D)$ is obtained by applying the positive resolution to every crossing in $D$, and we denote the graph obtained from this state by $G_+(D)$ (known as the all-positive or all-A state graph of $D$). Similarly, we define a state $s_-(D)$ with all negative resolutions along with its graph $G_-(D)$.
\begin{figure}[h]
\centering
\includegraphics[scale = 0.7]{knotsmoothings2.pdf}
\caption{The Kauffman state $s_+(D)$ and the graph $G_+(D)$.}
\end{figure}
We give a definition of the Jones polynomial using Kauffman states as in \cite{SaSco}.
\begin{defn} \label{JonesDef}
Let $L$ be a link and $D$ a diagram of $L$ with $c_+$ positive crossings and $c_-$ negative crossings. The unnormalized Jones polynomial of $L$ is given by:
\begin{equation*}
\hat{J}_L(q) = (-1)^{c_{-}}q^{c_{+}-2c_{-}}\sum_{i=0}^{c_++c_-} (-1)^i \sum_{\{s~:~n_-(s) = i\}} q^{i}(q+q^{-1})^{|s|}
\end{equation*}
where $s$ is a Kauffman state of $D$ with $n_{-}(s)$ negative smoothings and $|s|$ connected components.
The normalized version of the Jones polynomial is
\begin{equation*}
J_L(q) = \hat{J}_L(q)/(q+q^{-1})
\end{equation*}
where $q+q^{-1}$ represents evaluation on the unknot, $\hat{J}_{\Circle}(q) = q+q^{-1}$.
\end{defn}
Next we introduce some notation that will be useful when discussing graphs.
\begin{defn}\label{p1}
The cyclomatic number $p_1(G)$ of a connected graph $G$ with $v$ vertices and $E$ edges is equal to $p_1(G) = E-v+1$. For planar graphs such as $G_+(D)$, $G_-(D)$ $p_1$ is equal to the number of bounded faces of the graph.
\end{defn}
\begin{defn}[\cite{DasLin}, \cite{LowSpy}]\label{DLreducedgraph}
Let $D$ be a knot diagram with corresponding all-positive graph $G=G_+(D)$. The simplification $G'$ of $G$ is the graph obtained by deleting any loops
in $G$ and replacing each set of multiple edges with a single edge.
Define $\mu$ to be the number of edges in $G'$ which correspond to multiple edges in $G$.
\end{defn}
We consider the normalized version of the Jones polynomial and denote the coefficients as follows:
\begin{equation} \label{JonesCoef}
J_L(q) = \beta_0q^{C} + \beta_1q^{C+2} + \beta_2q^{C+4} + \beta_3q^{C+6}+ \ldots + \beta_iq^{C+2i} + \ldots
\end{equation}
where $C$, the minimal degree of $J_L(q)$, depends on the link $L$.
For a reduced alternating knot, Dasbach and Lin \cite{DasLin} showed that the first three coefficients of the normalized Jones polynomial may be stated in terms of the all-positive graph $G_{+}(D)$. This result is restated in Theorem \ref{DLJones}.
\begin{thm}[\cite{DasLin}] \label{DLJones}
Let $K$ be a knot with reduced alternating diagram $D$. Let $p_1$ and $t_1$ be the cyclomatic number and the number of triangles in $G_+(D)'$, and let $\mu$ be defined as above. Then the first three coefficients of $J_K(q)$ (up to an overall change in sign) are $\beta_0 = 1, \beta_1 = -p_1,$ and $\beta_2 = \binom{p_1+1}{2} + \mu - t_1$.
\end{thm}
The lowest-degree terms of the Jones polynomial are often referred to as the ``tail," while the highest-degree terms are referred to as the ``head." Note that if the all-positive graph obtained from $D$ is replaced by the all-negative graph in Theorem \ref{DLJones}, a similar result applies to the three extremal coefficients in the head of the Jones polynomial.
\subsection{Chromatic polynomial}
We now define the chromatic polynomial of a graph. Let $G$ be a finite, undirected graph with vertex set $V(G)$ and edge set $E(G)$. We will often denote the cardinalities of these sets by $v = |V(G)|$ and $E = |E(G)|$. If $G$ has an edge between vertices $x, y \in V(G)$, we write the corresponding element in $E(G)$ as $\{x,y\}$.
\begin{defn}[\cite{DKT}] \label{ChromDef}
A mapping $f:V(G) \to \{1, \ldots, \lambda\}$ is called a $\lambda$-coloring of $G$ if for any pair of vertices $x,y \in V(G)$ such that $\{x,y\} \in E(G)$, $f(x) \neq f(y)$. The chromatic polynomial of the graph $G$, denoted $P_G(\lambda)$, is equal to the number of distinct $\lambda$-colorings of $G$.
\end{defn}
For any graph $G$, the degree of $P_G(\lambda)$ is equal to $v$. We will represent the terms of the polynomial as follows:
\begin{equation} \label{ChromCoef}
P_G(\lambda) = c_v\lambda^v + c_{v-1}\lambda^{v-1} + c_{v-2} \lambda^{v-2} + \ldots + c_{v-i} \lambda^{v-i} + \ldots+ c_{1} \lambda
\end{equation}
The first few coefficients of $P_G(\lambda)$ can be described in terms of cycles and subgraphs found in $G$.
\begin{defn}
The girth of a graph $G$, denoted $\ell(G)$, is the number of edges in the shortest cycle in $G$.
\end{defn}
\begin{defn}
Let $H$ be a subgraph of graph $G$. We say $H$ is an induced subgraph if for every $\{x,y\} \in E(G)$ with $x, y \in V(H)$, the edge $\{x,y\}$ is in $E(H)$.
\end{defn}
We adopt the convention that the girth of a tree is zero, but it is worth noting that there are different conventions considering girth of a tree to be infinite \cite{Boll1, Diestel}.
\begin{thm}\cite{Meredith1} \label{Meredith}
If $G$ is a graph with girth $\ell>2$ and $n_{\ell}$ cycles of length $\ell$, then the first $\ell$ coefficients of the chromatic polynomial $P_G(\lambda)$ are:
$$c_{v-i} = \begin{cases}
(-1)^i \displaystyle\binom{E}{i} & 0 \le i < \ell - 1\\
(-1)^{\ell-1} \left( \displaystyle\binom{E}{\ell-1} - n_{\ell}\right) & i = \ell - 1\\
\end{cases}$$
\end{thm}
\begin{rem}
The statement of this result in \cite[Theorem 2]{Meredith1} is not explicitly restricted to graphs with $\ell>2$. In the case $i = \ell-1$, the proof contains an assumption that the number of cycle-containing subgraphs with $v-1$ connected components and $t$ edges is zero for $t > 2$; this is not true for graphs with edge multiplicities greater or equal to 3.
\end{rem}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph1-eps-converted-to.pdf}
\\
$T_1$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph2-eps-converted-to.pdf}
\\
$T_2$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph3-eps-converted-to.pdf}
\\
$T_3$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph4-eps-converted-to.pdf}
\\
$T_4$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph5-eps-converted-to.pdf}
\\
$T_5$
\end{subfigure}
\vspace{5mm}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph6-eps-converted-to.pdf}
\\
$T_6$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph7-eps-converted-to.pdf}
\\
$T_7$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph8-eps-converted-to.pdf}
\\
$T_8$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph9-eps-converted-to.pdf}
\\
$T_9$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.56\linewidth]{tgraph10-eps-converted-to.pdf}
\\
$T_{10}$
\end{subfigure}
\vspace{5mm}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph11-eps-converted-to.pdf}
\\
$T_{11}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph12-eps-converted-to.pdf}
\\
$T_{12}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph13-eps-converted-to.pdf}
\\
$T_{13}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{tgraph14-eps-converted-to.pdf}
\\
$T_{14}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph15-eps-converted-to.pdf}
\\
$T_{15}$
\end{subfigure}
\vspace{5mm}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph16-eps-converted-to.pdf}
\\
$T_{16}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph17-eps-converted-to.pdf}
\\
$T_{17}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph18-eps-converted-to.pdf}
\\
$T_{18}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph19-eps-converted-to.pdf}
\\
$T_{19}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph20-eps-converted-to.pdf}
\\
$T_{20}$
\end{subfigure}
\caption{Graphs $T_{1}$ through $T_{20}$ involved in the computation of the 5th and 6th coefficients of the chromatic polynomial \cite{Bielak1}.}\label{tgraphs1}
\end{figure}
\begin{thm}\label{FarBie}\cite{Farrell, Bielak1}
Let $G$ be a graph with $v$ vertices, $E$ edges, $t_1$ triangles, $t_2$ induced 4-cycles, and $t_3$ complete graphs of order 4. The first four coefficients of the chromatic polynomial $P_G(\lambda)$ are given by the following formulas: $c_v = 1$, $c_{v-1} = -E$, $c_{v-2} = \displaystyle\binom{E}{2}-t_1$, and
\begin{equation*}\label{chC3}
c_{v-3} = -\displaystyle\binom{E}{3}+(E-2)t_1+t_2-2t_3
\end{equation*}
The 5th and 6th coefficients are given by the following formulas, where $t_i$ is the number of induced subgraphs of $G$ isomorphic to graphs $T_i$ as shown in Figures \ref{tgraphs1} and \ref{tgraphs2}.
\begin{equation*}\label{chC4}
c_{v-4} =\binom{E}{4} - \binom{E-2}{2}t_1 + \binom{t_1}{2} - (E-3)t_2 -(2E-9)t_3
- t_4 +t_5 + 2t_6+3t_7-6t_8 \end{equation*}
\begin{eqnarray*}\label{chC5}
c_{v-5} &=& -\binom{E}{5} + \binom{E-2}{3}t_1 - (E-4)\binom{t_1}{2} + \binom{E-3}{2}t_2 - (t_2-2t_3)t_1 - (E^2-10E+30)t_3
\\&&+t_4 -(E-3)t_5-2(E-5)t_6-3(q-6)t_7+6(E-8)t_8+t_9-t_{10}
-2t_{11}-2t_{12}-t_{13}\\&&+t_{14}
-t_{15}-3t_{16}-4t_{17}-4t_{18}+2t_{19}-4t_{20}-t_{21}+4t_{22}
+3t_{23}+4t_{24}+5t_{25}+4t_{26}\\&&+6t_{27}+8t_{28}
+16t_{29}+12t_{30}-24t_{31}
\end{eqnarray*}
\end{thm}
\begin{figure}[!ht]
\centering
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph21-eps-converted-to.pdf}
\\
$T_{21}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{tgraph22-eps-converted-to.pdf}
\\
$T_{22}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{tgraph23-eps-converted-to.pdf}
\\
$T_{23}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph24-eps-converted-to.pdf}
\\
$T_{24}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{tgraph25-eps-converted-to.pdf}
\\
$T_{25}$
\end{subfigure}
\vspace{5mm}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph26-eps-converted-to.pdf}
\\
$T_{26}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph27-eps-converted-to.pdf}
\\
$T_{27}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph28-eps-converted-to.pdf}
\\
$T_{28}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph29-eps-converted-to.pdf}
\\
$T_{29}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph30-eps-converted-to.pdf}
\\
$T_{30}$
\end{subfigure}
\begin{subfigure}[b]{0.15\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{tgraph31-eps-converted-to.pdf}
\\
$T_{31}$
\end{subfigure}
\caption{Graphs $T_{21}$ through $T_{31}$ involved in the computation of the 5th and 6th coefficients of the chromatic polynomial \cite{Bielak1}.}\label{tgraphs2}
\end{figure}
\subsection{Khovanov and chromatic homology and their relations}\label{KhChCoeff}
The Jones polynomial has been categorified as the Euler characteristic of a bigraded homology theory known as Khovanov homology. We denote the Khovanov homology of a link by $Kh(L)$. The chromatic polynomial has a similar categorification known as chromatic graph homology. An overview of these homologies and their construction can be found in \cite{LS}, \cite{SaSco}. In this paper, we will use only the version of chromatic homology defined over $\mathcal{A}_2 = \mathbb{Z}[x]/(x^2)$ and will refer to it as $H_{\mathcal{A}_2}(G)$. Since $H_{\mathcal{A}_2}$ contains only $\mathbb{Z}_2$ torsion \cite{LS}, we introduce the following notation.
\begin{defn}\label{2torsion}
If $H$ is a subgroup of either Khovanov or chromatic homology, $\textnormal{tor$_2$} H$ denotes the order 2 torsion subgroup of $H$. We use $\textnormal{rk tor$_2$} H$ to indicate the number of copies of $\mathbb{Z}_2$.
\end{defn}
There is a partial correspondence between Khovanov homology of a link and the chromatic homology $H_{\mathcal{A}_2}$ of an associated graph.
\begin{thm}\cite{Przy1, PS} \label{Correspondence}
Let $D$ be an oriented diagram of link $L$ with $c_-$ negative crossings and $c_+$ positive crossings. Suppose $G_+(D)$ has $v$ vertices and positive girth $\ell$. Let $p = i - c_-$ and $q = v - 2j + c_+ - 2c_-$. For $0 \le i < \ell$ and $j \in \mathbb{Z}$, there is an isomorphism $$H_{\mathcal{A}_2}^{i,j}(G_+(D)) \cong Kh^{p,q}(L).$$ Additionally, for all $j \in \mathbb{Z}$, there is an isomorphism of torsion: $\textnormal{tor$_2$} H_{\mathcal{A}_2}^{\ell,j}(G_+(D)) \cong \textnormal{tor$_2$} Kh^{\ell-c_-,q}(L).$
\end{thm}
Chromatic homology $H_{\mathcal{A}_2}(G)$ is always homologically thin (all non-trivial homology lies on two diagonals). If $Kh(L)$ is homologically thin, then $Kh(L)$ also contains only $\mathbb{Z}_2$ torsion \cite{Shum2}.
\begin{thm}\cite{LS} \label{ChromaticDetermined}
The chromatic homology $H_{\mathcal{A}_2}(G)$ with coefficients in $\mathbb{Z}$ is entirely determined by the chromatic polynomial $P_G(\lambda)$.
\end{thm}
Note that $P_G(\lambda)$ and $H_{\mathcal{A}_2}(G)$ are both trivial if $G$ contains a loop, and both ignore the presence of multiple edges in $G$. If $G$ is a loopless graph ($\ell(G)>1$) then both $G$ and its simplification $G'$ have the same chromatic invariants: $P_G(\lambda) = P_{G'}(\lambda)$ and $H_{\mathcal{A}_2}(G) = H_{\mathcal{A}_2}(G')$. If $\ell(G)>2$, then we also have $G=G'$.
Theorem \ref{Correspondence} allows us to compute explicit formulae for extremal gradings of Khovanov homology, subject to combinatorial conditions on the Kauffman state of a link diagram. In \cite{AP,PPS,PS}, the following gradings of Khovanov homology are explicitly computed for diagrams when the isomorphism theorem holds.
\begin{prop}[\cite{PPS,PS}] \label{PS}
Let $D$ be a diagram of $L$ with $c_+$ positive crossings, $c_-$ negative crossings, and $|s_+|$ circles in the all-positive Kauffman state of $D$. Let $N = -|s_+|+c_{+}-2c_{-}$ and let $p_1, t_1$ denote the cyclomatic number and number of triangles in $G_+(D)'$, respectively.
If the girth of $G_+(D)$ is at least 2, then extreme Khovanov homology groups are given by:
\begin{eqnarray*}
&&Kh^{-c_{-},N}(L) = \mathbb{Z}\hspace{1cm}
Kh^{-c_{-},N+2}(L) = \begin{cases}
\mathbb{Z} & G_+(D) \text{ bipartite}\\
0 & \text{ otherwise}
\end{cases} \\
&& Kh^{1-c_{-},N+2}(L) =\begin{cases}
\mathbb{Z}^{p_1} & G_+(D) \text{ bipartite}\\
\mathbb{Z}^{p_1-1} \oplus \mathbb{Z}_2 & \text{ otherwise}
\end{cases}
\end{eqnarray*}
If, in addition, the girth of $G_+(D)$ is at least 3, then we have an additional grading in Khovanov homology:
\begin{eqnarray*}
Kh^{2-c_{-},N+4}(L) &=& \begin{cases}
\mathbb{Z}^{\binom{p_1}{2}} \oplus \mathbb{Z}_2^{p_1} & G_+(D) \text{ bipartite}\\
\mathbb{Z}^{\binom{p_1}{2}-t_1+1} \oplus \mathbb{Z}_2^{p_1-1}& \text{ otherwise}
\end{cases}
\end{eqnarray*}
\end{prop}
The following result is a restatement of \cite[Thm. 5.4]{SaSco}, describing the fourth and fifth homological gradings of $Kh(L)$ in terms of the associated graph.
\begin{thm}\cite{SaSco}\label{rankKhold}
Let $D$ be a diagram of $L$ as in Proposition \ref{PS}. Using conventions from Theorem \ref{FarBie} under the assumption that $G_+(D)$ has girth at least 4 we have the following gradings in Khovanov homology:
\begin{eqnarray*}
\textnormal{rk} Kh^{3-c_-,N+6}(L) = \textnormal{rk} \textnormal{tor$_2$} Kh^{4-c_-,N+8}(L) = \begin{cases}
p_1 + \binom{p_1+1}{3}-t_2 & G_+(D) \text{ bipartite,}\\
p_1 + \binom{p_1+1}{3}-t_2-1 & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
\end{thm}
\section{Extremal Khovanov homology computations}
In this section we rely on ideas used in Theorem \ref{rankKhold} to obtain explicit formulas for Khovanov homology in several additional extremal gradings using the formulas found in Theorem \ref{FarBie}. This approach can theoretically be extended to further groups on the diagonal using the method of \cite{Bielak1} to find formulas for additional chromatic coefficients, although this appears computationally challenging.
\begin{thm}\label{rankKhnew}
Let $D$ be a diagram of $L$ as in Proposition \ref{PS}. Suppose also that $G_+(D)$ has girth at least 5 with cyclomatic number $p_1$ and subgraphs $T_i$ denoted as in Theorem \ref{FarBie}. Let the coefficients $a_{v-4}$ and $a_{v-5}$ be as in Theorem \ref{456HGR}. Then we have the following relations in the Khovanov homology of $L$:
\begin{equation*}
\textnormal{rk} Kh^{4-c_-,N+8}(L)=
\textnormal{rk} \textnormal{tor$_2$} Kh^{5-c_-,N+10}(L) = \begin{cases}
\binom{p_1}{2}+ a_{v-4} & G_+(D) \text{ bipartite} \\
\binom{p_1}{2}+ a_{v-4} + 1 & \text{ otherwise}
\end{cases}
\end{equation*}
If in addition, the girth of $G_+(D)$ is at least 6, then we also have the following:
\begin{eqnarray*}
\textnormal{rk} Kh^{5-c_-,N+10}(L) &=& \textnormal{rk} \textnormal{tor$_2$} Kh^{6-c_-,N+12}(L) = \begin{cases}
p_1 + \binom{p_1+1}{3} - a_{v-5} & G_+(D) \text{ bipartite} \\
p_1 + \binom{p_1+1}{3}- a_{v-5} - 1 & \text{ otherwise}
\end{cases}
\end{eqnarray*}
\end{thm}
Theorem \ref{rankKhnew} is an immediate consequence of Theorem \ref{456HGR} and the isomorphism theorem for diagrams whose all-positive graphs have girth at least 5 or 6.
\begin{thm}\label{456HGR}
Let $G$ be a simple graph with cyclomatic number $p_1$ and subgraphs $T_i$ denoted as in Theorem \ref{FarBie}. Then we have the following groups in the chromatic homology of $G$:
\begin{eqnarray*}\label{RkTor45}
\textnormal{rk} H_{\mathcal{A}_2}^{4,v-4}(G) &=& \textnormal{rk} \textnormal{tor$_2$} H_{\mathcal{A}_2}^{5,v-5}(G) = \begin{cases}
\binom{p_1}{2}+ a_{v-4} & G \text{ bipartite} \\
\binom{p_1}{2}-t_1+1+ a_{v-4} & \text{ otherwise}
\end{cases}\\
\textnormal{rk} H_{\mathcal{A}_2}^{5,v-5}(G) &=& \textnormal{rk} \textnormal{tor$_2$} H_{\mathcal{A}_2}^{6,v-6}(G) = \begin{cases}
p_1 + \binom{p_1+1}{3}-t_2 - a_{v-5} & G \text{ bipartite} \\
p_1 + \binom{p_1+1}{3}-t_1(p_1-1)-t_2+2t_3-1 - a_{v-5} & \text{ otherwise}
\end{cases}
\end{eqnarray*}
The coefficients $a_{v-4}$ and $a_{v-5}$ are given by:
\begin{align*}
a_{v-4} &= \binom{v}{4}-E\binom{v-1}{3}+ \left(\binom{E}{2}-t_1\right)\binom{v-2}{2}+c_{v-3}(v-3) + c_{v-4}\\
a_{v-5} &= \binom{v}{5}-E\binom{v-1}{4}+ \left(\binom{E}{2}-t_1\right)\binom{v-2}{3}+c_{v-3}\binom{v-3}{2}+ c_{v-4}(v-4)+c_{v-5}
\end{align*}
\end{thm}
\begin{proof}
Let the chromatic polynomial of $G$ have the form given in Equation \ref{ChromCoef}. We change variables to $\lambda = q+1$ to match the graded Euler characteristic of $H_{\mathcal{A}_2}(G)$. The coefficient of $q^{v-i}$ in this polynomial will be denoted $a_i$.
\begin{align*}
P_G(q) &= (q+1)^v + c_{v-1}(q+1)^{v-1} + \ldots + c_2 (q+1)^2 + c_1 (q+1)\\
&= q^v + a_{v-1}q^{v-1} + \ldots + a_2 q^2 + a_1q + a_0.
\end{align*}
We proceed as in the proof of \cite[Thm. 5.3]{SaSco}, using the formulas for the $c_i$s in Theorem \ref{FarBie} and the equivalence of $P_G(q)$ with chromatic homology. Note that $t_1 = t_3 = 0$ if $G$ is bipartite.
\end{proof}
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1}
\begin{tabular}{|c| P{0.6cm} | P{0.6cm} | P{0.6cm} | P{0.7cm} | P{0.6cm} | P{0.6cm} | P{1.7cm} }
\cline{1-7}
$\mathbf{j/i}$ & 0 & 1 & $\ldots$ & $\ell-1$ & $\ell$ & $\cdots$ \\ \cline{1-7}
$v$ & $\blacksquare$ & & & & & & $\rightarrow a_v$\\ \cline{1-7}
$v-1$ & $\blacksquare$ & $\blacksquare$ & & & & & $\rightarrow a_{v-1}$\\ \cline{1-7}
$\vdots$ & & $\ddots$ & $\ddots$ & & & & $\vdots$ \\ \cline{1-7}
$v-\ell+1$ & & & $\blacksquare$ & $\blacksquare$ & & & $\rightarrow a_{v-\ell+1}$ \\ \cline{1-7}
$v-\ell$ & & & & $\blacksquare$ & $\square$ & \\ \cline{1-7}
$\vdots$ & & & & & $\square$ & $\ddots$ \\ \cline{1-7}
\end{tabular}
\caption{Chromatic homology $H_{\mathcal{A}_2}(G_+(D))$ with coefficients $a_{v-i}$ for the chromatic polynomial on the right. $\square$ indicates possible homology. $\blacksquare$ indicates isomorphism with Khovanov homology.} \label{tablechromproof}
\end{table}
The following theorem completely describes the part of Khovanov homology which is obtained from the isomorphism in Theorem \ref{Correspondence}.
\begin{thm}\label{rankKhovanovgirth}
Let $D$ be a diagram of a link $L$ with $c_+$ positive crossings, $c_-$ negative crossings, and $|s_+|$ circles in the all-positive Kauffman state, and let $N = -|s_+|+c_{+}-2c_{-}$. Suppose that $G_+(D)$ satisfies the conditions of Theorem \ref{rankHGRgirth} (in particular, the girth $\ell$ of $G_+(D)$ is greater than $2$). For $0 < i < \ell$, we have the following ranks of the Khovanov homology of $L$:
\begin{equation} \label{KhFormula}
\textnormal{rk} Kh^{i-c_-,N+2i}(L) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{p_1-2+k}{k} \right)-n_{i+1} +(-1)^{i+1} \delta^b
\end{equation}
where $\delta^b = 1$ if $G_+(D)$ is bipartite and 0 otherwise.
\end{thm}
Based on Theorem \ref{Correspondence} and \cite{LS}, formula \eqref{KhFormula} also gives the number of $\mathbb{Z}_2$-torsion groups on the next grading of this diagonal: $\textnormal{rk} \textnormal{tor$_2$} Kh^{(i+1)-c_-,-|s_+|+c_+-2c_-+2(i+1)}(L)$. If we consider the all-negative state graph $G_-(D)$, an analogous statement holds for the highest homological gradings in $Kh(L)$.
\begin{cor} \label{RankGenFun}
Let $D$ be a reduced diagram of $L$ that satisfies the conditions of Theorem \ref{rankKhovanovgirth}.
If in addition, $G_+(D)$ is a non-bipartite graph, then the sequence of ranks
\begin{equation}
\{S_0, \ldots, S_{\ell-2}\} = \{\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G_+(D))\}_{0\le i \le \ell-2} = \{\textnormal{rk} Kh^{i-c_-,N+2i}(L)\}_{0 \le i \le \ell-2}
\end{equation} is given by the first $\ell-1$ coefficients of the generating function $\dfrac{1}{(1+x)(1-x)^{p_1}}$.
\end{cor}
For graphs of girth $\ell$, Theorem \ref{Meredith} provides a succinct description of the first $\ell$ coefficients of the chromatic polynomial. We first translate this statement into a description of the ranks of chromatic homology in the first $\ell$ homological gradings. As a corollary, we obtain the entire part of Khovanov homology that is determined by the all-positive or all-negative state graph as in Theorem \ref{rankKhovanovgirth}.
\begin{thm}\label{rankHGRgirth}
Let $G$ be a simple graph with girth $\ell>2$, cyclomatic number $p_1$, and $n_{i}$ denoting the number of $i$-cycles in $G$. Then, for $0 < i < \ell$, we have the following ranks of the chromatic homology of $G$:
\begin{equation} \label{eqnrkHGR}
\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{p_1-2+k}{k} \right)-n_{i+1} +(-1)^{i+1} \delta^{b}
\end{equation}
where $\delta^{b} = 1$ if $G$ is bipartite and 0 otherwise.
\end{thm}
\begin{proof}
For $i=1,2,3$, this statement follows from \cite{PS}, \cite{SaSco}. We show by induction that it holds for $i>3$.
As above, let $a_{v-i}$ denote the coefficient of $P_G(q)$ derived from the quantum grading $j=v-i$ (see Table \ref{tablechromproof}). For $0 < i < \ell-1$, we use Theorem \ref{Meredith} to compute:
\begin{align*}
a_{v-i} &= \binom{v}{v-i} + c_{v-1}\binom{v-1}{v-i} + \ldots + c_{v-i}\binom{v-i}{v-i} = \displaystyle\sum_{k=0}^i c_{v-k} \binom{v-k}{v-i}\\
&= \displaystyle\sum_{k=0}^i (-1)^{k} \displaystyle\binom{E}{k} \binom{v-k}{v-i} = (-1)^i \binom{p_1-2+i}{i}
\end{align*}
and for $i= \ell-1$, a similar computation shows that $a_{v-(\ell-1)} = (-1)^{\ell-1} \left( \binom{p_1-2+(\ell-1)}{\ell-1} - n_{\ell} \right).$ For $i < \ell-1$, we have $n_{i+1} = 0$, and thus we can say $$a_{v-i} = (-1)^i\left( \binom{p_1-2+i}{i} - n_{i+1} \right)$$ for $0<i \le \ell-1$.
Suppose that $3 < i < \ell$ and that Equation \ref{eqnrkHGR} holds for all homological gradings less than $i$. We show that Equation \ref{eqnrkHGR} also holds for $\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G)$. Since chromatic homology is thin, each coefficient is the difference of the ranks of the two homology groups in the grading $j=v-i$.
\begin{equation} \label{RankDiff}
a_{v-i} = (-1)^{i-1}\textnormal{rk} H_{\mathcal{A}_2}^{i-1,v-i}(G) + (-1)^{i}\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G)
\end{equation}
By the knight move isomorphism of \cite{CCR}, $\textnormal{rk} H_{\mathcal{A}_2}^{i-1,v-i}(G) = \textnormal{rk} H_{\mathcal{A}_2}^{i-2,v-(i-2)}(G)$. We make this substitution into Equation \ref{RankDiff}, along with the value of $a_{v-i}$ derived above.
\begin{align*}
(-1)^i\left( \binom{p_1-2+i}{i} - n_{i+1} \right) &= (-1)^{i-1}\textnormal{rk} H_{\mathcal{A}_2}^{i-2,v-(i-2)}(G) + (-1)^{i}\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G)\\
\binom{p_1-2+i}{i} - n_{i+1} &= -\textnormal{rk} H_{\mathcal{A}_2}^{i-2,v-(i-2)}(G) + \textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G)\\
\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G) &= \textnormal{rk} H_{\mathcal{A}_2}^{i-2,v-(i-2)}(G) + \binom{p_1-2+i}{i}- n_{i+1}
\end{align*}
By the induction assumption
\begin{align*}
\textnormal{rk} H_{\mathcal{A}_2}^{i-2,v-(i-2)}(G) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0 \le k = (i-2)-2r \le i-2}} \binom{p_1-2+k}{k} \right)-n_{i-1} +(-1)^{i-1} \delta^b
\end{align*}
We may drop the term $n_{i-1} = 0$ since we are assuming $i < \ell$. Finally, we collect all binomial coefficients into the summation and note that $i-1$ has the same parity as $i+1$.
\begin{align*}
\textnormal{rk} H_{\mathcal{A}_2}^{i,v-i}(G) &= \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0 \le k = (i-2)-2r \le i-2}} \binom{p_1-2+k}{k} \right) +(-1)^{i-1} \delta^b + \binom{p_1-2+i}{i} - n_{i+1}\\
&= \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0 \le k = i-r \le i}} \binom{p_1-2+k}{k} \right)- n_{i+1} + (-1)^{i+1} \delta^b \qedhere
\end{align*}
\end{proof}
\begin{exa}\label{KhFormulaEx}
Let $K$ be the knot $11a362$ (Dowker-Thistlethwaite notation) with diagram $D$ depicted in Figure \ref{diagram11}. The all-positive state graph $G_+(D)$ has girth $\ell = 6$ and cyclomatic number $p_1 = 2$. The Khovanov homology $Kh(K)$ is shown in Table \ref{exampletable1} and the chromatic homology $H_{\mathcal{A}_2}(G_+(D))$ in Table \ref{exampletable2}.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{knot11a362withg.pdf}
\caption{Diagram of $11a362$ with all-positive state graph $G_+(D)$}
\label{diagram11}
\end{figure}
The graph $G_+(D)$ is bipartite with $c_- = 11$ and $N = -32$. The groups shown in bold in Table \ref{exampletable1} are those which correspond to chromatic homology groups in $H_{\mathcal{A}_2}(G_+(D))$. Theorem \ref{rankKhovanovgirth} describes the ranks of these Khovanov homology groups which are located on the lower diagonal. For $i=1$ through $i=\ell-2=4$:
\begin{align*}
\textnormal{rk} Kh^{i-c_-,N+2i}(L) &= \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{2-2+k}{k} \right) +(-1)^{i+1} \delta^b \\ &= \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} 1 \right) +(-1)^{i+1} = \left \lfloor \dfrac{i}{2}+1 \right \rfloor +(-1)^{i+1}
\end{align*}
while for $i=\ell-1 = 5$:
\begin{align*}
\textnormal{rk} Kh^{5-c_-,N+10}(L) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} 1 \right) -n_6 +(-1)^{5+1} = \left \lfloor \dfrac{5}{2}+1 \right \rfloor -1 +(-1)^{6} = 3
\end{align*}
Observe that if one ignores the $\delta^b$ term that keeps track of the bipartite property, the first $\ell-1$ ranks given by the formula are $1, 1, 2, 2, 3$ which are the first 5 coefficients of the generating function $\dfrac{1}{(1+x)(1-x)^{2}}$ (see Corollary \ref{RankGenFun}).
\begin{center}
\begin{table}
\renewcommand{\arraystretch}{1}
\begin{center}
\begin{tabular}{|c| P{0.6cm} | P{0.6cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.9cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.5cm} | P{0.5cm} | P{0.5cm} }
\cline{1-13}
$\mathbf{q/p}$ & $-11$ & $-10$ & $-9$ & $-8$ & $-7$ & $-6$ & $-5$ & $-4$ & $-3$ & $-2$ & $-1$ & $0$ & \\ \cline{1-13}
$-8$ & & & & & & & & & & & & $1$ & \\ \cline{1-13}
$-10$ & & & & & & & & & & & & $1_2$ & \\ \cline{1-13}
$-12$ & & & & & & & & & & $3$ & $1$ & & \\ \cline{1-13}
$-14$ & & & & & & & & & $1$ & $3_2$ & & & \\ \cline{1-13}
$-16$ & & & & & & & & $3$ & $3$, $1_2$ && & & \\ \cline{1-13}
$-18$ & & & & & & & $3$ & $1$, $3_2$ & && & & \\ \cline{1-13}
$-20$ & & & & & & $\fakebold 2$ & $3$, $\fakebold 3_{\textbf{2}} $ & & & & & & \\ \cline{1-13}
$-22$ & & & & & $\fakebold 3$ & $\fakebold 3, \fakebold 2_{\textbf{2}} $ & & && & & & \\ \cline{1-13}
$-24$ & & & & $\fakebold 1$ & $\fakebold 2, \fakebold 3_{\textbf{2}} $ & & & && & & & \\ \cline{1-13}
$-26$ & & & $\fakebold 2$ & $\fakebold 3, \fakebold 1_{\textbf{2}}$ & & & & & && & & \\ \cline{1-13}
$-28$ & & & $\fakebold 1, \fakebold 2_{\textbf{2}}$ & & & & & & & & & & \\ \cline{1-13}
$-30$ & $\fakebold 1$ & $\fakebold 2$ & & & & & & && & & & \\ \cline{1-13}
$-32$ & $\fakebold 1$ & & & & & & & && & & & \\ \cline{1-13}
\end{tabular}
\caption{Khovanov homology $Kh(11a362)$. An entry of $k$ represents a summand $\mathbb{Z}^k$, and $k_{2}$ represents a summand of $\mathbb{Z}_2^k$. Entries in bold, from -11 to -5, are isomorphic to gradings in chromatic homology of $G_+(D)$ via Theorem \ref{Correspondence}.} \label{exampletable1}
\end{center}
\end{table}\end{center}
\begin{center}
\begin{table}
\renewcommand{\arraystretch}{1}
\begin{center}
\begin{tabular}{|c| P{0.6cm} | P{0.6cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.9cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} }
\cline{1-10}
$\mathbf{j/i}$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & \\ \cline{1-10}
$10$ & $\fakebold 1$ & & & & & & & & & \\ \cline{1-10}
$9$ & $\fakebold 1$ & $\fakebold 2$ & & & & & & & & \\ \cline{1-10}
$8$ & & & $\fakebold 1, \fakebold 2_{\textbf{2}}$ & & & & & & & \\ \cline{1-10}
$7$ & & & $\fakebold 2$ & $\fakebold 3, \fakebold 1_2$ & & & & & & \\ \cline{1-10}
$6$ & & & & $\fakebold 1$ & $\fakebold 2, \fakebold 3_{\textbf{2}}$ & & & && \\ \cline{1-10}
$5$ & & & & & $\fakebold 3$ & $\fakebold 3, \fakebold 2_{\textbf{2}}$ & & & & \\ \cline{1-10}
$4$ & & & & & & $\fakebold 2$ & $2, \fakebold 3_{\textbf{2}}$ & & & \\ \cline{1-10}
$3$ & & & & & & & $3$ & $1, 2_2$ & & \\ \cline{1-10}
$2$ & & & & & & & & $2$ & $1_2$ & \\ \cline{1-10}
$1$ & & & & & & & & & $1$ & \\ \cline{1-10}
\end{tabular}
\caption{Chromatic homology $H_{\mathcal{A}_2}(G_+(D))$ for the graph in Example \ref{KhFormulaEx}. An entry of $k$ represents a summand $\mathbb{Z}^k$, and $k_{2}$ represents a summand of $\mathbb{Z}_2^k$. Entries in bold are isomorphic to gradings in Khovanov homology (see Table \ref{exampletable1}).} \label{exampletable2}
\end{center}
\end{table}\end{center}
\end{exa}
\section{More on the head and tail of the Jones polynomial}
Theorem \ref{rankKhovanovgirth} can be used to compute the $\ell$ extremal coefficients in the head or tail of the Jones polynomial, subject to certain conditions on the Khovanov homology of $L$. As shown in Table \ref{table2}, if $Kh^{i,j}(L)$ is trivial for $i > (\ell-1) - c_-$ and $j < N+2\ell$, then the extremal coefficients of the unnormalized Jones polynomial are determined by precisely by the ranks described in Theorem \ref{rankKhovanovgirth}. We conjecture that these gradings are always trivial in Khovanov homology, so that Theorem \ref{Jones1} is true for all links and not only those which are Khovanov thin.
\begin{table}
\centering
\renewcommand{\arraystretch}{1}
\begin{tabular}{|c|c | c | c | c | c|c |c }
\cline{1-7}
$\mathbf{q/p}$ & $-c_-$ & $1-c_-$ & $\ldots$ & $(\ell-1)-c_-$ & $\ell-c_-$ & $\cdots$ \\ \cline{1-7}
$\vdots$ & & & & & $\square$ & $\iddots$& \\ \cline{1-7}
$N+2\ell$ & & & & $\blacksquare$ &$\square$ & & \\ \cline{1-7}
$N+2(\ell-1)$ & & & $\blacksquare$ & $\blacksquare$ & ? & ? & $\rightarrow \alpha_{\ell-1} = a_{v-\ell+1}$ \\ \cline{1-7}
$\vdots$ & & $\iddots$ & $\iddots$ & & ? & ? & $\vdots$ \\ \cline{1-7}
$N+2$ & $\blacksquare$ & $\blacksquare$ & & & ? & ? & $\rightarrow \alpha_1 = a_{v-1}$ \\ \cline{1-7}
$N$ & $\blacksquare$ & & & & ? & ? & $\rightarrow \alpha_0 = a_v$ \\ \cline{1-7}
\end{tabular}
\caption{Khovanov homology $Kh(L)$ with coefficients $\alpha_i$ for the unnormalized Jones polynomial on the right (compare with chromatic coefficients $a_{v-i}$ in Table \ref{tablechromproof}). $\square$ indicates possible homology. $\blacksquare$ indicates isomorphism with chromatic homology. ? indicates gradings that must be zero for results to hold for all links} \label{table2}
\end{table}
\begin{thm} \label{Jones1}
Let $D$ be a diagram of a link $L$ such that $Kh(L)$ is homologically thin and $G_+(D)$ has girth $\ell > 2$, with cyclomatic number $p_1$ and number of $\ell$-cycles $n_{\ell}$. Then the first $\ell$ coefficients in the tail of $J_L$ are given by the formula:
\begin{equation}\label{JC}
\beta_{i} = \begin{cases}
(-1)^{i-c_-(D)}\binom{p_1-1+i}{i} & 0 \le i \le \ell - 2\\
(-1)^{\ell-1-c_-(D)}\left(\binom{p_1-1+(\ell-1)}{\ell-1} - n_{\ell}\right) & i = \ell-1\\
\end{cases}\end{equation}
If we consider the all-negative state graph $G_-(D)$, an analogous statement holds for the first $\ell(G_-(D))$ coefficients in the head of $J_L$.
\end{thm}
\begin{proof} The normalized Jones polynomial $J_L$ is the graded Euler characteristic of the reduced Khovanov homology $\Widetilde{Kh}(L)$, so we prove this result by describing $\Widetilde{Kh}(L)$. Let $R_i = \textnormal{rk} Kh^{i-c_-, N+2i}(L)$ given by the formula in Theorem \ref{rankKhovanovgirth}:
\begin{equation}
R_i = \textnormal{rk} Kh^{i-c_-,N+2i}(L) = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{p_1-2+k}{k} \right)-n_{i+1} +(-1)^{i+1} \delta^b
\end{equation}
for $0 < i < \ell$. We also assign $R_0 = 1$, the rank of $Kh^{-c_-, N}(L)$ for any $L$ satisfying the above condition on girth. For notational convenience we let $R_i = 0$ for $i<0$.
Since the extreme gradings of $Kh(L)$ from $i=c_-$ through $i=(\ell-1)-c_-$ are isomorphic to chromatic homology, this part of $Kh(L)$ is thin with only $\mathbb{Z}_2$ torsion. The knight move isomorphism gives us the ranks of the additional groups on the second diagonal:
\begin{equation}
R_i = \textnormal{rk} Kh^{i-c_-,N+2i}(L) = \textnormal{rk} Kh^{(i+1)-c_-,N+2(i+2)}(L)
\end{equation}
which is valid for $1 \le i \le \ell-2$ if $G_+(D)$ is bipartite and $0 \le i \le \ell-2$ otherwise. Passing to Khovanov homology over $\mathbb{Z}_2$ coefficients, we can replace ``knight move" pairs of $\mathbb{Z}$s with ``tetrominos" of $\mathbb{Z}_2$s, see \cite{Shum2, LS}:
\begin{equation}
\textnormal{rk} \textnormal{tor$_2$} Kh_{\mathbb{Z}_2}^{i-c_-,N+2i}(L) = \textnormal{rk} \textnormal{tor$_2$} Kh_{\mathbb{Z}_2}^{i-c_-,N+2(i+1)}(L) = \begin{cases}
R_1 & i=1 \text{ and $G_+(D)$ bipartite} \\
R_{i-1}+R_i & \text{ otherwise, } 0 \le i < \ell
\end{cases}
\end{equation}
Using \cite[Cor.3.2C]{Shum1} and the fact that $\textnormal{rk} \Widetilde{Kh^{i,j}}(L) = \textnormal{rk} \Widetilde{Kh^{i,j}_{\mathbb{Z}_2}}(L)$ for thin links, we obtain the reduced integer Khovanov homology:
\begin{equation}
\textnormal{rk} \Widetilde{Kh}^{i-c_-,N+2i+1}(L) = \begin{cases}
R_1 & i=1 \text{ and $G_+(D)$ bipartite} \\
R_{i-1}+R_i & \text{ otherwise, } 0 \le i < \ell
\end{cases}
\end{equation}
Taking the graded Euler characteristic of $\Widetilde{Kh}(L)$ yields the following coefficients in the tail of $J_L(q)$:
\begin{equation}
\beta_{i} = \begin{cases}
(-1)^{1-c_-}R_1 & i = 1 \text{ and $G_+(D)$ bipartite}\\
(-1)^{i-c_-}(R_{i-1}+R_i) & \text{ otherwise, } 0 \le i < \ell
\end{cases}\end{equation}
The formula in Equation \ref{JC} clearly holds for $i=0$, and for the bipartite case when $i=1$ by direct calculation.
\begin{align*}
R_1 = \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = 1-2r \le 1}} \binom{p_1-2+k}{k} +(-1)^2\right) = \binom{p_1-1}{1} + 1 = \binom{p_1-1+1}{1}
\end{align*}
For all other cases with $1 \le i < \ell-1$, we compute $R_{i-1} + R_i $:
\begin{align*}
&
\left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = (i-1)-2r \le i-1}} \binom{p_1-2+k}{k} +(-1)^{i}\delta_b\right)+ \left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{p_1-2+k}{k} +(-1)^{i+1}\delta_b\right)\\
&= \displaystyle\sum_{k=0}^i \binom{p_1-2+k}{k}= \binom{p_1-1+i}{i}=R_{i-1} + R_i
\end{align*}
where $\delta^{b} = 1$ if $G_+(D)$ is bipartite and 0 otherwise.
For $i=\ell-1$, $n_{i+1} = n_{\ell}$ is non-zero so that $R_{\ell-2} + R_{\ell-1} = \binom{p_1-1+(\ell-1)}{\ell-1} - n_{\ell}$.
\end{proof}
\begin{exa}\label{JonesFormulaEx}
Let $K$ be the knot $11a362$ as in Example \ref{KhFormulaEx}.
The unnormalized Jones polynomial $\hat{J}_K(q)$ is the graded Euler characteristic of $Kh(K)$:
\begin{align*}
\hat{J}_K(q) &= -q^{-32}+q^{-30}-q^{-28}+q^{-26}-q^{-24}-q^{-20}-2q^{-18}-q^{-14}+2q^{-12}+q^{-8}
\end{align*}
Table \ref{exampletablejones} illustrates how the first six terms in the tail (including one zero term) arise from the part of $Kh(K)$ that agrees with the chromatic homology of $G_+(D)$.
The normalized Jones polynomial is given by:
\begin{align*}
&J_K(q) = \hat{J}_K(q)/(q+q^{-1})= \beta_0 q^{-31} + \beta_1 q^{-29} + \beta_2 q^{-27} + \beta_3 q^{-25} + \beta_4 q^{-23} + \beta_5 q^{-21} + \ldots\\
&= -q^{-31} + 2q^{-29} - 3q^{-27} + 4q^{-25} - 5q^{-23} + 5q^{-21} - 6q^{-19} + 4q^{-17} - 4q^{-15} + 3q^{-13} - q^{-11} + q^{-9}
\end{align*}
Any A-adequate diagram of $K$ must have the same number of negative crossings $(c_- = 11)$ as diagram $D$ \cite{Lick}. The coefficients $\beta_i$ agree with the formulas given in Theorem \ref{Jones1} for $0 \le i \le \ell-1 = 5$.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1}
\begin{tabular}{|c| P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.8cm} | P{0.9cm} | P{0.8cm} | l }
\cline{1-9}
$\mathbf{q/p}$ & $-11$ & $-10$ & $-9$ & $-8$ & $-7$ & $-6$ & $-5$ & $-4$ & \\ \cline{1-9}
$-16$ & & & & & & & & $\iddots$ & \\ \cline{1-9}
$-18$ & & & & & & & $3$ & $\iddots$ & \\ \cline{1-9}
$-20$ & & & & & & $\fakebold 2$ & $3$, $\fakebold 3_2$ & & \\ \cline{1-9}
$-22$ & & & & & $\fakebold 3$ & $\fakebold 3, \fakebold 2_2$ & & & $\rightarrow 0$ \\ \cline{1-9}
$-24$ & & & & $\fakebold 1$ & $\fakebold 2, \fakebold 3_2$ & & & & $\rightarrow -q^{-24}$ \\ \cline{1-9}
$-26$ & & & $\fakebold 2$ & $\fakebold 3, \fakebold 1_2$ & & & & & $\rightarrow q^{-26}$ \\ \cline{1-9}
$-28$ & & & $\fakebold 1, \fakebold 2_2$ & & & & & & $\rightarrow -q^{-28}$ \\ \cline{1-9}
$-30$ & $\fakebold 1$ & $\fakebold 2$ & & & & & & & $\rightarrow q^{-30}$ \\ \cline{1-9}
$-32$ & $\fakebold 1$ & & & & & & & & $\rightarrow -q^{-32}$ \\ \cline{1-9}
\end{tabular}
\caption{Lowest gradings of the Khovanov homology $Kh(11a362)$. Entries in bold are isomorphic to gradings in chromatic homology of $G_+(D)$ via Theorem \ref{Correspondence}. The rightmost column contains the terms of the graded Euler characteristic which agree with the chromatic polynomial of $G_+(D)$.} \label{exampletablejones}
\end{table}
\end{exa}
\section{Girth of a link}\label{Girth1}
Each planar diagram $D$ of a link $L$ has an associated state graph $G_+(D)$ whose chromatic homology is related to Khovanov homology by the correspondence described in Theorem \ref{Correspondence}.
Notice that the girth of $G_+(D)$ depends on the diagram of a knot. For example, adding a right-hand twist to any strand of $D$ using a Reidemeister I move creates a loop in $G_+(D)$, reducing the girth of this graph to $1$.
The applicability of Theorem \ref{rankKhovanovgirth} depends on the girth of $G_+(D)$; therefore, given any link $L$, we are interested in finding a diagram $D$ that maximizes the contribution of chromatic homology to Khovanov homology.
The largest such contribution made to $Kh(L)$ comes from the diagram for which $G_+(D)$ has the largest possible girth. This fact motivated the following definition that allows us to explicitly state the extent of the correspondence between Khovanov homology and the Jones polynomial with chromatic homology and the chromatic polynomial, respectively.
\begin{defn}[\cite{SaSco}]
The girth of a link $L$ is $\textnormal{gr}(L) = \max\{\ell(G_+(D))~|~\text{$D$ is a diagram of L} \}$ where $G_+(D)$ is the graph obtained from the all-positive Kauffman state of diagram $D$, and $\ell(G_+(D))$ is the girth of graph $G_+(D)$.
\end{defn}
\begin{prop}[\cite{SaSco}] \label{GirthFinite}
The girth $\textnormal{gr}(L)$ of any link $L$ is finite.
\end{prop}
\begin{proof}
The non-trivial groups in $H_{\mathcal{A}_2}(G_+(D))$ span at least $\ell(G_+(D))-1$ homological gradings \cite{SaSco}, which are isomorphic to $\ell(G_+(D))-1$ corresponding gradings in $Kh(L)$. Since the span of $Kh(L)$ is bounded above, so are the possible girths of $G_+(D)$.
\end{proof}
Girth can alternatively be defined by taking the maximum value of $\ell(G_-(D))$ over all diagrams of $L$, or both values can be considered. A proof similar to that of Proposition \ref{GirthFinite} shows this invariant is also finite for any $L$.
In the rest of this section we analyze properties and bounds on girth coming from the Jones polynomial and Khovanov homology, as well as types of graphs that can appear as state graphs for diagrams of a given link.
\subsection{Bounds on the girth}
As with many knot invariants defined as a maximum or a minimum over all diagrams of a given knot, one bound is much easier to prove than the other.
In the case of girth, any knot diagram gives a lower bound. Theorems \ref{rankKhovanovgirth} and \ref{Jones1} provide some insight into what the upper bound on the girth of a link might be and the properties of a graph $G$ which realizes the girth.
\begin{cor} \label{KhUpper}
Let $L$ be a link and let $M_K$ be the greatest number such that
\begin{equation} \label{KhUpperEq}
\textnormal{rk} Kh^{P+i, Q+2i}(L) =
\left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = i-2r \le i}} \binom{b-2+k}{k} \right) +(-1)^{i+1} \delta
\end{equation} for all $0 < i \leq M_K-2$, where $P$ and $Q$ are the lowest homological and quantum gradings in $Kh(L)$, $b>0$, and either $\delta = 0$ or $\delta = 1$ for all $i$. Then \begin{equation}\label{UpKhov}
\textnormal{gr}(L) \le M_K.
\end{equation}
\end{cor}
\begin{cor} \label{JonesUpper}
Suppose that link $L$ is Khovanov thin with Jones polynomial $J_L(q)$ as in Definition \ref{JonesCoef}. Let $M_J$ be the greatest number such that $|\beta_i| = \binom{b-1+i}{i}$ for some b, with signs alternating, for all $0\leq i\leq M_J-2$. Then \begin{equation}\label{UpJon}
\textnormal{gr}(L) \le M_J.
\end{equation}
\end{cor}
It is worth noting that with the above notation, $\textnormal{gr}(L) \le M_J \le M_K$. Moreover, $M_J = M_K$ for homologically thin links and we conjecture this will be true in general.
In Example \ref{InequalityEx}, we demonstrate that the upper bounds for $\textnormal{gr}(K)$ provided by Khovanov homology and the Jones coefficients are not necessarily achieved by any diagram of $K$.
\subsection{On all-positive state graphs}
The following corollary of Theorem \ref{rankKhovanovgirth} states that if Khovanov and chromatic homology agree on 3 or more gradings, this agreement imposes a restriction on the type of graphs that realize the isomorphism.
\begin{cor} \label{Khgraphproperty}
Suppose that the link $L$ has
a diagram $D$ such that $G_+(D)$ has girth $\ell > 2$. Then:
\begin{enumerate}
\item \cite{PPS} $G_+(D)$ is bipartite if and only if $\textnormal{rk} Kh^{-c_-, N+2}(L) = 1$.
\item \cite{PPS} the cyclomatic number of $G_+(D)$ is $$p_1 = \textnormal{rk} Kh^{1-c_-, N+2}(L)-\textnormal{rk} Kh^{-c_-, N+2}(L)+1$$
\item
the number of $\ell$-cycles in $G_+(D)$ is equal to
\begin{align*}
n_{\ell} = \left(\sum_{\substack{r \ge 0,\\ 0 \le k = (\ell-1)-2r \le (\ell-1)}} \binom{p_1-2+k}{k} \right) +(-1)^{\ell}\textnormal{rk} Kh^{-c_-,N+2}(L) - \textnormal{rk} Kh^{(\ell-1)-c_-, N+2(\ell-1)}(L)
\end{align*}
\end{enumerate}
\end{cor}
A similar result exists for thin links via the Jones polynomial and Theorem \ref{Jones1}.
\begin{cor} \label{girthGraph}
Suppose that link $L$ is homologically thin with Jones polynomial:
$$J_L(q) = \beta_0q^{C} + \beta_1q^{C+2} + \beta_2q^{C+4} + \beta_3q^{C+6} + \ldots$$
and that $D$ is a diagram of $L$ such that $G_+(D)$ has girth $\ell > 2$. Then the cyclomatic number of $G_+(D)$ is equal to $|\beta_1|$ and the number of $\ell$-cycles in $G_+(D)$ is equal to $\binom{|\beta_1|-1+(\ell-1)}{\ell-1} - |\beta_{\ell}|.$
\end{cor}
\begin{exa}
Let $K = 11a362$ as in Examples \ref{KhFormulaEx} and \ref{JonesFormulaEx}. From the Khovanov homology in Table \ref{exampletable1}, we see that $\textnormal{rk} Kh^{-10, -30}(K) = 2 = \binom{1}{1}+1, \textnormal{rk} Kh^{-9, -28}(K) = 1 = \binom{2}{2}+\binom{2-2}{0}-1, \textnormal{rk} Kh^{-8, -26}(K) = 3 = \binom{3}{3}+\binom{2-1}{1}+1$, and
$\textnormal{rk} Kh^{-7, -24}(K) = 2 = \binom{4}{4}+\binom{2}{2}+\binom{2-2}{0}-1$.
These ranks agree with Equation \ref{KhUpperEq} for the values $b = 2$, $\delta = 1$, and $0 < i \le 4$. However, there is no agreement for $i=5$, since
\begin{equation*}
\textnormal{rk} Kh^{-6, -22}(K) = 3
\end{equation*}
but
\begin{equation*}
\left(\displaystyle\sum_{\substack{r \ge 0,\\ 0\le k = 5-2r \le 5}} \binom{2-2+k}{k} \right) +(-1)^{5+1} = \binom{5}{5}+\binom{3}{3}+\binom{2-1}{1}+1 = 4.
\end{equation*}
Using Corollary \ref{KhUpper} we conclude that $M_K = 4+2 = 6$ is an upper bound for $\textnormal{gr}(K)$.
Since $K$ is Khovanov thin, we can obtain the same upper bound for $\textnormal{gr}(K)$ using the Jones coefficients and Corollary \ref{JonesUpper}. From Example \ref{JonesFormulaEx} we have
\begin{equation*}
\beta_0 = -1, \beta_1= 2, \beta_2 = -3, \beta_3 = 4, \beta_4 = -5
\end{equation*}
These coefficients alternate in sign and their absolute values satisfy the formula $\binom{b+i-1}{i}$ for $b=2$, $0\leq i\leq 4$. From Corollary \ref{JonesUpper} we derive the upper bound of $M_J = 4+2 = 6$.
Suppose $D$ is a diagram of $K$ such that $G_+(D)$ realizes the maximum girth of 6. Using Corollary \ref{Khgraphproperty}, we determine $G_+(D)$ must be bipartite since $\textnormal{rk} Kh^{-11, -30}(K) = 1.$ The cyclomatic number of $G_+(D)$ must be
\begin{equation*}
p_1 = \textnormal{rk} Kh^{-10, 30}(L)-\textnormal{rk} Kh^{-11, 30}(L)+1 = 2
\end{equation*}
while the number of 6-cycles in $G_+(D)$ must be
\begin{align*}
n_{6} &= \left(\sum_{\substack{r \ge 0,\\ 0 \le k = 5-2r \le 5}} \binom{2-2+k}{k} \right) +(-1)^{6}\textnormal{rk} Kh^{-11, -30}(K) - \textnormal{rk} Kh^{-6, -22}(K)\\
&= 3 + 1 - 3 = 1.
\end{align*}
Together, these statements imply that $G_+(D)$ must be a bipartite graph containing exactly 2 cycles: one cycle of length 6 and another cycle of length $n$ where $n>6$ is even.
\end{exa}
The following example demonstrates that inequalities \eqref{UpKhov} and \eqref{UpJon} may be strict; i.e., there may be no diagram for a link that realizes either of these upper bounds for girth.
\begin{exa} \label{InequalityEx}
The knot $K=12n821$ is both non-alternating and homologically thin, with Jones polynomial $q^{-5}- 2q^{-4}+3q^{-3}- 4q^{-2}+5q^{-1}- 5+5q+4q^{2}- 3q^3+2q^4 + q^5$. By Corollary \ref{JonesUpper} applied to the first 5 coefficients, we find an upper bound $M_J = 6$ for the girth of $K$. Similarly, we can apply Corollary \ref{KhUpper} to ranks on the main diagonal of $Kh(L)$ to obtain the same upper bound for girth, $M_K = 6$. However, we can show that no diagram of $K$ exists which achieves this upper bound.\\
Suppose that $K$ has a diagram $D$ such that $G_+(D)$ has girth greater than 2. Then this diagram is both plus-adequate and non-alternating, so $G$ must have a cut-vertex \cite{Stoimenow1}. By Corollary \ref{Khgraphproperty}, $p_1(G_+(D)) = 2$. Since $G_+(D)$ has no loops or multiple edges, it must be a vertex join of two cycles, $P_n*P_m$. Any knot diagram with all-positive graph $P_n*P_m$ is a diagram of an alternating knot: either a connected sum of torus links, or a rational knot. But $K$ has no alternating diagram, so the girth of $12n821$ must be less than or equal to 2.\\
In addition, the same argument can be applied to show that the girth of any all-negative state graph for this knot is less than or equal to 2.
\end{exa}
The 2nd coefficient of the Jones polynomial, which captures the cyclomatic number of $G_+(D)$, uniquely determines the first $\textnormal{gr}(L)-1$ coefficients. In a similar fashion, the first two homological gradings of $Kh(L)$ determine the first $\textnormal{gr}(L)$ gradings. This leads to a somewhat surprising result.
\begin{thm} \label{girthPossibleKhovanov}
Let $L$ be a non-trivial link. If $D$ is a diagram of $L$ such that $\ell(G_+(D)) < \textnormal{gr}(L)$, then $\ell(G_+(D)) = 1$ or $\ell(G_+(D)) = 2$.
\end{thm}
\begin{proof}
The result holds for $1 \le \textnormal{gr}(L) \le 3$, since no non-trivial diagram can have $\ell(G_+(D)) = 0$. For $\textnormal{gr}(L)=M>3$, there exists some diagram $D_{max}$ such that $G_{max} = G_+(D_{max})$ has girth $M$. Suppose that there exists another diagram $D$ of $L$ such that $G = G_+(D)$ has girth $h$, with $2 < h < M$. Applying Theorem \ref{rankKhovanovgirth} to diagram $D_{max}$ for $i = h-1$, we find that the rank of the Khovanov homology group $Kh^{(h-1)-c_-(D_{max}),N(D_{max})+2(h-1)}(L)$ is equal to
\begin{equation} \label{Compare1}
\left(\displaystyle\sum_{\substack{r \ge 0,\\ 0 \le k = (h-1)-2r \le (h-1)}} \binom{p_1(G_{max})-2+k}{k} \right)-n_{h}(G_{max}) +(-1)^{h} \delta^b(G_{max})
\end{equation} where $N(D_{max}) = -|s_+(D_{max})|+c_{+}(D_{max})-2c_{-}(D_{max})$ and $\delta^b(G_{max})$ is 1 if $G_{max}$ is bipartite, 0 otherwise.
Now we also apply Theorem \ref{rankKhovanovgirth} to diagram $D$ for $i = h-1$. Since $D_{max}$ and $D$ are both plus-adequate, $c_-(D_{max}) = c_-(D)$. In addition, $-|s_+(D_{max})|+c_+(D_{max}) = -|s_+(D)|+c_+(D)$ and thus $N(D_{max}) = N(D)$ (see \cite{Lick}). Thus Equation \ref{Compare2} describes the rank of the same group in Khovanov homology as in Equation \ref{Compare1}:
\begin{equation} \label{Compare2}
\left(\displaystyle\sum_{\substack{r \ge 0,\\ 0 \le k = (h-1)-2r \le (h-1)}} \binom{p_1(G)-2+k}{k} \right)-n_{h}(G) +(-1)^{h} \delta^b(G)
\end{equation} with $N(D)$ and $\delta^b(G)$ defined as above.
Since $\ell(G_{max}) = M > 2$ and $\ell(G) = h > 2$, we can use Corollary \ref{Khgraphproperty} to calculate $p_1(G_{max}) = p_1(G)$ and $\delta^b(G_{max}) = \delta^b(G)$. Setting Equation \ref{Compare1} equal to Equation \ref{Compare2}, we see that $n_h(G_{max})$ must be equal to $n_h(G)$. However, $n_h(G_{max}) = 0$ by assumption (because $h<M$) while $n_h(G) > 0$ (because $G$ must contain at least one cycle of length $h$). Thus we have a contradiction and no such diagram $D$ may exist.
\end{proof}
\begin{cor}\label{girth3more}
If $L$ has a diagram $D$ such that $ \ell(G_+(D)) = p \ge 3$, then $\textnormal{gr}(L) = p$.
\end{cor}
\begin{exa}[Alternating pretzel links]
Let $L$ be an alternating pretzel link with twist parameters $(-a_1, -a_2, \ldots, -a_n)$ where $a_i > 1$ for all $i$. The girth of the graph obtained from the standard diagram is $\min\{a_i+a_j ~|~1 \le i \neq j \le n \}$. Since this number is at least 4, it is equal to $\textnormal{gr}(L)$ by Corollary \ref{girth3more}.
\end{exa}
\begin{exa}[3-braids]
Suppose $L$ is the closure of a negative 3-braid $\gamma = \sigma_{i_1}^{a_1}\sigma_{i_2}^{a_2}\ldots\sigma_{i_k}^{a_k}$, $a_j \le -1$. If $i_j \in \{1,2\}$ and $i_j \neq i_{j+1}$ for all $j$, then $\ell(G_+(\gamma)) \ge 3$ \cite[Proposition 5.1]{PS} and so $\textnormal{gr}(L) = \ell(G_+(\gamma))$ by Corollary \ref{girth3more}.
\end{exa}
\subsection{Girth and related knot invariants}
It turns out that girth behaves well under connected sum. Recall that the connected sum of two oriented knots $K_1, K_2$ is well-defined for any choice of planar diagrams for these two knots.
\begin{thm} \label{connectLowerBound}
The girth of a connect sum $ K_1 \# K_2$ of two knots $K_1, K_2$ is equal to the minimum of the girths of these knots:
$ \textnormal{gr}(K_1 \# K_2) = \min\{\textnormal{gr}(K_1), \textnormal{gr}(K_2)\}.$
\end{thm}
\begin{proof}
First we show that $ \min\{\textnormal{gr}(K_1), \textnormal{gr}(K_2)\} \leq \textnormal{gr}(K_1 \# K_2) $. Let $D_1$ be a diagram of $K_1$, $D_2$ be a diagram of $K_2$ such that $\ell(G_+(D_1)) = \textnormal{gr}(K_1)$ and $\ell(G_+(D_2)) = \textnormal{gr}(K_2)$. When we perform the connected sum operation on $D_1$ and $D_2$, the all-positive state graph of the new diagram consists of $G_+(D_1)$ and $G_+(D_2)$ joined at a single vertex, with girth $\min\{\textnormal{gr}(K_1), \textnormal{gr}(K_2)\}$. This gives a lower bound for the girth of $K_1 \# K_2$.
Now we use Corollary \ref{JonesUpper} to prove that $\textnormal{gr}(K_1 \# K_2) \le \min\{\textnormal{gr}(K_1), \textnormal{gr}(K_2)\}$ and thus show equality. Recall that $J_{K_1 \# K_2} = J_{K_1}J_{K_2}$ \cite{Lick}. Let $g_1 = \textnormal{gr}(K_1)$, $g_2 = \textnormal{gr}(K_2)$ and assume without loss of generality that $g_1 \le g_2$. Then the tail of $J_{K_1}$ has the form:
\begin{align} \label{Tail1}
J_{K_1}(q) &= (-1)^{s_1}\left(P_0q^{C_1} - P_1q^{C_1+2} + \ldots +(-1)^{g_1-1} P_{g_1-1}q^{C_1+2(g_1-1)} + \ldots\right)\\
&= (-1)^{s_1}\left(\sum_{i=0}^{g_1-1}(-1)^i P_i q^{C_1+2i} + \ldots\right)
\end{align}
where $P_i = \binom{b_1-1+i}{i}$ for $0 \le i \le g_1-2$, $P_{g_1-1} = \binom{b_1-1+(g_1-1)}{g_1-1} - n_{g_1}$ (Theorem \ref{Jones1}) and $s_1$, $C_1, b_1 > 0, n_{g_1}>0$ all depend on $K_1$.
Similarly, the tail of the Jones polynomial $J_{K_2}$ has the form:
\begin{equation} \label{Tail2}
J_{K_2}(q) = (-1)^{s_2}\left(\sum_{i=0}^{g_2-1}(-1)^i Q_i q^{C_2+2i} + \ldots\right)
\end{equation}
where $Q_i = \binom{b_2-1+i}{i}$ for $0 \le i \le g_2-2$, $Q_{g_2-1} = \binom{b_2-1+(g_2-1)}{g_2-1} - n_{g_2}$, and $s_2$, $C_2, b_2 > 0, n_{g_2}>0$ all depend on $K_2$.
Theorem \ref{Jones1} describes the first $g_1$ coefficients of $J_{K_1}$ and the first $g_2$ coefficients of $J_{K_2}$. The tail coefficients of $J_{K_1 \# K_2} = J_{K_1}J_{K_2}$ result from combinations of the coefficients in Equations \ref{Tail1} and \ref{Tail2}. We write the product as:
\begin{equation} \label{Tail3}
J_{K_1 \# K_2}(q) = (-1)^{s_1+s_2}\left(R_0q^{C_1+C_2} - R_1q^{C_1+C_2+2} + \ldots + (-1)^{g_1-1} R_{g_1-1}q^{C_1+C_2+2(g_1-1)} + \ldots\right)
\end{equation}
where $R_i = \sum_{n=0}^i P_nQ_{i-n}$ for $0 \le i \le g_1-1$. Recalling our assumption that $g_1 \le g_2$, observe that we cannot say anything about the coefficients that follow $R_{g_1-1}$ because we only have $P_0$ through $P_{g_1-1}$ for the first polynomial.
For $0 \le i \le g_1-2$, we compute the following, using a modified form of the Chu-Vandermonde identity (see \cite[Table 3]{Gould1}):
\begin{equation} \label{sumcoeff}
R_i = \sum_{n=0}^i P_nQ_{i-n} = \sum_{n=0}^i \binom{b_1-1+n}{n} \binom{b_2-1+(i-n)}{i-n} = \binom{(b_1+b_2)-1+i}{i}
\end{equation}
while on the other hand for $i = g_1-1$:
\begin{align} \label{sumcoeffLast}
R_{g_1-1} &= \sum_{n=0}^{g_1-1} P_nQ_{(g_1-1)-n} = \sum_{n=0}^{g_1-2} P_nQ_{(g_1-1)-n} + P_{g_1-1}Q_0
\end{align}
Since $P_{g_1-1} = \binom{b_1-1+(g_1-1)}{g_1-1}-n_{g_1}$ with $n_{g_1} > 0$, $R_{g_1-1}$ does not agree with the formula in Equation \ref{sumcoeff} that describes the coefficients from $i=0$ to $i=g_1-2$. Thus the sequence of coefficients $R_i$ for $J_{K_1 \# K_2}$ along with the alternating signs in Equation \ref{Tail3} satisfy the conditions of Corollary \ref{JonesUpper} for $0 \le i \le g_1-2$. Hence the upper bound for $\textnormal{gr}(K_1\#K_2)$ given by the tail of $J_{K_1 \# K_2}$ is $M_J = (g_1 - 2) + 2 = g_1 = \min\{\textnormal{gr}(K_1), \textnormal{gr}(K_2)\}$.
\end{proof}
For a reduced knot diagram $D$, the knot signature and numbers of crossings give upper bounds on the girths of $G_+(D)$ and $G_-(D)$, the graphs related to the all-positive and all-negative Kauffman states $s_+(D)$ and $s_-(D)$. Recall that $\sigma(L)$ is a link invariant given by the signature of the Seifert matrix obtained from any diagram of $L$. We denote the number of crossings in a diagram by $c(D)$ and the numbers of positive and negative crossings by $c_+(D), c_-(D)$ respectively, using the crossing conventions from \cite{Lick}.
\begin{thm} \label{SignatureBound}
Let $K$ be a non-trivial knot with an oriented, reduced diagram $D$. Then $$\ell(G_+(D)) \le \displaystyle\frac{2c(D)}{c_-(D)-\sigma(K)+1}.$$
\end{thm}
\begin{proof}
The graph $G_+(D)$ is planar and connected, so the girth $\ell(G_+(D))$ is related to the numbers of edges and vertices by the following inequality: $E \le \displaystyle\frac{\ell(G_+(D))}{\ell(G_+(D))-2}(v-2)$ (see e.g. \cite{Diestel}). Since $D$ is reduced, we may assume that $\ell(G_+(D)) \ge 2$ and rearrange the inequality as $\ell(G_+(D)) \le \displaystyle\frac{2E}{E-v+2}$. The number of edges $E$ is the number of crossings $c(D)$, and the number of vertices is the number of connected components $s_+$ in the all-positive smoothing of $D$. Using the inequality $\sigma(K) \le s_+(D)-c_+(D)-1$ \cite{DasLow2} we obtain the result:
\begin{equation}
\ell(G_+(D)) \le \displaystyle\frac{2E}{E-v+2} = \displaystyle\frac{2c(D)}{c(D)-s_+(D)+2} \le \displaystyle\frac{2c(D)}{c_-(D)-\sigma(K)+1} \qedhere
\end{equation}
\end{proof}
A similar proof using $\sigma(K) \le -s_-(D)+c_-(D)+1$ gives an upper bound for the girth of $G_-(D)$.
\begin{cor}\label{SignatureBoundCor}
Given an oriented, reduced, positive knot diagram $D$ then: $\ell(G_+(D)) \le \dfrac{2}{3}c(D)$. Note that this inequality is sharp.
\end{cor}
\begin{proof}
Since $D$ is positive, $c_-(D) = 0$ and $\sigma(K) \le -2$ \cite{Przy2}. The standard diagram of the $(-3,-3,-3)$ pretzel knot, which has 9 positive crossings, $\sigma = -2$, and $G_+(D)$ with girth 6 provides an example where the equality $\ell(G_+(D)) = \dfrac{2}{3}c(D)$ is achieved.
\end{proof}
\begin{exa} As an application, we observe that a positive alternating knot can never have a diagram $D$ such that $G_+(D)$ is a cycle graph. Such a diagram would have $\ell(G_+(D)) = c(D)$, a contradiction by Corollary \ref{SignatureBoundCor}.
\end{exa}
If we restrict our attention to alternating links, we can get more specific results about girths of alternating diagrams. In particular, the following theorem states that any two reduced diagrams of a prime alternating link have the same girth.
\begin{thm} \label{AltGirth}
Let $L$ be a prime alternating link. If $D$, $D'$ are two reduced alternating diagrams of $L$, then $G_+(D)$, $G_+(D')$ have the same girth.
\end{thm}
\begin{proof}
The Tait flyping conjecture states that any two reduced alternating diagrams of $L$ are related by flypes \cite{MT1}. Flypes may be expressed as a series of mutations, which induce Whitney flips on the corresponding graph \cite{Greene2}. Thus $G_+(D)$ and $G_+(D')$ are 2-isomorphic graphs. Girth is an invariant of the cycle matroid \cite{Oxley}, so $G_+(D)$ and $G_+(D')$ have the same girth.
\end{proof}
\clearpage
\bibliographystyle{abbrv} |
2103.15960 | \section{Methods}\label{sec:methods}
The described system is the result of tightly coupled interdisciplinary work ranging from machine learning to chip design.
The following sections describe different aspects of the \gls{bss2} mobile system from the perspective of the different technological areas.
\subsection{BrainScaleS-2 Mobile System}\label{subsec:system}
The hardware system is based on the combination of a \gls{fpga} and the \acrlong{bss2} mixed-signal \gls{asic}.
It has been introduced in \cref{fig:system overview}.
Further information about the \gls{fpga} base board can be found in~\citet{avnet2020ultra96}.
It can be used unaltered in conjunction with the \gls{asicab}.
In addition to the power monitoring capabilities of the \gls{fpga} base board, the individual supply currents of the \acrlong{bss} \gls{asic} can be monitored by several shunt-based power monitoring \glspl{ic}~\citep{TexasInstruments2020INA219} located on the \gls{asicab}\@.
The readout process has been optimized and the \glspl{ic} were configured for maximum sampling frequency.
This enables accurate calculation of the energy consumption by integrating the power samples.
Results of these measurements are shown in \cref{sec:results}.
The \gls{asicab} provides power supplies, reference voltages and a reference current to the \gls{asic}, which can be accessed via micro-SMT coaxial connectors for debugging purposes.
Additional coaxial connectors are available for monitoring the analog outputs from the \gls{bss2} \gls{asic}.
The \gls{asic} itself is directly bonded to a carrier \gls{pcb} using a zero-insertion force \gls{sodimm} board edge connector for an optimum combination of simplicity and reliability.
\Cref{fig:system photo} shows the die bonded to the \gls{asic} carrier \gls{pcb}\@.
The \gls{asic} provides eight independent bi-directional source-synchronous \gls{lvds} data channels operated at up to \SI{2}{\giga\bit\per\second} each.
Due to I/O limitations of the \gls{fpga} board, only five are routed through the \gls{asicab} to the \gls{fpga}.
\subsection{Neuromorphic ASIC}\label{subsec:neuromorphic-asic}
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{generated/hicann_layout.pdf}
\caption{%
\label{fig:hicann_layout}
\emph{Left:} internal structure of the \gls{bss2} \gls{asic}.
The analog network core consists of four quadrants, each containing \num{128} neurons and \num{128}\texttimes\num{256} synapses (red).
A total of \num{512} parallel ADC channels allow for readout of various analog parameters by two embedded \gls{simd} processors (yellow).
\emph{Right:} position of the described functional units on a layout drawing of the \gls{bss2} \gls{asic}.
}
\end{figure*}
The \gls{bss2} neuromorphic \gls{asic}\footnote{%
The \gls{asic} has been manufactured in a standard \SI{65}{\nano\meter} CMOS technology.
It was conceived and designed at Heidelberg University.
The link layer of the high-speed serial links has been developed in collaboration with the TU Dresden, who also contributed the PLL.
The fast ADC is a result of a collaboration with the EPFL Lausanne.}
is the key component of the inference system.
\Cref{fig:system overview} shows its major internal functional blocks:
\paragraph*{Analog Network Core}
This is the part of the system performing the inference calculations.
It will be described in more detail below.
\paragraph*{Digital Core Logic}
The core control and network logic handles all off-chip communication from the embedded processors and the event router.
It also bidirectionally converts between real-time and time-stamped event packets.
\paragraph*{Top and Bottom \glspl{ppu}}
The chip includes two custom \SI{32}{\bit} CPUs compatible with the embedded PowerPC \gls{isa}~\citep{powerisa_206}.
They feature \gls{simd} extensions for fast vector processing.
These embedded cores are primarily intended to support learning and plasticity algorithms in \glspl{snn}.
They can access most of the internal digital resources of the \gls{asic} and -- as in this case -- serve as experiment controllers.
\paragraph*{Event Router}
Digital logic responsible for the distribution of the real-time vector inputs or spike events to and from the analog network core.
The right side of \cref{fig:hicann_layout} shows a layout drawing of the \gls{asic}.
The embedded processors are highlighted by the yellow rectangles.
The red frame depicts one of the four identical quadrants of the analog core.
The left side of the figures illustrates the neuromorphic processing loop through the analog cores, together with the arrangement of neurons and synapses within the quadrants.
In the inference experiment, the flow of data is as follows:
first, weight data is written into the synapse array via the wide datapaths from the embedded processors.
The neurons are configured as linear integrators without any long-term internal dynamics.
This is done by setting the appropriate bias values in the analog parameter memories~\citep{hock13analogmemory}.
Inference calculation starts when the network logic transmits the events it has received from the \gls{fpga} to the center event routing logic.
All neurons are reset to an initial membrane value before the arrival of the first event of the input vector of a vector-matrix multiplication.
The event router distributes the events to the synapse drivers, which in turn transmit them into the synapse array.
\begin{figure*}
\center{
\includegraphics[width=.9\linewidth, page=2,viewport=0 0 20.5cm 5cm, clip]{Figures_from_report.pdf}
\caption{%
\label{fig:synapse}
Operation principle of the analog computation: the bottom half depicts the main functional blocks of a synapse circuit.
For the inference calculation only the shaded area is used.
The top half shows the analog operations taking place: each synapse generates a current pulse $I_\text{syn}$ in response to a pre-synaptic input event.
During the calculation period $T_\text{input}$ they are integrated on the membrane capacitance.
The final voltage $V_\text{out}$ of a single neuron represents the result of the analog dot-product calculation.
}
}
\end{figure*}
To perform the analog multiplication, the events are converted from binary coding to a pulse length representation.
\Cref{fig:synapse} illustrates the principle of analog computation used for the vector-matrix multiplication.
The synapses produce a current proportional to their stored weights $\omega_x$ for the duration of the input signal they receive from the synapse drivers $\Delta t$, thereby performing an analog multiplication of the pulse length representing the input rate value with their stored weights.
The input line of the neuron subsequently receives the sum of all output currents generated by the synapses within a vertical column.
For reasons of printing space the column is shown horizontally in the figure.
Each synapse array can process back-to-back activations within \SI{8}{\nano\second}, the maximum continuous input data rate is therefore \SI{125}{\mega\hertz}.
There are 256\texttimes{}512 synapses altogether, which can all simultaneously process input activations at the full input data rate. This equals to a maximum of
\begin{equation}
\SI{125}{\mega\hertz}\cdot 256\cdot512\cdot \SI{2}{\op} = \SI{32.8} {\tera\op\per\second}, \label{eq:maxrate}
\end{equation}
counting multiplication and addition as individual operations, see~\cref{fig:synapse}.
A transconductance amplifier in each neuron generates a current equivalent to the charge received from the synapses.
Each column's current is integrated on the membrane capacitance of its associated neuron circuit.
Each neuron has two separate inputs for excitatory (A) and inhibitory (B) synaptic inputs.
For the inference calculation, they are used to represent positive and negative weight values.
The full integration cycle within the neurons, including the necessary time to reset the neuron membrane voltages, takes about \SI{5}{\micro\second}.
This reduces the MAC frequency to \SI{200}{\kilo\hertz} and the resulting speed to approximately
\begin{equation}
\frac{1}{\SI{5}{\micro\second}}\cdot 256\cdot 512\cdot\SI{2}{\op} \approx\SI{52}{\giga\op\per\second} \label{eq:actrate}
\end{equation}
(see \cref{sec:discussion}).
After all input events have been received by the synapses, the membrane voltages of the neurons represent the result of the vector-matrix multiplication.
The parallel analog-to-digital converter between the synapse array and the embedded processor can also digitize the membrane voltages.
For \gls{relu} operation, its programmable input range can be set in a way that its minimum input value represents the initial reset voltage $V_\text{reset}$ the neurons have been initialized to, thereby cutting of the negative branch of the neurons transfer function.
After the ADC conversion is completed, the result of the vector-matrix multiplication is transferred into a vector register of the processor and the array is ready for the next operation.
For more details of the \acrlong{bss2} architecture refer to~\citet{schemmel2017nmda,aamir2018dls2neuron,schemmel2020accelerated_nourl,billaudelle2020versatile,friedmann2016hybridlearning}, for the rate-based operation mode see~\citet{weis2020inference,weis2020msc}.
\begin{figure*}
\center{%
\includegraphics[width=0.7\linewidth]{generated/fpga_logic_rainbow.pdf}
\caption{
Block diagram of the major functional units of the \gls{fpga}, the part inside the logic fabric has been realized as custom \gls{rtl} in SystemVerilog.
The \gls{dma} controller, preprocessing chain elements and vector event generator create the input activation events representing the vector in the vector-matrix multiplication.
Some of the preprocessing (blue) is problem-specific for the medical \gls{ecg} dataset.
To the right side the major blocks of the \gls{bss2} \gls{asic} are shown as well to illustrate the complete communication path from the embedded \glspl{ppu} to the \gls{dram} memory.
The arrows denote the control flow direction from initiator to follower of the internal (hollow) and external (filled) data buses shown in the figure.
\label{fig:fpga_logic} }}
\end{figure*}
\subsection{FPGA Fabric}\label{subsec:fpga}
\Cref{fig:fpga_logic} depicts the internal structure of the logic fabric.
Main components are the link control and physical layer that implement the high-speed serial links to the \gls{asic}.
The playback-buffer contains a list of commands to send to the \gls{asic}, while the trace buffer collects events sent back from the \gls{asic}.
Memory-mapped write and read commands can also be issued from the \gls{asic} to the \gls{fpga}.
This allows the \glspl{ppu} to access the \gls{dram} memory connected to the \gls{fpga} via a memory switch.
The \gls{dma} controller is programmed by the \gls{ppu} on the \gls{asic} to transfer the raw signal data, an \gls{ecg} trace composed of \SI{12}{\bit} values, from memory.
The \gls{asic} requires specially formatted event data packets encoding \SI{5}{\bit} input activations for the vector-matrix multiplication.
This demands a preprocessing chain inside the \gls{fpga}, which is problem-specific to some extent.
Its function will be explained in \cref{subsec:model}.
After the raw signal data is converted into \SI{5}{\bit} values, the vector event generator attaches an event address from a lookup table.
This event is sent to the \gls{asic} via the serial links.
In the \gls{asic} the attached addresses are used to forward the events to their target inputs of the analog neuromorphic core.
The use of a lookup table inside the \gls{fpga} allows arbitrary mapping of input vector elements onto the synapse matrix.
During the inference process the \gls{ppu} inside the \gls{asic} synchronizes the vector event generator inside the \gls{fpga} using multiple handshake signals to control the timing of the sent events.
The selected \gls{fpga} contains four \SI{64}{\bit} ARM processor cores in addition to the logic fabric ~\citep{xilinx2019zynqultra}.
They do not participate in the inner loop of the inference calculation and only perform initialization and calibration procedures beforehand.
\subsection{Model}\label{subsec:model}
The model selection for the \gls{ecg} classification was governed by minimizing runtime to optimize the energy efficiency while achieving reasonable classification accuracy using the available accelerator.
As a trade-off between accuracy and model complexity, a minimum \gls{afib} detection rate of \SI{90}{\percent} at a maximum of \SI{20}{\percent} false positives was targeted.
Due to the nature of our processing hardware, the best results in both aspects, energy efficiency and detection accuracy, can be obtained with a large batch size, i.e., processing the dataset as a whole.
Larger networks promise superior classification accuracy, but require reconfiguration of the \gls{asic} during execution.
Since the reconfiguration is only needed once per batch, with a larger batch size its time penalty diminishes.
The best accuracies achieved with larger networks on the \gls{bss2} \gls{asic} have been \SI{95.5}{\percent} for \gls{afib} with \SI{8.0}{\percent} false-positives~\citep{emmel2020msc}.
However, sequential processing of the data, i.e., batch size one, is preferable for low-energy edge applications.
Evaluation of network models showed that even a smaller network that fits on a single chip and does not require reconfiguration can achieve decent accuracy.
It is very plausible that, given the nature of the \gls{ecg} data, a hand-crafted signal-processing based algorithm running fully in the \gls{fpga} would also be feasible.
Nevertheless, we aimed to perform as much processing as possible in an automatically trained \gls{cdnn}, since this approach is also suitable for different kinds of data.
As a result, the developed model architecture and software integration for \gls{bss2} can be easily applied to other machine learning tasks and larger networks, as already demonstrated for human activity recognition~\citep{spilger2020hxtorch} and the MNIST handwritten digits~\citep{weis2020inference}.
It has been further demonstrated that our trained model can be directly applied to other \gls{ecg} datasets used in the machine learning community without any manual changes~\citep{emmel2020msc}.
The \gls{bss2} \gls{asic} supports acceleration of both spike-based and rate-based neural networks.
The main advantage of the spike-based operation modes are the possibility to combine it with online, biologically inspired, learning algorithms executed fully on-chip.
Some early tests with spike-based processing concepts to classify \gls{afib} have been performed on the hardware~\citep{weis2020msc}.
For future network concepts, the combination of both modes seems promising.
For this publication, however, the \gls{bss2} \gls{asic} is configured to be used as an analog inference accelerator for vector-matrix multiplication in rate-based \glspl{cdnn}.
A decisive advantage of this mode is that it is based on a stateless network, is very generic and supports multiplexing of larger networks.
\begin{figure}[ht]
\resizebox{\linewidth}{!}{\input{content/tikz/model}}
\caption{\label{fig:model}%
Layer structure (left) and on-chip arrangement (right) of the used deep convolutional neural network model.
The convolutional layer (green) is processed in the upper synapse array,
the identical weight is arranged \num{32} times on the substrate to enable parallel processing.
All \glspl{relu} (red) are performed in digital logic by the two \glspl{ppu}.
The further processing takes place on the lower synapse array with a fully connected layer and \num{123} hidden neurons (orange).
To ensure efficient use of the substrate, it is divided into two parts and placed side by side.
The dotted part of the layer receives the second half of inputs at the same time and is processed in parallel.
The actual classification is then achieved in the last layer (blue) with \num{10} neurons on the right,
which are combined into two logical neurons by average pooling, effectively reducing analog noise.
}
\end{figure}
The network as used in this publication is depicted in \cref{fig:model}.
The model operates on \SI{13.5}{\second} of the \SI{120}{\second} long \gls{ecg} records, as this has turned out to be sufficient for classification of \gls{afib}.
To the left, the graph of the model is shown.
It consists of one convolutional and two linear layers.
The small size of the network allows it to be completely realized on the \gls{asic}.
The mapping of the network layers to the two halves of the \gls{bss2} \gls{asic} is shown in the right side of the figure.
The \gls{relu} and the final argmax operations are preformed in the embedded \glspl{ppu} after digital readout of the analog neuron membrane voltages\@.
\begin{figure}
\center{%
\resizebox{\linewidth}{!}{\sansmath\pgfplot{ecg_preprocessing}}
\caption{Preprocessing steps performed in the \gls{fpga} fabric (from top to bottom): The raw, i.e. unprocessed, input sample is transformed by taking a discrete derivative to reduce baseline fluctuations. Subsequent maximum-minimum difference pooling reduces the sample rate and provides positive activations, which form the final input signal to the \gls{cdnn} in the \gls{asic}.
Original data taken from~\citet{clifford2017af}.
\label{fig:preprocessing} }}
\end{figure}
The \gls{asic} operates on positive activations with \SI{5}{\bit} resolution.
Since the raw data samples as input for the inference calculation are provided as \SI{12}{\bit} values with quite large dynamic range, some preprocessing is required.
\Cref{fig:preprocessing} illustrates the performed steps.
To avoid unnecessary data movement, the preprocessing is done in the \gls{fpga} fabric by a custom processing chain. %
In the first step of the preprocessing, a discrete derivative of the original signal is calculated to suppress the large baseline fluctuations of the signal.
In a second step, the data rate is reduced by calculating the difference between the maximum and the minimum of \num{32} samples.
This operation is performed with a stride of \num{16}, resulting in an output sample rate of \SI{32}{\hertz}.
The resulting samples are quantized to \SI{5}{\bit} and used as inputs to the analog vector-matrix multiplications performed within the \gls{asic}.
Considering the target of maximum energy efficiency, the chosen network is optimized to the resources provided by the \gls{bss2} \gls{asic}.
The calculations in its convolutional first layer can be performed fully in parallel, as well as those in the second and third layers.
The total time for processing of all layers of the network is in the order of microseconds.
The system has to be active only during this very small time window, plus a few microseconds at the beginning and the end to transfer data back and forth.
\subsection{Training}\label{subsec:training}
Training relies on the proven backpropagation algorithm for \glspl{cdnn}~\citep{rumelhart86backprop}. %
In order to fast-protoype and train the network described in \cref{subsec:model}, a mathematical abstraction of the hardware operations was implemented on top of PyTorch~\cite{paszke2019pytorch} in \emph{hxtorch}~\citep{spilger2020hxtorch}.
Incorporating hardware related constraints, like fixed-pattern noise and limited dynamic range, it enables training initial models off-chip and provides gradient information for the backward-path when training on hardware~\citep{emmel2020msc}.
Final model parameters as presented in \cref{sec:results} were trained on the real \gls{asic} from scratch with no pre-training in software.
The provided training data was randomly split into local training and validation datasets.
To prevent overfitting validation data was never used during training.
We follow the in the loop paradigm~\citep{schmitt2017hwitl}:
The forward path is evaluated using real hardware whereas the backwards path and parameter updates are calculated on the host computer using \emph{hxtorch}.
Tensor data structures are seamlessly converted to hardware resolution and back.
Data partitioning and experiment control is handled by both on-chip \glspl{ppu} (see \cref{subsec:software}).
To the user, the training procedure is completely embedded within PyTorch.
To increase robustness and decrease sensitivity to hardware variations, we replace the average pooling in the last layer by a max pooling operation during training.
Training ends when no substantial improvement is observed between training epochs.
\subsection{Software}\label{subsec:software}
Although the energy efficiency of the inference task relies on analog computing and digital support infrastructure,
software plays an essential role.
Similarly to other digital hardware platforms software is an essential component to make complex hardware systems accessible to users~\citep{kacher2020graphcore,rueckauer2021nxtf}.
In each phase -- from hardware commissioning, model design, training to validation phase -- users can take advantage from a software environment that provides appropriate abstraction levels, access to hardware debugging information as well as robust and transparent platform operation.
For the \gls{bss2} architecture, -- and, in particular the mobile system -- we provide software support for different system aspects:
\subsubsection*{User Interface}
The PyTorch toolkit~\cite{paszke2019pytorch} is a commonly used workhorse in the field.
Particularly, it simplifies many aspects of \gls{cdnn} modeling.
We developed a custom extension for PyTorch, \emph{hxtorch}~\cite{spilger2020hxtorch,spilger2021from}, providing support for the \gls{bss2} architecture.
\subsubsection*{Training}
Forward propagation is dispatched to the \gls{bss2} \gls{asic} while backward propagation is performed in software.
Hence, \emph{hxtorch} enables using the \acrlong{bss2} system as an inference accelerator in PyTorch while adopting a hardware-in-the-loop-based training approach.
The trained model can be serialized, stored to disk and used in an \emph{standalone inference mode} to increase energy efficiency.
In addition, a ``mock mode'' enables the simulation of certain hardware properties in software.
This facilitates migrating from the training of a pure software model to a hardware-in-the-loop-based training.
\subsubsection*{Hardware Resources}
\emph{hxtorch} provides support for the execution of neural network graphs on an arbitrary number of \gls{bss2} \glspl{asic}.
Individual layers are partitioned into chip-sized chunks and executed either in parallel, serially or in the appropriate mixture needed to fit on the available hardware resources.
Finally, each \gls{asic} receives and executes a stream of instructions and data.
\subsubsection*{Data-Flow Graph Execution}
Internally, \emph{hxtorch} model layers build up a data-flow graph.
A \gls{jit} compiler traverses the graph and partitions individual layers into chunks fitting onto the available hardware resources.
Partitioned layers are converted into configuration data and corresponding control flow statements;
Both of which are transferred to the \gls{bss2} hardware system and result data is read back.
Regarding control flow, the hardware execution engine supports two modes:
the first mode uses the \gls{fpga} to handle control flow;
the second mode, which is also largely used in the standalone inference mode, hands over the control flow to the embedded \glspl{ppu} of the \gls{asic}.
\subsubsection*{Memory Management}
Data input as well as output locations are precomputed by the \gls{bss2} software stack allowing for static memory management on the system.
The \glspl{ppu} use the communication link to the \gls{fpga} to program the \gls{dma} engine inside the \gls{fpga} to automatically deliver the input activations from \gls{dram} to the analog processing cores.
Analog operation results are read out by the processors, either held in \gls{sram} for temporary data, or stored back into \gls{dram} for output data.
\subsubsection*{Standalone Inference Mode} %
The \gls{bss2} software layers are written in C++ and provide faster execution speeds compared to an interpreted top-level language such as Python.
To create a lightweight inference flow for the energy measurements, a stand-alone version of the \emph{hxtorch} hardware graph executor was developed.
This executor is implemented as a standalone binary and builds upon the same internal software layers and data formats as the \emph{hxtorch} extension.
In contrast to the \gls{jit}-based execution flow, the standalone inference mode requires control flow to be handled by the embedded \glspl{ppu}.
The processors process a instruction stream representing:
data load and store operations,
trigger operations for delivery of input activations from the \gls{fpga},
reading out the neuron membrane values,
or performing digital operations that are not supported by the analog substrate.
\subsubsection*{Embedded System Environment}
The \gls{bss2} mobile system includes a Linux environment\footnote{Petalinux; the build flow of the embedded Linux distribution is provided by the \gls{fpga} manufacturer.} running on an embedded ARM64 processor.
We take advantage of a fully containerized software environment based on singularity~\citep{kurtzer2017singularity} and spack~\cite{gamblin2015spack} to provide a cross-compiler environment on the host computer as well as on the embedded Linux system.
Standard Linux drivers (xHCI, mass storage, FAT32) are used to readout test data from an USB mass storage device;
additionally, support for USB-based Ethernet networking hardware is enabled to facilitate remote system usage.
An experiment execution service enables users to run Python-based interfaces on host computers that exchange serialized experiment configurations and result data with the mobile system.
Details of \emph{hxtorch} and the lower levels of the \gls{bss2} software stack are described in~\citet{spilger2021from,mueller2020bss2ll_nourl}.
\subsection{Scope and Evaluation Procedure}
\label{subsec:bmbf2020-competition}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{generated/eval_setup.pdf}
\caption{%
\label{fig:eval_setup}
Evaluation setup for the \gls{bss2} mobile system:
power is supplied using a single \SI{5}{\volt} supply, the power delivery of which is measured at the system's input.
Test data can optionally be read from and results can be written back to a USB mass storage device.
GPIO pins are available for orchestrating different phases of the experiment from an optional external controller.}
\end{figure}
As introduced in \cref{sec:introduction}, the \gls{bss2} mobile system participated -- as project HD-BIO-AI -- in a competition for low-energy classification of \gls{ecg} data that has been organized by the \gls{bmbf}.
To ensure comparability of the contenders, all competing designs had to adhere to a common interface for experiment orchestration.
It specifies a total of four phases for system initialization, transfer of input data from a USB mass storage device to the internal \gls{dram}, inference and the transfer of classification results back to the attached storage device.
Final measurements were taken for multiple blocks per \num{500} \gls{ecg} traces.
\Cref{fig:eval_setup} depicts this evaluation setup.
The included \gls{i2c} chain is not part of this protocol and can optionally be used for fine-grained power measurements of the sensors described in \cref{subsec:system}.
All official measurements for comparing the competition's contenders were conducted with off-system equipment supplied by the external power supply.%
\footnote{The experiment interface has been designed and final evaluation measurements were taken at the \gls{dfki} Kaiserslautern under the direction of Prof.\ Dr.\ Hans Dieter Schotten.}
\section{Introduction}\label{sec:introduction}
\PARstart{I}{nference} with \glspl{cdnn} will become one of the most important tasks for upcoming edge computing devices.
Their energy and cost efficiency will determine the complexity of possible applications.
This expected demand has induced interest in novel computing architectures -- mostly in the form of specialized digital \glspl{asic}~\citep{google-edge-tpu}.
An alternative approach to these digital solutions is based on analog computing.
In the area of event-based neuromorphic computing, analog solutions are not new.
Over the last four decades several research groups have presented analog neuromorphic architectures, but most of them are not meant for the implementation of \glspl{cdnn} due to a lack of fast memory access and multi-valued synaptic inputs~\citep{moradi2018dynaps,benjamin2014neurogrid}.
The \acrlong{bss} neuromorphic architecture developed at Heidelberg University is one exception.
It supports fully analog vector-matrix multiplication and \gls{relu} functionality, in addition to its normal event-based operation.
The only other neuromorphic architecture supporting simultaneously rate- and spike-based models is the digital Tianjic architecture~\citep{pei2019tianjic}.
We present a demonstrator system for the energy-efficient analog inference capabilities of the \acrlong{bss2} architecture (\cref{fig:system overview,fig:system photo}).
It features a combination of a commercially available \gls{fpga} module and the most recent \acrlong{bss2} \gls{asic}.
The \gls{fpga} contains an embedded CPU which is used for standalone experiment control and I/O\@.
It is not used during the actual inference process.
The logic fabric in the \gls{fpga} acts as a memory interface and data format converter for the \gls{asic}.
\begin{figure}
\center{%
\includegraphics[width=\linewidth, page=4,viewport=0 0 25cm 14.5cm, clip]{Figures_from_report.pdf}
\caption{Overview of the \acrlong{bss2} mobile system (from left to right): \gls{fpga}-based controller, \gls{asic} interface consisting of two \glspl{pcb} -- the \gls{asicab} and the \gls{asic} carrier board --, and \gls{bss2} \gls{asic}.
The system controller provides the USB interface for the USB mass storage device with test and result data as well as the run control handshake signals for the energy measurement protocol of the inference competition.
\label{fig:system overview}}}
\end{figure}
\Cref{fig:system overview} depicts the three main components of the system:
\begin{itemize}
\item a base board consisting of a low-power \gls{fpga} with an embedded quad-core microprocessor~\citep{xilinx2019zynqultra} and \SI{2}{\gibi\byte} of LPDDR4 \gls{dram}, USB~3.0 (device \& host), SDXC, 802.11b/g/n Wi-Fi as well as Bluetooth 4.2 (BLE) communication circuits (see ``system controller'' in \cref{fig:system overview}),
\item a custom adapter \gls{pcb} (see ``\gls{asic} interface''), interfacing the different connectors of the \gls{fpga} board to a single \gls{sodimm} connector for the \gls{asic} carrier board,
\item the \acrlong{bss2} \gls{asic} directly bonded to a carrier \gls{pcb} using a \gls{sodimm} edge connector.
\end{itemize}
The \gls{asic} uses a bundle of high-speed serial links for bi-directional access to the \gls{fpga} fabric.
A \gls{dma} controller reads the input data from memory, converts it into input events and sends them to the \gls{asic}.
The event router on the \gls{asic} transports the events to inputs of the analog network core, where they are interpreted as the input vector of a vector-matrix multiplication operation.
For up to \num{65536} signed matrix elements this operation is carried out in parallel within the analog core.
The result vector of the vector-matrix multiplication is calculated by the analog neurons, which are configured as linear integrators.
Subsequently, the neuron voltages are converted back to the digital domain by a parallel ADC with \SI{8}{\bit} resolution.
The \gls{relu} operation can be performed automatically during this conversion.
Alternatively, the embedded \gls{ppu} can apply an activation function after digitizing the analog result.
These results, representing the output activations of a network layer, are passed to the \gls{fpga} fabric and either stored in \gls{dram} or used as inputs for the next layer.
This loop is repeatedly executed until all layers have been processed and the final results of the inference operation have been stored in the \gls{dram} memory.
\Cref{sec:methods} will provide a detailed description of the individual operations involved as well as more insights into the different components of the presented system.
The \acrlong{bss2} mobile system has participated in a competition for low-energy classification of \gls{afib} within a medical \gls{ecg} dataset.\footnote{Due to the fact that the dataset contains sensitive patient information it is not publicly available.}
We showcase the system's performance on the training samples of this dataset, consisting of \num{16000} data records from the same patient group.
The data has been recorded with two channels only, mimicking the signal quality to be expected from consumer-grade medical wearables.
\begin{figure}
\center{%
\includegraphics[width=\linewidth]{generated/hdbioai_nc_dl_aif_lowquality.jpg}
\caption{Photo of the \acrlong{bss2} mobile system (from bottom to top): \gls{fpga}-based system controller, \gls{asicab}, \gls{asic} carrier board with the latest \gls{bss2} \gls{asic} directly wire-bonded to the \gls{pcb}.
The system has the mechanical footprint of a credit card (\SI{84x55}{\milli\meter}) at a height of approximately \SI{40}{\milli\meter}.
It weighs roughly \SI{155}{\gram} with and \SI{70}{\gram} without the \gls{fpga}'s heatsink respectively.
\label{fig:system photo}}}
\end{figure}
\section{Results}\label{sec:results}
\input{content/tables/competition_results}
The performance of the presented system has been evaluated by assigning a set of \gls{ecg} traces to two classes: patients with sinus rhythm and patients showing \acrfull{afib}.
A description of the used dataset is given in \cref{sec:introduction}.
Mimicking the expected workload in a low-energy edge application, all data has been processed with a batch size of one.
To increase the accuracy of all measurements, data was processed in blocks of \num{500} traces per uninterrupted measurement cycle.
For each block, runtime and energy consumption have been measured using the characterization infrastructure described in \cref{subsec:system}.
The temporal resolution of the power measurements was approximately \SI{294}{\hertz} for sensors on the \gls{fpga} base \gls{pcb} and \SI{4.4}{\kilo\hertz} for sensors on the \gls{asic} adapter board.
\begin{figure}
\center{%
\resizebox{\linewidth}{!}{\sansmath\pgfplot{ecg_training_statistics}}
\caption{\label{fig:ecg_training_statistics}%
Training and validation metrics of the model presented in \cref{fig:model} performed with the \gls{bss2} \gls{asic}.
The dashed lines represent the targeted accuracy of \SI{90}{\percent} detected \acrlong{afib} at a maximum of \SI{20}{\percent} false positives.
The test set of 500 records was split from the provided \gls{ecg} dataset prior to training.}}
\end{figure}
Classification accuracy has been evaluated by selecting different test sets of \num{500} records prior to training.
Metrics of such a training course on the presented system are shown in \cref{fig:ecg_training_statistics}.
The trained model allows correct classification of \SI{93.7 \pm 0.7}{\percent} of the \gls{afib} patients at a false positive rate of \SI{14.0 \pm 1.0}{\percent}.
Each block of \num{500} input traces was found to be processed in \SI{138}{\milli\second};
starting with raw \gls{ecg} data in the \gls{dram} on the \gls{fpga} \gls{pcb} and ending with binary classification results ibidem.
\Cref{tab:competition_results} gives an overview over the achieved results:
The power consumption of the \gls{bss2} \gls{asic} -- where all matrix multiplications take place -- is below \SI{15}{\percent} of the total system power consumption, leaving significant room for optimization of the auxiliary circuitry.
Initial measurements indicate that by disabling all unused components during inference the total power consumption can be reduced by about \SIrange{50}{70}{\percent}.
In its current state, the \gls{bss2} mobile system used a total of \SI{0.78}{\joule} for classifying the data of all \num{500} patients in one evaluation block.
These results have been independently verified by the German Research Center for Artificial Intelligence.
\section{Contributions}%
\label{sec:contributions}
Yannik Stradmann directed the development and modeling efforts for the presented experiment and hardware setup.
He contributed to all components.
Sebastian Billaudelle contributed to the chip design, chip commissioning and implementation of the experiment.
Oliver Breitwieser contributed to the software stack, is the main architect of the preemptive experiment scheduling service and contributed to modeling and model verification.
Falk Ebert is a main contributor to the energy measurement system.
Arne Emmel developed and implemented the model, designed the preprocessing, adapted the training to the hardware platform and contributed to the software integration.
Dan Husmann developed the \gls{asicab}\@.
Joscha Ilmberger is the main system developer contributing to \gls{pcb} design, porting of the \gls{fpga} design to the new platform and adding functionality such as preprocessing and the vector event generator.
Eric Müller is the lead developer and architect of the \gls{bss2} software stack;
he commissioned the embedded platform, ported the software development environment as well as the \gls{bss2} software stack to the embedded \gls{fpga} platform.
Philipp Spilger is the main developer of the software for the non-spiking operation mode of the \gls{bss2} \gls{asic} and a contributor to the software stack.
Johannes Weis is the main developer of calibration routines for the analog network core, commissioned the first non-spiking experiments on the hardware platform and contributed to the model.
Johannes Schemmel is the lead designer and architect of the \gls{bss2} neuromorphic system. He wrote the initial version of the paper.
All authors contributed to and edited the final manuscript.
\section*{Acknowledgments}
\label{sec:acknowledgements}
The authors wish to thank all present and former members of the Electronic Vision(s) research group contributing to the BrainScaleS-2 neuromorphic platform.
\section{Discussion}\label{sec:discussion}
We have presented the \gls{bss2} mobile system as an analog inference platform and demonstrated medical \gls{ecg} data classification as one possible application.
With the shown combination of model, software and hardware, this system classified \gls{afib} with a detection rate of \SI{93.7 \pm 0.7}{\percent} at \SI{14.0 \pm 1.0}{\percent} false positives.
During the inference phase, it achieved \SI{477}{\mega\op\per\second} with a mean power consumption below \SI{6}{\watt}, of which below \SI{700}{\milli\watt} are consumed by the \gls{bss2} \gls{asic}, see~\cref{tab:competition_results}.
Every patient sample took approximately \SI{276}{\micro\second} to analyze, resulting in an energy efficiency of \SI{84}{\mega\op\per\joule} for the full system, or \SI{689}{\mega\op\per\joule} for \gls{bss2} respectively.
The small system is mobile by design and has proven to operate reliably under various environmental conditions.
Despite its early prototype stage, it is therefore directly applicable to inference tasks on the edge:
The results we have achieved demonstrate that our technology is sufficiently energy efficient such that a hypothetical medical device using it could run on battery while monitoring the health of the patient.
Assuming a common CR2032 lithium button battery with an approximated energy content of \SI{200}{\milli\ampere\hour}, the system could perform the inference calculations for detecting atrial fibrillation in two-minute intervals for five years.
The analog inference technology is however not limited to edge applications.
Due to its energy efficiency and low cost it is also suitable as a densely packed accelerator in datacenter applications, where the energy cost of inference workloads is expected to become dominant in the upcoming decade~\citep{park2018deep,dongarra2019race,chen2019race}.
Our analog solution could be an important step to reduce the energy demands as well as the carbon footprint linked to it.
In addition to the presented multiply-accumulate functionality, \gls{bss2} is designed to operate as an analog emulator for \glspl{snn}.
To the best of our knowledge, it is the first and only available system to accelerate both, multiply-accumulate operations and \glspl{snn} in the analog domain.
Due to the stateful nature of the necessary time-continuous operations, multiplexing of analog resources is not possible in most \gls{snn} accelerators, therefore limiting the maximum model size to the available hardware resources.
In contrast, rate-based stateless operation using our analog neuromorphic core as a parallel vector-matrix multiplier allows for multiplexing hardware resources in time and therefore has the advantage of supporting arbitrarily large model sizes.
Such networks are only limited by the available memory.
Most models that are capable of performing real-world tasks, like video analysis or speech translation, need model sizes in the orders of \numrange{e7}{e9} parameters~\citep{aharoni2019massively}.
These network sizes are feasible with the presented system, as neither the hardware platform nor the \emph{hxtorch} software environment described in \cref{subsec:software} impose size limitations on the model in use.
The combination of spiking and conventional neural networks on a single substrate therefore greatly widens the application of \gls{snn} in edge applications:
it allows features to be extracted by conventional high dimensional \gls{cdnn} layers on multiplexed hardware resources, while sparse spiking layers can simultaneously be used for their final classification.
Using the embedded \glspl{ppu}, \gls{bss2} can utilize online learning for the \gls{snn} layers and thereby improve classification performance and adapt to environmental changes in the field.
Given its early prototyping stage, the system as well as the \gls{bss2} chip itself contain a large potential for optimization.
Currently, the \gls{fpga} is primarily used as a memory controller for the \gls{asic} -- functionality that could be incorporated into the chip's digital core.
This would remove the power consumption of the \gls{fpga} from the system's energy balance and could increase the bandwidth between memory and analog core by up to two orders of magnitude.
Furthermore, the speed of the analog calculation has not yet been optimized.
While the synapse arrays that perform the multiply-accumulate operation already support \SI{32.8}{\tera\op\per\second}, see~\cref{eq:maxrate}, the usage of the spike-based neurons for the integration of the summation currents limits the actual speed to approximately \SI{52}{\giga\op\per\second}, see~\cref{eq:actrate}.
By incorporating specialized circuits for the integration of the synapses' output currents in the non-spiking operation mode of the \gls{asic}, a single chip could perform well above \SI{10}{\tera\op\per\second}.
To reach the maximum throughput of \SI{32.8}{\tera\op\per\second} achievable at the current clock frequency, the I/O bandwidth and the ADC conversion speed have to be increased as well.
Higher synaptic operating frequencies could be achieved by splitting the synapse arrays further in smaller banks to shorten the signal wires.
The current area efficiency of the analog MAC in the synapse arrays is
\begin{equation}
\frac{\SI{32.8}{\tera\op\per\second}}
{256\cdot 512\cdot \SI{8}{\micro\meter}\cdot \SI{12}{\micro\meter}} = \SI{2.6}{\tera\op\per\second\per\square\milli\meter}.
\end{equation}
Since the synapses contain the correlation sensors used only for plasticity in the event-based operation modes, the area efficiency could be increased by a factor of up to three.
The shortening of the columns following from the removal of the correlation sensor circuits would also allow a higher operating frequency, further increasing the area efficiency.
Values larger than \SI{10}{\tera\op\per\second\per\square\milli\meter} seem to be well achievable with the current technology node.
Energy efficiency would not increase by the same amount, since making better usage of the synapses' inherent analog computing speed would increase the dynamic power consumption of the I/O circuits and the synapse arrays themselves.
Since only a very small part of the power reported in \cref{tab:competition_results} is used for the analog MAC operations, effective total power for the chip performing at its full speed can be estimated to be about \SI{50}{\tera\op\per\second\per\watt}.
A fully optimized chip that keeps the analog MAC technology as it has been demonstrated in operation in this paper, while having all circuits optimized for energy and area efficiency, could be expected to perform better than \SI{100}{\tera\op\per\second\per\watt}. |
1006.5462 | \section{Introduction}
Lasers emit light over a range of wavelengths
described by the laser line shape
function.\cite{Csele,Milonni,Pedrotti} For a HeNe laser operating
under normal conditions, the main source of laser line shape
broadening is Doppler broadening in the lasing medium, resulting in
a Gaussian gain profile (see Fig.~\ref{fig:long_modes2}). The laser does not
emit a continuous spectrum of wavelengths over this Gaussian
gain-permitted wavelength range; rather, it can only lase when there
is resonance in the lasing cavity. For the TEM$_{00}$ mode there
exists an integer number, $N$, of half wavelengths between the mirrors
of the laser cavity, resulting in the allowed resonance wavelengths
\begin{equation}
\lambda_N = \frac{2nL}{N},
\end{equation}
where $L$ is the length of the laser cavity and $n$ is the index of refraction of the medium filling the laser cavity.
The laser output consists of discrete wavelength peaks with power dictated by the Gaussian line shape envelope and the unsaturated gain threshold (see Fig.~\ref{fig:long_modes2}). These peaks are called longitudinal cavity modes. When the laser cavity supports more than one peak (that is, where the gain is greater than the losses for those peaks), the laser output consists of multiple discrete wavelengths. If the light from these multiple modes is projected onto a detector (for example, a photodiode), then the photocurrent will oscillate at the difference frequency, producing a beat signal.
The beat frequency of interest is at the frequency due to the spacing between adjacent longitudinal modes. The frequency of the $N$th mode can be derived from Eq.~(1) to be $f_N = N(c/2nL)$. Thus the beat frequency is given by
\begin{equation}
\Delta f = {c\over{\lambda_{N+1}}} - {c\over{\lambda_{N}}} = {c \over 2nL},
\end{equation}
and therefore $L = c/2n\Delta f$, indicating that the cavity length is directly proportional to the reciprocal of the beat frequency.\cite{Razdan} Observing the variation in beat frequency between adjacent longitudinal modes with the cavity length $L$ gives the speed of light.
\begin{figure}[h!]
\centering
\includegraphics[width=0.37\textwidth]{DOrazio_Fig01}
\caption{Schematic illustration of the longitudinal cavity modes and gain bandwidth of a laser. In the situation shown, the net gain minus losses is sufficient for laser output at only two longitudinal cavity modes. The beat frequency that we observe to measure the speed of light is the spacing between these adjacent modes.}
\label{fig:long_modes2}
\end{figure}
Accurate measurements of the beat frequency are accomplished
inexpensively by directing the output of the laser onto a
high-speed photodetector\cite{Detectors} monitored with an RF
spectrum analyzer or frequency counter.\cite{CSA,Phillips,Conroy} This
approach has been demonstrated in Ref.~\onlinecite{Brickner} in an undergraduate experiment with the goal of measuring the speed of light using the relation in Eq.~(2) for a
single laser cavity length and single corresponding beat
frequency. The method is easily understood because it is analogous to investigations of waves on a string. It has a drawback, however; the inability to obtain a precise
measurement of the cavity length (from the inner-cavity
side of the output coupler to inner-cavity side of the back mirror)
inevitably leads to results that are only marginally better than those
obtained with standard time-of-flight or Foucault methods commonly
used in undergraduate physics laboratories, which typically yield
measurements accurate to within $\approx \pm 1 \%$.\cite{Bates,Fiber,Foucault} Minor
improvements on this method can be made by collecting
data for multiple lasers of different lengths and plotting the beat
frequency as a function of cavity length. In addition to the
uncertainty in length between the mirrors, there is also the problem
of not knowing a precise (and constant) value for the index of
refraction inside the gas tube. These
obstacles can be overcome by using the laser as a simple light source,
amplitude modulated at the intermode beat frequency, and measuring the phase
difference between detectors placed at two different locations along
the laser path.\cite{Barr} This modulation technique improves the measurement of
the speed of light by an order of magnitude, but
at the cost of increasing the conceptual complexity. The introduction of the
adjustable-length HeNe laser significantly reduces the consequences of
uncertainty in mirror location and the index of refraction, and improves the
measurement by a order of magnitude over the modulation technique,
while retaining the conceptual simplicity of the original
study of Ref.~\onlinecite{Brickner}.
\section{Methods}
Figure~\ref{fig:HeNe_set-up} represents a schematic of the
experimental set-up. The laser has an adjustable, open-cavity design
with a 28\,cm HeNe plasma tube terminated on one side with a mirror and
on the other with a Brewster window. The Brewster window suppresses
modes with polarization orthogonal to the Brewster plane, so that all
supported modes have the same polarization and thus mix
effectively in the photodetector.\cite{Csele, Milonni}
The experiment can be conducted without a Brewster window, but due to mode competition, adjacent longitudinal modes
are typically polarized orthogonal to each other and do not mix
in the photodetector, resulting in an observed signal with twice the
expected frequency.\cite{Tang} If a Brewster window is not present,
the situation can be remedied by placing a linear polarizer in front
of the photodetector to project the polarizations of adjacent modes
onto a common axis.
\begin{figure}[h!]
\centering
\includegraphics[width=0.37\textwidth]{DOrazio_Fig02}
\caption{A schematic of the experimental set-up. The length of the cavity can be adjusted over a range of approximately 16\,cm by sliding the output coupler along an optical track. The mode structure of the laser output is monitored using a scanning Fabry-Perot interferometer with a free spectral range of 1.5\,GHz and a Finesse of 250. The mode structure is controlled via an adjustable iris in the cavity. The portion of the beam that is not analyzed by the Fabry-Perot is incident on a fast photodetector (1\,ns rise time), which is coupled to an RF spectrum analyzer on which the beat signal between adjacent longitudinal modes is observed. (NPBS = non-polarizing beam splitter.)}
\label{fig:HeNe_set-up}
\end{figure}
The variable-length cavity system has been reported and
widely used in undergraduate labs to
explore laser cavity modes and
stability.\cite{Brandenberger,Polik,Jackson,Melles}
The output coupler is a 0.60\,m radius-of-curvature mirror held in
a gimbal mount. It is attached to a
sliding track, allowing the cavity length to be changed from
$\approx 38$\,cm (lower bound limited by the length of the plasma tube) up to $\approx 54$\,cm (upper bound restricted by laser losses). Typically we see two or three longitudinal modes separated by about 300\,MHz within the 1.5\,GHz gain bandwidth of the HeNe medium.\cite{Pedrotti} Inside the
cavity, between the output coupler and the plasma tube, is an iris used to restrict gain in the region away from the
optical axis of the cavity and thus force the laser to emit in the
TEM$_{00}$ (Gaussian) mode. Restricting the laser to a single transverse mode is necessary because higher-order modes produce additional beat frequencies that complicate the RF
spectrum. The allowed frequencies for the TEM$_{00}$ mode are given
by Eq.~(1), and the allowed frequencies for higher-order
TEM$_{ij}$ modes are given by
\begin{equation}
f_{Nij} = {c \over{2L}} \left[ N + {1 \over{\pi}}(i + j + 1)\cos^{-1}(\sqrt{g_1g_2}) \right],
\end{equation}
where $N$ is the same mode number as in Eq.~(1) and $g_1g_2$ is the
resonator stability.\cite{Milonni, Goldsborough} Thus if TEM$_{00}$
and TEM$_{ij}$ are allowed to exist simultaneously in the cavity, beat
frequencies will exist at ${c/2L}$ and ${c/2L} \pm {(1/\pi)} (i + j +
1)\cos^{-1}\sqrt{g_1g_2}$. These additional beat
frequencies could provide an interesting method for measuring the
resonator stability, $g_1g_2$, for a fixed cavity length.
\subsection{Cavity Length Measurement}
As noted in Sec.~I, we cannot accurately measure the entire
laser cavity length due to the uncertainty of the position of the
mirror in the HeNe tube. In addition, the index of refraction within
the He- and Ne-filled tube is different from that in the rest of the
cavity, which is filled with air (and a small length of glass at
the window). Because we do not know the index of refraction inside the
laser plasma tube, we modify Eq.~(2) by splitting $L$ into the
two main regions within the laser cavity that have different indices of
refraction. Let $n_{\mathrm{HeNe}}$ be the index of refraction inside
the laser plasma tube and $n_{\rm air}$ be the index of refraction of
air between the Brewster window and the output coupler. Then, $nL =
n_\mathrm{HeNe}L_{\mathrm{HeNe}} +
n_{\mathrm{\rm air}}L_{\mathrm{\rm air}}$, where additional fixed components such as the glass window and dielectric mirror coatings are assumed in the first term. In practice neither of these $L$
values is simple to measure accurately, and thus we split $L_{\rm air}$ further
into two arbitrary pieces (a fixed length and a measured variable
length) such that $nL = n_{\mathrm{HeNe}}L_{\mathrm{HeNe}} +
n_{\mathrm{\rm air}}[L_{\mathrm{fixed}} + \Delta L]$ (see Fig.~\ref{fig:HeNe_set-up}). We substitute this expression into
Eq.~(2) and obtain
\begin{equation}
\Delta L = {c \over{2n_{\mathrm{\rm air}} \Delta f}} - {\frac{n_{\mathrm{HeNe}}}{n_{\mathrm{\rm air}}}}L_{\mathrm{HeNe}} - L_{\mathrm{fixed}},
\end{equation}
which is the equation of a line with slope $c/
2n_{\rm air}$. Equation (4) allows us to measure the
cavity length to an arbitrarily chosen reference point fixed between
the laser plasma tube output and the output coupler. In practice we
measure $\Delta L$ from a fixed block near the sliding track to the
base of the output coupler using digital vernier calipers. The speed
of light is then found from the slope of a $\Delta L$ versus $1/\Delta f$ plot. The unknown details of $n_{\mathrm HeNe}$,
$L_{\mathrm HeNe}$, and similar terms for the glass window are
gathered in the $y$-intercept. This algebraic trick
works only when the laser is in the TEM$_{00}$ mode, and does not work if
the laser were in transverse TEM$_{Nij}$ modes (where $i$ and $j$ are
nonzero), as represented in Eq.~(3). More
elegantly, we are taking the derivative of Eq.~(2) in the region of
air where we are free to move the output coupler as shown:
\begin{equation}
\frac{dL}{d(\frac{1}{\Delta f})} = \frac{c}{2n_{\rm air}}.
\end{equation}
\subsection{Frequency Measurement}
For the range of laser cavity lengths in the set-up ($\approx
0.54$\,m to 0.38\,m), the beat frequency varies from $\approx
280$\,MHz to 390\,MHz, a change of 110\,MHz over 16\,cm. The signal from the photodetector was analyzed with an RF spectrum analyzer with a maximum span of 3\,GHz and a minimum resolution bandwidth of $10$\,Hz.\cite{Detectors,CSA} A frequency counter could in
principle be used, but would not provide insight into additional beat
frequencies from transverse mode contributions. In addition to
analyzing the laser output with the photodetector and spectrum
analyzer, we split off a portion of the laser output to a scanning
Fabry-Perot interferometer to observe its longitudinal mode
structure.\cite{Fabry} The Fabry-Perot spectrum shows the
number of modes and their amplitudes (and therefore the amplitude of
the gain curve).
The amplitude of the modes provides information on frequency pulling
and pushing, which cause small but statistically significant shifts in
the beat frequency. \textit{Frequency pulling} refers to a change in the
spacing of longitudinal modes under a gain curve resulting from the
different indices of refraction experienced by each mode. Across the
range of frequencies that lie within the laser gain curve, the index
of refraction varies steeply near the resonance transition, being
lower or higher for frequencies below or above the resonance
transition. From Eq.~(2) we see that the allowed
frequencies below the gain peak occur at higher frequencies than would
be expected and vice versa. The result is a ``pulling" of the
longitudinal modes toward the center of the gain curve, effectively
decreasing the difference frequency between the two. The amount by
which the modes are pulled together and the beat frequency is lowered
is a function of the relative intensity of the two heterodyning
modes. For a given gain curve amplitude we find that the beat
frequency varies over $\approx 30$ to 40\,kHz for the full range of
mode relative intensities, in agreement with other
studies.\cite{Lindberg}
\textit{Frequency pushing} refers to the increase of the difference frequency
between longitudinal modes as the field intensity in the laser cavity
increases.\cite{Siegman, Shimoda} As the gain in the cavity is
increased, the beat frequency also increases. We observe this increase in our
set-up; when two adjacent longitudinal modes are observed with
identical intensities, for a $\approx 10$\% change in total amplitude of
the gain curve, there is a $\approx 9$\,kHz change in beat frequency.
Figure~\ref{fig:pushing} shows this effect over a wide range of
amplitudes, showing a linear relation between
the change in the intensity of the modes and the frequency pushing
effect. When taking data to measure the speed of light, we are
able to hold our amplitude fluctuation to a variation of $\pm 10$\%.
To minimize inconsistencies due to frequency pulling effects, we use the Fabry-Perot to ensure that each measurement (that is, the beat frequency at each cavity length) is taken for two longitudinal modes at the same relative intensities (see Fig.~\ref{fig:lmodes_neq}). The refractive index within the laser tube is then the same for both modes and very similar for all beat frequency measurements, reducing the pulling effect. More complex methods of ensuring that the two longitudinal modes are symmetric about the frequency of the emission line have been implemented in other studies.\cite{Balhorn} These involve using a non-Brewster window laser and subtracting the outputs of the orthogonal modes detected with two photodetectors and a polarizing beam splitter. This difference is used to control the electronic feedback to make slight adjustments to the length of the cavity. We have not attempted such elaborate feedback schemes. Instead, students make the necessary adjustments by applying gentle pressure to the optical table, which affects the cavity length on the micron scale.
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{DOrazio_Fig03}
\caption{A sample plot of beat frequency as a function of gain curve amplitude as read from the Fabry-Perot transmission showing the effects of frequency pushing. The uncertainty in the gain curve amplitude of $\pm 10\%$ corresponds to a 18\,kHz frequency variation equivalent to a $\pm 9$\,kHz uncertainty in the beat frequency. The $0\%$ mark in this figure refers to the desired amplitude at which the frequency measurement is to be taken.}
\label{fig:pushing}
\end{figure}
To counteract inconsistencies due to frequency pushing effects, we use the Fabry-Perot to ensure that each measurement is taken with the longitudinal modes at the same total amplitude and thus at the same laser intensity (see Fig. ~\ref{fig:lmodes_eq}). The laser power is controlled by changing the cavity loss by adjusting the intra-cavity iris.
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{DOrazio_Fig04}
\caption{Screen shots from the oscilloscope showing transmission of the scanning Fabry-Perot interferometer. The laser output power is the same in both cases. (a) An instance where the two mode intensities are asymmetrical around the center of the gain curve, whereas (b) shows the two modes when they have equal intensities. Due to frequency pulling, the two instances will produce beat frequency values differing by a few kHz. }
\label{fig:lmodes_neq}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{DOrazio_Fig05}
\caption{Screen shots from the oscilloscope showing transmission of the scanning Fabry-Perot interferometer. Both show the existence of two longitudinal modes at the same relative intensity and thus each exhibit the same frequency pulling induced effects. (a) Two modes when the laser is operating at a higher gain setting than is present in (b). Due to frequency pushing, the beat frequency produced by the modes in (a) is higher than the beat frequency produced by the modes in (b). }
\label{fig:lmodes_eq}
\end{figure}
\section{Data Analysis and Results}
Figure~\ref{fig:HeNe_Data_Plot} represents experimental data for 28 cavity lengths. The
uncertainty in our $\Delta L$ measurement is $\pm 1 \times 10^{-5}$\,m, dictated by the measurement limit of the digital vernier calipers.
The uncertainty in our beat frequencies is dominated by frequency
variability due to frequency pulling and pushing and has been
minimized with the use of the Fabry-Perot interferometer. Due to
frequency pulling and pushing, a change in the relative or total
intensities of the heterodyning longitudinal modes corresponds to a
change in the beat frequency. Thus the uncertainty in the beat frequency
is found by estimating the precision to which we can achieve both the
desired mode relative intensity and desired gain curve
amplitude. Using the Fabry-Perot interferometer, we find that we can steadily hold
the two longitudinal modes at equal relative intensities, resulting in
a negligible uncertainty of $\approx \pm 2$\,kHz due to frequency pulling.
Most of the uncertainty comes from frequency
pushing, for it is not as simple to hold the total amplitude of the
gain curve at a fixed value. To estimate this uncertainty, the
precision to which the amplitudes of the modes can be held constant is
converted into an uncertainty in frequency from the spread of beat
frequencies observed simultaneously on the spectrum analyzer. We
observe that by adjusting the position and aperture size of the iris
in the resonator, we can manipulate the output to have two
longitudinal modes with equal intensity and an overall gain amplitude
that is constant to within $\pm 10\%$. Figure~\ref{fig:pushing}
shows the beat frequency as a function of the total mode amplitude
for our system. A $\pm 10\%$ variation in the total mode amplitude
corresponds to an uncertainty in a single measurement of the beat
frequency of $\pm9$\,kHz.
The uncertainty in the frequency measurement, $\sigma_{\Delta f}$, and the uncertainty in the length measurement, $\sigma_{\Delta L}$, are fixed for each data point, but the uncertainty in the reciprocal beat frequency, $\sigma_{{1/\Delta f}}$, is a function of $\Delta f$ (which varies for each data point). Hence the uncertainty in $1/{\Delta f}$ is not fixed for each data point: $\sigma_{{1/\Delta f}}=\sigma_{\Delta f}/(\Delta f)^2$. Additionally the equivalent uncertainty in $\Delta L$ due to
the uncertainty in $\Delta f$ is of the same order of magnitude as
$\sigma_{\Delta L}$. That is,
\begin{equation}
{{d (\Delta L)} \over{d ({1 \over{\Delta f}})}} \sigma_{1/\Delta f} \approx \sigma_{\Delta L}.
\end{equation}
For this reason, a weighted least squares regression incorporating uncertainty in both variables is performed for the $\Delta L$ versus $1/\Delta f$ data.\cite{Bevington}
The final result for the speed of light in air based on the data plotted in Fig.~6 is
\begin{equation}
c = (2.9972 \pm 0.0002)\times10^8\,\mbox{m/s}.
\end{equation}
The uncertainty of $\pm 0.0002$ is small enough to discriminate
between the speed of light in air ($2.9971 \times 10^8$\,m/s
for $n_{\rm air} = 1.00027$) and the speed of light in a vacuum
($2.9979 \times 10^8$\,m/s).
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{DOrazio_Fig06}
\caption{The plot of 28 data points are fit using a weighted least squares regression. Errors are too small to display on this scale. We find a slope of $c /2n_{\rm air} = (1.4986 \pm 0.0001) \times 10^8$\,m/s.}
\label{fig:HeNe_Data_Plot}
\end{figure}
The measured speed of light yields an index of refraction for air in
our lab of $n_{\rm air} = 1.00024 \pm 0.00006$. We compare this value
to the index of refraction of air as a function of temperature,
wavelength, pressure, and humidity. At conditions of $20^{\circ}$C,
632.8\,nm, 1 atm, and 40\% relative humidity, the
accepted index of refraction of air is 1.00027.\cite{NIST} No
realistic changes in relative humidity, room temperature, or
atmospheric pressure significantly affect the result. Therefore,
the method described here does not have the necessary precision to
demonstrate the effects of atmospheric fluctuations on the index of
refraction.
\section{Conclusion}
This experiment exposes students to a variety of experimental and mathematical techniques, demonstrates the importance of uncertainty in measurement, provides a meaningful context for using weighted regression, and familiarizes the student with three ubiquitous instruments: the laser, the Fabry-Perot interferometer, and the RF spectrum analyzer. In addition the experiment yields satisfying results, allowing measurement of the speed of light to a precision which differentiates between the speed of light in air and the speed of light in a vacuum. The precision to which the measurement is taken is limited by both the precision of our length measurement and our ability to minimize uncertainties due to the frequency pushing and pulling. One could improve length measurements with a precision linear stage and one could lock the HeNe laser so that the longitudinal modes are held to the same amplitude, but both of these improvements would be beyond the necessary scope of an intermediate physics laboratory course.
\begin{acknowledgments}
The authors would like to thank all of the Juniata College students who have
performed this measurement in the Advanced Physics Lab over the past
five years. The students that have been particularly instrumental in
improving the experimental technique or data analysis have been
included as authors. We also thank the reviewers for their insightful comments. This work has been supported by the von Liebig
Foundation and NSF PHY-0653518.
\end{acknowledgments} |
2106.00683 | \section{Introduction}
\label{sec:intro}
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{./figures/m87_snapshot-crop.pdf}
\caption{
(Left) Snapshot image from a magnetically arrested radiative GRMHD simulation of {M87$^{\ast}\xspace$} \citep[Model R17;][]{Chael_19}, convolved with a circular Gaussian blurring kernel with a full width at half maximum (FWHM) of $15\,\mu$as.
The features of the simulated image at this resolution qualitatively match those seen in the first images of {M87$^{\ast}\xspace$} from the EHT \citep{PaperIV}.
(Middle) The simulation snapshot at native resolution. The simulation is viewed at an inclination $\theta_{\rm o}=163\deg$ \citep{Mertens2016,PaperV}; the black hole spin vector is oriented to the left and into the page.
The snapshot image shows filamentary, turbulent structures, a central brightness depression, and a narrow, bright photon ring that closely tracks the theoretical critical curve (cyan curve).
(Right) The same simulation snapshot in a gamma color scale that accentuates low-brightness features.
In this scale, the central brightness depression corresponds to the black hole's \emph{inner shadow}, or the direct lensed image of the equatorial event horizon (white curve).
The EHT images released thus far do not resolve the inner shadow of {M87$^{\ast}\xspace$}, as they lack the requisite resolution and dynamic range.
These requirements may be met with a next-generation EHT.
\label{fig:Summary}
}
\end{figure*}
The Event Horizon Telescope (EHT) has recently produced the first resolved images of a black hole \citep{PaperI,PaperII,PaperIII,PaperIV,PaperV,PaperVI,PaperVII,PaperVIII}.
These 230~GHz images resolve the emission surrounding the supermassive black hole {M87$^{\ast}\xspace$} \citep[$M=6.5\pm 0.7\times10^9 M_\odot$;][]{PaperVI} at the center of the giant elliptical galaxy M87.
The EHT resolution of $\approx\!20\,\mu$as ($\approx\!5\,GM/Dc^2$ for {M87$^{\ast}\xspace$} at a distance $D\approx\!16.8$~Mpc) only just reveals the horizon-scale structure in {M87$^{\ast}\xspace$}.
The EHT images display a ring with a diameter of $\approx\!40\,\mu$as with a North-South brightness asymmetry and a relatively dim interior.
In models where the accretion flow onto a Kerr black hole is spherically symmetric and the emission is optically thin, the central brightness depression in the observed image coincides precisely with those light rays that terminate on the event horizon when traced backwards from the observer's image plane into the black hole spacetime \citep{Falcke2000,Narayan_Shadow}.
This dark region---the ``black hole shadow''--- is bounded by a ``critical curve'' consisting of light rays that asymptote to unstably bound photon orbits around the black hole \citep{Bardeen1973a}.
Motivated by these models, the critical curve is sometimes also called the ``shadow edge.''
Approaching the shadow edge, the path length through the emission region diverges logarithmically as null geodesics wrap around the black hole multiple times \citep{Luminet1979a,Ohanian_1987,GHW,Johnson_Ring,GrallaLupsasca2020}.
Hence, in models featuring a spherically symmetric and optically thin emission region, the image brightness also diverges logarithmically at the critical curve, resulting in a bright ``photon ring'' encircling the black hole shadow.
By contrast, in models where the emission region is confined to an equatorial disk that extends down to the event horizon, the edge of the observed central brightness depression does not generically correspond to the critical curve \citep[e.g.,][]{Beckwith,Broderick06,GHW}.
Nevertheless, as long as the emission is optically thin, these models still feature a photon ring with logarithmically divergent brightness at the critical curve.
Contrary to the case of spherical accretion, however, the brightness increase is not continuous; rather, it is broken up into a sequence of strongly lensed images of the disk stacked on top of each other.
These images arise from rays with deflection angles $>180\deg$ that execute an increasing number of half-orbits around the black hole \citep{Luminet1979a,GHW,Johnson_Ring}.
In reality, the hot ($T>10^{10}$~K), collisionless plasma that produces the submillimeter emission in {M87$^{\ast}\xspace$} is expected to be turbulent, with a more complex structure than can be captured in either of these simple geometric pictures (\autoref{fig:Summary}).
The primary numerical tools for investigating the structure and dynamics of hot accretion flows are general relativistic magnetohydrodynamic (GRMHD) simulations \citep[e.g.,][]{Komissarov99,Gammie03}.
To constrain the properties of {M87$^{\ast}\xspace$}, analyses of EHT images in both total intensity \citep{PaperV,PaperVI} and in polarization \citep{PaperVIII} made use of a library of these GRMHD simulations spanning a range of different parameters, including the black hole spin, accumulated magnetic flux on the black hole, and ion-to-electron temperature ratio.
Significantly, \citet{PaperVIII} found that, among the GRMHD simulation models in the EHT library, the currently favored models for {M87$^{\ast}\xspace$} all fall into the class of magnetically arrested disks \citep[MADs;][]{NarayanMAD,Igumenschchev2003}.
In addition to producing images that are consistent with those observed by the EHT, MAD simulations naturally produce powerful jets \citep[e.g.,][]{Tchekhovskoy11,Chael_19} similar in both observed shape and total power to the prominent jet in {M87$^{\ast}\xspace$} \citep[e.g.,][]{Junor99,Stawarz06,Abramowski2012,Hada2016,Walker18,Mwl}.
Analyses of GRMHD simulation images have generally focused on the mathematical shadow edge, i.e., the critical curve \citep[e.g.,][]{Dexter2012,Psaltis_2015,Moscibrodzka_16,Bronzwaer_2021}.
Because this curve only depends on the black hole mass and spin vector, inferring its size and shape would provide information about the black hole's intrinsic parameters and enable tests of the validity of the Kerr metric \citep[e.g.,][]{Takahashi,JohannsenPsaltis,PaperVI}.
However, in performing these tests with limited-resolution observations, it is critical to account for the systematic uncertainty in relating observed image features such as the emission ring and central brightness depression to gravitational properties such as the size and shape of the critical curve \citep[e.g.,][]{PaperVI,Bronzwaer_2021}.
These systematic uncertainties may be dramatically reduced via future observations using an enhanced ground or space-based array capable of distinguishing lensed subrings within the photon ring \citep[e.g.,][]{Johnson_Ring,Astro2020Ground,Astro2020Space,Astro2020Space2,GLM_20,Broderick21}.
In this paper, we show that MAD models of {M87$^{\ast}\xspace$} naturally exhibit a deep flux depression whose edge is contained well within the photon ring and critical curve.
This darkest region in a MAD simulation image corresponds to rays that terminate on the event horizon before crossing the equatorial plane even once (\autoref{fig:Rings}).
We refer to this feature as the ``inner shadow'' of the black hole.
This lensing feature was previously studied by \citet{DN19b,DN20,DN20b}.
As long as the emission is equatorial and extends all the way to the horizon, the darkest region in the observed image will correspond to the inner shadow, with a boundary defined by the direct, lensed image of the event horizon's intersection with the equatorial plane.
The MAD GRMHD models that we consider satisfy these criteria, with their submillimeter emission originating in the equatorial plane close to the event horizon \citep[as seen in][]{PaperV}.
Due to the effects of increasing gravitational redshift, the image brightness falls off rapidly near the edge of the inner shadow.
As a result, the correspondence between the lensed image of the equatorial horizon and the edge of the central brightness depression in an image is only apparent in faint image features viewed at high dynamic range.
The inner shadow of a Kerr black hole has a significantly different dependence on its parameters than the critical curve \citep{Takahashi}.
For instance, the photon ring and critical curve of a Schwarzschild black hole are circular and independent of the viewing inclination, while the inner shadow is only circular when viewed face-on and has a size, shape, and centroid that are highly sensitive to the viewing inclination.
The photon ring and inner shadow provide complementary information.
When considered independently, each is subject to degeneracies in its size and shape as a function of black hole mass, spin, and viewing angle, yet these degeneracies can be broken via simultaneous observations of both features.
In simple toy models, spherical accretion flows produce a central brightness depression that completely fills the critical curve, but they do not give rise to an inner shadow \citep[e.g.,][]{Falcke2000,Narayan_Shadow}.
By contrast, thin-disk accretion models with emission extending to the horizon and a large optical depth present a precisely observable inner shadow, but they do not display any visible feature near the critical curve, since the lensed images that would produce a photon ring are blocked by the optically thick disk \citep[e.g.,][Figure 5.]{Beckwith}.
As a result, past work has generally analyzed these two features independently under the expectation that only one or the other will be relevant to the observed image \citep[see, e.g.,][]{Takahashi,DN20b}.
Remarkably, we find that in both GRMHD simulations with strong magnetic fields and in semi-analytic, optically thin disk models with a radially dependent emissivity, the photon ring and the inner shadow are both prominent as potentially observable features (\autoref{fig:Summary}).
Thus, in the future, it may become possible to simultaneously measure both features in images of a black hole and thereby derive tighter, joint constraints on its parameters.
In this paper, we explore how the inner shadow may appear in images from realistic simulations and models of {M87$^{\ast}\xspace$}, we assess the information contained in the relative size and shape of the inner shadow compared to the critical curve, and we discuss the prospects for direct observation of this feature using submillimeter very-long-baseline interferometry (VLBI).
In \autoref{sec:BlackHoleImages}, we review the basic properties of null geodesics and radiative transfer in the Kerr spacetime that give rise to the photon ring and inner shadow.
In \autoref{sec:Models}, we discuss the appearance of the inner shadow in images simulated from GRMHD and semi-analytic models.
In \autoref{sec:GeometricDescription}, we discuss geometric properties of both the critical curve and inner shadow, including their relative size, shape, and centroid positions, and we provide convenient analytic approximations for these quantities.
Throughout the paper, we only consider the inner shadow arising from equatorial emission near the event horizon of a Kerr black hole; in \autoref{sec:discussion} we discuss some of the factors---including jet emission, disk thickness and tilt, and alternative spacetime geometries---that could affect whether or how this feature appears in black hole images.
We summarize our conclusions in \autoref{sec:Conclusions}.
\begin{figure*}[t]
\raisebox{3ex}{\includegraphics[height=0.425\textwidth,trim=0 0 0 0]{./figures/Conch-crop.pdf}}\hfill
\includegraphics[height=0.45\textwidth]{./figures/m87_nrings-crop.pdf}
\caption{
(Left) Photon trajectories around the black hole that reach a distant observer located to the far right \citep[see also][]{JohannsenPsaltis,GHW}.
The black hole is nonrotating (${a^\ast}=0$), with an event horizon at $r_+=2M$ (black disk) and a photon sphere at $r_{\rm c}=3M$ (dashed yellow circle).
Photon trajectories are colored according to the number of times they cross the equatorial plane (green line), which is inclined at $\theta_{\rm o}=17\deg$ from the observer.
Most trajectories cross the equatorial plane once (blue), but photons that appear close to the critical curve on the image plane wrap around the black hole and cross the plane twice (purple), three times (red), or more, with photons appearing exactly on the critical curve describing trajectories that asymptote to unstably bound orbits ruling the photon sphere.
The inner shadow is defined by the trajectories that do not cross the equatorial plane ($N_{\rm max}=0$) before intersecting the event horizon (black disk).
(Right) The maximum number of equatorial crossings $N_{\rm max}$ for null geodesics as a function of the coordinates $(\alpha,\beta)$ in the image plane for a black hole of spin ${a^\ast}=0$ observed at an inclination angle of $\theta_{\rm o}=17\deg$.
Rings with an increasing number of equatorial crossings $N_{\rm max}$ become increasingly narrow and exponentially approach the critical curve (dashed yellow circle).
Inside the contour of the lensed equatorial horizon (solid white line), light rays do not cross the equatorial plane even once.
In models of equatorial emission extending to the horizon, these rays are therefore dark, resulting in an ``inner shadow'' feature.
The ``unlensed'' outline of the event horizon $r_+=2M$ is indicated with the dotted white line.
}
\label{fig:Rings}
\end{figure*}
\section{Black hole images}
\label{sec:BlackHoleImages}
In this section, we review key features of the Kerr metric and the multiple lensed images of emission surrounding a black hole.
We argue that the curve marking the direct image of the equatorial event horizon should be visible as the edge of an ``inner shadow'' if the emission region is sufficiently equatorial and extends down to the event horizon.
From here on, we work in units normalized such that $G=c=1$.
\subsection{Kerr metric}
In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$, the metric of a Kerr black hole of mass $M$ and angular momentum $J=aM$ ($0\leq a\leq M$) is
\begin{align}
ds^2&=-\frac{\Delta}{\Sigma}\pa{\mathop{}\!\mathrm{d} t-a\sin^2{\theta}\mathop{}\!\mathrm{d}\phi}^2+\frac{\Sigma}{\Delta}\mathop{}\!\mathrm{d} r^2\notag\\
&\quad+\Sigma\mathop{}\!\mathrm{d}\theta^2+\frac{\sin^2{\theta}}{\Sigma}\br{\pa{r^2+a^2}\mathop{}\!\mathrm{d}\phi-a\mathop{}\!\mathrm{d} t}^2,
\end{align}
where
\begin{align}
\Delta\equiv r^2-2Mr+a^2,\quad
\Sigma\equiv r^2+a^2\cos^2{\theta}.
\end{align}
We frequently use the dimensionless spin $0\leq{a^\ast}\equiv a/M\leq1$.
The (outer) event horizon is located at radius
\begin{align}
r_+=M+\sqrt{M^2-a^2}.
\end{align}
Unstable bound null geodesics, which neither escape to infinity nor intersect the event horizon, form a ``photon shell'' \citep{Bardeen1973a,Teo_2003,Johnson_Ring} outside of the outer event horizon.
Each bound orbit exists at a fixed Boyer-Lindquist radius $r_{\rm c}$ in the range $r_{\rm c,-}\leq r_{\rm c}\leq r_{\rm c,+}$, where
\begin{align}
r_{\rm c,\pm}=2M\br{1+\cos\pa{\frac{2}{3}\arccos\pa{\pm{a^\ast}}}}.
\end{align}
The bound orbits at $r=r_{\rm c,\pm}$ are confined to the equatorial plane ($\theta=\pi/2$).
At intermediate radii $r_{\rm c,-}<r_{\rm c}<r_{\rm c,+}$, the bound orbits oscillate between two fixed polar angles $\theta_\pm$ (see \autoref{eq:TurningPoints}).
In the case of a nonrotating Schwarzschild black hole ($a=0$), the photon shell reduces to a single ``photon sphere'' at $r_{\rm c}=3M$.
There exist timelike, equatorial geodesics forming stable prograde circular orbits around the black hole for all radii $r\geq r_{\rm ISCO}$, where $r_{\rm ISCO}$ denotes the radius of the ``Innermost Stable Circular Orbit,''
\begin{align}
r_{\rm ISCO}=M\br{3+Z_2-\sqrt{(3-Z_1)(3+Z_1+2Z_2)}},
\end{align}
with
\begin{subequations}
\begin{align}
Z_1&=1+\pa{1-{a^\ast}^2}^{1/3}\br{\pa{1+{a^\ast}}^{1/3}+\pa{1-{a^\ast}}^{1/3}},\\
Z_2&=\sqrt{3{a^\ast}^2+Z_1^2}.
\end{align}
\end{subequations}
For Schwarzschild, $r_{\rm ISCO}=6M$.
\subsection{Lensed images and the critical curve}
\label{sec:LensedImages}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth]{./figures/disk_grid-crop.pdf}
\caption{Analytic models of emission from a Kerr black hole's equatorial plane covering a range of black hole spins and observer inclinations.
Columns from left to right display models with dimensionless spins ${a^\ast}=(0,0.5,0.75,0.99)$ and rows from top to bottom display inclinations $\theta_{\rm o}=(0,30,60,89)\deg$.
The intensity in each panel is normalized independently: all images are plotted in a gamma color scale with index $\gamma=1/4$.
For each model, we show the critical curve (cyan), the direct (primary, $n=0$) lensed image of the equatorial horizon (solid white) and the backside (secondary, $n=1$) lensed image of the equatorial horizon (dashed white).
\label{fig:imgrid}
}
\end{figure*}
We consider a distant observer ($r_{\rm o}\to\infty$) viewing the black hole at an inclination angle $0\leq\theta_{\rm o}<\pi$ with respect to its spin axis.
We parameterize the observer's image plane using ``Bardeen coordinates'' $(\alpha,\beta)$, given in units of $M$, defined such that the $\beta$ axis corresponds to the black hole spin axis projected onto the plane perpendicular to the ``line of sight.''
Each point in the image plane is associated with a null geodesic extending into the Kerr spacetime and labeled by two conserved quantities: the energy-rescaled angular momentum $\lambda$ and Carter constant $\eta$.
For a point $(\alpha,\beta)$ in the image plane, these constants are
\begin{subequations}
\label{eq:ConservedQuantities}
\begin{align}
\lambda&=-\alpha\sin{\theta_{\rm o}},\\
\eta&=\pa{\alpha^2-a^2}\cos^2{\theta_{\rm o}}+\beta^2.
\end{align}
\end{subequations}
The covariant four-momentum $k_\mu$ of the null geodesic at any point in the spacetime is given in terms of $\lambda$, $\eta$ and the photon energy-at-infinity $E$ as
\begin{subequations}
\label{eq:NullGeodesics}
\begin{align}
k_t&=-E,\quad
k_\phi=E\lambda,\\
k_r&=\pm E\sqrt{\mathcal{R}}/\Delta,\\
k_\theta&=\pm E\sqrt{\Theta},
\end{align}
\end{subequations}
where $\mathcal R(r)$ and $\Theta(\theta)$ are the radial and angular potentials
\begin{align}
\label{eq:RadialPotential}
\mathcal{R}(r)&=\pa{r^2+a^2-a\lambda}^2-\Delta\br{\eta+\pa{\lambda-a}^2},\\
\Theta(\theta)&=\eta+a^2\cos^2{\theta}-\lambda^2\cot^2{\theta}.
\end{align}
By integrating the null geodesic equation \ref{eq:NullGeodesics}, we can solve for the trajectory $x^\mu(\tau)$ through the Kerr spacetime of a photon shot back from position $(\alpha,\beta)$ on the observer's image plane.
Such trajectories can be divided into three classes: those that eventually cross the event horizon (photon capture), those that are deflected by the black hole but return to infinity (photon escape), and those that asymptote to unstable bound orbits around the black hole.
The latter form a closed curve $(\alpha_{\rm c},\beta_{\rm c})$ in the image plane---the \emph{critical curve}---delineating the region of photon capture (the curve's interior) from that of photon escape (its exterior).
Critical photons have conserved quantities $(\lambda_{\rm c},\eta_{\rm c})$ equal to those of a photon on a bound orbit. For a given photon orbit radius $r_{\rm c,-}\leq r_{\rm c}\leq r_{\rm c,+}$, these are
\begin{subequations}
\begin{align}
\lambda_{\rm c}&=a+\frac{r_{\rm c}}{a}\br{r_{\rm c} -\frac{2\Delta(r_{\rm c})}{r_{\rm c}-M}},\\
\eta_{\rm c}&=\frac{r_{\rm c}^3}{a^2}\br{\frac{4M\Delta(r_{\rm c})}{(r_{\rm c}-M)^2}-r_{\rm c}}.
\end{align}
\end{subequations}
The critical curve in the image plane is obtained by inverting \autoref{eq:ConservedQuantities} to find $(\alpha,\beta)$ for all $r_{\rm c,-}\leq r_{\rm c}\leq r_{\rm c,+}$.
Each bound photon orbit at constant radius $r_{\rm c}$ maps to \emph{two} points in the image plane corresponding to the two signs $\pm\beta$ allowed for a given pair $(\lambda,\eta)$.
As a result, the critical curve is symmetric about the $\alpha$ axis perpendicular to the projected spin.
The interior of the critical curve corresponds to geodesics that connect the observer to the event horizon and is often referred to as the ``black hole shadow.''
This name is motivated by the observation that, for a black hole that is immersed within an optically thin accretion flow with a spherically symmetric emissivity, light rays inside the critical curve (which terminate on the horizon) have a shorter path length along which to accumulate brightness than those in the exterior (which extend to infinity and can pick up more photons as they pass through the emission region); as a result, in such configurations, the critical curve's interior displays a brightness depression \citep{Falcke2000,Narayan_Shadow}.
Tracing back from the image plane, light rays that originate very near the critical curve approach the photon shell of bound orbits and execute many oscillations in $\theta$ between the turning points $\theta_\pm$ (\autoref{eq:TurningPoints}) before either terminating on the event horizon or escaping to infinity.
The number of oscillations (and the path length of the null geodesic) diverge logarithmically as the image-plane coordinate approaches a point on the critical curve.
If the black hole is surrounded by a uniform, optically thin emission region, this divergence in path length manifests as a logarithmic increase in the image brightness surrounding the critical curve: the ``photon ring'' \citep{Johnson_Ring,GrallaLupsasca2020}.
If instead the black hole has an optically thin emission region that does not fully surround it (e.g., one concentrated near the equatorial plane, in a tilted plane, or in a ``jet sheath'' region), then each oscillation in $\theta$ corresponds to an additional pass of the null geodesic through the emission region.
In this case, the photon ring is still present but exhibits additional substructure: its brightness profile increases in steps, forming exponentially narrow subrings that converge to the critical curve, with each ring assigned a label $n$ corresponding to the number $n-1$ of passes its light rays execute through the emission region \citep{Johnson_Ring}.\footnote{
Following \citet{Johnson_Ring}, we assign a number $n$ to each subring such that light rays appearing on that ring describe at least $n$ librations in $\theta$, or at least $n+1$ passes through the emission region.
Thus, $n=0$ refers to the ``direct image'' formed by rays passing through the emission region once.
}
\subsection{Equatorial images and the lensed horizon}
\label{sec:ComputingCurves}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{./figures/emis_m87-crop.pdf}
\caption{
(Left) Map of 230~GHz synchrotron emissivity proxy (\autoref{eq:EmissivityProxy}) in the poloidal plane for time- and azimuth-averaged data from a MAD radiative GRMHD simulation of {M87$^{\ast}\xspace$}.
(Right) Equatorial slice of the simulation emissivity proxy (solid), compared with the emissivity profile used in the analytic model (dashed), both normalized to unity at the horizon.
The simulation emissivity proxy is concentrated in the equatorial plane and does not truncate at the ISCO, but rather continues increasing with decreasing radius all the way to the event horizon.
}
\label{fig:EmissivityProfile}
\end{figure*}
We now focus on emission that is concentrated in the black hole's equatorial plane ($\theta=\pi/2$).
A geodesic ending at position ($\alpha,\beta$) in the image plane crosses the equatorial plane a maximum number of times $N_{\rm max}(\alpha,\beta)$ outside of the event horizon (an analytic procedure from \citet{GrallaLupsasca2020} for calculating the equatorial crossings is reviewed in \autoref{app:RayTracing}; in particular, see \autoref{eq:Nmax}).
In most of the image plane, $N_{\rm max}=1$; that is, geodesics cross the equator only once and project a direct (but still lensed) image of the equatorial emission on the observer sky.
In parts of the image plane that form increasingly narrow rings around the black hole, we instead have $N_{\rm max}=2,3,\ldots$
These concentric regions are the ``lensed subrings'' carrying contributions from geodesics that wrap around the black hole and cross its equatorial plane multiple times.
\autoref{fig:Rings} shows how $N_\mathrm{max}$ varies across the image plane for the case of a Schwarzschild black hole viewed at $\theta_{\rm o}=17\deg$.
For each $0\leq n<N_{\rm max}(\alpha,\beta)$, one can calculate the radius $r_{\rm eq}(\alpha,\beta;n)$ where the geodesic impinging on the observer's image plane at position $(\alpha,\beta)$ crosses the equatorial plane for the $(n+1)^\text{th}$ time.
This computation can be done either analytically (e.g., using the analytic method described in \cite{GrallaLupsasca2020} and reviewed in \autoref{app:RayTracing}) or numerically (e.g., using a GR ray tracing code like \texttt{grtrans} \citep{Dexter16} or \texttt{ipole} \citep{MoscibrodzkaIPole}).
One can also invert $r_{\rm eq}(\alpha,\beta;n)$ to determine the successive lensed images of equatorial circles of constant source radius $r_{\rm s}=r_{\rm eq}$.
These contours are convex curves in the image plane and can be described in image-plane polar coordinates $(\rho,\varphi)$ as curves $\rho(\varphi;r_{\rm s},n)$ with $-\pi\leq\varphi<\pi$ defined by\footnote{
We take the polar angle $\varphi=0$ in the image plane to lie along the $+\alpha$ axis: $\alpha=\rho\cos{\varphi}$, $\beta=\rho\sin{\varphi}$.
Because the lensed images of equatorial rings are convex curves containing the origin, there is a unique $\rho$ satisfying this equation for each $\varphi\in[-\pi,\pi)$, so the curves $\rho(\varphi;r_{\rm s},n)$ are well-defined.
}
\begin{align}
\label{eq:Contours}
r_{\rm eq}\pa{\alpha=\rho\cos{\varphi},\beta=\rho\sin{\varphi};n=0}=r_{\rm s},
\end{align}
For any fixed radius $r_{\rm s}\ge r_+$, the curves $\rho(\varphi;r_{\rm s},n)$ approach the critical curve exponentially fast with increasing $n$.
For small observing angles $\theta_{\rm o}\approx0$, the $n=0$ image of an equatorial ring of constant radius $r_{\rm s}$ is lensed by approximately one gravitational radius; that is, $\rho\approx r_{\rm s}+M$ \citep{GrallaLupsasca2020,Gates2020}.
While most of the image plane has $N_{\rm max}\geq1$, it also has a small region with $N_{\rm max}=0$ wherein geodesics do not cross the equatorial plane even once, but instead pierce the event horizon before ever reaching $\theta=\pi/2$ (right panel of \autoref{fig:Rings}).
This $N_{\rm max}=0$ region corresponds exactly to the interior of the direct ($n=0$) lensed image of the equatorial event horizon, and is therefore bounded by the curve
\begin{align}
\rho_{\rm h}(\varphi)=\rho(\varphi;r_+,0),
\end{align}
defined by \autoref{eq:Contours} with $r_{\rm s}=r_+$.
Like the critical curve, this curve divides the image plane into two qualitatively distinct regions.
Inside the critical curve, all geodesics terminate on the event horizon, while inside $\rho_{\rm h}(\varphi)$, all geodesics terminate on the horizon without crossing the equator (left panel of \autoref{fig:Rings}).
Thus, if the black hole is surrounded by an emission region that is predominantly equatorial and extends all the way down to the horizon, we should expect the interior of $\rho_{\rm h}(\varphi)$ to show up as a dark region in the image, thereby forming an ``inner shadow'' of low brightness.
In \autoref{fig:imgrid}, we plot the critical curve $\rho_{\rm c}(\varphi)$ and the direct, lensed equatorial horizon image $\rho_{\rm h}(\varphi)$ for a range of black hole spins and observer inclinations, on top of an image generated from the analytic model described in \autoref{sec:EmissionModel} below.
In this model, the emission is purely equatorial and extends to the horizon; thus, the interior of $\rho_{\rm h}(\varphi)$---the black hole's ``inner shadow''--- is visible in each image as a deep brightness depression contained within the critical curve.
\section{Models for \texorpdfstring{{M87$^{\ast}\xspace$}}{M87*}}
\label{sec:Models}
In this section, we investigate the appearance of the lensed equatorial horizon in images of synchrotron emission from a radiative GRMHD simulation of {M87$^{\ast}\xspace$}.
We find that the lensed horizon image is visible in GRMHD simulation images of this magnetically arrested disk model for {M87$^{\ast}\xspace$} because its emission region is primarily equatorial.
We compare images from the simulation with images from an analytic model that assumes all emission originates in the equatorial plane.
We scale all images of {M87$^{\ast}\xspace$} throughout this paper so that the angular gravitational size is \citep{PaperVI}
\begin{align}
\frac{M}{D}=3.78\,\mu\text{as}.
\end{align}
We also scale the total flux density at 230~GHz to 0.6~Jy \citep{PaperIII}.\footnote{
Note that the simulation images used here originally had an average flux density of $\approx\!1$~Jy based on observations of {M87$^{\ast}\xspace$} prior to 2017 \citep{Akiyama15}.
Here, we have scaled down the simulation's total flux density to match the updated value that better fits the 2017 EHT images.
}
\subsection{Radiative GRMHD simulation}
\begin{figure*}[t]
\centering
\includegraphics[height=0.475\textwidth]{./figures/uphi.pdf}
\includegraphics[height=0.475\textwidth]{./figures/ur.pdf}
\caption{
(Left) Equatorial value of the specific angular momentum $\ell\equiv u_\phi/u_t$ taken from the time- and azimuth-averaged simulation data (black curve).
At all radii, the averaged simulation angular momentum is well below the Keplerian value (red dashed curve).
We use a simple power-law fit to the GRMHD data in our analytic equatorial model (dotted green curve, \autoref{eq:Fit1}).
(Right) Equatorial value of the covariant infall velocity $v_r=u_r/u_t$ taken from the averaged simulation data (black curve).
We fit the average simulation data with a broken power law (dotted green curve, \autoref{eq:Fit2}).
This broken power-law fit is safely below the maximum infall velocity permitted for timelike geodesics by our power-law fit for $\ell$ (dashed green line).
}
\label{fig:SubKeplerian}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{./figures/m87_examples-crop.pdf}
\caption{
Top: (Left) 230~GHz image from an analytic, equatorial model for emission from {M87$^{\ast}\xspace$}.
(Middle left) Time-averaged 230~GHz GRMHD image.
(Middle right) Equatorial model tuned to match the 86~GHz simulation image, with higher-order image subrings suppressed to mimic optical depth.
(Right) Time-averaged 86~GHz GRMHD image.
Bottom: The same images in a gamma color scale with index $\gamma=0.25$.
All images were generated with dimensionless black hole spin ${a^\ast}=0.9375$ and observer inclination $\theta_{\rm o}=163\deg$.
Both the 230~GHz and 86~GHz analytic model images had the parameters in their emissivity profile (\autoref{eq:EmissivityProfile}) separately fixed to best match the corresponding simulation images.
The black hole spin (positive $\beta$ axis) points to the left, as indicated by the arrow in the left column images.
Each panel displays the critical curve (cyan) and the direct image of the equatorial horizon (white line).
In the left column, we also indicate the centroid of the critical curve (cyan square marker), the centroid of the direct equatorial horizon image (white circular marker), and the origin ($\alpha=0$, $\beta=0$) of the Bardeen coordinate system (white cross).
}
\label{fig:ModelSimulationComparison}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.95\textwidth]{./figures/ring_profile_230.pdf}
\caption{
Slices along the $\beta=0$ (red) and $\alpha=0$ axes (blue) of the time-averaged 230~GHz images from the {M87$^{\ast}\xspace$} GRMHD simulation (solid curves) and the corresponding analytic equatorial disk model (dashed curves).
Solid vertical lines indicate the exact location of the direct image of the equatorial horizon on these slices.
An approximate dynamic range for the EHT2017 array is indicated by the cyan horizontal line, while the width of the cyan rectangle shows one-half the nominal resolution of the 2017 array.
Likewise, the dynamic range for an ngEHT concept array is indicated by the magenta line, and the magenta rectangle indicates one-half of the ngEHT concept's nominal resolution.
In the time-averaged simulation image, the brightness inside the lensed horizon contour levels off at a finite floor value produce by foreground jet emission.
In this simulation, the position of this dark depression inside the main emission ring---the ``inner shadow''---coincides with the image of the lensed equatorial horizon to within a microarcsecond.
In this model, the photon ring contains $\approx\!10\%$ of the total flux in the image.
}
\label{fig:Slices}
\end{figure*}
In this paper, we consider images of a radiative GRMHD simulation of {M87$^{\ast}\xspace$}; specifically, we use
simulation \texttt{R17} from \citet{Chael_19}.
This simulation was performed using the radiative GRMHD code \texttt{KORAL} \citep{KORAL13,KORAL14,KORAL16}.
Unlike most GRMHD codes, which evolve a single combined electron-ion fluid and must apply a model for the electron-to-ion temperature ratio in post-processing, the \texttt{KORAL} code directly evolves the temperature of the emitting electrons under radiative cooling and dissipative heating.
The primary cooling mechanism in the simulation is from synchrotron radiation at submillimeter wavelengths.
The electron heating fraction in the simulation is provided by the \citet{Rowan17} subgrid prescription fit to simulations of collisionless, transrelativistic magnetic reconnection.
The simulation used a black hole mass of $M=6.2\times10^9\,M_\odot$ and spin ${a^\ast}=0.9375$.
The initial magnetic field was set up so that the magnetic flux saturates on the black hole, putting the system in the magnetically arrested (MAD) accretion state \citep{Igumenschchev2003,NarayanMAD,Tchekhovskoy11}.
Polarimetric EHT observations of {M87$^{\ast}\xspace$} favor this accretion state over one with weaker, turbulent magnetic fields \citep{PaperVIII}.
The simulation produces a relativistic jet of power $\approx\!10^{43}$~erg~s$^{-1}$, satisfying measurements of the jet power from M87 \citep[e.g.,][]{Stawarz06}.
Furthermore, the jet opening angle in 43~GHz images from this simulation is large.
When observed at an inclination angle $\theta_{\rm o}=163\deg$ \citep{Mertens2016}, the apparent opening angle is $\approx50\deg$, similar to that observed in VLBI images of M87 \citep{Walker18}.
The extended jet in the simulation is in steady-state out to $\approx\!2500\,M\approx1$~pc, while the disk in the midplane is in steady-state out to $\approx\!40\,M$.
The GRMHD simulation is turbulent and time-variable.
To investigate the persistent features of the GRMHD fluid data, we computed profiles of the key plasma quantities (e.g., the density $\rho$, electron temperature $T_{\rm e}$, magnetic field $B^i$, and velocity $u^\mu$) in the poloidal $(r,\theta)$ plane after averaging in time and azimuth.
We also generated images of the 230~GHz and 86~GHz synchrotron emission from this simulation using the GR ray tracing and radiative transfer code \texttt{grtrans} \citep{Dexter16}.
The images were generated at an observer inclination angle of $\theta_{\rm o}=163\deg$ and rotated so that the black hole spin points to the East, opposite to the direction of the approaching jet \citep{PaperV}.
The snapshot images from this simulation exhibit small-scale structure from plasma turbulence and magnetic filaments (\autoref{fig:Summary}).
In this paper, we focus on time-averaged images generated from the collection of snapshot images of the simulation; both the time-averaged images and the time-averaged simulation data were produced from simulation snapshots spanning $5000\,M$ in time at a cadence of $10\,M$.
Radiatively inefficient simulations with weak magnetic flux form geometrically thick disks supported by the gas pressure.
By contrast, in the high-magnetic-flux MAD state, the magnetic pressure exceeds the gas pressure in the ``disk'' near the black hole.
In the time-averaged simulation data, the near-horizon material forms a thin, highly magnetized structure in the equatorial plane; this thin structure is the source of the observed 230~GHz emission.
Note that the thickness of the equatorial ``disk'' in these simulations is limited by the resolution; in very-high-resolution simulations, the emission region is even thinner, and it occasionally collapses to a current sheet that may source very high energy emission \citep[e.g.,][]{Ripperda20}.
In \autoref{fig:EmissivityProfile}, we investigate the 230~GHz emissivity from the time- and azimuth-averaged \texttt{R17} simulation data.
The true rest frame emissivity used in the radiative transfer (e.g., that given in Appendix A1 of \citealt{Dexter16}) depends on the combined special-relativistic and gravitational redshift of the geodesic at the source, as well as the orientation of the magnetic field with respect to the wavevector in the fluid rest frame.
As a result, it is nontrivial to directly extract an emissivity profile from the time-averaged simulation data that would correspond meaningfully to the time-averaged images generated by \texttt{grtrans}.
Here, we use the proxy for the 230~GHz emissivity defined in the EHT GRMHD code comparison project \citep{CodeComparison}.
This function follows the characteristic behavior of the true synchrotron emissivity \citep[e.g.,][]{Leung11} with the density $\rho$, electron pressure $p_{\rm e}$, and magnetic field strength $|B|$.
The emissivity proxy is
\begin{align}
\label{eq:EmissivityProxy}
j_{\rm sim}=\frac{\rho^3}{p_{\rm e}^2}\exp\br{-C\pa{\frac{\rho^2}{|B|p_{\rm_e}^2}}^{1/3}}.
\end{align}
We follow \citet{CodeComparison} in setting the free constant $C=0.2$ so that the 230~GHz emission is contained within a characteristic radius $r_{\rm em}\leq5\,M$.
From the left panel of \autoref{fig:EmissivityProfile}, it is apparent that near the black hole, the emissivity proxy predicts that emission from the \citet{Chael_19} simulation is predominately located in the equatorial plane.
In the right panel, we extract the simulation emissivity in the equatorial plane $(\theta=\pi/2)$ and compare with the emissivity function we use in the analytic model described in the next section, \autoref{eq:EmissivityProfile}.
The simulation emissivity satisfies two criteria necessary for the lensed equatorial event horizon, or black hole inner shadow, to be visible as an image feature at 230~GHz:
\begin{enumerate}
\item The simulation emissivity is predominately concentrated in the equatorial plane (\autoref{fig:EmissivityProfile}, left panel).
\item The simulation emissivity extends to the event horizon, and is not truncated at any earlier radius such as the innermost stable circular orbit (\autoref{fig:EmissivityProfile}, right panel).
\end{enumerate}
We also investigate the time-averaged simulation velocity profile in the equatorial plane. \autoref{fig:SubKeplerian} shows profiles of the specific angular momentum $\ell\equiv u_\phi/u_t$ and covariant infall velocity $v_r=u_r/u_t$ computed from the average simulation data.
Notably, the average angular momentum in the equatorial plane is significantly below the Keplerian value at all radii \citep[as also seen in, e.g.,][]{NarayanMAD}.
These sub-Keplerian velocities significantly reduce the total Doppler+gravitational redshift factor for emission close to the event horizon.
As a result of this reduced redshift factor, the brightness of the emission falls off less severely near the lensed horizon curve than it would in a Keplerian model with infall inside the ISCO \citep[e.g,][]{Cunningham_75}.
\subsection{Equatorial emission model}
\label{sec:EmissionModel}
Because the time-averaged emissivity of the GRMHD simulation is predominantly equatorial (\autoref{fig:EmissivityProfile}), it is reasonable to compare time-averaged images from this simulation with those from a simple model with the emission confined to the equatorial plane.
\citet{GLM_20} introduced a convenient, analytic model for computing images of equatorial emission around a black hole.
These images are specified by the black hole spin ${a^\ast}$ and observer inclination $\theta_{\rm o}$ (which determine the lensed subring structure),
the four-velocity of the emitting material in the equatorial plane $u^\mu(r)$ (which determines the redshift of the emission), and
the rest-frame emissivity in the equatorial plane $j_{\rm model}(r)$.
The emissivity and velocity are assumed to be constant in azimuth.
In this model, the observed intensity at a point $(\alpha,\beta)$ on the image plane is
\begin{align}
\label{eq:EquatorialModel}
I(\alpha,\beta)=\sum_{n=0}^{N_{\rm max}-1}f_n\,j_{\rm model}(r_n)\,g^3(r_n,\alpha,\beta),
\end{align}
where $r_n=r_\mathrm{eq}(\alpha,\beta;n)$ is the radius at which the geodesic crosses the equatorial plane for the $(n+1)^\text{th}$ time (see \autoref{sec:ComputingCurves}), $N_{\rm max}=N_{\rm max}(\alpha,\beta)$ is the maximum number of equatorial crossings, $j_{\rm model}(r_n)$ is the equatorial emissivity at $r_n$, and $g$ is the redshift factor computed from the emitted photon wavevector $k_\mu$ and the four-velocity $u^\mu$ of the emitting material at radius $r_n$.
The factor $f_n$ is a ``fudge'' that can enhance or diminish the brightness of higher-order rings: we set $f_0=1$ and $f_n=2/3$ for $n>0$ to best match the time-averaged images from the radiative GRMHD simulation.
Note that while \autoref{eq:EquatorialModel} is of the same general form introduced in \citet{GLM_20}, we use a factor of $g^3$ to represent the redshift of the specific 230~GHz intensity (assuming a flat emssion spectrum in \autoref{eq:EmissivityProfile}) rather than the $g^4$ redshift factor they use for bolometric intensity.
We also use a ``fudge'' factor $f_n<1$ for $n>0$, while \citet{GLM_20} uses $f_n=1.5$, as we find it necessary to slightly suppress the contributions from higher-order subrings to match our model images to the time-averaged simulation images used here.
For the emissivity $j_{\rm model}(r)$, we use a second-order polynomial in log-space; that is,
\begin{align}
\label{eq:EmissivityProfile}
\log\br{j_{\rm model}(r)}=p_1\log[r/r_+]+p_2\pa{\log[r/r_+]}^2.
\end{align}
For the 230~GHz images shown throughout this paper, we set $p_1=-2$, and $p_2=-1/2$.\footnote{
Note that for the analytic model to match the change in the image structure with frequency observed in the GRMHD simulation, the emissivity profile parameters must change with the observation frequency; in the 86~GHz images in \autoref{fig:ModelSimulationComparison}, we set $p_1=0,p_2=-3/4$.
}
The overall scale of the emissivity in \autoref{eq:EmissivityProfile} is arbitrary; in computing images of {M87$^{\ast}\xspace$}, we normalize the emission so that the 230~GHz flux density is 0.6~Jy \citep{PaperIII}. The right panel of \autoref{fig:EmissivityProfile} compares the parametrization from \autoref{eq:EmissivityProfile} to the equatorial emissivity profile from the time-averaged GRMHD simulation (\autoref{eq:EmissivityProxy}).
The redshift factor is given by
\begin{align}
g=\frac{-k_t}{k_\mu u^\mu}
=\frac{1}{u^t-\lambda u^\phi\pm u^r\sqrt{\mathcal{R}}/\Delta},
\end{align}
where we assume $u^\theta=0$.
The sign of the $\pm\sqrt{\mathcal{R}}$ term is equal to the sign of the radial component of the null wavevector, $k^r$.
The factor of $g^3$ in \autoref{eq:EquatorialModel} suppresses the $n=0$ emission rapidly with decreasing radius toward the lensed horizon image.
Different models for the velocity $u^\mu$ will feature different rates of suppression, with different implications for how close the observable brightness depression on the sky is to the analytic solution for the inner shadow edge $\rho_{\rm h}(\varphi)$.
While \citet{GLM_20} follow \citet{Cunningham_75} and define the velocity $u^\mu$ to be on Keplerian circular orbits for $r>r_{\rm ISCO}$ and infalling for $r<r_{\rm ISCO}$, we instead model $u^\mu$ with sub-Keplerian angular velocities so as to mimic the characteristic behavior of magnetically arrested disks in GRMHD simulations.
In particular, we use a simple power-law fitting function to the covariant angular momentum $\ell\equiv u_\phi/u_t$, and a broken power-law fitting function to the infall velocity $v_r\equiv u_r/u_t$ derived from the GRMHD simulation:
\begin{align}
\label{eq:Fit1}
\ell&=\ell_{\rm ISCO}\pa{\frac{r}{r_{\rm ISCO}}}^{1/2},\\
\label{eq:Fit2}
v_r&=-V_{\rm ISCO}\pa{\frac{r}{r_{\rm ISCO}}}^{q_1}\br{\frac{1}{2}+\frac{1}{2}\pa{\frac{r}{r_{\rm ISCO}}}^{1/\delta}}^{\delta(q_2-q_1)}.
\end{align}
We set $\ell_{\rm ISCO}=1$, $V_{\rm ISCO}=2$, $q_1=-6$, $q_2=-2$, and $\delta=1/5$.
\autoref{fig:SubKeplerian} compares these fitting functions to the values obtained from the time-averaged GRMHD data.
\subsection{230~GHz and 86~GHz images}
\autoref{fig:ModelSimulationComparison} compares time-averaged images from the {M87$^{\ast}\xspace$} simulation at 230 and 86~GHz with images generated using the modified analytic model described in \autoref{sec:EmissionModel}.
We see that the direct lensed image of the equatorial event horizon is apparent as a deep central brightness depression---the inner shadow---in both the time-averaged images from the simulation and the equatorial model.
The brightness of the emission surrounding the horizon image is suppressed by the gravitational redshift; nonetheless, when looking at the image in a gamma color scale\footnote{
In the gamma scale, $I^\gamma$ is plotted in the linear color scale instead of $I$, where $I$ is the image brightness and we set $\gamma=1/4$.
}
(bottom row of \autoref{fig:ModelSimulationComparison}), the apparent edge of the central brightness depression in the simulation and model image approaches the exact location of the lensed equatorial horizon contour within a microarcsecond.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{./figures/m87_reconstructions-crop.pdf}
\caption{
(Left) Time-averaged GRMHD images blurred to an approximate ngEHT imaging resolution of $10\,\mu$as.
(Middle) Reconstruction of the simulation model from synthetic data generated on EHT2017 baselines.
(Right) Reconstruction of the simulation model from synthetic data generated from an example ngEHT array.
The top row shows images in a linear color scale and the bottom row shows the same images in gamma scale.
In all images, the white curve corresponds to the lensed equatorial horizon, while the cyan contour is the critical curve.
}
\label{fig:Reconstructions}
\end{figure*}
At 86~GHz, the increasing optical depth of the accretion flow washes out the images of the higher-order $(n=1,2,\ldots)$ subrings in the simulation image, except for part of the $n=1$ ring on the north half of the image.
We mimic this effect in the image from the analytic equatorial model by suppressing the higher-order rings and only showing the direct, $n=0$ emission.
Despite the optical depth suppressing the appearance of the lensed subrings, the central ``inner shadow'' depression is still visible at 86~GHz.
This is because in the simulation, the emitting material contributing to the increased total optical depth is still contained within the equatorial plane; the optical depth through the jet material in front of the event horizon remains low.
As a result, the direct geometrical effect of the equatorial emission being truncated at the event horizon is still visible at this frequency.
At lower frequencies ($<\!40$~GHz in this simulation), the jet material becomes optically thick and obscures both the equatorial emission and the inner shadow.
The transition between the optically thick opaque jet and optically thin transparent jet regimes occurs at the frequency above which the image ``core'' no longer moves along the jet, but rather stabilizes at the location of the black hole \citep[][Figure 11]{Hada2016,Chael_19}.
\subsection{Observability with the EHT}
\label{sec:observability}
In \autoref{fig:Slices}, we show profiles from the simulation and analytic model 230~GHz images in \autoref{fig:ModelSimulationComparison} extracted along the $\beta=0$ (red; North-South) and $\alpha=0$ (blue; East-West) axes.
The time-averaged simulation image and the equatorial model image show the same characteristic features: a ring of direct $n=0$ emission that peaks at a radius of $\approx\!15-20\,\mu$as from the origin, $n=1$ and $n=2$ subring images that approach the critical curve at a radius of $\approx\!20\,\mu$as, and a central brightness depression corresponding to the lensed image of the equatorial horizon, i.e., an inner shadow.
The exact position of the horizon image on these slices is indicated by the vertical lines.
The equatorial analytic model has no emission outside the equatorial plane; its brightness plunges toward zero with increasing redshift as the projected radius approaches the direct lensed image $\rho_{\rm h}(\varphi)$ of the equatorial horizon on the sky.
The simulation image features faint foreground emission from the approaching relativistic jet which lies in front of the bulk of the emission in the equatorial plane.
The approaching jet provides a finite brightness ``floor'' inside the main $n=0$ emission ring.
In this simulation, the edges of the floor correspond to the analytic location of the horizon image to within about a microarcsecond.
In other simulations, the exact location of the emission floor will depend on the equatorial emissivity profile, the velocity/redshift of the equatorial fluid, and the intensity of the foreground emission.
The cyan line on \autoref{fig:Slices} indicates the dynamic range of the EHT in 2017; the limited interferometric $(u,v)$ coverage of the array in this first observation of {M87$^{\ast}\xspace$} makes it impossible to extract dim features below $\approx\!10\%$ of the peak brightness \citep{PaperIV}.
The magenta line is an approximate forecast for the dynamic range of the next-generation EHT (ngEHT) array \citep{Astro2020Ground,Raymond21}.
With the addition of new sites and short interferometric baselines, the dynamic range of the ngEHT array should improve to be sensitive to emission that is a factor $10^{-3}$ dimmer than the beam emission.
In this simulation, the emission ``floor'' that fills the lensed horizon image is a factor of $10^{-2}$ dimmer than the peak of the emission.
As a result, in this scenario, we would expect an ngEHT array with improved coverage to be able to directly image the inner shadow feature down to the floor set by the foreground emission.
In \autoref{fig:Reconstructions}, we investigate the ability of the EHT and ngEHT arrays to recover the inner shadow feature with simulated image reconstructions.
We generate synthetic VLBI data from the time-averaged 230~GHz simulation image using the $(u,v)$ coverage on 2017 April 11 \citep{PaperIV}.
We also generate a synthetic ngEHT observation using an example array explored in \citet{Raymond21}.
This ngEHT concept array adds 12 telescopes to the current EHT, dramatically filling in the EHT's $(u,v)$ coverage and increasing its imaging dynamic range.
In both cases, we generated synthetic data including thermal noise and completely randomized station phases from atmospheric turbulence.
We did not include the time-variable amplitude gain errors that complicate real EHT imaging \citep{PaperIII,PaperIV}.
The left column of \autoref{fig:Reconstructions} shows the simulation image blurred to half of the nominal ngEHT resolution at 230~GHz (using a circular Gaussian blurring kernel of $10\,\mu$as FWHM).
The middle column shows the reconstruction from EHT2017 synthetic data, and the right column shows the ngEHT reconstruction.
Both reconstructions were performed using the \texttt{eht-imaging} library \citep{Chael_18}; in particular, the settings used in imaging the 2017 data were the same as those used in \texttt{eht-imaging} in the first publication of the M87 results in \citet{PaperIV}.
While the EHT2017 reconstruction shows a central brightness depression, its size and brightness contrast cannot be constrained or associated with the inner shadow.
However, the increased $(u,v)$ coverage of the ngEHT array dramatically increases the dynamic range, and the image reconstruction recovers the position and size of the high-dynamic-range ``inner shadow'' depression that is visible in the simulation image blurred to the equivalent resolution.
This imaging test is idealized.
First, neither the ngEHT nor EHT2017 directly image the time-averaged structure in {M87$^{\ast}\xspace$}, so an imaging test using a GRMHD snapshot would be more realistic.
However, the inner shadow is prominent in simulation snapshots as well as in the time-averaged image (\autoref{fig:Summary}).
Furthermore, we neglect realistic station amplitude gains and polarimetric leakage factors that complicate image inversion from EHT data.
However, {M87$^{\ast}\xspace$} is weakly polarized, making accurate recovery of the total intensity image possible with no leakage correction \citep{PaperIV,PaperVII}, and image reconstruction of EHT data with even very large amplitude gain factors is possible with a relatively small degradation of the reconstruction quality using \texttt{eht-imaging} \citep{Chael_18}.
This example demonstrates that the candidate ngEHT array from \citet{Raymond21} \emph{could} constrain the presence of an inner shadow in {M87$^{\ast}\xspace$} if it is indeed present in the image.
In particular, detecting this feature does not require dramatic increases in imaging \emph{resolution} (which, in the absence of a 230~GHz VLBI satellite, is limited by the size of the Earth) but by the imaging \emph{dynamic range}, which is limited by the sparse number of baselines in the EHT array.
Once its presence is established via imaging, parametric visibility domain modeling could recover the size and shape of the inner shadow to higher accuracy than is possible from imaging alone \citep[e.g.,][]{PaperVI}.
\section{Geometric description of the lensed horizon image}
\label{sec:GeometricDescription}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./figures/centroid-crop.pdf}
\caption{
Relative centroid displacement $(\Delta\mu_\alpha,\Delta\mu_\beta)$ of the direct equatorial horizon image with respect to the critical curve.
For all values of black hole spin and inclination, $\sign(\Delta\mu_\alpha)=\sign({a^\ast})$ and $\sign(\Delta\mu_\beta)=\sign(\cos{\theta_{\rm o}})$.
An abrupt transition occurs at $\theta_{\rm o}=90\deg$, where the $n=0$ and $n=1$ equatorial horizon images are degenerate.
The mapping $({a^\ast},\theta_{\rm o})\rightarrow (\Delta\mu_\alpha,\Delta\mu_\beta)$ is one-to-one and fairly linear up to high spin and nearly edge-on inclination.
}
\label{fig:Centroid}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.32\textwidth]{./figures/radius2-crop.pdf}
\includegraphics[width=0.32\textwidth]{./figures/orientation2-crop.pdf}
\includegraphics[width=0.32\textwidth]{./figures/eccentricity2-crop.pdf}
\caption{
(Left) The mean radius $\bar{r}_{\rm h}$ of the inner shadow as a function of inclination $\theta_{\rm o}$ for several values of the black hole spin: from top to bottom, ${a^\ast}=(0.01,0.25,0.5,0.75,0.99)$.
(Middle) The orientation angle $\chi_{\rm h}$ with respect to the $+\alpha$ axis as a function of inclination for the same spin values; the orientation angle increases with spin at low and moderate inclination.
(Right) The eccentricity $e_{\rm h}$ for the same spin values; the eccentricity of the inner shadow is nearly independent of spin and depends primarily on the inclination.
}
\label{fig:2ndMoment}
\end{figure*}
In this section, we describe the behavior of the lensed equatorial horizon contour as a function of black hole spin and observer inclination using image moments.
While not a complete description of the horizon image shape (particularly at high inclination), the moment description captures important properties of the horizon image that may be observable by the EHT or future VLBI experiments.
In the procedure outlined in \autoref{sec:ComputingCurves}, we compute the $n=0$ lensed horizon image as a closed curve $\rho_{\rm h}(\varphi)$ in the $(\alpha,\beta)$ image plane.
Given $\rho_{\rm h}$, we can compute image moments in a standard way (explicitly described in \autoref{app:Moments}).
The zeroth moment is the area $A_{\rm h}$ of the inner shadow.
The first moment is the centroid vector $\boldsymbol{\mu}_{\rm h}$ defined with respect to the $(\alpha,\beta)$ axes.
The second central moment is the covariance matrix $\boldsymbol\Sigma_{\rm h}$.
By diagonalizing $\boldsymbol\Sigma_{\rm h}$, we can compute the lengths $a_{\rm h}$ and $b_{\rm h}$ of the principal axes (where $a_{\rm h}\geq b_{\rm h}$), as well as the orientation angle $\chi_{\rm h}$ between the first principal axis and the positive $\alpha$ axis.
We can then define the mean radius $\bar{r}_{\rm h}$ and the eccentricity $e_{\rm h}$ of the lensed horizon as
\begin{align}
\label{eq:MeanRadius}
\bar{r}_{\rm h}=\sqrt{\frac{a_{\rm h}^2+b_{\rm h}^2}{2}},\quad
e_{\rm h}=\sqrt{1-\frac{b_{\rm h}^2}{a_{\rm h}^2}}.
\end{align}
In addition to computing these image moments for the lensed horizon, we also compute the area, centroid, average radius, eccentricity, and orientation angle of the critical curve ($A_{\rm c},\boldsymbol\mu_{\rm c},\bar{r}_{\rm c},e_{\rm c},\chi_{\rm c}$).
Note that our definition of the average critical curve radius $\bar{r}_{\rm c}$ differs from that introduced in \citet{JohannsenPsaltis}.
In \autoref{app:Moments}, \autoref{fig:RadiusComparison}, we compare results for the average critical curve radius from these two methods and find that they agree within one percent for all values of black hole spin and observer inclination.
\subsection{Centroid}
\label{sec:Centroid}
In \autoref{fig:Centroid}, we plot the \emph{relative} centroid displacement between the lensed horizon and the critical curve $\Delta\boldsymbol\mu=\boldsymbol\mu_{\rm h}-\boldsymbol\mu_{\rm c}$.
Measuring the absolute centroid displacement of either the lensed horizon or the critical curve would require precise prior knowledge of the location of the black hole on the sky; by contrast, the relative centroid displacement $\Delta\boldsymbol\mu$ could in principle be observed by simply measuring the two curves and determining the direction of the black hole spin to set the orientation of the $+\beta$ axis (in M87, for instance, these axes can be inferred from the direction of the large-scale jet).
The critical curve is symmetric about the $\beta=0$ axis for all values of spin and inclination, so the vertical displacement $\Delta\boldsymbol\mu_\beta$ is purely due to the offset of the inner shadow's centroid.
The direction of the vertical offset is set by the hemisphere that the observer lies in: $\sign(\Delta\mu_\beta)=\sign(\cos{\theta_{\rm o}})$.
Both the critical curve and the lensed horizon image have a horizontal displacement $\Delta\boldsymbol\mu_\alpha$ that is approximately linear with spin.
The sign of this displacement follows that of the spin: $\sign(\Delta\mu_\alpha)=\sign({a^\ast})$.\footnote{
The projected spin direction is along the $\beta$ axis: a positive spin is aligned with the $+\beta$ axis and a negative spin with the $-\beta$ axis.
}
In general, the mapping $({a^\ast},\theta_{\rm o})\rightarrow(\Delta\mu_\alpha,\Delta\mu_\beta)$ is one-to-one and fairly linear up to high spin and nearly edge-on inclination.
There is an abrupt transition in $\Delta\boldsymbol\mu$ at $\theta_{\rm o}=90\deg$, where the $n=0$ and $n=1$ images are degenerate.
The geometric centroids of the inner shadow and the critical curve are well approximated by
\begin{align}
\mu_{\alpha}&\approx
\begin{cases}
2M{a^\ast}\sin{\theta_{\rm o}}
&\text{critical curve},\\
\frac{1}{2}M{a^\ast}\sin{\theta_{\rm o}}
&\text{equatorial horizon},
\end{cases}\\
\mu_{\beta}&\approx
\begin{cases}
0
&\text{critical curve},\\
\pm\frac{3}{2}M\pa{1-\frac{1}{4}{a^\ast}^2}\sin{\theta_{\rm o}}
&\text{equatorial horizon},
\end{cases}
\end{align}
where $\sign(\mu_\beta)=\sign(\cos{\theta_{\rm o}})$.
For $\sin{\theta_{\rm o}}<1/2$ and $\ab{{a^\ast}}<1/2$, these centroid approximations have a maximum absolute error less than $0.03\,M$.
If the inclination and mass are known a~priori, then it is possible to estimate the spin by
\begin{align}
{a^\ast}\approx-\frac{2}{3\sin{\theta_{\rm o}}}\frac{\Delta\mu_\alpha}{M}.
\end{align}
For instance, if the spin of {M87$^{\ast}\xspace$} is aligned with its large-scale jet, then ${a^\ast}\approx-2.3\Delta\mu_\alpha/M\approx -\frac{\Delta\mu_\alpha}{0.6\,\mu\text{as}}$.
Indeed, based on the jet inclination, we expect that $\ab{\Delta\mu_\alpha}<0.53\,M$ and $0.31\,M<\ab{\Delta\mu_\beta}<0.44\,M$ for {M87$^{\ast}\xspace$}.
The narrow range in allowed $\Delta \mu_\beta$ is useful to assess whether features detected in the image can be associated with the equatorial horizon image or critical curve.
Likewise, if the inclination is unknown but there is an a priori spin estimate, then it is possible to estimate the inclination using
\begin{align}
\sin{\theta_{\rm o}}&\approx\frac{2}{3\pa{1-\frac{1}{4}{a^\ast}^2}}\frac{\ab{\Delta\mu_\beta}}{M}.
\end{align}
Hence, a measured centroid offset along the spin direction $\Delta\mu_\beta$ determines a narrow range of possible inclinations: $\sin{\theta_{\rm o}}\in\br{\frac{2}{3}, \frac{8}{9}}\ab{\Delta \mu_\beta}/M$.
For instance, measuring an offset $\Delta\mu_\beta=1.5\,\mu$as in {M87$^{\ast}\xspace$} would give a constraint $15\deg<\theta_{\rm o}<21\deg$.
\subsection{Radius, orientation, eccentricity}
\autoref{fig:2ndMoment} shows the variation of the quantities that define the second moment of the lensed horizon image---the mean radius $\bar{r}_{\rm h}$, the orientation angle $\chi_{\rm h}$, and the eccentricity $e_{\rm h}$---with the inclination $\theta_{\rm o}$ (in the range $0\leq\theta_{\rm o}\leq\pi/2$) for several values of the black hole spin ${a^\ast}$.
At low inclinations ($\theta_{\rm o}\lesssim30\deg$), the mean radius $r_{\rm h}$ and image orientation angle $\chi_{\rm h}$ are approximately independent of the inclination and hence directly probe the spin.
By contrast, the eccentricity $e_{\rm h}$ is almost entirely independent of spin over the whole range and thus provides a direct probe of the inclination.
Measuring $e_{\rm h}$ or $\chi_{\rm h}$ at inclinations $\theta_{\rm o}\lesssim30\deg$ would require extremely high precision measurements of the lensed horizon shape; since the eccentricity $e_{\rm h}<0.4$ for these inclinations, the relative sizes of the major and minor image axes differ by $\lesssim 8$\%.
At these low inclinations, the mean image radius varies by $\approx\!20$\% from zero to maximal spin.
\autoref{fig:RelativeRadius} shows, for several fixed values of the black hole spin, the dependence on inclination of the ratio $\bar{r}_{\rm h}/\bar{r}_{\rm c}$ of the lensed horizon mean radius to the critical curve mean radius.
Again, for $\theta_{\rm o}\lesssim30\deg$, $\bar{r}_{\rm h}/\bar{r}_{\rm c}$ is approximately independent of the inclination and hence provides a direct measurement of the spin.
In the low inclination case, $\bar{r}_{\rm h}/\bar{r}_{\rm c}$ shrinks from $\approx\!55$\% at zero spin to $\approx\!45$\% at maximal spin.
Importantly, measuring $\bar{r}_{\rm h}/\bar{r}_{\rm c}$ for an astrophysical black hole would \emph{not} require accurate measurements of the black hole mass $M$ or distance $D$.
\begin{figure}[t]
\includegraphics[width=0.4\textwidth]{./figures/radius_comparison-crop.pdf}
\caption{
The ratio of the mean radius of the lensed horizon $\bar{r}_{\rm h}$ to the mean radius of the critical curve $\bar{r}_{\rm c}$. In the low-inclination case, $\bar{r}_{\rm h}/\bar{r}_{\rm c}$ shrinks from $\approx\!55$\% at zero spin to $\approx\!45$\% at maximal spin.
}
\label{fig:RelativeRadius}
\end{figure}
Referencing the lensed horizon image directly to the critical curve would require detecting a lensed subring of order $n\gtrsim1$, which for {M87$^{\ast}\xspace$} only becomes visible on very high resolution baselines $\gtrsim\!20$~G$\lambda$ \citep{Johnson_Ring}.
However, it should be possible to constrain the location of the critical curve with measurements of the $n=0$ and $n=1$ rings by determining systematic calibration factors (and associated systematic uncertanties) that relate size of the EHT image to the critical curve size in a library of astrophysical models, as was done in \citet{PaperV,PaperVI} to measure the mass of {M87$^{\ast}\xspace$}.
Alternatively, parametric modeling with priors on the image structure may constrain the $n=1$ and higher subring images from measurements at lower spatial frequencies rings using parametric model fits \citep{Broderick20}.
For all spins at low and moderate inclinations, the lensed horizon image is well approximated by an ellipse, and the first three image moments give a fairly complete description of the curve.
At higher inclinations, the structure becomes more complex, with more information in the full curve shape than is captured in the first three moments.
In particular, at ${a^\ast}=0$ and $\theta_{\rm o}=90\deg$, the horizon image becomes a semicircle, degenerate with the $n=1$ horizon image semicircle in the other half plane (at higher spins, the image is not a perfect semicircle, but is still a mirror image of the degenerate $n=1$ image; see \autoref{fig:imgrid}).
In this case, the curve is probably better described by the single radius of the combined $n=0$ and $n=1$ circle, rather than the second moment parameters of just the $n=0$ image.
\begin{figure*}[t]
\centering
\includegraphics[width=0.45\textwidth]{./figures/rh_rc-crop.pdf}\hfill
\includegraphics[width=0.45\textwidth]{./figures/da_db-crop.pdf} \\
\caption{
(Left) Simultaneous constraints on the black hole mass-to-distance ratio $M/D$ and spin ${a^\ast}$ enabled by measuring the mean radius of the lensed horizon (blue, $\bar{r}_{\rm h}$) and critical curve (red $\bar{r}_{\rm c}$) in {M87$^{\ast}\xspace$}, when the inclination is fixed $\theta_{\rm o}=17\deg$ \citep{Mertens2016}.
Without fixing the mass, multiple values of ${a^\ast}$ provide the same result for the size of each feature, but combining a measurement of both features breaks the degeneracy.
The shaded regions show errors on the radius measurement of $0.1$, $0.5$, and $1\,\mu$as.
The input mass scale and spin are $M/D=3.78\,\mu$as and ${a^\ast}=0.94$.
(Right) Simultaneous constraints on $M/D$ and ${a^\ast}$ from measurements of the centroid offset $\boldsymbol\mu_{\rm h}-\boldsymbol\mu_{\rm c}$ in the $\alpha$ direction (green) and the $\beta$ direction (orange).
}
\label{fig:SpinMassConstraint}
\bigskip
\centering
\includegraphics[width=0.45\textwidth]{./figures/rh_rc_SgrA-crop.pdf}\hfill
\includegraphics[width=0.45\textwidth]{./figures/da_db_SgrA-crop.pdf}
\caption{
(Left) Simultaneous constraints on the black hole spin ${a^\ast}$ and inclination $\theta_{\rm o}$ enabled by measuring the mean radius of the lensed horizon (blue, $\bar{r}_{\rm h}$) and critical curve (red $\bar{r}_{\rm c}$) in {Sgr~A$^{\ast}\xspace$}, fixing the mass-to-distance ratio $M/D=5.01\,\mu$as \citep{Gravity2019}.
The shaded regions show errors on the radius measurement of $0.1$, $0.5$, and $1\,\mu$as.
The input spin and inclination are ${a^\ast}=0.75$ and $\theta_{\rm o}=30\deg$.
(Right) Simultaneous constraints on ${a^\ast}$ and $\theta_{\rm o}$ from measurements of the centroid offset $\boldsymbol\mu_{\rm h}-\boldsymbol\mu_{\rm c}$ in the $\alpha$ direction (green) and the $\beta$ direction (orange).
The absolute errors in the centroid measurement depicted by the shaded regions are the same as in the left panel.
When $M/D$ is fixed, the relative sizes of the lensed horizon and critical curve poorly constrain $\theta_{\rm o}$, but the centroid offset strongly constrains both ${a^\ast}$ and $\theta_{\rm o}$.
}
\label{fig:SpinInclinationConstraint}
\end{figure*}
\autoref{fig:SpinMassConstraint} demonstrates how a simultaneous measurement of the radius of the critical curve and the lensed horizon could be used to constrain the mass and spin in {M87$^{\ast}\xspace$} when the inclination is fixed at $\theta_{\rm o}=17\deg$ \citep{Mertens2016}.
These simultaneous constraints are analogous to those discussed in \citet{Broderick21}, which considers constraints from measuring multiple lensed images from a single face-on emitting ring.
The blue line shows the space of mass-to-distance ratios $M/D$ and spins ${a^\ast}$ that give the same mean lensed horizon radius for an image of {M87$^{\ast}\xspace$}; the red line shows the same for the critical curve.
The red and blue lines intersect in only one location corresponding to the input black hole mass $M/D=3.78\,\mu$as and spin ${a^\ast}=0.94$.
The shaded bands around the intersecting lines show absolute errors in the radius measurements of $0.1,0.5,1\,\mu$as.
Given a reported EHT radius measurement uncertainty of $1.5\,\mu$as from geometric modeling of the EHT2017 data in \citet{PaperVI}, measurements of the ring and inner shadow radius and centroid locations at $\lesssim\!1\,\mu$as precision may be feasible with the ngEHT.
In addition to reducing uncertainty in the image size measurement itself, precisely constraining both features will depend on reducing systematic uncertainty in the relationship between the gravitational features and images from a set of plausible astrophysical models \citep[e.g.,][]{PaperVI}.
The right panel of \autoref{fig:SpinMassConstraint} shows a similar figure for a simultaneous measurement of the centroid offset along the $\alpha$ ($\Delta\mu_\alpha$, in green) and $\beta$ axes ($\Delta\mu_\beta$, in orange).
Because the centroid offsets are relatively small, an absolute error of $1\,\mu$as in the measurement of the centroid offsets is less constraining than the corresponding radius measurement.
However, measuring the offset $\Delta\mu_\alpha$ to $1\,\mu$as precision could put a lower limit on the spin ${a^\ast}\gtrsim0.5$ independent of the mass.
Measurements that jointly constrain the image size and eccentricity could even more precisely constrain the mass and spin in {M87$^{\ast}\xspace$}.
In {Sgr~A$^{\ast}\xspace$}, the mass-to-distance ratio is known to high precision, $M/D=5.011\pm0.016\,\mu$as \citep{Gravity2019}, but the inclination is unconstrained.
In \autoref{fig:SpinInclinationConstraint}, we show similar simultaneous constraints on $\theta_{\rm o}$ and ${a^\ast}$ for {Sgr~A$^{\ast}\xspace$} from measurements of the inner shadow and critical curve radii (left panel) and centroid offset (right panel).
The inner shadow and critical curve radii poorly constrain the inclination, and constraining the spin with these radii and an unknown inclination requires a measurement precision finer than $1\,\mu$as.
By contrast, as discussed in \autoref{sec:Centroid}, the centroid offset is highly constraining of both $\theta_{\rm o}$ and ${a^\ast}$ if $M/D$ is known (right panel of \autoref{fig:SpinInclinationConstraint}).
Note that the constraints shown in the right panels of \autoref{fig:SpinMassConstraint} and \autoref{fig:SpinInclinationConstraint} require the $\alpha$ and $\beta$ axes to be known a priori; in practice, this could be informed with reference to the jet at large scales in {M87$^{\ast}\xspace$}, or by reference to the location of the brightness asymmetry from Doppler beaming in 230~GHz images of {M87$^{\ast}\xspace$} or {Sgr~A$^{\ast}\xspace$}.
\autoref{fig:SpinMassConstraint} and \autoref{fig:SpinInclinationConstraint} are idealized examples; in practice, the orientation of the $\alpha$ and $\beta$ axes would likely have to be fit to observations along with ${a^\ast}$, $\theta_{\rm o}$, and other model parameters that relate the critical curve and lensed equatorial horizon shapes to their corresponding features in the observed image (e.g., the emissivity and redshift profiles in the model described in \autoref{sec:EmissionModel}).
\subsection{Semi-analytic description of the lensed horizon}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.33\textwidth]{./figures/fit1-crop.pdf}
\includegraphics[width=0.33\textwidth]{./figures/fit2-crop.pdf}
\includegraphics[width=0.33\textwidth]{./figures/fit3-crop.pdf}
\caption{
Examples of the semi-analytic fitting function $\rho_{\rm h, fit}(x)$ (\autoref{eq:FittingFunction}) to the lensed equatorial horizon curve for a black hole with spin ${a^\ast}=0.5$ viewed at inclination $\theta_{\rm o}=10\deg$ (left), $45\deg$ (middle), and $80\deg$ (right).
In each panel, we indicate the ``unlensed'' image of the black hole event horizon $r=r_+$ (black dashed line), the direct lensed image of the equatorial horizon (black solid line), the critical curve (cyan solid line), and the approximate parameterization of the horizon image $\rho_{\rm h, fit}(x)$ (dashed magenta line).
The approximate parameterization matches the horizon image well at low inclination, but deviates from the truth in the $\beta<0$ part of the curve at high inclination.
}
\label{fig:FitCurve}
\end{figure*}
Here, we attempt to derive an approximate parameterization for the general shape of the lensed equatorial event horizon, starting with a nonrotating (${a^\ast}=0$) Schwarzschild black hole.
As is the case for the critical curve, such approximations are useful for easily exploring the behavior of the shape with spin and inclination and to identify parameter degeneracies \citep[e.g.,][]{devries_2003,Cunha_2018,Farah_2020,GL20}.
\cite{Gates2020} provide an analytic formula for the equatorial radius $r_{\rm eq}(\rho,\varphi,\theta_{\rm o};n)$ that a light ray shot back from position $(\rho,\varphi)$ on the image plane will cross on its $(n+1)^\text{th}$ equatorial crossing.
While the dependence of this transfer function on the impact radius $\rho$ is rather complicated, its dependence on the impact angle $\varphi$ and observer inclination $\theta_{\rm o}$ only enter through the particular combination
\begin{align}
x=\arctan\br{\sin\pa{\varphi}\tan\pa{\theta_{\rm o}}}.
\end{align}
In this paper, we are only interested in the direct image $n=0$.\footnote{
Lensed images with higher $n$ lie in the photon ring, and one may set up a large-$n$ expansion as in \cite{GrallaLupsasca2020} to obtain a simple asymptotic formula for $r_{\rm eq}(\rho,\varphi;n)$ given in App.~A of \cite{Hadar2020}.}
It is tempting to expand
\begin{align}
\label{eq:EquatorialRadius}
r_{\rm eq}(\rho,x;n=0)=Mf_0(\rho)-Mf_1(\rho)x+\O{x^2},
\end{align}
where $Mf_0(\rho)=r_{\rm eq}(\rho,x=0;n=0)$ is the exact, axisymmetric transfer function for a polar observer at $x=\theta_{\rm o}=0$.
\cite{Gates2020} derive its asymptotic expansion in large impact radius $\rho\to\infty$,
\begin{align}
\label{eq:f0}
f_0(\rho)=\frac{\rho}{M}-1+\frac{1}{2}\frac{M}{\rho}+\frac{3(5\pi-16)}{4}\frac{M^2}{\rho^2}+\O{\frac{M^3}{\rho^3}}.
\end{align}
\autoref{eq:f0} is already an excellent approximation even when truncated after the first two terms \citep{GrallaLupsasca2020}.
Likewise, $f_1(\rho)$ admits the expansion
\begin{align}
f_1(\rho)=2+\frac{(15\pi-64)}{8}\frac{M}{\rho}-\frac{(15\pi-50)}{2}\frac{M^2}{\rho^2}+\O{\frac{M^3}{\rho^3}}.
\end{align}
Inverting Eq.~\eqref{eq:EquatorialRadius} results in an expression of the form
\begin{align}
\rho(x; r_{\rm eq},n=0)=g_0(r_{\rm eq})+g_1(r_{\rm eq})x+\O{x^2},
\end{align}
which unfortunately is not as good an approximation as its inverse \eqref{eq:EquatorialRadius}.
In particular, this expansion breaks down at large inclination, where the higher-order terms in $x$ grow more relevant.
Nonetheless, we empirically observe that an excellent semi-analytic fit to the lensed equatorial horizon ($r_{\rm eq}=r_+$) for Schwarzschild ($r_+=2M$) is provided by the expression
\begin{align}
\rho_{\mathrm{h},{a^\ast}=0}(x)\approx M\br{2\sqrt{2}+\pa{1+\frac{1}{2}\cos^2{\theta_{\rm o}}}x}.
\end{align}
Note that the Schwarzschild approximation $g_0(r_+)=2\sqrt{2}M$ follows from an analytic approximation for Schwarzschild geodesics given by \cite{Beloborodov2002}.
Finally, we note that this expression can be simply extended to the rotating Kerr case with nonzero spin ${a^\ast}>0$ to obtain a general fitting function $\rho_{\rm h, fit}(\varphi)$ for the shape of the direct lensed equatorial horizon image:
\begin{align}
\label{eq:FittingFunction}
\rho_{\rm h,fit}(x)=M\left[2\sqrt{\frac{r_+}{M}}+\pa{1+\frac{1}{2}\cos^2{\theta_{\rm o}}}x\right].
\end{align}
In particular, for a face-on observer, the radius of the lensed equatorial horizon is well fit by
\begin{align}
\rho_{\rm h}(\theta_{\rm o}=0)\approx2M\sqrt{r_+/M}.
\end{align}
\autoref{fig:FitCurve} shows the quality of the approximation $\rho_{\rm h, fit}(x)$ (\autoref{eq:FittingFunction}) to the lensed equatorial horizon image for a black hole of spin ${a^\ast}=0.5$ at several inclinations.
The fitting function breaks down at high inclinations, where it develops an artifical ``bump'' on the part of the curve below the $\alpha$ axis.
\section{Discussion}
\label{sec:discussion}
\begin{figure*}[t]
\centering
\begin{tabular}{ccc}
\includegraphics[width=\textwidth]{./figures/sgra_panels-crop.pdf}
\end{tabular}
\caption{
Time-averaged 230~GHz images of a {Sgr~A$^{\ast}\xspace$} simulation with weak magnetic flux and ${a^\ast}=0.9375$ at different inclinations: from left to right, $\theta_{\rm o}=(10,30,50,70,90)\deg$.
The top row shows images in a linear scale, and the bottom row displays images in a gamma scale with $\gamma=1/4$.
The different panels are each normalized to their own peak brightness, since the total flux density increases with increasing inclination.
The critical curve is indicated in cyan, the direct image of the equatorial horizon is indicated by the solid white curve, and the $n=1$ image of the equatorial horizon is displayed with the dashed white curve.
In this low-magnetic-flux simulation, the 230~GHz emission is concentrated in a geometrially thick disk extending to the horizon.
The lensed equatorial horizon image is visible at low inclinations, but is blocked by direct emission from the disk when viewed at edge-on inclinations.
}
\label{fig:SANE}
\end{figure*}
Images from the radiative GRMHD simulations of {M87$^{\ast}\xspace$} from \citet{Chael_19} display a deep central brightness depression.
For these simulations of synchrotron-emitting plasma around a Kerr black hole, the edge of this image feature is the lensed image of the event horizon's intersection with the equatorial plane; the brightness depression is the ``inner shadow'' of the black hole.
Because the gravitational redshift diverges for emission approaching the event horizon, this inner shadow is only visible at high dynamic range in the simulation images.
Nonetheless, if it is present in {M87$^{\ast}\xspace$} images, this feature could be observable with the next-generation Event Horizon Telescope, which will increase the dynamic range of the current EHT by a factor of $\approx\!100$.
An inner shadow bounded by the image of the equatorial event horizon is only visible in images of a black hole if the emission region
\begin{enumerate}
\item is concentrated in the equatorial plane,
\item is not obscured by foreground emission (e.g., from the approaching jet), and
\item extends to the event horizon.
\end{enumerate}
This scenario is realized by 230~GHz emission from the MAD simulations of {M87$^{\ast}\xspace$} considered in this paper; however, there are other plausible scenarios for the distribution of emitting plasma.
If the emission is spherically symmetric and extends to the horizon, the direct outline of the horizon is not visible; the only brightness depression in this case occurs inside the critical curve \citep[e.g.,][]{Falcke2000,Narayan_Shadow}.
Because GRMHD simulations of hot accretion flows produce geometrically thick accretion disks, one might think that these simulations are better approximated by a spherically symmetric emissivity distribution than an equatorial model.
On the contrary, however, even geometrically thick accretion disks in GRMHD simulations are better approximated by the equatorial emission model than a spherically symmetric one.
\autoref{fig:SANE} shows time-averaged images from a GRMHD simulation of {Sgr~A$^{\ast}\xspace$} in the low-magnetic-flux accretion state at several observer inclinations $\theta_{\rm o}$.
At low inclination, the images from this simulation look qualitatively similar to the magnetically arrested {M87$^{\ast}\xspace$} simulation images in \autoref{fig:ModelSimulationComparison}; they display both a bright photon ring near the critical curve, and an inner shadow marking the equatorial event horizon.
Despite the effects of significant disk thickness, the horizon image is still visible up to moderate inclinations ($\theta_{\rm o}\approx50\deg$) in this simulation.
At higher inclinations, emission from the disk in front of the black hole blocks the appearance of the horizon inner shadow, and it is not visible.
Some GRMHD simulations produce 230~GHz emission predominantly in the black hole jet or along the ``jet sheath'' \citep[e.g.,][Figure 4]{PaperV}.
Because the observed emission in these simulations is not predominantly equatorial, the inner shadow seen in the simulations presented in this paper will not be present in this scenario.
Funnel emission in \citet{PaperV} is most often seen in simulations with weak magnetic flux, while \citet{PaperVIII} shows that polarimetric EHT observations strongly prefer MAD models with strong magnetic flux for {M87$^{\ast}\xspace$}.
While the MAD models in this paper and in \citet{PaperV} show 230~GHz emission that originates predominantly from the equatorial plane, a larger survey over different GRMHD images should be performed to assess how generic this behavior is with respect to different simulation parameters and electron heating/acceleration models.
Even in the \citet{Chael_19} MAD models explored here, in which the 230~GHz emission is predominantly equatorial, nonzero brightness from the forward jet adds a finite ``floor'' to the inner shadow (\autoref{fig:Slices}).
While this emission is dim at 230~GHz and even 86~GHz in these simulations, at even lower frequencies, the forward jet becomes optically thick and eventually obscures the equatorial emission, and hence the inner shadow.
Furthermore, the precise size of the observed brightness depression in these models depends on where the foreground emission becomes as bright as emission from the equatorial plane, which falls off rapidly due to the increasing gravitational redshift incurred as the emission radius approaches the horizon.
As a result, in these models, the area of the observed brightness depression is larger than the inner shadow of the equatorial emission.
By contrast, emission from a thick disk without forward jet emission could produce a central brightness depression with a slightly smaller area than the equatorial inner shadow, as the emission region intersects with the event horizon at a higher latitude above the equatorial plane.
The GRMHD simulation considered in this paper assumes that the plasma angular momentum is aligned with the black hole spin; however, alignment may not be generic in black hole accretion flows.
Relatively few ``tilted'' or misaligned disk GRMHD simulations have been conducted \citep[e.g.,][]{DexFrag,White19,Chaterjee20}.
Both \citet{White19} and \citet{Chaterjee20} found that in misaligned simulations, new image features can emerge due to shocks and gravitational lensing that change the appearance of the 230~GHz image relative to the images seen from aligned-disk simulations.
In particular, tilted simulations can show peak brightness contours that are farther from the black hole (and at shifted azimuthal angles) compared to what is typically seen in aligned simulations.
Because our definition of the inner shadow in this paper requires equatorial emission, the feature as discussed in this paper would not be visible if the accretion disk feeding the black hole is tilted.
However, if the emission extends to the horizon, a similar inner shadow---corresponding to the intersection of the event horizon with an inclined plane---may be present in images from tilted disks.
\citet{Chaterjee20} show time-averaged simulation images at several observing frequencies from their tilted-disk MAD simulations (their figure 6).
In all of these images from \citet{Chaterjee20}, there is a prominent central brightness depression inside the photon ring that most likely marks the event horizon image from emission concentrated in an inclined plane.
In a forthcoming work, we will investigate the visibility of the lensed horizon image in tilted disk simulations, and explore the dependence of the shape and size of the horizon image when varying the orientation $\theta$ of the emission plane.
While we have focused on the Kerr metric, exotic compact objects with horizons will also produce an analogous inner shadow.
For instance, \citet{Mizuno_2018} present GRMHD simulations of a dilaton black hole, which exhibits a prominent inner shadow (their figure 2).
The parametrized non-Kerr metrics explored in \citet{Medeiros20} all have well-defined horizons and so will likely display an inner shadow feature when illuminated by emitting plasma in the equatorial plane.
Thus, observing an inner shadow feature from {M87$^{\ast}\xspace$} or {Sgr~A$^{\ast}\xspace$} would not constrain the metric to be Kerr; nevertheless, if the properties of the emission region are well-constrained, then the relationship between the inner shadow and the photon ring can be used as a null hypothesis test of the Kerr metric.
For instance, in Kerr, an inner shadow must have diameter $0.45<\bar{r}_{\rm h}/\bar{r}_{\rm c}<0.7$ (see \autoref{fig:RelativeRadius}).
Non-Kerr metrics such as those explored in \citet{Medeiros20} may show different relationships between the inner shadow and critical curve sizes and shapes that depend on the deviation parameters, enabling these parameters to be constrained in fits to future observations.
In addition, horizonless compact objects may also exhibit a feature analogous to the inner shadow if their interior is evacuated \citep[see, e.g.,][]{Vincent_2016,Olivares_2020}; however, in this case, the properties of the inner shadow will be primarily determined by the structure of the emission region rather than the gravitational lensing of the compact object.
\section{Conclusions}
\label{sec:Conclusions}
In this paper, we have examined the appearance of the direct image of a Kerr black hole's equatorial event horizon.
This lensed feature always lies within the critical curve and is sensitive to the black hole spin and viewing inclination \citep{Takahashi,DN20b}.
Using GRMHD and analytic models of the submillimeter emission from {M87$^{\ast}\xspace$}, we have shown that:
\begin{itemize}
\item The direct ($n=0$) lensed image of the equatorial event horizon marks the boundary of an ``inner shadow'' that should be observable in future images of {M87$^{\ast}\xspace$} if the submillimeter or radio emission is both equatorial and extends to the horizon.
\item The radiative GRMHD simulations of {M87$^{\ast}\xspace$} from \citet{Chael_19} have emissivity profiles that extend to the horizon and sub-Keplerian flows that result in a relatively weak redshift near the horizon. The ``inner shadow'' is prominent in images of these simulations.
\item Analytic equatorial emission models show some of the main features we see in time-averaged images from these GRMHD simulations, including the photon ring structure and the inner shadow.
\item The ngEHT should have the dynamic range necessary to observe the inner shadow in {M87$^{\ast}\xspace$}, if it is present.
This feature could also be visible at other frequencies, even if the optical depth of the accretion disk is high at those frequencies (as it is in the GRMHD simulation we consider at 86~GHz).
\item The radius and centroid offset of the direct lensed equatorial horizon image can be used to measure the black hole spin and inclination, and to break degeneracies in estimating both the black hole mass and spin from only one image feature.
\item The presence and observability of this feature in {M87$^{\ast}\xspace$} is contingent on the emission being predominantly equatorial and extending to the event horizon.
If instead the accretion disk is tilted, then there may be an analogous low-brightness feature corresponding to an image of the event horizon's intersection with an inclined plane.
Non-Kerr spacetimes may produce an inner shadow with a different size relative to the photon ring than predicted in Kerr, potentially enabling tests of the spacetime using both features.
\end{itemize}
\acknowledgments
The authors thank Pierre Christian for serving as the EHT collaboration internal referee for this paper; his comments significantly improved the manuscript.
AC is supported by Hubble Fellowship grant HST-HF2-51431.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.
MDJ was supported by the National Science Foundation (AST-1716536, AST-1935980) and the Gordon and Betty Moore Foundation (GBMF-5278).
AL gratefully acknowledges support from the Jacob Goldfield Foundation and from Will and Kacie Snellings.
This work used the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF grant ACI-1548562. XSEDE Stampede2 resource at TACC was allocated through grant TG-AST190053 and TG-AST080026N. |
1801.00299 | \section{Notation and preliminaries}
Lower indices will denote different matrices, while upper indices will denote elements of a matrix. \emph{Bar} as in $\ov{\sigma}$ will denote the complex conjugate, upper index $T$ as in $\sigma^T$ will denote transpose, and $\dag$ as in $\sigma^\dag$ will denote conjugate transpose. $\partial_i\equiv\partial_{\epsilon_i}$ denotes partial derivative with respect to $i$'th element of the vector of estimated parameters $\be=(\epsilon_1,\epsilon_2,\dots)$, $\otimes$ denotes the Kronecker product, $[\cdot,\cdot]$ denotes the commutator, $\{\cdot,\cdot\}$ denotes the anti-commutator, $\tr[\cdot]$ denotes trace of a matrix, and $\vectorization{\cdot}$ denotes vectorization of a matrix, which is defined as a column vector constructed from columns of a matrix as
\[\label{eq:vectorizationdef}
A=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}, \quad
\vectorization{A}=\begin{pmatrix}
a \\
c \\
b \\
d \\
\end{pmatrix}.
\]
We consider a Bosonic system with a set annihilation and creation operators $\{\hat{a}_n,\hat{a}_n^\dag\}$. We collect them into a vector of operators $\bA:=(\hat{a}_1,\dots,\hat{a}_N,\hat{a}_1^\dag,\dots,\hat{a}_N^\dag)^T$, where $N$ denotes the number of modes. Now we can write commutation relations in an elegant form, $[\bA^{m},\bA^{n\dag}]=K^{mn}\mathrm{id}$, where $\mathrm{id}$ denotes the identity operator and
\[\label{def:K}
K=
\begin{bmatrix}
I & 0 \\
0 & -I
\end{bmatrix}
\]
is a constant matrix called \emph{the symplectic form}, and $I$ denotes the identity matrix.
In a Bosonic system\footnote{Although Gaussian states are also defined for Fermionic systems, see e.g. Ref.~\cite{carollo2018symmetric}.} one can define a special class of continuous variable states called \emph{Gaussian states}~\cite{Weedbrook2012a}, $\hat{\rho}$, which are fully characterized by its first moments $\bd^m=\mathrm{tr}\big[\R\bA^m\big]$ (\emph{the displacement vector}) and the second moments $\sigma^{mn}=\mathrm{tr}\big[\hat{\rho}\,\{\dbA^m,\dbA^{n\dag}\}\big]$ (\emph{the covariance matrix}), where $\dbA:=\bA-\bd$. In this form, $\sigma^\dag=\sigma$, and the moments have the following structure,
\[\label{def:first_and_second_moments}
\bd=
\begin{bmatrix}
\boldsymbol{\bg} \\ \overline{\boldsymbol{\bg}}
\end{bmatrix},\quad
\sigma\,=\,\begin{bmatrix}
X & Y \\
\overline{Y} & \overline{X}
\end{bmatrix}.
\]
These definitions are known as the complex form, and we will use this convention throughout this paper. This description is equivalent to the real form used by some authors~\cite{braunstein2005quantum,Weedbrook2012a,Adesso2014a}. We show how to switch between these descriptions in Appendix~\ref{app:mixed_state}. Both the real and the complex form phase-space represenations of common Gaussian unitaries and Gaussian states are shown in Appendix~\ref{sec:The_phase_space_Gaussian_unitaries}. For more information on the real and the complex form see for example~\cite{Arvind1995a,Safranek2015b,safranek2016optimal,vsafranek2016gaussian}.
The QFIM is defined as~\cite{Paris2009a}
\[\label{eq:qfi}
H^{ij}\equiv\tfrac{1}{2}\tr[\R\{\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j\}],
\]
where \emph{symmetric logarithmic derivatives} (SLDs) $\hat{\mathscr{L}}_i$ are defined as operator solutions to equations
\[\label{eq:sld}
\tfrac{1}{2}\{\R,\hat{\mathscr{L}}_i\}=\partial_i\R.
\]
The quantum Cram\'er-Rao bound gives a lower bound on the covariance matrix of estimators,
\[
\mathrm{Cov}(\hat{\be})\geq H^{-1},
\]
meaning that matrix $\mathrm{Cov}(\hat{\be})- H^{-1}$ is a positive semi-definite matrix. This bound is can be saturated when~\cite{szczykulska2016multi,ragy2016compatibility}
\[\label{eq:condition_for_CRbound}
\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]=0,
\]
and the optimal measurement basis is given by the eigenvectors of the SLDs.
\section{Results}
\textit{\textbf{Mixed states}}
In tasks where we know the covariance matrix and displacement vector of a Gaussian state, it is possible to use results derived in~\cite{Gao2014a} to calculate the QFIM and the symmetric logarithmic derivatives. These results can be expressed in an elegant matrix form: for a Gaussian state $(\bd,\sigma)$, the QFIM and symmetric logarithmic derivatives can be calculated as
\begin{align}
H^{ij}(\be)&=\frac{1}{2}\vectorization{\partial_i\sigma}^\dag\mathfrak{M}^{-1}\vectorization{\partial_j\sigma}+2\partial_i\bd^\dag\sigma^{-1}\partial_j\bd,\label{eq:mixed_QFI}\\
\hat{\mathscr{L}}_i(\be)&=\dbA^\dag\cA_i\dbA-\frac{1}{2}\tr[\sigma\cA_i]+2\dbA^\dag\sigma^{-1}\partial_i{\boldsymbol{d}},\label{eq:SLD_mixed}
\end{align}
where
\[
\mathfrak{M}=\ov{\sigma}\otimes\sigma-K\otimes K,\quad
\vectorization{\cA_i}=\mathfrak{M}^{-1}\vectorization{\partial_i\sigma}.
\]
These results represent the Gaussian version of the expressions published in Ref.~\cite{vsafranek2018simple}.
The above formulas require differentiating the displacement vector and the covariance matrix, and inverting two matrices. However, they cannot be used when at least one of the modes is in a pure state (at least not without any modification), because $\mathfrak{M}$ is not invertible in that case. We will discuss this issue later, and show how this can be resolved (see Eq.~\eqref{eq:any_state_QFIM}).
Proof of how results of~\cite{Gao2014a} transform into this elegant matrix form can be found in Appendix~\ref{app:mixed_state}.
Further, we derive expression that determines saturability of the quantum Cram\'er-Rao bound,
\[\label{eq:condition_QCR}
\begin{split}
\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]&\!=\!\vectorization{\partial_i\sigma}^\dag\mathfrak{M}^{-1}(\ov{\sigma}\!\otimes\! K\!-\!K\!\otimes\!\sigma)\mathfrak{M}^{-1}\vectorization{\partial_j\sigma}\\
&+4\partial_i\bd^\dag\sigma^{-1}K\sigma^{-1}\partial_j\bd.
\end{split}
\]
The derivation can be found in Appendix~\ref{app:saturability}.
\textit{\textbf{When the Williamson's decomposition of the covariance matrix is known}}
There are situations when we know the Williamson's decomposition of the covariance matrix. This is for example when we have control over preparation of the initial state, and we are planning to estimate parameters encoded into this state by a Gaussian unitary channel. This is a case when it is useful to simplify our calculations of QFIM by making use of this decomposition, without the actual need of calculating this decomposition. In other cases, it might be still numerically more efficient to calculate this decomposition, instead of inverting the large matrix $\mathfrak{M}$ needed for Eq.~\eqref{eq:mixed_QFI}.
According to the Williamson's theorem~\cite{Williamson1936a,deGosson2006a,simon1999congruences} any positive-definite matrix can be diagonalized by symplectic matrices, $\sigma=SDS^\dag$. We show how to do that explicitly in Appendix~\ref{app:diagonalization}. $D=\mathrm{diag}(\lambda_1,\dots,\lambda_N,\lambda_1,\dots,\lambda_N)$ is a diagonal matrix consisting of \emph{symplectic eigenvalues}, which are defined as the positive eigenvalues of the matrix $K\sigma$. It follows from Heisenberg uncertainty relations that for all $k$, $\lambda_k\geq 1$. $S$ is a \emph{symplectic matrix} satisfying the defining property of the complex form of the real symplectic group $Sp(2N,\mathbb{R})$,
\[\label{def:structure_of_S}
S=
\begin{bmatrix}
\A & \B \\
\ov{\B} & \ov{\A}
\end{bmatrix},\ \ SKS^\dag=K.
\]
We define matrices $P_i:=S^{-1}\partial_i{S}$, which are elements of the Lie algebra associated with the symplectic group, satisfying the defining properties of this algebra,
\[\label{def:P_1}
P_i=
\begin{bmatrix}
R_i & Q_i \\
\ov{Q}_i & \ov{R}_i
\end{bmatrix},\ \ P_iK+KP_i^\dag=0.
\]
Common symplectic matrices in the complex form, representing for example a squeezing operation, phase-change, or a beam-splitter, can be found in Appendix~\ref{sec:The_phase_space_Gaussian_unitaries} or in more detail in Ref.~\cite{safranek2016optimal} and Section II of Ref.~\cite{vsafranek2016gaussian}.
Rewriting Eq.~\eqref{eq:mixed_QFI} in terms of the Williamson's decomposition of the covariance matrix, switching to element-wise notation, and simplifying using identities~\eqref{def:structure_of_S} and~\eqref{def:P_1}, we derive we derive an analytical expression for the quantum Fisher information matrix of Gaussian states in terms of the Williamson's decomposition of the covariance matrix,
\begin{align}
H^{ij}(\be)&\!\!=\!\!\!\!\sum_{k,l=1}^{N}\!\!\frac{(\lambda_k\!-\!\lambda_l)^2}{\lambda_k\lambda_l\!-\!1}\Re[\ov{R_{i}}^{kl}\!R_{j}^{kl}]+\frac{(\lambda_k\!+\!\lambda_l)^2}{\lambda_k\lambda_l\!+\!1}\Re[\ov{Q_{i}}^{kl}\!Q_{j}^{kl}]\nonumber\\
&\!+\sum_{k=1}^{N}\frac{\partial_i\lambda_k\partial_j\lambda_k}{\lambda_k^2-1}+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d}.\label{eq:multimode_QFI}
\end{align}
$\Re$ denotes the real part, $R_i=\A^\dag\partial_i{\A}-\overline{\B^\dag\partial_i{\B}}$ is a skew-Hermitian and $Q_i=\A^\dag\partial_i{\B}-\overline{\B^\dag\partial_i{\A}}$ a (complex) symmetric matrix. We note that $\sigma^{-1}=KSD^{-1}S^\dag K$, which follows from Eq.~\eqref{def:structure_of_S} and properties of $K$. The above formula represents a multi-parameter generalization of the result for a single-parameter estimation published in Ref.~\cite{Safranek2015b}. The full derivation can be found in Appendix~\ref{app:williamson_QFI}.
The above formula can be used even for states with some pure modes, defined by $\lambda_k=1$ for some mode $k$. We will show how soon.
When none of the modes are pure, i.e., all symplectic eigenvalues are larger than one, we can rewrite Eq.~\eqref{eq:multimode_QFI} in a very elegant way. Defining Hermitian matrix $\widetilde{R}_i^{kl}:=\frac{\lambda_k-\lambda_l}{\sqrt{\lambda_k\lambda_l-1}}R_i^{kl}$, symmetric matrix $\widetilde{Q}_i^{kl}:=\frac{\lambda_k+\lambda_l}{\sqrt{\lambda_k\lambda_l+1}}Q_i^{kl}$, and diagonal matrix $L:=\mathrm{diag}(\lambda_1,\dots,\lambda_N)$, QFIM can be written as
\[\label{eq:exact_multimode_compact}
\begin{split}
H^{ij}(\be)&=\frac{1}{2}\tr\big[\widetilde{R}_i\widetilde{R}_j^\dag+\widetilde{R}_j\widetilde{R}_i^\dag+\widetilde{Q}_i\widetilde{Q}_j^\dag+\widetilde{Q}_j\widetilde{Q}_i^\dag\big]\\
&+\tr\big[(L^2-I)^{-1}\partial_iL\partial_jL\big]+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d}.
\end{split}
\]
Looking at the derivatives in each term, we can conclude that QFIM consists of three qualitatively different terms: the first term is connected to the change of orientation and squeezing of the Gaussian state with small variations in $\be$, the second to the change of purity, and the third to the change of displacement.
Similarly, we derive expression for the symmetric logarithmic derivative,
\[\label{eq:SLD_mixed_williamson}
\hat{\mathscr{L}}_i(\be)=\dbA^\dag \!(S^{-1})^\dag W_i S^{-1}\!\dbA-\sum_{k=1}^N\frac{\lambda_k\partial_i\lambda_k}{\lambda_k^2-1}+2\dbA^\dag\!\sigma^{-1}\partial_i{\boldsymbol{d}},
\]
where $W_i$ is a Hermitian matrix of form
\[
\begin{split}
W_i&=\begin{bmatrix}
W_{Xi} & W_{Yi} \\
\overline{W_{Yi}} & \overline{W_{Xi}}
\end{bmatrix},\\
W_{Xi}^{kl
&=-\frac{\lambda_k-\lambda_l}{\lambda_k\lambda_l-1}R_i^{kl}+\frac{\partial_i\lambda_k}{\lambda_k^2-1}\delta^{kl},\\
W_{Yi}^{kl}&=\frac{\lambda_k+\lambda_l}{\lambda_k\lambda_l+1}Q_i^{kl}.
\end{split}
\]
Further, we derive expression that determines saturability of the quantum Cram\'er-Rao bound,
\[\label{eq:multimode_condition}
\begin{split}
&\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]=4\partial_i\bd^\dag\sigma^{-1}K\sigma^{-1}\partial_j\bd\\
&+\!\!\sum_{k,l=1}^{N}\!\!\frac{2i(\lambda_k\!+\!\lambda_l)^3}{(\lambda_k\lambda_l\!+\!1)^2}\Im[\ov{Q_{i}}^{kl}\!Q_{j}^{kl}]-\frac{2i(\lambda_k\!-\!\lambda_l)^3}{(\lambda_k\lambda_l\!-\!1)^2}\Im[\ov{R_{i}}^{kl}\!R_{j}^{kl}].\\
\end{split}
\]
The derivation can be found in Appendix~\ref{app:williamson_QFI}. Alternative but equivalent forms of the above expressions are also published in Ref.~\cite{nichols2018multiparameter}.
\textit{\textbf{When some of the modes are pure}}
Eqs.~(\ref{eq:mixed_QFI},\ref{eq:SLD_mixed},\ref{eq:condition_QCR},\ref{eq:multimode_QFI},\ref{eq:SLD_mixed_williamson},\ref{eq:multimode_condition}) are not well defined for states that have at least one mode in a pure state, i.e., $\lambda_k=1$ for some mode $k$, which also results in matrix $\mathfrak{M}$ not being invertible. It has been shown~\cite{vsafranek2017discontinuities}, and we explain it in detail in Appendix~\ref{app:pops}, that there are two unique ways of defining QFIM at these problematic points, depending on the quantity we want to obtain.
First, we will illustrate this on Eq.~\eqref{eq:multimode_QFI}. To obtain the QFIM, which is defined~\cite{Paris2009a} through the symmetric logarithmic derivatives $\mathscr{L}_i$ as $H^{ij}\equiv\tfrac{1}{2}\tr[\R\{\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j\}]$, we define problematic terms as
\[\label{def:problematic_points}
\frac{(\lambda_k\!-\!\lambda_l)^2}{\lambda_k\lambda_l\!-\!1}=0,\quad
\frac{\partial_i\lambda_k\partial_j\lambda_k}{\lambda_k^2-1}=0,
\]
for $\be$ such that $\lambda_k(\be)=\lambda_l(\be)=1$.
The continuous quantum Fisher information matrix (cQFIM) is defined~\cite{vsafranek2017discontinuities} as four-times the Bures metric as $H_c^{ij}\equiv 4g^{ij}$, where the Bures metric is defined by $\sum_{i,j}g^{ij}(\be)\mathrm{d}\epsilon_i\mathrm{d}\epsilon_j\equiv 2\big(1-\sqrt{\mathcal{F}(\R_\be,\R_{\be+\bde})}\big)$, and ${\mathcal{F}({\R}_{1},{\R}_{2})=\big(\tr\sqrt{\sqrt{{\R}_{1}}\,{\R}_{2}\,\sqrt{{\R}_{1}}}\big)^{2}}$ denotes the Uhlmann's fidelity~\cite{Uhlmann1976a}.
To obtain the cQFIM, we define
\[\label{def:problematic_points2}
\frac{(\lambda_k\!-\!\lambda_l)^2}{\lambda_k\lambda_l\!-\!1}=0,\quad
\frac{\partial_i\lambda_k\partial_j\lambda_k}{\lambda_k^2-1}=\partial_i\partial_j\lambda_k,
\]
for $\be$ such that $\lambda_k(\be)=\lambda_l(\be)=1$.
QFIM and cQFIM are identical everywhere, apart from those problematic points~\cite{vsafranek2017discontinuities}. At those points, QFIM can be discontinuous, while cQFIM is in some sense continuous~\cite{vsafranek2017discontinuities}. Defining Hessian matrix $\mathcal{H}_k^{ij}:=\partial_i\partial_j\lambda_k$, we can use the above equations to write relation
\[\label{eq:QFIM_cQFIM_relation}
H_c(\be)=H(\be)+\!\!\!\!\!\!\sum_{k:\lambda_k(\be)=1}\!\!\!\!\!\!\mathcal{H}_k(\be).
\]
By writing $k:\lambda_k(\be)=1$ we mean that the sum goes only over values of $k$ for which $\lambda_k(\be)=1$. For any $k$ such that $\lambda_k(\epsilon)=1$, $\mathcal{H}_k(\be)$ is positive semi-definite, and we can therefore write $H_c\geq H$. $H_c= H$ if and only if for all $k$ such that $\lambda_k=1$, $\mathcal{H}_k=0$.
Similarly, in Eqs.~\eqref{eq:SLD_mixed_williamson} and~\eqref{eq:multimode_condition} we define $\frac{\lambda_k\!-\!\lambda_l}{\lambda_k\lambda_l\!-\!1}=\frac{(\lambda_k\!-\!\lambda_l)^3}{(\lambda_k\lambda_l\!-\!1)^2}=\frac{\partial_i\lambda_k}{\lambda_k^2-1}:=0$ for $\be$ such that $\lambda_k(\be)=\lambda_l(\be)=1$.
\textit{\textbf{Regularization procedure}}
Now we will show how to treat cases when some modes are pure in general, not limiting ourselves to already resolved case of Eq.~\eqref{eq:multimode_QFI}. We can devise a regularization procedure that will allow us to use expressions that work only for states where all the modes are mixed, such as Eq.~\eqref{eq:mixed_QFI}, to compute the QFIM for any state. Similar method has been already used for regularizing QFIM for non-Gaussian states~\cite{vsafranek2017discontinuities}.
It goes as follows. First we multiply the covariance matrix by regularization parameter $\nu>1$, and use some expression, such as Eq.~\eqref{eq:mixed_QFI}, to calculate the QFIM for state $(\bd,\nu\sigma)$. Then we perform limit $\nu\rightarrow 1$. The resulting value will represent the correct QFIM for state $(\bd,\sigma)$.
To prove that, however, we have to check that this limit leads to the proper definition of the problematic points, as given by Eq.~\eqref{def:problematic_points}. We take Eq.~\eqref{eq:multimode_QFI} as a study case, but because this formula is general, the result will be valid for any other expression for QFIM. When covariance matrix $\sigma$ has symplectic eigenvalues $\lambda_k$, covariance matrix $\nu\sigma$ has symplectic eigenvalues $\nu\lambda_k$. Sympletic matrices from the decompositions of $\sigma$ and $\nu\sigma$ are identical. Parameter $\nu$ therefore appears only as a modification of symplectic eigenvalues, which we will take advantage of. Assuming $\lambda_k(\be)=\lambda_l(\be)=1$ and performing the limit, both problematic terms are set to zero by taking the limit, $\lim_{\nu\rightarrow1}\frac{(\nu\lambda_k-\nu\lambda_l)^2}{\nu\lambda_k\nu\lambda_l-1}=\lim_{\nu\rightarrow1}\frac{0}{\nu^2-1}=0$, $\lim_{\nu\rightarrow1}\frac{(\nu\partial_i{\lambda}_k)^2}{(\nu\lambda_k)^2-1}=\lim_{\nu\rightarrow1}\frac{0}{\nu^2-1}=0$, which is exactly the definition, Eq.~\eqref{def:problematic_points}, that we wanted. $\partial_i{\lambda}_k(\be)=0$, because ${\lambda}_k(\be)$ achieves a local minimum at point $\be$ when $\lambda_k(\be)=1$.
This method therefore leads to the correct value of the QFIM, and we can write expression for the QFIM for any Gaussian quantum state as
\[\label{eq:regularization_procedure}
H(\be)\equiv H(\bd(\be),\sigma(\be))=\lim_{\nu\rightarrow1}H\big(\bd(\be),\nu\sigma(\be)\big).
\]
Applying this result to Eq.~\eqref{eq:mixed_QFI}, QFIM for any state can be computed as
\[\label{eq:any_state_QFIM}
\begin{split}
H^{ij}(\be)&=\lim_{\nu\rightarrow1} \frac{1}{2}\vectorization{\partial_i\sigma}^\dag(\nu^2\ov{\sigma}\otimes\sigma-K\otimes K)^{-1}\vectorization{\partial_j\sigma}\\
&+2\partial_i\bd^\dag\sigma^{-1}\partial_j\bd.
\end{split}
\]
Expression for the cQFIM can obtained by combining Eq.~\eqref{eq:regularization_procedure} with Eq.~\eqref{eq:QFIM_cQFIM_relation}.
Similarly, we have
\[\label{eq:SLD_regularization}
\hat{\mathscr{L}}_i(\be)=\lim_{\nu\rightarrow1}\dbA^\dag\cA_{i\nu}\dbA-\frac{1}{2}\tr[\sigma\cA_{i\nu}]+2\dbA^\dag\sigma^{-1}\partial_i{\boldsymbol{d}},
\]
and
\[\label{eq:condition_QCRregularization_procedure2}
\begin{split}
&\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]=4\partial_i\bd^\dag\sigma^{-1}K\sigma^{-1}\partial_j\bd\\
&+\lim_{\nu\rightarrow1}\vectorization{\partial_i\sigma}^\dag\mathfrak{M}_{\nu}^{-1}(\ov{\sigma}\!\otimes\! K\!-\!K\!\otimes\!\sigma)\mathfrak{M}_{\nu}^{-1}\vectorization{\partial_j\sigma},
\end{split}
\]
where $\vectorization{\cA_{i\nu}}=\mathfrak{M}_{\nu}^{-1}\vectorization{\partial_i\sigma}$ and $\mathfrak{M}_{\nu}=\nu^2\ov{\sigma}\otimes\sigma-K\otimes K$.
\textit{\textbf{Limit formula}}
We presented exact analytical expressions for QFIM, however, in some cases a numerical value that approximates the exact value to any desired precision is enough. Defining matrix $A:=K\sigma$, and generalizing procedure derived in Ref.~\cite{Safranek2015b} for a single-parameter estimation, we derive the limit expression for the QFIM,
\[\label{eq:limit_formula}
H^{ij}(\be)=\frac{1}{2}\sum_{n=1}^\infty\tr\big[A^{-n}\partial_i{A}A^{-n}\partial_j{A}]+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d}.
\]
This expression is proved in the next section, by showing its relation to Eq.~\eqref{eq:mixed_QFI}.
Note that although the above infinite series converges even when some symplectic eigenvalue are equal to one, for such cases it is not absolutely convergent and it does not give the correct expression for the QFIM. This can be shown by careful analysis of the elements in the series given by Eq.~\eqref{eq:expression_in_terms_of_P}, and it was explained in detail in Ref.~\cite{Safranek2015b}. The correct expression for cases when some of the modes are pure can be obtained by combining the above equation with the regularization procedure, Eq.~\eqref{eq:regularization_procedure}.
In applications, we would like to take just a few elements of the series, and believe that their sum well approximates the QFIM. To estimate the error when doing this, we define remainder of the series as $R_M^{ij}:=\frac{1}{2}\sum_{n=M+1}^\infty\tr\big[A^{-n}\!\partial_i{A}A^{-n}\!\partial_j{A}]$. As shown in Appendix~\ref{app:Remainder}, this remainder is bounded,
\[\label{eq:remainder}
|R_M^{ij}|\leq\frac{\sqrt{\mathrm{tr}[(A\partial_iA)^2]}\sqrt{\mathrm{tr}[(A\partial_jA)^2]}}{2\lambda_{\mathrm{min}}^{2(M+1)}(\lambda_{\mathrm{min}}^2-1)},
\]
where $\lambda_{\mathrm{min}}:=\min_{k}\{\lambda_k\}$ is the smallest symplectic eigenvalue of the covariance matrix $\sigma$. The right hand side therefore represents the maximal error when calculating the QFIM by using the first $M$ elements of the series, Eq.~\eqref{eq:limit_formula}.
We can derive similar limit expressions for the SLD and for $\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]$ by using expression
\[
\cA_i=\sum_{n=1}^\infty A^{-n}\partial_iA A^{-n}K,
\]
derived towards the end of Appendix~\ref{app:mixed_state}.
\textit{\textbf{Relations between different expressions for QFIM}}
Here we show how the limit expression, Eq.~\eqref{eq:limit_formula}, relates to Eqs.~\eqref{eq:mixed_QFI} and~\eqref{eq:multimode_QFI}. Relation between Eqs.~\eqref{eq:mixed_QFI} and~\eqref{eq:multimode_QFI} is shown in Appendix~\ref{app:williamson_QFI}.
To obtain Eq.~\eqref{eq:mixed_QFI}, we use $\sigma^\dag=\sigma$, properties of vectorization, $\tr[A^\dag B]=\vectorization{A}^\dag\vectorization{B}$, and properties of Kronecker product, $(AB)\otimes(A'B)=(A\otimes A')(B\otimes B')$, $(C^T\otimes A)\vectorization{B}=\vectorization{ABC}$, to transform the infinite sum in Eq.~\eqref{eq:limit_formula} into a Neumann series that can be evaluated,
\[
\begin{split}
&\sum_{n=1}^\infty\tr\big[A^{-n}\partial_i{A}A^{-n}\partial_j{A}]\\
&=\vectorization{\partial_i\sigma}^\dag\bigg(\sum_{n=0}^\infty(\ov{A}\otimes A)^{-n}\bigg)\big(\ov{\sigma}\otimes\sigma\big)^{-1}\vectorization{\partial_j\sigma}\\
&=\vectorization{\partial_i\sigma}^\dag(I-\ov{A}^{-1}\otimes A^{-1})^{-1}\big(\ov{\sigma}\otimes\sigma\big)^{-1}\vectorization{\partial_j\sigma}\\
&=\vectorization{\partial_i\sigma}^\dag(\ov{\sigma}\otimes\sigma-K\otimes K)^{-1}\vectorization{\partial_j\sigma},
\end{split}
\]
which gives Eq.~\eqref{eq:mixed_QFI}.
Using identity
\[\label{eq:expression_in_terms_of_P}
\begin{split}
&\tr[A^{-n}\partial_i{A}A^{-n}\partial_j{A}]=2\tr[D^{-n+1}\!K^{-n+1}\!P_{i}D^{-n+1}\!K^{-n+1}\!P_{j}]\\
&-\tr[D^{-n+2}\!K^n\!P_{i}D^{-n}\!K^n\!P_{j})]
-\tr[D^{-n+2}\!K^n\!P_{j}D^{-n}\!K^n\!P_{i})]\\
&+\tr[D^{-n}\partial_i{D}D^{-n}\partial_j{D}]
\end{split}
\]
and changing to element-wise notation, the infinite sum~\eqref{eq:limit_formula} turns out to be geometric series in powers of $\lambda_k$'s, which can be evaluated. Then, using $R_i^{kl}=-\ov{R}_i^{lk}$, $Q_i^{kl}=Q_i^{lk}$ which follows from Eq.~\eqref{def:P_1}, we prove that Eq.~\eqref{eq:limit_formula} simplifies to Eq.~\eqref{eq:multimode_QFI}.
\textit{\textbf{Pure states}}
Combining Eq.~\eqref{eq:limit_formula}, the regularization procedure~\eqref{eq:regularization_procedure}, and $A^2(\be)=I$ (which holds for pure states because for them, $\lambda_k(\be)=1$ for all $k$), we obtain the well-known result for pure states~\cite{Pinel2012a},
\[\label{eq:pure_non-elegant}
H^{ij}(\be)=\frac{1}{4}\mathrm{tr}[\sigma^{-1}\partial_i\sigma\sigma^{-1}\partial_j\sigma]+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d}.
\]
It is important to stress that although this expression is defined for any state, it can be applied only for states that are pure at point $\be$. If the state becomes mixed when $\be$ is slightly varied, i.e., when $\partial_i\partial_j\lambda_k(\be)\neq 0$ for some $k$ (see Appendix~\ref{app:pops}, and Ref.~\cite{vsafranek2017discontinuities}), QFIM at this varied parameter $H(\be+\bdeps)$ has to be calculated using some other formula (for example, Eqs.~\eqref{eq:mixed_QFI}, \eqref{eq:multimode_QFI}, or \eqref{eq:any_state_QFIM}), and one finds that in that case, function $H^{ij}$ is discontinuous at point $\be$.
To obtain cQFIM for states that are pure at point $\be$, i.e., $A^2(\be)=I$, we can use expression
\[\label{eq:pure_elegant}
\begin{split}
H_c^{ij}(\be)&=\frac{1}{4}\big(2\mathrm{tr}[\sigma^{-1}\partial_i\partial_j\sigma]-\mathrm{tr}[\sigma^{-1}\partial_i\sigma\sigma^{-1}\partial_j\sigma]\big)\\
&+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d},
\end{split}
\]
To prove this expression, one needs to utilize the Williamson's decomposition of the covariance matrix, find~$\mathrm{tr}[\sigma^{-1}\partial_i\partial_j\sigma]$ and $\mathrm{tr}[\sigma^{-1}\partial_i\sigma\sigma^{-1}\partial_j\sigma]$ in terms of matrices $K$, $P_i=S^{-1}\partial_i S,$ and $P_{ij}:=S^{-1}\partial_i\partial_jS$ (which will give expressions similar to Eq.~\eqref{eq:expression_in_terms_of_P}), and use Eqs.~\eqref{def:structure_of_S},~\eqref{def:P_1}, and $P_{ij}K+P_iKP_j^\dag+P_jKP_i^\dag+KP_{ji}^\dag=0$. When applied to both Eq.~\eqref{eq:pure_non-elegant} and Eq.~\eqref{eq:pure_elegant}, we find that Eq. ~\eqref{eq:pure_elegant} gives the same expression as Eq.~\eqref{eq:pure_non-elegant}, plus an additional factor given by the second part of Eq.~\eqref{eq:QFIM_cQFIM_relation}. This proves that Eq.~\eqref{eq:pure_elegant} represents the cQFIM for pure states.
Note that Eqs.~\eqref{eq:pure_non-elegant} and~\eqref{eq:pure_elegant} can be further simplified by using $\sigma^{-1}=K\sigma K$ and $\partial_i AA=-A\partial_i A$, which follows from $A^2(\be)=I$.
Finally, we derive symmetric logarithmic derivatives for pure states
\[\label{eq:sld_pure}
\hat{\mathscr{L}}_i(\be)=\frac{1}{2}\dbA^\dag\!\sigma^{-1}\! \partial_i\sigma \sigma^{-1}\dbA+2\dbA^\dag\!\sigma^{-1}\partial_i{\boldsymbol{d}},
\]
and
\[\label{eq:pure_state_condition}
\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]\!=\!\frac{1}{4}\tr[K\!\sigma[K\partial_i\sigma,K\partial_j\sigma]]
+4\partial_i\bd^\dag\!\sigma^{-1}\!K\!\sigma^{-1}\!\partial_j\bd.
\]
The derivation can be found in Appendix~\ref{app:pure_state_condition}.
\section{Examples}
Here we illustrate the derived formulas on several examples. As shown in~\cite{safranek2016optimal}, Gaussian unitary operations, which are generated via an exponential map with the exponent at most quadratic in the field operators~\cite{Weedbrook2012a}, can be parameterized by matrix $W$ and vector $\ba$ as
\[\label{def:Gaussian_unitary}
\hat{U}=\exp\big(\tfrac{i}{2}\bA^\dag W \bA+\bA^\dag K \ba\big).
\]
In case of $W=0$, this operator corresponds to the Weyl displacement operator, while for $\ba=0$ we obtain purely quadratic transformations such as the phase-changing operator, one- and two-mode squeezing operators, or mode-mixing operators, depending on the particular structure of $W$. The first and the second moments of the transformed density matrix $\hat{\rho}'=\hat{U}\hat{\rho}\hat{U}^\dag$ are computed as
\[\label{def:transformation}
\bd'=S\bd+\bb,\ \ \sigma'=S\sigma S^\dag,
\]
where the symplectic matrix and the displacement are given by
\[\label{eq:S_and_b}
S=e^{iKW},\ \
\bb=\Big(\!\int_0^1e^{iKWt}\mathrm{d}t\!\Big)\ \!\ba.
\]
The states in the following examples are generated using the above transformations, usually applied on a thermal state. Their form in the phase-space formalism is explicitly computed in Appendix~\ref{sec:The_phase_space_Gaussian_unitaries}, or in more detail in Ref.~\cite{safranek2016optimal} and in Chapter II~of Ref.~\cite{vsafranek2016gaussian}.
Although every formula demonstrated here can be used to calculate QFIM for Gaussian states of any number of modes, for simplicity we choose only single- and two-mode states. However, we point out that for a single- and two-mode Gaussian states it is often better to use expressions valid for these specific number of modes~\cite{Pinel2013b,Safranek2015b}.
\textit{\textbf{Mixed states}}
Let us consider estimation of a squeezing parameter $r$ and inverse temperature $\beta$ from a squeezed thermal state, $\R=\hat{S}(r)\R_{\mathrm{th}}(\beta)\hat{S}^\dag(r)$, where $\hat{S}(r)$ denotes the squeezing operator and $\R_{\mathrm{th}}(\beta)=\frac{1}{Z}\exp(-\beta\hat{n})$ is the thermal state. $Z$ denotes the partition function and $\hat{n}$ denotes the number operator. The final state can be expressed via the first and the second moment as
\[\label{eq:first_example_moments}
\bd=\begin{pmatrix}
0 \\
0 \\
\end{pmatrix}
,
\quad \sigma=\lambda\begin{pmatrix}
\cosh 2r & -\sinh 2r \\
-\sinh 2r & \cosh 2r
\end{pmatrix},
\]
where $\lambda=\coth\tfrac{\beta}{2}$. We compute
\begin{widetext}
\begin{gather}
\mathfrak{M}^{-1}=\frac{\lambda^2}{2(\lambda^4-1)}\begin{pmatrix}
\cosh 4r+1+\tfrac{2}{\lambda^2} & \sinh
4r & \sinh
4r & \cosh 4r-1 \\
\sinh
4r & \cosh 4r+1-\tfrac{2}{\lambda^2} & \cosh 4r-1 & \sinh
4r \\
\sinh
4r & \cosh 4r-1& \cosh 4r+1-\tfrac{2}{\lambda^2} & \sinh
4r \\
\cosh 4r-1 & \sinh
4r & \sinh
4r & \cosh 4r+1+\tfrac{2}{\lambda^2} \\
\end{pmatrix},\\
\vectorization{\partial_\beta\sigma}=\frac{\lambda^2-1}{2}\begin{pmatrix}
-\cosh
2r \\
\sinh
2r \\
\sinh
2r \\
-\cosh
2r \\
\end{pmatrix}, \quad
\vectorization{\partial_r\sigma}=2\lambda\begin{pmatrix}
\sinh
2r \\
-\cosh
2r \\
-\cosh
2r \\
\sinh
2r \\
\end{pmatrix}.\nonumber
\end{gather}
QFIM is calculated from Eq.~\eqref{eq:mixed_QFI},
\[\label{eq:example1exact}
H(\beta,r)=\frac{1}{2}\begin{pmatrix}
\vectorization{\partial_\beta\R}^\dag\mathfrak{M}^{-1}\vectorization{\partial_\beta\R} &\vectorization{\partial_\beta\R}^\dag\mathfrak{M}^{-1}\vectorization{\partial_r\R} \\
\vectorization{\partial_r\R}^\dag\mathfrak{M}^{-1}\vectorization{\partial_\beta\R} & \vectorization{\partial_r\R}^\dag\mathfrak{M}^{-1}\vectorization{\partial_r\R} \\
\end{pmatrix}=
\begin{pmatrix}
\frac{\lambda^2-1}{4} & 0 \\
0 & \frac{4\lambda^2}{\lambda^2+1} \\
\end{pmatrix}.
\]
\end{widetext}
From Eq.~\eqref{eq:condition_QCR} we derive $\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]=0$ for all $i,j=\beta,r$, which according to Eq.~\eqref{eq:condition_for_CRbound} means that the quantum Cram\'er-Rao bound is achievable.
\textit{\textbf{When the Williamson's decomposition of the covariance matrix is known}}
Let the initial state be a coherent state $\ket{\alpha}$, which is given by
\[
\bd_0=\begin{pmatrix}
\alpha \\
\ov{\alpha} \\
\end{pmatrix}
,
\quad \sigma_0=\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},
\]
and we use this state to probe squeezing channel with subsequent phase-change, which transforms the state to $\R=\hat{R}(\theta)\hat{S}(r)\pro{\alpha}{\alpha}\hat{S}^\dag(r)\hat{R}^\dag(\theta)$. Action of this channel corresponds to symplectic matrix
\[
S(r,\theta)=R(\theta)S(r)=\begin{pmatrix}
e^{- i \theta}\cosh r & -e^{- i \theta}\sinh r \\
-e^{ i \theta}\sinh r & e^{ i \theta}\cosh r
\end{pmatrix},
\]
which leads to moments for $\R$,
\[
\bd=S(r,\theta)\bd_0
,
\quad \sigma=S(r,\theta)\sigma_0S^\dag(r,\theta).
\]
Since $\sigma$ is already written in the form of its symplectic decomposition, we can immediately use Eq.~\eqref{eq:multimode_QFI} to compute the QFIM. We compute $P_r=S^{-1}(r,\theta)\partial_r{S}(r,\theta)$, and $P_\theta=S^{-1}(r,\theta)\partial_\theta{S}(r,\theta)$, from which we obtain
\[
R_r=0,\ \,
Q_r=-1,\ \,
R_\theta=-i \cosh 2r,\ \,
Q_\theta=i \sinh 2r.
\]
Moreover,
\begin{align}
\sigma^{-1}&=\begin{pmatrix}
\cosh 2r & e^{-2 i \theta}\sinh 2r \\
e^{2 i \theta}\sinh 2r & \cosh 2r
\end{pmatrix},\nonumber\\
\partial_r\boldsymbol{d}&=\begin{pmatrix}
e^{- i \theta}(-\ov{\alpha}\cosh r+\alpha \sinh r )\\
e^{ i \theta}(-\alpha\cosh r+\ov{\alpha} \sinh r )\\
\end{pmatrix},\\
\partial_\theta\boldsymbol{d}&=\begin{pmatrix}
-ie^{- i \theta}(\alpha\cosh r-\ov{\alpha} \sinh r )\\
ie^{ i \theta}(\ov{\alpha}\cosh r-\alpha \sinh r )\\
\end{pmatrix}.\nonumber
\end{align}
\break
Since the symplectic eigenvalue $\lambda=1$, according to Eq.~\eqref{def:problematic_points}, the first and the third term in the sum, Eq.~\eqref{eq:multimode_QFI}, disappears, which yields
\[
H^{ij}(\be)=2\Re[\ov{Q_{i}}Q_{j}]
+2\partial_i\boldsymbol{d}^\dag\sigma^{-1}\partial_j\boldsymbol{d},
\]
from which we compute QFIM as
\begin{widetext}
\[
H(r,\theta)=
\begin{pmatrix}
2+4\abs{\alpha}^2 & -4\Im[\alpha^2]\cosh 2r \\
-4\Im[\alpha^2]\cosh 2r & 2\sinh^2 2r\!+\!4e^{4r}\Im[\alpha]^2\!+\!4e^{-4r}\Re[\alpha]^2 \\
\end{pmatrix}.
\]
\end{widetext}
Further, applying $\lambda=1$ to Eq.~\eqref{eq:multimode_condition} we derive
\[
\tr[\R[\hat{\mathscr{L}}_i,\hat{\mathscr{L}}_j]]=4i\Im[\ov{Q_{i}}Q_{j}]+4\partial_i\bd^\dag\sigma^{-1}K\sigma^{-1}\partial_j\bd,
\]
which yields $\tr[\R[\hat{\mathscr{L}}_r,\hat{\mathscr{L}}_r]]=\tr[\R[\hat{\mathscr{L}}_\theta,\hat{\mathscr{L}}_\theta]]=0$, and
\[
\tr[\R[\hat{\mathscr{L}}_r,\hat{\mathscr{L}}_\theta]]=4i(-\sinh 2r+2e^{2r}\Im[\alpha]^2-2e^{-2r}\Re[\alpha]^2),
\]
which means that the quantum Cram\'er-Rao bound is not in general achievable for simultaneous estimation of $r$ and $\theta$ encoded into a coherent state.
\textit{\textbf{Limit formula}}
Here we are going to use Eqs.~\eqref{eq:limit_formula} and~\eqref{eq:remainder} to numerically estimate QFIM of the Gaussian state from the first example, and then compared it to the analytical result.
From the first and the second moments, Eq.~\eqref{eq:first_example_moments}, we calculate
\[
\begin{split}
A&=K\sigma=\lambda\begin{pmatrix}
\cosh 2r & -\sinh 2r \\
\sinh 2r & -\cosh 2r
\end{pmatrix},\\
A^{-1}&=\frac{1}{\lambda}\begin{pmatrix}
\cosh 2r & -\sinh 2r \\
\sinh 2r & -\cosh 2r
\end{pmatrix},\\
\partial_\beta A&=\frac{\lambda^2-1}{2}\begin{pmatrix}
-\cosh 2r & \sinh 2r \\
-\sinh 2r & \cosh 2r
\end{pmatrix},\\
\partial_r A&=2\lambda\begin{pmatrix}
\sinh 2r & -\cosh 2r \\
\cosh 2r & -\sinh 2r
\end{pmatrix}.
\end{split}
\]
In order to calculate QFIM for $n$ decimal places, we require $M$ to be such that
\[
|R_M^{\beta,\beta}|< \frac{1}{10^n},\quad |R_M^{\beta,r}|< \frac{1}{10^n},\quad|R_M^{rr}|< \frac{1}{10^n},
\]
which, using Eq.~\eqref{eq:remainder}, leads to
\[
M>\frac{n+\log_{10}\frac{\max_{i\in\{\beta,r\}}\mathrm{tr}[(A\partial_iA)^2]}{2(\lambda_{\mathrm{min}}^2-1)}}{2\log_{10}\lambda_{\min}}-1.
\]
To calculate QFIM for $\lambda=2$ and $r=1$ with precision for two decimal places, we insert $n=2$. In our example $\lambda_{\mathrm{min}}=\lambda$, which gives
\[
M>4.529,
\]
meaning that we need $M=5$ terms in the sum~\eqref{eq:limit_formula}. Summing these terms, we obtain an estimate for the QFIM,
\[
H(\beta,r)\approx
\begin{pmatrix}
0.749268 & 0 \\
0 & 3.20313 \\
\end{pmatrix}.
\]
Comparing this to the analytical result calculated by inserting values $\lambda=2$ and $r=1$ into Eq.~\eqref{eq:example1exact},
\[
H(\beta,r)=
\begin{pmatrix}
0.75 & 0 \\
0 & 3.2 \\
\end{pmatrix},
\]
shows that we are within the limit of two decimal places of precision.
\textit{\textbf{Pure states and discontinuity of QFIM}}
Here we show the difference between QFIM and cQFIM and show how it is connected to the discontinuity of the quantum Fisher information. We consider a task of estimating the squeezing parameter from the two-mode squeezed vacuum, $\R=\hat{S}_T(r)\pro{0}{0}\hat{S}_T^\dag(r)$, given by moments
\[\label{eq:full_state}
\bd=\begin{pmatrix}
0 \\
0 \\
0 \\
0 \\
\end{pmatrix}
,
\quad \sigma=\begin{pmatrix}
\cosh 2r & 0 & 0 & -\sinh 2r \\
0 & \cosh 2r & -\sinh 2r & 0 \\
0 & -\sinh 2r & \cosh 2r & 0 \\
-\sinh 2r & 0 & 0 & \cosh 2r
\end{pmatrix}.
\]
Since this state is a pure state, we can use Eq.~\eqref{eq:pure_non-elegant} to compute the QFIM, and Eq.~\eqref{eq:pure_elegant} to compute the cQFIM. Variation of the parameter $r$ does not change the purity of the state, i.e., both symplectic eigenvalues $\lambda_1=\lambda_2=1$ for all $r$, thus, according to Eq.~\eqref{eq:QFIM_cQFIM_relation}, we should find $H=H_c$. Indeed, we have
\[
H(r)=H_c(r)=4.
\]
Now, let us say that experimenter does not have an access to the second mode, so they have to trace over it. In the phase-space formalism, this is done simply by taking out the rows and columns representing that mode~\cite{Weedbrook2012a} (in our notation, the second and the fourth row and column), resulting in state
\[\label{eq:reduced_state}
\bd_1=\begin{pmatrix}
0 \\
0 \\
\end{pmatrix}
,
\quad \sigma_1=\begin{pmatrix}
\cosh 2r & 0 \\
0 & \cosh 2r
\end{pmatrix}.
\]
For $r=0$ the state is pure, so we can again use Eqs.~\eqref{eq:pure_non-elegant} and~\eqref{eq:pure_elegant} to compute QFIM and cQFIM. However, now we find
\[\label{eq:H_and_Hc}
H(0)=0,\quad H_c(0)=4.
\]
For $r>0$ the state is mixed, $\lambda_1=\cosh 2r$, according to Eq.~\eqref{eq:QFIM_cQFIM_relation}, $H=H_c$, and we can use any formula for mixed states (such as Eq.~\eqref{eq:multimode_QFI}) to compute the QFIM and cQFIM. Put together with Eq.~\eqref{eq:H_and_Hc}, we find
\[
H(r)=\begin{cases}
0, & r=0, \\
4, & \mathrm{otherwise},
\end{cases}\quad
\quad H_c(r)=4.
\]
Clearly, QFIM is discontinuous at point $r=0$, as expected from the theory~\cite{vsafranek2017discontinuities}. Intuitively, this can be explained as follows: QFIM measures the amount of identifiability of parameter $dr$ from state $\R_{r+dr}$. As we can see from Eq.~\eqref{eq:reduced_state}, if $r=0$, then states $\R_{+dr}$ and $\R_{-dr}$ correspond to the same density matrix, $\R_{+dr}=\R_{-dr}$. Therefore, $dr$ is not identifiable around point $r=0$, because there is no physical experiment that experimenter could apply on the system to distinguish parameter $-dr$ from parameter $+dr$. It is therefore reasonable to expect that QFIM, which measures the ability to estimate $dr$, is zero at point $r=0$. Experimenter does not have the same problem when they have access to the full state, Eq.~\eqref{eq:full_state}, because in there $\R_{+dr}\neq\R_{-dr}$, so $dr$ is identifiable at point $r=0$. On the other hand, cQFIM, which is defined as four times the Bures metric, measures the infinitesimal distance between states $\R_{r}$ and $\R_{r+dr}$. This distance is always positive, no matter what $r$ is. Identifiability does not play any role in the measure of distance. It is therefore not surprising, that in this case, cQFIM is a continuous function in $r$.
\section{Conclusion}
In this paper we derived several expressions for the quantum Fisher information matrix (Eqs.~(\ref{eq:mixed_QFI},\ref{eq:multimode_QFI},\ref{eq:limit_formula},\ref{eq:pure_non-elegant})) for the multi-parameter estimation of multi-mode Gaussian states, associated symmetric logarithmic derivatives (Eqs.~(\ref{eq:SLD_mixed},\ref{eq:SLD_mixed_williamson},\ref{eq:sld_pure})), and expressions that determine saturability of the quantum Cram\'er-Rao bound (Eqs.~(\ref{eq:condition_QCR},\ref{eq:multimode_condition},\ref{eq:pure_state_condition})). We then illustrated their use on several examples.
As our main results, we consider expression for the QFIM when the Williamson's decomposition of the covariance matrix is known, Eq.~\eqref{eq:multimode_QFI}, which can be used for example for finding optimal Gaussian probe states for Gaussian unitary channels; the limit formula together with the estimate of the remainder, Eqs.~\eqref{eq:limit_formula} and \eqref{eq:remainder}, which can be used for efficient numerical calculations, to any given precision; and expressions for SLDs, Eqs.~(\ref{eq:SLD_mixed},\ref{eq:SLD_mixed_williamson},\ref{eq:sld_pure}), which can be studied to provide the optimal measurement schemes.
In addition, we discussed and resolved problematic behavior of QFIM at the points of purity, and we devised a regularization procedure (Eqs.~(\ref{eq:any_state_QFIM},\ref{eq:SLD_regularization},\ref{eq:condition_QCRregularization_procedure2})) that allows to use expressions for mixed states to calculate quantities for Gaussian states with any number of pure modes. Altogether, we provided a useful set of tools for Gaussian quantum metrology.
\textit{\textbf{Acknowledgements}} I thank Tanja Fabsits and Karishma Hathlia, for reading the first version of the manuscript, and for useful feedback. This research was supported by the Foundational Questions Institute (FQXi.org). |
2202.00598 | \section*{Background}
Embedded feature selection in high-dimensional data to reduce the feature dimension can be time-consuming or even unmanageable in a reasonable time without acceleration. Such high-dimensional data with a very small sample size occur, for example, in biomarker pilot studies\footnote{Biomarkers are measurable indicators (1) for medical risk factors, (2) for a biological condition, (3) to study a disease, (4) to predict a diagnosis, (5) to determine the state of a disease or (6) the effectiveness of a treatment \cite{Califf2018}. Because of their usefulness and broad applicability, the search for biomarkers has gained considerable interest. Small pilot studies can be a first step when searching for new biomarkers or biomarker combinations to save effort, money, and time.
If data within a biomarker pilot study is based on high-throughput technologies, sample generation can be both expensive and time-consuming. Therefore, the sample size in these pilot studies is usually very limited while the number of potential biomarker candidates is easily in the thousands. \cite{Klawonn2016}}. For this kind of research, feature selection aims to exclude irrelevant features. When analyzing high-dimensional data with a statistically insufficient sample size, random effects cannot be eliminated \cite{Al-Mekhlafi2022}, overfitting is difficult to avoid, and outliers can have a high impact. Therefore, it is unlikely to directly find a final biomarker (feature) combination for clinical use within a pilot study. However, substantial feature reduction is highly relevant for extended studies with a larger sample size \cite{Al-Mekhlafi2022}. For example, only a massively reduced feature dimension allows the application of targeted "omics" \cite{Coman2020}, such as targeted proteomics \cite{Marx2013}, to detect proteins of interest with high sensitivity, quantitative accuracy, and reproducibility.\\
One approach to achieve a reduction of the feature dimension is embedded feature selection \cite{Ma2008} \cite{Torres2019} \cite{Qi2012}. How well the model fits the data is measured with a performance evaluation metric. For small sample sizes this is usually determined using cross-validation. But if only a very limited number of samples is available, k-fold cross-validation produces strongly biased performance estimates \cite{Vabalas2019}, \cite{Cawley2010}. Vabalas et. al \cite{Vabalas2019} therefore suggest applying nested cross-validation instead to obtain robust and unbiased performance estimates regardless of the sample size. The nested cross-validation has an outer and an inner loop. Thus, a complete nested cross validation has the number of outer folds multiplied by the number of inner folds (\textit{folds}$_{outer-cv} * {\textit{folds}}_{inner-cv}$) steps. In each of these steps a new model is built. The runtime of the model building process increases with the number of features. Consequently, building a large number of models with a high feature dimension leads to a long overall computation time.\\
To better adapt a machine learning model to its specific task, its hyperparameters must be tuned. Choosing tuned hyperparameters can directly impact the performance of the model \cite{Yang2020}. In order to reduce overfitting it is advisable to regularize the model building process by adapting the corresponding hyperparameters \cite{Nordhausen2009}. Given the multitude of hyperparameter dimensions that can be optimized, the search space for hyperparameter combinations can be large, making it challenging to evaluate every optimization configuration. Hence, finding a good combination of hyperparameter values often requires many trials. For each trial, nested cross-validation has to be applied. In addition, further time-consuming analyses may be necessary, such as computing SHAP values \cite{Lundberg2020} to determine feature importances \cite{Marcilio2020} based on the trained models. Hence, long computation times for each trial are a bottleneck for the required hyperparameter optimization.\\
One solution to this problem is to stop trials with unfavourable hyperparameter values at an early stage.
The strategy of early stopping in hyperparameter optimization - also called pruning - is different from the early stopping used by machine learning algorithms against overfitting. In this context, it means identifying and terminating unpromising trials of hyperparameter values as early as possible while continuing to calculate with the most promising combinations. By stopping the least promising trials early during training, better feature selection can be achieved with the same computational power. With the ability to compute more trials in less time, the hyperparameter space can be explored faster or deeper. This acceleration saves time and energy - and thus also money. Therefore, pruning is essential when budget is limited and calculation times are long.\\
However, standard pruning algorithms cannot handle the considered data in the usual way. They are unreliable at an early stage of nested cross-validation due to the high variance of the performance evaluation metrics (see Figure \ref{cumulated_mean}).
To solve this problem, we propose a new combined pruning strategy to accelerate this specific hyperparameter optimization massively. To the best of our current knowledge, no specific or adapted pruner currently exists suitable for nested cross-validation with high variance in the performance metric. Our proposed pruning strategy is the first to consider the treatment of outliers in small validation sets and the semantics of the results. In addition, we incorporate prior knowledge and combine different pruning approaches. Wang et. al \cite{Wang2021} also consider prior knowledge for pruning. They gain procedural knowledge from previous configurations and determine important hyperparameters to derive optimal configurations for neural networks. In contrast to Wang et. al \cite{Wang2021}, our approach refers to the minimum requirements for the performance evaluation metric. Moreover, we adjust the use of standard pruning strategies. Finally, we combine the three different approaches.\\
\section*{Pruning Automated Hyperparameter Optimization} \label{pruner}
We propose a three-layer pruning algorithm (see Algorithm \ref{alg: three-layer}) that accelerates the hyperparameter optimization.
\begin{algorithm}
\begin{algorithmic}[1]
\For{\textit{every iteration of the outer cross-validation}} \Comment{nested cross-validation}
\For{\textit{every iteration of the inner cross-validation}}
\State build model
\State evaluate model on validation set
\\
\Comment{pruning}
\If{\textit{no feature is important}}
\State stop trial for current combination of hyperparemter values
\State (see \textit{\nameref{semantic pruning}})
\\
\ElsIf {\textit{extrapolated performance evaluation metric\\ \hspace*{23mm} is not better than a user defined threshold }}
\State stop trial for current combination of hyperparemter values
\State (see \textit{\nameref{threshold-pruner}})
\\
\ElsIf {\textit{comparison-based standard pruner would prune }}
\State stop trial for current combination of hyperparemter values
\State (see \textit{\nameref{standard pruning strategy}})
\\
\Else
\State continue
\EndIf
\\
\State evaluate model on test set of outer loop
\EndFor \label{inner feature selection loop}
\EndFor \label{test loop}
\caption{Combined three-layer pruning.}
\label{alg: three-layer}
\end{algorithmic}
\end{algorithm}
\begin{samepage}
The three combined parts prune trials if they
\begin{itemize}
\item are worse compared to previously calculated trials\\
(see \textit{\nameref{standard pruning strategy}}),
\item are semantically irrelevant due to lack of information for feature selection\\
(see \textit{\nameref{semantic pruning}}), or
\item have insufficient prediction quality for the specific use case\\
(see \textit{\nameref{threshold-pruner}}).
\end{itemize}
\end{samepage}
In the following subsections, we present the three different concepts in detail. We then describe the combined three-layer pruner.
\subsection*{Adapting Intermediate Values for Standard Pruning Algorithms} \label{standard pruning strategy}
The first part of the three-layer pruner is based on the comparison of intermediate results. The intermediate value in cross-validation is usually the average of the previously calculated performance measures. However, using this directly as intermediate value is unreliable for this particular use case especially at the beginning of the nested cross-validation. Due to the very small sample size and thus a tiny validation set, the inner k-fold cross-validation tends to produce a high variance in the performance evaluation metric (see Figure \ref{figure_reduced_variance}). Outliers in tiny validation sets are one reason for that. The high variance leads to an unstable mean (see Figure \ref{cumulated_mean}) and increases the probability of aborting promising trials. For this, it is necessary to adapt the calculation of intermediate values.
\begin{figure}
[width=122mm]{cumulated-mean.pdf}
\caption{State-of-the-art cross-validation: Average of each previously calculated performance evaluation metric of a nested cross-validation as intermediate values. The figure shows a single trial calculated with colon cancer data \cite{Alon1999} (see \nameref{experimental-setup}. Experimental Setup).}
\label{cumulated_mean}
\end{figure}
\begin{figure}
[width=122mm]{Colon-reduced-variance-val.pdf}
\caption{Intermediate results of a nested cross-validation: The blue line shows the smoothed 20\% trimmed mean of the previously calculated performance evaluation metrics of a full inner cross-validation. Therefore, it starts after the first complete inner cross-validation has been fully calculated. The orange dots represent the performance evaluation metrics of every single step. This figure shows the same trial as Figure \ref{cumulated_mean}, which was also calculated with colon cancer data \cite{Alon1999} (see \nameref{experimental-setup}).}
\label{figure_reduced_variance}
\end{figure}
Hence, we do not consider each individual step of the nested cross-validation. Instead, an intermediate value always covers a complete inner k-fold cross-validation. In addition, we use the 20\% trimmed mean to smooth the high variance and eliminate the influence of outliers. These two interventions prevent overly aggressive pruning and result in more stable intermediate values (see Figure \ref{figure_reduced_variance}). Subsequently, a more stable intermediate value leads to an enhanced comparability of trials. To further increase the confidence of correct pruning, the pruner should have a high patience and prune
late. Patience in this context defines a number of iterations of the nested cross-validation
that are calculated before pruning is considered. That enables to decide based on a considerable amount of model performance evaluation data.\\
Generally, parallel processing can accelerate this computationally intensive hyperparameter optimization for high-dimensional data. In particular, asynchronous computation of the trials in a distributed environment is advantageous to reduce overhead. In this case, a pruner does not have to wait for the final results of other trials computed in parallel. Thus, multiple trials can be processed simultaneously without delay. The state-of-the-art Asynchronous Successive Halving pruner (ASHA) \cite{Li2020} applies such a strategy. It periodically monitors and ranks intermediate values and cancels the worst trials. As one component of the three-layer pruner, we have chosen a variant of ASHA implemented in Optuna \cite{Akiba2019}. This pruner scales linearly in a distributed environment. Nevertheless, other pruning strategies could also be applied that rely on the comparison of intermediate values. For example, median- or percentile-based pruning as less sophisticated pruning strategies \cite{Akiba2019}.
\subsection*{Pruning Semantically Irrelevant Trials of Hyperparameter Combinations} \label{semantic pruning}
Supplementary to the adapted calculation of intermediate values for a comparison-based pruning strategy (see \textit{\nameref{standard pruning strategy}}) the second part of the three-layer pruner considers the semantics. We distinguish between semantically irrelevant and possibly relevant trials. A trial containing a model without any selected features, as the importance of all features is zero, we define semantically irrelevant. Such a model is meaningless for the goal of feature selection. For example, a model might not include any selected features if the regularization is too strong due to inappropriate values for the hyperparameters. Trials containing those meaningless models are immediately terminated when they are detected. This pruning strategy is already active in the very first step of the nested cross-validation.
\subsection*{Pruning Trials with an Insufficient Performance Evaluation Metric} \label{threshold-pruner}
Integrating prior knowledge and accounting for variance provides an additional pruning opportunity. The third part of the tree-layer pruner does not focus on terminating inferior or semantically irrelevant trials. Instead, it prunes trials if they have an insufficient validation result compared to a user-defined threshold based on prior or domain knowledge. While comparison-based pruning requires a longer patience (see \textit{\nameref{standard pruning strategy}}), pruning against this threshold can already start earlier after half the folds of the first inner cross-validation or at least four steps. With this, we ensure to have at least four performance measurements to calculate a median. The threshold-based pruning strategy aims to close the gap at the beginning of a comparison-based standard pruning strategy (see Figure \ref{three-layer-pruner-patience}) by starting earlier. In addition, it returns intermediate results of the nested cross-validation. This has the advantage that the hyperparameter optimization algorithm can already orient itself in the hyperparameter fitness landscape without fully calculating unpromising trials.
However, we apply this pruning strategy only for the first third of all complete inner loops of a nested cross-validation. If a trial has not stopped at this point, the comparison-based pruner can make more reliable decisions.\\
As mentioned for the standard pruning strategy (see \textit{\nameref{standard pruning strategy}}), a trial should normally be pruned after considering a complete inner cross-validation to avoid overly aggressive pruning due to outliers and variance. But with this part of the three-layer pruner we already prune during the first inner k-fold cross validation if the given threshold is unlikely to be matched by its end. To achieve this, we assume that
\begin{itemize}
\item omitting the first sample in the outer cross-validation does not massively change the median of the corresponding inner k-fold cross-validation,
\item the variance of the validation performance evaluation metrics is similar for all inner cross-validations, and
\item the first complete inner cross-validation indicates the rough level of the overall validation performance evaluation metric.\\
\end{itemize}
For this part of the combined pruning strategy, we use the median instead of the 20\% trimmed mean, as it may not be possible to omit 40\% of a very small number of results. Nevertheless, it is not useful for early pruning to directly compare the current median to the threshold, as this leads to a higher number of falsely pruned trials with respect to this threshold (see Figure \ref{percentage}). Instead, we additionally consider extrapolated values for the missing steps of the current inner cross-validation. Extrapolating all remaining missing values of the full nested cross-validation is not reasonable. This is because an overall too optimistic extrapolation could lead to a later termination of a trial, which in turn would increase the computation time.\\
For an optimistic extrapolation in order to keep promising trials, only the values that could lead to a better metric are relevant. For this reason, we only consider the deviation from the median $\tilde{x}$ in direction to optimize rather than the variance, the absolute mean or median deviation.\\
To extrapolate the missing values \textit{e}, we explored three different approaches:\\
\begin{itemize}
\item the \textbf{optimal value of the evaluation metric} (for example 0 for logloss or 1 for AUC)\\
\textit{e = optimal performance evaluation metric}\\
\item median $+/-$ \textbf{maximum deviation from the median} in direction to optimize\\
$e = \tilde{x}\begin{cases} -\text{ } max(x_1, x_2, .. , x_s) & \text{in case of minimizing}\\ +\text{ } max(x_1, x_2, .. , x_s) & \text{in case of maximizing} \end{cases}$\\
\\
\item median $+/-$ \textbf{mean deviation from the median} in direction to optimize\\
$e = \tilde{x}\begin{cases} -\text{ } mean(x_1, x_2, .. , x_s) & \text{in case of minimizing}\\ +\text{ } mean(x_1, x_2, .. , x_s) & \text{in case of maximizing} \end{cases}$\\
\\
\end{itemize}
The value, which is finally compared with the threshold, is composed of the median $\tilde{x}$ for all steps $s$ calculated so far and an optimistically extrapolated value $e$ for all steps $m$ of the current inner cross-validation that are still missing (see Equation \ref{trial-pruned}). We do not extrapolate any values for any following inner cross-validations. The number of steps already calculated $s$ is multiplied by their median $\tilde{x}$. In case of maximizing, the extrapolated value $e$ multiplied by the number of missing steps until the next complete inner cross-validation loop is added, where $m$ is the number of these steps. In case of minimization, that value is subtracted. This result divided by the number of all steps until the next complete inner cross validation $s + m$ is compared to the threshold in order to decide if a trial is stopped.
\begin{equation}\label{trial-pruned}
\text{prune trial} = \begin{cases} threshold < \frac{(\tilde{x}*s)-(e*m)}{s + m} & \text{in case of minimizing}\\ \\threshold > \frac{(\tilde{x}*s)+(e*m)}{s + m} & \text{in case of maximizing} \end{cases}
\end{equation}
This part of the pruner is useful to speed up the rough search in the relevant range of the respective hyperparameters at the beginning of the optimization process. For example, Optuna \cite{Akiba2019} uses a few (10 by default) randomly chosen combinations of hyperparameter values for its start up. Therefore, those trials can often be pruned early.
\subsection*{Combined Three-Layer Pruner}\label{three-layer_complete}
The elements of the combined three-layer pruner operate within different ranges of the nested cross-validation (see Figure \ref{three-layer-pruner-patience}). Applying the semantic pruning strategy, a trial is pruned immediately if no features are selected, starting with the very first model created. In contrast, the threshold-based pruning strategy becomes active only after at least 4 steps or half of the folds of the first inner cross-validation loop. It is only active within the first three iterations of the outer cross-validation loop. After that, the more reliable standard comparison-based pruning strategy takes over. It removes all trials worse than previous ones or less promising than those currently running in parallel. This comparison-based strategy requires computing significantly more steps if there is a high variance in individual results in order to allow a valid comparison of different trials. A decision based on more validation results reduces the risk of stopping a promising trial.
\begin{figure}
\begin{center}
[width=122mm]{combined-pruner-patience.pdf}
\caption{Operating times of the different parts of the three-layer pruner.}
\label{three-layer-pruner-patience}
\end{center}
\end{figure}
The two pruning strategies, based on semantic and prior knowledge, respectively, complement this standard pruning strategy to further speed up hyperparameter optimization (see Figure \ref{percentage}).\\
When a comparison-based or an extrapolating threshold based pruning strategy terminates a trial, the current evaluation metric should be returned. This has the advantage that the hyperparameter optimization algorithm receives a representative signal in order to be able to adapt better.
\section*{Experimental Setup} \label{experimental-setup}
We repeated and evaluated three experiments to compare the pruning strategies. They are based on three different biological data sets. Whereas other high-dimensional biological datasets with a small sample size most likely yield similar results.
Each individual experiment includes 30 repetitions of the same hyperparameter optimization with 40 trials each. All trials are fully computed to allow simulations and direct comparisons.\\
In contrast, the fourth experiment implements the principle of returning the current performance evaluation metric when stopping. It not only simulates, but shows the interaction of the three-layer pruner as a whole. We use only one biological data set and one hyperparameter optimization for this experiment since other examples follow a similar pattern.\\
All computations were performed on a cluster platform (Sun Grid Engine) using twenty cores working in parallel. Results and intermediate results were saved in a local SQLite database with exclusive access for the current experiment.
\subsection*{Data Sets used for Empirical Evaluation} \label{data}
The first biological data set from Alon et al. includes 40 tumor and 22 normal samples from colon-cancer patients with 2000 measured genes \cite{Alon1999}. The second data set is based on a prostate cancer study comparing 52 patients with 50 healthy controls. The genetic activity was measured for 6033 genes \cite{Efron2016}. Golub et al. provide the third data set based on gene expression measurements on 72 leukemia patients with 7128 genes \cite{Golub1999}. 47 patients with acute lymphoblastic leukemia (ALL) and 25 patients with acute myeloid leukemia (AML) \cite{Camargo2020}.\\
We transformed all data sets so that each row corresponds to a patient, each column to a gene, and the label as an integer 0 or 1 to the respective classes. In addition, to ensure comparability and to simulate pilot studies with very small sample sizes, we included only the first 15 samples of each class. Thus, all data sets are balanced and contain 30 samples. Since there are no missing values and scaling is not required for tree-based models, no further preprocessing was performed.
\subsection*{Scoring Metric}
For our experiments with balanced binary classes, we decided to use logloss as a performance evaluation metric. We chose this way because metrics based on the confusion matrix \cite{Tharwat2021}, such as F1, accuracy, average precision value, and even AUC, have the disadvantage that the results are not continuous when minimal validation sets are used. Logloss, on the other hand, provides a more fine-grained signal \cite{Bishop2006} even for these tiny validation sets.
\subsection*{Hyperparameter Optimization Setup}\label{hyperparameter-optimization}
We used Optuna v2.9.1 \cite{Akiba2019} as automatic hyperparameter optimization software for our experimental setup. It is easy to use, scales well in distributed systems, integrates with Sun Grid Engine, and already provides an implementation of the Asynchronous Successive Halving (ASHA) pruner \cite{Li2020}. We opt for the multivariate Tree-structured Parzen Estimator (TPE) sampler, which outperforms the independent TPE sampler, as Falkner et al. have shown \cite{Falkner2018}.\\
We evaluate each combination of hyperparameter values with nested cross-validation to obtain an unbiased performance evaluation. For the outer loop, we use leave-one-out cross-validation to save as much data as possible for the training set.
As inner loop, we use a stratified 10-fold cross-validation\footnote{The calculation time would further increase if the even more suitable leave-pair-out cross-validation \cite{Airola2009} was used instead of the standard 10-fold inner cross-validation. If regression was chosen instead of classification, nested leave one out cross-validation would also be possible to save even more data for the validation set.}
including at least one sample of each class in each validation set.
All intermediate validation results of each fold are stored for later analysis. While we did not calculate the test metric for our pruning simulations, obviously in practice this is mandatory. To predict the associated class of an outer folds test set, each model of an inner fold should be used.\\
We use random forest for our simulations of tree-based embedded feature selection \cite{Qi2012}. Tree-based supervised learning performs well with unstandardized raw data and is robust to outliers. The models for our simulation we have trained with the fast C++ random forest implementation of lightGBM v3.2.1 \cite{Ke2017}. We set a fixed number of 100 trees to enhance comparability of trial durations. As further hyperparameters for regularization we choose \verb|lambda_l1|, \verb|min_gain_to_split| and \verb|min_data_in_leave| to combat overfitting. The complete hyperparameter space and the parameters for the asynchronous successive halfing pruner are listed in \nameref{app:A}. For semantic pruning (see \textit{\nameref{semantic pruning}}), we determine the feature importances based on information gain \cite{Quinlan1986}, the total gain of the splits that use the feature \cite{lightgbmBoosterDoc}.
\subsection*{Comparison of Pruning Strategies}
Due to the random initialization of hyperparameter optimization and training, it is not realistic to take accurate measurements for a direct comparison of the pruners. Therefore, on the one hand, we repeat each hyperparameter optimization thirty times. On the other hand, we simulate optimizations based on previously stored performance evaluation metrics. To compare the different pruning strategies with consistent data, all respective trials were calculated to completion. The principle of returning the current performance evaluation metric instead of terminating the experiment was not implemented in these experiments. In this way, we exclude the random part of the model building process. This allows to directly compare different extrapolating threshold pruning strategies on the same basis.\\
Pruning deals with a trade-off between the early termination of trials and the risk of pruning a promising trial. Accordingly, we compare pruning strategies regarding
\begin{itemize}
\item[a)] the number of steps until pruning,
\item[b)] the number of incorrectly stopped trials, and
\item[c)] their margin of error.
\end{itemize}
With the margin of error we evaluate the falsely pruned trials not only quantitatively but also qualitatively.
For the direct comparison, however, we did not consider time measurements because the experiments ran on different hardware, which leads to a lack of comparability. However, the number of models built is proportional to the calculation time and allows an objective comparison in this context. Therefore, we assume that the calculation of fewer steps within a nested cross-validation requires proportionally less time.
\section*{Results}
\begin{figure}[h!]
\begin{center}
[width=122mm]{colon-percent.pdf}
[width=122mm]{prostate-percent.pdf}
[width=122mm]{leukemia-percent.pdf}
\caption{The different extrapolation strategies within the three-layer pruner compared to using the ASHA pruner alone as a single pruner component applied to three different biological data sets. The error bars show the deviation between the repetitions of each experiment (see \textit{\nameref{experimental-setup}} for further details)}.
\label{percentage}
\end{center}
\end{figure}
\begin{figure}[h!]
[width=106mm]{combined-colon-rotated.pdf}
\caption{\textbf{Comparison of different extrapolation strategies}: Combined parts of the three-layer-pruner for 30 repetitions of hyperparameter optimization for colon cancer data analysis containing 40 trials each.}
\label{combination1}
\end{figure}
\begin{figure}[h!]
[width=106mm]{combined-prostate-rotated.pdf}
\caption{\textbf{Comparison of different extrapolation strategies}: Combined parts of the three-layer-pruner for 30 repetitions of hyperparameter optimization for prostate cancer data analysis containing 40 trials each.}
\label{combination2}
\end{figure}
\begin{figure}[h!]
[width=106mm]{combined-leukemia-rotated.pdf}
\caption{\textbf{Comparison of different extrapolation strategies}: Combined parts of the three-layer-pruner for 30 repetitions of hyperparameter optimization for leukemia data analysis containing 40 trials each.}
\label{combination3}
\end{figure}
\begin{figure}[h!]
\begin{center}
[width=122mm]{three-layer-pruner-small.pdf}
\caption{Pruning unpromising trials with the combined three-layer pruner: Terminated trials based on the intermediate values of a hyperparameter optimization (colon cancer data \cite{Alon1999}). The steps represent the iterations of the outer cross-validation.}
\label{three-layer-pruner}
\end{center}
\end{figure}
\begin{figure}[h!]
[width=122mm]{colon-cv-pruner-hist.pdf}
\\
[width=122mm]{prostate-cv-pruner-hist.pdf}
\\
[width=122mm]{leukemia-cv-pruner-hist.pdf}
\caption{\textbf{Margin of error for incorrectly pruned trials compared to the set threshold}: On the x-axis the difference between the 20\% trimmed mean of the performance evaluation metrics (logloss) of a full nested cross-validation and the threshold for the extrapolating pruner is shown. Only falsely pruned trials are included. Their respective number is plotted on the y-axes. For one hyperparameter optimization, only the globally largest margin of error was considered. Four different extrapolation methods are compared:
1. no extrapolation only the current median 2. mean deviation from the median subtracted from the current median 3. maximum deviation from the median subtracted from the current median 4. the optimal performance evaluation metric.
The figures apply to a complete experiment with 30 repeated hyperparameter optimizations including 40 trials each. Three biological data sets were examined using the same experimental setup. Further experimental details are specified in \textit{\nameref{experimental-setup}}.}
\label{qualitative_comparison}
\end{figure}
In our repeated experiments, the best overall trial was never pruned. At the same time, considerably fewer models had to be trained (see Figure \ref{percentage}) proportionally saving a remarkable amount of time.
All three parts of the combined three-layer pruner contributed to this acceleration (see Figures \ref{combination1},\ref{combination2} and \ref{combination3}). Not only the adapted usage of the state-of-the-art comparison-based ASHA pruner \cite{Akiba2019} was responsible for this (see \textit{\nameref{standard pruning strategy}}).
We can show that the additional pruning strategies based on semantic or prior knowledge can prevent computing complete trials or larger parts of trials that might not lead to a useful result. Figure \ref{three-layer-pruner} illustrates the early termination of unpromising trials within the first inner cross-validation combined with the comparison-based pruning of acceptable trials.\\
All three extrapolation methods for the threshold pruning strategy accelerate and enhance the hyperparameter optimization. They result in more correctly completed trials and fewer trained models than comparing a threshold directly to the median (see Figure \ref{percentage}). Extrapolation compensates the uncertainty due to the lack of robustness of the results at the beginning of the nested cross-validation. The \textit{optimal value of the evaluation metric} as a basis for extrapolation leads to the smallest number of falsely pruned trials and to the highest number of models to be trained. On the other hand, the \textit{maximum deviation from the median} in direction to optimize leads to a slight increase in the number of incorrectly pruned trials but, in turn, reduces the calculation time. The \textit{mean deviation from the median} in direction to optimize finally requires the least number of trained models for the hyperparameter optimization but has another tiny increase in the number of falsely pruned trials.
\textit{Mean deviation from median} as extrapolation method for the threshold pruning strategy was the approach with the highest acceleration.
Using this extrapolation method, the combined three-layer pruner could still achieve the same optimization result as the standard ASHA pruner \cite{Akiba2019} as single pruning strategy. But on average, 81.3\% fewer models for the colon cancer data \cite{Alon1999}, 64.2\% fewer models for the prostate cancer data \cite{Efron2016} and 63.3\% fewer models for the leukemia data \cite{Golub1999} had to be trained. The corresponding error bars in Figure \ref{percentage} show that the deviations of the acceleration between the repetitions of the individual experiments are relatively small. This indicates a repeatable and stable acceleration even though the hyperparameter optimization is initialized randomly.\\
Furthermore, the three extrapolation strategies and the threshold pruner without extrapolation were qualitatively compared. This qualitative analysis of the incorrectly pruned trials for three different biological datasets is shown in Figure \ref{qualitative_comparison}. It is again noticeable that larger errors occur when only the current median without any extrapolation is compared against a threshold. As expected, the correctness is the highest for the optimal performance evaluation metric as basis for the extrapolated values. \\
Despite any (expected) errors, a threshold with a clear distance to a likely best performance evaluation outcome always led to the completion of the most promising trial of the hyperparameter optimization. Due to this distance, small errors related to the threshold were not relevant for the global outcome of the complete hyperparameter optimization.
\section*{Discussion}
The overall goal of the combined pruning strategy is calculating many parallel trials, stop unpromising ones early, and continue the best. Termination of unpromising trials based on the principle of extrapolation and semantic pruning can also accelerate simple cross-validation. However, speedup might be smaller in this case, since normally the overall number of iterations in cross-validation is lower compared to nested cross-validation. For nested cross-validation these two pruning strategies based on semantic or prior knowledge help to roughly tune the hyperparameters fast. They increase the speed-up compared to using a comparison-based pruning strategy alone. Especially at the beginning of the hyperparameter optimization, comparison-based standard pruners compute complete trials or larger parts of trials that do not lead to a useful result. Otherwise, they would not have a stable baseline for comparison, which in turn would risk stopping promising trials. The advantage of the comparison-based pruning strategy is the ability to finely compare superior trials. Comparison-based pruning is, of course, not limited to tree-based methods. It can also accelerate the hyperparameter optimization of any embedded feature selection or supervised learning method that provides intermediate results, such as LASSO regression \cite{Muthukrishnan2016}.\\
However, saving time by early termination of trials must always be balanced against the risk of pruning a promising trial. For this reason, the pruning strategy based on a user-defined threshold takes prior knowledge into account. The threshold itself can influence the global error probability, but finding the best threshold is a challenge. The closer it is to the optimum, the more aggressive the pruning strategy and the higher the probability of error, and vice versa. The threshold should be an unacceptable value or a minimum requirement for the evaluation metric in the specific domain. Therefore, it optimally should be chosen with human experience. Likewise, knowledge from previous analyses related to the expected evaluation metric can serve as a base value. If determining a threshold in this way is not possible, it could be set slightly better than the naive classification baseline for the given data. Another option to get orientation regarding the expected evaluation metric could be applying an appropriate filter method for feature selection \cite{Bommert2020}. However, a single good biomarker candidate or feature might appear only by chance in high-dimensional data with tiny sample size. To better estimate this probability, the HAUCA method \cite{Klawonn2016} could be applied. It provides information on whether more informative features are likely to be present in the data than would be expected by chance for a given value of an evaluation metric.\\
Another conceivable option would be to automatically adjust the threshold to the current best performance evaluation metric after each successful trial. In practice, however, this can lead to pruning of promising trials, as it is likely that the threshold will be slightly mismatched (see Figure \ref{qualitative_comparison}). These mismatches can occur due to a high variance of the performance evaluation metric. We therefore suggest choosing a defensive threshold that has a sufficient distance to the expected best value of the performance evaluation metric.\\
In addition to the threshold, the method of extrapolation affects the acceleration as well as the probability of error. The more "aggressive" (e.g. \textit{mean deviation from the median}) the method of extrapolation, the less models are built but the higher the risk of falsely pruned trials. In contrast, the more "defensive" (e.g. \textit{optimal value of the evaluation metric}) the method of extrapolation, the more models must be calculated and the lower the risk of error.\\
Additional speed-up can be achieved by parallelization. As the combined asynchronous pruning strategy scales very well in distributed systems, massive parallelization of trials can further reduce the computation time.
\section*{Conclusion and Outlook}
Hyperparameter optimization with high-dimensional data and nested cross\hyp{}validation leads to long computation times. Thus, its speed-up is a key factor. Combining three different specifically adapted pruning strategies enables a substantially acceleration of the hyperparameter optimization. The part of the combined pruning strategy based on a user-defined threshold incorporates prior knowledge, which might be challenging to determine.\\
A complementary, time-saving early stopping approach is proposed by Makarova et al. \cite{Makarova} which focuses on early stopping of the complete hyperparameter optimization to avoid overfitting due to too extensive hyperparameter optimization. Combining both approaches is a promising future research topic.\\
In bioinformatics, for example, only accelerated hyperparameter optimization enables the analysis of larger multi-omics or very large high-throughput datasets in a reasonable time. In particular, faster computation allows the application of advanced ensemble techniques - the combination of different feature selection approaches. These ensemble techniques, in turn, can provide more reliable results \cite{Luftinger2021}. From the infection research perspective, the improved reliability increases the likelihood of finding more relevant biomarker candidates. And this allows cutting-edge research to gain new insights.\\
Reduced computation time can not only enable or speed up research in general, but also reduce costs such as personnel, hardware, management, cloud service, or energy costs. To slow down global warming, it is desirable to contribute to the reduction of CO$_2$ emissions by improving algorithms.\\
\section*{Availability and Requirements}\label{sec:Requirements}
Project name: cv-pruner\\
Project home page: https://github.com/sigrun-may/cv-pruner\\
Operating system(s): Platform independent\\
Programming language: Python\\
Other requirements: Python 3.8 or higher\\
License: MIT\\
Any restrictions to use by non-academics: -
\section*{Appendix A.}\label{app:A}
In this appendix, first we describe the tuned hyperparameters for regulation to combat overfitting. Following this, we present the complete hyperparameter space for the experiments (Table \ref{tab:hyperparameter-space}) and the paramenters for the asynchronous successive halfing pruner (Table \ref{tab:hyperparameters-asha}) from Chapter \textit{\nameref{hyperparameter-optimization}}.\\
\noindent In addition to the parameters required for lightGBM, we tuned the following supplemental hyperparameters for regulation to prevent overfitting:
\begin{itemize}
\item \verb|lambda_l1|: L1 regularization.
\item \verb|min_gain_to_split|: Gain is basically the reduction in training loss that results from adding a split point.
\item \verb|min_data_in_leave|: Minimum number of observations that must fall into a tree node for it to be added.
\end{itemize}
Further details can be found at \url{https://github.com/microsoft/LightGBM}\\
\begin{table}[h!]
\caption{Hyperparameters of experiments taken from uniform distribution.}
\label{tab:hyperparameter-space}
\begin{tabular}{lcc}
\hline
hyperparameter & min value & max value \\ \hline
min data in leaf & 2 & $\left \lceil{sample \ size / 2}\right \rceil$ \\
lambda l1 & 0.0 & 3.0 \\
min gain to split & 0 & 5 \\
max depth & 2 & 15 \\
bagging fraction & 0.1 & 1.0 \\
bagging freq & 1 & 10 \\
num leaves & 2 & $\min(2^{max \ depth} - 1, \quad 80)$ \\ \hline
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{Parameters for the asynchronous successive halving pruner.}
\label{tab:hyperparameters-asha}
\begin{tabular}{lcc}
\hline
hyperparameter & value \\ \hline
min resource & auto \\
reduction factor & 3 \\
min early stopping rate & 2 \\
bootstrap count & 0 \\ \hline
\end{tabular}
\end{table}
\begin{backmatter}
\section*{Acknowledgements
Diana Zimper, Philip May, Juliane Hoffmann
\section*{Funding
We would like to acknowledge funding for this project
from "Paving the way towards individualized vaccination (i.Vacc) - Exploring multi-omics Big Data in the general population based on a digital mHealth cohort". The funding agency did not influence the design of the experiments, analysis and interpretation of the data, or writing of the manuscript.
\section*{Abbreviations
ASHA - Asynchronous Successive Halving: "ASHA is an asynchronous pruning strategy suitable for massive parallelism \cite{Li2020}. Successive halving is a bandit-based algorithm to determine the best among multiple configurations \cite{Akiba2019}.
\section*{Availability of data and materials
All data analysed are included in this published articles \cite{Alon1999}, \cite{Efron2016}, \cite{Golub1999}.
\section*{Ethics approval and consent to participate
Not applicable.
\section*{Competing interests}
The authors declare that they have no competing interests.
\section*{Consent for publication
Not applicable.
\section*{Authors' contributions}
S.M. designed the algorithm, analyzed and interpreted the data and created new Software used in the work. S.H. and F.K. substantively revised the manuscript. All authors read and approved the final manuscript.
\bibliographystyle{bmc-mathphys} |
1608.01719 | \section{Introduction}
Conductometric chemical sensors are known to be very sensitive to
humidity levels in the environment
~\cite{barsan2001conduction,barsan2003understanding,Hubner2011347,morante2013chemical,buehler1997temperature,yamazoe2005toward,romain1997situ,hossein2010compensation,fine2010metal,oprea2009temperature,wang2010metal}.
This cross-sensitivity challenges the tasks of identification and
quantification of volatiles in uncontrolled scenarios. For example,
electronic noses can be used for human monitoring
purposes~\cite{romain2005monitoring,romain2008complementary,B307905H,ogawa2000monitoring,oyabu2001odor,oyabu2003proposition}.
In fact, they have been successfully used to quantify the number of
people working in a space-craft
simulator~\cite{fonollosa2014human}. In this case, it is likely that
the primary signal used by the algorithm to estimate the number of
people present at some given time is the humidity level in the
chamber. If we filter the sensor responses by the humidity and
temperature changes, a clearer chemical signature of the chamber can
be obtained, and this can facilitate more complex monitoring tasks
like identifying individuals \cite{rodriguez2013analysis}. A possible
solution to this sensitivity problem is the design of a special
sensing chamber that controls humidity and delivers the gas to the
sensors under predefined conditions
~\cite{chatonnet1999using,shevade2007development,ryan2004monitoring,fonollosa2014human,hossein2010compensation}.
Such preconditioning chambers are effective for signal improvement,
but their use increases the costs of electronic nose design for
applications in continuous monitoring of the environment
~\cite{B307905H}. A different approach is to build a model that
predicts the changes in the sensor conductance as a function of
humidity and temperature
variations~\cite{buehler1997temperature,hossein2010compensation,fort2011modeling,piedrahita2014next}.
The prevailing phenomenological model of sensor sensitivity is that
the ratio of the sensor resistance depends on a power law of the gas
concentration~\cite{windischmann1979model}. The model provides
accurate predictions when the gas is known and under controlled
conditions. However, it is rendered inaccurate with changes in the
environment. Correction methods based on artificial neural
networks~\cite{hossein2010compensation} using present and past values
of the input features are proven to be successful despite lacking an
explanation of the underlying processes. Fundamental models, on the
other hand, can capture the dynamical changes of resistance under
humidity variations accurately~\cite{fort2011modeling}. In these
models, the number of parameters is not large, but the model
parameters depend on the presented gas to the sensors. Therefore, in
continuous monitoring systems, where there can be a complex mixture of
gases present in the air, it is indeed challenging to make proper
corrections on the sensor readings based on humidity and temperature
variations.
In this work, we propose an online methodology to subtract the changes
driven by humidity and temperature from the MOX sensor responses, and
demonstrate that this procedure enhances the performance of pattern
recognition algorithms in discriminating different chemical
signatures. We first develop a model based on the energy bands of
n-type semiconductors that is suitable for low-power micro-controllers
(Texas Instruments MSP430F247). We then make use of the predictions of
this model to subtract changes expected to be due to humidity and
temperature variation. Using a wireless electronic nose composed of 8
MOX sensors, we collected 537 days of data in the residence of one of
the authors and showed that our model is capable of predicting all MOX
sensors with a coefficient of determination $R^2$ larger than 0.9.
Because the electronic nose was subject to several unpredictable
conditions (house cleaning, wireless connectivity issues, etc), this
data set represents a wide variety of events present in home
monitoring scenarios. To evaluate the impact to online discrimination
of volatiles identities, we created a small data set consisting of
exposing the electronic nose to two distinct stimuli: wine and
banana. We show that the discrimination performance is significantly
enhanced using the decorrelated data combined with the raw time
series. This is a crucial task for any electronic nose system if one
wants to characterize or detect events based on their chemical
signatures in the presence of varying environmental conditions.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Fig1_sensors.pdf}
\caption{Illustrative example of recording during one day using the
wireless electronic nose composed of 8 MOX sensors, including a
humidity and temperature sensor. The first panel presents the
humidity values, the second panel is the external temperature, and
then resistance values for 4 different MOX sensors in the
board. \label{fig:example} }
\end{figure}
\section{Example of sensors correlation with humidity and temperature}
In Fig.~\ref{fig:example}, we show a representative example of the humidity
problem using chemical sensors for continuous monitoring purposes. The
electronic nose in our setup is composed of 8 metal oxide (MOX) sensors, along
with temperature and humidity sensors. Such platform was previously used in our
wind tunnel studies to identify 10 gases at different
locations~\cite{vergara2013performance}. As a result of this previous
investigation, we know that we can discriminate between gases accurately, and
estimate gas concentrations in the ppm range ~\cite{fonollosa2014chemical}. The
time series shown in Fig. \ref{fig:example} were obtained in October 2014 in a
regular working day, in the residence of one of the authors.
The top panel shows the humidity levels throughout a complete day, where the
$x$-axis indicates the hour of the day. For example, the first rise in humidity
at about 5:30 AM corresponds to the morning shower. The sudden drop in humidity
at about 6:30 AM indicates opening the bathroom window, and the changes observed
at 5 PM are associated with the moment at which the family was returning home and the
door to the backyard was being opened. The second panel presents the temperature of
the electronic nose location that we denote by $T_E$ to differentiate it from
the temperature of the sensor heater, $T$. This residence did not have any air
conditioning system or heater operating during this period.
It is clear from this graph that the environmental changes in humidity
and temperature are often correlated. The measured resistance values
of the MOX sensors are presented in the four bottom panels. Although
the sensor board is made of 8 MOX sensors, here we present recordings
of only 4 of them because the remaining sensors are highly correlated
with those shown. Changes in the sensors resistance are strongly
affected by changes in humidity and temperature, as expected from the
extensive literature on the
topic~\cite{barsan2001conduction,barsan2003understanding,Hubner2011347,morante2013chemical,buehler1997temperature,yamazoe2005toward,romain1997situ,hossein2010compensation,fine2010metal,oprea2009temperature,wang2010metal}.
Nevertheless, the whole data set also includes examples where MOX
sensor changes cannot be explained only in terms of variations in
humidity and temperature as there also exist chemical variations in
the environment that have effects on sensors' responses. As exposed
before, our goal is to find a way to decorrelate the MOX sensors from
humidity and temperature, and show that this improves pattern
recognition tasks such as discrimination of gas identity.
\section{Design of the wireless electronic nose}
\begin{table}[t]
\centering
\begin{tabular}{@{}|l|c|l|}
\hline
Sensor type & Number of units &Target gases\\
\hline
TGS2611&1&Methane\\
TGS2612&1&Methane, Propane, Butane\\
TGS2610&1&Propane\\
TGS2600&1&Hydrogen, Carbon Monoxide\\
TGS2602&2&Ammonia, H2S, Volatile Organic Compounds (VOC)\\
TGS2620&2&Carbon Monoxide, combustible gases, VOC\\
\hline \hline
\end{tabular}
\caption{Sensor devices selected for the wireless electronic nose
(provided by Figaro Inc.)}
\label{tablesensorts}
\end{table}
In this section, we describe the electronic nose designed for home
monitoring purposes. The sensor array is based on eight metal oxide
gas sensors provided by Figaro Inc. The sensors are based on six
different sensitive surfaces, which are selected to enhance the system
selectivity and sensitivity. Table 1 shows the selected sensing
elements along with the corresponding target compounds. In order to
control the variability between the sensing elements and increase the
flexibility of the sensing platform, the operating temperature of the
sensors can be adjusted by applying a voltage to the built-in,
independently reachable heating element available in each sensor. The
humidity and temperature sensors are integrated in the board using the
Sensirion SHT75. The device is very similar to the
M-Pod~\cite{piedrahita2014next}, except that ours is directly powered
by any electrical outlet to record continuously over long periods of
time.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{Figures/complete_system.png}
\caption{The electronic nose made of the sensor board (right) and a
wireless communication board.\label{fig:enose}}
\end{figure}
The sensor array is integrated with a customized board that includes a
microprocessor MSP430F247 (Texas Instruments Inc.). In
Fig. \ref{fig:enose} we show the operating electronic nose. The
microcontroller was programmed to perform the following actions: i)
Continuous data collection from the eight chemical sensors through a
12-bit resolution analog-to-digital converter (ADC) device at a
sampling rate of 100 Hz; ii) Control of the sensor heater temperature
by means of 10 ms period and 6 V amplitude Pulse-Width-Modulated (PWM)
driving signals; iii) A two-way communication with another device to
transmit the acquired data from the sensors and control the voltage in
the sensors' heaters. The sensor board provides serial data
communication to another device via either a USB and/or a 4-pin
connector (Tx, Rx, Gnd, Vcc).
A wireless communication module acts as a bridge between the
MSP430F247 microcontroller and the network. The communication with the
MSP430F247 microcontroller is done via the UART port, whereas the
communication with the network is performed wirelessly. The board is
based on a WiFly RN-131G radio module included in a RN-134 SuRF board
(Roving Networks Inc). The WiFly module incorporates a 2.4GHz radio,
processor, full TCP/IP stack, real-time clock, FTP, DHCP, DNS, and web
server.
The module can be accessed via a RS-232 serial port (9600 default baud
rate) or a 802.11 wireless network so that its configuration can be
modified. The wireless communication module is configured such that it
accepts UDP and TCP connections, the baud rate of the microprocessor is
set to 115200 so that it can exchange data with the MSP430F247
microcontroller, and working with an external 4" reverse polarity
antenna to increase the power of the transmission.
\section{Online model for sensors response}
\label{sec:model}
An energy band model for n-type semiconductors describes the changes
in the resistance of the sensor before exposure, $R_I$, and after
exposure, $R_F$, as a nonlinear expression of the changes in the
semiconductor's energy
bands~\cite{barsan2001conduction,barsan2003understanding}. Energy
bands changes depend on variations in humidity and gas external
temperature, which modulates the overall transduction. If we denote by
$\Delta\Phi=\Phi_F-\Phi_I$ the work function change computed as the
difference between the work function after and before exposure, and we
express the electron affinity change as $\Delta \chi=\chi_F-\chi_I$,
the overall transduction can be expressed (following
~\cite{barsan2003understanding}) as:
\begin{linenomath}
\begin{equation}
\ln\left(\frac{R_F}{R_I}\right)=\frac{1}{k_B T}\left(\Delta\Phi-\Delta \chi\right),\label{eq1}
\end{equation}
\end{linenomath}
where $k_B$ is the Boltzmann constant, and $T$ is the sensor operating
temperature controlled by the built-in sensor heater. The sensor
temperature is not constant because it is modulated by the external
temperature, $T_E$. To be able to build a basic model to be fitted to
the data, we make the following assumptions. We assume that
relative changes in the external humidity, $\Delta H= h$, and changes
in external temperature, $\Delta T_E=t$, are small enough. We
also assume that the chemical content
remains unchanged during the environmental changes. This assumption is important because it is known that humidity
changes induce nonlinear changes in the energy depending on the
chemical agent (see ~\cite{morante2013chemical}). Under these
assumptions, we can rewrite the transduction in equation~\ref{eq1} as
\begin{linenomath}
\begin{equation}
\ln\left(\frac{R_F}{R_I}\right)=\frac{1}{k_B (T+\mu
t)}\left(\Delta\Phi(h)-\Delta \chi(h)\right),\label{eq2}
\end{equation}
\end{linenomath}
where $\mu > 0$ is a dimensionless factor that reflects the impact of
the external temperature into the sensor.
Because the sensor board is based on a Texas Instruments MSP430F247
micro-controller, which can only perform simple mathematical operations, we now
consider in equation \ref{eq2} terms up to second order in $\Delta H$ and
$\Delta T$. This removes
most of the non-linearities from equation \ref{eq2}, but without oversimplifying
the model. We investigate the validity of this approximation in section
\ref{sec:performancemodel} in each of the sensors separately. Thus,
\begin{linenomath}
\begin{multline}
\ln\left(\frac{R_F}{R_I}\right) = \left(\frac{1}{k_B
T}-\frac{\mu}{k_BT^2} t+\frac{\mu^2}{k_BT^3}
t^2+O(t^3)\right)\times \\ \left(\Delta\Phi(0)-\Delta \chi(0)+\left[
\left. \frac{\partial \Delta\Phi}{\partial h}\right|_{h=0}-
\left. \frac{\partial \Delta\chi}{\partial h}\right|_{h=0}\right]
h \quad + \right. \\ \left. \frac{1}{2}\left[\left. \frac{\partial^2 \Delta\Phi}{\partial
h^2}\right|_{h=0}-\left. \frac{\partial^2 \Delta\chi}{\partial
h^2}\right|_{h=0}\right] h^2 +O(h^3)\right) .
\label{eq:app}
\end{multline}
\end{linenomath}
Note that $\Delta\Phi(0)-\Delta \chi(0)=0$ because there are not
changes in humidity and temperature on our sampling time scale. The
simplified model is
\begin{linenomath}
\begin{eqnarray}
\nonumber \ln\left(\frac{R_F}{R_I}\right)&=&\frac{1}{k_B
T}\left[\left. \frac{\partial \Delta\Phi}{\partial
h}\right|_{h=0}-\left. \frac{\partial \Delta\chi}{\partial
h}\right|_{h=0}\right] h + \frac{1}{2k_B
T}\left[\left. \frac{\partial^2 \Delta\Phi}{\partial h^2}
\right|_{h=0}- \left. \frac{\partial^2 \Delta\chi}{\partial
h^2}\right|_{h=0} \right] h^2 \\ & &-\frac{\mu}{k_B
T^2}\left[\left. \frac{\partial \Delta\Phi}{\partial
h}\right|_{h=0}- \left. \frac{\partial \Delta\chi}{\partial
h}\right|_{h=0}\right] h t \; .
\label{prediction}
\end{eqnarray}
\end{linenomath}
Therefore, we fit the following model to the data
\begin{linenomath}
\begin{equation}
\ln\left(\frac{R_F}{R_I}\right)=\beta_1 \Delta H+ \beta_2
\left(\Delta H\right)^2+ \beta_3 \Delta H \Delta
T_E,\label{linearmodel}
\end{equation}
\end{linenomath}
where
\begin{linenomath}
\begin{eqnarray*}
\beta_1 &=& \frac{1}{k_B T}\left[\left. \frac{\partial \Delta\Phi}{\partial h}\right|_{h=0}-\left. \frac{\partial \Delta\chi}{\partial h}\right|_{h=0}\right] \\
\beta_2 &=& \frac{1}{2k_B T}\left[\left. \frac{\partial^2 \Delta\Phi}{\partial h^2}\right|_{h=0}-\left. \frac{\partial^2 \Delta\chi}{\partial h^2}\right|_{h=0}\right] \\
\beta_3 &=& -\frac{\mu}{k_B T^2}\left[\left. \frac{\partial \Delta\Phi}{\partial h}\right|_{h=0}-\left. \frac{\partial \Delta\chi}{\partial h}\right|_{h=0}\right] \; .
\end{eqnarray*}
\end{linenomath}
Thus, our model has only three parameters to be fitted: $\beta_1$, $\beta_2$ and
$\beta_3$. In particular, $\beta_1$ and $\beta_3$ have opposite sign and they
are related by $\beta_3/\beta_1=-\mu / T$. This means that the ratio
$|\beta_3/\beta_1|$ becomes smaller with increasing sensor temperature.
\section{Results}
\label{sec:performancemodel}
\begin{table}[t]
\centering
\begin{tabular}{@{}|l|cc|cccc|}
\hline
Sensor & RMS & $R^2$ & $\beta_{1}\,(\beta_{1}/se(\beta_{1})$) & $\beta_{2}\,(\beta_{2}/se(\beta_{2})$) & $\beta_{3}\,(\beta_{3}/se(\beta_{3}))$ & $\beta_3/\beta_1$ \\
\hline
1 & 0.06 & 1.00 & -0.0044 (-128.14)$^*$& 0.00014 (38.40)$^*$& 0.0110 (58.41)$^*$ & -2.61 \\
2 & 0.12 & 1.00 & -0.0110 (-186.04)$^*$& 0.00034 (54.11)$^*$& 0.0240 (71.75)$^*$ & -2.21 \\
3 & 0.12 & 1.00 & -0.0110 (-187.12)$^*$& 0.00034 (53.57)$^*$& 0.0230 (69.60)$^*$ & -2.18 \\
4 & 0.14 & 1.00 & -0.0110 (-190.95)$^*$& 0.00033 (55.31)$^*$& 0.0230 (73.06)$^*$ & -2.19 \\
5 & 1.24 & 0.98 & -0.0056 ( -41.48)$^*$& 0.00018 (12.23)$^*$& 0.0086 (11.15)$^*$ & -1.54 \\
6 & 0.48 & 0.99 & -0.0039 (-104.94)$^*$& 0.00012 (30.29)$^*$& 0.0071 (33.71)$^*$ & -1.84 \\
7 & 2.06 & 0.90 & -0.0070 ( -99.24)$^*$& 0.00022 (28.94)$^*$& 0.0095 (23.57)$^*$ & -1.36 \\
8 & 2.09 & 0.91 & -0.0057 ( -70.75)$^*$& 0.00020 (22.94)$^*$& 0.0029 ( 6.43)$^*$ & -0.52 \\
\hline \hline
\end{tabular}
\caption{Results of fitting the model defined in equation (\ref{linearmodel}). The
Root Mean Square (RMS) of the error in the predictions always remained below
3.0, and the coefficient of determination $R^2$ was always above 0.9. We also
show the coefficients $\beta_1$, $\beta_2$, and $\beta_3$ fitted for each
sensor, along with their signal-to-noise ratio ($se(X)$ stands for standard
error of $X$). All $\beta$ parameters are statistically significant
(indicated with a *), with a $p$-value below $10^{-10}$.}
\label{tableresults}
\end{table}
We fit the model defined in equation (\ref{linearmodel}) to data of 537 days
(from Feb 17, 2013 until June 5 2015) by down-sampling the time series to one
data point per minute and per sensor. Heaters for sensors 1-4 are always kept at
the same operating voltage, while sensors 5 to 8 are controlled under a protocol
that guarantees that the sensor responses always remain within a the same range
of values. Results summarized in Table~\ref{tableresults} prove the
effectiveness and statistical significance of the energy band model: the
accuracy rates achieved by the model, measured by the coefficient of
determination $R^2$, are above $90$\% for all sensors, and all the model
coefficients are statistically significant. Sensors with a fixed heater
temperature (i.e., sensors 1-4) outperformed sensors that operate with their
heater temperature actively changed (i.e., sensors 5-8). In the worst case
(sensor 8), the difference in $R^2$ is close to $10$\%. This probably suggests
that higher order terms become important in the approximation of equation
(\ref{eq:app}) when the heater temperature is actively changed. Moreover, as
predicted by equation (\ref{prediction}), the parameters $\beta_1$ and $\beta_3$
have opposite signs for all the sensors in the electronic nose. The ratios
$\beta_3/\beta_1$ estimated for the eight MOX sensors by our fitting (see
Table~\ref{tableresults}) are consistent with the voltage applied on the
sensors' heaters: obtained ratios for sensors 1-4 are similar as the sensors are
kept under the same heating conditions, and ratios for sensors 5-8 are lower as,
due to the active temperature control, they tend to be at higher temperature.
To filter the signal components due to changes in humidity and
temperature, we subtract the model prediction in equation~(\ref{linearmodel}) from the raw sensor
output. This operation is recognized as a method that searches signals
independent of environmental
conditions~\cite{cikeszczyk2013sensors}.
This is typically the case for continuous monitoring devices that are not
intended to measure the concentration of a particular gas. The resulting signal
is
\begin{linenomath}
\begin{equation}
R^*_i(t)=R_i(t)-\overline{R}_i(t)=R_i(t)-R_i(t-1)e^{\left(\beta_{1i}
\Delta H+ \beta_{2i} \left(\Delta H\right)^2+ \beta_{3i} \Delta
H \Delta T_E\right)},
\label{eq:decorrmethod}
\end{equation}
\end{linenomath}
where $R_i$ denotes the resistance values of the sensor $i$, and
$\beta_{1i}$, $\beta_{2i}$, and $\beta_{3i}$ are the adjusted values
for $\beta_1$, $\beta_2$, and $\beta_3$ for the i-th sensor. In
Fig. \ref{fig:filter}, we show the result of applying this
transformation on sensor 1. On the left panel, we present the
humidity, temperature, and sensor output. After applying the
transformation, the decorrelated output of the sensor is shown on the
right panel. The sensor drift due to the temperature and humidity
changes is filtered out. However, because we are subtracting from the
sensors signal ${R}_i(t)$ their predicted value $\overline{R}_i(t)$
according to our model, the resulting filtered signal $R^*_i(t)$ often
has zero mean and the relationship among the sensors is partially
lost. This is important for gas discrimination
\cite{vergara2013performance}, and we deal with this issue in section
\ref{sec:performance}.
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{Figures/Filter_sensoropaper.eps}
\caption{Result of applying the humidity and temperature filter provided by
equation (\ref{linearmodel}) on sensor 1. First, the resistance is is predicted
using the variation in humidity, and then this predicted resistance is subtracted
from the original signal}
\label{fig:filter}
\end{figure}
\subsection{Parameter Stability}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/Fig3_stability.eps}
\caption{Time evolution of the out-of-sample performance measured by
evaluating $R^2$ on the first sensor of the electronic nose. The
three bottom panels represent the evolution of the parameters,
$\beta_1,\beta_2$ and $\beta_3$ of the model over time.}
\label{fig:stability}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Fig5_Histograms.pdf}
\caption{Histograms of performance $R^2$ (a) and values of $\beta$ parameters
(b-d) for all the sensors using 3 months of training and testing in the
following month.}
\label{fig:stability2}
\end{figure}
To test the stability of the parameters over time, we trained the
model over a short period of time of 3 months of data and tested its
performance in the following month (i.e., forward testing
methodology). In Fig. \ref{fig:stability}, we show the time evolution
of the model performance and parameters $\beta$ of sensor 1 based on
humidity and temperature changes. The window of 3 months was chosen
in order to guarantee $R^2 > 0.9$ for all sensors throughout the year
(Fig. \ref{fig:stability2}a) and to avoid longer time scales, where
sensor drifting and seasonal changes in the environment may influence
sensors response. We also show the histogram of all values assumed by
$\beta$ parameters throughout this period
(Fig. \ref{fig:stability2}b--d).
Finally, the model is robust to failures in the sensors due to number
of reasons. For instance, in some instances the electronic nose
stopped transmitting due issues in the wireless connectivity; in other
events, sensors were displaced from their location during house
cleaning, and stopped working. Because algorithms need to be as robust
as possible given the uncontrolled conditions under which they
operate, our $R^2$ already takes it into account. In summary, there
are many possible reasons in daily operations that hinder the
operation of the electronic nose, and they reproduce uncontrolled
conditions that such sensors face.
\subsection{Sampling rate}
Another important question is determining an acceptable sampling rate
on the electronic nose to be able to filter the humidity and the
temperature. We estimate the effect in terms of regression accuracy of
different sampling rates by computing the average $R^2$ values for all
the sensors modifying the sampling period from 5 to 500 seconds. In
Fig. \ref{fig:sampling}, we can see that beyond the 2 minute sampling
period, the filter performance drops below 0.9. Beyond this point, the
approximations made in the band-based model in equations
(\ref{eq:app}-\ref{prediction}) fail.
Faster sampling rates may still be required to implement for some
strategies that use sensor heater control in an active manner
\cite{herrero2015active} or in fast changing environments. However,
further work is still needed to consider highly ventilated scenarios
in which temperature and humidity change in time at the same rate as
the atmosphere chemical composition.. Comparatively, an empirical
approach can be found in \cite{piedrahita2014next}, where a similar
model is fitted to a linear dependence on temperature and humidity,
but not on the changes of the temperature and humidity.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{Figures/sampling.eps}
\caption{Average $R^2$ performance for increasing values of the
sampling rate using 3 months of training and testing in the
following month. Beyond the two minute sampling rate the $R^2$ drops
below 0.9. }\label{fig:sampling}
\end{figure}
\section{Impact on online discrimination of gas identity}
\label{sec:performance}
To investigate whether a predictive model can potentially benefit from
filtering temperature and humidity sensors, we constructed a data set
from recordings of two distinct stimuli: wine and banana
(Fig. \ref{fig:bananawineExample}). We compared the impact of using
the raw data and the filtered data in terms of classification
performance when discriminating among presence of banana, wine and
lack of stimulus (i.e., background activity). Signals recorded with
banana or wine evoked different responses in the sensors. In
particular, responses to banana were often weaker and returned to the
baseline activity much faster than those of wine (compare for instance
$R_4$ in Fig. \ref{fig:bananawineExample}). Rather than using the
particular chemical signatures of compounds from bananas and wines,
our goal is to construct a model that learns to predict presence of
banana/wine based on the multivariate response of the sensors. The
chemical signature of bananas changes, for instance, as they ripen
\cite{llobet1999non}, and wine's signature depends on alcohol content
(ethanol), origin of the grape, among other factors
\cite{lozano2005classification,di1996electronic}. Thus, our approach
attempts at building a model that does not rely on wine type and
banana ripeness.
These data were collected over the course of 2 months by placing a
sample of either a banana or wine next to the electronic nose for a
period of time ranging from 10 minutes to 1 hour. Baseline signals
were taken from 2PM to 3PM to avoid additional noise due to home
activity. The time of the day when the stimulus was presented varied,
except between 12AM and 6AM. On total, our dataset comprises the time
series of 34 banana presentations, 36 wine presentations, and 30
baseline samples. To implement online discrimination, the data was
organized in moving windows with lengths of $10$ minutes. For
instance, for a presentation of length 60 min we create a total of 60
- 10 = 50 windows to be used during the classification.
To solve the classification problem, we used a nonlinear classifier
called Inhibitory Support Vector Machine (ISVM) \cite{isvm}, which, in
contrast to other multiclass SVM methods, is Bayes consistent for
three classes. ISVM is a particular case of the $\lambda$-SVM
classifier, a pointwise Fisher consistent multiclass classifier
\cite{rodriguez2015fisher}. ISVMs have been successfully applied to
arrays of electronic noses (identical to the one used in the present
paper) in controlled
conditions~\cite{rodriguez2014calibration,rodriguez2015fisher}, in
wind tunnel testing~\cite{vergara2013performance}, and for ethylene
discrimination in binary gas mixtures
\cite{fonollosa2014chemical}. Inspired by the learning mechanisms
present the insect brain \cite{201480:curropinion}, Inhibitory SVMs
use a large-margin classifier framework coupled to a mechanism of
mutual and unselective inhibition among classes. This mutual
inhibition creates a competition, from which only one class
emerges. The decision function of Inhibitory SVMs associated with the
$j$-th class and the input pattern $\vect{x}_i$ is defined as
$f_j(\vect{x_i}) = \langle \vect{w_j} , \Phi(\vect{x_i}) \rangle - \mu
\sum_{k=1}^L {\langle \vect{w_k} , \Phi(\vect{x_i}) \rangle}$, where
$L$ is the number of classes and $\mu$ scales how strong each class
will inhibit each other. If $\mu=0$, the decision function for
standard SVMs is recovered. It can be analytically shown that the
optimal value for $\mu$ is $1/L$. The predicted class of a data point
$\vect{x}_i$ is determined by the maximum among the decision functions
for each class: $y(\vect{x}_i)=\arg \max_j f_j(\vect{x}_i)$. Because
we used Radial Basis Functions (RBF) as the kernel of the inhibitory
SVM, our classifier had two meta-parameters: the soft margin
penalization $C$, and the inverse of the scale of the RBF function
$\gamma$. For more details about the ISVM model, see
\cite{isvm,rodriguez2015fisher}.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\textwidth]{Figures/Figure_example_banana_wine.pdf}
\caption{Example of response of all sensors due to the presentation
of our stimuli: banana and wine. Sensors are indexed according to
table \ref{tablesensorts}. Vertical blue lines delimit the period
of time that the stimulus remained close to the electronic
nose. These time series were recorded on September 22nd,
2015. \label{fig:bananawineExample} }
\end{figure}
To evaluate the impact on discrimination performance due to decorrelating the
signals from temperature and humidity sensors, we tested $4$ different feature
sets: raw sensor time series (RS), raw sensor data with humidity and temperature
(RS,T,H), filtered data (FS) by decorrelating sensors using equation
\ref{eq:decorrmethod}, and raw sensor data with filtered sensor data (RS,FS). To
properly estimate the generalization ability of the model, we used standard
procedures in machine learning to evaluate the performance of our classifier
when discriminating samples not used for training the classifier
\cite{duda2012pattern}. We first divided our data set into two groups: a
training set with 4/5th of the experimental presentations, and a test set with
1/5th of the data. All moving windows associated with the same presentations
were kept in the same group. We used 4-fold cross-validation on the training set
to estimate the classifier meta-parameters ($C$ and $\gamma$). Using these
meta-parameters, we re-trained the model using the whole training set and, then,
assessed the performance using the test set. The range of values for the
meta-parameters explored during the 4-fold cross-validation in the training set
were $\gamma=\lbrace 0.5, 1, 5, 10, 50, 100\rbrace$, and $C=\lbrace 10^4, 10^5,
10^6, 10^7, 10^8, 10^9 \rbrace$. To obtain a good statistical estimate of the
classification accuracy, we re-shuffled our data and repeated this procedure
$50$ times, which was enough for the average and variance to converge.
Using the raw sensor data combined with the filtered signals (RS,FS)
improved significantly (Kolmogorov-Smirnov, $p<0.025$) the performance
in online discrimination (Table~\ref{tableresultsclass}). The raw
sensors data (RS) alone reached $76\%$ of accuracy, and including the
temperature and humidity information (RS,T,H) did not improve. This
shows that the additional features are likely redundant. Probably due
to loss of inter-dependencies among sensors (as anticipated in section
\ref{sec:performancemodel}), the filtered sensor data (FS) by itself
underperformed RS. Still, the model becomes more consistent, with
lower variance in its performance, than the models trained on (RS) and
(RS,T,H). Indeed, using both raw and filtered time series (RS,FS)
improved significantly the model performance and its
consistency. Thus, this experiment illustrates that temperature and
humidity filters can not only improve pattern recognition performance,
but they can also improve model stability, which is especially
challenging in chemical
sensing~\cite{romain2010long,padilla2010,carlo2011,Vergara2012,martinelli2013adaptive}.
\begin{table}[t]
\centering
\begin{tabular}{@{}|l|ccc|c|}
\hline
Feature set & Cross-validated accuracy & Accuracy in test & Std & p-value\\
\hline
RS & 78.5\% & 76.5\% & 6.8\% & $0.02^*$ \\
RS,T,H & 73.3\% & 71.1\% & 6.8\% & $1\cdot10^{-12}$ $^{**}$ \\
FS & 72.4\% & 71.2\% & {\bf 4.8}\% & $2\cdot10^{-12}$ $^{**}$\\
RS,FS & 82.6\% & {\bf 80.9}\% & 6.3\% & 1\\
\hline \hline
\end{tabular}
\caption{Classification accuracies in four feature sets
(abbreviations are defined in the text) derived from our dataset
with three classes: wine, banana, and baseline activity.
The meta-parameters of the final Inhibitory SVM model were selected
as those with the best cross-validated accuracies in the training set
(second column), and the generalization error of the final model was
evaluated in the test set (third column). The standard deviation
(\textit{std}) for the test dataset is estimated over $50$
random partitions. Accuracy results from (RS,FS) are significantly
different from all other feature sets (p-values from
Kolmogorov-Smirnoff tests, $**$ passes at 1\%, $*$ passes at
5\%).}
\label{tableresultsclass}
\end{table}
\section{Conclusions}
Changes in humidity and temperature shape the responses of arrays of
MOX sensors, which in turn modifies nonlinearly chemical signatures of
different volatiles. Filtering changes in the sensor responses due to
changes in both humidity and temperature during sampling represents a
major improvement for complex machine learning and monitoring
tasks. We used a model based on semiconductor energy bands to express
the nonlinear dependence of sensor resistance with humidity and
temperature variations in an electronic nose. The model was designed
to fit in simpler micro-controllers, removing all possible
non-linearities up to second order in the change of humidity and
temperature, envisioning applications to cost-efficient devices. We
found that the most dominant terms are the change in humidity, the
quadratic term of the change in humidity, and the correlated
variations of humidity and temperature. We showed that the model
provides robust corrections to the distortions caused by environmental
changes. Therefore, our level of approximation on the semiconductor
energy band is an inexpensive solution for applications in online and
continuous home monitoring using chemical sensors.
Specifically, the coefficient of determination $R^2$ of our model when
fitted to all the 537 days of sampling is remarkably close to
$100$\%. The model predicts a particular dependence between two of the
coefficients that is consistently verified in all the tested
sensors. We also showed that the maximum sampling period to obtain a
reliable filter of humidity and temperature is of the order of 1
minute. The accuracy achieved with faster sampling rates provides
small gains, and it would require some overhead in wireless
communication when the corrections are done at the base
station. Additionally, 3-month training window was selected to ensure
that $R^2$ is larger than $90$\% for all sensors and throughout the
whole year. With 3 months, the training dataset likely included enough
number of training examples (events and background) while the effect
of long-term drift in the sensors was still weak to degrade the
trained models. Previous work using similar sensing units showed that
models trained in two-month windows keep high accuracy during the
following two months \cite{vergara2012chemical}. Stability could
probably be improved further if one selects longer training windows or
by coupling our strategy with already proposed strategies to
counteract long-term sensor drift
\cite{vergara2012chemical,padilla2010,ziyatdinov2010drift}.
We verified empirically the benefits of decorrelating humidity and
temperature from the sensors' response by applying it to a task of gas
discrimination. We recorded the response of the sensors when presented
with either a banana or glass of wine. Then, we used a
Bayes-consistent classifier \cite{rodriguez2015fisher,isvm} to
discriminate between the presence of banana, presence of wine, and
baseline activity. To compare the performance of the classifier with
and without the decorrelation of humidity-temperature, four different
subsets of data were created by combining raw sensor responses,
filtered sensor data, and temperature and humidity. Experimental
results show that including the filtered data in the classification
model improves not only the discrimination capability of the model,
but also its stability.
In summary, we have shown that simultaneous recordings of the humidity
and the temperature can be used to help extracting relevant chemical
signatures. The online decorrelation model proposed in this work was
designed for online operation even in the simpler micro-controllers
available in the market, which is essential for cost-efficient
devices. Additionally, humidity sensors are extremely appealing due to
a high correlation between humidity levels and human perception of air
quality \cite{wolkoff2007dichotomy,fang1998impact}. Thus, when
combined with other techniques
\cite{fonollosa2014human,rodriguez2014calibration,reservoir,fonollosa2014chemical,diamond2016classifying,mosqueiro2016CISS},
our model is likely to significantly enhance the performance of
chemical detection systems, as for instance of home monitoring
tasks. Our contribution thus emphasizes the importance of simultaneous
recordings of humidity and temperature, and that their use is
computationally amenable in sensor boards using low-energy
micro-controllers.
\section*{Acknowledgments}
This work has been supported by the California Institute for
Telecommunications and Information Technology (CALIT2) under Grant
Number 2014CSRO 136. JF acknowledges the support of the Marie Curie
Actions and the Agency for Business Competitiveness of the Government
of Catalonia (ACCI\'O) for the grant TECSPR15-1-0031; and the Spanish
MINECO program, grant TEC2014-59229-R (SIGVOL). RH, TM, and IR-L
acknowledge the partial support by 3\textordfeminine~Convocatoria de
Proyectos de Cooperacion Interuniversitaria UAM--Banco Santander con
EEUU. NR would like to acknowledge partial support by ONR grant
N000141612252. TM acknowledges CNPq grant 234817/2014- 3 for partial
support. We are also thankful to Flavia Huerta who collected data
examples during the summer of 2015.
\bibliographystyle{unsrt} |
1608.01898 | \section{Introduction}
\subsection{Motivation}
In this paper we study bifurcation equations and transversality conditions for
local bifurcations of $p$-periodic points in one-dimensional discrete
dynamical systems defined implicitly, with $p$ a positive integer. In
particular, we focus our attention on the fold, transcritical, pitchfork and
flip, i.e., the most frequently found in applications. The main result of the
paper is to obtain expressions for the general bifurcation equations and
transversality conditions in dynamical systems defined implicitly.
Implicitly defined discrete and continuous dynamical systems are not very well
studied, only very recently Albert Luo published \textquotedblleft\textit{the
first monograph to discuss the implicit mapping dynamics of periodic flows to
chaos\textquotedblright} \cite{luo2015}. The singularities of some implicit
continuous dynamical systems in dimension two have been addressed in
\cite{Dav2008}, namely the Clairaut system. Nevertheless, it is an interesting
and open field of research. This type of dynamical system appears in
applications, namely in the theory of PDE in the works of Sharkovsky and
co-workers \cite{CE,LSS,SSSV1,SSSV2,SP}, in Mathematical Economics directly
\cite{Me2} or in the context of backward dynamics \cite{KSY,ME}. It appears
also in the context of Control Theory \cite{Holl2005}. These implicit
dynamical systems appear also in numerical methods for ordinary differential
equations, v.g., the backward Euler, the trapezoid method
\cite{Ush1986,Hir194} and the Runge-Kutta implicit method, see the recent
article \cite{Zhang2016}. Implicit numeric methods are very useful when the
original equations exhibit stiffness, see for instance \cite{Hai2009,Hai1996}.
In \cite{Lamarque2000} the implicit Euler method was used in a concrete
mechanical problem. Some implicit iterative schemes were transformed in
forward dynamical systems using numerical methods, v.g., Newton method,
\cite{Ghe2003}. In implicit numerical schemes it is possible to prove the
existence of period doubling when the step size parameter increases as we do
with a simple example at the end of this article. It is also interesting to
see the existence of chaos when the parameter $h$ is big enough, but still
relatively small.
The case of $p$-periodic points with $p>1$, is very intricate, the
computations increase its complexity extraordinary with the powers of the
normal form, as we can see in this paper in the case of the pitchfork. For
that reason, we study codimension $1$ cases, the most common in applications.
The study of one-dimensional bifurcations makes sense, since many higher order
systems can be reduced \cite{Rega2005} to lower order dimensional dynamics
via center manifold and Poincar\'e map techniques as in \cite{Kle1995}, using
spectral properties and quasi-periodicity \cite{Mezic2005}, and in periodic
non-autonomous systems using Floquet theory \cite{david2000}.
It is completely open and would be interesting to investigate the invariance
of the bifurcation equations for periodic non-autonomous systems defined
implicitly in the line of work of \cite{HO1}.
One of the main reasons of this paper is to provide computational tools for
the applied researcher dealing with implicitly defined dynamical systems. It
is possible to study the bifurcations that can occur without the knowledge of
an explicit difference equation. All the formulae are programmable using the
usual platforms available for mathematicians. The examples where prepared
using Wolfram Mathematica 10.0.
We follow the terminology of \cite{KU}.
\subsection{Overview}
We organized this paper in four sections. In Section \ref{SectionPrelim} we
introduce basic concepts.
In Section 3, the core of this work, we study in detail the equations of
bifurcation for $p$-periodic orbits of implicitly defined dynamical systems.
In Section 4 we present examples, namely on the Euler method for numerical
solutions of ordinary differential equations. In the implicit difference
equations of numerical methods we show the existence of bifurcation depending
on the step size parameter $h$, and the existence of chaos even in very simple examples.
\section{\label{SectionPrelim}Preliminaries}
\subsection{Basic definitions and notation}
We define implicitly a discrete dynamical system using instead of the classic
definitio
\begin{equation}
x_{n+1}=f(x_{n})\text{, }x_{n}\in I\text{, with }n\in\mathbb
\mathbb{N}
}\text{,} \label{pp
\end{equation}
the alternative on
\[
F\left( x_{n},x_{n+1}\right) =0\text{, for }x_{n},x_{+1}\in I\text{, with
}n\in\mathbb
\mathbb{N}
}\text{,
\]
where $I$ is a real interval (not necessarily compact and maybe
\mathbb{R}
$), where we input $x_{n}$ and solve for $x_{n+1}$ giving an initial condition
$x_{0}\in I$. The usual Euclidean distance is defined in $I$. The map $F$ is
sufficiently differentiable for the purposes of bifurcation theory, assumption
that we keep in this paper. We suppose that given $F\left( x,y\right) =0$,
there exists the solution $\left( x_{0},y_{0}\right) $, and an implicit
function $y=f\left( x\right) $ with $y_{0}=f\left( x_{0}\right) $ such
tha
\[
F\left( x,f\left( x\right) \right) =0\text{,
\]
in a suitable neighborhood of $\left( x_{0},y_{0}\right) $. We follow
\cite{KP} concerning the implicit function theorem. For the purposes of this
article we admit the existence of the necessary solutions in the appropriate
neighborhoods of the bifurcation points. Obviously, each particular dynamical
system defined implicitly must be studied to ensure the existence of the
iteration function $f\left( x\right) $.
In the sequel, by ${\mathcal{C}}\left( D\right) $ we denote the collection
of all continuous maps in its domain $D$, by ${\mathcal{C}}^{1}\left(
D\right) $ the collection of all continuously differentiable elements of
${\mathcal{C}}\left( D\right) $ and, in general by ${\mathcal{C}}^{s}\left(
D\right) ,$ $s\geq1,$ the collection of all elements of ${\mathcal{C}}\left(
D\right) $ having continuous derivatives up to order $s$ in $D.$
The $p$ composition of $f$ a real function of real variable is denoted by
$f^{p}$, the usual power is denoted by $\left( f\right) ^{p}$.
Let $f\in{\mathcal{C}}^{1}\left( D\right) $, and let $x_{0}$ be a periodic
point of period $p$, $x_{0}$ is called a \emph{hyperbolic attractor} if
$|\frac{df^{p}(x_{0})}{dx}|<1$, a \emph{hyperbolic repeller} if $|\frac
{df^{p}(x_{0})}{dx}|>1$, and \emph{non-hyperbolic} if $|\frac{df^{p}(x_{0
)}{dx}|=1$.
\begin{defn}
We say that two continuous maps $f:I\rightarrow I$ and $g:J\rightarrow J$, are
topologically conjugate, if there exists a homeomorphism $h:I\rightarrow J$,
such that $h\circ f=g\circ h$. We call $h$ the topological conjugacy of $f$
and $g$.
\end{defn}
We use $\alpha$ for a real parameter.
\begin{defn}
\label{BifSet}If $f\left( \cdot,\alpha\right) $ is a family of maps, then
the regular values $\alpha$ of the parameters are those which have the
property that $f\left( \cdot,\widetilde{\alpha}\right) $ is topologically
conjugate to $f\left( \cdot,\alpha\right) $ for all $\widetilde{\alpha}$ in
some open neighbourhood of $\alpha$. If $\alpha$ is not a regular value, it is
a \emph{bifurcation value}. The collection of all the bifurcation values is
the \emph{bifurcation set}, $\Omega\subset\mathbb{R}$, in the parameter space.
\end{defn}
Let $f\left( \cdot,\alpha_{0}\right) $ be a parameter dependent family of
maps in ${\mathcal{C}}^{s}\left( D\right) $. Let $\alpha_{0}$ be a
particular parameter and $a\in D$ be a fixed point of the $p$ composition map
$f\left( \cdot,\alpha_{0}\right) $, with $p$ a minimal positive integer,
i.e.
\[
a=f^{p}\left( a,\alpha_{0}\right) \text{,
\]
$a$ is a periodic point of the dynamical system. The condition of $a$ being
non-hyperbolic is necessary for the existence of a local bifurcation. The
existence and nature of that bifurcation depends on other symmetry and
differentiable conditions that we will see bellow. If there exists a local
bifurcation we say that $(a,\alpha_{0})$ is a \emph{bifurcation point }(when
there is no risk of confusion, we say that $a$ is a \emph{bifurcation point}).
\begin{notat}
For notational simplicity we consider the real parameter $\alpha$ as a
standard variable along with the dynamic variable $x$, i.e., we write
$F\left( x,y,\alpha\right) $ instead of $F_{\alpha}\left( x,y\right) $,
reserving the last slot for the parameter, keeping in mind that the
compositions are always in the dynamic variables $x$ and $y$. In this paper we
never use $f_{\alpha}$ to mean dependence on the parameter.
When there is no danger of confusion and no operations regarding the
parameter, we denote the evaluation of functions depending on the dynamic
variable and the parameter omitting the later, for instance $F\left(
x,y,\alpha\right) $ or $f\left( x,\alpha\right) $ will be denoted by
$F\left( x,y\right) $ or $f\left( x\right) $ in order to avoid to overload
the complicated notation needed for the computations of chain rules.
Nevertheless, all the maps in this paper depend on the parameter as well on
the dynamic variable. We deal with parameter depending families of maps, even
when that dependence is not visible in some formulas or expressions.
We denote the derivatives relative to some variable $y$ by $\partial_{y}$.
Repeated differentiation relative to the same variable is denoted by
$\partial_{y^{n}}$, for instance $\partial_{yyy}=\partial_{y^{3}}$. When there
is no danger of confusion, we denote strict partial derivatives, i.e., not
seeing composed functions, by a subscript. For instance, the third partial
derivative of $f$ relative to $y$ is, in that case, denoted by $f_{yyy}$ or
$f_{y^{3}}$.
\end{notat}
This means, in particular, that when dealing with the composition of real
scalar functions $F\left( x,y\right) $ with $g\left( x,y\right) $ and
$h\left( x,y\right) $, we have the usual chain rule
\begin{multline*}
\partial_{x}F\left( g\left( x,t\right) ,h\left( x,t\right) ,\alpha
\right) =\\
F_{x}\left( g\left( x,t\right) ,h\left( x,t\right) ,\alpha\right)
g_{x}\left( x,t\right) +F_{y}\left( g\left( x,t\right) ,h\left(
x,t\right) ,\alpha\right) h_{x}\left( x,t\right) \text{,
\end{multline*}
\subsection{Classic conditions for fold, transcritical, pitchfork and flip
bifurcations}
In this paragraph, we recall briefly the conditions of codimension $1$ local
bifurcations with derivatives $\partial_{x}f^{p}(x_{0})=\pm1$.
We first consider the case $\partial_{x}f^{p}(x_{0})=+1$. Giving a discrete
dynamical system generated by the iteration of $f$ in its domain $D$, and a
real parameter $\alpha$, in order to compute the bifurcation points one has to
solve the bifurcation equations \cite{KU
\begin{equation
\begin{array}
[c]{l
{f}^{p}(x,\alpha)=x\text{, fixed point equation}\\
{f}_{x}^{p}(x,\alpha)=1\text{, non-hyperbolicity condition.
\end{array}
\label{Fold
\end{equation}
\subsubsection{Fold}
The simplest of such local bifurcations is the fold or saddle node
bifurcation. One assumes, in this case, the non-degeneracy condition
\begin{equation}
{f_{x^{2}}}(x,\alpha)\not =0 \label{Nondeg
\end{equation}
and the transversality condition \cite{KU
\begin{equation}
f_{\alpha}\left( x,\alpha\right) \not =0. \label{unfold1
\end{equation}
We set generically that $\alpha\i
\mathbb{R}
$, since one needs only one parameter to unfold locally this singularity
\cite{A,AR,CH,GS,GU,KU}. The normalized germ of this bifurcation i
\[
x\pm x^{2}\text{,
\]
with principal famil
\[
x\pm x^{2}+\alpha\text{,
\]
which is locally weak topologically conjugated to any other family
\cite{AR,KU} satisfying the bifurcation conditions.
\subsubsection{Transcritical}
Another simple bifurcation is the transcritical, in this case is a bifurcation
with symmetry. One assumes, in this case, the non-degeneracy condition
\begin{equation}
{f_{x^{2}}}(x,\alpha)\not =0,
\end{equation}
the transversality condition of the fold fail
\begin{equation}
f_{\alpha}\left( x,\alpha\right) =0,
\end{equation}
becoming a new degeneracy condition. The symmetry condition states that the
fixed point of $f$ persists. Without loss of generality we consider that $0$
is that fixed point. The new transversality condition i
\[
f_{x\alpha}\left( x,\alpha\right) \not =0\text{.
\]
Again, we set generically that $\alpha\i
\mathbb{R}
$, since one needs only one parameter to unfold locally this singularity
\cite{A,AR,CH,GS,GU}. The principal family is no
\[
\left( 1+\alpha\right) x\pm x^{2}\text{,
\]
which is weak topologically conjugated to any other family \cite{AR,KU}
satisfying the bifurcation conditions.
\subsubsection{Pitchfork}
The last type of bifurcation we consider with derivative $\partial_{x
f^{p}(x_{0})=+1$ is the pitchfork, another bifurcation with the same symmetry
on the fixed point as the transcritical. One assumes, in this case, the extra
degeneracy conditio
\[
{f_{x^{2}}}(x,\alpha)=0,
\]
and the new non-degeneracy condition
\begin{equation}
{f_{x^{3}}}(x,\alpha)\not =0.
\end{equation}
The transversality condition of the fold fails agai
\begin{equation}
f_{\alpha}\left( x,\alpha\right) =0\text{,
\end{equation}
and the transversality condition is assumed again to b
\[
f_{x\alpha}\left( x,\alpha\right) \not =0\text{.
\]
We set generically that $\alpha\i
\mathbb{R}
$ \cite{A,AR,CH,GS,GU}. The principal family is no
\[
\left( 1+\alpha\right) x\pm x^{3}\text{,
\]
which is weak topologically conjugated to any other family \cite{AR,KU}
satisfying the bifurcation conditions.
\subsubsection{Flip}
We consider now the conditions of codimension $1$ local bifurcations with
derivative $\partial_{x}f^{p}(x_{0})=-1$.
One has to solve the bifurcation equations \cite{KU
\begin{equation
\begin{array}
[c]{l
{f}^{p}(x,\alpha)=x\text{, fixed point equation}\\
{f}_{x}^{p}(x,\alpha)=-1\text{, non-hyperbolicity condition.
\end{array}
\label{Flip
\end{equation}
One assumes, in this case, the generic non-degeneracy condition
\begin{equation}
\frac{1}{2}\left( f_{x^{2}}\left( x,\alpha\right) \right) ^{2}+\frac{1
{3}f_{x^{3}}\left( x,\alpha\right) \not =0, \label{Schwarz
\end{equation}
which is equivalent to say that the Schwarzian derivative
\[
Sf\left( x,\alpha\right) =\frac{f_{x^{3}}\left( x,\alpha\right)
{f_{x}\left( x,\alpha\right) }-\frac{3}{2}\left( \frac{f_{x^{2}}\left(
x,\alpha\right) }{f_{x}\left( x,\alpha\right) }\right) ^{2
\]
of $f$ is not zero at the bifurcation point where $f_{x}\left( x,\alpha
\right) =-1$. The transversality condition \cite{KU} i
\begin{equation}
f_{x\alpha}(0,0)\not =0\text{.} \label{fliptransverse
\end{equation}
We set generically that $\alpha\i
\mathbb{R}
$ \cite{A,AR,CH,GS,GU,KU}. The normalized germ of this bifurcation i
\[
-x\pm x^{3}\text{,
\]
with principal famil
\[
-\left( 1+\alpha\right) x\pm x^{3}\text{,
\]
which is again locally weak topologically conjugated to any other family
\cite{AR,KU} satisfying the bifurcation conditions.
Adding degeneracy conditions, one obtains higher degeneracy (higher
codimension) local bifurcations. In this paper we keep it simple and do not
consider higher codimension.
\section{Implicit discrete dynamical systems}
\subsection{Bifurcation equations}
Let us now consider the case of implicit DDS. Given the parameter depend
family $F\in{\mathcal{C}}^{s}\left(
\mathbb{R}
^{2}\right) $, with $s\geq1$, enough for our results, such that
\
\begin{array}
[c]{cccc
F: &
\mathbb{R}
^{2} & \longrightarrow &
\mathbb{R}
\text{,}\\
& \left( x,y\right) & \longmapsto & F\left( x,y\right) \text{.
\end{array}
\]
We start by the example of dynamics near fixed points. So, consider $F\left(
x_{f},x_{f}\right) =0$, with derivative $F_{y}\left( x_{f},x_{f}\right)
\not =0$. We have the implicit discrete dynamical system near the fixed point
$\left( x_{f},x_{f}\right) $ defined b
\[
F\left( x_{n},x_{n+1},\alpha\right) =0\text{, for }x_{n},x_{+1}\in I\text{,
with }n\in\mathbb
\mathbb{N}
}\text{.
\]
Along this work we always consider the independent variable in the first slot
of $F\left( \cdot,\cdot,\cdot\right) $, being the dependent variable, or
implicit function, at the second slot and the parameter at the third slot. One
instance of this type of systems is obtained by Sharkovsky and coauthors
\cite{CE,LSS,SSSV1,SSSV2,SP} in some boundary value problems. The classic
counterpart of this scheme i
\[
x_{n+1}-f\left( x_{n},\alpha\right) =0\text{, for }x_{n},x_{+1}\in I\text{,
with }n\in\mathbb
\mathbb{N}
}\text{,
\]
with a fixed point $x_{f}$ and with $F\left( x,y,\alpha\right) =y-f\left(
x,\alpha\right) $. The classic bifurcation equations are relative to
$y=f\left( x,\alpha\right) $. The bifurcation equations in the implicit case
ar
\begin{align*}
F\left( x,y\left( x\right) ,\alpha\right) & =0\text{,}\\
F_{x}\left( x,y\left( x\right) ,\alpha\right) +F_{y}\left( x,y\left(
x\right) ,\alpha\right) y_{x}\left( x\right) & =0\text{.
\end{align*}
At the bifurcation point $y=x=x_{f}$, we have $y_{x}\left( x_{f}\right)
=f_{x}\left( x_{f},\alpha\right) =\pm1$, the equations becom
\begin{align*}
F\left( x_{f},x_{f},\alpha\right) & =0\text{,}\\
F_{x}\left( x_{f},x_{f},\alpha\right) \pm F_{y}\left( x_{f},x_{f
,\alpha\right) & =0\text{,
\end{align*}
with non-degeneracy conditio
\[
f_{x^{2}}\left( x_{f},\alpha\right) =-\frac{F_{x^{2}}\left( x_{f
,x_{f},\alpha\right) \pm2F_{xy}\left( x_{f},x_{f},\alpha\right) +F_{y^{2
}\left( x_{f},x_{f},\alpha\right) }{F_{y}\left( x_{f},x_{f},\alpha\right)
}\not =0\text{.
\]
The case of periodic points is more involved, the orbit of $x$ is obtained by
successive substitution at the function $F\left( x,y,\alpha\right) $,
accordingly to the schem
\begin{equation}
\left\{
\begin{array}
[c]{l
F\left( x,f\left( x\right) \right) =0,\\
F\left( f\left( x\right) ,f^{2}\left( x\right) \right) =0,\\
\cdots\\
F\left( f^{j-2}\left( x\right) ,f^{j-1}\left( x\right) \right) =0,\\
F\left( f^{j-1}\left( x\right) ,f^{j}\left( x\right) \right) =0,\\
\cdots
\end{array}
\right. \label{iterative
\end{equation}
or, with initial condition $x_{0}
\begin{equation}
\left\{
\begin{array}
[c]{l
F\left( x_{0},x_{1}\right) =0,\\
F\left( x_{1},x_{2}\right) =0,\\
\cdots\\
F\left( x_{j-2},x_{j-1}\right) =0,\\
F\left( x_{j-1},x_{j}\right) =0,\\
\cdots
\end{array}
\right.
\end{equation}
where we omitted $\alpha$ for the sake notational simplicity. In this case, we
suppose that there exists an implicit solution of $F\left( x,y\right) =0$,
such that $y=f\left( x\right) $ is well defined for all the points $x_{0},$
$\ldots,$ $x_{j}$, $\ldots$ meaning that $F_{y}\left( x_{0},x_{1}\right)
\not =0$, $F_{y}\left( x_{1},x_{2}\right) \not =0$, $\ldots$, $F_{y}\left(
x_{p-1},x_{0}\right) \not =0$, $\ldots$.
Naturally, $x_{0}$ is a periodic point of the implicit dynamical system i
\begin{equation}
\left\{
\begin{array}
[c]{l
F\left( x_{0},x_{1}\right) =0,\\
F\left( x_{1},x_{2}\right) =0,\\
\cdots\\
F\left( x_{p-2},x_{p-1}\right) =0,\\
F\left( x_{p-1},x_{0}\right) =0\text{.
\end{array}
\right. \label{implic
\end{equation}
To obtain the bifurcation equations for periodic points we compute the
derivatives of the system (\ref{implic}). The next two lemmas \ref{chain rule}
and \ref{chainparameter} are fundamental in the study of the bifurcation
conditions, giving explicit formulas for the computation of derivatives
relative to $x$ and the parameter $\alpha$. All the other derivatives used in
this paper in the bifurcation conditions whatsoever are obtained recursively
using the results of this two lemmas. The next Lemma \ref{chain rule
\ establishes the chain rule for the first derivative of $f$, the iteration
function defined implicitly by $F\left( x,y\right) =0$.
\begin{lem}
\label{chain rule}Chain rule for implicit orbits. The derivative of $f^{j}$
defined using the system (\ref{implic}) is given by
\begin{equation}
\partial_{x}f^{j}\left( x\right) =\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\frac{F_{x}\left( f^{i}\left( x\right) ,f^{i+1}\left( x\right) \right)
}{F_{y}\left( f^{i}\left( x\right) ,f^{i+1}\left( x\right) \right) }.
\label{chain
\end{equation}
Equivalently, given the initial condition $x_{0}$
\begin{equation}
\partial_{x}f^{j}\left( x_{0}\right) =\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\frac{F_{x}\left( x_{i},x_{i+1}\right) }{F_{y}\left( x_{i},x_{i+1}\right)
}. \label{chain2
\end{equation}
\end{lem}
\begin{pf}
We differentiate the system (\ref{iterative}) relative to $x$, noticing that
the zeroth order composition is the identity $f^{0}\left( x\right) =x$, and
$f_{x}^{0}\left( x\right) =1$, with the simplifying notation $f^{j}\left(
x\right) =f^{j}$ for $j=0,1,2\ldots.
\begin{align*}
F_{x}\left( f^{0},f\right) +F_{y}\left( f^{0},f\right) f_{x}\left(
f^{0}\right) & =0,\\
F_{x}\left( f,f^{2}\right) f_{x}\left( f^{0}\right) +F_{y}\left(
f,f^{2}\right) f_{x}\left( f^{0}\right) f_{x}\left( f\right) & =0,\\
& \cdots\\
F_{x}\left( f^{j-1},f^{j}\right)
{\displaystyle\prod\limits_{i=0}^{j-2}}
f_{x}\left( f^{i}\right) +F_{y}\left( f^{j-1},f^{j}\right)
{\displaystyle\prod\limits_{i=0}^{j-1}}
f_{x}\left( f^{i}\right) & =0,\\
& \cdots
\end{align*}
cancelling the common factors we ge
\begin{align*}
F_{x}\left( f^{0},f\right) +F_{y}\left( f^{0},f\right) f_{x}\left(
f^{0}\right) & =0,\\
F_{x}\left( f,f^{2}\right) +F_{y}\left( f,f^{2}\right) f_{x}\left(
f\right) & =0,\\
& \cdots\\
F_{x}\left( f^{j-1},f^{j}\right) +F_{y}\left( f^{j-1},f^{j}\right)
f_{x}\left( f^{j-1}\right) & =0,\\
& \cdots
\end{align*}
solving for $f_{x}\left( f^{j}\right) $ we obtai
\begin{align*}
f_{x}\left( f^{0}\right) & =-\frac{F_{x}\left( f^{0},f\right)
{F_{y}\left( f^{0},f\right) },\\
f_{x}\left( f\right) & =-\frac{F_{x}\left( f,f^{2}\right) }{F_{y}\left(
f,f^{2}\right) },\\
& \cdots\\
f_{x}\left( f^{j-1}\right) & =-\frac{F_{x}\left( f^{j-1},f^{j}\right)
}{F_{y}\left( f^{j-1},f^{j}\right) },\\
& \cdots.
\end{align*}
Using the chain rule along the orbit, one obtains the product
\[
\partial_{x}f^{j}\left( x\right)
{\displaystyle\prod\limits_{i=0}^{j-1}}
f_{x}\left( f^{i}\right) =\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) }{F_{y}\left( f^{i},f^{i+1}\right)
}.
\]
The second relation (\ref{chain2}) is a simple reformulation of the first one
(\ref{chain}).
\end{pf}
\begin{cor}
We have the first bifurcation equatio
\begin{subequations}
\begin{align}
\partial_{x}f^{p}\left( x_{0}\right) &
{\displaystyle\prod\limits_{j=0}^{p-1}}
f_{x}\left( x_{j}\right) =\pm1\label{a}\\
& =\left( -1\right) ^{p
{\displaystyle\prod\limits_{j=0}^{p-1}}
\frac{F_{x}\left( x_{j},x_{j+1\left( \operatorname{mod}p\right) }\right)
}{F_{y}\left( x_{j},x_{j+1\left( \operatorname{mod}p\right) }\right)
=\pm1. \label{b
\end{align}
\end{subequations}
\end{cor}
\begin{pf}
We consider that $x_{p}=x_{0}$ and substitute in the chain rule (\ref{chain2})
of Lemma \ref{chain rule}. The non-hyperbolicity condition is $\partial
_{x}f^{p}\left( x_{0}\right) =\pm1$.
\end{pf}
To decide if there is a bifurcation and its type is necessary to obtain the
transversality conditions using the parameter derivative. The first possible
condition involves $\partial_{\alpha}f^{p}$. The next Lemma
\ref{chainparameter} is fundamental in that concern.
\begin{lem}
\label{chainparameter}The derivative of $f^{j}$ relative to the parameter
$\alpha$ defined using the system (\ref{implic}) is given by
\begin{equation}
\partial_{\alpha}f^{j}=\left( -1\right) ^{j
{\displaystyle\sum\limits_{k=0}^{j-1}}
\frac{\left( -1\right) ^{k}F_{\alpha}\left( f^{k},f^{k+1}\right)
{F_{y}\left( f^{k},f^{k+1}\right)
{\displaystyle\prod\limits_{i>k}^{j-1}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) }{F_{y}\left( f^{i},f^{i+1}\right)
}. \label{derivalfa
\end{equation}
\end{lem}
\begin{pf}
Similar to the proof of Lemma \ref{chain rule}. We have now the general rul
\[
\partial_{\alpha}F\left( g,h\right) =F_{x}\left( g,h\right) \partial
_{\alpha}g+F_{y}\left( g,h\right) \partial_{\alpha}h+F_{\alpha}\left(
g,h\right) =0,
\]
the first derivative is
\[
F_{y}\left( f^{0},f\right) f_{\alpha}+F_{\alpha}\left( f^{0},f\right) =0,
\]
solving for $f_{\alpha}
\[
f_{\alpha}=-\frac{F_{\alpha}\left( f^{0},f\right) }{F_{y}\left(
f^{0},f\right) }.
\]
Doing the same for the second composition we obtai
\[
\partial_{\alpha}f^{2}=\frac{F_{\alpha}\left( f^{0},f\right) F_{x}\left(
f,f^{2}\right) }{F_{y}\left( f^{0},f\right) F_{y}\left( f,f^{2}\right)
}-\frac{F_{\alpha}\left( f,f^{2}\right) }{F_{y}\left( f,f^{2}\right) },
\]
for the third compositio
\[
\partial_{\alpha}f^{3}=-\frac{F_{\alpha}\left( f^{0},f\right) F_{x}\left(
f,f^{2}\right) F_{x}\left( f^{2},f^{3}\right) }{F_{y}\left( f^{0
,f\right) F_{y}\left( f,f^{2}\right) F_{y}\left( f^{2},f^{3}\right)
}+\frac{F_{\alpha}\left( f,f^{2}\right) F_{x}\left( f^{2},f^{3}\right)
}{F_{y}\left( f,f^{2}\right) F_{y}\left( f^{2},f^{3}\right)
-\frac{F_{\alpha}\left( f^{2},f^{3}\right) }{F_{y}\left( f^{2
,f^{3}\right) }.
\]
The previous expressions suggest the general formula for the derivatives
relative to $\alpha
\[
\partial_{\alpha}f^{k}=\left( -1\right) ^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{\left( -1\right) ^{j}F_{\alpha}\left( f^{j},f^{j+1}\right)
{F_{y}\left( f^{j},f^{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{k-1}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) }{F_{y}\left( f^{i},f^{i+1}\right)
},
\]
which is the induction hypothesis. Consider the general formula
\[
\partial_{\alpha}F\left( f^{k},f^{k+1}\right) =F_{x}\left( f^{k
,f^{k+1}\right) \partial_{\alpha}f^{k}+F_{y}\left( f^{k},f^{k+1}\right)
\partial_{\alpha}f^{k+1}+F_{\alpha}\left( f^{k},f^{k+1}\right) =0,
\]
solving for $\partial_{\alpha}f^{k+1}$ we hav
\[
\partial_{\alpha}f^{k+1}=\frac{-F_{x}\left( f^{k},f^{k+1}\right)
\partial_{\alpha}f^{k}-F_{\alpha}\left( f^{k},f^{k+1}\right) }{F_{y}\left(
f^{k},f^{k+1}\right)
\
\[
=\scriptstyle\frac{-F_{x}\left( f^{k},f^{k+1}\right) \left( \left(
-1\right) ^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{\left( -1\right) ^{j}F_{\alpha}\left( f^{j},f^{j+1}\right)
{F_{y}\left( f^{j},f^{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{k-1}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) }{F_{y}\left( f^{i},f^{i+1}\right)
}\right) }{F_{y}\left( f^{k},f^{k+1}\right) }-\frac{F_{\alpha}\left(
f^{k},f^{k+1}\right) }{F_{y}\left( f^{k},f^{k+1}\right)
\
\[
=\scriptstyle\left( -1\right) ^{k+1
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{\left( -1\right) ^{j}F_{\alpha}\left( f^{j},f^{j+1}\right)
{F_{y}\left( f^{j},f^{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{k-1}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) F_{x}\left( f^{k},f^{k+1}\right)
}{F_{y}\left( f^{i},f^{i+1}\right) F_{y}\left( f^{k},f^{k+1}\right)
}+\left( -1\right) ^{2k+1}\frac{F_{\alpha}\left( f^{k},f^{k+1}\right)
}{F_{y}\left( f^{k},f^{k+1}\right)
\
\[
=\left( -1\right) ^{k+1
{\displaystyle\sum\limits_{j=0}^{k}}
\frac{\left( -1\right) ^{j}F_{\alpha}\left( f^{j},f^{j+1}\right)
{F_{y}\left( f^{j},f^{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{k}}
\frac{F_{x}\left( f^{i},f^{i+1}\right) }{F_{y}\left( f^{i},f^{i+1}\right)
},
\]
as desired.
\end{pf}
\begin{cor}
In particular, at the bifurcation point the derivative relative to the
parameter takes the for
\begin{equation}
\partial_{\alpha}f^{p}\left( x_{0}\right) =\left( -1\right) ^{p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\left( -1\right) ^{j}F_{\alpha}\left( x_{j},x_{j+1}\right)
{F_{y}\left( x_{j},x_{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{p-1}}
\frac{F_{x}\left( x_{i},x_{i+1}\right) }{F_{y}\left( x_{i},x_{i+1}\right)
}\text{.} \label{Dalphabifpoint
\end{equation}
\end{cor}
To obtain the non-degeneracy conditions we have to compute the second
derivative of $F$. In the next proposition we obtain an explicit expression
for the second derivative.
For the next results we introduce the notation $F\left( f^{j},f^{j+1}\right)
=F^{j}$, $F_{x}\left( f^{j},f^{j+1}\right) =F_{x}^{j}$, $F_{x^{2}}\left(
f^{j},f^{j+1}\right) =F_{x^{2}}^{j}$, $F_{y}\left( f^{j},f^{j+1}\right)
=F_{y}^{j}$, $F_{y^{2}}\left( f^{j},f^{j+1}\right) =F_{y^{2}}^{j}$,
$F_{xy}\left( f^{j},f^{j+1}\right) =F_{xy}^{j}$, $F_{\alpha}\left(
f^{j},f^{j+1}\right) =F_{\alpha}^{j}$, $F_{x\alpha}\left( f^{j
,f^{j+1}\right) =F_{x\alpha}^{j}$, $F_{y\alpha}\left( f^{j},f^{j+1}\right)
=F_{y\alpha}^{j}$. At the bifurcation point we use the notation $F\left(
x_{j},x_{j+1}\right) =\widetilde{F}^{j}$, $F_{x}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{x}^{j}$, $F_{x^{2}}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{x^{2}}^{j}$, $F_{y}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{y}^{j}$, $F_{y^{2}}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{y^{2}}^{j}$, $F_{xy}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{xy}^{j}$, $F_{\alpha}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{\alpha}^{j}$, $F_{x\alpha}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{x\alpha}^{j}$, $F_{y\alpha}\left( x_{j},x_{j+1}\right)
=\widetilde{F}_{y\alpha}^{j}$ and the abbreviation
\[
\nu_{j}=\frac{F_{x}^{j}}{F_{y}^{j}}\text{ and }\widetilde{\nu}_{j
=\frac{\widetilde{F}_{x}^{j}}{\widetilde{F}_{y}^{j}}.
\]
\begin{prop}
\label{secd}The second derivative of $f^{k}$ defined using the system
(\ref{implic}) along the orbit i
\begin{equation}
\partial_{x^{2}}f^{k}=\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j
}\partial_{x}f^{j}. \label{SEC
\end{equation}
At the bifurcation point where $\partial_{x}f^{p}\left( x_{0}\right) =\pm1$,
with $x_{p}=x_{0}$, the second derivative takes the for
\begin{equation}
\partial_{x^{2}}f^{p}\left( x_{0}\right) =\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\widetilde{F}_{x^{2}}^{j}-2\widetilde{F}_{xy}^{j}\widetilde{\nu
_{j}+\widetilde{F}_{y^{2}}^{j}\widetilde{\nu}_{j}^{2}}{\widetilde{F}_{x}^{j
}\partial_{x}f^{j}.
\end{equation}
\end{prop}
\begin{pf}
We recall (\ref{chain}
\[
\partial_{x}f^{k}=\left( -1\right) ^{k
{\displaystyle\prod\limits_{j=0}^{k-1}}
\frac{F_{x}^{j}}{F_{y}^{j}}.
\]
The second derivative i
\[
\partial_{x^{2}}f^{k}
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( -1\right) ^{k}\partial_{x}F_{x}^{j
{\displaystyle\prod\limits_{i\not =j=0}^{k-1}}
\frac{F_{x}^{i}}{F_{y}^{i}}
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( -1\right) ^{k}\frac{\partial_{x}F_{y}^{j}}{\left( F_{y}^{j}\right)
^{2}
{\displaystyle\prod\limits_{i\not =j=0}^{k-1}}
\frac{F_{x}^{i}}{F_{y}^{i}},
\]
i.e.
\[
\partial_{x^{2}}f^{k}=\scriptstyl
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}\partial_{x}f^{j}+F_{xy}^{j}\partial_{x}f^{j+1}}{F_{x}^{j
}\left( -1\right) ^{k
{\displaystyle\prod\limits_{i=0}^{k-1}}
\frac{F_{x}^{i}}{F_{y}^{i}}
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{xy}^{j}\partial_{x}f^{j}+F_{y^{2}}^{j}\partial_{x}f^{j+1}}{F_{y}^{j
}\left( -1\right) ^{k
{\displaystyle\prod\limits_{i=0}^{k-1}}
\frac{F_{x}^{i}}{F_{y}^{i}},
\]
which i
\begin{align*}
\partial_{x^{2}}f^{k} &
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( \frac{F_{x^{2}}^{j}\partial_{x}f^{j}+F_{xy}^{j}\partial_{x}f^{j+1
}{F_{x}^{j}}-\frac{F_{xy}^{j}\partial_{x}f^{j}+F_{y^{2}}^{j}\partial
_{x}f^{j+1}}{F_{y}^{j}}\right) \partial_{x}f^{k}\\
&
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( \frac{F_{x^{2}}^{j}+F_{xy}^{j}\partial_{x}f^{j+1}}{F_{x}^{j}
-\frac{F_{xy}^{j}\partial_{x}f^{j}+F_{y^{2}}^{j}\partial_{x}f^{j+1}}{F_{y
^{j}}\right) \partial_{x}f^{k},
\end{align*}
substituting in the above expression the values of $\partial_{x}f^{j}$ and
$\partial_{x}f^{j+1}$, such that
\[
\partial_{x}f^{j}=\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\frac{F_{x}^{i}}{F_{y}^{i}}=\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\nu_{i
\]
and
\[
\partial_{x}f^{j+1}=\left( -1\right) ^{j+1
{\displaystyle\prod\limits_{i=0}^{j}}
\frac{F_{x}^{i}}{F_{y}^{i}}=\left( -1\right) ^{j
{\displaystyle\prod\limits_{i=0}^{j-1}}
\nu_{i},
\]
we obtain
\begin{align*}
\partial_{x^{2}}f^{k} & =\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{\left( -1\right) ^{j}}{F_{x}^{j}}\left(
{\displaystyle\prod\limits_{i=0}^{j-1}}
\frac{F_{x}^{i}}{F_{y}^{i}}\right) \left( F_{x^{2}}^{j}-2F_{xy}^{j
\frac{F_{x}^{j}}{F_{y}^{j}}+F_{y^{2}}^{j}\left( \frac{F_{x}^{j}}{F_{y}^{j
}\right) ^{2}\right) \\
& =\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j
}\partial_{x}f^{j},
\end{align*}
as desired.
The second statement is immediate.
\end{pf}
The mixed derivative $\partial_{\alpha x}f^{p}$ is also necessary for some
computations in the case of transcritical, pitchfork and flip.
\begin{prop}
At the bifurcation point we hav
\[
\partial_{\alpha x}f^{p}\left( x_{0}\right) =\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\widetilde{F}_{x^{2}}^{j}-2\widetilde{F}_{xy}^{j}\widetilde{\nu
}_{j}+\widetilde{F}_{y^{2}}^{j}\widetilde{\nu}_{j}^{2}}{\widetilde{F}_{x}^{j
}\partial_{\alpha}f^{j}+\left( \frac{\widetilde{F}_{y^{2}}^{j}}{\widetilde
{F}_{y}^{j}}-\frac{\widetilde{F}_{xy}^{j}}{\widetilde{F}_{x}^{j}}\right)
\frac{\widetilde{F}_{\alpha}^{j}}{\widetilde{F}_{y}^{j}}+\frac{\widetilde
{F}_{x\alpha}^{j}}{\widetilde{F}_{x}^{j}}-\frac{\widetilde{F}_{y\alpha}^{j
}{\widetilde{F}_{y}^{j}}\right) ,
\]
where
\[
\partial_{\alpha}f^{j}=\left( -1\right) ^{j
{\displaystyle\sum\limits_{k=0}^{j-1}}
\frac{\left( -1\right) ^{k}\widetilde{F}_{\alpha}^{k}}{\widetilde{F}_{y
^{k}
{\displaystyle\prod\limits_{i>j}^{p-1}}
\widetilde{\nu}_{i}.
\]
\end{prop}
\begin{pf}
We have now the derivative of (\ref{chain})
\begin{align*}
\partial_{\alpha x}f^{p} & =\left( -1\right) ^{p}\partial_{\alpha}\left(
\frac
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{x}^{j}}
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{y}^{j}}\right) \\
& =\left( -1\right) ^{p}\frac{\partial_{\alpha}\left(
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{x}^{j}\right) }
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{y}^{j}}-\left( -1\right) ^{p}\frac{\partial_{\alpha}\left(
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{y}^{j}\right)
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{x}^{j}}{\left(
{\displaystyle\prod\limits_{j=0}^{p-1}}
F_{y}^{j}\right) ^{2}},
\end{align*}
at the bifurcation point we hav
\[
\left( -1\right) ^{p}\frac
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{x}^{j}}
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{y}^{j}}=\pm1.
\]
Therefore
\[
\partial_{\alpha x}f^{p}=\frac{\left( -1\right) ^{p}\partial_{\alpha}\left(
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{x}^{j}\right) \mp\partial_{\alpha}\left(
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{y}^{j}\right) }
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{y}^{j}},
\
\[
=\frac{\left( -1\right) ^{p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\partial_{\alpha}\widetilde{F}_{x}^{j}\left(
{\displaystyle\prod\limits_{i\not =j=0}^{p-1}}
\widetilde{F}_{x}^{i}\right) \m
{\displaystyle\sum\limits_{j=0}^{p-1}}
\partial_{\alpha}\widetilde{F}_{y}^{j}\left(
{\displaystyle\prod\limits_{i\not =j=0}^{p-1}}
\widetilde{F}_{y}^{i}\right) }
{\displaystyle\prod\limits_{j=0}^{p-1}}
\widetilde{F}_{y}^{j}
\]
and at the bifurcation point this i
\begin{align*}
\partial_{\alpha x}f^{p} & =\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\partial_{\alpha}\widetilde{F}_{x}^{j}}{\widetilde{F}_{x}^{j}}\m
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\partial_{\alpha}\widetilde{F}_{y}^{j}}{\widetilde{F}_{y}^{j}}\\
& =\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\widetilde{F}_{x^{2}}^{j}\partial_{\alpha}f^{j}+\widetilde
{F}_{xy}^{j}\partial_{\alpha}f^{j+1}+\widetilde{F}_{x\alpha}^{j}
{\widetilde{F}_{x}^{j}}-\frac{\widetilde{F}_{xy}^{j}\partial_{\alpha
f^{j}+\widetilde{F}_{y^{2}}^{j}\partial_{\alpha}f^{j+1}+\widetilde{F
_{y\alpha}^{j}}{\widetilde{F}_{y}^{j}}\right) .
\end{align*}
Knowing tha
\begin{align*}
\partial_{\alpha}f^{j+1} & =\left( -1\right) ^{j+1
{\displaystyle\sum\limits_{k=0}^{j}}
\frac{\left( -1\right) ^{k}F_{\alpha}^{k}}{F_{y}^{k}
{\displaystyle\prod\limits_{i>k}^{j}}
\frac{F_{x}^{i}}{F_{y}^{i}}\\
& =-\frac{F_{x}^{j}\partial_{\alpha}f^{j}+F_{\alpha}^{j}}{F_{y}^{j}},
\end{align*}
we hav
\begin{align*}
\partial_{\alpha x}f^{p} & =\\
& =\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\widetilde{F}_{x^{2}}^{j}\partial_{\alpha}f^{j}-\widetilde
{F}_{xy}^{j}\frac{\widetilde{F}_{x}^{j}\partial_{\alpha}f^{j}+\widetilde
{F}_{\alpha}^{j}}{\widetilde{F}_{y}^{j}}+\widetilde{F}_{x\alpha}^{j
}{\widetilde{F}_{x}^{j}}-\frac{\widetilde{F}_{xy}^{j}\partial_{\alpha
f^{j}-\widetilde{F}_{y^{2}}^{j}\frac{\widetilde{F}_{x}^{j}\partial_{\alpha
}f^{j}+\widetilde{F}_{\alpha}^{j}}{\widetilde{F}_{y}^{j}}+\widetilde
{F}_{y\alpha}^{j}}{\widetilde{F}_{y}^{j}}\right) \\
& =\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\left( \widetilde{F}_{x^{2}}^{j}\frac{\widetilde{F}_{y}^{j
}{\widetilde{F}_{x}^{j}}+\widetilde{F}_{y^{2}}^{j}\frac{\widetilde{F}_{x}^{j
}{\widetilde{F}_{y}^{j}}-2\widetilde{F}_{xy}^{j}\right) \partial_{\alpha
}f^{j}}{\widetilde{F}_{y}^{j}}+\left( \frac{\widetilde{F}_{y^{2}}^{j
}{\widetilde{F}_{y}^{j}}-\frac{\widetilde{F}_{xy}^{j}}{\widetilde{F}_{x}^{j
}\right) \widetilde{F}_{\alpha}^{j}+\left( \frac{\widetilde{F}_{x\alpha
^{j}}{\widetilde{F}_{x}^{j}}-\frac{\widetilde{F}_{y\alpha}^{j}}{\widetilde
{F}_{y}^{j}}\right) \right) \\
& =\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\widetilde{F}_{x^{2}}^{j}-2\widetilde{F}_{xy}^{j}\widetilde{\nu
}_{j}+\widetilde{F}_{y^{2}}^{j}\widetilde{\nu}_{j}^{2}}{\widetilde{F}_{x}^{j
}\partial_{\alpha}f^{j}+\left( \frac{\widetilde{F}_{y^{2}}^{j}}{\widetilde
{F}_{y}^{j}}-\frac{\widetilde{F}_{xy}^{j}}{\widetilde{F}_{x}^{j}}\right)
\frac{\widetilde{F}_{\alpha}^{j}}{\widetilde{F}_{y}^{j}}+\frac{\widetilde
{F}_{x\alpha}^{j}}{\widetilde{F}_{x}^{j}}-\frac{\widetilde{F}_{y\alpha}^{j
}{\widetilde{F}_{y}^{j}}\right)
\end{align*}
as desired.
\end{pf}
\begin{prop}
\label{third}The third derivative of $f^{k}$ defined using the system
(\ref{implic}) along the orbit is given b
\begin{multline*}
\partial_{x^{3}}f^{k}=\scriptstyle\frac{\left( \partial_{x^{2}}f^{k}\right)
^{2}}{\partial_{x}f^{k}}+\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j
}\partial_{x^{2}}f^{j}\\
\scriptstyle+\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{3}}^{j}-3F_{x^{2}y}^{j}\nu_{j}+3F_{xy^{2}}^{j}\nu_{j}^{2
-F_{y^{3}}^{j}\nu_{j}^{3}}{F_{x}^{j}}\left( \partial_{x}f^{j}\right) ^{2}\\
\scriptstyle+\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( -\left( F_{x^{2}}^{j}\right) ^{2}+F_{x^{2}}^{j}F_{xy}^{j}\nu
_{j}+\left( F_{x^{2}}^{j}{}F_{y^{2}}^{j}+2\left( F_{xy}^{j}\right)
^{2}\right) \nu_{j}^{2}-5F_{xy}^{j}F_{y^{2}}^{j}\nu_{j}^{3}+2\left(
F_{y^{2}}^{j}\right) ^{2}\nu_{j}^{4}\right) \left( \frac{\partial_{x}f^{j
}{F_{x}^{j}}\right) ^{2
\end{multline*}
with $\partial_{x}f^{j}$, $\partial_{x^{2}}f^{j}$ known from the previous results.
At the bifurcation point we obtai
\begin{multline*}
\partial_{x^{3}}f^{p}=\scriptstyle\pm\left( \partial_{x^{2}}f^{p}\right)
^{2}\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\widetilde{F}_{x^{2}}^{j}-2\widetilde{F}_{xy}^{j}\widetilde{\nu
_{j}+\widetilde{F}_{y^{2}}^{j}\widetilde{\nu}_{j}^{2}}{\widetilde{F}_{x}^{j
}\partial_{x^{2}}f^{j}\\
\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{\widetilde{F}_{x^{3}}^{j}-3\widetilde{F}_{x^{2}y}^{j
\widetilde{\nu}_{j}+3\widetilde{F}_{xy^{2}}^{j}\widetilde{\nu}_{j
^{2}-\widetilde{F}_{y^{3}}^{j}\widetilde{\nu}_{j}^{3}}{\widetilde{F}_{x}^{j
}\right) \left( \partial_{x}f^{j}\right) ^{2}\\
\scriptstyle\p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\left( \frac{-\left( \widetilde{F}_{x^{2}}^{j}\right) ^{2}+\widetilde
{F}_{x^{2}}^{j}\widetilde{F}_{xy}^{j}\widetilde{\nu}_{j}+\left( \widetilde
{F}_{x^{2}}^{j}\widetilde{F}_{y^{2}}^{j}+2\left( \widetilde{F}_{xy
^{j}\right) ^{2}\right) \widetilde{\nu}_{j}^{2}}{\left( \widetilde{F
_{x}^{j}\right) ^{2}}+\frac{-5\widetilde{F}_{xy}^{j}\widetilde{F}_{y^{2}
^{j}\widetilde{\nu}_{j}^{3}+2\left( \widetilde{F}_{y^{2}}^{j}\right)
^{2}\widetilde{\nu}_{j}^{4}}{\left( \widetilde{F}_{x}^{j}\right) ^{2
}\right) \partial_{x}f^{j},
\end{multline*}
\end{prop}
\begin{pf}
We recall the second derivative from (\ref{SEC})
\begin{equation}
\partial_{x^{2}}f^{k}=\partial_{x}f^{k
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}\left( F_{y}^{j}\right) ^{2}-2F_{xy}^{j}F_{x}^{j
F_{y}^{j}+F_{y^{2}}^{j}\left( F_{x}^{j}\right) ^{2}}{F_{x}^{j}\left(
F_{y}^{j}\right) ^{2}}\partial_{x}f^{j}. \label{second
\end{equation}
We have als
\[
\partial_{x}f^{j+1}=-\frac{F_{x}^{j}}{F_{y}^{j}}\partial_{x}f^{j}=-\nu
_{j}\partial_{x}f^{j}\text{,
\]
an
\begin{align*}
\partial_{x^{2}}f^{j+1} & =-\frac{F_{x}^{j}}{F_{y}^{j}}\partial_{x^{2}
f^{j}+\frac{F_{x^{2}}^{j}\left( F_{y}^{j}\right) ^{2}-2F_{xy}^{j}F_{x
^{j}F_{y}^{j}+F_{y^{2}}^{j}\left( F_{x}^{j}\right) ^{2}}{\left( F_{y
^{j}\right) ^{3}}\left( \partial_{x}f^{j}\right) ^{2}\\
& =-\nu_{j}\partial_{x^{2}}f^{j}+\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu
_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{y}^{j}}\left( \partial_{x}f^{j}\right)
^{2}\text{.
\end{align*}
The result is obtained differentiating (\ref{second}) and substituting
$\partial_{x}f^{j+1}$ and $\partial_{x^{2}}f^{j+1}$ by the expressions above
and simplifying. After some painful but straightforward computations we arrive
at the result.
At the bifurcation point we have $\partial_{x}f^{p}=\pm1$. Therefore, we get
easily the second statement.
\end{pf}
The classic Schwarzian derivative takes the for
\[
Sf^{p}=\frac{\partial_{x}^{3}f^{p}}{\partial_{x}f^{p}}-\frac{3}{2}\left(
\frac{\partial_{x}^{2}f^{p}}{\partial_{x}f^{p}}\right) ^{2}.
\]
In the case of implicitly defined dynamical systems, the Schwarzian derivative
can be computed using the previous results, givin
\begin{multline*}
Sf^{k}\left( x_{0}\right) =\scriptstyl
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j
}\partial_{x^{2}}f^{j}\\
\scriptstyle
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{3}}^{j}-3F_{x^{2}y}^{j}\nu_{j}+3F_{xy^{2}}^{j}\nu_{j}^{2
-F_{y^{3}}^{j}\nu_{j}^{3}}{F_{x}^{j}}\left( \partial_{x}f^{j}\right) ^{2}\\
\scriptstyle
{\displaystyle\sum\limits_{j=0}^{k-1}}
\left( -\left( F_{x^{2}}^{j}\right) ^{2}+F_{x^{2}}^{j}F_{xy}^{j}\nu
_{j}+\left( F_{x^{2}}^{j}{}F_{y^{2}}^{j}+2\left( F_{xy}^{j}\right)
^{2}\right) \nu_{j}^{2}-5F_{xy}^{j}F_{y^{2}}^{j}\nu_{j}^{3}+2\left(
F_{y^{2}}^{j}\right) ^{2}\nu_{j}^{4}\right) \left( \frac{\partial_{x}f^{j
}{F_{x}^{j}}\right) ^{2}\\
\scriptstyle-\frac{1}{2}\left(
{\displaystyle\sum\limits_{j=0}^{k-1}}
\frac{F_{x^{2}}^{j}-2F_{xy}^{j}\nu_{j}+F_{y^{2}}^{j}\nu_{j}^{2}}{F_{x}^{j
}\partial_{x}f^{j}\right) ^{2}\text{.
\end{multline*}
Although the rather long expression, the Schwarzian derivative can be easily
computed. In the case of the pitchfork, the last term vanishes.
Combining all the results in this section, we are able to study the
codimension one bifurcations of implicitly defined one-dimensional discrete
dynamical systems.
\section{Examples}
In this section we give examples for fold, transcritical, pitchfork and flip
bifurcations for periodic orbits of implicitly defined dynamical discrete
dynamical systems
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5676in,
width=2.5797in
{implicit1.eps
\caption{Period three saddle orbit generated by the fold bifurcation when
$f^{3}$ crosses the diagonal in three points.
\label{FIG1
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5676in,
width=2.5797in
{implicit2.eps
\caption{Triple tangency of the fold bifurcation for the implicit defined
modified logistic.
\label{FIG2
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5668in,
width=2.6187in
{implicit5.eps
\caption{The period two orbit at the transcritical bifurcation point. This
non-hyperbolic orbit is a saddle, attracting from the outside and repelling to
the inside of the interval.
\label{FIG5
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5668in,
width=2.5668in
{implicit6.eps
\caption{Double tangency of the transcritical bifurcation of period $2$ for
the implicit dynamical system of example \ref{ex2}. We can see $f^{2}$ for
values of the parameter near $2$. The periodic points cross stabilities.
\label{FIG6
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5668in,
width=2.6187in
{implicit3.eps
\caption{Non-hyperbolic orbit, although topologically stable, of period $2$ at
the pitchfork bifurcation for the modified bimodal implicit dynamical system.
\label{FIG3
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.5668in,
width=2.6187in
{implicit4.eps
\caption{Double tangency of the pitchfork bifurcation of period $2$ for the
implicit defined modified bimodal map. We can see $f$ and $f^{2}$.
\label{FIG4
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.2523in,
width=2.2628in
{implicit7.eps
\caption{Flip bifurcation. We see the double intersection of the map $f^{2}$
with the diagonal. The map has slope $-1$ at the relevant intersections.
\label{FIG7
\end{center}
\end{figure}
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.4534in,
width=2.4647in
{implicit8.eps
\caption{Flip bifurcation. We can see the attracting period $4$ orbit
generated by the flip bifurcation and in the center the repelling period $2$
orbit obtained from the original attracting period $2$ orbit.
\label{FIG8
\end{center}
\end{figure}
\begin{exmp}
\label{ex1}\textbf{Fold case, period 3. }Let be the implicitly defined
discrete dynamical system for $x_{n}\in\left[ 0,1\right] $ and $\alpha
\in\left[ 0,4\right] $, we call to the following model a modified implicit
logistic ma
\[
F\left( x_{n},x_{n+1},\alpha\right) =x_{n+1}-\alpha x_{n}(1-x_{n
+\frac{x_{n+1}^{P}}{B})=0.
\]
With $P=5$ and $B=100$, an explicit solution for $x_{n+1}$ is not possible,
since the map
\[
F\left( x,y,\alpha\right) =y-\alpha x(1-x+\frac{y^{5}}{100})=0,
\]
does not admit a closed formula for the solution $y$. The derivatives of $F$
ar
\begin{align*}
F_{x}\left( x,y,\alpha\right) & =\alpha(-1+2x-\frac{y^{5}}{100}),\\
F_{y}\left( x,y,\alpha\right) & =1-\frac{1}{20}\alpha xy^{4},\\
F_{\alpha}\left( x,y,\alpha\right) & =-x(1-x+\frac{y^{5}}{100}),\\
F_{x^{2}}\left( x,y,\alpha\right) & =2\alpha,\\
F_{y^{2}}\left( x,y,\alpha\right) & =-\frac{\alpha xy^{3}}{5},\\
F_{xy}\left( x,y,\alpha\right) & =-\frac{\alpha y^{4}}{20}.
\end{align*}
We are looking for a period $3$ fold, the bifurcation equations ar
\[
\left\{
\begin{array}
[c]{l
F\left( x_{0},x_{1},\alpha\right) =0,\\
F\left( x_{1},x_{2},\alpha\right) =0,\\
F\left( x_{2},x_{0},\alpha\right) =0,\\
\partial_{x}f^{3}\left( x_{0}\right) =\left( -1\right) ^{3
{\displaystyle\prod\limits_{j=0}^{2}}
\frac{F_{x}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right)
}{F_{y}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right) }=1.
\end{array}
\right.
\]
A solution found numerically i
\begin{align*}
x_{0} & =0.16498\ldots,x_{1}=0.51813\ldots,\\
x_{2} & =0.954\ldots,\alpha=3.75938\ldots.
\end{align*}
The previous computations show that there exists locally the implicitly
defined discrete dynamical system, since the derivative $F_{y}\left(
x,y,\alpha\right) $ does not vanish in the the interval $\left[ 0,1\right]
$ containing the orbit. The non-degeneracy condition holds at the periodic
orbit where $f=y$ is the implicitly defined iteration functio
\[
\partial_{x^{2}}f^{3}\left( x_{0}\right) =23.5\ldots\text{.
\]
The transversality condition gives
\[
\partial_{\alpha}f^{3}\left( x_{0}\right) =-0.844\ldots\text{.
\]
Therefore, the bifurcation is a supercritical fold with period three,
generating one period three attracting and one period three repelling orbits.
The saddle orbit at the bifurcation point can be seen in Figure \ref{FIG1}.
The bifurcation is via a simultaneous triple tangency at the diagonal, and can
be seen in Figure \ref{FIG2}.
\end{exmp}
\begin{exmp}
\label{ex2}\textbf{Transcritical case, period 2. }Let be the implicitly
defined discrete dynamical system for $x_{n}\in\left[ 0,1\right] $ and
$\alpha\in\left[ 0,4\right]
\[
F\left( x_{n},x_{n+1},\alpha\right) =x_{n+1}+x_{n}+\alpha x_{n}\left(
\left( x_{n}-\frac{x_{n+1}^{P}}{B}\right) ^{2}-1\right) -x_{n}\left(
\left( x_{n}-\frac{x_{n+1}^{P}}{B}\right) ^{4}-1\right) =0.
\]
With $P=3$ and $B=100$, an explicit solution for $x_{n+1}$ is not possible,
since the equation
\[
F\left( x,y,\alpha\right) =y+x-\alpha x\left( \left( x-\frac{y^{3}
{100}\right) ^{2}-1\right) -x\left( \left( x-\frac{y^{3}}{100}\right)
^{4}-1\right) =0,
\]
does not admit a closed solution for $y$. At this point we omit the long
computations needed and the list of derivatives, for sake of brevity. The
reader can confirm our conclusions easily.
A period two solution for the transcritical bifurcation is found numerically
to b
\[
x_{0}=0.9903\ldots,x_{1}=-0.9903\ldots,\alpha=2.
\]
There exists locally the implicitly defined discrete dynamical system, since
the derivative $F_{y}\left( x,y,\alpha\right) $ does not vanish in the the
interval $\left[ 0,1\right] $ containing the orbit. The non-degeneracy
condition holds at the periodic orbi
\[
\partial_{x^{2}}f^{2}\left( x_{0}\right) =-16.79\ldots\text{,
\]
the derivative relative to the parameter is
\[
\partial_{\alpha}f^{2}\left( x_{0}\right) =0\text{.
\]
Therefore, the transversality condition is no
\[
\partial_{\alpha x}f^{2}\left( x_{0}\right) =4.07769\text{.
\]
The conditions indicate a classical transcritical bifurcation, similar to the
one that happens for the logistic map at the origin, but for a period two
orbit. See Figures \ref{FIG5} and \ref{FIG6} for a graphical perspective of
this type of bifurcation.
\end{exmp}
\begin{exmp}
\label{ex3}\textbf{Pitchfork case, period 2. }Let be the implicitly defined
discrete dynamical system for $x_{n}\in\left[ 0,1\right] $ and $\alpha
\in\left[ 0,4\right] $, we call to the following model a modified implicit
bimodal ma
\[
F\left( x_{n},x_{n+1},\alpha\right) =x_{n+1}-\alpha\left( x_{n
+\frac{x_{n+1}^{P}}{B}\right) ^{3}-\left( 1-\alpha\right) \left(
x_{n}+\frac{x_{n+1}^{P}}{B}\right) =0.
\]
With $P=5$ and $B=100$, an explicit solution for $x_{n+1}$ is not possible,
since the map
\[
F\left( x,y,\alpha\right) =y-\alpha\left( x+\frac{y^{5}}{B}\right)
^{3}-\left( 1-\alpha\right) \left( x+\frac{y^{5}}{B}\right) =0,
\]
does not admit a closed formula for the solution $y$. We are looking for a
period $2$ pitchfork. The bifurcation equations ar
\[
\left\{
\begin{array}
[c]{l
F\left( x_{0},x_{1},\alpha\right) =0,\\
F\left( x_{1},x_{0},\alpha\right) =0,\\
\partial_{x}y^{2}\left( x_{0}\right) =\left( -1\right) ^{2
{\displaystyle\prod\limits_{j=0}^{1}}
\frac{F_{x}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right)
}{F_{y}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right)
}=1,\\
\partial_{x^{2}}y^{2}\left( x_{0}\right) =0.
\end{array}
\right.
\]
For sake of brevity we do not present here the derivatives of $F$ but only the
final results. A solution found numerically i
\[
x_{0}=-0.5774599\ldots,x_{1}=0.5774599\ldots,\alpha=2.9989\ldots.
\]
meaning that there exists a periodic orbit with period two that bifurcates.
The first non-degeneracy condition holds at the periodic orbi
\[
\partial_{x^{3}}y^{2}\left( x_{0}\right) =-295,6\ldots\text{.
\]
the first derivative in order to the parameter gives naturall
\[
\partial_{\alpha}f^{2}\left( x_{0}\right) =0
\]
and the transversality condition is no
\[
\partial_{\alpha x}f^{2}\left( x_{0}\right) =4.05\ldots\text{.
\]
Therefore, the bifurcation is a supercritical pitchfork with period two,
generating two new period two attracting orbits and the original period two
attracting orbit becomes repelling. The orbit at the bifurcation point can be
seen in Figure \ref{FIG3}. The bifurcation is via a simultaneous double
unfolding at the diagonal, and can be seen in Figure \ref{FIG4}.
\end{exmp}
\begin{exmp}
\label{ex4}\textbf{Flip case, period 2 into period 4. }Let be the implicitly
defined discrete dynamical system of example \ref{ex1} for $x_{n}\in\left[
0,1\right] $ and $\alpha\in\left[ 0,4\right] $
\[
F\left( x,y,\alpha\right) =y-\alpha x(1-x+\frac{y^{5}}{100})=0,
\]
The new derivatives of $F$ that matter ar
\begin{align*}
F_{\alpha x}\left( x,y,\alpha\right) & =-1+2x-\frac{y^{5}}{100},\\
F_{\alpha y}\left( x,y,\alpha\right) & =-\frac{xy^{4}}{20},\\
F_{x^{3}}\left( x,y,\alpha\right) & =0,\\
F_{x^{2}y}\left( x,y,\alpha\right) & =0,\\
F_{xy^{2}}\left( x,y,\alpha\right) & =-\frac{\alpha y^{3}}{5}\\
F_{y^{3}}\left( x,y,\alpha\right) & =-\frac{3\alpha xy^{2}}{5}.
\end{align*}
We are looking for a period $2$ flip that bifurcates in a period $4$, the
bifurcation equations are for $x_{0}\not =x_{1}
\[
\left\{
\begin{array}
[c]{l
F\left( x_{0},x_{1},\alpha\right) =0,\\
F\left( x_{1},x_{0},\alpha\right) =0,\\
\partial_{x}f^{2}\left( x_{0}\right) =\left( -1\right) ^{2
{\displaystyle\prod\limits_{j=0}^{1}}
\frac{F_{x}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right)
}{F_{y}\left( x_{j},x_{j+1\left( \operatorname{mod}3\right) }\right) }=-1.
\end{array}
\right.
\]
A solution found numerically i
\[
x_{0}=0.8466\ldots,x_{1}=0.4427\ldots,\alpha=3.405\ldots.
\]
The previous computations show that there exists locally the implicitly
defined discrete dynamical system, since the derivative $F_{y}\left(
x,y,\alpha\right) $ does not vanish in the the interval $\left[ 0,1\right]
$ containing the orbit. The non-degeneracy condition (\ref{Schwarz}) holds at
the periodic orbit where $f$ is the implicitly defined iteration functio
\begin{equation}
\frac{1}{2}\left( f_{x^{2}}^{2}\left( x_{0}\right) \right) ^{2}+\frac
{1}{3}f_{x^{3}}^{2}\left( x_{0}\right) =1383.1\not =0,
\end{equation}
which is equivalent to say that the Schwarzian derivative of $f^{2}$ is not
zero at $x_{0}$. The transversality condition (\ref{fliptransverse}) i
\begin{equation}
f_{x\alpha}^{2}(x_{0})=1.45122\not =0\text{.
\end{equation}
Therefore, the bifurcation is a supercritical flip from period two to period
four, generating one period four attracting and one period two repelling
orbits. The bifurcation is via a simultaneous double $-1$ derivative for
$f^{2}$ at the diagonal, and can be seen in Figure \ref{FIG7}, finally in
Figure \ref{FIG8} we can see the orbits after the bifurcation, the dotted line
is the period two repelling orbit.
\end{exmp}
\subsubsection{Bifurcations in backward Euler and trapezoid methods}
Consider the autonomous differential equatio
\begin{equation}
x^{\prime}\left( t\right) =G\left( x\left( t\right) \right) \text{,
}x\left( 0\right) =x_{0}\text{.} \label{diff
\end{equation}
In the usual Euler method the integral is estimated at the leftmost point of
each interval givin
\begin{equation}
x_{n+1}-x_{n}=hG\left( x_{n}\right) , \label{Euler
\end{equation}
where $h$ is a positive real number, possibly very small. The backward, or
implicit Euler method \cite{Hai2009,Hai1996}, where the integral is estimated
using the rightmost point of each interval $x_{n+1}$ gives the iterative
schem
\begin{equation}
x_{n+1}-x_{n}=hG\left( x_{n+1}\right) . \label{Eulerb
\end{equation}
Actually this is a very simple one-dimensional discrete dynamical system,
obviously it can depend on internal parameters in $G$, but we are interested
in considering $h$ as the bifurcation parameter.
The iterative scheme is given b
\begin{equation}
F\left( x_{n},x_{n+1}\right) =x_{n+1}-x_{n}-hG\left( x_{n+1}\right) =0.
\label{FEulerb
\end{equation}
Our function $F$ i
\begin{equation}
F\left( x,y\right) =y-x-hG\left( y\right) .
\end{equation}
The original Euler method is considered explicit since in that case
$y=x+hG\left( x\right) $ and $x_{n+1}=x_{n}+hG\left( x_{n}\right) $.
In the case of the trapezoid method \cite{Hai2009} (which is a second order
method) we have for the same differential equation the iterative schem
\begin{equation}
F\left( x_{n},x_{n+1}\right) =x_{n+1}-x_{n}-\frac{h}{2}\left( G\left(
x_{n+1}\right) +G\left( x_{n}\right) \right) =0 \label{Ftrapezoid
\end{equation}
and the function $F$ i
\begin{equation}
F\left( x,y\right) =y-x-\frac{h}{2}\left( G\left( y\right) +G\left(
x\right) \right) \text{,} \label{trapez
\end{equation}
this method is intrinsically implicit, since there is no immediate solution of
$F\left( x,y\right) =0$ for $y$. We consider now the existence of periodic
orbits in the Euler iteration, the period is $p$, the simplest case is the
asymptotic stable fixed point, which indicates that the solution of the
original differential equation has a limit when $t$ goes to infinity for a set
of initial conditions. Obviously the non-hyperbolic condition (\ref{b})\ for
the backward Euler method simplifie
\[
\partial_{x}f^{p}\left( x_{0}\right) =\frac{1}
{\displaystyle\prod\limits_{j=0}^{p-1}}
\left( 1-hG^{\prime}\left( x_{j+1\left( \operatorname{mod}p\right)
}\right) \right) }=\pm1,
\]
this gives the non hyperbolic conditions for the backward Euler metho
\begin{equation
{\displaystyle\prod\limits_{j=0}^{p-1}}
\left( 1-hG^{\prime}\left( x_{j}\right) \right) =\pm1\text{.}
\label{bEuler-nh
\end{equation}
For the trapezoid method the non-hyperbolic condition (\ref{b}) i
\
{\displaystyle\prod\limits_{j=0}^{p-1}}
\frac{1+\frac{h}{2}G^{\prime}\left( x_{j}\right) }{1-\frac{h}{2}G^{\prime
}\left( x_{j}\right) }=\pm1\text{.
\]
The non-degeneracy condition (\ref{Dalphabifpoint}) i
\[
\partial_{h}f^{p}\left( x_{0}\right) =\left( -1\right) ^{p
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\left( -1\right) ^{j}F_{h}\left( x_{j},x_{j+1}\right) }{F_{y}\left(
x_{j},x_{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{p-1}}
\frac{F_{x}\left( x_{i},x_{i+1}\right) }{F_{y}\left( x_{i},x_{i+1}\right)
}\not =0.
\]
For the backward Euler method it give
\
{\displaystyle\sum\limits_{j=1}^{p}}
{\displaystyle\prod\limits_{i\geq j}^{p}}
\frac{G\left( x_{j}\right) }{\left( 1-hG^{\prime}\left( x_{i}\right)
\right) }\not =0\text{, with }x_{p}=x_{0}.
\]
For the trapezoid method we hav
\[
\frac{1}{2
{\displaystyle\sum\limits_{j=0}^{p-1}}
\frac{\left( -1\right) ^{j}\left( G\left( x_{j}\right) +G\left(
x_{j+1}\right) \right) }{1-\frac{h}{2}G^{\prime}\left( x_{j+1}\right)
{\displaystyle\prod\limits_{i>j}^{p-1}}
\frac{1+\frac{h}{2}G^{\prime}\left( x_{i}\right) }{1-\frac{h}{2}G^{\prime
}\left( x_{i+1}\right) }\not =0\text{, with }x_{p}=x_{0}.
\]
We study a simple example for the backward Euler method. Similar examples can
be constructed for the trapezoid method.
\begin{exmp}
Consider the simple differential equatio
\begin{equation}
x^{\prime}=x^{5}-1\text{, }x_{0}=0\text{.} \label{differential
\end{equation}
This equation can be solved by quadratures but it is impossible to obtain an
explicit expression for the solution. Applying Euler backward method we ge
\[
x_{n+1}-x_{n}=hx_{n+1}^{5}-h\text{,
\]
i.e.,
\[
F\left( x_{n},x_{n+1}\right) =x_{n+1}-x_{n}-h\left( x_{n+1}^{5}-1\right)
=0\text{.
\]
Naturally, $G\left( y\right) =y^{5}-1$. We have $G^{\prime}\left( y\right)
=-5y^{4}$. Equation (\ref{bEuler-nh}) i
\begin{equation
{\displaystyle\prod\limits_{j=0}^{p-1}}
\left( 1-5hx_{j}^{4}\right) =\pm1\text{.} \label{BackEulerH
\end{equation}
We have to solve (\ref{implic}) together with (\ref{BackEulerH}), we start by
the fixed poin
\[
\left\{
\begin{array}
[c]{l
x_{0}^{5}-1=0,\\
\left( 1-5hx_{0}^{4}\right) =\pm1\text{.
\end{array}
\right.
\]
Excluding the trivial case $x=1$, $h=0$, there are no solutions for the fold
case. The solution is fairly simple for the flip case
\[
x_{0}=1\text{, }h=0.4\text{,
\]
This means that when $h=0.4$ the fixed point $x_{0}=1$ duplicates. When $h$ is
greater than $0.4$ the fixed point becomes attracting and is generated a
period two repelling orbit. Below $0.4$ the fixed point $x_{0}=1$ is repelling.
Now let us consider a period two orbit, the bifurcation equations are no
\[
\left\{
\begin{array}
[c]{l
x_{1}-x_{0}=hx_{1}^{5}-h\\
x_{0}-x_{1}=hx_{0}^{5}-h\\
\left( 1-5hx_{0}^{4}\right) \left( 1-5hx_{1}^{4}\right) =\pm1\text{.
\end{array}
\right.
\]
We get, among complex solutions not considered here, the non trivial
($h\not =0$) real solutions for the fold cas
\begin{align*}
x_{0} & =x_{1}=1\text{, }h=0.4\text{, degenerate and obtained previously }\\
x_{0} & =1.15767\text{, }x_{1}=-0.602341\text{, }h=1.63071\text{.
\end{align*}
obviously $x_{0}=-0.602341$ and $x_{1}=1.15767$ is also a solution.
For the flip case we get the period doubling point where a period two orbit
duplicates its perio
\begin{align*}
x_{0} & =1.12579,\text{ }x_{1}=0.718620,\text{ }h=0.503700,\\
x_{0} & =-0.580682,\text{ }x_{1}=1.15618,\text{ }h=1.62930.
\end{align*}
This means that the previously created at $h=0.4$ repelling period two
solution, bifurcates again when $h=0.503700$ to a period $4$ orbit.
Finally, among other period three solutions, there is a period three fold at a
low value of $h$
\[
x_{0}=0.784072,\text{ }x_{1}=0.16453,\text{ }x_{2}=1.22008,\text{
}h=0.619616.
\]
Due to the continuity of all the functions involved this implies the existence
of chaos for low values of the parameters, even in the case of the backward
Euler method of a very simple first order differential equation. It is a well
known fact that one-dimensional discrete dynamical systems are more complex
than one-dimensional continuous dynamical systems. Nevertheless, the existence
of chaos for small values of the parameter $h$ is still exciting.
\end{exmp}
The previous example suggests the existence of a plethora of phenomena
deserving further research in implicit numeric methods. By force, the more
general cases of implicit discrete dynamical systems, which are very scarce in
the literature, are a vast field of research totally open.
\textbf{Acknowledgement} Partially funded by FCT/Portugal through
UID/MAT/04459/ 2013 for CMAGDS. |
1608.01921 | \section{Introduction}
Let $P\subset\ensuremath{\mathbb{R}}^d$ be a $d$-dimensional point set. We say $P$
\emph{embraces} a point $\ve{p} \in \ensuremath{\mathbb{R}}^d$ or $P$ is
\emph{$\ve{p}$-embracing} if $\ve{p} \in \conv(P)$, and we
say $P$ \emph{ray-embraces} $\ve{p}$ if $\ve{p} \in \pos(C)$, where $\pos(P)
= \big\{ \sum_{\ve{p} \in P} \alpha_{\ve{p}} \ve{p}
\mid \alpha_{\ve{p}} \geq 0 \text{ for all $\ve{p} \in P$} \big\}$.
\Caratheodory's
theorem~\cite[Theorem~1.2.3]{Matouvsek2002} states that if $P$
embraces the origin, then there exists a subset $P' \subseteq P$ of
size $d+1$ that also embraces the origin. This was generalized by
\Barany~\cite{Barany1982} to the \emph{colorful} setting: let
$C_1,\dots,C_{d+1} \subset \ensuremath{\mathbb{R}}^d$ be point sets that each embrace the
origin. We call a set $C = \{\ve{c}_1, \dots, \ve{c}_{d+1}\}$
a \emph{colorful choice} (or
\emph{rainbow}) for $C_1, \dots, C_{d+1}$,
if $\ve{c}_i \in C_i$, for $i = 1, \dots, d+1$. The
\emph{colorful} \Caratheodory theorem states that there always exists
a $\ve{0}$-embracing colorful choice that contains the origin in its
convex hull. \Barany also gave the following generalization.
\begingroup
\newcommand{0.6}{0.6}
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:colcara:conv}\includegraphics[scale=0.6]{colcara.pdf}}
\hspace{2cm}
\subfloat[]{\label{fig:colcara:cone}\includegraphics[scale=0.6,page=2]{colcara.pdf}}
\caption{\protect\subref{fig:colcara:conv} Example of the convex version of
Theorem~\ref{thm:colcara} in two dimensions.
\protect\subref{fig:colcara:cone} Example of the cone version of
Theorem~\ref{thm:colcara} in two dimensions.}
\label{fig:colcara}
\end{figure}
\endgroup
\begin{theorem}[Colorful \Caratheodory Theorem, Cone Version
\cite{Barany1982}]
\label{thm:colcara}
Let $C_1,\dots,C_d \subset \ensuremath{\mathbb{R}}^d$ be point sets and $\ve{b} \in \ensuremath{\mathbb{R}}^d$ a
point with $\ve{b} \in \pos(C_i)$, for $i = 1, \dots, d$. Then, there is a
colorful choice $C$ for $C_1, \dots, C_d$ that ray-embraces $\ve{b}$. \qed
\end{theorem}
The classic (convex) version of the
colorful \Caratheodory theorem follows easily from
Theorem~\ref{thm:colcara}: lift the sets $C_1,\dots,C_{d+1}
\subset \ensuremath{\mathbb{R}}^d$ to $\ensuremath{\mathbb{R}}^{d+1}$ by appending a $1$ to each element, and set
$\ve{b}=(0,\dots,0,1)^T$. See Figure~\ref{fig:colcara} for an example of both
versions in two dimensions.
Even though the cone version of the colorful \Caratheodory theorem
guarantees the existence of a colorful choice that ray-embraces the
point $\ve{b}$, it is far from clear how to find it efficiently. We call
this computational problem the \emph{colorful \Caratheodory problem}
(\CCP). To this day, settling the complexity of \CCP remains an
intriguing open problem, with a potentially wide range of
consequences. We can use linear programming to check in polynomial
time whether a
given colorful choice ray-embraces a point, so \CCP
lies in \emph{total function \NP}
(\TFNP)~\cite{Papadimitriou1994}, the complexity class of total
search problems that
can be solved in non-deterministic polynomial time. This implies that
\CCP cannot be \NP-hard unless
$\NP=\coNP$~\cite{JohnsonPaYa1988}.
However, the complexity landscape
inside \TFNP is far from understood, and there exists a rich body of
work that studies subclasses of \TFNP meant to capture
different aspects of mathematical existence proofs,
such as the pigeonhole principle (\PPP), potential function
arguments (\PLS,
\CLS), or various parity arguments (\PPAD, \PPA,
\PPADS)~\cite{DaskalakisPa11,JohnsonPaYa1988,Papadimitriou1994}.
While the
complexity of \CCP remains elusive, related problems are known
to be complete for \PPAD or for
\PLS. For example, given $d+1$ point sets
$C_1,\dots,C_{d+1} \subset \ensuremath{\mathbb{Q}}^d$ consisting of two points each and a
colorful choice $C$ for $C_1, \dots, C_{d+1}$ that embraces the
origin, it is \PPAD-complete to find another
colorful choice that embraces the origin~\cite{MeunierSa2014}.
Furthermore, given $d+1$ point sets $C_1,\dots,C_{d+1} \subset \ensuremath{\mathbb{Q}}^d$,
we call a colorful
choice $C$ for $C_1, \dots, C_{d+1}$
\emph{locally optimal} if the $L_1$-distance of $\conv(C)$ to the
origin cannot be decreased by swapping a point of color $i$ in $C$
with another
point from the same color. Then, computing a locally optimal
colorful choice is \PLS-complete~\cite{MulzerSt2015}.
Understanding the complexity of \CCP becomes even more interesting in the
light of the fact that the colorful \Caratheodory theorem plays a
crucial role in proving several other prominent theorems in convex
geometry, such as Tverberg's theorem~\cite{Sarkaria1992} (and hence
the centerpoint theorem~\cite{Rado1946}) and the first
selection lemma~\cite{Matouvsek2002,Barany1982}. In fact, these proofs
can be
interpreted as polynomial time reductions from the respective
computational
problems, \Tverberg, \Centerpoint, and \SimCenter,
to \CCP. See Section~\ref{sec:app:reductions} for more details.
Several approximation algorithms have been proposed for \CCP.
\Barany and Onn~\cite{BaranyOn1997} describe an
exact algorithm that can be stopped early to find a colorful choice
whose convex hull is ``close'' to the origin. More precisely, let
$\ensuremath{\varepsilon}, \rho > 0$ be
parameters. We call a set \emph{$\ensuremath{\varepsilon}$-close} if its convex hull has
$L_2$-distance at most $\ensuremath{\varepsilon}$ to the origin. Given sets
$C_1,\dots,C_{d+1}\subset \ensuremath{\mathbb{R}}^d$ such that (i) each $C_i$ contains a ball
of radius $\rho$ centered at the origin in its convex hull; and (ii) all
points $\ve{p} \in \bigcup_{i = 1}^{d+1} C_i$ fulfill $1\leq \|\ve{p}\| \leq 2$
and can be encoded using $L$ bits, one can find an
$\ensuremath{\varepsilon}$-close colorful choice in time
$O(\text{poly}(L, \log(1/\ensuremath{\varepsilon}),1/\rho))$ on the \textsc{Word-Ram} with
logarithmic costs. For $\ensuremath{\varepsilon}=0$, the algorithm actually finds a
solution to \CCP in finite time, and, more interestingly, if $1/\rho =
O(\text{poly}(L))$, the algorithm finds a solution to \CCP in
polynomial time. In the same spirit, Barman~\cite{barman2015} showed
that if the points have constant norm, an $\ensuremath{\varepsilon}$-close colorful choice
can be found by solving $d^{O(1/\ensuremath{\varepsilon}^2)}$ convex programs.
Mulzer and Stein~\cite{MulzerSt2015}
considered a different notion of approximation: a set is called
\emph{$m$-colorful} if it contains at most $m$ points from each $C_i$.
They showed that for all fixed $\ensuremath{\varepsilon}>0$, an $\lceil \ensuremath{\varepsilon}
d\rceil$-colorful choice that contains the origin in its convex hull
can be found in polynomial time.
\paragraph{Our Results.}
We provide a new upper bound on the complexity of \CCP by showing that
the problem is contained in $\PPAD \cap \PLS$, implying the first
nontrivial upper bound on the computational complexity of computing
centerpoints or finding Tverberg partitions.
The traditional proofs of the colorful \Caratheodory theorem all
proceed through a potential function argument. Thus, it may not be
surprising that \CCP lies in \PLS, even though a detailed proof
that can deal with degenerate instances requires
some care (see Section~\ref{sec:app:pls}).
On the other hand, showing that \CCP lies in \PPAD calls for
a completely new approach. Even though there are proofs of the
colorful \Caratheodory theorem that use topological methods usually
associated with \PPAD (such as certain variants of Sperner's
lemma)~\cite{holmsen2013,kalai2005}, these proofs
involve existential
arguments that have no clear algorithmic interpretation.
Thus, we present a new proof of the
colorful \Caratheodory theorem that proceeds similarly as the usual
proof for Sperner's lemma~\cite{Cohen1967}. This new proof has
an algorithmic interpretation that leads to a
formulation of \CCP as a \PPAD-problem.
Finally, we consider the special case of \CCP that we are given
two color classes
$C_1, C_2 \subset \ensuremath{\mathbb{R}}^d$ of $d$ points each and a vector $\ve{b} \in \ensuremath{\mathbb{R}}^d$
such that both $C_1$ and $C_2$ ray-embrace $\ve{b}$.
We describe an algorithm that solves the following problem in
polynomial time: given $k \in [d]$, find a set $C \subseteq C_1 \cup
C_2$ with $|C \cap C_1| = k$ and $|C \cap C_2| = d-k$ such that
$C$ ray-embraces $\ve{b}$. Note that this is a special
case of \CCP since we can just take $k$ copies of $C_1$ and
$d-k$ copies of $C_2$ in a problem instance for \CCP.
\section{Preliminaries}
\label{sec:prelim}
\paragraph{The Complexity Class \PPAD.}
The complexity class \emph{polynomial parity argument in a directed
graph} (\PPAD)~\cite{Papadimitriou1994} is a subclass of \TFNP that contains
search problems that can be modeled as follows: let $G=(V,E)$ be
a directed graph in which each node has indegree and outdegree at most one. That
is, $G$ consists of paths and cycles.
We call a node $v \in V$ a \emph{source} if $v$ has indegree $0$ and we call $v$ a \emph{sink} if it has outdegree
$0$. Given a source in $G$, we want to find another source
or sink. By a parity argument, there is an even number of sources and
sinks in $G$ and hence another source or sink must exist.
However, finding this sink or source is nontrivial since $G$ is defined
implicitly and the total number of nodes may be exponential.
More formally, a problem in \PPAD is a relation $\mc{R}$ between
a set $\mc{I} \subseteq \{0,1\}^\star$ of \emph{problem instances}
and a set $\mc{S} \subset \{0,1\}^\star$ of \emph{candidate solutions}. Assume
further the following.
\begin{itemize}
\item The set $\mc{I}$ is polynomial-time verifiable.
Furthermore, there is an algorithm that on input $I \in \mc{I}$ and $s
\in \mc{S}$ decides in time $\poly(|I|)$ whether $s$ is a \emph{valid}
candidate solution for $I$. We denote with $\mc{S}_I \subseteq
\mc{S}$ the set of all valid candidate solutions for a fixed instance
$I$.
\item There exist two polynomial-time computable functions
$\pred$ and $\suc$ that define the edge set of $G$ as follows:
on input $I \in \mc{I}$ and $s \in \mc{S}_I$,
$\pred$ and $\suc$ return a valid candidate solution from $\mc{S}_I$
or $\bot$. Here, $\bot$ means
that $v$ has no predecessor/successor.
\item There is a polynomial-time algorithm that returns for each
instance $I$ a valid candidate solution $s \in \mc{S}_I$ with $\pred(s)
= \bot$. We call $s$ the \emph{standard source}.
\end{itemize}
Now, each instance $I \in \mc{I}$ defines a graph $G_I = (V,E)$ as follows. The
set of nodes $V$ is the set of all valid candidate solutions $\mc{S}_I$ and
there is a directed edge from $u$ to $v$ if and
only if $v = \suc(u)$ and $u = \pred(v)$. Clearly, each node in $G_I$ has
indegree and outdegree at most one. The relation $\mc{R}$ consists of all tuples
$(I,s)$ such that $s$ is a sink or source other than the standard source in $G_I$.
The definition of a \PPAD-problem suggests a simple algorithm, called the
\emph{standard algorithm}: start at the standard source and follow the path
until a sink is reached. This algorithm always finds a solution but the length
of the traversed path may be exponential in the size of the input instance.
\paragraph{Polyhedral Complexes and Subdivisions.}
We call a finite set of polyhedra $\mc{P}$ in $\ensuremath{\mathbb{R}}^d$ a \emph{polyhedral
complex} if and only if
(i) for all polyhedra $f \in \mc{P}$, all faces of $f$
are contained in $\mc{P}$; and (ii)
for all $f,f' \in \mc{P}$, the intersection $f \cap f'$ is a face of both.
Note that the first requirement implies that $\emptyset \in
\mc{P}$. Furthermore, we say $\mc{P}$ has \emph{dimension} $k$ if there
exists some polyhedron $f \in \mc{P}$ with $\dim f = k$ and all other
polyhedra in $\mc{P}$
have dimension at most $k$. We call $\mc{P}$ a \emph{polytopal complex}
if it is a polyhedral complex and all elements are polytopes. Similarly,
we say
$\mc{P}$ is a \emph{simplicial complex} if it is a polytopal complex whose
elements are simplices. Finally, we say $\mc{P}$ \emph{subdivides} a set $Q
\subseteq \ensuremath{\mathbb{R}}^d$ if $\bigcup_{f \in \mc{P}} f = Q$. For more details,
see~\cite[Section~5.1]{Ziegler1995}.
\paragraph{Linear Programming.}
Let $A \in \ensuremath{\mathbb{R}}^{d \times n}$ be a matrix and $F$ a set of column
vectors from $A$. Then, we denote with $\ind{F} \subseteq [n]$ the set
of column indices in $F$ and for an index set $I \subseteq [n]$, we denote
with $A_I$ the submatrix of $A$ that consists of the columns indexed
by
$I$. Similarly, for a vector $\ve{c} \in \ensuremath{\mathbb{R}}^n$ and an index set
$I \subset [n]$, we
denote with $\ve{c}_I$ the subvector of $\ve{c}$ with the coordinates
indexed
by $I$. Now, let $L'$ denote a system of linear equations
\begin{align*}
L': A\ve{x} =\ve{b},
\end{align*}
where $A \in \ensuremath{\mathbb{Q}}^{d \times n}$, $\ve{b} \in \ensuremath{\mathbb{Q}}^d$
and $\rank(A) = k$.
By multiplying with the least common denominator,
we may assume in the following that $A\in \ensuremath{\mathbb{Z}}^{d \times n}$ and
$\ve{b} \in \ensuremath{\mathbb{Z}}^d$.
We call a set of $k$ linearly independent
column vectors $B$ of $A$ a \emph{basis} and we say that $A$ is
\emph{non-degenerate} if $k = d$ and for all bases $B$ of $A$, no
coordinate of the corresponding solution $\ve{x}_{\ind{B}}$ is $0$.
In particular, if $L'$ is non-degenerate, then $\ve{b}$ is not contained
in the linear span of any set of $d' < d$ column vectors from $A$
and hence if $d > n$, the linear system $L'$ has no solution. In the
following, we assume that $L'$ is non-degenerate and that
$d\leq n$.
We denote with $L$ the linear program obtained by extending the
linear system $L'$ with the constraints $\ve{x} \geq \ve{0}$ and with a
cost vector $\ve{c} \in \ensuremath{\mathbb{Q}}^n$:
\[
L: \min \ve{c}^T \ve{x} \text{ subject to }
A\ve{x} =\ve{b}, \, \ve{x} \geq \ve{0}.
\]
We say a set of column vectors $B$ is a \emph{basis}
for $L$ if $B$ is a basis for $L'$. Let $\ve{x} \in \ensuremath{\mathbb{R}}^n$
be the corresponding solution, i.e., let $\ve{x}$ be
such that $A \ve{x} = \ve{b}$ and $x_i = 0$ for $i \in [n] \setminus
\ind{B}$. We call $\ve{x}$ a \emph{basic feasible solution}, and
$B$ a \emph{feasible basis}, if $\ve{x} \geq
\ve{0}$.
Furthermore, we say $L$ is \emph{non-degenerate} if for
all feasible bases $B$, the corresponding basic feasible
solutions have strictly positive values in the coordinates
of $B$.
Now, let $R = [n] \setminus \ind{B}$ be the column indices
not in $B$. The \emph{reduced cost vector} $\ve{r}_{B,\ve{c}} \in
\ensuremath{\mathbb{Q}}^{n-d}$ with respect to $B$ and $\ve{c}$ is then defined as
\begin{equation}
\ve{r}_{B,\ve{c}} = \ve{c}_{R} - \left(A^{-1}_\ind{B} A_R\right)^T \ve{c}_\ind{B}.
\label{eq:redcosts}
\end{equation}
It is well-known that $B$ is optimal for $\ve{c}$ if and only if
$\ve{r}_{B,\ve{c}}$ is non-negative in all
coordinates~\cite{MatousekGa2007}. For technical reasons,
we consider in the
following the \emph{extended reduced cost vector} $\up{\ve{r}}_{B,\ve{c}} \in \ensuremath{\mathbb{Q}}^n$
that has a $0$ in dimensions $\ind{B}$ and otherwise
equals $\ve{r}_{B,\ve{c}}$ to align the coordinates of the reduced cost vector with the column indices in $A$.
More formally, we set
\[
\left(\up{\ve{r}}_{B,\ve{c}} \right)_j = \begin{cases}
0 & \text{if } j \in \ind{B}\text{, and} \\
(\ve{r}_{B,\ve{c}})_{j'} & \text{otherwise,}
\end{cases}
\]
where $j'$ is the rank of $j$ in $R$, that is, $(\ve{r}_{B,\ve{c}})_{j'}$ is the
coordinate of $\ve{r}_{B,\ve{c}}$ that corresponds to the $j'$th non-basis column with
column index $j$ in $A$.
Geometrically, the feasible solutions for the linear program $L$ define
an $(n-d)$-dimensional polyhedron $\pc{P}$ in $\ensuremath{\mathbb{R}}^n$.
Since $L$ is non-degenerate, $\pc{P}$ is simple.
Let $f \subseteq \pc{P}$ be a $k$-face of $\pc{P}$. Then, $f$ has an
associated set $\supp{f} \subseteq [n]$ of $k$ column indices
such that $f$ consists precisely of the feasible solutions for
the linear program
$A_\supp{f} \ve{x}' = \ve{b},\, \ve{x}' \geq \ve{0}$, lifted to
$\ensuremath{\mathbb{R}}^n$ by setting the coordinates with indices not in $\supp{f}$
to $0$.
We call $\supp{f}$ the \emph{support} of $f$ and we say the columns in
$A_\supp{f}$ \emph{define} $f$.
Furthermore,
for all subfaces $\down{f} \subseteq f$, we have $\supp{\down{f}} \subseteq
\supp{f}$ and in particular, all bases that define vertices of $f$ are
$d$-subsets of columns from $A_\supp{f}$.
Moreover, we say a nonempty face $f \subseteq \pc{P}$ is \emph{optimal}
for a cost vector $\ve{c}$ if all points in $f$ are optimal for $\ve{c}$. We
can express this
condition using the reduced cost vector. Let $B$ be a basis for a vertex in
$f$. Then $f$ is optimal for $\ve{c}$ if and only if
\[
(\up{\ve{r}}_{B,\ve{c}})_j = 0 \text{~for $j \in \supp{f}$, and }
(\up{\ve{r}}_{B,\ve{c}})_j \leq 0 \text{~otherwise.}
\]
\section{Overview of the PPAD-Formulation}
\label{sec:ppad}
We give a new constructive proof of
the cone version of the colorful \Caratheodory theorem based on
Sperner's lemma. Using this, we can obtain a \PPAD-formulation
of \CCP, by adapting Papadimitriou's formulation of
Sperner's lemma as a \PPAD problem.
Recall the statement of Sperner's lemma: let $\mc{S}$ be a simplicial
subdivision of the $d$-dimensional standard simplex
$\Delta^{d} = \conv(\ve{e}_1, \dots, \ve{e}_{d+1}) \subset
\ensuremath{\mathbb{R}}^{d+1}$, where $\ve{e}_i$ is the $i$th canonical basis vector.
We call a function $\lambda$ that assigns to
each vertex in $\mc{S}$ a label from $[d+1]$ a \emph{Sperner labeling}
if for each vertex $\ve{v}$ of $\mc{S}$ contained in
$\conv(\ve{e}_{i_1}, \dots, \ve{e}_{i_k})$, we
have $\lambda(\ve{v}) \in \{i_1, \dots, i_k\}$, for all
$\{i_1, \dots, i_k\} \subseteq [d+1], k \in [d+1]$.
For a simplex $\sigma \in \mc{S}$, we set $\lambda(\sigma)$
to be the set of labels of the vertices of $\sigma$.
We call $\sigma$ \emph{fully-labeled} if $\lambda(\sigma) = [d+1]$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.7]{sperner.pdf}
\end{center}
\caption{An example of Sperner's lemma in two dimensions. The fully-labeled
simplices are marked yellow.}
\label{fig:sperner}
\end{figure}
\begin{theorem}[Strong Sperner's Lemma~\cite{Cohen1967}]
\label{thm:sperner}
The number of fully-labeled simplices is odd.
\end{theorem}
Now suppose we are given an instance $I = (C_1, \dots, C_d, \ve{b})$ of
(the cone version of)
\CCP, where $\ve{b} \in \ensuremath{\mathbb{R}}^d$, $\ve{b} \neq \ve{0}$, and each $C_i \subset
\ensuremath{\mathbb{Q}}^d$, $i \in [d]$, ray-embraces $\ve{b}$. In Section~\ref{sec:app:eqccp}, we
show that we can assume w.l.o.g.\ that each set $C_i$ has size $d$.
We now describe how to define a simplicial complex $\mc{S}$
and a Sperner labeling $\lambda$ for $I$ such that a fully labeled
simplex will encode a colorful choice that contains the
vector $\ve{b}$ in its positive span.
In the following, we call $\ensuremath{\mathbb{R}}^d$ the \emph{parameter space} and a
vector $\ve{\mu} \in \ensuremath{\mathbb{R}}^d$ a \emph{parameter vector}.
We define a family of linear programs $\{L^\CC_\ve{\mu} \mid \ve{\mu} \in
\ensuremath{\mathbb{R}}^d\}$, where each linear program $L^\CC_\ve{\mu}$ has the same constraints
and differs only in its cost vector $\ve{c}_\ve{\mu}$. The cost vector
$\ve{c}_\ve{\mu}$ is defined by a linear function in $\ve{\mu} \in \ensuremath{\mathbb{R}}^d$.
Let $A = ( C_1\; C_2 \;\dots\; C_d ) \in \ensuremath{\mathbb{Q}}^{d \times d^2}$ be
the matrix that has the vectors from $C_1$ in the first $d$ columns,
the vectors from $C_2$ in the second $d$ columns, and so on.
Then, we denote with $L^\CC_{\ve{\mu}}$ the linear program
\begin{equation}\label{eq:lp}
L^\CC_{\ve{\mu}}:
\min \ve{c}^T_{\ve{\mu}} \ve{x},\,
\text{ subject to }
A \ve{x} = \ve{b},
\ve{x} \geq \ve{0},
\end{equation}
and we denote with $\pc{P}^\CC \subset \ensuremath{\mathbb{R}}^{d^2}$ the polyhedron that is
defined by the linear system $L^\CC$.
We can think of the $i$th coordinate of the parameter vector
$\ve{\mu} \in \ensuremath{\mathbb{R}}^d$ as the weight of color $i$, i.e., the costs of columns
from $A$ with color $i$ decrease if $(\ve{\mu})_i$ increases.
To each face $f$ of $P$, we assign the set of parameter vectors
$\Phi(f) \subset \ensuremath{\mathbb{R}}^d$ such that for all $\ve{\mu} \in \Phi(f)$, the face
$f$ is optimal for the linear program $L^\CC_\ve{\mu}$ that has $L^\CC$ as
constraints and $\ve{c}_\ve{\mu}$ as cost vector. We call $\Phi(f)$ the
\emph{parameter region} of $f$. The cost vector is designed to
control the colors that appear in the support of optimal faces for a
specific subset of parameter vectors. Let $\pc{M} = \left\{ \ve{\mu} \in
\ensuremath{\mathbb{R}}^d \,\middle\vert\, \ve{\mu} \geq \ve{0},\, \|\ve{\mu}\|_\infty =1 \right\}$ denote the
faces of the unit cube in which at least one coordinate is set to
$1$. Then, no face $f$ that is assigned to a parameter vector $\ve{\mu}
\in \pc{M}$ with $(\ve{\mu})_{i^\times}=0$ has a column from $A$ with color
$i^\times$ in its defining set $A_\supp{f}$. This property will become
crucial when we define a Sperner labeling later on.
Now, we define a polyhedral subcomplex $\pc{F}$ of $\pc{P}^\CC$ that consists
of all faces $f$ of $\pc{P}^\CC$
such that $\Phi(f) \cap \pc{M} \neq \emptyset$. Furthermore,
the
intersections of the parameter regions with $\pc{M}$ induce a polytopal
complex $\pc{Q}$ that is in a dual relationship to $\pc{F}$.
By performing a central projection with the origin as
center of $\pc{Q}$ onto the standard simplex $\Delta^{d-1}$, we obtain a
polytopal subdivision $\pc{Q}_\Delta$ of $\Delta^{d-1}$.
To get the desired \emph{simplicial} subdivision of
$\Delta^{d-1}$, we take the
barycentric subdivision $\sd \pc{Q}_\Delta$
of $\pc{Q}_\Delta$.
We construct a Sperner labeling $\lambda$ for $\sd \pc{Q}_\Delta$
as follows:
let $\ve{v}$ be a vertex in $\sd \pc{Q}_\Delta$, and let $f$
be the face of $\pc{F}$ that corresponds to $\ve{v}$. Then,
we set $\lambda(\ve{v}) = i$ if the
$i$th color appears most often in the support of $f$.
The color controlling property of the
cost function $c_\ve{\mu}$ then implies that $\lambda$ is a Sperner labeling.
Furthermore,
using the properties of the barycentric subdivision and the
correspondence between $\pc{Q}_\Delta$ and $\pc{F}$, we can
show that one vertex of a fully-labeled
$(d-1)$-simplex in $\sd \pc{Q}_\Delta$ encodes a colorful feasible basis
of the \CCP instance $I$. This
concludes a new constructive proof of the colorful \Caratheodory
theorem using Sperner's lemma.
To show
that \CCP is in \PPAD however, we need to be able to traverse $\sd
\pc{Q}_\Delta$ efficiently. For this, we introduce a combinatorial encoding
of the simplices in $\pc{Q}_\Delta$ that represents neighboring simplices
in a similar manner.
Furthermore, we describe how
to generalize the orientation used in the \PPAD
formulation of 2D-Sperner~\cite{Papadimitriou1994} to our
setting.
This finally shows that \CCP is in \PPAD.
To ensure that the complexes that appear in our algorithms are
sufficiently generic, we prove several perturbation lemmas
that give a deterministic way of achieving this.
Our \PPAD-formulation also shows that the special case of \CCP
involving two colors can be solved in polynomial time. Indeed,
we will see that in this case the polytopal complex $\pc{Q}_\Delta$
can be made $1$-dimensional. Then,
binary search can be used to find a fully-labeled simplex in
$\pc{Q}_\Delta$. In order to prove that
the binary search terminates after a polynomial number of steps,
we use methods similar to our perturbation techniques to
obtain a bound on the length of the $1$-dimensional fully-labeled simplex.
\section{The Colorful \Caratheodory Problem is in PPAD}
\label{sec:ccpppad}
As before, let $I=(C_1,\dots,C_d,\ve{b})$ denote an instance for the cone version of
\CCP. Our formulation of \CCP as a \PPAD-problem requires $I$ to be in
general position. In particular, we assume that (P1) all color classes
$C_i \subset \ensuremath{\mathbb{Z}}^d$ consist of $d$ points and all points have integer
coordinates. Furthermore, we assume that (P2) there exist no subset $P
\subset \bigcup_{i=1}^d C_i$ of size $d-1$ that ray-embraces $\ve{b}$. We
show in Section~\ref{sec:app:eqccp} how to ensure the
properties by an explicit deterministic perturbation of polynomial
bit-complexity.
\subsection{The Polytopal Complex}
\label{sec:ppad:para}
Let $N = d!m^d$, where
$m$ is the largest absolute value that appears in $A$ and $\ve{b}$
(see~Lemma~\ref{lem:lpsol}).
Then, we define $\ve{c}_{\ve{\mu}} \in \ensuremath{\mathbb{R}}^{d^2}$ as
\begin{equation}\label{eq:costs}
(\ve{c}_{\ve{\mu}})_j = 1 + \left(1 - (\ve{\mu})_i\right) d N^2 + \ensuremath{\varepsilon}^j,
\end{equation}
where $j \in [d^2]$,
$i$ is the color of the $j$th column in $A$, and $0 < \ensuremath{\varepsilon} \leq N^{-3}$
is a suitable perturbation that ensures non-degeneracy of the
reduced costs (see~\cite{chvatal1983}).
As stated in the overview, the cost function
controls the colors in the support of the optimal faces for
parameter vectors in $\pc{M}$. The proof of the following lemma
can be found in Section~\ref{sec:app:polycompl}.
\begin{lemma}\label{lem:colors}
Let $i^\times \in [d]$ be a color and let $\ve{\mu} \in \pc{M}$ be
a parameter vector
with $\ve{\mu}_{i^\times} = 0$. Furthermore, let $B^\star$ be
an optimal feasible
basis for $L^\CC_\ve{\mu}$. Then, $B^\star \cap C_{i^\times} = \emptyset$.
\end{lemma}
We denote for a face
$f \subseteq \pc{P}^\CC$, $f \neq \emptyset$,
with
$
\Phi(f) = \big\{\ve{\mu} \in \ensuremath{\mathbb{R}}^d \mid \text{$f$ is optimal for
$L_{\ve{\mu}}$}\big\}
$
the set of all parameter vectors for which $f$ is optimal. We call
this the
\emph{parameter region} for $f$. Using the reduced cost vector, we can
express $\Phi(f)$ as solution space to the
following linear system, where $B$ is a feasible basis of some vertex of
$f$ and the $d$ coordinates of the parameter vector $\ve{\mu}$ are the variables:
\begin{equation}
L^\Phi_{B,f}:
(\up{\ve{r}}_{B,\ve{c}_\ve{\mu}})_j = 0 \text{\ for } j \in \supp{f} \setminus
\ind{B} \text{ and }
(\up{\ve{r}}_{B,\ve{c}_\ve{\mu}})_j \leq 0 \text{\ for }\big[d^2\big] \setminus \supp{f}.
\end{equation}
Then, we define $\pc{F}$ as the set of all faces that are optimal for some
parameter vector in $\pc{M}$:
\[
\pc{F} = \left\{ f \,\middle\vert\, \text{$f$ is a face of $\pc{P}^\CC$},\,
\Phi(f) \cap \pc{M} \neq
\emptyset \right\}.
\]
By definition, $\pc{F} \cup \{\emptyset\}$ is a
polyhedral subcomplex of $\pc{P}^\CC$.
The intersections of the parameter regions with faces of $\pc{M}$ induce
a subdivision
$\pc{Q}$ of $\pc{M}$:
\[
\pc{Q} = \left\{ \Phi(f) \cap g \,\middle\vert\, f \in \pc{F},\,
\text{$g$ is a face of $\pc{M}$} \right\}.
\]
In Section~\ref{sec:app:polycompl}, we show that $\pc{Q}$ is a
$(d-1)$-dimensional polytopal complex.
Next, we construct $\pc{Q}_\Delta$ through a central projection with
the origin as center of
$\pc{Q}$ onto the $(d-1)$-dimensional standard simplex $\Delta \subset \ensuremath{\mathbb{R}}^d$.
It is
easy to see that this projection is a bijection. For a
parameter vector $\ve{\mu} \in \ensuremath{\mathbb{R}}^d$, we denote with
$
\Delta(\ve{\mu}) = \ve{\mu}/\|\ve{\mu}\|_1
$
its projection onto $\Delta$. Similarly, we
denote with
$
\pc{M}(\ve{\mu}) = \ve{\mu}/\|\ve{\mu}\|_\infty
$
the
projection of $\ve{\mu}$ onto $\pc{M}$ and we use the same notation to denote the
element-wise projection of sets. Then, we can write the projection
$\pc{Q}_\Delta$ of
$\pc{Q}$ onto
$\Delta$ as $\pc{Q}_\Delta = \{\Delta(q) \mid q \in \pc{Q}\}$.
Furthermore, let $\SS = \{ \Delta(g) \mid \text{$g$ is a face of $\pc{M}$}\}$
denote the projections
of the faces of $\pc{M}$ onto $\Delta$. For $f \in \pc{F}$, let
$\Phi_\Delta(f) = \Delta(\Phi(f) \cap \pc{M})$ denote the projection of all
parameter vectors in
$\pc{M}$ for which $f$ is optimal onto $\Delta$. Please refer to
Table~\ref{tab:notation} on Page~\pageref{tab:notation} for an
overview of the current and future notation.
The following results are proved in Section~\ref{sec:app:polycompl}.
\begin{lemma}\label{stm:face_dim}\label{stm:unique}
Let $q \neq \emptyset$ be an element from
$\pc{Q}_\Delta$. Then, there exists unique pair
$(f, g)$ where $f$ is a face of $\pc{F}$
and $g$ is a face of $\SS$ such that $q = \Phi_{\Delta}(f) \cap g$.
Moreover, $q$ is a simple polytope of dimension $\dim g - \dim f$ and,
if $\dim q > 0$, the set of facets of $q$ can be written as
\[
\left.\Big\{ \Phi_{\Delta}\left(f\right) \cap \facet{g} \neq \emptyset \,\middle\vert\,
\text{$\facet{g}$ is a facet of $g$} \Big\}\right. \cup
\left\{ \Phi_{\Delta}\left(\up{f}\right) \cap g \neq \emptyset \,\middle\vert\, \text{$f$ is a
facet of $\up{f} \in \pc{F}$}\right\}.
\]
\end{lemma}
\begin{lemma}\label{stm:polycompl}
The set $\pc{Q}_\Delta$ is a $(d-1)$-dimensional polytopal
complex that decomposes $\Delta$. \qed
\end{lemma}
\subsection{The Barycentric Subdivision}
\label{sec:ppad:sd}
The \emph{barycentric subdivision}~\cite[Definition~1.7.2]{Matousek2008}
is a
well-known method to subdivide a polytopal complex into
simplices. We define $\sd \pc{Q}_\Delta$
as the set of all simplices $\conv(\ve{v}_0,\dots,\ve{v}_k)$, $k \in [d]$,
such that
there exists a chain $q_0 \subset \dots \subset q_k$ of polytopes in
$\pc{Q}_\Delta$ with $\dim q_{i-1} < \dim q_i$ and such that $\ve{v}_i$ is
the barycenter of
$q_i$ for $i \in [k]$. We define the label of a
vertex $\ve{v} \in \sd \pc{Q}_\Delta$ as follows. By Lemma~\ref{stm:unique}, there
exists a
unique pair $f \in \pc{F}$ and $g \in \SS$ with $\ve{v} = \Phi_\Delta(f) \cap g$. Then, the
label $\lambda(\ve{v})$ of $\ve{v}$ is defined as
\begin{equation}
\label{eq:ppad:labeling}
\lambda(\ve{v}) = \argmax_{i \in [d]} \left|\ind{C}_i \cap \supp{f}\right|.
\end{equation}
In case of a tie, we take the smallest $i \in [d]$ that achieves the
maximum. Lemma~\ref{lem:colors} implies that $\lambda(\cdot)$ is a Sperner
labeling
of $\sd \pc{Q}_\Delta$.
In fact, $\lambda$ is
a Sperner labeling for any fixed simplicial subdivision of $\Delta$.
Now, Theorem~\ref{thm:sperner} guarantees the existence of a $(d-1)$-simplex
$\sigma
\in \sd \pc{Q}_\Delta$ whose vertices have all $d$ possible labels.
The next lemma
shows that then one of the vertices of $\sigma$ defines a solution to the
$\CCP$ instance. Here, we use specific properties of the barycentric
subdivision.
\begin{lemma}\label{lem:fullycolored_colbasis}
Let $\sigma \in \sd \pc{Q}_\Delta$ be a fully-labeled
$(d-1)$-simplex and let $\ve{v}_{d-1}$ denote the vertex of $\sigma$ that
is the
barycenter of a $(d-1)$-face $q_{d-1} = \Phi_\Delta(f_{d-1}) \cap g_{d-1} \in
\pc{Q}_\Delta$, where $f_{d-1} \in \pc{F}$ and $g_{d-1} \in \SS$. Then, the
columns from
$A_{\supp{f_{d-1}}}$ are a colorful choice that ray-embraces $\ve{b}$.
\end{lemma}
Our discussion up to now already yields a new Sperner-based proof
of the colorful \Caratheodory theorem.
However, in order to show that $\CCP \in \PPAD$, we need to replace
the invocation of Theorem~\ref{thm:sperner} by a \PPAD-problem. Note that it is
not possible to use the formulation of Sperner
from~\cite[Theorem~2]{Papadimitriou1994} directly, since
it is defined for a fixed simplicial subdivision of the standard
simplex. In our
case, the simplicial subdivision of $\Delta$ depends on the input instance.
In the following, we generalize the \PPAD formulation of
Sperner in \cite{Papadimitriou1994} to $\pc{Q}_\Delta$ by mimicking the proof of
Theorem~\ref{thm:sperner}. For this, we need to be able to find simplices
in $\sd \pc{Q}_\Delta$ that share a given facet. We begin with a
simple encoding of simplices in $\sd \pc{Q}_\Delta$ that allows us to
solve this problem completely combinatorially.
We first show how to encode a polytope $q \in \pc{Q}_\Delta$. By
Lemma~\ref{stm:unique},
there exists a unique pair of faces $f \in \pc{F}$ and $g \in S$ such
that $q = \Phi_\Delta(f) \cap g$. Since $\pc{M}(g)$ is a face of the unit
cube, the value of $d - \dim g$ coordinates in $\pc{M}(g)$ is fixed
to either $0$ or $1$. Let $I_j
\subseteq [d]$,
$j=0,1$, denote the indices of the coordinates that are fixed to $j$.
Then, the encoding of $q$ is defined as
$
\en{q} = \left(\supp{f}, I_0, I_1\right)$.
We use this to define an encoding of the simplices in $\pc{Q}_\Delta$ as follows.
Let $\sigma \in \pc{Q}_\Delta$ be a $k$-simplex and let $q_0 \subset\dots
\subset q_k$ be the corresponding face chain in $\pc{Q}_\Delta$ such that the
$i$th vertex of $\sigma$ is the barycenter of $q_i$. Then, the
encoding $\en{\sigma}$ is defined as
\begin{equation}\label{eq:enc:simplex}
\en{\sigma} = \left(\en{q_0},\dots,\en{q_k}\right).
\end{equation}
In the proof of Theorem~\ref{thm:sperner}, we traverse only a subset of simplices
in the simplicial subdivision, namely $(k-1)$-simplices that are
contained in the face $\Delta_{[k]} = \conv\{\ve{e}_i \mid i \in [k]\}$ of $\Delta$ for $k \in [d]$.
Let
$
\Sigma_k = \left\{ \sigma \in \sd \pc{Q}_\Delta \,\middle\vert\, \dim(\sigma) = k-1,\ \sigma
\subseteq \Delta_{[k]} \right\}
$
denote the set
of $(k-1)$-simplices in $\sd \pc{Q}_\Delta$ that are contained in the $(k-1)$-face,
where $k \in [d]$, and let $\Sigma = \bigcup_{k=1}^d
\Sigma_k$ be the collection of all those simplices. In the following, we give a
precise characterization of the encodings of the simplices in $\Sigma_k$.
For two disjoint index sets $I_0,I_1 \subseteq [d]$, we denote with
$g(I_0,I_1)= \left\{ \ve{\mu} \in
\pc{M} \,\middle\vert\, j=0,1,\, (\ve{\mu})_i = j \text{ for $i \in I_j$}\right\}$ the
face of
$\pc{M}$ that we obtain by fixing the coordinates in dimensions $I_0\cup I_1$.
Let now $T=\left(Q_0,\dots,Q_{k-1}\right)$, $k \in [d-1]$, be a tuple, where $Q_i =
\left(S^{(i)}, I^{(i)}_0,
I^{(i)}_1\right)$, $S^{(i)} \subset \left[d^2\right]$, and $I^{(i)}_0, I^{(i)}_1$ are
disjoint subsets of $[d]$ with $I^{(i)}_1 \neq \emptyset$ for $i \in [k-1]_0$.
We say $T$ is \emph{valid} if and only if $T$ has the following
properties.
\begin{enumerate}[label=(\roman{enumi})]
\item\label{valid:ini} We have $I^{(k-1)}_0 = [d] \setminus [k]$,
$\left|I^{(k-1)}_1\right| = 1$,
and the columns in $A_{S^{(k-1)}}$ are a feasible basis for a vertex $f$.
Moreover, the intersection $\Phi(f) \cap g\left(I^{(k-1)}_0 \cup
I^{(k-1)}_1\right)$ is nonempty.
\item\label{valid:facet} For all $i \in [k-1]$, we either have
\begin{enumerate}[label=(\roman{enumi}.\alph{enumii})]
\item\label{valid:facet:f} $I^{(i-1)}_0 = I^{(i)}_0$,
$I^{(i-1)}_1 = I^{(i)}_1$, and $S^{(i-1)} = S^{(i)} \cup \left\{ a_{i-1}
\right\}$ for some index $a_{i-1} \in \left[d^2\right] \setminus S^{(i)}$,
\item\label{valid:facet:g} or $S^{(i-1)} = S^{(i)}$ and there is an index
$j_{i-1} \in [d] \setminus \left(I^{(i)}_0 \cup I^{(i)}_1\right)$ such that
either $I^{(i-1)}_0 = I^{(i)}_0$ and $I^{(i-1)}_1 = I^{(i)}_1 \cup
\left\{ j_{i-1} \right\}$, or $I^{(i-1)}_1 = I^{(i)}_1$ and $I^{(i-1)}_0
= I^{(i)}_0 \cup \left\{ j_{i-1} \right\}$.
\end{enumerate}
\end{enumerate}
\begin{lemma}\label{lem:enc_bij}
For $k \in [d]$, the function $\en{\cdot}$ restricted to the simplices in
$\Sigma_k$ is a bijection from $\Sigma_k$ to the set of valid $k$-tuples.
\end{lemma}
Using our characterization of encodings as valid tuples, it becomes
an easy task to check whether a given candidate encoding corresponds
to a simplex in $\Sigma$.
\begin{lemma}\label{lem:enc_verify}
Let $T=\left(Q_0,\dots,Q_{k-1}\right)$, $k \in [d-1]$, be a tuple,
where $Q_i =
\left(S^{(i)}, I^{(i)}_0,
I^{(i)}_1\right)$, $S^{(i)} \subset \left[d^2\right]$, and $I^{(i)}_0, I^{(i)}_1$ are
disjoint subsets of $[d]$ with $I^{(i)}_1 \neq \emptyset$ for $i \in [k-1]_0$.
Then, we can check in polynomial time whether $T$ is a valid $k$-tuple.
\end{lemma}
In Section~\ref{sec:app:barycentric}, we show that simplices in $\Sigma$ that
share a facet have similar encodings that differ only in one element
of the encoding tuples.
Using this fact, we can traverse $\Sigma$ efficiently by manipulating the
respective encodings.
\begin{lemma}\label{lem:enc_algs}
Let $\sigma \in\Sigma_k$ be a simplex and let
$q_0 \subset \dots \subset q_{k-1}$ be the
corresponding face chain in $\pc{Q}_\Delta$ such
that the $i$th vertex $\ve{v}_i$ of $\sigma$ is
the barycenter of $q_i$, where $k \in[d]$ and $i \in
[k-1]_0$. Then, we can solve the following problems
in polynomial time:
(i) Given $\en{\sigma}$ and $i$, compute the encoding of the simplex
$\sigma' \in \Sigma_k$ that shares the facet
$\conv\left\{ \ve{v}_j \,\middle\vert\, j \in [k-1]_0,\, j \neq i \right\}$ with
$\sigma$ or
state that there is none; (ii) Assuming that $k<d$ and
given $\en{\sigma}$, compute the encoding of
the simplex $\up{\sigma} \in \Sigma_{k+1}$ that has $\sigma$ as facet;
and (iii) Assuming that $k>1$ and given $\en{\sigma}$, compute
the encoding of the simplex $\down{\sigma} \in \Sigma_{k-1}$ that
is a facet of $\sigma$ or state that there is none.
\end{lemma}
\subsection{The PPAD graph}
\label{sec:ppad:graph}
Using our tools from the previous sections, we now describe the \PPAD
graph $G=(V,E)$ for the \CCP instance.
The definition of $G$ follows mainly the ideas
from the formulation of Sperner as a
PPAD-problem~\cite[Theorem~2]{Papadimitriou1994} and the proof of
Theorem~\ref{thm:sperner}.
The graph has one node per simplex in $\Sigma$ that has all labels or
all but the largest possible label. That is, we have one node for
each $(k-1)$-simplex $\sigma$ in
$\Sigma_k$ with $[k-1] \subseteq \lambda(\sigma)$. Two
simplices are connected by an edge if one simplex is the facet of
the other or
if both simplices share a facet that has all but the largest
possible label.
More formally,
for $k \in [d]$,
we set
$
V_k = \left\{ \en{\sigma} \,\middle\vert\, \sigma \in \Sigma_k,\, [k-1] \subseteq
\lambda(\sigma) \right\}
$,
the set of all encodings for $(k-1)$-simplices in $\Sigma_k$ whose vertices
have
all or all but the largest possible label.
Then, $V$ is the union of all $V_k$
for $k \in [d]$.
There are
two types of edges: edges within a set $V_k$, $k \in [d]$, and
edges connecting nodes from $V_k$ to nodes in $V_{k-1}$
and $V_{k+1}$. Let $\en{\sigma}, \en{\sigma'}$ be two vertices in
$V_k$ for some $k \in [d]$. Then, there is an edge between
$\en{\sigma}$ and $\en{\sigma'}$ if the
encoded simplices $\sigma,\, \sigma' \in \Sigma_k$ share a facet
$\facet{\sigma}$ with $\lambda(\facet{\sigma}) = [k-1]$, i.e., both
simplices are connected by a facet that has all but the largest possible
label. Now, let $\en{\sigma} \in V_k$ and $\en{\sigma'} \in
V_{k+1}$ for some $k \in [d-1]$. Then, there is an edge between
$\en{\sigma}$ and $\en{\sigma'}$ if $\lambda(\sigma) = [k]$ and
$\sigma$ is a facet of $\sigma'$.
In the next lemma, we show that
$G$ consists only of paths and cycles. Please see
Section~\ref{sec:app:ppadgraph} for the proof.
\begin{lemma}\label{stm:deg}
Let $\en{\sigma} \in V$ be a node. If $\en{\sigma} \in V_1$ or
$\en{\sigma} \in V_d$ with $\lambda(\sigma) = [d]$, then $\deg \en{\sigma}
= 1$. Otherwise, $\deg \en{\sigma} = 2$.
\end{lemma}
This already shows that $\CCP \in \PPA$. By generalizing the
orientation from \cite{Papadimitriou1994} to our setting, we obtain a
function $\dir$ that orients the edges of $G$ such that only vertices
with degree one in $G$ are sinks or sources in the oriented graph. In
Section~\ref{sec:app:ppadgraph}, we show how to compute this
function in polynomial time.
This finally yields our main result.
\begin{theorem}\label{thm:ppad}
\CCP, \Centerpoint, \Tverberg, and \SimCenter are in $\PPAD \cap \PLS$.
\end{theorem}
\begin{proof}
We give a formulation of \CCP as \PPAD-problem. See Section~\ref{sec:app:pls}
for a formulation of \CCP as \PLS-problem. Using the
classic proofs discussed in Section~\ref{sec:app:reductions},
this then also implies the statement for the other problems.
The set of problem instances $\mc{I}$ consists of all tuples
$I=(C_1,\dots,C_d,\ve{b})$, where $d \in \ensuremath{\mathbb{N}}$, the set $C_i \subset \ensuremath{\mathbb{Q}}^d$
ray-embraces $\ve{b} \in \ensuremath{\mathbb{Q}}^d$ and $\ve{b} \neq \ve{0}$. Let $I^\approx=
(C^\approx_1,\dots,C^\approx_d,\ve{b}^\approx)$ denote then the \CCP instance that we
obtain by applying our perturbation techniques to $I$ (see
Section~\ref{sec:app:eqccp}). Then, $I^\approx$ has the general position
properties
(P1) and (P2). The set of candidate
solutions $\mc{S}$ consists of all tuples $(Q_0,\dots,Q_{k-1})$,
where $k \in \ensuremath{\mathbb{N}}$ and $Q_i$ is a tuple
$\left(S^{(i)},I^{(i)}_0,I^{(i)}_1\right)$
with $S^{(i)},I^{(i)}_0,I^{(i)}_1 \subset \ensuremath{\mathbb{N}}$. Furthermore,
$\mc{S}$ contains
all $d$-subsets $C \subset \ensuremath{\mathbb{Q}}^d$ for $d \in \ensuremath{\mathbb{N}}$.
We define the set of valid
candidate solutions $\mc{S}_I$ for the instance $I$ to be the set of all valid
$k$-tuples with
respect to the instance $I^\approx$ and the set of all colorful choices with
respect to $I$ that
ray-embrace $\ve{b}$, where $k \in [d]$. Let $s \in \mc{S}$ be a candidate
solution. If it is a tuple, we first use the algorithm from
Lemma~\ref{lem:enc_verify} to check in polynomial time in the length of
$I^\approx$ and hence in the length of $I$ whether $s \in \mc{S}_I$.
If affirmative, we check whether the simplex has all or all but the
largest possible label. Using the encoding, this can be carried
out in polynomial time. If $s$ is a set of points, we can determine
in polynomial time with linear programming whether the points in $s$
ray-embrace $\ve{b}$.
We set as standard source the $0$-simplex $\{\ve{e}_1\}$. We can assume
without loss of generality that $\{\ve{e}_1\}$ is a source (otherwise we
invert the orientation).
Given a valid candidate solution $s \in \mc{S}_I$, we compute its
predecessor and successor with the algorithms from
Lemma~\ref{lem:enc_algs} and the orientation function discussed
above, with one
modification: if a node $s \in V$ is a source different from the
standard source in the graph $G$, it encodes by
the above discussion
a colorful choice $C^\approx$ that
ray-embraces $\ve{b}^\approx$. Let $C$ be the corresponding colorful
choice for $I$ that ray-embraces $\ve{b}$. Then, we set the
predecessor of $s$ to $C$. The properties of our perturbation
ensure that we can compute $C$ in
polynomial time. Similarly, if $s$ is a sink in $G$, we set its
successor to the corresponding solution for the instance $I$.
\end{proof}
\section{A Polynomial-Time Case}
\label{sec:halfhalf}
We show that for a special case of \CCP, our formulation
of \CCP as a \PPAD problem has algorithmic implications.
Let $C_1,C_2 \in
\ensuremath{\mathbb{R}}^d$ be two color classes and let $C \subseteq
C_1 \cup C_2$ be a set. We call $C$ an
\emph{$(k,d-k)$-colorful
choice} for $C_1$ and $C_2$ if there are two subsets
$C'_1 \subseteq C_1$, $C_2' \subseteq C_2$ with $|C'_1| \leq k$
and $|C'_2| \leq d-k$.
Now, given two color classes $C_1, C_2$ that each
ray-embrace a point $\ve{b} \in \ensuremath{\mathbb{R}}^d$ and a number $k \in [d]_0$,
we want to find an $(k,d-k)$-colorful choice that ray-embraces $\ve{b}$.
It is a straightforward
consequence of the colorful \Caratheodory theorem that such
a colorful choice always exists.
Using our techniques from Section~\ref{sec:ccpppad}, we present a weakly
polynomial-time algorithm for this case. As described in
Section~\ref{sec:ppad:para}, we construct
implicitly a $1$-dimensional polytopal complex, where at least one edge
corresponds to a solution. Then, we apply binary search to
find this edge. Since the length of the edges can be exponentially
small in the
length of the input, this results in a weakly polynomial-time algorithm.
\begin{theorem}
Let $\ve{b} \in \ensuremath{\mathbb{Q}}^d$ be a point and let $C_1, C_2 \subset \ensuremath{\mathbb{Q}}^d$ be
two sets of size $d$ that ray-embrace $\ve{b}$. Furthermore,
let $k \in [d-1]$ be a parameter. Then, there is an algorithm that
computes a $(k, d-k)$-colorful choice $C$ that ray-embraces $\ve{b}$
in weakly-polynomial time.
\end{theorem}
For Sperner's lemma, it is well-known that a fully-labeled simplex
can be found if there are only two labels by binary search.
Essentially, this is also what the presented algorithm
does: reducing the problem to Sperner's lemma and then applying
binary search to find the right simplex. Since the computational
problem Sperner is \PPAD-complete even for $d=2$, a polynomial-time
generalization of this approach to three colors must use specific
properties of the colorful \Caratheodory instance
under the assumption that no \PPAD-complete problem can be solved in
polynomial time.
\section{Conclusion}
\label{sec:conclusion}
We have shown that \CCP lies in the intersection of \PPAD and \PLS.
This also immediately implies that several
illustrious problems associated with \CCP, such as finding centerpoints
or Tverberg partitions, belong to $\PPAD \cap \PLS$.
Previously, the intersection $\PPAD \cap \PLS$ has
been studied in the context of \emph{continuous
local search}: Daskalakis and Papadimitriou~\cite{DaskalakisPa11}
define a subclass $\CLS \subseteq \PPAD \cap \PLS$ that
``captures a particularly benign kind of local optimization''.
Daskalakis and Papadimitriou describe several interesting
problems that lie in \CLS but are not known to
be solvable in polynomial time.
Unfortunately, our results do not show that
\CCP lies in \CLS, since we reduce \CCP in $d$ dimensions to
Sperner in $d-1$ dimensions, and since Sperner is not known
to be in \CLS. Indeed, if Sperner's lemma could
be shown to be in \CLS, this would imply that
$\PPAD = \CLS \subseteq \PLS$, solving a major
open problem. Thus, showing that \CCP lies
in \CLS would require fundamentally new ideas, maybe
exploiting the special structure of the resulting
Sperner instance. On the other hand,
it appears that Sperner is a more difficult problem than
\CCP, since Sperner is \PPAD-complete for every fixed dimension
larger than $1$, whereas \CCP becomes hard only in unbounded
dimension.
On the positive side, our perturbation results show that
a polynomial-time algorithm for \CCP, even under strong
general position assumptions, would
lead to polynomial-time algorithms for several well-studied problems
in high-dimensional computational geometry.
Finally, it would also be interesting to find further special
cases of \CCP that are amenable to polynomial-time solutions.
For example, can we extend our algorithm for two color classes to
\emph{three} color classes? We expect this to be difficult, due to
an analogy between 1D-Sperner, which is in \textsf{P}, and 2D-Sperner,
which
is \PPAD-complete. However, there seems to be
no formal justification for this intuition.
\bibliographystyle{plain} |
1508.00890 | \section{Introduction}
\subsection{The thin-film equation as a classical free-boundary problem}
We are interested in the thin-film equation with \emph{quadratic mobility}
\begin{subequations}\label{tfe_free}
\begin{equation}\label{tfe}
\partial_t h + \partial_z\left(h^2 \partial_z^3 h\right) = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z > Z_0(t).
\end{equation}
This is a \emph{fourth-order} \emph{degenerate-parabolic} equation modeling the evolution of the height $h$ of a two-dimensional thin viscous film on a one-dimensional flat substrate as a function of time $t > 0$ and base point $z > Z_0(t)$ \cite{beimr.2009,d.1985,odb.1997}. For simplicity we assume that the droplet extends infinitely to positive $z$ and has a free boundary $Z_0(t)$ denoting the \emph{contact line}, that is, the \emph{triple junction} between the three phases liquid, gas, and solid (cf.~Fig.~\ref{fig:hodograph}).
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=1]
\path[fill=lightblue] (1.5,0) to [out=0,in=195] (9,4) -- (9,0) -- (1.5,0);
\draw [very thick,->] (-1.2,0) -- (10,0);
\draw [very thick,->] (0,-1.2) -- (0,5.2);
\draw[very thick,red,dashed] (1.5,0) to [out=0,in=240] (9,5);
\draw[very thick,blue] (1.5,0) to [out=0,in=195] (9,4);
\draw [gray,dashed] (5.7,-.2) -- (5.7,2);
\draw [gray,dashed] (6.7,-.8) -- (6.7,2);
\draw [gray,dashed] (1.5,0) -- (1.5,-.8);
\draw [gray,dashed] (0,2) -- (6.7,2);
\draw [thick,blue,dashed,<->] (0,-.2) -- (5.7,-.2);
\draw [blue] (2.85,-.2) node[anchor=north] {$Z(t,x)$};
\draw [thick,red,dashed,<->] (1.5,-.8) -- (6.7,-.8);
\draw [red] (4.1,-.8) node[anchor=north] {$x$};
\draw (0,2) node[anchor=east] {$\color{blue}h(t,Z(t,x))\color{black} = \color{red}x^{\frac 32}\color{black}$};
\draw (2,3) node[anchor=east] {gas};
\draw [blue] (8.5,1.5) node[anchor=east] {liquid};
\draw [thick,violet,->] (1.8,.8) -- (1.5,0);
\draw [violet] (1.8,.8) node[anchor=south] {triple junction};
\draw (0,4.9) node[anchor=east] {film height $h$};
\draw (8.7,0) node[anchor=north] {base point $z$ or $x$};
\end{tikzpicture}
\caption{Schematic of a liquid thin film and the hodograph transform \eqref{hodograph}.}
\label{fig:hodograph}
\end{figure}
Necessarily
\begin{equation}\label{tfe_bdry_1a}
h = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z = Z_0(t),
\end{equation}
which simply sets the location of the free boundary to be $Z_0(t)$. Secondly, we assume
\begin{equation}\label{tfe_bdry_1b}
\partial_z h = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z = Z_0(t),
\end{equation}
leading to a \emph{zero contact angle} at the triple junction, known as \emph{complete} (or \emph{perfect}) \emph{wetting}. The notion can be explained if one considers \emph{quasi-static} droplet motion in which the contact angle is determined by a balance between the surface tensions of the three interfaces at the contact line (\emph{Young's law}). The condition $\partial_z h = 0$ at $z = Z_0(t)$ implies that an equilibrium is generically not achieved and therefore the film will ultimately cover the whole surface. Finally, a condition determining the evolution of $Z_0(t)$ is imposed:
\begin{equation}\label{tfe_bdry_2}
\lim_{z \searrow Z_0(t)} h \partial_z^3 h = \frac{\d Z_0}{\d t}(t) \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z = Z_0(t).\end{equation}
\end{subequations}
This condition may be viewed as a \emph{Rankine-Hugoniot condition} for a viscous shock wave: Since \eqref{tfe} has the form of a (nonlinear) continuity equation
\begin{subequations}\label{continuity_eq}
\begin{equation}
\partial_t h + \partial_z (V h) = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z > Z_0(t),
\end{equation}
where
\begin{equation}\label{def_vel}
V = h \partial_z^3 h
\end{equation}
\end{subequations}
is the transport velocity of $h$, by compatibility necessarily \eqref{tfe_bdry_2} holds true.
\medskip
Equation~\eqref{tfe} is a particular version of the general class of thin-film equations
\begin{equation}\label{tfe_general}
\partial_t h + \partial_z \left(h^n \partial_z^3 h\right) = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z \in \{h > 0\},
\end{equation}
where $n$ is a real parameter. Apparently for $n \le 0$ equation~\eqref{tfe_general} has infinite speed of propagation and non-negativity of $h$ is not ensured. On the other hand, for $n = 3$ (corresponding to the \emph{no-slip condition} at the liquid-solid interface) equation~\eqref{tfe_general} exhibits unphysical features as well: The solution is singular at the free boundary, which does not move unless an infinite amount of energy to overcome dissipation is inserted into the system \cite{dd.1974,hs.1971,m.1964}. Since for $n > 3$ equation~\eqref{tfe_general} is even more degenerate, for a model relevant for the motion of fluid films (in which contact lines move with finite and in general nonzero velocity) necessarily $n \in (0,3$). The most important cases are $n = 1$ and $n = 2$, corresponding to the lubrication approximation of \emph{Darcy's flow} in the \emph{Hele-Shaw cell} ($n = 1$)\footnote{For an extensive well-posedness and regularity theory of \eqref{tfe_general} for $n = 1$ in the complete wetting case, we refer to \cite{gk.2010,gko.2008,g.2014,j.2015}. The partial wetting case is addressed in \cite{km.2015,km.2013}.} or the lubrication approximation of the \emph{Navier-Stokes equations} with a (linear) \emph{Navier-slip condition} ($n = 2$)\footnote{We refer to \cite{ggko.2014} for a well-posedness and partial regularity result of \eqref{tfe_general} in the complete wetting regime with $n = 2$ and to \cite{k.2011} for a result in the partial wetting regime.} at the liquid-solid interface, respectively \cite{d.1985,go.2003,km.2015,km.2013,odb.1997}. More detailed discussions of the literature, also addressing the well-established existence theory of weak solutions, can be found in \cite{ag.2004,b.1998,gs.2005}.
\medskip
This note addresses the regularity of solutions at the free boundary and may therefore be considered as a contribution to a regularity theory of higher-order degenerate-parabolic equations, an only insufficiently explored field. The author hopes that the present study is also relevant regarding an existence, uniqueness, and regularity theory for the (Navier-)Stokes equations with a moving contact line, an essentially open problem. Here, a thorough understanding of the singular behavior at the free boundary can potentially help in the construction of suitable function spaces. Additionally, in view of the aforementioned no-slip paradox, the question of how the Navier-slip condition (or even general nonlinear slip conditions) de-singularizes the solution at the contact line $z = Z_0(t)$ is also of interest from the applied view point. Loosely speaking, the following arguments will demonstrate that higher spatial regularity (i.e., regularity in the variable $z$) goes hand in hand with higher regularity in the time variable $t$. In numerical studies in \cite{p.2015}, this observation is used to test numerical schemes regarding their precision in resolving the dynamics of moving contact lines.
\subsection{Transformations}
We review the transformations discussed in detail in \cite[Sec.~1.2, Sec.~1.3, Sec.~A.1]{ggko.2014}: Generic solutions to \eqref{tfe_free} are traveling-wave solutions, that is, solutions of the form $h(t,z) = H_\mathrm{TW}(x)$, where $x = z - V_0 t$ and $V_0 > 0$ is the speed of the traveling wave. By rescaling, we may without loss of generality assume $V_0 = \frac 3 8$ and one may conclude that in the case of a moving infinite cusp a strictly monotone similarity solution is given by\footnote{A discussion of traveling-wave solutions can be found in \cite{bko.1993}.} $H_\mathrm{TW}(x) = x^{\frac 3 2}$. Considering perturbations of this profile, we set
\begin{equation}\label{hodograph}
h(t,Z(t,x)) = x^{\frac 3 2} \quad \mbox{for } \, t, x > 0.
\end{equation}
Under the assumption that $h(t,z)$ is strictly monotone in $z$ for $z > Z_0(t)$, equation~\eqref{hodograph} uniquely determines the function $Z = Z(t,x)$, thus interchanging dependent and independent variables and fixing the boundary to $x = 0$. The transformation \eqref{hodograph} is known as the \emph{hodograph} transform (cf.~Fig.~\ref{fig:hodograph}). The related \emph{von Mises} transform has been applied already in the context of the porous-medium equation and the thin-film equation with linear mobility in higher dimensions \cite{j.2015, k.1999}. Plugging expression~\eqref{hodograph} into \eqref{tfe} and noting that the chain rule (applied to \eqref{hodograph}) gives the transformations
\begin{equation}\label{tfe_trafo_0}
\partial_t h = - Z_t \partial_z h \qquad \mbox{and} \qquad \partial_z = Z_x^{-1} \partial_x,
\end{equation}
equation \eqref{tfe_free} now reads\footnote{Here and in what follows, differential operators act on everything to their right, that is, e.g.
\[
Z_x^{-1} \partial_x x^3 \left(Z_x^{-1} \partial_x\right)^3 x^{\frac 3 2} \equiv Z_x^{-1} \partial_x \left(x^3 \left(Z_x^{-1} \partial_x \left(Z_x^{-1} \partial_x \left(Z_x^{-1} \partial_x\right)\right)\right)\right).
\]
}
\begin{equation}\label{tfe_trafo_1}
- Z_t Z_x^{-1} \partial_x x^{\frac 3 2} + Z_x^{-1} \partial_x x^3 \left(Z_x^{-1} \partial_x\right)^3 x^{\frac 3 2} = 0 \quad \mbox{for } \, t, x > 0.
\end{equation}
Defining
\begin{equation}\label{trafo}
F := Z_x^{-1}
\end{equation}
and noting that $F_t = - F^2 Z_{xt}$, we observe that \eqref{tfe_trafo_1} alters to
\[
x \partial_t F + F^2 x \partial_x x^{- \frac 1 2} \partial_x x^3 F \partial_x F \partial_x F x^{\frac 1 2} = 0 \quad \mbox{for } \, t, x > 0.
\]
By commuting the powers of $x$ with the differential operators $\partial_x$, we then arrive at the equation
\begin{equation}\label{tfe_trafo_2}
x \partial_t F + \ensuremath{\mathcal M}(F,\cdots,F) = 0 \quad \mbox{for } \, t, x > 0,
\end{equation}
where $\ensuremath{\mathcal M}$ is a 5-linear form explicitly given by
\begin{equation}\label{5linear}
\ensuremath{\mathcal M}(F_1,\cdots,F_5) := F_1 F_2 D \left(D + \frac 3 2\right) F_3 \left(D - \frac 1 2\right) F_4 \left(D + \frac 1 2\right) F_5
\end{equation}
and $D := x \partial_x$ denotes the scaling-invariant (logarithmic) derivative in space (setting $s := \ln x$ we have $D = \partial_s$). We observe $Z_\mathrm{TW} \stackrel{\eqref{hodograph}}{=} x - \frac 3 8 t$ and $F_\mathrm{TW} \stackrel{\eqref{trafo}}{\equiv} 1$ for the traveling-wave profile (indeed $\ensuremath{\mathcal M}(1,\cdots,1) \stackrel{\eqref{5linear}}{=} 0$), so that by setting
\begin{equation}\label{trafo_2}
u := F-1
\end{equation}
in fact $u_\mathrm{TW} = 0$ and we are lead to study the Cauchy problem
\begin{subequations}\label{cauchy}
\begin{align}
x \partial_t u + p(D) u &= \ensuremath{\mathcal N}(u) \quad \mbox{for } \, t, x > 0, \label{pde}\\
u_{|t = 0} &= u^{(0)} \quad \mbox{for } \, x > 0.
\end{align}
\end{subequations}
Here we use the following notation:
\begin{itemize}
\item[$\bullet$] $p(\zeta)$ is a fourth-order polynomial
\begin{equation}\label{poly}
p(\zeta) := \zeta \left(\zeta^2 + \frac 1 2 \zeta - \frac 3 4\right) \left(\zeta + \frac 3 2\right) = \zeta (\zeta - \beta) \left(\zeta + \beta + \frac 1 2\right) \left(\zeta + \frac 3 2\right)
\end{equation}
with the irrational root $\beta := \frac{\sqrt{13} - 1}{4} \approx 0.65$. The operator $p(D)$ can be derived from the 5-linear form $\ensuremath{\mathcal M}$ (cf.~\eqref{5linear}) by noting that
\[
p(D) u = \ensuremath{\mathcal M}(1,\cdots,1,u) + \cdots + \ensuremath{\mathcal M}(u,1,\cdots,1).
\]
\item[$\bullet$] $\ensuremath{\mathcal N}(u)$ stands for the nonlinearity of the equation and has the structure
\begin{equation}\label{nonlinearity}
\ensuremath{\mathcal N}(u) := p(D) u - \ensuremath{\mathcal M}(1+u,\cdots,1+u).
\end{equation}
\end{itemize}
The structure of \eqref{pde} is such that the left-hand side is linear in $u$, whereas the right-hand side contains terms that are at least quadratic and at most quintic in $\{u, Du, \cdots, D^4 u\}$. The form of \eqref{pde} can be guessed from \eqref{tfe} immediately: By \eqref{hodograph} the linearization of the spatial part $\partial_z \left(h^2 \partial_z^3 h\right)$ of \eqref{tfe} scales as $x^{-1}$ and therefore the spatial operator $x^{-1} p(D)$ appears, where $p(\zeta)$ has to be of order $4$. This implies the space-time scaling $x \sim t$, unlike $x \sim t^{\frac 1 4}$ for non-degenerate fourth-order parabolic equations. Two of the roots of the polynomial $p(\zeta)$ are immediate as well: The root $-\frac 3 2$ is directly related to the exponent of the right-hand side of the hodograph transform \eqref{hodograph}. The root $0$ appears as the nonlinear equation \eqref{tfe} has divergence form and this feature is preserved in a linearization. In other words, one may understand the occurrence of this root by noting that the traveling wave $F_{\mathrm{TW}} = 1$ is simultaneously a solution of the respective linear and nonlinear equations, and the equations for $u$ and $F$ yield the same linear operator. In contrast to these two roots, the emergence of the roots $\beta$ and $-\beta-\frac 1 2$ in the polynomial $p(\zeta)$ in \eqref{poly} is a genuinely non-trivial feature, specific to higher-order degenerate-parabolic equations: In the second-order case, $p(\zeta)$ would be a second-order polynomial and the only two roots of $p(\zeta)$ would be immediate by the same arguments as above. An accessible way to understand this uncommon phenomenon in the fourth-order case for source-type self-similar solutions using the language of dynamical systems theory is explained in \cite{ggo.2013,bgk.2016}: There the roots discussed above turn out to be the eigenvalues of a linearized dynamical system at a stationary point (representing the contact line) and the situation can be reduced due to known regularity results of the corresponding invariant manifolds (Hartman-Grobman theorem).
\medskip
The rest of the note will be mainly concerned with the analysis of \eqref{cauchy}.
\subsection{Notation}
For $f, g \ge 0$ we write $f \lesssim_S g$ if there exists a constant $C > 0$ depending on parameters $S$ such that $f \le C g$. In this case we say that $f$ can be estimated by $g$ or equivalently that $g$ bounds/controls $f$. Furthermore, we write $f \sim_S g$ if $f \lesssim_S g$ and $f \gtrsim_S g$. For a non-negative quantity $r$ we say that a property is true for $r \ll_S 1$ (or $r \gg_S 1$, respectively) if there exists a (sufficiently large) constant $C > 0$ depending on $S$ such that the property is true for $r \le C^{-1}$ ($r \ge C$). Then we say that $r \ge 0$ has to be sufficiently small (large). If $S = \emptyset$ or if the dependence is specified elsewhere, we just write $f \lesssim g$ etc. The space $C_0^\infty((0,\infty))$ denotes the space of test functions, i.e., the space of all $u: (0,\infty) \to \ensuremath{\mathbb{R}}$ which are infinitely differentiable with compact support contained in $(0,\infty)$. For $\alpha \in \ensuremath{\mathbb{R}}$, we denote by $\floor{\alpha} := \max\{k \in \ensuremath{\mathbb{Z}}: \, k \le \alpha\}$ the integer part (floor) of $\alpha$. We write $\verti{A}$ for the number of elements (cardinality) of a finite set $A$.
\subsection{Weighted $L^2$-norms}
For the subsequent results, we introduce a scale of weighted $L^2$-norms $\verti{\cdot}_\alpha$. These are given by
\begin{equation}\label{l2_norm}
\verti{u}_\alpha^2 := \int_0^\infty x^{-2 \alpha} (u(x))^2 \frac{\d x}{x} = \int_{-\infty}^\infty e^{-2 \alpha s} \left(u(e^s)\right)^2 \d s \quad \mbox{with } \, \alpha \in \ensuremath{\mathbb{R}}.
\end{equation}
The larger $\alpha$ is, the better the decay of $u(x)$ as $x \searrow 0$ if $\verti{u}_\alpha < \infty$. To make this a point-wise statement, it is convenient to define Sobolev norms
\begin{equation}\label{sobolev_norm}
\verti{u}_{k,\alpha}^2 := \sum_{\ell = 0}^k \int_0^\infty x^{-2 \alpha} \left(D^\ell u(x)\right)^2 \frac{\d x}{x} = \sum_{\ell = 0}^k \int_{-\infty}^\infty e^{-2 \alpha s} \left(\partial_s^\ell u(e^s)\right)^2 \d s \quad \mbox{with } \, k \in \ensuremath{\mathbb{N}}_0, \; \, \alpha \in \ensuremath{\mathbb{R}}.
\end{equation}
We also use the notation $(\cdot,\cdot)_{k,\alpha}$ and $(\cdot,\cdot)_\alpha$ for the induced inner products. Setting $v(s) := e^{- \alpha s} u\left(e^s\right)$, it is elementary to see that $\verti{u}_{k,\alpha} \sim_\alpha \vertii{v}_{W^{k,2}(\ensuremath{\mathbb{R}})}$. In particular $\verti{u}_{1,\alpha} < \infty$ implies $u(x) = o\left(x^\alpha\right)$ as $x \searrow 0$. Note, however, that control of an increasing number of $D$-derivatives does not lead to better regularity properties of $u(x)$ in $x = 0$ as $D$ is scaling-invariant in $x$. Essentially the index $k$ in the norm $\verti{u}_{k,\alpha}$ controls the \emph{interior regularity} of $u$, whereas $\alpha$ yields control on the regularity at the boundary $x = 0$. The identification with the standard Sobolev norms also guarantees that the test functions $C_0^\infty((0,\infty))$ are dense in the spaces $\{u \mbox{ locally integrable}: \, \verti{u}_{k,\alpha} < \infty\}$. More details are contained in \cite[Sec.~4]{ggko.2014}.
\medskip
The well-posedness proof of \cite{ggko.2014} relied on the control of the initial data in the norm $\vertiii{\cdot}_0$, where
\begin{equation}\label{norm_initial}
\vertiii{u^{(0)}}_0^2 := \verti{u^{(0)}}_{k+6,-\delta}^2 + \verti{u^{(0)} - u^{(0)}_0}_{k+6,\delta}^2,
\end{equation}
$k \ge 3$, $u^{(0)}_0 = u^{(0)}\left(x = 0\right)$ (the boundary value of $u^{(0)}$), and $\delta > 0$ is chosen sufficiently small. Indeed, the subsequent result will use the same norm with sufficiently large $k \in \ensuremath{\mathbb{N}}$ and sufficiently small $\delta > 0$ depending on the value of $N_0$. We notice that by quite elementary arguments (cf.~\cite[Eq.~(8.5)]{ggko.2014}) the Lipschitz norm of the hodograph coordinates $Z(0,x)-x$ is controlled by this norm, i.e., $\sup_{x > 0} \verti{u^{(0)}(x)} \lesssim \vertiii{u^{(0)}}_0$. Smallness of the Lipschitz norm of $Z(0,x)-x$ ensures strict monotonicity and thus invertibility of transformation~\eqref{hodograph}. This was in fact the crucial assumption in \cite{j.2015, k.1999}.
\section{The main result}
\subsection{A regularity result\label{ssec:regularity}}
For motivating our main result, we may have a heuristic look at the properties of problem~\eqref{cauchy}: Since we are interested in a perturbative result, that is, $u^{(0)}$ and $u$ are assumed to be small in suitable norms (cf.~e.g.~\eqref{norm_initial}), the precise understanding of the linearized problem
\begin{subequations}\label{lin_cauchy}
\begin{align}
x \partial_t u + p(D) u &= f \quad \mbox{for } \, t, x > 0, \label{lin_pde}\\
u_{|t = 0} &= u^{(0)} \quad \mbox{for } \, x > 0
\end{align}
\end{subequations}
with a right-hand side $f(t,x)$, turns out the be essential. For $x \ll 1$, the term $p(D) u$ will be dominant. Considering $p(D) u \approx f$ for $x \ll 1$, all solutions of this ordinary differential equation (ODE) are given by the sum of a particular solution and a linear combination of solutions to the homogeneous equation $p(D) u \equiv 0$. The solution space of $p(D) u \equiv 0$ is spanned by $x^0$, $x^\beta$, $x^{-\beta-\frac 1 2}$, and $x^{-\frac 3 2}$. Clearly, the last two powers are ruled out by compatibility with transformations~\eqref{hodograph} and \eqref{trafo}. Nonetheless, the other two, $x^0$ and $x^\beta$, generically appear in the expansion of the solution $u$ close to the boundary $x = 0$ as a detailed analysis of the resolvent equation shows \cite[Sec.~6]{ggko.2014}. Due to the perturbation $x \partial_t u$, one may then expect that $u = v + x^\beta w$, where both $v$ and $w$ are smooth functions in $x$ up to the boundary $x = 0$. While it is in principle possible to construct solutions of this form for the linear problem \eqref{lin_cauchy}, such a structure is incompatible with the nonlinear equation \eqref{pde} that mixes the powers $x$ and $x^\beta$. What we are instead aiming at is
\begin{equation}\label{structure_u}
u(t,x) = \bar u\left(t,x,x^\beta\right),
\end{equation}
where $\bar u = \bar u(t,x,y)$ is a smooth function in $(x,y) \in \left\{\left(x^\prime,y^\prime\right) \in \ensuremath{\mathbb{R}}^2: \, x^\prime, y^\prime \ge 0\right\}$. Indeed, such a result can be expected in view of the analysis of source-type self-similar solutions by the author joint with Giacomelli and Otto \cite{ggo.2013}, where a stronger result is shown: There
\[
h(t,z) = t^{-\frac 1 6} H(x) \quad \mbox{with } \, x = z t^{-\frac 1 6} \, \mbox{ and } \, H(x) = C x^{\frac 3 2}\left(1 + \bar u\left(x,x^\beta\right)\right),
\]
where $C$ is a positive constant and $\bar u(x,y)$ is analytic in a neighborhood of $(x,y) = (0,0)$. Not surprisingly, as in \eqref{cauchy} or \eqref{lin_cauchy}, $\beta = \frac{\sqrt{13}-1}{4}$ turns out to be the root of a polynomial that determines the linearized problem for $H$.
\medskip
We denote by
\begin{equation}\label{def_kn}
K_{N_0} := \left\{n_1 + \beta n_2: \, (n_1, n_2) \in \ensuremath{\mathbb{N}}_0^2, \; n_1 + \beta n_2 < N_0\right\}
\end{equation}
the set of admissible exponents up to order $O\left(x^{N_0}\right)$ (cf.~Fig.~\ref{fig:power}). The main result of the note reads as follows:
\begin{theorem}\label{th:main}
For any $N_0 \in \ensuremath{\mathbb{N}}_0$ there exist $\ensuremath{\varepsilon} > 0$, $k \in \ensuremath{\mathbb{N}}$, and $\delta > 0$ such that if
\[
\vertiii{u^{(0)}}_0 \stackrel{\eqref{norm_initial}}{=} \sqrt{\verti{u^{(0)}}_{k+6,-\delta}^2 + \verti{u^{(0)} - u^{(0)}_0}_{k+6,\delta}^2} \le \ensuremath{\varepsilon},
\]
the unique solution $u$ of \eqref{cauchy} (constructed in \cite[Th.~3.1]{ggko.2014}) fulfills the point-wise expansion (cf.~Fig.~\ref{fig:power})
\begin{equation}\label{expansion}
u(t,x) = \sum_{i \in K_{N_0}} u_i(t) x^i + R_{N_0}(x,t) x^{N_0} \quad \mbox{as } \, x \searrow 0 \, \mbox{ and } \, t > 0,
\end{equation}
where the coefficients $u_i = u_i(t)$ are continuous functions in $t > 0$, and the correction $R_{N_0} = R_{N_0}(x,t)$ is again continuous and for every $t > 0$ uniformly bounded in $x > 0$. Furthermore,
\begin{subequations}\label{decay_as}
\begin{equation}
u_i(t) = o\left(t^{-i}\right) \quad \mbox{as } \, t \to \infty
\end{equation}
and
\begin{equation}
R_{N_0}(x,t) = o\left(t^{-N_0}\right) \quad \mbox{as } \, t \to \infty \, \mbox{ uniformly in } \, x > 0.
\end{equation}
\end{subequations}
\end{theorem}
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=1]
\draw[very thick,->] (-4,0) -- (5,0);
\draw (4.7,0) node[anchor=north] {$n_1$};
\draw (-4,0) node[anchor=east] {$0$};
\draw [gray,dashed] (-4,1.3028) -- (4,1.3028);
\draw (-4,1.3028) node[anchor=east] {$\beta$};
\draw [gray,dashed] (-4,2.6056) -- (4,2.6056);
\draw (-4,2.6056) node[anchor=east] {$2\beta$};
\draw [gray,dashed] (-4,3.9083) -- (4,3.9083);
\draw (-4,3.9083) node[anchor=east] {$3\beta$};
\draw [gray,dashed] (-4,5.2111) -- (4,5.2111);
\draw (-4,5.2111) node[anchor=east] {$4\beta$};
\draw [gray,dashed] (-4,6.5139) -- (4,6.5139);
\draw (-4,6.5139) node[anchor=east] {$5\beta$};
\draw [gray,dashed] (-4,7.8167) -- (4,7.8167);
\draw (-4,7.6167) node[anchor=east] {$6\beta$};
\draw[very thick,->] (-4,0) -- (-4,8.8167);
\draw (-4,8.5167) node[anchor=east] {$n_2$};
\draw [gray,dashed] (-2,0) -- (-2,7.8167);
\draw [gray,dashed] (0,0) -- (0,7.8167);
\draw [gray,dashed] (2,0) -- (2,7.8167);
\draw [gray,dashed] (4,0) -- (4,7.8167);
\draw [blue] (-4,1.3028) -- (-2.6972,0);
\draw [blue] (-2,1.3028) -- (-.6972,0);
\draw [blue] (0,1.3028) -- (1.3028,0);
\draw [blue] (2,1.3028) -- (3.3028,0);
\draw [blue] (-4,2.6056) -- (-1.3944,0);
\draw [blue] (-2,2.6056) -- (.6056,0);
\draw [blue] (0,2.6056) -- (2.6056,0);
\draw [blue] (-4,3.9083) -- (-.0917,0);
\draw [blue] (-2,3.9083) -- (1.90832,0);
\draw [blue] (0,3.9083) -- (3.9083,0);
\draw [blue] (-4,5.2111) -- (1.2111,0);
\draw [blue] (-2,5.2111) -- (3.2111,0);
\draw [blue] (-4,6.5139) -- (2.5139,0);
\draw [blue] (-4,7.8167) -- (3.8167,0);
\draw[blue,->] (-4,0) -- (-4,-.95);
\draw[blue,->] (-2,0) -- (-2,-.95);
\draw[blue,->] (0,0) -- (0,-.95);
\draw[blue,->] (2,0) -- (2,-.95);
\draw[blue,->] (4,0) -- (4,-.95);
\draw[blue,->] (-2.6972,0) -- (-2.6972,-.95);
\draw[blue,->] (-.6972,0) -- (-.6972,-.95);
\draw[blue,->] (1.3028,0) -- (1.3028,-.95);
\draw[blue,->] (3.3028,0) -- (3.3028,-.95);
\draw[blue,->] (-1.3944,0) -- (-1.3944,-.95);
\draw[blue,->] (.6056,0) -- (.6056,-.95);
\draw[blue,->] (2.6056,0) -- (2.6056,-.95);
\draw[blue,->] (-.0917,0) -- (-.0917,-.95);
\draw[blue,->] (1.90832,0) -- (1.90832,-.95);
\draw[blue,->] (3.90832,0) -- (3.90832,-.95);
\draw[blue,->] (1.2111,0) -- (1.2111,-.95);
\draw[blue,->] (3.2111,0) -- (3.2111,-.95);
\draw[blue,->] (2.5139,0) -- (2.5139,-.95);
\draw[blue,->] (3.8167,0) -- (3.8167,-.95);
\draw[very thick,red,->] (-4,-1) -- (5,-1);
\draw (-4,-1) node[anchor=north] {$0$};
\draw (-2,-1) node[anchor=north] {$1$};
\draw (0,-1) node[anchor=north] {$2$};
\draw (2,-1) node[anchor=north] {$3$};
\draw (4,-1) node[anchor=north] {$4$};
\draw (4.7,-1) node[anchor=north] {$i$};
\end{tikzpicture}
\caption{Schematic of the admissible exponents $i \in K_{N_0}$ (cf.~\eqref{def_kn}) in the generalized power series \eqref{expansion} of the solution $u$ up to $i \le 4$.}
\label{fig:power}
\end{figure}
Actually, we are able to prove further regularity properties for the coefficients $u_i(t)$ and the correction $R_{N_0} = R_{N_0}(x,t)$ and also interior-regularity bounds, the presentation of which we postpone to later sections (cf. Lemma~\ref{lem:est_coeff} and Theorem~\ref{th:regularity}). Note that in \cite{ggko.2014} it was only proven that $u(t,x) = u_0(t) + u_\beta(t) x^\beta + o(x^\beta)$ as $x \searrow 0$ almost everywhere, where $u_0 = u_0(t)$ is bounded and continuous with $u_0(t) \to 0$ as $t \to \infty$ and $t^{\beta - \frac 1 2} u_\beta = t^{\beta - \frac 1 2} u_\beta(t)$ is square integrable with $u_\beta(t) = o\left(t^{\frac 1 2 - \beta}\right)$ as $t \to \infty$ for a subsequence.
\subsection{Discussion}
The result captures the regularizing effect of the degenerate-parabolic equation \eqref{pde}. Unlike in the case of the porous-medium equation
\[
\partial_t h - \partial_z^2 h^m = 0 \quad \mbox{for } \, t > 0 \, \mbox{ and } \, z \in \{h > 0\},
\]
with a constant $m > 1$ -- the second-order counterpart of \eqref{pde} for which a comparison principle holds and solutions become smooth for positive times \cite{a.1988,dh.1998,k.1999} -- here the solution only becomes smooth in the generalized sense \eqref{expansion}.
\medskip
We emphasize that in the case of partial wetting, that is, without loss of generality $\verti{\partial_z h} = 1$ at $z = Z_0(t)$, Kn\"upfer found that generically such singular expansions do appear as well: The solutions turn out to have a polynomial expansion in $x$ and $x \ln x$ for $n = 2$ \cite{k.2011}, and $x$ and $x^{3-n}$ for $n \in \left(0,\frac{14}{5}\right) \setminus \{1,2,\frac 5 2, \frac 8 3, \frac{11}{4}\}$ \cite{k.2012,k.2012.err} (cf.~also \cite{bgk.2016} for a discussion of source-type self-similar solutions with nonzero dynamic contact angle). In these cases, however, Kn\"upfer assumes such an expansion already for the initial data which also need to fulfill additional compatibility conditions. Hence in this case there is no proof available that the solution acquires additional regularity at the contact line for positive times, except for a smoothing effect in \cite[Cor.~4.3]{k.2012,k.2012.err} if one waits sufficiently long. This is due to the different techniques in \cite{k.2011,k.2012} relying on the Mellin transform and a suitable subtraction of the singular expansion multiplied with a cut off at $x = \infty$: The resulting estimates contain terms with different scaling in $x$ and consequently have no distinct scaling in time $t$. Thus it seems impossible to introduce time weights in order to capture the smoothing effect of the (non-)linear equation in this setting. On the other hand, it is well-known in the theory of non-degenerate parabolic equations (in domains with smooth boundary for instance) that solutions typically become smooth for positive times without having to assume well-prepared initial data.
\medskip
This (generalized) smoothing effect is the essentially new insight of Theorem~\ref{th:main}. The fact that we need control of an increasing number $k+6$ of logarithmic derivatives $D$ the larger $N_0$, essentially amounts to having sufficient interior regularity of the initial data. Since the degeneracy of \eqref{pde} is immaterial away from $x = 0$, the interior regularity of problem~\eqref{cauchy} can be treated by standard theory.
\medskip
We further remark that $\ensuremath{\varepsilon}$ has to be chosen dependent on $N_0$. With our method it appears to be unavoidable to assume this dependence\footnote{As we will observe in Section~\ref{sec:linear}, the coercivity constant of $p(D-N_0)$ with respect to the inner product $(\cdot,\cdot)_\alpha$ vanishes as $\alpha \nearrow N_0$ or $\alpha \searrow N_0-1$, and as $N_0 \to \infty$ the admissible exponents in the interval $(N_0-1,N_0)$ become dense. Thus the constant in the maximal-regularity estimate \eqref{mr_main} blows up as $N_0 \to \infty$.}, so that we cannot determine whether the series $\sum_{i \in K_\infty} u_i(t) x^i$ converges.
\medskip
We also notice that there are connections of this work with the theory of elliptic boundary problems in domains with isolated point singularities at the boundary, for which it is known that singular expansions of solutions occur \cite{kmr.1997}. This is not surprising, as the underlying fluid model, the Stokes (or Navier-Stokes) system, has to be solved on a moving infinite-cusp domain. In the partial wetting case, instead, a moving infinite wedge may be considered and in fact, Kn\"upfer's analysis in \cite{k.2011,k.2012} strongly relies on this analogy.
\subsection{Transformation into the original set of variables}
In \cite[Rm.~3.2, Rm.~3.3, App.~A]{ggko.2014} it was explained that analogous leading-order expansions also hold for the function $h$ and the velocity $V$ (the vertically-averaged horizontal velocity within lubrication theory, cf.~\eqref{continuity_eq}). Indeed, in the expression $V = h \partial_z^3 h$ for the velocity (cf.~\eqref{def_vel}), we may employ the hodograph transform \eqref{hodograph}, i.e., $h = x^{\frac 3 2}$ as well as $\partial_z = F \partial_x$ (cf.~\eqref{tfe_trafo_0}\&\eqref{trafo}), so that
\begin{subequations}\label{vel_mult}
\begin{equation}\label{velocity}
V = \frac 3 2 x^{\frac 3 2} F \partial_x F \partial_x F x^{\frac 1 2} = \tilde \ensuremath{\mathcal M}(F,F,F)
\end{equation}
with
\begin{equation}\label{multi_vel}
\tilde \ensuremath{\mathcal M}(F_1,F_2,F_3) := \frac 3 2 F_1 \left(D - \frac 1 2\right) F_2 \left(D + \frac 1 2\right) F_3.
\end{equation}
\end{subequations}
Equations~\eqref{vel_mult} can be used to derive
\begin{subequations}\label{vel_non}
\begin{equation}
V = A(t) + B(t,x) + C(t,x)
\end{equation}
with
\begin{align}
A &= - \frac 3 8 (1+u_0)^3, \\
B &= 3 (1+u_0)^2 \tilde\ensuremath{\mathcal M}_\mathrm{sym}(u-u_0,1,1) = \frac 3 2 (1+u_0)^2 \tilde p(D) (1+u_0) \quad \mbox{where } \, \tilde p(\zeta) = \left(\zeta+\beta+\frac 1 2\right) (\zeta-\beta), \\
C &= 3 (1+u_0) \tilde\ensuremath{\mathcal M}_\mathrm{sym}(u-u_0,u-u_0,1) + \tilde\ensuremath{\mathcal M}_\mathrm{sym}(u-u_0,u-u_0,u-u_0).
\end{align}
\end{subequations}
With help of expansion~\eqref{expansion}, equations~\eqref{vel_non} upgrade to
\begin{equation}\label{expansion_v}
V(t,Z(t,x)) = - \frac 3 8 \left(1+u_0(t)\right)^3 + \sum_{\substack{i \in K_{N_0}\\ 1 \le i < N_0}} V_i(t) x^i + O\left(x^{N_0}\right) \quad \mbox{as } \, x \searrow 0 \, \mbox{ and } \, t > 0,
\end{equation}
where the coefficients $V_i(t)$ and the correction $O\left(x^{N_0}\right)$ fulfill decay estimates as in \eqref{decay_as}. From \eqref{expansion_v} we can also read off the velocity $V_0(t)$ and the position $Z_0(t)$ of the contact line as
\[
V_0(t) = \frac 3 8 \left(1+u_0(t)\right)^3 \qquad \mbox{and} \qquad Z_0(t) = z_0 - \frac 3 8 \int_0^t \left(1 + u_0\left(t^\prime\right)\right)^3 \, \d t^\prime,
\]
where $z_0 \in \ensuremath{\mathbb{R}}$ is a free parameter.
\medskip
Finally, we can derive an expression for the expansion of the film height $h$ and the velocity $V$ in the vicinity of the contact line in terms of the original variables $t$ and $z$: Employing \eqref{expansion} in \eqref{trafo} in conjunction with \eqref{trafo_2}, we conclude
\[
Z_x = \left(1+u_0(t)\right)^{-1} + \sum_{\substack{i \in K_{N_0}\\ \beta \le i < N_0}} (Z_x)_i(t) x^i + O\left(x^{N_0}\right) \quad \mbox{as } \, x \searrow 0 \, \mbox{ and } \, t > 0,
\]
where the coefficients and remainder fulfill estimates analogous to \eqref{decay_as}. Integration of this expansion gives
\begin{eqnarray}\nonumber
\tilde x &:=& Z(t,x) - Z_0(t) \\
&=& x \left(\left(1+u_0(t)\right)^{-1} + \sum_{\substack{i \in K_{N_0}\\ \beta \le i < N_0}} (1+i)^{-1} (Z_x)_i(t) x^i + O\left(x^{N_0}\right)\right) \quad \mbox{as } \, x \searrow 0 \, \mbox{ and } \, t > 0. \label{def_tx}
\end{eqnarray}
Inversion yields
\begin{equation}\label{exp_x}
x = \left(1+u_0(t)\right) \tilde x \left(1 + \sum_{\substack{i \in K_{N_0}\\ \beta \le i < N_0}} c_i(t) \tilde x^i + O\left(\tilde x^{N_0}\right)\right) \quad \mbox{as } \, \tilde x \searrow 0 \, \mbox{ and } \, t > 0,
\end{equation}
with coefficients $c_i(t)$ and remainder $O\left(\tilde x^{N_0}\right)$ obeying estimates analogous to those in \eqref{decay_as}. Utilizing expansion~\eqref{exp_x} in the hodograph transform \eqref{hodograph}, we end up with
\[
h(t,z) = \tilde x^{\frac 3 2} \left(1 + \sum_{i \in K_N} \tilde u_i(t) \tilde x^i + O\left(\tilde x^{N_0}\right)\right) \quad \mbox{as } \, \tilde x \searrow 0 \, \mbox{ and } \, t > 0,
\]
where the $\tilde u_i(t)$ are (at least) continuous in time $t$, $\tilde x := z - Z_0(t)$ (cf.~\eqref{def_tx}), and decay estimates as in \eqref{decay_as} for the $\tilde u_i(t)$ and the remainder $O\left(\tilde x^{N_0}\right)$ hold true. We leave it up to the reader to directly relate the coefficients $\tilde u_i(t)$ to the $u_i(t)$ appearing in \eqref{expansion}. Furthermore, we can also derive an expansion for the velocity $V = V(t,z)$ by using \eqref{exp_x} in \eqref{expansion_v}:
\[
V(t,z) = - \frac 3 8 \left(1+u_0(t)\right)^3 + \sum_{\substack{i \in K_{N_0}\\ 1 \le i < N_0}} \tilde V_i(t) \tilde x^i + O\left(\tilde x^{N_0}\right) \quad \mbox{as } \, \tilde x \searrow 0 \, \mbox{ and } \, t > 0,
\]
where again $\tilde x := z - Z_0(t)$ and the $\tilde V_i(t)$ and the remainder term $O\left(\tilde x^{N_0}\right)$ meet decay estimates as in \eqref{decay_as}. We remark that it is open, whether a similar expansion for the velocity occurs for the (Navier-)Stokes equations on a moving infinite-cusp domain.
\subsection{Outline}
The paper consists of two parts: the linear theory (discussed in Section~\ref{sec:linear}) and the nonlinear theory (contained in Section~\ref{sec:nonlinear}). After recalling some of the basic notions on coercivity and maximal regularity in Section~\ref{ssec:coerc}, Section~\ref{ssec:formal} deals with the formal structure of the linear degenerate-parabolic equation \eqref{lin_cauchy}. Here we demonstrate that by applying an appropriate combination of \emph{scaling-invariant} operators $D-\gamma$ with $\gamma \in \ensuremath{\mathbb{R}}$ to the linear equation \eqref{lin_pde}, we are able to derive maximal-regularity estimates for \eqref{lin_cauchy} that control the singular expansion of $u$ at $x = 0$ to arbitrary orders (cf.~Section~\ref{ssec:heur_par}). Since all higher-order equations for $u$ are \emph{scaling-invariant}, the corresponding maximal-regularity estimates have a distinct scaling in $x$. Thus, when multiplied with an appropriate time weight, we can combine them to a quasi scaling-invariant maximal-regularity estimate for $u$ in terms of the initial data $u^{(0)}$ in the \emph{quasi-minimal} norm \eqref{norm_initial} and the right-hand side $f$. These arguments are made rigorous in Sections~\ref{ssec:func} and \ref{ssec:rigorous} (cf.~Proposition~\ref{prop:main_lin}) without going into all details. In particular Section~\ref{ssec:rigorous} is not essential for the understanding of the main ideas.
\medskip
Finally, in Section~\ref{sec:nonlinear} we prove our main regularity result, Theorem~\ref{th:regularity}, from which Theorem~\ref{th:main} follows as a special case. The proof strategy is standard and requires two ingredients: maximal-regularity estimates for the linearized problem \eqref{lin_cauchy} and the factorization of the nonlinearity $\ensuremath{\mathcal N}(u)$ (cf.~\eqref{nonlinearity}) given in Proposition~\ref{prop:nonlinear_est}. The latter is in fact the non-trivial part of the proof (cf.~Section~\ref{ssec:nonest}) and requires detailed estimates that rely on the symmetry properties of the multi-linear form $\ensuremath{\mathcal M}$ (cf.~\eqref{5linear}).
\section{The linear problem\label{sec:linear}}
\subsection{Coercivity and parabolic maximal regularity\label{ssec:coerc}}
Again, we briefly repeat some of the linear theory in \cite{ggko.2014}. Consider the linear problem
\begin{subequations}\label{lin_cauchy_alt}
\begin{align}
x \partial_t u + P(D) u &= f \quad \mbox{for } \, t, x > 0, \label{lin_pde_alt}\\
u_{|t = 0} &= u^{(0)},
\end{align}
\end{subequations}
which is structurally the same as \eqref{lin_cauchy} but with a general fourth-order polynomial $P(\zeta)$. We assume that the zeros $\gamma_1\le \cdots \le \gamma_4$ of $P(\zeta)$ are real. Then we know from \cite[Prop.~5.3]{ggko.2014} that there is a range of weights $\alpha$ -- which we may call \emph{coercivity range} -- such that $P(D)$ is formally coercive, i.e.,~$(u,P(D)u)_\alpha \gtrsim_\alpha \verti{u}_{2,\alpha}^2$ for all $u \in C_0^\infty((0,\infty))$. A sufficient criterion (which can be elementarily computed) is
\begin{equation}\label{coerc_cond}
\alpha \in (-\infty, \gamma_1) \cap (\gamma_2,\gamma_3) \cap (\gamma_4,\infty) \quad \mbox{and} \quad (\alpha-m(\gamma))^2 \le \frac{\sigma^2(\gamma)}{3},
\end{equation}
where $m(\gamma) := \frac 1 4 \sum_{j = 1}^4 \gamma_j$ (mean of the zeros $\gamma_j$) and $\sigma^2(\gamma) := \frac 1 4 \sum_{j = 1}^4 (\gamma_j - m(\gamma))^2$ (variance of the zeros $\gamma_j$). In the particular case of $P(D) = p(D)$, \eqref{coerc_cond} yields that the coercivity range contains the interval $(-1,0)$.
\medskip
Now suppose that $\alpha \in \ensuremath{\mathbb{R}}$ is in the coercivity range of $P(D)$. Then at least formally by quite elementary arguments (cf.~\cite[Sec.~2, Sec.~7.1]{ggko.2014}), we can derive a differential version of a maximal-regularity estimate for \eqref{lin_pde_alt} that reads
\begin{equation}\label{maxreg_diff}
\frac{\d}{\d t} \verti{u}_{\ell+2,\alpha-\frac 1 2}^2 + \verti{\partial_t u}_{\ell,\alpha-1}^2 + \verti{u}_{\ell+4,\alpha}^2 \lesssim_{\ell,\alpha} \verti{f}_{\ell,\alpha}^2, \quad \mbox{where } \, \ell \in \ensuremath{\mathbb{N}}_0.
\end{equation}
Indeed, in \eqref{maxreg_diff} derivatives $f, \cdots, D^\ell f$ control $u, \cdots, D^{\ell+4} u$ in the same norm, which is the maximal control in space one can expect as \eqref{lin_pde_alt} is fourth-order in $D$. The additional control of the time derivative $\partial_t u, \cdots, D^\ell \partial_t u$ (with reduced weight due to the degeneracy in \eqref{lin_cauchy_alt}) can be obtained by using control of $u, \cdots, D^{\ell+4} u$ and the fact that \eqref{lin_pde_alt} is fulfilled. This also yields control of the trace $\frac{\d}{\d t} \verti{u}_{\ell+2,\alpha-\frac 1 2}^2$ by interpolation. We refer to Section~\ref{ssec:rigorous} for more details.
\medskip
By multiplying \eqref{maxreg_diff} with a time weight $t^\sigma$ (where $\sigma \ge 0$), we obtain the integrated version of \eqref{maxreg_diff}, i.e.,
\begin{equation}\label{maxreg_int}
\begin{aligned}
&\sup_{t \ge 0} t^{2 \sigma} \verti{u}_{\ell+2,\alpha-\frac 1 2}^2 + \int_0^\infty t^{2 \sigma} \left(\verti{\partial_t u}_{\ell,\alpha-1}^2 + \verti{u}_{\ell+4,\alpha}^2\right) \d t \\
& \quad \lesssim_{\ell,\alpha} \delta_{\sigma,0} \verti{u^{(0)}}_{\ell+2,\alpha-\frac 1 2}^2 + \int_0^\infty t^{2 \sigma} \verti{f}_{\ell,\alpha}^2 \d t + 2 \sigma \int_0^\infty t^{2 \sigma - 1} \verti{u}_{\ell+2,\alpha-\frac 1 2}^2 \d t, \quad \mbox{where } \, \ell \in \ensuremath{\mathbb{N}}_0.
\end{aligned}
\end{equation}
The purpose of introducing time weights is two-fold: On the one hand they will enable us to prove the decay estimates as $t \to \infty$ in \eqref{decay_as} for the coefficients $u_i$ and the remainder $R_{N_0}$. While this would be irrelevant on a finite time interval, they also make it possible to prove regularity immediately after time $t = 0$, whereas without them the time after which regularity is obtained is unknown (cf.~\cite[Cor.~4.3]{k.2012,k.2012.err} for a similar case).
\medskip
Estimate~\eqref{maxreg_int} will be the basis of all linear estimates in the sequel.
\subsection{The formal structure of the linear equation\label{ssec:formal}}
As pointed out in \cite[Sec.~2]{ggko.2014}, just applying maximal regularity of the form \eqref{maxreg_int} with $\sigma = 0$ to the linear equation \eqref{lin_pde} is not sufficient in order to obtain well-posedness of the corresponding nonlinear problem \eqref{cauchy}. This is so, since only negative weights $\alpha$ are admissible (viz.~in the coercivity range of $p(D)$) and hence no control of the boundary value $u_0$ or the sup-norm $\sup_{t,x > 0} \verti{u(t,x)}$ can be achieved. On the other hand, as products of up to five factors in $\{u, D u, \cdots, D^4 u\}$ appear in the nonlinearity $\ensuremath{\mathcal N}(u)$ (cf.~\eqref{5linear}~and~\eqref{nonlinearity}), the control of $\sup_{t,x > 0} \verti{u(t,x)}$ appears to be necessary for proving well-posedness by a contraction argument. In order to circumvent this problem, it was convenient to apply $p(D-1)$ to the linear equation \eqref{lin_pde} that, by using the commutation relation $D x = x (D+1)$, transforms into
\begin{equation}\label{lin_v1}
x \partial_t v^{(1)} + p(D-1) v^{(1)} = g^{(1)} \quad \mbox{for } \, t, x > 0,
\end{equation}
where $v^{(1)} := p(D) u$ and $g^{(1)} := p(D-1) f$. Since the coercivity range has translated to the positive interval $(0,1)$ (cf.~e.g.~\eqref{coerc_cond}), one obtains better control on the regularity of $v$ at the boundary $x = 0$. This is not surprising as in view of \eqref{expansion} we expect $u(t,x) = u_0(t) + u_\beta(t) x^\beta + O(x)$ as $x \searrow 0$ and $t > 0$ and the powers $x^0$ and $x^\beta$ are in the kernel of $p(D)$, whence $v^{(1)}(t,x) = O(x)$ as $x \searrow 0$ and $t > 0$. Furthermore, by compatibility $f(t,x) = O(x)$ as $x \searrow 0$ and $t > 0$ and therefore also $g^{(1)}(t,x) = O(x)$ as $x \searrow 0$ and $t > 0$. As a second step one can then retrieve regularity information on $u$ from regularity information on $v^{(1)}$ using elliptic estimates for the operator $p(D)$ (cf.~\cite[Sec.~2, Lem.~7.2]{ggko.2014}).
\medskip
Before reviewing the arguments, we point out the limitations of the ansatz: As a natural second step, we apply the operator $p(D-2)$ to equation \eqref{lin_v1} and obtain
\begin{equation}\label{lin_v2}
x \partial_t v^{(2)} + p(D-2) v^{(2)} = g^{(2)} \quad \mbox{for } \, t, x > 0,
\end{equation}
with $v^{(2)} := p(D-1) v^{(1)}$ and $g^{(2)} := p(D-2) g^{(1)}$. Following the argumentation above and noting that the coercivity range of the operator $p(D-2)$ contains the interval $(1,2)$, we apply the maximal-regularity estimate \eqref{maxreg_int} to \eqref{lin_v2} and seemingly obtain even better control on the boundary regularity of $v^{(2)}$ which formally suggests $v(t,x) = O\left(x^{2-\ensuremath{\varepsilon}}\right)$ as $x \searrow 0$ and $t > 0$, where $\ensuremath{\varepsilon} > 0$ is arbitrarily small. Apparently, such a claim is too strong as generically also the term $x^{2 \beta}$ appears in the expansion of $u$ (cf.~Section~\ref{ssec:regularity}), this term is not in the kernel of $p(D-1) p(D)$, and thus in general $v(t,x) = O\left(x^{2\beta}\right)$ as $x \searrow 0$ and $t > 0$.
\medskip
In order to work around this problem, we set
\[
I_2 := \left\{n_1 + \beta n_2: \, (n_1,n_2) \in \ensuremath{\mathbb{N}}_0^2, \; 1 < n_1 + \beta n_2 < 2\right\} \setminus \{1 + \beta\} = \{2 \beta, 3 \beta\}
\]
and apply the operator $\prod_{i \in I_2} (D-i) = (D-2\beta) (D-3\beta)$ to \eqref{lin_v2}. Setting $w^{(2)} := \left(\prod_{i \in I_2} (D-i)\right) v^{(2)}$ and $r^{(2)} := \left(\prod_{i \in I_2} (D-i)\right) g^{(2)}$, equation~\eqref{lin_v2} turns into
\begin{equation}\label{lin_w2}
x \partial_t w^{(2)} + p(D-2) w^{(2)} = r^{(2)} + x q_2(D) \partial_t v^{(2)} \quad \mbox{for } \, t, x > 0,
\end{equation}
where $q_2(D) = \prod_{i \in I_2} (D-i) - \prod_{i \in I_2} (D-i+1) = - 2 D + 5 \beta - 1$ is a polynomial of degree $\verti{I_2}-1 = 1$. The additional term $x q_2(D) \partial_t v^{(2)}$ appears as the commutator of $\prod_{i \in I_2} (D-i)$ and the multiplication with $x$ does not vanish. Nevertheless, \eqref{lin_w2} is structurally advantageous compared to \eqref{lin_v2} as now we may indeed expect $w^{(2)}(t,x) = O(x^2)$, $r^{(2)}(t,x) =O(x^2)$, and $x q_2(D) \partial_t v^{(2)}(t,x) = o(x^2)$ as $x \searrow 0$ and $t > 0$. Hence applying the maximal-regularity estimate \eqref{maxreg_int} with $\alpha \in (1,2)$, we obtain better control on $w^{(2)}$ (assuming $\sigma > 0$):
\begin{equation}\label{mr_w2}
\begin{aligned}
&\sup_{t \ge 0} t^{2 \sigma} \verti{w^{(2)}}_{\ell+2,\alpha-\frac 1 2}^2 + \int_0^\infty t^{2 \sigma} \left(\verti{\partial_t w^{(2)}}_{\ell,\alpha-1}^2 + \verti{w^{(2)}}_{\ell+4,\alpha}^2\right) \d t \\
& \quad \lesssim_{\ell,\alpha} \int_0^\infty t^{2 \sigma} \verti{r^{(2)}}_{\ell,\alpha}^2 \d t + 2 \sigma \int_0^\infty t^{2 \sigma - 1} \verti{w^{(2)}}_{\ell+2,\alpha-\frac 1 2}^2 \d t + \int_0^\infty t^{2 \sigma} \verti{q_2(D) \partial_t v^{(2)}}_{\ell, \alpha-1}^2 \d t.
\end{aligned}
\end{equation}
Yet, the additional term $\int_0^\infty t^{2 \sigma} \verti{q_2(D) \partial_t v^{(2)}}_{\ell, \alpha-1}^2 \d t$ in \eqref{mr_w2} has to be treated: Since $\alpha-1 \in (0,1)$ is in the coercivity range of $p(D-1)$, we can apply maximal regularity of the form \eqref{maxreg_int} to the \emph{time-differentiated} version of \eqref{lin_v1}, i.e.,
\begin{equation}\label{lin_tv1}
x \partial_t^2 v^{(1)} + p(D-1) \partial_t v^{(1)} = \partial_t g^{(1)} \quad \mbox{for } \, t, x > 0,
\end{equation}
which leaves us with
\begin{equation}\label{mr_tv1}
\begin{aligned}
& \sup_{t \ge 0} t^{2 \sigma} \verti{\partial_t v^{(1)}}_{\ell^\prime+2,\alpha-\frac 3 2}^2 + \int_0^\infty t^{2 \sigma} \left(\verti{\partial_t^2 v^{(1)}}_{\ell^\prime,\alpha-2}^2 + \verti{\partial_t v^{(1)}}_{\ell^\prime+4,\alpha-1}^2\right) \d t \\
& \quad \lesssim_{\ell^\prime,\alpha} \int_0^\infty t^{2 \sigma} \verti{\partial_t g^{(1)}}_{\ell^\prime,\alpha-1}^2 \d t + 2 \sigma \int_0^\infty t^{2 \sigma - 1} \verti{\partial_t v^{(1)}}_{\ell^\prime+2,\alpha-\frac 3 2}^2 \d t.
\end{aligned}
\end{equation}
Assuming $\ell^\prime \ge \ell + \verti{I_2} - 5 = \ell - 3$ and noting that trivially $\verti{q_2(D) \partial_t v^{(2)}}_{\ell, \alpha-1}^2 \lesssim \verti{\partial_t v^{(1)}}_{\ell^\prime+4,\alpha}^2$, the combination of \eqref{mr_w2} and \eqref{mr_tv1} yields
\begin{equation}\label{mr_w2_tv1}
\begin{aligned}
&\sup_{t \ge 0} t^{2 \sigma} \left(\verti{w^{(2)}}_{\ell+2,\alpha-\frac 1 2}^2 + \verti{\partial_t v^{(1)}}_{\ell^\prime+2,\alpha-\frac 3 2}^2\right) \\
&+ \int_0^\infty t^{2 \sigma} \left(\verti{\partial_t w^{(2)}}_{\ell,\alpha-1}^2 + \verti{w^{(2)}}_{\ell+4,\alpha}^2 + \verti{\partial_t^2 v^{(1)}}_{\ell^\prime,\alpha-2}^2 + \verti{\partial_t v^{(1)}}_{\ell^\prime+4,\alpha-1}^2\right) \d t \\
& \quad \lesssim_{\ell,\ell^\prime,\alpha} \int_0^\infty t^{2 \sigma} \left(\verti{r^{(2)}}_{\ell,\alpha}^2 + \verti{\partial_t g^{(1)}}_{\ell^\prime,\alpha-1}^2\right) \d t \\
& \qquad\qquad + 2 \sigma \int_0^\infty t^{2 \sigma - 1} \left(\verti{w^{(2)}}_{\ell+2,\alpha-\frac 1 2}^2 + \verti{\partial_t v^{(1)}}_{\ell^\prime+2,\alpha-\frac 3 2}^2 \d t\right) \d t.
\end{aligned}
\end{equation}
Now the solutions $w^{(2)}$ and $v^{(1)}$ are estimated in sufficiently strong norms by $\partial_t g^{(1)}$ and $r^{(2)}$ in respective norms and the integral
\begin{equation}\label{remainder1}
\int_0^\infty t^{2 \sigma - 1} \left(\verti{w^{(2)}}_{\ell+2,\alpha-\frac 1 2}^2 + \verti{\partial_t v^{(1)}}_{\ell^\prime+2,\alpha-\frac 1 2}^2 \right) \d t.
\end{equation}
The norms in \eqref{remainder1} have reduced spatial and temporal weights and thus it is possible to absorb these terms by lower-order estimates\footnote{Here ``lower order" is meant in the sense of using norms with lowered weights and the same number of time derivatives.}. We will detail the arguments in Section~\ref{ssec:heur_par}. Notably it seems unavoidable to combine higher-regularity estimates in space with higher-regularity estimates in time as opposed to the case of the thin-film equation \eqref{tfe_general} with $n = 1$, that is, the lubrication approximation of the Hele-Shaw cell \cite[Sec.~8, Sec.~9]{gko.2008}. This has already been observed by Kn\"upfer in \cite{k.2011,k.2012} for the partial wetting case using rather different techniques.
\medskip
Before addressing the issue of dealing with the lower-order terms in \eqref{mr_w2_tv1}, we will first systematize our observations: In order to obtain better control on the solution, we apply the operator $p(D-3)$ to equation \eqref{lin_w2}. Thus we arrive at
\begin{equation}\label{lin_v3}
x \partial_t v^{(3)} + p(D-3) v^{(3)} = g^{(3)} + x q_2(D) p(D-2) \partial_t v^{(2)} \quad \mbox{for } \, t, x > 0,
\end{equation}
where we have set $v^{(3)} := p(D-2) w^{(2)}$ and $g^{(3)} := p(D-3) r^{(2)}$. Again, we cannot expect to have $v^{(3)}(t,x) = O(x^3)$ as $x \searrow 0$ and $t > 0$ as the set
\[
I_3 := \left\{n_1 + \beta n_2: \, (n_1,n_2) \in \ensuremath{\mathbb{N}}_0^2, \; 2 < n_1 + \beta n_2 < 3\right\} \setminus \{2 + \beta\}
\]
is non-empty (cf.~Fig.~\ref{fig:power}). Applying $\prod_{i \in I_3} (D-i)$ to \eqref{lin_w2}, we obtain
\begin{equation}\label{lin_w3_0}
\begin{aligned}
& x \partial_t w^{(3)} + p(D-3) w^{(3)} \\
& \quad = r^{(3)} + x \tilde q_3(D) \partial_t v^{(3)} + x q_2(D) p(D-2) \left(\prod_{i \in I_3} (D-i+1)\right) \partial_t v^{(2)} \quad \mbox{for } \, t, x > 0,
\end{aligned}
\end{equation}
where again $w^{(3)} := \left(\prod_{i \in I_3} (D-i)\right) v^{(3)}$, $r^{(3)} := \left(\prod_{i \in I_3} (D-i)\right) g^{(3)}$, and $\tilde q_3(D)= \prod_{i \in I_3} (D-i) - \prod_{i \in I_3} (D-i+1)$ is a polynomial in $D$ of degree $\verti{I_3}-1$ that originates from the commutator of $x$ and $\prod_{i \in I_3} (D-i)$. Then we make the rather trivial observation $I_2 \subset I_3 - 1 := \{i-1: \, i \in I_3\}$ (cf.~Fig.~\ref{fig:power}). By exploiting
\begin{align*}
p(D-2) \left(\prod_{i \in I_3} (D-i+1)\right) \partial_t v^{(2)} &= \left(\prod_{i \in (I_3-1) \setminus I_2} (D-i)\right) p(D-2) \underbrace{\left(\prod_{i \in I_2} (D-i)\right) v^{(2)}}_{= w^{(2)}} \\
&= \left(\prod_{i \in (I_3-1) \setminus I_2} (D-i)\right) v^{(3)}
\end{align*}
and setting
\[
q_3(D) := \tilde q_3(D) + q_2(D) \prod_{i \in (I_3-1) \setminus I_2} (D-i),
\]
which is a polynomial of degree less or equal to $\verti{I_3}-1$, we can rewrite the last two terms in \eqref{lin_w3_0} and obtain:
\begin{equation}\label{lin_w3}
x \partial_t w^{(3)} + p(D-3) w^{(3)} = r^{(3)} + x q_3(D) \partial_t v^{(3)} \quad \mbox{for } \, t, x > 0.
\end{equation}
As \eqref{lin_w3} is structurally the same as \eqref{lin_w2}, this procedure can be iterated, and we arrive at the following set of equations
\begin{equation}\label{lin_wnm}
(x \partial_t + p(D-n)) \partial_t^m w^{(n)} = \partial_t^m r^{(n)} + x q_n(D) \partial_t^{m+1} v^{(n)} \quad \mbox{for } \, t, x > 0,
\end{equation}
where we define the sets of indices
\begin{subequations}\label{def_jnin}
\begin{align}
I_n &:= \left\{n_1 + \beta n_2: \, (n_1,n_2) \in \ensuremath{\mathbb{N}}_0^2, \; n-1 < n_1 + \beta n_2 < n\right\} \setminus \{n-1+\beta\},\\
J_n &:= \left\{n_1 + \beta n_2: \, (n_1,n_2) \in \ensuremath{\mathbb{N}}_0^2, \; 0 < n_1 + \beta n_2 < n\right\} \setminus \left(\ensuremath{\mathbb{N}} \cup (\ensuremath{\mathbb{N}}_0 + \beta)\right) = \cup_{n^\prime = 1}^n I_{n^\prime}, \label{def_jn}
\end{align}
\end{subequations}
introduce the functions
\begin{subequations}\label{def_vwr}
\begin{align}
v^{(n)} &:= \left(\prod_{n^\prime = 0}^{n-1} p(D-n^\prime)\right) \left(\prod_{i \in J_{n-1}} (D-i)\right) u, \label{def_vn}\\
w^{(n)} &:= \left(\prod_{n^\prime = 0}^{n-1} p(D-n^\prime)\right) \left(\prod_{i \in J_n} (D-i)\right) u = \left(\prod_{i \in I_n} (D-i)\right) v^{(n)}, \label{def_wn}\\
r^{(n)} &:= \left(\prod_{n^\prime = 1}^n p(D-n^\prime)\right) \left(\prod_{i \in J_n} (D-i)\right) f, \label{def_rn}
\end{align}
\end{subequations}
and denote by $q_n(D)$ a polynomial of degree less or equal to $\verti{I_n}-1$. The numbers $n \in \ensuremath{\mathbb{N}}$ and $m \in \ensuremath{\mathbb{N}}_0$ are arbitrary.
\subsection{Heuristics for parabolic maximal regularity\label{ssec:heur_par}}
In this section we systematize the ideas of the previous section, leading to estimate~\eqref{mr_w2_tv1}. Throughout the section, all estimates may depend on $N$, $n$, $m$, $\alpha$, or $\delta$. Applying the maximal-regularity estimate \eqref{maxreg_int} to equation~\eqref{lin_w3}, we obtain
\begin{equation}\label{mr_wnm}
\begin{aligned}
&\sup_{t \ge 0} t^{2 (\alpha+n+m) - 3} \verti{\partial_t^m w^{(n)}}_{k(n,m,\alpha^\prime) + 2,\alpha^\prime + n - \frac 3 2}^2 \\
&+ \int_0^\infty t^{2 (\alpha+n+m) - 3} \left(\verti{\partial_t^{m+1} w^{(n)}}_{k(n,m,\alpha^\prime),\alpha^\prime+n-2}^2 + \verti{\partial_t^m w^{(n)}}_{k(n,m,\alpha^\prime)+4,\alpha^\prime+n-1}^2\right) \d t \\
& \quad \lesssim \delta_{2 (\alpha+n+m-1),1} \verti{\partial_t^m w^{(n)}_{| t = 0}}_{k(n,m,\alpha^\prime) + 2,\alpha^\prime + n - \frac 3 2}^2 + \int_0^\infty t^{2 (\alpha+n+m) - 3} \verti{\partial_t^m r^{(n)}}_{k(n,m,\alpha^\prime),\alpha^\prime+n-1}^2 \d t \\
& \qquad + (1-\delta_{n,1}) \int_0^\infty t^{2 (\alpha+n+m) - 3} \verti{\partial_t^{m+1} v^{(n)}}_{k(n,m,\alpha^\prime)+\verti{I_n}-1,\alpha^\prime+n-2}^2 \d t \\
& \qquad + (2 (\alpha+n+m) - 3) \int_0^\infty t^{2 (\alpha+n+m) - 4} \verti{\partial_t^m w^{(n)}}_{k(n,m,\alpha^\prime)+2,\alpha^\prime+n-\frac 3 2}^2 \d t,
\end{aligned}
\end{equation}
where we assume and use the following:
\begin{itemize}
\item[$\bullet$] We take a finite number of weights $\alpha \in [0,1]$ and we choose $\delta > 0$ sufficiently small such that
\begin{itemize}
\item[$\star$] $\alpha^\prime := \alpha \pm \delta \in (0,1)$ if $\alpha \in (0,1)$,
\item[$\star$] $\delta \in (0,1)$ and thus also $1-\delta \in (0,1)$.
\end{itemize}
Thus $\alpha^\prime$ is in the coercivity range of $p(D-1)$ (i.e.,~$\alpha+n-1 \in (n-1,n)$ is in the coercivity range of $p(D-n)$).
\item[$\bullet$] $\delta > 0$ has to be chosen small enough such that if $\alpha_1 < \alpha_2$, then also $\alpha_1 +\delta < \alpha_2 -\delta$. Further smallness conditions on $\delta$ will be specified when necessary.
\item[$\bullet$] We need to assume $2(\alpha+n+m)-3 \ge 0$ so that all time weights are integrable at $t = 0$ (the last term in \eqref{mr_wnm} vanishes for $2(\alpha+n+m)-3 = 0$).
\item[$\bullet$] The indices $k(n,m,\alpha) \in \ensuremath{\mathbb{N}}_0$, determining the number of $D$-derivatives in the norms appearing in \eqref{mr_wnm}, will be chosen later.
\end{itemize}
The specific choice of the weights $\alpha$ will be explained further below. Choosing proper weights turns out to be essential for obtaining control on the coefficients $u_i(t)$ of the generalized power series \eqref{expansion} of $u$ and for being able to absorb the last line of \eqref{mr_wnm}.
\medskip
Note that estimate~\eqref{mr_wnm} itself is insufficient as the solution $u$ still appears on the estimate's right-hand side in the last two lines.
\subsubsection{Absorption of remnant terms I}
We will first concentrate on absorbing the second but last line of \eqref{mr_wnm} by exploiting the additional time regularity: Therefore we start with estimate~\eqref{mr_wnm} with $n = N$ and $m = 0$. For $N > 1$, the term
\[
\int_0^\infty t^{2 (\alpha+N) - 3} \verti{\partial_t v^{(N)}}_{k(N,0,\alpha^\prime)+\verti{I_N}-1,\alpha^\prime+N-2}^2 \d t
\]
has to be absorbed. This term can be estimated by the left-hand side of \eqref{mr_wnm} for $n = N-1$ and $m = 1$, provided that the indices $k(n,m,\alpha^\prime)$ obey $k(N-1,1,\alpha^\prime) \ge k(N,0,\alpha^\prime) + \verti{I_N} - 1$. Then indeed
\[
\verti{\partial_t v^{(N)}}_{k(N,0,\alpha^\prime)+\verti{I_N}-1,\alpha^\prime+N-2} \lesssim \verti{\partial_t w^{(N-1)}}_{k(N-1,1,\alpha^\prime)+4,\alpha^\prime+N-2}.
\]
Next, supposed that $N > 2$, the term
\[
\int_0^\infty t^{2 (\alpha+N) - 3} \verti{\partial_t^2 v^{(N-1)}}_{k(N-1,1,\alpha^\prime)+\verti{I_{N-1}}-1,\alpha^\prime+N-3}^2 \d t
\]
has to be absorbed, which can be achieved by combining it with estimate~\eqref{mr_wnm} for $n = N - 2$ and $m = 2$, provided that $k(N-2,2,\alpha^\prime) \ge k(N-1,1,\alpha^\prime) + \verti{I_{N-1}} - 1$. Apparently, this procedure can be iterated (cf.~Fig.~\ref{fig:absorb1})
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=1]
\draw[very thick,->] (.5,1) -- (.5,5);
\draw[very thick,->] (1,.5) -- (5,.5);
\draw [gray] (.4,1) -- (.6,1);
\draw [gray] (1,.4) -- (1,.6);
\draw [gray] (.4,2) -- (.6,2);
\draw [gray] (2,.4) -- (2,.6);
\draw [gray] (.4,3) -- (.6,3);
\draw [gray] (3,.4) -- (3,.6);
\draw [gray] (.4,4) -- (.6,4);
\draw [gray] (4,.4) -- (4,.6);
\draw (.3,1) node[anchor=east] {$1$};
\draw (1,.3) node[anchor=north] {$0$};
\draw (.3,2) node[anchor=east] {$2$};
\draw (2,.3) node[anchor=north] {$1$};
\draw (.3,3) node[anchor=east] {$3$};
\draw (3,.3) node[anchor=north] {$2$};
\draw (.3,4) node[anchor=east] {$4$};
\draw (4,.3) node[anchor=north] {$3$};
\draw [gray] (.9,1) -- (1.1,1);
\draw [gray] (1,.9) -- (1,1.1);
\draw [gray] (.9,2) -- (1.1,2);
\draw [gray] (1,1.9) -- (1,2.1);
\draw [gray] (.9,3) -- (1.1,3);
\draw [gray] (1,2.9) -- (1,3.1);
\draw [gray] (.9,4) -- (1.1,4);
\draw [gray] (1,3.9) -- (1,4.1);
\draw [gray] (1.9,1) -- (2.1,1);
\draw [gray] (2,.9) -- (2,1.1);
\draw [gray] (1.9,2) -- (2.1,2);
\draw [gray] (2,1.9) -- (2,2.1);
\draw [gray] (1.9,3) -- (2.1,3);
\draw [gray] (2,2.9) -- (2,3.1);
\draw [gray] (1.9,4) -- (2.1,4);
\draw [gray] (2,3.9) -- (2,4.1);
\draw [gray] (2.9,1) -- (3.1,1);
\draw [gray] (3,.9) -- (3,1.1);
\draw [gray] (2.9,2) -- (3.1,2);
\draw [gray] (3,1.9) -- (3,2.1);
\draw [gray] (2.9,3) -- (3.1,3);
\draw [gray] (3,2.9) -- (3,3.1);
\draw [gray] (2.9,4) -- (3.1,4);
\draw [gray] (3,3.9) -- (3,4.1);
\draw [gray] (3.9,1) -- (4.1,1);
\draw [gray] (4,.9) -- (4,1.1);
\draw [gray] (3.9,2) -- (4.1,2);
\draw [gray] (4,1.9) -- (4,2.1);
\draw [gray] (3.9,3) -- (4.1,3);
\draw [gray] (4,2.9) -- (4,3.1);
\draw [gray] (3.9,4) -- (4.1,4);
\draw [gray] (4,3.9) -- (4,4.1);
\draw[blue,->] (1.1,3.9) -- (1.9,3.1);
\draw[blue,->] (2.1,2.9) -- (2.9,2.1);
\draw[blue,->] (3.1,1.9) -- (3.9,1.1);
\draw (.5,4.7) node[anchor=east] {$n$};
\draw (4.7,.5) node[anchor=north] {$m$};
\end{tikzpicture}
\caption{Schematic: absorption of remnant terms I. Each node $+$ corresponds to an estimate of the form \eqref{mr_wnm}. The displayed arrows visualize the absorption mechanism for $N = 4$, that is, the remnant at the base of the arrow (forming the second but last line in \eqref{mr_wnm}) is absorbed by the corresponding estimate \eqref{mr_wnm} at the tip of the arrow under the assumption that \eqref{same_alpha} holds true.}
\label{fig:absorb1}
\end{figure}
and by merely summing \eqref{mr_wnm} for indices
\[
(n,m) \in \left\{(N,0), (N-1,1), \cdots, (1,N-1)\right\},
\]
we end up with
\begin{equation}\label{wnm_sum}
\begin{aligned}
&\sup_{t \ge 0} t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime) + 2,\alpha^\prime + N-m - \frac 3 2}^2 \\
&+ \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^{m+1} w^{(N-m)}}_{k(N-m,m,\alpha^\prime),\alpha^\prime+N-m-2}^2 \d t \\
&+ \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+4,\alpha^\prime+N-m-1}^2 \d t \\
& \quad \lesssim \delta_{2 \alpha,1} \delta_{N,1} \verti{w^{(1)}_{| t = 0}}_{k\left(1,0,\frac 1 2 \pm \delta\right) + 2,\pm \delta}^2 + \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m r^{(N-m)}}_{k(N-m,m,\alpha^\prime),\alpha^\prime+N-m-1}^2 \d t \\
& \qquad + (2 \alpha + 2 N - 3) \int_0^\infty t^{2 (\alpha+N) - 4} \sum_{m = 0}^{N-1} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+2,\alpha^\prime+N-m-\frac 3 2}^2 \d t,
\end{aligned}
\end{equation}
provided that we have
\[
k(N-m-1,m+1,\alpha^\prime) \ge k(N-m,m,\alpha^\prime) + \verti{I_{N-m}} - 1 \quad \mbox{for } \, m = 0, \cdots, N-2
\]
and
\begin{equation}\label{inequ_an}
2 \alpha + 2 N - 3 \ge 0.
\end{equation}
For later purpose we require the stronger assumption
\begin{equation}\label{same_alpha}
k(N-m-1,m+1,\alpha^\prime) \ge k(N-m,m,\alpha^\prime) + \verti{I_{N-m}} \quad \mbox{for } \, m = 0, \cdots, N-2.
\end{equation}
\subsubsection{Absorption of remnant terms II}
Estimate~\eqref{wnm_sum} is still insufficient as yet the solution appears on the right-hand side of the estimate (forming the last line). The absorption of this last line is indeed more complicated and demands to specifically choose the weights $\alpha$. In order to understand the choices made, we first make additional comments on elliptic maximal regularity: In fact, one can get control on $u$ from control on the functions $w^{(n)}$.
\begin{proposition}\label{prop:elliptic}
Suppose that $k \in \ensuremath{\mathbb{N}}_0$, $\varrho \in \ensuremath{\mathbb{R}} \setminus K_\infty$, and $u: (0,\infty) \to \ensuremath{\mathbb{R}}$ is smooth satisfying
\begin{equation}\label{ell_assume}
D^\ell u(x) = \sum_{i \in K_\varrho} u_i i^\ell x^i + o(x^\varrho) \quad \mbox{as } \, x \searrow 0
\end{equation}
for all $\ell = 0, \cdots, k + \verti{K_\varrho}$ (cf.~\eqref{def_kn} for the definition of $K_\varrho$). Then for any polynomial $Q(\zeta) = \prod_{\ell = 1}^m (\zeta - \zeta_\ell)$ with real zeros $\zeta_1, \cdots, \zeta_m$ such that $K_\varrho \subset \{\zeta_1,\cdots,\zeta_m\}$ and $\varrho \notin \{\zeta_1,\cdots,\zeta_m\}$, we have
\begin{equation}\label{elliptic_mr}
\verti{u - \sum_{i \in K_\varrho} u_i x^i}_{k+m,\varrho} \lesssim_{k,\varrho} \verti{Q(D) u}_{k,\varrho}.
\end{equation}
\end{proposition}
A proof in a similar case can be found in \cite[Lem.~7.2]{ggko.2014}. It is an immediate consequence of a version of Hardy's inequality:
\begin{lemma}\label{lem:hardy}
For any $w \in C^\infty((0,\infty))$, $\gamma, \varrho \in \ensuremath{\mathbb{R}}$ with $\gamma \ne \varrho$, $\verti{w}_{1,\varrho} < \infty$, and $w(x) = o(x^\varrho)$ as $x \searrow 0$ ($x \nearrow \infty$) if $\gamma < \varrho$ ($\gamma > \varrho$), we have
\begin{equation}\label{hardy}
\verti{w}_{1,\varrho} \lesssim_{\gamma,\varrho} \verti{(D-\gamma) w}_\varrho.
\end{equation}
\end{lemma}
For a particular admissible exponent $i \in K_\infty \setminus \{0\}$, we may choose the weights $i \pm \delta$. Both $i + \delta$ and $i-\delta$ appear in the coercivity range of $p(D-\floor{i})$ or $p(D-\floor{i}-1)$ (these are the intervals $(\floor{i}-1,\floor{i})$ and $(\floor{i},\floor{i}+1)$). Having control on
\[
\int_0^\infty t^{2 i - 1} \verti{w^{(\floor{i})}}_{k(\floor{i},0,i-\floor{i}+1-\delta)+4,i - \delta}^2 \d t \quad \mbox{if } \, i \in \ensuremath{\mathbb{N}}_0
\]
or
\[
\int_0^\infty t^{2 i + 1} \verti{w^{(\floor{i}+1)}}_{k(\floor{i}+1,0,i-\floor{i}\pm\delta)+4,i \pm \delta}^2 \d t \quad \mbox{else},
\]
we obtain control on
\[
\int_0^\infty t^{2 i - 1} \verti{u - \sum_{j \in K_i} u_j x^j}_{k(\floor{i},0,i-\floor{i}+1-\delta)+4+ 4 \floor{i} + \verti{J_{\floor{i}}},i - \delta}^2 \d t \quad \mbox{for } \, i \in \ensuremath{\mathbb{N}}_0
\]
or else
\[
\int_0^\infty t^{2 i - 1} \verti{u - \sum_{j \in K_i} u_j x^j}_{k(\floor{i}+1,0,i-\floor{i}-\delta)+4+ 4 \left(\floor{i}+1\right) + \verti{J_{\floor{i}+1}},i - \delta}^2 \d t,
\]
and
\[
\int_0^\infty t^{2 i - 1} \verti{u - \sum_{j \in K_i} u_j x^j - u_i x^i}_{k(\floor{i}+1,0,i-\floor{i}+\delta)+4+ 4 \left(\floor{i}+1\right) + \verti{J_{\floor{i}+1}},i + \delta}^2 \d t,
\]
respectively, by using elliptic maximal regularity given by \eqref{elliptic_mr}. It is quite apparent that, by applying the triangle inequality, the last three terms yield control on the coefficient $u_i(t)$ of the form: $\int_0^\infty t^{2 i - 1} \verti{u_i(t)}^2 \d t$. As we have already noted in the introduction, for $\alpha = \frac 1 2$ and $N = 1$ the first term in estimate~\eqref{wnm_sum} also yields control on $\sup_{t \ge 0} \verti{u_0(t)}^2$ and analogous estimates of higher-order coefficients are possible. We will postpone the details on how to extract further control on the coefficients to Section~\ref{ssec:func} (cf.~Lemma~\ref{lem:est_coeff})\footnote{Indeed, sufficiently strong estimates on them are essential for proving appropriate estimates for the nonlinearity (cf.~Proposition~\ref{prop:nonlinear_est}).}. For the moment we just note that for controlling the singular expansion of $u$, it is convenient to use spatial weights $i \pm \delta$ with $i \in K_\infty \setminus \{0\}$ in our norm.
\medskip
Without further ado, we start with the choice of weights by recalling that
\begin{equation}\label{index_sub}
I_{N-1} \subset I_N - 1 = \{i-1: \, i \in I_N\} \quad \mbox{(cf.~Fig.~\ref{fig:power})}.
\end{equation}
We fix $N_0 \in \ensuremath{\mathbb{N}}$ and are aiming at controlling expansion~\eqref{expansion} up to order $O\left(x^{N_0}\right)$. In view of \eqref{index_sub} it is reasonable to view the set of indices $\ensuremath{\mathcal A}$ as a subset of $[0,1] \times \{1,\cdots,N_0\}$. For each $(\alpha,N) \in \ensuremath{\mathcal A}$ we may use estimate~\eqref{wnm_sum} where $\alpha^\prime = \alpha \pm \delta \in (0,1)$. We distinguish between two classes of weights:
\begin{itemize}
\item[(a)] By the above considerations on estimating the coefficients and in view of the fact that \eqref{index_sub} holds true, we start by including $(\alpha,N)$ with
\begin{itemize}
\item[$\star$] $\alpha \in \left(I_{N_0} - N_0 + 1\right) \cup \{\beta\}$ and $N = 2, \cdots, N_0$,
\item[$\star$] $\alpha \in \left(\left(I_{N_0} - N_0 + 1\right) \cup \{\beta\}\right) \cap \left(\frac 1 2,1\right)$ and $N = 1$,
\item[$\star$] as well as $(\alpha,N) = (0,N)$ with $N = 2,\cdots,N_0$
\item[$\star$] and $(\alpha,N) = (1,N)$ with $N = 1,\cdots,N_0-1$,
\end{itemize}
in the set $\ensuremath{\mathcal A}$. Thus we are already able to control all coefficients $u_i(t)$ with $N_0 > i > \frac 1 2$.
\item[(b)] In all these cases, we need to be able to absorb the respective remnant terms forming the last line of \eqref{wnm_sum}. Since the weight in the norm is shifted by $-\frac 1 2$, this requires to include $(\alpha,N)$ with
\begin{itemize}
\item[$\star$] $\alpha \in \left(\left(I_{N_0} - N_0+\frac 1 2\right) \cup \{\beta - \frac 1 2\}\right) \cap \left(0,\frac 1 2\right)$ and $N = 2, \cdots, N_0$,
\item[$\star$] $\alpha \in \left(I_{N_0} - N_0+\frac 3 2\right) \cap \left(\frac 1 2,1\right)$ and $N = 1,\cdots,N_0-1$,
\item[$\star$] as well as $\left(\alpha,N\right) = \left(\frac 1 2,N\right)$ with $N = 1,\cdots,N_0$,
\end{itemize}
in $\ensuremath{\mathcal A}$.
\end{itemize}
Now that we have chosen the set of weights $\ensuremath{\mathcal A}$, we need to ensure that the absorption mechanism for the last line of \eqref{wnm_sum} works. This requires some additional conditions on the number of derivatives $k(n,m,\alpha^\prime)$. We note that since $\beta$ is irrational, by choosing $\delta > 0$ sufficiently small we have $\alpha^\prime \ne \frac 1 2$ for all $(\alpha,N) \in \ensuremath{\mathcal A}$. For $(\alpha,N) \in \ensuremath{\mathcal A}$ such that $\alpha + N \ge 2$, we need to distinguish between three cases (cf.~Fig.~\ref{fig:absorb2}):
\begin{itemize}
\item[(a)] If $\alpha^\prime > \frac 1 2$, we can absorb the remnant in the last line of \eqref{wnm_sum} through
\begin{align*}
& \int_0^\infty t^{2 (\alpha+N) - 4} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+2,\alpha^\prime+N-m-\frac 3 2}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 \left(\alpha - \frac 1 2 + N\right) - 3} \verti{\partial_t^m w^{(N-m)}}_{k\left(N-m,m,\alpha^\prime-\frac 1 2\right)+4,\alpha^\prime-\frac 1 2 + N - m - 1}^2 \d t,
\end{align*}
where the second line of the inequality appears on the left-hand side of \eqref{wnm_sum} with $\alpha$ replaced by $\alpha - \frac 1 2$. This requires that the indices obey
\begin{equation}\label{alpha-12}
k\left(N-m,m,\alpha^\prime-\frac 1 2\right) \ge k\left(N-m,m,\alpha^\prime\right) - 2 \quad \mbox{if } \, \alpha^\prime \in \left(\frac 1 2, 1\right).
\end{equation}
Indeed one may verify that by construction in all such cases $\left(\alpha-\frac 1 2,N\right) \in \ensuremath{\mathcal A}$.
\item[(b)] If $\alpha^\prime < \frac 1 2$ and $N-m \ge 2$, we can absorb the remnant term in \eqref{wnm_sum} through
\begin{align*}
& \int_0^\infty t^{2 (\alpha+N) - 4} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+2,\alpha^\prime+N-m-\frac 3 2}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 \left((\alpha + \frac 1 2) + (N - 1)\right) - 3} \verti{\partial_t^m w^{((N-1)-m)}}_{k\left((N-1)-m,m,\alpha^\prime+\frac 1 2\right)+4,\alpha^\prime+\frac 1 2 + (N-1) - m - 1}^2 \d t,
\end{align*}
where the last line of the estimate appears on the left-hand side of \eqref{wnm_sum} with $\alpha$ replaced by $\alpha + \frac 1 2$ and $N$ replaced by $N-1$. In view of \eqref{def_vwr}, this requires the constraint
\begin{equation}\label{alpha+12}
k\left(N-m-1,m,\alpha^\prime+\frac 1 2\right) \ge k\left(N-m,m,\alpha^\prime\right) + 2 + \verti{I_{N-m}} \quad \mbox{if } \, \alpha^\prime \in \left(0, \frac 1 2\right).
\end{equation}
Again, by construction we have $\left(\alpha+\frac 1 2, N-1\right) \in \ensuremath{\mathcal A}$.
\item[(c)] If $\alpha^\prime < \frac 1 2$ and $N-m = 1$, necessarily $m \ge 1$ and we can estimate the remnant in \eqref{wnm_sum} by
\begin{align*}
& \int_0^\infty t^{2 (\alpha+N) - 4} \verti{\partial_t^m w^{(1)}}_{k(1,m,\alpha^\prime)+2,\alpha^\prime-\frac 1 2}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 \left((\alpha + \frac 1 2) + N - 1\right) - 3} \verti{\partial_t^{(m-1)+1} w^{(1)}}_{k\left(1,m-1,\alpha^\prime+\frac 1 2\right),\alpha^\prime + \frac 1 2 - 1}^2 \d t.
\end{align*}
Here, the second line in the estimate is controlled in \eqref{wnm_sum} with $\alpha + \frac 1 2$ instead of $\alpha$ and $m$ replaced by $m-1$. Yet, the absorption only works if the indices obey
\begin{equation}\label{alpha+12_alt}
k\left(1,m-1,\alpha^\prime+\frac 1 2\right) \ge k\left(1,m,\alpha^\prime\right) + 2 \quad \mbox{if } \, \alpha^\prime \in \left(0, \frac 1 2\right).
\end{equation}
\end{itemize}
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=1]
\draw[very thick,->] (.5,1) -- (.5,5);
\draw[very thick,->] (1,.5) -- (5,.5);
\draw [gray] (.4,1) -- (.6,1);
\draw [gray] (1,.4) -- (1,.6);
\draw [gray] (.4,2) -- (.6,2);
\draw [gray] (2,.4) -- (2,.6);
\draw [gray] (.4,3) -- (.6,3);
\draw [gray] (3,.4) -- (3,.6);
\draw [gray] (.4,4) -- (.6,4);
\draw [gray] (4,.4) -- (4,.6);
\draw (.3,1) node[anchor=east] {$1$};
\draw (1,.3) node[anchor=north] {$0$};
\draw (.3,2) node[anchor=east] {$2$};
\draw (2,.3) node[anchor=north] {$1$};
\draw (.3,3) node[anchor=east] {$3$};
\draw (3,.3) node[anchor=north] {$2$};
\draw (.3,4) node[anchor=east] {$4$};
\draw (4,.3) node[anchor=north] {$3$};
\draw [gray] (.9,1) -- (1.1,1);
\draw [gray] (1,.9) -- (1,1.1);
\draw [gray] (.9,2) -- (1.1,2);
\draw [gray] (1,1.9) -- (1,2.1);
\draw [gray] (.9,3) -- (1.1,3);
\draw [gray] (1,2.9) -- (1,3.1);
\draw [gray] (.9,4) -- (1.1,4);
\draw [gray] (1,3.9) -- (1,4.1);
\draw [red] (2,1) node {$\otimes$};
\draw [gray] (1.9,2) -- (2.1,2);
\draw [gray] (2,1.9) -- (2,2.1);
\draw [gray] (1.9,3) -- (2.1,3);
\draw [gray] (2,2.9) -- (2,3.1);
\draw [gray] (1.9,4) -- (2.1,4);
\draw [gray] (2,3.9) -- (2,4.1);
\draw [red] (3,1) node {$\otimes$};
\draw [gray] (2.9,2) -- (3.1,2);
\draw [gray] (3,1.9) -- (3,2.1);
\draw [gray] (2.9,3) -- (3.1,3);
\draw [gray] (3,2.9) -- (3,3.1);
\draw [gray] (2.9,4) -- (3.1,4);
\draw [gray] (3,3.9) -- (3,4.1);
\draw [red] (4,1) node {$\otimes$};
\draw [red] (4,2) node {$\otimes$};
\draw [red] (4,3) node {$\otimes$};
\draw [red] (4,4) node {$\otimes$};
\draw[blue,->] (4,3.8) -- (4,3.2);
\draw[blue,->] (4,2.8) -- (4,2.2);
\draw[blue,->] (4,1.8) -- (4,1.2);
\draw[violet,->] (3.8,1) -- (3.2,1);
\draw[violet,->] (2.8,1) -- (2.2,1);
\draw[violet,->] (1.8,1) -- (1.2,1);
\draw (4.1,4) node[anchor=west] {a};
\draw (4,3.5) node[anchor=west] {b};
\draw (4.1,3) node[anchor=west] {a};
\draw (4,2.5) node[anchor=west] {b};
\draw (4.1,2) node[anchor=west] {a};
\draw (4,1.5) node[anchor=west] {b};
\draw (4.1,1) node[anchor=west] {a};
\draw (3.5,1) node[anchor=south] {c};
\draw (3,1.1) node[anchor=south] {a};
\draw (2.5,1) node[anchor=south] {c};
\draw (2,1.1) node[anchor=south] {a};
\draw (1.5,1) node[anchor=south] {c};
\draw (.5,4.7) node[anchor=east] {$n = N-m$};
\draw (4.7,.5) node[anchor=north] {$m$};
\end{tikzpicture}
\caption{Schematic: absorption of remnant terms II. Each node $+$ corresponds to an estimate of the form \eqref{wnm_sum} (with $n = N-m$). The displayed arrows visualize the absorption mechanism starting with $N = 7$, $m = 3$, and $\alpha^\prime > \frac 1 2$. The symbol $\otimes$ denotes a shift of $\alpha$ by $- \frac 1 2$ (keeping $n$ and $m$ fixed). The remnant at the base of the arrow (forming the last line in \eqref{wnm_sum}) is absorbed by the corresponding estimate \eqref{wnm_sum} at the tip of the arrow under the assumption that \eqref{alpha-12}, \eqref{alpha+12}, and \eqref{alpha+12_alt}, respectively, holds true.}
\label{fig:absorb2}
\end{figure}
The above argumentation shows that we can restrict our considerations to the cases in which $\alpha \in \left(\frac 1 2, 1\right)$, $N = 1$, and $m = 0$ since in the case $\alpha = \frac 1 2$, the remnant term in \eqref{wnm_sum} disappears. The remaining terms can be treated by applying an anisotropic version of Hardy's inequality:
\begin{lemma}\label{lem:anisotropic}
Suppose $v: \, (0,\infty)^2 \to \ensuremath{\mathbb{R}}$ is smooth, $\ell \in \ensuremath{\mathbb{N}}_0$, and $\alpha \in \left(\frac 1 2, 1\right)$. Then
\begin{equation}\label{anisotropic}
\int_0^\infty t^{2 \alpha - 2} \verti{v}_{\ell,\alpha-\frac 1 2\pm \delta}^2 \d t \lesssim \int_0^\infty \verti{\partial_t v}_{\ell,-\frac 1 2\pm \delta}^2 \d t + \int_0^\infty \verti{v}_{\ell+1,\frac 1 2\pm \delta}^2
\end{equation}
\end{lemma}
A proof can be found in \cite[Lem.~7.5]{ggko.2014}. The remnant in \eqref{wnm_sum} for $(N,m) = (1,0)$ is of the form
\[
\int_0^\infty t^{2 \alpha - 2} \verti{w^{(1)}}_{k(1,0,\alpha^\prime)+2,\alpha^\prime-\frac 1 2}^2 \d t
\]
and can be absorbed by estimate~\eqref{wnm_sum} with $\alpha = \frac 1 2$ and $N = 1$, that is,
\begin{equation}\label{aniso_appl}
\int_0^\infty t^{2 \alpha - 2} \verti{w^{(1)}}_{k\left(1,0,\alpha^\prime\right)+2,\alpha^\prime - \frac 1 2}^2 \, \d t \lesssim \int_0^\infty \verti{\partial_t w^{(1)}}_{k,-\frac 1 2 \pm \delta}^2 \, \d t + \int_0^\infty \verti{w^{(1)}}_{k+4,\frac 1 2 \pm \delta}^2 \, \d t,
\end{equation}
where we write $k := k\left(1,0,\frac 1 2 \pm \delta\right)$ (assuming that the values for $+$ and $-$ coincide), provided we have
\begin{equation}\label{alpha_12_1}
k \ge k\left(1,0,\alpha^\prime\right) + 2 \quad \mbox{if } \, \alpha^\prime \in \left(\frac 1 2, 1\right) \, \mbox{ and } \, \alpha \ne \frac 1 2.
\end{equation}
\bigskip
By summing over all estimates~\eqref{wnm_sum} with $(\alpha,N) \in \ensuremath{\mathcal A}$ and $\alpha^\prime = \alpha \pm \delta \in (0,1)$, and fulfilling conditions~\eqref{same_alpha}, \eqref{alpha-12}, \eqref{alpha+12}, \eqref{alpha+12_alt}, and \eqref{alpha_12_1}, we obtain
\begin{equation}\label{mr_star}
\vertiii{u}_* \lesssim \vertiii{u^{(0)}}_{*,0} + \vertiii{f}_{*,1},
\end{equation}
where
\begin{equation}\label{norm_0star}
\vertiii{u^{(0)}}_{*,0}^2 := \verti{w^{(0,0)}}_{k+2,\frac 1 2 - \delta}^2 + \verti{w^{(0,0)}}_{k+2,\frac 1 2 + \delta}^2 \quad \mbox{with } \, w^{(0,0)} := p(D) u^{(0)}
\end{equation}
is the norm for the initial data,
\begin{equation}\label{norm_sol_star}
\begin{aligned}
\vertiii{u}_*^2 := \, & \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \sup_{t \ge 0} t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime) + 2,\alpha^\prime + N-m - \frac 3 2}^2 \\
&+ \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^{m+1} w^{(N-m)}}_{k(N-m,m,\alpha^\prime),\alpha^\prime+N-m-2}^2 \d t \\
&+ \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+4,\alpha^\prime+N-m-1}^2 \d t
\end{aligned}
\end{equation}
is the norm for the solution $u$ (with $w^{(n)}$ defined in \eqref{def_wn}), and
\begin{equation}\label{norm_rhs_star}
\vertiii{f}_{*,1}^2 := \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m r^{(N-m)}}_{k(N-m,m,\alpha^\prime),\alpha^\prime+N-m-1}^2 \d t
\end{equation}
is the norm for the right-hand side $f$ (with $r^{(n)}$ defined in \eqref{def_rn}).
\medskip
Now we make a further assumption, that is, we assume
\begin{equation}\label{alpha_equiv}
\begin{aligned}
& k(n,m,\alpha^\prime) \, \mbox{ is constant for } \, \alpha^\prime \in \left(0, \frac 1 2\right), \, \mbox{ and } \, \alpha^\prime \in \left(\frac 1 2, 1\right) \, \mbox{ respectively}, \\
&\mbox{except for half integers } \, \alpha \in \left\{0,\frac 1 2, 1\right\}.
\end{aligned}
\end{equation}
One may verify that thus still conditions~\eqref{same_alpha}, \eqref{alpha-12}, \eqref{alpha+12}, \eqref{alpha+12_alt}, and \eqref{alpha_12_1} can be satisfied (see below).
\subsubsection{Maximal regularity for the linear equation}
Further applying elliptic maximal regularity given by Proposition~\ref{prop:elliptic}, we infer that the norms $\vertiii{\cdot}_0$ and $\vertiii{\cdot}_{*,0}$, $\vertiii{\cdot}$ and $\vertiii{\cdot}_*$, as well as $\vertiii{\cdot}_1$ and $\vertiii{\cdot}_{*,1}$, respectively, are equivalent, where $\vertiii{\cdot}_0$ is given by \eqref{norm_initial}, i.e.,
\[
\vertiii{u^{(0)}}_0^2 := \verti{u^{(0)}}_{k+6,-\delta}^2 + \verti{u^{(0)} - u^{(0)}_0}_{k+6,\delta}^2,
\]
the norm $\vertiii{\cdot}$ for the solution $u$ is given by
\begin{equation}\label{norm_sol}
\begin{aligned}
\vertiii{u}^2 := \, & \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \sup_{t \ge 0} t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m u - \sum_{i < \alpha^\prime + N-m - \frac 3 2} \frac{\d^m u_i}{\d t^m} x^i}_{\ell(N-m,m,\alpha^\prime) + 2,\alpha^\prime + N-m - \frac 3 2}^2 \\
&+ \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^{m+1} u - \sum_{i < \alpha^\prime + N-m - 2} \frac{\d^{m+1} u_i}{\d t^{m+1}} x^i}_{\ell(N-m,m,\alpha^\prime),\alpha^\prime+N-m-2}^2 \d t \\
&+ \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m u - \sum_{i < \alpha^\prime + N-m - 1} \frac{\d^m u_i}{\d t^m} x^i}_{\ell(N-m,m,\alpha^\prime)+4,\alpha^\prime+N-m-1}^2 \d t
\end{aligned}
\end{equation}
with (cf.~\eqref{def_jnin} and \eqref{def_vwr})
\begin{equation}\label{def_lnm}
\ell(n,m,\alpha^\prime) := k(n,m,\alpha^\prime) + \verti{J_n} + 4 n,
\end{equation}
and the norm $\vertiii{\cdot}_1$ for the right-hand side $f$ reads
\begin{equation}\label{norm_rhs}
\vertiii{f}_1^2 := \sum_{\substack{(\alpha,N) \in \ensuremath{\mathcal A} \\ \alpha^\prime = \alpha \pm \delta \in (0,1)}} \int_0^\infty t^{2 (\alpha+N) - 3} \sum_{m = 0}^{N-1} \verti{\partial_t^m f - \sum_{\beta < i < \alpha^\prime + N-m - 1} \frac{\d^m f_i}{\d t^m} x^i}_{\ell(N-m,m,\alpha^\prime),\alpha^\prime+N-m-1}^2 \d t.
\end{equation}
Consequently, \eqref{mr_star} turns into the maximal-regularity estimate
\begin{equation}\label{mr_main}
\vertiii{u} \lesssim \vertiii{u^{(0)}}_0 + \vertiii{f}_1.
\end{equation}
For convenience, we summarize the conditions on the numbers $\ell(n,m,\alpha^\prime)$: $\ell(n,m,\alpha^\prime)$ is constant for all $\alpha^\prime \in \left(0,\frac 1 2\right)$ and $\alpha^\prime \in \left(\frac 1 2, 1\right)$, respectively, except for $\alpha \in \left\{0,\frac 1 2, 1\right\}$. Furthermore, the following inequalities (through \eqref{def_lnm} equivalent to \eqref{same_alpha}, \eqref{alpha-12}, \eqref{alpha+12}, \eqref{alpha+12_alt}, and \eqref{alpha_12_1}) must hold:
\begin{subequations}\label{linear_cond}
\begin{align}
\ell(N-m-1,m+1,\alpha^\prime) &\ge \ell(N-m,m,\alpha^\prime) - 4 \quad \mbox{for } \, N-m \ge 2,\label{linear_cond1}\\
\ell\left(N-m,m,\alpha^\prime-\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) - 2 \quad \mbox{if } \, \alpha^\prime \in \left(\frac 1 2, 1\right) \, \mbox{ and } \, \alpha + N \ge 2,\label{linear_cond2}\\
\ell\left(N-m-1,m,\alpha^\prime+\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) - 2 \quad \mbox{if } \, \alpha^\prime \in \left(0, \frac 1 2\right), \; N-m \ge 2,\label{linear_cond3}\\
\ell\left(1,m-1,\alpha^\prime+\frac 1 2\right) &\ge \ell\left(1,m,\alpha^\prime\right) + 2 \quad \mbox{if } \, \alpha^\prime \in \left(0, \frac 1 2\right) \, \mbox{ and } \, m \ge 1,\label{linear_cond3_alt}\\
k &\ge \ell\left(1,0,\alpha^\prime\right) - 2 \quad \mbox{if } \, \alpha^\prime \in \left(\frac 1 2, 1\right) \, \mbox{ and } \, \alpha \ne \frac 1 2.\label{linear_cond4}
\end{align}
\end{subequations}
It is apparent that conditions~\eqref{linear_cond} can be fulfilled and non-negativity of $k\left(n,m,\alpha^\prime\right)$ can be ensured (cf.~\eqref{def_lnm}) if we explicitly choose (cf.~\eqref{def_jn} for the definition of $J_n$)
\begin{subequations}\label{explicit}
\begin{align}
\ell\left(n,m,\alpha^\prime\right) &:= 8 N_0 + \verti{J_{N_0}} - 2 \floor{2 \left(n+m+\alpha^\prime\right)} \quad \mbox{for } \, \alpha \notin \left\{0,\frac 1 2,1\right\}, \\
\ell\left(n,m,\alpha^\prime\right) &:= 8 N_0 + \verti{J_{N_0}} + 3 - 4 \left(n+m+\alpha\right) \quad \mbox{for } \, \alpha \in \left\{0,\frac 1 2,1\right\}, \;\, (n,m,\alpha) \ne \left(1,0,\frac 1 2\right), \\
k &:= 8 N_0 + \verti{J_{N_0}} - 5.
\end{align}
\end{subequations}
This choice is also compatible with the ``nonlinear" conditions \eqref{cond_l_non}, which are necessary for the treatment of the nonlinearity $\ensuremath{\mathcal N}(u)$ in Section~\ref{sec:nonlinear}.
\subsection{Properties of the parabolic norms and definition of function spaces\label{ssec:func}}
In this subsection we summarize some of the properties of the parabolic norms $\vertiii{\cdot}$, $\vertiii{\cdot}_0$, and $\vertiii{\cdot}_1$ (cf.~\eqref{norm_initial}, \eqref{norm_sol}, \eqref{norm_rhs}).
\begin{lemma}\label{lem:est_coeff}
For given $N_0 \in \ensuremath{\mathbb{N}}$ and locally integrable $u, f: \, (0,\infty)^2 \to \ensuremath{\mathbb{R}}$ such that the generalized power series \eqref{expansion} exists to order $O(x^{N_0})$, the following estimates (with constants independent of $f$ and $u$) hold true:
\begin{subequations}\label{est_coeff}
\begin{align}
\int_0^\infty t^{2 i + 2 m - 1} \verti{\frac{\d^m u_i}{\d t^m}(t)}^2 \d t &\lesssim \vertiii{u}^2 \quad \mbox{for } \, i \in K_{N_0-m} \setminus \{0\} \, \mbox{ and } \, m \in \ensuremath{\mathbb{N}}_0, \label{est_coeff1}\\
\sup_{t \ge 0} t^{2 i + 2 m} \verti{\frac{\d^m u_i}{\d t^m}(t)}^2 &\lesssim \vertiii{u}^2 \quad \mbox{for } \, i \in K_{N_0-m-\frac 1 2} \, \mbox{ and } \, m \in \ensuremath{\mathbb{N}}_0, \label{est_coeff2}\\
\int_0^\infty t^{2 i + 2 m - 1} \verti{\frac{\d^m f_i}{\d t^m}(t)}^2 \d t &\lesssim \vertiii{f}_1^2 \quad \mbox{for } \, i \in K_{N_0-m} \setminus \{0,\beta\} \, \mbox{ and } \, m \in \ensuremath{\mathbb{N}}_0. \label{est_coeff3}
\end{align}
\end{subequations}
Furthermore, for any locally integrable $u^{(0)}: \, (0,\infty) \to \ensuremath{\mathbb{R}}$ such that $u^{(0)}_0 = \lim_{x \searrow 0} u^{(0)}(x)$ exists, we have $\verti{u^{(0)}_0} \lesssim \vertiii{u^{(0)}}_0$ (where the constant is independent of $u^{(0)}$).
\end{lemma}
\begin{proof}
The estimate for $u^{(0)}$ has already been proven in \cite[Lem.~4.3~(a)]{ggko.2014}. Estimates~\eqref{est_coeff} for the coefficients follow quite elementarily by the same reasoning as in the proof of \cite[Lem.~4.3~(b)]{ggko.2014}:
\medskip
For estimate~\eqref{est_coeff1} we take $(\alpha,N) \in \ensuremath{\mathcal A}$ with $\alpha + N - m - 1 = i$ and $i \notin \ensuremath{\mathbb{N}}_0$ and obtain\footnote{The assumption $i \notin \ensuremath{\mathbb{N}}_0$ is merely for notational simplicity. The reader may verify that the reasoning works in the same way for $i \in \ensuremath{\mathbb{N}}_0$.}:
\begin{align*}
\verti{\frac{\d^m u_i}{\d t^m}}^2 \lesssim \, & \int_{\frac 1 2}^2 \verti{\frac{\d^m u_i}{\d t^m}}^2 \d x \\
\lesssim \, & \int_{\frac 1 2}^2 \left(\verti{\partial_t^m u - \sum_{j < \alpha + N - m - 1 - \delta} \frac{\d^m u_j}{\d t^m} x^j}^2 + \verti{\partial_t^m u - \sum_{j < \alpha + N - m - 1 - \delta} \frac{\d^m u_j}{\d t^m} x^j - \frac{\d^m u_i}{\d t^m} x^i}^2\right) \d x \\
\lesssim \, & \verti{\partial_t^m u - \sum_{j < \alpha + N - m - 1 - \delta} \frac{\d^m u_j}{\d t^m} x^j}_{\ell(N-m,m,\alpha^\prime)+4,\alpha+N-m-1-\delta}^2 \\
&+ \verti{\partial_t^m u - \sum_{j < \alpha + N - m - 1 + \delta} \frac{\d^m u_j}{\d t^m} x^j}_{\ell(N-m,m,\alpha^\prime)+4,\alpha+N-m-1+\delta}^2
\end{align*}
Multiplying with the time weight $t^{2 (\alpha+N)-3}$ and integrating in time, we obtain \eqref{est_coeff1} (cf.~\eqref{norm_sol}). For proving estimate~\eqref{est_coeff2}, we take $(\alpha,N) \in \ensuremath{\mathcal A}$ with $\alpha + N - m - \frac 3 2 = i$ and the same reasoning as above (taking the $\sup$ in time instead of integrating) leads to estimate~\eqref{est_coeff2}. The proof of \eqref{est_coeff3} is the same as for \eqref{est_coeff1}.
\end{proof}
Estimates~\eqref{est_coeff} will turn out to be relevant for estimating the nonlinearity. However, they are also convenient in order to define appropriate spaces for our solution and the right-hand side. As in the case of standard Sobolev spaces there are two possible approaches:
\medskip
On the one hand, one may define for a locally integrable function $u: (0,\infty)^2 \to \ensuremath{\mathbb{R}}$ its distributional derivatives and take the infimum over all possible coefficients $u_i : (0,\infty) \to \ensuremath{\mathbb{R}}$ in the definition of the norm $\vertiii{u}$ in \eqref{norm_sol}. It is apparent from \eqref{norm_sol} that if $\vertiii{u} < \infty$, the coefficients $u_i$ are uniquely defined almost everywhere. This corresponds to the definition of the $W$-scale in the standard theory of Sobolev spaces.
\medskip
On the other hand, it was shown in \cite[Lem.~B.4]{ggko.2014} for $N_0 = 1$ that one can approximate any locally integrable $u: (0,\infty)^2 \to \ensuremath{\mathbb{R}}$ with $\vertiii{u} < \infty$ by a sequence $\left(u^{(\nu)}: (0,\infty)^2 \to \ensuremath{\mathbb{R}}\right)_{\nu \in \ensuremath{\mathbb{N}}}$ of more regular functions $u^{(\nu)}$ fulfilling:
\begin{itemize}
\item[(a)] $u^{(\nu)} \in C^\infty((0,\infty)^2) \cap C^0_0([0,\infty)^2)$;
\item[(b)] for every $t \in [0,\infty)$ we have $u^{(\nu)}(t,x) = u^{(\nu)}_0(t) + u^{(\nu)}_\beta(t) x^\beta$ for $x \ll_\nu 1$, where $u^{(\nu)}_0, u^{(\nu)}_{\beta}: (0,\infty) \to \ensuremath{\mathbb{R}}$ are smooth;
\item[(c)] $\vertiii{u-u^{(\nu)}} \to 0$ as $\nu \to \infty$.
\end{itemize}
Taking the closure of all $u: (0,\infty)^2 \to \ensuremath{\mathbb{R}}$ with (a), (b), and (c) with respect to $\vertiii{\cdot}$ (with $N_0 = 1$), we end up with the analogue of what one commonly refers to as the $H$-scale of Sobolev spaces. Lemma~B.4 of reference \cite{ggko.2014} then states that for $N_0 = 1$ indeed $H = W$ holds\footnote{This is also true for the norm $\vertiii{\cdot}_0$ as shown in \cite[Lem.~B.3]{ggko.2014}.}.
\medskip
Unlike in \cite{ggko.2014}, where it turned out to be more convenient to rely on the $W$-approach of Sobolev spaces, we will employ the $H$-approach in what follows:
\begin{definition}\label{def:spaces}
Suppose $N_0 \in \ensuremath{\mathbb{N}}$ and $\delta > 0$ is chosen sufficiently small. Furthermore suppose that conditions~\eqref{linear_cond} are fulfilled (cf.~\eqref{explicit} for a specific choice).
\begin{itemize}
\item[(a)] We define the space $U_0$ of initial data $u^{(0)}$ as the closure of all $u^{(0)} \in C^\infty((0,\infty)) \cap C^0_0([0,\infty))$ with $u^{(0)}(x) = u^{(0)}_0 + u^{(0)}_\beta x^\beta$ for $x \ll 1$ with respect to $\vertiii{\cdot}_0$ (cf.~\eqref{norm_initial}).
\item[(b)] The solution space $U$ is defined as the closure with respect to $\vertiii{\cdot}$ (cf.~\eqref{norm_sol}) of all $u \in C^\infty\left((0,\infty)^2\right) \cap C^0_0\left([0,\infty)^2\right)$ such that $u(t,x) = \sum_{i \in K_{N_0}} u_i(t) x^i$ for $x \ll 1$, where the $u_i: (0,\infty) \to \ensuremath{\mathbb{R}}$ are smooth functions of time.
\item[(c)] The space $F$ of right-hand sides $f$ is defined as the closure with respect to $\vertiii{\cdot}_1$ (cf.~\eqref{norm_rhs}) of all $f \in C^\infty\left((0,\infty)^2\right) \cap C^0_0\left([0,\infty)^2\right)$ such that $f(t,x) = \sum_{i \in K_{N_0} \setminus \{0,\beta\}} f_i(t) x^i$ for $x \ll 1$, where the $f_i: (0,\infty) \to \ensuremath{\mathbb{R}}$ are smooth.
\end{itemize}
\end{definition}
We mark that in view of Lemma~\ref{lem:est_coeff} the coefficients $u_i$, $u_0^{(0)}$, and $f_i$ of the generalized power series are also defined (at least almost everywhere in time $t$) for functions $u$, $u^{(0)}$, and $f$, respectively, for which $\vertiii{u}$, $\vertiii{u^{(0)}}_0$, and $\vertiii{f}_1$, respectively, is finite.
\medskip
Next we also provide another estimate that in particular guarantees control of the norm $\sup_{t \ge 0} \vertii{u}$, where
\begin{equation}\label{def_c0}
\vertii{u} := \sup_{x > 0} \verti{u(x)}
\end{equation}
denotes the $\infty$-norm in space $x$. By approximation this also implies continuity of $u$ and derivatives if $u \in U$.
\begin{lemma}\label{lem:c0control}
Suppose that $N_0 \in \ensuremath{\mathbb{N}}$, $m \in \{0,\cdots,N_0-1\}$, and $\ell \in \left\{0,\cdots,\ell\left(1,m,\frac 1 2\pm\delta\right)+1\right\}$. Then we have
\begin{equation}\label{est_c0norm}
\sup_{t \ge 0} t^{2 m} \vertii{\partial_t^m D^\ell u}^2 + \sup_{t \ge 0} t^{2 m} \vertii{\partial_t^m D^\ell (u-u_0)}^2 \lesssim \vertiii{u}^2 \quad \mbox{for all } \, u \in U,
\end{equation}
where the constant in the estimate is independent of $u$.
\end{lemma}
\begin{proof}
A proof for an analogous estimate is contained in \cite[est.~(8.5)]{ggko.2014} so that we only sketch the arguments here. First we may show that, passing to the logarithmic variable $s := \ln x$ and using a cut-off argument in combination with the standard embedding $H^1(\ensuremath{\mathbb{R}}) \hookrightarrow C^0(\ensuremath{\mathbb{R}})$, the following estimates hold
\begin{equation}\label{est_interval}
\vertii{v-v_0}_{(0,1]} \lesssim \verti{v-v_0}_{1,\delta} \quad \mbox{and} \quad \vertii{v}_{[1,\infty)} \lesssim \verti{v}_{1,-\delta} \quad \mbox{for any locally integrable } \, v,
\end{equation}
where the constants are independent of $\delta$ and $\vertii{v}_A := \sup_{x \in A} \verti{v(x)}$ for any set $A \subset (0,\infty)$. estimate~\eqref{est_interval} in combination with Lemma~\ref{lem:est_coeff} shows
\begin{eqnarray*}
\sup_{t \ge 0} t^{2 m} \vertii{\partial_t^m D^\ell (u-u_0)}^2 &\lesssim& \sup_{t \ge 0} t^{2 m} \vertii{\partial_t^m D^\ell u - \delta_{\ell,0} \frac{\d^m u_0}{\d t^m}}_{(0,1]}^2 + \sup_{t \ge 0} t^{2 m} \verti{\frac{\d^m u_0}{\d t^m}}^2 \\
&& + \sup_{t \ge 0} t^{2 m} \vertii{\partial_t^m D^\ell u}_{[1,\infty)}^2 \\
&\stackrel{\eqref{est_coeff2}, \,\eqref{est_interval}}{\lesssim}& \sup_{t \ge 0} t^{2 m} \left(\verti{\partial_t^m u - \frac{\d^m u_0}{\d t^m}}_{\ell+1,\delta}^2 + \verti{\partial_t^m u}_{\ell+1,-\delta}^2\right) + \vertiii{u}^2\\
&\stackrel{\eqref{norm_sol}}{\lesssim}& \vertiii{u}^2.
\end{eqnarray*}
By the triangle inequality and again using estimate~\eqref{est_coeff2}, we obtain \eqref{est_c0norm}.
\end{proof}
\subsection{Rigorous treatment of the linear equation\label{ssec:rigorous}}
In this section we prove our main result for the linear equation:
\begin{proposition}\label{prop:main_lin}
Suppose $N_0 \in \ensuremath{\mathbb{N}}$ and $\delta > 0$ is chosen sufficiently small. Furthermore, suppose that conditions~\eqref{linear_cond} are fulfilled (cf.~\eqref{explicit} for explicitly chosen indices). Then for any $f \in F$ and $u^{(0)} \in U_0$ there exists exactly one solution $u = S\left[u^{(0)},f\right] \in U$ of the linear degenerate-parabolic problem \eqref{lin_cauchy}. This solution fulfills the maximal-regularity estimate \eqref{mr_main}.
\end{proposition}
\begin{proof}[of Proposition~\ref{prop:main_lin}]
The statement is the generalization of \cite[Prop.~7.6]{ggko.2014} for $N_0 = 1$ to arbitrary $N_0 \in \ensuremath{\mathbb{N}}$. Since our Banach spaces $U$ and $F$ are nested for increasing $N_0$, uniqueness is already clear and it remains to prove existence. As Proposition~\ref{prop:main_lin} for $N_0 = 1$ is already proven, there exists a unique solution $u$ of \eqref{lin_cauchy} lying in the space $U$ for $N_0 = 1$. By approximation (cf.~Definition~\ref{def:spaces}), we may without loss of generality assume that $f \in C^\infty\left((0,\infty)^2\right) \cap C_0^0\left([0,\infty)\right)$ and $u^{(0)} \in C^\infty\left((0,\infty)\right) \cap C_0^0\left([0,\infty)\right)$, with the expansions $u^{(0)}(x) = u^{(0)}_0 + u^{(0)}_\beta x^\beta$ for $x \ll 1$, and $f(t,x) = \sum_{i \in K_{N_0} \setminus \{0,\beta\}} f_i(t) x^i$ for $x \ll 1$, where the $f_i: (0,\infty) \to \ensuremath{\mathbb{R}}$ are smooth functions of time. By standard parabolic theory in the bulk and using \cite[Prop.~7.6]{ggko.2014} also $u \in C^\infty\left((0,\infty)^2\right) \cap C^0_0\left([0,\infty)^2\right)$ with $u(t,x) = u_0(t) + u_\beta(t) x^\beta + o\left(x^\beta\right)$ as $x \searrow 0$.
\subsubsection*{A qualitative argument for regularity}
Here we argue why the solution indeed has additional spatial regularity. As we have observed before, this requires higher regularity in time. At the basis of our reasoning is the existence and uniqueness result for the linear problem \eqref{lin_cauchy} given by \cite[Prop.~7.6]{ggko.2014}. In particular we have finiteness of
\[
\int_0^\infty t^{2 \beta - 1} \verti{\partial_t w^{(1)}}_{k(1,0,\beta\pm\delta),\beta - 1 \pm \delta}^2 \d t \stackrel{\eqref{def_wn}}{\sim} \int_0^\infty t^{2 \beta - 1} \verti{\partial_t u}_{\ell(1,0,\beta\pm\delta),\beta - 1 \pm \delta}^2 \d t
\]
and thus we may use finiteness of the norms $\verti{\partial_t u}_{\ell(1,0,\beta\pm\delta),\beta - 1 \pm \delta}^2$ for some time $t = \tau > 0$ to solve
\[
x \partial_t (\partial_t u) + p(D) (\partial_t u) = \partial_t f \quad \mbox{for } \, t > \tau \, \mbox{ and } \, x > 0
\]
and infer that $\partial_t u$ has additional regularity for times $t > \tau$. In principle such a reasoning is possible but as it was noted already in \cite[Sec.~2]{ggko.2014}, the arguments there\footnote{We refer to the discussion in \cite[Sec.~2, Eqs.~(2.8)--(2.12)]{ggko.2014} and the (rigorous) time discretization in \cite[Prop.~7.6]{ggko.2014}. Utilizing the linear equations \eqref{lin_v1} and \eqref{lin_v2} (where $v^{(1)} \stackrel{\eqref{def_vwr}}{=} w^{(1)}$ and the difference between $v^{(2)}$ and $w^{(2)}$ is immaterial due to the choice of the weight $1 + \delta < 2 \beta$), the integrals in \eqref{prt_w12} are finite through control of the spatial integrals $\int_0^\infty t \left(\verti{v^{(1)}}_{k(1,0,1-\delta)+4,1-\delta}^2 + \verti{g^{(1)}}_{k(1,0,1-\delta),1-\delta}^2\right) \d t$ and $\int_0^\infty t \left(\verti{w^{(2)}}_{k(2,0,\delta)+4,1+\delta}^2 + \verti{r^{(2)}}_{k(2,0,\delta),1+\delta}^2\right) \d t$, respectively.} were also suitable to obtain finiteness of the integrals
\begin{subequations}\label{prt_w12}
\begin{equation}
\int_0^\infty t \verti{\partial_t w^{(1)}}_{k(1,0,1-\delta),-\delta}^2 \d t \sim \int_0^\infty t \verti{\partial_t u}_{\ell(1,0,1-\delta),-\delta}^2 \d t
\end{equation}
and
\begin{equation}
\int_0^\infty t \verti{\partial_t w^{(2)}}_{k(2,0,\delta),\delta}^2 \d t \sim \int_0^\infty t \verti{\partial_t u - \frac{\d u_0}{\d t}}_{\ell(2,0,\delta),\delta}^2 \d t.
\end{equation}
\end{subequations}
Now we may use finiteness of the norms $\verti{\partial_t u}_{\ell(1,0,1-\delta),-\delta}$ and $\verti{\partial_t u - \frac{\d u_0}{\d t}}_{\ell(2,0,\delta),\delta}$ for some time $\tau_\nu > 0$ with $\tau_\nu \searrow 0$ as $\nu \to \infty$. Applying \cite[Prop.~7.6]{ggko.2014} implies that also $\vertiii{\partial_t u(\cdot+\tau_\nu)}$ for $N_0 = 1$ (and with appropriately chosen derivative numbers) is finite\footnote{Notice that the solution of \cite[Prop.~7.6]{ggko.2014} coincides with $\partial_t u$, because uniqueness under the assumption $\int_{\tau_\nu}^\infty \verti{\partial_t u}_{\ell\left(1,0,1-\delta\right),- \delta} \, \d t < \infty$ holds true as can be seen in the uniqueness proof for \cite[Prop.~7.6]{ggko.2014}.}. In fact, since the arguments in \cite[Sec.~7]{ggko.2014} were also applicable for general weights in the interval $\left(\frac 1 2, 1\right)$, we also obtain
\[
\sup_{t \ge \tau_\nu} (t - \tau_\nu)^{2 \alpha - 1} \verti{\partial_t w^{(1)}}_{k(1,1,\alpha^\prime),\alpha^\prime-1}^2 \d t < \infty
\]
and
\[
\int_{\tau_\nu}^\infty \left(t - \tau_\nu\right)^{2 \alpha - 1} \left(\verti{\partial_t^2 w^{(1)}}_{k(1,1,\alpha^\prime),\alpha^\prime-1}^2 + \verti{\partial_t w^{(1)}}_{k(1,1,\alpha^\prime),\alpha^\prime}^2\right) \d t < \infty
\]
for all admissible weights $\alpha$. In particular the integral
\[
\int_{\tau_\nu}^\infty \left(t - \tau_\nu\right) \verti{\partial_t^2 w^{(1)}}_{k(1,1,1-\delta),-\delta}^2 \d t \sim \int_{\tau_\nu}^\infty \left(t - \tau_\nu\right) \verti{\partial_t^2 u}_{\ell(1,1,1-\delta),-\delta}^2 \d t
\]
is finite and we may argue as above that
\[
\int_{\tau_\nu}^\infty \left(t - \tau_\nu\right) \verti{\partial_t^2 w^{(2)}}_{k(2,1,\delta),\delta}^2 \d t \sim \int_{\tau_\nu}^\infty \left(t - \tau_\nu\right) \verti{\partial_t^2 u - \frac{\d^2 u_0}{\d t^2}}_{\ell(2,1,\delta),\delta}^2 \d t
\]
is finite. Hence $\verti{\partial_t^2 u}_{\ell(1,1,1-\delta),-\delta} < \infty$ and $\verti{\partial_t^2 u - \frac{\d^2 u_0}{\d t^2}}_{\ell(2,1,\delta),\delta} < \infty$ for some $t = \tilde \tau_\nu > \tau_\nu$, where $\tilde \tau_\nu \searrow 0$ as $\nu \to \infty$. Apparently the reasoning can be boot strapped and (taking the limit $\nu \to \infty$) we obtain
\begin{subequations}
\begin{equation}
\sup_{t \ge \tau} \verti{\partial_t^m w^{(1)}}_{k(1,m,\alpha^\prime)+2,\alpha^\prime-\frac 1 2}^2 < \infty, \quad \int_{\tau}^\infty \verti{\partial_t^{m+1} w^{(1)}}_{k(1,m,\alpha^\prime),\alpha^\prime-1}^2 \d t< \infty,
\end{equation}
and
\begin{equation}\label{dtmw1_fin}
\int_{\tau}^\infty \verti{\partial_t^m w^{(1)}}_{k(1,m,\alpha^\prime)+4,\alpha^\prime}^2 \d t < \infty
\end{equation}
\end{subequations}
for all $\tau > 0$, all $m = 0, \cdots, N_0-1$, and all admissible $\alpha$.
\medskip
In order to obtain additional spatial regularity, we first observe
\[
\int_\tau^\infty \verti{\partial_t^{m+1} w^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime}^2 \d t \stackrel{\eqref{same_alpha}}{\lesssim} \int_\tau^\infty \verti{\partial_t^{m+1} w^{(1)}}_{k\left(1,m+1,\alpha^\prime\right)+4,\alpha^\prime}^2 \d t < \infty \quad \mbox{for all } \, \tau > 0.
\]
Then we may use
\[
\partial_t^m v^{(3)} \stackrel{\eqref{def_vwr}}{=} p(D-2) \partial_t^m w^{(2)} \stackrel{\eqref{lin_wnm}}{=} \partial_t^m r^{(2)} + x q_2(D) \partial_t^{m+1} v^{(2)} - x \partial_t^{m+1} w^{(2)} \quad \mbox{for } \, t, x > 0
\]
and therefore
\begin{equation}\label{est_iter}
\begin{aligned}
\verti{\partial_t^m v^{(3)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1} &\lesssim \verti{\partial_t^m r^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime + 1} + \verti{q_2(D) \partial_t^{m+1} v^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime} + \verti{\partial_t^{m+1} w^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime} \nonumber\\
&\hspace{-18pt}\stackrel{\eqref{def_vwr}, \eqref{same_alpha}}{\lesssim} \verti{\partial_t^m r^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime + 1} + \verti{\partial_t^{m+1} w^{(1)}}_{k(1,m+1,\alpha^\prime)+4,\alpha^\prime}.
\end{aligned}
\end{equation}
Since by construction $\int_\tau^\infty \verti{\partial_t^m r^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime + 1}^2 \d t < \infty$ (cf.~\eqref{norm_rhs_star}) and by \eqref{dtmw1_fin}, we also have
\[
\int_\tau^\infty \verti{\partial_t^m v^{(3)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1}^2 \d t < \infty \quad \mbox{for all } \, \tau > 0.
\]
Our aim is to show
\begin{equation}\label{w2_est_v3}
\verti{\partial_t^m w^{(2)}}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} \lesssim \verti{\partial_t^m v^{(3)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1} \quad \mbox{for } \, t > 0.
\end{equation}
This follows by applying elliptic regularity of $p(D-2)$ (Proposition~\ref{prop:elliptic}), but care has to be taken as the decay assumptions \eqref{ell_assume} have to be satisfied: As in the proof of \cite[Lem.~B.3]{ggko.2014} we may approximate $\partial_t^m v^{(3)}$ by a sequence of functions $\psi^{(\nu)} \in C^\infty_0((0,\infty))$ in the norm $\verti{\cdot}_{k(2,m,\alpha^\prime),\alpha^\prime+1}$. Solving $p(D-2) \omega^{(\nu)} = \psi^{(\nu)}$ as in \cite[Lem.~B.3]{ggko.2014}, we find a smooth solution $\omega^{(\nu)}$ such that $D^\ell \omega^{(\nu)} = O(x^2)$ as $x \searrow 0$ and $D^\ell \omega^{(\nu)} = O\left(x^{\frac 3 2 - \beta}\right)$ as $x \nearrow \infty$ for all $\ell \ge 0$. By Proposition~\ref{prop:elliptic}, $\verti{\omega^{(\nu)} - \omega^{\left(\nu^\prime\right)}}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} \lesssim \verti{\psi^{(\nu)} - \psi^{\left(\nu^\prime\right)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1} \to 0$ as $\nu, \nu^\prime \to \infty$ and hence $\verti{\omega^{(\nu)} - \omega}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} \to 0$ as $\nu \to \infty$ for some locally integrable $\omega$. Since $\verti{\omega^{(\nu)}}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} \lesssim \verti{\psi^{(\nu)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1}$, in particular $\verti{\omega}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} \lesssim \verti{\partial_t^m v^{(3)}}_{k(2,m,\alpha^\prime),\alpha^\prime+1}$ for $t > 0$ by continuity of the norms. Hence, for establishing \eqref{w2_est_v3} it remains to prove
\begin{equation}\label{om_eq_w}
\omega = \partial_t^m w^{(2)} \quad \mbox{almost everywhere.}
\end{equation}
Note that
\[
p(D-2) \left(\omega - \partial_t^m w^{(2)}\right) = p(D-2) \lim_{\nu \to \infty} \omega^{(\nu)} - \partial_t^m v^{(3)} = \lim_{\nu \to \infty} \psi^{(\nu)} - \partial_t^m v^{(3)} = 0
\]
almost everywhere and hence
\begin{equation}\label{spanned}
\omega - \partial_t^m w^{(2)} \in \ker p(D-2) = \ensuremath{{\rm span}} \left\{x^2, x^{\beta+2},x^{\frac 3 2-\beta}, x^{\frac 1 2}\right\}.
\end{equation}
Because of $\verti{\partial_t^m w^{(2)}}_{1-\delta} \lesssim \verti{\partial_t^m w^{(1)}}_{4+\verti{I_2},1-\delta} < \infty$, necessarily $\partial_t^m w^{(2)} = o\left(x^{1-\delta}\right)$ as $x \searrow 0$ and $x \nearrow \infty$ for $t > 0$. Together with $\verti{\omega}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1} < \infty$ therefore
\[
\omega - \partial_t^m w^{(2)} = o\left(x^{1-\delta}\right) \quad \mbox{as } \, x \searrow 0 \quad \mbox{and} \quad \omega - \partial_t^m w^{(2)} = o\left(x^{2-\delta}\right) \quad \mbox{as } \, x \nearrow \infty
\]
almost everywhere in $t > 0$. In view of \eqref{spanned} this implies \eqref{om_eq_w}.
\medskip
Finiteness of $\sup_{t \ge \tau} \verti{\partial_t^m w^{(2)}}_{k(2,m,\alpha^\prime)+2,\alpha^\prime+\frac 1 2}^2$ for $\tau > 0$ can be obtained by the standard trace estimate
\[
\sup_{t \ge \tau} \verti{\partial_t^m w^{(2)}}_{k(2,m,\alpha^\prime)+2,\alpha^\prime+\frac 1 2}^2 \lesssim \int_\tau^\infty \left(\verti{\partial_t^{m+1} w^{(2)}}_{k(2,m,\alpha^\prime),\alpha^\prime}^2 + \verti{\partial_t^m w^{(2)}}_{k(2,m,\alpha^\prime)+4,\alpha^\prime+1}^2\right) \d t.
\]
A proof is contained e.g.~in \cite[Lem.~B.2]{ggko.2014}.
\medskip
Apparently the argument can be boot strapped and we obtain
\begin{subequations}\label{finite}
\begin{align}
&\sup_{t \ge \tau} \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+2,\alpha^\prime-N-m-\frac 3 2}^2 < \infty,\\
&\int_\tau^\infty \verti{\partial_t^{m+1} w^{(N-m)}}_{k(N-m,m,\alpha^\prime),\alpha^\prime+N-m-2}^2 \d t < \infty,
\end{align}
and
\begin{equation}
\int_\tau^\infty \verti{\partial_t^m w^{(N-m)}}_{k(N-m,m,\alpha^\prime)+4,\alpha^\prime+N-m-1}^2 \d t < \infty
\end{equation}
\end{subequations}
for all $\tau > 0$, all $N = 1, \cdots, N_0$, $m = 0, \cdots, N-1$, and all admissible $\alpha$.
\subsubsection*{Proof of estimate~\eqref{mr_star}}
Now we consider equation~\eqref{lin_wnm}, where $r^{(n)}$, $v^{(n)}$, and $w^{(n)}$ -- defined through \eqref{def_vwr} -- are smooth. We are aiming at deriving estimate~\eqref{mr_wnm}. Therefore it is more convenient to use the logarithmic variable $s := \ln x$, for which equation~\eqref{lin_wnm} reads
\begin{equation}\label{eq_wnm_s}
e^s \partial_t^{m+1} w^{(n)} + p(\partial_s - n) \partial_t^m w^{(n)} = \partial_t^m r^{(n)} + e^s q_n(\partial_s) \partial_t^{m+1} v^{(n)} \quad \mbox{for $t > 0$ and $s \in \ensuremath{\mathbb{R}}$.}
\end{equation}
In order to simplify the notation, we write
\begin{equation}\label{notation}
\omega := \partial_t^m w^{(n)}, \quad P(\zeta) := p(\zeta - n), \quad \mbox{and} \quad \varphi := \partial_t^m r^{(n)} + e^s q_n(\partial_s) \partial_t^{m+1} v^{(n)},
\end{equation}
that is, equation~\eqref{eq_wnm_s} can be rephrased as
\begin{equation}\label{eq_om_phi}
e^s \partial_t \omega + P(\partial_s) \omega = \varphi \quad \mbox{for $t > 0$ and $s \in \ensuremath{\mathbb{R}}$.}
\end{equation}
We take a cut off $\eta \in C^\infty(\ensuremath{\mathbb{R}})$ with $\eta \ge 0$, $\eta(s) \equiv 1$ for $\verti{s} \le 1$, and $\eta(s) \equiv 0$ for $\verti{s} \ge 2$. Let $\eta_R(s) := \eta(s/R)$ and test equation~\eqref{eq_om_phi} with $e^{- 2 \mu s} \eta_R^2 w^{(n)}$ in $L^2(\ensuremath{\mathbb{R}}_s)$, where $\mu$ is in the coercivity range of $P(\zeta)$ (We specifically choose $\mu := \alpha^\prime+n-1$ later on.). Thus we arrive at
\[
\frac 1 2 \frac{\d}{\d t} \int_\ensuremath{\mathbb{R}} e^{- 2 \left(\mu - \frac 1 2\right) s} \eta_R^2 \omega^2 \d s + \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} \eta_R^2 \omega P(\partial_s) \omega \, \d s = \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} \eta_R^2 \varphi \omega \, \d s.
\]
Now we commute one factor $\eta_R$ with the differential operator $P(\partial_s)$. Since in the commutator of $\eta_R$ and $P(\partial_s)$ at least one derivative acts on $\eta_R$, we obtain through integration by parts
\begin{align*}
& \frac 1 2 \frac{\d}{\d t} \int_\ensuremath{\mathbb{R}} e^{- 2 \left(\mu - \frac 1 2\right) s} (\eta_R \omega)^2 \d s + \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} (\eta_R \omega) P(\partial_s) (\eta_R \omega) \, \d s \\
& \le \frac{1}{2 \ensuremath{\varepsilon}} \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} (\eta_R \varphi)^2 \, \d s + \frac{\ensuremath{\varepsilon}}{2} \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} (\eta_R \omega)^2 \, \d s + \frac C R \int_{-2R}^{2R} e^{- 2 \mu s} \left(\omega^2 + (\partial_s\omega)^2 + (\partial_s^2\omega)^2\right) \d s,
\end{align*}
where $C > 0$ is independent of $R$ and $\ensuremath{\varepsilon} > 0$ is arbitrary\footnote{All estimates and constants in this part only depend on $\kappa$ and $\mu$.}. As a next step, we use coercivity of $P(\partial_s)$ and obtain
\begin{equation}\label{basic}
\begin{aligned}
& \frac{\d}{\d t} \int_\ensuremath{\mathbb{R}} e^{- 2 \left(\mu - \frac 1 2\right) s} (\eta_R \omega)^2 \d s + \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} \left((\eta_R \omega)^2 + (\partial_s \eta_R \omega)^2 + (\partial_s^2 \eta_R \omega)^2\right) \, \d s\\
&\quad \lesssim \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} (\eta_R \varphi)^2 \, \d s + \frac C R \int_{-2R}^{2R} e^{- 2 \mu s} \left(\omega^2 + (\partial_s\omega)^2 + (\partial_s^2\omega)^2\right) \d s.
\end{aligned}
\end{equation}
Estimate~\eqref{basic} is a basic estimate that has to be combined with a higher-order estimate to arrive at \eqref{mr_wnm}. Therefore, we again consider equation~\eqref{eq_om_phi} and apply the operator $e^{- 2 \mu s} (\partial_s-1)^{\kappa+2} \eta_R$ to it (where $\kappa = k(n,m,\alpha^\prime)$). Testing with $\partial_s^{\kappa+2} (\eta_R \omega)$ in $L^2(\ensuremath{\mathbb{R}}_s)$, we obtain after integrating by parts and using a standard interpolation estimate
\begin{equation}\label{est_higher}
\begin{aligned}
&\frac{\d}{\d t} \int_\ensuremath{\mathbb{R}} e^{-2 \left(\mu - \frac 1 2\right) s} \left(\partial_s^{\kappa+2}(\eta_R \omega)\right)^2 \d s + \int_\ensuremath{\mathbb{R}} e^{-2 \mu s} \left(\partial_s^{\kappa+4} (\eta_R \omega)\right)^2 \d s \\
&\lesssim \ensuremath{\varepsilon}^{-1} \sum_{j = 0}^\kappa \int_\ensuremath{\mathbb{R}} e^{- 2 \mu s} \verti{\partial_s^j \left(\eta_R \varphi\right)}^2 \, \d s + \ensuremath{\varepsilon} \sum_{j = 0}^{\kappa+4} \int_\ensuremath{\mathbb{R}} e^{-2 \mu s} \left(\partial_s^j \left(\eta_R \omega\right)\right)^2 \d s + \frac 1 R \int_{-2 R}^{2 R} e^{-2 \mu s} \left(\omega^2 + (\partial_s^{\kappa+4} \omega)^2\right) \d s,
\end{aligned}
\end{equation}
where $\ensuremath{\varepsilon} > 0$ is arbitrary. By interpolation and increasing $C$, the second term in the second line can be absorbed. Using \eqref{basic}, \eqref{est_higher}, and interpolating once more, we obtain, after undoing the transformation $s = \ln x$,
\[
\frac{\d}{\d t} \verti{\eta_R \omega}_{\kappa+2,\mu-\frac 1 2}^2 + \verti{\eta_R \omega}_{\kappa+4,\mu}^2 \lesssim \verti{\eta_R \varphi}_{\kappa,\mu}^2 + \frac 1 R \verti{\eta_{2 R} \omega}_{\kappa+4,\mu}^2.
\]
Multiplying with $t^{2 \sigma}$, where $\sigma := \alpha + n + m - \frac 3 2$ and integrating in time (assuming $\tau^\prime > \tau > 0$), we obtain
\begin{eqnarray}\nonumber
\lefteqn{\sup_{t \in \left[\tau, \tau^\prime\right]} t^{2 \sigma} \verti{\eta_R \omega}_{\kappa+2,\mu-\frac 1 2}^2 + \int_{\tau}^{\tau^\prime} t^{2 \sigma} \verti{\eta_R \omega}_{\kappa+4,\mu}^2 \d t} \\
&\lesssim& \tau^{2 \sigma} \verti{\eta_R \omega_{|t=\tau}}_{\kappa+2,\mu-\frac 1 2}^2 + \int_{\tau}^{\tau^\prime} t^{2 \sigma} \verti{\eta_R \varphi}_{\kappa,\mu}^2 \d t + 2 \sigma \int_{\tau}^{\tau^\prime} t^{2 \sigma - 1} \verti{\eta_R \omega}_{\kappa+2,\mu-\frac 1 2}^2 \d t \nonumber\\
&&+ \frac 1 R \int_{\tau}^{\tau^\prime} t^{2 \sigma} \verti{\eta_{2 R} \omega}_{\kappa+4,\mu}^2 \d t. \label{est_s_var}
\end{eqnarray}
Due to \eqref{finite} and \eqref{notation} all terms appearing in \eqref{est_s_var} remain finite in the limit $R \to \infty$ (note that $\verti{\eta_{2 R} \omega}_{\kappa+4,\mu} \lesssim \verti{\partial_t^m w^{(n)}}_{k(n,m,\alpha^\prime)+4,\alpha^\prime+n-1}$ for $t > 0$) so that we obtain
\begin{align*}
& \sup_{t \in \left[\tau, \tau^\prime\right]} t^{2 \sigma} \verti{\omega}_{\kappa+2,\mu-\frac 1 2}^2 + \int_{\tau}^{\tau^\prime} t^{2 \sigma} \verti{\omega}_{\kappa+4,\mu}^2 \d t \\
& \quad \lesssim \tau^{2 \sigma} \verti{\omega_{|t=\tau}}_{\kappa+2,\mu-\frac 1 2}^2 + \int_{\tau}^{\tau^\prime} t^{2 \sigma} \verti{\varphi}_{\kappa,\mu}^2 \d t + 2 \sigma \int_{\tau}^{\tau^\prime} t^{2 \sigma - 1} \verti{\omega}_{\kappa+2,\mu-\frac 1 2}^2 \d t.
\end{align*}
Undoing the notational change \eqref{notation} and increasing the set $\left[\tau,\tau^\prime\right]$, we end up with \eqref{mr_wnm}, provided that for $\alpha + n + m > \frac 3 2$
\begin{equation}\label{conv_init}
\tau^{2 \sigma} \verti{\omega_{|t=\tau}}_{\kappa+2,\mu-\frac 1 2}^2 = \tau^{2 (\alpha+n+m)-3} \verti{\partial_t^m w^{(n)}_{|t=\tau}}_{k\left(n,m,\alpha^\prime\right)+2,\alpha^\prime+n-\frac 3 2}^2 \to 0 \quad \mbox{as } \, \tau \searrow 0
\end{equation}
holds true at least for a subsequence. This is already clear for $(n,m) = (1,0)$ and $\alpha^\prime \in \left(\frac 1 2, 1\right)$ from the extension of the discretization argument in \cite[Sec.~7]{ggko.2014}. From \eqref{mr_wnm} for $(n,m) = (1,0)$, we get
\begin{equation}\label{lim1}
\int_0^\infty t^{2 \alpha-1} \left(\verti{\partial_t w^{(1)}}_{k\left(1,0,\alpha^\prime\right),\alpha^\prime-1}^2 + \verti{w^{(1)}}_{k\left(1,0,\alpha^\prime\right)+4,\alpha^\prime}^2\right) \, \d t < \infty \quad \mbox{for } \, \alpha^\prime \in \left(\frac 1 2, 1\right).
\end{equation}
Utilizing
\[
\int_0^\infty t^{2 \alpha} \verti{w^{(2)}}_{k\left(2,0,\alpha^\prime\right)+2,\alpha^\prime+\frac 1 2}^2 \, \d t \stackrel{\eqref{alpha+12}}{\lesssim} \int_0^\infty t^{2 \alpha} \verti{w^{(1)}}_{k\left(1,0,\alpha^\prime + \frac 1 2\right)+4,\alpha^\prime+\frac 1 2}^2 \, \d t < \infty \quad \mbox{for } \, \alpha^\prime \in \left(0,\frac 1 2\right),
\]
we conclude that \eqref{conv_init} holds for $(n,m) = (2,0)$ and $\alpha^\prime \in \left(0,\frac 1 2\right)$, too. Additionally, we have
\[
\int_0^\infty t^{2 \alpha} \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime\right)+2,\alpha^\prime-\frac 1 2}^2 \, \d t \stackrel{\eqref{alpha+12_alt}}{\lesssim} \int_0^\infty t^{2 \left(\alpha + \frac 1 2\right)-1} \verti{\partial_t w^{(1)}}_{k\left(1,0,\alpha^\prime+\frac 1 2\right), \left(\alpha^\prime+\frac 1 2\right) - 1}^2 \, \d t \stackrel{\eqref{lim1}}{<} \infty
\]
for $\alpha^\prime \in \left(0,\frac 1 2\right)$, that is, \eqref{conv_init} also holds for $(n,m) = (1,1)$ and $\alpha^\prime \in \left(0,\frac 1 2\right)$.
\medskip
Next, \eqref{mr_wnm} yields
\begin{equation}\label{lim2}
\int_0^\infty t^{2 \alpha+1} \left(\verti{\partial_t^2 w^{(1)}}_{k\left(1,1,\alpha^\prime\right),\alpha^\prime-1}^2 + \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime\right)+4,\alpha^\prime}^2\right) \, \d t < \infty \quad \mbox{for } \, \alpha^\prime \in \left(0,\frac 1 2\right).
\end{equation}
As
\[
\int_0^\infty t^{2 \alpha + 1} \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime\right)+2,\alpha^\prime-\frac 1 2}^2 \, \d t \stackrel{\eqref{alpha-12}}{\lesssim} \int_0^\infty t^{2 \left(\alpha - \frac 1 2\right) + 2} \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime-\frac 1 2\right)+4,\alpha^\prime - \frac 1 2}^2 \, \d t \stackrel{\eqref{lim2}}{<} \infty
\]
for $\alpha^\prime \in \left(\frac 1 2, 1\right)$, \eqref{conv_init} is true for $(n,m) = (1,1)$ and $\alpha^\prime \in \left(\frac 1 2,1\right)$ and \eqref{mr_wnm} yields
\begin{equation}\label{lim3}
\int_0^\infty t^{2 \alpha+1} \left(\verti{\partial_t^2 w^{(1)}}_{k\left(1,1,\alpha^\prime\right),\alpha^\prime-1}^2 + \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime\right)+4,\alpha^\prime}^2\right) \, \d t < \infty \quad \mbox{for } \, \alpha^\prime \in \left(\frac 1 2,1\right).
\end{equation}
Since we also have
\[
\int_0^\infty t^{2 \alpha + 1} \verti{\partial_t v^{(2)}}_{k\left(2,0,\alpha^\prime\right) + \verti{I_2}-1,\alpha^\prime}^2 \, \d t \stackrel{\eqref{same_alpha}}{\lesssim} \int_0^\infty t^{2 \alpha + 1} \verti{\partial_t w^{(1)}}_{k\left(1,1,\alpha^\prime\right)+4,\alpha^\prime}^2 \, \d t \stackrel{\eqref{lim2}}{<} \infty
\]
for $\alpha^\prime \in \left(0,\frac 1 2\right)$, we get from \eqref{mr_wnm}
\begin{equation}\label{lim4}
\int_0^\infty t^{2 \alpha+1} \left(\verti{\partial_t w^{(2)}}_{k\left(2,0,\alpha^\prime\right),\alpha^\prime}^2 + \verti{w^{(2)}}_{k\left(2,0,\alpha^\prime\right)+4,\alpha^\prime}^2\right) \, \d t < \infty \quad \mbox{for } \, \alpha^\prime \in \left(0,\frac 1 2\right).
\end{equation}
This enables us to estimate
\[
\int_0^\infty t^{2 \alpha} \verti{w^{(2)}}_{k\left(2,0,\alpha^\prime\right)+2,\alpha^\prime+\frac 12}^2 \, \d t \stackrel{\eqref{alpha-12}}{\lesssim} \int_0^\infty t^{2 \left(\alpha - \frac 1 2\right) + 1} \verti{w^{(2)}}_{k\left(2,0,\alpha^\prime-\frac 1 2\right)+4,\left(\alpha^\prime-\frac 12\right)+1}^2 \, \d t \stackrel{\eqref{lim4}}{<} \infty
\]
for $\alpha^\prime \in \left(\frac 1 2, 1\right)$, thus proving \eqref{conv_init} with $(n,m) = (2,0)$ and $\alpha^\prime \in \left(\frac 1 2,1\right)$.
\medskip
Induction on $n+m$ verifies \eqref{conv_init} for all $\alpha+n+m > \frac 3 2$.
\medskip
Now, starting from \eqref{mr_wnm}, we follow the reasoning of Section~\ref{ssec:heur_par} and arrive at estimate~\eqref{mr_star}.
\subsubsection*{Proof of estimate~\eqref{mr_main}}
To obtain the desired maximal-regularity estimate \eqref{mr_main}, we only need to be able to apply Proposition~\ref{prop:elliptic}, that is, we need to verify \eqref{ell_assume}. We exemplify the reasoning by treating the last line of \eqref{norm_sol_star}. For a fixed value $\left(\alpha,N\right) \in \ensuremath{\mathcal A}$ it suffices to show
\begin{equation}\label{dellu}
D^\ell \partial_t^m u = \sum_{i < \varrho} \frac{\d^m u_i}{\d t^m} i^\ell x^i + o\left(x^\varrho\right) \quad \mbox{as } \, x \searrow 0,
\end{equation}
where $\varrho := \alpha^\prime+N-m-1$, $\alpha^\prime = \alpha \pm \delta \in (0,1)$, and $\ell = 0,\cdots, k\left(N-m,m,\alpha^\prime\right)+4$.
\medskip
Due to $\vertiii{u}_* < \infty$, we have $D^\ell \partial_t^m w^{(N-m)}(x) = o(x^\varrho)$ as $x \searrow 0$. This implies $D^\ell \partial_t^m u(x) = D^\ell \pi^{(m)}(x) + D^\ell \tilde u(x)$, where $D^\ell \tilde u(x) = o(x^\varrho)$ as $x \searrow 0$ and $\pi^{(m)}(x)$ is a solution of the homogeneous equation $P(D) \pi^{(m)} = 0$ with $P(\zeta) := \left(\prod_{k = 0}^{N-m-1} p(\zeta-k)\right) \left(\prod_{i \in J_{N-m}} (\zeta-i)\right)$ (cf.~\eqref{def_wn}). Hence $\pi^{(m)} = \sum_j \frac{\d^m u_j}{\d t^m} x^j$, where the $u_j$ are smooth functions of time $t$ and $j$ is a zero of the polynomial $P(\zeta)$. In fact, the set of admissible exponents $j$ is smaller: Since we know that Proposition~\ref{prop:main_lin} also holds for $N_0 = 1$, we need to have $\pi^{(m)}(x) = \frac{\d^m u_0}{\d t^m} + \frac{\d^m u_\beta}{\d t^m} x^\beta + o(x^\beta)$ as $x \searrow 0$. Hence $j \in \{0,\beta\}$ or $\beta < j < N_0$.
\medskip
Now suppose that $\beta < j < N_0$ and $j \notin K_{N_0}$. Since $x^j$ does not appear in the expansion of $f$, we obtain from the linear equation \eqref{lin_pde} $\frac{\d u_{j-1}}{\d t} + p(j) u_j \equiv 0$, that is, if $\frac{\d^m u_j}{\d t^m} \not\equiv 0$ necessarily $\frac{\d^m u_{j-1}}{\d t^m} \not\equiv 0$. This implies $j - \floor{j} \in \{0,\beta\}$ (as otherwise negative $j$ would appear) thus contradicting the assumption $j \notin K_{N_0}$ (cf.~\eqref{def_kn}).
\medskip
Hence indeed \eqref{dellu} holds true and Proposition~\ref{prop:elliptic} yields estimate~\eqref{mr_main}.
\end{proof}
\section{The nonlinear problem\label{sec:nonlinear}}
\subsection{The main results\label{ssec:nonmain}}
In this section we prove our main theorem. For the treatment of the nonlinear problem \eqref{cauchy}, we need additional assumptions on the numbers $\ell(n,m,\alpha)$ which we already state here:
\begin{subequations}\label{cond_l_non}
\begin{align}
\ell\left(N-m,m^\prime,\alpha^\prime\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) \quad \mbox{for } \, m^\prime \le m-1,\label{cond_l_non1} \\
\ell\left(1,m^\prime,\frac 1 2\pm\delta\right) &\ge \frac{\ell\left(N-m,m,\alpha^\prime\right)}{2} + 1 \quad \mbox{where } \begin{cases} 0 \le m^\prime \le m, \\ \alpha^\prime+N-m-1 < \beta, \end{cases} \label{cond_l_non20}\\
\ell\left(1,m^\prime,\frac 1 2\pm\delta\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) + 3 \quad \mbox{where } \, \begin{cases} 0 \le m^\prime \le m, \\ \alpha^\prime+N-m-1 > \beta,\end{cases} \label{cond_l_non2}\\
\ell\left(N-m,m,\alpha^\prime-\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) + 2 \quad \mbox{for all } \alpha^\prime \in \left(\frac 1 2, 1\right), \label{cond_l_non3}\\
\ell\left(N-m-1,m,\alpha^\prime+\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) + 2 \quad \mbox{for all } \alpha^\prime\in \left(0, \frac 1 2\right), \label{cond_l_non4} \\
k &\ge \ell\left(1,m,\alpha^\prime\right)-2 \quad \mbox{for all } \, \alpha^\prime > \beta \, \mbox{ and } \, m \ge 0, \label{cond_l_non5} \\
\begin{split} \ell\left(1,0,1-\delta\right) &\ge \ell\left(1,m,\alpha^\prime\right)+2, \\ \ell\left(2,0,\delta\right) &\ge \ell\left(1,m,\alpha^\prime\right)+2 \end{split} \Bigg\} \quad \mbox{for all } \, \alpha^\prime > \beta \, \mbox{ and } \, m \ge 1. \label{cond_l_non6}
\end{align}
\end{subequations}
For the explicit choice \eqref{explicit}, we are indeed able to fulfill the ``linear" conditions \eqref{linear_cond} and the ``nonlinear" conditions \eqref{cond_l_non}. The relevance of conditions~\eqref{cond_l_non} will become clear in the proof of the main estimate for the nonlinearity (cf.~Proposition~\ref{prop:nonlinear_est}). The aim of this section is to show the following statement:
\begin{theorem}\label{th:regularity}
Suppose $N_0 \in \ensuremath{\mathbb{N}}_0$ and further suppose that conditions~\eqref{linear_cond} and \eqref{cond_l_non} are fulfilled (cf.~\eqref{explicit} for an explicit choice). Then there exist $\ensuremath{\varepsilon} > 0$ and $\delta > 0$ such that for any $u^{(0)} \in U_0$ with $\vertiii{u^{(0)}}_0 \le \ensuremath{\varepsilon}$ (cf.~\eqref{norm_initial}) problem~\eqref{cauchy} has a unique solution $u \in U$. This solution obeys the \emph{a-priori estimate} $\vertiii{u} \lesssim \vertiii{u^{(0)}}_0$ (cf.~\eqref{norm_sol}) with a constant independent of $u^{(0)}$.
\end{theorem}
It is quite apparent that Theorem~\ref{th:regularity} (with $N_0$ replaced by $N_0 + 1$) yields Theorem~\ref{th:main} (the regularity and decay properties of $u_i = u_i(t)$ and $R_{N_0} = R_{N_0}(x,t)$ in \eqref{expansion} follow from Lemma~\ref{lem:est_coeff} and the definition of $\vertiii{\cdot}$, cf.~\eqref{norm_sol}, by approximation with smooth functions as given in Definition~\ref{def:spaces}). Theorem~\ref{th:regularity} in turn has already been proven for $N_0 = 1$ in \cite[Th.~3.1]{ggko.2014} so that in particular uniqueness holds true. The fact that existence also holds in a smaller space follows by applying the following proposition:
\begin{proposition}\label{prop:nonlinear_est}
Suppose that $N_0 \in \ensuremath{\mathbb{N}}$ and $\delta > 0$ is chosen sufficiently small. Furthermore suppose that conditions~\eqref{linear_cond}~and~\eqref{cond_l_non} are fulfilled (cf.~\eqref{explicit} for an explicit choice). Then the following estimate for the nonlinearity $\ensuremath{\mathcal N}(u)$ (defined in \eqref{nonlinearity} and \eqref{5linear}) holds true:
\begin{equation}\label{non_main}
\vertiii{\ensuremath{\mathcal N}(u)}_1 \lesssim \max_{m = 2, 5} \vertiii{u}^m \quad \mbox{for any } \, u \in U,
\end{equation}
where the constant in the estimate is independent of $u$.
\end{proposition}
Before proving Proposition~\ref{prop:nonlinear_est}, we show how it can be used to prove Theorem~\ref{th:regularity}:
\begin{proof}[of Theorem~\ref{th:regularity}]
In Proposition~\ref{prop:main_lin} we have constructed a solution operator $S: \, U_0 \times F \to U$ to the linear problem \eqref{lin_cauchy}, so that the nonlinear problem \eqref{cauchy} is equivalent to
\begin{equation}\label{fixed}
u = S\left[u^{(0)},\ensuremath{\mathcal N}(u)\right].
\end{equation}
This fixed-point problem can be solved by applying the contraction-mapping theorem, which for $N_0 = 1$ has indeed been carried out in the proof of \cite[Th.~3.1]{ggko.2014} under the assumption of small data $u^{(0)}$. This implies that for $\vertiii{u^{(0)}}_0 \ll 1$ the sequence $\left(u^{(\nu)}\right)_{\nu \in \ensuremath{\mathbb{N}}}$ defined through
\begin{equation}\label{def_sequence}
u^{(1)} := S[u^{(0)},0] \quad \mbox{and} \quad u^{(\nu+1)} := S\left[u^{(0)},\ensuremath{\mathcal N}\left(u^{(\nu)}\right)\right] \quad \mbox{for } \, \nu \ge 1
\end{equation}
converges in $U$ for $N_0 = 1$ to the unique solution $u$. In particular $u^{(\nu)}$ converges point-wise to $u$ (cf.~Lemma~\ref{lem:c0control}). For arbitrary $N_0 \ge 1$ we note that since $u^{(\nu)} \in U$, applying the maximal-regularity estimate \eqref{mr_main} (cf.~Proposition~\ref{prop:main_lin}) and the nonlinear estimate \eqref{non_main} (cf.~Proposition~\ref{prop:nonlinear_est}), we obtain
\begin{equation}\label{est_sequence_1}
\vertiii{u^{(1)}} \le C \vertiii{u^{(0)}}_0 \quad \mbox{and} \quad \vertiii{u^{(\nu+1)}} \le C \left(\vertiii{u^{(0)}}_0 + \max_{m = 2,5} \vertiii{u^{(\nu)}}^m\right) \quad \mbox{for } \, \nu \ge 1,
\end{equation}
where $C > 0$ is independent of $u^{(\nu)}$. Assuming $\ensuremath{\varepsilon} \le (4 C^2)^{-1}$ and $2 C \ge 1$, estimates~\eqref{est_sequence_1} upgrade to
\begin{equation}\label{est_sequence_2}
\vertiii{u^{(1)}} \le \frac{1}{4 C} \quad \mbox{and} \quad \vertiii{u^{(\nu+1)}} \le \frac{1}{4 C} + C \max_{m = 2,5} \vertiii{u^{(\nu)}}^m \quad \mbox{for } \, \nu \ge 1.
\end{equation}
Inequalities~\eqref{est_sequence_2} in turn imply
\begin{equation}\label{est_sequence_3}
\vertiii{u^{(\nu)}} \le \frac{1}{2 C} \quad \mbox{for all } \, \nu \ge 1.
\end{equation}
By weak-$*$-compactness, $u^{(\nu)}$ has a subsequence that converges in the weak-$*$-topology of $U$ to the unique limit $u$ and therefore $u \in U$. By weak-$*$ lower semi-continuity of the norm $\vertiii{\cdot}$, estimate~\eqref{est_sequence_3} implies
\begin{equation}\label{est_lim_u}
\vertiii{u} \le \frac{1}{2 C}.
\end{equation}
Applying the maximal-regularity estimate \eqref{mr_main} (cf.~Proposition~\ref{prop:main_lin}) and the nonlinear estimate \eqref{non_main} (cf.~Proposition~\ref{prop:nonlinear_est}) once more to the fixed-point equation \eqref{fixed}, we obtain
\[
\vertiii{u} \le C \left(\vertiii{u^{(0)}}_0 + \max_{m = 2,5} \vertiii{u}^m\right),
\]
which, in view of \eqref{est_lim_u} and since $2 C \ge 1$, upgrades to $\vertiii{u} \lesssim \vertiii{u^{(0)}}_0$.
\end{proof}
\subsection{Nonlinear estimates\label{ssec:nonest}}
We conclude the paper with estimating the nonlinear part $\ensuremath{\mathcal N}(u)$ of equation~\eqref{lin_pde}:
\begin{proof}[of Proposition~\ref{prop:nonlinear_est}]
We repeat some of the observations on the formal structure of the nonlinearity $\ensuremath{\mathcal N}(u)$ that are contained in \cite[Sec.~8]{ggko.2014}:
\subsubsection*{The formal structure of the nonlinearity}
We recall that the nonlinearity $\ensuremath{\mathcal N}(u)$ can be written as
\[
\ensuremath{\mathcal N}(u) \stackrel{\eqref{nonlinearity}}{=} p(D) u - \ensuremath{\mathcal M}_\mathrm{sym}(1+u,\cdots,1+u),
\]
where $p(D) u = 5 \ensuremath{\mathcal M}_\mathrm{sym}(1+u,1,\cdots,1)$ and $\ensuremath{\mathcal M}_\mathrm{sym}$ is the symmetrization of $\ensuremath{\mathcal M}$. Since we know\footnote{The identity \eqref{vanish_m_1} holds, as the single factor $D$ appears once, because the traveling wave $u_\mathrm{TW} \equiv 0$ is a solution of the nonlinear problem \eqref{cauchy}.}
\begin{equation}\label{vanish_m_1}
\ensuremath{\mathcal M}_\mathrm{sym}(1,\cdots,1) \stackrel{\eqref{5linear}}{=} 0,
\end{equation}
we can conclude that by multi-linearity $\ensuremath{\mathcal N}(u)$ is a linear combination of terms of the form
\begin{equation}\label{struct_nu}
\ensuremath{\mathcal M}_\mathrm{sym}\left(u,u,\omega^{(3)}, \omega^{(4)}, \omega^{(5)}\right) \quad \mbox{with } \, \omega^{(3)}, \omega^{(4)}, \omega^{(5)} \in \{1,u\}.
\end{equation}
As $u$ can be decomposed into $u = (u-u_0) + u_0$, once more appealing to \eqref{vanish_m_1} and using multi-linearity, we infer from \eqref{struct_nu} that $\ensuremath{\mathcal N}(u)$ is a linear combination (with constant coefficients) of terms of the form
\begin{equation}\label{struct_nu_2}
\ensuremath{\mathcal M}_\mathrm{sym}\left(u-u_0,\omega^{(2)}, \cdots, \omega^{(5)}\right) \quad \mbox{with } \, \omega^{(2)} \in \{u_0,u-u_0\}, \; \omega^{(3)},\omega^{(4)}, \omega^{(5)} \in \{1,u_0,u-u_0\}.
\end{equation}
For the first entry of $\ensuremath{\mathcal M}_\mathrm{sym}$ in \eqref{struct_nu_2} we may further use $u-u_0 = (u-u_0-u_\beta x^\beta) + u_\beta x^\beta$ and the fact that $\ensuremath{\mathcal M}_\mathrm{sym}(x^\beta,1,\cdots,1) = \frac 1 5 p(D) x^\beta \stackrel{\eqref{poly}}{=} 0$. Thus $\ensuremath{\mathcal N}(u)$ is a linear combination (with constant coefficients) of terms of the form
\begin{subequations}\label{struct_nu_3}
\begin{equation}\label{struct_nu_31}
\ensuremath{\mathcal M}_\mathrm{sym}\left(u-u_0-u_\beta x^\beta,\omega^{(2)}, \cdots, \omega^{(5)}\right) \quad \mbox{with } \, \omega^{(2)} \in \{u_0,u-u_0\}, \; \omega^{(3)},\omega^{(4)}, \omega^{(5)} \in \{1,u_0,u-u_0\}.
\end{equation}
or
\begin{equation}\label{struct_nu_32}
\ensuremath{\mathcal M}_\mathrm{sym}\left(u_\beta x^\beta,u-u_0, \omega^{(3)}, \omega^{(4)}, \omega^{(5)}\right) \quad \mbox{with } \, \omega^{(3)},\omega^{(4)}, \omega^{(5)} \in \{1,u_0,u-u_0\}.
\end{equation}
\end{subequations}
\subsubsection*{The norms}
In view of the definition of the norm $\vertiii{\cdot}_1$ in \eqref{norm_rhs}, it suffices to estimate the expression
\begin{equation}\label{rhs_term}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\partial_t^m \ensuremath{\mathcal N}(u) - \sum_{\beta < i < \alpha^\prime+N-m-1} \frac{\d^m (\ensuremath{\mathcal N}(u))_i}{\d t^m} x^i}_{\ell,\alpha^\prime+N-m-1}^2 \d t,
\end{equation}
where $\ell = \ell(N-m,m,\alpha^\prime)$ and as in \eqref{expansion}
\[
\ensuremath{\mathcal N}(u) = \sum_{i \in K_{N_0}} \left(\ensuremath{\mathcal N}(u)\right)_i x^i + O\left(x^{N_0}\right) \quad \mbox{as } \, x \searrow 0 \, \mbox{ and } \, t > 0,
\]
for each $(\alpha,N) \in \ensuremath{\mathcal A}$ with $\alpha^\prime = \alpha \pm \delta \in (0,1)$ and $m \in \{0,\cdots, N-1\}$ individually. We distinguish between sub- and super-critical terms, that is, terms for which either $\alpha^\prime+N-m-1 < \beta$ or $\alpha^\prime+N-m-1 > \beta$.
\subsubsection*{Estimating the sub-critical terms}
In this part of the proof, we concentrate on the \emph{sub-critical} terms, that is, the terms for which $\alpha^\prime+N-m-1 < \beta$. Then the sum $\sum_{\beta < i < \alpha^\prime+N-m-1} \frac{\d^m (\ensuremath{\mathcal N}(u))_i}{\d t^m} x^i$ is identically zero. It is convenient to use the decomposition \eqref{struct_nu_2} for the nonlinearity $\ensuremath{\mathcal N}(u)$, so that by distributing the $D$-derivatives and applying the triangle inequality, it suffices to estimate
\[
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\partial_t^m \left(\omega^{(1)}\cdots\omega^{(5)}\right)}_{\alpha^\prime+N-m-1}^2 \d t,
\]
where
\begin{align*}
\omega^{(1)} &= D^{\ell_1} (u-u_0) \quad \mbox{with } \, 0 \le \ell_1 \le \ell+4,\\
\omega^{(2)} &\in \left\{u_0, D^{\ell_2} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_2 \le \frac{\ell+4}{2},\\
\omega^{(j)} &\in \left\{1, u_0, D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{3} \, \mbox{ for } \, j \ge 3.
\end{align*}
Now we distribute the time derivatives on the individual factors, so that it suffices to estimate
\[
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}\cdots\omega^{(5)}}_{\alpha^\prime+N-m-1}^2 \d t,
\]
with new factors $\omega^{(j)}$ obeying
\begin{align*}
\omega^{(1)} &= \partial_t^{m_1} D^{\ell_1} (u-u_0) \quad \mbox{with } \, 0 \le \ell_1 \le \ell+4 \, \mbox{ and } \, 0 \le m_1 \le N-1,\\
\omega^{(2)} &\in \left\{\frac{\d^{m_2} u_0}{\d t^{m_2}}, \partial_t^{m_2} D^{\ell_2} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_2 \le \frac{\ell+4}{2} \, \mbox{ and } \, 0 \le m_2 \le N-1,\\
\omega^{(j)} &\in \left\{\delta_{m_j,0}, \frac{\d^{m_j} u_0}{\d t^{m_j}}, \partial_t^{m_j} D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{3} \, \mbox{ and } \, 0 \le m_j \le N-1 \, \mbox{ for } \, j \ge 3,
\end{align*}
where $\sum_{j = 1}^5 m_j = m$. This enables us to obtain the following bound:
\begin{equation}\label{est_sub_1}
\begin{aligned}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}\cdots\omega^{(5)}}_{\alpha^\prime+N-m-1}^2 \d t \lesssim \, & \int_0^\infty t^{2 (\alpha+N-m+m_1)-3} \verti{\omega^{(1)}}_{\alpha^\prime+N-m-1}^2 \d t \\
&\times \prod_{j = 2}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2,
\end{aligned}
\end{equation}
where $\vertii{v} := \sup_{x < 0} \verti{v(x)}$. Then we note that
\begin{align*}
\int_0^\infty t^{2 (\alpha+N-m+m_1)-3} \verti{\omega^{(1)}}_{\alpha^\prime+N-m-1}^2 \d t &\lesssim \int_0^\infty t^{2 (\alpha+N-m+m_1)-3} \verti{\partial_t^{m_1} u}_{\ell\left(N-m,m_1,\alpha^\prime\right)+4,\alpha^\prime+N-m-1}^2 \d t \\
&\le \vertiii{u}^2
\end{align*}
for the first term on the right-hand side of \eqref{est_sub_1}, provided that condition~\eqref{cond_l_non1} is fulfilled.
\medskip
We now turn our attention to the terms $\omega^{(j)}$ with $j \ge 2$ in \eqref{est_sub_1}. If $\omega^{(j)} = \frac{\d^{m_j} u_0}{\d t^{m_j}}$ we have $\sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2 \lesssim \vertiii{u}^2$ by estimate~\eqref{est_coeff2} of Lemma~\ref{lem:est_coeff}. Finally, if $\omega^{(j)} = \partial_t^{m_j} D^{\ell_j} (u-u_0)$, Lemma~\ref{lem:c0control} yields $\sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2 \lesssim \vertiii{u}^2$, provided that condition~\eqref{cond_l_non20} is fulfilled.
\medskip
In summary we can estimate the term in \eqref{rhs_term} in the sub-critical case $\alpha^\prime+N-m-1 < \beta$ as follows:
\begin{equation}\label{main_subcrit}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\partial_t^m \ensuremath{\mathcal N}(u) - \sum_{\beta < i < \alpha^\prime+N-m-1} \frac{\d^m (\ensuremath{\mathcal N}(u))_i}{\d t^m} x^i}_{\ell,\alpha^\prime+N-m-1}^2 \d t \lesssim \vertiii{u}^4 \times \left(1 + \vertiii{u}^6\right),
\end{equation}
where $\ell = \ell(N-m,m,\alpha^\prime)$.
\subsubsection*{Estimating the super-critical terms}
We now aim at estimating terms of the form \eqref{rhs_term} with $\ell = \ell(N-m,m,\alpha^\prime)$ for each $(\alpha,N) \in \ensuremath{\mathcal A}$ with $\alpha^\prime = \alpha \pm \delta \in (0,1)$ and $m \in \{0,\cdots, N-1\}$, where we assume $\alpha^\prime+N-m-1 > \beta$ (the ``super-critical" terms). Here we use the decomposition of $\ensuremath{\mathcal N}(u)$ in the form \eqref{struct_nu_3}, that is, the super-critical terms \eqref{rhs_term} can be estimated by a sum of terms of the form
\begin{equation}\label{omegas}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\partial_t^m \left(\omega^{(1)}\cdots\omega^{(5)}\right) - \sum_{i < \alpha^\prime+N-m-1} \frac{\d^m}{\d t^m}\left(\omega^{(1)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1}^2 \d t,
\end{equation}
where one of the following two situations occurs:
\begin{itemize}
\item[(a)] The $\omega^{(j)}$ obey (cf.~\eqref{struct_nu_31})
\begin{align*}
\omega^{(1)} &= D^{\ell_1} (u - u_0 - u_\beta x^\beta) \quad \mbox{with } \, 0 \le \ell_1 \le \ell+4,\\
\omega^{(2)} &\in \left\{u_0, D^{\ell_2} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_2 \le \ell+4,\\
\omega^{(j)} &\in \left\{1, u_0, D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{2} \, \mbox{ for } \, j \ge 3.
\end{align*}
\item[(b)] The $\omega^{(j)}$ obey (cf.~\eqref{struct_nu_32})
\begin{align*}
\omega^{(1)} &= u_\beta x^\beta,\\
\omega^{(2)} &= D^{\ell_2} (u-u_0) \quad \mbox{with } \, 0 \le \ell_2 \le \ell+4,\\
\omega^{(j)} &\in \left\{1, u_0, D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{2} \, \mbox{ for } \, j \ge 3.
\end{align*}
\end{itemize}
\subsubsection*{Estimating \eqref{omegas} for terms of the form {\rm (a)}}
As a first step, we distribute the time derivatives on the individual factors $\omega^{(j)}$ so that terms of the form \eqref{omegas} fulfilling (a) can be estimated by a sum of terms of the form
\begin{equation}\label{omegas_a}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1} \left(\omega^{(1)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1}^2 \d t,
\end{equation}
where the $\omega^{(j)}$ are given by
\begin{align*}
\omega^{(1)} &= \partial_t^{m_1} D^{\ell_1} (u - u_0 - u_\beta x^\beta) \quad \mbox{with } \, 0 \le \ell_1 \le \ell+4 \, \mbox{ and } \, 0 \le m_1 \le m,\\
\omega^{(2)} &\in \left\{\frac{\d^{m_2} u_0}{\d t^{m_2}}, \partial_t^{m_2} D^{\ell_2} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_2 \le \ell+4 \, \mbox{ and } \, 0 \le m_2 \le m,\\
\omega^{(j)} &\in \left\{\delta_{m_j,0}, \frac{\d^{m_j} u_0}{\d t^{m_j}}, \partial_t^{m_j} D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{2} \, \mbox{ and } \, 0 \le m_j \le m \, \mbox{ for } \, j \ge 3,
\end{align*}
with $\sum_{j = 1}^5 m_j = m$. Since $\beta > \frac 1 2$ we have $\omega_i^{(1)} \equiv 0$ for $i < 1$ (cf.~\eqref{def_kn} for the definition of admissible exponents $i$). Hence we may decompose $\omega^{(1)}$ according to
\[
\omega^{(1)} = \left(\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}\right) + \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}
\]
and instead of terms of the form \eqref{omegas_a}, we may estimate terms of the structure
\begin{subequations}\label{terms_a12}
\begin{equation}\label{terms_a1}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\left(\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}\right) \omega^{(2)} \cdots\omega^{(5)}}_{\alpha^\prime+N-m-1}^2 \d t,
\end{equation}
and
\begin{equation}\label{terms_a2}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}_{i_1}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-i_1} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-i_1}^2 \d t,
\end{equation}
\end{subequations}
respectively, where in the latter case $1 \le i_1 < \alpha^\prime+N-m-1$. Furthermore, the term \eqref{terms_a2} identically vanishes if $\omega^{(j)} \in \left\{\delta_{m_j,0}, \frac{\d^{m_j} u_0}{\d t^{m_j}}\right\}$ for all $j \ge 2$, so that we may assume $\omega^{(2)} = \partial_t^{m_2} D^{\ell_2} (u-u_0)$ in this case.
\medskip
We begin by estimating terms of the form \eqref{terms_a1} through
\begin{align*}
& \int_0^\infty t^{2 (\alpha+N)-3} \verti{\left(\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}\right) \omega^{(2)} \cdots\omega^{(5)}}_{\alpha^\prime+N-m-1}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 (\alpha+N-m+m_1)-3} \verti{\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}}_{\alpha^\prime+N-m-1}^2 \d t \times \prod_{j = 2}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2.
\end{align*}
Then we note that
\begin{align*}
& \int_0^\infty t^{2 (\alpha+N-m+m_1)-3} \verti{\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}}_{\alpha^\prime+N-m-1}^2 \d t \\
& \quad \lesssim t^{2 (\alpha+N-m+m_1)-3} \verti{\partial_t^{m_1} u - \sum_{i_1 < \alpha^\prime+N-m-1} \frac{\d^{m_1} u_{i_1}}{\d x^{m_1}} x^{i_1}}_{\ell+4,\alpha^\prime+N-m-1}^2 \d t \lesssim \vertiii{u}^2,
\end{align*}
provided that the indices meet condition~\eqref{cond_l_non1}. By Lemma~\ref{lem:est_coeff} and Lemma~\ref{lem:c0control}, respectively, we can further bound $\sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2 \lesssim \vertiii{u}^2$ for $j \ge 2$ if $\omega^{(j)} \not\equiv 1$ and we fulfill condition~\eqref{cond_l_non2}.
\medskip
Altogether, terms of the form \eqref{terms_a1} can be bound as follows:
\begin{equation}\label{super_1}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\left(\omega^{(1)} - \sum_{1 \le i_1 < \alpha^\prime+N-m-1} \omega^{(1)}_{i_1} x^{i_1}\right) \omega^{(2)} \cdots\omega^{(5)}}_{\alpha^\prime+N-m-1}^2 \d t \lesssim \vertiii{u}^4 \times \left(1 + \vertiii{u}^6\right).
\end{equation}
\medskip
For estimating the term \eqref{terms_a2}, we use the $L^2$-bound on $\omega_{i_1}^{(1)}$, i.e.,
\begin{equation}\label{fc_1}
\begin{aligned}
& \int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}_{i_1}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-i_1} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-i_1}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 m_1 + 2 i_1 - 1} \verti{\omega^{(1)}_{i_1}}^2 \d t \\
& \qquad \times \sup_{t \ge 0} t^{2 (\alpha+N-m_1-i_1)-2} \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-i_1} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-i_1}^2.
\end{aligned}
\end{equation}
Then we observe that by estimate~\eqref{est_coeff1} of Lemma~\ref{lem:est_coeff} we have for the second line in \eqref{fc_1}
\[
\int_0^\infty t^{2 m_1 + 2 i_1 - 1} \verti{\omega^{(1)}_{i_1}}^2 \d t = \int_0^\infty t^{2 m_1 + 2 i_1 - 1} \verti{\frac{\d^{m_1} u_{i_1}}{\d t^{m_1}}}^2 \d t \lesssim \vertiii{u}^2
\]
due to $i_1 + m_1 < N \le N_0$. Furthermore, we may use the decomposition
\[
\omega^{(2)} = \left(\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1} \omega^{(2)}_{i_2} x^{i_2}\right) + \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1} \omega^{(2)}_{i_2} x^{i_2}
\]
and therefore the third line in \eqref{fc_1} reduces to estimating
\begin{subequations}\label{fc_3}
\begin{equation}\label{fc_31}
\sup_{t \ge 0} t^{2 (\alpha+N-m+m_2-i_1)-2} \verti{\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1} \omega^{(2)}_{i_2} x^{i_2}}_{\alpha^\prime+N-m-1-i_1}^2 \times \prod_{j = 3}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2
\end{equation}
and
\begin{equation}\label{fc_32}
\begin{aligned}
& \sup_{t \ge 0} t^{2 m_2 + 2 i_2} \verti{\omega^{(2)}_{i_2}}^2 \\
& \quad \times \sup_{t \ge 0} t^{2 (\alpha+N-m_1-m_2-i_1-i_2)-2} \verti{\omega^{(3)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-i_1-i_2} \left(\omega^{(3)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-i_1-i_2}^2,
\end{aligned}
\end{equation}
\end{subequations}
where the term \eqref{fc_32} identically vanishes unless $\omega^{(j)} = \partial_t^{m_j} D^{\ell_j} (u-u_0)$ for at least one $j \ge 3$. We will assume without loss of generality $\omega^{(3)} = \partial_t^{m_3} D^{\ell_3} (u-u_0)$ in this case. Then we can decompose $\omega^{(3)}$ into
\[
\omega^{(3)} = \left(\omega^{(3)} - \sum_{\beta \le i_3 < \alpha^\prime+N-m-1-i_1-i_2} \omega^{(3)}_{i_3} x^{i_3}\right) + \sum_{\beta \le i_3 < \alpha^\prime+N-m-1-i_1-i_2} \omega^{(3)}_{i_3} x^{i_3}
\]
and argue as in the previous step. Hence, in order to close the argument, we need to estimate terms of the from \eqref{fc_31}. Note that not necessarily $\alpha+N-m-\frac 1 2-i_1 \in K_{N_0}$ but we do know that there exist $\left(\alpha_1,\tilde N\right), \left(\alpha_2,\tilde N\right) \in \ensuremath{\mathcal A}$ such that
\begin{equation}\label{indices_eq}
\alpha_1 + \tilde N - m_2 - \frac 1 2 \le \alpha+N-m-i_1 \le \alpha_2+\tilde N - m_2 - \frac 1 2
\end{equation}
and $\alpha_j + \tilde N$ is maximal ($j = 1$) or minimal ($j = 2$) with this property. Since $i_1 \ge 1$, in particular $\alpha_j \le \alpha + \frac 1 2$ and $\tilde N \le N-1$ or $\alpha_j \le \alpha - \frac 1 2$ and $\tilde N \le N$ for $j = 1, 2$. Further noting that
\[
(t/x)^{2 (\alpha+N-m-1-i_1)} \le (t/x)^{2 \left(\alpha_1+\tilde N-m_2\right)-3} + (t/x)^{2 \left(\alpha_2+\tilde N-m_2\right)-3},
\]
we can estimate
\begin{eqnarray*}
\lefteqn{\sup_{t \ge 0} t^{2 (\alpha+N-m+m_2-i_1)-2} \verti{\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1} \omega^{(2)}_{i_2} x^{i_2}}_{\alpha^\prime+N-m-1-i_1}^2} \\
&\lesssim& \sup_{t \ge 0} t^{2 (\alpha+N-m+m_2-i_1)-2} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha^\prime+N-m-1-i_1} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell+4,\alpha^\prime+N-m-1-i_1}^2 \\
&\stackrel{\eqref{def_wn},\eqref{def_lnm}}{\lesssim}& \sup_{t \ge 0} t^{2 (\alpha+N-m+m_2-i_1)-2} \verti{\partial_t^{m_2} w^{(N-m)}}_{k\left(N-m,m,\alpha^\prime\right)+4,\alpha^\prime+N-m-1-i_1}^2 \\
&\stackrel{\eqref{cond_l_non3}, \eqref{cond_l_non4}}{\lesssim}& \sup_{t \ge 0} t^{2 \left(\alpha_1+\tilde N\right)-3} \verti{\partial_t^{m_2} w^{\left(\tilde N-m_2\right)}}_{k\left(\tilde N-m_2,m_2,\alpha_1^\prime\right)+2,\alpha_1^\prime+\tilde N-m_2-\frac 32}^2 \\
&& + \sup_{t \ge 0} t^{2 \left(\alpha_2+\tilde N\right)-3} \verti{\partial_t^{m_2} w^{\left(\tilde N-m_2\right)}}_{k\left(\tilde N-m_2,m_2,\alpha_2^\prime\right)+2,\alpha_2^\prime+\tilde N-m_2-\frac 32}^2\\
&\stackrel{\eqref{def_wn},\eqref{def_lnm}}{\lesssim}& \sup_{t \ge 0} t^{2 \left(\alpha_1+\tilde N\right)-3} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha_1^\prime+\tilde N-m_2-\frac 3 2} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell\left(\tilde N-m_2,m_2,\alpha_1^\prime\right)+2,\alpha_1^\prime+\tilde N-m_2-\frac 32}^2 \\
&& + \sup_{t \ge 0} t^{2 \left(\alpha_2+\tilde N\right)-3} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha_2^\prime+\tilde N-m_2-\frac 3 2} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell\left(\tilde N-m_2,m_2,\alpha_2^\prime\right)+2,\alpha_2^\prime+\tilde N-m_2-\frac 32}^2 \\
&\stackrel{\eqref{norm_sol}}{\lesssim}& \vertiii{u}^2,
\end{eqnarray*}
where Proposition~\ref{prop:elliptic} (elliptic maximal regularity) as well as conditions~\eqref{cond_l_non1}, \eqref{cond_l_non3}, and \eqref{cond_l_non4} have been used.
\medskip
Finally, the product $\prod_{j = 3}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2$ in \eqref{fc_3} can be estimated as before by using estimate~\eqref{est_coeff2} of Lemma~\ref{lem:est_coeff} and Lemma~\ref{lem:c0control}, respectively, provided condition~\eqref{cond_l_non2} is satisfied. This shows that we can bound terms of the form \eqref{terms_a2} as
\begin{equation}\label{super_2}
\begin{aligned}
&\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}_{i_1}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-i_1} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-i_1}^2 \d t \\
& \quad \lesssim \vertiii{u}^4 \times \left(1 + \vertiii{u}^6\right).
\end{aligned}
\end{equation}
The combination of \eqref{super_1} and \eqref{super_2} gives
\begin{equation}\label{super_a}
\int_0^\infty t^{2 (\alpha+N)-3} \verti{\omega^{(1)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1} \left(\omega^{(1)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1}^2 \d t \lesssim \vertiii{u}^4 \times \left(1 + \vertiii{u}^6\right)
\end{equation}
for super-critical terms of type \eqref{omegas_a}.
\subsubsection*{Estimating \eqref{omegas} for terms of the form {\rm (b)}}
Distributing the time derivatives on the factors $\omega^{(j)}$ in \eqref{omegas} meeting (b), it suffices to estimate terms of the form
\begin{equation}\label{omegas_b}
\int_0^\infty t^{2(\alpha+N)-3} \verti{\omega^{(1)}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta}^2,
\end{equation}
where the $\omega^{(j)}$ obey
\begin{align*}
\omega^{(1)} &= \frac{\d^{m_1} u_\beta}{\d t^{m_1}} \quad \mbox{with } \, 0 \le m_1 \le m,\\
\omega^{(2)} &= \partial_t^{m_2} D^{\ell_2} (u-u_0) \quad \mbox{with } \, 0 \le \ell_2 \le \ell+4 \, \mbox{ and } \, 0 \le m_2 \le m,\\
\omega^{(j)} &\in \left\{\delta_{m_j,0}, \frac{\d^{m_j} u_0}{\d t^{m_j}}, \partial_t^{m_j} D^{\ell_j} (u-u_0)\right\} \quad \mbox{with } \, 0 \le \ell_j \le \frac{\ell+4}{2} \, \mbox{ and } \, 0 \le m_j \le m \, \mbox{ for } \, j \ge 3,
\end{align*}
where $\sum_{j = 1}^5 m_j = m$. If $m_1 = m = N-1 = N_0-1$, we may use the $L^2$-bound on $\omega^{(1)}$ and obtain for the term \eqref{omegas_b}:
\begin{equation}\label{case_b1}
\begin{aligned}
&\int_0^\infty t^{2(\alpha+N)-3} \verti{\omega^{(1)}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta}^2 \d t \\
& \quad \lesssim \int_0^\infty t^{2 (\beta+N)-3} \verti{\omega^{(1)}}^2 \d t \times \sup_{t \ge 0} t^{2(\alpha-\beta)} \verti{\omega^{(2)}}_{\alpha^\prime-\beta}^2 \times \prod_{j = 3}^5 \sup_{t \ge 0} \vertii{\omega^{(j)}}^2.
\end{aligned}
\end{equation}
Then we note that we can estimate the first term in the second line of \eqref{case_b1} through
\[
\int_0^\infty t^{2 (\beta+N)-3} \verti{\omega^{(1)}}^2 \d t = \int_0^\infty t^{2 (\beta+N)-3} \verti{\frac{\d^{N-1} u_\beta}{\d t^{N-1}}}^2 \d t \lesssim \vertiii{u}^2
\]
by estimate~\eqref{est_coeff1} of Lemma~\ref{lem:est_coeff}. Furthermore, since we are in the super-critical case, we have $(t/x)^{2(\alpha-\beta)} \le 1 + t/x$ and we can bound as follows\footnote{In the special case $N = 1$ (i.e.,~$N_0 = 1$ by assumption) we have $\alpha = \beta$ and the second summand on the right-hand side can be dropped.}
\begin{align*}
\sup_{t \ge 0} t^{2(\alpha-\beta)} \verti{\omega^{(2)}}_{\alpha^\prime-\beta}^2 &\lesssim \sup_{t \ge 0} t^{2(\alpha-\beta)} \verti{D (u - u_0)}_{\ell + 3,\alpha^\prime-\beta}^2 \\
&\lesssim \sup_{t \ge 0} \verti{u}_{\ell+4,-\delta}^2 + \sup_{t \ge 0} \verti{u-u_0}_{\ell+4,\delta}^2 + \left(1 - \delta_{1,N}\right) \sup_{t \ge 0} t \verti{u-u_0}_{\ell+4,\frac 1 2 \pm \delta}^2 \lesssim \vertiii{u}^2,
\end{align*}
where Proposition~\ref{prop:elliptic} has been used and conditions~\eqref{cond_l_non5} and \eqref{cond_l_non6} need to be satisfied.
\medskip
The product $\prod_{j = 3}^5 \sup_{t \ge 0} \vertii{\omega^{(j)}}^2$ in \eqref{case_b1} can be estimated by employing estimate~\eqref{est_coeff2} of Lemma~\ref{lem:est_coeff} and Lemma~\ref{lem:c0control}, respectively, if condition~\eqref{cond_l_non2} holds true. In summary we obtain
\begin{equation}\label{super_3}
\begin{aligned}
&\int_0^\infty t^{2(\alpha+N)-3} \verti{\omega^{(1)}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta}^2 \\
&\quad\lesssim \vertiii{u}^4 \times \left(1 + \vertiii{u}^6\right)
\end{aligned}
\end{equation}
in the case $m_1 = m = N-1 = N_0-1$.
\medskip
In all other cases, we can estimate \eqref{omegas_b} by taking the $C^0$-bound on $\omega^{(1)}$ and get:
\begin{equation}\label{case_b2}
\begin{aligned}
&\int_0^\infty t^{2(\alpha+N)-3} \verti{\omega^{(1)}}^2 \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta}^2 \\
& \quad \lesssim \sup_{t \ge 0} t^{2 m_1 + 2 \beta} \verti{\omega^{(1)}}^2 \\
& \qquad \times \int_0^\infty t^{2 (\alpha+N-m_1-\beta)-3} \verti{\omega^{(2)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta} \left(\omega^{(2)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta}^2 \d t.
\end{aligned}
\end{equation}
Then we notice
\[
\sup_{t \ge 0} t^{2 m_1 + 2 \beta} \verti{\omega^{(1)}}^2 = \sup_{t \ge 0} t^{2 m_1 + 2 \beta} \verti{\frac{\d^{m_1} u_\beta}{\d t^{m_1}}}^2 \lesssim \vertiii{u}^2
\]
by estimate~\eqref{est_coeff2} of Lemma~\ref{lem:est_coeff}. For the last line of \eqref{case_b2} we may use the decomposition
\[
\omega^{(2)} = \left(\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta} \omega^{(2)}_{i_2} x^{i_2}\right) + \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta} \omega^{(2)}_{i_2} x^{i_2},
\]
so that it suffices to estimate
\begin{subequations}\label{lc}
\begin{equation}\label{lc_1}
\int_0^\infty t^{2 (\alpha+N-m+m_2-\beta)-3} \verti{\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta} \omega^{(2)}_{i_2} x^{i_2}}_{\alpha^\prime+N-m-1-\beta}^2 \d t \times \prod_{j = 3}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2
\end{equation}
and
\begin{equation}\label{lc_2}
\begin{aligned}
& \int_0^\infty t^{2 m_2 + 2 i_2 - 1} \verti{\omega^{(2)}_{i_2}}^2 \d t \\
& \quad \times \sup_{t \ge 0} t^{2 (\alpha+N-m_1-m_2-\beta-i_2)-2} \verti{\omega^{(3)}\cdots\omega^{(5)} - \sum_{i < \alpha^\prime+N-m-1-\beta-i_2} \left(\omega^{(3)}\cdots\omega^{(5)}\right)_i x^i}_{\alpha^\prime+N-m-1-\beta-i_2}^2,
\end{aligned}
\end{equation}
\end{subequations}
where the term \eqref{lc_2} is identically zero unless $\omega^{(j)} = \partial_t^{m_j} D^{\ell_j} (u-u_0)$ for at least one $j \ge 3$. Since $\beta + i_2 \ge 1$, the expression \eqref{lc_2} can be treated in the same way as the second and third line of \eqref{fc_1}.
\medskip
For the term \eqref{lc_1} we can treat the product $\prod_{j = 3}^5 \sup_{t \ge 0} t^{2 m_j} \vertii{\omega^{(j)}}^2$ as before (by using Lemma~\ref{lem:c0control} and estimate~\eqref{est_coeff2} of Lemma~\ref{lem:est_coeff}, respectively) given that we fulfill condition~\eqref{cond_l_non2}. For treating the $L^2$-part in \eqref{lc_1} we pick $\left(\alpha_1,\tilde N\right), \left(\alpha_2,\tilde N\right) \in \ensuremath{\mathcal A}$ such that
\begin{equation}\label{indices_eq2}
\alpha_1 + \tilde N - m_2 \le \alpha+N-m-\beta \le \alpha_2+\tilde N - m_2
\end{equation}
and $\alpha_j + \tilde N$ is maximal ($j = 1$) or minimal ($j = 2$) with this property. Since $\beta > \frac 1 2$ we have in particular $\alpha_j \le \alpha + \frac 1 2$ and $\tilde N - m_2 \le N-m-1$ or $\alpha_j \le \alpha - \frac 1 2$ and $\tilde N - m_2 \le N-m$ for $j = 1, 2$ under the assumption of a sufficiently small $\delta > 0$. Utilizing
\[
(t/x)^{2 (\alpha+N-m-\beta-1)} \le (t/x)^{2 \left(\alpha_1+\tilde N-m_2-1\right)} + (t/x)^{2 \left(\alpha_2+\tilde N-m_2-1\right)},
\]
we obtain the estimate
\begin{eqnarray*}
\lefteqn{\int_0^\infty t^{2 (\alpha+N-m+m_2-\beta)-3} \verti{\omega^{(2)} - \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-\beta} \omega^{(2)}_{i_2} x^{i_2}}_{\alpha^\prime+N-m-1-\beta}^2 \d t}\\
&\lesssim& \int_0^\infty t^{2 (\alpha+N-m+m_2-\beta)-3} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha^\prime+N-m-1-\beta} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell+4,\alpha^\prime+N-m-1-\beta}^2 \d t\\
&\stackrel{\eqref{def_wn},\eqref{def_lnm}}{\lesssim}& \int_0^\infty t^{2 (\alpha+N-m+m_2-\beta)-3} \verti{\partial_t^{m_2} w^{(N-m)}}_{k\left(N-m,m,\alpha^\prime\right)+4,\alpha^\prime+N-m-1-\beta}^2 \d t\\
&\stackrel{\eqref{cond_last}}{\lesssim}& \int_0^\infty t^{2 (\alpha_1+\tilde N)-3} \verti{\partial_t^{m_2} w^{\left(\tilde N-m_2\right)}}_{k\left(\tilde N-m_2,m_2,\alpha_1^\prime\right)+4,\alpha_1^\prime+\tilde N-m_2-1}^2 \d t \\
&& + \int_0^\infty t^{2 (\alpha_2+\tilde N)-3} \verti{\partial_t^{m_2} w^{\left(\tilde N-m_2\right)}}_{k\left(\tilde N-m_2,m_2,\alpha_2^\prime\right)+4,\alpha_2^\prime+\tilde N-m_2-1}^2 \d t\\
&\stackrel{\eqref{def_wn},\eqref{def_lnm}}{\lesssim}& \int_0^\infty t^{2 (\alpha_1+\tilde N)-3} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha_1^\prime+\tilde N-m_2-1} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell\left(\tilde N-m_2,m_2,\alpha_1^\prime\right)+4,\alpha_1^\prime+\tilde N-m_2-1}^2 \d t \\
&& + \int_0^\infty t^{2 (\alpha_2+\tilde N)-3} \verti{\partial_t^{m_2} u - \sum_{i_2 < \alpha_2^\prime+\tilde N-m_2-1} \frac{\d^{m_2} u_{i_2}}{\d t^{m_2}} x^{i_2}}_{\ell\left(\tilde N-m_2,m_2,\alpha_2^\prime\right)+4,\alpha_2^\prime+\tilde N-m_2-1}^2 \d t \\
&\stackrel{\eqref{norm_sol}}{\lesssim}& \vertiii{u}^2,
\end{eqnarray*}
where we have used elliptic maximal regularity in the form of Proposition~\ref{prop:elliptic}, \eqref{cond_l_non1}, and the conditions
\begin{subequations}\label{cond_last}
\begin{align}
\ell\left(N-m,m,\alpha-\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) \quad \mbox{for all } \, \alpha \in \left(\frac 1 2, 1\right),\\
\ell\left(N-m-1,m,\alpha+\frac 1 2\right) &\ge \ell\left(N-m,m,\alpha^\prime\right) \quad \mbox{for all } \, \alpha \in \left(0, \frac 1 2\right).
\end{align}
\end{subequations}
These conditions in particular hold if the stronger conditions~\eqref{cond_l_non3}~and~\eqref{cond_l_non4} are fulfilled.
\medskip
In summary we obtain \eqref{super_3} for the case of arbitrary $m_1$ and $N$ as well.
\subsubsection*{Conclusion}
Gathering inequalities~\eqref{main_subcrit},~\eqref{super_a},~and~\eqref{super_3}, we obtain inequality~\eqref{non_main}, thus finishing the proof of Proposition~\ref{prop:nonlinear_est}.
\end{proof} |
2008.05629 | \section{Introduction}\label{sec:introduction}
Network security is one of the most important problems faced by enterprises and countries today \cite{Goel16}.
Before launching a network attack,
malicious attackers often scan systems in a network to identify vulnerabilities
that can be exploited to intrude into the network \cite{Carroll11}.
The aim of this scanning is to understand the configurations of these systems,
including the operating systems they are running,
and their IP/MAC addresses on the network.
Once these questions are answered,
attackers can efficiently formulate plans to attack the network.
In order to prevent attackers from receiving true answers to these questions
and thus reduce the likelihood of successful attacks,
cyber deception techniques are employed \cite{Alme16}.
Instead of stopping an attack or identifying an attacker,
cyber deception techniques aim to mislead an attacker and induce him to attack non-critical systems
by hiding or lying about the configurations of the systems in a network \cite{Albanese16}.
For example, an important system $s$ with configuration $k$ is obfuscated by the defender to appear as a less important system with configuration $k'$.
Thus, when an attacker scans the network,
he observes system $s$ with configuration $k'$ rather than $k$, as shown in Fig. \ref{fig:simpleExample}.
Since configuration $k'$ is less important than $k$,
the attacker may skip over system $s$.
\begin{figure}[ht]
\vspace{-2mm}
\centering
\includegraphics[scale=0.7]{config.pdf}
\vspace{-2mm}
\caption{System $s$ with configuration $k$ is obfuscated to and appears as configuration $k'$}
\label{fig:simpleExample}
\vspace{-1mm}
\end{figure}
To model such an interaction between an attacker and a defender,
game theory has been adopted as a means of studying cyber deception \cite{Kiennert18,Shaer19}.
Game theory is a theoretical framework to study the decision-making strategies of competing players,
where each player aims to maximize her or his own utility.
In cyber deception, defenders and attackers can be modelled as players.
Game theory, thus, can be used to investigate how a defender reacts to an attacker and vice versa.
Game theoretic formulation overcomes traditional solutions to cyber deception in many aspects,
such as proven mathematics, reliable defense mechanisms and timely actions \cite{Do17}.
Existing game theoretic approaches, however, have two common limitations.
The first limitation is that
the number of systems in a network is assumed to be fixed.
This assumption hinders the applicability of existing approaches to real-world applications,
because in the real world, the number of systems in a network may be dynamically changed.
Although some existing approaches, like \cite{Jajodia17}, are computationally feasible to recompute the strategy
when new systems are added, this change in the number of systems may affect the security of a network \cite{Modi13,Zarp17}.
For example, a network has three systems: $s_1$, $s_2$ and $s_3$.
The three systems are associated with two configurations,
where $s_1$ and $s_2$ are associated with configuration $k_1$,
and $s_3$ is associated with configuration $k_2$.
The attacker knows that the network has three systems and two configurations.
He also knows that configuration $k_1$ has two systems and configuration $k_2$ has one system.
He, however, is unaware of which system is associated with which configuration.
Now when system $s_3$ is removed by the defender,
the attacker knows that configuration $k_1$ has two systems and configuration $k_2$ has zero system.
By comparing the two results,
the attacker can deduce that system $s_3$ is associated with configuration $k_2$.
After deducing system $s_3$'s configuration,
the attacker can also deduce that systems $s_1$ and $s_2$ are associated with configuration $k_1$.
Similarly, when a new system $s_4$ is added and associated with configuration $k_2$,
the attacker can immediately deduce the configuration of system $s_4$
because he realizes that the number of systems in configuration $k_2$ increases by $1$.
Therefore, the problem of how to strategically change the number of systems in a network without sacrificing its security is challenging.
The second limitation is that
attackers' approaches are simplified in the literature through the use of only greedy-like approaches.
Greedy-like approaches create deterministic strategies which are highly predictable to defenders.
Hence, using these approaches may underestimate the ability of real-world attackers
who can use more complex approaches that are much less predictable to defenders.
If a defender's approach is developed based on simplified attackers' approaches,
the defender may be easily compromised by real-world attackers.
Thus, the problem of how to develop an efficient defender's approach against powerful attackers is also challenging \cite{Ding2018}.
Accordingly, to overcome the above two limitations,
in this paper, we propose a novel differentially private game theoretic approach to cyber deception.
By using the differential privacy technique \cite{Dwork06},
the change in the number of systems does not affect the security of a network,
as an attacker cannot determine whether a given system is inside or outside of the network.
Thus, the attacker cannot deduce each system's true configuration.
Moreover, by using the differential privacy-based approach,
a defender's strategy is unpredictable to an attacker, irrespective of his reasoning power.
In summary, the contribution of this paper is two-fold.
1) To the best of our knowledge, we are the first to apply differential privacy to the cyber deception game in order to overcome the two common limitations mentioned above.
2) We theoretically illustrate the properties of the proposed approach,
and experimentally demonstrate its effectiveness.
\section{Related Work}\label{sec:related work}
In this section, we first review related works about game theory for cyber deception
and next discuss their limitations.
A thorough survey on game theory for general network and cyber security can be found in \cite{Man13,Liang13,Do17,Kiennert18}.
\subsection{Review of related works}
Albanese et al. \cite{Alba15} propose a deceptive approach to defeat an attacker's effort to fingerprint operating systems and services.
For operating system fingerprinting, they manipulate the outgoing traffic to make it resemble traffic generated by a different system.
For service fingerprinting, they modify the service banner by
intercepting and manipulating certain packets before they leave the network.
Albanese et al. \cite{Albanese16} present a graph-based approach to manipulate the attacker's view of a system's attack surface.
They formalize system views and define the distance between these different views.
Based on this formalization and definition, they develop an approach to confuse the attacker by manipulating responses to his probes
in order to induce an external view of the system, which can minimize the costs incurred by the defender.
Jajodia et al. \cite{Jajodia17} develop a probabilistic logic of deception
and demonstrate that these associated computations are NP-hard.
Specifically, they propose two algorithms that allow the defender to generate faked scan results
in different states to minimize the damage caused by the attacker.
Wang and Zeng \cite{Wang18c} propose a two-stage deception game for network protection.
In the first stage, the attacker scans the whole network
and the defender decides how to respond these scan queries
to distract the attacker from high value hosts.
In the second stage, the attacker selects some hosts for further probing
and the defender decides how to answer these probes.
The defender's decisions are formulated as optimization problems solved by heuristic algorithms.
Schlenker et al. \cite{Schlenker18} propose a game theoretic approach to deceive cyber adversaries.
Their approach involves two types of attackers: a rational attacker and a naive attacker.
The rational attacker knows the defender's deception strategy,
while the naive attacker does not.
Two optimal deception algorithms are developed to counter these two types of attackers.
Bilinski et al. \cite{Bilinski19} propose a simplified cyber deception game model.
In their model, there are only two machines,
where one is real and the other is fake.
At each round, the attacker is allowed to ask one machine about its type,
and the machine can either lie or tell the truth about its type.
After $N$ rounds, the attacker chooses one machine to attack.
The attacker will receive a positive payoff if he attacks the real machine.
Otherwise, he will receive a negative payoff.
Other game theoretic approaches have also been developed \cite{Huang19,Cho19}.
These, however, are not closely related to our work,
as they do not particularly focus on obfuscating configurations of systems.
Some game theoretic approaches \cite{Nguyen19,Pawlick19} mainly focus on dealing with the interactions between defenders and attackers,
while other approaches \cite{Thakoor19,Gan19} aim at investigating the behaivors of defenders and attackers.
Another branch of related work deals with the repeated interactions of defender and attacker
over time following the ``FlipIt'' model \cite{Dijk13,Bowers12,Laszka14}.
Their research focuses on stealthy takeovers,
where both the defender and the attacker want to take control of a set of resources by flipping them.
By comparison, our research focuses on how a defender modifies the configuration of systems to deceive an attacker.
\subsection{Discussion of related works}
The above-reviewed works have two common limitations:
1) the number of systems in a network is assumed to be fixed,
and 2) the attacker's approaches are often simplified.
\subsubsection{Fixed number of systems}
In the real-world networks, the number of systems is often changed in a dynamic manner.
Most of existing works, however, are conducted on the basis of fixed number of systems,
though some works are computationally feasible to accommodate the increase of the number of systems.
An intuitive solution is to extend existing works by enabling systems to be dynamically introduced or removed.
However,
arbitrarily introducing or removing systems
may compromise the security of a network.
\subsubsection{Simplified attackers}
In existing works, attackers are considered to be either naive \cite{Jajodia17}
or to use only greedy-like approaches \cite{Schlenker18}.
A naive attacker always attacks the observed configuration which has the highest utility.
A greedy attacker knows the defender's deception scheme
and attacks the configuration which has the expected highest utility.
The drawback of naive and greedy-like approaches is that
the selection of a configuration is deterministic
and highly predictable to a defender.
As real-world attackers are more powerful than the simplified naive and greedy attackers,
defenders' approaches developed based on simplified attackers' approaches
may not be applicable to the real world.
In this paper, we develop a novel differentially private game theoretic approach
which can strategically change the number of systems in a network without sacrificing its security.
Moreover, to model a powerful attacker, we adopt a Bayesian inference approach.
By using Bayesian inference, the attacker can infer the probability
with which an observed configuration $k'$ could be a real configuration $k$,
and selects a configuration to attack based on the probability distribution over the observed configurations.
Therefore, the selection of a configuration using the Bayesian inference approach is non-deterministic
and hardly predictable to the defender.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Cyber deception game}\label{sub:game}
The cyber deception game (CDG) is an imperfect-information Stacklberg game
between a defender and an attacker \cite{Schlenker18},
where a player cannot accurately observe the actions of the other player \cite{Do17}.
The defender takes the first actions.
She decides how the systems should respond to scan queries from an attacker by obfuscating the configurations of these systems.
The attacker subsequently follows by choosing which systems to attack based on his scan query results.
The game continues in this alternating manner,
until a pre-defined number of rounds is reached.
We use the imperfect information game because in this cyber deception game,
the aim of the defender is to hide the configurations of systems from the attacker
rather than stopping an attack or identifying the attacker.
If we use a perfect information game,
the defender’s previous strategies of obfuscating configurations can be accurately observed by the attacker.
Then, the attacker can immediately deduce the real configurations of all the systems.
In this game, we use $N$ to denote the set of systems in a network protected by the defender.
The number of systems is denoted by $|N|$.
Each system has a set of attributes: an operating system, a file system, services hosted, and so on.
These attributes constitute a system configuration.
We use $K$ to denote the set of configurations,
and the number of configurations is denoted by $|K|$.
According to the configurations, the system set $N$ can be partitioned into a number of subsets: $N_1,...,N_{|K|}$,
where $N_i\cap N_j=\emptyset$ for any $i\not= j$, and $N_1\cup...\cup N_{|K|}=N$.
Each of the systems in subset $N_k$ has configuration $k$ and an associated utility $u_k$.
If a system with configuration $k$ is attacked by the attacker,
the attacker receives utility $u_k$ and the defender loses utility $u_k$.
The utility of a system depends on not only the importance of the system
but also on its security level.
A well-chronicled finding of information security is that
attackers may go for the weakest link, i.e., the system with the weakest security level, to have a foothold
and then try to progress from there via privilege escalation \cite{Arce03}.
Detailed theoretical discussion on the weakest link can be found in \cite{Varian03,Bohme10}.
To mislead the attacker, the defender may obfuscate the configurations of these systems.
Obfuscating a system with configuration $k$ to appear as $k'$ incurs the defender a cost $c(k',k)$.
When scanning, the attacker observes the obfuscated configurations of these systems.
If a system with configuration $k$ is obfuscated to appear as $k'$,
the system's real configuration is still $k$ rather than $k'$.
Thus, when this system is attacked by the attacker,
the attacker receives utility $u_k$ instead of $u_{k'}$.
After obfuscation, the system set $N$ becomes $N'$,
which can be partitioned into: $N'_1,...,N'_{|K|}$.
Since $|N|$ may not be equal to $|N'|$,
there may be void systems ($|N'|>|N|$), or some systems may be taken offline ($|N'|<|N|$).
Void systems can be interpreted as honeypots\footnote{Although our work also uses honeypots, these do not permanently exist in the network, but are rather randomly introduced by our differentially private approach. Our main aim is still to focus on the cyber deception.}.
Deploying a honeypot with configuration $k$ incurs the defender a cost $h_k$.
If the attacker attacks the honeypot,
he receives a negative utility $-u_k$.
Usually, $u_k>h_k$, as the defender would not otherwise have the motivation to deploy a honeypot.
By taking a system offline, the defender loses utility $l_k$,
but this system will not be attacked by the attacker.
Again, generally, $u_k>l_k$, as the defender would not otherwise have the motivation to take a system offline.
While the defender aims to minimize her utility loss,
the attacker aims to maximize his utility gain.
The attacker is aware of the defender's obfuscation approach,
and knows the statistical information pertaining to the network:
the number of systems $|N|$, the number of configurations $|K|$, and the number of the systems associated with each configuration $|N_1|,...,|N_{|K|}|$.
The attacker, however, does not know which system is associated with which configuration.
Moreover, as the aim of the defender is to hide the configurations of systems,
the defender's strategies, i.e., when and how to make $k$ appear as $k'$,
are exactly what the defender wants to hide
and cannot be accurately observed by the attacker.
However, the attacker can partially observe the defender’s previous actions.
As in the above example, during each round, the attacker attacks a system which appears as configuration $k'$,
but after the attack, the attacker receives utility $u_k$.
The attacker, thus, knows that the attacked system was obfuscated from $k$ to $k'$ in this round.
The attacker accumulates this information and
uses Bayesian inference to make decisions in future rounds.
Specifically, in Bayesian inference, the posterior probability $q(k'|k)$ takes this partial observation into account by computing how many times $k$ appears as $k'$ in previous rounds.
The details will be given in Section \ref{sec:method}.
\subsection{Differential privacy}\label{sub:DP}
Differential privacy is a prevalent privacy model capable of guaranteeing that
any individual record being stored in or removed from a dataset makes little difference to the analytical output of the dataset \cite{Dwork06}.
Differential privacy has been successfully applied to cyber physical systems \cite{Zhu20}, machine learning \cite{Ye19} and artificial intelligence \cite{Zhu19,Zhu20TKDE}.
In differential privacy, two datasets $D$ and $D'$ are neighboring datasets if they differ by only one record.
A query $f$ is a function that maps dataset $D$ to an abstract range $\mathbb{R}$, $f: D\rightarrow\mathbb{R}$.
The maximal difference in the results of query $f$ is defined as sensitivity $\Delta S$,
which determines how much perturbation is required for the privacy-preserving answer.
The formal definition of differential privacy is presented as follows.
\begin{definition}[$\epsilon$-Differential Privacy \cite{Dwork14}]\label{Def-DP}
A mechanism $\mathcal{M}$ provides $\epsilon$-differential privacy for any pair of neighboring datasets $D$ and $D'$, and for every set of outcomes $\Omega$, if $\mathcal{M}$ satisfies:
\begin{equation}
Pr[\mathcal{M}(D) \in \Omega] \leq \exp(\epsilon) \cdot Pr[\mathcal{M}(D') \in \Omega]
\end{equation}
\end{definition}
\begin{definition}[Sensitivity \cite{Dwork14}]\label{Def-GS} For a query $f:D\rightarrow\mathbb{R}$, the sensitivity of $f$ is defined as
\begin{equation}
\Delta S=\max_{D,D'} ||f(D)-f(D')||_{1}
\end{equation}
\end{definition}
Two of the most widely used differential privacy mechanisms are the Laplace mechanism and the exponential mechanism \cite{Zhu17}.
The Laplace mechanism adds Laplace noise to the true answer.
We use $Lap(b)$ to represent the noise
sampled from the Laplace distribution with scaling $b$.
\begin{definition}[The Laplace Mechanism \cite{Dwork14}]\label{Def-LA}
Given a function $f: D \rightarrow \mathbb{R}$ over a dataset $D$, Equation~\ref{eq-lap} is the Laplace mechanism.
\begin{equation}
\widehat{f}(D)=f(D)+Lap(\frac{\Delta S}{\epsilon})
\end{equation}\label{eq-lap}
\end{definition}
\begin{definition}[The Exponential Mechanism \cite{Dwork14}]\label{Def-Ex}
The exponential mechanism $\mathcal{M}_E$ selects and outputs an element $r\in\mathcal{R}$
with probability proportional to $exp(\frac{\epsilon u(D,r)}{2\Delta u})$,
where $u(D,r)$ is the utility of a pair of dataset and output,
and $\Delta u=\max\limits_{r\in\mathcal{R}} \max\limits_{D,D':||D-D'||_1\leq 1} |u(D,r)-u(D',r)|$
is the sensitivity of utility.
\end{definition}
Table \ref{tab:notation} presents the notations and terms used in this paper.
\begin{table}[!ht]\scriptsize
\vspace{-3mm}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\centering
\caption{The meaning of notations used in this paper}
\begin{tabular} {|p{1cm}|p{5cm}|} \hline
\textbf{Notations}&\textbf{Meaning}\\ \hline
$N$ & a set of systems\\\hline
$K$ & a set of configurations\\\hline
$N'$ & a set of obfuscated systems\\\hline
$u_k$ & the utility of a system with configuration $k$\\\hline
$c(k',k)$ & cost to the defender of obfuscating a system from configuration $k$ to appear as $k'$\\\hline
$h_k$ & cost to the defender's of deploying a honeypot with configuration $k$\\\hline
$l_k$ & utility loss incurred by the defender to take a system with configuration $k$ offline\\\hline
$U_d$ & the defender's total expected utility loss\\\hline
$C_d$ & the defender's total cost\\\hline
$B_d$ & the defender's deploy budget\\\hline
$U_a$ & the attacker's total expected utility gain\\\hline
$\Delta S$ & the sensitivity of a query \\\hline
$\Delta u$ & the sensitivity of utility \\\hline
$\epsilon$ & privacy budget \\\hline
\end{tabular}
\label{tab:notation}
\vspace{-2mm}
\end{table}
\vspace{-2mm}
\section{The differentially private approach}\label{sec:method}
\subsection{Overview of the approach}
We provide an example here to describe the workflow of our approach.
In a network, there are three systems $N=\{s_1,s_2,s_3\}$ and three configurations $K=\{k_1,k_2,k_3\}$.
System $1$, $2$ and $3$ are associated with configuration $1$, $2$ and $3$, respectively:
$N_1=\{s_1\}$, $N_2=\{s_2\}$ and $N_3=\{s_3\}$.
Each configuration is associated with a utility,
such that systems with the same configuration have the same utility.
Once a system is attacked, the defender loses a corresponding utility,
while the attacker gains this utility.
The aim of our approach is to minimize the defender's utility loss
by using the differential privacy technique to hide the real configurations of systems.
At each round of the game, our approach consists of four steps:
Steps 1 and 2 are carried out by the defender,
while Steps 3 and 4 are conducted by the attacker.
\emph{Step 1}: The defender obfuscates the configurations.
The obfuscation is conducted on the statistical information of the network according to Algorithm \ref{alg:DP}.
In this example, the number of systems in each configuration is $1$, i.e., $|N_1|=|N_2|=|N_3|=1$,
while the total number of systems in the network is $|N|=3$.
After obfuscation, the number of systems in each configuration could be: $|N'_1|=2$, $|N'_2|=1$, $|N'_3|=1$,
while the total number of systems in the network becomes $|N'|=|N'_1|+|N'_2|+|N'_3|=4$.
\emph{Step 2}: As $|N'|-|N|=1$, an extra system is introduced into the network.
This extra system is interpreted as a honeypot, denoted as $hp_4$.
The defender uses Algorithm \ref{alg:deployment2} to deploy the systems and the honeypot to the configurations.
The deployment result could be: $N'_1=\{s_3,hp_4\}$, $N'_2=\{s_1\}$ and $N'_3=\{s_2\}$,
which is shown to the attacker as demonstrated in Fig. \ref{fig:obfuscation}.
\begin{figure}[ht]
\vspace{-3mm}
\includegraphics[width=0.49\textwidth, height=2.5cm]{obfuscation.pdf}
\caption{The defender obfuscates configurations of systems}
\label{fig:obfuscation}
\vspace{-2mm}
\end{figure}
\emph{Step 3}: The attacker estimates the probability with which an observed configuration $k'$ could be a real configuration $k$ using Bayesian inference (Equation \ref{eq:Bayes}),
i.e., estimating:
$q(k_1|k_1)$, $q(k_2|k_1)$ and $q(k_3|k_1)$;
$q(k_1|k_2)$, $q(k_2|k_2)$ and $q(k_3|k_2)$;
$q(k_1|k_3)$, $q(k_2|k_3)$ and $q(k_3|k_3)$.
\emph{Step 4}: Based on the estimation, the attacker uses Equation \ref{eq:utilitygain} to calculate the expected utility gain of selecting each configuration.
Finally, the attacker selects a specific configuration as the target
based on a probability distribution over configurations (Equation \ref{eq:selection}).
\subsection{The defender's strategy}
At each round of the game, the defender first obfuscates the configurations using the differentially private Laplace mechanism,
and deploys systems according to these configurations.
\subsubsection{Step 1: obfuscating configurations}
The obfuscation is described in Algorithm \ref{alg:DP}.
In Line 5, Laplace noise is added to each $|N_k|$ to obfuscate the number of systems with configuration $k$.
In Line 5, $\Delta S$ denotes the sensitivity of the maximum number of systems with a specific configuration.
As this maximum number has a direct impact on the configuration arrangement of these systems,
the sensitivity $\Delta S$ is determined by the maximum number.
Based on the definition of sensitivity, $\Delta S=1$ in this algorithm.
The rationale underpinning Algorithm \ref{alg:DP} is as follows.
Differential privacy is originally designed to preserve data privacy in datasets.
It can guarantee that an individual data record in or out of a dataset has little impact on the analytical output of the dataset.
In other words, an attacker cannot infer whether a data record is in the dataset by making queries to the dataset.
Here we use differential privacy to preserve the privacy of systems in networks.
The privacy of systems in this paper means the configurations of systems.
To map from differential privacy to its meaning in the network configuration,
we treat a network as a dataset and treat a system in the network as a data record in the dataset.
Since differential privacy can guarantee that
an attacker cannot infer whether a given data record is in a dataset,
it can also guarantee that an attacker cannot infer whether a given system is in a network.
Specifically, as we add differentially private Laplace noise on the number of systems in each configuration,
the attacker cannot infer whether a given system is really associated with the shown configuration.
Thus, the system's real configuration is preserved.
\vspace{-2mm}
\begin{algorithm}
\caption{The obfuscation of configurations}
\label{alg:DP}
\textbf{Input}: $|N_1|,...,|N_{|K|}|$;\\
Partition the system set $N$ into $N_1,...,N_{|K|}$ based on the configurations;\\
Calculate the size of each subset: $|N_1|,...,|N_{|K|}|$;\\
\For{$k=1$ to $|K|$}{
$|N'_k|\leftarrow |N_k|+\lceil Lap(\frac{\Delta S\cdot |K|}{\epsilon})\rceil$;\\
}
\textbf{Output}: $|N'_1|,...,|N'_{|K|}|$;\\
\end{algorithm}
\vspace{-2mm}
After Algorithm \ref{alg:DP} is executed,
the defender receives an output: $|N'_1|,...,|N'_{|K|}|$.
Let $|N'|=\sum_{1\leq k\leq |K|}|N'_k|$,
so that there are three possible situations: $|N'|=|N|$, $|N'|>|N|$ or $|N'|<|N|$.
When $|N'|>|N|$, there are $|N'|-|N|$ void systems,
which can be interpreted as honeypots.
When $|N'|<|N|$, there are $|N|-|N'|$ systems taken offline.
\subsubsection{Step 2: deploying systems}
System deployment will be conducted according to three possible situations: $|N'|=|N|$, $|N'|>|N|$ or $|N'|<|N|$.
This deployment is based not only on the utility of these systems,
but also on the attacker's attack strategy in the previous rounds.
The aim of this deployment is to minimize the defender's expected utility loss
while satisfying the defender's budget constraint $B_d$.
\emph{Situation 1}: When $|N'|=|N|$, the defender's expected utility loss is shown in Equation \ref{eq:utilityloss1},
\begin{equation}\label{eq:utilityloss1}
U_d=\sum_{1\leq k\leq |K|}(p(k)\cdot\sum_{1\leq i\leq |N_k|} u_i),
\end{equation}
where $p(k)$ is the estimated probability that the attacker will attack the systems with configuration $k$ in the next round.
The estimated probability is the ratio between the number of times that $k$ was attacked to the total number of game rounds.
For example, if the game has been played for $10$ rounds and systems with configuration $k$ have been attacked $3$ times,
the estimated probability is $p(k)=\frac{3}{10}=0.3$.
Initially, before the first round, the probability distribution over the configurations is uniform.
The probability distribution is then updated every $10$ rounds using the method described in the example.
Since the game is an imperfect information game,
the defender is unaware of the real attack probability of the attacker.
The defender can use only her past experience to make a strategy, i.e., past observation of the attacker’s strategies.
During the deployment, the defender's cost is shown in Equation \ref{eq:dcost1},
\begin{equation}\label{eq:dcost1}
C_d=\sum_{1\leq i\leq |N|}I_i\cdot(c(k',k)),
\end{equation}
where $I_i=1$ if system $i$ is obfuscated from configuration $k$ to be perceived as $k'$, and $I_i=0$ otherwise.
The problem faced by the defender is to identify the deployment configuration
that can minimize her utility loss $U_d$ while satisfying her budget constraint: $C_d\leq B_d$.
Algorithm \ref{alg:deployment1} is developed to solve this problem.
\begin{algorithm}
\caption{Deployment of configurations, $|N'|=|N|$}
\label{alg:deployment1}
\textbf{Input}: $N$ and $|N'_1|,...,|N'_{|K|}|$;\\
Let $|N'_1|=n'_1$, ..., $|N'_{|K|}|=n'_{|K|}$;\\
Initialize $N'_1=...=N'_{|K|}=\emptyset$;\\
Initialize $C_d=0$;\\
Rank $u_1$, ..., $u_{|N|}$ in an increasing order, and supposing that the result is $u_1\leq...\leq u_{|N|}$;\\
\For{$i=1$ to $|N|$}{
Select a configuration $k'$ with probability proportional to $exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})$;\\
\If{$C_d+c(k',k)>B_d$}{
\textbf{continue};\\
}
\If{$|N'_{k'}|<n'_{k'}$}{
$N'_{k'}\leftarrow N'_{k'}\cup \{i\}$;\\
$C_d\leftarrow C_d+c(k',k)$;\\
}\Else{
\textbf{goto} Line 7;\\
}
}
\textbf{Output}: $N'_1,...,N'_{|K|}$;\\
\end{algorithm}
The idea behind Algorithm \ref{alg:deployment1} involves preferentially obfuscating low-utility systems, with specific probabilities,
to be perceived as those configurations that have a high probability of being attacked.
To a large extent, this idea can prevent high-utility systems from being attacked.
It seems that the high-utility systems may not be protected by the exponential mechanism,
because 1) the exponential mechanism preferentially obfuscates low-utility systems
and 2) the budget constraint $B_d$ may limit the number of obfuscated systems.
However, the exponential mechanism still gives high-utility systems the probability to be obfuscated,
since the exponential mechanism is probabilistic rather than deterministic.
In addition, the budget constraint is a parameter which is set by users.
A larger budget constraint means a larger number of obfuscated systems.
The algorithm starts by ranking the utility of the systems in increasing order (Line 5).
Next, each of the ranked systems $i$ is obfuscated to appear as a configuration, $k'$,
selected using the exponential mechanism,
until all systems are obfuscated or the defender's budget is used up (Lines 6-14).
In the exponential mechanism in Line 7,
$U_{k'}=p(k')\cdot\sum_{1\leq i\leq |N_{k'}|} u_i$ is the defender's expected utility loss on the systems with configuration $k'$.
Moreover, $\Delta U_d=max_{1\leq i\leq |N|}u_i$ is the sensitivity of the defender's expected utility loss,
which is used to to obfuscate system configurations.
A system configuration with a higher expected utility loss has a larger probability to be obfuscated.
The expected utility loss is used only as a parameter in the exponential mechanism,
and it does not change any properties of the exponential mechanism.
Therefore, the use of the expected utility loss does not violate differential privacy.
The rationale of using the exponential mechanism is described as follows.
According to the definitions of differential privacy,
an obfuscated network $N'$ is interpreted as a dataset $D$,
while a configuration $k'$ in network $N'$ is interpreted as an output $r$ of dataset $D$.
Thus, the utility of a pair of dataset and output, $u(D,r)$,
is equivalent to the defender's expected utility loss of selecting configuration $k'$ from network $N'$, i.e., $U(N',k')$ denoted as $U_{k'}$.
By comparing Definition \ref{Def-Ex} to Line 7 in Algorithm \ref{alg:deployment1},
we can conclude that as the exponential mechanism can guarantee the privacy of the output of a dataset,
it can also guarantee the privacy of the configurations of the systems in a network.
\emph{Situation 2}. When $|N'|>|N|$, there are $|N'|-|N|$ honeypots in the network.
The defender's expected utility loss is shown in Equation \ref{eq:utilityloss2},
\begin{equation}\label{eq:utilityloss2}
U_d=\sum_{1\leq k\leq |K|}(p(k)\cdot\sum_{1\leq i\leq |N_k|} u_i)-\sum_{1\leq j\leq |N'|-|N|} p(k)\cdot u_j,
\end{equation}
where the second part, $\sum_{1\leq j\leq |N'|-|N|} p(k)\cdot u_j$, indicates the expected utility gain obtained by using honeypots.
The defender's cost is shown in Equation \ref{eq:dcost2},
\begin{equation}\label{eq:dcost2}
C_d=\sum_{1\leq i\leq |N|}I_i\cdot(c(k',k))+\sum_{1\leq j\leq |N'|-|N|}I_j\cdot h_{k'},
\end{equation}
where the second part, $\sum_{1\leq j\leq |N'|-|N|}I_j\cdot h_{k'}$, indicates the extra cost incurred by deploying honeypots.
Moreover, the privacy budget $\epsilon$ is proportionally increased to $\frac{|N'|}{|N|}\cdot\epsilon$ to cover the extra honeypots.
We use this proportional change of privacy budget, because in Algorithm \ref{alg:deployment2},
the amount of privacy budget is consumed identically in each iteration.
\begin{algorithm}
\caption{Deployment of configurations, $|N'|>|N|$}
\label{alg:deployment2}
\textbf{Input}: $N$ and $|N'_1|,...,|N'_{|K|}|$;\\
Let $|N'_1|=n'_1$, ..., $|N'_{|K|}|=n'_{|K|}$;\\
Let $x_k=p(k)\cdot\sum_{1\leq i\leq |N_k|} u_i$;\\
Initialize $N'_1=...=N'_{|K|}=\emptyset$;\\
Initialize $C_d=0$;\\
Rank $u_1$, ..., $u_{|N|}$ in increasing order, and supposing that the result is $u_1\leq...\leq u_{|N|}$;\\
\For{$i=1$ to $|N'|-|N|$}{
Select a configuration $k'$ with probability proportional to $exp(\frac{\frac{\epsilon}{|N'|-|N|} U_{k'}}{2\Delta U_d})$;\\
$C_d\leftarrow C_d+h_{k'}$;\\
\If{$C_d\leq B_d$}{
Create a honeypot $i$ with configuration $k'$;\\
$N'_{k'}\leftarrow N'_{k'}\cup\{i\}$;\\
}\Else{
\textbf{goto} Line 8;\\
}
}
\For{$i=1$ to $|N|$}{
Select a configuration $k'$ with probability proportional to $exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})$;\\
\If{$C_d+c(k',k)>B_d$}{
\textbf{continue};\\
}
\If{$|N'_{k'}|<n'_{k'}$}{
$N'_{k'}\leftarrow N'_{k'}\cup \{i\}$;\\
$C_d\leftarrow C_d+c(k',k)$;\\
}\Else{
\textbf{goto} Line 16;\\
}
}
\textbf{Output}: $N'_1,...,N'_{|K|}$;\\
\end{algorithm}
The problem faced by the defender is the same as in \emph{Situation 1}.
Algorithm \ref{alg:deployment2} is developed to solve this problem.
In Algorithm \ref{alg:deployment2}, the defender first creates honeypots (Lines 7-14)
and then obfuscates systems (Lines 15-23).
In Line 8, $U_{k'}=p(k')u_{k'}$ while in Line 16, $U_{k'}=p(k')\cdot\sum_{1\leq i\leq |N_{k'}|} u_i$.
In both lines, $\Delta U_d=max_{1\leq i\leq |N|}u_i$.
The defender favors honeypots over obfuscation,
as the use of the former may result in her incurring a utility gain.
Honeypots are created based on the estimated probability with which a configuration will be attacked.
Configurations with the highest estimated probability of being attacked will take priority for use to configure honeypots.
\emph{Situation 3}. When $|N'|<|N|$, there are $|N|-|N'|$ systems taken offline.
The defender's expected utility loss is shown in Equation \ref{eq:utilityloss3},
\begin{equation}\label{eq:utilityloss3}
U_d=\sum_{1\leq k\leq |K|}\sum_{1\leq i\leq |N_k|}[(p(k)\cdot J_i\cdot u_i)+(J_i-1)^2\cdot l_k],
\end{equation}
where $J_i=0$ if system $i$ is taken offline, and $J_i=1$ otherwise.
The second part, $\sum_{1\leq i\leq |N|-|N'|} (J_i-1)^2\cdot l_k$, indicates the extra utility loss incurred by taking systems offline.
The defender's cost is shown in Equation \ref{eq:dcost3},
\begin{equation}\label{eq:dcost3}
C_d=\sum_{1\leq i\leq |N|}I_i\cdot(c(k',k)).
\end{equation}
Moreover, akin to \emph{Situation 2}, the privacy budget $\epsilon$ is proportionally decreased to $\frac{|N'|}{|N|}\cdot\epsilon$,
as the systems taken offline need not be protected.
Thus, this part of the privacy budget can be conserved.
The problem for the defender is the same as in \emph{Situation 1}.
Algorithm \ref{alg:deployment3} is developed to solve this problem.
In Algorithm \ref{alg:deployment3}, the defender first takes $|N|-|N'|$ systems offline (Lines 7-9)
and next obfuscates the remaining systems (Lines 10-18).
In Line 11, $U_{k'}=\sum_{1\leq i\leq |N_k|}[(p(k)\cdot J_i\cdot u_i)+(J_i-1)^2\cdot l_k]$
and $\Delta U_d=max_{1\leq i\leq |N|}u_i$.
The defender will be inclined to take those systems offline
that will incur the highest utility loss if attacked.
\begin{algorithm}
\caption{Deployment of configurations, $|N'|<|N|$}
\label{alg:deployment3}
\textbf{Input}: $N$ and $|N'_1|,...,|N'_{|K|}|$;\\
Let $|N'_1|=n'_1$, ..., $|N'_{|K|}|=n'_{|K|}$;\\
Let $x_k=p(k)\cdot\sum_{1\leq i\leq |N_k|} u_i$;\\
Initialize $N'_1=...=N'_{|K|}=\emptyset$;\\
Initialize $C_d=0$;\\
Rank $u_1$, ..., $u_{|N|}$ in increasing order, and supposing that the result is $u_1\leq...\leq u_{|N|}$;\\
\For{$j=1$ to $|N|-|N'|$}{
$s\leftarrow argMax_{i} [p(k)\cdot u_i-l_k]$;\\
Take system $s$ offline;\\
}
\For{$i=1$ to $|N'|$}{
Select a configuration $k'$ with probability proportional to $exp(\frac{\frac{\epsilon}{|N'|} U_{k'}}{2\Delta U_d})$;\\
\If{$C_d+c(k',k)>B_d$}{
\textbf{continue};\\
}
\If{$|N'_{k'}|<n'_{k'}$}{
$N'_{k'}\leftarrow N'_{k'}\cup \{i\}$;\\
$C_d\leftarrow C_d+c(k',k)$;\\
}\Else{
\textbf{goto} Line 11;\\
}
}
\textbf{Output}: $N'_1,...,N'_{|K|}$;\\
\end{algorithm}
\subsection{The attacker's strategy}\label{sub:Bayesian}
In each round of the game, the attacker first estimates the probability
with which an observed configuration $k'$ is a real configuration $k$ using Bayesian inference.
Next, he decides which observed configuration $k'$ should be attacked,
i.e., he will attack those systems associated with configuration $k'$.
\subsubsection{Step 3: probability estimation}
To estimate the probability that an observed configuration $k'$ is a real configuration $k$: $q(k|k')$,
the attacker uses Bayesian inference:
\begin{equation}\label{eq:Bayes}
q(k|k')=\frac{q(k)\cdot q(k'|k)}{q(k')}.
\end{equation}
In Equation \ref{eq:Bayes}, $q(k)=\frac{|N_k|}{|N|}$, and $q(k')=\frac{|N'_{k'}|}{|N'|}$,
where $q(k)$ is calculated using the attacker's prior knowledge
and $q(k')$ is computed with reference to the attacker's current observations.
In addition, $q(k'|k)$ can be obtained from the attacker's experience, i.e., posterior knowledge.
For example, in previous game rounds, the attacker attacked $10$ systems with configuration $k$.
Among these $10$ systems, four systems are obfuscated to appear as configuration $k'$.
Thus, $q(k'|k)=\frac{4}{10}=0.4$.
\subsubsection{Step 4: target selection}
Based on Equation \ref{eq:Bayes}, the attacker can calculate the expected utility gain of attacking each configuration $k'$:
\begin{equation}\label{eq:utilitygain}
U^{(k')}_a=\sum_{1\leq k\leq |K|}[q(k|k')\cdot u_k].
\end{equation}
According to Equation \ref{eq:utilitygain}, the attacker uses an $epsilon$-greedy strategy to distribute selection probabilities over the observed configurations:
\begin{equation}\label{eq:selection}
\mathcal{P}_{k'}=\begin{cases}(1-e)+\frac{e}{|K|}, \mbox{if $U^{(k')}_a$ is the largest}\\
\frac{e}{|K|}, \mbox{otherwise}\end{cases}.
\end{equation}
where $e$ is a small positive number in $[0,1]$.
The reason for using the $epsilon$-greedy strategy will be described in \textbf{Remark 2} in Section \ref{sub:attacker analysis}.
Finally, the attacker selects a target $k'$ based on the probability distribution $\boldsymbol{\mathcal{P}}=\{\mathcal{P}_{1},...,\mathcal{P}_{|K|}\}$.
\subsection{Potential application of our approach}
Our approach can be applied against powerful attackers in the real world.
An attacker may execute a suite of scan queries on a network using tools such as NMAP \cite{Lyon09}.
Our DP-based approach returns a mix of true and false results to the attacker’s scan to confuse him.
The false results include not only obfuscated configurations of systems but also additional fake systems, i.e., honey pots.
Moreover, our approach can also take vulnerable systems offline to avoid the attacker’s scan.
For example, if an attacker makes a scan query on one system many times,
he may combine the query results to reason the true configuration of this system.
Thus, this system should be taken offline temporarily.
In summary, our approach can significantly increase the complexity on attackers
to formulate their attack, irrespective of their reasoning power,
so that administrators can have additional time to build defense policies to interdict attackers.
A specific real-world example is an advanced persistent threat (APT) attacker \cite{Tankard11}.
Our approach can be adopted to model interactions between a system manager and an APT attacker.
Typically, an APT attacker continuously scans the vulnerability of the target systems
and studies the defense policy of these systems
so as to steal sensitive information from these systems \cite{Yang19}.
Accordingly, the system manager misleads the APT attacker by obfuscating the true information of these systems,
and formulates a complex defense policy for these systems
which is hardly predictable to the APT attacker.
\section{Theoretical analysis}\label{sec:analysis}
Our theoretical analysis consists of two parts:
analysis of the defender's strategy and analysis of the attacker's strategy.
In the analysis of the defender's strategy, we first analyze the privacy preservation of the defender's strategy.
Then, we analyze the defender's expected utility loss in different situations.
After that, the complexity of the defender's strategy is also analyzed.
In the analysis of the attacker's strategy, we analyze the attacker's optimal strategy and the lower bound of his expected utility gain.
Finally, we give a discussion on the equalibria between the defender and the attacker.
\subsection{Analysis of the defender's strategy}
\subsubsection{Analysis of privacy preservation}
We first prove that the defender's strategy satisfies differential privacy.
Then, we give an upper bound on the number of rounds of the game.
Exceeding this bound, the privacy level of the defender cannot be guaranteed.
\begin{lem}[Sequential Composition Theorem \cite{McSherry200794}]\label{comp1}
Suppose a set of privacy steps
$\mathcal{M}=\{\mathcal{M}_1,...,\mathcal{M}_m\}$
are sequentially performed on a dataset,
and each $\mathcal{M}_i$ provides $\epsilon_i$ privacy guarantee,
then $\mathcal{M}$ will provide
$\sum_{1\leq i\leq m}\epsilon_i$-\emph{differential privacy}.
\end{lem}
\begin{lem}[Parallel Composition Theorem \cite{McSherry200794}]\label{comp2}
Suppose we have a set of privacy step
$\mathcal{M}=\{\mathcal{M}_1,...,\mathcal{M}_m\}$,
if each $\mathcal{M}_i$ provides $\epsilon_{i}$ privacy guarantee on
a disjoint subset of the entire dataset,
then the parallel composition of $\mathcal{M}$ will provide
$\max_{1\leq i\leq m}\epsilon_{i}$-\emph{differential privacy}.
\end{lem}
\begin{thm}\label{thm:DP1}
Algorithm \ref{alg:DP} satisfies $\epsilon$-differential privacy.
\end{thm}
\begin{proof}
Algorithm \ref{alg:DP} consumes a privacy budget $\epsilon$.
The Laplace noise sampled from $Lap(\frac{\Delta S\cdot |K|}{\epsilon})$ is added to $|K|$ steps.
At each step, the average allocated privacy budget is $\frac{\epsilon}{|K|}$.
Thus, for each step, Algorithm \ref{alg:DP} satisfies $\frac{\epsilon}{|K|}$-differential privacy.
As there are $|K|$ steps, based on Lemma \ref{comp1},
Algorithm \ref{alg:DP} satisfies $\epsilon$-differential privacy.
\end{proof}
\begin{thm}\label{thm:DP2}
Algorithms \ref{alg:deployment1}, \ref{alg:deployment2} and \ref{alg:deployment3} satisfy $\epsilon$-differential privacy.
\end{thm}
\begin{proof}
We prove that Algorithm \ref{alg:deployment1} satisfies $\epsilon$-differential privacy.
The proof of Algorithms \ref{alg:deployment2} and \ref{alg:deployment3} is similar.
In Line 7 of Algorithm \ref{alg:deployment1}, the selection of a configuration takes place $|N|$ times.
The privacy budget $\epsilon$ is, thus, consumed in $|N|$ steps.
At each step, the average allocated privacy budget is $\frac{\epsilon}{|N|}$.
Hence, for each step, Algorithm \ref{alg:deployment1} satisfies $\frac{\epsilon}{|N|}$-differential privacy.
As there are $|N|$ steps, based on the sequential composition theorem (Lemma \ref{comp1}),
Algorithm \ref{alg:deployment1} satisfies $\epsilon$-differential privacy.
\end{proof}
The processes of Algorithms \ref{alg:deployment1}, \ref{alg:deployment2} and \ref{alg:deployment3} are similar to the NoisyAverage sampling developed in McSherry’s PINQ framework \cite{McSherry09}.
They, however, are different.
The output value of the NoisyAverage sampling is: $Pr[NoisyAvg(A)=x]\propto\max\limits_{avg(B)=x}exp(-\epsilon\times\frac{|A\oplus B|}{2})$,
where $A$ and $B$ are two datasets and $\oplus$ is for symmetric difference.
In our algorithms, the output value is: $Pr[output=k']\propto exp(\frac{\frac{\epsilon}{|N|}U_{k'}}{2\Delta U_d})$.
The major difference between the NoisyAverage sampling and ours is that
in the NoisyAverage sampling, the output of a value $x$ is based on an operation of $\max\limits_{avg(B)=x}$,
while in our algorithm, the output of a value $k'$ is based on its utility $U_{k'}$.
The operation of $\max\limits_{avg(B)=x}$ may violate the definition of the exponential mechanism and
thus lead the NoisyAverage sampling to fail to satisfy differential privacy.
By comparison, our algorithms strictly follow the definition of the exponential mechanism (ref. page 39, Definition 3.4 in \cite{Dwork14}).
Thus, our algorithms satisfy differential privacy.
\begin{thm}\label{thm:DP}
The defender is guaranteed $\epsilon$-differential privacy.
\end{thm}
\begin{proof}
The defender uses Algorithms \ref{alg:deployment1}, \ref{alg:deployment2} and \ref{alg:deployment3} for deployment of configurations.
The three algorithms are disjoint
because they are applied to three mutually exclusive situations:
$|N'|=|N|$, $|N'|>|N|$ and $|N'|<|N|$.
Therefore, the parallel composition theorem (Lemma \ref{comp2}) can be used here.
Since all of these algorithms satisfy $\epsilon$-differential privacy,
the defender is guaranteed $\epsilon$-differential privacy.
\end{proof}
\begin{cor}\label{cor:strong}
The attacker cannot deduce the exact configurations of the systems.
\end{cor}
\begin{proof}
According to Theorem \ref{thm:DP}, the defender is guaranteed $\epsilon$-differential privacy.
Based on the description of differential privacy in Section \ref{sub:DP},
differential privacy guarantees that the attacker cannot tell whether a particular system is associated with configuration $k$,
because whether or not the system is associated with configuration $k$ makes little difference to the output observations.
Since the attacker cannot deduce the configuration of each individual system,
he cannot deduce the exact configurations of the systems in the network.
\end{proof}
\textbf{Remark 1.} Corollary \ref{cor:strong} guarantees the privacy of the defender in a single game round.
However, if no bound on the number of rounds is set,
the defender's privacy may be leaked \cite{Dwork14},
i.e., that the attacker figures out the real configurations of the systems.
This is because in differential privacy,
a privacy budget is used to control the privacy level.
Every time the system configuration is obfuscated and released,
the privacy budget is partially consumed.
Once the privacy budget is used up,
differential privacy cannot guarantee the privacy of the defender anymore.
To guarantee the privacy level of the defender,
we need to set a bound on the number of rounds.
\begin{definition}[KL-Divergence \cite{Dwork14}]\label{Def:KL}
The KL-Divergence between two random variables $Y$ and $Z$ taking values from the same domain is defined to be:
\begin{equation}
D(Y||Z)=\mathbb{E}_{y\sim Y}\left[ln\frac{Pr(Y=y)}{Pr(Z=y)}\right].
\end{equation}
\end{definition}
\begin{definition}[Max Divergence \cite{Dwork14}]\label{Def:Max}
The Max Divergence between two random variables $Y$ and $Z$ taking values from the same domain is defined to be:
\begin{equation}
D_{\infty}(Y||Z)=\max\limits_{S\subseteq Supp(Y)}\left[ln\frac{Pr(Y\in S)}{Pr(Z\in S)}\right].
\end{equation}
\end{definition}
\begin{lem}[\cite{Dwork14}]\label{lem:DPDivergence}
A mechanism $\mathcal{M}$ is $\epsilon$-differentially private
if and only if on every two neighboring datasets $x$ and $x'$,
$D_{\infty}(\mathcal{M}(x)||\mathcal{M}(x'))\leq\epsilon$
and $D_{\infty}(\mathcal{M}(x')||\mathcal{M}(x))\leq\epsilon$.
\end{lem}
\begin{lem}[\cite{Dwork14}]\label{lem:KLMax}
Suppose that random variables $Y$ and $Z$ satisfy $D_{\infty}(Y||Z)\leq\epsilon$
and $D_{\infty}(Z||Y)\leq\epsilon$.
Then, $D(Y||Z)\leq\epsilon\cdot (e^{\epsilon}-1)$.
\end{lem}
\begin{thm}\label{thm:bound}
Given that the defender's privacy level is $\epsilon$ at each single round,
to guarantee the defender's overall privacy level to be $\epsilon'$,
the upper bound of the number of rounds is $\frac{\epsilon '\cdot(e^{\epsilon'}-1)}{\epsilon\cdot(e^{\epsilon}-1)}$.
\end{thm}
\begin{proof}
Let the upper bound of the number of rounds be $k$ and
the view of the attacker in the $k$ rounds be $v=(v_1,...,v_k)$.
We have $D(Y||Z)=ln\left[\frac{Pr(Y=v)}{Pr(Z=v)}\right]=ln\left[\prod^k_{i=1}\frac{Pr(Y_i=v_i)}{Pr(Z_i=v_i)}\right]=\sum^k_{i=1}ln\left[\frac{Pr(Y_i=v_i)}{Pr(Z_i=v_i)}\right]=\sum^k_{i=1}D(Y_i||Z_i)$.
As the defender is guaranteed $\epsilon$-differential privacy at each single round,
based on Lemma \ref{lem:DPDivergence},
we have $D_{\infty}(Y_i||Z_i)\leq\epsilon$ and $D_{\infty}(Z_i||Y_i)\leq\epsilon$.
Based on this result, according to Lemma \ref{lem:KLMax},
we have $D(Y_i||Z_i)\leq\epsilon\cdot(e^{\epsilon}-1)$.
Thus, we have $D(Y||Z)=\sum^k_{i=1}D(Y_i||Z_i)\leq k\cdot\epsilon\cdot(e^{\epsilon}-1)$.
To guarantee the defender's overall privacy level to be $\epsilon '$,
according to Lemma \ref{lem:DPDivergence},
we have $D_{\infty}(Y||Z)\leq\epsilon'$ and $D_{\infty}(Z||Y)\leq\epsilon'$.
By using Lemma \ref{lem:KLMax},
we have $D(Y||Z)\leq\epsilon'\cdot(e^{\epsilon'}-1)$.
Finally, since $D(Y||Z)\leq k\cdot\epsilon\cdot(e^{\epsilon}-1)$ and $D(Y||Z)\leq\epsilon'\cdot(e^{\epsilon'}-1)$,
the upper bound $k$ is limited by $\frac{\epsilon'\cdot(e^{\epsilon'}-1)}{\epsilon\cdot(e^{\epsilon}-1)}$.
\end{proof}
\subsubsection{Analysis of the deployment algorithms}
Here, we compute the expected utility loss of the defender in three situations:
$|N'|=|N|$, $|N'|>|N|$ and $|N'|<|N|$.
\begin{thm}\label{thm:DUtility1}
In Situation 1 ($|N'|=|N|$), the defender's expected utility loss is $\sum_{1\leq i\leq |N|}[u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'}]$,
where $p_{k'}=\frac{exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})}{\sum_{1\leq k\leq |K|}exp(\frac{\frac{\epsilon}{|N|} U_{k}}{2\Delta U_d})}$,
if the defender uses Algorithm \ref{alg:deployment1} to deploy configurations
and the attacker uses Equation \ref{eq:selection} to select the target.
\end{thm}
\begin{proof}
In Algorithm \ref{alg:deployment1}, the defender deploys the systems based on 1) the utility of each system
and 2) the estimated probability that the attacker will select each configuration to attack.
By using the exponential mechanism,
each system $i$ is obfuscated to appear as configuration $k'$ with probability
$\frac{exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})}{\sum_{1\leq k\leq |K|}exp(\frac{\frac{\epsilon}{|N|} U_{k}}{2\Delta U_d})}$.
The attacker thus selects configuration $k'$ as a target with probability $\mathcal{P}_{k'}$.
System $i$ is attacked only when the defender obfuscates $i$ to appear as configuration $k'$ and
the attacker selects $k'$ as a target.
As there are $|K|$ configurations,
the defender's expected utility loss on system $i$ is $u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'}$.
Since there are $|N|$ systems,
the defender's total expected utility loss is $\sum_{1\leq i\leq |N|}[u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'}]$.
\end{proof}
\begin{cor}\label{cor:DUtility1}
In Situation 1, the upper bound of the defender's utility loss is $\sum_{1\leq i\leq |N'_l|}u_i$,
where $|N'_l|=max_{1\leq k'\leq |K|}|N'_{k'}|$. \\
$\sum_{1\leq i\leq |N'_l|}u_i$ represents the utility sum of $|N'_l|$ systems, which have the highest utility among all the systems.
\end{cor}
\begin{proof}
In Algorithm \ref{alg:deployment1}, the defender deploys each system in order of increasing utility.
Therefore, it is possible for the defender to deploy the $|N'_l|$ systems in configuration $l$,
where $|N'_l|=max_{1\leq k'\leq |K|}|N'_{k'}|$ and the $|N'_l|$ systems have the highest utility among all the systems.
When configuration $l$ is selected by the attacker as a target,
the utility of all systems in $N'_l$ is lost;
this amounts to $\sum_{1\leq i\leq |N'_l|}u_i$.
As this is the worst case for the defender in Situation 1,
the utility loss in this case is the upper bound.
\end{proof}
Similarly, we can also draw the following conclusions.
\begin{thm}\label{thm:DUtility2}
In Situation 2 ($|N'|>|N|$), the defender's expected utility loss is $\sum_{1\leq i\leq |N'|}[\frac{2|N|-|N'|}{|N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})]$,
where $p_{k'}=\frac{exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})}{\sum_{1\leq k\leq |K|}exp(\frac{\frac{\epsilon}{|N|} U_{k}}{2\Delta U_d})}$,
if the defender uses Algorithm \ref{alg:deployment2} to deploy configurations
and the attacker uses Equation \ref{eq:selection} to select the target.
\end{thm}
\begin{proof}
By comparing the results of Theorems \ref{thm:DUtility1} and \ref{thm:DUtility2},
the difference is that in Theorem \ref{thm:DUtility2}, there is an extra coefficient, $\frac{2|N|-|N'|}{|N'|}$.
In Situation 2, there are $|N'|-|N|$ honeypots.
Hence, a system $i$ could be a honeypot, with probability $\frac{|N'|-|N|}{|N'|}$,
or a genuine system, with probability $\frac{|N|}{|N'|}$.
When system $i$ is attacked,
the expected utility loss on system $i$ is $\frac{|N|}{|N'|}[u_i\cdot\sum_{1\leq k'\leq |K|}p(k')\mathcal{P}(k')]-\frac{|N'|-|N|}{|N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})$,
where $\frac{|N'|-|N|}{|N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})$ means a utility gain if system $i$ is a honeypot.
As there are $|N'|$ systems, including honeypots,
the expected utility loss is\\ $\sum_{1\leq i\leq |N'|}[\frac{2|N|-|N'|}{|N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})]$.
\end{proof}
\begin{cor}\label{cor:DUtility2}
In Situation 2, the upper bound of the defender's utility loss is $\sum_{1\leq i\leq |N'_l|}u_i$,
where $|N'_l|=max_{1\leq k'\leq |K|}|N'_{k'}|$. \\
$\sum_{1\leq i\leq |N'_l|}u_i$ represents the utility sum of the $|N'_l|$ systems that have the highest utility among all the systems.
\end{cor}
\begin{proof}
Although Situation 2 is different from Situation 1,
the worst case in Situation 2 is the same as that in Situation 1.
This is because in Situation 2, all genuine systems are still in the network.
In the worst case, the attacker attacks all of the systems with the highest utility.
The result, thus, is the same as that in Situation 1.
\end{proof}
\begin{thm}\label{thm:DUtility3}
In Situation 3 ($|N'|<|N|$), the defender's expected utility loss is
\\$\sum_{1\leq i\leq |N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})-\sum_{1\leq j\leq |N|-|N'|}(u_j-l_j)$,
where $p_{k'}=\frac{exp(\frac{\frac{\epsilon}{|N|} U_{k'}}{2\Delta U_d})}{\sum_{1\leq k\leq |K|}exp(\frac{\frac{\epsilon}{|N|} U_{k}}{2\Delta U_d})}$,
if the defender uses Algorithm \ref{alg:deployment3} to deploy configurations
and the attacker uses Equation \ref{eq:selection} to select the target.
\end{thm}
\begin{proof}
By comparing the results of Theorems \ref{thm:DUtility1} and \ref{thm:DUtility3},
we can see that there is an extra part in Theorem \ref{thm:DUtility3}, $\sum_{1\leq j\leq |N|-|N'|}(u_j-l_j)$.
This part means that $|N|-|N'|$ systems are taken offline.
Thus, the utility of these systems will not be lost.
\end{proof}
\begin{cor}\label{cor:DUtility3}
In Situation 3, the upper bound of the defender's utility loss is $\sum_{1\leq i\leq |N'_l|}u_i$,
where $|N'_l|=max_{1\leq k'\leq |K|}|N'_{k'}|$.\\
$\sum_{1\leq i\leq |N'_l|}u_i$ represents the utility sum of the $|N'_l|$ systems
that have the highest utility among the remaining $|N'|$ systems.
\end{cor}
\begin{proof}
In Situation 3, $|N|-|N'|$ systems are taken offline.
These systems, therefore, will not be attacked.
Hence, in the worst case, the $|N'_l|$ systems with the highest utility among the remaining $|N'|$ systems are attacked.
\end{proof}
\subsubsection{Analysis of the complexity of the algorithms}
The proposed approach includes four algorithms.
The analysis of their complexity is given as follows.
\begin{thm}\label{thm:Agl1Complexity}
The computational complexity of Algorithm \ref{alg:DP} is $O(|N|\cdot|K|)$,
where $|N|$ is the number of systems and $|K|$ is the number of configurations.
\end{thm}
\begin{proof}
In Line 3 of Algorithm \ref{alg:DP}, $|N|$ systems are partitioned into $|K|$ categories.
This partition implicitly involves a nested loop,
where the number of iterations is $|N|\cdot|K|$.
In Lines 4-5 of Algorithm \ref{alg:DP}, the number of iterations in this loop is $|K|$.
Thus, the overall number of iterations is $|N|\cdot|K|+|K|=(|N|+1)\cdot|K|$.
The complexity of Algorithm \ref{alg:DP} is $O(|N|\cdot|K|)$.
\end{proof}
\begin{thm}\label{thm:Agl2Complexity}
The computational complexity of Algorithm \ref{alg:deployment1} is $O(|N|^2)$,
where $|N|$ is the number of systems.
\end{thm}
\begin{proof}
In Line 5 of Algorithm \ref{alg:deployment1}, $|N|$ systems are ranked in an increasing order based on their utilities.
The number of iterations in this ranking is $\frac{|N|\cdot(|N|-1)}{2}$.
Moreover, in Lines 7 to 14, the number of iterations in this loop is $|N|$.
Thus, the overall number of iterations is $\frac{|N|\cdot(|N|-1)}{2}+|N|=\frac{|N|\cdot(|N|+1)}{2}$.
The complexity of Algorithm \ref{alg:deployment1}, therefore, is $O(|N|^2)$.
\end{proof}
\begin{thm}\label{thm:Agl3Complexity}
The computational complexity of Algorithm \ref{alg:deployment2} is $O(|N|^2)$,
where $|N|$ is the number of systems.
\end{thm}
\begin{proof}
In Line 6 of Algorithm \ref{alg:deployment2}, $|N|$ systems are ranked in an increasing order based on their utilities.
The number of iterations in this ranking is $\frac{|N|\cdot(|N|-1)}{2}$.
Moreover, in Lines 7 to 14, the number of iterations in this loop is $|N'|-|N|$,
where $|N'|$ is the number of systems after obfuscation.
Also, in Lines 15 to 23, the number of iterations in this loop is $|N|$.
Thus, the overall number of iterations is $\frac{|N|\cdot(|N|-1)}{2}+|N'|$.
Since $|N'|$ and $|N|$ are in the same scale,
the complexity of Algorithm \ref{alg:deployment1} is $O(|N|^2)$.
\end{proof}
\begin{thm}\label{thm:Agl4Complexity}
The computational complexity of Algorithm \ref{alg:deployment3} is $O(|N|^2)$,
where $|N|$ is the number of systems.
\end{thm}
\begin{proof}
In Line 6 of Algorithm \ref{alg:deployment3}, $|N|$ systems are ranked in an increasing order based on their utilities.
The number of iterations in this ranking is $\frac{|N|\cdot(|N|-1)}{2}$.
Moreover, in Lines 7 to 9, the number of iterations in this loop is $|N|-|N'|$,
where $|N'|$ is the number of systems after obfuscation.
Also, in Lines 10 to 18, the number of iterations in this loop is $|N'|$.
Thus, the overall number of iterations is $\frac{|N|\cdot(|N|-1)}{2}+|N|=\frac{|N|\cdot(|N|+1)}{2}$.
The complexity of Algorithm \ref{alg:deployment1}, therefore, is $O(|N|^2)$.
\end{proof}
\subsection{Analysis of the attacker's strategy}\label{sub:attacker analysis}
We first analyze the attacker's optimal strategy
and then compute the lower bound of his expected utility gain.
\textbf{Remark 2} (the attacker's expected utility gain).
As the cyber deception game is a zero-sum game in Situations 1 ($|N'|=|N|$) and 2 ($|N'|>|N|$),
the defender's expected utility loss is the attacker's expected utility gain.
In Situation 3 ($|N'|<|N|$), since the attacker does not attack the offline systems,
his expected utility gain is based only on the utility of the remaining $|N'|$ systems,
which is $\sum_{1\leq i\leq |N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})$.
\begin{thm}\label{thm:AStrategy}
The attacker's optimal strategy is the solution to the following problem:
given $U^{(1)}_a$, ..., $U^{(|K|)}_a$, find a probability distribution, $\mathcal{P}_1$, ..., $\mathcal{P}_{|K|}$,
which can maximize $\sum_{1\leq k'\leq |K|}\mathcal{P}_{k'}\cdot U^{(k')}_a$.
\end{thm}
\begin{proof}
According to the discussion in \textbf{Remark 2},
the major component of the attacker's expected utility gain is $\sum_{1\leq i\leq |N'|}(u_i\cdot\sum_{1\leq k'\leq |K|}p_{k'}\mathcal{P}_{k'})$.
In this component, the only uncertain part is $\mathcal{P}(k')$.
The calculation of $\mathcal{P}(k')$ is based on Equations \ref{eq:Bayes} and \ref{eq:utilitygain}.
In Equation \ref{eq:Bayes}, $q(k|k')=\frac{q(k)\cdot q(k'|k)}{q(k')}$, $q(k)$ and $q(k')$ are known to the attacker,
while $q(k'|k)$ can be calculated by the attacker based on the defender's strategy.
Hence, Equation \ref{eq:utilitygain}, $U^{(k')}_a=\sum_{1\leq k\leq |K|}[q(k|k')\cdot u_k]$, can also be computed by the attacker.
Since the attacker's aim is to maximize his expected utility gain,
he needs to identify the appropriate probability distribution, $\mathcal{P}_{1}$, ..., $\mathcal{P}_{|K|}$, for use as his attack strategy
in order to maximize $\sum_{1\leq k'\leq |K|}\mathcal{P}_{k'}\cdot U^{(k')}_a$.
\end{proof}
\textbf{Remark 3} (The attacker's risk).
According to Theorem \ref{thm:AStrategy}, the attacker's optimal strategy should be to attack configuration $k'$ with probability $1$,
where $U^{(k')}_a$ is the maximum among $U^{(1)}_a$, ..., $U^{(|K|)}_a$.
However, such a deterministic strategy is highly predictable to the defender.
The attacker has to use a mixed strategy
that is as close as possible to the deterministic strategy.
The $epsilon$-greedy strategy is a good solution,
as it is similar to the deterministic strategy while also exhibits a reasonable degree of randomness.
In the $epsilon$-greedy strategy, the maximum $U^{(k')}_a$ is given a high probability,
while the remainder shares the remaining probability.
\begin{thm}\label{thm:AUtiligy}
Let $U^{(1)}_a\geq U^{(2)}_a\geq... U^{(|K|)}_a$,
the lower bound of the attacker's utility gain is $\frac{1}{|K|}\sum_{1\leq k'\leq |K|}U^{k'}_a$.
\end{thm}
\begin{proof}
By using the $epsilon$-greedy strategy,
the attacker's expected utility gain is \\
$U^{(1)}_a[(1-e)+\frac{e}{|K|}]+\frac{e}{|K|}\sum_{2\leq k'\leq |K|}U^{(k')}_a$\\
$=(1-e)U^{(1)}_a+\frac{e}{|K|}\sum_{1\leq k'\leq |K|}U^{(k')}_a$.
This expected utility gain decreases monotonically with the increase of the value of $e$.
Thus, when $e=1$, the expected utility reaches the minimum: $\frac{1}{|K|}\sum_{1\leq k'\leq |K|}U^{k'}_a$.
\end{proof}
\subsection{Analysis of the equilibria}
We provide a characterization of equilibria
and leave the deep investigation of equilibria in this highly dynamic security game as one of our future studies.
In our approach, the number of systems may change at each round
due to the introduction of Laplace noise (recall Algorithm \ref{alg:DP}).
Therefore, the available strategies to the defender also change at each round,
where each strategy is interpreted as an assignment of systems to configurations, i.e., obfuscation.
By comparison, the available strategies to the attacker is fixed in the game,
because 1) the number of configurations is fixed;
and 2) the attacker selects only one configuration to attack at each round,
i.e., attacking the systems associated with a configuration.
Formally, let $S$ be the set of all potentially attainable strategies of the defender,
and $S^t\subset S$ be the set of the defender's strategies at round $t$.
Correspondingly, let $A^t\in\mathbb{R}^{m_t\times n}$ be the payoff matrix at round $t$,
where $m_t=|S^t|$ is the number of the defender's strategies at round $t$
and $n$ is the number of the attacker's strategies.
Let $\Delta^t_d$ be the set of probability distributions over the defender's available strategies at round $t$
and $\Delta_a$ be the set of probability distributions over the attacker's available strategies.
Then, at round $t$, if the defender uses a probability distribution $\mathbf{x}\in\Delta^t_d$
and the attacker uses a probability distribution $\mathbf{y}\in\Delta_a$,
the defender's utility loss is $\mathbf{x}A^t\mathbf{y}$.
The defender wishes to minimize the quantity of $\mathbf{x}A^t\mathbf{y}$,
while the attacker wants to maximize it.
This is a \emph{minimax} problem.
The solution of this problem is an equilibrium of the game \cite{Gilpin08}.
However, the defender's strategy set $S^t$ at each round $t$ is randomly generated using the differentially private Laplace mechanism
and $S^t$ is unknown to the attacker.
Hence, the attacker is difficult to precisely compute the solution
which can force the defender not to unilaterally change her strategy.
This finding has also been demonstrated in our experimental results
which will be given in detail in Section \ref{sec:experiment}.
\section{Experiment and Analysis}\label{sec:experiment}
\subsection{Experimental setup}
In the experiments, three defender approaches are evaluated,
which are our approach, referred to as \emph{DP-based},
a deterministic greedy approach \cite{Schlenker18}, referred to as \emph{Greedy},
and a mixed greedy approach, referred to as \emph{Greedy-Mixed}.
The \emph{Greedy} approach obfuscates systems using a greedy and deterministic strategy to minimize the defender's expected utility loss.
The \emph{Greedy-Mixed} approach obfuscates systems using a greedy and mixed strategy,
where a high utility system is obfuscated to a low utility system with a certain probability.
Two metrics are used to evaluate the three approaches: the attacker's utility gain and the defender's cost.
For the first metric, the attacker's utility gain, its amount is the same as the amount of the defender's utility loss.
We use the metric attacker’s utility gain by following reference \cite{Schlenker18}.
Certainly, using the defender’s utility loss as a metric will reach the same results.
For the second metric, the defender's cost, it is the defender’s deployment cost
used to obfuscate configurations of systems.
Against the three defender approaches, two attacker approaches are adopted,
which are the Bayesian inference approach and the deep reinforcement learning (DRL) approach.
The Bayesian inference approach has been described in Section \ref{sub:Bayesian}.
The DRL approach involves a deep neural network
which consists of an input layer, two hidden layers and an output layer.
The input is the state of the attacker,
which is defined as the selected configurations in the last eight rounds
and the corresponding utility received by attacking the systems in these configurations.
The output is the selected configuration
that the attacker will attack in this round.
Moreover, each of the hidden layer has ten neurons.
The hyper-parameters used in DRL are listed as follows:
learning rate $\alpha=0.1$, discount factor $\gamma=0.9$,
$epsilon$-greedy selection probability $epsilon=0.9$,
$batch\_size=32$ and $memory\_size=2000$.
These two techniques, Bayesian inference and deep reinforcement learning,
are powerful enough and prevalent in cybersecurity \cite{Xie10,Nguyen19b,Chivukula20}.
By comparison, deep reinforcement learning is more powerful than Bayesian inference,
but it is more complex than Bayesian inference and it needs a long training period.
Therefore, we take both of them into our evaluation.
The experiments are conducted in four scenarios.
\noindent\textbf{Scenario 1}: The numbers of systems and configurations vary,
but the average number of systems associated with each configuration is fixed.
This scenario is used to evaluate the scalability of the three approaches.
It simulates the real-world situation that
different organizations have different scales of networks.
\noindent\textbf{Scenario 2}: The number of systems is fixed,
but the number of configurations varies.
Thus, the average number of systems associated with each configuration also varies.
This scenario is used to evaluate the adaptivity of the three approaches.
It simulates the real-world situation that
an organization updates the configurations of systems from time to time.
\noindent\textbf{Scenario 3}: The number of both systems and configurations is fixed,
but the value of the defender's deployment budget $B_d$ varies.
This scenario is used to evaluate the impact of deployment budget value on the performance of the three approaches.
It simulates the real-world situation that
different organizations have different deployment budgets
or an organization adjusts its deployment budget sometimes.
\noindent\textbf{Scenario 4}: The number of both the systems and configurations is fixed,
but the value of the privacy budget $\epsilon$ varies.
This scenario is used to evaluate the impact of the privacy budget value on the performance of our approach.
It simulates the real-world situation that
different organizations have different requirements of privacy level
and thus requires different privacy budgets.
In a given experimental setting, as arbitrarily changing the number of systems may incur security issues,
in each round we use the differential privacy mechanism (Algorithm \ref{alg:DP})
to compute the number of systems in each configuration.
Therefore, the number of systems is strategically determined and their privacy can be guaranteed.
The value of $\Delta S$ is set to $1$.
The utility of a configuration, $u_k$, is uniformly drawn from $[1,20]$.
The defender's obfuscation cost, $c(k',k)$, is set to $0.1\cdot u'_k$.
The defender's honeypot deployment cost, $h_k$, is set to $0.2\cdot u_k$.
The defender's utility loss, $l_k$, for taking a system with configuration $k$ offline is set to $0.15\cdot u_k$.
These settings are experimentally set up to yield the best results.
Here, ``best'' means the most representative.
Increasing these costs will increase both the cost and utility loss of the defender.
On one hand, these costs are used by the defender to obfuscate configurations of systems and deploy honeypots.
Thus, increasing these costs will increase the defender’s cost.
On the other hand, increasing these costs will result in the defender using up her deployment budget too early.
The defender then cannot perform obfuscation or deployment.
Thus, her utility loss will increase.
On the contrary, decreasing these costs will decrease both the cost and utility loss of the defender.
However, excessively decreasing these costs will render the experimental results meaningless.
Therefore, we set representative parameter values to
balance the defender’s performance and the meanings of the experimental results.
The experimental results are obtained by averaging $1000$ rounds of the game.
\subsection{Experimental results}
\subsubsection{Scenario 1}
\begin{figure}[ht]
\vspace{-6mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility in different scales of networks}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1Utility.pdf}
\label{fig:S1Utility}}
\subfigure[\scriptsize{The defender's cost in different scales of networks}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1Cost.pdf}
\label{fig:S1Cost}}\\[2ex]
\end{minipage}
\vspace{-5mm}
\caption{The three approaches' performance in Scenario 1}
\vspace{-2mm}
\label{fig:S1}
\end{figure}
Fig. \ref{fig:S1} demonstrates the performance of the three approaches in Scenario 1.
The number of systems varies from $50$ to $250$,
and the number of configurations varies from $5$ to $25$, accordingly.
The privacy budget $\epsilon$ is fixed at $0.3$.
The cost budget $B_d$ is fixed at $1000$ for each round.
As the number of systems increases, across the three approaches,
the attacker's utility gain increases slightly and linearly
while the defender's cost increases sub-linearly.
Along with the increase in the number of systems, the number of configurations also increases.
More configurations gives the attacker more choices.
By using Bayesian inference, the attacker can choose a configuration with high utility, to attack.
Thus, the attacker's utility gain increases linearly.
As the number of systems increases, according to Algorithms \ref{alg:deployment1}, \ref{alg:deployment2} and \ref{alg:deployment3},
the defender deals with more systems.
Thus, the defender's cost inevitably increases provided that the budget $B_d$ is large enough.
Comparing our \emph{DP-based} approach to the \emph{Greedy} and \emph{Greedy-Mixed} approaches,
the attacker in the \emph{DP-based} approach achieves about $30\%$ and $12\%$ less utility than in the \emph{Greedy} and \emph{Greedy-Mixed} approaches, respectively.
Moreover, the defender in the \emph{DP-based} approach incurs about $3\%$ and $5\%$ more cost than in the \emph{Greedy} and \emph{Greedy-Mixed} approaches, respectively.
In the \emph{DP-based} approach, the defender can not only obfuscate systems,
but can also deploy honeypots to attract the attacker.
The cost of deploying a honeypot exceeds that of obfuscating a system.
However, a honeypot results in a negative utility to the attacker,
while obfuscating a system can only reduce the attacker's utility gain.
By comparison, the defender in the \emph{Greedy} and \emph{Greedy-Mixed} approaches only obfuscates systems.
The difference between defender's costs among the three approaches is negligible.
This demonstrates that in our \emph{DP-based} approach,
the defender uses almost the same cost as the other two approaches
to achieve much better results, i.e., lowering the attacker’s utility gain.
\begin{figure}[ht]
\vspace{-6mm}
\centering
\begin{minipage}{0.51\textwidth}
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{DP-based}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1DPUtility.pdf}
\label{fig:S1DPUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{DP-based}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1DPCost.pdf}
\label{fig:S1DPCost}}\\[2ex]
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{Greedy}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyUtility.pdf}
\label{fig:S1GreedyUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{Greedy}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyCost.pdf}
\label{fig:S1GreedyCost}}\\[2ex]
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyMixedUtility.pdf}
\label{fig:S1GreedyMixedUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyMixedCost.pdf}
\label{fig:S1GreedyMixedCost}}\\%[2ex]
\end{minipage}
\vspace{-4mm}
\caption{The three approaches' performance as game progress in Scenario 1}
\vspace{-2mm}
\label{fig:S1Round}
\end{figure}
Fig. \ref{fig:S1Round} shows the variation in the attacker's utility gain
and the defender's cost in the three approaches as the game progresses in Scenario 1.
The number of systems is fixed at $100$,
and the number of configurations is fixed at $10$.
In Fig. \ref{fig:S1GreedyUtility}, which depicts the \emph{Greedy} approach, there are a number of plateaus.
These plateaus indicate the steadiness of the attacker's utility gain,
which implies that the attacker has predicted the defender's strategy.
The attacker can thus adopt an optimal strategy to maximize his utility gain.
By comparison, in Fig. \ref{fig:S1DPUtility} and Fig. \ref{fig:S1GreedyMixedUtility},
which depicts the \emph{DP-based} and \emph{Greedy-Mixed} approaches, respectively, there is no plateau.
This means that the attacker cannot predict the defender's strategy
and adopts only random strategies.
This finding also demonstrates that equilibria may not exist between the defender and attacker
if the defender uses our \emph{DP-based} approach,
as the attacker’s utility fluctuates all the time with no equilibrium.
However, when the defender uses the \emph{Greedy} approach
which does not change the available strategies of the defender,
the attacker can predict the defender’s strategy and may reach an equilibrium.
Particularly, by comparing Figs. \ref{fig:S1DPUtility}, \ref{fig:S1GreedyUtility} and \ref{fig:S1GreedyMixedUtility},
we can see that the shape of Fig. \ref{fig:S1GreedyMixedUtility} is more similar to Fig. \ref{fig:S1GreedyUtility} than Fig. \ref{fig:S1DPUtility}.
This implies that although the defender in the \emph{Greedy-Mixed} approach uses a mixed strategy,
the attacker can still predict the defender's strategy to some extent.
By comparing Figs. \ref{fig:S1DPCost}, \ref{fig:S1GreedyCost} and \ref{fig:S1GreedyMixedCost},
we can see that the variation of the defender's cost in the \emph{Greedy} and \emph{Greedy-Mixed} approaches
is more stable than the \emph{DP-based} approach.
This is due to the fact that in the \emph{DP-based} approach,
the defender can deploy honeypots which incurs extra cost.
However, as shown in Fig. \ref{fig:S1Cost},
the defender's average cost in the \emph{DP-based} approach still stays at a relatively low level
in comparison with the \emph{Greedy} and \emph{Greedy-Mixed} approaches.
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.51\textwidth}
\subfigure[\scriptsize{The attacker's utility with DRL in different scales of networks}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1UtilityDRL.pdf}
\label{fig:S1DRL}}
\subfigure[\scriptsize{The attacker's utility with DRL as game progresses in \emph{DP-based}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1DPUtilityDRL.pdf}
\label{fig:S1DPUtilityDRL}}\\[2ex]
\subfigure[\scriptsize{The attacker's utility with DRL as game progresses in \emph{Greedy}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyUtilityDRL.pdf}
\label{fig:S1GreedyUtilityDRL}}
\subfigure[\scriptsize{The attacker's utility with DRL as game progresses in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S1GreedyMixedUtilityDRL.pdf}
\label{fig:S1GreedyMixedUtilityDRL}}
\end{minipage}
\vspace{-4mm}
\caption{The three approaches against the DRL attacker in Scenario 1}
\vspace{-2mm}
\label{fig:S1RoundDRL}
\end{figure}
Fig. \ref{fig:S1RoundDRL} demonstrates the performance of the three defender approaches against the DRL attacker.
Compared to the Bayesian inference attacker (Figs. \ref{fig:S1} and \ref{fig:S1Round}), the DRL attacker can obtain more utility,
when the defender adopts either the \emph{Greedy} or the \emph{Greedy-Mixed} approach.
This is because the \emph{Greedy} approach is deterministic
and thus the \emph{Greedy} defender's strategies are easy to be learned by the DRL attacker.
Although the \emph{Greedy-Mixed} approach introduces randomization to some extent,
the major component of the approach is still greedy.
Therefore, it is still not difficult for a powerful DRL attacker to learn
the \emph{Greedy-Mixed} defender's strategies.
However, when the defender employs our \emph{DP-based} approach,
the two types of attackers obtain almost the same utility.
To explain, our \emph{DP-based} approach introduces differentially private random noise into the configurations.
As analyzed in Section V-A, with the protection of differential privacy,
it is hard for an attacker to deduce the \emph{DP-based} defender's strategies
irrespective of the attacker's reasoning power.
Against the two types of attackers,
the defender uses almost the same cost.
This is because in the \emph{Greedy} and \emph{Greedy-Mixed} approaches,
defender's strategies are independent of the attacker's strategies.
Thus, the defender's cost is independent of the attacker's types.
In our \emph{DP-based} approach, the defender's strategies do take the attacker's strategies into consideration.
However, due to the use of differential privacy mechanisms,
the utility loss of the defender against the two attacker approaches is almost the same as shown in Figs. \ref{fig:S1Utility} and \ref{fig:S1DRL},
given that the defender's utility loss is identical to the attacker's utility gain.
Hence, according to Equations \ref{eq:utilityloss1}, \ref{eq:utilityloss2}, \ref{eq:utilityloss3}
and Algorithms \ref{alg:deployment1}, \ref{alg:deployment2}, \ref{alg:deployment3},
as the utility loss of the defender stays steady,
the defender's strategies are not affected much.
Thus, the defender's cost remains almost the same.
For simplicity, the figures regarding the defender's cost spent against the DRL attacker are not presented.
Moreover, in the remaining scenarios, the performance variation tendency of the three defender approaches against the DLR attacker
is similar to the tendency shown in Fig. \ref{fig:S1RoundDRL}.
For simplicity and clarity, they are not included in the paper.
\subsubsection{Scenario 2}
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility with different numbers of configurations}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2Utility.pdf}
\label{fig:S2Utility}}
\subfigure[\scriptsize{The defender's cost with different numbers of configurations}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2Cost.pdf}
\label{fig:S2Cost}}\\[2ex]
\end{minipage}
\vspace{-4mm}
\caption{The three approaches' performance in Scenario 2}
\vspace{-2mm}
\label{fig:S2}
\end{figure}
Fig. \ref{fig:S2} demonstrates the performance of the three approaches in Scenario 2.
The number of systems is fixed at $150$,
and the number of configurations varies from $5$ to $25$.
The privacy budget $\epsilon$ is fixed at $0.3$.
The cost budget $B_d$ is fixed at $1000$ for each round.
With the increase of the number of configurations,
the attacker's utility gain decreases while the defender's cost increases.
In Scenario 2, since the number of systems is fixed,
as the number of configurations increases, the average number of systems associated with each configuration decreases.
As discussed in Fig. \ref{fig:S1}, the attacker's utility gain is based on the number of systems associated with the attacked configuration.
Thus, the attacker's utility gain decreases.
Moreover, as the number of configurations increases, the defender is more likely to obfuscate a system.
Thus, the defender's cost increases.
For example, in our \emph{DP-based} approach, when there are two configurations: $1$ and $2$,
the defender may obfuscate a system from configuration $1$ to appear as $2$ with probability $0.5$.
However, when there are three configurations: $1$, $2$ and $3$,
the defender may obfuscate a system from configuration $1$ to appear as $2$ or $3$ with the same probability of $0.33$, or $0.66$ altogether.
This example shows that as the number of configurations increases,
the probability that the defender will obfuscate a system will increase.
\begin{figure}[ht]
\vspace{-6mm}
\centering
\begin{minipage}{0.51\textwidth}
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{DP-based}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2DPUtility.pdf}
\label{fig:S2DPUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{DP-based}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2DPCost.pdf}
\label{fig:S2DPCost}}\\[2ex]
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{Greedy}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2GreedyUtility.pdf}
\label{fig:S2GreedyUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{Greedy}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2GreedyCost.pdf}
\label{fig:S2GreedyCost}}\\[2ex]
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2GreedyMixedUtility.pdf}
\label{fig:S2GreedyMixedUtility}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S2GreedyMixedCost.pdf}
\label{fig:S2GreedyMixedCost}}\\%[2ex]
\end{minipage}
\vspace{-4mm}
\caption{The three approaches' performance as game progress in Scenario 2}
\vspace{-2mm}
\label{fig:S2Round}
\end{figure}
Fig. \ref{fig:S2Round} shows the variation in the attacker's utility gain
and the defender's cost in the three approaches as the game progresses in Scenario 2.
The number of systems is fixed at $150$,
and the number of configurations is fixed at $25$.
Fig. \ref{fig:S2Round} has a similar trend to Fig. \ref{fig:S1Round},
because Scenario 2 has a similar setting to Scenario 1,
where budget $B_d$ is large enough to cover the whole obfuscation process.
\subsubsection{Scenario 3}
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility with different values of budget $B_d$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Utility.pdf}
\label{fig:S3Utility}}
\subfigure[\scriptsize{The defender's cost with different values of budget $B_d$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Cost.pdf}
\label{fig:S3Cost}}\\[2ex]
\end{minipage}
\vspace{-4mm}
\caption{The three approaches' performance in Scenario 3}
\vspace{-4mm}
\label{fig:S3}
\end{figure}
Fig. \ref{fig:S3} demonstrates the performance of the three approaches in Scenario 3.
The numbers of systems and configurations are fixed at $100$ and $10$, respectively.
The privacy budget $\epsilon$ is fixed at $0.3$.
The cost budget $B_d$ for each round varies from $30$ to $150$.
When the defender's budget $B_d$ is tight (less than $90$),
the defender's cost can be properly controlled (Fig. \ref{fig:S3Cost}).
However, the attacker will gain high utility (Fig. \ref{fig:S3Utility}).
Due to the budget limitation (less than $90$), the defender obfuscates only a few systems,
meaning that the attacker can easily select high-utility systems.
By contrast, when the defender's budget $B_d$ is large enough (greater than $90$),
she can obfuscate almost all necessary systems,
which increases the difficulty experienced by the attacker when attempting to select high-utility systems.
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{DP-based}, $B_d=60$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Utility60.pdf}
\label{fig:S3Utility60}}
\subfigure[\scriptsize{The attacker's utility as game progresses in \emph{Greedy}, $B_d=150$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Utility150.pdf}
\label{fig:S3Utility150}}\\[2ex]
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{DP-based}, $B_d=60$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Cost60.pdf}
\label{fig:S3Cost60}}
\subfigure[\scriptsize{The defender's cost as game progresses in \emph{Greedy}, $B_d=150$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S3Cost150.pdf}
\label{fig:S3Cost150}}\\[2ex]
\end{minipage}
\vspace{-4mm}
\caption{Performance of the \emph{DP-based} approach with different cost budgets}
\vspace{-2mm}
\label{fig:S3Budget}
\end{figure}
Fig. \ref{fig:S3Budget} shows the details of the situations of $B_d=60$ and $B_d=150$ in our \emph{DP-based} approach.
The details in the \emph{Greedy} and \emph{Greedy-Mixed} approaches are similar to the \emph{DP-based} approach,
which thus are not presented.
By comparing Fig. \ref{fig:S3Utility60} and Fig. \ref{fig:S3Utility150},
the attacker's utility significantly reduces from $88$ to $57$ on average,
when increasing the defender's budget from $60$ to $150$.
Moreover, in Fig. \ref{fig:S3Cost60}, the defender's cost is confined to $60$,
while in Fig. \ref{fig:S3Cost150}, there is no confinement on the defender's cost.
This is because budget $B_d=60$ is insufficient for the defender to obfuscate all the necessary systems in most rounds.
Therefore, budget $B_d=60$ becomes a confinement for the defender.
When the budget is increased to $150$,
budget $B_d=150$ is enough to cover the obfuscation of all the necessary systems.
Thus, the confinement disappears.
\subsubsection{Scenario 4}
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility with different values of privacy budget $\epsilon$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S4Utility.pdf}
\label{fig:S4Utility}}
\subfigure[\scriptsize{The defender's cost with different values of privacy budget $\epsilon$}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S4Cost.pdf}
\label{fig:S4Cost}}\\[2ex]
\end{minipage}
\vspace{-4mm}
\caption{Our approach's performance in Scenario 4}
\vspace{-2mm}
\label{fig:S4}
\end{figure}
Fig. \ref{fig:S4} demonstrates the performance of our approach in Scenario 4.
The number of systems is fixed at $150$,
and the number of configurations is fixed at $15$.
The privacy budget $\epsilon$ various from $0.1$ to $0.5$.
The cost budget $B_d$ is fixed at $1000$ for each round.
According to Definition \ref{Def-LA} in Sub-section \ref{sub:DP}, a smaller $\epsilon$ denotes a larger Laplace noise.
In Line 5 of Algorithm \ref{alg:DP}, the number of systems associated with each configuration is adjusted by adding Laplace noise.
Therefore, a larger Laplace noise implies a larger change in the number of systems.
On one hand, if the Laplace noise is positive,
the number of systems increases ($|N'|>|N|$).
The defender thus incurs extra cost to deploy the added systems, i.e., honeypots.
These honeypots will attract the attacker and decrease his utility gain.
On the other hand, if the Laplace noise is negative,
the number of systems decreases ($|N'|<|N|$).
The defender sacrifices some utility to take $|N|-|N'|$ systems offline.
These systems can avoid the attacker,
and the attacker's utility gain reduces.
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.45\textwidth}
\subfigure[\scriptsize{The attacker's utility as game progresses}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S4UtilityRounds.pdf}
\label{fig:S4UtilityRounds}}
\subfigure[\scriptsize{The defender's cost as game progresses}]{
\includegraphics[width=0.45\textwidth, height=3cm]{S4CostRounds.pdf}
\label{fig:S4CostRounds}}\\[2ex]
\end{minipage}
\vspace{-4mm}
\caption{Our approach with different number of game rounds in Scenario 4}
\vspace{-2mm}
\label{fig:S4Rounds}
\end{figure}
Fig. \ref{fig:S4Rounds} demonstrates the performance of our approach with different number of game rounds,
where the privacy budget $\epsilon$ is fixed at $0.3$.
Based on the discussion in Theorem \ref{thm:bound}, to guarantee a given privacy level,
an upper bound of the number of rounds must be set.
In Fig. \ref{fig:S4UtilityRounds}, with the increasing number of rounds,
the attacker's utility gain increases gradually,
especially after $2000$ rounds.
This implies that when the game is played in a large number of rounds,
the attacker can infer the true configurations of systems.
This experimental result empirically prove our theoretical result.
In Fig. \ref{fig:S4CostRounds}, the defender's cost is not affected by the number of rounds,
since both the number of systems and configurations is fixed.
\iffalse
\subsubsection{Scenario 5}
\begin{figure}[ht]
\vspace{-4mm}
\centering
\begin{minipage}{0.51\textwidth}
\subfigure[\scriptsize{The attacker's utility in \emph{DP-based}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5DPUtility.pdf}
\label{fig:S5DPUtility}}
\subfigure[\scriptsize{The attacker's utility in \emph{Greedy}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5GreedyUtility.pdf}
\label{fig:S5GreedyUtility}}
\subfigure[\scriptsize{The attacker's utility in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5GreedyMixedUtility.pdf}
\label{fig:S5GreedyMixedUtility}}\\%[2ex]
\subfigure[\scriptsize{The defender's cost in \emph{DP-based}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5DPCost.pdf}
\label{fig:S5DPCost}}
\subfigure[\scriptsize{The defender's cost in \emph{Greedy}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5GreedyCost.pdf}
\label{fig:S5GreedyCost}}
\subfigure[\scriptsize{The defender's cost in \emph{Greedy-Mixed}}]{
\includegraphics[width=0.3\textwidth, height=2.2cm]{S5GreedyMixedCost.pdf}
\label{fig:S5GreedyMixedCost}}\\%[2ex]
\end{minipage}
\vspace{-4mm}
\caption{The three approaches' performance as game progress in Scenario 5}
\vspace{-4mm}
\label{fig:S5Round}
\end{figure}
Fig. \ref{fig:S5Round} demonstrates the performance of the three approaches in a dynamic network.
The number of configurations is fixed at $15$.
The number of systems starts with $150$.
The privacy budget $\epsilon$ is fixed at $0.3$.
The cost budget $B_d$ is fixed at $1000$ for each round.
In every $50$ rounds, a new system is introduced into the network
and randomly associated with a configuration.
In Fig. \ref{fig:S5Round}, in both the \emph{Greedy} and \emph{Greedy-Mixed} approaches,
as game progresses, the attacker's utility increases (Figs. \ref{fig:S5GreedyUtility} and \ref{fig:S5GreedyMixedUtility})
and the defender's cost also increases (Figs. \ref{fig:S5GreedyCost} and \ref{fig:S5GreedyMixedCost}).
As discussed in Scenario 1, both the attacker's utility gain and the defender's cost are based on
the number of configurations and the number of systems associated with each configuration.
As game progresses, new systems are introduced.
Since the number of configurations is fixed,
the number of systems in each configuration may increase.
Thus, both the attacker's utility gain and the defender's cost increase.
By comparison, our \emph{DP-based} approach is more stable than the those approaches (Figs. \ref{fig:S5DPUtility} and \ref{fig:S5DPCost}).
As our \emph{DP-based} approach has already been dynamic,
e.g., adding honeypots and taking existing systems offline,
the introduction of new systems does not affect much on its performance.
\fi
\subsection{Summary}
According to the experimental results, due to the adoption of the differential privacy technique,
our \emph{DP-based} approach outperforms the \emph{Greedy} and \emph{Greedy-mixed} approaches in various scenarios
by significantly reducing the attacker's utility gain by about $30\%\sim 40\%$.
In terms of the overhead resulting from adopting the differential privacy technique,
the defender in our \emph{DP-based} approach incurs about $5\%\sim 8\%$ more cost than in the \emph{Greedy} and \emph{Greedy-mixed} approaches.
Moreover, the privacy budget $\epsilon$ can be used to balance the attacker's utility gain and the defender's cost.
A small value of $\epsilon$ means a low utility gain for the attacker, but a high cost to the defender.
The setting of the $\epsilon$ value is left to users.
Furthermore, the running times of the three approaches are almost the same.
Thus, they are not shown graphically in the experiments.
Based on the theoretical analysis in Section \ref{sec:analysis}, the complexity of our \emph{DP-based} approach is $O(|N|^2)$,
where $|N|$ is the number of systems.
In the \emph{Greedy} and \emph{Greedy-mixed} approaches, there is a ranking process to rank the utility of systems.
Since we use bubble sort to implement the \emph{Greedy} and \emph{Greedy-mixed} approaches,
the complexity of both approaches is $O(|N|^2)$
which is the same as our \emph{DP-based} approach.
\vspace{-1mm}
\section{Conclusion and Future Work}\label{sec:conclusion}
This paper proposes a novel differentially private game theoretic approach to cyber deception.
Our approach is the first to adopt the differential privacy technique,
which can efficiently handle any change in the number of systems
and complex strategies potentially adopted by an attacker.
Compared to the benchmark approaches,
our approach is more stable and achieves much better performance
in various scenarios with slightly higher cost.
In the future, as mentioned in Section \ref{sec:analysis},
we will deeply investigate the equilibria in highly dynamic security games.
We also intend to improve our approach by introducing multiple defenders and multiple attackers.
In this paper, we consider only one defender and one attacker.
Future games will be very interesting when multiple defenders and attackers are involved.
Moreover, we will develop a general prototype of our approach to allow the state of the art to progress faster.
Specifically, to develop a prototype, we intend to follow Jajodia et al.’s work \cite{Jajodia17}.
We will first use Cauldron software \cite{Jajodia16} in a real network environment
to scan for vulnerabilities on each node in the network.
Then we will replicate the network into a larger network with the NS2 network simulator \cite{Issariyakul08}.
After that, our approach will be implemented on the simulated network.
\bibliographystyle{IEEEtran}
{\small |
2203.03623 | \section{Introduction}
\label{sec:introduction}
Reconstruction from under-sampled measurements in medical imaging has been deeply studied over the years, including reconstruction of accelerated magnetic resonance imaging (MRI) \citep{aggarwal2018modl,hammernik2018learning,eo2018kiki,han2019k}, sparse view or limited angles computed tomography (CT) \citep{han2018framing,zhang2019jsr,wang2019admm} and digital breast tomosynthesis (DBT). Most of works aim to obtain one sample of the posterior distribution $p\left(\mathbf{x} \mid \mathbf{y} \right)$ where $\mathbf{x}$ is the reconstructed target image and $\mathbf{y}$ is the under-sampled measurements.
Recently, denoising diffusion probabilistic models (DDPM) \citep{sohl2015deep,ho2020denoising}, which is a new class of unconditional generative model, have demonstrated superior performance and have been widely used in various image processing tasks. DDPM utilizes a latent variable model to reverse a diffusion process, where the data distribution is perturbed to the noise distribution by gradually adding Gaussian noise. Similar to DDPM, score-based generative models \citep{hyvarinen2005estimation,song2019generative} also generate data samples by reversing a diffusion process. Both DDPM and score-based models are proved to be discretizations of different continuous stochastic differential equations by \citep{song2020score}. The difference between them lies in the specific setting of diffusion process and sampling algorithms. They have been applied to the generation of image \citep{song2020score,nichol2021improved,dhariwal2021diffusion}, audio \citep{kong2020diffwave} or graph \citep{niu2020permutation}, and to conditional generation tasks such as in in-painting \citep{song2019generative,song2020score}, super-resolution \citep{choi2021ilvr,saharia2021image} and image editing \citep{meng2021sdedit}. In these applications, the diffusion process of DDPM or score-based generative model is defined in data domain, and is unconditioned although the reverse process could be conditioned given certain downstream task. Particularly, the score-based generative model has been used for under-sampled medical image reconstruction \citep{jalal2021robust,song2021solving,chung2021score}, where the diffusion process is defined in the domain of image $\mathbf{x}$ and is irrelevant to under-sampled measurements $\mathbf{y}$.
In this paper, We design our method based on DDPM rather than score-based generative model because DDPM is more flexible to control the noise distribution. We propose a novel and unified method, measurement-conditioned DDPM (MC-DDPM) for under-sampled medical image reconstruction based on DDPM (Fig.\ref{fig:method} illustrates the method by the example of under-sampled MRI reconstruction), where the under-sampling is in the measurement space (e.g. k-space in MRI reconstruction) and thus the conditional diffusion process is also defined in the measurement sapce. Two points distinguish our method from previous works \citep{jalal2021robust,song2021solving,chung2021score}: (1) the diffusion and sampling process are defined in measurement domain rather than image domain; (2) the diffusion process is conditioned on under-sampling mask so that data consistency is contained in the model naturally and inherently, and there is no need to execute extra data consistency when sampling. The proposed method allows us to sample multiple reconstruction results from the same measurements $\mathbf{y}$. Thus, we are able to quantify uncertainty for $q\left(\mathbf{x}\mid \mathbf{y}\right)$, such as pixel-variance. Our experiments on accelerated MRI reconstruction show MC-DDPM can generate samples of high quality of $q\left(\mathbf{x}\mid \mathbf{y}\right)$ and it outperforms baseline models and proposed method by \citep{chung2021score} in evaluation metrics.
This paper is organized as follows: relevant background on DDPM and the under-sampled medical image reconstruction task is in Sect.~\ref{sec:background}; details of the proposed method MC-DDPM is presented in Sect~\ref{sec:method}; specifications about the implementation of the application to accelerated MRI reconstruction, experimental results and discussion are given in Sect.~\ref{sec:experiments};
we conclude our work in Sect.~\ref{sec:conclusion}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{method.png}
\caption{Overview of the proposed method illustrated by the example of under-sampled MRI reconstruction. Diffusion process: starting from the non-sampled k-space $\mathbf{y}_{\mathbf{M}^c, 0}$, Gaussian noise is gradually added until time $T$. Reverse process: starting from total noise, $\mathbf{y}_{\mathbf{M}^c, 0}$ is generated step by step. The details of notations is presented in Sect.~\ref{sec:method}.}
\label{fig:method}
\end{figure}
\section{Background}
\label{sec:background}
\subsection{Denoising Diffusion Probabilistic Model}
\label{sec:background_ddpm}
DDPM \citep{ho2020denoising} is a certain parameterization of diffusion models \citep{sohl2015deep}, which is a class of latent variable models using a Markov chain to convert the noise distribution to the data distribution. It has the form of $p_{\theta}\left(\mathbf{x}_{0}\right):= \int p_{\theta} \left( \mathbf{x}_{0:T} \right) \mathrm{d} \mathbf{x}_{1:T} $, where $\mathbf{x}_{0}$ follows the data distribution $q\left(\mathbf{x}_{0} \right)$ and $\mathbf{x}_{1}, ..., \mathbf{x}_{T}$ are latent variables of the same dimensionality as $\mathbf{x}_{0}$. The joint distribution $p_{\theta}\left( \mathbf{x}_{0:T} \right)$ is defined as a Markov chain with learned Gaussian transitions starting from $p\left(\mathbf{x}_{T} \right)=\mathcal{N}\left(\mathbf{0}, \mathbf{I} \right)$:
\begin{equation}
\label{eq:ddpm_p}
p_{\theta}\left(\mathbf{x}_{0: T}\right) := p\left(\mathbf{x}_{T}\right) \prod_{t=1}^{T} p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right), p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right):=\mathcal{N} \left(\boldsymbol{\mu}_{\theta}\left(\mathbf{x}_{t}, t\right), \sigma_{t}^{2} \mathbf{I}\right).
\end{equation}
The sampling process of $p_{\theta}\left( \mathbf{x}_{0}\right)$ is: to sample $\mathbf{x}_{T}$ from $\mathcal{N}\left(\mathbf{0}, \mathbf{I} \right)$ firstly; then, to sample $\mathbf{x}_{t-1}$ from $p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right)$ until $\mathbf{x}_{0}$ is obtained. It can be regarded as a reverse process of the diffusion process, which converts the data distribution to the noise distribution $\mathcal{N} \left(\mathbf{0}, \mathbf{I} \right)$. In DDPM the diffusion process is fixed to a Markov chain that gradually adds Gaussian noise to the data according to a variance schedule $\beta_1$, $...$, $\beta_T$:
\begin{equation}
\label{eq:ddpm_diffusion_process}
q\left(\mathbf{x}_{1:T} \mid \mathbf{x}_{0}\right) := \prod_{t=1}^T q\left( \mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right), \quad q\left(\mathbf{x}_{t} \mid \mathbf{x}_{t-1} \right):= \mathcal{N}\left( \alpha_{t} \mathbf{x}_{t-1}, \beta_{t}^2 \mathbf{I} \right),
\end{equation}
where $\alpha_t^2 + \beta_t^2 = 1$ for all $t$ and $\beta_1$, $...$, $\beta_T$ are fixed to constants and their value are set specially so that $q\left( \mathbf{x}_{T} \mid \mathbf{x}_{0} \right) \approx \mathcal{N} \left(\mathbf{0}, \mathbf{I} \right)$.
\subsection{Under-sampled Medical Image Reconstruction}
\label{sec:background_med_recon}
Suppose $\mathbf{x} \in \mathbb{R}^n$ represents a medical image and $\mathbf{y} \in \mathbb{R}^m, m < n$ is the under-sampled measurements which is obtained by the following forward model:
\begin{equation}
\label{eq:undersampled_forward}
\mathbf{y} =\mathbf{P}_{\Omega} \mathbf{A} \mathbf{x} + \boldsymbol{\epsilon},
\end{equation}
where $\mathbf{A} \in \mathbb{R}^{n \times n}$ is the measuring matrix and usually is invertible, $\mathbf{P}_{\Omega} \in \mathbb{R}^{m\times n}$ is the under-sampling matrix with the given sampling pattern $\Omega$,\footnote{Assuming the sampling pattern $\Omega$ is $\{s_1, ..., s_m\} \subseteqq \{1, ..., n\}$, the element of $\mathbf{P}_{\Omega}$ at position $(i, s_i)$, $i = 1, ..., m$, is $1$ and other elements are all $0$.} and $\boldsymbol{\epsilon}$ is the noise. For example, $\mathbf{x}$ is a CT image, $\mathbf{A}$ is the Radon transform matrix and $\mathbf{y}$ is the sinogram of limited angles. Under-sampled medical image reconstruction is to reconstruct $\mathbf{x}$ from $\mathbf{y}$ as possible. Assuming $\mathbf{x}$ follows a distribution of $q\left( \mathbf{x} \right)$ and given $\mathbf{P}_{\Omega}$, according to Bayesian Formula, we can derive the posterior distribution as follows (usually $\mathbf{P}_{\Omega}$ is neglected):
\begin{equation}
\label{eq:bayesian_formula}
q \left( \mathbf{x} \mid \mathbf{y}, \mathbf{P}_{\Omega} \right) = \frac{q \left(\mathbf{x}, \mathbf{y} \mid \mathbf{P}_{\Omega} \right)}{q\left( \mathbf{y} \right)} = \frac{q\left(\mathbf{y} \mid \mathbf{x}, \mathbf{P}_{\Omega} \right) q \left(\mathbf{x} \right)}{q\left( \mathbf{y} \right)}.
\end{equation}
Therefore, the task of under-sampled medical image reconstruction to reconstruct the posterior distribution.
\section{Method: Measurement-conditioned DDPM}
\label{sec:method}
Inspired by DDPM, we propose measurement-conditioned DDPM (MC-DDPM) which is designed for under-sampled medical image reconstruction. In this section, we formulate the MC-DDPM, including the diffusion process and its reverse process, training objective and sampling algorithm. For the convenience, we use new notations different from Eq.~\ref{eq:undersampled_forward} to represent the under-sampled forward model:
\begin{equation}
\label{eq:undersampled_forward_new}
\mathbf{y}_{\mathbf{M}} = \mathbf{M} \mathbf{A} \mathbf{x} + \boldsymbol{\epsilon}_{\mathbf{M}},
\end{equation}
where $\mathbf{M} \in \mathbb{R}^{n \times n}$ is a diagonal matrix whose diagonal elements are either $1$ or $0$ depending on the sampling pattern $\Omega$.\footnote{Specifically, $M_{i, i} = 1$ if $i \in \Omega$. Otherwise its value is $0$.} $\mathbf{y}_{\mathbf{M}}$ and $\boldsymbol{\epsilon}_{\mathbf{M}}$ are both $n$-dimension vectors and their components at non-sampled positions are $0$. The merit of the new notations is that we can further define $\mathbf{M}^{c} = \mathbf{I} - \mathbf{M}$ (the superscript $c$ means complement) and $\mathbf{y}_{\mathbf{M}^c} = \mathbf{M}^{c} \mathbf{A} \mathbf{x}$ which represents the non-sampled measurements. In this paper, we assume $\boldsymbol{\epsilon}_{\mathbf{M}} = \mathbf{0}$. Then, we have $\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c} = \mathbf{A} \mathbf{x} $, i.e. $\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c}$ is the full-sampled measurements. In addition, the posterior distribution of reconstruction can be rewritten as $q \left(\mathbf{x} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}} \right)$. Through this paper, the subscript $\mathbf{M}$ or $\mathbf{M}^c$ in notations indicates that only components at under-sampled or non-sampled positions are not $0$.
The purpose of reconstruction task is to estimate $q \left( \mathbf{x} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$. Since $\mathbf{y}_{\mathbf{M}}$ is known and $\mathbf{x} = \mathbf{A}^{-1} \left( \mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c}\right)$ , the problem is transformed to estimate $q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$. Because $\mathbf{M}$ and $\mathbf{M}^c$ are equivalent as the condition, we can replace $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$ by $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. Based on this observation, we propose MC-DDPM which solves the reconstruction problem by generating samples of $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. MC-DDPM is defined in measurement domain, instead of image domain as usual DDPM, and is conditioned on the non-sampling matrix $\mathbf{M}^c$ and sampled measurements $\mathbf{y}_{\mathbf{M}}$. It has the following form:
\begin{equation}
\label{eq:MC-DDPM_model}
p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right):= \int p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0:T} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right) \mathrm{d} \mathbf{y}_{\mathbf{M}^c, 1:T},
\end{equation}
where $\mathbf{y}_{\mathbf{M}^c, 0}$ = $\mathbf{y}_{\mathbf{M}^c}$. $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0:T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$ is defined as follows:
\begin{equation*}
\label{eq:MC-DDPM_reverse_process}
\begin{split}
p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0: T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) := p\left(\mathbf{y}_{\mathbf{M}^c, T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) \prod_{t=1}^{T} p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \\ p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right):=\mathcal{N} \left(\boldsymbol{\mu}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \sigma_{t}^{2} \mathbf{M}^c\right),
\end{split}
\end{equation*}
where $\sigma_{t}^{2} \mathbf{M}^c$ is the covariance matrix and it means the noise is only added at non-sampled positions because for all $t$ the components of $\mathbf{y}_{\mathbf{M}^c,t}$ at under-sampled positions are always $0$. If the conditions $\left( \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$ in equations above is removed, they degrade to the form of Eq.~\ref{eq:ddpm_p}.
Similar to DDPM, the sampling process of $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ is a reverse process of the diffusion process which is also defined in measurement domain. Specifically, the Gaussian noise is gradually added to the non-sampled measurements $\mathbf{y}_{\mathbf{M}^c, 0}$. The diffusion process has the following form:
\begin{equation}
\label{eq:MC-DDPM_diffusion_process}
\begin{split}
q\left(\mathbf{y}_{\mathbf{M}^c,1:T} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) := \prod_{t=1}^T q\left( \mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \\ q\left(\mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right):= \mathcal{N}\left(\alpha_{t} \mathbf{y}_{\mathbf{M}^c,t-1}, \beta_{t}^2 \mathbf{M}^c \right),
\end{split}
\end{equation}
There are two points worthy of noting: (1) $\alpha_t, \beta_{t}$ are not restricted to satisfy $\alpha_t^2 + \beta_t^2 = 1$; (2) formally, we add $\mathbf{y}_{\mathbf{M}}$ as one of the conditions, but it has no effect on the diffusion process in fact. Let $\bar{\alpha}_{t}=\prod_{i=i}^{t} \alpha_{i}, \bar{\beta}_{t}^2={\sum_{i=1}^{t} \frac{\bar{\alpha}_{t}^{2}}{\bar{\alpha}_{i}^{2}} \beta_{i}^{2}}$, and we additionally define $\bar{\alpha}_{0} = 1, \bar{\beta}_{0} = 0$. Then, we can derive that:
\begin{equation}
\label{eq:MC-DDPM_q_t}
q\left(\mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right) = \mathcal{N}\left(\bar{\alpha}_{t} \mathbf{y}_{\mathbf{M}^c,0}, \bar{\beta}_{t}^{2} \mathbf{M}^c\right),
\end{equation}
\begin{equation}
\label{eq:MC-DDPM_q_t-1_t_0}
q\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c, t}, \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right) = \mathcal{N}\left(\tilde{\boldsymbol{\mu}}_{t}, \tilde{\beta}_{t}^{2} \mathbf{M}^c\right),
\end{equation}
where $\tilde{\boldsymbol{\mu}}_{t}=\frac{\alpha_{t} \bar{\beta}_{t-1}^{2}}{\bar{\beta}_{t}^{2}} \mathbf{y}_{\mathbf{M}^c,t}+\frac{\bar{\alpha}_{t-1} \beta_{t}^{2}}{\bar{\beta}_{t}^{2}} \mathbf{y}_{\mathbf{M}^c,0}, \tilde{\beta}_{t}=\frac{\beta_{t} \bar{\beta}_{t-1}}{\bar{\beta}_{t}}$. In MC-DDPM, we assume that $\alpha_{t}$ is set specially so that $\bar{\alpha}_{T} \approx 0$, i.e. $q\left( \mathbf{y}_{\mathbf{M}^c,T} \mid \mathbf{y}_{\mathbf{M}^c,0} \right) \approx \mathcal{N}\left(\mathbf{0}, \bar{\beta}_{T}^2\mathbf{M}^c \right)$ is a noise distribution independent of $\mathbf{y}_{M^c, 0}$.
Next, we discuss how to train MC-DDPM $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. Firstly, let $p\left(\mathbf{y}_{\mathbf{M}^c, T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) = \mathcal{N}\left(\mathbf{0}, \bar{\beta}_{T}^2\mathbf{M}^c \right)$ so that it is nearly equal to $q\left( \mathbf{y}_{\mathbf{M}^c,T} \mid \mathbf{y}_{\mathbf{M}^c,0} \right)$. Training of $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ is performed by optimizing the variational bound on negative log likelihood:
\begin{align*}
& \mathbb{E}\left[-\log p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right] \leq \mathbb{E}_{q}\left[-\log \frac{p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0:T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}{q\left(\mathbf{y}_{\mathbf{M}^c,1:T} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)}\right] \notag \\
=& \mathbb{E}_{q}\left[-\log p\left(\mathbf{y}_{\mathbf{M}^c,T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)-\sum_{t \geq 1} \log \frac{p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}{q\left(\mathbf{y}_{\mathbf{M}^c, t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}\right]=: L.
\end{align*}
Assuming that
\begin{equation}
\boldsymbol{\mu}_{\theta}\left( \mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)=\frac{1}{\alpha_{t}}\left(\mathbf{y}_{\mathbf{M}^c, t}-\frac{\beta_{t}^{2}}{\bar{\beta}_{t}} \boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right),
\end{equation}
and supposing $\mathbf{y}_{\mathbf{M}^c, t} = \bar{\alpha}_{t} \mathbf{y}_{\mathbf{M}^c,0} + \boldsymbol{\varepsilon}, \boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \bar{\beta}_{t}^{2} \mathbf{M}^c \right)$ (Eq.~\ref{eq:MC-DDPM_q_t}), after reweighting $L$ can be simplified as follows:
\begin{equation}
L_{\mathrm{simple}}=\mathbb{E}_{\mathbf{y}^c_{\mathbf{M},0}, t, \boldsymbol{\varepsilon}} \left\| \boldsymbol{\varepsilon} -\boldsymbol{\varepsilon}_{\theta}\left(\bar{\alpha}_{t} \mathbf{y}^c_{\mathbf{M},0} + \bar{\beta}_{t} \boldsymbol{\varepsilon}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right\|_{2}^{2}, \boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \mathbf{M}^c \right),
\end{equation}
where $t$ is uniform between $1$ and $T$. The details of derivation is in supplementary materials.
Algorithm~\ref{alg:MC-DDPM_training} displays the complete training procedure with this simplified objective and Algorithm~\ref{alg:MC-DDPM_sampling} shows the sampling process.
Since MC-DDPM can produce multiple samples of the posterior distribution $q\left(\mathbf{x} \mid \mathbf{y}_{\mathbf{M}}, \mathbf{M} \right)$, the pixel-variance can be computed by Monte Carlo approach which is used to quantify uncertainty of reconstruction.
\begin{figure}[t]
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{MC-DDPM Training}
\label{alg:MC-DDPM_training}
\begin{algorithmic}[1]
\Repeat
\State $\mathbf{x} \sim q \left( \mathbf{x} \right)$, obtain $\mathbf{M}$ and $\mathbf{M}^c$
\State $\mathbf{y}_{\mathbf{M}} = \mathbf{M} \mathbf{A} \mathbf{x} $, $\mathbf{y}_{\mathbf{M}^c} = \mathbf{M}^c \mathbf{A} \mathbf{x} $
\State $\boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \mathbf{M}^c \right)$
\State $t \sim \operatorname{Uniform}(\{1, \ldots, T\})$
\State $\mathbf{y}^c_{\mathbf{M},t} = \bar{\alpha}_{t} \mathbf{y}^c_{\mathbf{M},0} + \bar{\beta}_{t} \boldsymbol{\varepsilon}$
\State Take gradient descent step on
\Statex $\quad\quad\quad \nabla_{\theta} \left\| \boldsymbol{\varepsilon} -\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}^c_{\mathbf{M},t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right\|_{2}^{2}$
\Until{converged}
\end{algorithmic}
\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{MC-DDPM Sampling}
\label{alg:MC-DDPM_sampling}
\begin{algorithmic}[1]
\State Given $\mathbf{M}^c$ and $\mathbf{y}_{\mathbf{M}}$
\State $\mathbf{y}_{\mathbf{M}^c, T} \sim \mathcal{N} \left(\mathbf{0}, \bar{\beta}_{T}^2 \mathbf{M}^c \right)$
\For{$t = T, ..., 1$}
\State $\mathbf{z}_{t} \sim \mathcal{N}(\mathbf{0}, \mathbf{M}^c)$ if $t > 1$, else $\mathbf{z}_{t} = \mathbf{0}$
\State $\boldsymbol{\mu}_{t} = \boldsymbol{\mu}_{\theta}\left( \mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$
\State $\mathbf{y}_{\mathbf{M}^c, t-1}= \boldsymbol{\mu}_{t} + \sigma_{t} \mathbf{z}_{t}$
\EndFor
\State \textbf{return} $\mathbf{x} = \mathbf{A}^{-1} \left(\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c, 0}\right)$
\end{algorithmic}
\end{algorithm}
\end{minipage}
\end{figure}
\section{Experiments}
\label{sec:experiments}
We apply MC-DDPPM to accelerated MRI reconstruction where $\mathbf{A}$ is 2d Fourier transform and $\mathbf{y}_{\mathbf{M}}$ is the under-sampled k-space data. The specific design for $\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ in our experiments is given as follows:
\begin{equation}
\label{eq:MC-DDPM_mri_recon}
\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) = \mathbf{M}^c f\left(g\left(\mathbf{A}^{-1}\left(\mathbf{y}_{\mathbf{M}^c, t} + \mathbf{y}_{M} \right), \mathbf{A}^{-1}\mathbf{y}_{M}\right), t; \theta \right),
\end{equation}
where $f$ is a deep neural network and $g\left(\cdot,\cdot \right)$ is the concatenation operation. Because MR image $\mathbf{x}$ is in complex filed, we use $\left|\mathbf{x}\right|$, the magnitude of it, as the final image. Pixel-wise variance is also computed using magnitude images.
\subsection{Experimental Setting}
\label{sec:experiments_setting}
All experiments are performed with fastMRI single-coil knee dataset \citep{zbontar2018fastmri}, which is publicly available\footnote{https://fastmri.org} and is divided into two parts, proton-density with (PDFS) and without fat suppression (PD). We trained the network with k-space data which were computed from $320 \times 320$ size complex images. We base the implementation of guided-DDPM \citep{dhariwal2021diffusion} and also follow similar setting for the diffusion process in \citep{dhariwal2021diffusion} but multiply $\beta_{t}$ by $0.5$ so that $\bar{\beta}_{T} \approx 0.5$. All networks were trained with learning rate of $0.0001$ using AdamW optimizer. More details of experiments is in supplementary materials.
To verify superiority, we perform comparison studies with baseline methods (U-Net \citep{ronneberger2015u}) used in \citep{zbontar2018fastmri}. The evaluation metrics, peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), of score-based reconstruction method proposed in \citep{chung2021score} are also used for comparison\footnote{They are provided in the paper of \citep{chung2021score}.} since their experiments are conducted on the same dataset.
\subsection{Experimental Results}
\label{sec:experiments_results}
We show the results of PD with $4 \times$ (the first row) and $8\times$ (the second row) acceleration in Fig.~\ref{fig:reconstruction}. More results are shown in supplementary materials. We compare our method to zero-filled reconstruction (ZF) and U-Net. Since MC-DDPM can produce multiple reconstruction samples, we use the mean of 20 samples as the object for comparison. We observe that the proposed method performs best both in $4 \times$ and $8 \times$ accelerations, where we see virtually more realistic structures and less error in the zoomed-in image than ZF and U-Net. In the last column of Fig.~\ref{fig:reconstruction}, we show the standard deviation of the samples. As the acceleration factor is increased, we see that the uncertainty increases correspondingly. Quantitative metrics in Table.~\ref{tab:evaluation_metrics} also confirm the superiority of our method. In the last two columns of Table.~\ref{tab:evaluation_metrics}, we compare MC-DDPM to the score-based reconstruction method proposed in \citep{chung2021score}. Because the testing volumes are randomly selected both in our experiments and in \citep{chung2021score}, it is impossible to compare directly. As a substitution, we compare the enhancement of evaluation metrics which is computed by the result of proposed method subtracting the result of U-Net. Due to the experiments in \citep{chung2021score} were conducted on the whole dataset (both PD and PDFS), we compute the average enhancement of PD and PDFS as our final
result. Our method outperforms \citep{chung2021score} by 3.62/0.077 ($4 \times$) and 2.96/0.089 ($8 \times$) in PSNR/SSIM.
We also explore the effects of sampling steps and number of samples on reconstruction quality, which are illustrated in Fig.~\ref{fig:steps} and Fig.~\ref{fig:num_samples}. The two experiments are conducted on one volume of PDFS with $4 \times$ and $8 \times$ acceleration, and PSNR is computed on the mean of generated samples. We discover that: (1) even the sampling steps decrease to $250$, PSNR only reduces a little; (2) the quality of the mean of samples is enhanced when the number of samples increases and seems to converge. Taking the efficiency into account, 20 samples with 250 sampling steps may be a good choice.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{merged_images.png}
\caption{Reconstruction results of $4 \times$ (the first row) and $8 \times$ (the second row) on PD data: (a) ground-truth, (b) zero-filled reconstruction, (c) U-Net, (d) the mean of samples generated by proposed method, (e) the standard deviation of the samples: range is set to [0, 0.2]. Blue box: Zoom in version of the indicated red box. Green box: Difference magnitude of the inset. White numbers indicate PSNR and SSIM, respectively.}
\label{fig:reconstruction}
\end{figure}
\begin{table}[t]
\caption{Quantitative metrics. Numbers in bold face indicate the best metric out of all the methods. The enhancement in the last two columns is computed based on U-Net.}
\centering
\begin{tabular}{c|c|p{0.9cm}<{\centering}p{0.9cm}<{\centering}p{0.9cm}<{\centering}|p{0.9cm}<{\centering}p{0.9cm}<{\centering}p{0.9cm}<{\centering}|p{1.7cm}<{\centering}p{0.9cm}<{\centering}}
\hline
~ & ~ & \multicolumn{3}{c|}{PD} & \multicolumn{3}{c|}{PDFS} & \multicolumn{2}{c}{Enhancement} \\
~ & ~ & ZF & U-Net & Ours & ZF & U-Net & Ours & \cite{chung2021score} & Ours \\
\hline
\multirow{2}{*}{$\times$ 4} & PSNR & 29.62 & 34.04 & \textbf{36.69} & 26.32 & 28.30 & \textbf{33.00} & +0.06 & \textbf{+3.68} \\
~ & SSIM & 0.745 & 0.834 & \textbf{0.905} & 0.545 & 0.648 & \textbf{0.735} & +0.002 & \textbf{+0.079} \\
\hline
\multirow{2}{*}{$\times$ 8} & PSNR & 25.94 & 31.13 & \textbf{33.49} & 24.90 & 26.17 & \textbf{31.75} & +1.01 & \textbf{+3.97} \\
~ & SSIM & 0.667 & 0.750 & \textbf{0.862} & 0.513 & 0.580 & \textbf{0.702} & +0.028 & \textbf{+0.117} \\
\hline
\end{tabular}
\label{tab:evaluation_metrics}
\end{table}
\begin{figure}[ht]
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{steps.pdf}
\caption{Tradeoff between number of sampling steps vs. PSNR testing on one volume of PDFS.}
\label{fig:steps}
\end{minipage}
\hfill
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{num_samples.pdf}
\caption{Tradeoff between number of samples vs. PSNR testing on one volume of PDFS.}
\label{fig:num_samples}
\end{minipage}
\end{figure}
\subsection{Discussion}
\label{sec:experiments_discussion}
It is very common in medical imaging that the measurement is under sampled to reduce the cost or dosage. Therefore, it is important to define the conditional diffusion process in the measurement space for a reconstruction task. In this project, although our experiments are conducted using MR images, our method can be applied to other under-sampled medical image reconstruction tasks, such as limited angle or spare view CT Reconstruction. In our MC-DDPM, the variance schedule $\left\{\beta_{t}\right\}$ is an important hyper-parameter that is potentially related to the sampling quality and efficiency. Further investigation on hyper-parameter $\left\{\beta_{t}\right\}$ is planned in our future study.
\section{Conclusion}
\label{sec:conclusion}
In this paper we present a novel and unified mathematical framework, MC-DDPM, for medical image reconstruction using under-sampled measurements. Our method applies diffusion process in measurement domain with conditioned under-sampling mask, and provides an estimate of uncertainty as output. The superior performance of our method is demonstrated using accelerated MRI reconstruction, although MC-DDPM should potentially work for other under-sampled medical image reconstruction tasks.
\section{Acknowledgements} Thank Zifan Chen for his discussion and advice for paper writing.
\section{Introduction}
\label{sec:introduction}
Reconstruction from under-sampled measurements in medical imaging has been deeply studied over the years, including reconstruction of accelerated magnetic resonance imaging (MRI) \citep{aggarwal2018modl,hammernik2018learning,eo2018kiki,han2019k}, sparse view or limited angles computed tomography (CT) \citep{han2018framing,zhang2019jsr,wang2019admm} and digital breast tomosynthesis (DBT). Most of works aim to obtain one sample of the posterior distribution $p\left(\mathbf{x} \mid \mathbf{y} \right)$ where $\mathbf{x}$ is the reconstructed target image and $\mathbf{y}$ is the under-sampled measurements.
Recently, denoising diffusion probabilistic models (DDPM) \citep{sohl2015deep,ho2020denoising}, which is a new class of unconditional generative model, have demonstrated superior performance and have been widely used in various image processing tasks. DDPM utilizes a latent variable model to reverse a diffusion process, where the data distribution is perturbed to the noise distribution by gradually adding Gaussian noise. Similar to DDPM, score-based generative models \citep{hyvarinen2005estimation,song2019generative} also generate data samples by reversing a diffusion process. Both DDPM and score-based models are proved to be discretizations of different continuous stochastic differential equations by \citep{song2020score}. The difference between them lies in the specific setting of diffusion process and sampling algorithms. They have been applied to the generation of image \citep{song2020score,nichol2021improved,dhariwal2021diffusion}, audio \citep{kong2020diffwave} or graph \citep{niu2020permutation}, and to conditional generation tasks such as in in-painting \citep{song2019generative,song2020score}, super-resolution \citep{choi2021ilvr,saharia2021image} and image editing \citep{meng2021sdedit}. In these applications, the diffusion process of DDPM or score-based generative model is defined in data domain, and is unconditioned although the reverse process could be conditioned given certain downstream task. Particularly, the score-based generative model has been used for under-sampled medical image reconstruction \citep{jalal2021robust,song2021solving,chung2021score}, where the diffusion process is defined in the domain of image $\mathbf{x}$ and is irrelevant to under-sampled measurements $\mathbf{y}$.
In this paper, We design our method based on DDPM rather than score-based generative model because DDPM is more flexible to control the noise distribution. We propose a novel and unified method, measurement-conditioned DDPM (MC-DDPM) for under-sampled medical image reconstruction based on DDPM (Fig.\ref{fig:method} illustrates the method by the example of under-sampled MRI reconstruction), where the under-sampling is in the measurement space (e.g. k-space in MRI reconstruction) and thus the conditional diffusion process is also defined in the measurement sapce. Two points distinguish our method from previous works \citep{jalal2021robust,song2021solving,chung2021score}: (1) the diffusion and sampling process are defined in measurement domain rather than image domain; (2) the diffusion process is conditioned on under-sampling mask so that data consistency is contained in the model naturally and inherently, and there is no need to execute extra data consistency when sampling. The proposed method allows us to sample multiple reconstruction results from the same measurements $\mathbf{y}$. Thus, we are able to quantify uncertainty for $q\left(\mathbf{x}\mid \mathbf{y}\right)$, such as pixel-variance. Our experiments on accelerated MRI reconstruction show MC-DDPM can generate samples of high quality of $q\left(\mathbf{x}\mid \mathbf{y}\right)$ and it outperforms baseline models and proposed method by \citep{chung2021score} in evaluation metrics.
This paper is organized as follows: relevant background on DDPM and the under-sampled medical image reconstruction task is in Sect.~\ref{sec:background}; details of the proposed method MC-DDPM is presented in Sect~\ref{sec:method}; specifications about the implementation of the application to accelerated MRI reconstruction, experimental results and discussion are given in Sect.~\ref{sec:experiments};
we conclude our work in Sect.~\ref{sec:conclusion}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{method.pdf}
\caption{Overview of the proposed method illustrated by the example of under-sampled MRI reconstruction. Diffusion process: starting from the non-sampled k-space $\mathbf{y}_{\mathbf{M}^c, 0}$, Gaussian noise is gradually added until time $T$. Reverse process: starting from total noise, $\mathbf{y}_{\mathbf{M}^c, 0}$ is generated step by step. The details of notations is presented in Sect.~\ref{sec:method}.}
\label{fig:method}
\end{figure}
\section{Background}
\label{sec:background}
\subsection{Denoising Diffusion Probabilistic Model}
\label{sec:background_ddpm}
DDPM \citep{ho2020denoising} is a certain parameterization of diffusion models \citep{sohl2015deep}, which is a class of latent variable models using a Markov chain to convert the noise distribution to the data distribution. It has the form of $p_{\theta}\left(\mathbf{x}_{0}\right):= \int p_{\theta} \left( \mathbf{x}_{0:T} \right) \mathrm{d} \mathbf{x}_{1:T} $, where $\mathbf{x}_{0}$ follows the data distribution $q\left(\mathbf{x}_{0} \right)$ and $\mathbf{x}_{1}, ..., \mathbf{x}_{T}$ are latent variables of the same dimensionality as $\mathbf{x}_{0}$. The joint distribution $p_{\theta}\left( \mathbf{x}_{0:T} \right)$ is defined as a Markov chain with learned Gaussian transitions starting from $p\left(\mathbf{x}_{T} \right)=\mathcal{N}\left(\mathbf{0}, \mathbf{I} \right)$:
\begin{equation}
\label{eq:ddpm_p}
p_{\theta}\left(\mathbf{x}_{0: T}\right) := p\left(\mathbf{x}_{T}\right) \prod_{t=1}^{T} p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right), p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right):=\mathcal{N} \left(\boldsymbol{\mu}_{\theta}\left(\mathbf{x}_{t}, t\right), \sigma_{t}^{2} \mathbf{I}\right).
\end{equation}
The sampling process of $p_{\theta}\left( \mathbf{x}_{0}\right)$ is: to sample $\mathbf{x}_{T}$ from $\mathcal{N}\left(\mathbf{0}, \mathbf{I} \right)$ firstly; then, to sample $\mathbf{x}_{t-1}$ from $p_{\theta}\left(\mathbf{x}_{t-1} \mid \mathbf{x}_{t}\right)$ until $\mathbf{x}_{0}$ is obtained. It can be regarded as a reverse process of the diffusion process, which converts the data distribution to the noise distribution $\mathcal{N} \left(\mathbf{0}, \mathbf{I} \right)$. In DDPM the diffusion process is fixed to a Markov chain that gradually adds Gaussian noise to the data according to a variance schedule $\beta_1$, $...$, $\beta_T$:
\begin{equation}
\label{eq:ddpm_diffusion_process}
q\left(\mathbf{x}_{1:T} \mid \mathbf{x}_{0}\right) := \prod_{t=1}^T q\left( \mathbf{x}_{t} \mid \mathbf{x}_{t-1}\right), \quad q\left(\mathbf{x}_{t} \mid \mathbf{x}_{t-1} \right):= \mathcal{N}\left( \alpha_{t} \mathbf{x}_{t-1}, \beta_{t}^2 \mathbf{I} \right),
\end{equation}
where $\alpha_t^2 + \beta_t^2 = 1$ for all $t$ and $\beta_1$, $...$, $\beta_T$ are fixed to constants and their value are set specially so that $q\left( \mathbf{x}_{T} \mid \mathbf{x}_{0} \right) \approx \mathcal{N} \left(\mathbf{0}, \mathbf{I} \right)$.
\subsection{Under-sampled Medical Image Reconstruction}
\label{sec:background_med_recon}
Suppose $\mathbf{x} \in \mathbb{R}^n$ represents a medical image and $\mathbf{y} \in \mathbb{R}^m, m < n$ is the under-sampled measurements which is obtained by the following forward model:
\begin{equation}
\label{eq:undersampled_forward}
\mathbf{y} =\mathbf{P}_{\Omega} \mathbf{A} \mathbf{x} + \boldsymbol{\epsilon},
\end{equation}
where $\mathbf{A} \in \mathbb{R}^{n \times n}$ is the measuring matrix and usually is invertible, $\mathbf{P}_{\Omega} \in \mathbb{R}^{m\times n}$ is the under-sampling matrix with the given sampling pattern $\Omega$,\footnote{Assuming the sampling pattern $\Omega$ is $\{s_1, ..., s_m\} \subseteqq \{1, ..., n\}$, the element of $\mathbf{P}_{\Omega}$ at position $(i, s_i)$, $i = 1, ..., m$, is $1$ and other elements are all $0$.} and $\boldsymbol{\epsilon}$ is the noise. For example, $\mathbf{x}$ is a CT image, $\mathbf{A}$ is the Radon transform matrix and $\mathbf{y}$ is the sinogram of limited angles. Under-sampled medical image reconstruction is to reconstruct $\mathbf{x}$ from $\mathbf{y}$ as possible. Assuming $\mathbf{x}$ follows a distribution of $q\left( \mathbf{x} \right)$ and given $\mathbf{P}_{\Omega}$, according to Bayesian Formula, we can derive the posterior distribution as follows (usually $\mathbf{P}_{\Omega}$ is neglected):
\begin{equation}
\label{eq:bayesian_formula}
q \left( \mathbf{x} \mid \mathbf{y}, \mathbf{P}_{\Omega} \right) = \frac{q \left(\mathbf{x}, \mathbf{y} \mid \mathbf{P}_{\Omega} \right)}{q\left( \mathbf{y} \right)} = \frac{q\left(\mathbf{y} \mid \mathbf{x}, \mathbf{P}_{\Omega} \right) q \left(\mathbf{x} \right)}{q\left( \mathbf{y} \right)}.
\end{equation}
Therefore, the task of under-sampled medical image reconstruction to reconstruct the posterior distribution.
\section{Method: Measurement-conditioned DDPM}
\label{sec:method}
Inspired by DDPM, we propose measurement-conditioned DDPM (MC-DDPM) which is designed for under-sampled medical image reconstruction. In this section, we formulate the MC-DDPM, including the diffusion process and its reverse process, training objective and sampling algorithm. For the convenience, we use new notations different from Eq.~\ref{eq:undersampled_forward} to represent the under-sampled forward model:
\begin{equation}
\label{eq:undersampled_forward_new}
\mathbf{y}_{\mathbf{M}} = \mathbf{M} \mathbf{A} \mathbf{x} + \boldsymbol{\epsilon}_{\mathbf{M}},
\end{equation}
where $\mathbf{M} \in \mathbb{R}^{n \times n}$ is a diagonal matrix whose diagonal elements are either $1$ or $0$ depending on the sampling pattern $\Omega$.\footnote{Specifically, $M_{i, i} = 1$ if $i \in \Omega$. Otherwise its value is $0$.} $\mathbf{y}_{\mathbf{M}}$ and $\boldsymbol{\epsilon}_{\mathbf{M}}$ are both $n$-dimension vectors and their components at non-sampled positions are $0$. The merit of the new notations is that we can further define $\mathbf{M}^{c} = \mathbf{I} - \mathbf{M}$ (the superscript $c$ means complement) and $\mathbf{y}_{\mathbf{M}^c} = \mathbf{M}^{c} \mathbf{A} \mathbf{x}$ which represents the non-sampled measurements. In this paper, we assume $\boldsymbol{\epsilon}_{\mathbf{M}} = \mathbf{0}$. Then, we have $\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c} = \mathbf{A} \mathbf{x} $, i.e. $\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c}$ is the full-sampled measurements. In addition, the posterior distribution of reconstruction can be rewritten as $q \left(\mathbf{x} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}} \right)$. Through this paper, the subscript $\mathbf{M}$ or $\mathbf{M}^c$ in notations indicates that only components at under-sampled or non-sampled positions are not $0$.
The purpose of reconstruction task is to estimate $q \left( \mathbf{x} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$. Since $\mathbf{y}_{\mathbf{M}}$ is known and $\mathbf{x} = \mathbf{A}^{-1} \left( \mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c}\right)$ , the problem is transformed to estimate $q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$. Because $\mathbf{M}$ and $\mathbf{M}^c$ are equivalent as the condition, we can replace $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right)$ by $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. Based on this observation, we propose MC-DDPM which solves the reconstruction problem by generating samples of $ q \left(\mathbf{y}_{\mathbf{M}^c} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. MC-DDPM is defined in measurement domain, instead of image domain as usual DDPM, and is conditioned on the non-sampling matrix $\mathbf{M}^c$ and sampled measurements $\mathbf{y}_{\mathbf{M}}$. It has the following form:
\begin{equation}
\label{eq:MC-DDPM_model}
p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right):= \int p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0:T} \mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right) \mathrm{d} \mathbf{y}_{\mathbf{M}^c, 1:T},
\end{equation}
where $\mathbf{y}_{\mathbf{M}^c, 0}$ = $\mathbf{y}_{\mathbf{M}^c}$. $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, 0:T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$ is defined as follows:
\begin{equation*}
\label{eq:MC-DDPM_reverse_process}
\begin{split}
p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0: T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) := p\left(\mathbf{y}_{\mathbf{M}^c, T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) \prod_{t=1}^{T} p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \\ p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right):=\mathcal{N} \left(\boldsymbol{\mu}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \sigma_{t}^{2} \mathbf{M}^c\right),
\end{split}
\end{equation*}
where $\sigma_{t}^{2} \mathbf{M}^c$ is the covariance matrix and it means the noise is only added at non-sampled positions because for all $t$ the components of $\mathbf{y}_{\mathbf{M}^c,t}$ at under-sampled positions are always $0$. If the conditions $\left( \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$ in equations above is removed, they degrade to the form of Eq.~\ref{eq:ddpm_p}.
Similar to DDPM, the sampling process of $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ is a reverse process of the diffusion process which is also defined in measurement domain. Specifically, the Gaussian noise is gradually added to the non-sampled measurements $\mathbf{y}_{\mathbf{M}^c, 0}$. The diffusion process has the following form:
\begin{equation}
\label{eq:MC-DDPM_diffusion_process}
\begin{split}
q\left(\mathbf{y}_{\mathbf{M}^c,1:T} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) := \prod_{t=1}^T q\left( \mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right), \\ q\left(\mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right):= \mathcal{N}\left(\alpha_{t} \mathbf{y}_{\mathbf{M}^c,t-1}, \beta_{t}^2 \mathbf{M}^c \right),
\end{split}
\end{equation}
There are two points worthy of noting: (1) $\alpha_t, \beta_{t}$ are not restricted to satisfy $\alpha_t^2 + \beta_t^2 = 1$; (2) formally, we add $\mathbf{y}_{\mathbf{M}}$ as one of the conditions, but it has no effect on the diffusion process in fact. Let $\bar{\alpha}_{t}=\prod_{i=i}^{t} \alpha_{i}, \bar{\beta}_{t}^2={\sum_{i=1}^{t} \frac{\bar{\alpha}_{t}^{2}}{\bar{\alpha}_{i}^{2}} \beta_{i}^{2}}$, and we additionally define $\bar{\alpha}_{0} = 1, \bar{\beta}_{0} = 0$. Then, we can derive that:
\begin{equation}
\label{eq:MC-DDPM_q_t}
q\left(\mathbf{y}_{\mathbf{M}^c,t} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right) = \mathcal{N}\left(\bar{\alpha}_{t} \mathbf{y}_{\mathbf{M}^c,0}, \bar{\beta}_{t}^{2} \mathbf{M}^c\right),
\end{equation}
\begin{equation}
\label{eq:MC-DDPM_q_t-1_t_0}
q\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c, t}, \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}, \mathbf{y}_{\mathbf{M}}\right) = \mathcal{N}\left(\tilde{\boldsymbol{\mu}}_{t}, \tilde{\beta}_{t}^{2} \mathbf{M}^c\right),
\end{equation}
where $\tilde{\boldsymbol{\mu}}_{t}=\frac{\alpha_{t} \bar{\beta}_{t-1}^{2}}{\bar{\beta}_{t}^{2}} \mathbf{y}_{\mathbf{M}^c,t}+\frac{\bar{\alpha}_{t-1} \beta_{t}^{2}}{\bar{\beta}_{t}^{2}} \mathbf{y}_{\mathbf{M}^c,0}, \tilde{\beta}_{t}=\frac{\beta_{t} \bar{\beta}_{t-1}}{\bar{\beta}_{t}}$. In MC-DDPM, we assume that $\alpha_{t}$ is set specially so that $\bar{\alpha}_{T} \approx 0$, i.e. $q\left( \mathbf{y}_{\mathbf{M}^c,T} \mid \mathbf{y}_{\mathbf{M}^c,0} \right) \approx \mathcal{N}\left(\mathbf{0}, \bar{\beta}_{T}^2\mathbf{M}^c \right)$ is a noise distribution independent of $\mathbf{y}_{M^c, 0}$.
Next, we discuss how to train MC-DDPM $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$. Firstly, let $p\left(\mathbf{y}_{\mathbf{M}^c, T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) = \mathcal{N}\left(\mathbf{0}, \bar{\beta}_{T}^2\mathbf{M}^c \right)$ so that it is nearly equal to $q\left( \mathbf{y}_{\mathbf{M}^c,T} \mid \mathbf{y}_{\mathbf{M}^c,0} \right)$. Training of $p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ is performed by optimizing the variational bound on negative log likelihood:
\begin{align*}
& \mathbb{E}\left[-\log p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right] \leq \mathbb{E}_{q}\left[-\log \frac{p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,0:T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}{q\left(\mathbf{y}_{\mathbf{M}^c,1:T} \mid \mathbf{y}_{\mathbf{M}^c,0}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)}\right] \notag \\
=& \mathbb{E}_{q}\left[-\log p\left(\mathbf{y}_{\mathbf{M}^c,T}\mid \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)-\sum_{t \geq 1} \log \frac{p_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t-1} \mid \mathbf{y}_{\mathbf{M}^c,t}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}{q\left(\mathbf{y}_{\mathbf{M}^c, t} \mid \mathbf{y}_{\mathbf{M}^c,t-1}, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)}\right]=: L.
\end{align*}
Assuming that
\begin{equation}
\boldsymbol{\mu}_{\theta}\left( \mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)=\frac{1}{\alpha_{t}}\left(\mathbf{y}_{\mathbf{M}^c, t}-\frac{\beta_{t}^{2}}{\bar{\beta}_{t}} \boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right),
\end{equation}
and supposing $\mathbf{y}_{\mathbf{M}^c, t} = \bar{\alpha}_{t} \mathbf{y}_{\mathbf{M}^c,0} + \boldsymbol{\varepsilon}, \boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \bar{\beta}_{t}^{2} \mathbf{M}^c \right)$ (Eq.~\ref{eq:MC-DDPM_q_t}), after reweighting $L$ can be simplified as follows:
\begin{equation}
L_{\mathrm{simple}}=\mathbb{E}_{\mathbf{y}^c_{\mathbf{M},0}, t, \boldsymbol{\varepsilon}} \left\| \boldsymbol{\varepsilon} -\boldsymbol{\varepsilon}_{\theta}\left(\bar{\alpha}_{t} \mathbf{y}^c_{\mathbf{M},0} + \bar{\beta}_{t} \boldsymbol{\varepsilon}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right\|_{2}^{2}, \boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \mathbf{M}^c \right),
\end{equation}
where $t$ is uniform between $1$ and $T$. The details of derivation is in supplementary materials.
Algorithm~\ref{alg:MC-DDPM_training} displays the complete training procedure with this simplified objective and Algorithm~\ref{alg:MC-DDPM_sampling} shows the sampling process.
Since MC-DDPM can produce multiple samples of the posterior distribution $q\left(\mathbf{x} \mid \mathbf{y}_{\mathbf{M}}, \mathbf{M} \right)$, the pixel-variance can be computed by Monte Carlo approach which is used to quantify uncertainty of reconstruction.
\begin{figure}[t]
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{MC-DDPM Training}
\label{alg:MC-DDPM_training}
\begin{algorithmic}[1]
\Repeat
\State $\mathbf{x} \sim q \left( \mathbf{x} \right)$, obtain $\mathbf{M}$ and $\mathbf{M}^c$
\State $\mathbf{y}_{\mathbf{M}} = \mathbf{M} \mathbf{A} \mathbf{x} $, $\mathbf{y}_{\mathbf{M}^c} = \mathbf{M}^c \mathbf{A} \mathbf{x} $
\State $\boldsymbol{\varepsilon} \sim \mathcal{N} \left(\mathbf{0}, \mathbf{M}^c \right)$
\State $t \sim \operatorname{Uniform}(\{1, \ldots, T\})$
\State $\mathbf{y}^c_{\mathbf{M},t} = \bar{\alpha}_{t} \mathbf{y}^c_{\mathbf{M},0} + \bar{\beta}_{t} \boldsymbol{\varepsilon}$
\State Take gradient descent step on
\Statex $\quad\quad\quad \nabla_{\theta} \left\| \boldsymbol{\varepsilon} -\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}^c_{\mathbf{M},t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)\right\|_{2}^{2}$
\Until{converged}
\end{algorithmic}
\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{MC-DDPM Sampling}
\label{alg:MC-DDPM_sampling}
\begin{algorithmic}[1]
\State Given $\mathbf{M}^c$ and $\mathbf{y}_{\mathbf{M}}$
\State $\mathbf{y}_{\mathbf{M}^c, T} \sim \mathcal{N} \left(\mathbf{0}, \bar{\beta}_{T}^2 \mathbf{M}^c \right)$
\For{$t = T, ..., 1$}
\State $\mathbf{z}_{t} \sim \mathcal{N}(\mathbf{0}, \mathbf{M}^c)$ if $t > 1$, else $\mathbf{z}_{t} = \mathbf{0}$
\State $\boldsymbol{\mu}_{t} = \boldsymbol{\mu}_{\theta}\left( \mathbf{y}_{\mathbf{M}^c,t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}} \right)$
\State $\mathbf{y}_{\mathbf{M}^c, t-1}= \boldsymbol{\mu}_{t} + \sigma_{t} \mathbf{z}_{t}$
\EndFor
\State \textbf{return} $\mathbf{x} = \mathbf{A}^{-1} \left(\mathbf{y}_{\mathbf{M}} + \mathbf{y}_{\mathbf{M}^c, 0}\right)$
\end{algorithmic}
\end{algorithm}
\end{minipage}
\end{figure}
\section{Experiments}
\label{sec:experiments}
We apply MC-DDPPM to accelerated MRI reconstruction where $\mathbf{A}$ is 2d Fourier transform and $\mathbf{y}_{\mathbf{M}}$ is the under-sampled k-space data. The specific design for $\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right)$ in our experiments is given as follows:
\begin{equation}
\label{eq:MC-DDPM_mri_recon}
\boldsymbol{\varepsilon}_{\theta}\left(\mathbf{y}_{\mathbf{M}^c, t}, t, \mathbf{M}^c, \mathbf{y}_{\mathbf{M}}\right) = \mathbf{M}^c f\left(g\left(\mathbf{A}^{-1}\left(\mathbf{y}_{\mathbf{M}^c, t} + \mathbf{y}_{M} \right), \mathbf{A}^{-1}\mathbf{y}_{M}\right), t; \theta \right),
\end{equation}
where $f$ is a deep neural network and $g\left(\cdot,\cdot \right)$ is the concatenation operation. Because MR image $\mathbf{x}$ is in complex filed, we use $\left|\mathbf{x}\right|$, the magnitude of it, as the final image. Pixel-wise variance is also computed using magnitude images.
\subsection{Experimental Setting}
\label{sec:experiments_setting}
All experiments are performed with fastMRI single-coil knee dataset \citep{zbontar2018fastmri}, which is publicly available\footnote{https://fastmri.org} and is divided into two parts, proton-density with (PDFS) and without fat suppression (PD). We trained the network with k-space data which were computed from $320 \times 320$ size complex images. We base the implementation of guided-DDPM \citep{dhariwal2021diffusion} and also follow similar setting for the diffusion process in \citep{dhariwal2021diffusion} but multiply $\beta_{t}$ by $0.5$ so that $\bar{\beta}_{T} \approx 0.5$. All networks were trained with learning rate of $0.0001$ using AdamW optimizer. More details of experiments is in supplementary materials.
To verify superiority, we perform comparison studies with baseline methods (U-Net \citep{ronneberger2015u}) used in \citep{zbontar2018fastmri}. The evaluation metrics, peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), of score-based reconstruction method proposed in \citep{chung2021score} are also used for comparison\footnote{They are provided in the paper of \citep{chung2021score}.} since their experiments are conducted on the same dataset.
\subsection{Experimental Results}
\label{sec:experiments_results}
We show the results of PD with $4 \times$ (the first row) and $8\times$ (the second row) acceleration in Fig.~\ref{fig:reconstruction}. More results are shown in supplementary materials. We compare our method to zero-filled reconstruction (ZF) and U-Net. Since MC-DDPM can produce multiple reconstruction samples, we use the mean of 20 samples as the object for comparison. We observe that the proposed method performs best both in $4 \times$ and $8 \times$ accelerations, where we see virtually more realistic structures and less error in the zoomed-in image than ZF and U-Net. In the last column of Fig.~\ref{fig:reconstruction}, we show the standard deviation of the samples. As the acceleration factor is increased, we see that the uncertainty increases correspondingly. Quantitative metrics in Table.~\ref{tab:evaluation_metrics} also confirm the superiority of our method. In the last two columns of Table.~\ref{tab:evaluation_metrics}, we compare MC-DDPM to the score-based reconstruction method proposed in \citep{chung2021score}. Because the testing volumes are randomly selected both in our experiments and in \citep{chung2021score}, it is impossible to compare directly. As a substitution, we compare the enhancement of evaluation metrics which is computed by the result of proposed method subtracting the result of U-Net. Due to the experiments in \citep{chung2021score} were conducted on the whole dataset (both PD and PDFS), we compute the average enhancement of PD and PDFS as our final
result. Our method outperforms \citep{chung2021score} by 3.62/0.077 ($4 \times$) and 2.96/0.089 ($8 \times$) in PSNR/SSIM.
We also explore the effects of sampling steps and number of samples on reconstruction quality, which are illustrated in Fig.~\ref{fig:steps} and Fig.~\ref{fig:num_samples}. The two experiments are conducted on one volume of PDFS with $4 \times$ and $8 \times$ acceleration, and PSNR is computed on the mean of generated samples. We discover that: (1) even the sampling steps decrease to $250$, PSNR only reduces a little; (2) the quality of the mean of samples is enhanced when the number of samples increases and seems to converge. Taking the efficiency into account, 20 samples with 250 sampling steps may be a good choice.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{merged_images.png}
\caption{Reconstruction results of $4 \times$ (the first row) and $8 \times$ (the second row) on PD data: (a) ground-truth, (b) zero-filled reconstruction, (c) U-Net, (d) the mean of samples generated by proposed method, (e) the standard deviation of the samples: range is set to [0, 0.2]. Blue box: Zoom in version of the indicated red box. Green box: Difference magnitude of the inset. White numbers indicate PSNR and SSIM, respectively.}
\label{fig:reconstruction}
\end{figure}
\begin{table}[t]
\caption{Quantitative metrics. Numbers in bold face indicate the best metric out of all the methods. The enhancement in the last two columns is computed based on U-Net.}
\centering
\begin{tabular}{c|c|p{0.9cm}<{\centering}p{0.9cm}<{\centering}p{0.9cm}<{\centering}|p{0.9cm}<{\centering}p{0.9cm}<{\centering}p{0.9cm}<{\centering}|p{1.7cm}<{\centering}p{0.9cm}<{\centering}}
\hline
~ & ~ & \multicolumn{3}{c|}{PD} & \multicolumn{3}{c|}{PDFS} & \multicolumn{2}{c}{Enhancement} \\
~ & ~ & ZF & U-Net & Ours & ZF & U-Net & Ours & \cite{chung2021score} & Ours \\
\hline
\multirow{2}{*}{$\times$ 4} & PSNR & 29.62 & 34.04 & \textbf{36.69} & 26.32 & 28.30 & \textbf{33.00} & +0.06 & \textbf{+3.68} \\
~ & SSIM & 0.745 & 0.834 & \textbf{0.905} & 0.545 & 0.648 & \textbf{0.735} & +0.002 & \textbf{+0.079} \\
\hline
\multirow{2}{*}{$\times$ 8} & PSNR & 25.94 & 31.13 & \textbf{33.49} & 24.90 & 26.17 & \textbf{31.75} & +1.01 & \textbf{+3.97} \\
~ & SSIM & 0.667 & 0.750 & \textbf{0.862} & 0.513 & 0.580 & \textbf{0.702} & +0.028 & \textbf{+0.117} \\
\hline
\end{tabular}
\label{tab:evaluation_metrics}
\end{table}
\begin{figure}[ht]
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{steps.pdf}
\caption{Tradeoff between number of sampling steps vs. PSNR testing on one volume of PDFS.}
\label{fig:steps}
\end{minipage}
\hfill
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{num_samples.pdf}
\caption{Tradeoff between number of samples vs. PSNR testing on one volume of PDFS.}
\label{fig:num_samples}
\end{minipage}
\end{figure}
\subsection{Discussion}
\label{sec:experiments_discussion}
It is very common in medical imaging that the measurement is under sampled to reduce the cost or dosage. Therefore, it is important to define the conditional diffusion process in the measurement space for a reconstruction task. In this project, although our experiments are conducted using MR images, our method can be applied to other under-sampled medical image reconstruction tasks, such as limited angle or spare view CT Reconstruction. In our MC-DDPM, the variance schedule $\left\{\beta_{t}\right\}$ is an important hyper-parameter that is potentially related to the sampling quality and efficiency. Further investigation on hyper-parameter $\left\{\beta_{t}\right\}$ is planned in our future study.
\section{Conclusion}
\label{sec:conclusion}
In this paper we present a novel and unified mathematical framework, MC-DDPM, for medical image reconstruction using under-sampled measurements. Our method applies diffusion process in measurement domain with conditioned under-sampling mask, and provides an estimate of uncertainty as output. The superior performance of our method is demonstrated using accelerated MRI reconstruction, although MC-DDPM should potentially work for other under-sampled medical image reconstruction tasks.
\section{Acknowledgements} Thank Zifan Chen for his discussion and advice for paper writing. |
2009.08780 | \section{Introduction}
The Arimoto-Blahut algorithm \cite{ari}, \cite{bla} is an algorithm for calculating a sequence of input distributions $\{\bm\lambda^N\}_{N=0,1,\ldots}$ converging to $\bm\lambda^\ast$ that achieves the capacity $C$ of a discrete memoryless channel. In the algorithm, $\bm\lambda^{N+1}$ is obtained by a function of $\bm\lambda^N$, i.e., $\bm\lambda^{N+1}=F(\bm\lambda^N)$, where $F$ is a differentiable function from the set $\Delta(\cal X)$ of the input distributions to itself. $\bm\lambda^\ast$ is the fixed point of $F$. In this paper and \cite{nak4}, the convergence speed of the Arimoto-Blahut algorithm is analyzed by the Taylor expansion of $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$.
Previous researches \cite{ari}, \cite{mat}, \cite{yu} deal with the case where $\bm\lambda^\ast=(\lambda^\ast_1,\ldots,\lambda^\ast_m)$ is an interior point of $\Delta(\cal X)$, i.e., $\lambda^\ast_i>0,\,i=1,\ldots,m$. Let $J(\bm\lambda^\ast)$ be the Jacobian matrix of the function $F(\bm\lambda)$ at $\bm\lambda^\ast$. If $\bm\lambda^\ast$ is an interior point of $\Delta(\cal X)$, all the eigenvalues $\theta_1,\ldots,\theta_m$ of $J(\bm\lambda^\ast)$ satisfy $0\leq\theta_i<1,\,i=1,\ldots,m$, then $\bm\lambda^N\to\bm\lambda^\ast$ is the exponential convergence, so this case is relatively easy.
If the convergence of $\bm\lambda^N\to\bm\lambda^\ast$ is slower than exponential, then $\bm\lambda^\ast$ is on the boundary of $\Delta(\cal X)$. Therefore, we need to examine the behavior of a differentiable function at the boundary, however, this is a difficult problem in general. We obtained \cite{nak4} a necessary and sufficient condition for the $O(1/N)$ convergence in the case that the input alphabet size is $m=3$. We classified the input symbols, or equivalently, the indices of the input alphabet into three types, and showed that the existence of type-II indices is a necessary and sufficient condition for the $O(1/N)$ convergence. In this paper, we will generalize the results of \cite{nak4} to arbitrary $m\geq3$.
We will analyze the recurrence formula obtained by truncating the Taylor expansion of the function $F(\bm\lambda)$ up to the second order term, which we call the {\it second order recurrence formula}. The results of this paper can be stated as follows. If there are no type-II indices, the convergence speed of the Arimoto-Blahut algorithm is exponential. If there are type-II indices, the convergence speed of the second order recurrence formula is $O(1/N)$ for some initial vector. We will consider the condition on the $O(1/N)$ convergence for any initial vector. Furthermore, we will consider the speed that the mutual information converges to the channel capacity $C$. We will prove that if type-III indices do not exist, then the convergence speed of $I(\bm\lambda^N,\Phi)\to C$ is faster than that of $\bm\lambda^N\to\bm\lambda^\ast$. Especially, if $\bm\lambda^N\to\bm\lambda^\ast$ is $O(1/N)$, then $I(\bm\lambda^N,\Phi)\to C$ is $O(1/N^2)$.
The reason of the slow convergence of the Arimoto-Blahut algorithm will be made clear by this paper. Since $\lambda_i^\ast=0$ for the type-II index $i$, if those input symbols with type-II indices are removed from the input alphabet, the channel capacity does not change, and the convergence of the Arimoto-Blahut algorithm becomes exponential. Based on the results of this study, it is expected to be applied to the acceleration of the Arimoto-Blahut algorithm.
\section{Related works}
There have been many related works on the Arimoto-Blahut algorithm. For example, the extension to different kinds of channels \cite{nai}, \cite{rez}, \cite{von}, the acceleration of the Arimoto-Blahut algorithm \cite{mat}, \cite{yu}, and the characterization of the Arimoto-Blahut algorithm by the divergence geometry \cite{csi2}, \cite{mat}, \cite{naj}, \cite{nak1},\,etc. If we focus on the analysis of the convergence speed of the Arimoto-Blahut algorithm, we see in \cite{ari}, \cite{mat}, \cite{yu} where the eigenvalues of the Jacobian matrix are calculated and the convergence speed is investigated in the case that $\bm\lambda^\ast$ is in the interior of $\Delta(\cal X)$.
In this paper, we consider the Taylor expansion of the defining function $F(\bm\lambda)$ of the Arimoto-Blahut algorithm. We will calculate not only the Jacobian matrix of the first order term of the Taylor expansion, but also the Hessian matrix of the second order term, and examine the convergence speed of the exponential or $O(1/N)$ based on the Jacobian and Hessian matrices. Investigation of the Hessian matrix is new in this paper and \cite{nak4}.
\section{Channel matrix and channel capacity}
Consider a discrete memoryless channel $X\rightarrow Y$ with the input source $X$ and the output source $Y$. Let ${\cal X}=\{x_1,\ldots,x_m\}$ be the input alphabet and ${\cal Y}=\{y_1,\ldots,y_n\}$ be the output alphabet. The conditional probability that the output symbol $y_j$ is received when the input symbol $x_i$ was transmitted is denoted by $P^i_j=P(Y=y_j|X=x_i),\,i=1,\ldots,m, j=1,\ldots,n,$ and the row vector $P^i$ is defined by $P^i=(P^i_1,\ldots,P^i_n),\,i=1,\ldots,m$. The channel matrix $\Phi$ is defined by\\[-3mm]
\begin{align}
\label{eqn:thechannelmatrix}
\Phi=\begin{pmatrix}
\,P^1\,\\
\vdots\\
\,P^m\,
\end{pmatrix}
=\begin{pmatrix}
\,P^1_1 & \ldots & P^1_n\,\\
\vdots & & \vdots\\
\,P^m_1 & \ldots & P^m_n\,
\end{pmatrix}.
\end{align}
We assume that for any $j\,(j=1,\ldots,n)$ there exist at least one $i\,(i=1,\ldots,m)$ with $P^i_j>0$. This means that there are no useless output symbols.
The set of input probability distributions on the input alphabet ${\cal X}$ is denoted by $\Delta({\cal X})\equiv\{\bm\lambda=(\lambda_1,\ldots,\lambda_m)|\lambda_i\geq0,i=1,\ldots,m,\sum_{i=1}^m\lambda_i=1\}$. The interior of $\Delta({\cal X})$ is denoted by $\Delta({\cal X})^\circ\equiv\{\bm\lambda=(\lambda_1,\ldots,\lambda_m)\in\Delta({\cal X})\,|\,\lambda_i>0,\,i=1,\ldots,m\}$. Similarly, the set of output probability distributions on the output alphabet ${\cal Y}$ is denoted by $\Delta({\cal Y})\equiv\{Q=(Q_1,\ldots,Q_n)|Q_j\geq0,j=1,\ldots,n,\sum_{j=1}^nQ_j=1\}$, and its interior $\Delta({\cal Y})^\circ$ is similarly defined.
Let $Q=\bm\lambda\Phi$ be the output distribution for the input distribution $\bm\lambda\in\Delta(\cal X)$ and write its components as $Q_j=\sum_{i=1}^m\lambda_iP^i_j,\,j=1,\ldots,n$, then the mutual information is defined by $I(\bm\lambda,\Phi)=\sum_{i=1}^m\sum_{j=1}^n\lambda_iP^i_j\log\left({P^i_j}/{Q_j}\right)$. The channel capacity $C$ is defined by
\begin{align}
\label{eqn:Cdefinition}
C=\max_{\bm\lambda\in\Delta({\cal X})}I(\bm\lambda,\Phi).
\end{align}
The Kullback-Leibler divergence $D(Q\|Q')$ for two output distributions $Q=(Q_1,\ldots,Q_n)$, $Q'=(Q'_1,\ldots,Q'_n)\in\Delta(\cal Y)$ is defined \cite{csi1} by
\begin{align}
D(Q\|Q')=\sum_{j=1}^nQ_j\log\displaystyle\frac{Q_j}{Q'_j}.
\end{align}
An important proposition for investigating the convergence speed of the Arimoto-Blahut algorithm is the Kuhn-Tucker condition on the input distribution $\bm\lambda=\bm\lambda^\ast$ that achieves the maximum of (\ref{eqn:Cdefinition}).
\medskip
\noindent{\bf Theorem}\ (Kuhn-Tucker condition \cite{cov}) In the maximization problem (\ref{eqn:Cdefinition}), a necessary and sufficient condition for the input distribution $\bm\lambda^\ast=(\lambda^\ast_1,\ldots,\lambda^\ast_m)\in\Delta({\cal X})$ to achieve the maximum is that there is a certain constant $\tilde{C}$ with
\begin{align}
\label{eqn:Kuhn-Tucker}
D(P^i\|\bm\lambda^\ast\Phi)\left\{\begin{array}{ll}=\tilde{C}, & {\mbox{\rm for}}\ i\ {\mbox{\rm with}}\ \lambda^\ast_i>0,\\
\leq \tilde{C}, & {\mbox{\rm for}}\ i\ {\mbox{\rm with}}\ \lambda^\ast_i=0.
\end{array}\right.
\end{align}
In (\ref{eqn:Kuhn-Tucker}), $\tilde{C}$ is equal to the channel capacity $C$.
\medskip
Since this Kuhn-Tucker condition is a necessary and sufficient condition, all the information about the capacity-achieving input distribution $\bm\lambda^\ast$ can be derived from this condition.
\section{Arimoto-Blahut algorithm}
\subsection{Definition of the algorithm}
A sequence of input distributions $\{\bm\lambda^N=(\lambda^N_1,\ldots,\lambda^N_m)\}_ {N=0,1,\ldots}\subset\Delta({\cal X})$ is defined by the Arimoto-Blahut algorithm as follows \cite{ari}, \cite{bla}. First, let $\bm\lambda^0=(\lambda^0_1,\ldots,\lambda^0_m)$ be an initial distribution taken in $\Delta(\cal X)^\circ$, i.e., $\lambda^0_i>0,\,i=1,\ldots,m$. Then, the Arimoto-Blahut algorithm is given by the recurrence formula
\begin{align}
\lambda^{N+1}_i=\displaystyle\frac{\lambda^N_i\exp D(P^i\|\bm\lambda^N\Phi)}{\displaystyle\sum_{k=1}^m\lambda^N_k\exp D(P^k\|\bm\lambda^N\Phi)},\,i=1,\ldots,m,\,N=0,1,\ldots.\label{eqn:arimotoalgorithm}
\end{align}
On the convergence of this Arimoto-Blahut algorithm, the following results were obtained in Arimoto \cite{ari}, \cite{ari2}.
By defining
\begin{align}
C(N+1,N)\equiv-\displaystyle\sum_{i=1}^m\lambda^{N+1}_i\log\lambda^{N+1}_i+\displaystyle\sum_{i=1}^m\sum_{j=1}^n\lambda^{N+1}_iP^i_j\log\displaystyle\frac{\lambda^N_iP^i_j}{\displaystyle\sum_{k=1}^m\lambda^N_kP^k_j},
\end{align}
he obtained the following theorems.
\noindent{\bf Theorem A1}\,\cite{ari} If the initial input distribution $\bm\lambda^0$ is in $\Delta({\cal X})^\circ$, then
\begin{align}
\lim_{N\to\infty}C(N+1,N)=C.
\end{align}
\noindent{\bf Theorem A2}\,\cite{ari} If $\bm\lambda^0$ is the uniform distribution, then
\begin{align}
0\leq C-C(N+1,N)\leq\displaystyle\frac{\log m-h(\bm\lambda^\ast)}{N},
\end{align}
where $\bm\lambda^\ast$ is the capacity-achieving input distribution and $h(\bm\lambda^\ast)$ is the entropy of $\bm\lambda^\ast$.
\noindent{\bf Theorem A3}\,\cite{ari2} Assume that $\bm\lambda^\ast$ is unique and belongs to $\Delta({\cal X})^\circ$. Then, for sufficiently small arbitrary $\epsilon>0$, there exists $N_0=N_0(\epsilon)$ such that
\begin{align}
0\leq C-C(N+1,N)\leq\epsilon(\theta)^{N-N_0},\,N\geq N_0,
\end{align}
where $\theta$ is a constant with $0\leq\theta<1$ and is unrelated to $\epsilon$ and $N_0$, further, $(\theta)^N$ denotes the $N$th power of $\theta$.
\medskip
In \cite{ari}, he considered the Taylor expansion of $D(\bm\lambda^\ast\|\bm\lambda)$ by $\bm\lambda$, and that of $D(Q^\ast\|Q)$ by $Q$, however he did not consider the Taylor expansion of the function $F(\bm\lambda)$. Further, in the above Theorem A3, he considered only the case $\bm\lambda^\ast\in\Delta({\cal X})^\circ$, where the convergence is exponential.
In Yu\,\cite{yu}, he considered the function $F(\bm\lambda)$ and the Taylor expansion of $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$. He calculated the eigenvalues of the Jacobian matrix $J(\bm\lambda^\ast)$, however he did not consider the Hessian matrix. Further, he considered only the case $\bm\lambda^\ast\in\Delta({\cal X})^\circ$ as in \cite{ari}, \cite{ari2}.
\subsection{Function from $\Delta({\cal X})$ to $\Delta({\cal X})$}
Let $F_i(\bm\lambda)$ be the defining function of the Arimoto-Blahut algorithm (\ref{eqn:arimotoalgorithm}), i.e.,
\begin{align}
F_i(\bm\lambda)=\displaystyle\frac{\lambda_i\exp D(P^i\|\bm\lambda\Phi)}{\displaystyle\sum_{k=1}^m\lambda_k\exp D(P^k\|\bm\lambda\Phi)},\,i=1,\ldots,m.\label{eqn:Arimotofunction}
\end{align}
Define $F(\bm\lambda)\equiv(F_1(\bm\lambda),\ldots,F_m(\bm\lambda))$, then $F(\bm\lambda)$ is a differentiable function from $\Delta(\cal X)$ to $\Delta(\cal X)$, and (\ref{eqn:arimotoalgorithm}) is represented by $\bm\lambda^{N+1}=F(\bm\lambda^N)$.
In this paper, for the analysis of the convergence speed, we assume
\begin{align}
{\rm rank}\,\Phi=m.\label{eqn:rankmdefinition}
\end{align}
Concerning this assumption, we see that in \cite{ari}, \cite{ari2}, for the analysis of the convergence speed, the uniqueness of the capacity-achieving $\bm\lambda^\ast$ is assumed, which is a necessary condition for (\ref{eqn:rankmdefinition}), in fact, we have
\begin{lemma}
\label{lem:capacityachievinglambdaisunique}
The capacity-achieving input distribution $\bm\lambda^\ast$ is unique.
\end{lemma}
{\bf Proof:} By Csisz\'{a}r\cite{csi1},\,p.137,\,eq.\,(37), for arbitrary $Q\in\Delta(\cal Y)$,%
\begin{align}
\displaystyle\sum_{i=1}^m\lambda_iD(P^i\|Q)=I(\bm\lambda,\Phi)+D(\bm\lambda\Phi\|Q).\label{eqn:CKequality}
\end{align}
By the assumption (\ref{eqn:rankmdefinition}), we see that there exists $Q^0\in\Delta(\cal Y)$ \cite{nak2} with
\begin{align}
D(P^1\|Q^0)=\ldots=D(P^m\|Q^0)\equiv C^0.
\end{align}
Substituting $Q=Q^0$ into (\ref{eqn:CKequality}), we have $C^0=I(\bm\lambda,\Phi)+D(\bm\lambda\Phi\|Q^0)$. Because $C^0$ is a constant,
\begin{align}
\max_{\bm\lambda\in\Delta(\cal X)}I(\bm\lambda,\Phi)\Longleftrightarrow\min_{\bm\lambda\in\Delta(\cal X)}D(\bm\lambda\Phi\|Q^0).\label{eqn:maxequalmin}
\end{align}
Define $W\equiv\{\bm\lambda\Phi\,|\,\bm\lambda\in\Delta(\cal X)\}$, then $W$ is a closed convex set, thus by Cover\,\cite{cov},\,p.297,\, Theorem 12.6.1, $Q=Q^\ast$ that achieves $\min_{Q\in W}D(Q\|Q^0)$ exists and is unique. By the assumption (\ref{eqn:rankmdefinition}), the mapping $\Delta\ni\bm\lambda\mapsto\bm\lambda\Phi\in W$ is one to one, therefore, $\bm\lambda^\ast$ with $Q^\ast=\bm\lambda^\ast\Phi$ is unique.\hfill$\blacksquare$
\begin{remark}
\rm Due to the equivalence (\ref{eqn:maxequalmin}), the Arimoto-Blahut algorithm can be obtained by Csisz\'{a}r \cite{csi2}, Chapter 4, ``Minimizing information distance from a single measure'', Theorem 5.
\end{remark}
\begin{lemma}
\label{lem:capacity_achieving_lambda_is_the_fixed_point}
The capacity-achieving input distribution $\bm\lambda^\ast$ is the fixed point of the function $F(\bm\lambda)$. That is, $\bm\lambda^\ast=F(\bm\lambda^\ast)$.
\end{lemma}
{\bf Proof:} In the Kuhn-Tucker condition (\ref{eqn:Kuhn-Tucker}), let us define $m_1$ as the number of indices $i$ with $\lambda^\ast_i>0$, i.e.,
\begin{align}
\lambda^\ast_i\left\{\begin{array}{ll}>0, & i=1,\ldots,m_1,\\=0, & i=m_1+1,\ldots,m,\end{array}\right.\label{eqn:m1definition}
\end{align}
by reordering the input symbols (if necessary), then
\begin{align}
D(P^i\|\bm\lambda^\ast\Phi)\left\{\begin{array}{ll}=C, & i=1,\ldots,m_1,\\\leq C, & i=m_1+1,\ldots,m.\end{array}\right.
\end{align}
We have
\begin{align}
\displaystyle\sum_{k=1}^m\lambda^\ast_k\exp D(P^k\|\bm\lambda^\ast\Phi)=\displaystyle\sum_{k=1}^{m_1}\lambda^\ast_ke^C=e^C,\label{eqn:yobitekikeisan}
\end{align}
hence by (\ref{eqn:Arimotofunction}),\,(\ref{eqn:m1definition}),\,(\ref{eqn:yobitekikeisan}),
\begin{align}
F_i(\bm\lambda^\ast)&=\left\{\begin{array}{ll}e^{-C}\lambda^\ast_ie^C, & i=1,\ldots,m_1,\\0, & i=m_1+1,\ldots,m,\end{array}\right.\\
&=\lambda^\ast_i,\,i=1,\ldots,m,
\end{align}
which shows $F(\bm\lambda^\ast)=\bm\lambda^\ast$.\hfill$\blacksquare$
\medskip
The sequence $\left\{\bm\lambda^N\right\}_{N=0,1,\ldots}$ of the Arimoto-Blahut algorithm converges to the fixed point $\bm\lambda^\ast$, i.e., $\bm\lambda^N\to\bm\lambda^\ast,\,N\to\infty$.
We will investigate the convergence speed by using the Taylor expansion of $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$.
Now, we define two kinds of convergence speed for investigating $\bm\lambda^N\to\bm\lambda^\ast$.
\begin{itemize}
\item[(i)] Exponential convergence\\
$\bm\lambda^N\to\bm\lambda^\ast$ is the {\it exponential convergence} if
\begin{align}
\|\bm\lambda^N-\bm\lambda^\ast\|<K(\theta)^N,\,K>0,\,0\leq\theta<1,\,N=0,1,\ldots,
\end{align}
where $\|\bm\lambda\|$ denotes the Euclidean norm $\|\bm\lambda\|=\left(\lambda_1^2+\ldots+\lambda_m^2\right)^{1/2}$.
\item[(ii)] $O(1/N)$ convergence\\
$\bm\lambda^N\to\bm\lambda^\ast$ is the {\it $O(1/N)$ convergence} if
\begin{align}
\lim_{N\to\infty}N\left(\lambda^N_i-\lambda^\ast_i\right)=K_i\neq0,\,i=1,\ldots,m.
\end{align}
\end{itemize}
\subsection{Type of index}
Now, we classify the indices $i\,(i=1,\ldots,m)$ in the Kuhn-Tucker condition (\ref{eqn:Kuhn-Tucker}) in more detail into the following 3 types.
\begin{align}
\label{eqn:Kuhn-Tucker2}
D(P^i\|\bm\lambda^\ast\Phi)\left\{\begin{array}{ll}=C, & {\mbox{\rm for}}\ i\ {\mbox{\rm with}}\ \lambda^\ast_i>0\ \mbox{\rm [type-I]},\\
=C, & {\mbox{\rm for}}\ i\ {\mbox{\rm with}}\ \lambda^\ast_i=0\ \mbox{\rm [type-II]},\\
<C, & {\mbox{\rm for}}\ i\ {\mbox{\rm with}}\ \lambda^\ast_i=0\ \mbox{\rm [type-III]}.
\end{array}\right.
\end{align}
Let us define the sets of indices as follows.
\begin{align}
&{\rm all\ the\ indices:}\ {\cal I}\equiv\{1,\ldots,m\},\label{eqn:allset}\\
&\mbox{\rm type-I\ indices:}\ {\cal I}_{\rm I}\equiv\{1,\ldots,m_1\},\label{eqn:type1set}\\
&\mbox{\rm type-II\ indices:}\ {\cal I}_{\rm II}\equiv\{m_1+1,\ldots,m_1+m_2\},\label{eqn:type2set}\\
&\mbox{\rm type-III\ indices:}\ {\cal I}_{\rm III}\equiv\{m_1+m_2+1,\ldots,m\}.\label{eqn:type3set}
\end{align}
We have $|{\cal I}|=m$, $|{\cal I}_{\rm I}|=m_1$, $|{\cal I}_{\rm II}|=m_2$, $|{\cal I}_{\rm III}|=m-m_1-m_2\equiv m_3$, further, ${\cal I}={\cal I}_{\rm I}\cup{\cal I}_{\rm II}\cup{\cal I}_{\rm III}$ and $m=m_1+m_2+m_3$. ${\cal I}_{\rm I}$ is not empty and $|{\cal I}_{\rm I}|=m_1\geq2$ for any channel matrix, but ${\cal I}_{\rm II}$ and ${\cal I}_{\rm III}$ may be empty for some channel matrix.
\subsection{Examples of convergence speed}
Let us consider the difference of convergence speed of the Arimoto-Blahut algorithm depending on the channel matrices.
For many channel matrices $\Phi$, the convergence is exponential, but for some special $\Phi$ the convergence is very slow. Let us consider the following examples with input alphabet size $m=3$ and output alphabet size $n=3$ taking types-I, II and III into account.
\begin{example}
\label{exa:CM_Phi(1)}
\rm (only type-I) If only type-I indices exist, then $\lambda^\ast_i>0,\,i=1,2,3$, hence $Q^\ast\equiv\bm\lambda^\ast\Phi$ is in the interior of $\triangle P^1P^2P^3$. See Fig. \ref{fig:1}. As a concrete channel matrix of this example, let us consider
\begin{align}
\label{eqn:Phi1}
\Phi^{(1)}=\begin{pmatrix}
\,0.800 & 0.100 & 0.100\,\\
\,0.100 & 0.800 & 0.100\,\\
\,0.250 & 0.250 & 0.500\,
\end{pmatrix}.
\end{align}
For this $\Phi^{(1)}$, we have $\bm\lambda^\ast=(0.431,0.431,0.138)$ and $Q^\ast=(0.422,0.422,0.156)$. The vertices of the large triangle in Fig. \ref{fig:1} are $\bm{e}_1=(1,0,0),\,\bm{e}_2=(0,1,0),\,\bm{e}_3=(0,0,1)$. We have $D(P^i\|Q^\ast)=C,\,i=1,2,3$, then considering the analogy to Euclidean geometry, $\triangle P^1P^2P^3$ can be regarded as an ``acute triangle''.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8.8cm]{figure1.eps}
\put(-5,-4){$\bm{e}_1$}
\put(101,-4){$\bm{e}_2$}
\put(48,89){$\bm{e}_3$}
\put(52,16){$Q^\ast$}
\put(9,5){$P^1$}
\put(86,5){$P^2$}
\put(48,46){$P^3$}
\end{overpic}
\medskip
\caption{Positional relation of row vectors $P^1,P^2,P^3$ of $\Phi^{(1)}$ and $Q^\ast$ in Example \ref{exa:CM_Phi(1)}.}
\label{fig:1}
\end{center}
\end{figure}
\end{example}
\begin{example}
\label{exa:CM_Phi(2)}
\rm (types-I and II) If there are type-I and type-II indices, we can assume $\lambda^\ast_1>0,\lambda^\ast_2>0,\lambda^\ast_3=0$ without loss of generality, hence $Q^\ast$ is on the side $P^1P^2$ and $D(P^i\|Q^\ast)=C,\,i=1,2,3$. See Fig. \ref{fig:2}. As a concrete channel matrix of this example, let us consider
\begin{align}
\label{eqn:Phi2}
\Phi^{(2)}=\begin{pmatrix}
\,0.800 & 0.100 & 0.100\,\\
\,0.100 & 0.800 & 0.100\,\\
\,0.300 & 0.300 & 0.400\,
\end{pmatrix}.
\end{align}
For this $\Phi^{(2)}$, we have $\bm\lambda^\ast=(0.500,0.500,0.000)$ and $Q^\ast=(0.450,0.450,0.100)$. Considering the analogy to Euclidean geometry, $\triangle P^1P^2P^3$ can be regarded as a ``right triangle''.
\begin{figure}[t]
\begin{center}
\medskip
\begin{overpic}[width=8.8cm]{figure2.eps}
\put(-5,-4){$\bm{e}_1$}
\put(101,-4){$\bm{e}_2$}
\put(48,89){$\bm{e}_3$}
\put(48,4){$Q^\ast$}
\put(9,5){$P^1$}
\put(86,5){$P^2$}
\put(48,37){$P^3$}
\end{overpic}
\medskip
\caption{Positional relation of row vectors $P^1,P^2,P^3$ of $\Phi^{(2)}$ and $Q^\ast$ in Example \ref{exa:CM_Phi(2)}.}
\label{fig:2}
\end{center}
\end{figure}
\end{example}
\begin{example}
\label{exa:CM_Phi(3)}
\rm (types-I and III) If there are type-I and type-III indices, we can assume $\lambda^\ast_1>0,\lambda^\ast_2>0,\lambda^\ast_3=0$ without loss of generality, hence $Q^\ast$ is on the side $P^1P^2$ and $C=D(P^1\|Q^\ast)=D(P^2\|Q^\ast)>D(P^3\|Q^\ast)$. See Fig. \ref{fig:3}. As a concrete channel matrix of this example, let us consider
\begin{align}
\label{eqn:Phi3}
\Phi^{(3)}=\begin{pmatrix}
\,0.800 & 0.100 & 0.100\,\\
\,0.100 & 0.800 & 0.100\,\\
\,0.350 & 0.350 & 0.300\,
\end{pmatrix}.
\end{align}
For this $\Phi^{(3)}$, we have $\bm\lambda^\ast=(0.500,0.500,0.000)$ and $Q^\ast=(0.450,0.450,0.100)$. Considering the analogy to Euclidean geometry, $\triangle P^1P^2P^3$ can be regarded as an ``obtuse triangle''.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8.8cm]{figure3.eps}
\put(-5,-4){$\bm{e}_1$}
\put(101,-4){$\bm{e}_2$}
\put(48,89){$\bm{e}_3$}
\put(48,4){$Q^\ast$}
\put(9,5){$P^1$}
\put(86,5){$P^2$}
\put(48,28.5){$P^3$}
\end{overpic}
\medskip
\caption{Positional relation of row vectors $P^1,P^2,P^3$ of $\Phi^{(3)}$ and $Q^\ast$ in Example \ref{exa:CM_Phi(3)}.}
\label{fig:3}
\end{center}
\end{figure}
\end{example}
For the above $\Phi^{(1)},\Phi^{(2)},\Phi^{(3)}$, we show in Fig. \ref{fig:lambda1} the state of convergence of $|\lambda^N_1-\lambda^\ast_1|\to0$. By Fig. \ref{fig:lambda1}, we see that in Examples 1 and 3 the convergence is exponential, while in Example 2 the convergence is slower than exponential.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=9cm]{figure4.eps}
\put(57,74){\rotatebox{60}{$\leftarrow$}}
\put(62,78){Example 2}
\put(62,74){(types-I and II)}
\put(58,47.5){\rotatebox{60}{$\leftarrow$}}
\put(62,50){Example 3}
\put(62,46){(types-I and III)}
\put(52,40){\rotatebox{60}{$\rightarrow$}}
\put(39,36){Example 1}
\put(39,31.5){(only type-I)}
\put(54,-3){$N$}
\put(-6,45){\rotatebox{90}{$|\lambda^N_1-\lambda^\ast_1|$}}
\end{overpic}
\caption{Comparison of the convergence speed in Examples \ref{exa:CM_Phi(1)},\ref{exa:CM_Phi(2)},\ref{exa:CM_Phi(3)}.}
\label{fig:lambda1}
\end{center}
\end{figure}
From the above three examples, it is inferred that the Arimoto-Blahut algorithm converges very slowly when type-II indices exist, and converges exponentially when type-II indices do not exist. We will analyze this phenomenon in the following.
\section{Taylor expansion of $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$}
We will examine the convergence speed of the Arimoto-Blahut algorithm by the Taylor expansion of $F(\bm\lambda)$ about the fixed point $\bm\lambda=\bm\lambda^\ast$. Taylor expansion of the function $F(\bm\lambda)=(F_1(\bm\lambda),\ldots,F_m(\bm\lambda))$ about $\bm\lambda=\bm\lambda^\ast$ is
\begin{align}
F(\bm\lambda)=F(\bm\lambda^\ast)+(\bm\lambda-\bm\lambda^\ast)J(\bm\lambda^\ast)+\displaystyle\frac{1}{2!}(\bm\lambda-\bm\lambda^\ast)H(\bm\lambda^\ast)\,^t(\bm\lambda-\bm\lambda^\ast)+o\left(\|\bm\lambda-\bm\lambda^\ast\|^2\right),\label{eqn:Taylortenkai1}
\end{align}
where ${^t}\bm\lambda$ denotes the transpose of $\bm\lambda$, and $o\left(\|\bm\lambda\|^2\right)$ means $\lim_{\|\bm\lambda\|\to0}o\left(\|\bm\lambda\|^2\right)/\|\bm\lambda\|^2=0$.
In (\ref{eqn:Taylortenkai1}), $J(\bm\lambda^\ast)$ is the Jacobian matrix of $F(\bm\lambda)$ at $\bm\lambda=\bm\lambda^\ast$, i.e.,
\begin{align}
J(\bm\lambda^\ast)&=\left(\left.\displaystyle\frac{\partial F_i}{\partial\lambda_{i'}}\right|_{\bm\lambda=\bm\lambda^\ast}\right)_{i',i=1,\ldots,m}.\label{eqn:Jacobiseibun}
\end{align}
In this paper, we assume that the input probability distribution $\bm\lambda$ is a row vector, thus the Jacobian matrix $J(\bm\lambda^\ast)$ is\\[-7mm]
\begin{align}
&\hspace{30mm}\leftarrow i\rightarrow\nonumber\\[0mm]
J(\bm\lambda^\ast)&=\begin{array}{c}\uparrow\\ i'\\\downarrow\end{array}
\hspace{-1mm}\begin{pmatrix}
\,\left.\displaystyle\frac{\partial F_1}{\partial\lambda_1}\right|_{\bm\lambda=\bm\lambda^\ast} & \ldots & \left.\displaystyle\frac{\partial F_m}{\partial\lambda_1}\right|_{\bm\lambda=\bm\lambda^\ast}\,\\
\vdots & & \vdots\\
\,\left.\displaystyle\frac{\partial F_1}{\partial\lambda_m}\right|_{\bm\lambda=\bm\lambda^\ast} & \ldots & \left.\displaystyle\frac{\partial F_m}{\partial\lambda_m}\right|_{\bm\lambda=\bm\lambda^\ast}\,
\end{pmatrix}\in\mathbb R^{m\times m},\label{eqn:rowvectorJacobimatrix}
\end{align}
i.e., $\partial F_i/\partial\lambda_{i'}|_{\bm\lambda=\bm\lambda^\ast}$ is the $(i',i)$ component. Note that our $J(\bm\lambda^\ast)$ is the transpose of the usual Jacobian matrix corresponding to column vector.
Because $\sum_{i=1}^mF_i(\bm\lambda)=1$ by (\ref{eqn:Arimotofunction}), we have by (\ref{eqn:Jacobiseibun}),
\begin{lemma}
\label{lem:rowsumofJis0}
Every row sum of $J(\bm\lambda^\ast)$ is equal to $0$.
\end{lemma}
\medskip
In (\ref{eqn:Taylortenkai1}), $H(\bm\lambda^\ast)\equiv(H_1(\bm\lambda^\ast),\ldots,H_m(\bm\lambda^\ast))$, where $H_i(\bm\lambda^\ast)$ is the Hessian matrix of $F_i(\bm\lambda)$ at $\bm\lambda=\bm\lambda^\ast$, i.e.,
\begin{align}
H_i(\bm\lambda^\ast)=\left(\left.\displaystyle\frac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}\right)_{i',i''=1,\ldots,m},\label{eqn:Hesseseibun}
\end{align}
and $(\bm\lambda-\bm\lambda^\ast)H(\bm\lambda^\ast)\,^t(\bm\lambda-\bm\lambda^\ast)$ is an abbreviated expression of the $m$ dimensional row vector $\left((\bm\lambda-\bm\lambda^\ast)H_1(\bm\lambda^\ast)\,^t(\bm\lambda-\bm\lambda^\ast),\ldots,(\bm\lambda-\bm\lambda^\ast)H_m(\bm\lambda^\ast)\,^t(\bm\lambda-\bm\lambda^\ast)\right).$
\begin{remark}
\label{rem:justify}
\rm $\lambda_1,\ldots,\lambda_m$ satisfy the constraint $\sum_{i=1}^m\lambda_i=1$, but in (\ref{eqn:Taylortenkai1}),\,(\ref{eqn:Jacobiseibun}),\,(\ref{eqn:Hesseseibun}) we consider $\lambda_1,\ldots,\lambda_m$ as independent (or constraint free) variables to have the Taylor series approximation (\ref{eqn:Taylortenkai1}). This approximation is justified as follows. By the Kuhn-Tucker condition (\ref{eqn:Kuhn-Tucker}), $D(P^i\|Q^\ast)\leq C<\infty,\,i=1,\ldots,m$, hence by the assumption put below (\ref{eqn:thechannelmatrix}), we have $Q^\ast_j>0,\,j=1,\ldots,n$. See \cite{ari}. For $\epsilon>0$, define ${\cal Q}^\ast_\epsilon\equiv\{Q=(Q_1,\ldots,Q_n)\in{\mathbb R}^n\,|\,\|Q-Q^\ast\|<\epsilon\}$, i.e., ${\cal Q}^\ast_\epsilon$ is an open ball in $\mathbb{R}^n$ centered at $Q^\ast$ with radius $\epsilon$. Note that $Q\in{\cal Q}^\ast_\epsilon$ is free from the constraint $\sum_{j=1}^nQ_j=1$. Taking $\epsilon>0$ sufficiently small, we can have $Q_j>0,j=1,\ldots,n$, for any $Q\in{\cal Q}^\ast_\epsilon$. The function $F(\bm\lambda)$ is defined for $\bm\lambda$ with $Q_j=\left(\bm\lambda\Phi\right)_j>0,\,j=1,\ldots,n$, even if some $\lambda_i<0$. Therefore, the domain of definition of $F(\bm\lambda)$ can be extended to $\Phi^{-1}\left({\cal Q}^\ast_\epsilon\right)\subset\mathbb{R}^m$, where $\Phi^{-1}\left({\cal Q}^\ast_\epsilon\right)$ is the inverse image of ${\cal Q}^\ast_\epsilon$ by the mapping $\mathbb{R}^m\ni\bm\lambda\mapsto\bm\lambda\Phi\in\mathbb{R}^n$. $\Phi^{-1}\left({\cal Q}^\ast_\epsilon\right)$ is an open neighborhood of $\bm\lambda^\ast$ in $\mathbb{R}^m$. Then $F(\bm\lambda)$ is a function of $\bm\lambda=(\lambda_1,\ldots,\lambda_m)\in\Phi^{-1}\left({\cal Q}^\ast_\epsilon\right)$ as independent variables (free from the constraint $\sum_{i=1}^m\lambda_i=1$). We can consider (\ref{eqn:Taylortenkai1}) to be the Taylor expansion by independent variables $\lambda_1,\ldots,\lambda_m$, then substituting $\bm\lambda\in\Delta({\cal X})\cap\Phi^{-1}\left({\cal Q}^\ast_\epsilon\right)$ into (\ref{eqn:Taylortenkai1}) to obtain the approximation for $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$.
\end{remark}
\medskip
Now, substituting $\bm\lambda=\bm\lambda^N$ into (\ref{eqn:Taylortenkai1}), then by $F(\bm\lambda^\ast)=\bm\lambda^\ast$ and $F(\bm\lambda^N)=\bm\lambda^{N+1}$, we have
\begin{align}
\bm\lambda^{N+1}=\bm\lambda^\ast+(\bm\lambda^N-\bm\lambda^\ast)J(\bm\lambda^\ast)+\displaystyle\frac{1}{2!}(\bm\lambda^N-\bm\lambda^\ast)H(\bm\lambda^\ast)\,^t(\bm\lambda^N-\bm\lambda^\ast)+o\left(\|\bm\lambda^N-\bm\lambda^\ast\|^2\right).\label{eqn:Taylortenkai2}
\end{align}
Then, by putting $\bm\mu^N\equiv\bm\lambda^N-\bm\lambda^\ast$, (\ref{eqn:Taylortenkai2}) becomes
\begin{align}
\bm\mu^{N+1}=\bm\mu^NJ(\bm\lambda^\ast)+\displaystyle\frac{1}{2!}\bm\mu^NH(\bm\lambda^\ast)\,{^t}\bm\mu^N+o\left(\|\bm\mu^N\|^2\right).\label{eqn:Taylortenkai3}
\end{align}
Then, we will investigate the convergence $\bm\mu^N\to\bm0,\,N\to\infty$, based on the Taylor expansion (\ref{eqn:Taylortenkai3}). Let $\mu^N_i\equiv\lambda^N_i-\lambda^\ast_i,\,i=1,\ldots,m$, denote the components of $\bm\mu^N=\bm\lambda^N-\bm\lambda^\ast$, and write $\bm\mu^N$ by components as $\bm\mu^N=(\mu^N_1,\ldots,\mu^N_m)$, then we have $\sum_{i=1}^m\mu^N_i=0,\,N=0,1,\ldots$, because $\sum_{i=1}^m\lambda^N_i=\sum_{i=1}^m\lambda^\ast_i=1$.
\subsection{The Jacobian matrix $J(\bm\lambda^\ast)$}
Let us consider the Jacobian matrix $J(\bm\lambda^\ast)$. We are assuming ${\rm rank}\,\Phi=m$ in (\ref{eqn:rankmdefinition}), hence $m\leq n$. We will calculate the components (\ref{eqn:Jacobiseibun}) of $J(\bm\lambda^\ast)$.
Defining $D_i\equiv D(P^i\|\bm\lambda\Phi)$ and $F_i\equiv F_i(\bm\lambda),\,i=1,\ldots,m$, we can write (\ref{eqn:Arimotofunction}) as
\begin{align}
F_i=\displaystyle\frac{\lambda_ie^{D_i}}{\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}},\,i=1,\ldots,m.\label{eqn:teigikansuFi}
\end{align}
From (\ref{eqn:teigikansuFi}) it follows that
\begin{align}
F_i\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}=\lambda_ie^{D_i},\label{eqn:Fibunboharau}
\end{align}
then differentiating both sides of (\ref{eqn:Fibunboharau}) with respect to $\lambda_{i'}$, we have
\begin{align}
\displaystyle\frac{\partial F_i}{\partial\lambda_{i'}}\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}+F_i\displaystyle\frac{\partial}{\partial\lambda_{i'}}\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}=\delta_{i'i}e^{D_i}+\lambda_ie^{D_i}\displaystyle\frac{\partial D_i}{\partial\lambda_{i'}},\label{eqn:dFi}
\end{align}
where $\delta_{i'i}$ is the Kronecker delta.
Before substituting $\bm\lambda=\bm\lambda^\ast=(\lambda^\ast_1,\ldots,\lambda^\ast_m)$ into the both sides of (\ref{eqn:dFi}), we define the following symbols. Remember that the integer $m_1$ was defined in (\ref{eqn:m1definition}). See also (\ref{eqn:type1set}).
Let us define
\begin{align}
Q^\ast&\equiv Q(\bm\lambda^\ast)=\bm\lambda^\ast\Phi,\\
Q_j^\ast&\equiv Q(\bm\lambda^\ast)_j=\displaystyle\sum_{i=1}^m\lambda_i^\ast P_j^i=\displaystyle\sum_{i=1}^{m_1}\lambda_i^\ast P_j^i,\,j=1,\ldots,n,\\
D_i^\ast&\equiv D(P^i\|Q^\ast),\,i=1,\ldots,m,\\
D_{i',i}^\ast&\left.\equiv\displaystyle\frac{\partial D_i}{\partial\lambda_{i'}}\right|_{\bm\lambda=\bm\lambda^\ast},\,i',i=1,\ldots,m,\label{eqn:Diidefinition}\\
F_i^\ast&\equiv F_i(\bm\lambda^\ast),\,i=1,\ldots,m.
\end{align}
\begin{lemma}
\label{lem:shoryou}We have
\begin{align}
&\left.\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}\right|_{\bm\lambda=\bm\lambda^\ast}=e^C,\label{eqn:lem3-1}\\[2mm]
&\displaystyle\frac{\partial D_i}{\partial\lambda_{i'}}=-\displaystyle\sum_{j=1}^n\displaystyle\frac{P_j^{i'}P_j^i}{Q_j},\,i',i=1,\ldots,m,\label{eqn:lem3-2}\\[2mm]
&\left.\displaystyle\frac{\partial}{\partial\lambda_{i'}}\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}\right|_{\bm\lambda=\bm\lambda^\ast}=e^{D_{i'}^\ast}-e^C,\,i'=1,\ldots,m,\label{eqn:lem3-3}\\[2mm]
&F^\ast_i=\lambda^\ast_i,\,i=1,\ldots,m.\label{eqn:lem3-4}
\end{align}
\end{lemma}
{\bf Proof:} Eq. (\ref{eqn:lem3-1}) was proved in (\ref{eqn:yobitekikeisan}). Eq. (\ref{eqn:lem3-2}) is proved by simple calculation. Eq. (\ref{eqn:lem3-3}) is proved as follows.
\begin{align}
\left.\displaystyle\frac{\partial}{\partial\lambda_{i'}}\displaystyle\sum_{k=1}^m\lambda_ke^{D_k}\right|_{\bm\lambda=\bm\lambda^\ast}
&=\left.\displaystyle\sum_{k=1}^m\left(\delta_{i'k}e^{D_k}+\lambda_ke^{D_k}\displaystyle\frac{\partial D_k}{\partial\lambda_{i'}}\right)\right|_{\bm\lambda=\bm\lambda^\ast}\\
&=e^{D_{i'}^\ast}+\displaystyle\sum_{k=1}^{m_1}\lambda_k^\ast e^C\left(-\displaystyle\sum_{j=1}^n\displaystyle\frac{P_j^kP_j^{i'}}{Q_j^\ast}\right)\\
&=e^{D_{i'}^\ast}-e^C\displaystyle\sum_{j=1}^nP_j^{i'}\displaystyle\frac{1}{Q_j^\ast}\displaystyle\sum_{k=1}^{m_1}\lambda_k^\ast P_j^k\\
&=e^{D_{i'}^\ast}-e^C.
\end{align}
Note that $Q^\ast_j>0,\,j=1,\ldots,n$, from Remark \ref{rem:justify}. Eq. (\ref{eqn:lem3-4}) is the result of Lemma \ref{lem:capacity_achieving_lambda_is_the_fixed_point}.\hfill$\blacksquare$
\medskip
Substituting the results of Lemma \ref{lem:shoryou} into (\ref{eqn:dFi}), we have
\begin{align}
\left.\displaystyle\frac{\partial F_i}{\partial\lambda_{i'}}\right|_{\bm\lambda=\bm\lambda^\ast}e^C+\lambda_i^\ast\left(e^{D_{i'}^\ast}-e^C\right)=\delta_{i'i}e^{D^\ast_i}+\lambda_i^\ast e^{D^\ast_i}D^\ast_{i',i}.
\end{align}
Consequently, we have
\begin{theorem}
\label{the:1}The components of the Jacobian matrix $J(\bm\lambda^\ast)$ are given as follows.
\begin{align}
\left.\displaystyle\frac{\partial F_i}{\partial\lambda_{i'}}\right|_{\bm\lambda=\bm\lambda^\ast}&=e^{D_i^\ast-C}\left(\delta_{i'i}+\lambda_i^\ast D^\ast_{i',i}\right)+\lambda^\ast_i\left(1-e^{D^\ast_{i'}-C}\right),\,i',i\in{\cal I},\label{eqn:theorem1-0}\\[1mm]
&=\left\{\begin{array}{l}
\delta_{i'i}+\lambda_i^\ast\left(D^\ast_{i',i}+1-e^{D_{i'}^\ast-C}\right),\,i'\in{\cal I},\,i\in{\cal I}_{\rm I},\\[2mm]
\delta_{i'i},\,i'\in{\cal I},\,i\in{\cal I}_{\rm II},\\[2mm]
e^{D^\ast_i-C}\delta_{i'i},\,i'\in{\cal I},\,i\in{\cal I}_{\rm III},
\end{array}\right.\label{eqn:theorem1-1}
\end{align}
where the sets of indices ${\cal I}$, ${\cal I}_{\rm I}$, ${\cal I}_{\rm II}$, ${\cal I}_{\rm III}$ were defined in $(\ref{eqn:allset})$-$(\ref{eqn:type3set})$. Note that $D^\ast_i=C$ for $i\in{\cal I}_{\rm I}\cup{\cal I}_{\rm II}$ and $\lambda^\ast_i=0$ for $i\in{\cal I}_{\rm II}\cup{\cal I}_{\rm III}$.
\end{theorem}
\subsection{Eigenvalues of the Jacobian matrix $J(\bm\lambda^\ast)$}
From (\ref{eqn:theorem1-1}), we see that the Jacobian matrix $J(\bm\lambda^\ast)$ is of the form
\begin{align}
&J(\bm\lambda^\ast)\equiv\begin{pmatrix}
\,J^{\rm I} & O & O\,\\[1mm]
\,\ast & J^{\rm II} & O\,\\[1mm]
\,\ast & O & J^{\rm III}
\end{pmatrix},\label{eqn:J1AJ2}\\
&J^{\rm I}\equiv\left(\partial F_i/\partial\lambda_{i'}|_{\bm\lambda=\bm\lambda^\ast}\right)_{i,i'\in{\cal I}_{\rm I}}\in{\mathbb R}^{m_1\times m_1},\label{eqn:Jstructure1}\\
&J^{\rm II}\equiv\left(\partial F_i/\partial\lambda_{i'}|_{\bm\lambda=\bm\lambda^\ast}\right)_{i,i'\in{\cal I}_{\rm II}}=I\ ({\rm the\ identity\ matrix})\in{\mathbb R}^{m_2\times m_2},\label{eqn:Jstructure2}\\
&J^{\rm III}\equiv\left(\partial F_i/\partial\lambda_{i'}|_{\bm\lambda=\bm\lambda^\ast}\right)_{i,i'\in{\cal I}_{\rm III}}={\rm diag}\left(e^{D_i^\ast-C},\,i\in{\cal I}_{\rm III}\right)\in{\mathbb R}^{m_3\times m_3},\label{eqn:Jstructure3}\\
&O\ \mbox{\rm denotes\ the\ all-zero\ matrix\ of\ appropriate\ size.}\nonumber
\end{align}
\noindent In (\ref{eqn:Jstructure3}), ${\rm diag}\left(e^{D_i^\ast-C},\,i\in{\cal I}_{\rm III}\right)$ denotes the diagonal matrix with diagonal components $e^{D^\ast_i-C},\,i\in{\cal I}_{\rm III}$. Here, $e^{D^\ast_i-C}<1,\,i\in{\cal I}_{\rm III}$ holds from type-III in (\ref{eqn:Kuhn-Tucker2}).
\bigskip
Let $\{\theta_1,\ldots,\theta_m\}\equiv\{\theta_i\,|\,i\in{\cal I}\}$ be the set of eigenvalues of $J(\bm\lambda^\ast)$. By (\ref{eqn:J1AJ2}), the eigenvalues of $J(\bm\lambda^\ast)$ are the eigenvalues of $J^{\rm I}$, $J^{\rm II}$, $J^{\rm III}$, hence we can put
$\{\theta_i\,|\,i\in{\cal I}_{\rm I}\}$: the set of eigenvalues of $J^{\rm I}$,
$\{\theta_i\,|\,i\in{\cal I}_{\rm II}\}$: the set of eigenvalues of $J^{\rm II}$,
$\{\theta_i\,|\,i\in{\cal I}_{\rm III}\}$: the set of eigenvalues of $J^{\rm III}$.
\bigskip
We will evaluate the eigenvalues of $J^{\rm I}$, $J^{\rm II}$ and $J^{\rm III}$ as follows.
\subsection{Eigenvalues of $J^{\rm I}$}
First, we consider the eigenvalues of $J^{\rm I}$. Let $J^{\rm I}_{i'i}$ be the $(i',i)$ component of $J^{\rm I}$, then by (\ref{eqn:theorem1-1}),
\begin{align}
J^{\rm I}_{i'i}=\delta_{i'i}+\lambda_i^\ast D_{i',i}^\ast,\ i',i\in{\cal I}_{\rm I}.\label{eqn:J1seibun}
\end{align}
Let $I\in{\mathbb R}^{m_1\times m_1}$ denote the identity matrix and define $B\equiv I-J^{\rm I}$. Let $B_{i'i}$ be the $(i',i)$ component of $B$, then from (\ref{eqn:J1seibun}),
\begin{align}
B_{i'i}&=-\lambda^\ast_iD^\ast_{i',i}\\[1mm]
&=\lambda^\ast_i\displaystyle\sum_{j=1}^n\displaystyle\frac{P_j^{i'}P_j^i}{Q_j^\ast},\,i',i\in{\cal I}_{\rm I}.\label{eqn:componentofB}
\end{align}
Let $\{\beta_i\,|\,i\in{\cal I}_{\rm I}\}$ be the set of eigenvalues of $B$, then we have $\theta_i=1-\beta_i,\,i\in{\cal I}_{\rm I}$. In order to calculate the eigenvalues of $B$, we will define the following matrices. Similar calculations are performed in \cite{yu}.
Let us define
\begin{align}
\Phi_1&\equiv\begin{pmatrix}P^1\\\vdots\\P^{m_1}\end{pmatrix}\in\mathbb {R}^{m_1\times n},\\[3mm]
\Gamma&\equiv\left(-D_{i',i}^\ast\right)=\left(\displaystyle\sum_{j=1}^n\displaystyle\frac{P_j^{i'}P_j^i}{Q_j^\ast}\right)_{i',i\in{\cal I}_{\rm I}}\in\mathbb {R}^{m_1\times m_1},\\[3mm]
\Lambda&\equiv{\rm diag}\left(\lambda_1^\ast,\ldots,\lambda_{m_1}^\ast\right)\in\mathbb {R}^{m_1\times m_1}.\label{eqn:diagonalLambda}
\end{align}
Furthermore, define
\begin{align}
\sqrt{\Lambda}&\equiv{\rm diag}\left(\sqrt{\lambda_1^\ast},\ldots,\sqrt{\lambda_{m_1}^\ast}\right)\in\mathbb {R}^{m_1\times m_1},\\
\Omega&\equiv{\rm diag}\left((Q_1^\ast)^{-1},\ldots,(Q_n^\ast)^{-1}\right)\in\mathbb {R}^{n\times n},\\
\sqrt\Omega&\equiv{\rm diag}\left((Q_1^\ast)^{-1/2},\ldots,(Q_n^\ast)^{-1/2}\right)\in\mathbb {R}^{n\times n}.
\end{align}
Then, we have, by calculation,
\begin{align}
\sqrt\Lambda B\sqrt\Lambda^{-1}&=\sqrt\Lambda\Gamma\sqrt\Lambda\\
&=\sqrt\Lambda\Phi_1\Omega\ {^t}\Phi_1\ {^t}\sqrt\Lambda\\
&=\sqrt\Lambda\Phi_1\sqrt\Omega\ {^t}\sqrt\Omega\ {^t}\Phi_1\ {^t}\sqrt\Lambda\\
&=\sqrt\Lambda\Phi_1\sqrt\Omega\ {^t}\!\left(\sqrt\Lambda\Phi_1\sqrt\Omega\right).\label{eqn:L-1BL}
\end{align}
From (\ref{eqn:m1definition}), $\sqrt\Lambda$ is a regular matrix and from the assumption (\ref{eqn:rankmdefinition}), ${\rm rank}\,\Phi_1=m_1$. Therefore, by $m_1\leq m\leq n$, we have ${\rm rank}\,\sqrt\Lambda\Phi_1\sqrt\Omega=m_1$, and thus from (\ref{eqn:L-1BL}), $\sqrt\Lambda B\sqrt\Lambda^{-1}$ is symmetric and positive\ definite. In particular, all the eigenvalues $\beta_1,\ldots,\beta_{m_1}$ of $B$ are positive. Without loss of generality, let $\beta_1\geq\ldots\geq\beta_{m_1}>0$.
By (\ref{eqn:componentofB}), every component of $B$ is non-negative and by Lemma \ref{lem:rowsumofJis0}, every row sum of $B$ is equal to 1, hence by the Perron-Frobenius theorem \cite{hor}
\begin{align}
1=\beta_1\geq\beta_2\geq\ldots\geq\beta_{m_1}>0.
\end{align}
Because $\theta_i=1-\beta_i,\,i\in{\cal I}_{\rm I}$, we have
\begin{align}
0=\theta_1\leq\theta_2\leq\ldots\leq\theta_{m_1}<1,
\end{align}
therefore,
\begin{theorem}
\label{the:eigenvaluesofJ1}
The eigenvalues of $J^{\rm I}$ satisfy
\begin{align}
0\leq\theta_i<1,\,i\in{\cal I}_{\rm I}.\label{eqn:J1nokoyuchi}
\end{align}
\end{theorem}
\subsection{Eigenvalues of $J^{\rm II}$}
Second, we consider the eigenvalues of $J^{\rm II}$. From (\ref{eqn:J1AJ2}),\,(\ref{eqn:Jstructure2}), we have
\begin{theorem}
\label{the:eigenvaluesofJ2}
The eigenvalues of $J^{\rm II}$ satisfy
\begin{align}
\theta_i=1,\,i\in{\cal I}_{\rm II}.\label{eqn:J2nokoyuchi}
\end{align}
\end{theorem}
\subsection{Eigenvalues of $J^{\rm III}$}
Third, we consider the eigenvalues of $J^{\rm III}$. From (\ref{eqn:J1AJ2}),\,(\ref{eqn:Jstructure3}), we have
\begin{theorem}
\label{the:eigenvaluesofJ3}
The eigenvalues of $J^{\rm III}$ are $\theta_i=e^{D^\ast_i-C},\,D^\ast_i<C,\,i\in{\cal I}_{\rm III}$, hence
\begin{align}
0<\theta_i<1,\,i\in{\cal I}_{\rm III}.\label{eqn:J3nokoyuchi}
\end{align}
\end{theorem}
\begin{remark}
\rm From the above consideration, we know that all the eigenvalues of the Jacobian matrix $J(\bm\lambda^\ast)$ are real.
\end{remark}
\begin{lemma}
\label{lem:Jlambdadiagonalizable}
Assume that the eigenvalues of $J^{\rm I}$ and $J^{\rm III}$ are distinct, i.e.,
\begin{align}
\theta_i\neq\theta_{i'},\,i\in{\cal I}_{\rm I},\,i'\in{\cal I}_{\rm III},\label{eqn:eigenvaluesaredistinct}
\end{align}
then $J(\bm\lambda^\ast)$ is diagonalizable. Especially, if ${\cal I}_{\rm III}=\emptyset$ then $(\ref{eqn:eigenvaluesaredistinct})$ holds, and thus $J(\bm\lambda^\ast)$ is diagonalizable.
\end{lemma}
\noindent{\bf Proof:} See Appendix \ref{sec:proooflemmaJlambdadiagonalizable}.\hfill$\blacksquare$
\section{On the exponential convergence}
We obtained in Theorems \ref{the:eigenvaluesofJ1}, \ref{the:eigenvaluesofJ2} and \ref{the:eigenvaluesofJ3} the evaluation for the eigenvalues of $J(\bm\lambda^\ast)$. Let $\theta_{\rm max}\equiv\max_{i\in{\cal I}}\theta_i$ be the maximum eigenvalue of $J(\bm\lambda^\ast)$, then by Theorems \ref{the:eigenvaluesofJ1}, \ref{the:eigenvaluesofJ2} and \ref{the:eigenvaluesofJ3}, we have $0\leq\theta_{\rm max}<1$ if ${\cal I}_{\rm II}$ is empty and $\theta_{\rm max}=1$ if ${\cal I}_{\rm II}$ is not empty. First, we show that the convergence is exponential if ${\cal I}_{\rm II}$ is empty.
\begin{theorem}
\label{the:exponentialconvergence}
Assume ${\cal I}_{\rm II}=\emptyset$, then for any $\theta$ with $\theta_{\rm max}<\theta<1$, there exist $\delta>0$ and $K>0$, such that for arbitrary initial distribution $\bm\lambda^0$ with $\|\bm\lambda^0-\bm\lambda^\ast\|<\delta$, we have
\begin{align}
\|\bm\mu^N\|=\|\bm\lambda^N-\bm\lambda^\ast\|<K(\theta)^N,\,N=0,1,\ldots,
\end{align}
i.e., the convergence is exponential.
\end{theorem}
{\bf Proof:} See Appendix \ref{sec:exponentialconvergence}.\hfill$\blacksquare$
\section{On the $O(1/N)$ convergence}
We will consider the second order recurrence formula obtained by truncating the Taylor expansion of $F(\bm\lambda)$ up to the second order term and analyze the $O(1/N)$ convergence of the sequence defined by the second order recurrence formula.
\subsection{The Hessian matrix $H_i(\bm\lambda^\ast)$}
If $0\leq\theta_{\rm max}<1$, then the convergence speed of $\bm\lambda^N\to\bm\lambda^\ast$ is determined by the Jacobian matrix $J(\bm\lambda^\ast)$ due to Theorem \ref{the:exponentialconvergence}. But, if $\theta_{\rm max}=1$, the convergence speed is not determined only by $J(\bm\lambda^\ast)$, hence we must investigate the Hessian matrix. In \cite{ari}, \cite{yu}, the Jacobian matrix is considered, but the Hessian matrix is not considered in the past literature.
Now, we will calculate the Hessian matrix
\begin{align}
H_i(\bm\lambda^\ast)=\left(\left.\displaystyle\frac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}\right)_{i',i''\in{\cal I}},\,i\in{\cal I}
\end{align}
of $F_i(\bm\lambda)$ at $\bm\lambda=\bm\lambda^\ast$. We have
\begin{theorem}
\label{the:Hessecomponents}The components of the Hessian matrix $H_i(\bm\lambda^\ast),\,i=1,\ldots,m$, are given as follows.
\begin{align}
\left.\dfrac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}&=e^{D^\ast_i-C}\Big\{\delta_{ii'}D^\ast_{i,i''}+\delta_{ii''}D^\ast_{i,i'}+\lambda^\ast_i\left(D^\ast_{i,i'}D^\ast_{i,i''}+D^\ast_{i,i',i''}\right)\nonumber\\
&\ \ \ +\left(\delta_{ii'}+\lambda^\ast_iD^\ast_{i,i'}\right)\left(1-e^{D^\ast_{i''}-C}\right)+\left(\delta_{ii''}+\lambda^\ast_iD^\ast_{i,i''}\right)\left(1-e^{D^\ast_{i'}-C}\right)\Big\}\nonumber\\
&\ \ \ +2\lambda^\ast_i\left(1-e^{D^\ast_{i'}-C}\right)\left(1-e^{D^\ast_{i''}-C}\right)-\lambda^\ast_i\Big(e^{D^\ast_{i'}-C}D^\ast_{i',i''}+e^{D^\ast_{i''}-C}D^\ast_{i',i''}\nonumber\\
&\ \ \ +E_{i',i''}-D^\ast_{i',i''}\Big),\,i,i',i''\in{\cal I}.
\end{align}
where $D^\ast_{i,i',i''}\equiv\partial^2D_i/\partial\lambda_{i'}\partial\lambda_{i{''}}|_{\bm\lambda=\bm\lambda^\ast}$ and $E_{i',i''}\equiv\sum_{k=1}^{m_1}\lambda_k^\ast D_{k,i'}^\ast D_{k,i''}^\ast$.
Especially, if $i\in{\cal I}_{\rm II}$, then $\lambda^\ast_i=0$ by $(\ref{eqn:Kuhn-Tucker2})$, thus
\begin{align}
\left.\displaystyle\frac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}
=\delta_{ii''}\left(1-e^{D_{i'}^\ast-C}+D_{i,i'}^\ast\right)+\delta_{ii'}\left(1-e^{D_{i''}^\ast-C}+D_{i,i''}^\ast\right).\label{eqn:Hesse1}
\end{align}
Further, if $i\in{\cal I}_{\rm II}$ and $D^\ast_{i'}=C$ holds for arbitrary $i'\in{\cal I}$, then by $(\ref{eqn:Hesse1})$, we have
\begin{align}
\left.\displaystyle\frac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}=\delta_{ii'}D_{i,i''}^\ast+\delta_{ii''}D_{i,i'}^\ast,\ i',i''\in{\cal I}.
\end{align}
\end{theorem}
\noindent{\bf Proof:} See Appendix \ref{sec:Hessiancomponents}.\hfill$\blacksquare$
\subsection{Analysis of the $O(1/N)$ convergence}
We consider a recurrence formula obtained by truncating the Taylor expansion (\ref{eqn:Taylortenkai3}) up to the second order term and write the variables as $\bar{\bm\mu}^N=(\bar{\mu} _1^N,\ldots,\bar{\mu}_m^N)$. That is, we have
\begin{align}
\bar{\bm\mu}^{N+1}=\bar{\bm\mu}^NJ(\bm\lambda^\ast)+\displaystyle\frac{1}{2!}\bar{\bm\mu}^NH(\bm\lambda^\ast)\,{^t}\bar{\bm\mu}^N.\label{eqn:second_order_recurrence_formula}
\end{align}
The recurrence formula (\ref{eqn:second_order_recurrence_formula}) is called the {\it second order recurrence formula} of the Taylor expansion (\ref{eqn:Taylortenkai3}). We investigate the convergence speed of $\bar{\bm\mu}^N\to\bm0$. The convergence speed of $\bar{\bm\mu}^N\to\bm0$ seems to be the same as that of the original $\bm\mu^N\to\bm0$, but the proof is not obtained. Numerical comparison will be done in Chapter \ref{sec:numerical_evaluation}. In this chapter we will prove that, if ${\cal I}_{\rm II}\neq\emptyset$, there exists an initial vector $\bar{\bm\mu}^0$ such that $\bar{\bm\mu}^ N\to\bm0$ is the $O(1/N)$ convergence. Furthermore, we will consider the condition that $\bar{\bm\mu}^N\to\bm0$ is the $O(1/N)$ convergence for arbitrary initial vector $\bar{\bm\mu}^0$. The $O(1/N)$ convergence will be proved by the following three steps.
\medskip
\begin{description}
\item[Step 1:] Represent $\bar\mu^N_i$ with types-I and III indices by $\bar\mu^N_i$ with type-II indices.
\item[Step 2:] Obtain the recurrence formula satisfied by $\bar\mu^N_i$ with type-II indices.
\item[Step 3:] Prove that the convergence of $\bar\mu^N_i$ with type-II indices is $O(1/N)$ for some initial vector $\bar{\bm\mu}^0$.
\end{description}
\subsection{Step 1}
Here, we consider the types-I and III indices together. Then, put ${\cal I}_{\rm I}\cup{\cal I}_{\rm III}=\{1,\ldots,m'\}$ and ${\cal I}_{\rm II}=\{m'+1,\ldots,m\}$. We have $m_2=m-m'$ and $|{\cal I}_{\rm II}|=m_2$. The purpose of the step 1 is to represent $\bar{\bm\mu}^N_{\rm I,III}\equiv(\bar\mu^N_1,\ldots,\bar\mu^N_{m'})$ by $\bar{\bm\mu}^N_{\rm II}\equiv(\bar{\mu}^N_{m'+1},\ldots,\bar{\mu}^N_m)$.
In the Jacobian matrix, by changing the order of $J^{\rm II}$ and $J^{\rm III}$ we have
\begin{align}
J(\bm\lambda^\ast)=\begin{pmatrix}
\,J^{\rm I} & O & O\,\\[1mm]
\,\ast & J^{\rm III} & O\,\\[1mm]
\,\ast & O & J^{\rm II}
\end{pmatrix}.
\end{align}
Then by defining
\begin{align}
\label{eqn:J'JIJIII}
J'\equiv\begin{pmatrix}
\,J^{\rm I} & O \,\\[1mm]
\,\ast & J^{\rm III}
\end{pmatrix},
\end{align}
we have
\begin{align}
\label{eqn:JJ'JII}
J(\bm\lambda^\ast)=\begin{pmatrix}
\,J' & O \,\\[1mm]
\,\ast & J^{\rm II}
\end{pmatrix}.
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $\theta_i,\,i=1,\ldots,m$, and then the eigenvalues of $J'$ are $\theta_i,\,i=1,\ldots,m'$ with $0\leq\theta_i<1$, and those of $J^{\rm II}$ are $\theta_i,\,i=m'+1,\ldots,m$ with $\theta_i=1$.
\medskip
Now, let $\bm a_i$ be a right eigenvector of $J(\bm\lambda^\ast)$ for $\theta_i$ and define
\begin{align}
A\equiv(\bm a_1,\ldots,\bm a_m)\in{\mathbb R}^{m\times m}.
\end{align}
Under the assumption (\ref{eqn:eigenvaluesaredistinct}), by choosing the eigenvectors $\bm a_1,\ldots,\bm a_m$ appropriately we can make $A$ a regular matrix. In fact, because $J(\bm\lambda^\ast)$ is diagonalizable by Lemma \ref{lem:Jlambdadiagonalizable}, the direct sum of all the eigenspaces spans the whole ${\mathbb R}^m$ (See \cite{sat},\,p.161,\,Example 4).
For $i=m'+1,\ldots,m$, define
\begin{align}
\bm{e}_i=(0,\ldots,0,\stackrel{i\,\text{th}}{\stackrel{\vee}{1}}\hspace{-1mm},\ 0,\ldots,0)\in{\mathbb R}^m,\ i=m'+1,\ldots,m,
\end{align}
then because $\theta_i=1$, we can take
\begin{align}
{\bm a}_i={^t}\bm{e}_i,\,i=m'+1,\ldots,m.\label{eqn:eigenvectorfor1}
\end{align}
Therefore, we have
\begin{align}
A&=\left(\bm a_1,\ldots,\bm a_{m'},{^t}\bm e_{m'+1},\ldots,{^t}\bm e_m\right)\\
&=\begin{pmatrix}
\begin{array}{l:c}
\begin{matrix}\hspace{1mm}a_{11}&\hspace{1mm}\ldots&\hspace{3mm}a_{m'1}\\
\hspace{3.5mm}\vdots&&\vdots\\
\hspace{2mm}a_{1m'}&\hspace{2mm}\ldots&\hspace{2mm}a_{m'm'}
\end{matrix}&O\\[7mm]
\hdashline\\[-4mm]
\begin{matrix}a_{1,m'+1}&\ldots&a_{m',m'+1}\\\vdots&&\vdots\\a_{1m}&\ldots&a_{m'm}\end{matrix}&
\begin{matrix}1&\ldots&0\\\vdots&\ddots&\vdots\\0&\ldots&1\end{matrix}
\end{array}
\end{pmatrix}\\
&\equiv\begin{pmatrix}A_1&O\\A_2&I\end{pmatrix},\label{eqn:A1OA2I}
\end{align}
where
\begin{align}
A_1\equiv\begin{pmatrix}a_{11}&\ldots&a_{m'1}\\\vdots&&\vdots\\a_{1m'}&\ldots&a_{m'm'}\end{pmatrix},\ \ A_2\equiv\begin{pmatrix}a_{1,m'+1}&\ldots&a_{m',m'+1}\\\vdots&&\vdots\\a_{1m}&\ldots&a_{m'm}\end{pmatrix}.\label{eqn:A1A2definition}
\end{align}
Because $A$ is regular, $A_1$ is also regular by (\ref{eqn:A1OA2I}). $J(\bm\lambda^\ast)$ is diagonalized by $A$, i.e., $A^{-1}J(\bm\lambda^\ast)A=\Theta$, where
\begin{align}
\Theta=\begin{pmatrix}\Theta_1&O\\O&I\end{pmatrix},\ \Theta_1=\begin{pmatrix}\theta_1&&O\\&\ddots&\\O&&\theta_{m'}\end{pmatrix},\,0\leq\theta_i<1,\,i=1,\ldots,m'.\label{eqn:Theta_and_Theta1}
\end{align}
Calculating by using only the first order term of the Taylor expansion (\ref{eqn:Taylortenkai3}), we have
\begin{align}
\bm\mu^{N+1}A&=\bm\mu^NJ(\bm\lambda^\ast)A\\
&=\bm\mu^NA\Theta,\label{eqn:bmmu{N+1}A=bmmuNATheta}
\end{align}
thus by $\bm\mu^N=(\bm\mu^N_{\rm I,III},\bm\mu^N_{\rm II})$, (\ref{eqn:bmmu{N+1}A=bmmuNATheta}) and (\ref{eqn:A1OA2I}),
\begin{align}
\bm\mu^{N+1}_{\rm I,III}A_1+\bm\mu^{N+1}_{\rm II}A_2&=\left(\bm\mu^N_{\rm I,III}A_1+\bm\mu^N_{\rm II}A_2\right)\Theta_1.\label{eqn:bmmu{N+1}{I,III}A1+bmmu{N+1}{II}A2}
\end{align}
Hence, if the second and higher order terms are negligible, we have
\begin{align}
\bm\mu^N_{\rm I,III}A_1+\bm\mu^N_{\rm II}A_2\to\bm0\ ({\mbox{\rm exponentially}}),\ N\to\infty.\label{eqn:bmmuN{I,III}A1+bmmuN{II}A2tobm0}
\end{align}
To show that the second and higher terms are negligible, let us consider $\bm\mu^{N+1}\bm{a}_i$ as a function of $\bm\mu^N\bm{a}_1,\ldots,\bm\mu^N\bm{a}_{m'}$. If the Taylor expansion (\ref{eqn:Taylortenkai3}) satisfies that
\begin{align}
\bm\mu^{N+1}\bm{a}_i\ {\mbox{\rm is\ divisible\ by}}\ \bm\mu^N\bm{a}_i,\label{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}
\end{align}
i.e., if $\bm\mu^N\bm{a}_i=0$ implies that $\bm\mu^{N+1}\bm{a}_i=0$, then, we have
\begin{align}
\bm\mu^{N+1}\bm{a}_i=\theta_i\bm\mu^N\bm{a}_i\left(1+o(|\bm\mu^N\bm{a}_i|)\right),\ N\to\infty,\ i=1,\ldots,m',
\end{align}
and hence (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2tobm0}) holds. However, in general, it is difficult to prove (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}). We will show later in Examples \ref{exa:CM_Phi(2)_again} and \ref{exa:CM_Phi(5)} that (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}) holds.
In what follows, we assume (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2tobm0}) and regard it as $\bm\mu^N_{\rm I,III}A_1+\bm\mu^N_{\rm II}A_2=\bm0.$ Then, we replace $\bm\mu^N$ by $\bar{\bm\mu}^N$ to have $\bar{\bm\mu}^N_{\rm I,III}A_1+\bar{\bm\mu}^N_{\rm II}A_2=\bm0$ and hence
\begin{align}
\bar{\bm\mu}^N_{\rm I,III}=-\bar{\bm\mu}^N_{\rm II}A_2A_1^{-1}.\label{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}
\end{align}
The validity of (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}) will be checked by numerical examples.
\subsection{Step 2}
The purpose of the step 2 is to obtain a recurrence formula satisfied by $\bar{\mu}_i^N,\,i=m'+1,\ldots,m$.
The $i$-th component of (\ref{eqn:second_order_recurrence_formula}) for $i=m'+1,\ldots,m$ is
\begin{align}
\bar{\mu}^{N+1}_i=\bar{\mu}^N_i+\dfrac{1}{2!}\bar{\bm\mu}^NH_i(\bm\lambda^\ast)\,{^t}\bar{\bm\mu}^N,\,i=m'+1,\ldots,m.\label{eqn:Taylor2nd}
\end{align}
We will represent the second term of the right hand side of (\ref{eqn:Taylor2nd}) by $\bar{\bm\mu}^N_{\rm II}$.
Let $H_{i,i'i''}$ be the $(i',i'')$ component of the Hessian matrix $H_i(\bm\lambda^\ast)$, then by Theorem \ref{the:Hessecomponents}, we have
\begin{align}
H_{i,i'i''}&=\left.\dfrac{\partial^2F_i}{\partial\lambda_{i'}\partial\lambda_{i''}}\right|_{\bm\lambda=\bm\lambda^\ast}\\
&=\delta_{ii''}\left(1-e^{D^\ast_{i'}-C}+D^\ast_{i,i'}\right)+\delta_{ii'}\left(1-e^{D^\ast_{i''}-C}+D^\ast_{i,i''}\right),\label{eqn:Hessiancomponent}\\
i&=m'+1,\ldots,m,\,i',i''=1,\ldots,m.\nonumber
\end{align}
Here, for the simplicity of symbols, define
\begin{align}
S_{ii'}\equiv1-e^{D^\ast_{i'}-C}+D^\ast_{i,i'},\,i=m'+1,\ldots,m,i'=1,\ldots,m,
\end{align}
then, we can write (\ref{eqn:Hessiancomponent}) as
\begin{align}
H_{i,i'i''}=\delta_{ii''}S_{ii'}+\delta_{ii'}S_{ii''}.\label{eqn:Hessiancomponent2}
\end{align}
Further, writing $A_1^{-1}\equiv\left(\zeta_{i'i''}\right),\,i',i''=1,\ldots,m'$, $T_{ii'}\equiv-\sum_{k,k'=1}^{m'}a_{i'k}\zeta_{kk'}S_{ik'},\,i,i'=m'+1,\ldots,m$, $r_{ii'}\equiv T_{ii'}+D^\ast_{i,i'},\,i,i'=m'+1,\ldots,m$, we have the following theorem.
\begin{theorem}
\label{the:1/Norderkihonzenkashiki}
$\{\bar{\mu}^N_i\},\,i=m'+1,\ldots,m$, satisfies the recurrence formula
\begin{align}
\bar{\mu}^{N+1}_i=\bar{\mu}^N_i+\bar{\mu}^N_i\displaystyle\sum_{i'=m'+1}^mr_{ii'}\bar{\mu}^N_{i'},\,i=m'+1,\ldots,m.\label{eqn:1/Norderzenkashiki}
\end{align}
\end{theorem}
\noindent{\bf Proof:} See Appendix \ref{sec:proofofstep2}.\hfill$\blacksquare$
\medskip
The step 2 is achieved by (\ref{eqn:1/Norderzenkashiki}).
\subsection{Step 3}
\label{sec:step3}
The purpose of the step 3 is to prove that $\bar{\mu}_i^N\to0,\,i=1,\ldots,m$, is the $O(1/N)$ convergence.
We will define the canonical form of the recurrence formula (\ref{eqn:1/Norderzenkashiki}). Writing $R\equiv(r_{ii'}),\,i,i'=m'+1,\ldots,m$, and $\bm1=(1,\ldots,1)\in{\mathbb R}^{m_2}$, we consider the equation $R\,{^t}\bm\sigma=-{^t}\bm1$ for the variables $\bm\sigma=(\sigma_{m'+1},\ldots,\sigma_m)$. Assuming that $R$ is regular, we have
\begin{align}
{^t}\bm\sigma=-R^{-1}{^t}\bm1,\label{eqn:sigmadefinition}
\end{align}
and then we assume $\bm\sigma>0$, i.e., $\sigma_i>0, i=m'+1,\ldots,m$. Further, by putting
\begin{align}
\nu_i^N&\equiv\bar{\mu}_i^N/\sigma_i,\,i=m'+1,\ldots,m,\label{eqn:nuiNequivmuiN/sigmai}\\
p_{ii'}&\equiv-r_{ii'}\sigma_{i'},\,i,i'=m'+1,\ldots,m,
\end{align}
the recurrence formula (\ref{eqn:1/Norderzenkashiki}) becomes
\begin{align}
\nu_i^{N+1}&=\nu_i^N-\nu_i^N\sum_{i'=1}^mp_{ii'}\nu_{i'}^N,\,i=m'+1,\ldots,m,\label{eqn:canonical_recursion_formula}\\
{\rm where}\ {\bm p}_i&\equiv(p_{i,m'+1},\ldots,p_{i,m})\,{\rm\ is\ a\ probability\ vector}.
\end{align}
The recurrence formula (\ref{eqn:canonical_recursion_formula}) is called the {\it canonical form} of (\ref{eqn:1/Norderzenkashiki}).
\bigskip
For the analysis of (\ref{eqn:canonical_recursion_formula}), we prepare the following lemma.
\begin{lemma}
\label{lem:onevariablereccurence}
Let us define a positive sequence $\{\nu^N\}_{N=0,1\ldots}$ by the recurrence formula
\begin{align}
&\nu^{N+1}=\nu^N-\left(\nu^N\right)^2,\,N=0,1,\ldots,\label{eqn:1hensuzenkashiki}\\
&0<\nu^0\leq1/2.
\end{align}
Then we have
\begin{align}
\lim_{N\to\infty}N\nu^N=1.
\end{align}
\end{lemma}
\noindent{\bf Proof:} Since the function $g(\nu)\equiv\nu-\nu^2$ satisfies $0<g(\nu)<\nu$ for $0<\nu\leq1/2$, we see $0<\nu^{N+1}<\nu^N,\,N=0,1,\ldots$ by mathematical induction. Thus, $\nu^\infty\equiv\lim_{N\to\infty}\nu^N\geq0$ exists and by (\ref{eqn:1hensuzenkashiki}) $\nu^\infty=\nu^\infty-\left(\nu^\infty\right)^2$ holds, then we have $\nu^\infty=0$.
Next, by (\ref{eqn:1hensuzenkashiki}) we have
\begin{align}
\dfrac{1}{N}\left(\dfrac{1}{\nu^N}-\dfrac{1}{\nu^0}\right)&=\dfrac{1}{N}\sum_{l=0}^{N-1}\left(\dfrac{1}{\nu^{l+1}}-\dfrac{1}{\nu^l}\right)\\
&=\dfrac{1}{N}\sum_{l=0}^{N-1}\dfrac{1}{1-\nu^l}.\label{eqn:soukaheikin}
\end{align}
Applying the proposition that ``the arithmetic mean of a convergent sequence converges to the same limit as the original sequence'' (\cite{ahl},\,p.37) to the right hand side of (\ref{eqn:soukaheikin}), we have
\begin{align}
\lim_{N\to\infty}\dfrac{1}{N\nu^N}&=\lim_{N\to\infty}\dfrac{1}{1-\nu^N}\\
&=1,
\end{align}
which proves the lemma.\hfill$\blacksquare$
\begin{lemma}
\label{lem:lim_{Ntoinfty}Nnu_iN=1}
In the canonical form $(\ref{eqn:canonical_recursion_formula})$, for the initial values $\nu_i^0=1/2,\,i=m'+1,\ldots,m$, we have
\begin{align}
\lim_{N\to\infty}N\nu_i^N=1,\,i=m'+1,\ldots,m.\label{eqn:lim_{Ntoinfty}Nnu_iN=1}
\end{align}
\end{lemma}
\noindent{\bf Proof:} By mathematical induction, we see that $\nu_{m'+1}^N=\ldots=\nu_m^N$ holds for $N=0,1,\ldots$, thus (\ref{eqn:canonical_recursion_formula}) becomes $\nu_i^{N+1}=\nu_i^N-\left(\nu_i^N\right)^2,\,i=m'+1,\ldots,m$. Therefore (\ref{eqn:lim_{Ntoinfty}Nnu_iN=1}) holds by Lemma \ref{lem:onevariablereccurence}.\hfill$\blacksquare$
\begin{theorem}
\label{the:1/Norderconvergence_specialinitial}
In $(\ref{eqn:1/Norderzenkashiki})$, for the initial values $\bar{\mu}_i^0=\sigma_i/2,\ i=m'+1,\ldots,m$, we have
\begin{align}
\lim_{N\to\infty}N\bar{\mu}_i^N=\sigma_i,\ i=m'+1,\ldots,m.\label{eqn:lim_{Ntoinfty}Nmu_iN=sigma_i}
\end{align}
Further, under the assumption $(\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0})$, we have
\begin{align}
\lim_{N\to\infty}N\bar{\mu}^N_i=-\left(\bm\sigma A_2A_1^{-1}\right)_i,\ i=1,\ldots,m',\label{eqn:NmuNi1...m'}
\end{align}
where $\bm\sigma$, $A_1$, $A_2$ were defined by $(\ref{eqn:sigmadefinition}),(\ref{eqn:A1A2definition})$.
\end{theorem}
\noindent{\bf Proof:} From $\nu_i^N=\bar{\mu}_i^N/\sigma_i,\ i=m'+1,\ldots,m$, and Lemma \ref{lem:lim_{Ntoinfty}Nnu_iN=1}, we obtain (\ref{eqn:lim_{Ntoinfty}Nmu_iN=sigma_i}). Further, by (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}), we obtain (\ref{eqn:NmuNi1...m'}).\hfill$\blacksquare$
\medskip
The step 3 is achieved by Theorem \ref{the:1/Norderconvergence_specialinitial}. Summarizing above, we have the following theorem.
\begin{theorem}
\label{the:maintheorem}
If type-II indices exist, then there exists an initial vector $\bar{\bm\mu}^0$ such that $\bar{\bm\mu}^N\to\bm0$ is the $O(1/N)$ convergence.
\end{theorem}
\begin{remark} In Theorem $\ref{the:maintheorem}$, we must choose a specific initial vector $\bar{\bm\mu}^0$, but we want to prove it for arbitrary initial distribution. If the existence of $\lim_{N\to\infty}N\mu^N_i,\,i=1,\ldots,m$, is proved for arbitrary $\bar{\bm\mu}^0$, then we have $(\ref{eqn:lim_{Ntoinfty}Nmu_iN=sigma_i})$ and $(\ref{eqn:NmuNi1...m'})$. However, the analysis for the convergence speed of the recurrence formula $(\ref{eqn:1/Norderzenkashiki})$ for arbitrary initial distribution is very difficult, so it has not been achieved. We will give a proof under the assumption that a conjecture holds.
\end{remark}
\subsection{On the initial distribution}
In this section, we will investigate the convergence of (\ref{eqn:canonical_recursion_formula}) for arbitrary initial distribution.
Now, for the sake of simplicity, we will change the indices and symbols of the canonical form (\ref{eqn:canonical_recursion_formula}). Noting $m-m'=m_2$, we change the indices from $m'+1,\ldots,m$ to $1,\ldots,m_2$, and define
\begin{align}
\xi_i^N&\equiv\nu_{m'+i},\,i=1,\ldots,m_2,\\
q_{ii'}&\equiv p_{m'+i,m'+i'},\,i,i'=1,\ldots,m_2,
\end{align}
which are just shifting the indices. By the above change, the canonical form (\ref{eqn:canonical_recursion_formula}) becomes
\begin{align}
\xi_i^{N+1}&=\xi_i^N-\xi_i^N\sum_{i'=1}^{m_2}q_{ii'}\xi_{i'}^N,\,i=1,\ldots,m_2,\label{eqn:modified_standardform}\\
{\rm where}\ {\bm q}_i&\equiv\left(q_{i1},\ldots,q_{i,m_2}\right)\in{\mathbb R^{m_2}}\ \mbox{\rm is\ a\ probability\ vector}.
\end{align}
\subsection{Diagonally dominant condition}
For the probability vectors $\bm q_i,\,i=1,\cdots,m_2$, we assume
\begin{align}
q_{ii}>\sum_{i'=1,i'\neq i}^{m_2}q_{ii'},\,i=1,\ldots,m_2.\label{eqn:taikakuyuui2}
\end{align}
This assumption means that the matrix made by arranging the row vectors ${\bm q}_1,\ldots,{\bm q}_{m_2}$ vertically is diagonally dominant. We call (\ref{eqn:taikakuyuui2}) the {\it diagonally dominant condition}. By numerical calculation, we confirmed that (\ref{eqn:taikakuyuui2}) holds in all the cases of our examples.
\subsection{Conjecture}
Our goal is to prove that $\lim_{N\to\infty}N\xi^N_i=1,\,i=1,\ldots,m_2$ for any initial vector $\bm\xi^0$. We found that we can prove it if the following conjecture holds, however, the proof of this conjecture has not yet been obtained.
\bigskip
\noindent{\bf Conjecture}\ \ By reordering the indices if necessary, there exists an $N_0$ such that the following inequalities hold.
\begin{align}
\xi_1^N\geq\xi_2^N\geq\ldots\geq\xi_{m_2}^N,\,N\geq N_0.\label{eqn:daishoukankei}
\end{align}
Many numerical examples we observed seem to support this conjecture.
We have
\begin{theorem}
\label{the:1/Norderconvergenceinitialfree}
We assume that the sequence defined by the recurrence formula $(\ref{eqn:modified_standardform})$ satisfies $(\ref{eqn:taikakuyuui2})$ and $(\ref{eqn:daishoukankei})$. Then, for arbitrary initial values with $0<\xi_i^0\leq1/2,\,i=1,\ldots,m_2$, we have
\begin{align}
\lim_{N\to\infty}N\xi_i^N&=1,\,i=1,\ldots,m_2.
\end{align}
\end{theorem}
\noindent{\bf Proof:} See Appendix \ref{sec:proofof1/Norderconvergenceinitialfree}.\hfill$\blacksquare$
\subsection{Special cases where the conjecture holds}
Now, we consider some special cases where the conjecture (\ref{eqn:daishoukankei}) holds.
If $m_2=1$, then the variable is only $\xi_1^N$, hence we do not need to consider the inequality condition.
If $m_2=2$, the canonical form (\ref{eqn:modified_standardform}) becomes
\begin{align}
\xi_1^{N+1}&=\xi_1^N-q_{11}\left(\xi_1^N\right)^2-q_{12}\xi_1^N\xi_2^N,\\
\xi_2^{N+1}&=\xi_2^N-q_{21}\xi_2^N\xi_1^N-q_{22}\left(\xi_2^N\right)^2,\\
q_{11}&>0,\,q_{12}>0,\,q_{11}+q_{12}=1,\\
q_{21}&>0,\,q_{22}>0,\,q_{21}+q_{22}=1.
\end{align}
By calculation, we have
\begin{align}
\xi_1^{N+1}-\xi_2^{N+1}=\left(\xi_1^N-\xi_2^N\right)\left(1-q_{11}\xi_1^N-q_{22}\xi_2^N\right).
\end{align}
By Lemma \ref{lem:0<xiiN<=1/2} in Appendix \ref{sec:proofof1/Norderconvergenceinitialfree}, we have $1-q_{11}\xi_1^N-q_{22}\xi_2^N>0$ for any $N=0,1,\ldots$, hence if $\xi_1^0\geq\xi_2^0$ then $\xi_1^N\geq\xi_2^N,\,N=0,1,\ldots$. Thus, the conjecture (\ref{eqn:daishoukankei}) holds with $N_0=0$.
In the case of $m_2\geq3$, this problem is very difficult. We have a sufficient condition for (\ref{eqn:daishoukankei}) in the following lemma.
\begin{lemma}
\label{lem:daishou_kenkei_hozon_under_special_condition}
For $i'$, consider $q_{ii'},\,i=1,\ldots,m_2$. Assume that $q_{ii'}$ are equal for all $i$ except $i'$, i.e., $q_{1i'}=\ldots=q_{i'-1,i'}=q_{i'+1,i'}=\ldots=q_{m_2,i'}$. Then, the conjecture $(\ref{eqn:daishoukankei})$ holds.
\end{lemma}
\noindent{\bf Proof:} By calculation, we have
\begin{align}
\xi_1^{N+1}-\xi_2^{N+1}=\left(\xi_1^N-\xi_2^N\right)\left(1-q_{11}\xi_1^N-q_{22}\xi_2^N-\sum_{i'=3}^{m_2}q_{1i'}\xi_{i'}^N\right).\label{eqn:tahensuinsubunkai}
\end{align}
The second factor of the right hand side of (\ref{eqn:tahensuinsubunkai}) is positive for all $N=0,1,\ldots$ by Lemma \ref{lem:0<xiiN<=1/2}, hence if $\xi^0_1\geq\xi^0_2$ then $\xi_1^N\geq\xi_2^N$ holds for $N=0,1,\ldots$. Similarly, by considering any pair $\xi_i^N$ and $\xi_{i'}^N$, we see that (\ref{eqn:daishoukankei}) holds. \hfill$\blacksquare$
\bigskip
Summarizing above, we have
\begin{theorem}
\label{the:1/N_order_convergence_under_special_condition}
Under the assumptions of Lemma $\ref{lem:daishou_kenkei_hozon_under_special_condition}$ and the diagonally dominant condition $(\ref{eqn:taikakuyuui2})$, the convegence of the recurrence formula $(\ref{eqn:modified_standardform})$ is $O(1/N)$ for arbitrary initial vector $\bm\xi^0=(\xi_1^0,\ldots,\xi_{m_2}^0)$ with $0<\xi_i\leq1/2,\,i=1,\ldots,m_2$, and
\begin{align}
\lim_{N\to\infty}N\xi_i^N=1,\,i=1,\ldots,m_2.
\end{align}
\end{theorem}
\section{Numerical Evaluation}
\label{sec:numerical_evaluation}
Based on the analysis in the previous sections, we will evaluate numerically the convergence speed of the Arimoto-Blahut algorithm for several channel matrices.
In Examples \ref{exa:CM_Phi(1)_again} and \ref{exa:CM_Phi(4)} below, we will investigate the exponential convergence, where the capacity-achieving $\bm\lambda^\ast$ is in $\Delta({\cal X})^\circ$ (the interior of $\Delta({\cal X})$). In Example \ref{exa:CM_Phi(4)}, we will discuss how the convergence speed depends on the choice of the initial distribution $\bm\lambda^0$. Next, in Examples \ref{exa:CM_Phi(2)_again} and \ref{exa:CM_Phi(5)}, we will consider the $O(1/N)$ convergence. It will be confirmed that the convergence speed is accurately approximated by the values obtained in Theorem \ref{the:1/Norderconvergence_specialinitial}. In Example \ref{exa:CM_Phi(3)_again}, we will investigate the exponential convergence, where $\bm\lambda^\ast$ is on $\partial\Delta({\cal X})$ (the boundary of $\Delta({\cal X})$).
Here, in the case of exponential convergence, we will evaluate the values of the function
\begin{align}
L(N)\equiv-\displaystyle\frac{1}{N}\log\|\bm\mu^N\|.
\end{align}
Based on the results of Theorem \ref{the:exponentialconvergence}, i.e., $\|\bm\mu^N\|=\|\bm\lambda^N-\bm\lambda^\ast\|<K(\theta)^N$, $\theta\doteqdot\theta_{\rm max}$, we will compare $L(N)$ for large $N$ with $-\log\theta_{\rm max}$ (or other values).
On the other hand, in the case of $O(1/N)$ convergence, we will evaluate
\begin{align}
N\bm\mu^N=(N\mu^N_1,\ldots,N\mu_m^N).
\end{align}
We will compare $N\bm\mu^N$ for large $N$ with the values obtained in Theorem \ref{the:1/Norderconvergence_specialinitial}.
\subsection{Exponential convergence where $\bm\lambda^\ast\in\Delta({\cal X})^\circ$}
\begin{example}
\label{exa:CM_Phi(1)_again}
\rm Consider the channel matrix $\Phi^{(1)}$ of (\ref{eqn:Phi1}), i.e.,
\begin{align}
\Phi^{(1)}=
\begin{pmatrix}
0.800 & 0.100 & 0.100\\
0.100 & 0.800 & 0.100\\
0.250 & 0.250 & 0.500
\end{pmatrix},
\end{align}
and an initial distribution $\bm\lambda^0=(1/3,1/3,1/3)$.
We have
\begin{align}
\bm\lambda^\ast&=(0.431,0.431,0.138),\\
Q^\ast&=(0.422,0.422,0.156),\\
J(\bm\lambda^\ast)&=
\begin{pmatrix}
\,0.308 & -0.191 & -0.117\,\cr
\,-0.191 & 0.308 & -0.117\,\cr
\,-0.369 & -0.369 & 0.738\,\cr
\end{pmatrix}.
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $(\theta_1,\theta_2,\theta_3)=(0.000,0.500,$ $0.855)$. Then, $\theta_{\rm max}=\theta_3=0.855$. We have, for $N=500$,
\begin{align}
L(500)=0.161\doteqdot-\log\theta_{\rm max}=0.157.\label{eqn:example4speedcomparison}
\end{align}
We can see from Fig. \ref{fig:Phi(1)graphexpconvergence} that $L(N)$ for large $N$ is accurately approximated by the value $-\log\theta_{\rm max}$.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8cm]{figure5.eps}
\put(87,23){\vector(1,1){6}}
\put(53,18){$-\log\theta_{\rm max}=0.157$}
\put(51,-4){$N$}
\put(24,40){$L(N)\ \text{\rm with}\,\bm\lambda^0=(1/3,1/3,1/3)$}
\end{overpic}
\caption{Convergence of $L(N)$ in Example \ref{exa:CM_Phi(1)_again} with initial distribution $\bm\lambda^0=(1/3,1/3,1/3)$.}
\label{fig:Phi(1)graphexpconvergence}
\end{center}
\end{figure}
\end{example}
\begin{example}
\label{exa:CM_Phi(4)}
\rm Let us consider another channel matrix. Define
\begin{align}
\Phi^{(4)}\equiv
\begin{pmatrix}
\,0.793 & 0.196 & 0.011\,\\
0.196 & 0.793 & 0.011 \\
0.250 & 0.250 & 0.500
\end{pmatrix}.
\end{align}
We have
\begin{align}
\bm\lambda^\ast&=(0.352,0.352,0.296),\\
Q^\ast&=(0.422,0.422,0.156),\\
J(\bm\lambda^\ast)&=
\begin{pmatrix}
0.443 & -0.260 & -0.183\,\cr
-0.260 & 0.443 & -0.183\cr
\,-0.218 & -0.218 & 0.436
\end{pmatrix}.\label{eqn:example5J}
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $(\theta_1,\theta_2,\theta_3)=(0.000,0.618,$ $0.702)$. Then, $\theta_{\rm max}=\theta_3=0.702$. Write the second largest eigenvalue as $\theta_{\rm sec}$, then $\theta_{\rm sec}=\theta_2=0.618$.
We show in Fig. \ref{fig:2initialsgraphexpconvergence} the graph of $L(N)$ with initial distribution ${\bm\lambda}^0_1\equiv(1/3,1/3,1/3)$ by the solid line, and the graph with initial distribution ${\bm\lambda}^0_2\equiv(1/2,1/3,1/6)$ by the dotted line.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8cm]{figure6.eps}
\put(23,54){$L(N)\ \text{\rm with}\,\bm\lambda^0_1=(1/3,1/3,1/3)$}
\put(20,29){$L(N)\ \text{\rm with}\,\bm\lambda^0_2=(1/2,1/3,1/6)$}
\put(87,43){\vector(4,3){6}}
\put(49,42){$-\log\theta_{\rm sec}=0.481$}
\put(86,25){\vector(2,3){7}}
\put(54,20){$-\log\theta_{\text{\rm max}}=0.353$}
\put(51,-4){$N$}
\end{overpic}
\caption{Convergence of $L(N)$ in Example \ref{exa:CM_Phi(4)} with initial distribution $\bm\lambda^0_1=(1/3,1/3,1/3)$ and $\bm\lambda^0_2=(1/2,1/3,1/6)$.}
\label{fig:2initialsgraphexpconvergence}
\end{center}
\end{figure}
The larger $L(N)$ the faster the convergence, hence the convergence with $\bm\lambda^0_1$ is faster than with $\bm\lambda^0_2$. The convergence speed varies depending on the choice of initial distribution. What kind of initial distribution yields faster convergence? We will investigate it below.
Let us define
\begin{align}
\bm\mu^0_1&\equiv\bm\lambda^0_1-\bm\lambda^\ast=(-0.019,-0.019,0.038),\label{eqn:initialmu0}\\
\bm\mu^0_2&\equiv\bm\lambda^0_2-\bm\lambda^\ast=(0.148,-0.019,-0.129).\label{eqn:initialbarmu0}
\end{align}
We will execute the following calculation by regarding $\bm\mu^{N+1}=\bm\mu^NJ(\bm\lambda^\ast),\,N=0,1,\ldots$ holds exactly.
Here, we will investigate for general $m,n$. We assume (\ref{eqn:eigenvaluesaredistinct}). Let $\bm b_{\rm max}$ be the left eigenvector of $J(\bm\lambda^\ast)$ for $\theta_{\rm max}$, and let $\bm b_{\rm max}^\perp$ be the orthogonal complement of $\bm b_{\rm max}$, i.e., $\bm b_{\rm max}^\perp\equiv\{\bm\mu\in\mathbb{R}^m\,|\,\bm\mu\,{^t}\bm b_{\rm max}=0\}.$
\begin{lemma}
\label{lem:thetasec}
If
\begin{align}
\bm\mu^N\in\bm b_{\rm max}^\perp,\,N=0,1,\ldots,\label{eqn:innuperp}
\end{align}
then for any $\theta$ with $\theta_{\rm sec}<\theta<1$, we have $\|\bm\mu^N\|<K\left(\theta\right)^N,\,K>0,\,N=0,1,\ldots$.
\end{lemma}
{\bf Proof:}
See Appendix \ref{sec:proofofthetasec}.\hfill$\blacksquare$
\bigskip
Because $\theta_{\text{\rm sec}}<\theta_{\rm max}$, if (\ref{eqn:innuperp}) holds then the convergence speed is faster than $\theta_{\rm max}$ by Lemma \ref{lem:thetasec}. Next lemma gives a necessary and sufficient condition for guaranteeing (\ref{eqn:innuperp}).
\begin{lemma}
\label{lem:migikoyuuvector}
$\bm\mu J(\bm\lambda^\ast)\in\bm b_{\rm max}^\perp$ holds for any $\bm\mu\in\bm b_{\rm max}^\perp$ if and only if ${^t}\bm b_{\rm max}$ is a right eigenvector for $\theta_{\rm max}$.
\end{lemma}
{\bf Proof:} See Appendix \ref{sec:proofofmigikoyuuvector}.\hfill$\blacksquare$
\bigskip
If ${^t}\bm b_{\rm max}$ is a right eigenvector, then by Lemma \ref{lem:migikoyuuvector}, any $\bm\mu^0\in\bm b_{\rm max}^\perp$ yields (\ref{eqn:innuperp}), hence the convergence becomes faster.
Now, we will evaluate the convergence speed for the initial distributions (\ref{eqn:initialmu0}) and (\ref{eqn:initialbarmu0}). For $J(\bm\lambda^\ast)$ of (\ref{eqn:example5J}), $\theta_{\rm max}=0.702$ and $\theta_{\rm sec}=0.618$. The left eigenvector for $\theta_{\rm max}$ is $\bm b_{\rm max}=(-0.500,0.500,0.000)$. We can confirm that ${^t}\bm b_{\rm max}$ is a right eigenvector for $\theta_{\rm max}$ and $\bm\mu^0_1\,{^t}\bm b_{\rm max}=0$, thus by Lemmas \ref{lem:thetasec} and \ref{lem:migikoyuuvector}, we have $\lim_{N\to\infty}L(N)\doteqdot-\log\theta_{\rm sec}$. Then by the solid line in Fig. \ref{fig:2initialsgraphexpconvergence}, we have, for $N=500$,
\begin{align}
L(500)=0.489\doteqdot-\log\theta_{\rm sec}=0.481.
\end{align}
On the other hand, we have $\bm\mu^0_2\,{^t}\bm b_{\rm max}\neq0$, thus by Lemma \ref{lem:migikoyuuvector}, we have $\lim_{N\to\infty}L(N)\doteqdot-\log\theta_{\rm max}$. Then by the dotted line, we have, for $N=500$,
\begin{align}
L(500)=0.360\doteqdot-\log\theta_{\rm max}=0.353.
\end{align}
Checking Example \ref{exa:CM_Phi(1)_again} in this way, we can see that $\bm b_{\rm max}=(-0.431,-0.431,0.862)$ is a left eigenvector for $\theta_{\rm max}=0.855$, but ${^t}\bm b_{\rm max}$ is not a right eigenvector. Thus, by Lemma \ref{lem:migikoyuuvector}, we have $\lim_{N\to\infty}L(N)\doteqdot-\log\theta_{\rm max}$ and (\ref{eqn:example4speedcomparison}).
\end{example}
\subsection{$O(1/N)$ convergence}
\begin{example}
\label{exa:CM_Phi(2)_again}
\rm Consider the channel matrix $\Phi^{(2)}$ of (\ref{eqn:Phi2}), i.e.,
\begin{align}
\Phi^{(2)}&=\begin{pmatrix}
\,0.800 & 0.100 & 0.100\,\\
\,0.100 & 0.800 & 0.100\,\\
\,0.300 & 0.300 & 0.400\,
\end{pmatrix},
\end{align}
and an initial distribution $\bm\lambda^0=(1/3,1/3,1/3)$. We have
\begin{align}
\bm\lambda^\ast&=(0.500,0.500,0.000),\\
Q^\ast&=(0.450,0.450,0.100),\\
D_{1,1}^\ast&=-1.544,\,D_{1,2}^\ast=-0.456,\,D_{1,3}^\ast=-1,\\
D_{2,2}^\ast&=-1.544,\,D_{2,3}^\ast=-1,\,D_{3,3}^\ast=-2,\\
J(\bm\lambda^\ast)
&=\begin{pmatrix}
\,1+\lambda_1^\ast D_{1,1}^\ast & \lambda_2^\ast D_{1,2}^\ast & 0\,\\
\,\lambda_1^\ast D_{2,1}^\ast & 1+\lambda_2^\ast D_{2,2}^\ast & 0 \\
\,\lambda_1^\ast D_{3,1}^\ast & \lambda_2^\ast D_{3,2}^\ast & 1\end{pmatrix}\\
&=\begin{pmatrix}
\,0.228 & -0.228 & 0.000\,\\
\,-0.228 & 0.228 & 0.000\,\\
\,-0.500 & -0.500 & 1.000\,
\end{pmatrix}.
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $(\theta_1,\theta_2,\theta_3)=(0.000,0.456,1.000)$.
\begin{align}
A=\begin{pmatrix}
\,1 & 1 & 0\,\\
\,1 & -1 & 0\,\\
\,1 & 0 & 1\,
\end{pmatrix},\
A_1=\begin{pmatrix}
\,1 & 1\,\\
\,1 & -1\,
\end{pmatrix},\
A_2=\begin{pmatrix}
\,1 & 0\,
\end{pmatrix}.\label{eqn:AA1A2}
\end{align}
By (\ref{eqn:AA1A2}), the eigenvectors $\bm{a}_1$ for $\theta_1=0$ and $\bm{a}_2$ for $\theta_2=0.456$ are
\begin{align}
\bm{a}_1=\begin{pmatrix}1\\1\\1\end{pmatrix},\ \bm{a}_2=\begin{pmatrix}1\\-1\\0\end{pmatrix}.
\end{align}
We will prove (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}). First, for $\bm{a}_1$, $\bm\mu^{N+1}\bm{a}_1=\bm\mu^N\bm{a}_1=\bm0$, thus (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}) is trivial. Next, for $\bm{a}_2$, we will prove that $\bm\mu^{N+1}\bm{a}_2=\mu^{N+1}_1-\mu^{N+1}_2$ is divisible by $\bm\mu^N\bm{a}_2=\mu^N_1-\mu^N_2$. In fact, by $\lambda^\ast_1=\lambda^\ast_2$ we have
\begin{align}
\mu^{N+1}_1-\mu^{N+1}_2&=\lambda^{N+1}_1-\lambda^{N+1}_2\\
&=F_1(\bm\lambda^N)-F_2(\bm\lambda^N)\\
&=\dfrac{\lambda^N_1\exp D(P^1\|\bm\lambda^N\Phi)-\lambda^N_2\exp D(P^2\|\bm\lambda^N\Phi)}{\displaystyle\sum_{k=1}^3\lambda^N_k\exp D(P^k\|\bm\lambda^N\Phi)}.\label{eqn:F1-F2}
\end{align}
If we put $\lambda^N_1=\lambda^N_2$ in (\ref{eqn:F1-F2}), we have by calculation, $\mu^{N+1}_1-\mu^{N+1}_2=0$. Because $\lambda^N_1=\lambda^N_2$ is equivalent to $\mu^N_1=\mu^N_2$, we see that $\mu^N_1-\mu^N_2=0$ implies $\mu^{N+1}_1-\mu^{N+1}_2=0$, thus (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}) holds and then we can consider (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}).
For $\bar{\bm\mu}^N_{\rm I,\,III}=(\bar{\mu}^N_1,\,\bar{\mu}^N_2),\ \bar{\bm\mu}^N_{\rm II}=(\bar{\mu}^N_3)$, we have $\bar{\bm\mu}^N_{\rm I,\,III}=-\bar{\bm\mu}^N_{\rm II}A_2A_1^{-1}$ by (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}), hence
\begin{align}
\bar{\mu}^N_1=\bar{\mu}^N_2=-(1/2)\bar{\mu}^N_3.\label{eqn:mu_1=mu_2=-(1/2)mu_3}
\end{align}
Further, the Hessian matrix $H_3(\bm\lambda^\ast)$ is
\begin{align}
H_3(\bm\lambda^\ast)&=\begin{pmatrix}\,0 & 0 & D_{3,1}^\ast \,\\
0 & 0 & D_{3,2}^\ast \\
\,D_{3,1}^\ast & D_{3,2}^\ast & 2D_{3,3}^\ast\,\end{pmatrix}.\\
&=\begin{pmatrix}
\,0.000 & 0.000 & -1.000\,\\
\,0.000 & 0.000 & -1.000\,\\
\,-1.000 & -1.000 & -4.000\,\end{pmatrix},
\end{align}
then, we have by (\ref{eqn:mu_1=mu_2=-(1/2)mu_3}),
\begin{align}
\dfrac{1}{2}\bar{\bm\mu}^NH_3{^t}\bar{\bm\mu}^N=-\left(\bar{\mu}^N_3\right)^2,
\end{align}
and the second order recurrence formula
\begin{align}
\bar{\mu}_3^{N+1}=\bar{\mu}_3^N-\left(\bar{\mu}_3^N\right)^2.
\end{align}
By Lemma \ref{lem:onevariablereccurence} and (\ref{eqn:mu_1=mu_2=-(1/2)mu_3}), we have $\lim_{N\to\infty}N\bar{\mu}_3^N=1,\,\lim_{N\to\infty}N\bar{\mu}_1^N=\lim_{N\to\infty}N\bar{\mu}_2^N=-1/2$.
By the numerical simulation, $N\bm\mu^N$ for $N=500$ is
\begin{align}
&N\bm\mu^N=(-0.510,-0.510,1.019)\\
&\doteqdot\displaystyle\lim_{N\to\infty}N\bar{\bm\mu}^N=(-1/2,-1/2,1).\label{eqn:rironchiexample6}
\end{align}
See Fig. \ref{fig:Phi(2)graph1Nconvergence}. We can confirm that $N\bm\mu^N$ for large $N$ is close to the values obtained in Theorem \ref{the:1/Norderconvergence_specialinitial}.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8cm]{figure7.eps}
\put(89,69){\vector(1,1){5}}
\put(85,67){$1$}
\put(89,29){\vector(1,-1){5}}
\put(80,30){$-1/2$}
\put(51,-4){$N$}
\put(45,80){$N\mu^N_3$}
\put(43,27){$N\mu^N_1=N\mu^N_2$}
\end{overpic}
\caption{Convergence of $N\mu^N_i$ in Example \ref{exa:CM_Phi(2)_again}.}
\label{fig:Phi(2)graph1Nconvergence}
\end{center}
\end{figure}
\end{example}
\begin{example}
\label{exa:CM_Phi(5)}
\rm Consider a channel matrix
\begin{align}
\Phi^{(5)}&=\begin{pmatrix}
0.6 & 0.1 & 0.1 & 0.1 & 0.1\\
0.1 & 0.6 & 0.1 & 0.1 & 0.1\\
s & s & t & 0.1 & 0.1\\
s & s & 0.1 & t & 0.1\\
s & s & 0.1 & 0.1 & t\\
\end{pmatrix},\\
s&\equiv0.238,\,t\equiv0.324,\,(2s+t+0.2=1),
\end{align}
and an initial distribution $\bm\lambda^0=(1/5,1/5,1/5,1/5,1/5)$. For this $\Phi^{(5)}$, we have
\begin{align}
\bm\lambda^\ast&=(0.5,\,0.5,\,0,\,0,\,0),\\
Q^\ast&=(0.35,\,0.35,\,0.1,\,0.1,\,0.1),\\
D^\ast_{1,1}&=-19/14,\,D^\ast_{1,2}=-9/14,\,D^\ast_{1,3}=-1,\,D^\ast_{1,4}=-1,\,D^\ast_{1,5}=-1,\\
D^\ast_{2,2}&=-19/14,\,D^\ast_{2,3}=-1,\,D^\ast_{2,4}=-1,\,D^\ast_{2,5}=-1,\\
D^\ast_{3,3}&=-1.576\equiv-\alpha,\,D^\ast_{3,4}=-1.072\equiv-\beta,\,D^\ast_{3,5}=-\beta,\\
D^\ast_{4,4}&=-\alpha,\,D^\ast_{4,5}=-\beta,\,D^\ast_{5,5}=-\alpha,\\[2mm]
J(\bm\lambda^\ast)
&=\begin{pmatrix}
\,9/28 & -9/28 & 0 & 0 & 0\,\\
\,-9/28 & 9/28 & 0 & 0 & 0\,\\
\,-1/2 & -1/2 & 1 & 0 & 0\,\\
\,-1/2 & -1/2 & 0 & 1 & 0\,\\
\,-1/2 & -1/2 & 0 & 0 & 1\,
\end{pmatrix}.
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $(\theta_1,\theta_2,\theta_3,\theta_4,\theta_5)=(0,9/14,1,1,1)$ and
\begin{align}
A=\begin{pmatrix}
\,1 & 1 & 0 & 0 & 0\,\\
\,1 & -1 & 0 & 0 & 0\,\\
\,1 & 0 & 1 & 0 & 0\,\\
\,1 & 0 & 0 & 1 & 0\,\\
\,1 & 0 & 0 & 0 & 1\,
\end{pmatrix},\
A_1=\begin{pmatrix}
\,1 & 1\,\\
\,1 & -1\,
\end{pmatrix},\
A_2=\begin{pmatrix}
\,1 & 0\,\\
\,1 & 0\,\\
\,1 & 0\,
\end{pmatrix}.
\end{align}
We can prove (\ref{eqn:bmmu{N+1}bm{a}isdivisiblebybmmuNbm{a}i}) in a similar way as in the Example \ref{exa:CM_Phi(2)_again}.
For $\bar{\bm\mu}^N_{\rm I,\,III}=(\bar{\mu}^N_1,\,\bar{\mu}^N_2),\ \bar{\bm\mu}^N_{\rm II}=(\bar{\mu}^N_3,\,\bar{\mu}^N_4,\,\bar{\mu}^N_5)$, we have $\bar{\bm\mu}^N_{\rm I,\,III}=-\bar{\bm\mu}^N_{\rm II}A_2A_1^{-1}$ by (\ref{eqn:bmmuN{I,III}A1+bmmuN{II}A2=bm0}), hence
\begin{align}
\bar{\mu}^N_1=\bar{\mu}^N_2=-(\bar{\mu}^N_3+\bar{\mu}^N_4+\bar{\mu}^N_5)/2.\label{eqn:mu1=mu2=-(mu3+mu4+mu5)/2}
\end{align}
Further, the Hessian matrix $H_3(\bm\lambda^\ast)$ is
\begin{align}
H_3(\bm\lambda^\ast)
&=\begin{pmatrix}
\,0 & 0 & -1 & 0 & 0\,\\
\,0 & 0 & -1 & 0 & 0\,\\
\,-1 & -1 & -2\alpha & -\beta & -\beta\,\\
\,0 & 0 & -\beta & 0 & 0\,\\
\,0 & 0 & -\beta & 0 & 0\,
\end{pmatrix},
\end{align}
thus, we have by (\ref{eqn:mu1=mu2=-(mu3+mu4+mu5)/2})
\begin{align}
\dfrac{1}{2}\bar{\bm\mu}^NH_3(\bm\lambda^\ast)\,{^t}\bar{\bm\mu}^N=-(\alpha-1)\left(\bar{\mu}^N_3\right)^2-(\beta-1)\bar{\mu}^N_3\bar{\mu}^N_4-(1-\beta)\bar{\mu}^N_3\bar{\mu}^N_5.
\end{align}
Similarly,
\begin{align}
\dfrac{1}{2}\bar{\bm\mu}^NH_4(\bm\lambda^\ast)\,{^t}\bar{\bm\mu}^N&=-(\beta-1)\bar{\mu}^N_3\bar{\mu}^N_4-(\alpha-1)\left(\bar{\mu}^N_4\right)^2-(\beta-1)\bar{\mu}^N_4\bar{\mu}^N_5,\\
\dfrac{1}{2}\bar{\bm\mu}^NH_5(\bm\lambda^\ast)\,{^t}\bar{\bm\mu}^N&=-(\beta-1)\bar{\mu}^N_3\bar{\mu}^N_5-(\beta-1)\bar{\mu}^N_4\bar{\mu}^N_5-(\alpha-1)\left(\bar{\mu}^N_5\right)^2.
\end{align}
Therefore, by putting $\alpha'\equiv\alpha-1,\,\beta'\equiv\beta-1$, we have
\begin{align}
\bar{\mu}_3^{N+1}&=\bar{\mu}_3^N-\alpha'\left(\bar{\mu}_3^N\right)^2-\beta'\bar{\mu}_3^N\bar{\mu}_4^N-\beta'\bar{\mu}_3^N\bar{\mu}_5^N,\label{eqn:Phi6zenkashiki1}\\
\bar{\mu}_4^{N+1}&=\bar{\mu}_4^N-\beta'\bar{\mu}_3^N\bar{\mu}_4^N-\alpha'\left(\bar{\mu}_4^N\right)^2-\beta'\bar{\mu}_4^N\bar{\mu}_5^N,\label{eqn:Phi6zenkashiki2}\\
\bar{\mu}_5^{N+1}&=\bar{\mu}_5^N-\beta'\bar{\mu}_3^N\bar{\mu}_5^N-\beta'\bar{\mu}_4^N\bar{\mu}_5^N-\alpha'\left(\bar{\mu}_5^N\right)^2.\label{eqn:Phi6zenkashiki3}
\end{align}
From $(\ref{eqn:sigmadefinition})$, we have $\bm\sigma=(\sigma_3,\sigma_4,\sigma_5)=(1.389,1.389,1.389)$, so the canonical form for (\ref{eqn:Phi6zenkashiki1}), (\ref{eqn:Phi6zenkashiki2}), (\ref{eqn:Phi6zenkashiki3}) is
\begin{align}
\nu_3^{N+1}&=\nu_3^N-0.8\left(\nu_3^N\right)^2-0.1\nu_3^N\mu_4^N-0.1\nu_3^N\nu_5^N,\label{eqn:Phi6canonical1}\\
\nu_4^{N+1}&=\nu_4^N-0.1\nu_3^N\nu_4^N-0.8\left(\nu_4^N\right)^2-0.1\nu_4^N\nu_5^N,\label{eqn:Phi6canonical2}\\
\nu_5^{N+1}&=\nu_5^N-0.1\nu_3^N\nu_5^N-0.1\nu_4^N\nu_5^N-0.8\left(\nu_5^N\right)^2.\label{eqn:Phi6canonical3}
\end{align}
Eqs. (\ref{eqn:Phi6canonical1}), (\ref{eqn:Phi6canonical2}), (\ref{eqn:Phi6canonical3}) satisfy the assumptions of Lemma $\ref{lem:daishou_kenkei_hozon_under_special_condition}$ and the diagonally dominant condition $(\ref{eqn:taikakuyuui2})$, then by Theorem $\ref{the:1/N_order_convergence_under_special_condition}$, they converge for arbitrary initial values and
\begin{align}
\lim_{N\to\infty}N\nu_i^N=1,\,i=3,4,5.
\end{align}
Therefore, by (\ref{eqn:nuiNequivmuiN/sigmai})
\begin{align}
\lim_{N\to\infty}N\bar{\mu}_i^N=\sigma_i=1.389,\,i=3,4,5,\label{eqn:lim{Ntoinfty}NmuiN=sigmai=1.389}
\end{align}
and by $(\ref{eqn:mu1=mu2=-(mu3+mu4+mu5)/2})$,
\begin{align}
\lim_{N\to\infty}N\bar{\mu}_i^N=-3\sigma_3/2=-2.083,\,i=1,2.\label{eqn:lim{Ntoinfty}NmuiN=-3sigma3/2=-2.083}
\end{align}
We will show in Fig. $\ref{fig:5-5}$ the comparison of the numerical results and the values of (\ref{eqn:lim{Ntoinfty}NmuiN=sigmai=1.389}), (\ref{eqn:lim{Ntoinfty}NmuiN=-3sigma3/2=-2.083}).
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8cm]{figure8.eps}
\put(86,71.5){\vector(1,1){8}}
\put(70,67){$\sigma_3=1.389$}
\put(86.5,22.5){\vector(1,-1){8}}
\put(55.5,25){$-3\sigma_3/2=-2.083$}
\put(51,-4){$N$}
\put(25,74){$N\mu^N_3=N\mu^N_4=N\mu^N_5$}
\put(30,17){$N\mu^N_1=N\mu^N_2$}
\end{overpic}
\caption{Convergence of $N\mu^N_i$ in Example \ref{exa:CM_Phi(5)}.}
\label{fig:5-5}
\end{center}
\end{figure}
\end{example}
\subsection{Exponential convergence where $\bm\lambda^\ast\in\partial\Delta({\cal X})$}
\begin{example}
\label{exa:CM_Phi(3)_again}
\rm Consider the channel matrix $\Phi^{(3)}$ of (\ref{eqn:Phi3})
\begin{align}
\Phi^{(3)}&=\begin{pmatrix}
\,0.800 & 0.100 & 0.100\,\\
\,0.100 & 0.800 & 0.100\,\\
\,0.350 & 0.350 & 0.300\,
\end{pmatrix},
\end{align}
and an initial distribution $\bm\lambda^0=(1/3,1/3,1/3)$. We have
\begin{align}
\bm\lambda^\ast&=(0.500,0.500,0.000),\\
Q^\ast&=(0.450,0.450,0.100),\\
J(\bm\lambda^\ast)&=
\begin{pmatrix}
\,0.228 & -0.228 & 0.000\,\cr
\,-0.228 & 0.228 & 0.000\,\cr
\,-0.428 & -0.428 & 0.856\,\cr
\end{pmatrix}.\label{eqn:example8Jacobimatrix}
\end{align}
The eigenvalues of $J(\bm\lambda^\ast)$ are $(\theta_1,\theta_2,\theta_3)=(0.000,0.456,$ $0.856)$. Then, $\theta_{\rm max}=\theta_3=0.856$. We have for $N=500$
\begin{align}
L(500)=0.159\doteqdot-\log\theta_{\rm max}=0.155.
\end{align}
See Fig. \ref{fig:Phi(3)graphexpconvergence}.
\begin{figure}[t]
\begin{center}
\begin{overpic}[width=8cm]{figure9.eps}
\put(85,25){\vector(1,1){8}}
\put(53,20){$-\log\theta_{\rm max}=0.155$}
\put(51,-4){$N$}
\put(23,43){$L(N)\ \text{\rm with}\,\bm\lambda^0=(1/3,1/3,1/3)$}
\end{overpic}
\caption{Convergence of $L(N)$ in Example \ref{exa:CM_Phi(3)_again} with initial distribution $\bm\lambda^0=(1/3,1/3,1/3)$.}
\label{fig:Phi(3)graphexpconvergence}
\end{center}
\end{figure}
Extending this result, we have the following lemma.
\begin{lemma}
\label{lem:thetamaxisinI3}
Assume that type-II indices do not exist and $\theta_{\rm max}=\max_{i\in{\cal I}_{\rm III}}\theta_i$, i.e., the maximum eigenvalue of $J(\bm\lambda^\ast)$ is achieved in $J^{\rm III}$. Then, the convergence speed does not depend on the choice of initial distribution. In other words, $\lim_{N\to\infty}L(N)=-\log\theta_{\rm max}$ holds for arbitrary initial distribution, hence the convergence speed cannot be increased any more.
\end{lemma}
\noindent{\bf Proof:} Let $\theta_{\rm max}=\theta_{i^\ast},\,i^\ast\in{\cal I}_{\rm III}$. $J^{\rm III}$ is diagonal by (\ref{eqn:Jstructure3}), thus we can take ${^t}{\bm e}_{i^\ast}={^t}(0,\ldots,0,\stackrel{i^\ast\,\text{th}}{\stackrel{\vee}{1}}\hspace{-1mm},\ 0,\ldots,0)$ as a right eigenvector for $\theta_{i^\ast}$. However, ${\bm e}_{i^\ast}$ is not a left eigenvector for $\theta_{i^\ast}$. In fact, since every row sum of $J(\bm\lambda^\ast)$ is 0 by Lemma \ref{lem:rowsumofJis0}, putting $\bm1\equiv(1,\ldots,1)\in{\mathbb R}^m$, we have $J(\bm\lambda^\ast){^t}\bm1=\bm0$. If ${\bm e}_{i^\ast}$ were a left eigenvector for $\theta_{i^\ast}$, then $0={\bm e}_{i^\ast}J(\bm\lambda^\ast){^t}\bm1=\theta_{i^\ast}{\bm e}_{i^\ast}{^t}\bm1=\theta_{i^\ast}>0$, which is a contradiction. $\theta_{i^\ast}>0$ is due to Theorem \ref{the:eigenvaluesofJ3}. Therefore, by Theorems \ref{lem:thetasec} and \ref{lem:migikoyuuvector}, the convergence speed does not depend on the choice of initial distribution and $\lim_{N\to\infty}L(N)=-\log\theta_{\rm max}$.\hfill$\blacksquare$
\end{example}
\section{Convergence speed of $I(\bm\lambda^N,\Phi)\to C$}
Based on the results obtained so far, we will consider the convergence speed that the mutual information $I(\bm\lambda^N,\Phi)$ tends to $C$ as $N\to\infty$. We will show that if ${\cal I}_{\rm III}=\emptyset$ and $\bm\lambda^N\to\bm\lambda^\ast$ is the $O(1/N)$ convergence, then $I(\bm\lambda^N,\Phi)\to C$ is $O(1/N^2)$. Including this fact, we have the following theorem.
\begin{theorem}
\label{the:I(lambdaN,Phi)toC} Let ${\cal I}_{\rm III}=\emptyset$.
If $\|\bm\mu^N\|<K_1\left(\theta\right)^N,\,0\leq\theta_{\rm max}<\theta<1,\,K_1>0,\,N=0,1,\ldots$, then
\begin{align}
0<C-I(\bm\lambda^N,\Phi)<K_2\left(\theta\right)^{2N},\,K_2>0,\,N=0,1,\ldots.\label{eqn:IlambdaPhitoC_1}
\end{align}
While, if $\lim_{N\to\infty}N\mu^N_i=\sigma_i\neq0,\,i=1,\ldots,m$, then
\begin{align}
\lim_{N\to\infty}N^2\left(C-I(\bm\lambda^N,\Phi)\right)=\dfrac{1}{2}\sum_{j=1}^n\dfrac{1}{Q^\ast_j}\left(\sum_{i=1}^m\sigma_iP^i_j\right)^2.\label{eqn:IlambdaPhitoC_2}
\end{align}
Next, let ${\cal I}_{\rm III}\neq\emptyset$.
If $\|\bm\mu^N\|<K_1\left(\theta\right)^N,\,0\leq\theta_{\rm max}<\theta<1,\,K_1>0,\,N=0,1,\ldots$, then
\begin{align}
0<C-I(\bm\lambda^N,\Phi)<K_2\left(\theta\right)^N,\,K_2>0,\,N=0,1,\ldots.\label{eqn:IlambdaPhitoC_3}
\end{align}
While, if $\lim_{N\to\infty}N\mu^N_i=\sigma_i\neq0,\,i=1,\ldots,m$, then
\begin{align}
0<C-I(\bm\lambda^N,\Phi)<K/N,\,K>0,\,N=0,1,\ldots.\label{eqn:IlambdaPhitoC_4}
\end{align}
\end{theorem}
\noindent{\bf Proof:} See Appendix \ref{sec:proofofI(lambdaN,Phi)toC}.\hfill$\blacksquare$
\bigskip
\noindent We will show in the following tables the evaluation of the convergence speed of $I(\bm\lambda^N,\Phi^{(k)})\to C$ for $k=1,2,3$, where $\Phi^{(1)}$, $\Phi^{(2)}$ and $\Phi^{(3)}$ were defined in Examples \ref{exa:CM_Phi(1)}, \ref{exa:CM_Phi(2)} and \ref{exa:CM_Phi(3)}, and examined in Examples \ref{exa:CM_Phi(1)_again}, \ref{exa:CM_Phi(2)_again} and \ref{exa:CM_Phi(3)_again}, respectively.
\begin{table}[H]
\begin{center}
\caption{Convergence speed of $I(\bm\lambda^N,\Phi^{(1)})\to C$.}
\medskip
\begin{tabular}{|l|c|}
\hline
$-(1/N)\log\left(C-I(\bm\lambda^N,\Phi^{(1)})\right)\big|_{N=500}$ & 0.324\rule[-3mm]{0mm}{8.2mm}\\
\hline
$-2\log\theta_{\rm max}$ & 0.313\rule[-2mm]{0mm}{7mm}\\
\hline
\end{tabular}
\end{center}
\end{table}
\vspace{-12mm}
\begin{table}[H]
\begin{center}
\caption{Convergence speed of $I(\bm\lambda^N,\Phi^{(2)})\to C$.}
\medskip
\begin{tabular}{|l|c|}
\hline
$N^2\left(C-I(\bm\lambda^N,\Phi^{(2)})\right)\big|_{N=500}$ & 0.516\rule[-3mm]{0mm}{8.2mm}\\
\hline
eq. (\ref{eqn:IlambdaPhitoC_2}) & 0.500\rule[-2mm]{0mm}{7mm}\\
\hline
\end{tabular}
\end{center}
\end{table}
\vspace{-12mm}
\begin{table}[H]
\begin{center}
\caption{Convergence speed of $I(\bm\lambda^N,\Phi^{(3)})\to C$.}
\medskip
\begin{tabular}{|l|c|}
\hline
$-(1/N)\log\left(C-I(\bm\lambda^N,\Phi^{(3)})\right)\big|_{N=500}$ & 0.163 \rule[-3mm]{0mm}{8.2mm}\\
\hline
$-\log\theta_{\rm max}$ & 0.155 \rule[-2mm]{0mm}{7mm}\\
\hline
\end{tabular}
\end{center}
\end{table}
We can see from these tables that the convergence speed of $I\left(\bm\lambda^N,\Phi\right)\to C$ is accurately approximated by the results of Theorem \ref{the:I(lambdaN,Phi)toC}.
\section{Conclusion}
In this paper, we investigated the convergence speed of the Arimoto-Blahut algorithm. We showed that the capacity-achieving input distribution $\bm\lambda^\ast$ is the fixed point of $F(\bm\lambda)$, and analyzed the convergence speed by the Taylor expansion of $F(\bm\lambda)$ about $\bm\lambda=\bm\lambda^\ast$. We concretely calculated the Jacobian matrix $J$ of the first order term of the Taylor expansion and the Hessian matrix $H$ of the second order term. The analysis of the convergence speed by the Hessian matrix $H$ was done for the first time in this paper.
We showed that if type-II indices do not exist then the convergence of $\bm\lambda^N\to\bm\lambda^\ast$ is exponential and if type-II indices exist then the convergence of the second order recurrence formula obtained by truncating the Taylor expansion is $O(1/N)$ for some initial vector. Further, we considered the condition for the $O(1/N)$ convergence for arbitrary initial vector. Next, we considered the convergence speed of $I(\bm\lambda^N,\Phi)\to C$ and showed that the type-III indices concern the convergence speed. Especially, if there exist no type-III indices and $\bm\lambda^N\to\bm\lambda^\ast$ is $O(1/N)$, then $I(\bm\lambda^N,\Phi)\to C$ is $O(1/N^2)$.
Based on these analysis, the convergence speeds for several channel matrices were numerically evaluated. As a result, it was confirmed that the convergence speed of the Arimoto-Blahut algorithm is very accurately approximated by the values obtained by our theorems.
\newpage |
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