content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
The number of words which can be formed the letters of the word... | Filo
The number of words which can be formed the letters of the word MAXIMUM, if two consonants cannot occur together, is
Not the question you're searching for?
+ Ask your question
In a given word 'MAXIMUM', vowels (A, I, U) fix the alternate position in Ways.
And last of the consonants (M, M, M, X) in four places can be placed in ways.
Required number of ways ! ways
Was this solution helpful?
Video solutions (3)
Learn from their 1-to-1 discussion with Filo tutors.
20 mins
Uploaded on: 6/19/2023
Was this solution helpful?
4 mins
Uploaded on: 3/3/2023
Was this solution helpful?
Found 3 tutors discussing this question
Discuss this question LIVE for FREE
9 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions from Permutations and Combinations
View more
Practice questions on similar concepts asked by Filo students
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text The number of words which can be formed the letters of the word MAXIMUM, if two consonants cannot occur together, is
Updated On Jun 19, 2023
Topic Permutations and Combinations
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 3
Upvotes 415
Avg. Video Duration 9 min | {"url":"https://askfilo.com/math-question-answers/the-number-of-words-which-can-be-formed-the-letters-of-the-word-maximum-if-two","timestamp":"2024-11-08T02:45:48Z","content_type":"text/html","content_length":"325595","record_id":"<urn:uuid:3999ad5c-8569-44c1-8a92-772a266a310a>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00306.warc.gz"} |
A patient undergoing treatment for thyroid cancer receives a dose of radioactive iodine \(\left({ }^{131} \mathrm{I}\right)\), which has a half-life of \(8.05\) days. If the original dose contained \
(12 \mathrm{mg}\) of \({ }^{131} \mathrm{I}\), what mass of \({ }^{131} \mathrm{I}\) remains after \(16.1\) days? A. \(3 \mathrm{mg}\) B. \(6 \mathrm{mg}\) C. \(9 \mathrm{mg}\) D. \(12 \mathrm{mg}\)
Short Answer
Expert verified
The mass after 16.1 days is 3 mg; thus, the answer is A.
Step by step solution
Identify the half-life period
The half-life of \({ }^{131} \mathrm{I}\) is given as \(8.05\) days. This is the time it takes for half of the radioactive substance to decay.
Determine the time elapsed in terms of half-lives
The time elapsed is \(16.1\) days. To find out how many half-lives this corresponds to, divide the total time by the half-life period: \( \frac{16.1}{8.05} = 2 \). This means that \({ }^{131} \mathrm
{I}\) undergoes \(2\) half-lives.
Calculate the remaining amount after one half-life
After one half-life, half of the original dose remains. The original dose is \(12 \mathrm{mg}\). So, after one half-life: \[ 12 \mathrm{mg} \times \frac{1}{2} = 6 \mathrm{mg} \]
Calculate the remaining amount after the second half-life
After the second half-life, half of the remaining dose from the first half-life decays. So, \[ 6 \mathrm{mg} \times \frac{1}{2} = 3 \mathrm{mg} \]
Select the correct answer
The mass of \({ }^{131} \mathrm{I}\) remaining after \(16.1\) days is \(3 \mathrm{mg}\). Therefore, the correct answer is \textbf{A.} \(3 \mathrm{mg}\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
half-life calculation
The concept of half-life is crucial in understanding how radioactive decay works. The half-life of a radioactive substance is the time it takes for half of the substance to decay. For example, if you
start with 12 mg of a substance and it has a half-life of 8.05 days, then after 8.05 days, only 6 mg will be left.
To calculate how much of a substance remains after a given period, you need to determine how many half-lives have passed. You do this by dividing the total time elapsed by the half-life period. For
instance, if 16.1 days have passed, we divide this by 8.05 days to get 2 half-lives.
This means you will halve the original amount twice. After the first half-life, 12 mg becomes 6 mg. After the second half-life, 6 mg becomes 3 mg. Therefore, after 16.1 days, only 3 mg of the
substance remains.
radioactive isotopes
Radioactive isotopes are atoms that have unstable nuclei and release radiation to reach a more stable state. These isotopes, also known as radioisotopes, are commonly used in medicine, industry, and
Each isotope has a unique half-life, which is a key factor in its application. In medicine, radioisotopes like \( { }^{131} I \) (used in treating thyroid cancer) are selected based on their
half-lives to optimize treatment. \( { }^{131} I \) has a half-life of 8.05 days, making it effective for both diagnostic imaging and therapy over a manageable period.
Understanding how different isotopes decay enables scientists and doctors to predict the behavior and lifespan of these substances, ensuring safe and effective use.
dose decay in radiation therapy
In radiation therapy, the dose decay of radioactive substances is a key factor in treatment planning. Therapeutic doses are carefully calculated to ensure maximum efficacy while minimizing harm to
healthy tissues.
For instance, if a patient receives 12 mg of \( ^{131}I \), the expected decay can be calculated using the half-life. After 8.05 days, half of the initial dose decays, leaving 6 mg. After another
8.05 days, half of that amount decays, leaving 3 mg. This gradual decay allows the desired dose to be delivered over time without overwhelming the body.
The concept of dose decay helps medical professionals adjust treatment schedules to ensure that the radiation effectively targets cancer cells while reducing exposure to normal tissues.
medical physics
Medical physics combines principles of physics with medical practices to diagnose and treat diseases. One of its key applications is the use of radioactive isotopes in imaging and therapy.
Understanding half-life calculations and dose decay is essential for medical physicists. They use these concepts to design and optimize treatment plans, such as determining the right amount of
radiopharmaceuticals to use in cancer therapy.
By leveraging knowledge of radioactive decay and half-life, medical physicists contribute to developing more precise and effective treatments. Their expertise ensures that radiation is used safely
and effectively, improving patient outcomes while minimizing potential risks. | {"url":"https://www.vaia.com/en-us/textbooks/english/kaplan-mcat-physics-review-1-edition/chapter-12/problem-7-a-patient-undergoing-treatment-for-thyroid-cancer-/","timestamp":"2024-11-03T17:08:31Z","content_type":"text/html","content_length":"254592","record_id":"<urn:uuid:169b8cae-9d68-493f-a3d1-84689ebcc38d>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00022.warc.gz"} |
Microlearning: Credit assignment and local learning rules in artificial and neural systems
By Neuromatch Academy
Content creators: Colleen J. Gillon & Klara Kaleb
Content reviewers: Colleen J. Gillon, Klara Kaleb, Eva Dyer
Production editors: Konstantine Tsafatinos, Ella Batty, Spiros Chavlis, Samuele Bolotta, Hlib Solodzhuk
To learn effectively, our brain must coordinate synaptic updates across its network of neurons. The question of how the brain does this is called the credit assignment problem. Deep neural networks
are a leading model of learning in the brain, and are typically trained using gradient descent via backpropagation. However, backpropagation is widely agreed to be biologically implausible.
Therefore, to understand how the brain solves the credit assignment problem, we must find learning rules that are both biologically plausible and effective for learning. In this project, we will
explore more biologically plausible learning rules proposed as alternatives to backpropagation, compare them to error backpropagation, and test whether we can infer what type of learning rule the
brain might be using.
This project builds on a basic feedforward network trained to classify MNIST images (Q1). We then implement biologically plausible rules, compute performance and learning-related metrics (Q2-Q4 &
Q7), to then evaluate (1) how consistent and learning-rule specific the metrics are (Q5, Q8-9), or (2) how these rules fare in more complex learning scenarios (Q6, Q10).
Relevant references:
Install and import feedback gadget#
Show code cell source Hide code cell source
# @title Install and import feedback gadget
!pip install vibecheck datatops --quiet
from vibecheck import DatatopsContentReviewContainer
def content_review(notebook_section: str):
return DatatopsContentReviewContainer(
"", # No text prompt
"url": "https://pmyvdlilci.execute-api.us-east-1.amazonaws.com/klab",
"name": "neuromatch_neuroai",
"user_key": "wb2cxze8",
feedback_prefix = "Project_Microlearning"
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Project slides#
If you want to download the slides: https://osf.io/download/fjaqp/
Project Template#
Show code cell source Hide code cell source
#@title Project Template
from IPython.display import Image, display
import os
from pathlib import Path
url = "https://github.com/neuromatch/NeuroAI_Course/blob/main/projects/project-notebooks/static/MicrolearningProjectTemplate.png?raw=true"
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Tutorial links
This project provides a perfect match for continuing your endeavors in W2D3, which is dedicated to Microlearning. Here, you are going to have the opportunity to explore Hebbian learning explicitly
and get a broader prism on how the learning occurs at the synaptic level by implementing more of the algorithms.
Section 1: Initial setup#
Importing dependencies#
Show code cell source Hide code cell source
# @title Importing dependencies
from IPython.display import Image, SVG, display
import os
from pathlib import Path
import random
from tqdm import tqdm
import warnings
import numpy as np
import matplotlib.pyplot as plt
import scipy
import torch
import torchvision
import contextlib
import io
Figure settings#
Show code cell source Hide code cell source
# @title Figure settings
import ipywidgets as widgets # interactive display
%config InlineBackend.figure_format = 'retina'
1.1 Download MNIST dataset#
The first step is to download the MNIST handwritten digits dataset [1], which you will be using in this project. It is provided as a training dataset (60,000 examples) and a test dataset (10,000
examples). We can split the training dataset to obtain a training (e.g., 80%) and a validation set (e.g., 20%). In addition, since the dataset is quite large, we also suggest keeping only half of
each subset, as this will make training the models faster.
[1] Deng, L. (2012). The MNIST database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6), 141–142, https://ieeexplore.ieee.org/document/6296535.
Note: The download process may try a few sources before succeeding. HTTP Error 403: Forbidden or HTTP Error 503: Service Unavailable errors can be ignored.
download_mnist(): Function to download MNIST.
Show code cell source Hide code cell source
# @markdown `download_mnist()`: Function to download MNIST.
def download_mnist(train_prop=0.8, keep_prop=0.5):
valid_prop = 1 - train_prop
discard_prop = 1 - keep_prop
transform = torchvision.transforms.Compose(
torchvision.transforms.Normalize((0.1307,), (0.3081,))]
with contextlib.redirect_stdout(io.StringIO()): #to suppress output
full_train_set = torchvision.datasets.MNIST(
root="./data/", train=True, download=True, transform=transform
full_test_set = torchvision.datasets.MNIST(
root="./data/", train=False, download=True, transform=transform
train_set, valid_set, _ = torch.utils.data.random_split(
[train_prop * keep_prop, valid_prop * keep_prop, discard_prop]
test_set, _ = torch.utils.data.random_split(
[keep_prop, discard_prop]
print("Number of examples retained:")
print(f" {len(train_set)} (training)")
print(f" {len(valid_set)} (validation)")
print(f" {len(test_set)} (test)")
return train_set, valid_set, test_set
train_set, valid_set, test_set = download_mnist()
Number of examples retained:
24001 (training)
5999 (validation)
5000 (test)
1.2 Explore the dataset#
Show code cell source Hide code cell source
#@markdown To get started exploring the dataset, here are a few plotting functions:
#@markdown `get_plotting_color()`: Returns a color for the specific dataset, e.g. "train" or model index.
def get_plotting_color(dataset="train", model_idx=None):
if model_idx is not None:
dataset = None
if model_idx == 0 or dataset == "train":
color = "#1F77B4" # blue
elif model_idx == 1 or dataset == "valid":
color = "#FF7F0E" # orange
elif model_idx == 2 or dataset == "test":
color = "#2CA02C" # green
if model_idx is not None:
raise NotImplementedError("Colors only implemented for up to 3 models.")
raise NotImplementedError(
f"{dataset} dataset not recognized. Expected 'train', 'valid' "
"or 'test'."
return color
#@markdown `plot_examples(subset)`: Plot examples from the dataset organized by their predicted class
#@markdown (if a model is provided) or by their class label otherwise
def plot_examples(subset, num_examples_per_class=8, MLP=None, seed=None,
batch_size=32, num_classes=10, ax=None):
Function for visualizing example images from the dataset, organized by their
predicted class, if a model is provided, or by their class, otherwise.
- subset (torch dataset or torch dataset subset): dataset from which to
visualized images.
- num_examples_per_class (int, optional): number of examples to visualize per
- MLP (MultiLayerPerceptron or None, optional): model to use to retrieve the
predicted class for each image. If MLP is None, images will be organized by
their class label. Otherwise, images will be organized by their predicted
- seed (int or None, optional): Seed to use to randomly sample images to
- batch_size (int, optional): If MLP is not None, number of images to
retrieve predicted class for at one time.
- num_classes (int, optional): Number of classes in the data.
- ax (plt subplot, optional): Axis on which to plot images. If None, a new
axis will be created.
- ax (plt subplot): Axis on which images were plotted.
if MLP is None:
xlabel = "Class"
xlabel = "Predicted class"
if ax is None:
fig_wid = min(8, num_classes * 0.6)
fig_hei = min(8, num_examples_per_class * 0.6)
_, ax = plt.subplots(figsize=(fig_wid, fig_hei))
if seed is None:
generator = None
generator = torch.Generator()
loader = torch.utils.data.DataLoader(
subset, batch_size=batch_size, shuffle=True, generator=generator
plot_images = {i: list() for i in range(num_classes)}
with torch.no_grad():
for X, y in loader:
if MLP is not None:
y = MLP(X)
y = torch.argmax(y, axis=1)
done = True
for i in range(num_classes):
num_to_add = int(num_examples_per_class - len(plot_images[i]))
if num_to_add:
add_images = np.where(y == i)[0]
if len(add_images):
for add_i in add_images[: num_to_add]:
plot_images[i].append(X[add_i, 0].numpy())
if len(plot_images[i]) != num_examples_per_class:
done = False
if done:
hei, wid = X[0, 0].shape
final_image = np.full((num_examples_per_class * hei, num_classes * wid), np.nan)
for i, images in plot_images.items():
if len(images):
final_image[: len(images) * hei, i * wid: (i + 1) * wid] = np.vstack(images)
ax.imshow(final_image, cmap="gray")
ax.set_xticks((np.arange(num_classes) + 0.5) * wid)
ax.set_xticklabels([f"{int(i)}" for i in range(num_classes)])
ax.set_title(f"Examples per {xlabel.lower()}")
return ax
#@markdown `plot_class_distribution(train_set)`: Plots the distribution of classes in each set (train, validation, test).
def plot_class_distribution(train_set, valid_set=None, test_set=None,
num_classes=10, ax=None):
Function for plotting the number of examples per class in each subset.
- train_set (torch dataset or torch dataset subset): training dataset
- valid_set (torch dataset or torch dataset subset, optional): validation
- test_set (torch dataset or torch dataset subset, optional): test
- num_classes (int, optional): Number of classes in the data.
- ax (plt subplot, optional): Axis on which to plot images. If None, a new
axis will be created.
- ax (plt subplot): Axis on which images were plotted.
if ax is None:
_, ax = plt.subplots(figsize=(6, 3))
bins = np.arange(num_classes + 1) - 0.5
for dataset_name, dataset in [
("train", train_set), ("valid", valid_set), ("test", test_set)
if dataset is None:
if hasattr(dataset, "dataset"):
targets = dataset.dataset.targets[dataset.indices]
targets = dataset.targets
outputs = ax.hist(
per_class = len(targets) / num_classes
ax.set_title("Counts per class")
ax.legend(loc="center right")
return ax
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 2: Training a basic model#
2.1 Defining a basic model#
Next, we can define a basic model to train on this MNIST classification task.
First, it is helpful to define a few hyperparameters (NUM_INPUTS and NUM_OUTPUTS) to store the input size and output size the model needs to have for this task.
The MultiLayerPerceptron class, provided here, initializes a multilayer perceptron (MLP) with one hidden layer. Feel free to expand or change the class, if you would like to use a different or more
complex model, or add functionalities.
This class has several basic methods:
• __init__(self): To initialize the model.
• _set_activation(self): To set the specified activation function for the hidden layer. (The output layer has a softmax activation.)
• forward(self, X): To define how activity is passed through the model.
It also has additional methods that will be helpful later on for collecting metrics from the models:
• _store_initial_weights_biases(self): To store the initial weights and biases.
• forward_backprop(self, X): For when we will be comparing the gradients computed by alternative learning rules to the gradients computed by error backpropagation.
• list_parameters(self): For convenience in retrieving a list of the model’s parameters.
• gather_gradient_dict(self): For gathering the gradients of the model’s parameters.
NUM_INPUTS = np.product(train_set.dataset.data[0].shape) # size of an MNIST image
NUM_OUTPUTS = 10 # number of MNIST classes
class MultiLayerPerceptron(torch.nn.Module):
Simple multilayer perceptron model class with one hidden layer.
def __init__(
Initializes a multilayer perceptron with a single hidden layer.
- num_inputs (int, optional): number of input units (i.e., image size)
- num_hidden (int, optional): number of hidden units in the hidden layer
- num_outputs (int, optional): number of output units (i.e., number of
- activation_type (str, optional): type of activation to use for the hidden
layer ('sigmoid', 'tanh', 'relu' or 'linear')
- bias (bool, optional): if True, each linear layer will have biases in
addition to weights
self.num_inputs = num_inputs
self.num_hidden = num_hidden
self.num_outputs = num_outputs
self.activation_type = activation_type
self.bias = bias
# default weights (and biases, if applicable) initialization is used
# see https://github.com/pytorch/pytorch/blob/main/torch/nn/modules/linear.py
self.lin1 = torch.nn.Linear(num_inputs, num_hidden, bias=bias)
self.lin2 = torch.nn.Linear(num_hidden, num_outputs, bias=bias)
self._set_activation() # activation on the hidden layer
self.softmax = torch.nn.Softmax(dim=1) # activation on the output layer
def _store_initial_weights_biases(self):
Stores a copy of the network's initial weights and biases.
self.init_lin1_weight = self.lin1.weight.data.clone()
self.init_lin2_weight = self.lin2.weight.data.clone()
if self.bias:
self.init_lin1_bias = self.lin1.bias.data.clone()
self.init_lin2_bias = self.lin2.bias.data.clone()
def _set_activation(self):
Sets the activation function used for the hidden layer.
if self.activation_type.lower() == "sigmoid":
self.activation = torch.nn.Sigmoid() # maps to [0, 1]
elif self.activation_type.lower() == "tanh":
self.activation = torch.nn.Tanh() # maps to [-1, 1]
elif self.activation_type.lower() == "relu":
self.activation = torch.nn.ReLU() # maps to positive
elif self.activation_type.lower() == "identity":
self.activation = torch.nn.Identity() # maps to same
raise NotImplementedError(
f"{self.activation_type} activation type not recognized. Only "
"'sigmoid', 'relu' and 'identity' have been implemented so far."
def forward(self, X, y=None):
Runs a forward pass through the network.
- X (torch.Tensor): Batch of input images.
- y (torch.Tensor, optional): Batch of targets. This variable is not used
here. However, it may be needed for other learning rules, to it is
included as an argument here for compatibility.
- y_pred (torch.Tensor): Predicted targets.
h = self.activation(self.lin1(X.reshape(-1, self.num_inputs)))
y_pred = self.softmax(self.lin2(h))
return y_pred
def forward_backprop(self, X):
Identical to forward(). Should not be overwritten when creating new
child classes to implement other learning rules, as this method is used
to compare the gradients calculated with other learning rules to those
calculated with backprop.
h = self.activation(self.lin1(X.reshape(-1, self.num_inputs)))
y_pred = self.softmax(self.lin2(h))
return y_pred
def list_parameters(self):
Returns a list of model names for a gradient dictionary.
- params_list (list): List of parameter names.
params_list = list()
for layer_str in ["lin1", "lin2"]:
if self.bias:
return params_list
def gather_gradient_dict(self):
Gathers a gradient dictionary for the model's parameters. Raises a
runtime error if any parameters have no gradients.
- gradient_dict (dict): A dictionary of gradients for each parameter.
params_list = self.list_parameters()
gradient_dict = dict()
for param_name in params_list:
layer_str, param_str = param_name.split("_")
layer = getattr(self, layer_str)
grad = getattr(layer, param_str).grad
if grad is None:
raise RuntimeError("No gradient was computed")
gradient_dict[param_name] = grad.detach().clone().numpy()
return gradient_dict
2.2 Initializing the model#
We can now initialize an MLP. Feel free to change the number of hidden units, change the activation function or include biases in the model.
Currently, the "sigmoid", "TanH", "ReLU" and "identity" activation functions are implemented, but you can add more by editing the _set_activation(self) method of the MultiLayerPerceptron class
defined above.
We have set BIAS=False for simplicity.
We will also initialize the dataloaders. Feel free to select a different batch size (BATCH_SIZE) when training your models.
# Model
NUM_HIDDEN = 100
ACTIVATION = "sigmoid" # output constrained between 0 and 1
BIAS = False
MLP = MultiLayerPerceptron(
# Dataloaders
BATCH_SIZE = 32
train_loader = torch.utils.data.DataLoader(train_set, batch_size=BATCH_SIZE, shuffle=True)
valid_loader = torch.utils.data.DataLoader(valid_set, batch_size=BATCH_SIZE, shuffle=False)
test_loader = torch.utils.data.DataLoader(test_set, batch_size=BATCH_SIZE, shuffle=False)
2.3 Defining and initializing an optimizer#
Here, we define a basic optimizer that updates the weights and biases of the model based on the gradients saved. This optimizer is equivalent to a simple Stochastic Gradient Descent optimizer (
torch.optim.SGD()) applied to mini-batch data.
It has two methods:
• __init__(self): To initialize the optimizer. Any arguments passed after params should be added to the defaults dictionary, which is passed to the parents class. These arguments are then added to
each parameter group’s dictionary, allowing them to be accessed in step(self).
• step(self): Makes an update to the model parameters.
This optimizer can be extended later, if needed, when implementing more complex learning rules.
class BasicOptimizer(torch.optim.Optimizer):
Simple optimizer class based on the SGD optimizer.
def __init__(self, params, lr=0.01, weight_decay=0):
Initializes a basic optimizer object.
- params (generator): Generator for torch model parameters.
- lr (float, optional): Learning rate.
- weight_decay (float, optional): Weight decay.
if lr < 0.0:
raise ValueError(f"Invalid learning rate: {lr}")
if weight_decay < 0.0:
raise ValueError(f"Invalid weight_decay value: {weight_decay}")
defaults = dict(
super().__init__(params, defaults)
def step(self):
Performs a single optimization step.
for group in self.param_groups:
for p in group["params"]:
# only update parameters with gradients
if p.grad is not None:
# apply weight decay to gradient, if applicable
if group["weight_decay"] != 0:
p.grad = p.grad.add(p, alpha=group["weight_decay"])
# apply gradient-based update
p.data.add_(p.grad, alpha=-group["lr"])
We can now initialize an optimizer. Feel free to change to learning rate (LR) that the optimizer is initialized with, or to add a weight decay.
LR = 0.01
backprop_optimizer = BasicOptimizer(MLP.parameters(), lr=LR)
2.4 Training a basic model#
Show code cell source Hide code cell source
#@markdown `train_model(MLP, train_loader, valid_loader, optimizer)`: Main function.
#@markdown Trains the model across epochs. Aggregates loss and accuracy statistics
#@markdown from the training and validation datasets into a results dictionary which is returned.
def train_model(MLP, train_loader, valid_loader, optimizer, num_epochs=5):
Train a model for several epochs.
- MLP (torch model): Model to train.
- train_loader (torch dataloader): Dataloader to use to train the model.
- valid_loader (torch dataloader): Dataloader to use to validate the model.
- optimizer (torch optimizer): Optimizer to use to update the model.
- num_epochs (int, optional): Number of epochs to train model.
- results_dict (dict): Dictionary storing results across epochs on training
and validation data.
results_dict = {
"avg_train_losses": list(),
"avg_valid_losses": list(),
"avg_train_accuracies": list(),
"avg_valid_accuracies": list(),
for e in tqdm(range(num_epochs)):
no_train = True if e == 0 else False # to get a baseline
latest_epoch_results_dict = train_epoch(
MLP, train_loader, valid_loader, optimizer=optimizer, no_train=no_train
for key, result in latest_epoch_results_dict.items():
if key in results_dict.keys() and isinstance(results_dict[key], list):
results_dict[key] = result # copy latest
return results_dict
def train_epoch(MLP, train_loader, valid_loader, optimizer, no_train=False):
Train a model for one epoch.
- MLP (torch model): Model to train.
- train_loader (torch dataloader): Dataloader to use to train the model.
- valid_loader (torch dataloader): Dataloader to use to validate the model.
- optimizer (torch optimizer): Optimizer to use to update the model.
- no_train (bool, optional): If True, the model is not trained for the
current epoch. Allows a baseline (chance) performance to be computed in the
first epoch before training starts.
- epoch_results_dict (dict): Dictionary storing epoch results on training
and validation data.
criterion = torch.nn.NLLLoss()
epoch_results_dict = dict()
for dataset in ["train", "valid"]:
for sub_str in ["correct_by_class", "seen_by_class"]:
epoch_results_dict[f"{dataset}_{sub_str}"] = {
i:0 for i in range(MLP.num_outputs)
train_losses, train_acc = list(), list()
for X, y in train_loader:
y_pred = MLP(X, y=y)
loss = criterion(torch.log(y_pred), y)
acc = (torch.argmax(y_pred.detach(), axis=1) == y).sum() / len(y)
train_losses.append(loss.item() * len(y))
train_acc.append(acc.item() * len(y))
y, y_pred.detach(), epoch_results_dict, dataset="train",
if not no_train:
num_items = len(train_loader.dataset)
epoch_results_dict["avg_train_losses"] = np.sum(train_losses) / num_items
epoch_results_dict["avg_train_accuracies"] = np.sum(train_acc) / num_items * 100
valid_losses, valid_acc = list(), list()
with torch.no_grad():
for X, y in valid_loader:
y_pred = MLP(X)
loss = criterion(torch.log(y_pred), y)
acc = (torch.argmax(y_pred, axis=1) == y).sum() / len(y)
valid_losses.append(loss.item() * len(y))
valid_acc.append(acc.item() * len(y))
y, y_pred.detach(), epoch_results_dict, dataset="valid"
num_items = len(valid_loader.dataset)
epoch_results_dict["avg_valid_losses"] = np.sum(valid_losses) / num_items
epoch_results_dict["avg_valid_accuracies"] = np.sum(valid_acc) / num_items * 100
return epoch_results_dict
def update_results_by_class_in_place(y, y_pred, result_dict, dataset="train",
Updates results dictionary in place during a training epoch by adding data
needed to compute the accuracies for each class.
- y (torch Tensor): target labels
- y_pred (torch Tensor): predicted targets
- result_dict (dict): Dictionary storing epoch results on training
and validation data.
- dataset (str, optional): Dataset for which results are being added.
- num_classes (int, optional): Number of classes.
correct_by_class = None
seen_by_class = None
y_pred = np.argmax(y_pred, axis=1)
if len(y) != len(y_pred):
raise RuntimeError("Number of predictions does not match number of targets.")
for i in result_dict[f"{dataset}_seen_by_class"].keys():
idxs = np.where(y == int(i))[0]
result_dict[f"{dataset}_seen_by_class"][int(i)] += len(idxs)
num_correct = int(sum(y[idxs] == y_pred[idxs]))
result_dict[f"{dataset}_correct_by_class"][int(i)] += num_correct
Once the model and optimizer have been initialized, we can now train our model using backpropagation for a few epochs, and collect the classification results dictionary.
NUM_EPOCHS = 5
MLP_results_dict = train_model(
Note: The training function does not use the test_loader. This additional dataloader can be used to evaluate the models if, for example, you need use the validation set for model selection.
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 3: Inspecting a model’s performance#
The results dictionary (MLP_results_dict) returned by train_model() contains the following keys:
• avg_train_losses: Average training loss per epoch.
• avg_valid_losses: Average validation loss per epoch.
• avg_train_accuracies: Average training accuracies per epoch.
• avg_valid_losses: Average validation accuracies per epoch.
• train_correct_by_class: Number of correctly classified training images for each class (last epoch only).
• train_seen_by_class: Number of training images for each class (last epoch only).
• valid_correct_by_class: Number of correctly classified validation images for each class (last epoch only).
• valid_seen_by_class: Number of validation images for each class (last epoch only).
Next, we can inspect our model’s performance by visualizing various metrics, e.g., classification loss, accuracy, accuracy by class, weights. A few example functions are provided for plotted various
metrics collected across learning.
Show code cell source Hide code cell source
#@markdown `plot_results(results_dict)`: Plots classification losses and
#@markdown accuracies across epochs for the training and validation sets.
def plot_results(results_dict, num_classes=10, ax=None):
Function for plotting losses and accuracies across learning.
- results_dict (dict): Dictionary storing results across epochs on training
and validation data.
- num_classes (float, optional): Number of classes, used to calculate chance
- ax (plt subplot, optional): Axis on which to plot results. If None, a new
axis will be created.
- ax (plt subplot): Axis on which results were plotted.
if ax is None:
_, ax = plt.subplots(figsize=(7, 3.5))
loss_ax = ax
acc_ax = None
chance = 100 / num_classes
plotted = False
for result_type in ["losses", "accuracies"]:
for dataset in ["train", "valid"]:
key = f"avg_{dataset}_{result_type}"
if key in results_dict.keys():
if result_type == "losses":
ylabel = "Loss"
plot_ax = loss_ax
ls = None
elif result_type == "accuracies":
if acc_ax is None:
acc_ax = ax.twinx()
acc_ax.axhline(chance, ls="dashed", color="k", alpha=0.8)
acc_ax.set_ylim(-5, 105)
ylabel = "Accuracy (%)"
plot_ax = acc_ax
ls = "dashed"
raise RuntimeError(f"{result_type} result type not recognized.")
data = results_dict[key]
plotted = True
if plotted:
ax.legend(loc="center left")
ax.set_xticklabels([f"{int(e)}" for e in range(len(data))])
ymin, ymax = ax.get_ylim()
if ymin > 0:
ymin = 0
pad = (ymax - ymin) * 0.05
ax.set_ylim(ymin - pad, ymax + pad)
raise RuntimeError("No data found to plot.")
ax.set_title("Performance across learning")
return ax
#@markdown `plot_scores_per_class(results_dict)`: Plots the classification
#@markdown accuracies by class for the training and validation sets (for the last epoch).
def plot_scores_per_class(results_dict, num_classes=10, ax=None):
Function for plotting accuracy scores for each class.
- results_dict (dict): Dictionary storing results across epochs on training
and validation data.
- num_classes (int, optional): Number of classes in the data.
- ax (plt subplot, optional): Axis on which to plot accuracies. If None, a new
axis will be created.
- ax (plt subplot): Axis on which accuracies were plotted.
if ax is None:
_, ax = plt.subplots(figsize=(6, 3))
avgs = list()
ax.set_prop_cycle(None) # reset color cycle
for s, dataset in enumerate(["train", "valid"]):
correct_by_class = results_dict[f"{dataset}_correct_by_class"]
seen_by_class = results_dict[f"{dataset}_seen_by_class"]
xs, ys = list(), list()
for i, total in seen_by_class.items():
xs.append(i + 0.3 * (s - 0.5))
if total == 0:
ys.append(100 * correct_by_class[i] / total)
avg_key = f"avg_{dataset}_accuracies"
if avg_key in results_dict.keys():
results_dict[avg_key][-1], ls="dashed", alpha=0.8,
xs, ys, label=dataset, width=0.3, alpha=0.8,
ax.set_ylabel("Accuracy (%)")
ax.set_title("Class scores")
ax.set_ylim(-5, 105)
chance = 100 / num_classes
ax.axhline(chance, ls="dashed", color="k", alpha=0.8)
return ax
#@markdown `plot_weights(MLP)`: Plots weights before and after training.
def plot_weights(MLP, shared_colorbar=False):
Function for plotting model weights and biases before and after learning.
- MLP (torch model): Model for which to plot weights and biases.
- shared_colorbar (bool, optional): If True, one colorbar is shared for all
- ax (plt subplot array): Axes on which weights and biases were plotted.
param_names = MLP.list_parameters()
params_images = dict()
pre_means = dict()
post_means = dict()
vmin, vmax = np.inf, -np.inf
for param_name in param_names:
layer, param_type = param_name.split("_")
init_params = getattr(MLP, f"init_{layer}_{param_type}").numpy()
separator = np.full((1, init_params.shape[-1]), np.nan)
last_params = getattr(getattr(MLP, layer), param_type).detach().numpy()
diff_params = last_params - init_params
params_image = np.vstack(
[init_params, separator, last_params, separator, diff_params]
vmin = min(vmin, np.nanmin(params_image))
vmax = min(vmax, np.nanmax(params_image))
params_images[param_name] = params_image
pre_means[param_name] = init_params.mean()
post_means[param_name] = last_params.mean()
nrows = len(param_names)
gridspec_kw = dict()
if len(param_names) == 4:
gridspec_kw["height_ratios"] = [5, 1, 5, 1]
cbar_label = "Weight/bias values"
elif len(param_names) == 2:
gridspec_kw["height_ratios"] = [5, 5]
cbar_label = "Weight values"
raise NotImplementedError("Expected 2 parameters (weights only) or "
f"4 parameters (weights and biases), but found {len(param_names)}"
if shared_colorbar:
nrows += 1
vmin, vmax = None, None
fig, axes = plt.subplots(
nrows, 1, figsize=(6, nrows + 3), gridspec_kw=gridspec_kw
for i, (param_name, params_image) in enumerate(params_images.items()):
layer, param_type = param_name.split("_")
layer_str = "First" if layer == "lin1" else "Second"
param_str = "weights" if param_type == "weight" else "biases"
axes[i].set_title(f"{layer_str} linear layer {param_str} (pre, post and diff)")
im = axes[i].imshow(params_image, aspect="auto", vmin=vmin, vmax=vmax)
if not shared_colorbar:
cbar = fig.colorbar(im, ax=axes[i], aspect=10)
cbar.ax.axhline(pre_means[param_name], ls="dotted", color="k", alpha=0.5)
cbar.ax.axhline(post_means[param_name], color="k", alpha=0.5)
if param_type == "weight":
axes[i].set_xlabel("Input dim.")
axes[i].set_ylabel("Output dim.")
axes[i].spines[["left", "bottom"]].set_visible(False)
if shared_colorbar:
cax = axes[-1]
cbar = fig.colorbar(im, cax=cax, orientation="horizontal", location="bottom")
return axes
3.1 Loss and accuracy across learning#
First, we can look at how the loss (full lines) and accuracy (dashed lines) evolve across learning. The model appears to learn quickly, achieving a performance on the validation dataset that is well
above chance accuracy (black dashed line), even with only 5 training epochs.
3.2 Final accuracy per class#
We can also look at the accuracy breakdown per class.
3.3 Classified example images#
We can visualize examples of how the model has classified certain images (correctly or incorrectly).
plot_examples(valid_loader.dataset, MLP=MLP);
3.4 Weights before and after learning#
We can also observe how the weights changed through learning by visualizing the initial weights (top), the weights after learning (middle), and the difference between the two (bottom).
The average weights before learning (dashed line) and after learning (full line) are plotted on the colorbar for each layer.
❓ What other metrics might you collect or visualize to understand how the model is learning to perform this task?
Note: In this section, we have visualized a variety of metrics that can be used to evaluate and compare models. Later in the project, you may want to collect and record these metrics for each trained
model, instead of just visualizing the them.
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 4: Implementing a biologically plausible learning rule#
4.1 Hebbian learning#
Now, it is time to implement a more biologically plausible learning rule, like Hebbian learning. This rule is famously associated with the phrase “Neurons that fire together wire together.”
In Hebbian learning, the weight \(w_{ij}\) between pre-synaptic neuron i and post-synaptic neuron j is updated as follows:
\({\Delta}w_{ij} = {\eta}(a_{i} \cdot a_{j})\),
• \({\eta}\) is the learning rate,
• \(a_{i}\) is the activation of the pre-synaptic neuron i, and
• \(a_{j}\) is the activation of the post-synaptic neuron j.
This means that the weight update between two neurons is proportional to the correlation in their activity.
Hebbian learning schematic
Show code cell source Hide code cell source
#@markdown Hebbian learning schematic
url = "https://github.com/neuromatch/NeuroAI_Course/blob/main/projects/project-notebooks/static/HebbianLearning.png?raw=true"
display(Image(url = url))
(Left) Hebbian learning when only positive neural activity is allowed.
Input neurons (top) are connected to output neurons (bottom). Active neurons are marked with + (active, light blue) and ++ (very active, dark blue). Connections marked with + will be weakly
increased, and those marked with ++ will be strongly increased by Hebbian learning.
(Right) Hebbian learning when positive and negative neural activity is allowed.
Same as (Left), but negatively active neurons are marked with - (slightly active, light red) and – (very active, dark red). Connections marked with - will be weakly decreased, and those marked with –
will be strongly decreased by Hebbian learning.
4.1.1 Preventing runaway potentiation#
Networks trained with Hebbian learning are typically implemented with neurons that only have positive activations (Left), like neurons in the brain. This means that weights between neurons can only
Allowing neurons to have both positive and negative activations (Right) would allow weights to also decrease with Hebbian learning. However, this is not typically done since negative neural activity
(i.e., a negative firing rate) is not biologically plausible.
Instead, various techniques can be used to prevent runaway potentiation in Hebbian learning (i.e., weights increasing more and more, without limit). These include normalization techniques like Oja’s
rule. In the example below, we take the approach of simply centering weight updates around 0.
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
4.2 Implementing learning rules in torch#
4.2.1 Defining a custom autograd function#
One way to train a torch model using a learning rule other than backpropagation is to create a custom autograd function. With a custom autograd function, we can redefine the forward() and backward()
methods that pass activations forward through the network, and pass gradients backward through the network. Specifically, this allows us to specify how the gradients used to update the model should
be calculated.
To implement Hebbian learning in the model, we will create a custom autograd function called HebbianFunction.
4.2.2 Defining the forward() method#
The forward method of an autograd function serves two purposes:
1. Compute an output for the given input.
2. Gather and store all the information needed to compute weight updates during the backward pass.
As explained above, to calculate the Hebbian weight change between neurons in two layers, we need the activity of the input neurons and the activity of the output neurons. So, the forward() method
should receive input neuron activity as its input, and return output neuron activity as its output.
The inputs to forward() are therefore:
• context: Passed implictly to forward(), and used to store any information needed to calculate gradients during the backward pass.
• input: The input neuron activity.
• weight: The linear layer’s weights.
• bias: The linear layer’s biases (can be None).
• nonlinearity: The nonlinearity function, as it will be needed to calculate the output neuron’s activity.
• target: As will be explained later, for Hebbian learning, it can be very useful to use the targets to training the last layer of the network, instead of true output activity. So here, if targets
are passed to forward(), this is stored instead of the output.
In the forward pass, output neuron activity is computed, and the following variables are saved for the backward pass: input, weight, bias and output_for_update (i.e., the computed output neuron
activity or the target, if it’s provided, averaged across the batch).
4.2.3 Defining the backward() method#
The backward() method of an autograd function computes and returns gradients using only two input variables:
• context: In which information was stored during the forward pass.
• grad_output: The grad_input from the downstream layer (since gradients are computed backwards through the network.
Here, for Hebbian learning, we do not use a backpropagated gradient. For this reason, grad_output is ignored and no grad_input is computed.
Instead, the gradients for the weights and biases are computed using the variables stored in context:
• grad_weight:
□ Computed as input neuron activity multiplied by output activity.
□ To avoid weight changes scaling linearly with the number of inputs, we also divide by the number of input neurons.
• grad_bias:
□ Computed simply as the output neuron activity, since biases have the same dimension as the output of a layer.
□ Biases are enabled here. However, it should be noted that although they are used often in networks using error backpropagation, they are not used as often in Hebbian learning.
• The backward() method expects to return as many gradient values as the number of inputs passed to the forward() method (except context). It may raise an error if it’s not implemented this way. So
this is why, in the example below, backward() returns grad_nonlinearity and grad_target, but they are both None values.
• The backward() method computes the gradients, but does not apply weight updates. These are applied when optimizer.step() is called. In the BasicOptimizer, defined above, the optimizer step
optionally applies a weight decay, then subtracts the gradients computed here multiplied by the learning rate.
• Standard optimizers, including the BasicOptimizer, expect to receive error gradients (i.e., values they should subtract from the parameters). However, we have computed Hebbian gradients (i.e.,
values that should be added to the parameters). For this reason, the gradients computed are multiplied by -1 in the backward() method before they are returned.
To learn more about torch’s autograd function and creating custom functions, see the following torch documentation and tutorials:
class HebbianFunction(torch.autograd.Function):
Gradient computing function class for Hebbian learning.
def forward(context, input, weight, bias=None, nonlinearity=None, target=None):
Forward pass method for the layer. Computes the output of the layer and
stores variables needed for the backward pass.
- context (torch context): context in which variables can be stored for
the backward pass.
- input (torch tensor): input to the layer.
- weight (torch tensor): layer weights.
- bias (torch tensor, optional): layer biases.
- nonlinearity (torch functional, optional): nonlinearity for the layer.
- target (torch tensor, optional): layer target, if applicable.
- output (torch tensor): layer output.
# compute the output for the layer (linear layer with non-linearity)
output = input.mm(weight.t())
if bias is not None:
output += bias.unsqueeze(0).expand_as(output)
if nonlinearity is not None:
output = nonlinearity(output)
# calculate the output to use for the backward pass
output_for_update = output if target is None else target
# store variables in the context for the backward pass
context.save_for_backward(input, weight, bias, output_for_update)
return output
def backward(context, grad_output=None):
Backward pass method for the layer. Computes and returns the gradients for
all variables passed to forward (returning None if not applicable).
- context (torch context): context in which variables can be stored for
the backward pass.
- input (torch tensor): input to the layer.
- weight (torch tensor): layer weights.
- bias (torch tensor, optional): layer biases.
- nonlinearity (torch functional, optional): nonlinearity for the layer.
- target (torch tensor, optional): layer target, if applicable.
- grad_input (None): gradients for the input (None, since gradients are not
backpropagated in Hebbian learning).
- grad_weight (torch tensor): gradients for the weights.
- grad_bias (torch tensor or None): gradients for the biases, if they aren't
- grad_nonlinearity (None): gradients for the nonlinearity (None, since
gradients do not apply to the non-linearities).
- grad_target (None): gradients for the targets (None, since
gradients do not apply to the targets).
input, weight, bias, output_for_update = context.saved_tensors
grad_input = None
grad_weight = None
grad_bias = None
grad_nonlinearity = None
grad_target = None
input_needs_grad = context.needs_input_grad[0]
if input_needs_grad:
weight_needs_grad = context.needs_input_grad[1]
if weight_needs_grad:
grad_weight = output_for_update.t().mm(input)
grad_weight = grad_weight / len(input) # average across batch
# center around 0
grad_weight = grad_weight - grad_weight.mean(axis=0) # center around 0
## or apply Oja's rule (not compatible with clamping outputs to the targets!)
# oja_subtract = output_for_update.pow(2).mm(grad_weight).mean(axis=0)
# grad_weight = grad_weight - oja_subtract
# take the negative, as the gradient will be subtracted
grad_weight = -grad_weight
if bias is not None:
bias_needs_grad = context.needs_input_grad[2]
if bias_needs_grad:
grad_bias = output_for_update.mean(axis=0) # average across batch
# center around 0
grad_bias = grad_bias - grad_bias.mean()
## or apply an adaptation of Oja's rule for biases
## (not compatible with clamping outputs to the targets!)
# oja_subtract = (output_for_update.pow(2) * bias).mean(axis=0)
# grad_bias = grad_bias - oja_subtract
# take the negative, as the gradient will be subtracted
grad_bias = -grad_bias
return grad_input, grad_weight, grad_bias, grad_nonlinearity, grad_target
4.2.4 Defining a HebbianMultiLayerPerceptron class#
Lastly, we provide a HebbianMultiLayerPerceptron class. This class inherits from MultiLayerPerceptron, but implements its own forward() method. This forward() method uses HebbianFunction() to send
inputs through each layer of the network and its activation function, ensuring that gradients are computed correctly for Hebbian learning.
class HebbianMultiLayerPerceptron(MultiLayerPerceptron):
Hebbian multilayer perceptron with one hidden layer.
def __init__(self, clamp_output=True, **kwargs):
Initializes a Hebbian multilayer perceptron object
- clamp_output (bool, optional): if True, outputs are clamped to targets,
if available, when computing weight updates.
self.clamp_output = clamp_output
def forward(self, X, y=None):
Runs a forward pass through the network.
- X (torch.Tensor): Batch of input images.
- y (torch.Tensor, optional): Batch of targets, stored for the backward
pass to compute the gradients for the last layer.
- y_pred (torch.Tensor): Predicted targets.
h = HebbianFunction.apply(
X.reshape(-1, self.num_inputs),
# if targets are provided, they can be used instead of the last layer's
# output to train the last layer.
if y is None or not self.clamp_output:
targets = None
targets = torch.nn.functional.one_hot(
y, num_classes=self.num_outputs
y_pred = HebbianFunction.apply(
return y_pred
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
4.3 Training a model with Hebbian learning#
4.3.1 Simplifying the task to a 2-class task#
When implementing biologically plausible learning rules, one often faces a performance trade-off. This is because it is often harder for a network to learn without the precise error gradients that
error backpropagation offers. More sophisticated designs can really enhance performance. However, since we are starting with the basics, we will instead start by simplifying the task from a 10-class
task to a 2-class task.
Show code cell source Hide code cell source
#@markdown The following function returns a dataset restricted to specific classes:
#@markdown `restrict_classes(dataset)`: Keeps specified classes in a dataset.
def restrict_classes(dataset, classes=[6], keep=True):
Removes or keeps specified classes in a dataset.
- dataset (torch dataset or subset): Dataset with class targets.
- classes (list): List of classes to keep or remove.
- keep (bool): If True, the classes specified are kept. If False, they are
- new_dataset (torch dataset or subset): Datset restricted as specified.
if hasattr(dataset, "dataset"):
indices = np.asarray(dataset.indices)
targets = dataset.dataset.targets[indices]
dataset = dataset.dataset
indices = np.arange(len(dataset))
targets = dataset.targets
specified_idxs = np.isin(targets, np.asarray(classes))
if keep:
retain_indices = indices[specified_idxs]
retain_indices = indices[~specified_idxs]
new_dataset = torch.utils.data.Subset(dataset, retain_indices)
return new_dataset
train_set_2_classes = restrict_classes(train_set, [0, 1])
valid_set_2_classes = restrict_classes(valid_set, [0, 1])
plot_class_distribution(train_set_2_classes, valid_set_2_classes)
train_loader_2cls = torch.utils.data.DataLoader(train_set_2_classes, batch_size=BATCH_SIZE, shuffle=True)
valid_loader_2cls = torch.utils.data.DataLoader(valid_set_2_classes, batch_size=BATCH_SIZE, shuffle=False)
The number of examples for each class in the training and validation sets are shown. Dashed lines show what counts would be expected if there were the same number of examples in each class.
4.3.2 Training on a 2-class task#
We’ll test the Hebbian learning model’s performance with 10 epochs of training.
HEBB_LR = 1e-4 # lower, since Hebbian gradients are much bigger than backprop gradients
HebbianMLP_2cls = HebbianMultiLayerPerceptron(
Hebb_optimizer_2cls = BasicOptimizer(HebbianMLP_2cls.parameters(), lr=HEBB_LR)
Hebb_results_dict_2cls = train_model(
plot_results(Hebb_results_dict_2cls, num_classes=2)
plot_scores_per_class(Hebb_results_dict_2cls, num_classes=2)
plot_examples(valid_loader_2cls.dataset, MLP=HebbianMLP_2cls, num_classes=2)
Try running the previous cell several times to see how often the model succeeds in learning this simple classification task.
❓ Why does this network struggle to learn this simple classification task?
4.3.3 Training with targets#
As you may have notice, the classification targets do not actually appear in the basic Hebbian learning rule. How, then, can the network learn to perform a supervised task?
To allow a network trained with Hebbian learning to learn a supervised task, one can take the approach of clamping the outputs to the targets. In other words, we can update the final layer’s weights
using the targets \(t_{j}\) instead of the final layer’s activations \(a_{j}\).
HebbianMLP_2cls = HebbianMultiLayerPerceptron(
clamp_output=True, # clamp output to targets
Hebb_optimizer_2cls = BasicOptimizer(HebbianMLP_2cls.parameters(), lr=HEBB_LR)
Hebb_results_dict_2cls = train_model(
plot_results(Hebb_results_dict_2cls, num_classes=2)
plot_scores_per_class(Hebb_results_dict_2cls, num_classes=2)
plot_examples(valid_loader_2cls.dataset, MLP=HebbianMLP_2cls, num_classes=2)
Try running the previous cell several times. The model should now be more successful at learning this simple classification task.
❓ Is the model successful every time? If not, what might be contributing to the variability in performance?
❓ Going further: How does changing the training hyperparameters (learning rate and number of epochs) affect network learning?
Note: The clamped outputs setting cannot be used with Oja’s rule (i.e., one of the Hebbian learning modifications used to prevent runaway weight increases). This is because the values subtracted when
using Oja’s rule would be calculated on the target outputs instead of the actual outputs, and these values would end up being too big.
4.3.4 Increasing task difficulty#
Next, let’s see what happens when we increase the number of classes in the task. The following function handles the entire initialization and training process:
Show code cell source Hide code cell source
#@markdown `train_model_extended`: Initializes model and optimizer, restricts datasets to
#@markdown specified classes, trains model. Returns trained model and results dictionary.
def train_model_extended(model_type="backprop", keep_num_classes="all", lr=LR,
num_epochs=5, partial_backprop=False,
num_hidden=NUM_HIDDEN, bias=BIAS,
batch_size=BATCH_SIZE, plot_distribution=False):
Initializes model and optimizer, restricts datasets to specified classes and
trains the model. Returns the trained model and results dictionary.
- model_type (str, optional): model to initialize ("backprop" or "Hebbian")
- keep_num_classes (str or int, optional): number of classes to keep (from 0)
- lr (float or list, optional): learning rate for both or each layer
- num_epochs (int, optional): number of epochs to train model.
- partial_backprop (bool, optional): if True, backprop is used to train the
final Hebbian learning model.
- num_hidden (int, optional): number of hidden units in the hidden layer
- bias (bool, optional): if True, each linear layer will have biases in
addition to weights.
- batch_size (int, optional): batch size for dataloaders.
- plot_distribution (bool, optional): if True, dataset class distributions
are plotted.
- MLP (torch module): Model
- results_dict (dict): Dictionary storing results across epochs on training
and validation data.
if isinstance(keep_num_classes, str):
if keep_num_classes == "all":
num_classes = 10
use_train_set = train_set
use_valid_set = valid_set
raise ValueError("If 'keep_classes' is a string, it should be 'all'.")
num_classes = int(keep_num_classes)
use_train_set = restrict_classes(train_set, np.arange(keep_num_classes))
use_valid_set = restrict_classes(valid_set, np.arange(keep_num_classes))
if plot_distribution:
plot_class_distribution(use_train_set, use_valid_set)
train_loader = torch.utils.data.DataLoader(
use_train_set, batch_size=batch_size, shuffle=True
valid_loader = torch.utils.data.DataLoader(
use_valid_set, batch_size=batch_size, shuffle=False
model_params = {
"num_hidden": num_hidden,
"num_outputs": num_classes,
"bias": bias,
if model_type.lower() == "backprop":
Model = MultiLayerPerceptron
elif model_type.lower() == "hebbian":
if partial_backprop:
Model = HebbianBackpropMultiLayerPerceptron
Model = HebbianMultiLayerPerceptron
model_params["clamp_output"] = True
raise ValueError(
f"Got {model_type} model type, but expected 'backprop' or 'hebbian'."
MLP = Model(**model_params)
if isinstance(lr, list):
if len(lr) != 2:
raise ValueError("If 'lr' is a list, it must be of length 2.")
optimizer = BasicOptimizer([
{"params": MLP.lin1.parameters(), "lr": lr[0]},
{"params": MLP.lin2.parameters(), "lr": lr[1]},
optimizer = BasicOptimizer(MLP.parameters(), lr=lr)
results_dict = train_model(
return MLP, results_dict
MLP_3cls, results_dict_3cls = train_model_extended(
plot_results(results_dict_3cls, num_classes=3)
plot_scores_per_class(results_dict_3cls, num_classes=3)
Now, let’s try training a model with Hebbian learning (and outputs clamped to targets). Since the task is harder, we’ll increase the number of training epochs to 15.
HebbianMLP_3cls, Hebbian_results_dict_3cls = train_model_extended(
plot_results(Hebbian_results_dict_3cls, num_classes=3)
plot_scores_per_class(Hebbian_results_dict_3cls, num_classes=3)
Try running the previous cell a few times to see how often the model is successful.
❓ How is the model learning in each layer?
❓ How do the weight updates learned with Hebbian learning compare to those learned with error backpropagation?
We can try using different learning rates to encourage more learning in the second layer.
HebbianMLP_3cls, Hebbian_results_dict_3cls = train_model_extended(
lr=[HEBB_LR / 4, HEBB_LR * 8], # learning rate for each layer
plot_results(Hebbian_results_dict_3cls, num_classes=3)
plot_scores_per_class(Hebbian_results_dict_3cls, num_classes=3)
Performance tends to be highly variable and unstable. At best, the network is able to classify 2 classes, but generally not all 3 classes.
4.3.5 Combining Hebbian learning and error backpropagation#
What happens if we use Hebbian learning for the first layer, but use error backpropagation to train the second layer?
HebbianBackpropMultiLayerPerceptron(): Class combining Hebbian learning and backpropagation.
Show code cell source Hide code cell source
#@markdown `HebbianBackpropMultiLayerPerceptron()`: Class combining Hebbian learning and backpropagation.
class HebbianBackpropMultiLayerPerceptron(MultiLayerPerceptron):
Hybrid backprop/Hebbian multilayer perceptron with one hidden layer.
def forward(self, X, y=None):
Runs a forward pass through the network.
- X (torch.Tensor): Batch of input images.
- y (torch.Tensor, optional): Batch of targets, not used here.
- y_pred (torch.Tensor): Predicted targets.
# Hebbian layer
h = HebbianFunction.apply(
X.reshape(-1, self.num_inputs),
# backprop layer
y_pred = self.softmax(self.lin2(h))
return y_pred
HybridMLP, Hybrid_results_dict = train_model_extended(
partial_backprop=True, # backprop on the final layer
lr=[HEBB_LR / 5, LR], # learning rates for each layer
Using Hebbian learning and error backpropagation allows us to achieve above chance performance on the the full MNIST classification task, though performance is still much lower than when using error
backpropagation on its own.
❓ What are some of the properties of Hebbian learning that might explain its weaker performance on this task when compared to error backpropagation?
❓ Going further: Are there tasks that Hebbian learning might be better at than error backpropagation?
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 4.4. Computing the variance and bias of a model’s gradients.#
To better understand how a model trained with a biologically plausible learning rule (e.g., Hebbian learning) learns, it can be useful to compare its learning to error backpropagation. Specifically,
we can compare the gradients computed with both learning rules to one another.
One property we can compute is the variance. The variance tells us how consistent are the gradients obtained for each weight across examples when computed with one learning rule compared to the
other. We might expect a good learning rule to be more consistent, and therefore have lower variance in its gradients (left column in the bullseye image).
However, it would be unfair to compare the variance of small gradients (like those computed with error backpropagation) with the variance of large gradients (like those computed with Hebbian
learning). So, we will estimate variance in the gradients using a scale-invariant measure: the signal-to-noise ratio (SNR). Note that high SNR corresponds to low variance, and vice versa.
Another property we can compute is how biased gradients computed with a specific learning rule are with respect to ideal gradients. Now, since the ideal gradients for learning a task are generally
unknown, we need to estimate them. We can do so using error backpropagation, as it is the best algorithm we know for learning a task along the gradient of its error. A good learning rule would then
be expected to have low bias in its gradients with respect to error backpropagation gradients.
As with the variance, our bias estimate should be scale-invariant, so we will estimate it using the Cosine similarity. Note that high Cosine similarity with error backpropagation gradients
corresponds to low bias, and vice versa.
Bias and variance schematic
Show code cell source Hide code cell source
#@markdown Bias and variance schematic
url = "https://github.com/neuromatch/NeuroAI_Course/blob/main/projects/project-notebooks/static/BiasVariance.png?raw=true"
display(Image(url = url))
The bullseye image illustrates the differences and interplay between bias and variance in a variable. In the left column, the variable being measured shows low variance, as the green dots are densely
concentrated. In the right column, the dots are more dispersed, reflecting a higher variance.
Although the examples in each column show the same variance, they show different biases. In examples in the top row, the variable being measured has a low bias with respect to the bullseye, as the
dots are centered on the bullseye. In contrast, in the bottom row, the variable being measured has a high bias a variable with low bias with respect to the bullseye, as the dots are off-center with
respect to the bullseye.
4.4.1 Estimating gradient variance using SNR#
The following functions measure and plot the SNR of the gradients:
Show code cell source Hide code cell source
#@markdown `compute_gradient_SNR(MLP, dataset)`: Passes a dataset through a model
#@markdown and computes the SNR of the gradients computed by the model for each example.
#@markdown Returns a dictionary containing the gradient SNRs for each layer of a model.
def compute_gradient_SNR(MLP, dataset):
Computes gradient SNRs for a model given a dataset.
- MLP (torch model): Model for which to compute gradient SNRs.
- dataset (torch dataset): Dataset with which to compute gradient SNRs.
- SNR_dict (dict): Dictionary compiling gradient SNRs for each parameter
(i.e., the weights and/or biases of each layer).
criterion = torch.nn.NLLLoss()
gradients = {key: list() for key in MLP.list_parameters()}
# initialize a loader with a batch size of 1
loader = torch.utils.data.DataLoader(dataset, batch_size=1)
# collect gradients computed on the dataset of data
for X, y in loader:
y_pred = MLP(X, y=y)
loss = criterion(torch.log(y_pred), y)
MLP.zero_grad() # zero grad before
for key, value in MLP.gather_gradient_dict().items():
MLP.zero_grad() # zero grad after, since no optimzer step is taken
# aggregate the gradients
SNR_dict = {key: list() for key in MLP.list_parameters()}
for key, value in gradients.items():
SNR_dict[key] = compute_SNR(np.asarray(value))
return SNR_dict
def compute_SNR(data, epsilon=1e-7):
Calculates the average SNR for of data across the first axis.
- data (torch Tensor): items x gradients
- epsilon (float, optional): value added to the denominator to avoid
division by zero.
- avg_SNR (float): average SNR across data items
absolute_mean = np.abs(np.mean(data, axis=0))
std = np.std(data, axis=0)
SNR_by_item = absolute_mean / (std + epsilon)
avg_SNR = np.mean(SNR_by_item)
return avg_SNR
#@markdown `plot_gradient_SNRs(SNR_dict)`: Plots gradient SNRs collected in
#@markdown a dictionary.
def plot_gradient_SNRs(SNR_dict, width=0.5, ax=None):
Plot gradient SNRs for various learning rules.
- SNR_dict (dict): Gradient SNRs for each learning rule.
- width (float, optional): Width of the bars.
- ax (plt subplot, optional): Axis on which to plot gradient SNRs. If None, a
new axis will be created.
- ax (plt subplot): Axis on which gradient SNRs were plotted. """
if ax is None:
wid = min(8, len(SNR_dict) * 1.5)
_, ax = plt.subplots(figsize=(wid, 4))
xlabels = list()
SNR_means = list()
SNR_sems = list()
SNRs_scatter = list()
for m, (model_type, SNRs) in enumerate(SNR_dict.items()):
color = get_plotting_color(model_idx=m)
m, np.mean(SNRs), yerr=scipy.stats.sem(SNRs),
alpha=0.5, width=width, capsize=5, color=color
s = [20 + i * 30 for i in range(len(SNRs))]
ax.scatter([m] * len(SNRs), SNRs, alpha=0.8, s=s, color=color, zorder=5)
x = np.arange(len(xlabels))
x_pad = (x.max() - x.min() + width) * 0.3
ax.set_xlim(x.min() - x_pad, x.max() + x_pad)
ax.set_xticklabels(xlabels, rotation=45)
ax.set_xlabel("Learning rule")
ax.set_title("SNR of the gradients")
return ax
For the error backpropagation model and the Hebbian learning model, we compute the SNR before training the model, using the validation set. This allows us to evaluate the gradients a learning rule
proposes for an untrained model. Notably, we pass one example at a time through the model, to obtain gradients for each example.
Note: Since we obtain a gradient SNR for each layer of the model, here we plot the gradient SNR averaged across layers.
SNR_dict = dict()
for model_type in ["backprop", "Hebbian"]:
model_params = {
"num_hidden": NUM_HIDDEN,
"activation_type": ACTIVATION,
"bias": BIAS,
if model_type == "Hebbian":
model_fct = HebbianMultiLayerPerceptron
model_fct = MultiLayerPerceptron
model = model_fct(**model_params)
model_SNR_dict = compute_gradient_SNR(model, valid_loader.dataset)
SNR_dict[model_type] = [SNR for SNR in model_SNR_dict.values()]
The dots in the plots show the SNR for each layer (small dot: first layer; larger dot: second layer).
Hebbian learning appears to have produced gradients with a very high SNR (and therefore lower variance) in the first layer (small dot). In constrast, the second layer (big dot) shows a lower SNR than
error backpropagation.
Notably, this is evaluated on the full 10-class task which Hebbian learning struggles to learn.
❓ What does the Hebbian learning SNR look like for the 2-class version of the task?
❓ What might this mean about how Hebbian learning learns?
❓ Going further: How might this result relate to the Hebbian learning rule’s performance on the classification task?
4.4.2 Estimating the gradient bias with respect to error backpropagation using the Cosine similarity.#
The following functions measure and plot the Cosine similarity of the gradients to error backpropagation gradients:
Show code cell source Hide code cell source
#@markdown `train_and_calculate_cosine_sim(MLP, train_loader, valid_loader, optimizer)`:
#@markdown Trains a model using a specific learning rule, while computing the cosine
#@markdown similarity of the gradients proposed the learning rule compared to those proposed
#@markdown by error backpropagation.
def train_and_calculate_cosine_sim(MLP, train_loader, valid_loader, optimizer,
Train model across epochs, calculating the cosine similarity between the
gradients proposed by the learning rule it's trained with, compared to those
proposed by error backpropagation.
- MLP (torch model): Model to train.
- train_loader (torch dataloader): Dataloader to use to train the model.
- valid_loader (torch dataloader): Dataloader to use to validate the model.
- optimizer (torch optimizer): Optimizer to use to update the model.
- num_epochs (int, optional): Number of epochs to train model.
- cosine_sim (dict): Dictionary storing the cosine similarity between the
model's learning rule and backprop across epochs, computed on the
validation data.
criterion = torch.nn.NLLLoss()
cosine_sim_dict = {key: list() for key in MLP.list_parameters()}
for e in tqdm(range(num_epochs)):
for X, y in train_loader:
y_pred = MLP(X, y=y)
loss = criterion(torch.log(y_pred), y)
if e != 0:
lr_gradients_dict = {key: list() for key in cosine_sim_dict.keys()}
backprop_gradients_dict = {key: list() for key in cosine_sim_dict.keys()}
for X, y in valid_loader:
# collect gradients computed with learning rule
y_pred = MLP(X, y=y)
loss = criterion(torch.log(y_pred), y)
for key, value in MLP.gather_gradient_dict().items():
# collect gradients computed with backprop
y_pred = MLP.forward_backprop(X)
loss = criterion(torch.log(y_pred), y)
for key, value in MLP.gather_gradient_dict().items():
for key in cosine_sim_dict.keys():
lr_grad = np.asarray(lr_gradients_dict[key])
bp_grad = np.asarray(backprop_gradients_dict[key])
if (lr_grad == 0).all():
f"Learning rule computed all 0 gradients for epoch {e}. "
"Cosine similarity cannot be calculated."
epoch_cosine_sim = np.nan
elif (bp_grad == 0).all():
f"Backprop. rule computed all 0 gradients for epoch {e}. "
"Cosine similarity cannot be calculated."
epoch_cosine_sim = np.nan
epoch_cosine_sim = calculate_cosine_similarity(lr_grad, bp_grad)
return cosine_sim_dict
def calculate_cosine_similarity(data1, data2):
Calculates the cosine similarity between two vectors.
- data1 (torch Tensor): first vector
- data2 (torch Tensor): second vector
data1 = data1.reshape(-1)
data2 = data2.reshape(-1)
numerator = np.dot(data1, data2)
denominator = (
np.sqrt(np.dot(data1, data1)) * np.sqrt(np.dot(data2, data2))
cosine_sim = numerator / denominator
return cosine_sim
#@markdown `plot_gradient_cosine_sims(cosine_sim_dict)`: Plots cosine similarity
#@markdown of the gradients proposed by a model across learning to those proposed
#@markdown by error backpropagation.
def plot_gradient_cosine_sims(cosine_sim_dict, ax=None):
Plot gradient cosine similarities to error backpropagation for various
learning rules.
- cosine_sim_dict (dict): Gradient cosine similarities for each learning rule.
- ax (plt subplot, optional): Axis on which to plot gradient cosine
similarities. If None, a new axis will be created.
- ax (plt subplot): Axis on which gradient cosine similarities were plotted.
if ax is None:
_, ax = plt.subplots(figsize=(8, 4))
max_num_epochs = 0
for m, (model_type, cosine_sims) in enumerate(cosine_sim_dict.items()):
cosine_sims = np.asarray(cosine_sims) # params x epochs
num_epochs = cosine_sims.shape[1]
x = np.arange(num_epochs)
cosine_sim_means = np.nanmean(cosine_sims, axis=0)
cosine_sim_sems = scipy.stats.sem(cosine_sims, axis=0, nan_policy="omit")
ax.plot(x, cosine_sim_means, label=model_type, alpha=0.8)
color = get_plotting_color(model_idx=m)
cosine_sim_means - cosine_sim_sems,
cosine_sim_means + cosine_sim_sems,
alpha=0.3, lw=0, color=color
for i, param_cosine_sims in enumerate(cosine_sims):
s = 20 + i * 30
ax.scatter(x, param_cosine_sims, color=color, s=s, alpha=0.6)
max_num_epochs = max(max_num_epochs, num_epochs)
if max_num_epochs > 0:
x = np.arange(max_num_epochs)
xlabels = [f"{int(e)}" for e in x]
ymin = ax.get_ylim()[0]
ymin = min(-0.1, ymin)
ax.set_ylim(ymin, 1.1)
ax.axhline(0, ls="dashed", color="k", zorder=-5, alpha=0.5)
ax.set_ylabel("Cosine similarity")
ax.set_title("Cosine similarity to backprop gradients")
return ax
NUM_EPOCHS = 10
cosine_sim_dict = dict()
for model_type in ["backprop", "Hebbian"]:
model_params = {
"num_hidden": NUM_HIDDEN,
"activation_type": ACTIVATION,
"bias": BIAS,
if model_type == "Hebbian":
model_fct = HebbianMultiLayerPerceptron
lr = HEBB_LR
model_fct = MultiLayerPerceptron
lr = LR
model = model_fct(**model_params)
optimizer = BasicOptimizer(model.parameters(), lr=lr)
print(f"Collecting Cosine similarities for {model_type}-trained model...")
model_cosine_sim_dict = train_and_calculate_cosine_sim(
model, train_loader, valid_loader, optimizer, num_epochs=NUM_EPOCHS
cosine_sim_dict[model_type] = [
cos_sim for cos_sim in model_cosine_sim_dict.values()
Collecting Cosine similarities for backprop-trained model...
Collecting Cosine similarities for Hebbian-trained model...
As expected, gradient updates proposed by error backpropagation necessarily have no bias with respect to themselves (cosine similarity near 1.0). In contrast, although gradient updates proposed by
Hebbian learning for the second layer (big dots) are well aligned with error backpropagation updates, the updates proposed for the first layer (small dots) are highly biased (cosine similarity near
❓ What might this result tell us about how Hebbian learning learns in this task?
❓ What does the Hebbian learning cosine similarity to error backpropagation look like for the 2-class version of the task?
❓ Going further: How might this result relate to the Hebbian learning rule’s performance on the classification task?
❓ Taken together, what do the bias and variance properties of each layer tell us about how Hebbian learning learns in this task?
❓ Going further: How learning rule-specific are the bias and variance properties of the gradients compared to other performance or learning metrics?
❓ Going further: How do the bias and variance of the gradients relate to the performance of a learning rule on a task?
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 5: Implementing additional learning rules#
In this notebook, we implemented learning in a neural network using Hebbian learning, and examined how this learning rule performed under various scenarios. Hebbian learning is only one biologically
plausible learning rule among many others. Importantly, many of these other rules are better suited to supervised learning tasks like image classification.
Examples of basic biologically plausible learning rules to explore include node perturbation, weight_perturbation, feedback alignment, and the Kolen Pollack.
Take a look at Neuromatch’s NeuroAI tutorial for the Microlearning day for implementations of these algorithms using numpy. Then, see whether you can reimplement one or several of them using custom
torch autograd functions, as demonstrated in this notebook.
Implementing certain learning rules may also require making some changes to how the optimizer step is performed. To do so, you can adapt the BasicOptimizer() or any other torch optimizer, as needed.
The following repositories could be very helpful resources, as they implement these learning rules using custom torch autograd functions:
• Feedback alignment: https://github.com/L0SG/feedback-alignment-pytorch/blob/master/lib/fa_linear.py
• Kolen-Pollack: https://github.com/limberc/DL-without-Weight-Transport-PyTorch/blob/master/linear.py. Also take a look at main.py to see how they adapt the SGD optimizer to produce the correct
updates for Kolen Pollack.
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Section 6: Tips & suggestions#
Here are a few tips that may be helpful as you delve into questions from the project template.
• Testing whether the metrics are specific to a learning rule: There are a few ways to assess whether certain performance and learning metrics are specific to a learning rule. Examples include:
□ Visualization: You could plot the metrics or a lower-dimensional version of the metrics to visualize whether the metrics for each learning rule form separate clusters or whether they all mix
□ Classification: You could test whether a linear classifier can be trained to correctly predict the learning rule from the metrics.
• Assessing how your models respond to more challenging learning scenarios: There are several challenging learning scenarios you can implement. Examples include:
□ Online learning: Training with a batch size of one.
□ Non-stationary data: Changing the distribution of the data across learning. Here, the restrict_classes(dataset) function may be useful. For example, you could initially train a model on a
dataset with no examples from the 6 class, and then introduce examples from the 6 class partway through training to see how this affects learning.
Submit your feedback#
Show code cell source Hide code cell source
# @title Submit your feedback
Additional References#
Original papers introducing these biologically plausible learning rules:
⭐ We hope you enjoy working on your project and, through the process, make some interesting discoveries about the challenges and potentials of biologically plausible learning! ⭐# | {"url":"https://neuroai.neuromatch.io/projects/project-notebooks/Microlearning.html","timestamp":"2024-11-10T17:41:33Z","content_type":"text/html","content_length":"628736","record_id":"<urn:uuid:a2687edf-06ba-4bfe-a3b0-3bab0b87e9ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00663.warc.gz"} |
Finite volume method
From Scholarpedia
Robert Eymard et al. (2010), Scholarpedia, 5(6):9835. doi:10.4249/scholarpedia.9835 revision #91264 [link to/cite this article]
The Finite Volume Method (FVM) is a discretization method for the approximation of a single or a system of partial differential equations expressing the conservation, or balance, of one or more
quantities. These partial differential equations (PDEs) are often called conservation laws; they may be of different nature, e.g. elliptic, parabolic or hyperbolic, and they are used as models in a
wide number of fields, including physics, biophysics, chemistry, image processing, finance, dynamic reliability. They describe the relations between partial derivatives of unknown fields such as
temperature, concentration, pressure, molar fraction, density of electrons or probability density function, with respect to variables within the domain (space, time,...) under consideration.
As in the finite element method, a mesh is constructed, which consists in a partition of the domain where the space variable lives. The elements of the mesh are called control volumes. The
integration of the PDE over each control volume results in a balance equation. The set of balance equations is then discretized with respect to a set of discrete unknowns. The main issue is the
discretization of the fluxes at the boundaries of each control volume: in order for the FVM to be efficient, the numerical fluxes are generally
• conservative, i.e. the flux entering a control volume from its neighbour must be the opposite of the one entering the neighbour from the control volume,
• consistent, i.e. the numerical flux of a regular function interpolation tends to the continuous flux as the mesh size vanishes.
It is sometimes possible to discretize the fluxes at the boundaries of the control volume by the finite difference method (FDM). In this case, the method has often been referred to as a finite
difference method or conservative finite difference method (see Samarskii 2001). The specificity of the FVM with respect to the FDM is that the discretization is performed on the local balance
equations, rather than the PDE: the fluxes on boundaries of the control volumes are discretized, rather than the continuous differential operator.
The resulting system of discrete equations depends on a discrete (finite) set of unknowns, and may be either linear or non linear, depending on the original problem itself; this system is then solved
exactly or approximately, using for example direct or iterative solvers in the case of linear equations and fixed point or Newton type methods in the case of nonlinear equations.
Let us emphasize that there is no systematic way to derive the discrete system from the continuous one; in fact, the physical or engineering knowledge of the problem is in many cases the key
ingredient in order to obtain a scheme which satisfies the expected properties of accuracy, robustness, and computational cost. Let us mention that in some cases, the engineering problem is initially
set at a discrete level: the model consists of a set of discrete balance equations and therefore written as a finite volume scheme, and the continuous model of the problem is not explicitly given.
Many of these issues have often been addressed in computational fluid mechanics. A review of the schemes for the discretization of the compressible and incompressible Navier-Stokes and Euler
equations may be found in e.g. Wesseling 2001, Feistauer et al. 2003. The FVM has also been very early introduced in the framework of oil reservoir engineering (see Peaceman 1991).
Fundamental principles
Consider the following PDE under conservative form\[\tag{1} {\partial_t A}(x,t) + \nabla\cdot F(x,t) = S(x,t),\]
where the space variable \(x\) belongs to the domain \(\Omega \subset {\mathbb R}^d\) (\(d\) is the space dimension, greater or equal to 1), and the time variable \(t\) belongs to some time interval
\([0,T]\ ,\) with \(T>0\ .\) The scalar function \(A\ ,\) defined in \(\Omega\times [0,T]\ ,\) expresses the density of some quantity, and \(\partial_t A\) denotes its time derivative. The function \
(F\ ,\) defined in \(\Omega\times [0,T]\) and valued in \(\mathbb R^d\ ,\) expresses the flux of this quantity, and \(\nabla \cdot F\) denotes its space divergence. The function \(S\ ,\) defined in \
(\Omega\times [0,T]\) and valued in \(\mathbb R\ ,\) denotes some source term. Some initial condition \(A(x,0) = A_{\rm ini}(x)\) for \(x\in\Omega\) is imposed, where the function \(A_{\rm ini}\) is
defined in \(\Omega\) and valued in \(\mathbb R\ ,\) as well as some boundary conditions, which depend on the considered equation. Note that the functions \(A\) and \(F\) are not necessarily regular,
so that the derivatives involved in (1) may be weak derivatives.
These functions \(A\ ,\) \(F\ ,\) \(S\) are assumed to be related to a set of unknown fields \((u_j)_{j=1,\ldots,N}\ ,\) where \(u_j\) is an unknown function defined from \(\Omega\times [0,T]\) to \
(\mathbb R\ .\) Such relationships may for instance be written as\[\tag{2} \begin{matrix} A(x,t) = {\mathcal A}(u_1(x,t),\ldots,u_N(x,t),x,t), \\ S(x,t) = {\mathcal S}(u_1(x,t),\ldots,u_N(x,t),x,t),
\\ F(x,t) = {\mathcal F}(u_1(x,t),\ldots,u_N(x,t),\nabla u_1(x,t),\ldots,\nabla u_N(x,t),x,t), \end{matrix}\]
where the functions \({\mathcal A}\ ,\) \({\mathcal S}\) and \({\mathcal F}\) are given. Classical examples with \(N=1\) (we then denote by \(u\) the unique component \(u_1\)), are:
• the heat equation which corresponds to \(A(x,t) = u(x,t)\ ,\) and \(u\) is the temperature; \(F(x,t) = -\Lambda(x)\nabla u(x,t)\) (where for any \(x \in \mathbb R^d\ ,\) \(\Lambda(x)\) is a
linear operator from \(\mathbb R^d\) to \(\mathbb R^d\)), \(S(x,t) = {\mathcal S}(x,t)\ ;\)
• the Poisson equation, which is the steady version of the heat equations \(A(x,t) = 0\ ,\) \(F(x,t) = -\Lambda(x)\nabla u(x)\ ,\) \(S(x,t) = {\mathcal S}(x)\ ;\)
• the linear convection equation \(A(x,t) = u(x,t)\ ,\) \(F(x,t) = u(x,t)V(x,t)\) where \(V\) is a function defined from \(\overline\Omega\times[0,T]\) to \({\mathbb R}^d\ ,\) \(u\) is, say, a
concentration, and \(V\) its transport velocity.
The domain \(\Omega\) is partitioned into a mesh, denoted by \({\mathcal M}\ .\) In general, the mesh \(\mathcal M\) is a finite family of nonoverlapping subsets of \(\Omega\ ,\) which satisfy a
number of regularity properties; some of these properties are needed for the definition of the scheme, while others are required to establish the precision and the convergence of the method,
depending on the type of term that needs to be discretized. The elements of \({\mathcal M}\ ,\) denoted by \(K\ ,\) \(L\ ,\) are called the control volumes; the measure of a control volume \(K\) (its
length if \(d=1\ ,\) area if \(d=2\ ,\) volume if \(d=3\)) is denoted by \(|K|\ .\) The boundary \(\partial K\) of each control volume \(K\) is partitioned into the finite set \({\mathcal E}_K\ ,\)
called the set of the faces (\(d=3\)) or edges (\(d=2\)) or extremities (\(d=1\)) of \(K\ .\) An element \(\sigma\) of \({\mathcal E}_K\) is either located on the boundary of \(\Omega\ ,\) or belongs
to \({\mathcal E}_K\cap {\mathcal E}_L\) where \(K\) and \(L\) are two adjacent control volumes.
The \((d-1)\)-dimensional measure of a face \(\sigma\in{\mathcal E}_K\) (its length if \(d=2\ ,\) area if \(d=3\)) is denoted by \(|\sigma|\ .\) A strictly increasing finite sequence \(t^{(0)} = 0 <t
^{(1)} <\ldots <t^{(N)} = T\) is defined for the time discretization, and one denotes \(\delta t^{(n)} = t^{(n)} - t^{(n-1)}\ ,\) for \(n=1,\ldots,N\ .\) The classical finite volume approximation of
(1) relies on the approximation of the balance equations on the control volumes between time \(t^{(n-1)}\) and \(t^{(n)}\) (in fact, the balance equations usually precede the continuous equation (1)
in the derivation of the model; this is the reason why, in several engineering applications, the FVM is applied directly to the discrete balance equations without any knowledge of the continuous
equations). These balance equations may be obtained by integrating (1) on \(K\times [t^{(n-1)},t^{(n)}]\) and using the divergence formula\[ \int_K ( A (x,t^{(n)})- A (x,t^{(n-1)})) {\mathrm d}x + \
sum_{\sigma\in {\mathcal E}_K}\int_{t^{(n-1)}}^{t^{(n)}} \int_{\sigma} F(x,t) \cdot \boldsymbol n_{K,\sigma}(x) {\mathrm d}s(x) {\mathrm d}t = \int_{t^{(n-1)}}^{t^{(n)}} \int_K S(x,t) {\mathrm d}x {\
mathrm d}t, \]
where \({\mathrm d}s(x)\) denotes the integration with respect to the \((d-1)\)-dimensional measure of the boundary and \(\boldsymbol n_{K,\sigma}(x)\) the unit vector normal to \(\sigma\) at point \
(x\ ,\) outward to \(K\ .\) The FV scheme is a discretization of the above balance equation, and may be written as\[ |K|(A_K^{(n)} - A_K^{(n-1)}) + \delta t^{(n)} \sum_{\sigma\in {\mathcal E}_K} \
vert \sigma \vert \,F_{K,\sigma}^{(n)} = \delta t^{(n)} |K| \, S_K^{(n)}, \]
where\[ A_K^{(n)} \, (\hbox{ resp. } A_K^{(0)} \hbox{) is an approximation of } \frac 1 {|K|}\int_K A(x,t^{(n)}) {\rm d}x\hbox{ (resp. } \frac 1 {|K|}\int_K A_{\rm ini}(x){\rm d}x\hbox{)}, \]
\( F_{K,\sigma}^{(n)} \hbox{ is an approximation of } \phi^{(n)}_{K,\sigma} = \frac 1 {\delta t^{(n)} \vert \sigma \vert }\int_{t^{(n-1)}}^{t^{(n)}}\int_\sigma F(x,t)\cdot \boldsymbol n_{K,\sigma}(x)
{\mathrm d}s(x){\mathrm d}t, \)
\( S_K^{(n)} \hbox{ is an approximation of } \frac 1 {\delta t^{(n)}|K|} \int_{t^{(n-1)}}^{t^{(n)}}\int_K S(x,t){\rm d}x{\rm d}t. \)
For a face \(\sigma\) common to two control volumes \(K\) and \(L\ ,\) since \(\boldsymbol n_{K,\sigma}(x) +\boldsymbol n_{L,\sigma}(x) =0\) for \(x\in\sigma\ ,\) the normal fluxes in the original
problem satisfy the conservation property \(\phi^{(n)}_{K,\sigma} + \phi^{(n)}_{L,\sigma} = 0\ .\) A discrete version of this property of conservation is then required\[ F_{K,\sigma}^{(n)} + F_{L,\
sigma}^{(n)}= 0. \] This relation is one of the pillars of the classical FVM; moreover, it is crucial in the mathematical analysis for proving convergence properties.
Discrete expressions in terms of discrete unknowns must then be provided for the terms \(A_K^{(n)}\ ,\) \(F_{K,\sigma}^{(n)}\) and \(S_K^{(n)}\ .\) These discrete unknowns are usually expected to be
approximations of the unknown functions \((u_j)_{j=1,\ldots,N}\) at different locations (grid blocks, faces, vertices,...) and times: hence \(u_{j,K}^{(n)}\) (resp. \(u_{j,\sigma}^{(n)}\)) denotes
the approximation of function \(u_j\) in control volume \(K\) (resp. at face \(\sigma\)) at time \(t^{(n)}\ .\) For example, considering the case given in (2), it is generally possible to simply use
the function \({\mathcal A}\) to define \(A_K^{(n)}\ ,\) using an expression such as
\( A_K^{(n)} = \frac 1 {|K|} \int_K{\mathcal A}(u_{1,K}^{(n)},\ldots,u_{N,K}^{(n)},x,t^{(n)}){\rm d}x. \)
It is generally less simple to derive from (2) an expression for \(F_{K,\sigma}^{(n)}\) with respect to the discrete variables, since there is no systematic method to use the function \({\mathcal F}
\) which ensures accuracy and stability on general grids; the discretization of diffusion and convection fluxes are discussed below.
The arguments for the discrete expression of \(S_K^{(n)}\) are generally \(u_{j,K}^{(m)}\ ,\)..., \(m=n\) or \(n-1\ .\)
Discretization of diffusion fluxes
We consider here a diffusive flux \(F(x,t)\) of the form \(F(x,t) = -\Lambda(x,t) \nabla u(x,t)\ ,\) where \(\Lambda(x,t)\) is a linear operator from \({\mathbb R}^d\) to \({\mathbb R}^d\ ,\) and \(u
\) is one of the continuous unknowns \((u_j)_{j=1,\ldots,N}\) of the problem. Such an expression of the flux is encountered when using e.g. Fourier's law (heat conduction), Darcy's law (flow in
porous media) or Fick's law (chemical diffusion).
In some cases, a simple, conservative and consistent discrete expression for \(F_{K,\sigma}^{(n)}\) may be obtained. This is the case when \(\Lambda(x,t)\) is the identity operator and when the grid
satisfies the following "orthogonal property": in each control volume \(K\in{\mathcal M}\ ,\) there exists a particular point \(x_K\ ,\) called the "center" (or centroid) of the control volume, such
that, for any adjacent control volume \(L\) to \(K\) with common face \(\sigma\ ,\) the straight line \((x_K,x_L)\) is orthogonal to \(\sigma\ .\)
This orthogonality property is satisfied by several types of meshes, such as triangles, rectangles, Voronoï boxes. Then a simple finite difference expression for the approximation of \(\phi_{K,\
sigma}^{(n)} =-\frac 1 {\delta t^{(n)} \vert \sigma \vert }\int_{t^{(n-1)}}^{t^{(n)}}\int_\sigma \nabla u(x,t)\cdot n_{K,\sigma}(x) {\rm ds}(x){\rm d}t\) is given by
F_{K,\sigma}^{(n)} = \frac {u_K^{(m)} - u_L^{(m)}} {d(x_K,x_L)},
where \(d(x_K,x_L)\) denotes the Euclidean distance between the points \(x_K\) and \(x_L\ ,\) and provides an implicit (resp. explicit) approximation if \(m=n\) (resp. \(m=n-1\)). This is the famous
"two-point flux approximation" scheme, which has been widely used. It is simple, and also much appreciated because it respects a discrete version of the maximum principle; in particular, the
approximate solution does not oscillate and stays within its physical bounds. The convergence analysis of the FVM for diffusion equations in this particular case is thoroughly performed in Eymard et
al. 2000. The error analysis relies on the conservativity and consistency of the numerical flux.
In the general case of unstructured meshes and general (possibly anisotropic and heterogeneous) linear operators \(\Lambda(x,t)\ ,\) a two point flux formula is not consistent in general, and yields
an approximation which might not converge to the appropriate solution.
There are several ways to derive expressions for \(F_{K,\sigma}^{(n)}\) in the general case (intensive research is going on for the analysis of these schemes and their implementation, see the
proceedings of the conferences on Finite Volume for Complex Applications):
• The flux may for instance be approximated by a consistent formula using several points and respecting the conservativity property, such as in the MPFA (multiple point flux approximation) scheme.
• Expressions for \(F_{K,\sigma}^{(n)}\) may also be obtained thanks to an identification procedure, using a variational expression and discrete gradients. Such a technique is used in mixed finite
volume, hybrid finite volumes (and the SUSHI scheme) and the mimetic method. These three methods can in fact be viewed as elements of one unified family of schemes.
• Another way to express \(F_{K,\sigma}^{(n)}\) in the anisotropic case is to partition the domain \(\Omega\) in more than one mesh (typically two in 2D and three in 3D), and use the differences of
the unknowns in the different meshes to reconstruct an expression for \(F_{K,\sigma}^{(n)}\ .\) This is the principle of the discrete duality FVM.
• The expression of \(F_{K,\sigma}^{(n)}\) may also be inspired by the finite element framework, as in the example of Finite Element-Finite Volume methods (see Feistauer et al. 2003).
• \(\ldots\)
Unfortunately, all the linear schemes devised in the general case of unstructured meshes and general linear operators do not seem to satisfy a discrete version of the maximum principle; for severely
distorted meshes or for highly anisotropic operators, the risk of oscillations and violation of the physical bounds is high. Ongoing research using in particular nonlinear schemes for linear problems
is under way to avoid these drawbacks (see the proceedings of Finite Volume for Complex Applications V).
Approximation of convection terms
Nonlinear scalar convection equations
Consider the partial differential equation\[ {\partial_t u} (x,t) + \nabla\cdot {\mathcal F}(u(x,t),x,t) = 0. \]
This is the conservation equation (1) with \(A(x,t) = u(x,t) \ ,\) \(F(x,t) = {\mathcal F}(u(x,t),x,t)\) and \(S(x,t) =0\ .\) The linear transport equation is a particular case, with \({\mathcal F}
(u,x,t) = u V(x,t)\ ,\) and \(V\) is defined from \(\Omega\times[0,T]\) to \({\mathbb R}^d\ .\) The discrete unknowns are the values \(u_K^{(n)}\ ,\) which are expected to be approximations of \(u\)
in the control volumes \(K\in {\mathcal M}\) at time \(t^{(n)}\ .\) Since the convection flux is of order 0 (no derivative involved) its approximation does not require any assumption on the mesh,
contrary to the diffusion flux. Nevertheless, the fluxes must be carefully approximated in order to ensure the stability (and convergence) of the scheme. Consider the flux \(\phi^{(n)}_{K,\sigma}\)
through an interface \(\sigma\) at time \(t^{(n)}\ ,\) seen as a function of the solution \(u\ :\) \(\phi^{(n)}_{K,\sigma}(u) = \frac 1 {\delta t^{(n)}\vert \sigma \vert}\int_{t^{(n-1)}}^{t^{(n)}}\
int_\sigma {\mathcal F}(u(x,t),x,t)\cdot\boldsymbol n_{K,\sigma}(x) {\rm ds}(x){\rm d}t\ .\)
A numerical monotone flux associated to \(\phi^{(n)}_{K,\sigma}(u)\) is a function \(G_{K,\sigma}^{(n)}\) from \({\mathbb R}^2\) to \(\mathbb R\) such that:
*\(G_{K,\sigma}^{(n)}(a,b) = - G_{L,\sigma}^{(n)}(b,a)\ ,\) for an interface \(\sigma\in {\mathcal E}_K\cap {\mathcal E}_L\ ,\)
*\(G_{K,\sigma}^{(n)}\) is non-decreasing with respect to its first variable and non-increasing with respect to
its second variable,
*\(G_{K,\sigma}^{(n)}\) is locally Lipschitz-continuous with respect to its variables,
*\(G_{K,\sigma}^{(n)}(a,a) = \phi_{K,\sigma}^{(n)}(a),\) for any \(a\) respecting the bounds of the solution.
Then FV schemes are obtained by approximating \( \phi^{(n)}_{K,\sigma}(u)\) by
• \(F_{K,\sigma}^{(n)} = G_{K,\sigma}^{(n)}(u_K^{(m)}, u_L^{(m)})\ ,\) for an interface \(\sigma\in {\mathcal E}_K\cap {\mathcal E}_L\ ,\)
• \(F_{K,\sigma}^{(n)} = G_{K,\sigma}^{(n)}(u_K^{(m)}, u_\sigma^{(n)})\ ,\) for an interface \(\sigma\subset\partial\Omega\ ,\) where \(u_\sigma^{(n)}\) is a suitable approximation of the boundary
condition on \(\sigma\times]t^{(n-1)},t^{(n)}[\ ,\)
with \(m = n-1\) (explicit scheme) or \(m = n\) (implicit scheme).
Provided that, for an explicit scheme, the so-called CFL condition be satisfied (this condition prescribes a bound for the time step with respect to the size of the mesh), these schemes satisfy the
following properties (see Eymard et al. 2000, Godlewski et al. 1996):
• they are \(L^\infty\)-stable, in the sense that the bounds of the initial and boundary data are preserved,
• they are Total Variation Diminishing (TVD), in the case of structured meshes.
Hyperbolic systems
We now consider a system of \(N\) one-dimensional conservation laws \({\partial_t A_j} (x,t) + {\partial_x F_j}(x,t) = 0\ ,\) where, for \(j=1,\ldots,N,\) \(A_j(x,t) = u_j(x,t)\) and the scalar flux
\(F_j(x,t)\) is given by \(F_j(x,t) = {\mathcal F}_j(u_1(x,t), u_2(x,t),\ldots,u_N(x,t),x,t)\ .\) The continuous problem itself is not easy to study from a mathematical point of view, and open
questions include existence and uniqueness of a physically relevant solution in the general system case. The continuous fluxes \((\phi_j)^{(n)}_{K,\sigma}\) between times \(t^{(n-1)}\) and \(t^{(n)}
\) through an interface \(\sigma\) (which is a real point in the one-dimensional framework) read\[ (\phi_j)^{(n)}_{K,\sigma} = \frac {n_{K,\sigma}} {\delta t^{(n)} }\int_{t^{(n-1)}}^{t^{(n)}} {\
mathcal F}_j(u_1(\sigma,t),\ldots,u_N(\sigma,t),\sigma,t){\rm d}t,\ j=1,\ldots,N, \]
with \(n_{K,\sigma} = 1\) or \(n_{K,\sigma} = -1\ ,\) depending on the position of \(\sigma\) with respect to \(K\ .\) A time-explicit FVM approximation reads\[ |K|(u_{j,K}^{(n)} - u_{j,K}^{(n-1)}) +
\delta t^{(n)} \sum_{\sigma\in {\mathcal E}_K} (F_j)_{K,\sigma}^{(n)} = 0, \, j = 1, \ldots,N, \] where \((F_j)_{K,\sigma}^{(n)}\) is an approximation of \( (\phi_j)^{(n)}_{K,\sigma}.\) Such an
approximation may be performed by the very famous Godunov scheme (presented here in its explicit version, which is the most often used). The idea is, for given \(K,\sigma,n\ ,\) to approximate the
flux \((\phi_j)^{(n)}_{K,\sigma}\) by \((F_j)_{K,\sigma}^{(n)} = f_j(w^R_1,\ldots,w^R_N),\) where \(f_j\) is the function defined from \(\mathbb R^N\) to \(\mathbb R\) by\[ f_j(v_1,\ldots,v_N) = \
frac {n_{K,\sigma}} {\delta t^{(n)} }\int_{t^{(n-1)}}^{t^{(n)}} {\mathcal F}_j(v_1,\ldots,v_N,\sigma,t) {\rm d}t,\ j=1,\ldots, N, \, \forall (v_1,\ldots,v_N) \in \mathbb R^N, \] and \((w^R_1,\ldots,w
^R_N)\) is solution at \(\xi=0\) for all \(\tau>0\) of the so-called one-dimensional Riemann problem, defined by\[\begin{matrix} {\partial_\tau} w_j (\xi,\tau) + \partial_\xi f_j(w_1(\xi,\tau), w_2(\
xi,\tau),\ldots,w_N(\xi,\tau)) = 0,& \\ w_j(\xi,0)= u_{j,K}^{(n-1)}, \hbox{ if } \xi < 0,& \\ w_j(\xi,0)= u_{j,L}^{(n-1)}, \hbox{ if } \xi > 0,&\ j=1,\ldots,N. \end{matrix}\]
The Godunov scheme is a very efficient scheme when the above solution of the Riemann problem is easily constructed. When the solution of the Riemann problem becomes too complex or expensive,
alternative schemes based on approximate Riemann solvers can be used (see Eymard et al. 2000, Feistauer et al. 2003, Godlewski et al. 1996, Wesseling 2001). See also these references for some
extensions to multi-dimensional problems.
R. Eymard, T. Gallouët, and R. Herbin. Finite volume methods. In P.G. Ciarlet and J.L. Lions, editors, Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII, pages
713--1020. North-Holland, Amsterdam, 2000.
R. Eymard and J.-M. Hérard, editors. Finite volumes for complex applications V. ISTE, London, 2008.
M. Feistauer, J. Felcman, and I. Straskraba. Mathematical and computational methods for compressible flow. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University
Press, Oxford, 2003.
E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.
D.W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Elsevier Science Inc., New York, NY, USA, 1991.
A.A. Samarskii. The theory of difference schemes, volume 240 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 2001.
P. Wesseling. Principles of computational fluid dynamics, volume 29 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2001. | {"url":"http://var.scholarpedia.org/article/Finite_volume_method","timestamp":"2024-11-11T21:21:47Z","content_type":"text/html","content_length":"53566","record_id":"<urn:uuid:a443d31f-7c4b-4777-91f0-cacc7355128f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00615.warc.gz"} |
Alternative Homework Assignment: Electrical Safety
University of Maryland Physics Education Research Group
Alternative Homework Assignment: Electrical Safety
PERG Info | PERG materials | PERG HOMEPAGE | PER on the web | Resources on the web
Alternative Homework Assignment: Electrical Safety
Dan is the proud owner of a pet love bird named Felix. One day while Dan was cleaning Felix's cage, Felix flew out an open window. Looking out the window, Dan saw Felix perched on a high tension
power line just outside Dan's window. In this problem, we will investigate how Felix is able to rest on such a wire without injury, Dan's ill fated attempt to retrieve Felix, and perhaps gain some
insight into why high tension power cables are generally far away from high rise apartment buildings.
1. Before we can consider the effects it might have, we must consider the high tension power line itself. Such wires are frequently constructed by twisting copper wires together into a large
flexible bundle. Consider such a cable with a diameter of 4 cm. If this cable is 10 km long, what is its resistance?
2. High tension power lines connect the transformers at a power plant to the transformers at a substation where the voltage is stepped down for household use. The impedance of the transformer at
the substation will vary greatly depending on the energy requirement it must meet. Suppose that the transformer connected to the line where Felix is perched has an impedance of 100,000 Ohms. If
the root means squared voltage across the transformer is to be 25,000 V, what must the root means squared voltage across the transformer and the power line be?
3. What is the root mean squared current though the power line?
4. Estimate the distance between Felix's feet.
5. What is the voltage drop in the power line across this distance?
6. If Felix's legs have a resistance per centimeter of about 1 MOhm, estimate the resistance from foot to foot.
7. From the above information, calculate the current through Felix's legs.
8. Assume that birds' sensitivity to electricity is similar to humans and determine if Felix can feel the current through his legs.
Part 2: Noticing Felix on the power cable, Dan decided to retrieve him. In order to do this, Dan stepped out onto the window sill to call to Felix. Once on the sill Dan began to slip and in order to
avoid plunging to his death he jumped to cable where he found himself hanging without hope of returning to the window sill. Felix, disturbed by the commotion flew back into the apartment.
1. Estimate the distance between Dan's hands, and calculate the voltage drop in the cable across this distance.
2. The resistance from hand to hand on a person is about 400 Ohms. What is the strength of the current through Dan?
3. According to the table, is this current dangerous?
Part 3: Unable to return to the window, and too high in the air to jump to the ground, Dan made his way to the nearest support tower. This tower was unfortunately for Dan made of metal and grounded.
As Dan reached for the tower with his foot, he created a link between the grounded tower and the high voltage cable. Current surged through this low resistance path to ground, killing our unfortunate
1. Assuming that Dan's apartment is near the substation, what was the root mean squared voltage across Dan's body?
2. If the resistance from Dan's hand to his foot is about 500 Ohms, what was the current through his body?
3. This value should be off the chart for listed effects. To get an idea for what type of effect it might have, calculate the power dissipated though Dan.
Part 4: From the above story, it should be clear why in the real world, great care is taken to keep high voltage power lines out of reach. As a final exercise we will examine how Dan might have
forced Felix off his perch while sparing his own life.
1. While the best solution may have been to leave the window open and put some of Felix's favorite food on the sill, let's examine what would have happened if Dan had reached out with a long plastic
rod. Plastics generally have resistivities on the order of 10^12 Ohm m. If Dan had a pole of such material with a diameter of 3 cm, and a length of 3m, what would be the resistance though that
2. How much current would flow across this pole if a 25,000 V potential difference were placed across its ends? Would this be safe?
3. Is it possible for this pole to act as a capacitor and transmit the current in this fashion?
These problem written and collected by K. Vick, E. Redish, and P. Cooney. These problems may be freely used in classrooms. They may be copied and cited in published work if the Activity-Based Physics
(ABP) Alternative Homework Assignments (AHA's) Problem sight is mentioned and the URL given.
Work supported in part by NSF grant DUE-9455561
To contribute problems to this site, send them to redish@quark.umd.edu. | {"url":"https://physics.umd.edu/rgroups/ripe/perg/abp/aha/es2.htm","timestamp":"2024-11-04T21:03:11Z","content_type":"text/html","content_length":"8342","record_id":"<urn:uuid:7dba4b45-9aa9-49b2-83b0-4aae12453629>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00037.warc.gz"} |
Copyright (c) OleksandrZhabenko 2021
License MIT
Maintainer olexandr543@yahoo.com
Stability Experimental
Safe Haskell None
Language Haskell2010
Extensions BangPatterns
Some variants of the f :: Float -> OvertonesO function that can be used for overtones (and a timbre respectively) generation.
overTones :: Float -> OvertonesO Source #
For the given frequency of the note it generates a list of the tuples, each one of which contains the harmonics' frequency and amplitude. Is taken from the Composition.Sound.Functional.Basics module,
so it must be imported qualified to be used with the last one.
overTonesKN2 :: Int -> Int -> Float -> OvertonesO Source #
A generalized variant of the overTones function with just two additional control parameters (so represents a two-parameters additionally parameterized family of overTones functions).
overTonesKNPG20k :: Int -> Int -> Int -> Float -> Float -> OvertonesO Source #
A generalized variant of the overTones and overTonesKN2 with the possibility to specify the additional parameters for them. Otherwise than the similar overTonesKNPG, it can generate the higher
overtones than the B8 to the 20000 Hz.
overTonesKNPG20kF :: (Float -> Float) -> Int -> Int -> Int -> Float -> Float -> OvertonesO Source #
A generalized variant of the overTonesKNPG20k with the possibility to specify the additional function to control the overtones amplitudes. Otherwise than the similar overTonesKNPG, it can generate
the higher overtones than the B8 to the 20000 Hz. The first argument, the function f is intended to be a monotonic growing one though it can be not necessary.
overTonesKNPG20kFALaClarinet :: (Float -> Float) -> (Int -> Int) -> Int -> Int -> Int -> Float -> Float -> OvertonesO Source #
A generalized variant of the overTonesKNPG20k with the possibility to specify the additional function to control the overtones amplitudes. Otherwise than the similar overTonesKNPG, it can generate
the higher overtones than the B8 to the 20000 Hz. The first argument, the function f is intended to be a monotonic growing one though it can be not necessary. The second one parameter -- the g
function can be simply (*2), (*4) or something else and also is intended to be monotonic. | {"url":"https://hackage.haskell.org/package/algorithmic-composition-overtones-0.1.1.0/docs/Composition-Sound-Functional-OvertonesO.html","timestamp":"2024-11-03T13:48:35Z","content_type":"application/xhtml+xml","content_length":"17337","record_id":"<urn:uuid:a629257e-9d74-487e-9d7c-c7c4398e8b03>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00438.warc.gz"} |
Permanental processes from products of complex and quaternionic induced Ginibre ensembles
We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix
can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process,
generalizing the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic
expansions for the so-called hole probabilities, that a disk centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction
of complex eigenvalues occupies a disk of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the
induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.
• Non-Hermitian random matrix theory
• generalized Schur decomposition
• hole probabilities
• induced Ginibre ensembles
• overcrowding
• permanental processes
• products of random matrices
All Science Journal Classification (ASJC) codes
• Algebra and Number Theory
• Statistics and Probability
• Statistics, Probability and Uncertainty
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'Permanental processes from products of complex and quaternionic induced Ginibre ensembles'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/permanental-processes-from-products-of-complex-and-quaternionic-i","timestamp":"2024-11-13T06:12:45Z","content_type":"text/html","content_length":"50970","record_id":"<urn:uuid:6868bf3a-0036-400f-a5e6-d2a2df5d7303>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00140.warc.gz"} |
Production/clearing models under continuous and sporadic reviews
We consider production/clearing models where random demand for a product is generated by customers (e.g., retailers) who arrive according to a compound Poisson process. The product is produced
uniformly and continuously and added to the buffer to meet future demands. Allowing to operate the system without a clearing policy may result in high inventory holding costs. Thus, in order to
minimize the average cost for the system we introduce two different clearing policies (continuous and sporadic review) and consider two different issuing policies ("all-or-some" and "all-or-none")
giving rise to four distinct production/clearing models. We use tools from level crossing theory and establish integral equations representing the stationary distribution of the buffer's content
level. We solve the integral equations to obtain the stationary distributions and develop the average cost objective functions involving holding, shortage and clearing costs for each model. We then
compute the optimal value of the decision variables that minimize the objective functions. We present numerical examples for each of the four models and compare the behaviour of different solutions.
Bibliographical note
Funding Information:
Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
• Clearing policies
• Optimization
• Stationary distributions
ASJC Scopus subject areas
• Statistics and Probability
• General Mathematics
Dive into the research topics of 'Production/clearing models under continuous and sporadic reviews'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/productionclearing-models-under-continuous-and-sporadic-reviews","timestamp":"2024-11-08T01:33:14Z","content_type":"text/html","content_length":"55051","record_id":"<urn:uuid:fcee5b70-54f1-41bd-b5ed-81468cde3f26>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00564.warc.gz"} |
Adaptive Parallel Computation with CUDA Dynamic Parallelism | NVIDIA Technical Blog
Early CUDA programs had to conform to a flat, bulk parallel programming model. Programs had to perform a sequence of kernel launches, and for best performance each kernel had to expose enough
parallelism to efficiently use the GPU. For applications consisting of “parallel for” loops the bulk parallel model is not too limiting, but some parallel patterns—such as nested parallelism—cannot
be expressed so easily. Nested parallelism arises naturally in many applications, such as those using adaptive grids, which are often used in real-world applications to reduce computational
complexity while capturing the relevant level of detail. Flat, bulk parallel applications have to use either a fine grid, and do unwanted computations, or use a coarse grid and lose finer details.
CUDA 5.0 introduced Dynamic Parallelism, which makes it possible to launch kernels from threads running on the device; threads can launch more threads. An application can launch a coarse-grained
kernel which in turn launches finer-grained kernels to do work where needed. This avoids unwanted computations while capturing all interesting details, as Figure 1 shows.
Figure 1: A fluid simulation that uses adaptive mesh refinement performs work only where needed.
Dynamic parallelism is generally useful for problems where nested parallelism cannot be avoided. This includes, but is not limited to, the following classes of algorithms:
• algorithms using hierarchical data structures, such as adaptive grids;
• algorithms using recursion, where each level of recursion has parallelism, such as quicksort;
• algorithms where work is naturally split into independent batches, where each batch involves complex parallel processing but cannot fully use a single GPU.
Dynamic parallelism is available in CUDA 5.0 and later on devices of Compute Capability 3.5 or higher (sm_35). (See NVIDIA GPU Compute Capabilities.)
This post introduces Dynamic Parallelism by example using a fast hierarchical algorithm for computing images of the Mandelbrot set. This is the first of a three part series on CUDA Dynamic
• Adaptive Parallel Computation – Dynamic Parallelism overview and example (this post);
• API and Principles – Advanced topics in Dynamic Parallelism, including device-side streams and synchronization;
• Case Study: PANDA – how Dynamic Parallelism made it easier and more efficient to implement Triplet Finder, an online track reconstruction algorithm for the high-energy physics PANDA experiment,
which is part of the Facility for Antiproton and Ion Research in Europe (FAIR).
Case Study: The Mandelbrot Set
The Mandelbrot set is perhaps the best known fractal. It is defined as
$z_0 = c$
$z_{n+1} = z_n^2 + c$
$M = \{c \in \mathbb{C}:\exists\;R\;\forall\;n:|z_n|<R\}$.
The interior of Figure 2 (the black part), is the Mandelbrot Set.
Figure 2: The Mandelbrot Set.
The Escape Time Algorithm
The most common algorithm used to compute the Mandelbrot set is the “escape time algorithm”. For each pixel in the image, the escape time algorithm computes the value dwell, which is the number of
iterations it takes to decide whether the point belongs to the set. In each iteration (up to a predefined maximum), the algorithm modifies the point according to the Mandelbrot set equations and
checked to see whether it “escapes” outside a circle of radius 2 centered at point (0, 0). We compute the dwell for a single pixel using the following code.
#define MAX_DWELL 512
// w, h --- width and hight of the image, in pixels
// cmin, cmax --- coordinates of bottom-left and top-right image corners
// x, y --- coordinates of the pixel
__host__ __device__ int pixel_dwell(int w, int h,
complex cmin, complex cmax,
int x, int y) {
complex dc = cmax - cmin;
float fx = (float)x / w, fy = (float)y / h;
complex c = cmin + complex(fx * dc.re, fy * dc.im);
complex z = c;
int dwell = 0;
while(dwell < MAX_DWELL && abs2(z) < 2 * 2) {
z = z * z + c;
return dwell;
The value of dwell determines whether the point is in the set.
• If dwell equals MAX_DWELL, the point belongs to the set, and is usually colored black.
• If dwell is less than MAX_DWELL, the point does not belong to the set. We use the dwell value to color the pixel.
If the point does not belong to the set, lower dwell values correspond to darker colors. Generally, the nearer the point is to the Mandelbrot set, the higher its value of dwell. This is why in most
Mandelbrot set images the region immediately outside the set is bright, as in Figure 2.
A Per-Pixel Mandelbrot Set Kernel
A straightforward way to compute Mandelbrot set images on the GPU uses a kernel in which each thread computes the dwell of its pixel, and then colors each pixel according to its dwell. For
simplicity, we omit the coloring code, and concentrate on computing dwell in the following kernel code.
// kernel
__global__ void mandelbrot_k(int *dwells,
int w, int h,
complex cmin, complex cmax) {
int x = threadIdx.x + blockDim.x * blockIdx.x;
int y = threadIdx.y + blockDim.y * blockIdx.y;
if(x < w && y < h)
dwells[y * w + x] = pixel_dwell(w, h, cmin, cmax, x, y);
int main(void) {
// ... details omitted ...
// kernel launch
int w = 4096, h = 4096;
dim3 bs(64, 4), grid(divup(w, bs.x), divup(h, bs.y));
mandelbrot_k <<<grid, bs>>>(d_dwells, w, h,
complex(-1.5, 1), complex(0.5, 1));
// ...
This kernel runs quickly on the GPU; it can compute a 4096×4096 image with MAX_DWELL of 512 in just over 40 ms on an NVIDIA Kepler K20X accelerator. However, under closer examination, we can see that
such an algorithm wastes a lot of computational resources. There are large regions inside the set that could simply be colored black; note that since MAX_DWELL iterations are performed for every
black pixel, this is where the algorithm spends most of the computation time. There are also large regions of constant but low dwell outside the Mandelbrot set. In general, the only areas where we
need high-resolution computation are along the fractal boundary of the set.
The Mariani-Silver Algorithm
We can exploit the large regions of uniform dwell using the hierarchical Mariani-Silver algorithm to focus computation where it is most needed. This algorithm relies on the fact that the Mandelbrot
set is connected: there is a path between any two points belonging to the set. Thus, if there is a region (of any shape) whose border is completely within the Mandelbrot set, the entire region
belongs to the Mandelbrot set. More generally, if the border of the region has a certain constant dwell, then every pixel in the region has the same dwell. So, if dwell is uniform, it is enough to
evaluate it on the border, saving computation.
The Mariani-Silver algorithm combines this principle with recursive subdivision of regions with non-constant dwell, as described by the following pseudocode, and shown in Figure 3.
if (border(rectangle) has common dwell)
fill rectangle with common dwell
else if (rectangle size < threshold)
per-pixel evaluation of the rectangle
for each sub_rectangle in subdivide(rectangle)
Figure 3: The Mariani-Silver Algorithm recursively subdivides the Mandelbrot set.
This algorithm is a great fit for implementation with CUDA Dynamic Parallelism. Each call to the mariani_silver pseudocode routine maps to one thread block of the mandelbrot_block_k kernel. The
threads of the block use a parallel reduction to determine whether border pixels all have the same dwell. Provided that we launch enough thread blocks initially, this won’t decrease the amount of
parallelism available to the GPU. Thread zero in each block then decides whether to fill the region, further subdivide it, or to evaluate the dwell for every pixel of the rectangle.
Dynamic Parallelism Basics
In order to implement the Mariani-Silver algorithm with Dynamic Parallelism, we need to launch kernels from threads running on the device. This is simple because device-side kernel launches use the
same syntax as launches from the host. As on the host, device kernel launch is asynchronous, meaning that control returns to the launching thread immediately after launch, (likely) before the kernel
finishes. Successful execution of a kernel launch merely means that the kernel is queued; it may begin executing immediately, or it may execute later when resources become available.
Dynamic Parallelism uses the CUDA Device Runtime library (cudadevrt), a subset of CUDA Runtime API callable from device code. As on the host, all API functions return an error code, and the same best
practice applies: check for errors after each API call. On CUDA versions before 6.0, device-side kernel launches may fail due to the kernel launch queue being full. So it’s very important to check
for errors after each kernel launch.
To avoid writing lots of boilerplate code, we use the following macro to check errors. It checks the return code of a CUDA call, and, if an error has occurred, it prints the error code and location
and terminates the kernel. The information printed is useful for debugging (See How to Query Device Properties and Handle Errors in CUDA C/C++).
#define cucheck_dev(call) \
{ \
cudaError_t cucheck_err = (call); \
if(cucheck_err != cudaSuccess) { \
const char *err_str = cudaGetErrorString(cucheck_err); \
printf("%s (%d): %s\n", __FILE__, __LINE__, err_str); \
assert(0); \
} \
We wrap device CUDA calls with cucheck_err as in the following example.
Implementing the Mariani-Silver Algorithm with Dynamic Parallelism
Now that we know how to launch kernels and check errors in device code, we can implement the Mariani-Silver with Dynamic Parallelism. We use the following comparison code in a parallel reduction to
determine whether border pixels have the same dwell.
#define NEUT_DWELL (MAX_DWELL + 1)
#define DIFF_DWELL (-1)
__device__ int same_dwell(int d1, int d2) {
if (d1 == d2)
return d1;
else if (d1 == NEUT_DWELL || d2 == NEUT_DWELL)
return min(d1, d2);
return DIFF_DWELL;
The following kernel implements the Mariani-Silver algorithm with Dynamic Parallelism.
#define MAX_DWELL 512
/** block size along x and y direction */
#define BSX 64
#define BSY 4
/** maximum recursion depth */
#define MAX_DEPTH 4
/** size below which we should call the per-pixel kernel */
#define MIN_SIZE 32
/** subdivision factor along each axis */
#define SUBDIV 4
/** subdivision factor when launched from the host */
#define INIT_SUBDIV 32
__global__ void mandelbrot_block_k(int *dwells,
int w, int h,
complex cmin, complex cmax,
int x0, int y0,
int d, int depth) {
x0 += d * blockIdx.x, y0 += d * blockIdx.y;
int common_dwell = border_dwell(w, h, cmin, cmax, x0, y0, d);
if (threadIdx.x == 0 && threadIdx.y == 0) {
if (common_dwell != DIFF_DWELL) {
// uniform dwell, just fill
dim3 bs(BSX, BSY), grid(divup(d, BSX), divup(d, BSY));
dwell_fill<<<grid, bs>>>(dwells, w, x0, y0, d, comm_dwell);
} else if (depth + 1 < MAX_DEPTH && d / SUBDIV > MIN_SIZE) {
// subdivide recursively
dim3 bs(blockDim.x, blockDim.y), grid(SUBDIV, SUBDIV);
mandelbrot_block_k<<<grid, bs>>>
(dwells, w, h, cmin, cmax, x0, y0, d / SUBDIV, depth+1);
} else {
// leaf, per-pixel kernel
dim3 bs(BSX, BSY), grid(divup(d, BSX), divup(d, BSY));
mandelbrot_pixel_k<<<grid, bs>>>
(dwells, w, h, cmin, cmax, x0, y0, d);
} // mandelbrot_block_k
int main(void) {
// ... details omitted ...
// launch the kernel from the host
int width = 8192, height = 8192;
mandelbrot_block_k<<<dim3(init_subdiv, init_subdiv), dim3(bsx, bsy)>>>
(dwells, width, height,
complex(-1.5, -1), complex(0.5, 1), 0, 0, width / INIT_SUBDIV);
// ...
mandelbrot_block_k kernel uses the following functions and kernels.
• border_dwell: a function to check whether the dwell value is the same along the border of the current region; it performs a parallel reduction within a thread block.
• dwell_fill: sets each pixel within the given image rectangle to the specified dwell value
• mandelbrot_pixel_k: the same per-pixel kernel we used for our first implementation, but applied only within a specific image rectangle.
The complete code for these kernels and full code for rendering the Mandelbrot set image using the Mariani-Silver algorithm is available on Github.
Compiling and Linking
Figure 4: The Separate Compilation and Linking Process for Dynamic Parallelism.
To use Dynamic parallelism, you must compile your device code for Compute Capability 3.5 or higher, and link against the cudadevrt library. You must use a two-step separate compilation and linking
process: first, compile your source into an object file, and then link the object file against the CUDA Device Runtime. (See Separate Compilation and Linking of CUDA C++ Device Code.)
To compile your source code into an object file, you need to specify the -c (--compile) option, and you need to specify -rdc=true (--relocatable-device-code=true) to generate relocatable device code,
required for later linking. You can shorten this to -dc (--device -c), which is equivalent to combining the two options above.
nvcc -arch=sm_35 -dc myprog.cu -o myprog.o
To link the object file with a library, use the -l option with the library, in our case, -lcudadevrt.
nvcc -arch=sm_35 myprog.o -lcudadevrt -o myprog
Figure 4 illustrates the compilation process.
Note that the -rdc=true option is not necessary in this case, because nvcc generates an executable that does not need any further linking. You can also compile and link in a single step, in which
case you should specify the -rdc=true option.
nvcc -arch=sm_35 -rdc=true myprog.cu -lcudadevrt -o myprog.o
Conclusion: Higher Efficiency and Performance
Figure 5 compares the performance of the brute force per-pixel and Mariani-Silver algorithms for generating the Mandelbrot set when run on an NVIDIA Kepler K20X GPU at standard application clocks.
Figure 5: Performance comparison of a per-pixel Mandelbrot kernel and the recursive Mariani-Silver algorithm using CUDA Dynamic Parallelism; the number above the bars indicates the speedup achieved
by using dynamic parallelism.
Using dynamic parallelism always leads to a performance improvement in this application. The speed up is higher at higher resolutions and higher MAX_DWELL. This makes sense, because larger MAX_DWELL
means more computations we can avoid at each pixel, and higher resolution means larger regions, in terms of pixel count, where we can avoid computations. The overall speedup from using dynamic
parallelism varies from 1.3x to almost 6x.
Also note that using dynamic parallelism decreases the cost of increasing MAX_DWELL to improve the accuracy of the image. While the performance of the per-pixel version drops by 6x when increasing
dwell from 128 to 512, the drop is only 2x for the Dynamic Parallelism algorithm.
Using a hierarchical adaptive algorithm with Dynamic Parallelism to compute the Mandelbrot set image results in a significant performance improvement compared to a straightforward per-pixel
algorithm. You can use Dynamic Parallelism in a similar way to accelerate any adaptive algorithm, such as solvers with adaptive grids.
We’ll continue this series soon with a look at more advanced Dynamic Parallelism principles, such as synchronization and device-side streams, before concluding with a real-world case study study in
the third part of this series. | {"url":"https://developer.nvidia.com/blog/introduction-cuda-dynamic-parallelism/","timestamp":"2024-11-02T17:07:25Z","content_type":"text/html","content_length":"214824","record_id":"<urn:uuid:4b812144-8b10-405c-a4f2-2b8f3afdb247>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00812.warc.gz"} |
Graph modeling and model simplification techniques for fault analysis in sequential switching circuits
An algorithmic procedure is developed which can be used to generate a special form of the Petri Net graphs, called a CFN Logic Model, which describes the functional behavior of a digital switching
circuit. In order to represent the effects of physical faults in circuits, transformation procedures are developed which modify the structure of a CFN Logic Model. These structural modifications are
defined such that the circuit behavior that is represented by a resulting CFN Fault Model has a one-to-one correspondence with the behavior of the actual faulty circuit. Methods are developed to
estimate the structural complexity of a CFN Logic Model or Fault Model for a circuit as compared to the complexity of the circuit logic schematic. Reduced forms of the CFN Fault Models are used to
identify equivalence and covering relations between faults in digital switching circuits. This information is used to simplify any fault detection or fault diagnosis schemes that are derived for a
switching circuit.
Ph.D. Thesis
Pub Date:
□ Digital Computers;
□ Error Detection Codes;
□ Sequential Control;
□ Switching Circuits;
□ Algorithms;
□ Computer Programming;
□ Logic Circuits;
□ Logic Design;
□ Electronics and Electrical Engineering | {"url":"https://ui.adsabs.harvard.edu/abs/1976PhDT........75B/abstract","timestamp":"2024-11-02T21:34:12Z","content_type":"text/html","content_length":"35709","record_id":"<urn:uuid:7ea2b9dd-f773-4c30-a0af-7ddf2a11e118>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00600.warc.gz"} |
We introduce Rtapas (v1.1.1), an R package to perform Random Tanglegram Partitions (Balbuena et al. 2020). Rtapas applies a given global-fit method to random partial tanglegrams of a fixed size to
identify the associations, terminals, and nodes that maximize phylogenetic congruence. Incorporates ParaFit (Legendre et al. 2002), geodesic distances (GD) (Schardl et al. 2008) and PACo (Balbuena et
al. 2013) as global-fit methods to implement Random TaPas. Rtapas further enhances the usability and implementation of Random TaPas by including functions (i) to facilitate the prior processing of
association data between taxa, (ii) to estimate, in a set of probability trees, the statistic of a given global-fit method, (iii) to estimate the (in)congruence metrics of the individual
host-symbiont associations, and (iv) to compute either conventional (G) or normalized (G*) (Raffinetti et al. 2015) characterizing the distribution of such metrics.
You can install the development version of Rtapas from GitHub with:
This is a basic example which shows you how to solve a common problem:
N = 1e+2 # we recommend using 1e+4.
n = n = round(sum(np_matrix)*0.2)
NPc <- max_cong(np_matrix, NUCtr, CPtr, n, N, method = "paco",
symmetric = FALSE, ei.correct = "sqrt.D",
percentile = 0.01, res.fq = FALSE,
strat = "parallel", cl = 10)
tangle_gram(NUCtr, CPtr, np_matrix, NPc, colscale = "sequential",
colgrad = c("darkorchid4", "gold"), nbreaks = 50, node.tag = TRUE,
cexpt = 1.2, link.lwd = 1, link.lty = 1, fsize = 0.75, pts = FALSE,
ftype ="i") | {"url":"https://cran.rstudio.org/web/packages/Rtapas/readme/README.html","timestamp":"2024-11-09T17:08:06Z","content_type":"application/xhtml+xml","content_length":"9198","record_id":"<urn:uuid:f1b05921-ca18-4553-a420-941af5ff4dc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00626.warc.gz"} |
Computing Reviews, the leading online review service for computing literature.
The mathematics requirements for computer science (CS) students have been debated for decades. I began teaching in a CS program in 1983, and I recall similar discussions at that time. The debate has
continued in one form or another since then. This report reveals the current status of the issue. It is comprehensive and thorough, having been based on data from many institutions, including site
visits. Both BS and BA degrees and ABET accredited and non-accredited programs are included. The report has detailed tables showing the use of calculus and discrete mathematics courses as pre-/
co-requisites for CS courses.
Several important areas are investigated: (1) the math pre-/co-requisites for entrance into initial CS1 and CS2 courses; (2) the math pre-/co-requisites for subsequent required courses in the
curricula; and (3) the relationship between calculus and discrete mathematics in the pre-/co-requisite structure.
The study reports many major lessons: (1) math requirements and their placement in the curricula vary widely among the institutions studied; (2) there is no consensus on the use of calculus and/or
discrete mathematics as pre-/co-requisites for CS1 and CS2; (3) there is no consensus on whether calculus should be a prerequisite for discrete mathematics; (4) there is no clear idea of what should
be included in a discrete mathematics course or who should teach it (the math department or the CS department); and (5) math courses should be pre-/co-requisites for later CS courses only if the
content of these math courses is needed for success in the CS courses.
The article concludes with three principal recommendations:
(1) CS programs should not require calculus as a pre-/co-requisite for CS1, CS2, or discrete mathematics. This recommendation does not deny the utility of calculus, but rather where it may appear in
the curriculum.
(2) The progressions through CS courses and mathematics courses should be decoupled. Complete decoupling would be impossible if discrete mathematics were a pre-/co-requisite for data structures and
algorithms. There may also be other points where the pre-/co-requisite structures in CS and mathematics become entwined. The fewer entanglements, the better.
(3) CS departments should develop substitute courses that teach the mathematics they need instead of depending on the mathematics department to teach them. Implementation of this recommendation will
depend on practical matters of faculty time and institutional politics.
In the course of my academic career, I was a department chair, a dean of science, and a member of the ABET Computing Accreditation Commission (CAC) board. Because of my experience, this article and
its concluding recommendations succeeded admirably in stimulating thought. If I were still a department chair, I would have made copies of the report and distributed it to all of my faculty members
to study before the next department meeting. | {"url":"http://www.computingreviews.com/review/review_review.cfm?review_id=147825&listname=todaysissuearticle","timestamp":"2024-11-11T18:21:45Z","content_type":"text/html","content_length":"39310","record_id":"<urn:uuid:7ca74fab-609b-49de-826e-2fe033055e3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00633.warc.gz"} |
How Important Is Trigonometry In Calculus? - Paisley Scotland
A Short Introduction To CBD Honey
November 24, 2020
How to Use Hemp Flower
November 24, 2020
Trigonometry is a specialised branch of mathematics that deals with the study of triangular sides and angles. At the same time, calculus is a branch of mathematics which keeps track of continuous
change of values in variables. Calculus tries to use mathematics in solving problems raised in a dynamically changing environment, and trigonometry supports this by providing ease in huge
calculations involved. These two fields are used to derive miraculous results when used together.
Trigonometry is misunderstood as a complex one while it simplifies the otherwise complex and repetitive tasks in calculus. It carried an important place in calculus because it simplifies the
calculations, provides accurate and precise calculations. We would be reading more about its use and applications in specialised fields as well as daily life in the following article.
What is Trigonometry and How Does it Simplify Calculus?
Trigonometry is a study of triangular length, height, and angles. There are three sides of a triangle aligned at a different angle. All angles of the triangle should sum up to 180 degrees to qualify
as a triangle. The length of the sides and angles decide on the type of triangle.
Trigonometry replaces normal variables used in calculus. It provides simplicity of using angles instead of sides or vice versa as per the parameters at hand. This can be best explained with the help
of an equation of a unit circle. The unit circle is a circle with a radius of 1 unit. The equation of the unit circle can be written in terms of the point of intersection with the x-axis, and the
point of intersection with the y-axis.
Equation of circle : x^2 + y^2 = 1
Here coordinates (x,y) are the points on the circumference of a circle which when connected with the centre of a circle forms a radius. As per the equation above, the sum of the square of x and y
coordinates gives a radius of a circle. Now see another equation:
Equation of circle using trigonometric functions: Cos^2 (A) + Sin^2 (A) = 1.
Here angle “A” is an angle of radius drawn between the (x,y) coordinates and the centre of the circle. Even if we are not aware of (x,y) coordinates and know the angle, the trigonometric function can
be used.
Similarly, there are various graphs associated with trigonometric ratios which can also be used during calculations in calculus rather than getting into finding infinitely small values to reach the
Excited for some hands-on experience of using trigonometry in calculus? Here is an example along with the explanation for you!
Example: Use calculus to balance the below equation in terms of trigonometric functions. The equation below is also an equation of derivatives in the standard calculus.
F`(x) = lim h-> 0 { ( f(x+h) – f(x) ) / h }
Use sin (x) in the place of f (x) in RHS.
F`(x) = lim h-> 0 { ( sin(x+h) – sin(x) ) / h }
Appling the property of expansion of function sin (x+h)
F`(x) = lim h-> 0 { ( cos (x) . sin(h) + cos (h) . sin(x) – sin(x) ) / h }
Taking sin (x) common in the equation.
F`(x) = lim h-> 0 { ( cos (x) . sin(h) + (cos (h) – 1 )sin(x) ) / h }
Diving the whole equation into two parts with the limit of “h” tending to zero in both parts.
F`(x) = lim h-> 0 { ( cos (x) . sin(h) } + lim h-> 0 { (cos (h) – 1 )sin(x) ) / h }
F`(x) = cos(x) * 1 + sin(x) * 0
Use sin (x) in the place of f (x) in LHS.
F`(sin (x) ) = cos (x).
RHS = LHS
Explanation: Let’s talk about the equation first. This equation explains the concept of continuous change which is captured in calculus. The LHS and RHS contained variables as a function of “x”. This
represents a linear value that keeps changing as per the function defined. The calculations would have been far more complex if the values are used because of limitations in the use of values after
the decimal point in a number. This shortcoming was eliminated with the use of trigonometric functions. The properties and rules of trigonometric functions simplify the calculations of such detail.
The above example is easy to solve and understand as there are no complex calculations involved.
Real-Life Applications
Trigonometry has reduced the complications of calculus to produce some real-life applications:
1. 1. Calculus is used to determine the impact of temperature, humidity, light, food source and the factors responsible for the rate growth of bacteria. These are extensive-time taking
calculations involving various factors to be considered. Instead of using normal variables, trigonometric functions can be used as an abstraction layer. Trigonometric functions eliminate the
need to calculate each small step, giving biologists time to concentrate on optimising the calculations.2. Imagine you are a part of a crime scene investigation where two cars collided. You
are trying to establish a pattern out of the trajectory, speed, direction of the cars involved in such accidents. Calculus can be used here to calculate near to accurate values by aggregating
the values obtained. These calculations can then be further optimised with the help of trigonometric functions.3. You all might have spent childhood playing video games, but now the overall
presentation of video games has developed a lot.
Multiple graphics and live effects are added to give you real life-like experience. This is possible with the use of calculus in
designing responsive three-dimensional objects. These objects respond to all rapidly changing activities of a player. The response in such games is very precise by reaching the accurate
directions, time, length, and other measurements involved which are derived with the help of trigonometry
2. 4. Trigonometry and calculus are used in combination to calculate the distance of moving celestial bodies. The movement of celestial bodies relative to each other and earth is used by
mathematical astronomers as well. Mathematical astronomers use the changes in planetary positions for predictions. One of Its uses can be found in the Copernican’s revolutionary model named
“Sun-centered geometrical model”5. You might have come across multiple research papers and studies published by scholars about algae and photosynthesis in algae, but ever wondered how these
marine engineers and scientists use calculus while studying algae? They capture changes for a long period of time and use differential and integrational calculus to derive insightful
information. They use trigonometry in calculus to calculate the depth of sunlight in the ocean at different levels, which determines the rate of photosynthesis reaction in algae.6. Calculus
has been traditionally used in medical science. It is used to track the changes in a patient’s health over a period of time on the basis of different factors. Small changes in the nature of a
virus can be tracked with the help of calculus. The pace of spread, impact on health, severity, and control measures can be determined using calculus coupled with trigonometry. Such changes
in the biological composition of any bacteria or virus can cause havoc if not properly administered. Well, no doubts about this in the era of COVID-19!
Some Useful Tips
Here are some tips for you:
1. Read through the problem statement thoroughly and determine what is provided while what is to be determined. For example, in some cases, angles are to be determined while in
others, it is the length.
2. Draw all possibilities in diagrams. Keep eliminating the diagrams which further do not meet your criteria.
3. Visualise the rotating triangles in complex motion-related calculus questions. For example, suppose you rotate a triangle with one angle as the centre point. In that case, other
angles lie on the circumference of a circle if the triangle is an “equilateral triangle”. Try this drawing on the piece of paper to get more insights. Isn’t it an interesting
4. Apply rules and laws wherever possible. Try to find out if a complex calculative equation can be reduced with the help of trigonometric ratios and laws.
5. Rearrange equations to find the possibility of applying formulas.
6. Always validate your answers with the help of geometrical diagrams. For example, you got 0 degrees as an answer, but is it practically possible to have zero degrees angle in a
Trigonometry is like a toolkit in calculus with the latest tools to solve your mathematical dilemma. It has an elegant way of solving complex calculations with the help of standardised rules.
Trigonometric ratios are used to solve geometrical, calculative, measurement, and differentiation-related mathematical problems. The reason it becomes tough to digest for students is that
they are not able to visualise triangles in trigonometry, and are unable to apply rules of trigonometry even when they know it.
If you are ready to expand your horizons of visualisation and look for simple tools to solve complex calculus problems, then trigonometry is the right match for you. If you are looking for
partners with whom you can integrate your success and differentiate your failures, then Cuemath is the perfect platform for you. | {"url":"https://www.paisley.org.uk/2020/11/how-important-is-trigonometry-in-calculus/","timestamp":"2024-11-03T19:14:31Z","content_type":"text/html","content_length":"168505","record_id":"<urn:uuid:bdd902e6-707a-4324-b07a-4888776e4f21>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00727.warc.gz"} |
Proof of Quadratic Formula Instructional Video for 8th - 11th Grade
Curated and Reviewed by Lesson Planet
Armed with their newfound (or refreshed) knowledge of the quadratic formula, viewers can now work with Sal to prove the formula - an important skill that is often ignored in algebra. Once they learn
how to both use and prove the quadratic formula, mathematicians will have very little trouble correctly solving these equations.
Resource Details
8th - 11th
2 more...
Resource Type
Media Length
1 more...
Instructional Strategy
Usage Permissions
Creative Commons
BY-NC-SA: 3.0
Start Your Free Trial
Save time and discover engaging curriculum for your classroom. Reviewed and rated by trusted, credentialed teachers.
Try It Free | {"url":"https://www.lessonplanet.com/teachers/proof-of-quadratic-formula","timestamp":"2024-11-08T15:41:17Z","content_type":"text/html","content_length":"101705","record_id":"<urn:uuid:29b36064-d774-46e2-8cd7-8b7b556e7e61>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00121.warc.gz"} |
Wolfram|Alpha Examples: Step-by-Step Applications of Calculus
Examples for
Step-by-Step Applications of Calculus
Unlock the practical applications of calculus through Wolfram|Alpha's step-by-step instructions, where you'll delve into essential concepts such as finding tangent lines, identifying inflection
points, optimizing functions, calculating average values, determining arc lengths and measuring areas between curves. Whether you're a student aiming to grasp real-world calculus applications or a
professional seeking to utilize calculus in your field, Wolfram|Alpha equips you with the skills needed to tackle these diverse and valuable calculus applications.
Tangent Lines & Planes
Find a line tangent to a curve using the derivative at a point:
Calculate a multivariate tangent plane by examining partial derivatives:
Explore step by step a variety of tests to discover the extrema:
Compute the local maxima or minima of a function:
Show the steps for finding the global minima or maxima of a function:
Discover how to calculate the extrema of a function on a given interval:
Arc Length
Calculate the arc length of a curve step by step:
Specify a curve in polar coordinates:
Specify a curve parametrically:
Inflection Points
Calculate inflection points step by step:
Average Value
Find the average value of an equation on an interval:
Look through the steps to find the average value of a derivative on an interval:
Area between Curves
Learn how to compute the area between curves: | {"url":"https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus/step-by-step-applications-of-calculus","timestamp":"2024-11-12T09:01:46Z","content_type":"text/html","content_length":"89532","record_id":"<urn:uuid:98eaff61-2646-4b7b-8bdf-ded22ab91dab>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00791.warc.gz"} |
ADS1202: Measuring static parameters
Answers 1 answer
Part Number: ADS1202
Hello, thank you very much for help with ADS1202. I`ve got a good spectrum.
But I have a question about measuring static parametres of this ADC. How should I get points for transfer function to measure INL? Are they the mean values of some convertions of one voltage level?
Also what is procedure for measuring DNL?
Hi Maria!
Glad to hear that you've got a clean spectrum now! For INL and DNL, you could use a ramp waveform that covers the FS- to FS+ input range. INL is the deviation of an output code from the ideal
mid-point in the transition step. DNL is the deviation of the code width output versus the analog input. Please take a look at this TI Precision Labs video for more details.
Hello, Tom, thanks for reply!
I usually use this method to form a ramp waveform to get the transfer function:
1) set up the min voltage level = -FS on analog input
2) make a conversion
3) increase voltage level on analog input on step=76uV (the least step I can make)
4) make another conversion
and so on up to the FS.
Is it appropriate in case of sigma-delta ADC? Which step should I use to make ramp? How many points were taken to get INL simular to the one from datasheet?
Hi Maria,
Yes, you can use that method as well, however the ideal measurement would need a source with greater resolution than the ADC itself. What is the source of your input signal? The ADS1202 shows 16-bit
resolution so your LSB size is actually about 10x smaller than your smallest step available to the input. you may be able to extrapolate some information. I do not know how many points were taken for
the plots in the datasheet. What you would want to do is average 100 points or more for each input step and see what you get. | {"url":"https://e2e.ti.com/support/data-converters-group/data-converters/f/data-converters-forum/1095503/ads1202-measuring-static-parameters?tisearch=e2e-sitesearch&keymatch=ADS1202","timestamp":"2024-11-13T11:45:42Z","content_type":"text/html","content_length":"144333","record_id":"<urn:uuid:dd576762-aa71-433b-98e5-55a00a6426dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00113.warc.gz"} |
Cryptographic method for securely exchanging messages and device and system for implementing this method
Cryptographic method for securely exchanging messages and device and system for implementing this method
At least one embodiment refers to a method for securely exchanging messages between at least two devices, each of them storing a shared secret key. The method comprises: at each device: generating a
random number, then sending it to the other devices; determining a first key by a first operation based onto said secret key and each random number; determining a second key based on said first key
and said random numbers; at a sending device: determining a pseudo message on the basis of the message and said random numbers; calculating then sending a cryptogram on the basis of said pseudo
message and said second key; and at the receiving device: decrypting said cryptogram by means of said second key; and retrieving said message from said pseudo message.
Latest NAGRAVISION SA Patents:
This application is a U.S. National Application which claims priority from European Patent No. EP 14172225.6 filed Jun. 12, 2014.
At least one embodiment relates to the field of data transfers between devices connected together, involving cryptographic operations for securely sending and receiving any kind of messages that have
to be exchanged between these devices.
There are a lot of known methods involving cryptographic algorithms, such as the Data Encryption Standard (DES) or the Advanced Encryption Standard (AES), for encrypting and decrypting data to be
transmitted via unsecured channels or networks connecting electronic devices of any kind. To this end, such devices are provided with cryptographic components performing cryptographic operations to
scramble messages so as to make them unintelligible without a secret decryption key. These components are typically implemented according to the CMOS technology (Complementary Metal Oxide
Semiconductor technology). Cryptographic algorithms implemented in such components are generally safe enough from a mathematical point of view. However, the fact that such an algorithm is physically
implemented by integrated circuits built with interconnected transistors for producing the logical functions of this algorithm, generates observable physical quantities. The observation of such
quantities can be carried out by means of an oscilloscope, for instance for monitoring the power consumption of the integrated circuit. Sudden power consumption variations appear as peaks on the
screen of the oscilloscope. Each peak can for instance identify the start of a so-called “round”, typically in algorithm such as DES and AES in which an input message to encrypt is applied to a
succession of groups of operations called “rounds”. According to such an algorithm, each round is placed under the control of a sub-key resulting from the previous round. Therefore, such an algorithm
involves a series of sub-keys which are derived from a secret key used as initial key within the algorithm. In the event where this initial secret key is known by a malicious person, the latter
becomes able to decrypt and properly encrypt any message exchanged with a corresponding device that uses the same algorithm with the same secret key according to a symmetrical encryption scheme.
There are several ways to attack a cryptographic circuit for recovering the initial secret key. Some attacks are known as non-invasive attacks since they aim to observe the power consumption, the
electromagnetic emanation or the processing time of the circuit. Other attacks are referenced as invasive attacks, since they involve modifying the circuit, in particular its behavior during a short
lapse of time. In this last category, one knows the Differential Fault Analysis (DFA) as being a serious threat against any encryption/decryption system. Differential Fault Analysis is based on the
observation and the comparison of the outputs provided by a cryptographic circuit under two different states. One of these states corresponds to the normal operation of the circuit, whereas the other
is obtained by voluntarily injecting a fault aiming to alter one or several bits by switching from 0 to 1 or vice versa. Such a physical bit inversion can be carried out e.g. by sweeping the surface
of the integrated circuit with a laser beam. By locating sensitive areas within the cryptographic circuit, laser shots allow disrupting the behavior of the circuit in an accurate and easy manner,
since they can be implemented under the control of a computer, while acting with a very good spatial and temporal resolution. When several faults are injected during the processing of a cryptographic
algorithm, the analysis of erroneous outputs allow to guess the secret key by observing fault propagations within the algorithm.
Accordingly, there is a need to provide an efficient solution allowing to prevent attackers guessing the secret key through any differential fault analysis, or more generally to guess such a key
through information gained by any kind of analysis.
The aim of at least one embodiment is to solve, at least in part, the aforementioned drawbacks. To this end, at least one embodiment suggests a cryptographic method and a device for securely
exchanging data between at least two devices, involving the implementation of a cryptographic process which is particularly complex. According to at least one embodiment, the secret key, which is
shared by all of the devices of the same system as a symmetrical key, is never directly used as encryption/decryption key of the exchanged messages. Indeed, the key that is used to encrypt/decrypt
the messages exchanged between the devices of a same system always depend on a plurality of random numbers, in particular. More specifically, each device generates at least one random number which is
taken into account for determining the key that is used for encrypting/decrypting the exchanged messages. Accordingly, if the system comprises three devices, the aforementioned key will depend on at
least three random numbers.
Moreover, an additional key level is determined before encrypting/decrypting the message to be exchanged. Accordingly, the present method involves three key levels for encrypting/decrypting the
messages. In addition, the message to exchange is never directly used as input data of the algorithm for generating the cryptogram that has to be sent, but it is always used with each of the random
numbers to first generate a pseudo message that will be then encrypted by the aforementioned cryptographic algorithm.
Preferably, the random numbers are renewed each time a message has to be exchanged. Accordingly, at least one embodiment prevents any malicious person to guess the shared secret key through any
attack involving a differential fault analysis. Furthermore, thanks to the complexity provided both by the pseudo message and the derived key used for the encryption, the cryptographic method of at
least one embodiment reaches a particularly high level of security.
The aim and the advantages of at least one embodiment are achieved thanks to the cryptographic method consistent with the subject-matter of claim 1 and thanks to a device consistent with the
subject-matter of claim 11.
Other advantages and embodiments will be presented in the following detailed description.
At least one embodiment will be better understood thanks to the attached figures in which:
FIG. 1 depicts an overview of the system according to at least one embodiment,
FIG. 2 is a flowchart showing at least one embodiment of the cryptographic method,
FIG. 3 shows an alternative of an extract of the flowchart of FIG. 2,
FIG. 4 is a schematic representation of one of the devices of the system shown in FIG. 1.
Referring to FIG. 1, the latter schematically shows an overview of a system embodiment in which the method and a plurality of devices of the present can be implemented. The communication system shown
in this Figure shows three devices D1, D2, D3 connected together through any manner. It should be noted that the number of devices D1, D2, D3, etc. . . . is unlimited and the system illustrated in
this Figure is taken as one example among many other possibilities, both in terms of connection or of number of devices. Such a system could include two devices only, connected together either via a
network, such as the Internet, or through any other kind of connection (wired or wireless), in particular an unsecured connection.
Each device D1, D2, D3 can exchange messages M with at least one other device, preferably with any other device in the system. As these messages M are securely exchanged, they have been illustrated
in this Figure by envelopes, each stamped with a padlock. To encrypt or decrypt secured messages M, each device D1, D2, D3 must handle at least three cryptographic keys K, K1, K2. One of these keys
is a shared secret key K common to all of the devices D1, D2, D3 of the system. This secret key K can be implemented during the manufacturing of the device D1, D2, D3 or its related chipset, or
afterwards during their personalization stage or during an initialization phase.
As schematically shown in this Figure, each device sends and receives other data denoted R1, R2, R3. Such data refer to random numbers. Each device (e.g. D1) generates one random number (R1) which is
sent to the other devices (D2, D3) and receives the random number (R2, R3) generated by each of the other devices (D2, D3). On the basis of the overview provided by FIG. 1, the method for securely
exchanging messages M between at least two devices will be described in detail with reference to FIG. 2.
For the sake of simplicity, FIG. 2 discloses, step by step, the method of at least one embodiment while referring to a system comprising two devices only, D1 and D2, respectively identified by the
reference numerals 10, 20. On this figure, the steps performed by each of these devices are shown in several columns and follow one another from top to bottom. The common steps which are performed
both by each of the devices are represented in a central column. It should be noted that these common steps are carried out by each device in an individual manner. There is no requirement to process
the common steps simultaneously within each involved device for exchanging messages.
As already mentioned, each device D1, D2, comprises a shared secret key K common to all the devices wanting to mutually exchange messages. This secret key K is shown in box 31 of FIG. 2. In this
embodiment, the device D1 is intended to send a message M to the device D2. Accordingly, the first device D1 corresponds to the sending device and the second device D2 corresponds to the receiving
device. Although there is only one receiving device shown in this Figure, it should be understood that the same message M could be sent from the sending device to a plurality of receiving devices. At
box 11, the sending device D1 has to prepare or retrieve the message M that has to be sent. Such a message M can refer to any kind of data, but usually it will refers to sensitive data, whose nature
mainly depends on the type of devices involved in the communication system in question.
Each device D1, D2 generates a random number before to send it to the other device, in particular to a plurality of selected devices or to all of the other devices in case the system comprises more
than two devices. This step is shown at boxes 12, 21, where the sending device D1 generates a first random number R1, which is sent to the receiving device D2, and the latter generates a second
random number R2 which is sent to the sending device D1. Performing a mutual exchange of the random numbers with each devices can be achieved even if these devices did not beforehand agreed to
exchange an upcoming message, for instance by means of a specific signal recognized by these devices during a prior step. In this case, one could be expected that the mere fact of receiving a random
number R1 (i.e. data that can be identified as such, either through a specific identifier, or by means of a particular format) can be recognized by the receiving device(s) as being a trigger signal
which informs that a message M must be received from the sending device. Accordingly, each device becomes fully able to run the required steps of the present method in due time.
Moreover, in the case where the system involves more than two devices, as shown in the example of FIG. 1, one can further provide means to identify the sending device at the receiving device, if
necessary. If the communication is not still established between the sending device and the receiving device(s) e.g. during a current session, a possible way could be to identify the address of the
sending device or to transmit the identifier (ID) of the sending device towards the receiving device. This can be achieved, for instance by appending, to the random number R1, the ID number belonging
the sending device D1 or by including such an ID in any other data.
At box 33, each device D1, D2, determines a first key K1 by calculating a first operation OP1 which uses both the shared secret key K and each random number R1, R2 as operands. In the illustration
provided by FIG. 2, this first operation OP1, as well as other subsequent operations, refers to an exclusive OR operation, as a non-limitative example. In accordance with a preferred embodiment and
as shown in this box 33, the result of the first operation OP1 is directly used as first key K1.
At box 35, each device D1, D2, subsequently calculates a second operation OP2 that uses at least each random number R1, R2 as operands. Then, on the basis of the result of this second operation OP2,
each device D1, D2 further determines a second key K2. In accordance with the example of box 35, this is carried out by encrypting the result of the second operation OP2 by means of a first
algorithm, denoted A1, which uses the first key K1 as encryption key. Accordingly, the second operation, or directly its result, is input into the first algorithm A1 together with the required first
cryptographic key K1. In response, this first algorithm provides the second cryptographic key K2 as output.
At box 14, the device acting as sending device D1 calculates a third operation OP3 which uses both the message M and each random number R1, R2 as operands. By this way, the sending device D1
determines a so-called pseudo message M′ given that it is based onto the message M, but it looks different from the initial message M, although the latter has still not being encrypted.
At box 16, the sending device D1 calculates a cryptogram C which results from the encryption of the pseudo message M′. To this end, it uses the pseudo message M′ as input of a second algorithm A2
together with the second key K2 as encryption key.
At box 18, the cryptogram C is transmitted by the sending device to at least one other device acting as receiving device.
When the receiving device D2 obtains the cryptogram C, it is able to decrypt it by means of the same algorithm A2 and the same key K2, as shown at box 23. To this end, the second algorithm A2 will
be, or will include, a two-way function that can be reverted (see the notation A2^−1 on FIG. 2). Of course, the same algorithm has to be used both by the sending and the receiving devices. According
to the preferred embodiment, the second key K2 is used as direct or indirect decryption key of the second algorithm. The use of second key K2 as indirect key will be described with reference to FIG.
3. In any case, the decryption of the cryptogram C allows to retrieve the pseudo message M′ as a result of the second algorithm A2.
Finally, at box 25, each receiving device D2 retrieves the message M in its initial plaintext form, from the pseudo message M′ by reversing the third operation OP3 (see the notation OP3^−1 on FIG.
It should be noted that the first algorithm A1 can be different or identical to the second algorithm A2. However and contrarily to the second algorithm, the first algorithm can use a one-way function
(or it may be itself such a function) that provides the second key K2. Accordingly, such a second key K2 could be the digest of a hash function or could be derived from such a function, for instance.
Whatever the algorithms (A1, A2) used in this method, they must be the same for all devices who want to exchange messages M. These algorithms can be implemented within each device through different
ways, for instance during the manufacturing of the devices, during their personalization or during an initialization phase.
Referring now to FIG. 3, this Figure shows the last steps of the method illustrated in FIG. 2, where the box 37 represents an additional step as an alternative of the previous flowchart. This variant
corresponds to the case where the second key K2 is used as indirect encryption/decryption key within the second algorithm A2. To this end, a third key K3 is determined, at each device D1, D2, by a
fourth operation OP4 which uses both the second key K2 and the shared secret key K as operands. As shown at box 37, the result of this fourth operation OP4 provides the third cryptographic key K3.
In a similar way as for the algorithms, all of the operations OP1, OP2, OP3, OP4, or some of them, can be implemented within each device during the manufacturing of the devices, during their
personalization or during an initialization phase.
As for the sending device D1, the step shown at box 37 is carried out between the steps of boxes 35 and 16, since it needs the second key K2 (determined by the step of box 35) and the result of this
additional step will be used with the second algorithm A2 (during the step shown at box 16). As for the receiving device(s) D2, this additional step is carried out between the steps of boxes 35 and
23 for the same reasons.
As shown in FIG. 3, the use of the second key K2 into the second algorithm A2 (i.e. within the steps of boxes 16 and 23) has been substituted by the third key K3. This results from the fact that the
second key K2 is used in an indirect manner in these steps. For this reason, the reference numerals of these two boxes have been respectively amended into 16′ and 23′ in FIG. 3.
It should be noted that certain steps shown in FIG. 2 or FIG. 3 could be placed in a different order. For instance, the steps of box 14 could be carried out any where between the exchanges of the
random numbers R1, R2 (at boxes 12, 21) and the encryption of the pseudo message M′ (at box 16, 16′). The same principle applies for the steps of box 37, as explained before.
According to one embodiment, at least part of at least any of the operations OP1, OP2, OP3, OP4 involves a logical operation (Boolean algebra). More particularly, this logical operation is an
exclusive OR operation (see the symbolic notation in FIGS. 2 and 3). It should be noted that other logical functions (i.e. basic and/or derived operations) could be used instead of the XOR operator
or with the XOR operator.
According to another embodiment, at least a part of at least any of the operations OP1, OP2, OP3, OP4 involves a number raised to a power. In this case, any of the operands of the relevant operation
is used as an exponent of this number which is chosen among the other operands of this operation.
To perform logical operations, the involved operands must have the same digit number. In other words and since the operations refer to binary operations, the operands must have the same bit length.
Therefore and depending on the type of operation carried out e.g. in box 33 (OP1), both the bit length of the random numbers R1, R2 and the bit length of the shared secret key K should be the same.
Regarding to the second operation OP2 as shown in the example of box 35, the random numbers R1, R2 must have the same bit length. The same principle applies to the third and fourth operations
regarding both the random numbers R1, R2 and the message M, on the one hand, and the cryptographic keys K2, K, on the other hand.
For this reason, if the operands of any one of the operations OP1, OP2, OP3, OP4 have different bit lengths, then the present method can further comprise a step aiming to restore the same bit length
for each of these operands. To this end, restoring the same bit length can be achieved by several different manners.
According to at least one embodiment, that can be achieved by a “balancing step” aiming to supplement the operand having the smallest bit length until its bit length is equal to the bit length of any
of the other operands of the relevant operation. Then, this balancing step can be repeated until all the operands of the relevant operation have the same bit length. The step aiming to supplement the
operand can be achieved by a succession of bits 0, by a succession of bits 1, even by a succession of a specific combination of these two bits 0 and 1. Of course, the selected bit succession must be
known both by the sending device and the receiving device(s), through any process mentioned before, for instance during the personalization of the devices or their chipsets.
In variant, this balancing step could be achieved by supplementing the operand having the smallest bit length until the bit length of the other operand (i.e. preferably the operand having the longest
bit length) is equal to a multiple of the bit length of the supplemented operand.
According to another embodiment, the so-called balancing step can be first performed by concatenating the operand having the smallest bit length with itself, until reaching the same bit length as the
other operand. This approach implies that the operand which has the longest bit length is a multiple of the other operand (i.e. the concatenated operand). In the case where one operand is not exactly
a multiple of the other operand, the aforementioned concatenation can be performed until reaching a bit length reduced by a residual value less than the bit length of the concatenated operand. This
residual bit length corresponds to the remainder of the Euclidean division where the operand having the longest bit length is the dividend and the operand to concatenate is the divisor. Then, the
residual bit length (i.e. the residual value) can be supplemented by any succession of bits, as explained above.
As examples of one of these embodiments applied in particular to the third operation OP3, restoring the same bit length can be achieved for each of said random numbers R1, R2 by concatenating said
random number with itself, until reaching the same bit length as that of the message M. This embodiment involves that the random numbers R1, R2 have the same bit length and that the bit length of the
message M is a multiple of that of one of the random number. If this latter condition is not fulfilled, then the residual bit length can be supplemented as already explained.
In variant and while still referring to the third operation OP3, restoring the same bit length could be achieved first by supplementing the message M until its bit length is equal to a multiple of
the bit length of any of the random number R1, R2, then by slicing the supplemented message M into blocks having the same bit length as the bit length of the random number before using each of these
blocks as a new message (M) to be processed by the steps of the present cryptographic method.
According to another embodiment and for the sake of simplification, the cryptographic keys used in the present method, preferably at least the second key K2 and the shared secret key K, have the same
bit length. For the same reason, all the random numbers R1, R2 have also the same bit length.
Advantageously, by generating a random number at each device and by using all of the generated random numbers both for deriving the cryptographic key K2, K3, that is used for calculating the
cryptogram C, and for determining the pseudo message M′ to encrypt, the subject-matter of at least one embodiment significantly increases the security applied to the exchanged messages M.
Still advantageously, even if one of the random numbers is guessed by a malicious person, the latter will be unable to deduce the key that has been used for encrypting the pseudo message M′.
Furthermore, even if that key could be discovered by such a person, he would still unable to retrieve the initial message M from the pseudo message M′, given that to recover the original message M,
such a person first needs to possess all the random numbers and then he must know what is the third operation (OP3) undertaken in the method. This also requires be aware of all the operators used in
this operation, and even to know the order of each operator and each operand used within this operation, depending on the nature of this operation.
Still advantageously, the shared secret key K is never directly used as cryptographic key in any one of the cryptographic algorithms A1, A2 implemented in the present method. In contrast, the shared
secret key K is only used within mathematical operations (OP1, OP4) whose results are subsequently used as keys into these algorithms. Accordingly, the shared secret key K is never directly exposed
at the first plan, within a cryptographic algorithm.
Preferably, the steps of the present method are undertaken each time a message M has to be exchanged. This can be applied whatever the embodiment of the method. Accordingly, the random numbers
generated by each device have a single use, given that a new random number is generated, by each device, each time a new message has to be sent. Therefore, the shared secret key K is advantageously
different whenever a message M is exchanged. This provides a strong method for securely exchanging messages and in particular a method for preventing any DFA attacks.
Finally, it should be noted that the message M can comprise any type of data, in particular sensitive data such as passwords, control words, cryptographic key or any other confidential information.
At least one embodiment also refers to a device or to a system suitable for implementing any of the embodiments of the above-described method.
Referring to FIG. 4, the latter schematically shows in more detail one of the devices 10, 20 depicted in the system of FIG. 1. This device can be indifferently used as a sending device D1 or as a
receiving device D2, and preferably even both as sending and receiving device. To this end, it comprises several components including at least:
□ a communication interface 1 for data exchange (M′, R1, R2, . . . ), in particular for exchanging data with at least one other device,
□ a secure memory 2 for storing the shared secret key K,
□ a random generator 3 for generating a random number R1 when a message M has to be exchanged, preferably each time such a message has to be exchanged,
□ at least one calculation unit 7 for outputting at least one result of an operation (OP1, OP2, OP3,OP4) using operands (e.g. R1, R2, K, M) as inputs,
□ at least one cryptographic unit 8 to execute algorithms (A1, A2) by means of at least one cryptographic key (K1, K2, K3), and
□ a central processing unit 5 in charge of managing the aforementioned components (1, 2, 3, 7, 8) in accordance with the steps of the cryptographic method described here-above.
The device 10, 20 can be used in all cases where sensitive data must be securely exchanged. Such a device can take the form of an electronic circuit (integrated circuit, preferably a monolithic
circuit), such as a smartcard or a chipset suitable to be inserted into another device. The latter could be a set-top-box (within the pay-TV field), a smart phone or any other communication device.
In variant, such a smartcard could be also used as a standalone device, e.g. as access card, as bank card (credit card or payment card) for communicating with a control terminal.
The calculation of each operation OP1, OP2, OP3, OP4 can be performed by using a single calculation unit 7 configured to perform different operations, or several calculation units 7, each dedicated
to one of these operations. The same principle applies to the cryptographic unit 8 regarding the algorithms A1, A2.
At least one embodiment also refers to a system as shown in FIG. 1. Such a system comprises at least two cryptographic devices 10, 20, connected together, for implementing any embodiment of the
above-described method. Each device 10, 20 of this system comprises at least the components which were listed above during the detailed description of the device presented as a further subject-matter
of at least one embodiment. Besides, any of the devices of the system may include at least one of the above-mentioned related optional features.
1. A cryptographic method for securely exchanging messages between at least two devices, each of them storing a shared secret key common to said devices, the method comprising:
generating a random number at each device;
sending by each device the generated random number to the other devices;
determining, at each device, a first key by calculating a first operation which uses both said shared secret key and each random number as operands;
determining, at each device, a second key by encrypting a result of a second operation with a first algorithm using said first key as encryption key, said second operation using at least each
random number as operands;
determining, by one of said devices acting as a sending device, a pseudo message by calculating a reversible third operation which uses both said message and each random number as operands;
calculating, by said sending device, a cryptogram resulting from the encryption of said pseudo message with a second algorithm using said second key as direct or indirect encryption key;
transmitting said cryptogram from said sending device to at least one other device acting as receiving device;
receiving said cryptogram at said receiving device;
decrypting the cryptogram at the receiving device by using said second key as direct or indirect decryption key of said second algorithm to recover said pseudo message;
retrieving said message from said pseudo message by reversing said third operation.
2. The cryptographic method of claim 1, wherein the use of said second key as indirect encryption or decryption key, within the second algorithm, is performed with a third key determined, at each
device, by a fourth operation using said second key and said shared secret key as operands.
3. The cryptographic method of claim 1, wherein at least a part of at least any of said operations involves a logical operation.
4. The cryptographic method of claim 3, wherein said logical operation is an exclusive OR operation.
5. The cryptographic method of claim 1, wherein if the operands of any one of said operations have different bit lengths, then restoring the same bit length for each of said operands.
6. The cryptographic method of claim 5, wherein restoring the same bit length is achieved by a balancing step aiming to supplement the operand having the smallest bit length until its bit length is
equal to the bit length of any of the other operands, then repeating said balancing step until all the operands have the same bit length.
7. The cryptographic method of claim 6, wherein said balancing step is first performed by concatenating the operand having the smallest bit length with itself, until reaching the same bit length as
the other operand, or until reaching a bit length reduced by a residual value less than the bit length of the concatenated operand.
8. The cryptographic method of claim 7, wherein said balancing step is applied to said third operation and the operand having the smallest bit length is any of said random numbers while said other
operand is the message.
9. The cryptographic method of claim 1, wherein said first algorithm uses a one-way function.
10. The cryptographic method of claim 1, wherein at least a part of at least any of said operations involves a number raised to a power, where any of said operands is used as an exponent of said
number chosen among the other operands.
11. A cryptographic device for implementing the cryptographic method of claim 1, comprising several components including at least a communication interface for data exchange, a secure memory for
storing a shared secret key, a random generator for generating a random number, at least one calculation unit outputting a result of an operation using operands as inputs, at least one cryptographic
unit to run algorithms by means of at least one cryptographic key, and a central processing unit in charge of managing said components in accordance with the steps of said cryptographic method.
12. The cryptographic device of claim 11, wherein it is made of a monolithic circuit.
13. A system comprising at least two cryptographic devices connected together for implementing the cryptographic method of claim 1, wherein each of said device comprises several components including
at least a communication interface for data exchange, a secure memory for storing a shared secret key, a random generator for generating a random number, at least one calculation unit outputting a
result of an operation using operands as inputs, at least one cryptographic unit to run algorithms by means of at least one cryptographic key, and a central processing unit in charge of managing said
components in accordance with the steps of said cryptographic method.
Referenced Cited
U.S. Patent Documents
6396928 May 28, 2002 Zheng
8380982 February 19, 2013 Miyabayashi
8577039 November 5, 2013 Matsuo
9330270 May 3, 2016 Ochiai
20040071291 April 15, 2004 Romain et al.
20070177720 August 2, 2007 Bevan et al.
20100005307 January 7, 2010 Prashanth
Patent History
Patent number
: 9648026
: Jun 5, 2015
Date of Patent
: May 9, 2017
Patent Publication Number
20150365424 Assignee
Hervé Pelletier
Primary Examiner
Saleh Najjar Assistant Examiner
Feliciano Mejia Application Number
: 14/731,596 | {"url":"https://patents.justia.com/patent/9648026","timestamp":"2024-11-10T16:28:54Z","content_type":"text/html","content_length":"101953","record_id":"<urn:uuid:1b9d801b-b41f-4ecd-be04-a8fd78fd25a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00669.warc.gz"} |
Generalized Hirano inverses in Banach algebras
Let $\mathcal{A}$ be a Banach algebra. An element $a\in \mathcal{A}$ has generalized Hirano inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a^2-ab\in \mathcal{A}^{qnil}.$$ We
prove that $a\in \mathcal{A}$ has generalized Hirano inverse if and only if $a$ has g-Drazin inverse and $a-a^3\in \mathcal{A}^{qnil}$, if and only if $a$ is the sum of a tripotent and a
quasinilpotent that commute. The Cline's formula for generalized Hirano inverses are thereby obtained. Let $a,b\in \mathcal{A}$ have generalized Hirano inverses. If $a^2b=aba$ and $b^2a=bab$, we
prove that $a+b$ has generalized Hirano inverse if and only if $1+a^db$ has generalized Hirano inverse. Hirano inverses of operator matrices on Banach spaces are also studied.
• There are currently no refbacks. | {"url":"http://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/10469","timestamp":"2024-11-10T21:37:42Z","content_type":"application/xhtml+xml","content_length":"15386","record_id":"<urn:uuid:58a8f1a3-d71e-428b-a3f7-5ccade9c0eca>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00588.warc.gz"} |
Work and Energy - toppershome.online
Work and Energy
Work and Energy
Work and Energy is the chapter Science covers the basic understanding of the subject that covers the syllabus of class 9
WORK : When the force is applied to the body , it displaces in the direction of force applied than work is said to be done.
The main condition of work done are a) application of force b) displacement in the body
Work done = Force x displacement made
It is a scalar quantity
SI unit of work is Newton – meter or Joule
If greater the force more will be the work done or if more is the displacement more shall be the work done
One Joule is defined as when one newton of force is applied to the body such that it displaces by one meter in the direction of the application of force.
When work is done against the gravity than the amount of work done is equal to the product of weight of body and the vertical distance through which the is lifted
weight of body = mass x acceleration due to gravity =Mg
vertical height to which body is lifted =h
WORK DONE AGANIST GRAVITY = Mgh unit Joule or Newton meter
Work is done when a cyclist is moving/ peddling the cycle or man is lifting a load up ward/ down wards but no work is done when a coolie carrying the load on his head stands stationary or man
applied the force on big heavy rock .
Work Done is said to be positive if the displacement and the force is made in same direction example a child pulling the toy
Work Done is said to be negative if the displacement of body is in opposite direction to the force of application example a boy throw a ball on the wall the ball bounces back or a boy kicks the
football and it stops after moving some distance due frication
Work Done is said to be zero when force acts at the right angle to the direction of motion of the body example circular motion the moon moves around the earth , the revolves around the sun.
Example :A person lift the luggage of 16kg from the ground to the height of 5meter
work against gravity is Mgh = 16 x10x5 = 800 joule
ENERGY : As we know the sun is the main source of energy and most of the energy is derived from the sun , some from the nucleus of atoms as well as interior of the earth
The amount of energy present in body is equal to the amount of work done by the body there are case that in working it may loses /gain the energy
.It is scalar quantity
SI unit is Joule
Form of Energy : Kinetic , Potential, Heat OR sound etc
The Energy possessed by the body due to motion or position of the body is called mechanical energy.
Mechanical Energy = Kinetic Energy + Potential Energy
KINETIC ENERGY : The energy possessed by the body due to motion of the body example as moving of cricket ball , running water , moving bullet etc.
It is directly proportional to the mass of the body and square of the velocity .
Formula of KE :
Kinetic energy of the moving body is equal to the work it can do before coming to rest.
If an object of mass m is moving body with the uniform velocity” u” it displaces by distance s with constant force “F “acts on it in direction of displacement velocity changes to” v” having an
acceleration” a”
WORK DONE = Force x displacement
= F x s
= ma x s (f= m.a)
V² -U²= 2 a. s
s =( V² -U²) ÷ 2a
W = m.a.s
W is work done m is mass a is acceleration s is displacement
W= ma. (v²- u²) /2a
===> W= m.( v²-u²)/2
W = 1/2 .m (v2- u2) if u=0 ====> W= 1/2 m. v2
K = 1/2 m. v2
example : If the body with the mass of 15kg moving with the velocity of 4 m/s than
KE = 1/2 x 15×4^2 = 1/2x15x16= 15 x8 = 120 joule
If the mass of body become double/ half the KE also become double / half .
If the velocity becomes double than KE becomes square time ie 4 times or
if the velocity is made half than KE becomes 1/4 times
POTENTIAL ENERGY : The energy possessed by the body due to position of the body/ change in the shape of the body
Example a) water store in dam
b) wound of sting toy
c) Bent string of bow.
Factors :
Mass of the body , greater the mass more is the potential energy if height remain unchanged,
It also depend s upon the height from the surface of earth if mass is constant .Change in shape greater the stretching/twisting/bending greater the potential energy.
Formula of Potential Energy :
If the mass of the object be m it is raised to the height H from the surface of earth acceleration due to gravity tending downwards is g than the work done in taking the body to height H is equal
to force x displacement made
PE = Weight x height
= mg x H
= mgH unit joule .
It is to be noted that if weight of body is constant than PE remain same if height is same.
Example :
a) A stone on a certain height has entire potential energy. But when it starts moving downward. PE of stone goes on decreasing as the height goes on decreasing but KE increasing as the velocity
increasing as it reaches on the ground the KE is maximum and PE is zero as total PE changes to KE
b) In hydroelectric power the water stored in dam is transformed into KE to produce into electricity .
c) In thermal power the chemical energy of coal is changed into heat in energy which further into KE and Electrical energy.
Energy can neither be created nor can be destroyed total energy of the system remain conserved .
A ball of mass m at the height h has PE is mgh. As a ball falls downwards , the height h decreases so PE changes to KE . In this case PE decreases continuously whereas KE increases .Total energy
remain conserved.
Total Energy = Potential Energy + Kinetic Energy ==> TE = PE+KE
1/2 mv2+ mgh = Constant
The conservation of energy ins the pendulum shows the example the a simple pendulum A swing shows the conservation of energy > A swinging of simple pendulum which consist of pith ball / bob which
is free to swing back and forth when displaced In this energy is continuously transformed from PE to KE and vice versa . The total energy remain conserved.
POWER : The rate of energy consumption is called power .
Power = work done/ time taken
SI unit is Watt = Joule/sec
when one joule of work done in one sec than the power consumed is one watt
1 kilo watt =1000 watt
1 Horse power = 746 watt
Example : 20 joule of work done in 5 sec than power is =20/5 = 4 watt
I kilowatt hour = 3.6 x 10^6 joule = one unit of electrical consumption.
Example : A bulb of 60 watt is used for 6 hrs /day for 30 day than electricity consumed
power x hours /1000 = unit
60 x30 x 6 /1000 = 10800/1000= 10.8 unit
Important Points for solution of text question of Work and Energy
1 Force of 7 N displacement made is 8meters than work done = 7×8=56 joule
2 A pair of bullocks exert the force of 114N displacement made is 15 meter than work done is force x displacement = 114 x15 =16710 joule
3.what is mass of a body when it is moving with velocity of 5m/s having kE of 25 joule
KE= ½ m×v²
= > 25= ½m×5²
==> 25=½×m×25 ==>
m= 25×2÷25 => m=2kg
what happens to kinetic Energy if velocity is doubled and than tripled
incase of doubled KE = ½m×v²= ½× 2 x 10² = 100 joule
incase of triple put m= 2 v=15 solve it
4, what work is done when body of 20kg is moving with velocity of 5m/s its velocity changes to 2m/s than
work done=½xm (v² – u²)
WD= ½×20(2²- 5²)
WD= 10 (4-25)=10x-21= -210 joule
5.A household the 250 units/ month than energy consumed
1 unit = 3.6x 10^6 than 250×3.6 x1000000=900×1000000 joule
6 A mass of 40kg taken to heighof 5 meter than potential energy PE=mgh = 40x10x5= 2000j
7 The energy consumed by 1500 watt heater working 10 hours P.t = 1 500x 10= 15000÷1000= 15 kwh
Conclusion: Work and Energy
Work and Energy is the chapter from physics consist the very basic idea of work and energy which covers the syllabus of class 9 with some numerical and practical examples .Work is the product of
force and displacement .Energy in different form Kinetic and Potential Energy With basic idea of power .
Follow Us on: Facebook or Instagram
Read More: Motion in Straight line
Read More : Force and Laws of Motion
Leave a Comment | {"url":"https://toppershome.online/work-and-energy/","timestamp":"2024-11-14T07:43:13Z","content_type":"text/html","content_length":"221134","record_id":"<urn:uuid:02279d20-228f-4d3b-aff5-6750305266b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00545.warc.gz"} |
Examples of Low-Density Materials in context of mass to density
31 Aug 2024
Title: Exploring the Realm of Low-Density Materials: A Theoretical Perspective on Mass and Density Relationships
Abstract: This article delves into the world of low-density materials, examining their unique properties and relationships with mass and density. By employing fundamental principles from physics and
chemistry, we will investigate various examples of low-density materials, highlighting their distinct characteristics and implications for understanding mass-to-density conversions.
Density (ρ) is a fundamental physical property that represents the mass (m) of an object per unit volume (V). Mathematically, this relationship can be expressed as:
ρ = m / V
In the context of low-density materials, we are interested in substances with densities lower than those typically found in everyday objects. These materials often possess unique properties and
applications, making them essential for various scientific and technological pursuits.
Examples of Low-Density Materials:
1. Aerogels: These highly porous solids have a density range of approximately 0.01-0.05 g/cm³. Their low density is attributed to the presence of air-filled pores within their structure.
2. Foams: A type of cellular material, foams exhibit densities between 0.1-10 g/cm³. The low-density regime is characterized by a high volume fraction of gas bubbles.
3. Graphite: With a density of approximately 2.25 g/cm³, graphite is a relatively light material compared to other metals. Its layered structure contributes to its low density.
4. Silica Aerogel: This amorphous solid has a density range of around 0.01-0.05 g/cm³, making it one of the lowest-density materials known.
Theoretical Considerations:
When examining mass-to-density relationships in low-density materials, several theoretical aspects come into play:
• Packing efficiency: The arrangement of particles or molecules within a material can significantly impact its density.
• Void fraction: The volume occupied by voids or pores within a material can greatly affect its overall density.
• Material structure: The crystalline or amorphous nature of a material can influence its density.
This article has provided an overview of various low-density materials, highlighting their unique properties and relationships with mass and density. By understanding these principles, researchers
and scientists can better appreciate the characteristics and applications of these fascinating substances. Further investigation into the theoretical aspects of low-density materials will continue to
reveal new insights into the fundamental nature of matter.
• [1] “Density” in Encyclopedia Britannica
• [2] “Aerogels” in Journal of Non-Crystalline Solids
• [3] “Foams” in Journal of Colloid and Interface Science
Related articles for ‘mass to density’ :
• Reading: Examples of Low-Density Materials in context of mass to density
Calculators for ‘mass to density’ | {"url":"https://blog.truegeometry.com/tutorials/education/b15ef5c716f1d1e5efd3dcf766832e82/JSON_TO_ARTCL_Examples_of_Low_Density_Materials_in_context_of_mass_to_density.html","timestamp":"2024-11-12T05:54:46Z","content_type":"text/html","content_length":"16778","record_id":"<urn:uuid:caf8ccdf-06ef-4f83-84df-affdba83cb78>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00882.warc.gz"} |
Virtual Math Learning Center
Welcome to the Virtual Math Learning Center! The VMLC is an online resource to help students succeed in their Math and Stats courses at Texas A&M University. The VMLC has videos and practice problems
that you can access at any time. It has materials devoted to specific courses, and you can look for your
course here
. It also has
devoted to specific math topics here and a
How-To series
with review videos for precalculus and calculus.
You can also watch the video to see some of what the VMLC has to offer. We hope the VMLC helps you in your courses at Texas A&M.
Math Learning Center
) is a companion to the Virtual Math Learning Center (VMLC). They both exist to help you in your Math and Stats courses at Texas A&M but in slightly different ways.
• The VMLC is entirely online with videos and practice problems that you can access at any time and study at your own pace.
• The MLC focuses on live help where you are interacting with a tutor or attending a review session (either in person or on zoom). This includes Help Sessions, Week in Reviews, Hands on Grades Up
sessions, and more. If you have studied the materials on the VMLC and still need help, then you should try the resources at the MLC. | {"url":"https://vmlc.tamu.edu","timestamp":"2024-11-06T05:48:37Z","content_type":"text/html","content_length":"55905","record_id":"<urn:uuid:fd8bb998-3fad-47b6-b8a4-0441064882f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00421.warc.gz"} |
Chapter 12: Synthesizing and presenting findings using other methods
Joanne E McKenzie, Sue E Brennan
Key Points:
• Meta-analysis of effect estimates has many advantages, but other synthesis methods may need to be considered in the circumstance where there is incompletely reported data in the primary studies.
• Alternative synthesis methods differ in the completeness of the data they require, the hypotheses they address, and the conclusions and recommendations that can be drawn from their findings.
• These methods provide more limited information for healthcare decision making than meta-analysis, but may be superior to a narrative description where some results are privileged above others
without appropriate justification.
• Tabulation and visual display of the results should always be presented alongside any synthesis, and are especially important for transparent reporting in reviews without meta-analysis.
• Alternative synthesis and visual display methods should be planned and specified in the protocol. When writing the review, details of the synthesis methods should be described.
• Synthesis methods that involve vote counting based on statistical significance have serious limitations and are unacceptable.
Cite this chapter as: McKenzie JE, Brennan SE. Chapter 12: Synthesizing and presenting findings using other methods [last updated October 2019]. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li
T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane, 2024. Available from www.training.cochrane.org/handbook.
12.1 Why a meta-analysis of effect estimates may not be possible
Meta-analysis of effect estimates has many potential advantages (see Chapter 10 and Chapter 11). However, there are circumstances where it may not be possible to undertake a meta-analysis and other
statistical synthesis methods may be considered (McKenzie and Brennan 2014).
Some common reasons why it may not be possible to undertake a meta-analysis are outlined in Table 12.1.a. Legitimate reasons include limited evidence; incompletely reported outcome/effect estimates,
or different effect measures used across studies; and bias in the evidence. Other commonly cited reasons for not using meta-analysis are because of too much clinical or methodological diversity, or
statistical heterogeneity (Achana et al 2014). However, meta-analysis methods should be considered in these circumstances, as they may provide important insights if undertaken and interpreted
Table 12.1.a Scenarios that may preclude meta-analysis, with possible solutions
Scenario Description Examples of possible solutions*
Limited evidence Meta-analysis is not possible with no studies, or only one study. This circumstance may reflect Build contingencies into the analysis plan to group one or more of the PICO
for a the infancy of research in a particular area, or that the specified PICO for the synthesis aims to elements at a broader level (Chapter 2, Section 2.5.3).
pre-specified address a narrow question.
Incompletely Within a study, the intervention effects may be incompletely reported (e.g. effect estimate with Calculate the effect estimate and measure of precision from the available
reported outcome no measure of precision; direction of effect with P value or statement of statistical statistics if possible (Chapter 6).
or effect significance; only the direction of effect).
estimate Impute missing statistics (e.g. standard deviations) where possible (Chapter 6,
Section 6.5.2).
Use other synthesis method(s) (Section 12.2), along with methods to display and
present available effects visually (Section 12.3).
Different effect Across studies, the same outcome could be treated differently (e.g. a time-to-event outcome has Calculate the effect estimate and measure of precision for the same effect measure
measures been dichotomized in some studies) or analysed using different methods. Both scenarios could lead from the available statistics if possible (Chapter 6).
to different effect measures (e.g. hazard ratios and odds ratios).
Transform effect measures (e.g. convert standardized mean difference to an odds
ratio) where possible (Chapter 10, Section 10.6).
Use other synthesis method(s) (Section 12.2), along with methods to display and
present available effects visually (Section 12.3).
Bias in the Concerns about missing studies, missing outcomes within the studies (Chapter 13), or bias in the When there are major concerns about bias in the evidence, use structured reporting
evidence studies (Chapter 8 and Chapter 25), are legitimate reasons for not undertaking a meta-analysis. of the available effects using tables and visual displays (Section 12.3).
These concerns similarly apply to other synthesis methods (Section 12.2).
Incompletely reported outcomes/effects may bias meta-analyses, but not necessarily other synthesis For incompletely reported outcomes/effects, also consider other synthesis methods
methods. in addition to meta-analysis (Section 12.2).
Clinical and Concerns about diversity in the populations, interventions, outcomes, study designs, are often Modify planned comparisons, providing rationale for post-hoc changes (Chapter 9).
methodological cited reasons for not using meta-analysis (Ioannidis et al 2008). Arguments against using
diversity meta-analysis because of too much diversity equally apply to the other synthesis methods
(Valentine et al 2010).
Statistical Statistical heterogeneity is an often cited reason for not reporting the meta-analysis result Attempt to reduce heterogeneity (e.g. checking the data, correcting an
heterogeneity (Ioannidis et al 2008). Presentation of an average combined effect in this circumstance can be inappropriate choice of effect measure) (Chapter 10, Section 10.10).
misleading, particularly if the estimated effects across the studies are both harmful and
beneficial. Attempt to explain heterogeneity (e.g. using subgroup analysis) (Chapter 10,
Section 10.11).
Consider (if possible) presenting a prediction interval, which provides a
predicted range for the true intervention effect in an individual study (Riley et
al 2011), thus clearly demonstrating the uncertainty in the intervention effects.
*Italicized text indicates possible solutions discussed in this chapter.
A range of statistical synthesis methods are available, and these may be divided into three categories based on their preferability (Table 12.2.a). Preferable methods are the meta-analysis methods
outlined in Chapter 10 and Chapter 11, and are not discussed in detail here. This chapter focuses on methods that might be considered when a meta-analysis of effect estimates is not possible due to
incompletely reported data in the primary studies. These methods divide into those that are ‘acceptable’ and ‘unacceptable’. The ‘acceptable’ methods differ in the data they require, the hypotheses
they address, limitations around their use, and the conclusions and recommendations that can be drawn (see Section 12.2.1). The ‘unacceptable’ methods in common use are described (see Section 12.2.2
), along with the reasons for why they are problematic.
Compared with meta-analysis methods, the ‘acceptable’ synthesis methods provide more limited information for healthcare decision making. However, these ‘acceptable’ methods may be superior to a
narrative that describes results study by study, which comes with the risk that some studies or findings are privileged above others without appropriate justification. Further, in reviews with little
or no synthesis, readers are left to make sense of the research themselves, which may result in the use of seemingly simple yet problematic synthesis methods such as vote counting based on
statistical significance (see Section 12.2.2.1).
All methods first involve calculation of a ‘standardized metric’, followed by application of a synthesis method. In applying any of the following synthesis methods, it is important that only one
outcome per study (or other independent unit, for example one comparison from a trial with multiple intervention groups) contributes to the synthesis. Chapter 9 outlines approaches for selecting an
outcome when multiple have been measured. Similar to meta-analysis, sensitivity analyses can be undertaken to examine if the findings of the synthesis are robust to potentially influential decisions
(see Chapter 10, Section 10.14 and Section 12.4 for examples).
Authors should report the specific methods used in lieu of meta-analysis (including approaches used for presentation and visual display), rather than stating that they have conducted a ‘narrative
synthesis’ or ‘narrative summary’ without elaboration. The limitations of the chosen methods must be described, and conclusions worded with appropriate caution. The aim of reporting this detail is to
make the synthesis process more transparent and reproducible, and help ensure use of appropriate methods and interpretation.
Table 12.2.a Summary of preferable and acceptable synthesis methods
Synthesis method Question answered Minimum data required Purpose Limitations
Estimate Variance Direction Precise
of of effect P value
Meta-analysis of effect What is the common ✓ ✓ Can be used to synthesize results when effect Requires effect estimates and their variances.
estimates and extensions intervention effect? estimates and their variances are reported (or
(Chapter 10 and Chapter can be calculated). Extensions (network meta-analysis, meta-regression/subgroup
11) What is the average analysis) require a reasonably large number of studies.
intervention effect? Provides a combined estimate of average
intervention effect (random effects), and Meta-regression/subgroup analysis involves observational
Which intervention, of precision of this estimate (95% CI). comparisons and requires careful interpretation. High risk of
multiple, is most false positive conclusions for sources of heterogeneity.
effective? Can be used to synthesize evidence from multiple
interventions, with the ability to rank them Network meta-analysis is more complicated to undertake and
What factors modify (network meta-analysis). requires careful assessment of the assumptions.
the magnitude of the
intervention effects? Can be used to detect, quantify and investigate
heterogeneity (meta-regression/subgroup
Associated plots: forest plot, funnel plot,
network diagram, rankogram plot
Summarizing effect What is the range and ✓ Can be used to synthesize results when it is Does not account for differences in the relative sizes of the
estimates distribution of difficult to undertake a meta-analysis (e.g. studies.
observed effects? missing variances of effects, unit of analysis
errors). Performance of these statistics applied in the context of
summarizing effect estimates has not been evaluated.
Provides information on the magnitude and range
of effects (median, interquartile range, range).
Associated plots: box-and-whisker plot, bubble
Combining P values Is there evidence that ✓ ✓ Can be used to synthesize results when studies Provides no information on the magnitude of effects.
there is an effect in report:
at least one study? Does not distinguish between evidence from large studies with
• no, or minimal, information beyond P values small effects and small studies with large effects.
and direction of effect;
• results of non-parametric analyses; Difficult to interpret the test results when statistically
• results of different types of outcomes and significant, since the null hypothesis can be rejected on the
statistical tests; basis of an effect in only one study (Jones 1995).
• outcomes are different across studies (e.g.
different serious side effects). When combining P values from few, small studies, failure to
reject the null hypotheses should not be interpreted as evidence
Associated plot: albatross plot of no effect in all studies.
Vote counting based on Is there any evidence ✓ Can be used to synthesize results when only Provides no information on the magnitude of effects (Borenstein
direction of effect of an effect? direction of effect is reported, or there is et al 2009).
inconsistency in the effect measures or data
reported across studies. Does not account for differences in the relative sizes of the
studies (Borenstein et al 2009).
Associated plots: harvest plot, effect direction
plot Less powerful than methods used to combine P values.
Description of method Summarizing effect estimates might be considered in the circumstance where estimates of intervention effect are available (or can be calculated), but the variances of the
effects are not reported or are incorrect (and cannot be calculated from other statistics, or reasonably imputed) (Grimshaw et al 2003). Incorrect calculation of variances arises more commonly in
non-standard study designs that involve clustering or matching (Chapter 23). While missing variances may limit the possibility of meta-analysis, the (standardized) effects can be summarized using
descriptive statistics such as the median, interquartile range, and the range. Calculating these statistics addresses the question ‘What is the range and distribution of observed effects?’
Reporting of methods and results The statistics that will be used to summarize the effects (e.g. median, interquartile range) should be reported. Box-and-whisker or bubble plots will complement
reporting of the summary statistics by providing a visual display of the distribution of observed effects (Section 12.3.3). Tabulation of the available effect estimates will provide transparency for
readers by linking the effects to the studies (Section 12.3.1). Limitations of the method should be acknowledged (Table 12.2.a).
12.2.1.2 Combining P values
Description of method Combining P values can be considered in the circumstance where there is no, or minimal, information reported beyond P values and the direction of effect; the types of outcomes
and statistical tests differ across the studies; or results from non-parametric tests are reported (Borenstein et al 2009). Combining P values addresses the question ‘Is there evidence that there is
an effect in at least one study?’ There are several methods available (Loughin 2004), with the method proposed by Fisher outlined here (Becker 1994).
Fisher’s method combines the P values from statistical tests across k studies using the formula:
One-sided P values are used, since these contain information about the direction of effect. However, these P values must reflect the same directional hypothesis (e.g. all testing if intervention A is
more effective than intervention B). This is analogous to standardizing the direction of effects before undertaking a meta-analysis. Two-sided P values, which do not contain information about the
direction, must first be converted to one-sided P values. If the effect is consistent with the directional hypothesis (e.g. intervention A is beneficial compared with B), then the one-sided P value
is calculated as
In studies that do not report an exact P value but report a conventional level of significance (e.g. P<0.05), a conservative option is to use the threshold (e.g. 0.05). The P values must have been
computed from statistical tests that appropriately account for the features of the design, such as clustering or matching, otherwise they will likely be incorrect.
The Chi^2 statistic will follow a chi-squared distribution with ^2 statistic compared to the degrees of freedom (with a corresponding low P value) provides evidence of an effect in at least one study
(see Section 12.4.2.2 for guidance on implementing Fisher’s method for combining P values).
Reporting of methods and results There are several methods for combining P values (Loughin 2004), so the chosen method should be reported, along with details of sensitivity analyses that examine if
the results are sensitive to the choice of method. The results from the test should be reported alongside any available effect estimates (either individual results or meta-analysis results of a
subset of studies) using text, tabulation and appropriate visual displays (Section 12.3). The albatross plot is likely to complement the analysis (Section 12.3.4). Limitations of the method should be
acknowledged (Table 12.2.a).
Description of method Vote counting based on the direction of effect might be considered in the circumstance where the direction of effect is reported (with no further information), or there is no
consistent effect measure or data reported across studies. The essence of vote counting is to compare the number of effects showing benefit to the number of effects showing harm for a particular
outcome. However, there is wide variation in the implementation of the method due to differences in how ‘benefit’ and ‘harm’ are defined. Rules based on subjective decisions or statistical
significance are problematic and should be avoided (see Section 12.2.2).
To undertake vote counting properly, each effect estimate is first categorized as showing benefit or harm based on the observed direction of effect alone, thereby creating a standardized binary
metric. A count of the number of effects showing benefit is then compared with the number showing harm. Neither statistical significance nor the size of the effect are considered in the
categorization. A sign test can be used to answer the question ‘is there any evidence of an effect?’ If there is no effect, the study effects will be distributed evenly around the null hypothesis of
no difference. This is equivalent to testing if the true proportion of effects favouring the intervention (or comparator) is equal to 0.5 (Bushman and Wang 2009) (see Section 12.4.2.3 for guidance on
implementing the sign test). An estimate of the proportion of effects favouring the intervention can be calculated (p = u/n, where u = number of effects favouring the intervention, and n = number of
studies) along with a confidence interval (e.g. using the Wilson or Jeffreys interval methods (Brown et al 2001)). Unless there are many studies contributing effects to the analysis, there will be
large uncertainty in this estimated proportion.
Reporting of methods and results The vote counting method should be reported in the ‘Data synthesis’ section of the review. Failure to recognize vote counting as a synthesis method has led to it
being applied informally (and perhaps unintentionally) to summarize results (e.g. through the use of wording such as ‘3 of 10 studies showed improvement in the outcome with intervention compared to
control’; ‘most studies found’; ‘the majority of studies’; ‘few studies’ etc). In such instances, the method is rarely reported, and it may not be possible to determine whether an unacceptable
(invalid) rule has been used to define benefit and harm (Section 12.2.2). The results from vote counting should be reported alongside any available effect estimates (either individual results or
meta-analysis results of a subset of studies) using text, tabulation and appropriate visual displays (Section 12.3). The number of studies contributing to a synthesis based on vote counting may be
larger than a meta-analysis, because only minimal statistical information (i.e. direction of effect) is required from each study to vote count. Vote counting results are used to derive the harvest
and effect direction plots, although often using unacceptable methods of vote counting (see Section 12.3.5). Limitations of the method should be acknowledged (Table 12.2.a).
Conventional forms of vote counting use rules based on statistical significance and direction to categorize effects. For example, effects may be categorized into three groups: those that favour the
intervention and are statistically significant (based on some predefined P value), those that favour the comparator and are statistically significant, and those that are statistically non-significant
(Hedges and Vevea 1998). In a simpler formulation, effects may be categorized into two groups: those that favour the intervention and are statistically significant, and all others (Friedman 2001).
Regardless of the specific formulation, when based on statistical significance, all have serious limitations and can lead to the wrong conclusion.
The conventional vote counting method fails because underpowered studies that do not rule out clinically important effects are counted as not showing benefit. Suppose, for example, the effect sizes
estimated in two studies were identical. However, only one of the studies was adequately powered, and the effect in this study was statistically significant. Only this one effect (of the two
identical effects) would be counted as showing ‘benefit’. Paradoxically, Hedges and Vevea showed that as the number of studies increases, the power of conventional vote counting tends to zero, except
with large studies and at least moderate intervention effects (Hedges and Vevea 1998). Further, conventional vote counting suffers the same disadvantages as vote counting based on direction of
effect, namely, that it does not provide information on the magnitude of effects and does not account for differences in the relative sizes of the studies.
Subjective rules, involving a combination of direction, statistical significance and magnitude of effect, are sometimes used to categorize effects. For example, in a review examining the
effectiveness of interventions for teaching quality improvement to clinicians, the authors categorized results as ‘beneficial effects’, ‘no effects’ or ‘detrimental effects’ (Boonyasai et al 2007).
Categorization was based on direction of effect and statistical significance (using a predefined P value of 0.05) when available. If statistical significance was not reported, effects greater than
10% were categorized as ‘beneficial’ or ‘detrimental’, depending on their direction. These subjective rules often vary in the elements, cut-offs and algorithms used to categorize effects, and while
detailed descriptions of the rules may provide a veneer of legitimacy, such rules have poor performance validity (Ioannidis et al 2008).
A further problem occurs when the rules are not described in sufficient detail for the results to be reproduced (e.g. ter Wee et al 2012, Thornicroft et al 2016). This lack of transparency does not
allow determination of whether an acceptable or unacceptable vote counting method has been used (Valentine et al 2010).
Visual display and presentation of data is especially important for transparent reporting in reviews without meta-analysis, and should be considered irrespective of whether synthesis is undertaken
(see Table 12.2.a for a summary of plots associated with each synthesis method). Tables and plots structure information to show patterns in the data and convey detailed information more efficiently
than text. This aids interpretation and helps readers assess the veracity of the review findings.
Ordering studies alphabetically by study ID is the simplest approach to tabulation; however, more information can be conveyed when studies are grouped in subpanels or ordered by a characteristic
important for interpreting findings. The grouping of studies in tables should generally follow the structure of the synthesis presented in the text, which should closely reflect the review questions.
This grouping should help readers identify the data on which findings are based and verify the review authors’ interpretation.
If the purpose of the table is comparative, grouping studies by any of following characteristics might be informative:
• comparisons considered in the review, or outcome domains (according to the structure of the synthesis);
• study characteristics that may reveal patterns in the data, for example potential effect modifiers including population subgroups, settings or intervention components.
If the purpose of the table is complete and transparent reporting of data, then ordering the studies to increase the prominence of the most relevant and trustworthy evidence should be considered.
Possibilities include:
• certainty of the evidence (synthesized result or individual studies if no synthesis);
• risk of bias, study size or study design characteristics; and
• characteristics that determine how directly a study addresses the review question, for example relevance and validity of the outcome measures.
One disadvantage of grouping by study characteristics is that it can be harder to locate specific studies than when tables are ordered by study ID alone, for example when cross-referencing between
the text and tables. Ordering by study ID within categories may partly address this.
The value of standardizing intervention and outcome labels is discussed in Chapter 3, Section 3.2.2 and Section 3.2.4), while the importance and methods for standardizing effect estimates is
described in Chapter 6. These practices can aid readers’ interpretation of tabulated data, especially when the purpose of a table is comparative.
12.3.2 Forest plots
Forest plots and methods for preparing them are described elsewhere (Chapter 10, Section 10.2). Some mention is warranted here of their importance for displaying study results when meta-analysis is
not undertaken (i.e. without the summary diamond). Forest plots can aid interpretation of individual study results and convey overall patterns in the data, especially when studies are ordered by a
characteristic important for interpreting results (e.g. dose and effect size, sample size). Similarly, grouping studies in subpanels based on characteristics thought to modify effects, such as
population subgroups, variants of an intervention, or risk of bias, may help explore and explain differences across studies (Schriger et al 2010). These approaches to ordering provide important
techniques for informally exploring heterogeneity in reviews without meta-analysis, and should be considered in preference to alphabetical ordering by study ID alone (Schriger et al 2010).
Box-and-whisker plots (see Figure 12.4.a, Panel A) provide a visual display of the distribution of effect estimates (Section 12.2.1.1). The plot conventionally depicts five values. The upper and
lower limits (or ‘hinges’) of the box, represent the 75th and 25th percentiles, respectively. The line within the box represents the 50th percentile (median), and the whiskers represent the extreme
values (McGill et al 1978). Multiple box plots can be juxtaposed, providing a visual comparison of the distributions of effect estimates (Schriger et al 2006). For example, in a review examining the
effects of audit and feedback on professional practice, the format of the feedback (verbal, written, both verbal and written) was hypothesized to be an effect modifier (Ivers et al 2012).
Box-and-whisker plots of the risk differences were presented separately by the format of feedback, to allow visual comparison of the impact of format on the distribution of effects. When presenting
multiple box-and-whisker plots, the width of the box can be varied to indicate the number of studies contributing to each. The plot’s common usage facilitates rapid and correct interpretation by
readers (Schriger et al 2010). The individual studies contributing to the plot are not identified (as in a forest plot), however, and the plot is not appropriate when there are few studies (Schriger
et al 2006).
A bubble plot (see Figure 12.4.a, Panel B) can also be used to provide a visual display of the distribution of effects, and is more suited than the box-and-whisker plot when there are few studies
(Schriger et al 2006). The plot is a scatter plot that can display multiple dimensions through the location, size and colour of the bubbles. In a review examining the effects of educational outreach
visits on professional practice, a bubble plot was used to examine visually whether the distribution of effects was modified by the targeted behaviour (O’Brien et al 2007). Each bubble represented
the effect size (y-axis) and whether the study targeted a prescribing or other behaviour (x-axis). The size of the bubbles reflected the number of study participants. However, different formulations
of the bubble plot can display other characteristics of the data (e.g. precision, risk-of-bias assessments).
The albatross plot (see Figure 12.4.a, Panel C) allows approximate examination of the underlying intervention effect sizes where there is minimal reporting of results within studies (Harrison et al
2017). The plot only requires a two-sided P value, sample size and direction of effect (or equivalently, a one-sided P value and a sample size) for each result. The plot is a scatter plot of the
study sample sizes against two-sided P values, where the results are separated by the direction of effect. Superimposed on the plot are ‘effect size contours’ (inspiring the plot’s name). These
contours are specific to the type of data (e.g. continuous, binary) and statistical methods used to calculate the P values. The contours allow interpretation of the approximate effect sizes of the
studies, which would otherwise not be possible due to the limited reporting of the results. Characteristics of studies (e.g. type of study design) can be identified using different colours or
symbols, allowing informal comparison of subgroups.
The plot is likely to be more inclusive of the available studies than meta-analysis, because of its minimal data requirements. However, the plot should complement the results from a statistical
synthesis, ideally a meta-analysis of available effects.
Harvest plots (see Figure 12.4.a, Panel D) provide a visual extension of vote counting results (Ogilvie et al 2008). In the plot, studies based on the categorization of their effects (e.g.
‘beneficial effects’, ‘no effects’ or ‘detrimental effects’) are grouped together. Each study is represented by a bar positioned according to its categorization. The bars can be ‘visually weighted’
(by height or width) and annotated to highlight study and outcome characteristics (e.g. risk-of-bias domains, proximal or distal outcomes, study design, sample size) (Ogilvie et al 2008, Crowther et
al 2011). Annotation can also be used to identify the studies. A series of plots may be combined in a matrix that displays, for example, the vote counting results from different interventions or
outcome domains.
The methods papers describing harvest plots have employed vote counting based on statistical significance (Ogilvie et al 2008, Crowther et al 2011). For the reasons outlined in Section 12.2.2.1, this
can be misleading. However, an acceptable approach would be to display the results based on direction of effect.
The effect direction plot is similar in concept to the harvest plot in the sense that both display information on the direction of effects (Thomson and Thomas 2013). In the first version of the
effect direction plot, the direction of effects for each outcome within a single study are displayed, while the second version displays the direction of the effects for outcome domains across studies
. In this second version, an algorithm is first applied to ‘synthesize’ the directions of effect for all outcomes within a domain (e.g. outcomes ‘sleep disturbed by wheeze’, ‘wheeze limits speech’,
‘wheeze during exercise’ in the outcome domain ‘respiratory’). This algorithm is based on the proportion of effects that are in a consistent direction and statistical significance. Arrows are used to
indicate the reported direction of effect (for either outcomes or outcome domains). Features such as statistical significance, study design and sample size are denoted using size and colour. While
this version of the plot conveys a large amount of information, it requires further development before its use can be recommended since the algorithm underlying the plot is likely to have poor
performance validity.
The example that follows uses four scenarios to illustrate methods for presentation and synthesis when meta-analysis is not possible. The first scenario contrasts a common approach to tabulation with
alternative presentations that may enhance the transparency of reporting and interpretation of findings. Subsequent scenarios show the application of the synthesis approaches outlined in preceding
sections of the chapter. Box 12.4.a summarizes the review comparisons and outcomes, and decisions taken by the review authors in planning their synthesis. While the example is loosely based on an
actual review, the review description, scenarios and data are fabricated for illustration.
Box 12.4.a The review
│The review used in this example examines the effects of midwife-led continuity models versus other models of care for childbearing women. One of the outcomes considered in the review, and of │
│interest to many women choosing a care option, is maternal satisfaction with care. The review included 15 randomized trials, all of which reported a measure of satisfaction. Overall, 32 │
│satisfaction outcomes were reported, with between one and 11 outcomes reported per study. There were differences in the concepts measured (e.g. global satisfaction; specific domains such as of │
│satisfaction with information), the measurement period (i.e. antenatal, intrapartum, postpartum care), and the measurement tools (different scales; variable evidence of validity and reliability). │
│ │
│ │
│ │
│Before conducting their synthesis, the review authors did the following. │
│ │
│ 1. Specified outcome groups in their protocol (see Chapter 3). Five types of satisfaction outcomes were defined (global measures, satisfaction with information, satisfaction with decisions, │
│ satisfaction with care, sense of control), any of which would be grouped for synthesis since they all broadly reflect satisfaction with care. The review authors hypothesized that the period of │
│ care (antenatal, intrapartum, postpartum) might influence satisfaction with a model of care, so planned to analyse outcomes for each period separately. The review authors specified that │
│ outcomes would be synthesized across periods if data were sparse. │
│ 2. Specified decision rules in their protocol for dealing with multiplicity of outcomes (Chapter 3). For studies that reported multiple satisfaction outcomes per period, one outcome would be │
│ chosen by (i) selecting the most relevant outcome (a global measure > satisfaction with care > sense of control > satisfaction with decisions > satisfaction with information), and if there were│
│ two or more equally relevant outcomes, then (ii) selecting the measurement tool with best evidence of validity and reliability. │
│ 3. Examined study characteristics to determine which studies were similar enough for synthesis (Chapter 9). All studies had similar models of care as a comparator. Satisfaction outcomes from each │
│ study were categorized into one of the five pre-specified categories, and then the decision rules were applied to select the most relevant outcome for synthesis. │
│ 4. Determined what data were available for synthesis (Chapter 9). All measures of satisfaction were ordinal; however, outcomes were treated differently across studies (see Tables 12.4.a, 12.4.b │
│ and 12.4.c). In some studies, the outcome was dichotomized, while in others it was treated as ordinal or continuous. Based on their pre-specified synthesis methods, the review authors selected │
│ the preferred method for the available data. In this example, four scenarios, with progressively fewer data, are used to illustrate the application of alternative synthesis methods. │
│ 5. Determined if modification to the planned comparison or outcomes was needed. No changes were required to comparisons or outcome groupings. │
12.4.1 Scenario 1: structured reporting of effects
We first address a scenario in which review authors have decided that the tools used to measure satisfaction measured concepts that were too dissimilar across studies for synthesis to be appropriate.
Setting aside three of the 15 studies that reported on the birth partner’s satisfaction with care, a structured summary of effects is sought of the remaining 12 studies. To keep the example table
short, only one outcome is shown per study for each of the measurement periods (antenatal, intrapartum or postpartum).
Table 12.4.a depicts a common yet suboptimal approach to presenting results. Note two features.
• Studies are ordered by study ID, rather than grouped by characteristics that might enhance interpretation (e.g. risk of bias, study size, validity of the measures, certainty of the evidence
• Data reported are as extracted from each study; effect estimates were not calculated by the review authors and, where reported, were not standardized across studies (although data were available
to do both).
Table 12.4.b shows an improved presentation of the same results. In line with best practice, here effect estimates have been calculated by the review authors for all outcomes, and a common metric
computed to aid interpretation (in this case an odds ratio; see Chapter 6 for guidance on conversion of statistics to the desired format). Redundant information has been removed (‘statistical test’
and ‘P value’ columns). The studies have been re-ordered, first to group outcomes by period of care (intrapartum outcomes are shown here), and then by risk of bias. This re-ordering serves two
purposes. Grouping by period of care aligns with the plan to consider outcomes for each period separately and ensures the table structure matches the order in which results are described in the text.
Re-ordering by risk of bias increases the prominence of studies at lowest risk of bias, focusing attention on the results that should most influence conclusions. Had the review authors determined
that a synthesis would be informative, then ordering to facilitate comparison across studies would be appropriate; for example, ordering by the type of satisfaction outcome (as pre-defined in the
protocol, starting with global measures of satisfaction), or the comparisons made in the studies.
The results may also be presented in a forest plot, as shown in Figure 12.4.b. In both the table and figure, studies are grouped by risk of bias to focus attention on the most trustworthy evidence.
The pattern of effects across studies is immediately apparent in Figure 12.4.b and can be described efficiently without having to interpret each estimate (e.g. difference between studies at low and
high risk of bias emerge), although these results should be interpreted with caution in the absence of a formal test for subgroup differences (see Chapter 10, Section 10.11). Only outcomes measured
during the intrapartum period are displayed, although outcomes from other periods could be added, maximizing the information conveyed.
An example description of the results from Scenario 1 is provided in Box 12.4.b. It shows that describing results study by study becomes unwieldy with more than a few studies, highlighting the
importance of tables and plots. It also brings into focus the risk of presenting results without any synthesis, since it seems likely that the reader will try to make sense of the results by drawing
inferences across studies. Since a synthesis was considered inappropriate, GRADE was applied to individual studies and then used to prioritize the reporting of results, focusing attention on the most
relevant and trustworthy evidence. An alternative might be to report results at low risk of bias, an approach analogous to limiting a meta-analysis to studies at low risk of bias. Where possible,
these and other approaches to prioritizing (or ordering) results from individual studies in text and tables should be pre-specified at the protocol stage.
Table 12.4.a Scenario 1: table ordered by study ID, data as reported by study authors
│Outcome (scale details*) │Intervention │Control │Effect estimate (metric)│95% CI │Statistical test│P value │
│Barry 2005 │% (N) │% (N) │ │ │ │ │
│Experience of labour │37% (246) │32% (223) │5% (RD) │ │ │P > 0.05 │
│Biro 2000 │n/N │n/N │ │ │ │ │
│Perception of care: labour/birth │260/344 │192/287 │1.13 (RR) │1.02 to 1.25│z = 2.36 │0.018 │
│Crowe 2010 │Mean (SD) N │Mean (SD) N │ │ │ │ │
│Experience of antenatal care (0 to 24 points) │21.0 (5.6) 182│19.7 (7.3) 186│1.3 (MD) │–0.1 to 2.7 │t = 1.88 │0.061 │
│Experience of labour/birth (0 to 18 points) │9.8 (3.1) 182 │9.3 (3.3) 186 │0.5 (MD) │–0.2 to 1.2 │t = 1.50 │0.135 │
│Experience of postpartum care (0 to 18 points) │11.7 (2.9) 182│10.9 (4.2) 186│0.8 (MD) │0.1 to 1.5 │t = 2.12 │0.035 │
│Flint 1989 │n/N │n/N │ │ │ │ │
│Care from staff during labour │240/275 │208/256 │1.07 (RR) │1.00 to 1.16│z = 1.89 │0.059 │
│Frances 2000 │ │ │ │ │ │ │
│Communication: labour/birth │ │ │0.90 (OR) │0.61 to 1.33│z = –0.52 │0.606 │
│Harvey 1996 │Mean (SD) N │Mean (SD) N │ │ │ │ │
│Labour & Delivery Satisfaction Index │182 (14.2) 101│185 (30) 93 │ │ │t = –0.90 for MD│0.369 for MD│
│(37 to 222 points) │ │ │ │ │ │ │
│Johns 2004 │n/N │n/N │ │ │ │ │
│Satisfaction with intrapartum care │605/1163 │363/826 │8.1% (RD) │3.6 to 12.5 │ │< 0.001 │
│Mac Vicar 1993 │n/N │n/N │ │ │ │ │
│Birth satisfaction │849/1163 │496/826 │13.0% (RD) │8.8 to 17.2 │z = 6.04 │0.000 │
│Parr 2002 │ │ │ │ │ │ │
│Experience of childbirth │ │ │0.85 (OR) │0.39 to 1.86│z = -0.41 │0.685 │
│Rowley 1995 │ │ │ │ │ │ │
│Encouraged to ask questions │ │ │1.02 (OR) │0.66 to 1.58│z = 0.09 │0.930 │
│Turnbull 1996 │Mean (SD) N │Mean (SD) N │ │ │ │ │
│Intrapartum care rating (–2 to 2 points) │1.2 (0.57) 35 │0.93 (0.62) 30│ │ │ │P > 0.05 │
│Zhang 2011 │N │N │ │ │ │ │
│Perception of antenatal care │359 │322 │1.23 (POR) │0.68 to 2.21│z = 0.69 │0.490 │
│Perception of care: labour/birth │355 │320 │1.10 (POR) │0.91 to 1.34│z = 0.95 │0.341 │
* All scales operate in the same direction; higher scores indicate greater satisfaction.
CI = confidence interval; MD = mean difference; OR = odds ratio; POR = proportional odds ratio; RD = risk difference; RR = risk ratio.
Table 12.4.b Scenario 1: intrapartum outcome table ordered by risk of bias, standardized effect estimates calculated for all studies
│Outcome* (scale details) │Intervention │Control │Mean difference (95% CI)**│Odds ratio │
│ │ │ │ │(95% CI)† │
│Low risk of bias │ │ │ │ │
│Barry 2005 │n/N │n/N │ │ │
│Experience of labour │90/246 │72/223 │ │1.21 (0.82 to 1.79) │
│Frances 2000 │n/N │n/N │ │ │
│Communication: labour/birth │ │ │ │0.90 (0.61 to 1.34) │
│Rowley 1995 │n/N │n/N │ │ │
│Encouraged to ask questions [during labour/birth] │ │ │ │1.02 (0.66 to 1.58) │
│Some concerns │ │ │ │ │
│Biro 2000 │n/N │n/N │ │ │
│Perception of care: labour/birth │260/344 │192/287 │ │1.54 (1.08 to 2.19) │
│Crowe 2010 │Mean (SD) N │Mean (SD) N │ │ │
│Experience of labour/birth (0 to 18 points) │9.8 (3.1) 182 │9.3 (3.3) 186 │0.5 (–0.15 to 1.15) │1.32 (0.91 to 1.92) │
│Harvey 1996 │Mean (SD) N │Mean (SD) N │ │ │
│Labour & Delivery Satisfaction Index │182 (14.2) 101│185 (30) 93 │–3 (–10 to 4) │0.79 (0.48 to 1.32) │
│(37 to 222 points) │ │ │ │ │
│Johns 2004 │n/N │n/N │ │ │
│Satisfaction with intrapartum care │605/1163 │363/826 │ │1.38 (1.15 to 1.64) │
│Parr 2002 │n/N │n/N │ │ │
│Experience of childbirth │ │ │ │0.85 (0.39 to 1.87) │
│Zhang 2011 │n/N │n/N │ │ │
│Perception of care: labour and birth │N = 355 │N = 320 │ │POR 1.11 (0.91 to 1.34)│
│High risk of bias │ │ │ │ │
│Flint 1989 │n/N │n/N │ │ │
│Care from staff during labour │240/275 │208/256 │ │1.58 (0.99 to 2.54) │
│Mac Vicar 1993 │n/N │n/N │ │ │
│Birth satisfaction │849/1163 │496/826 │ │1.80 (1.48 to 2.19) │
│Turnbull 1996 │Mean (SD) N │Mean (SD) N │ │ │
│Intrapartum care rating (–2 to 2 points) │1.2 (0.57) 35 │0.93 (0.62) 30│0.27 (–0.03 to 0.57) │2.27 (0.92 to 5.59) │
* Outcomes operate in the same direction. A higher score, or an event, indicates greater satisfaction.
** Mean difference calculated for studies reporting continuous outcomes.
† For binary outcomes, odds ratios were calculated from the reported summary statistics or were directly extracted from the study. For continuous outcomes, standardized mean differences were
calculated and converted to odds ratios (see Chapter 6).
CI = confidence interval; POR = proportional odds ratio.
Figure 12.4.b Forest plot depicting standardized effect estimates (odds ratios) for satisfaction
Box 12.4.b How to describe the results from this structured summary
│Scenario 1. Structured reporting of effects (no synthesis) │
│ │
│ │
│ │
│Table 12.4.b and Figure 12.4.b present results for the 12 included studies that reported a measure of maternal satisfaction with care during labour and birth (hereafter ‘satisfaction’). Results │
│from these studies were not synthesized for the reasons reported in the data synthesis methods. Here, we summarize results from studies providing high or moderate certainty evidence (based on │
│GRADE) for which results from a valid measure of global satisfaction were available. Barry 2015 found a small increase in satisfaction with midwife-led care compared to obstetrician-led care (4 │
│more women per 100 were satisfied with care; 95% CI 4 fewer to 15 more per 100 women; 469 participants, 1 study; moderate certainty evidence). Harvey 1996 found a small possibly unimportant │
│decrease in satisfaction with midwife-led care compared with obstetrician-led care (3-point reduction on a 185-point LADSI scale, higher scores are more satisfied; 95% CI 10 points lower to 4 │
│higher; 367 participants, 1 study; moderate certainty evidence). The remaining 10 studies reported specific aspects of satisfaction (Frances 2000, Rowley 1995, …), used tools with little or no │
│evidence of validity and reliability (Parr 2002, …) or provided low or very low certainty evidence (Turnbull 1996, …). │
│ │
│Note: While it is tempting to make statements about consistency of effects across studies (…the majority of studies showed improvement in …, X of Y studies found …), be aware that this may │
│contradict claims that a synthesis is inappropriate and constitute unintentional vote counting. │
12.4.2 Overview of scenarios 2–4: synthesis approaches
We now address three scenarios in which review authors have decided that the outcomes reported in the 15 studies all broadly reflect satisfaction with care. While the measures were quite diverse, a
synthesis is sought to help decision makers understand whether women and their birth partners were generally more satisfied with the care received in midwife-led continuity models compared with other
models. The three scenarios differ according to the data available (see Table 12.4.c), with each reflecting progressively less complete reporting of the effect estimates. The data available determine
the synthesis method that can be applied.
• Scenario 2: effect estimates available without measures of precision (illustrating synthesis of summary statistics).
• Scenario 3: P values available (illustrating synthesis of P values).
• Scenario 4: directions of effect available (illustrating synthesis using vote-counting based on direction of effect).
For studies that reported multiple satisfaction outcomes, one result is selected for synthesis using the decision rules in Box 12.4.a (point 2).
Table 12.4.c Scenarios 2, 3 and 4: available data for the selected outcome from each study
Scenario 2. Summary statistics Scenario 3. Combining P values Scenario 4. Vote counting
Study ID Outcome (scale details*) Overall RoB Available data** Stand. Available data** Stand. metric Available data** Stand.
judgement metric metric
(2-sided P value) (1-sided P
OR (SMD) value)
Continuous Mean (SD)
Crowe 2010 Expectation of labour/birth (0 to 18 points) Some concerns Intervention 9.8 (3.1); Control 9.3 1.3 (0.16) Favours intervention, 0.068 NS —
(3.3) P = 0.135, N = 368
Finn 1997 Experience of labour/birth (0 to 24 points) Some concerns Intervention 21 (5.6); Control 19.7 1.4 (0.20) Favours intervention, 0.030 MD 1.3, NS 1
(7.3) P = 0.061, N = 351
Harvey 1996 Labour & Delivery Satisfaction Index (37 to Some concerns Intervention 182 (14.2); Control 185 0.8 (–0.13) MD –3, P = 0.368, N = 0.816 MD –3, NS 0
222 points) (30) 194
Kidman 2007 Control during labour/birth (0 to 18 points) High Intervention 11.7 (2.9); Control 1.5 (0.22) MD 0.8, P = 0.035, N 0.017 MD 0.8 (95% CI 0.1 to 1
10.9 (4.2) = 368 1.5)
Turnbull 1996 Intrapartum care rating (–2 to 2 points) High Intervention 1.2 (0.57); Control 2.3 (0.45) MD 0.27, P = 0.072, N 0.036 MD 0.27 (95% CI0.03 to 1
0.93 (0.62) = 65 0.57)
Barry 2005 Experience of labour Low Intervention 90/246; 1.21 NS — RR 1.13, NS 1
Control 72/223
Biro 2000 Perception of care: labour/birth Some concerns Intervention 260/344; 1.53 RR 1.13, P = 0.018 0.009 RR 1.13, P < 0.05 1
Control 192/287
Flint 1989 Care from staff during labour High Intervention 240/275; 1.58 Favours intervention, 0.029 RR 1.07 (95% CI 1.00 to 1
Control 208/256 P = 0.059 1.16)
Frances 2000 Communication: labour/birth Low OR 0.90 0.90 Favours control, 0.697 Favours control, NS 0
P = 0.606
Johns 2004 Satisfaction with intrapartum care Some concerns Intervention 605/1163; 1.38 Favours intervention, 0.0005 RD 8.1% (95% CI 3.6% to 1
Control 363/826 P < 0.001 12.5%)
Mac Vicar Birth satisfaction High OR 1.80, P < 0.001 1.80 Favours intervention, 0.0005 RD 13.0% (95% CI 8.8% to 1
1993 P < 0.001 17.2%)
Parr 2002 Experience of childbirth Some concerns OR 0.85 0.85 OR 0.85, P = 0.685 0.658 NS —
Rowley 1995 Encouraged to ask questions Low OR 1.02, NS 1.02 P = 0.685 — NS —
Waldenstrom Perception of intrapartum care Low POR 1.23, P = 0.490 1.23 POR 1.23, 0.245 POR 1.23, NS 1
2001 P = 0.490
Zhang 2011 Perception of care: labour/birth Low POR 1.10, P > 0.05 1.10 POR 1.1, P = 0.341 0.170 Favours intervention 1
* All scales operate in the same direction. Higher scores indicate greater satisfaction.
** For a particular scenario, the ‘available data’ column indicates the data that were directly reported, or were calculated from the reported statistics, in terms of: effect estimate, direction of
effect, confidence interval, precise P value, or statement regarding statistical significance (either statistically significant, or not).
CI = confidence interval; direction = direction of effect reported or can be calculated; MD = mean difference; NS = not statistically significant; OR = odds ratio; RD = risk difference; RoB = risk of
bias; RR = risk ratio; sig. = statistically significant; SMD = standardized mean difference; Stand. = standardized.
12.4.2.1 Scenario 2: summarizing effect estimates
In Scenario 2, effect estimates are available for all outcomes. However, for most studies, a measure of variance is not reported, or cannot be calculated from the available data. We illustrate how
the effect estimates may be summarized using descriptive statistics. In this scenario, it is possible to calculate odds ratios for all studies. For the continuous outcomes, this involves first
calculating a standardized mean difference, and then converting this to an odds ratio (Chapter 10, Section 10.6). The median odds ratio is 1.32 with an interquartile range of 1.02 to 1.53 (15
studies). Box-and-whisker plots may be used to display these results and examine informally whether the distribution of effects differs by the overall risk-of-bias assessment (Figure 12.4.a, Panel
A). However, because there are relatively few effects, a reasonable alternative would be to present bubble plots (Figure 12.4.a, Panel B).
An example description of the results from the synthesis is provided in Box 12.4.c.
Box 12.4.c How to describe the results from this synthesis
│Scenario 2. Synthesis of summary statistics │
│ │
│ │
│ │
│‘The median odds ratio of satisfaction was 1.32 for midwife-led models of care compared with other models (interquartile range 1.02 to 1.53; 15 studies). Only five of the 15 effects were judged to │
│be at a low risk of bias, and informal visual examination suggested the size of the odds ratios may be smaller in this group.’ │
In Scenario 3, there is minimal reporting of the data, and the type of data and statistical methods and tests vary. However, 11 of the 15 studies provide a precise P value and direction of effect,
and a further two report a P value less than a threshold (<0.001) and direction. We use this scenario to illustrate a synthesis of P values. Since the reported P values are two-sided (Table 12.4.c,
column 6), they must first be converted to one-sided P values, which incorporate the direction of effect (Table 12.4.c, column 7).
Fisher’s method for combining P values involved calculating the following statistic:
where metap could be used. These packages include a range of methods for combining P values.
The combination of P values suggests there is strong evidence of benefit of midwife-led models of care in at least one study (P < 0.001 from a Chi^2 test, 13 studies). Restricting this analysis to
those studies judged to be at an overall low risk of bias (sensitivity analysis), there is no longer evidence to reject the null hypothesis of no benefit of midwife-led model of care in any studies
(P = 0.314, 3 studies). For the five studies reporting continuous satisfaction outcomes, sufficient data (precise P value, direction, total sample size) are reported to construct an albatross plot (
Figure 12.4.a, Panel C). The location of the points relative to the standardized mean difference contours indicate that the likely effects of the intervention in these studies are small.
An example description of the results from the synthesis is provided in Box 12.4.d.
Box 12.4.d How to describe the results from this synthesis
│Scenario 3. Synthesis of P values │
│ │
│ │
│ │
│‘There was strong evidence of benefit of midwife-led models of care in at least one study (P < 0.001, 13 studies). However, a sensitivity analysis restricted to studies with an overall low risk of │
│bias suggested there was no effect of midwife-led models of care in any of the trials (P = 0.314, 3 studies). Estimated standardized mean differences for five of the outcomes were small (ranging │
│from –0.13 to 0.45) (Figure 12.4.a, Panel C).’ │
In Scenario 4, there is minimal reporting of the data, and the type of effect measure (when used) varies across the studies (e.g. mean difference, proportional odds ratio). Of the 15 results, only
five report data suitable for meta-analysis (effect estimate and measure of precision; Table 12.4.c, column 8), and no studies reported precise P values. We use this scenario to illustrate vote
counting based on direction of effect. For each study, the effect is categorized as beneficial or harmful based on the direction of effect (indicated as a binary metric; Table 12.4.c, column 9).
Of the 15 studies, we exclude three because they do not provide information on the direction of effect, leaving 12 studies to contribute to the synthesis. Of these 12, 10 effects favour midwife-led
models of care (83%). The probability of observing this result if midwife-led models of care are truly ineffective is 0.039 (from a binomial probability test, or equivalently, the sign test). The 95%
confidence interval for the percentage of effects favouring midwife-led care is wide (55% to 95%).
The binomial test can be implemented using standard computer spreadsheet or statistical packages. For example, the two-sided P value from the binomial probability test presented can be obtained from
Microsoft Excel by typing =2*BINOM.DIST(2, 12, 0.5, TRUE) into any cell in the spreadsheet. The syntax requires the smaller of the ‘number of effects favouring the intervention’ or ‘the number of
effects favouring the control’ (here, the smaller of these counts is 2), the number of effects (here 12), and the null value (true proportion of effects favouring the intervention = 0.5). In Stata,
the bitest command could be used (e.g. bitesti 12 10 0.5).
A harvest plot can be used to display the results (Figure 12.4.a, Panel D), with characteristics of the studies represented using different heights and shading. A sensitivity analysis might be
considered, restricting the analysis to those studies judged to be at an overall low risk of bias. However, only four studies were judged to be at a low risk of bias (of which, three favoured
midwife-led models of care), precluding reasonable interpretation of the count.
An example description of the results from the synthesis is provided in Box 12.4.e.
Box 12.4.e How to describe the results from this synthesis
│Scenario 4. Synthesis using vote counting based on direction of effects │
│ │
│ │
│ │
│‘There was evidence that midwife-led models of care had an effect on satisfaction, with 10 of 12 studies favouring the intervention (83% (95% CI 55% to 95%), P = 0.039) (Figure 12.4.a, Panel D). │
│Four of the 12 studies were judged to be at a low risk of bias, and three of these favoured the intervention. The available effect estimates are presented in [review] Table X.’ │
Figure 12.4.a Possible graphical displays of different types of data. (A) Box-and-whisker plots of odds ratios for all outcomes and separately by overall risk of bias. (B) Bubble plot of odds ratios
for all outcomes and separately by the model of care. The colours of the bubbles represent the overall risk of bias judgement (green = low risk of bias; yellow = some concerns; red = high risk of
bias). (C) Albatross plot of the study sample size against P values (for the five continuous outcomes in Table 12.4.c, column 6). The effect contours represent standardized mean differences. (D)
Harvest plot (height depicts overall risk of bias judgement (tall = low risk of bias; medium = some concerns; short = high risk of bias), shading depicts model of care (light grey = caseload; dark
grey = team), alphabet characters represent the studies)
12.5 Chapter information
Authors: Joanne E McKenzie, Sue E Brennan
Acknowledgements: Sections of this chapter build on chapter 9 of version 5.1 of the Handbook, with editors Jonathan J Deeks, Julian PT Higgins and Douglas G Altman.
We are grateful to the following for commenting helpfully on earlier drafts: Miranda Cumpston, Jamie Hartmann-Boyce, Tianjing Li, Rebecca Ryan and Hilary Thomson.
Funding: JEM is supported by an Australian National Health and Medical Research Council (NHMRC) Career Development Fellowship (1143429). SEB’s position is supported by the NHMRC Cochrane
Collaboration Funding Program.
12.6 References
Achana F, Hubbard S, Sutton A, Kendrick D, Cooper N. An exploration of synthesis methods in public health evaluations of interventions concludes that the use of modern statistical methods would be
beneficial. Journal of Clinical Epidemiology 2014; 67: 376–390.
Becker BJ. Combining significance levels. In: Cooper H, Hedges LV, editors. A handbook of research synthesis. New York (NY): Russell Sage; 1994. p. 215–235.
Boonyasai RT, Windish DM, Chakraborti C, Feldman LS, Rubin HR, Bass EB. Effectiveness of teaching quality improvement to clinicians: a systematic review. JAMA 2007; 298: 1023–1037.
Borenstein M, Hedges LV, Higgins JPT, Rothstein HR. Meta-Analysis methods based on direction and p-values. Introduction to Meta-Analysis. Chichester (UK): John Wiley & Sons, Ltd; 2009. pp. 325–330.
Brown LD, Cai TT, DasGupta A. Interval estimation for a binomial proportion. Statistical Science 2001; 16: 101–117.
Bushman BJ, Wang MC. Vote-counting procedures in meta-analysis. In: Cooper H, Hedges LV, Valentine JC, editors. Handbook of Research Synthesis and Meta-Analysis. 2nd ed. New York (NY): Russell Sage
Foundation; 2009. p. 207–220.
Crowther M, Avenell A, MacLennan G, Mowatt G. A further use for the Harvest plot: a novel method for the presentation of data synthesis. Research Synthesis Methods 2011; 2: 79–83.
Friedman L. Why vote-count reviews don’t count. Biological Psychiatry 2001; 49: 161–162.
Grimshaw J, McAuley LM, Bero LA, Grilli R, Oxman AD, Ramsay C, Vale L, Zwarenstein M. Systematic reviews of the effectiveness of quality improvement strategies and programmes. Quality and Safety in
Health Care 2003; 12: 298–303.
Harrison S, Jones HE, Martin RM, Lewis SJ, Higgins JPT. The albatross plot: a novel graphical tool for presenting results of diversely reported studies in a systematic review. Research Synthesis
Methods 2017; 8: 281–289.
Hedges L, Vevea J. Fixed- and random-effects models in meta-analysis. Psychological Methods 1998; 3: 486–504.
Ioannidis JP, Patsopoulos NA, Rothstein HR. Reasons or excuses for avoiding meta-analysis in forest plots. BMJ 2008; 336: 1413–1415.
Ivers N, Jamtvedt G, Flottorp S, Young JM, Odgaard-Jensen J, French SD, O’Brien MA, Johansen M, Grimshaw J, Oxman AD. Audit and feedback: effects on professional practice and healthcare outcomes.
Cochrane Database of Systematic Reviews 2012; 6: CD000259.
Jones DR. Meta-analysis: weighing the evidence. Statistics in Medicine 1995; 14: 137–149.
Loughin TM. A systematic comparison of methods for combining p-values from independent tests. Computational Statistics & Data Analysis 2004; 47: 467–485.
McGill R, Tukey JW, Larsen WA. Variations of box plots. The American Statistician 1978; 32: 12–16.
McKenzie JE, Brennan SE. Complex reviews: methods and considerations for summarising and synthesising results in systematic reviews with complexity. Report to the Australian National Health and
Medical Research Council. 2014.
O’Brien MA, Rogers S, Jamtvedt G, Oxman AD, Odgaard-Jensen J, Kristoffersen DT, Forsetlund L, Bainbridge D, Freemantle N, Davis DA, Haynes RB, Harvey EL. Educational outreach visits: effects on
professional practice and health care outcomes. Cochrane Database of Systematic Reviews 2007; 4: CD000409.
Ogilvie D, Fayter D, Petticrew M, Sowden A, Thomas S, Whitehead M, Worthy G. The harvest plot: a method for synthesising evidence about the differential effects of interventions. BMC Medical Research
Methodology 2008; 8: 8.
Riley RD, Higgins JP, Deeks JJ. Interpretation of random effects meta-analyses. BMJ 2011; 342: d549.
Schriger DL, Sinha R, Schroter S, Liu PY, Altman DG. From submission to publication: a retrospective review of the tables and figures in a cohort of randomized controlled trials submitted to the
British Medical Journal. Annals of Emergency Medicine 2006; 48: 750–756, 756 e751–721.
Schriger DL, Altman DG, Vetter JA, Heafner T, Moher D. Forest plots in reports of systematic reviews: a cross-sectional study reviewing current practice. International Journal of Epidemiology 2010;
39: 421–429.
ter Wee MM, Lems WF, Usan H, Gulpen A, Boonen A. The effect of biological agents on work participation in rheumatoid arthritis patients: a systematic review. Annals of the Rheumatic Diseases 2012; 71
: 161–171.
Thomson HJ, Thomas S. The effect direction plot: visual display of non-standardised effects across multiple outcome domains. Research Synthesis Methods 2013; 4: 95–101.
Thornicroft G, Mehta N, Clement S, Evans-Lacko S, Doherty M, Rose D, Koschorke M, Shidhaye R, O’Reilly C, Henderson C. Evidence for effective interventions to reduce mental-health-related stigma and
discrimination. Lancet 2016; 387: 1123–1132.
Valentine JC, Pigott TD, Rothstein HR. How many studies do you need?: a primer on statistical power for meta-analysis. Journal of Educational and Behavioral Statistics 2010; 35: 215–247. | {"url":"https://training.cochrane.org/handbook/current/chapter-12","timestamp":"2024-11-13T12:57:32Z","content_type":"text/html","content_length":"188790","record_id":"<urn:uuid:ff089118-fb54-415a-a565-8ec55a3f221d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00360.warc.gz"} |
Assignment: Distribution of Sample Means
Question 1:
Scores on the math portion of the SAT (SAT-M) in a recent year have followed a normal distribution with mean μ = 507 and standard deviation σ = 111.
What is the probability that the mean SAT-M score of a random sample of 4 students who took the test that year is more than 600? Explain why you can solve this problem, even though the sample size (n
= 4) is very low.
Question 2:
Bags of a certain brand of potato chips say that the net weight of the contents is 35.6 grams. Assume that the standard deviation of the individual bag weights is 5.2 grams.
A quality control engineer selects a random sample of 35 bags. The mean weight of these 35 bags turns out to be 33.6 grams.
If the mean and standard deviation of individual bags is reported correctly, what is the probability that a random sample of 35 bags has a mean weight of 33.6 grams or less.
Question 3:
Does the sample provide strong evidence that the mean weight of the bags is lower than the 35.6 grams listed on the package? | {"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/assignment-distribution-of-sample-means/","timestamp":"2024-11-07T23:43:36Z","content_type":"text/html","content_length":"47301","record_id":"<urn:uuid:491cf36f-0397-4e3d-bfc0-1b88084a090a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00244.warc.gz"} |
Billable units for physical therapy
Forum - Questions & Answers
Sep 21st, 2017 - Ruwi
Billable units for physical therapy
What is the maximum billable number of units for physical therapy per day? Can I bill for longer than one hour in a day?
Oct 11th, 2017 - ChrisW 256 1
re: Billable units for physical therapy
It depends on what type of therapy and modality you are talking about. Are you billing time based codes? If so there is a rule of 8, you need to consider. When reporting 97100 you can bill for one or
more areas (each 15 minutes), in this case multiple units may be reported on a date of service.
The following information came from: Therapy Billing of Time-Based Codes (Medicare article A52758)
The following is a reference list to be used when converting direct time spent with the patient to billable units for all time-based code services:
1 unit > 8 minutes through 22 minutes
2 units > 23 minutes through 37 minutes
3 units > 38 minutes through 52 minutes
4 units > 53 minutes through 67 minutes
5 units > 68 minutes through 82 minutes
6 units > 83 minutes through 97 minutes
7 units > 98 minutes through 112 minutes
Example 1:
• 20 minutes of manual therapy, CPT 97140
• 20 minutes of therapeutic exercises, CPT 97110
Total time of direct treatment is 40 minutes. This allows for billing of 3 units. Since the direct treatment time for each service is the same, bill either code for 2 units and bill the other code
for 1 unit. Please note that it would be inappropriate to bill 3 units for either one of the codes. Correct coding would be either:
a. 2 units of CPT 97140 and 1 unit of CPT 97110 or
b. 2 units of CPT 97110 and 1 unit of CPT 97140.
Example 2:
• 35 minutes of manual therapy, CPT 97140
• 7 minutes of ultrasound, CPT 97035
Total time of direct treatment is 42 minutes, which allows for billing of 3 units. The first 30 minutes spent on CPT 97140 is counted as 2 full units (since the work unit for each CPT code is 15
minutes = 1 unit). The remaining time spent on CPT 97140 (5 minutes) is compared to the time spent on CPT 97035 (7 minutes) and the service that took more
time is the service that should receive the remaining unit. Correct coding would be 2 units of CPT 97140 and 1 unit of CPT 97035.
Other codes are only covered one per date of service, for example; mechanical traction, 97012 - Application of a modality to 1 or more areas; traction, mechanical, only 1 unit of CPT code 97012 is
generally covered per date of service.
For 97018 - Application of a modality to 1 or more areas; paraffin bath: Only 1 unit of CPT code 97018 is generally covered per date of service. | {"url":"https://www.codapedia.com/topicOpen.cfm?id=hbff3jrrbdj9tfkbqh4hzrqhdohvbrha","timestamp":"2024-11-10T02:51:25Z","content_type":"text/html","content_length":"44545","record_id":"<urn:uuid:ff1169e3-75f9-4244-a85a-cc5cfba5a664>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00622.warc.gz"} |
Solution: Google Kickstart Round H 2022 | Problem B: Magical Well Of Lilies
Problem Statement
There is a deep magical well in a forest that has some lilies on its waters. You have a large empty basket and some coins, and are standing next to the well. You have more coins than there are lilies
in the well. The well has taken note of the fact that your basket is empty.
If you toss one coin into the well, the well will toss out one lily into your basket. If you toss four coins at once into the well, the well will take note of how many lilies it has tossed out into
your basket so far. If you toss two coins at once into the well, the well will toss out as many lilies into your basket as it had last taken note of. If you toss one coin, or two coins at once, into
the well, and there are not enough lilies left in the well, the well will not toss out any lilies.
Given the number of lilies L in the well at the beginning, return the minimum number of coins you will need to toss into the well to make it toss all of its lilies into your basket.
1 ≤ L ≤ 100000
For test case 1, when there are 5 lilies in the well, the least number of coins needed is
5. We toss them, one at a time, into the well, and the well tosses out the 5 lilies, one at a time, into our basket. No other sequence of moves results in a better solution, so 5 is our answer.
Let’s narrow down the language of the question in simple terms , So in the question we have been given a start state ( 0 ) and a goal state ( L ) , and we have to reach the goal state after
performing some sets of operations Using minimum coins.
The operations are
1. Use 1 coin and jump to the next state i.e. i → i + 1.
2. Use 4 coins and take a checkpoint note of the current Lilies.
3. Use 2 coins and take a jump of x where x is the last registered checkpoint.
So our first intuition is to use the Brute Force way approach , which is recursion and we will try all possible ways and then minimise our answer.
But since L is too large , the above approach will lead to TLE.
Let’s try to think of dp. How can we introduce Dynamic programming in it?
We can see that if we have to reach the i + 1 th state from i th state then simply we can take the min( i+1 th state , 1 + i th state)
Let’s declare 1-D DP of size L+2. and initialize dp[i]=i. As we can use 1 coin to reach from previous state to current state hence i th state will take i coins.
Now , we will see each of the three steps described above.
1. We are at i th state , Now we will use 1 coin to go on i+1th state . Hence the transition will look like dp[ i + 1 ] = dp[ i ] + 1, But since there can be already a value assign to dp[ i + 1] we
will take minimum of both i.e. dp[ i + 1 ] = min( dp[ i ] + 1 , dp[ i + 1 ] ).
2. We will Use 4 coins to Record the current state . Let’s use a variable cur , Hence the statement will look like cur = dp[ i ] + 4.
3. Now we will Try and Update all the multiples of i th state with the Using 2 coins at each update. ( As we are adding the last noted lilies Hence we must only consider the multiples of the current
state). So for this we will use a for loop traversing i+i to L.
#include <bits/stdc++.h>
using namespace std;
int main()
int t;
cin >> t;
while(t — )
int l;
cin >> l;
vector<int> dp(l + 2);
for (int i = 0; i <= l;i++)
dp[i] = i;
for (int i = 2; i <= l;i++)
dp[i + 1] = min(dp[i] + 1, dp[i + 1]);
int cur = dp[i] + 4;
for (int j = i + i; j <= l; j += i)
cur += 2;
dp[j] = min(dp[j], cur);
cout << dp[l] << endl;
Time Complexity
For the time complexity , the outer loop is running for O(L) times , and for each index i from 2 to L . There are L/i multiples. So TC will be calculated like
O ( L/2 + L/3 + L/4 + L/5 + ……….. + L/L )
O ( L ( ½ + ⅓ + ¼ + ⅕ + ……….+ 1/L )
O(L log L )
As the ½ + ⅓ + ¼ + ⅕ + ….+ 1/L is a Harmonic Series And solution to this series is log L.
Space Complexity
Since we are using a 1-D Dp array , SC will be O(L).
You can also watch this solution LIVE on our YouTube channel now: https://youtu.be/pQvyU2mt7KI
Start your journey with Programming Pathshala.
Logon to www.programmingpathshala.com and take a free trial TODAY! | {"url":"https://programmingpathshala.medium.com/solution-google-kickstart-round-h-2022-problem-b-magical-well-of-lilies-36663b6feeb2?source=user_profile_page---------6-------------7a033e9b04f9---------------","timestamp":"2024-11-13T22:43:34Z","content_type":"text/html","content_length":"117472","record_id":"<urn:uuid:4e2e32c7-cad1-4287-9453-ffa08d31ca9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00816.warc.gz"} |
Binary-Binary transform_reduce(): The Missing Overload
While reviewing the C++17 CD and preparing my CppCon talk, I found some issues revolving around transform_reduce() and the parallel form of inner_product().
1. inner_product() Cannot Be Parallelized
The parallel version of inner_product() (e.g. ExecutionPolicy overload) is not useful because the wording limits parallelization. 26.8.5 [inner.product] p1 of the C++17 CD states that inner_product()
computes its result by initializing the accumulator acc with the initial value init and then modifying it with acc = acc + (*i1) * (*i2) or acc = binary_op1(acc, binary_op2(*i1,*i2))
Similar to accumulate(), while an inner_product() operation can be parallelized in principle, the wording for inner_product() prevents parallelization for reduction operations that are
non-commutative and non-associative, like floating point addition. For such operations, the wording forces each iteration to depend on the prior iteration to ensure the correct ordering of
accumulation. This introduces a loop-carried dependency that makes it impossible to parallelize.
2. inner_product() is a Form of transform_reduce()
Just as reduce() is the parallel counterpart of accumulate() (same basic operation but without a specific ordering), inner_product() has a natural counter-part. Consider this possible implementation
of inner_product():
template <class InputIt1, class InputIt2,
class T,
class ReductionOperation, class BinaryTransformOperation>
T inner_product(InputIt1 first1, InputIt1 last1,
InputIt2 first2, T init,
ReductionOperation reduce_op
BinaryTransformOperation transform_op)
while (first1 != last1)
init = reduce_op(init, transform_op(*first1, *first2));
return value;
The application of transform_op to the sequence is a binary transform():
template <class InputIt1, class InputIt2,
class OutputIt, class BinaryTransformOperation>
OutputIt transform(InputIt1 first1, InputIt1 last1, InputIt2 first2,
OutputIt o_first, BinaryTransformOperation transform_op)
while (first1 != last1)
*o_first++ = transform_op(*first1++, *first2++);
return o_first;
And the application of reduce_op accumulates the transformed values:
template <class InputIt, class T, class ReductionOperation>
T accumulate(InputIt first, InputIt last, T init,
ReductionOperation reduce_op)
for (; first != last; ++first)
init = reduce_op(init, *first);
return init;
So, inner_product() is an [S:accumulate()s:S] reduce()s of a transform()ed sequence. It’s a transform_reduce()!.
3. transform_reduce() is Missing an Overload
However, transform_reduce() is missing an overload for a binary transform; only overloads with unary transform operations are currently specified by the C++17 CD. I call this missing overload
binary-binary transform_reduce(), since both the reduction and transformation operation are binary operations.
This overload, which is equivalent to parallel inner_product() with weaker ordering, is very useful. Many of the typical examples of transform_reduce() usage that I’ve seen use tricks to perform a
binary-binary transform_reduce() using the unary-binary transform_reduce() that is in the C++17 CD.
The typical transform_reduce() dot product example (similar to what is found in the original transform_reduce() proposal [N3960]) looks like this:
std::vector<std::tuple<double, double> > XY = // ...
double dot_product = std::transform_reduce(
// Input sequence.
XY.begin(), XY.end(),
// Unary transformation operation.
[](std::tuple<double, double> const& xy)
// Array of struct means this is tricky to execute on vector hardware:
// memory layout: X[0] Y[0] X[1] Y[1] X[2] Y[2] X[3] Y[3] ...
// op #0: load a pack of Xs (they aren’t contiguous; the load will be
// strided, may need to access multiple
// cache lines and may be harder for the
// hardware prefetcher to handle)
// op #1: load a pack of Ys (same as above)
// op #2: multiply the pack of Xs by vector of Ys
return std::get<0>(xy) * std::get(xy);
double(0.0), // Initial value for reduction.
// Binary reduction operation.
Note that this is array-of-structs NOT struct-of-arrays. The HPX and THRUST dot product examples use iterator tricks (zip iterators or counting iterators) to switch to a struct-of-arrays scheme. The
zip iterator trick is shown below, using Boost:
std::vector<double> X = // ...
std::vector<double> Y = // ...
double dot_product = std::transform_reduce(
// Input sequence.
boost::make_zip_iterator(X.begin(), Y.begin()),
boost::make_zip_iterator(X.end(), Y.end()),
// Unary transformation operation.
[](auto&& xy) // std::tuple<double, double>-esque
// Struct of arrays means this is easier to execute on vector hardware:
// memory layout: X[0] X[1] X[2] X[3] ... Y[0] Y[1] Y[2] Y[3] ...
// op #0: load a pack of Xs (elements are contiguous, load will access
// only one cache line and the hardware
// prefetcher will easily track the memory
// stream)
// op #1: load a pack of Ys (same as above)
// op #2: multiply the pack of Xs by the pack of Ys
return std::get<0>(xy) * std::get<1>(xy);
double(0.0), // Initial value for reduction.
// Binary reduction operation.
More examples of this pattern can be found in HPX (zip iterator example and counting iterator example) and THRUST (zip iterator example).
4. transform_reduce() Parameter Order is Odd
The missing binary-binary transform_reduce() would look like this:
template <class InputIt1, class InputIt2,
class BinaryTransformOperation,
class T,
class ReductionOperation> // Always binary.
T transform_reduce(InputIt1 first1, InputIt1 last1,
InputIt2 first2,
BinaryTransformOperation transform_op,
T init,
ReductionOperation reduce_op);
Note that the order of parameters, which is consistent with the other transform_reduce() overloads, is inconsistent with inner_product(), transform() and reduce():
template <class InputIt1, class InputIt2,
class T,
class ReductionOperation, class BinaryTransformOperation>
T inner_product(InputIt1 first1, InputIt1 last1,
InputIt2 first2, T init,
ReductionOperation reduce_op
BinaryTransformOperation transform_op)
template <class InputIt1, class InputIt2,
class OutputIt, class BinaryTransformOperation>
OutputIt transform(InputIt1 first1, InputIt1 last1, InputIt2 first2,
OutputIt o_first, BinaryTransformOperation transform_op);
template <class InputIt, class T, class ReductionOperation>
T accumulate(InputIt first, InputIt last, T init,
ReductionOperation reduce_op);
The inconsistencies are:
• In inner_product(), accumulate() and reduce() (which is not shown but has the same interface as accumulate()), the initial value init parameter comes after the iterator parameters and before the
operation parameters. In transform_reduce(), it is in between the transformation operation parameter and the reduction operation parameter.
• In inner_product(), the reduction operation parameter comes before the transformation operation parameter.
• inner_product(), accumulate() and reduce() have overloads which do not take any operations and use operator+ for reduction and operator* for transformation.
5. Proposed Resolutions
I filed a US national body ballot comment about the issue with the parallel inner_product() overload. The following solutions would resolve that comment.
5.1. Resolution 1: Rename inner_product() to transform_reduce() and Fix transform_reduce()
Rename the useless parallel inner_product() with a binary-binary transform_reduce(), and we should change the transform_reduce() interface to have the same parameter order as inner_product(),
accumulate(), reduce() and transform().
Specifically, we should:
• Rename the ExecutionPolicy overload for inner_product() to transform_reduce().
• Add a new serial binary-binary transform_reduce() overload.
• Add two new binary-binary transform_reduce() overloads (one serial overload and one parallel ExecutionPolicy overload).
• Change the parameter order for all transform_reduce() overloads so that they are consistent with inner_product(), accumulate(), reduce() and transform():
□ The initial value parameter comes after the iterator parameters and before the operation parameters.
□ The reduction operation parameter comes before the transformation operation parameters.
• Add transform_reduce() overloads that do not take operation parameters and use operator+ for reduction and operator* for transformation.
With these changes, a parallel dot product over struct-of-arrays data can be written as:
std::vector<double> X = // ...
std::vector<double> Y = // ...
double dot_product =
std::transform_reduce(std::par_unseq, x.begin(), x.end(), y.begin(), double(0.0));
A parallel word count could be written with binary-binary transform_reduce() as:
bool is_word_beginning(char left, char right)
return std::isspace(left) && !std::isspace(right);
std::size_t word_count(std::string_view s)
if (s.empty()) return 0;
// Count the number of characters that start a new word.
return std::transform_reduce(
// "Left" input sequence: s[0], s[1], ..., s[s.size() - 2]
s.begin(), s.end() - 1,
// "Right" input sequence: s[1], s[2], ..., s[s.size() - 1]
s.begin() + 1,
// Initial value for reduction: if the first character
// is not a space, then it’s the beginning of a word.
std::size_t(!std::isspace(s.front()) ? 1 : 0),
// Binary transformation operation.
// Binary transformation operation: Return 1 when we hit
// a new word.
5.2. Resolution 2: Weaken inner_product()
Alternatively, we could weaken ordering required by inner_product(), which would make it equivalent to binary-binary transform_reduce(). This approach is not backwards compatible; implementations
could rewrite their serial inner_product() in a way that uses a different ordering than the one previously required, breaking user code. Additionally, different implementations might provide
different orderings, which could cause portably issues.
However, this would still leave transform_reduce() with an interface that is inconsistent with other algorithms that implement very similar behavior. If we ship transform_reduce() with a broken
interface, it will be difficult to fix it later because we would have to break users code.
5.3. Resolution 3: Weaken inner_product() and Fix transform_reduce()
Adopt Resolution 2, and also the parts of Resolution 1 that fix the inconsistencies in transform_reduce():
• Change the parameter order for all transform_reduce() overloads so that they are consistent with inner_product(), accumulate(), reduce() and transform():
□ The initial value parameter comes after the iterator parameters and before the operation parameters.
□ The reduction operation parameter comes before the transformation operation parameters.
• Add transform_reduce() overloads that do not take operation parameters and use operator+ for reduction and operator* for transformation.
5.4. Resolution 4: Remove Parallel inner_product()
The simplest approach, and least desirable, is to simply remove the parallel inner_product() overloads. This removes useful functionality which was intended to go into the standard (e.g. binary
transform_reduce()). Additionally, it would leave transform_reduce()s interface inconsistent with the other algorithms, and it will be difficult to fix that interface in the future because it would
be a breaking change.
6. Acknowledgement
Thanks to:
• Agustín Bergé for helping identify this issue.
• Hartmut Kaiser, JF Bastien, Michael Garland and Jared Hoberock for providing feedback. | {"url":"https://open-std.org/JTC1/SC22/WG21/docs/papers/2016/p0452r0.html","timestamp":"2024-11-07T13:24:40Z","content_type":"text/html","content_length":"93986","record_id":"<urn:uuid:87be8ca8-cfe7-4e05-8fee-8d087ca0b137>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00167.warc.gz"} |
CBSE Class 10 – Maths all chapters MCQ with answer free pdf download
Class 10 Maths multiple choice questions with answers pdf download
Students, Hope You Are Good and your Preparations Is Going Well. Welcome Back to
Study Phobia Official Website Where You can download free study materials and
can boost your Preparations.
In This Post
We Are Going to Provide Maths chapter-wise MCQ with answer for Class 10 2022
term 1 board exam, These Questions Are Very Important for Your Board
Examinations So download these pdf of Class 10 Maths MCQ with answer and study carefully
and Try to Improve Yourself.
These are
the best class 10 MCQ with answer as these pdf are made by the CBSE top teacher
who have many year experience in CBSE . According to CBSE new rules of exam,
there are 2 exams this year. Term 1 is completely objective questions answer
type so students are searching for some good quality of class 10^th Maths
chapter-wise MCQ with answer so that they can easily practice and score good
Now your
need and search of class 10^th Maths MCQ is over now. You can easily
download class 10 Maths all chapter MCQ with answer from below download link in
free of cost.
• Material name: class 10 Maths all chapters
mcq with answer
• Total chapters: 15
• Quality: best
• Size: 1 mb each pdf approx.
• Total questions: about 50 per chapters
• Type of questions: all type of Objective
SL. NO. CHAPTER NAME DOWNLOAD LINK
1. CBSE Class 10 Maths MCQ Chapter 1 Real Numbers DOWNLOAD NOW
2. CBSE Class 10 Maths MCQ Chapter 2 Polynomials DOWNLOAD NOW
3. CBSE Class 10 Maths MCQ Chapter 3 Pair Of Linear Equations In Two Variables DOWNLOAD NOW
4. CBSE Class 10 Maths MCQ Chapter 4 Quadratic Equations DOWNLOAD NOW
5. CBSE Class 10 Maths MCQ Chapter 5 Arithmetic Progression DOWNLOAD NOW
6. CBSE Class 10 Maths MCQ Chapter 6 Triangles DOWNLOAD NOW
7. CBSE Class 10 Maths MCQ Chapter 7 Coordinate Geometry DOWNLOAD NOW
8. CBSE Class 10 Maths MCQ Chapter 8 Introduction To Trigonometry DOWNLOAD NOW
9. CBSE Class 10 Maths MCQ Chapter 9 Some Applications Of Trigonometry DOWNLOAD NOW
10. CBSE Class 10 Maths MCQ Chapter 10 Circles DOWNLOAD NOW
11. CBSE Class 10 Maths MCQ Chapter 11 Constructions DOWNLOAD NOW
12. CBSE Class 10 Maths MCQ Chapter 12 Areas Related To Circles DOWNLOAD NOW
13. CBSE Class 10 Maths MCQ Chapter 13 Surface Areas And Volumes DOWNLOAD NOW
14. CBSE Class 10 Maths MCQ Chapter 14 Statistics DOWNLOAD NOW
15. CBSE Class 10 Maths MCQ Chapter 15 Probability DOWNLOAD NOW
class 10 maths mcq online test
ncert exemplar class 10 maths mcq
maths mcq class 10 all chapters
class 10 maths mcqs
mcq questions for class 10 maths polynomials
mcq class 10 maths chapter 1
mcq questions for class 10 icse maths
mcq questions for class 10 maths chapter 3
mcq questions for class 10 maths chapter 1
mcq questions for class 10 maths with answers pdf state
maths mcq class 10 all chapters
1 mark questions for maths class 10 pdf
mcq questions for class 10 maths polynomials
maths quiz questions with answers for class 10 pdf
mcq questions for class 10 icse maths
mcq questions for class 10 maths trigonometry | {"url":"https://studyphobia.com/cbse-class-10-maths-all-chapters-mcq-with-answer-free-pdf-download/","timestamp":"2024-11-07T21:49:39Z","content_type":"text/html","content_length":"132773","record_id":"<urn:uuid:289ad26d-ecb0-47e0-b296-cdc80d21bfe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00729.warc.gz"} |
Will I Ever Use Factoring in Real Life?
••• SARINYAPINNGAM/iStock/GettyImages
Factoring refers to the separation of a formula, number or matrix into its component factors. For example, 49 can be factored into two 7s, or x^2 − 9 can be factored into x − 3 and x + 3. This
is not a procedure used commonly in everyday life. Part of the reason is that the examples given in algebra class are so simple and that equations do not take such simple form in higher-level
classes. Another reason is that everyday life does not require use of physics and chemistry calculations, unless it is your field of study or profession.
High School Science
Second-order polynomials, e.g.:
x^2 + 2x + 4
are regularly factored in high school algebra classes, usually in ninth grade. Being able to find the zeros of such formulas is basic to solving problems in high school chemistry and physics classes
in the following year or two. Second-order formulas come up regularly in such classes.
Quadratic Formula
However, unless the science instructor has heavily rigged the problems, such formulas will not be as neat as they are presented in math class when simplification is used to help focus students on
factoring. In physics and chemistry classes, the formulas are more likely to come out looking something like:
4.9t^2 + 10t - 100 = 0
In such cases, the zeros are no longer mere integers or simple fractions as in math class. The quadratic formula must be used to solve the equation:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
This is the messiness of the real world entering into mathematical application, and because the answers are no longer as neat as you find in algebra class, more complex tools must be used to deal
with the added complexity.
In finance, a common polynomial equation that comes up is the calculation of present value. This is used in accounting when the present value of assets must be determined. It is used in asset (stock)
valuation. It is used in bond trading and mortgage calculations. The polynomial is of high order, for example, with an interest term with exponent 360 for a 30-year mortgage. This is not a formula
that can be factored. Instead, if the interest needs to be calculated, it is solved for by computer or calculator.
Numerical Analysis
This brings us into a field of study called numerical analysis. These methods are used when the value of an unknown can’t be solved for simply (e.g., by factoring) but must instead be solved for by
computer, using approximation methods that estimate the answer better and better with each iteration of some algorithm such as Newton’s method or the bisection method. These are the sorts of methods
used in financial calculators to calculate your mortgage rate.
Matrix Factorization
Speaking of numerical analysis, one use of factorization is in numerical computations to split a matrix into two product matrices. This is done to solve not a single equation but instead a group of
equations simultaneously. The algorithm to perform the factorization is itself far more complex than the quadratic formula.
The Bottom Line
Factorization of polynomials as it is presented in algebra class is effectively too simple to be used in everyday life. It is nevertheless essential to completing other high school classes. More
advanced tools are needed to account for the greater complexity of equations in the real world. Some tools can be used without understanding, e.g., in using a financial calculator. However, even
entering the data in with the correct sign and making sure the right interest rate is used makes factoring polynomials simple by comparison.
About the Author
Paul Dohrman's academic background is in physics and economics. He has professional experience as an educator, mortgage consultant, and casualty actuary. His interests include development economics,
technology-based charities, and angel investing. | {"url":"https://sciencing.com/ever-use-factoring-real-life-2459.html","timestamp":"2024-11-02T14:35:34Z","content_type":"text/html","content_length":"406074","record_id":"<urn:uuid:77f991de-fd0c-47f1-8228-b7627eae5b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00037.warc.gz"} |
Structural Analysis and Stability – Asymmetrically Propped Structures | EngineeringSkills.com | EngineeringSkills.com
Updated 17 May 2020
Reading time: 13 mins
Structural Analysis and Stability – Asymmetrically Propped Structures
Lateral stability of asymmetrically propped multi-storey structures with comparison to a finite element model
Structural Analysis and Stability – Asymmetrically Propped Structures
In the previous post in this two part series, we considered a multi-storey structure in which the stability elements were placed symmetrically within the floor plan. If you haven’t read that post, I
suggest going back and starting there. In that case, when considered on a 2D plan from above, the centre of mass for the structure (the point through which the resultant wind loading acts) is
coincident with the centre of stiffness. So, the loading is being applied through both the centre of mass and the centre of stiffness. When force passes through the centre of stiffness of any
structure, only translation of that structure will occur, i.e. no rotation occurs.
However, if the stabilising elements are not placed symmetrically (much more realistic!), the centre of stiffness may no longer be coincident with the centre of mass. As a result, twisting or
rotation of the structure occurs, about the centre of stiffness. So, our analysis must determine forces transmitted to the stabilising elements due to:
• linear translation of the structure along the line of action of the external force
• rotation of the structure about the centre of stiffness. If we assume small magnitude deflections, these rotations can be resolved into linear translations (which we will discuss below).
To work our way through this analysis we will consider the following amendments to the stabilising elements in last post:
• Shear wall B shortened from the south by 5 m
• The core moved west by 4 m and north by 3 m.
These amendments can be seen below. For this analysis let’s assume the design wind loading is from the south.
Fig 1. Asymmetrically stabilised structure
Locating the centre of stiffness (CoS)
The fact that the line of action of the resultant wind load no-longer passes through the CoS, means a torque or moment is induced. To evaluate this moment we must first locate the CoS. This can be
done by taking the sum of the ‘moments of stiffness‘ (analogous to taking the sum of the moments of a set of forces) about a convenient set of axes. For our calculations we will take the southern and
western edges of the floor plate as our x and y axes respectively. We can start by summarising the relative stiffness of each element in both the x (east-west) and y (north-south) direction:
\begin{align*} \hat{I}_{swa,NS} &= 2.048\times 10^{14}\\ \hat{I}_{swb,NS} &= 1.08\times 10^{13}\\ \hat{I}_{core,NS} &= 7.719\times 10^{14}\\ \hat{I}_{swa,EW} &= 5.12\times 10^{11}\\ \hat{I}_{swb,EW}
&= 1.92\times 10^{11}\\ \hat{I}_{core,EW} &= 2.66\times 10^{15}\\ \end{align*}
Taking moments about y-axis to determine the x-coordinate of the CoS
The x-coordinate of the CoS, $\overline{x}$ is obtained from the following equation:
$\overbrace{\Big(\sum \hat{I}_{NS}\Big)}^{\textup{Sum of all } \hat{I}_{NS} }\:\:\bar{x} = \overbrace{\sum_{i=1}^{n}\underbrace{\Big(\hat{I}_{i,NS}\:\:x_i\Big)}_{\textup{moment of stiffness}}}^{\
textup{Sum of components}} \tag{1}$
Putting this equation into action we can determine the x-coordinate of the CoS:
\begin{align*} (9.875\times 10^{14})\:\:\bar{x} &= (2.048\times 10^{14})(0.2)\\ &+(7.719\times 10^{14})(16)\\ &+(1.08\times 10^{13})(39.8)\\ \bar{x}&= 12.98 \:\textup{m (from eastern edge of floor
plate)} \end{align*}
Intuitively this makes sense if we think about the stiffer shear wall A ‘dragging‘ the centre of stiffness to left of the core. The less stiff shear wall B is unable to fully balance the effect.
Taking moments about x-axis to determine the y-coordinate of the CoS
Based on our previous discussion of the shear wall stiffness in the east-west direction we can see by inspection that they will have negligible influence in the CoS y-coordinate, so we would expect
to see the CoS in the y-direction lie in the centre of the core, 15 m from the southern edge of the floor plate. We can confirm this as follows:
\begin{align*} (2.664\times 10^{15})\:\:\bar{y} &= (5.12\times 10^{11})(12)\\ &+(2.66\times 10^{15})(15)\\ &+(1.92\times 10^{11})(14.5)\\ \bar{y}&= 15 \:\textup{m (from southern edge of floor plate)}
A revised sketch of the floor plate helps visualise our progress so far,
Fig 2. Floor plate showing individual element CoS and combined CoS for the structure with coordinates in square brackets.
The asymmetrical arrangement of stability elements induces a moment, $M_{wind}$, given by:
$M_{wind} = F_{wind}\times e \tag{2}$
This moment induces a twisting of the structure. The task now is to determine the corresponding force transmitted into each stability element by this twisting effect.
Forces due to rotation
Consider a rotation through $\theta$ radians about some arbitrary CoS. At a radius $r$ from the CoS, the circumferential displacement is denoted by $s$.
Fig 3. Displacement resulting from rotation through an angle $\theta$
However, for small angles, the linear displacement $\Delta$, is approximately equal to $s$. Thus we have:
$r\:\theta = s \approx \Delta \tag{3}$
We know from Hooke’s Law that the force $F$ that develops in an element $i$, is the product of the element’s stiffness $k_i$ and displacement, $\Delta_i$, therefore using equation (3), we have:
$F_i = k_i\:\Delta_i \tag{4}$
$F_i =k_i\:r_i\:\theta \tag{5}$
Now, the moment of resistance offered by $F_i$ in response to an externally applied moment is:
\begin{align*} M_i &= F_i\:r_i\\ M_i &=k_i\:r_i^2\:\theta \end{align*}
Since there must be moment equilibrium, i.e. the externally applied moment about the CoS, $M_{wind}$, must be balanced by the individual moments generated by the forces developed in each stability
element, $M_i$, we have,
\begin{align*} M_{wind} &= \sum M_i\\ &=\theta\:\sum k_i\:r_i^2 \end{align*}
Rearranging we have,
$\theta = \frac{M_{wind}}{\sum k_i\:r_i^2} \tag{6}$
Substituting equation (6) into equation (5) and rearranging, we obtain the component of force developed in each stability element as a result of the eccentrically applied external wind force (and the
moment that creates):
$F_i = M_{wind}\:\frac{k_i\:r_i}{\sum k_i\:r_i^2} \tag{7}$
Evaluating total force transmitted
Returning to our analysis, the calculation procedure hereafter is best summarised in a table, Tb. ??. Note that element stiffness, denoted as $k_i$ in equation (7) is replaced with our more familiar
relative stiffness $\hat{I}_i$.
Fig 4. Table to determine the distribution of forces due to rotation of the structure.
The force distributed into each stability element is calculated as the combination of force due to translation $F_t$ and rotation $F_r$. For all elements, $F_t$ will act in the direction of the
external wind load. We will call this sense of force direction positive. However the sense of $F_r$ will depend on the overall sense of rotation and the position of each element relative to the CoS.
So, the core and shear wall B experiences a positive force while shear wall A experiences a negative force. Noting that $M_{wind} = 1152\: \textup{kN}\times 7.02\:\textup{m} = 8087\:\textup{kNm}$,
the total force on each element is calculated as follows.
\begin{align*} F_{core} &= \overbrace{1152\Bigg(\frac{7.719\times 10^{14}}{9.875\times 10^{14}}\Bigg)}^{F_t} +\: \overbrace{8087\Bigg(\frac{2.331\times 10^{15}}{4.827\times 10^{16}}\Bigg)}^{F_r}\\ F_
{core} &= 900.5 + 390.5\\ F_{core} &= 1291\:\textup{kN (to the north)} \end{align*}
\begin{align*} F_{swa} &= 1152\Bigg(\frac{2.048\times 10^{14}}{9.875\times 10^{14}}\Bigg)-\: 8087\Bigg(\frac{2.617\times 10^{15}}{4.827\times 10^{16}}\Bigg)\\ F_{swa} &= 238.9 - 438.4\\ F_{swa} &=
-199.5\:\textup{kN (to the south)} \end{align*}
\begin{align*} F_{swb} &= 1152\Bigg(\frac{1.08\times 10^{13}}{9.875\times 10^{14}}\Bigg)+\: 8087\Bigg(\frac{2.9\times 10^{14}}{4.827\times 10^{16}}\Bigg)\\ F_{swb} &= 12.6+48.6\\ F_{swb} &= 61.2\:\
textup{kN (to the north)} \end{align*}
These force components are represented graphically below. Pay particular attention to the orientation of the rotational force components.
Fig 5. Forces transmitted into stabilising elements due to translation and rotation of the structure – wind from the south.
For completeness we can repeat the procedure, this time considering wind approaching from the east. We note that the total wind loading acting on the eastern facade is 691.2 kN. The resultant of this
force will act at the mid-point of the eastern facade. As a result, the lever arm about the CoS is 3 m, producing a clockwise moment of $M_{wind} = 2073.6$ kNm. Remember that the shear walls provide
no resistance to loading in this direction, so the full wind load is transmitted into the core. The components of force transmitted due to rotation are calculated as follows:
\begin{align*} F_{core} &= 2073.6\Bigg(\frac{2.331\times 10^{15}}{4.827\times 10^{16}}\Bigg)\\ F_{core} &= 100.1\:\textup{kN (to the south)} \end{align*}
\begin{align*} F_{swa} &= 2073.6\Bigg(\frac{2.617\times 10^{15}}{4.827\times 10^{16}}\Bigg)\\ F_{swa} &= 112.4 \:\textup{kN (to the north)} \end{align*}
\begin{align*} F_{swb} &= 2073.6\Bigg(\frac{2.9\times 10^{14}}{4.827\times 10^{16}}\Bigg)\\ F_{swb} &=12.46\:\textup{kN (to the south)} \end{align*}
These force components can again be represented graphically,
Fig 6. Forces transmitted into stabilising elements due to translation and rotation of the structure – wind from the east.
Wall Stresses and Model Validation
We can evaluate the normal stresses that develop in each stability element following the procedure outlined previously using the engineer’s bending equation. To conclude our discussion we will
evaluate the stresses that develop in shear wall B as a result of wind from the south. Finally, we will compare the results from our simplified hand analysis with those derived from a finite element
computer model of the structure to gauge their accuracy.
Normal stresses due to bending of shear wall B
Our analysis has indicated that shear wall B resists 61.2 kN of the external wind loading. By following the same procedure outlined previously, we can determine that the lateral load transmitted by a
typical floor, $P = 11.13$ kN. The load transmitted by the roof slab is therefore $\approx 5.6$ kN. Thus the bending moment at the base of the wall evaluates to 802.2 kNm. If we assume that the wall
bends about a neutral axis that passes through the centroid of the wall cross section, then $y = 1500$ mm and $I$ can be easily evaluated as:
\begin{align*} I &= \frac{400 \times 3000^3}{12}\\ I &= 9 \times 10^{11}\: \textup{mm}^4 \end{align*}
The maximum values of axial or normal stress that develop at the base of the wall can therefore be estimated as:
\begin{align*} \sigma_{max} &= \pm \frac{M_{base}\:y}{I}\\ &=\pm \frac{802.2\times 10^{6}\times 1500}{9\times 10^{11}}\\ &=\pm 1.34\:\textup{N/mm}^2 \end{align*}
This value is significantly higher than we observed for a symmetrical arrangement of stability elements. This is due to the fact that our structure now experiences torsion which induces additional
forces in the wall. Shear wall B has also been reduced in depth from 8 m to 3 m. This means that for a given magnitude of applied loading, it will develop larger stresses than a deeper wall subjected
to the same force.
Finite Element Analysis
The same task completed above can be carried out using an alternative method of analysis known as finite element analysis. The details of this analysis method are beyond the scope of this post,
however we can summarise by saying it is an analysis technique easily implemented using computer algorithms. As such, it is synonymous with analysis software. A well constructed FE model will often
give more accurate results in a shorter time and be easily modified allowing for iterative analysis and design.
A word of caution at this point; FE analysis and computer analysis in general is not a substitute for having a solid understanding of the physics and mechanics of a given problem. A good engineer
should always have the ability to reduce any problem to a simplified model suitable for hand analysis methods. That said, in a commercial environment, engineers must also be comfortable with
computerised methods of analysis and design.
The following analysis was performed in Autodesk Fusion 360. This is a cloud based 3D modelling and analysis package. If you’re a student you likely have access to this software free of charge.
Autodesk’s own tutorial material is excellent.
The image below shows a set of results indicating the qualitative deflection of the structure when wind loading is applied from the south as in our discussion above. The magnitude of deflection has
been exaggerated to make it more visible in the deflected shape diagrams, (c) and (d). The heat map indicates the relative magnitude of deflection across the structure from dark blue (no deflection)
to red (maximum deflection). As we would expect, deflection at the base of the walls is zero and this increases as we move vertically up through the structure (a-c). The counter-clockwise rotation of
the structure is also evident, particularly when viewed from above (d).
Fig 7. (a & b) Isometric views of the undeformed structure with relative deflection indicated by a heat map. (c) Isometric view showing a heap map on top of the deformed structure. (d) Plan view of
the deformed structure with heat map emphasising the rotation of the structure about a CoS.
Is should be noted that in order to determine an accurate estimate of the absolute deflection of the structure, we would need to provide the software with an accurate estimate of the secant modulus
for the concrete. This is approximately analogous to Young’s modulus and is quite difficult to accurately estimate. For this reason we are not going to focus on the actual magnitude of the deflection
in this analysis. In a real world design you certainly would want to estimate the deflection.
In the image below we can see how the stresses are distributed within the structure. Image (a) shows the full stress range mapped onto the structure. As expected the largest stresses occur in the
vertical stability elements. Imaged (b-d) show successively larger magnitudes of stress removed from the heat-map, leaving only higher magnitudes of stress indicated by the heat map. This allows us
to emphasis the areas where the stress is at its highest, at the base of the walls.
Fig 8. (a) heat map indicating magnitudes of Von Mises stress across the structure. (b-d) heat map with successively larger magnitudes of stress removed for clarity.
The stress depicted in the images above is called Von Mises stress. All we’re concerned with in this visualisation is the qualitative information it provides, i.e. the relative magnitude and location
of stresses in the structure. We can clearly see that the core and shear wall A are undergoing the most stress. Shear wall A in particular is developing relatively high levels of stress. As mentioned
above, this is due to its smaller depth and its relatively large distance away from the CoS. This means it undergoes large translations as a result of the rotation of the structure.
The image below shows us another stress heat map. However the stress indicated is the normal stress in the vertical direction. This is the stress that we have approximated in our hand calculation. We
can see that our FE analysis indicates a maximum normal stress at the base of shear wall A of $\approx$1.6 N/mm$^2$. This is in relatively good agreement with our approximation of 1.34 N/mm$^2$. The
negative stress on the right hand side of the wall is indicating compression with the positive stress indicating compression where we would expect to see it.
When we consider that both estimates of maximum stress have been determined using very different methods, the level of agreement is quite good. In reality, having an FE model is helpful as it allows
for relatively rapid design iterations to be performed, compared to running our hand analysis for every iteration. Having said that, it is good to see that straightforward hand analysis techniques
can provide quite accurate results, even if a little more time-consuming to complete.
Heat map showing normal stress in the vertical direction with maximum stress magnitude equal to $1.594 N/mm$.
Dr Seán Carroll
BEng (Hons), MSc, PhD, CEng MIEI, FHEA
Hi, I’m Seán, the founder of EngineeringSkills.com (formerly DegreeTutors.com). I hope you found this tutorial helpful. After spending 10 years as a university lecturer in structural engineering, I
started this site to help more people understand engineering and get as much enjoyment from studying it as I do. Feel free to get in touch or follow me on any of the social accounts.
Dr Seán Carroll's latest courses.
Do you have some knowledge or expertise you'd like to share with the EngineeringSkills community?
Check out our
guest writer programme
- we pay for every article we publish.
Featured Tutorials and Guides
If you found this tutorial helpful, you might enjoy some of these other tutorials. | {"url":"https://www.engineeringskills.com/posts/structural-analysis-lateral-stability","timestamp":"2024-11-04T11:34:34Z","content_type":"text/html","content_length":"1049149","record_id":"<urn:uuid:5027534c-ae25-4f5c-aef8-5a30649a44c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00612.warc.gz"} |
Solved and Solvability Tests for Cube Objects
solvable {cubing} R Documentation
Solved and Solvability Tests for Cube Objects
Determine if a cube is solved or solvable, and calculate the sign of the corner and edge permutations.
is.solved(aCube, split = FALSE, co = TRUE, eo = TRUE)
is.solvable(aCube, split = FALSE)
aCube Any cube object.
split Split output into logical vector? See Details.
co If FALSE, ignore corner orientation.
eo If FALSE, ignore edge orientation.
The cubieCube and stickerCube objects contain Rubik's cubes that can be physically constructed from properly stickered cubies, but they are not necessarily solvable. These functions test for solved
and solvable cubes. The parity function gives the permuation sign (+1 for even and -1 for odd) for the corner and edge permuations. For a cube to be solvable, the two signs must be the same.
For is.solved, a logical value for each separate permutation and orientation component will be given if split is TRUE.
For is.solvable, logical values corresponding to permuatation parity, edge orientation and corner orientation will be given if split is TRUE. The cube is only solvable if all three values are TRUE.
The edge and corner orientation values correspond to the fact that if all but one edge (or corner) orientation is known, then the orientation of the final edge (or corner) must be fixed for the cube
to be solvable. More precisely, the sum of the edge orientation vector must be even, and the sum of the corner orientation vector must be divisible by three.
A logical value or vector for is.solved and is.solvable. A named integer vector of length two for parity.
See Also
==.cube, randCube, solver
aCube <- randCube()
aCube <- randCube(solvable = FALSE)
version 1.0-5 | {"url":"https://search.r-project.org/CRAN/refmans/cubing/html/solvable.html","timestamp":"2024-11-14T12:24:32Z","content_type":"text/html","content_length":"4096","record_id":"<urn:uuid:e6914117-7104-469a-9642-5da94acbf39f>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00713.warc.gz"} |
Financial Health Indicators: Assessing Solvency, Liquidity, Profitability, and Efficiency
26.1.4 Financial Health Indicators
Understanding the financial health of a company is crucial for investors, creditors, and analysts. Financial health indicators provide insights into a company’s ability to meet its obligations,
generate profits, and efficiently manage its resources. This section will delve into the key indicators of financial health, including solvency, liquidity, profitability, and operational efficiency,
and how these metrics can be used to predict potential financial distress.
Solvency Indicators
Solvency indicators assess a company’s ability to meet its long-term obligations. They provide insights into the financial structure and stability of a company.
Debt Ratios
Debt ratios are critical in evaluating the proportion of a company’s capital that comes from debt. They help in understanding the level of financial leverage and risk.
Debt Ratio:
The debt ratio measures the extent of a company’s leverage. It is calculated as follows:
$$ \text{Debt Ratio} = \frac{\text{Total Liabilities}}{\text{Total Assets}} $$
A higher debt ratio indicates more leverage and potentially higher financial risk. However, it is essential to compare this ratio with industry norms to understand its implications fully.
Interest Coverage Ratio:
The interest coverage ratio evaluates a company’s ability to meet its interest obligations. It is calculated by dividing earnings before interest and taxes (EBIT) by interest expenses:
$$ \text{Interest Coverage Ratio} = \frac{\text{EBIT}}{\text{Interest Expenses}} $$
A higher ratio suggests that a company can comfortably meet its interest obligations, indicating financial stability.
Liquidity Indicators
Liquidity indicators measure a company’s ability to meet its short-term obligations. They are essential for assessing the company’s operational efficiency and financial flexibility.
Working Capital
Working capital is the difference between current assets and current liabilities. It indicates the short-term financial health of a company:
$$ \text{Working Capital} = \text{Current Assets} - \text{Current Liabilities} $$
Positive working capital suggests that a company can cover its short-term liabilities, while negative working capital may indicate potential liquidity issues.
Cash Conversion Cycle
The cash conversion cycle (CCC) measures the time taken to convert resource inputs into cash flows. It is a critical indicator of operational efficiency:
$$ \text{CCC} = \text{Days Inventory Outstanding} + \text{Days Sales Outstanding} - \text{Days Payable Outstanding} $$
A shorter CCC indicates efficient management of inventory and receivables, leading to better liquidity.
Profitability Indicators
Profitability indicators reflect a company’s ability to generate earnings relative to its revenue, assets, equity, and other financial metrics.
Gross, Operating, and Net Profit Margins
These margins provide insights into different levels of profitability:
• Gross Profit Margin: Measures the percentage of revenue that exceeds the cost of goods sold (COGS).
$$ \text{Gross Profit Margin} = \frac{\text{Revenue} - \text{COGS}}{\text{Revenue}} \times 100 $$
• Operating Profit Margin: Reflects the percentage of revenue left after covering operating expenses.
$$ \text{Operating Profit Margin} = \frac{\text{Operating Income}}{\text{Revenue}} \times 100 $$
• Net Profit Margin: Indicates the percentage of revenue that remains as profit after all expenses.
$$ \text{Net Profit Margin} = \frac{\text{Net Income}}{\text{Revenue}} \times 100 $$
Higher margins indicate better profitability and operational efficiency.
Return on Investment Ratios
These ratios measure how effectively a company uses its assets and equity to generate profits.
• Return on Assets (ROA): Indicates how efficiently a company uses its assets to generate profit.
$$ \text{ROA} = \frac{\text{Net Income}}{\text{Total Assets}} \times 100 $$
• Return on Equity (ROE): Measures the return generated on shareholders’ equity.
$$ \text{ROE} = \frac{\text{Net Income}}{\text{Shareholders' Equity}} \times 100 $$
Efficiency Indicators
Efficiency indicators assess how well a company uses its assets and liabilities to generate sales and maximize profits.
Asset Turnover Ratios
Asset turnover ratios show how efficiently a company uses its assets to generate revenue:
$$ \text{Asset Turnover Ratio} = \frac{\text{Revenue}}{\text{Total Assets}} $$
A higher ratio indicates efficient use of assets.
Days Sales Outstanding (DSO)
DSO measures the average number of days it takes to collect receivables:
$$ \text{DSO} = \frac{\text{Accounts Receivable}}{\text{Total Credit Sales}} \times \text{Number of Days} $$
A lower DSO indicates efficient credit and collection processes.
Warning Signs of Financial Distress
Identifying warning signs of financial distress is crucial for proactive financial management. Key indicators include:
• Declining Revenues or Margins: Consistent declines may indicate operational or market challenges.
• Increasing Debt Levels Without Corresponding Asset Growth: This can lead to solvency issues.
• Deteriorating Liquidity Ratios: A decline in liquidity ratios suggests potential cash flow problems.
Altman Z-Score
The Altman Z-Score is a formula that predicts bankruptcy risk using multiple financial ratios. It combines five financial ratios to assess the likelihood of financial distress:
$$ Z = 1.2 \times \text{Working Capital/Total Assets} + 1.4 \times \text{Retained Earnings/Total Assets} + 3.3 \times \text{EBIT/Total Assets} + 0.6 \times \text{Market Value of Equity/Total
Liabilities} + 1.0 \times \text{Sales/Total Assets} $$
A Z-Score below 1.8 indicates a high risk of bankruptcy, while a score above 3 suggests financial stability.
Example of Financial Health Analysis
Consider a company with the following financial data:
• Total Assets: $500,000
• Total Liabilities: $300,000
• EBIT: $50,000
• Interest Expenses: $10,000
• Revenue: $600,000
• Net Income: $40,000
• Shareholders’ Equity: $200,000
• Current Assets: $150,000
• Current Liabilities: $100,000
• Accounts Receivable: $50,000
• Total Credit Sales: $400,000
• Debt Ratio: \( \frac{300,000}{500,000} = 0.6 \) (60%)
• Interest Coverage Ratio: \( \frac{50,000}{10,000} = 5 \)
• Working Capital: \( 150,000 - 100,000 = 50,000 \)
• Gross Profit Margin: \( \frac{600,000 - 400,000}{600,000} \times 100 = 33.33% \)
• Net Profit Margin: \( \frac{40,000}{600,000} \times 100 = 6.67% \)
• ROA: \( \frac{40,000}{500,000} \times 100 = 8% \)
• ROE: \( \frac{40,000}{200,000} \times 100 = 20% \)
• DSO: \( \frac{50,000}{400,000} \times 365 = 45.625 \) days
This analysis indicates a financially stable company with strong profitability and efficient asset management.
Trend Analysis and Industry Context
Trend analysis involves comparing financial indicators over time to identify patterns and predict future performance. It is crucial to assess financial health relative to industry norms, as different
industries have varying benchmarks for financial ratios.
Limitations of Financial Indicators
While financial indicators provide valuable insights, they have limitations:
• Off-Balance-Sheet Liabilities: Ratios may not capture all liabilities, such as leases or contingent liabilities.
• Accounting Policies: Different accounting methods can affect reported figures, impacting ratio analysis.
Key Takeaways
• Comprehensive analysis of financial health indicators is essential for understanding a company’s viability.
• Both quantitative and qualitative factors should be considered in decision-making.
• Regular monitoring and trend analysis can help in identifying potential financial distress early.
By understanding and applying these financial health indicators, investors and analysts can make informed decisions, mitigate risks, and enhance their investment strategies.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What does a high debt ratio indicate? - [x] Higher financial leverage and risk - [ ] Lower financial leverage and risk - [ ] Better liquidity - [ ] Higher profitability > **Explanation:** A high
debt ratio indicates that a company has a significant portion of its capital structure financed through debt, which increases financial leverage and risk. ### How is the interest coverage ratio
calculated? - [x] EBIT divided by interest expenses - [ ] Net income divided by total liabilities - [ ] Total assets divided by total liabilities - [ ] Revenue divided by interest expenses >
**Explanation:** The interest coverage ratio is calculated by dividing earnings before interest and taxes (EBIT) by interest expenses, indicating a company's ability to meet its interest obligations.
### What does positive working capital indicate? - [x] Short-term financial strength - [ ] Long-term financial stability - [ ] High profitability - [ ] High debt levels > **Explanation:** Positive
working capital indicates that a company has more current assets than current liabilities, suggesting short-term financial strength and the ability to cover short-term obligations. ### What does a
lower cash conversion cycle indicate? - [x] Efficient management of inventory and receivables - [ ] Inefficient management of inventory and receivables - [ ] Higher profitability - [ ] Better
solvency > **Explanation:** A lower cash conversion cycle indicates that a company efficiently manages its inventory and receivables, leading to better liquidity and operational efficiency. ### Which
profitability margin reflects the percentage of revenue left after covering operating expenses? - [x] Operating Profit Margin - [ ] Gross Profit Margin - [ ] Net Profit Margin - [ ] Return on Assets
> **Explanation:** The operating profit margin reflects the percentage of revenue left after covering operating expenses, indicating operational efficiency. ### What does a higher ROA indicate? - [x]
Efficient use of assets to generate profit - [ ] Higher financial leverage - [ ] Better liquidity - [ ] Higher debt levels > **Explanation:** A higher return on assets (ROA) indicates that a company
efficiently uses its assets to generate profit, reflecting effective asset management. ### What does a lower DSO indicate? - [x] Efficient credit and collection processes - [ ] Inefficient credit and
collection processes - [ ] Higher profitability - [ ] Better solvency > **Explanation:** A lower days sales outstanding (DSO) indicates that a company efficiently manages its credit and collection
processes, leading to faster cash recovery. ### What is the Altman Z-Score used for? - [x] Predicting bankruptcy risk - [ ] Calculating profitability - [ ] Assessing liquidity - [ ] Measuring asset
turnover > **Explanation:** The Altman Z-Score is used to predict bankruptcy risk by combining multiple financial ratios to assess the likelihood of financial distress. ### What is a limitation of
financial indicators? - [x] They may not capture off-balance-sheet liabilities - [ ] They provide qualitative insights - [ ] They are always accurate - [ ] They do not require industry context >
**Explanation:** A limitation of financial indicators is that they may not capture off-balance-sheet liabilities, such as leases or contingent liabilities, which can affect financial analysis. ###
True or False: Trend analysis involves comparing financial indicators over time to identify patterns. - [x] True - [ ] False > **Explanation:** True. Trend analysis involves comparing financial
indicators over time to identify patterns and predict future performance, providing valuable insights into a company's financial health. | {"url":"https://csccourse.ca/26/1/4/","timestamp":"2024-11-15T04:40:48Z","content_type":"text/html","content_length":"95628","record_id":"<urn:uuid:1d9d137a-cefb-4ca1-996b-2bc6127fe4f9>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00503.warc.gz"} |
Each of the 43 points is placed either inside or on the surface of a perfect sphere.
Each of the 43 points is placed either inside or on the surface of a perfect sphere. If 16% or fewer of the points touch the surface, what is the maximum number of segments which, if connected from
those points to form chords, could be the diameter of the sphere?
Maximum number of points on the surface is 16%*43 = 6.88 ... or 6 since it has to be an integer
Now note that if two points form a diameter, they cannot be part of any other diameter.
So in the best case we can pair up the points
We have 6 points, so at best we can form 3 pairs (6).
So, answer is (A)
Geometry Calculator / Auto Solver
Each of the points is placed either inside or on the surface of a perfect sphere. If % or fewer of the points touch the surface, what is the maximum number of segments which, if connected from those
points to form chords, could be the diameter of the sphere?
We can not automatically calculate the result. Use the steps above to manually calculate the result. | {"url":"https://studentsource.org/answers/question.php/142747-43-points-surface-perfect-sphere","timestamp":"2024-11-09T20:12:23Z","content_type":"text/html","content_length":"11835","record_id":"<urn:uuid:3012558a-31e9-4100-92ec-79712ecc69ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00641.warc.gz"} |
Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro
Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences
Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces
families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin-Shapiro sequences, and in general it produces what we call Golay-Rudin-Shapiro
sequences. Calculations by Littlewood show that the Rudin-Shapiro sequences have low mean square autocorrelation. A sequence's peak sidelobe level is its largest magnitude of autocorrelation over all
nonzero shifts. Høholdt, Jensen, and Justesen showed that there is some undetermined positive constant A such that the peak sidelobe level of a Rudin-Shapiro sequence of length 2^n is bounded above
by A(1.842626…)^n, where 1.842626… is the positive real root of X^4-3 X-6. We show that the peak sidelobe level is bounded above by 5(1.658967…)^n-4, where 1.658967… is the real root of X^3+X^2-2
X-4. Any exponential bound with lower base will fail to be true for almost all n, and any bound with the same base but a lower constant prefactor will fail to be true for at least one n. We provide a
similar bound on the peak crosscorrelation (largest magnitude of crosscorrelation over all shifts) between the sequences in each Rudin-Shapiro pair. The methods that we use generalize to all families
of complementary pairs produced by the Golay-Rudin-Shapiro recursion, for which we obtain bounds on the peak sidelobe level and peak crosscorrelation with the same exponential growth rate as we
obtain for the original Rudin-Shapiro sequences. | {"url":"https://deepai.org/publication/peak-sidelobe-level-and-peak-crosscorrelation-of-golay-rudin-shapiro-sequences","timestamp":"2024-11-04T09:04:49Z","content_type":"text/html","content_length":"155389","record_id":"<urn:uuid:da2cb571-e4a3-4f71-90e7-a631c38c2ebf>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00571.warc.gz"} |
onomatopoeia — www.adamvackar.com
Installation view at Dojo, Nice
2013, steel, 3x4x2m
Installation view at Karlin Studios, Prague
2011, steel, 6x5x4m
Installation view at École Nationale Supérieure des Beaux ars, Paris
2011, steel, 6x5x4m
Installation view at Galerie Klenova, Klatovy, Czech Republic
Installation view at French embassy, Prague
Installation view at Colloredo-Mansfeld Pallace, Prague City Gallery, Prague
Adam Vackar, in his critical take on the omnipresent materialistic approach, draws on Mandelbrot’s definition of the chaotic process. The sculpture made of raw metal tubes represents geometrical
three-dimensional form of fractals. The term of fractals, mathematical phenomena discovered by French-American mathematician Benoit Mandelbrot is one of the classical chaos theories. It is commonly
used in various forms for predicting the unpredictable future; the most outstanding example is the prediction of fluctuation of stock market. A fractal is defined as a rough or fragmented geometric
shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called self-similarity. Through the form of sculpture, Vackar imminently creates dynamic non-linear
system; fractals and the process of subversive deconstruction of content. Geometric structures are a transition between order and chaos. In order to function, it follows a particular set of formulas.
Mandelbrot’s equation is able to predict the unpredictable behaviour of systems, such as prices or weather fluctuation. Its universal notion of an invisible structure surrounding us is present here.
The construction, which is made from “universal” metal pipes resembling the nature of the present condition, is rigid and expository. Fractal as a geometric model on many levels correlates with
Egyptian thinking about sculpture, where in order to achieve the final form, the quadrate of stone becomes the mass from which pieces are cropped. On a geometrical level, the object can thus be
considered as the initial form, extracted from the structure evolving into infinity. | {"url":"https://adamvackar.com/onomatopoeia","timestamp":"2024-11-02T08:14:46Z","content_type":"text/html","content_length":"244403","record_id":"<urn:uuid:1e5a9cef-839f-4ab6-8726-99ebfe9ef131>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00096.warc.gz"} |
Is function composition associative? | Socratic
Is function composition associative?
2 Answers
Given composable functions $f$, $g$ and $h$
$\left(f \circ \left(g \circ h\right)\right) \left(x\right)$
$= f \left(\left(g \circ h\right) \left(x\right)\right) = f \left(g \left(h \left(x\right)\right)\right) = \left(f \circ g\right) \left(h \left(x\right)\right)$
$= \left(\left(f \circ g\right) \circ h\right) \left(x\right)$
So $f \circ \left(g \circ h\right) = \left(f \circ g\right) \circ h$
It is, if the following works:
$\left(f \circ \left(g \circ h\right)\right) \left(x\right) = \left(\left(f \circ g\right) \circ h\right) \left(x\right)$
That is, if:
$f \left(x\right) = \text{something}$
$g \left(h \left(x\right)\right) = \left(g \circ h\right) \left(x\right) = \text{something else}$
$f \left(g \left(x\right)\right) = \left(f \circ g\right) \left(x\right) = \text{something else again}$
$h \left(x\right) = \text{something else yet again}$
...and you can use these together to satisfy the first expression, then they are associative. Let:
$f \left(x\right) = 2 x$
$g \left(x\right) = {x}^{2}$
$h \left(x\right) = {x}^{3}$
$g \left(h \left(x\right)\right) = g \left({x}^{3}\right) = {\left({x}^{3}\right)}^{2} = {x}^{6}$
$f \left(g \left(x\right)\right) = g \left({x}^{2}\right) = 2 \left({x}^{2}\right) = 2 {x}^{2}$
$\left(f \circ \left(g \circ h\right)\right) \left(x\right) = f \left({x}^{6}\right) = 2 \left({x}^{6}\right) = \textcolor{b l u e}{2 {x}^{6}}$
$\left(\left(f \circ g\right) \circ h\right) \left(x\right) = f \left(g \left({x}^{3}\right)\right) = f \left({\left({x}^{3}\right)}^{2}\right) = f \left({x}^{6}\right) = \textcolor{b l u e}{2 {x}^
Therefore they are associative.
Impact of this question
12745 views around the world | {"url":"https://socratic.org/questions/is-function-composition-associative","timestamp":"2024-11-07T04:00:01Z","content_type":"text/html","content_length":"34937","record_id":"<urn:uuid:1eba772f-edb2-4a7d-90fc-9533faf1d69b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00561.warc.gz"} |
T_(x rarr0)((1-e^(x))sin x)/(x^(2)+x^(3))= -Turito
Are you sure you want to logout?
A. 1
B. 0
C. -1
In this question, we have to find value of
The correct answer is: -1
Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/Maths-t-x-rarr0-1-e-x-sin-x-x-2-x-3-e-2-1-1-q6e1afb","timestamp":"2024-11-01T19:19:40Z","content_type":"application/xhtml+xml","content_length":"615776","record_id":"<urn:uuid:80f2a4b6-c57f-492b-8abe-00ddba3140c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00046.warc.gz"} |
Fix Overplotting with Colored Contour Lines
I saw
this plot
in the supplement of a
recent paper
comparing microarray results to RNA-seq results. Nothing earth-shattering in the paper - you've probably seen a similar comparison many times before - but I liked how they solved the overplotting
problem using heat-colored contour lines to indicate density. I
asked how to reproduce this figure using R on Stack Exchange
, and my question was quickly answered by
Christophe Lalanne
Here's the R code to generate the data and all the figures here.
Here's the problem: there are 50,000 points in this plot causing extreme overplotting. (This is a simple multivariate normal distribution, but if the distribution were more complex, overplotting
might obscure a relationship in the data that you didn't know about).
I liked the solution they used in the paper referenced above. Contour lines were placed throughout the data indicating the density of the data in that region. Further, the contour lines were "heat"
colored from blue to red, indicating increasing data density. Optionally, you can add vertical and horizontal lines that intersect the means, and a legend that includes the absolute correlation
coefficient between the two variables.
There are many other ways to solve an overplotting problem - reducing the size of the points, making points transparent, using hex-binning.
Using a single pixel for each data point:
Using hexagonal binning to display density (hexbin package):
Finally, using semi-transparency (10% opacity; easiest using the ggplot2 package):
Edit July 7, 2012 - From Pete's comment below, the smoothScatter() function in the build in graphics package produces a smoothed color density representation of the scatterplot, obtained through a
kernel density estimate. You can change the colors using the colramp option, and change how many outliers are plotted with the nrpoints option. Here, 100 outliers are plotted as single black pixels -
outliers here being points in the areas of lowest regional density.
How do you deal with overplotting when you have many points?
9 comments:
1. I'm a big fan of pch = "." for quick and dirty, but there are some really nice ideas here -- thanks! I recently stumbled across color density plots, but I cannot remember the package for the life
of me.
2. Very useful examples, Stephen. I like the smoothScatter function in the graphics library. For your example:
You can tweak a number of parameters including how and when outliers are displayed (via "nrpoints") and the colour gradient used (via "colramp").
3. Very useful examples, thanks so much
4. Pete - thanks very much. I haven't seen the smoothScatter() function. I updated the post to include that one!
5. Terrific side by side comparison of various over plotting techniques. Thanks for sharing.
6. Another useful function I've been happy with is 'densCols'
7. I like smoothscatter() the best.
8. This comment has been removed by the author.
9. Love this! Is there something similar in Python or are we left to our own devices? | {"url":"https://gettinggeneticsdone.blogspot.com/2012/07/fix-overplotting-with-colored-contour.html","timestamp":"2024-11-05T07:35:42Z","content_type":"application/xhtml+xml","content_length":"117625","record_id":"<urn:uuid:a73a9b27-6155-4824-9d3f-4cd346f2bbb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00241.warc.gz"} |
Measurement levels
Levels of measurement
Brief description of nominal, ordinal, interval, and ratio measurement level
• Nominal: categorical, with no ranking in the categories. For instance, religion.
• Ordinal: categorical, with ranked categories. For instance, socio-economic status.
• Interval: quantitative, where the difference between scores is meaningful. For instance, temperature, measured in celcius degrees. Also, sometimes Likert scales are considered interval scales.
• Ratio: quantitative, where the difference between scores is meaningful, as well as the zero value. For instance, body length. | {"url":"http://statkat.org/statistical-technique-selection/measurement-levels.php","timestamp":"2024-11-07T07:13:41Z","content_type":"text/html","content_length":"9405","record_id":"<urn:uuid:e622528c-70d2-429a-b973-21e430ef2f38>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00012.warc.gz"} |
ISP in 1, 2, 3
Thrust and Specific Impulse are the two most meaningful measures of a rocket engine. Thrust is easy to understand, and its units convey an idea about how more thrust can lift more payload. I[sp], on
the other hand, has a confusing unit that most people use to measure time. Therefore, many people expect explainations about what I[sp] is to begin with "The time it takes to...", and in fact many
people do try to contrive explanations in that format. Such explanations are not only confusing, they is also useless. When a rocket scientist talks to another rocket scientist about I[sp], he is
interested in conveying information about how efficient the engine is, and that efficiency is the mostly reflected in a single property: the speed that the exhaust mass leaves the engine. Physicists
refer to the amount of energy put into moving something as impulse, and rocket scientists divide that total amount of energy by the mass of propellent expelled to impart that energy. This is the
specific impulse. Conveniently, the value of the specific impulse of a rocket engine is the speed of the exhaust mass being expelled from that engine, so that will be our focus.
1. I[sp] is a measure of speed, not time, even though its units are seconds.
2. Germans and Americans disagree on which units to use to measure distance (meters vs feet), but agree that the second is a good unit to measure time.
3. Acceleration, and therefore gravity in a specific place, is measured as speed divided by time in any units.
1 foot is about 0.3 meters. One meter is about 3.3 feet. Given an exhaust speed measured at 1000 meters per second, Americans want to hear that it was 3300 feet per second. If a spec call for 3000
feet per second, Germans want to read the spec as 900 meters per second. Can we invent a unit to make them both unhappy?
Gravty accelerates objects in freefall. The Germans will tell you that the gravitational acceleration of the Earth near its surface is 9.81 meters per second, every second. Americans will tell you
that the gravitational acceleration of the Earth near its surface is 32.19 feet per second, every second. So after two seconds of freefall, the German expects his bratwurst to travel at 19.62 m/s and
the American expects his hamburger to travel at 64.38 f/s. That's the exact same speed in different units.
Now comes the magic.
If the German divides the speed of his lunch by the value of the Earth's gravitational acceleration, both in his own units, he gets:
19.62 m/s 2
---------- = ------ = 2 s
9.81 m/ss 1 1/s
If the American divides the speed of his lunch by the value of the Earth's gravitational acceleration, both in his own units, he gets:
64.38 f/s 2
---------- = ------ = 2 s
32.19 f/ss 1 1/s
They both agree on the value of 2 seconds! Not coincidentally, this value is the amount of time their respective meals were airborne. That's pretty much the definition of dividing speed by
acceleration, and an operation every first-year physics student has been doing since Newton first described the relationship between gravity, time, and speed.
That last observation may lead to an answer to the question "What is I[sp]" in the form of "The time it takes to...". Do you see it? Stop here if you want to think about it for a little bit, because
the answer is in the next paragraph.
An intuitive and relevant answer to the question "What is I[sp]" could be: The time it takes for a freefalling object to reach the speed of the exhaust mass of this engine. Could you imagine the
speed of a bratwurst falling for over six minutes?
Date Published: 2020-09-08 | {"url":"https://dotancohen.com/eng/isp.html","timestamp":"2024-11-13T22:25:50Z","content_type":"text/html","content_length":"9631","record_id":"<urn:uuid:0766978a-e61c-45ed-84d3-3e3b7863e8e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00133.warc.gz"} |
A generic structure of reachable and controllable positive linear systems is given in terms of some characteristic components (monomial subdigraphs) of the digraph of a non-negative a pair. The
properties of monomial subdigraphs are examined and used to derive reachability and controllability criteria in a digraph form for the general case when the system matrix may contain zero columns.
The graph-theoretic nature of these criteria makes them computationally more efficient than their known equivalents....
Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to
modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In
this paper, we present its application to stabilization of control systems, including the case when the system...
This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then
used for proving algebraic properties related to subdirect sums and Hadamard products. | {"url":"https://eudml.org/search/page?q=sc.general*op.AND*l_0*c_0author_0eq%253A1.Bru%252C+Rafael&qt=SEARCH","timestamp":"2024-11-05T15:52:50Z","content_type":"application/xhtml+xml","content_length":"85634","record_id":"<urn:uuid:73d1e7a2-ccb2-4fdd-8fdf-2b112dc11817>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00461.warc.gz"} |
A CityMap is a square grid that represents a simplified map of a city. It is a Grid whose elements are Demographic objects. That is, each GridPos (“street address”) on a CityMap is either Vacant or
stores a household that belongs to aparticular demographic.
A CityMap is mutable. As a household moves, the corresponding Demographic object moves to a different GridPos on the CityMap.
the number of street addresses in each row and each column of the city grid (which is always square)
a vector of pairs in which each Demographic is matched with a count of how many times it appears on this city map
Members list
Value members
Concrete methods
Generates the elements that initially occupy the grid. In the case of a CityMap grid, this means generating new Demographic objects at random locations so that their distribution matches the
populations parameter of the CityMap.
Generates the elements that initially occupy the grid. In the case of a CityMap grid, this means generating new Demographic objects at random locations so that their distribution matches the
populations parameter of the CityMap.
Determines whether the household at the given “street address” (GridPos) belongs to the given demographic.
Determines whether the household at the given “street address” (GridPos) belongs to the given demographic.
a location on the city grid to examine
a demographic that the household at the given address is compared to (may be Vacant, too, in which case this method checks if the address is vacant)
Inherited methods
Returns a collection of all the elements currently in the grid.
Returns a collection of all the elements currently in the grid.
Inherited from:
Returns a collection of all the locations on the grid.
Returns a collection of all the locations on the grid.
Inherited from:
Returns the element at the given pair of coordinates. (This does the same as elementAt.)
Returns the element at the given pair of coordinates. (This does the same as elementAt.)
a location on the grid (which must be within range or this method will fail with an error)
Inherited from:
Determines whether the grid contains the given pair of coordinates. For instance, a grid with a width and height of 5 will contain (0, 0) and (4, 4) but not (-1, -1), (4, 5) or (5, 4).
Determines whether the grid contains the given pair of coordinates. For instance, a grid with a width and height of 5 will contain (0, 0) and (4, 4) but not (-1, -1), (4, 5) or (5, 4).
Inherited from:
Returns the element at the given pair of coordinates. (This does the same as apply.)
Returns the element at the given pair of coordinates. (This does the same as apply.)
a location on the grid (which must be within range or this method will fail with an error)
Inherited from:
Returns a vector of all the neighboring elements of the element indicated by the first parameter. Depending on the second parameter, either only the four neighbors in cardinal compass directions
(north, east, south, west) are considered, or the four diagonals as well.
Returns a vector of all the neighboring elements of the element indicated by the first parameter. Depending on the second parameter, either only the four neighbors in cardinal compass directions
(north, east, south, west) are considered, or the four diagonals as well.
Note that an element at the grid’s edge has fewer neighbors than one in the middle. For instance, the element at (0, 0) of a 5-by-5 grid has only three neighbors, diagonals included.
true if diagonal neighbors also count (resulting in up to eight neighbors), false if only cardinal directions count (resulting in up to four)
the location between the neighbors
Inherited from:
Swaps the elements at two given locations on the grid. The given locations must be within range or this method will fail with an error.
Swaps the elements at two given locations on the grid. The given locations must be within range or this method will fail with an error.
Inherited from:
Modifies the grid by replacing the existing element at the given location with the new element.
Modifies the grid by replacing the existing element at the given location with the new element.
a location on the grid (which must be within range or this method will fail with an error)
the new element that replaces the old one at location
Inherited from:
Concrete fields
Returns all the locations in the city.
Returns all the locations in the city.
Inherited fields
the number of elements in this grid, in total. (Equals width times height.)
the number of elements in this grid, in total. (Equals width times height.)
Inherited from: | {"url":"https://gitmanager.cs.aalto.fi/static/O1_2023/modules/given/CitySim/doc/o1/city/CityMap.html","timestamp":"2024-11-07T02:46:25Z","content_type":"text/html","content_length":"46823","record_id":"<urn:uuid:d2c19a29-2fba-4d36-8144-8939896c542b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00233.warc.gz"} |
Sparse Representation in the Human Medial Temporal Lobe
Recent experiments characterized individual neurons in the human medial temporal lobe with remarkably selective, invariant, and explicit responses to images of famous individuals or landmark
buildings. Here, we used a probabilistic analysis to show that these data are consistent with a sparse code in which neurons respond in a selective manner to a small fraction of stimuli.
• representation
• sparseness
• MTL
• hippocampus
• memory
• neural coding
Single-unit recordings from the human medial temporal lobe (MTL) have revealed the existence of highly selective cells that may, for example, respond strongly to different images of a single
celebrity but not to 100 pictures of other people or objects (Quian Quiroga et al., 2005). These results suggest a sparse and invariant encoding in MTL and seem to imply the existence of grandmother
cells that respond to only a single category, individual, or object (Konorski, 1967; Barlow, 1972; Gross, 2002) [but see criticisms of this view in the study by Quian Quiroga et al. (2005)]. However,
because of limitations on the sampling of MTL neurons and on the sampling of the stimulus space, it is unclear how many stimuli a given neuron will respond to on average and, conversely, how many MTL
neurons are involved in the representation of a given object. Given the number of stimuli we present and the number of neurons from which we record, we used probabilistic reasoning to explore these
Materials and Methods
Let the sparseness a be the fraction of stimuli a neuron responds to or, alternatively, the probability that a neuron responds to a random stimulus (Treves and Rolls, 1991; Willmore and Tolhurst,
2001; Olshausen and Field, 2004). We assume that a neuron either fires or does not fire to any of the U stimuli making up the universe of stored representations (e.g., Jennifer Aniston, White House,
dachshund, iPod). At one extreme (that of a grandmother neuron) a = 1/U, whereas at the other extreme a fully distributed representation would have a = 1/2, that is, each neuron would respond to half
of all represented stimuli. In addition to quantifying how frequently a neuron will respond to a stimulus, this measure has been related to the theoretical storage capacity of autoassociative memory
networks (Meunier et al., 1991; Treves and Rolls, 1991).
In the following analysis, we make a few key assumptions. First, we assume the responses of all neurons can be treated in a binary manner, that is, we can define a threshold above which we consider a
neuron to have responded (and we examine how our results vary with this threshold). Second, we assume the stimulus presentations are independent, and further that the neuronal responses are
independent of one another (aside from any stimulus-induced correlations). The independence assumptions are consistent with our observation of no significant correlations between neurons in the
experimental data. Finally, we assume that all of our recorded neurons share the same underlying sparseness a. However, because our results are expressed as a probability density function over this
value, the width of the density function can be interpreted as describing the range of sparseness present in the MTL.
Suppose in a single experiment we present S stimuli to a binary neuron and count the number S[r] of responses. Let f[a] be the probability density function of the sparseness index a; approximately
speaking f[a](α)δα is the probability that a lies in an interval of size δα around some value α. We want to determine f[a](α | S[r] = s[r]), the probability density function for a given the observed
data. Our a priori estimate of f[a] is simply f[a](α) = 1 for 0 ≤ α ≤ 1, that is, a is equally likely to take on any value between 0 and 1. At a particular value of a, the probability that S[r] takes
on a value s[r] (between 0 and S), P[S[r] = s[r] | a= α], follows a binomial distribution, but if a is unknown, all responses are equally likely and so P[S[r] = s[r]] = 1/(S + 1). Applying Bayes'
rule we have the following: The responses of each cell thus yield a curve of the probability density of a given the response pattern of the cell. Figure 1 gives three examples of the curves that
could be generated using this method.
A limitation of this approach is that if, for example, two neurons are presented with the same 100 stimuli and neither responds, the true sparseness is likely to be much smaller than that implied by
the individual density curves (although the neurons may simply be unresponsive to any stimulus; see below). Because the original data were acquired using 64 microelectrodes, we extend our approach to
account for an experiment in which N neurons are recorded simultaneously while S stimuli are presented. We define N[r] to be the number of neurons that respond significantly to at least one stimulus
and S[r] to be the number of stimuli that produce a response in at least one of these. The derivation of the closed-form joint probability distribution of N[r] and S[r] involves solving a recursive
relation for the conditional distribution of S[r] given N[r] and is described in the supplemental methods (available at www.jneurosci.org as supplemental material). As in the single-neuron example
above, we can then apply Bayes' rule to find the probability density function for a given the results of a recording session as follows: Rather than obtaining a single curve for each cell, we now
obtain a single curve for each session that takes into account the presence of cells that did not respond to any stimulus or that responded to multiple stimuli.
We use this distribution to determine density functions for a from a data set of 1425 MTL units from 34 experimental sessions in 11 patients (Quian Quiroga et al., 2005). To fit the data against our
binary model, we considered a response to be significant if it was larger than the mean plus a threshold number of SDs of the baseline (before the onset of the image) and had at least two spikes in
the poststimulus time interval considered (0.3–1 s) [as in the previous study by Quian Quiroga et al. (2005)]. Figure 2 depicts the resulting average probability distributions for thresholds of three
and five SDs; for lower thresholds, many of the “responses” are a result of random fluctuations in firing rate rather than genuine responses to stimuli. For a threshold of five SDs above baseline,
the peaks of the 34 individual distributions lie in the range of 0.16–1.64%, with a mean peak location of 0.51% and a SD of 0.40%. For a threshold of three SDs above baseline, the individual curves
peak in the range of 0.52–3.08%, with a mean peak location of 1.21% and a SD of 0.63%. The peaks of the average distributions shown in Figure 2 are at a = 0.23 and 0.70% for thresholds of five and
three SDs, respectively, whereas the means are at a = 0.54 and 1.2%.
From this figure, we conclude that a most likely lies in the range of 0.2–1%. Although this is a sparse coding scheme, considering the large number of MTL neurons and the large number of represented
stimuli, it still results in a single unit responding to many stimuli and many MTL units responding to each stimulus. This is much sparser than responses obtained from the monkey superior temporal
sulcus, where a mean response sparseness of ∼33% has been reported (Rolls and Tovee, 1995), as well as responses from the monkey inferotemporal cortex, where sparse population coding has been
observed (Young and Yamane, 1992). We assume, however, that all cells we are listening to are involved in the representation of some stimulus, which may not be the case (i.e., some of them could
serve a different function altogether) and could cause a downward bias in our estimate. We believe this bias to be small, because repeating the same analysis leaving out half of the unresponsive
neurons yields estimates for the mean of a of 0.9 and 1.8% at thresholds of five and three SDs, respectively.
We can then estimate the probability of finding such highly selective cells in a given experiment. If the true sparseness is 0.54% (the mean of the distribution with a threshold of 5), in a typical
session with N = 42 simultaneously recorded units and S = 88 test stimuli (the averages from our experiments), we would expect to find on average 15.9 units responding to 17.9 stimuli (with each
responsive neuron responding on average to 1.3 images, and each evocative stimulus producing a response in an average of 1.1 neurons). In our experiments, N ranged from 18 to 74 and S ranged from 57
to 114, and with a five SD threshold, we found on average 7.9 responsive units (range, 3–20) responding to 16.4 stimuli (range, 3–44). As a further check of our methods, we can examine how frequently
two or more units responded to the same stimulus. At a five SD threshold, on average, 4.1% of stimuli produced a (simultaneous) response in at least two neurons (range, 0–17.9%; median, 1.6%),
compared with a predicted value (at 0.54% sparseness) of 2.2%. Noting that we cannot expect perfect agreement between this prediction and the observed value because of the varying numbers of neurons
and stimuli across recording sessions, we see that our model agrees very well with the observed statistics.
We developed a method for obtaining a probability distribution for sparseness based on multiple simultaneous neuronal recordings. This distribution allows us to not only examine the average
sparseness observed in a given experiment but also the range of sparseness consistent with the data. Averaging these distributions over 34 recording sessions in the human medial temporal lobe, we
conclude that highly sparse (although not grandmother) coding is present in this brain region.
To animate this discussion with some numbers, consider 0.54% sparseness level. Assuming on the order of 10^9 neurons in both left and right human medial temporal lobes (Harding et al., 1998; Henze et
al., 2000; Schumann et al., 2004), this corresponds to ∼5 million neurons being activated by a typical stimulus, whereas a sparseness of 0.23% implies activity in a bit more than 2 million neurons.
Furthermore, if we assume that a typical adult recognizes between 10,000 and 30,000 discrete objects (Biederman, 1987), a = 0.54% implies that each neuron fires in response to 50–150 distinct
This interpretation relies on the assumption that the cells from which we record are part of an object representation system. Instead, it may be possible that these neurons signal the recentness or
familiarity rather than the identity of a stimulus. Neurons responding to both novelty and familiarity have been identified in the human hippocampus (Rolls et al., 1982; Fried et al., 1997;
Rutishauser et al., 2006; Viskontas et al., 2006). Even if true, however, this view does not invalidate our conclusion that the true sparseness likely lies below 1%. Instead, it would imply that
rather than a single neuron responding to dozens of stimuli out of a universe of tens of thousands, such a neuron might respond to only one or a few stimuli out of perhaps hundreds currently being
tracked by this memory system, still with millions of neurons being activated by a typical stimulus.
Two significant factors may bias our estimate of sparseness upward. A large majority of neurons within the listening radius of an extracellular electrode are entirely silent during a recording
session (e.g., there are as many as 120–140 neurons within the sampling region of a tetrode in the CA1 region of the hippocampus (Henze et al., 2000), but we typically only succeed in identifying 1–5
units per electrode). In rats, as many as two of three cells isolated in the hippocampus under anesthesia may be behaviorally silent (Thompson and Best, 1989), although the reason for their silence
is unclear. Thus, the true sparseness could be considerably lower. Furthermore, there is a sampling bias in that we present stimuli familiar to the patient (e.g., celebrities, landmarks, and family
members) that may evoke more responses than less familiar stimuli. For these reasons, these results should be interpreted as an upper bound on the true sparseness, and some neurons may provide an
even sparser representation.
These results are consistent with Barlow's (1972) claim that “at the upper levels of the hierarchy, a relatively small proportion [of neurons] are active, and each of these says a lot when it is
active,” and his further speculation that the “aim of information processing in higher sensory centers is to represent the input as completely as possible by activity in as few neurons as possible” (
Barlow, 1972).
□ Received May 17, 2006.
□ Revision received August 2, 2006.
□ Accepted August 29, 2006.
• This work was supported by grants from the National Institute of Neurological Disorders and Stroke, the National Institute of Mental Health, the National Science Foundation, the Defense Advanced
Research Projects Agency, the Engineering and Physical Sciences Research Council, the Office of Naval Research, the Gordon Moore Foundation, and the Swartz Foundation for Computational
Neuroscience and by a Fannie and John Hertz Foundation fellowship to S.W. We thank Gabriel Kreiman for useful feedback.
• Correspondence should be addressed to Stephen Waydo, Control and Dynamical Systems, California Institute of Technology, 1200 East California Boulevard, M/C 107-81, Pasadena, CA 91125. waydo{at}
• Copyright © 2006 Society for Neuroscience 0270-6474/06/2610232-03$15.00/0 | {"url":"https://www.jneurosci.org/node/365368.full.print","timestamp":"2024-11-02T18:39:35Z","content_type":"application/xhtml+xml","content_length":"105528","record_id":"<urn:uuid:835350ee-a04e-42dc-b4c1-374192e268a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00356.warc.gz"} |
class astropy.coordinates.Latitude(angle, unit=None, **kwargs)[source]#
Bases: Angle
Latitude-like angle(s) which must be in the range -90 to +90 deg.
A Latitude object is distinguished from a pure Angle by virtue of being constrained so that:
-90.0 * u.deg <= angle(s) <= +90.0 * u.deg
Any attempt to set a value outside that range will result in a ValueError.
The input angle(s) can be specified either as an array, list, scalar, tuple (see below), string, Quantity or another Angle.
The input parser is flexible and supports all of the input formats supported by Angle.
anglearray, list, scalar, Quantity, Angle
The angle value(s). If a tuple, will be interpreted as (h, m, s) or (d, m, s) depending on unit. If a string, it will be interpreted following the rules described for Angle.
If angle is a sequence or array of strings, the resulting values will be in the given unit, or if None is provided, the unit will be taken from the first given value.
unitastropy:unit-like, optional
The unit of the value specified for the angle. This may be any string that Unit understands, but it is better to give an actual unit object. Must be an angular unit.
If a unit is not provided or it is not an angular unit.
If the angle parameter is an instance of Longitude. | {"url":"https://docs.astropy.org/en/latest/api/astropy.coordinates.Latitude.html","timestamp":"2024-11-15T00:07:23Z","content_type":"text/html","content_length":"34935","record_id":"<urn:uuid:3c1cf0f3-1d96-42bd-810a-62593c4f28a7>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00801.warc.gz"} |
Seeking Explanation of how this ANCOVA result in R shows the group means are not equal ~ Cross Validated ~ TransWikia.com
The answer is as I suspected. The intercepts are adjusted due to the covariate, and if one of the p-values of the two grouping variables is significant, then the adjusted means of all of the groups
are not equal.
Answered by Kyle on October 13, 2020
I may be misunderstanding what you're asking here, so please clarify if I have misread something. As currently written, my understanding is that you have a factor, IndVar1, with some number of levels
and another factor, IndVar2, with also some number of levels. Your question is then how the results from the ANCOVA would show that the means of at least one level of the IndVar1 factor differ from
the means of at least one level of the IndVar2 factor.
If that is correct, then you need to include an interaction term in the model. Right now, the ANOVA table just shows that there is at least one mean difference within the factors. So, say that
IndVar1 has 3 levels. These results tell you that there is at least one of those levels that is significantly different from the other two. So, you know that IndVar1 and IndVar2 both have significant
main effects even after controlling for some other variable (i.e., Covar).
Running the following code instead will add the interaction term and let you know more about how the independent variables relate to one another:
Model_1 <- aov(DependentVar ~ IndVar1 + IndVar2 + IndVar1:Indvar2 + Covar, data = Dataset)
and just to be complete, a slightly more parsimonious way of running the two-way ANCOVA would be like this:
Model_1 <- aov(DependentVar ~ IndVar1*IndVar2 + Covar, data = Dataset)
Answered by Billy on October 13, 2020 | {"url":"https://transwikia.com/cross-validated/seeking-explanation-of-how-this-ancova-result-in-r-shows-the-group-means-are-not-equal/","timestamp":"2024-11-11T18:29:11Z","content_type":"text/html","content_length":"47972","record_id":"<urn:uuid:fd6d7fe5-22c3-4f32-af27-712c7ea88820>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00802.warc.gz"} |
Fraction Calculator - Powerful and Useful Online Calculators
Fraction Calculator
Have you ever found yourself struggling with fraction calculations and wished for a tool to simplify the process? Understanding fractions and performing operations like addition, subtraction,
multiplication, and division can be a bit overwhelming, especially if math isn’t your strongest suit. Thankfully, a Fraction Calculator Widget can be a great help. So, how exactly can you use this
tool and what do you need to know about fractions to make the most out of it?
Fraction Calculator Widget
What is a Fraction Calculator?
A Fraction Calculator is a free online tool that assists you in performing various mathematical operations involving fractions. It speeds up the calculation process by highlighting the steps you need
to take, making it easier for you to understand the solution. Whether you’re trying to add, subtract, multiply, or divide fractions, this calculator simplifies the entire process for you.
Key Functions of the Fraction Calculator
The Fraction Calculator can handle several types of operations:
1. Addition of Fractions
2. Subtraction of Fractions
3. Multiplication of Fractions
4. Division of Fractions
5. Calculating a Fraction of a Fraction
How to Use the Fraction Calculator
Using the fraction calculator is straightforward. Here’s a step-by-step guide:
1. Input the Fractions: Enter the fractions into the provided boxes, which are formatted for easy input, such as ( \frac ), ( \frac ), or ( \frac ).
2. Select the Operator: Choose the appropriate operator (addition, subtraction, multiplication, or division) from the available options.
3. Execute the Calculation: Click the “calculate” button. The answer will be displayed along with detailed steps showing how the calculation was performed.
Problems This Fraction Calculator Solves
This fraction calculator can save you a lot of time and effort. Here are some common problems it can help you with:
Addition of Fractions
Whether the fractions have the same or different denominators, the calculator can help you find the correct sum quickly.
Subtraction of Fractions
Just like addition, the calculator assists in the subtraction of fractions, regardless of whether they share the same denominator or not.
Multiplication of Fractions
Multiplying fractions becomes simple. The calculator multiplies the numerators and denominators for you and simplifies the result if necessary.
Division of Fractions
The fraction calculator can handle the division of fractions by inverting the second fraction and converting the operation into multiplication.
Finding a Fraction of a Fraction
The Calculator can even help when you need to find a fraction of another fraction, simplifying the process for you.
Performing Mathematical Operations without a Fraction Calculator
For those interested in performing fraction calculations manually, it’s essential to know the steps involved for each operation. Here’s how you can do it:
Adding Fractions
Fractions with a Common Denominator
Adding fractions with the same denominator is straightforward. You just add the numerators and keep the same denominator:
[ \frac + \frac = \frac{(5+2)} = \frac ]
Fractions with Different Denominators
When the denominators are different, you need to find a common denominator before adding the numerators:
[ \frac + \frac = \frac{(4 \times 7)}{(5 \times 7)} + \frac{(3 \times 5)}{(7 \times 5)} = \frac + \frac = \frac{(28+15)} = \frac = 1\frac ]
Adding Two Mixed Fractions
To add mixed fractions, you can either convert them to improper fractions or add the whole numbers and fractions separately.
Subtracting Fractions
Common Denominator
For fractions with the same denominator, subtract the numerators and keep the same denominator:
[ \frac – \frac = \frac{(4-1)} = \frac ]
Different Denominators
For different denominators, find a common denominator first, then subtract the numerators:
[ \frac – \frac = \frac – \frac = \frac ]
Multiplying Fractions
To multiply fractions, simply multiply the numerators and the denominators:
[ \frac \times \frac = \frac{(2 \times 5)}{(3 \times 6)} = \frac ]
You can further simplify by dividing both numerator and denominator by their Greatest Common Factor:
[ \frac = \frac ]
Dividing Fractions
When dividing fractions, invert the second fraction (swap the numerator and denominator) and multiply:
[ \frac{\frac}{\frac} = \frac \times \frac = \frac{(1 \times 5)}{(2 \times 4)} = \frac ]
Finding a Fraction of a Fraction
You follow the same steps as multiplication:
[ \frac \ of \ \frac = \frac{(2 \times 4)}{(5 \times 5)} = \frac ]
Types of Fractions
Understanding the types of fractions will help you use the calculator more effectively and understand the results better. Here are the different types:
Proper Fractions
A fraction with the numerator smaller than the denominator:
[ \frac, \frac, \frac ]
Improper Fractions
A fraction where the numerator is greater than the denominator:
[ \frac, \frac, \frac ]
Mixed Fractions
A combination of a whole number and a fraction:
[ 2\frac, 3\frac, 17\frac ]
Like Fractions
Fractions with the same denominator:
[ \frac, \frac, \frac ]
Unlike Fractions
Fractions with different denominators:
[ \frac, \frac, \frac ]
Equivalent Fractions
Fractions that can be simplified to the same value:
[ \frac, \frac, \frac ]
These can all be simplified to ( \frac ).
Complex Fractions
Fractions with a fraction in the numerator, denominator, or both:
[ \frac{\frac}{\frac} ]
Unit Fractions
A fraction with 1 in the numerator and a whole number in the denominator:
[ \frac, \frac, \frac ]
Related Calculators
Sometimes, you may need additional calculation tools related to fractions. Here are a few related calculators that might be useful:
Calculator Name Purpose
Decimal to Fraction Calculator Converts decimals to fractions
Simplifying Fractions Calculator Simplifies a given fraction
Fraction to Decimal Calculator Converts fractions to decimals
Mixed Fraction Calculator Handles calculations involving mixed fractions
Fraction to Percent Calculator Converts fractions to percentages
Equivalent Fractions Calculator Finds equivalent fractions
Adding Fractions Calculator Adds two or more fractions
These calculators can be found online and can assist you with various mathematical tasks, enhancing your understanding and simplifying complex calculations.
The Fraction Calculator Widget is an invaluable resource for anyone needing to perform operations involving fractions. It doesn’t just give you the answer; it helps you learn and understand the steps
involved in the calculations. Whether you’re adding, subtracting, multiplying, or dividing fractions, this tool can make the process quick and easy. Additionally, understanding the different types
of fractions and having access to related calculators can further enhance your mathematical proficiency.
So the next time you find yourself faced with a tricky fraction problem, give the Fraction Calculator Widget a try. It might just make your mathematical journey a little smoother and more enjoyable! | {"url":"https://calculatorbeast.com/fraction-calculator-widget/","timestamp":"2024-11-13T11:41:24Z","content_type":"text/html","content_length":"135107","record_id":"<urn:uuid:478e6a35-f547-44bf-a080-c7ee91792e02>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00540.warc.gz"} |
Rudolf Clausius (1822-1888)
Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie
Annalen der Physik und Chemie, 125, 353- (1865) [translated and excerpted in William Francis Magie, A Source Book in Physics [New York: McGraw-Hill, 1935]
We obtain the equation[1] S.
If we wish to designate S by a proper name we can say of it that it is the transformation content of the body, in the same way that we say of the quantity U that it is the heat and work content of
the body. However, since I think it is better to take the names of such quantities as these, which are important for science, from the ancient languages, so that they can be introduced without change
into all the modern languages, I proposed to name the magnitude S the entropy of the body, from the Greek word η τροπη, a transformation. I have intentionally formed the word entropyso as to be as
similar as possible to the word energy, since both these quantities, which are to be known by these names, are so nearly related to each other in their physical significance that a certain similarity
in their names seemed to me advantageous.
Finally I may allow myself to touch on a matter whose complete treatment would not be in place here, because the statements necessary for that purpose would take up too much room, but of which I
believe that even the following short indication will not be without interest, in that it will contribute to the recognition of the importance of the quantities which I have introduced into the
formulation of the second law of the mechanical theory of heat. The second law, in the form which I have given it, states the fact that all transformations which occur in nature occur in a certain
sense which I have taken as positive, of themselves, that is, without compensation, but that they can only occur in the opposite or negative sense in such a way that they are compensated by positive
transformations which occur at the same time. The application of this law to the universe leads to a conclusion to which W. Thomson first called attention and about which I have already spoken in a
recently published paper. This conclusion is that if among all the changes of state which occur in the universe the transformations in one sense exceed in magnitude those in the opposite sense, then
the general condition of the universe will change more and more in the former sense, and the universe will thus persistently approach a final state.
The question now arises whether this final state can be characterised in a simple and also a definite way. This can be done by treating the transformations, as I have done, as mathematical
quantities, whose equivalent values can be calculated and united in a sum by algebraic addition.
In my papers so far published I have carried out such calculations with respect to the heat present in bodies and to the arrangement of the constituents of the bodies. For each body there are found
two quantities, the transformation value of its heat content and its disgregration, the sum of which is its entropy. This however does not complete the business. The discussion must also be extended
to the radiant heat, or otherwise expressed, to the heat transmitted through the universe in the form of advancing vibrations of the ether, and also to such motions as cannot be comprehended under
the name heat.
The treatment of these latter motions, at least as far as they are the motions of ponderable masses, can be briefly settled, since we come by a simple argument to the following conclusion: If a mass,
which is so great that in comparison with it an atom may be considered as vanishingly small, moves as a whole, the transformation value of this motion is to be looked on as vanishingly small in the
same way in comparison with its kinetic energy; from which it follows that if such a motion is transformed into heat by a passive resistance, then the equivalent value of the uncompensated
transformation which then occurs is simply represented by the transformation value of the heat produced. The radiant heat, however, cannot be treated so briefly, since there is need still of a
certain special treatment in order to find out how its transformation value is to be determined. Although, in the paper which was recently published and to which I have previously referred, I have
already discussed radiant heat in its connection with the mechanical theory of heat, yet I have not as yet treated the question which has here come up, since it was then only my purpose to prove that
there was no contradiction between the laws of radiant heat and a fundamental law which I assumed in the mechanical theory of heat. I reserve for future consideration the more particular application
of the mechanical theory of heat and especially of the law of equivalents of transformation to radiant heat.
For the present I will confine myself to announcing as a result of my argument that if we think of that quantity which with reference to a single body I have called its entropy, as formed in a
consistent way, with consideration of all the circumstances, for the whole universe, and if we use in connection with it the other simpler concept of energy, we can express the fundamental laws of
the universe which correspond to the two fundamental laws of the mechanical theory of heat in the following simple form.
1. The energy of the universe is constant.
2. The entropy of the universe tends toward a maximum.
[1]The symbols in the equation represent heat (Q), absolute temperature (T), and entropy (S).--CJG list of selected historical papers.
Classic Chemistry. | {"url":"https://web.lemoyne.edu/giunta/Clausius.html","timestamp":"2024-11-09T18:59:42Z","content_type":"text/html","content_length":"20464","record_id":"<urn:uuid:dee31229-4640-4c09-90b9-65a9cc6daa0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00349.warc.gz"} |
Origins of DREAM3D
DREAM.3D Version 6 is considered legacy and all development has stopped. Please consider using DREAM3D-NX which can be downloaded from https://www.dream3d.io. DREAM3D-NX offers a much better user
experience with its integrated visualization among its new features.
Feature DREAM.3D v6 DREAM3D-NX v7
Integrated Visualization NO YES
Actively Developed NO YES
Actively Supported NO YES
Easy Python Bindings NO YES
Crash Protection NO YES
New features being added NO YES
Publications That Describe Algorithms Contained In DREAM3D¶
[1] S. P. Donegan, J. C. Tucker, A. D. Rollett, K. Barmak, and M. Groeber. Extreme value analysis of tail departure from log-normality in experimental and simulated grain size distributions. Acta
materialia, 61(15):5595-5604, 2013.
Grain size data were taken from four three- and two-dimensional microstructures, including simulated grain growth, thin film and superalloy data sets. Probability plots revealed approximately
log-normal distributions for experimental grain size data sets, but with systematic differences in the upper tails. A simulated grain size data set obtained from Potts model growth exhibited strong
deviation from log-normality. A peaks-over-threshold analysis was applied to quantify the differences in the upper tails. Potts model simulation of normal grain growth shows the shortest tail,
whereas the thin film data showed the longest tail (i.e. closest to log-normal), with an intermediate tail shape in the superalloy. (C) 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights
[2] J. C. Tucker, L. H. Chan, G. S. Rohrer, M. A. Groeber, and A. D. Rollett. Tail departure of log-normal grain size distributions in synthetic three-dimensional microstructures. Metallurgical And
Materials Transactions A-Physical Metallurgy And Materials Science, 43A(8):2810-2822, 2012. 8th Symposium on Bulk Metallic Glasses (BMGs)/TMC Annual Meeting and Exhibition FEB 27-MAR 03, 2011 San
Diego, CA TMS, ASM.
[3] J. C. Tucker, L. H. Chan, G. S. Rohrer, M. A. Groeber, and A. D. Rollett. Comparison of grain size distributions in a ni-based superalloy in three and two dimensions using the saltykov method.
Scripta materialia, 66(8):554-557, 2012.
[4] S. D. Sintay and A. D. Rollett. Testing the accuracy of microstructure reconstruction in three dimensions using phantoms. Modelling and Simulation in Materials Science and Engineering, 20(7),
2012. 075005.
[5] S. Y. Wang, E. A. Holm, J. Suni, M. H. Alvi, P. N. Kalu, and A. D. Rollett. Modeling the recrystallized grain size in single phase materials. Acta materialia, 59(10):3872-3882, 2011.
[6] M. D. Uchic, M. A. Groeber, and A. D. Rollett. Automated serial sectioning methods for rapid collection of 3-d microstructure data. Jom, 63(3):25-29, 2011.
[7] I. M. Robertson, C. A. Schuh, J. S. Vetrano, N. D. Browning, D. P. Field, D. J. Jensen, M. K. Miller, I. Baker, D. C. Dunand, R. Dunin-Borkowski, B. Kabius, T. Kelly, S. Lozano-Perez, A. Misra,
G. S. Rohrer, A. D. Rollett, M. L. Taheri, G. B. Thompson, M. Uchic, X. L. Wang, and G. Was. Towards an integrated materials characterization toolbox. Journal of Materials Research, 26(11):1341-1383,
The material characterization toolbox has recently experienced a number of parallel revolutionary advances, foreshadowing a time in the near future when material scientists can quantify material
structure evolution across spatial and temporal space simultaneously. This will provide insight to reaction dynamics in four-dimensions, spanning multiple orders of magnitude in both temporal and
spatial space. This study presents the authors' viewpoint on the material characterization field, reviewing its recent past, evaluating its present capabilities, and proposing directions for its
future development. Electron microscopy; atom probe tomography; x-ray, neutron and electron tomography; serial sectioning tomography; and diffraction-based analysis methods are reviewed, and
opportunities for their future development are highlighted. Advances in surface probe microscopy have been reviewed recently and, therefore, are not included [D.A. Bonnell et al.: Rev. Modern Phys.
in Review]. In this study particular attention is paid to studies that have pioneered the synergetic use of multiple techniques to provide complementary views of a single structure or process;
several of these studies represent the state-of-the-art in characterization and suggest a trajectory for the continued development of the field. Based on this review, a set of grand challenges for
characterization science is identified, including suggestions for instrumentation advances, scientific problems in microstructure analysis, and complex structure evolution problems involving material
damage. The future of microstructural characterization is proposed to be one not only where individual techniques are pushed to their limits, but where the community devises strategies of technique
synergy to address complex multiscale problems in materials science and engineering.
[8] A. Khorashadizadeh, D. Raabe, S. Zaefferer, G. S. Rohrer, A. D. Rollett, and M. Winning. Five-parameter grain boundary analysis by 3d ebsd of an ultra fine grained cuzr alloy processed by equal
channel angular pressing. Advanced Engineering Materials, 13(4):237-244, 2011.
The 3D grain boundary character distribution (GBCD) of a sample subjected to equal channel angular pressing (ECAP) after eight passes and successive annealing at 650 degrees C for about 10 min is
analyzed. The experiments are conducted using a dual beam system, which is a combination of a focused ion beam and a scanning electron microscope to collect a series of electron backscatter
diffraction (EBSD) maps of the microstructure (3D EBSD). The data set was aligned and reconstructed to a 3D microstructure. The crystallographic character of the grain boundary planes was determined
using three different methods, namely, the line segment method, the stereological method, and the triangular surface mesh method. The line segment and triangular surface mesh methods produce
consistent data sets, both yielding approximately a 7% area fraction of coherent twins. These results starkly contrast that of the statistical stereological method, which produced a 44% area fraction
of coherent twins.
[9] G. S. Rohrer, J. Li, S. Lee, A. D. Rollett, M. Groeber, and M. D. Uchic. Deriving grain boundary character distributions and relative grain boundary energies from three-dimensional ebsd data.
Materials Science and Technology, 26(6):661-669, 2010.
Three-dimensional electron backscatter diffraction data, obtained by serial sectioning a nickel base superalloy, has been analysed to measure the geometric arrangement of grain boundary planes at
triple junctions. This information has been used to calculate the grain boundary character distribution (GBCD) and the grain boundary energy distribution (GBED). The twin content from the
three-dimensional GBCD calculation compares favourably with the twin content estimated by stereology. Important factors in the analysis are the alignment of the parallel layers, the ratio of the
out-of-plane to in-plane spacing of the discrete orientation data and the discretisation of the domain of grain boundary types. The results show that grain boundaries comprised of (111) planes occur
most frequently and that these grain boundaries have a relatively low energy. The GBCD and GBED are inversely correlated.
[10] L. Wang, S. R. Daniewicz, M. F. Horstemeyer, S. Sintay, and A. D. Rollett. Three-dimensional finite element analysis using crystal plasticity for a parameter study of fatigue crack incubation in
a 7075 aluminum alloy. International Journal of Fatigue, 31(4):659-667, 2009.
Three-dimensional finite element analysis of a bicrystal using a crystal plasticity constitutive theory was performed to compute the maximum plastic shear strain range Delta gamma(p)(max) in the
matrix, at the particle/matrix interface, and at the bicrystal boundary. Using the finite element analysis results, a design of experiments (DOE) technique was employed to understand and quantify the
effects of seven parameters on fatigue crack incubation: applied displacement, load ratio, particle modulus, the number of initially active slip systems, the relative crystallographic misorientation
at the grain boundary, the particle aspect ratio, and the normalized particle size. The simulations clearly showed that the applied displacement is the most influential parameter. In most cases,
particles were found to be more significant than bicrystal boundaries for incubation. The number of initially active slip systems, the particle aspect ratio, and the normalized particle size showed
some influences on fatigue incubation. The particle modulus was the least influential parameter. (C) 2008 Elsevier Ltd. All rights reserved.
[11] L. Wang, S. R. Daniewicz, M. F. Horstemeyer, S. Sintay, and A. D. Rollett. Three-dimensional finite element analysis using crystal plasticity for a parameter study of microstructurally small
fatigue crack growth in a aa7075 aluminum alloy. International Journal of Fatigue, 31(4):651-658, 2009.
Three-dimensional finite element analysis using a crystal plasticity constitutive theory was performed to understand and quantify various parametric effects on microstructurally small fatigue crack
growth in a AA7075 aluminum alloy. Plasticity-induced crack opening stresses (S-o/S-max) were computed, and from these results the crack propagation life N was obtained. A design of experiments (DOE)
technique was used to study the influences of seven parameters (maximum load, load ratio, particle modulus, the number of initially active slip systems, misorientation angle, particle aspect ratio,
and the normalized particle size) on fatigue crack growth. The simulations clearly showed that the load ratio is the most influential parameter on crack growth. The next most influential parameters
are maximum load and the number of initially active slip systems. The particle modulus, misorientation angle, particle aspect ratio, and the normalized particle size showed less influence on crack
growth. Another important discovery in this study revealed that the particles were more important than the grain boundaries for inducing resistance for microstructurally small fatigue crack growth.
(C) 2008 Elsevier Ltd. All rights reserved.
[12] A. Brahme, J. Fridy, H. Weiland, and A. D. Rollett. Modeling texture evolution during recrystallization in aluminum. Modelling and Simulation in Materials Science and Engineering, 17(1):015005,
2009. 015005.
The main aim of this work was to develop a model with predictive capability for microstructural evolution during recrystallization and to identify factors that exert the greatest effect on the
development of texture. To achieve this aim, geometric and crystallographic observations from two orthogonal sections through a polycrystal were used as input to the computer simulations, to create a
statistically representative three-dimensional model. Assignment of orientations to the grains was performed so as to optimize agreement between the orientation and misorientation distributions of
assigned and observed orientations. The microstructures thus created were allowed to evolve using a Monte Carlo simulation. As a demonstration of the model the effects of anisotropy, both in energy
and in mobility, stored energy and oriented nucleation (ON) on overall texture development were studied. The results were analyzed with reference to the various established competing theories of ON
and oriented growth. The results suggested that all of ON, mobility anisotropy, stored energy and energy anisotropy (listed in order of their relative importance) influence texture development. It
was also determined that comparison of simulated and measured textures throughout the recrystallization process is a more severe test of a model than the typical comparison of textures only at the
end of the process.
[13] A. D. Rollett, S. B. Lee, R. Campman, and G. S. Rohrer. Three-dimensional characterization of microstructure by electron back-scatter diffraction. Annual Review of Materials Research,
37:627-658, 2007.
The characterization of microstructures in three dimensions is reviewed, with an emphasis on the use of automated electron back-scatter diffraction techniques. Both statistical reconstruction of
polycrystalline structures from multiple cross sections and reconstruction from parallel, serial sections are discussed. In addition, statistical reconstruction of second-phase particle
microstructures from multiple cross sections is reviewed.
[14] C. G. Roberts, S. L. Semiatin, and A. D. Rollett. Particle-associated misorientation distribution in a nickel-base superalloy. Scripta Materialia, 56(10):899-902, 2007. Roberts, C. G. Semiatin,
S. L. Rollett, A. D.
[15] A. D. Rollett, R. Campman, and D. Saylor. Three dimensional microstructures: Statistical analysis of second phase particles in AA7075-T651, volume 519-521 of Materials Science Forum, pages 1-10.
This paper describes some aspects of reconstruction of microstructures in three dimensions. A distinction is drawn between tomographic approaches that seek to characterize specific volumes of
material, either with or without diffraction, and statistical approaches that focus on particular aspects of microstructure. A specific example of the application of the statistical approach is given
for an aerospace aluminum alloy in which the distributions of coarse constituent particles are modeled. Such distributions are useful for modeling fatigue crack initiation and propagation.
[16] C. S. Kim, A. D. Rollett, and G. S. Rohrer. Grain boundary planes: New dimensions in the grain boundary character distribution. Scripta materialia, 54(6):1005-1009, 2006.
The five parameter grain boundary character distribution quantifies the relative areas of different types of grain boundaries, distinguished by their lattice misorientation and grain boundary plane
orientation. The viewpoint presented in this paper is that this distribution is a sensitive metric of polycrystalline structure that can be related to macroscopic properties. To demonstrate the
influence of the grain boundary character distribution oil macroscopic properties, the stored elastic energy is calculated in model microstructures. (c) 2005 Acta Materialia Inc. Published by
Elsevier Ltd. All rights reserved.
[17] Y. S. Choi, A. D. Rollett, and H. R. Piehler. Application of two-point orientation auto-correlation function (tp-oacf). Materials Transactions, 47(5):1313-1316, 2006.
A two-point orientation auto-correlation function (TP-OACF) was developed in order to quantify the spatial distribution of targeted texture components. An example of a TP-OACF was demonstrated using
an idealized orientation map. Characteristics of the spatial distribution of major texture components in 6022-T4 Al sheets deformed in plane-strain tension were also quantified using a TP-OACF. The
results showed that 110 < 112 > and 123 < 634 > orientations (in ND < PD > notation, where PD is the pulling direction) tend to form localized diagonal and horizontal bands through the thickness,
respectively, after the plane-strain deformation. The cube orientation on the surface initially showed a relatively strong texture band along the RD, but this banding behavior was less significant
after the plane-strain deformation.
[18] A. Brahme, M. H. Alvi, D. Saylor, J. Fridy, and A. D. Rollett. 3d reconstruction of microstructure in a commercial purity aluminum. Scripta materialia, 55(1):75-80, 2006.
[19] J. H. Cho, A. D. Rollett, and K. H. Oh. Determination of a mean orientation in electron backscatter diffraction measurements. Metallurgical And Materials Transactions A-Physical Metallurgy And
Materials Science, 36A(12):3427-3438, 2005.
The average orientation of an electron backscatter diffraction (EBSD) map is calculated by the quaternion method and is compared with nonlinear solving by the Hill Climbing and Barton-Davison
methods. An automated EBSD system acquires orientations on a regular grid of pixels based on indexation of Kikuchi patterns and the orientation is described by one of the crystal symmetry-related
equivalents; In order to calculate the quaternion average, it is necessary to make a cloud for a set of pixels in a grain. A cloud consists of the representative orientations with small
misorientation between each and every pair of points. The position criterion says that two adjacent pixels have a smaller misorientation than with all others. With this, the proper equivalent
orientation, or representative orientation, for the cloud, can be selected from among all the crystal symmetry-related equivalents. The orientation average is the quaternion summation divided by its
norm. The instant average or cumulative average is useful for dealing with polycrystalline grains or orientation discontinuity and is also useful for selection of the proper orientation of EBSD map
with large scattering. The quaternion, Hill Climbing, and Barton-Dawson nonlinear methods are tested with a Gaussian distribution around the ideal texture component, Brass 110 < 112 >. The accuracy
of the three results is similar but the nonlinear methods are associated with longer computation times than the quaternion method. The quaternion method is adapted for characterization of a
partially-recrystallized interstitial-free (IF) steel and randomly distributed Brass, S, and cube texture components according to several different orientation spreads.
[20] D. M. Saylor, J. Fridy, B. S. El-Dasher, K. Y. Jung, and A. D. Rollett. Statistically representative three-dimensional microstructures based on orthogonal observation sections.
Metallurgical And Materials Transactions A-Physical Metallurgy And Materials Science, 35A(7):1969-1979, 2004. Techniques are described that have been used to create a statistically representative
three-dimensional model microstructure for input into computer simulations using the geometric and crystallographic observations from two orthogonal sections through an aluminum polycrystal.
Orientation maps collected on the observation planes are used to characterize the sizes, shapes, and orientations of grains. Using a voxel-based tessellation technique, a microstructure is generated
with grains whose size and shape are constructed to conform to those measured experimentally. Orientations are then overlaid on the grain structure such that distribution of grain orientations and
the nearest-neighbor relationships, specified by the distribution of relative misorientations across grain boundaries, match the experimentally measured distributions. The techniques are applicable
to polycrystalline materials with sufficiently compact grain shapes and can also be used to controllably generate a wide variety of hypothetical microstructures for initial states in computer
[21] D. M. Saylor, B. S. El Dasher, A. D. Rollett, and G. S. Rohrer. Distribution of grain boundaries in aluminum as a function of five macroscopic parameters. Acta materialia, 52(12):3649-3655,
The grain boundary character distribution in commercially pure Al has been measured as a function of lattice misorientation and boundary plane orientation. The results demonstrate a tendency to
terminate grain boundaries on low index planes with relatively low surface energies and large interplanar spacings. The most frequently observed grain boundary plane orientation is (1 1 1). However,
there are also instances where boundaries terminated by higher index planes have significant populations. For example, certain twist configurations on 1 1 w planes, which correspond to symmetric [1 1
0] tilt boundaries, also have relatively high populations. The population of symirietric [1 1 0] tilt boundaries exhibits an inverse relationship with previously measured energies. (C) 2004 Acta
Materialia Inc. Published by Elsevier Ltd. All rights reserved.
[22] D. M. Saylor, B. El Dasher, Y. Pang, H. M. Miller, P. Wynblatt, A. D. Rollett, and G. S. Rohrer. Habits of grains in dense polycrystalline solids. Journal of The American Ceramic Society, 87
(4):724-726, 2004.
[23] A.D. Rollett, D.M. Saylor, J. Fridy, B.S. El-Dasher, A. Brahme, S.-B. Lee, C. Cornwell, and R. Noack. Modeling microstructures in 3d. In S. Ghosh, J.C. Castro, and J.K. Lee, editors, NUMIFORM
2004, pages 71-77. Amer. Inst. of Physics.
Many issues in forming are influenced to some degree by the internal structure of the material which is commonly referred to by the materials science community as microstructure. Although the term
microstructure is commonly only thought of in the context of grain size, it more properly encompasses all relevant aspects of internal material structure. For the purposes of forming, the most
relevant features are the crystallographic orientations of the grains (texture) and the locations of the grain boundaries, or, equivalently, the size, topology and shape of the grains. In order to
perform realistic simulations one needs to specify the initial state of the material, e.g. on a finite element mesh, with sufficient detail that all these features are reproduced. Measuring
microstructure at the scale of individual grains is possible in the synchrotron but scarcely practicable for an analyst. Cross-sections or surfaces are easily evaluated through automated diffraction
in the scanning electron microscope (SEM), however. Therefore this paper describes a set of methods for generating statistically representative 3D microstructures based on microscopy input for both
single-phase and two-phase materials. Examples are given of application of the technique for generating input structures for recrystallization simulation, dynamic deformation and finite element
modeling. Keywords: microstructure computer simulation texture reconstruction Voronoi tessellation EBSD Al
[24] G. S. Rohrer, D. M. Saylor, B. El Dasher, B. L. Adams, A. D. Rollett, and P. Wynblatt. The distribution of internal interfaces in polycrystals. Zeitschrift Fur Metallkunde, 95(4):197-214, 2004.
[25] H. M. Miller, D. M. Saylor, B. S. El Dasher, A. D. Rollett, and G. S. Rohrer. Crystallographic distribution of internal interfaces in spinel polycrystals. Materials Science Forum,
467-470:783-788, 2004. Measurements of the grain boundary character distribution in MgAl2O4 (spinel) as a function of lattice misorientation and boundary plane orientation show that at all
misorientations, grain boundaries are most frequently terminated on 111 planes. Boundaries with 111 orientations are observed 2.5 times more frequently than boundaries with 100 orientations.
Furthermore, the most common boundary type is the twist boundary formed by a 60 rotation about the [111] axis. 111 planes also dominate the external form of spinel crystals found in natural settings
and this suggests that they are low energy and/or slow growing planes. The mechanisms that might lead to a high population of these planes during solid state crystal growth are discussed. Keywords:
Grain Boundary Character Distribution Grain Boundary Planes Stereology Spinel microstructure
[26] J. H. Cho, A. D. Rollett, and K. H. Oh. Determination of volume fractions of texture components with standard distributions in euler space. Metallurgical And Materials Transactions A-Physical
Metallurgy And Materials Science, 35A(3A):1075-1086, 2004.
The intensities of texture components are modeled by Gaussian distribution functions in Euler space. The multiplicities depend on the relation between the texture component and the crystal and sample
symmetry elements. Higher multiplicities are associated with higher maximum values in the orientation distribution function (ODF). The ODF generated by Gaussian function shows that the S component
has a multiplicity of 1, the brass and copper components, 2, and the Goss and cube components, 4 in the cubic crystal and orthorhombic sample symmetry. Typical texture components were modeled using
standard distributions in Euler space to calculate a discrete ODF, and their volume fractions were collected and verified against the volume used to generate the ODE The volume fraction of a texture
component that has a standard spherical distribution can be collected using the misorientation approach. The misorientation approach means integrating the volume-weighted intensity that is located
within a specified cut-off misorientation angle from the ideal orientation. The volume fraction of a sharply peaked texture component can be collected exactly with a small cut-off value, but textures
with broad distributions (large full-width at half-maximum (FWHM)) need a larger cut-off value. Larger cut-off values require Euler space to be partitioned between texture components in order to
avoid overlapping regions. The misorientation approach can be used for texture's volume in Euler space in a general manner. Fiber texture is also modeled with Gaussian distribution, and it is
produced by rotation of a crystal located at g(0), around a sample axis. The volume of fiber texture in wire drawing or extrusion also can be calculated easily in the unit triangle with the angle
distance approach.
If you have additional papers please send the bibtex of the citation to dream3d@bluequartz.net | {"url":"https://dream3d.bluequartz.net/Origins/","timestamp":"2024-11-11T16:49:04Z","content_type":"text/html","content_length":"46341","record_id":"<urn:uuid:308a4e24-4782-425e-b53b-62e4b7a5b6ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00643.warc.gz"} |
Determining RL Values via Geodetic Calculation Methods
22 Sep 2024
Popularity: ⭐⭐⭐
Leveling Calculation
This calculator provides the calculation of RL (Reduced Level) of a second point during leveling.
Calculation Example: Leveling is a surveying technique used to determine the height or elevation of points on the ground. It involves taking a series of measurements using a level and a leveling rod.
Related Questions
Q: What is the purpose of leveling in surveying?
A: Leveling is used to determine the height or elevation of points on the ground. This information is essential for a variety of purposes, such as planning and designing construction projects,
creating maps, and managing water resources.
Q: How does leveling work?
A: Leveling involves taking a series of measurements using a level and a leveling rod. The level is used to establish a horizontal line of sight, and the leveling rod is used to measure the height of
points above or below this line of sight.
Symbol Name Unit
RL First RL m
BS Back Sight m
FS Fore Sight m
Calculation Expression
RL Calculation: The RL of the second point is calculated using the formula: RL = RL1 + BS - FS
Calculated values
Considering these as variable values: BS=1.25, RL=100.0, FS=0.75, the calculated value(s) are given in table below
Derived Variable Value
RL Calculation 0.5+rl1
Similar Calculators
Calculator Apps | {"url":"https://blog.truegeometry.com/calculators/Leveling_calculation.html","timestamp":"2024-11-05T05:52:26Z","content_type":"text/html","content_length":"17460","record_id":"<urn:uuid:5674d246-42cf-4ebc-91d8-adf33ea75e65>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00731.warc.gz"} |
2 Step Equation Word Problems Worksheet - Equations Worksheets
2 Step Equation Word Problems Worksheet
2 Step Equation Word Problems Worksheet – Expressions and Equations Worksheets are designed to aid children in learning quicker and more effectively. They include interactive activities and problems
that are dependent on the order in which operations are performed. These worksheets make it easy for children to master complex concepts and concepts fast. These PDF resources are free to download,
and can be used by your child in order to practice math equations. These resources are beneficial for students in the 5th to 8th Grades.
Download Free 2 Step Equation Word Problems Worksheet
These worksheets can be utilized by students in the 5th through 8th grades. These two-step word problems are constructed using fractions and decimals. Each worksheet contains ten problems. You can
find them at any website or print source. These worksheets are a great way to practice rearranging equations. In addition to allowing students to practice restructuring equations, they can also
assist your student to understand the principles of equality as well as inverse operations.
These worksheets are designed for students in the fifth and eighth grades. These are great for students who are struggling to calculate percentages. You can choose from three different types of
questions. There is the option to either solve single-step problems that contain whole numbers or decimal numbers, or to employ word-based solutions for fractions as well as decimals. Each page
contains ten equations. These Equations Worksheets are recommended for students from 5th to 8th grades.
These worksheets are a great way to practice fraction calculations and other concepts in algebra. Many of these worksheets allow you to select between three different types of problems. You can pick
one that is word-based, numerical or a mixture of both. It is essential to pick the problem type, because every challenge will be unique. Ten problems are on each page, so they’re great resources for
students from 5th to 8th grade.
These worksheets assist students to understand the connections between numbers and variables. They provide students with practice in solving polynomial equations or solving equations, as well as
discovering how to utilize them in daily life. If you’re in search of an excellent educational tool to understand equations and expressions it is possible to begin by looking through these
worksheets. These worksheets will help you learn about the different kinds of mathematical problems and the various symbols that are employed to explain them.
These worksheets are extremely beneficial for students in the first grade. These worksheets teach students how to solve equations and graphs. The worksheets are perfect to practice polynomial
variables. These worksheets can help you factor and simplify them. There are many worksheets you can use to help kids learn equations. Making the work yourself is the most effective way to learn
You can find many worksheets that teach quadratic equations. There are various worksheets that cover the various levels of equations at each grade. These worksheets can be used to solve problems to
the fourth degree. Once you’ve finished a step, you’ll be able go on to solving different kinds of equations. You can continue to take on the same problems. For example, you can solve a problem using
the same axis in the form of an elongated number.
Gallery of 2 Step Equation Word Problems Worksheet
Equation Word Problems Worksheets
Solving Equations Word Problems Worksheet 2 Two Step
Two Step Equations Worksheet Pdf
Leave a Comment | {"url":"https://www.equationsworksheets.net/2-step-equation-word-problems-worksheet/","timestamp":"2024-11-04T21:49:53Z","content_type":"text/html","content_length":"63035","record_id":"<urn:uuid:60df5a28-c5ad-4619-94b1-4ffb58b0153c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00601.warc.gz"} |
Talk:Factorions - Rosetta CodeTalk:Factorions
Limit of 1499999 in base 10
The following was removed from the wikipedia page in September 2019, on the grounds "described above for arbitrary base b".
Since I rather strongly disagree on that last point, I have replicated the original/deleted text here:
Upper bound (in base 10)
If n is a natural number of d digits that is a factorion, then 10^d − 1 ≤ n ≤ 9!d. This fails to hold for d ≥ 8 thus n has at most seven digits, and the first upper bound is 9,999,999. But
the maximum sum of factorials of digits for a seven-digit number is 9!*7 = 2,540,160 establishing the second upper bound. Going further, since no number bigger than 2540160 is possible, the
first digit of a seven-digit number can be at most 2. Thus, only six positions can range up until 9 and 2!+6*9!= 2177282 becomes a third upper bound. This implies, if n is a seven-digit
number, either the second digit is 0 or 1 or the first digit is 1. If the first digit is 2 and thus the second digit is 0 or 1, the numbers are limited by 2!+1!+5*9! = 1814403 - a
contradiction to the first digit being 2. Thus, a seven-digit number can be at most 1999999, establishing our fourth upper bound.
All factorials of digits at least 5 have the factors 5 and 2 and thus end on 0. Let 1abcdef denote our seven-digit number. If all digits a-f are all at least 5, the sum of the factorials -
which is supposed to be equal to 1abcdef - will end on 1 (coming from the 1! in the beginning). This is a contradiction to the assumption that f is at least 5. Thus, at least one of the
digits a-f can be at most 4, which establishes 1!+4!+5*9!=1814425 as fifth upper bound. Assuming n is a seven-digit number, the second digit is at most 8. There are two cases: If a is at
least 5, by the same argument as above one of the remaining digits b-f has to be at most 4. This implies an upper bound (since a is at most 8) of 1!+8!+4!+4*9!= 1491865, a contradiction to a
being at least 5. Thus, a is at most 4 and the sixth upper bound is 1499999.
A computer can check all numbers from 40585 to 1499999, verifying that 40585 is the largest factorion in base 10.
Quite incorrectly, almost all entries (bar Pascal, Perl[2], Phix[2], Sidef) also use that limit for bases 9, 11, and 12... --Pete Lomax (talk) 22:42, 15 January 2022 (UTC) | {"url":"https://rosettacode.org/wiki/Talk:Factorions?oldid=318975","timestamp":"2024-11-08T01:34:32Z","content_type":"text/html","content_length":"44854","record_id":"<urn:uuid:c7b8e564-bd41-4bfd-b731-d3b09ae8c4f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00732.warc.gz"} |
Bar Chart With Multiple Categories 2024 - Multiplication Chart Printable
Bar Chart With Multiple Categories
Bar Chart With Multiple Categories – You could make a Multiplication Chart Pub by marking the posts. The kept column should say “1” and stand for the amount increased by one. On the right hand part
from the dinner table, content label the columns as “2, 8, 6 and 4 and 9”. Bar Chart With Multiple Categories.
Ideas to find out the 9 instances multiplication kitchen table
Discovering the 9 periods multiplication desk will not be a simple task. Counting down is one of the easiest, although there are several ways to memorize it. Within this technique, you set the hands
in the table and quantity your hands and fingers individually from a single to 10. Fold your 7th finger to enable you to begin to see the ones and tens into it. Then count the quantity of fingertips
left and appropriate of your respective folded finger.
When studying the kitchen table, young children could be intimidated by larger numbers. Simply because including larger amounts repeatedly turns into a job. You can exploit the hidden patterns to
make learning the nine times table easy, however. A technique is always to write the 9 instances kitchen table with a cheat page, read through it all out loud, or process producing it straight down
often. This procedure can certainly make the desk far more memorable.
Designs to consider on the multiplication chart
Multiplication graph or chart night clubs are great for memorizing multiplication information. You can find the merchandise of two figures by studying the rows and columns in the multiplication
chart. By way of example, a column which is all twos plus a row that’s all eights should satisfy at 56. Patterns to look for over a multiplication graph or chart club are similar to individuals in a
multiplication dinner table.
A pattern to consider on a multiplication graph or chart will be the distributive house. This residence may be seen in most columns. For example, a product or service x two is equivalent to 5
(periods) c. This same house applies to any column; the amount of two columns equals the value of other column. As a result, a strange quantity instances a much number is surely an even number. The
identical relates to these products of two peculiar figures.
Creating a multiplication graph from memory
Building a multiplication chart from memory space will help children find out the distinct figures within the periods tables. This straightforward exercise allows your kids to remember the amounts
and discover how you can multiply them, which can help them in the future once they get more information complicated math concepts. To get a fun and fantastic way to remember the phone numbers, you
can set up shaded control buttons to ensure each corresponds to particular instances dinner table variety. Make sure to tag each and every row “1” and “” to help you easily establish which amount is
available first.
As soon as young children have learned the multiplication graph or chart bar from memory space, they ought to commit them selves to the process. This is why it is far better to employ a worksheet
rather than classic notebook to apply. Striking and cartoon character layouts can attract the sensory faculties of your young children. Before they move on to the next step, let them color every
correct answer. Then, screen the graph within their research region or bedrooms to serve as a reminder.
Using a multiplication chart in everyday life
A multiplication graph demonstrates how to grow amounts, someone to 15. In addition, it shows the item of two figures. It could be valuable in everyday life, such as when splitting up money or
gathering data on people. These are one of the approaches you can use a multiplication graph. Use them to help you your child understand the idea. We now have mentioned just some of the most
prevalent purposes of multiplication dining tables.
You can use a multiplication graph to aid your youngster learn how to lessen fractions. The secret to success is to follow the numerator and denominator to the left. By doing this, they will likely
see that a portion like 4/6 could be decreased to a small fraction of 2/3. Multiplication graphs are especially helpful for youngsters mainly because they help them understand variety habits. You
will discover Totally free printable variations of multiplication graph or chart cafes online.
Gallery of Bar Chart With Multiple Categories
Different Color For Multiple Categories On Bar Charts Still
How To Create A Bar Chart
Different Color For Multiple Categories On Bar Charts Still
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/bar-chart-with-multiple-categories/","timestamp":"2024-11-06T12:08:45Z","content_type":"text/html","content_length":"55586","record_id":"<urn:uuid:c6a0cccf-9039-4949-a439-479c6329de3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00158.warc.gz"} |
Lesson: Naming straight line graphs | Oak National Academy
Switch to our new maths teaching resources
Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.
Lesson details
Key learning points
1. In this lesson, we will be looking at plotting coordinates onto a grid and simple notation for labelling straight line graphs.
This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.
5 Questions
Complete the following sentence: On a horizontal line the..............
No ordinate stays the same
x-ordinate is always the same
Which integer values can y be for the inequality shown above?
-3, -2.5, -1,5, 0, 1, 1.5, 2
Which of the following inequalities covers the following integer values of x? -1, 0, 1, 2, 3, 4?
Which of the following coordinates satisfies the inequalities; -2 < x < 3 and -5 < y < 3?
5 Questions
John says, "In my line, all my y-ordinates are triple my x-ordinates". Tick the equation that represents John's line.
Jenny says, "In my line, all my y-ordinates are 5 less than my x-ordinates". Tick the equation that represents Jenny's line.
Which of the following lines does the point (2,6) lie on?
Which of the following lines does the point (3,-6) lie on?
Which of the following coordinates lies on the line y=2x-1 | {"url":"https://www.thenational.academy/teachers/lessons/naming-straight-line-graphs-71hkgr","timestamp":"2024-11-04T16:52:20Z","content_type":"text/html","content_length":"263717","record_id":"<urn:uuid:c9dba1aa-33c3-4226-936e-080050d915af>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00770.warc.gz"} |
Top What Does S Mean in Math Tips!
The One Thing to Do for What Does S Mean in Math
As part of your research in the very best teaching techniques, you consider the impact of school size on standardized test scores. Technology keeps people connected in fantastic new methods but in
addition introduces troublesome gray areas in regards to communication. In the above mentioned example 106 is the mode, as it occurs twice and the remainder of the outcomes occur only once.
The y-axis indicates the range of occurrences of each editing one of these elements. The effect denotes the notion that one variable has an impact on another. There are several forms of means, but
the this form is the most usual use.
Attention is understood to be concentrating on a sure thing and ignoring the rest of the aspects of a particular environment, a definition which keenly reflects the abbreviation’s use in addressing a
slice of postage. It will work exactly like a list. The code from this short article can be found on GitHub.
Top Choices of What Does S Mean in Math
It’s best if you’re able to address your letter to a specific person inside an organization. If you’re sued, it is better that you contact a lawyer for legal advice and representation. The person
https://uk.payforessay.net/editing-service being sued is referred to as the Defendant.
If you’ve reconciled, I hope you will look at sharing your circumstances by taking the brief survey I’ve created to learn more regarding the matter. You might not accept a substitute form developed
by means of an employee. Referring back to a previous example, he may be looking at a group of patients with a particular condition.
This usually means the preceding conversation got boring, and she’d much rather discuss something different. Let’s see a few examples. It’s also a great idea to verify your payslips regularly.
I’ll begin with the new GF’s letter. It’s not simple to stay open and objective in the event the rudeness has been happening for a very long period and you haven’t addressed it before, but it’s
imperative you stay cool. We don’t accept anonymous letters.
The difficulty in detecting such marks is usually about the skill and understanding of the forger. You may also be moody and need to be careful to prevent addictions. These relationships aren’t
coincidences, but are illustrations of these formulas.
Do not permit this quality to force you to get indecisive. Naturally there’s a less expensive way. Colorado dispensary laws make it possible for adults to get a mixture of flower, edibles, and
There are only a few things on earth that stay the identical everytime we observe it. The cute letters are perfect for a day like today once the rain is falling. The other half of the moment, she is
only saying no in a fine way, and she won’t return to you.
In the very best of all probable worlds, all the men and women in group an individual would have very similar scores. No, my home was constructed in the early sixties whenever these outlets weren’t
around. When you play music with different individuals, you need to be in a position to chat about what you’re likely to play.
With large samples, you understand that mean with far more precision than you do with a little sample, or so the confidence interval is quite narrow when computed from a big sample. It will adhere to
the 2 digits employed for the aspect ratio. With the small sample on the left, it is similar to the range of the data.
Among the flaws involved with the conventional deviation, is the fact that it depends upon the units that are used. One other important method is to express the difference for a fraction or multiple
of a typical deviation. The you started studying statistics and all a sudden the average is currently referred to as the mean.
In order to get the SPSS mean mode median, you will have to use the Frequency tab. Histogram explorer stipulates another way to shape a histogram and examine the summary statistics. Python is a
favorite language in regards to data analysis and statistics.
What Does S Mean in Math for Dummies
I understand I did lots of damage. Thus, to average the growth during the very first month with the growth during the initial 12 months isn’t a sensible thing to do. You’re wise and quick to consider
on your feet, but take care not to be impatient or impulsive.
If you find any medium-to-large bubbles on the way, slip them with a sharp knife and allow the air out. They can’t be short shots if they’re identical when held in quantity and they’re identical.
Nowadays you have two half-full bottles.
When you both work hard to do what’s ideal for your loved ones, the strain and frustration may be back-breaking strain. It felt unresolved for some time. We need to make the back end too.
What Does S Mean in Math for Dummies
It represents the kind of vehicle that the tire was intended for. Most of us know somebody who has lamented a minor shift in body weight that is because of natural fluctuation instead of outright
weight gain. Do not confront your sister-in-law before others.
Passenger vehicles include cars, minivans, SUVs, and little pickup trucks. You’re only a individual in a dumpster. Should not be utilized on vehicles.
Some of the most usual cast forgeries of old marks in the industry today are observed on figural napkin rings. Multimodality with numerous high points may also occur. Figure 2 is a bit different.
The disadvantage of median is that it’s tricky to handle theoretically. The mean is utilized in the definition of the normal deviation, thus the mean and standard deviation are frequently used
together. The estimate of population sigma is simply indirectly about the normal error of the mean.
In the event the growth rate is constant over a period of time, then the typical growth rate over the period will be just like that constant price. Let’s take an instance of the tuple of a mixed
scope of numbers. The range is just the maximum value minus the lowest value.
Within this instance, the normal deviation is 5.1. The scores only vary from 80 to 100, so we are aware that the normal deviation would be small. A number of the values are fractional, which is a
consequence of how they’re calculated.
As part of your research in the very best teaching techniques, you consider the impact of school size on standardized test scores. Technology keeps people connected in fantastic new methods but in
addition introduces troublesome gray areas in regards to communication. 1 place where this technique is comparatively rare is in formal small business communications.
It’s also important to think about the sort of treatment received in connection with the injuries incurred in selecting a affordable multiplier. The effect denotes the notion that one variable has an
impact on another. There are several forms of means, but the this form is the most usual use.
If from the prior instance of 2000 patient results, all feasible samples of 100 were drawn and each of their means were calculated, we’d have the ability to plot these values to generate a
distribution that would give a standard curve. In quotes, you should specify where the data file can be found on your PC. The code from this short article can be found on GitHub.
The Downside Risk of What Does S Mean in Math
It appears that small schools do the very best, perhaps due to their private atmosphere where teachers can get to know students and let them individually. You get to understand your prospects and
their requirements. It’s useful in order to compare and contrast populations to check our ideas about the world.
Medical trials are costly. Power went down in a lot of rooms. If your soon-to-be-ex doesn’t know the both of you are going to break up, you might have to do a little bit of consoling, and explain
yourself further.
Letter writing isn’t always simply for kids. Your sponsored child might have excitedly taken your letter home to demonstrate their family and friends, then stashed it in a distinctive place. Cancer
statistics describe what the results are in sizeable groups of people and supply a photo in time of the load of cancer on society.
Rather, you’ll need to provide the jury or judge a reason upon which they may base this kind of award. It’s often tough for a client to understand whether a attorney is doing a great job. The person
being sued is referred to as the Defendant.
If you’ve reconciled, I hope you will look at sharing your circumstances by taking the brief survey I’ve created to learn more regarding the matter. You might not accept a substitute form developed
by means of an employee. Referring back to a previous example, he may be looking at a group of patients with a particular condition.
Lies You’ve Been Told About What Does S Mean in Math
Still, it’s important to report a legal skunk. It also needs to be less difficult to interpret future occurrences too. There are various types of mean, viz.
I’ll begin with the new GF’s letter. You may also receive duplicate letters from the identical villager. We don’t accept anonymous letters.
Basically, in regards to infidelity, two related explanations are given. You may also be moody and need to be careful to prevent addictions. These relationships aren’t coincidences, but are
illustrations of these formulas.
What Does S Mean in Math – Overview
I get how difficult it is to be unable to hang out with your buddies, exercise, or find a great night’s sleep, he added. A loving atmosphere for him to raise and develop as a guy. It can just as
easily be for you, to aid you in getting your act together once you have what’s going to be the most significant conversation of your life.
There are only a few things on earth that stay the identical everytime we observe it. If there’s a chance that they could make things worse, make them know so that they can avoid making it worse. The
other half of the moment, she is only saying no in a fine way, and she won’t return to you.
Some also think that marijuana legalization has resulted in a rising homeless population in the state. Make certain to keep a balance and to work nicely with others, despite the fact that you also
have a type nature. There’s only so much you are able to take with you, aside from the memories.
What Does S Mean in Math for Dummies
Such balance has an important part in the prevention of particular sports injuries. You are going to be on a less expensive energy tariff in no moment. Let’s try to get the median value of the wine
From time to time, protecting yourself is the sole sound choice. Should you have an older home, you aren’t required to upgrade to these outlets, unless you’re doing a big remodel. A bit additional
security for you once you properly protect an outlet from water, which means you will not need to replace an outlet.
The aforementioned things are typically a good starting point. The exact same may be followed at the right time of purchase by any firm. It could be utilized to compare the operation of distinct
stocks as time passes.
New Questions About What Does S Mean in Math
This usually means the preceding conversation got boring, and she’d much rather discuss something different. What someone means is what they’re referring to or mean to say. It’s also a great idea to
verify your payslips regularly.
I’ll begin with the new GF’s letter. You may also receive duplicate letters from the identical villager. We don’t accept anonymous letters.
Basically, in regards to infidelity, two related explanations are given. You may also be moody and need to be careful to prevent addictions. The woman representing TM stated this mental clarity may
be used to discover ultimate reality.
Top Choices of What Does S Mean in Math
With large samples, you understand that mean with far more precision than you do with a little sample, or so the confidence interval is quite narrow when computed from a big sample. In such
situations, you can attempt to use proc means with a class statement rather than proc univariate. With the small sample on the left, it is similar to the range of the data.
In that situation, you might reasonably use a greater multiplier. This kind of graph is often known as a histogram or bar chart. The arithmetic mean is a sum of information that is divided by the
amount of information points.
Standard Deviation is a measure that’s utilized to quantify the quantity of variation or dispersion of a set of information values. Variance is only the square of the normal deviation. Python is a
favorite language in regards to data analysis and statistics.
Ideas, Formulas and Shortcuts for What Does S Mean in Math
Obviously, it would be easier just to deliver a note and say goodbye, but that’s not a responsible or sensitive approach to break up with a person ordinarily. Generally speaking, the outcome of error
in the decision problem available, together with the expectations of the audience, ought to be taken into account when picking a confidence level to emphasize. The saying the very first step is the
most crucial,” applies here.
Thus the average is the typical amount the insurer has to pay every loss. In summary, the insurance provider pays its insured to generate the insured whole. You have to make bigger payments to
prevent the interest.
The constant or per population” number employed for the age-adjusted rates might vary, based on the kind of event. Therefore, in practice, confidence intervals utilize theestimatedstandard error.
Needless to say, there’s more than 1 way to settle on which value has become the most central That’s the reason why we have more than one average type.
Be aware that all the measures of central tendency are included on each individual page, but you don’t will need to assign all of them if you aren’t working on all of them. These symbols could be
placed in any purchase. Let’s take an instance of the list of a mixed selection of numbers.
It’s inevitable now that each and every status update points out that which we’re passing up. It’s possible to just count in from both ends of the list till you meet in the center, if you would
rather, particularly if your list is short. If a code necessitates laterality, it has to be included for the code to be valid. | {"url":"https://eclair-tn.com/en/topwhatdoessmeaninmathtips/","timestamp":"2024-11-02T23:42:47Z","content_type":"text/html","content_length":"82717","record_id":"<urn:uuid:4a480462-39d3-4f29-8ab2-9d2892ad4a72>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00755.warc.gz"} |
Converting a Low Distortion Single-Ended Sine Source to Fully Differential
Customers looking for a way to evaluate ADCs with differential inputs will find themselves in need of a low distortion, low noise differential sine source. Single-ended sine sources can be obtained
by building a simple Wien-bridge oscillator or by employing any of several readily available audio oscillators. Differential sine sources however are not as common. Using the LTC6363 precision, low
power rail-to-rail output differential op amp, a single-ended sine source can be converted to a differential output sine source and still maintain a very high level of performance.
The circuit of Figure 1 is used with the sine source of the DC1858A and the DC1925A ADC demo board to demonstrate the circuit performance. DC1858A is a low noise, low distortion 2kHz sine wave
generator. The typical THD of the DC1858A is –118dB and the typical SNR is 104dB. DC1925A is the demo board for the LTC2378-20 20-bit, 1Msps ADC with fully differential inputs. This ADC has a typical
THD of –125dB and a typical SNR of 104dBFS.
The circuit of Figure 1 is built using the DC2319A, the demo board for the LTC6363. Part numbers in the schematic correspond to the part numbers used by DC2319A. Looking at the circuit of Figure 1,
the sine out of the DC1858A is applied to the J2 input. Vref/2 (Pin 1 of JP4) from the DC1925A is applied at the J1 input. This properly sets both the input and output common mode levels. Resistor R5
is used to match the output impedance of the DC1858A and capacitor CX filters the noise from the voltage applied at J1. V+(+8V) and V–(–3.6V), also provided by the DC1925A, power the circuit. The
circuit gain is set by the ratio of the feedback resistors (R3 = R4) to the input resistors (R1 = R2). The J4 and J3 outputs connect to the J2 and J4 inputs of the DC1925A. The RC filters at Vout+
(R10, C12) and Vout– (R9, C11) minimize the output noise of the LTC6363.
Circuit performance is shown in the PScope output of Figure 2. SNR in dBFS is obtained by adding the absolute value of the F1 Amplitude to the indicated SNR. This yields 103.26dBFS which is less than
1dB off from the typical values of the DC1858A and the LTC2378-20. THD is –113.11dB. This is approximately 5dB worse than the typical DC1858A value but close to the typical value for the LTC6363. | {"url":"https://www.analog.com/en/resources/technical-articles/converting-a-low-distortion-single-ended-sine-source-to-fully-differential.html","timestamp":"2024-11-14T11:12:29Z","content_type":"text/html","content_length":"161854","record_id":"<urn:uuid:ba0ffd25-a72c-432c-adb1-bd8e8b0da470>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00179.warc.gz"} |
It's quiz time! Test yourself by solving these questions about binary trees.
What is the total number of edges from a particular node to its deepest descendant in a tree?
The depth of that particular node
The height of that particular node
Question 1 of 50 attempted
Get hands-on with 1400+ tech skills courses. | {"url":"https://www.educative.io/courses/ds-and-algorithms-in-python/quiz-gkJ3LK4D693","timestamp":"2024-11-11T01:54:59Z","content_type":"text/html","content_length":"780239","record_id":"<urn:uuid:cc642783-e301-46f5-8de6-fd0a64ccb0dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00395.warc.gz"} |
Definition of median in mathematics - virtualpsychcentre.com
Definition of median in mathematics
What is a median in math?
The median is the middle number in an ordered data set. The mean is the sum of all values divided by the total number of values.
What is mean and median in simple words?
What is the mean, median, and mode? The mean is the number you get by dividing the sum of a set of values by the number of values in the set. In contrast, the median is the middle number in a set of
values when those values are arranged from smallest to largest.
What is median give example?
Median, in statistics, is the middle value of the given list of data, when arranged in an order. The arrangement of data or observations can be done either in ascending order or descending order.
Example: The median of 2,3,4 is 3. In Maths, the median is also a type of average, which is used to find the center value.
What is median in math definition for kids?
Median – The median is the middle number of the data set. It is exactly like it sounds. To figure out the median you put all the numbers in order (highest to lowest or lowest to highest) and then
pick the middle number. If there is an odd number of data points, then you will have just one middle number.
Whats the definition of mean in math?
The mean is the mathematical average of a set of two or more numbers. The arithmetic mean and the geometric mean are two types of mean that can be calculated. The formula for calculating the
arithmetic mean is to add up the numbers in a set and divide by the total quantity of numbers in the set.
What is difference between median and average?
The average is calculated by adding up all of the individual values and dividing this total by the number of observations. The median is calculated by taking the “middle” value, the value for
which half of the observations are larger and half are smaller.
How median is calculated?
The median is calculated by arranging the scores in numerical order, dividing the total number of scores by two, then rounding that number up if using an odd number of scores to get the position of
the median or, if using an even number of scores, by averaging the number in that position and the next position.
What is the definition of median in statistics?
The statistical median is the middle number in a sequence of numbers. To find the median, organize each number in order by size; the number in the middle is the median.
How do you find median and range?
To find it, add together all of your values and divide by the number of addends. The median is the middle number of your data set when in order from least to greatest. The mode is the number that
occurred the most often. The range is the difference between the highest and lowest values.
What is the word mean in English?
1 : occupying a middle position : intermediate in space, order, time, kind, or degree. 2 : occupying a position about midway between extremes especially : being the mean of a set of values : average
the mean temperature.
How do you find the mean?
To calculate the mean, you first add all the numbers together (3 + 11 + 4 + 6 + 8 + 9 + 6 = 47). Then you divide the total sum by the number of scores used (47 / 7 = 6.7). In this example, the mean
or average of the number set is 6.7.
What is mean median mode and range?
Mean is the average of all of the numbers. Median is the middle number, when in order. Mode is the most common number. Range is the largest number minus the smallest number.
Does mean mean average?
A mean can be defined as an average of the set of values in a sample of data. In other words, an average is also called the arithmetic mean. Describing the average is called a mean.
What is the formula to find the median?
Median formula when a data set is even
Determine if the number of values, n, is even. Locate the two numbers in the middle of the data set. Find the average of the two middle numbers by adding them together and dividing the sum by two.
The result of this average is the median.
What is the median of the numbers?
Median is the middle number in a sorted list of numbers. To determine the median value in a sequence of numbers, the numbers must first be sorted, or arranged, in value order from lowest to highest
or highest to lowest.
How do you find the median value?
Example: There are 66 numbers
That means that the 33rd and 34th numbers in the sorted list are the two middle numbers. So to find the median: add the 33rd and 34th numbers together and divide by 2.
What is the median symbol?
List of Probability and Statistics Symbols
Symbol Symbol Name Meaning / definition
MR mid-range MR = (x[max]+x[min])/2
Q2 median / second quartile 50% of population are below this value = median of samples
Q[1] lower / first quartile 25% of population are below this value
x sample mean average / arithmetic mean
What are the two formulas of median?
In mean median mode formula the median formula is given for even as well as for odd number of observations (n). If the number of observations is even then the median formula is [Median = ((n/2)^th
term + ((n/2) + 1)^th term)/2] and if n = odd then the median formula is [Median = {(n + 1)/2} ^th term]. | {"url":"https://virtualpsychcentre.com/definition-of-median-in-mathematics/","timestamp":"2024-11-14T08:47:58Z","content_type":"text/html","content_length":"44484","record_id":"<urn:uuid:c851c71c-64e8-48c9-b8b4-3a7b7d84834d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00415.warc.gz"} |
Houston City Council approves safe-passing ordinance
Houston City Council approves safe-passing ordinance
Ordinance designed to protect 'vulnerable road users'
HOUSTON -
The Houston City Council on Wednesday approved a safe-passing ordinance aimed at protecting bicyclists and other vulnerable road users.
The ordinance requires drivers to keep a minimum of three feet between their vehicle and those vulnerable road users. Motorists also:
Cannot overtake and turn in front of a vulnerable road user unless it's safe.
Cannot maneuver their vehicle to intimidate or harass someone.
May not throw any object or substance at or against the vulnerable road user.
Vulnerable road users are defined as a walkers or runners; the physically disabled, such as someone in a wheelchair; a stranded motorist or passengers; highway construction, utility or
maintenance workers; tow truck operators; cyclists; moped, motor-driven cycle and scooter drivers; or horseback riders.
Mayor Annise Parker said more than 600 tickets have been issued to bicyclists who don't follow traffic laws.
“As a city, we need to protect everyone and anyone who uses our roads,” said Parker. “This ordinance will make our city even more attractive to those who want to enjoy traveling in forms other
than by car.”
“BikeHouston is pleased to see this ordinance pass and proud of the Mayor’s continued efforts on helping Houston become a more bicycle-friendly city,” says Kathryn Baumeister, Chair of
BikeHouston. “Houston is a city of cars, but also has a big population of people who rely on cycling for transportation and recreation. We feel it is important for cyclists and drivers of
automobiles to respect one another on the road. This ordinance will help provide a measure of safety for the vulnerable road users.”
Drivers who violate the ordinance will face a $500 fine. A similar law was passed statewide in 2009 but was vetoed by Gov. Rick Perry.
This takes effect today.
Edited by kylejack
This is welcomed news for the city of Houston, and something that is sorely needed. But even with the ordinance, our roads are still far from "safe" until we actually get them rebuilt. Too many
potholes, cracks and drop-offs!
I know a bunch of people in my Critical Mass group who were definitely excited about this. Now let's just see if the bikers will obey the law. (They won't).
I know a bunch of people in my Critical Mass group who were definitely excited about this. Now let's just see if the bikers will obey the law. (They won't).
No worries, the ordinance provides no protection for a cyclist breaking a law.
unless a cyclist rides in the debris right next to the curb, it's gonna be tough to maintain 3 ft in those places where COH shoehorned in a bike lane where the street is not really wide enough to
accomodate one.
still some regulation is better than no regulation at all.
unless a cyclist rides in the debris right next to the curb, it's gonna be tough to maintain 3 ft in those places where COH shoehorned in a bike lane where the street is not really wide enough
to accomodate one.
It's not tough. Just change lanes, or if not possible, follow behind.
Vulnerable road users are defined as a walkers or runners; the physically disabled, such as someone in a wheelchair; a stranded motorist or passengers; highway construction, utility or
maintenance workers; tow truck operators; cyclists; moped, motor-driven cycle and scooter drivers; or horseback riders.
Finally I can ride my horse to work without worry.
It's not tough. Just change lanes, or if not possible, follow behind.
Really? Follow behind a bicycle?
Really? Follow behind a bicycle?
Yes, really. If there is not space to pass safely (determined by ordinance as 3 feet), follow behind. If you don't like traveling on that road, turn off at the first opportunity and take another.
Really? Follow behind a bicycle?
It's been state law now for quite a while that a bicycle has all the rights and responsibilities of a motor vehicle. Bicyclists can take the whole lane if they want. It's just that most recognize the
reality of physics when it comes to collisions and most are also polite enough to not want to unnecessarily block traffic. But, they can make you follow behind if they really want to.
It's been state law now for quite a while that a bicycle has all the rights and responsibilities of a motor vehicle. Bicyclists can take the whole lane if they want. It's just that most recognize
the reality of physics when it comes to collisions and most are also polite enough to not want to unnecessarily block traffic. But, they can make you follow behind if they really want to.
not exactly...
Sec. 551.103. OPERATION ON ROADWAY. (a) Except as provided by Subsection (b ), a person operating a bicycle on a roadway who is moving slower than the other traffic on the roadway shall ride as
near as practicable to the right curb or edge of the roadway, unless:
(1) the person is passing another vehicle moving in the same direction;
(2) the person is preparing to turn left at an intersection or onto a private road or driveway;
(3) a condition on or of the roadway, including a fixed or moving object, parked or moving vehicle, pedestrian, animal, or surface hazard prevents the person from safely riding next to the right
curb or edge of the roadway; or
(4) the person is operating a bicycle in an outside lane that is:
(A) less than 14 feet in width and does not have a designated bicycle lane adjacent to that lane; or
(B ) too narrow for a bicycle and a motor vehicle to safely travel side by side.
(b ) A person operating a bicycle on a one-way roadway with two or more marked traffic lanes may ride as near as practicable to the left curb or edge of the roadway.
© Persons operating bicycles on a roadway may ride two abreast. Persons riding two abreast on a laned roadway shall ride in a single lane. Persons riding two abreast may not impede the normal
and reasonable flow of traffic on the roadway. Persons may not ride more than two abreast unless they are riding on a part of a roadway set aside for the exclusive operation of bicycles.
Edited by samagon
Really? Follow behind a bicycle?
Yep... It requires the drivers to actually pay attention to the road, instead of their phone, or their food. It will even occasionally require the removal of the lead foot. Houstonians drive too
damn FAST on city streets anyway. They're not freeways!!!
• 2
not exactly...
Ok...I stand corrected. I think you could argue that exceptions 3 and 4B frequently apply, though.
Yep... It requires the drivers to actually pay attention to the road, instead of their phone, or their food. It will even occasionally require the removal of the lead foot. Houstonians drive too
damn FAST on city streets anyway. They're not freeways!!!
So let's all go at 10 mph and follow behind bicycles (.....or just move over). I love the assumptions you and other posters made about my post.
So let's all go at 10 mph and follow behind bicycles (.....or just move over).
Edited by kylejack
What exactly are you suggesting as an alternative?
Driving less than 3ft from a cyclist is *not* safe, and it is an extremely unpleasant experience for the cyclist.
So let's all go at 10 mph and follow behind bicycles (.....or just move over). I love the assumptions you and other posters made about my post.
What exactly are you suggesting as an alternative?
Driving less than 3ft from a cyclist is *not* safe, and it is an extremely unpleasant experience for the cyclist.
I wasn't suggesting an alternative besides just moving over. I just don't know anyone who follows behind bicycle riders. If you have the patience to follow behind someone on a bike more power to you.
While I applaud the city's move on this, I'm not sure it's really going to have a proactive effect on cyclist safety. You'd need a really strong PR campaign behind it to have an impact. Three feet is
just over an arms length for me and that's way too close for comfort. Primarily, this will give HPD additional basis for tickets after an incident occurs.
I wasn't suggesting an alternative besides just moving over. I just don't know anyone who follows behind bicycle riders. If you have the patience to follow behind someone on a bike more power to
Come watch me ride on Main Street and you'll see some people who have no choice. Cars should be banned on Main Street downtown.
Edited by kylejack
• 1
While I applaud the city's move on this, I'm not sure it's really going to have a proactive effect on cyclist safety. You'd need a really strong PR campaign behind it to have an impact. Three
feet is just over an arms length for me and that's way too close for comfort. Primarily, this will give HPD additional basis for tickets after an incident occurs.
Yes, especially since right now you can kill someone and they can't find a charge to put on you. http://www.khou.com/news/local/Police-identify-woman-killed-in-auto-pedestrian-accident-194530351.html
Come watch me ride on Main Street and you'll see some people who have no choice. Cars should be banned on Main Street downtown.
Main Street's width has been reduced by light rail. If Main Street is dangerous for biking why not ride on the wider streets east & west of it?
As the City's bicycle trail building spree fuels an explosion of cyclists in the city, you will see drivers organically begin to give cyclists room. I have watched the number of aggressive drivers
steadily shrink over the last 15 years or so. Whereas in the 90s and early 2000s, you would see drivers running cyclists off the road, these days, the more common scenario is the tragic accident by
inattentive drivers as depicted in kylejack's link. That is a huge change in behavior from intentional aggression to accidents. It will continue as more cyclists take to the streets. It mirrors the
change in attitudes toward minority groups over time. As public acceptance towards a group grows, bias against the group is frowned upon.
We are slowly getting there. Those of us who have been here for 30 years or more can really see the difference, both in the behavior of Houstonians, and the perception of Houston by others.
• 2
Main Street's width has been reduced by light rail. If Main Street is dangerous for biking why not ride on the wider streets east & west of it?
I live on Main Street and feel I should be able to ride my bike on the street I live on. It's kinda useless for cars anyway. No left turns, and Main Street Square prevents passing through.
As the City's bicycle trail building spree fuels an explosion of cyclists in the city, you will see drivers organically begin to give cyclists room. I have watched the number of aggressive
drivers steadily shrink over the last 15 years or so. Whereas in the 90s and early 2000s, you would see drivers running cyclists off the road, these days, the more common scenario is the tragic
accident by inattentive drivers as depicted in kylejack's link. That is a huge change in behavior from intentional aggression to accidents. It will continue as more cyclists take to the streets.
It mirrors the change in attitudes toward minority groups over time. As public acceptance towards a group grows, bias against the group is frowned upon.
We are slowly getting there. Those of us who have been here for 30 years or more can really see the difference, both in the behavior of Houstonians, and the perception of Houston by others.
Very true. While there are still a few jerks and a few drunks in the city, I don't feel nearly the hostility I feel when I try to ride in, say, the Cypress area.
This is just another ordinance to appease a certain group. Status quo rules.
Groups like critical mass are the ones that will ruin it for all. Blatantly violating traffic rules and then claiming cars are the problem.
you're cynical, musicman. ill give you that.
Critical Mass is really not a problem in this city. In some places, yes, they break as many laws as they can, but Houston's Critical Mass has always been completely respectful of other people using
the road. I've never seen them take more than one lane (on roads with multiple lanes) and they have a good relationship with HPD.
This is just another ordinance to appease a certain group. Status quo rules.
Groups like critical mass are the ones that will ruin it for all. Blatantly violating traffic rules and then claiming cars are the problem.
The ordinance provides no new protection for a cyclist engaged in breaking a law.
Critical Mass is really not a problem in this city. In some places, yes, they break as many laws as they can, but Houston's Critical Mass has always been completely respectful of other people
using the road. I've never seen them take more than one lane (on roads with multiple lanes) and they have a good relationship with HPD.
And i've seen them go through red lights yelling obscenties at cars who have the green light and are honking at the cyclists. That's NOT being respectful to me. I personally almost hit a rider who
decided to cross in the middle of a block in front of a parked vehicle which made him impossible to see until he was already in my lane.
The ordinance provides no new protection for a cyclist engaged in breaking a law.
It provide no new protection period. Just passed to make cyclists feel safer. Our current administration loves these type of ordinances. Edited by musicman
I almost hit a motorist who ran a stop sign. I've had motorists yell obscenities at me as they attempt to cut me off in traffic.
The difference is that the people you complain of are operating vehicles that weigh 2 to 3 tons less than the ones doing it to me. Further, there are literally millions of motorists on the roads
daily versus literally thousands of cyclists.
Would you like some cream with your hot steaming cup of manufactured outrage?
• 1 | {"url":"https://www.houstonarchitecture.com/topic/28028-houston-city-council-approves-safe-passing-ordinance/","timestamp":"2024-11-14T11:27:35Z","content_type":"text/html","content_length":"491339","record_id":"<urn:uuid:0791f65d-a32d-47ba-b299-366ff45c292b>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00239.warc.gz"} |
SVE Module Support
Click to download Terminology for Module 1
Module 1 Topics
Topic A reactivates students' Kindergarten and Grade 1 learning. Student remember their "make ten" facts. They use ten-frame cards and number bonds of ten to move from concrete to pictorial to
abstract representations.They will use mental strategies to add and subtract within 20.
│ │ │
│ │ │
│mental strategies │number bonds │
say ten counting
Topic B- Students use decomposing strategies to add and subtract within 20. They will use the ten- structure to reason about making a ten to add and subtract. Students will use math drawings to solve
problems and understand the relationship between problems.
Topic C calls on students to review strategies to add and subtract within 100.
Decompose and add to make the ten
RDWW is a four step problem-solving process. The steps are as follows:
1. Read
2. Draw a picture
3. Write a number sentence
4. Write a word sentence to answer the question | {"url":"https://v2.toolboxpro.org/classrooms/template.cfm?ID=6254&P=116988","timestamp":"2024-11-03T21:32:15Z","content_type":"text/html","content_length":"32850","record_id":"<urn:uuid:0a05ed75-ae4a-447b-9df0-44a3ea94a716>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00600.warc.gz"} |
American Mathematical Society
A pair $(X,Y)$ of topological spaces is said to be a Blumberg pair ("BP") if for every $f:X \to Y$, there exists a dense subset $D$ of $X$ such that $f|D$ is continuous. $X$ is a Blumberg space if $
(X,R)$ is BP, where $R$ denotes the reals. $Y$ is co-Blumberg if $(R,Y)$ is BP. We survey the literature concerning the relationships between Blumberg spaces and Baire spaces and then study the
relationships between co-Blumberg spaces and separability properties. References
• Kevin A. Broughan, The intersection of a continuum of open dense sets, Bull. Austral. Math. Soc. 16 (1977), no. 2, 267–272. MR 448288, DOI 10.1017/S0004972700023297
• J. C. Bradford and Casper Goffman, Metric spaces in which Blumberg’s theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667–670. MR 146310, DOI 10.1090/S0002-9939-1960-0146310-1 J. C. Bradford,
Characterization of metric Blumberg pairs, unpublished manuscript.
• Henry Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), no. 2, 113–128. MR 1501216, DOI 10.1090/S0002-9947-1922-1501216-9
• Jack B. Brown, Metric spaces in which a strengthened form of Blumberg’s theorem holds, Fund. Math. 71 (1971), no. 3, 243–253. MR 292024, DOI 10.4064/fm-71-3-243-253 —, Variations on Blumberg’s
theorem, Real Analysis Exchange 9 (1983/84), 123-137.
• W. W. Comfort, A survey of cardinal invariants, General Topology and Appl. 1 (1971), no. 2, 163–199. MR 290326
• R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York,
1968. Translated from the Polish by K. Sieklucki. MR 0230273
• I. Juhász, Cardinal functions in topology, Mathematical Centre Tracts, No. 34, Mathematisch Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S. Kroonenberg. MR 0340021
• Władysław Kulpa and Andrzej Szymański, Decompositions into nowhere dense sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 1, 37–39 (English, with Russian summary). MR
• Ronnie Levy, A totally ordered Baire space for which Blumberg’s theorem fails, Proc. Amer. Math. Soc. 41 (1973), 304. MR 324630, DOI 10.1090/S0002-9939-1973-0324630-4
• Zbigniew Piotrowski and Andrzej Szymański, Concerning Blumberg’s theorem, Houston J. Math. 10 (1984), no. 1, 109–115. MR 736579
• Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446
• Petr Štěpánek and Petr Vopěnka, Decomposition of metric spaces into nowhere dense sets, Comment. Math. Univ. Carolinae 8 (1967), 387–404; correction, 8 (1967), 567–568. MR 225657
• Andrzej Szymański, On $m$-Baire and $m$-Blumberg spaces, Proceedings of the Conference Topology and Measure, II, Part 1 (Rostock/Warnemünde, 1977) Ernst-Moritz-Arndt Univ., Greifswald, 1980,
pp. 151–161. MR 646287
• William A. R. Weiss, The Blumberg problem, Trans. Amer. Math. Soc. 230 (1977), 71–85. MR 438280, DOI 10.1090/S0002-9947-1977-0438280-0
• H. E. White Jr., Topological spaces in which Blumberg’s theorem holds, Proc. Amer. Math. Soc. 44 (1974), 454–462. MR 341379, DOI 10.1090/S0002-9939-1974-0341379-3
• H. E. White Jr., Topological spaces in which Blumberg’s theorem holds. II, Illinois J. Math. 23 (1979), no. 3, 464–468. MR 537801
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C30, 26A03
• Retrieve articles in all journals with MSC: 54C30, 26A03
Bibliographic Information
• © Copyright 1986 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 96 (1986), 683-688
• MSC: Primary 54C30; Secondary 26A03
• DOI: https://doi.org/10.1090/S0002-9939-1986-0826502-4
• MathSciNet review: 826502 | {"url":"https://www.ams.org/journals/proc/1986-096-04/S0002-9939-1986-0826502-4/?active=current","timestamp":"2024-11-11T23:51:14Z","content_type":"text/html","content_length":"67646","record_id":"<urn:uuid:cb39a895-60a9-4a06-b64f-c9f93546b3ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00729.warc.gz"} |
Kansas Universities Might Scrap College Algebra Requirement Since 1 Out of 3 Students Fail It the First TimeKansas Universities Might Scrap College Algebra Requirement Since 1 Out of 3 Students Fail It the First Time
“We’re sending the majority of students down the college algebra road, which is really not necessary”
Wednesday, December 14, 2022 at 10:00am 32 Comments
Couldn’t Kansas try finding better professors? Isn’t that an option?
The College Fix reports:
Kansas universities may scrap algebra requirement because too many students fail it
Kansas universities may scrap their algebra graduation requirement because too many students fail the course, NPR Kansas reported.
“About one in three Kansas students fails college algebra the first time around. Some take it several times before they pass. Others get so frustrated that they drop out altogether. And that cuts
into university graduation rates,” the news outlet reported Dec. 12.
With that, the Kansas Board of Regents is considering alternative requirements such as statistics and quantitative reasoning under what’s called a Math Pathways program, it added.
“We’re sending the majority of students down the college algebra road, which is really not necessary,” said Daniel Archer, vice president of academic affairs for the Kansas Board of Regents.
“It’s not practical. It’s not really needed. And it’s not relevant for their fields.”
The pathways program aims to accelerate “students’ path through developmental math and enables them to take different paths through the math curriculum depending on their course of study,” the
Daily Caller reported.
According to NPR Kansas, Regent Wint Winter said investigating the new pathway program is critical because enrollment continues to decline.
Donations tax deductible
to the full extent allowed by law.
The Gentle Grizzly | December 14, 2022 at 10:17 am
I found algebra to be impossible. Then, I took it at Portland CC. It was taught by an engineer from one of the Silicon Forest companies, not a “real” teacher. He got through to me. I flunked the
first test and the make-up, but passed. With an 88% average.
John M | December 14, 2022 at 10:47 am
This is an interesting issue.
I’m a college educator and did about 4 years supervising STEM programs at a community college. A common question we wrestled with was, “what maths should all students have?”
Graduating any student into a world filled with problems and issues that are addressed with quantitative methods — that is, maths — would be criminal. But does that mean everyone needs algebra?
Well, at different levels, yes. Obviously, STEM students need the calculus, up through differential equations, as that’s the foundation of their discipline (I know; my background is civil engineering
and you can’t determine stresses like wind shear and whatnot without calc — of course, the computer does it for us now, but I’d like to not just trust the black box without knowing what the black box
is doing). And for anyone who’s done it, a lot of solving problems in calc involves the abstractions of algebra.
Social and Behavioral Science students, not to mention Business students, live — or should live — in a world of statistics, so Prob and Stats ought to be their bread and butter. There’s more to that
than just being able to play around in Excel, and some of that also involves some level of algebra (and analytic geometry) to understand what’s behind all of those distributions and extrapolations
and whatnot.
So what about humanities students? Well, philosophy majors, as Plato tells us, need to learn geometry. Its rigor and systems of proofs help us understand logic, and math logic (cool branch of math)
is also essential for the study of Philosophy today. But that discipline involves abstraction, which is taught in … algebra. Oh, and philosophy is nothing if not abstraction, and algebra, again,
teaches abstraction.
And for students in art, geometry is essential to understanding spatial relationships. History students need to understand statistics, and literature students need — well, they need to stop avoiding
rigorous math because a mathematical sensibility might temper their love for postmodern theory, which can be seriously challenged by maths. Euler’s number, Planck’s constant, and the humble number Pi
are not social constructions. If they were, we wouldn’t have cars or air conditioning. And that would suck.
For students in general, everyone needs some understanding of probability and statistics nowadays to negotiate the world. If nothing else, the little classic “How to Lie with Statistics” should be a
required read, but there are some other more recent books I’ve also liked, such as “Naked Statistics” and “How to be Never Wrong.” But also, algebra exercises our brains in a lot of good ways. It
helps with conceptual thinking and with exactness of thought, rigor and logic.
So I’m an upvote for everyone learning algebra at some level, and then focusing on the maths most relevant to their direction in life. Calculus for STEM, Stats for Business, Social Science, History,
Psychology, and Logic for Humanities.
But algebra, yes.
Sorry for the long rant.
henrybowman | December 14, 2022 at 11:50 am
I have no idea what “college algebra” even is. Our generation was done with algebra in HS, and those of us with an affinity for math took AP Calculus to get a leg up into college. Is this another
example of slipping stuff into college because the kids who were supposed to learn it in HS never did — like basic English grammar, spelling, and what have you?
Paula | December 14, 2022 at 12:23 pm
Eventually everything will be scrapped except the tuition.
Morning Sunshine | December 14, 2022 at 1:03 pm
I was bemoaning the fact that 3/5 of my homeschool kids will be graduating without leaving pre-algebra. I was one of those AP calc HS seniors, my husband made it to nationals in MathCounts at 13 – we
like math in this house. And yet my 2 oldest graduated not really finishing pre-algebra, and #3 is barley going to finish.
I was having a pity party about my bad math teaching skills, and my husband said something very profound (he took over math class 2 years ago). He said that the purpose of math is not just number
skills. Number skills are great and necessary.
But Algebra is also and at its base – problem solving. And that is why we go so slow on our math curriculum. We (they) take the time to problem solve
oh, and my oldest passed his math GED with an 87 last year, so clearly I didn’t do TOO bad…
and the second oldest got a math-tutoring job as a high school senior… so I guess I didn’t do too bad….
1073 | December 14, 2022 at 5:28 pm
Algebra is required for life.
Your grocery list is an Algebra problem.
2 tomatoes
3 Onions
1 head of lettuce
In my university experience, Algebra was taught in the largest classroom by the least experienced teacher (definitely not a professor.) But a professor was getting a kickback for using a specific
expensive book. Different book every few years so used wasn’t available.
I made a lot of money tutoring. Changing X to tomato or 3 point shots
Y to onion or field goals
In about 3 sessions they were caught up and got it.
Every cook, plumber and carpenter I’ve ever met is great at algebra unless you call it algebra.
Jack Klompus | December 14, 2022 at 5:51 pm
If you’re in college, you should’ve passed algebra in 9th grade at the latest.
I’m teaching in a remote location with a majority Native American population and my 7th graders who don’t come from anything remotely “privileged” are understanding algebra, or at least the
“pre-algebra” concepts of how to set up and solve equations with unknown quantities. They practice, practice, practice.
“According to NPR Kansas, Regent Wint Winter said investigating the new pathway program is critical because enrollment continues to decline.”
In other words, the higher ed gravy train is threatened because too many unqualified students are being admitted to college from schools that have lousy math teachers.
rleisenman | December 14, 2022 at 7:02 pm
“… investigating the new pathway program is critical because enrollment continues to decline.” This is absolutely, positively wrong. Problems with enrollment can never and should never be addressed
by curriculum changes, unless an actual fault is found in the curriculum apart from the fall in enrollment, i.e., unless there is an inherent fault that just happens to have the incidental effect of
lowering enrollment. To alter curriculum just to improve enrollment might just as well be followed by mailing diplomas to anyone who sends in a check, because otherwise “fewer people will send
Idonttweet | December 14, 2022 at 10:24 pm
So, the solution to students not being adequately prepared to meet a college-level mathematics standard is to lower the standard instead of requiring increased student preparedness. Is that what
you’re telling me? It appears that high schools are not adequately preparing students for a college career.
randian | December 15, 2022 at 7:29 am
The demographic breakdown of the failing students would be interesting.
randian | December 15, 2022 at 7:32 am
I thought college algebra was groups, rings, fields, vector spaces, and the like.
computer-mediator | December 15, 2022 at 8:30 am
I believe the ‘money shot’, as in it’s all about the money, in this higher education drama comes from of all places . . . . . . . NPR quoting regent Winter “. . . According to NPR Kansas, Regent Wint
Winter said investigating the new pathway program is critical because enrollment continues to decline.“
As a previous comment pointed out, it appears that the material being discussed here is what should be taught and learned thoroughly in 8th or 9th grade, maybe also 11th grade — things like solving
simultaneous equations, proficiency with numbers and quantities of widely varying orders of magnitude, interpretation of graphs of equations and at least some components of logic. Any student who
lacks these things should not be in college. Period.
Even to study humanities or social sciences or trendy majors the second word of which is “Studies.” Almost nothing worthwhile in academics has no dependence on logic and mathematical relationships,
and many life decisions require those things as well even if a day-to-day job does not. You can do without calculus, differential equations and linear algebra (as well as all the “advanced”
theoretical underpinnings of these subjects) if you are staying away from STEM, but even so you need to grasp basic concepts such as what it means for a quantity to tend toward a limit or toward
infinity and the rate of change of a quantity and the change in the rate of change (first and second derivatives), as well as to be able to take somewhat complex formulas and make your own
approximations so as to be able to make comparative judgments. Ignore those things, and you are at risk to manipulation by others, not to mention not going to be able to understand what your children
are learning as they become teenagers and young adults.
amatuerwrangler | December 15, 2022 at 11:43 am
From the article: “About one in three Kansas students fails college algebra the first time around. Some take it several times before they pass. Others get so frustrated that they drop out altogether.
And that cuts into university graduation rates,” the news outlet reported Dec. 12.
Graduation rates. They worry that people who cannot meet the standards will not graduate, so they want to water-down the standards. This sounds like “participation trophy” in the form of a diploma.
Maybe the university should install a test in the application process to assess the applicant’s math ability with passing the algebra course as the goal. That would screen out those without the
necessary background and stop the waste of funds (by both student and university) now consumed in trying to push them through.
However, the bottom line is that two out of three pass the course in the first taking. We haven’t been shown (here) what the numbers are for success by those who take the second run at it.
actionjksn | December 15, 2022 at 6:33 pm
Isn’t a college education supposed to be valuable because it shows you are elite and you can do something that most people cannot do? How easy are they going to continue making it? This is why we’re
seeing more and more people with college degrees though, and why they are less and less of impressive than they used to be. | {"url":"https://legalinsurrection.com/2022/12/kansas-board-of-regents-might-scrap-college-algebra-requirement-since-1-out-of-3-students-fail-it-the-first-time/","timestamp":"2024-11-10T01:46:00Z","content_type":"text/html","content_length":"296042","record_id":"<urn:uuid:4ae8cb17-5875-42a2-a200-0f076f2c708c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00133.warc.gz"} |
mill critical speed
The normal specific rates of breakage vary with mill speed in the same way. However, the maximum in power occurs at different fractions of critical speed from one mill to another, depending on
the mill diameter, the type of lifters, the ratio of ball-to-mill …
How to calculate cement mill critical speed attentions to cement ball mills critical speed ball mill wikipedia the free encyclopedia raw mills critical speed cement a ball mill is a type of the
grinding works on the principle of critical speed and it is widely used in production lines for powders such as cementball mill with full formula .
critical speed of a ball mill and ball size – Grinding Mill … Posted at: July 30, 2012. Ball mill – Wikipedia, the free encyclopedia The critical speed can be understood as that speed after which
the steel…
The Critical Speed is used for the determination of ball mill ideal operating speed. But for comparison, rod mills would operate between 50% to 95% of the critical speed. The faster the mill
speed, the greater the wear on the rods and liners. So, the …
Derived Equation For Estimating The Critical Speed In A. ball mill critical speed formula For instance a 90cm diameter ball mill with a critical speed of 44 rpm The tumbling of the as the cutting
speed is
Mill Critical Speed Determination. The Critical Speed for a grinding mill is defined as the rotational speed where centrifugal forces equal gravitational forces at ...
In Mineral Processing Design and Operations (Second Edition), 2016. 9.3.4 Mill Speed. During normal operation the mill speed tends to vary with mill charge. According to available literature, the
operating speeds of AG mills are much higher than conventional tumbling mills and are in the range of 80–85% of the critical speed.
The critical speed of ball mill is given by, where R = radius of ball mill; r = radius of ball. For R = 1000 mm and r = 50 mm, n c = 30.7 rpm. But the mill is operated at a speed of 15 rpm.
Therefore, the mill is operated at 100 x 15/30.7 = 48.86 % of critical speed. Get Price.
The mill was rotated at 63% of the critical speed. The position of the balls at the end of five revolutions is shown in Figure 2. It is seen that, using a low coefficient …
Mill Critical Speed Determination – SAGMILLING. Mill Critical Speed Determination. The "Critical Speed" for a grinding mill is defined as the rotational speed …
20201220 critical speed of ball mill formula derivation 3 apr 2018 semiautogenous grinding sag mill and a ball millapply bonds equation to industrial mills which differ from the standard for each
mill there is a critical speed that creates centrifuging figure 37c of the with the help of figure 38 the concepts used in derivation of.
Critical Speed Calculation Ball Mill. Ball mill critical speed mineral processing A ball mill critical speed actually ball rod ag or sag is the speed at which the centrifugal forces equal
gravitational forces at the mill shells inside surface and no balls will fall from its position onto the shell Read More Ball Mill Critical Speed Formula ...
The result of tests, the effect of fraction of mill critical speed on the grinding, it was found more different results from two different samples. I INTRODUCTION Comminution is extremely energy
intensive, consuming 3% to 4% of the electricity generated world-wide, and comprising up to 70% of all energy required in a typical cement plant.
A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from
its position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula. The mill speed is typically defined as the percent of the
Theoretical ...
The critical speed n rpm when the balls are attached to the wall due to centrifugation Figure 27 Displacement of balls in mill Conical Ball Mills differ in mill body construction which is
composed of two cones and a short cylindrical part located between them Fig 212 Such a ball mill body is expedient because efficiency is...
If speed of rotation is low, it results in insignificant milling action as balls only move in lower part of the drum. If speed of rotation is very high, the balls cling to the drum walls due to
centrifugal force. Therefore, intense grinding action is obtained only when drum rotates at critical speed and here the ball mill rotates at a speed ...
The critical speed formula of ball mill 4229 is a horizontal cylindrical equipment, which is used in grinding under critical speed. It is expressed by the formula NC 4229 VDD. The critical speed
is m, and the mill with diameter of 5 m rotates when reading more.
A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from
its position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula The mill speed is typically defined as the percent of the
Theoretical ...
Mill Speed - Critical Speed. Mill Speed . No matter how large or small a mill, ball mill, ceramic lined mill, pebble mill, jar mill or laboratory jar rolling mill, its rotational speed is
important to proper and efficient mill operation. Too low a speed and little energy is imparted on the product.
However, after reaching a critical speed, the mill charge clings to the inside perimeter of the mill. Under this conditions, the grinding rate is significant reduced or stopped. All mills must
operate less than Critical Speed; At position A, the media is held to the wall due …
calculate critical speed grinder – Grinding Mill China. Read Craftsman Professional 8 in. Bench Grinder, Variable Speed reviews and find out why …
Iron Ore Primary Ball Mill Critical Speed Of Ball Mill Formulae . Home; Products; Contact; What we do. We is a high-tech company integrating R&D, production and distribution, and provides
crusher, sand making, grinding equipment, mobile crushing station, etc. mature products and solutions used in aggregate, mining and waste recycling. At ...
Now, Click on Ball Mill Sizing under Materials and Metallurgical Now, Click on Critical Speed of Mill under Ball Mill Sizing. The screenshot below displays the page or activity to enter your
values, to get the answer for the critical speed of mill according to the respective parameters which is the Mill Diameter (D) and Diameter of Balls (d).. Now, enter the value appropriately and
accordingly ...
The critical speed n, of a ball mill is the rotational velocity at which the balls cease to grind, and are carried read more mechanisms of dry ball milling in mox fabrication is carried out in
the french plants in a ball milling and under dry conditions in order to avoid any he rotation speed is expressed in percentage of the critical. Read More.
Mill Speed is one variable that can often be easily changed with a variable frequency drive (VFD). The starting point for mill speed calculations is the critical speed. Critical speed (CS) is the
speed at which the grinding media will centrifuge against the wall of …
Critical Speed Of Mill Henan Mining Machinery Co Ltd. An analysis of the published experimental data on variation of the mill power with mill speed shows that in the range 5570 critical speed the
torque corresponding to the net power drawn by the mill either remains practically constant or it increases gradually with the mill speed by about 8 These data can be utilized for evaluating the
Live Chat
The terms high-speed vibration milling (HSVM), high-speed ball milling (HSBM), and planetary ball mill (PBM) are often used. The commercial apparatus are PBMs Fritsch P-5 and Fritsch
Pulverisettes 6 and 7 classic line, the Retsch shaker (or mixer) mills ZM1, MM200, MM400, AS200, the Spex 8000, 6750 freezer/mill SPEX CertiPrep, and the SWH-0.4 ...
The critical speed of the mill, & c, is defined as the speed at which a single ball will just remain against the wall for a full cycle. At the top of the cycle =0 and Fc Fg (8.5) mp & 2 cDm 2 mpg
(8.6) & c 2g Dm 1/2 (8.7) The critical speed is usually expressed in terms of the number of revolutions per second Nc & c 2 1 2 2g Dm 1/2 (2×9.81)1/2 ...
The image above represents critical mill of speed. To compute for critical mill of speed, two essential parameters are needed and these parameters are Mill Diameter (D) and Diameter of Balls (d).
The formula for calculating critical mill of speed: N c = 42.3 / √(D – d) Where: N c = Critical Speed of Mill D = Mill Diameter d = Diameter of Balls
Critical Speed_ When the ball mill cylinder is rotated, there is no relative slip between the grinding medium and the cylinder wall, and it just starts to run in a state of rotation with the
cylinder of the mill. This instantaneous speed of the mill is as follows: | {"url":"https://www.zielonadroga.edu.pl/Mar/23-17168.html","timestamp":"2024-11-10T20:26:17Z","content_type":"application/xhtml+xml","content_length":"23366","record_id":"<urn:uuid:28476ce5-2530-447b-a7ea-5cc26a26ff7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00308.warc.gz"} |
Graphs of Polynomial Functions – Practice and Tutorial
Quick Tutorial
A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). Or, an equation with many terms, usually
comprised of powers and a constant
Graphing a Polynomial
Write the polynomial in standard form (highest power first). Create an input-output table to determine points. Plot the points and connect the dots to draw the graph.
Ex. Graph the given equation.
y = x^3 – 2x^2 + 3x – 5
Noticing the highest degree is 3, we know that the general form of the graph should be a sideways “S.”
Here is the input – output table
• If the degree of the polynomial is even and the leading coefficient is positive, both ends of the graph point up.
• If the degree is even and the leading coefficient is negative, both ends of the graph point down.
• If the degree is odd and the leading coefficient is positive, the left side of the graph points down and the right side points up.
• If the degree is odd and the leading coefficient is negative, the left side of the graph points up and the right side points down.
Graphing Polynomial Practice Questions
1. Describe the end behavior for the function f(x) = – 2x^5 + 3x + 97.
a. Up on the left and right.
b. Down on the left and right.
c. Down on the left, up on the right.
d. Up on the left, down on the right.
2. Describe the end behavior of the polynomial P(x) = (2x – 5)(3 – x)(4x^2 + 7).
a. y → 0 as x → -∞ and y → -∞ as x → ∞
b. y → -∞ as x → -∞ and y → ∞ as x → ∞
c. y → -∞ as x → -∞ and y → -∞ as x → ∞
d. y → -∞ as x → -∞ and y → 0 as x → ∞
3. Determine the x-coordinate of the graph for the polynomial: y = x^2 + 2x – 1
a. 1
b. -1
c. 1/2
d. -1/2
4. What happens to the graph when the original equation y = x^2 – x – 2 is changed to y = 2x^2 – x – 2 ?
a. The graph shifts up, but the shape remains the same
b. The graph shifts down, but the shape remains the same
c. The graph becomes more narrow (compressed)
d. The graph becomes more wide (stretched)
5. Which equation fits the graph shown?
a. y = x^2 + x – 2
b. y = x^2 – x – 2
c. y = 2x^2 – x – 2
d. y = 2x^2 + x – 2
6. What value can x NOT have, in the polynomial: y = x^2 – x – 1
a. 0
b. 1
c. 2
d. x can be any real number
7. Which equation fits the graph shown?
a. y = (1/2)x^2 – x + 1
b. y = (1/2)x^2 + x + 1
c. y = (1/2)x^2 + x – 1
d. y = (1/2)x^2 – x – 1
8. What happens to the graph when the original equation y = x^2 – 2 is changed to y = -x^2 – 2 ?
a. The new graph shifts down, but the shape remains the same
b. The new graph shifts down, and the shape changes
c. The new graph flips (opens down), but the shape changes
d. The new graph flips (opens down), but the shape remains the same
9. What is the maximum or minimum for the equation: y = 2x^2 + 2x – 2 ?
a. Minimum at (-0.5, -2.5)
b. Minimum at (0, -2)
c. Maximum at (-0.5, -2.5)
d. Maximum at (0, -2)
10. Describe the general shape of the graph found by the equation: y = x^4 – x^2 .
a. A curved “W”
b. A curved “M”
c. A curved “S”
d. A parabola
Algebra Problem Answer Key
Answer Key
1. D
Drawing the graph of a 5^th degree polynomial is not a practical way to solve this problem. Instead; we need to remember some properties of polynomial graphs:
Notice that this is an odd degree polynomial. So; two ends of the graph head off in opposite directions. If the leading term is positive; the left end would be down and the right end would be up.
However, the leading term here is -2x^5 that is negative. So; the end behavior for this function is up on the left and down on the right.
2. C
The end behavior of a polynomial is determined by the degree of the polynomial and the sign of the leading term. We do not need to expand the polynomial; by multiplying the x values in three
parenthesis, it is easy to see that the leading term is – 8x^4. Then, the degree of the polynomial is 4 and the leading term is negative. This means that its graph is downwards with end points ® – ¥.
Even degree means that both left and right end points have the same characteristics; y → -∞ both as x → -∞ and x → ∞.
3. B
The x-coordinate of the vertex is found by (-b)/(2a) = (-2)/(2 * 1) = -2/2 = -1
4. C
The leading coefficient of 2 makes the curve more “steep”, giving the effect of compressing the sides of the parabola together.
5. B
The points (-1,0), (0,-2), and (2,0) are on the graph. Substituting the x-coordinates into the equations, only “y = x^2 – x – 2″ gives the correct corresponding y-coordinates.
6. D
All real numbers are valid for x.
7. B
The point (0,1) is on the graph and fits both A and B. However, the vertex is at (-1, 1/2) and works with B but does not work for A.
8. D
The negative with the x^2 term makes the parabola open down, showing a reflection across the horizontal line (y = -2).
9. A
The vertex identifies the vertical boundary for a parabola. The x-coordinate is found by (-b)/(2a) = (-2)/(2 * 2) = (-2/4) = -1/2. Substituting into the equation gives a y-coordinate (-2.5).
Because the leading coefficient is positive, the parabola opens up, so the vertex is a minimum.
10. A
Select x-coordinates of -1, -1/2, 0, 1/2, 1 and substitute into the equation gives points of (-1, 0), (-1/2, -0.19), (0, 0), (1/2, -0.19), and (1, 0), creating a curved “W”.
1 Comment
1. Can you enlarge some of the graphs? | {"url":"https://test-preparation.ca/graph-polynomial-functions/","timestamp":"2024-11-08T15:54:31Z","content_type":"text/html","content_length":"202359","record_id":"<urn:uuid:3d23bfd3-75dd-413c-80c5-d6b86b2e266f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00571.warc.gz"} |
Radiative vector meson decays: omega ->pi(0)eta gamma and rho ->pi(0)eta gamma
We study the omega and p decays into pi(0)eta gamma using phenomenological approach by adding to the amplitude calculated with in the framework of VMD, chiral loop, a(0)-meson intermediate state and
p-omega-mixing. a(0)-meson intermediate state amplitude makes a small contribution for two decays when considering to proceed by a two step mechanism.
A. Kucukarslan, “Radiative vector meson decays: omega ->pi(0)eta gamma and rho ->pi(0)eta gamma,” INTERNATIONAL JOURNAL OF MODERN PHYSICS A, pp. 712–714, 2005, Accessed: 00, 2020. [Online].
Available: https://hdl.handle.net/11511/63469. | {"url":"https://open.metu.edu.tr/handle/11511/63469","timestamp":"2024-11-02T02:50:47Z","content_type":"application/xhtml+xml","content_length":"53684","record_id":"<urn:uuid:d9bedd90-3cec-452f-8d80-940275a5a873>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00685.warc.gz"} |
GreeneMath.com | Ace your next Math Test!
Practice Objectives
• Demonstrate an understanding of multi-digit multiplication
Practice Multiplying Mult-Digit Whole Numbers
Answer 7/10 questions correctly to pass.
Find each product.
Not Correct!
Your answer was: 0
The correct answer was: 0
Multi-Digit Multiplication of Whole Numbers:
1. Arrange the numbers being multiplied into a vertical format
□ Place the number with more digits on top
□ If both numbers have the same number of digits either can be on top
□ Draw a horizontal line underneath the bottom number
□ Place a "×" to the left of the bottom number
2. Begin by selecting the rightmost digit of the bottom number
3. Multiply this digit by each digit in the top number working right to left
□ Place each answer directly below the horizontal line working right to left
□ Use the carrying process for any answer larger than 9
4. Continue the process for each digit working left in the bottom number
5. Record the answers on a new row and shift the start one place left
6. It may be helpful to write a zero in the skipped over places
7. Once the multiplication process is done, find the sum of the numbers below the horizonal line
You Have Missed 4 Questions...
Your answer should be a number!
Current Score: 0%
Correct Answers: 0 of 7
Wrong Answers: 0 of 3
Need Help?
Video Lesson | Written Lesson
Prefer Multiple Choice?
Multiple Choice Test
Wow! You have mastered Multi-Digit Multiplication!
Correct Answers: 0/0
Your Score: 0%
Restart | Next Lesson | {"url":"https://greenemath.com/Prealgebra/11/MultiplicationTest.html","timestamp":"2024-11-14T15:07:03Z","content_type":"application/xhtml+xml","content_length":"12467","record_id":"<urn:uuid:e8d2fe33-54c5-4626-87aa-a992b5f774cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00510.warc.gz"} |
Configure product ranking criteria
Inform ERP uses product ranks to give you additional insight into your products and categorize products for cycle count frequency. Please note that all Stock and Non Stock products will receive a
ranking. Inform will not update the rank of Consumable, Discontinued, or Inactive products through any automated processes.
By default, it calculates product rank based on Hits, which is the number of times that a product appears as a line item on an invoice. However, you can alternatively base rank on Sales Dollars or
GMROI, or a combination of the three variables. Product ranking is recalculated each night, so changes made to ranking calculations are reflected in the product rank the next day.
You can also configure Inform ERP to automatically set poorly performing Active products to non-stock status, based on a rank threshold. For example, any item that drops to D rank can be set to a
non-stock item, so you do not waste shelf space on poorly moving product. You can later change the status back to Active, and Inform will wait a set number of days before re-evaluating the item.
Note: By default, product ranking is recalculated each weeknight: changes made to ranking calculations are reflected in the product rank the next day
Inform ERP calculates a rank for each product based on the variables that you choose to include in the rank calculation. Each rank is assigned a numerical value ( A = 40, B= 30, C = 20, D = 10).
Those values are used with the weighted value assigned to each metric to calculate an overall numerical rank for the item, from 0-40. Letter ranks are assigned to these numbers using a sliding scale
to give a rank of the item.
1. Go to File > Product > Ranking Criteria.
2. Click Edit.
3. Calculation, and define the ranking metrics:
□ Sales ($): The total value of the products sales. Each night, the system will list the items from greatest value to least value. The top X% of the list will be A ranked, the next X% will be B
ranked, etc. The total must equal 100%.
□ Hits: The number of times the product appears on an invoice. As with sales, each night, the system list the items from greatest number of hits to the least. The top X% of the list will be A
ranked, the next X% will be B ranked, etc. the total must equal 100%.
□ GMROI: Gross Margin Return on Investment. An inventory profitability evaluation ratio that analyzes a firm's ability to turn inventory into cash above the cost of the inventory. It is
calculated by dividing the gross margin by the average inventory cost.
4. Weights section, assign the Weight each variable has in the calculation.
If you want to value all variables equally, set two of them to 33% and one to 34%. The weights must add up to 100%. If you do not want a metric to be used in the calculation, set the weight to
zero. This amount will be multiplied by the numerical rank value of the metric and then added together to calculate a total numerical rank.
5. Spreads.
When the system calculates the rank for a product, it assigns a numerical value to each letter rank (A = 40, B = 30, C = 20, D =10) and then uses that number, multiplied times the weight, to
calculate a total rank number for the product. The spread tells the system what letter rank that number corresponds to.
For Example:
Item Q is A ranked in Sales $, B ranked in Hits, and B ranked in GMROI. All variables are weighted equally. The system completes the calculation below to get a numerical value of the rank of the
The spread section defines what rank that number corresponds to on a sliding scale. In the example shown below, 21-40 is an A ranked product, so Item Q would be A ranked because its weighted
score of 33 is within the 21-40 Spread configured for A, below.
6. If you'd like to flag very slow moving items as non-stock, so that you do not bother tracking or stocking this product any longer, then from the Non-Stock Threshold list, choose the rank A-D that
will flag the product as Non-Stock. Typically, you would set this threshold to D, and make sure that you set the D calculations to be the lowest possible Sales, Hits, and/or GMROIs.
When the product’s rank by warehouse reaches this rank, the status of the product in that warehouse will change to non-stock.
If a user manually changes this product back to an Active item, the system will wait until the Re-evaluation #of days to change the product to non-stock again, if the rank does not improve. If
you select a Non-Stock Threshold, you must specify the Re-evaluation number of days.
If you do not want the system to change the stock status, set the Non-Stock Threshold to None.
Setting the Non-Stock Threshold changes the Status of the item by warehouse (File>Product: Setup, Purchasing–Warehouse Procurement, check Stock?); not the overall Status (File> Product: Setup ).
This is true even if your company has only one warehouse.
7. Click Save. The rank recalculation runs overnight, reflecting the new product ranks by the next day. Utilities>Phantom Monitor: Product Rank Update) | {"url":"https://ddisys.zendesk.com/hc/en-us/articles/360020734373-Configure-product-ranking-criteria","timestamp":"2024-11-02T09:30:48Z","content_type":"text/html","content_length":"43854","record_id":"<urn:uuid:79067d86-d878-452f-85f1-d3956ec91a48>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00865.warc.gz"} |
HEXA to HEX
HEXA to HEX FAQ
What is the difference between HEXA and HEX in computer science?
HEXA and HEX are often used interchangeably in computer science to refer to hexadecimal notation. HEXA typically stands for hexadecimal, which is a base-16 number system using digits 0-9 and letters
A-F. HEX is a shorthand for the same concept. Essentially, they both represent the same numerical system, but HEXA is just a longer form of referring to hexadecimal.
How do you convert a HEXA value to a HEX value?
Since HEXA and HEX are essentially the same in terms of representing hexadecimal values, there is no need for conversion. If you encounter a value labeled as HEXA or HEX, you can treat them both as
hexadecimal values without any conversion. For example, the HEXA value 1A3F is the same as the HEX value 1A3F.
Are there any differences in usage between HEXA and HEX in programming languages?
In most programming languages, HEX and HEXA are used synonymously to denote hexadecimal values. For example, in languages like C, C++, and Python, hexadecimal values are typically prefixed with 0x.
Both terms are understood to mean the same thing, but the term "HEX" is more commonly used. For instance:
int hexValue = 0x1A3F;
This line of code is using a hexadecimal value, which can be referred to as either HEX or HEXA.
Why is hexadecimal (HEXA/HEX) used in computer science?
Hexadecimal is used in computer science because it offers a more human-readable way to represent binary values. Since one hexadecimal digit represents four binary digits (bits), it is more concise
and easier to read than binary notation. For instance, the binary number 1010 0011 1110 can be represented as A3E in hexadecimal, making it simpler to read and debug.
How can you manually convert a decimal number to HEXA/HEX?
To manually convert a decimal number to hexadecimal (HEXA/HEX), follow these steps:
1. Divide the decimal number by 16.
2. Record the remainder (this will be the least significant digit of the HEX value).
3. Divide the quotient obtained in step 1 by 16.
4. Repeat steps 2 and 3 until the quotient is 0.
5. The HEX value is the remainders read in reverse order.
For example, to convert the decimal number 255 to HEXA/HEX:
1. 255 ÷ 16 = 15 remainder 15 (F in hexadecimal)
2. 15 ÷ 16 = 0 remainder 15 (F in hexadecimal)
Reading the remainders in reverse order, the HEXA/HEX value of 255 is FF. | {"url":"https://toolator.com/hexa-to-hex","timestamp":"2024-11-09T19:36:05Z","content_type":"text/html","content_length":"30795","record_id":"<urn:uuid:d9c72862-6f3e-4714-afcd-4945541dac28>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00066.warc.gz"} |
Are there options for adding appendices to the coursework? | Pay Someone Take My Coursework Writing
Are there options for adding appendices to the coursework? The more I look, the more my reading a knockout post come to an visit I have added appendices to the courses in this question, in each
example I have added a question explaining each of the attributes to add each course to. I want this in Kicci for students building new courses. With this approach, we lose students with learning
difficulties and their academic grades, while engaging in a creative exercise with no one figuring out what your courses are all about. When I look at this “experience”, I don’t have any problem
selecting courses from the list of examples. I seem to have found some where the way to select the exercises is to not look at the examples at the top of the list. My question is, visit this site
right here exactly is a course, and how do you specify where, instead of looking at its examples? The answer is, that the form of the list of examples needs to be a line containing a “;” (without
spaces) and a “in” (with spaces) Homepage with a }. However, is there a better way? Hi Amy, We’re looking to add some examples of interest to our team, but really just want “; and the lines….” as a
line character, ideally with spaces in it… Thanks for looking into this, we have a learning course for the students, in his response year 2000 – 2005 and they are quite inexperienced in the
application in their English courses. When you visit the courses that you choose as you choose them, they have large sections of English teaching experience. But they lack the experience of most of
the courses on that list. So you don’t need that extensive experience when you choose them. Would you recommend to all of our students, that they have English courses at a location away from English
classes? Firstly The textbook requirements – the official statement are roughly 800 papers in English as taught by University of Wuppia in Wuppia, North RhineAre there options for adding appendices
to the coursework? I can’t figure out something to explain in my code. For (Length i = 1; i < length; i++) { //create an ellipse pattern that is a circle with an upper bound on the bottom half of the
line plus the middle line.
First-hour Class
It should be something along the lines below. Also should be below along the top. The ellipse pattern should have a center of the ellipse but also be centered at the bottom. The center should not be
above the side. public static void LoopGt(int nNumberToFill, float distance, int maxSize) { float topLeft = Mathf.Max(Mathf.Cos(distance), Mathf.Min(distance.Length, maxSize – 1)); float right =
Mathf.Cos(distance); float bottomRight = Mathf.Cos(right); float leftRight = Mathf.Cos(left); float bottomLeft = Mathf.Sin(distance); //create center point Point centerX = new Point(Mathf.Clamp
(topLeft, center.X + bottomRight), 2); Point centerY = new Point(Mathf.Clamp(topLeft, center.Y – bottomLeft), 0); float fill = 0.0f; //create a line segment that is a circle with an upper bound on
the distance between the center point and the intersection of the center, and the segment. float centerEllipse = centerX + bottomRight – bottomLeft; //create a circle segment that has an upper bound
on the distance between the center point and the intersection of the center. if (centerX.
How Can I Legally Employ Someone?
Len() > 1) { float centerC = centerX.C; float centerEllipse = centerX.E; Point centerEllipseCenter = centerX.Center(); Point centerEllipseRegion = new Point(body.Width/2, body.Height); float x =
centerellipseCenter.X – centerEllipseCenter.X; float y = centerELLipseCenter.Y – centerEllipseCenter.Y; float square = 10d – square; float border = 1.4f – border; float radius = 1.0f; float
newCenterCount = 1.0f;Are there options for adding appendices to the coursework? | {"url":"https://dothecoursework.com/are-there-options-for-adding-appendices-to-the-coursework","timestamp":"2024-11-03T02:43:28Z","content_type":"text/html","content_length":"89472","record_id":"<urn:uuid:a73068a8-7642-45d6-8dcc-e2ffc29fc9a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00216.warc.gz"} |
T4T Sorting Problem Types
Material Type:
Lesson, Lesson Plan
Lower Primary
Media Formats:
Downloadable docs
Education Standards
T4T Sorting Problem Types
This resource is from Tools4NCTeachers.
In this lesson, students use their understanding of word problems to sort problems based on type. The focus for students in this lesson shifts from solving word problems to analyzing word problem
structures in order to group like problems. The purpose is to help students understand and visualize problems so they will know how to write equations that match and know what operation is needed to
solve. This lesson focuses on sorting addition and subtraction word problems with a specific focus on Take Apart and Put Together types. The lesson, however, can be adjusted to use with any problem
types you notice students need more experience with. You could also adjust the number size in the problems to use this lesson at different times in the year.
Here is a sample of this resource. Click the attachment to download the fully-formatted lesson and support materials.
Sorting Problem Types
In this lesson, students use their understanding of word problems to sort problems based on type. The focus for students in this lesson shifts from solving word problems to analyzing word problem
structures in order to group like problems. The purpose is to help students understand and visualize problems so they will know how to write equations that match and know what operation is needed to
solve. This lesson focuses on sorting addition and subtraction word problems with a specific focus on Take Apart and Put Together types. The lesson, however, can be adjusted to use with any problem
types you notice students need more experience with. You could also adjust the number size in the problems to use this lesson at different times in the year.
NC Mathematics Standard:
Represent and solve problems
NC.1.OA.1 Represent and solve addition and subtraction word problems, within 20, with unknowns, by using objects, drawings, and equations with a symbol for the unknown number to represent the problem
when solving:
• Add to / Take from – Change unknown
• Put together / Take apart – Addend unknown
• Compare – Difference unknown
Standards for Mathematical Practice:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
7. Look for and make use of structure.
Student Outcomes:
• I can sort word problems by determining whether situations involve taking apart or putting together groups.
• I can write equations to match word problem situations.
• I can analyze and sort word problems that involve change unknown situations.
• I can solve word problems.
Math Language:
What words or phrases do I expect students to talk about during this lesson?
equation, word problems, add to, put together, take from, take apart, sort, unknown, solution
• book, Elevator Magic by Stuart J. Murphy
• word problem cards for students to sort
• white boards or student math notebooks
• copies of Exit Ticket
• manipulatives to model problems to determine problem type
Advance Preparation:
• Make copies of word problem cards (Activity Sheets) and cut apart, unless students will cut apart during the lesson.
• Read aloud the book Elevator Magic by Stuart J. Murphy (or similar text). This could occur up to a few days prior to this lesson or earlier in the same day you plan to teach this lesson. As you
read, ask students to act out the subtraction situations through models or drawings. Briefly discuss the different situations with students throughout your reading of the text.
1. Introduce the task (5 minutes)
Remind students of all the work they have been doing and continue to do with solving problems, using the context of Elevator Magic or similar story as an example. Say: As we have learned, there are
many different situations or stories that show things being put together, taken apart, added to, or taken from other things. Tell students they will work with more of these stories or word problems
today. Instead of solving the problem as they normally do, students will analyze the problems and sort them into groups by thinking about the type of problem or situation.
Give students the first set of word problem cards (Activity Sheet 1). Read aloud the problem on each word problem card, asking students to follow along as you read.
*Note: You may choose to only use four word problem cards instead of all six. Just be sure both types are represented, and that all children receive the same word problem cards.
There were 15 fish in a fish tank. 8 of the fish were orange and 8 black cows and 7 brown cows are in a field There are 15 cupcakes. The first grade students ate 7 of the cupcakes. How many
the rest were white. How many fish were white? together. How many cows are in the field? cupcakes are left?
I have a bag of candy. There are 8 chocolate candies and 7 lemon We found 7 ladybugs and 8 crickets in the My dad bought 15 balloons for the party. 7 of the balloons are yellow. The rest of
candies. How many pieces of candy are in the bag? backyard. How many bugs did we find? the balloons are blue. How many blue balloons did my dad buy?
1. Sorting Word Problems (10-12 minutes)
Ask students to work in groups of 2 or 3 to sort the word problem cards as they deem appropriate based on their own criteria. For example, some students may say “I know the whole group [total] in
this problem” and then sort the rest based on that criteria (whether the total is known or unknown). Remind students they will need to explain their thinking and justify why they sorted the problems
as they did. Be available to read problems aloud to students again as needed.
As students are working, refrain from suggesting how students could sort. If groups are struggling, ask questions to help them explore and to spark their thinking (e.g. Do you notice any similarities
in these two problems?). Also encourage students to discuss their thinking while sorting and their rationales with each other.
You can use questions such as:
• What are you thinking about this problem? What makes you think that?
• Do you agree? Why? Why not?
• How did you decide to sort the problems?
• Can you explain what he/she just said?
• Why did you decide to put this problem in this group?
• What can you do since you do not agree on how to sort this problem?
• What do you think of his/her reasoning?
Students may not be able to distinguish action and operation without modeling the problem. Provide students with manipulatives to directly model the problem, but do not make finding the solution a
focus of your questioning or instruction. Move among the groups as children are working and discussing the problems. Observe their methods for sorting and notice the groups they create. Most likely,
students will sort into two groups: situations that show adding and situations that show subtracting. However, notice if any students or groups have a different way of thinking, even if it seems
incorrect or inefficient. This may be something you will need to address during the class discussion. Also notice meaningful conversation between students or groups as they talk about the problems
that you would like to highlight during the whole class discussion.
1. Explanation of Sorted Groups (15-20 minutes)
Before you have students move together as a large group, collect some of the word problem cards and keep them in the groups students placed them in when they sorted.
Bring the class together to talk through how students sorted the word problems. During this time of discussion, you will want to ask student groups to share their thinking for sorting but keep the
conversation open to allow students not in that particular group to ask questions and agree or disagree with what is being shared. Remember to showcase the thinking and discussion you observed during
the Explore activity by asking those students to share their experiences. Since all students sorted the same word problems but possibly in different ways, facilitate the discussion to allow for one
train of thought followed by another so at the end of the discussion the whole class will have sorted all problems and have agreed upon the criteria for sorting.
The end result should be word problem cards sorted into two groups: Take Apart problems and Put Together problems. To assist students as they interpret and analyze the problems, suggest writing
equations to match the problems. Working together as a class, ask students to write and explain equations that match the situation of the problems. Specifically be aware of the take apart problems.
It is possible students will see these as an addition situation with a missing addend. Help them to see the situation as having a whole and taking that whole apart to make two groups, thus using
subtraction to take away the known quantity to find the unknown group (e.g. the fish tank problem).
Additional Activities (if needed)
1. Sorting & Solving Problems including Change Unknown Problems (15-20 minutes)
Allow students opportunity to extend their thinking by repeating the process of analyzing and sorting word problems, this time including other problem types. Giving students opportunity to analyze
problems and recognize different types will allow them to better understand the problems, link them to accurate equations, and solve with increased consistency.
Give students the second set of word problem cards (Activity Sheet 2). Read aloud the problem on each word problem card, asking students to follow along as you read.
Maria saw three yellow butterflies. She also saw Annya has 8 pennies. She found more pennies on the Jose had 11 lollipops. He gave a lollipop to each Bill had 11 toy cars. He lost some, but he
eight orange butterflies. How many butterflies did sidewalk. Now she has 11. How many pennies did of his 8 friends. How many lollipops does Jose still has 3 toy cars. How many toy cars did
Maria see? Annya find? have now? Bill lose?
Two of the problems are subtraction and two of them are addition but the set does include a couple of change unknown problems as well. Listen in to conversations as children work together to sort
these four problems into groups. There is again opportunity for students to sort in various ways, including change unknown vs. result unknown or by operation (e.g. subtraction vs. addition). Notice
the elements of the problems students recognize easily and what they struggle to comprehend then make notes for future instruction.
For an additional challenge, you may ask students to find solutions to the word problems on the cards. You could give students specific word problems you would like them to solve or you can allow
students to choose two or three of the cards they have been working with and solve the problems on those cards.
Students can pair up with a partner to check each other’s work and discuss strategies for finding solutions to the problems.
Evaluation of Student Understanding
Informal Evaluation:
Observe students as they discuss the problems and sort. Make note of obstacles and/or misconceptions students face so they may be addressed in later lessons or small group meetings. Question students
to uncover their thinking, especially since they are not producing an actual product. Have conversations with students to encourage oral explanation and use of academic language.
Formal Evaluation/Exit Ticket:
Pose the following task for students to complete.
The answer to a word problem is 5 monsters. What could the problem be? Write a word problem with a question that can be answered by the solution “5 monsters.”
Meeting the Needs of the Range of Learners
• Reduce the number of word problem cards to sort for partners or small groups who have difficulty getting started or are struggling with the task.
• Encourage children to model the problem with manipulatives in order for them to more clearly see the situation and determine the action.
• Students could select a word problem from each sorted group and write the equation that matches the problem situation.
• Ask students to collaboratively write one more word problems that would match the type for each sorted group. Challenge them to use the same numbers or fact family as the given problems.
Possible Misconceptions/Suggestions:
Possible Misconceptions Suggestions
Select first a word problem card with noticeable
action. Read the problem aloud again asking
students to visualize what is happening, like
making a movie in their mind. Have them act out
the situation with cubes or counters. Repeat with a
Students sort the word problems simply by looking at the numbers used in the problem.
second problem then ask students to decide if the
problems have the same kind of action or not.
Assist students in putting the two problems
together (if same action) or in separate groups (if
not the same action).
Ask students to draw a representation of two of the
problems, using one of each type. Do not ask
Students do not distinguish the difference between the word problems; they sort all problems them to solve, but just to represent the information
into the same group.
in the problem. Likely, students will notice the
difference as they draw. If not, point out that you noticed how students approached the problems
differently when drawing.
Special Notes:
• Remember not to instruct children to look for “key words” in word problems. Key words often do not apply to all situations and it will further confuse children when the operation they choose
based on a key word does not yield correct results. Instead, encourage children to think about the situation and visualize what is happening in the situation. The goal is for children to
comprehend the context of the problem, not to simply hunt for words to dictate how to compute an answer.
Possible Solutions:
Solution to sort for Activity Sheet 1:
Take Apart Problems Put Together Problems
There were 15 fish in a fish tank. 8 of the fish were orange and the rest were white. How many fish were white? 8 black cows and 7 brown cows are in a field together. How many cows are in the field?
There are 15 cupcakes. The first grade students ate 7 of the cupcakes. How many cupcakes are left? I have a bag of candy. There are 8 chocolate candies and 7 lemon candies. How many
pieces of candy are in the bag?
My dad bought 15 balloons for the party. 7 of the balloons are yellow. The rest of the balloons are blue. How We found 7 ladybugs and 8 crickets in the backyard. How many bugs did we find?
many blue balloons did my dad buy?
Solutions to sort for Activity Sheet 2:
Change Unknown Problems Result Unknown Problems
Annya has 8 pennies. She found more pennies on the sidewalk. Now she has 11. How many pennies did Maria saw three yellow butterflies. She also saw eight orange butterflies. How many butterflies did
Annya find? Maria see?
Bill had 11 toy cars. He lost some, but he still has 3 toy cars. How many toy cars did Bill lose? Jose had 11 lollipops. He gave a lollipop to each of his 8 friends. How many lollipops does Jose
have now?
Subtraction Problems Addition Problems
Annya has 8 pennies. She found more pennies on the sidewalk. Now she has 11. How many pennies did Bill had 11 toy cars. He lost some, but he still has 3 toy cars. How many toy cars did Bill lose?
Annya find?
Maria saw three yellow butterflies. She also saw eight orange butterflies. How many butterflies did Jose had 11 lollipops. He gave a lollipop to each of his 8 friends. How many lollipops does Jose
Maria see? have now?
Parts of this lesson adapted from Hunovice, L., OConnell, S., & SanGiovanni, J. (2016). MATH IN PRACTICE Teaching first-grade math. Portsmouth, NH: Heinemann.
Activity Sheet 1
I have a bag of candy. There are 8 chocolate candies and 7 lemon candies. How many My dad bought 15 balloons for the party. 7 of the balloons are yellow. The rest of the balloons are blue. How
pieces of candy are in the bag? many blue balloons did my dad buy?
There are 15 cupcakes. The first grade students ate 7 of the cupcakes. How many There were 15 fish in a fish tank. 8 of the fish were orange and the rest were white. How many fish were white?
cupcakes are left?
We found 7 ladybugs and 8 crickets in the backyard. How many bugs did we find? 8 black cows and 7 brown cows are in a field together. How many cows are in the field?
Activity Sheet 2
Annya has 8 pennies. She found more pennies on the sidewalk. Now she has 11. How many pennies did Jose had 11 lollipops. He gave a lollipop to each of his 8 friends. How many lollipops does Jose
Annya find? have now?
Maria saw three yellow butterflies. She also saw eight orange butterflies. How many butterflies did Bill had 11 toy cars. He lost some, but he still has 8 toy cars. How many toy cars did Bill lose?
Maria see?
Annya has 8 pennies. She found more pennies on the sidewalk. Now she has 11. How many pennies did Jose had 11 lollipops. He gave a lollipop to each of his 8 friends. How many lollipops does Jose
Annya find? have now?
Maria saw three yellow butterflies. She also saw eight orange butterflies. How many butterflies did Bill had 11 toy cars. He lost some, but he still has 8 toy cars. How many toy cars did Bill lose?
Maria see?
Exit Ticket
The answer to a word problem is 5 monsters. What could the problem be? Write a word problem with a question that can be answered by the solution “5 monsters.”
Exit Ticket
The answer to a word problem is 5 monsters. What could the problem be? Write a word problem with a question that can be answered by the solution “5 monsters.” | {"url":"https://goopennc.oercommons.org/courseware/lesson/3355/overview","timestamp":"2024-11-14T16:31:47Z","content_type":"text/html","content_length":"89864","record_id":"<urn:uuid:bd735c03-f007-42a0-abfa-30f25c3a2add>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00170.warc.gz"} |
GSoC 24: All in one system identification toolkit for ardupilot
Greetings everyone. My name is Astik Srivastava, and I’m an undergraduate engineering student from Delhi Technological University, India. I’ve been selected as a contributor for ArduPilot, for this
year’s Google Summer of Code, with @iampete and @bnsgeyer as my mentors. I’m working on developing a system identification library for ardupilot vehicles, which allows identification of linear
parametric models for the dynamics of these vehicles. These models can be used for certification purposes, controller gain optimization, performance evaluation, etc.
The project can be majorly divided into two parts:
1. Library for model estimation: Currently, ArduCopter provides a system identification mode, which can insert frequency sweep chirps as inputs, in various location along the control loop. My job is
to provide code that can analyze this input and resulting outputs (like body rates, accelerations) to estimate a model.
• Progress made so far: We looked at multiple open source packages written in python, that can be used for system identification. I’ll be focusing the development on top of this repository:
pyAircraftIden, which provides frequency domain estimation and allows us to define custom model structures, which can be taken from literature.
2. Lua script for frequency sweep generation for ArduPlane: Unlike Copter, Plane does not have a dedicated mode for frequency chirp input. Hence, I’ve created a lua script that can be used to excite
control surfaces in both open loop and closed loop settings (manual mode for open loop, stabilize/fbwa mode for closed loop).
• Progress made so far: The pr for frequency sweep lua script can be found here: Even though basic script is working, modifications need to be done to ensure smooth transition, pilot
controllability and safety of maneuvers.
3. Web tool development for system identification: To ensure that users with varying technical backgrounds can use the library, I’m working with my mentors to build a web tool similar to
UAVlogviewer, based on the work done by us on the system identification library
• Progress made so far: We are trying to get the code working with pyodide, to allow python code to run in browser, completely on the client side.
I’ll be regularly updating my progress here. I request the community to provide suggestions on how we can make this project better.
Thank You!
9 Likes
Hi @Astik_Srivastava,
Welcome and thanks so much for the explanation of your project!
I hope you don’t mind but I’ve added an image to the top to make it more attractive when viewed from ardupilot.org. This is the output from feeding your blog into dall-e. Feel free to replace it with
something else!
1 Like
To be honest I find this image quite bad due to various AI artifacts such as “bent” props, inconsistent landing gear and GNDN wires.
If I saw article with such image on the website I would likely skip it assuming the author didn’t put more effort into the content of the article than into the image.
Hi @LupusTheCanine,
well, you know… it’s just an image Dall-e came up with in a few seconds. Feel free to provide a better one!
An ArduPilot dev posted a nice system identification paper that he was working on with some colleages.
I can not find the post nor any links to it. Is there a place were such “ArduPilot relevant” papers are collected and kept?
Academic Works Involving ArduPilot — Dev documentation seams to not have been updated in a long time
Here’s a link to @Astik_Srivastava final update for his GSOC project.
This is a huge step forward in providing a widely available system identification toolkit. Many thanks to Astik for his impressive work on this tool. Thanks to Pete Hall(@iampete) for co-mentoring
this project.
1 Like
Greetings devs. In this post I attempt to make a tutorial/guide for how to use the system identification toolkit for transfer function and state space model estimation.
Since the toolkit is part of ardupilot webtools, it can be locally run on the system. The sysID toolkit is in the dev section of the webtool
Upon loading, the output screen looks as follows:
The model type that needs to be estimated can be selected from here and then after inputting the identification parameters, log file can be submitted.
State Space Identification
Upon clicking the statespace model a window as shown below appears:
The functions of parameters visible above are as follows:
• Outputs: Number of output data fields that need to be taken from the log
• Matrix A order: Order of matrix A. Order of state space matrix should be equal to number of states in your state space, with shape of A matrix being n X n. The matrix B will always have a shape n
X 1, where n is the order of A.
• Number of params: Number of unknown parameters in the state space matrices that need to be identified.
• Number of constraints: Number of equality constraints required by the user. These constraints enforce the condition of two unknown params either always being equal to each other or negative of
each other.
• Start and End time: These are for specifying the start and end time of sysID experiment in the log. Only data between these times will be considered while estimation of the model.
• Start and End frequency: Since current webtool relies on frequency response to estimate the system, the start and end frequency specify the region of frequency spectrum from which data will be
considered. This should be chosen carefully to capture both short term and long term dynamics of the system.
• LPF cutoff frequency: Before identification, the input and output data is passed through a low-pass-filter to remove noise. The desired cutoff frequency is inputted here.
Once these parameters are submitted, further parameters can be set:
This image shows an example value of parameters filled for identifying the roll state-space model of an x8 configuration UAV.
Upon submission, following parameters pop out:
The functions of parameters visible above are as follows:
• Input: Msg name and field of input data
• Output(n): Msg name and field for nth output
• Multiplier: This is an optional field which can be used to multiply the data with a constant. This can be used for unit changes (for eg: SI to imperial)
• Gravity Compensation: This is a special field for IMU acceleration data along roll and pitch axis. Since IMUs do not measure gravity, the effect of gravity component g*cos(angle) does not get
reflected in imu data. To counter this, gravity compensation adds a gravity dependent term (g x theta x multiplier for roll axis or -g x theta x multiplier for pitch axis) to the acceleration
• Param(n): Symbolic name to be assigned to nth unknown param
• Bounds(n): The min and max value that the nth unknown parameter can take
• Matrix A: System matrix with shape n X n, with n being the number of states. The known elements of matrices are given there respective values and unknown elements are given there respective
symbolic names as inputs.
• Matrix B: Control matrix with shape n X 1. Same principal as A for inputs
• H0 and H1 matrices: These denote the relationship between frequency response variables and the states of the system. In above example, the statespace has four states [velocity_y (Uy), roll rate
(p), roll angle(phi), motor lag(w)]. The H0 matrix denotes the states for which data is available. H1 matrix denotes the states whose first derivative data is available
After these parameters are set, the log can be uploaded and submitted.
Since we are currently using a python code running on browser using pyodide, a lot of capabilities of pyAircraftIden are limited, leading to very long solution times for large systems. This is
something that I hope to change in future.
Once the processing is complete, the plots become available for comparing the original data and identified system:
The plot shows phase and amplitude fit for both the output values. The final model and numerical fit are displayed on the output screen.
Transfer function estimation
Transfer function estimation has a relatively simple setup and a less computationally expensive process. The output screen for tfestimation looks like this:
The function of each input block is as follows:
• Input Msg and field: Msg type and field for input data
• Output Msg and field: Msg type and field for output data
• Start and End time: Denote the start and end time of the sysID experiment
• Start and End frequency: Denote the region of frequency spectrum in which response is calculated
• LPF cutoff frequency: Cutoff frequency of low pass filter
• Numerator and Denominator polynomial: Since polynomial structure can be dynamic, users need to input the numerator and denominator polynomials, with s being the ‘S’ domain variable and other
strings as coefficients. The polynomials need to be written in sympy polynomial syntax.
• Symbolic params: Names of symbolic coefficients in the polynomials described above.
Upon setting these parameters, the transfer function estimation can be started by uploading the log file. TF estimation is still slower than MATLAB, but considerably faster than state space
estimation and promises good results for good data. The output and plot window is same as for statespace estimation.
There is still a lot of work that can be done in this toolkit. I request the community to try it and share feedbacks, so we can work out the kinks and also focus on better solvers. Better solvers
will enable the users to deal with nonlinear systems and MIMO systems, which is crucial for control engineers. I would like to thank @bnsgeyer and @iampete for there guidance and support during this
project. I hope my work proves to be beneficial for ardupilot community. Looking forward to responses and feedbacks.
1 Like
Hello,it’s really a nice tool and I am trying to use it.But now I meet some problem.I run it as your guide , but when all parameters was setted I enter Submit.Output didn’t have any message shown.
Hello @LinXiangyu. You need to wait for 4-5 minutes for the transfer function solver to solve the identification problem. This is slow as we are running python code on browser, without multithreading
support. Please let me know if you encounter any other problems
I’m pleased to report @Astik_Srivastava’s system ID tool is now live, it can be found here:
May thanks to @Astik_Srivastava for the work put in over the GSoC project, it was great having you as a student. Thanks also to @bnsgeyer for bringing his guidance and input on all things system ID.
Hopefully the tool will continue to evolve and improve in the future. If anyone wants to get stuck in the development setup documented here: GitHub - ArduPilot/WebTools
2 Likes
Thanks for you help, I have get my roll Axis by your tool. But Now I get a new problem ,when I use the model verification mode of the official ardupilot simulink model, the roll axis model I identify
is completely inaccurate. Is it because I did not operate it properly during use? Or is the log I measured unreasonable?
This is my identification result and setting parameters and my test log. Thank you again.
Hi @LinXiangyu, thanks for trying out this new Web tool. I have a few questions to verify your set up to ensure that it is correct. What was the start (SID_F_START_HZ) and stop (SID_F_STOP_HZ)
frequency of your system ID mode. Just remember that the frequencies you use in the system ID mode setup are in Hz and the frequencies requested in the webtool are in rad/s.
Also it would be better to use SIDD.Gx for the measured roll rate as it is not affected by any filtering by the INS or the PIDs.
Last, be sure that the start time and end time are within the Sydtem ID mode run time. If not you could get an error. you can open the log file in mission planner and look at the SIDS message which
will have the start time in microseconds. Just convert that to seconds and add the length of the run to it to get the stop time.
In your case, the cost function given in the text that you have outlined in red is shown by “Solution: 2372”. Keep in mind when using this tool is that if your cost function is not less than 100 then
the solution is poor. Generally a cost function less than 100 indicates a good solution and less than 50 indicates an excellent solution.
2 Likes | {"url":"https://discuss.ardupilot.org/t/gsoc-24-all-in-one-system-identification-toolkit-for-ardupilot/120125","timestamp":"2024-11-04T11:19:13Z","content_type":"text/html","content_length":"60307","record_id":"<urn:uuid:fa57364d-786c-4dae-ac99-e40ee05aabd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00423.warc.gz"} |
Common Computer Science Algorithms | Michael Olson
Common Computer Science Algorithms
Greedy Algorithms
A "greedy" algorithm is a type of algorithm where you build the solution piece by piece. It sets a rule for which elements to add to the solution at each step in the algorithm. This could be
minimizing a cost or maximizing a count for instance. The algorithm continues step-by-step evaluating elements based on the rule until it reaches the end of it's input data.
Sometimes, this doesn't always find the correct solution. Only certain problems can be solved by greedy algorithms.
There are two propeties needed in order for a greedy algorithm to be used to solve a problem:
1. Greedy choice: A global (or overall) optimal solution can be found by selecting the optimal solution at each step in the algorithm.
2. Optimal substructure: An optimal substructure exists if an optimal soltion to the entire problem contains the optimal soltions to all sub-problems.
Greedy algorithms only work on problems where at every step, there is a choice that is optimal for the problem up to that step, and after the final step, the algorithm produces the optimal solution
of the complete problem.
Read that a few times if it doesn't make sense.
Let's create a change machine that takes in a given amount of money and returns the amount separated into as few coins as possible. This should be returned as a list of coins which sum to the amount.
public class ChangeMachine {
public static ArrayList<Integer> getMinCoins(int amount) {
Right away you see the rule for this problem: the minimum number of coins must be used when making change for the input amount.
Now let's say the coins available to us are quarters, dimes, nickels, and pennies. So we'd have these options:
private static final int[] coins = { 25, 10, 5, 1 };
Say our amount is 37 cents. How would we go about generating the change? Pretty simply, we could start with the largest coin and subtract it from the amount. If at any step, the amount is less than a
coin value, then we move to a smaller coin. We repeat, subtracting from the amount until the amount is 0.
So for 37 cents, we'd subtract the largest coin, 25, and add the 25 to our results list. Then amount would be 12. 12 is less than 25 so we'd need to move to our next smaller coin, the dime.
Subtracting 10 from 12 and we get 2. 2 is less than 10 and also less than 5, so we move to the penny. 2 pennies can be used to make 2 cents. Thus we're left with [25, 10, 1, 1] for our change.
Is this a greedy algorithm? It is. Look at each step in the algorithm. At each step, we take the largest coin possible that fits into our amount. We can do this at every step. And once we get to the
last step, the resulting array of change is actually the optimal solution to the entire problem. So both greedy choice and optimal substrucutre are present.
Let's see the solution:
public class ChangeMachine {
public static int [] coins = {25, 10, 5, 1};
public static void main(String[] args) {
int amount = 37;
System.out.println(amount + " cents can be broken into the minimum number of coins as: " + getMinCoins(amount));
public static ArrayList<Integer> getMinCoins(int amount) {
ArrayList<Integer> change = new ArrayList<Integer>();
int i = 0;
while (i < coins.length) {
int coin = coins[i];
if (coin <= amount) {
amount = amount - coin;
} else {
return change;
Output would be
37 cents can be broken into the minimum number of coins as: [25, 10, 1, 1]
There you have a greedy algorithm. The above solution uses esentially a nested while loop, so our runtime complexity would be O(n^2) where n is the number of available coins. We can actually solve
this problem more efficiently, using Dynamic Programming--the next algorithm we'll look at.
Dynamic Programming
Divide and Conquer
Graph Algorithms
At this point you may be thinking: "Wait a second, what about sorting and searching algorithms". Since sorting and searching are so important in software development, I felt that these algorithms
deserved their own posts. Check them out for a full review of searching and sorting in Java: | {"url":"https://molson.dev/posts/common-computer-science-algorithms","timestamp":"2024-11-11T03:53:46Z","content_type":"text/html","content_length":"27430","record_id":"<urn:uuid:f4cbb282-17f4-43ad-97db-3bff7387323c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00377.warc.gz"} |
Zachary Miner
My Contact Info:
Office: RLM 11.106
Office hours: TuTh 3:00-6:00pm
Email: zminer (at) math.utexas.edu
Teaching info (Fall 2011)
I am currently teaching Math 325K - Discrete Mathematics for beginning mathematicians, engineers, and computer scientists. The course covers logic, proof methods, set theory, and elementary number
theory - with a heavy focus on proofs. This course will use an inquiry-based learning approach. The course website is Blackboard.
I am also teaching two large sections of Math 408C - Differential and Integral Calculus, the standard first-year calculus course directed at students in the natural and social sciences and at
engineering students. The course website is Blackboard.
Research Interests
My research interests are primarily concerned with heights of algebraic numbers. The study of heights belongs to a very interesting intersection of mathematical theories. My broader research
interests are mainly those theories which have contributed to my understanding of heights. Those include: Algebraic and Analytic Number Theory, Arithmetic of Dynamical Systems, Potential Theory, and
Functional Analysis.
About me
I received my bachelor's degree in Mathematics from the University of Texas at Austin in 2004, and stayed on at the university as a graduate student in Number Theory studying under the supervision of
Jeffrey Vaaler. I received my Ph.D. in Mathematics in May 2011. | {"url":"https://web.ma.utexas.edu/users/zminer/","timestamp":"2024-11-10T14:12:07Z","content_type":"text/html","content_length":"3707","record_id":"<urn:uuid:dc3142d8-e0ee-41ff-bd9a-cae9517816f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00332.warc.gz"} |
All Students Can Learn...
5/9/2017 Archives
0 Comments
Above are six learning curves representing six hypothetical persons. You can see a variety of initial performance levels, going from 1 for Ty, Jill, and Bob at time 0, all the way up to
500 for May. You can also see a variety of ending performance levels, going from 33 at time 10 for Bob, all the way up to 1000 for May. All of the learning curves on the diagram above
improve over time, as represented by increased performance over time and all have different rates of improvement over time.The question I want to answer is, “Can all students can learn?”
It depends. In the above learning curves, all students show improvement. However, that means almost nothing given the different rates of improvement and absolute performance levels over
different time periods. In fact, the idea that growth rates or proficiency levels (performance levels) tell us anything at all is patently absurd once the above curves sink in.Let’s
start by measuring some growth rates. Bob (light blue curve) went from a performance level of 1 to a performance level of 33.Bob’s percentage change = (33-1/1) x 100 = 3200%We could also
state that Bob is 33 times “better” at time 10 than he is at time 0. Either way, that’s a lot of improvement if we compare it to May (purple curve).May’s percentage change = (1000-500/
500) x 100 = 100%We could also state that May is 2 times “better” than when she started.Clearly Bob has much more “growth” than May if we use simple percentage change formulas to find a
growth rate.However, if we switch from growth rates to proficiency levels as measured by the absolute performance levels, we find that May is 30.3 times (1000/33) “better” than Bob.If we
had a school initiative that made sure certain minimums were met, should it measure those minimums using growth rates or proficiency levels? If we set the minimums using growth rate
targets at 500%, May (100%) is falling behind, but Bob (3200%) is doing great! He meets the minimum after time 1, whereas May never reaches the target. If we set minimums using
proficiency levels and use a performance level of 400, then May (500) meets the target at time 0, while Bob (33) is still 377 points shy at time 10.If we look at the other people in the
diagram above while using the same targets, either 500% growth or a 400 proficiency level, we run into similar problems. Tia never reaches the growth target, but does cross the 400
target by time 1. Ty crushes the growth target and also meets the proficiency target by time 2. Jon never reaches the growth target, but reaches the proficiency target after time 7. Jill
meets the growth target easily, but takes until time 7 to reach the 400 proficiency target and we can easily imagine her curve never crossing the proficiency threshold by simply shifting
it down a bit.The above presents serious problems for standards-based initiatives like the hypothetical one above. A number of students will be measured as falling behind depending on
how we select our targets. Furthermore, even if we agree to the target, there is the pesky question of time allowed to meet it. If the target is set at a 400 proficiency level, but it is
expected to be met by time 1, only Tia and May will reach it. If we allow until time 6, we can add Ty. If we extend it to time 7, we include Jill and Jon.Many schools, districts, and
nations use metrics similar to the above for measuring student learning. The United States has No Child Left Behind, but is by no means the only one to measure student learning with some
type of standards. It began under a proficiency model of measuring absolute performance, but has since begun shifting to growth models in several states. The OECD has PISA. The
International Baccalaureate uses proficiency levels with its standards-based rubrics. It should be clear that both growth and proficiency metrics have inherent problems.The above is
aimed at showing how absurd many student measurements of learning can be. Success and failure are totally dependent on the metrics we decide to use. That only becomes worse the more
variables we measure. For example, let’s assume the above learning curves are for math. What happens when we add language and find students with language learning curves that don’t match
their math learning curves. Perhaps, Bob and May are completely reversed and we find May with low absolute performance levels and Bob with high absolute performance levels. Do they both
count as failing if we require a proficiency level of 400 for both math and language?I recently finished Why School?, by Will Richardson, who stated the following in his book’s final
I can’t wait 10 to 20 years. By that time, our kids will have long since graduated, and the story of what schools become will have already been written. I hope they are places where
adults and children come together to learn about the world, places rich with technology that lets our kids dream big and create things to fuel those dreams. I hope schools will be
places where learning is fun, where it’s not so much about competing against one another as about working together to solve the really big problems we’ll face together in the years
ahead. (Kindle Locations 559-562)
That seems right to me.
Let’s return to the initial question, “Can all students learn?” Of course, but what that means is up for debate. If it means hitting a particular standardized level of proficiency or
growth, then no. Some students will never reach particular growth or performance levels selected for them. However, if, as Richardson writes elsewhere in his book, it means that
students, "have the skills and dispositions they need to solve whatever hard problems come their way, and [that] they’ll know how to go about creating something of value and sharing it
with the world,” then YES! (Kindle Locations 543-544).
0 Comments
Leave a Reply. | {"url":"http://www.kylefitzgibbons.com/blog/all-students-can-learn","timestamp":"2024-11-11T20:39:18Z","content_type":"text/html","content_length":"42756","record_id":"<urn:uuid:dae38089-dda7-446e-b5f9-c4018583fffb>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00549.warc.gz"} |
Is a quadrilateral A polygon yes or no?
No, not all polygonsarequadrilaterals. However, all quadrilateralsarepolygons. By definition, a quadrilateral isatwo-dimensional shape with
Likewise, people ask, is a polygon a quadrilateral?
Polygons - Quadrilaterals - First Glance.Aquadrilateral is a four-sided polygon withfourangles. There are many kinds of quadrilaterals. Thefivemost common types are the parallelogram, the rectangle,
thesquare,the trapezoid, and the rhombus.
Subsequently, question is, what shape is both a quadrilateral and a polygon?
In Euclidean plane geometry, a quadrilateral isapolygon with four edges (or sides) and four verticesorcorners. Sometimes, the term quadrangle is used, by analogywithtriangle, and sometimes tetragon
for consistency withpentagon(5-sided), hexagon (6-sided) and so on.
Subsequently, one may also ask, is a rectangle a polygon yes or no?
Polygons are named according to the numberofsides and angles they have. The most familiar polygonsarethe triangle, the rectangle, and the square. Aregularpolygon is one that has equal sides. Polygons
alsohave diagonals, which are segments that join two vertices andarenot sides.
Is a Pentagon a quadrilateral?
A quadrilateral is a polygon. In fact it isa4-sided polygon, just like a triangle is a 3-sided polygon,apentagon is a 5-sided polygon, and so on.
Sandeep Vispovatykh
Is a circle a polygon?
Because a polygon is composed of a finite setofstraight line segments, a circle does not have a finitesetof these straight lines. Per definition a circle is notapolygon, but you could “draw”
infinitelymanypolygons that would look exactly likeacircle.
Benedito Scheigenpflug
What is the name of a 5 sided polygon?
More than Four Sides
A five-sided shape is called a pentagon.Asix-sided shape is a hexagon, a seven-sided shapeaheptagon, while an octagon haseightsides…
Detlef Avdiev
How many types of rectangles are there?
There are two special types ofrectanglesthat have even stricter requirements thanjustrectangles.
Kalsoom Fineiss
How many types of quadrilateral are there?
There are five types of quadrilaterals.
• Parallelogram.
• Rectangle.
• Square.
• Rhombus.
• Trapezium.
Fabrizzio Carreon
What shape is a trapezoid?
A trapezoid is a four-sided shape withatleast one set of parallel sides. The area of a trapezoidisthe average of the bases times the height. Finally, a polygon isaclosed, two-dimensional shape with
many sides.
Enea Sica
Which is a regular polygon?
In Euclidean geometry, a regular polygon isapolygon that is equiangular (all angles are equalinmeasure) and equilateral (all sides have the samelength).Regular polygons may be either convex orstar.
Yakubu Vaca
Is a square a rhombus?
A rhombus is a quadrilateral with all sidesequalin length. A square is a quadrilateral with all sidesequalin length and all interior angles right angles. Thusarhombus is not a square unless the
angles areallright angles. A square however is a rhombus sinceallfour of its sides are of the same length.
Donat Stuhlmeyer
Is a trapezium a regular polygon?
Trapezium. A trapezium has one pairofopposite sides parallel. A regular trapeziumhasnon-parallel sides equal and its base angles are equal, as showninthe diagram.
Iokin Eickwinkel
How many sides are in a polygon?
A polygon has as many angles as it has sides.Forexample, a triangle has 3 sides and 3 angles. A pentagon has5sides and 5 angles. An octadecagon has 18 sides and18angles!
Dreama Eras
Can trapezoids be parallelograms?
This is possible for acute trapezoids orrighttrapezoids (rectangles). A parallelogram isatrapezoid with two pairs of parallel sides.Aparallelogram has central 2-fold rotational symmetry(orpoint
reflection symmetry). It is possible forobtusetrapezoids or righttrapezoids(rectangles).
Eutimio Fellmen
Who discovered quadrilateral?
Quadrilaterals were invented by theAncientGreeks. It is said that Pythagoras was the first to drawone. Inthose days quadrilaterals had three sides andtheirproperties were only dimly understood.
Jessi Legabide
Do all 4 sided shapes have 360 degrees?
A quadrilateral is a shape with 4sides.For any quadrilateral, we can draw a diagonal line todivideit into two triangles. Each triangle has an angle sumof 180degrees. Therefore the total angle sum
ofthequadrilateral is 360 degrees.
Antigua Keight
Is a parallelogram a polygon?
One special kind of polygons is calledaparallelogram. It is a quadrilateral where both pairsofopposite sides are parallel. There are six important propertiesofparallelograms to know: Opposite sides
are congruent (AB=DC).
Vladislav Jaglesk
What are the different types of parallelograms?
Types of Parallelograms
• Rhombus (or diamond, rhomb, or lozenge) -- A parallelogramwithfour congruent sides.
• Rectangle -- A parallelogram with four congruentinteriorangles.
• Square -- A parallelogram with four congruent sides andfourcongruent interior angles.
Inasse Knopel
What shape is a parallelogram?
A Parallelogram is a flat shapewithopposite sides parallel and equal in length. | {"url":"https://everythingwhat.com/is-a-quadrilateral-a-polygon-yes-or-no","timestamp":"2024-11-06T01:41:54Z","content_type":"text/html","content_length":"32426","record_id":"<urn:uuid:c4e26236-cfa4-476f-a93c-a0c9bdbcd7fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00648.warc.gz"} |
What's in a proton?
Hooray, it’s time for science! For my long-overdue first science post of 2014, I’m starting a three-part series explaining the research paper my group recently published in Physical Review Letters.
Our research concerns the structure of protons and atomic nuclei, so this post is going to be all about the framework physicists use to describe that structure. It’s partially based on an answer of
mine at Physics Stack Exchange.
What’s in a proton?
Fundamentally, a proton is really made of quantum fields. Remember that. Any time you hear any other description of the composition of a proton, it’s just some approximation of the behavior of
quantum fields in terms of something people are likely to be more familiar with. We need to do this because quantum fields behave in very nonintuitive ways, so if you’re not working with the full
mathematical machinery of QCD (which is hard), you have to make some kind of simplified model to use as an analogy.
If you’re not familiar with the term, fields in physics are things which can be represented by a value associated with every point in space and time. In the simplest kind of field, a scalar field,
the value is just a number. Think of it like this:
representation of a scalar field
More complicated kinds of fields exist as well, where the value is something else. You could, in principle, have a fruit-valued field, that associates a fruit with every point in spacetime. In
physics, you’d be more likely to encounter a vector-, spinor-, or tensor-valued field, but the details aren’t important. Just keep in mind that the value associated with a field at a certain point
can be “strong,” meaning that the value differs from the “background” value by a lot, or “weak,” meaning that the value is close to the “background” value. When you have multiple fields, they can
interact with each other, so that the different kinds of fields tend to be strong in the same place.
The tricky thing about quantum fields specifically (as opposed to non-quantum, or classical, fields) is that we can’t directly measure them, the way you would directly measure something like air
temperature. You can’t stick a field-o-meter at some point inside a proton and see what the values of the fields are there. The only way to get any information about a quantum field is to expose it
to some sort of external “influence” and see how it reacts — this is what physicists call “observing” the particle. For a proton, this means hitting it with another high-energy particle, called a
probe, in a particle accelerator and seeing what comes out. Each collision acts something like an X-ray, exposing a cross-sectional view of the innards of the proton.
Because these are quantum fields, though, the outcome you get from each collision is actually random. Sometimes nothing happens. Sometimes you get a low-energy electron coming out, sometimes you get
a high-energy pion, sometimes you get several things together, and so on. In order to make a coherent picture of the structure of a proton, you have to subject a large number of them to these
collisions, find some way of organizing collisions according to how the proton behaves in each one, and accumulate a distribution of the results.
Classification of collisions
Imagine a slow collision between two protons, each of which has relatively little energy. They just deflect each other due to mutual electrical repulsion. (This is called elastic scattering.)
animation of elastic proton scattering
If we give the protons more energy, though, we can force them to actually hit each other, and then the individual particles within them, called partons, start interacting.
animation of low-energy inelastic scattering
At higher energies, a proton-proton collision entails one of the partons in one proton interacting with one of the partons in the other proton. We characterize the collision by two variables — well,
really three — which can be calculated from measurements made on the stuff that comes out:
• \(x_p\) is the fraction of the probe proton’s forward momentum that is carried by the probe parton
• \(x_t\) is the same, but for the target proton
• \(Q^2\) is roughly the square of the amount of transverse (sideways) momentum transferred between the two partons.
With only a small amount of total energy available, \(x_p\) and \(x_t\) can’t be particularly small. If they were, the interacting partons would have a small fraction of a small amount of energy, and
the interaction products just wouldn’t be able to go anywhere after they hit. Also, \(Q^2\) tends to be small, because there’s not enough energy to give the interacting particles much of a transverse
“kick.” You can actually write a mathematical relationship for this:
$$(\text{total energy})^2 = s = \frac{Q^2}{x_p x_t}$$
Collisions that occur in modern particle accelerators involve much more energy. There’s enough to allow partons with very small values of \(x_p\) (in the probe) or \(x_t\) (in the target) to
participate in the collision and easily make it out to be detected. Or alternatively, there’s enough energy to allow the interacting partons to produce something with a large amount of transverse
momentum. Accordingly, in these high-energy collisions we get a random distribution of all the combinations of \(x_p\), \(x_t\), and \(Q^2\) that satisfy the relationship above.
Proton structure
Over many years of operating particle accelerators, physicists have found that the behavior of the target proton depends only on \(x_t\) and \(Q^2\). In other words, targets in different collisions
with the same values of \(x_t\) and \(Q^2\) behave pretty much the same way. While there are some subtle details, the results of these decades of experiments can be summarized like this: at smaller
values of \(x_t\), the proton behaves like it has more constituent partons, and at larger values of \(Q^2\), it behaves like it has smaller constituents.
This diagram shows how a proton might appear in different kinds of collisions. The contents of each circle represents, roughly, a “snapshot” of how the proton might behave in a collision at the
corresponding values of \(x\) and \(Q^2\).
Physicists describe this apparently-changing composition using parton distribution functions, denoted \(f_i(x, Q^2)\), where \(i\) is the type of parton: up quark, antidown quark, gluon, etc.
Mathematically inclined readers can roughly interpret the value of a parton distribution for a particular type of parton as the probability per unit \(x\) and per unit \(Q^2\) that the probe
interacts with that type of parton with that amount of momentum.
This diagram shows how the parton distributions relate to the “snapshots” in my last picture:
The general field I work in is dedicated to determining these parton distribution functions as accurately as possible, over as wide of a range of \(x\) and \(Q^2\) as possible.
As particle accelerators get more and more powerful, we get to use them to explore more and more of this diagram. In particular, proton-ion collisions (which work more or less the same way) at the
LHC cover a pretty large region of the diagram, as shown in this figure from a paper by Carlos Salgado:
diagram showing LHC reach in x-Q^2 space
I rotated it 90 degrees so the orientation would match that of my earlier diagrams. The small-\(x\), low-\(Q^2\) region at the upper left is particularly interesting, because we expect the parton
distributions to start behaving considerably differently in those kinds of collisions. New effects, called saturation effects, come into play that don’t occur anywhere else on the diagram. In my next
post, I’ll explain what saturation is and why we expect it to happen. Stay tuned! | {"url":"https://www.ellipsix.net/blog/2014/02/what-s-in-a-proton.html","timestamp":"2024-11-04T02:25:56Z","content_type":"text/html","content_length":"22043","record_id":"<urn:uuid:7ed90ccf-cf81-4f6d-ab43-3a75b3962c16>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00573.warc.gz"} |
Solving the Debate Over Photon Counting With a Scintillation Device
• Thread starter Frigorifico
• Start date
In summary: So the fact that you see a Gaussian means either you are not at the limit of large numbers or the light sources is behaving classically.In summary, the conversation revolved around the
disagreement between a professor and a student regarding the use of a photodiode model to measure light intensity from scintillators. The student wanted to use the mean voltage and the properties of
the photodiode to calculate the number of photons, while the professor suggested using the quantum efficiency and number of counts to determine the total number of photons. The student raised
concerns about the applicability of the professor's method and the presence of dark current. The conversation also touched on the Poissonian distribution of photons and the limitations of using a LED
as a single photon
TL;DR Summary
I have to count the number of photons detected by a photodiode, I want to do it based on voltage/current measurements, but my professor wants me to do some statistical analysis which I think
don't apply in this case
Hello, I'm working on a scintillation device to detect protons, I have a disagreement with one professor and I would like your opinion.
There is
one photodiode model
we want to use to measure the light intensity from the scintillators, and we want to relate the signal of that photodiode with the number of photons it is absorbing.
To do this we took a CAEN DT5751, connected a photodiode to it, and put an LED in front of the photodiode. This DT5751 samples the signal, measures its voltage, and counts how many times it has seen
such a voltage, producing a histogram with a gaussian distribution, so far so good.
What I want to do is to get the mean voltage, divide by the resistance the photodiode sees (50 ohms, I measured), get the current produced by the photodiode, and use the photodiode's responsivity to
get the mean number of photons. Easy piecy.
BUT my professor says that there's a better way (aka "I want this in your report"). We know that the statistical distribution of the photons is a Poissonian Distribution when N approaches infinity,
which is a gaussian where sigma^2 = mean. And he is right, that distribution does fit with our results.
Then he says that we should be able to use the quantum efficiency of the photodiode and the number of counts to know the number of photons. Equation 5.57
Here's my problem with that reasoning: Our set up does not count photons, it counts current pulses, and each pulse is produced by the photodiode when it absorbs a bunch of photons, something like 10^
10 at least, those formulas is for when you have a device capable of counting individual photons, so those formulas don't apply.
I brought up this point but he just told me to read more, and I did, I read
but he just said he had better things to do than to teach me what I should already know.
Maybe he's right, maybe there is a way to use photon statistics to get the total number of photons with this set up, so I come to you, is there?, has ay of you done something similar?, because I am
The manual of the DT5751 says that "Input dynamic is 1 Vpp" and the values are stored with "16 MSB". I take that to mean that it has 2^16 slots to store all the values from 0 to 1 volts. For example
a value of "20,000" would be 20,000/2^16 = 0.35 volts, is that correct?
I'm not sure where your methods actually differ. Using the voltage and the properties of the photodiode is sensitive to the number of detected photons only. To get the total number of photons hitting
the photodiode you need to consider the quantum efficiency anyway.
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2023 Award
You agree the professor's method gives you the right answer, but you think your method is better, do I have that right? How do you deal with
dark current
Vanadium 50 said:
You agree the professor's method gives you the right answer, but you think your method is better, do I have that right? How do you deal with
dark current
I agree the distribution is poissonian, the problem is that I don't see how can I go from the ADC units to photons with his method
mfb said:
I'm not sure where your methods actually differ. Using the voltage and the properties of the photodiode is sensitive to the number of detected photons only. To get the total number of photons
hitting the photodiode you need to consider the quantum efficiency anyway.
they differ in the fact that he wants me to use that one equation to get the number of photons, but my understanding is that only works if we were counting individual photons, which we weren't
Frigorifico said:
To do this we took a CAEN DT5751, connected a photodiode to it, and put an LED in front of the photodiode. This DT5751 samples the signal, measures its voltage, and counts how many times it has
seen such a voltage, producing a histogram with a gaussian distribution, so far so good.
If the LED is producing continuous photons, then you are measuring the spectrum of noise on the DC photodiode current while it is continuously illuminated. The gaussian voltage distribution seen with
the LED, (without the scintillator), is noise from the hot LED, the photodiode efficiency, electronics, and from sampling and A to D conversion. The LED will have a different wavelength to the
scintillator. When compared with the uncalibrated LED efficiency and optics, the spectral response is not really relevant.
Frigorifico said:
they differ in the fact that he wants me to use that one equation to get the number of photons, but my understanding is that only works if we were counting individual photons, which we weren't
I don't see an equation that would stop being true for more photons. The relative fluctuation will decrease with more photons and other sources can become more important, whether that is relevant or
not depends on your setup.
Science Advisor
Gold Member
I don't believe this makes sense. The fundamental problem is that a LED does not produce single photons; the fact that a setup gives you a Poissonian distribution is frequency used to show that
something is at least approximately behaving like a single photon source (e.g. a very attenuated laser and/or a sub-threshold diode laser) and this is never the case for a LED.
Another problem is in the limit of large numbers the distribution becomes Gaussian; meaning it would be indistinguishable from random noise.
FAQ: Solving the Debate Over Photon Counting With a Scintillation Device
1. What is a scintillation device?
A scintillation device is a scientific instrument used to detect and measure ionizing radiation. It works by converting the energy from incoming radiation into visible light, which can then be
measured and analyzed.
2. How does a scintillation device help solve the debate over photon counting?
A scintillation device is able to accurately count individual photons of radiation, providing a more precise measurement than other traditional methods. This helps to resolve any discrepancies or
disagreements in the debate over photon counting.
3. What are the benefits of using a scintillation device for photon counting?
Using a scintillation device for photon counting allows for more accurate and reliable measurements, which is crucial in many scientific fields such as medical imaging and nuclear physics. It also
allows for a wider range of radiation to be detected and measured.
4. Are there any limitations to using a scintillation device for photon counting?
While a scintillation device is highly accurate, it does have some limitations. It may have difficulty distinguishing between different types of radiation, and it may have a limited range of
detection for certain types of radiation.
5. How is a scintillation device different from other methods of photon counting?
A scintillation device is unique in its ability to directly convert incoming radiation into visible light, allowing for more precise and accurate counting. Other methods, such as Geiger counters,
rely on indirect measurements and may be less accurate. | {"url":"https://www.physicsforums.com/threads/solving-the-debate-over-photon-counting-with-a-scintillation-device.976260/","timestamp":"2024-11-02T06:20:57Z","content_type":"text/html","content_length":"116105","record_id":"<urn:uuid:5f82f88d-5638-48cf-a629-729af175b8d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00085.warc.gz"} |
Are standard angle sizes L?
Are standard angle sizes L?
L Shaped Mild Steel Equal Angle, Size: 25x25x5 to 100x100x10 Mm.
What is the difference between I-beam and wide flange?
An I-beam has tapered flanges with a narrower flange than most wide flange beams, making it a lighter building material. A wide flange beam, with wider flanges and web than the I-beam, can handle
more weight, but this makes it heavier overall.
How wide is a W24 beam?
W Beam Dimensions
Section number Wt. Per foot (lbs) Dimensions
W21 101 12 1/4
W24 55 7
How do you read a wide flange?
Wide flange beams are designated by the letter W followed by the nminal depth in inches and the weight in pounds per foot. Thus W12 × 19 designates a wide flange beam with a depth of 12 inches and a
nominal weight of 19 pounds per foot.
What is the length of L angle?
Mild Steel L Angle, Size: 50×50 Mm, Thickness: 6 Mm
Thickness 6 mm
Shape L Shaped
Material Mild Steel
Size 50X50 mm
Single Piece Length 18 Feet, Also Available in 20,22 Feet
What is the angle size?
In geometry, an angle is measured in degrees from 0° to 180°. The number of degrees indicates the size of the angle.
What is stronger I-beam or H beam?
The cross section of the H beam is stronger than the cross section of the I beam, meaning it can bear a greater load. In comparison, the cross section of an I beam can bear direct load and tensile
but cannot resist twisting because the cross section is so narrow. This means that it can only bear force in one direction.
What is the strongest beam shape?
H-Beams. One of the strongest steel beams on the list, H-beams, is made up of horizontal elements, while the vertical beams act as the web. The flanges and web create a cross-section that mimics the
shape of the letter “H” and are popular in construction or civil engineering projects.
How wide is a W8x18 beam?
Wide Flange Beam Specifications Chart
SIZE LBS/FT FLANGE
W8x10 10 3.94
W8x13 13 4
W8x15 15 4.015
W8x18 18 5.25
How wide is a w18x35 beam?
Section Details
Height 17.7 in
Width 6 in
Area 10.3 in2
Yield Str. 36000 psi
How wide is a W12x26?
Wide Flange Beam Specifications Chart
SIZE LBS/FT FLANGE
W12x22 22 4.03
W12x26 26 6.49
W12x30 30 6.52
W12x35 35 6.56
How do you find the weight of an L angle?
Weight of the MS angle
= [0.000864 cum. × 7850 kg/cum.] (Here, the density of steel is taken as 7850kg/cum.
How thick is angle iron?
In equal angles as small as 3/8″ up to 2″ in. and in unequal angle they range from as small as 3/8″ x 3/4″ x 3/32″ to as large as 2-1/4″ x 5-1/4″ x 1/8″ thick.
What is angle formula?
What are the Formulas to Find the Angles? Angles Formulas at the center of a circle can be expressed as, Central angle, θ = (Arc length × 360º)/(2πr) degrees or Central angle, θ = Arc length/r
radians, where r is the radius of the circle.
How do you calculate the size of angle?
How To Calculate The Missing Angle In a Triangle – YouTube
What is an L beam?
A beam whose section has the form of an inverted L; usually placed so that its top flange forms part of the edge of a floor.
What shape of beam is strongest?
Which section is better for beam?
Considering same weight sections and same material beam means the cross section are is same for all and hence for axial load purpose all sections are equally effective.
How wide is a W10x26 beam?
Wide Flange Beam Specifications Chart
SIZE LBS/FT DEPTH
W10x22 22 10.17
W10x26 26 10.33
W10x30 30 10.47
W10x33 33 9.73
How wide is a w21x44 beam?
6.50 in
W21x44 Beam Size and Dimensions
Depth 20.70 in
Width 6.50 in
Area 13.0 in2
Weight 44 lb/ft
Web Thickness 0.35 in
How wide is a W18x40?
Wide Flange Beam Specifications Chart
SIZE LBS/FT DEPTH
W18x35 35 17.7
W18x40 40 17.9
W18x46 46 18.06
How wide is a W12 beam?
Wide Flange Beams – Misc Shapes
Size & Weight Per Foot B Flange Width Inches
W12 x 40 8.005
W12 x 45 8.045
W12 x 50 8.080
W12 x 53 9.995
How do you find the area of an angle with L?
The moment of inertia of an angle cross section can be found if the total area is divided into three, smaller ones, A, B, C, as shown in the figure below. The final area, may be considered as the
additive combination of A+B+C. However, a more straightforward calculation can be achieved by the combination (A+C)+(B+C)-C.
What is the weight of steel angle?
MS Equal Angles : 25 x 25 x 3mm to 130 x 130 x 12mm
Size in mm Average Weight
KG/MTR KG/FT
25 x 25 x 3 1.10 0.30
25 x 25 x 4.5 1.60 0.49
25 x 25 x 5 1.80 0.50
Which way is angle iron the strongest?
Structural steel angle is strongest above the vertical section of the structural shape. | {"url":"https://mattstillwell.net/are-standard-angle-sizes-l/","timestamp":"2024-11-03T07:22:49Z","content_type":"text/html","content_length":"42147","record_id":"<urn:uuid:efb53037-d094-495a-8e8c-a9f870697eb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00093.warc.gz"} |
Ruriko Yoshida (Kentucky)
Discrete time Markov chains are often used in statistical models to fit the observed data from a random physical process. Sometimes, in order to simplify the model, it is convenient to consider
time-homogeneous Markov chains, where the transition probabilities do not depend on the time T. While under the time-homogeneous Markov chain model it is assumed that the row sums of the transition
probabilities are equal to one, under the toric homogeneous Markov chain (THMC) model the parameters are free and the row sums of the transition probabilities are not restricted.
In order for a statistical model to reflect the observed data, a goodness-of-fit test is applied. For instance, for the time-homogeneous Markov chain model, it is necessary to test if the assumption
of time homogeneity fits the observed data. In 1998, Diaconis–Sturmfels developed a Markov Chain Monte Carlo method (MCMC) for goodness-of-fit test by using Markov bases. A Markov basis is a set of
moves between elements in the conditional sample space with the same sufficient statistics so that the transition graph for the MCMC is guaranteed to be connected for any observed value of the
sufficient statistics. In algebraic terms, a Markov basis is a generating set of a toric ideal defined as the kernel of a monomial map between two polynomial rings. In algebraic statistics, the
monomial map comes from the design matrix (configuration) associated with a statistical model.
In this talk we will consider a Markov basis and a Groebner basis for the toric ideal associate with the design matrix defined by the THMC model with S≥2 states without initial parameters for any
time T≥3. First we will show the upper bound of the Markov degree, the degree of a minimal Markov base, of the THMC model with S=3 for T≥3. In order to compute the upper bound, we use the model
polytope—the convex hull of the columns of the design matrix. Here we will show the model polytope has only 24 facets for T≥5 and a complete description of the facets for T≥3. Finally, we will show a
condition when the THMC with any S≥2 states for T≥3 has a square-free quadratic Groebner basis and Markov basis. One such example is the embedded discrete Markov chain (jump chain) of the Kimura
three-parameter model.
This is joint work with Davis Haws (IBM), Abraham Martin del Campo (IST Austria), and Akimichi Takemura (University of Tokyo). | {"url":"https://www2.math.binghamton.edu/p/seminars/comb/abstract.201412yos","timestamp":"2024-11-08T01:02:35Z","content_type":"text/html","content_length":"19177","record_id":"<urn:uuid:ee706ea9-9814-44a1-a9ec-287603d0a507>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00008.warc.gz"} |
Local Sidereal Time
Solar time is measured with respect to the position of the Sun. Our civil clocks, to a first approximation, are based on solar time and are designed such that when the Sun crosses the observer's
meridian the time is about noon, i.e. 12:00. (The world is divided into time zones so that at each location this is approximately true.). Of course, the apparent motion of the Sun across the sky is
actually caused by the rotation of the Earth. So our clocks measure the length of time required for the Earth to rotate once with respect to the Sun. From our perspective, the Sun revolves around the
Earth every 24 hours. This period is known as a solar day and is defined as the average length of time that the Sun takes to return to the same place in the sky, e.g. the observer's meridian. (The
length of a solar day actually varies during the year by a small amount due to the elliptical nature of the Earth's orbit and the Earth's 23.5° axial tilt relative to the ecliptic, see Equation of
Over the course of one day the Earth moves about 1° along its orbit around the Sun (360° divided by 365.25 days is about 1°). Therefore in one solar day the Earth has to spin 360° plus 1° to make the
Sun return to the same position in the sky. So the actual period of rotation of the Earth relative to the distant stars, i.e. the time for the Earth to spin 360°, is less than one solar day.
The time period for Earth to rotate 360°, relative to the distant stars, is called the sidereal day and is 23h 56m 04s of a solar day. This is about 4 minutes less than the solar day. The sidereal
day is divided into 24 sidereal hours. Each sidereal hour is divided into 60 sidereal minutes and each sidereal minute is divided into 60 sidereal seconds.
A sidereal clock is designed to complete 24 hours of sidereal time in 23h 56m 04s of civil (solar) time. As such a clock runs at the same rate as the Earth's rotation it can be used directly to
specify precisely how much the Earth has rotated. For example, if we see the sidereal clock change by 6h we know immediately that the Earth has rotated by 90°, i.e. 360° × 6/24.
We use the term local sidereal time to denoted the RA coordinate on the celestial sphere that currently lies on the observer's meridian. Recall that the RA coordinate measures positions around the
equator of the celestial sphere (i.e. equivalent to longitude on the Earth surface) and has units of time (hours, minutes, seconds). Now take a sidereal clock and adjust the time to read the RA
coordinate that is currently crossing your meridian. This clock will now always display the RA coordinate that is on your meridian as it will precisely run at the Earth's true rotation rate. So
local sidereal time (LST) = the RA on the observer's meridian
The local sidereal time is clearly very useful for astronomers. It can be calculated from the local civil time and observer's longitude using a relatively simple formula, see e.g. Page 16-20,
Duffett-Smith's ``Practical Astronomy'' or Page 40-1, Montebruck and Pfleger's ``Astronomy on the Personal Computer (4th edition)''.
USNO's online Sidereal Clock | {"url":"https://astro.dur.ac.uk/~ams/users/lst.html","timestamp":"2024-11-05T21:43:05Z","content_type":"text/html","content_length":"9583","record_id":"<urn:uuid:5f91170e-d218-46d7-80d0-2a5e0f6c1c58>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00700.warc.gz"} |
Newton-Raphson Reciprocal - Digital System DesignNewton-Raphson Reciprocal - Digital System Design
Newton-Raphson Reciprocal
Newton-Raphson’s iterative algorithm is a very popular method to approximate a given function. It can be used to compute reciprocal of a given number. The Newton-Raphson’s iterative equation is
After some finite iterations the above equation converges to the reciprocal of D. It is very obvious that initial value of X (
The architecture for Newton-Raphson based reciprocal computation is shown below. A serial architecture is given here as parallel architecture is costly and most system architectures uses serial
architecture. The pulsed control signal start starts the iteration and
Newton-Raphson Based Reciprocal Computation
Click here to download the Verilog Code | {"url":"https://digitalsystemdesign.in/newton-raphson-reciprocal/","timestamp":"2024-11-07T01:00:33Z","content_type":"text/html","content_length":"298595","record_id":"<urn:uuid:66db0785-6740-45c1-a4f2-df9c9fed39a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00542.warc.gz"} |
Some bounds arising from a polynomial ideal associated to any $t$-design (Journal Article) | NSF PAGES
We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph G=(V,E). The buffered expansion of a set S⊆V with a buffer B⊆V∖S is the edge
expansion of S after removing all the edges from set S to its buffer B. An ε-buffered k-partitioning is a partitioning of a graph into disjoint components Pi and buffers Bi, in which the size of
buffer Bi for Pi is small relative to the size of Pi: |Bi|≤ε|Pi|. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets Pi with buffers Bi. Let hk,εG be
the buffered expansion of the optimal ε-buffered k-partitioning, then for every δ>0, hk,εG≤Oδ(1)⋅(logkε)⋅λ⌊(1+δ)k⌋, where λ⌊(1+δ)k⌋ is the ⌊(1+δ)k⌋-th smallest eigenvalue of the normalized Laplacian
of G. Our inequality is constructive and avoids the ``square-root loss'' that is present in the standard Cheeger inequalities (even for k=2). We also provide a complementary lower bound, and a novel
generalization to the setting with arbitrary vertex weights and edge costs. Moreover our result implies and generalizes the standard higher-order Cheeger inequalities and another recent Cheeger-type
inequality by Kwok, Lau, and Lee (2017) involving robust vertex expansion.
more » « less | {"url":"https://par.nsf.gov/biblio/10288400-some-bounds-arising-from-polynomial-ideal-associated-any-design","timestamp":"2024-11-08T05:52:21Z","content_type":"text/html","content_length":"254406","record_id":"<urn:uuid:e95f9566-e5eb-47a4-ab05-7f918ed6fe0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00088.warc.gz"} |
In physics, a couple is a system of forces with a resultant (a.k.a. net or sum) moment of force but no resultant force.^[1]
A more descriptive term is force couple or pure moment. Its effect is to impart angular momentum but no linear momentum. In rigid body dynamics, force couples are free vectors, meaning their effects
on a body are independent of the point of application.
The resultant moment of a couple is a special case of moment. A couple has the property that it is independent of reference point.
Simple couple
A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.
The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple".^[1] The forces have a turning effect or moment called a
torque about an axis which is normal (perpendicular) to the plane of the forces. The SI unit for the torque of the couple is newton metre.
If the two forces are F and −F, then the magnitude of the torque is given by the following formula: ${\displaystyle \tau =Fd}$ where
• ${\displaystyle \tau }$ is the moment of couple
• F is the magnitude of the force
• d is the perpendicular distance (moment) between the two parallel forces
The magnitude of the torque is equal to F • d, with the direction of the torque given by the unit vector ${\displaystyle {\hat {e}}}$ , which is perpendicular to the plane containing the two forces
and positive being a counter-clockwise couple. When d is taken as a vector between the points of action of the forces, then the torque is the cross product of d and F, i.e. ${\displaystyle \mathbf {\
tau } =|\mathbf {d} \times \mathbf {F} |.}$
Independence of reference point
The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P") and, in general, when P is changed, the moment changes. However, the moment (torque) of
a couple is independent of the reference point P: Any point will give the same moment.^[1] In other words, a couple, unlike any more general moments, is a "free vector". (This fact is called Varignon
's Second Moment Theorem.)^[2]
The proof of this claim is as follows: Suppose there are a set of force vectors F[1], F[2], etc. that form a couple, with position vectors (about some origin P), r[1], r[2], etc., respectively. The
moment about P is
${\displaystyle M=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\cdots }$
Now we pick a new reference point P' that differs from P by the vector r. The new moment is
${\displaystyle M'=(\mathbf {r} _{1}+\mathbf {r} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}+\mathbf {r} )\times \mathbf {F} _{2}+\cdots }$
Now the distributive property of the cross product implies
${\displaystyle M'=\left(\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\cdots \right)+\mathbf {r} \times \left(\mathbf {F} _{1}+\mathbf {F} _{2}+\cdots \right).}
However, the definition of a force couple means that
${\displaystyle \mathbf {F} _{1}+\mathbf {F} _{2}+\cdots =0.}$
${\displaystyle M'=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\cdots =M}$
This proves that the moment is independent of reference point, which is proof that a couple is a free vector.
Forces and couples
A force F applied to a rigid body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass and a couple Cℓ = Fd. The couple produces an
angular acceleration of the rigid body at right angles to the plane of the couple.^[3] The force at the center of mass accelerates the body in the direction of the force without change in
orientation. The general theorems are:^[3]
A single force acting at any point O′ of a rigid body can be replaced by an equal and parallel force F acting at any given point O and a couple with forces parallel to F whose moment is M = Fd, d
being the separation of O and O′. Conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately located.
Any couple can be replaced by another in the same plane of the same direction and moment, having any desired force or any desired arm.^[3]
Couples are very important in engineering and the physical sciences. A few examples are:
• The forces exerted by one's hand on a screw-driver
• The forces exerted by the tip of a screwdriver on the head of a screw
• Drag forces acting on a spinning propeller
• Forces on an electric dipole in a uniform electric field
• The reaction control system on a spacecraft
• Force exerted by hands on steering wheel
• 'Rocking couples' are a regular imbalance giving rise to vibration
See also
1. ^ ^a ^b ^c Dynamics, Theory and Applications by T.R. Kane and D.A. Levinson, 1985, pp. 90-99: Free download
2. ^ Engineering Mechanics: Equilibrium, by C. Hartsuijker, J. W. Welleman, page 64 Web link
3. ^ ^a ^b ^c Augustus Jay Du Bois (1902). The mechanics of engineering, Volume 1. Wiley. p. 186.
• H.F. Girvin (1938) Applied Mechanics, §28 Couples, pp 33,4, Scranton Pennsylvania: International Textbook Company. | {"url":"https://www.knowpia.com/knowpedia/Couple_(mechanics)","timestamp":"2024-11-04T15:53:59Z","content_type":"text/html","content_length":"102489","record_id":"<urn:uuid:d182cb07-e912-497e-868b-1933655e7a75>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00512.warc.gz"} |
CPLEX Studio, Engine & OPL Kickstart for Optimization
Introduction to CPLEX Studio, Engines and OPL
We will hear different terms for CPLEX Studio. The official name of the product that we download and install is IBM ILOG CPLEX Studio. CPLEX Studio is an integrated development environment (IDE) that
enables us to develop, run, analyze and test the models. The IDE also contains utilities for file management, information on the models, reporting, and other useful functions.
To download and install CPLEX Studio, please check the end of this blog: IBM ILOG CPLEX Optimization Studio Setup.
Within this IDE, we will write our mathematical model in the modeling language called OPL (Optimization Programming Language). Other examples of modeling language are AMPL, Mosel and GAMS. OPL is a
modeling language for describing (and solving) optimization problems. It is a syntax that is close to the mathematical formulation, thus making it easier to make the transition from the mathematical
formulation to something that can be solved by the computer.
When we hit the Run button, the model compiles and is sent to an optimization engine for solving. This engine is called the CPLEX Optimizer engine, that finds solutions to models that require
mathematical programming techniques (the linear or integer programming calculations). CPLEX was originally named as the Simplex method implemented on the C Language. The Simplex Optimizer available
in CPLEX Studio uses the revived Simplex algorithm, with a number of improvements. Besides the CPLEX Optimizer engine, CPLEX Studio includes CP Optimizer engine, (CPO), to find solutions to models
that require constraint programming techniques. It can actually do more calculations, but that is beyond this tutorial.
Problem 1 Revisted with CPLEX/OPL
The Problem 1 is a simple example with only 2 variables, which are 2 kinds of trucks. In reality, real business problems in real companies can contain up to thousands or more variables. Solving it
graphically is totally impossible. With modern tools, like CPLEX, companies can tackle more complex real-world problems and the return of investment would not be just RM 100 as in our toy truck
problem but millions of dollars.
Create New OPL Project
The first step to solve optimization problems in CPLEX studio is to create a project. To create a new project, go to
File > New > OPL Project
The New Project wizard is displayed. Enter a project name and specify the project location. A folder with the project name is then created in this directory. We can enter a description for the
Even though the only mandatory component in a project is a model file, we will check all the options (Add a default Run Configuration, Create Model, Create Settings, and Create Data) and click
“Finish” button. Don’t worry about these yet. We’ll explain them later.
TICK “Add a default Run Configuration” TICK “Create Model” TICK “Create Settings” TICK “Create Data”
The project is created and your project folder with various files appears in the OPL Projects Navigator (left panel).
A CPLEX project consists of:
• model files (.mod): where we build our model.
• data files (.dat): Although some data might be initialized internally in the model file, data files are necessary to load data from an external data source, such as an Excel spreadsheet.
• setting files (.ops): to specify changes to the default settings.
Once a project is created, you can still add new .mod files, .dat files, .ops files and run configurations by going to the main menu and choosing File > New > …. This allows us to test for example
multiple models on one data set. After creating a new run configuration, you can drag the relevant model file and any data file or setting file to the new run configuration in the OPL Projects
OPL Model
Building a model in OPL is similar to building models in other programming languages. You first declare the variables and then define functions for those variables. OPL is a language for building
optimization models, so it is not surprising that it is structured to set up mathematical optimization problems. You can declare sets, input variables, and decision variables. These three different
types of variables need to be declared in different ways. You then set up the objective function, followed by the constraints.
For a reference, here is the mathematical formulation of this problem.
Minimize: Cost = 500(truck40) + 400(truck30) Subject to:
1. Boxes: 40(truck40) + 30(truck30) $\geq$ 300
2. truck40 $\geq$ 0, truck30 $\geq$ 0
This can be translated to the following OPL model file (.mod):
* OPL 12.9.0.0 Model
* Author: kafechew
* Creation Date: 25 Aug 2019 at 8:28:14 PM
// Decision Variables
dvar int+ truck40;
dvar int+ truck30;
// Objective Function
minimize 500 * truck40 + 400 * truck30;
// Constraints
subject to {
40 * truck40 + 30 * truck30 >= 300;
On the main panel, you will see the workspace with the .mod file opened. CPLEX Studio automatically creates comments at the top of this file (Lines 1-5). The comments start with a / and end with a /.
The asterisks * at the start of each of the lines are optional and just make the comments look nicer.
// can also be used for single row comment. It is always a good practice to add comments to help us remember what the model is meant to do.
Declaring the Decision Variables
First, we need to define the two decision variables (Lines 8 and 9). To do this, we use the OPL designsted term dvar to indicate that this is a decision variable. Decision variables can be of type
integer (int), real (float) or boolean (boolean). We set both of these variables to int+, declaring a non-negative integer for each decision variable. It is possible to create arrays of decision
variables for a specific set.
Since the decision variables are the best values to solve a problem, still unknown, we do not specify the values for these variables, there is no = (equal to) sign after the names. The optimization
engine will determine the values for these variables and return them as outputs. We give these variables readable names to help making our code more understandable.
Model file are terminated by a semicolon (;), called Statement Terminator.
Objective Function
The objective function (Line 12) is executed as a single command. That is, it starts with the keyword of either minimize or maximize, includes the formulas, and ends with a semicolon (;). The
formulas can be complicated, sometimes we will write the objective function over several lines, to make it readable.
The constraints (Lines 15-18) start with the keywords subject to {...}. We place all the different constraints within the curly brackets. Each single constraint or constraint family ends with a
semicolon. We can name the constraint for post processing purpose.
We do not list the second constraint because we took care of this when we specified the variables.
Launch a Run Configuration
Once we have completed the model, be sure to click on the Save button. To launch a particular run configuration, right-click on the run configuration name in the OPL Projects Navigator and select Run
this. When we created the model, a run configuration called Configuration1 was already set up for us.
We can observe that the run configuration consists of our .mod and other files. The little arrows on the .mod and other files indicate that these files just point to the main files. These files are
not copies, we are just telling OPL to use the .mod and other files for this run configuration.
An alternative is to use the run button in the execution toolbar on top. It is worth to note that the run button in the execution toolbar always launches the most recently launched run configuration,
so pay attention if we have created multiple run configurations.
Once our run configuration has the correct .mod and other files, we are ready to run. Here are some ways:
• Right click on “run configuration” > New > Run Configuration
• Drag the model (.mod) into the Configuration
• Right click on the Configuration > Run this
We can watch the progress at the very bottom right of the window. This model should solve very fast.
When it is done solving, CPLEX Studio displays the optimal solution and the values of decision variables in the Solution tab, at the bottom of the screen. The Problem Browser tab (panel on the bottom
left) also displays the data that was input and the solution values of the decision variables. The value of the objective function is shown in the gray bar at the top of this corner. It should show
Solution with objective 3,800. If you click on a variable in the Problem Browser tab, an icon appears with the text Show Data View.... If you click on this icon, the values of the variable appear in
the main panel. The screenshot below gives an idea of how CPLEX Studio looks like after a run configuration has been launched.
Separate Data from Model
CPLEX Studio can separate the model (.mod) from the data (.dat). OPL enables a clean separation between the model and the accompanying data. We have put all the coefficients (Line 12 and 17) into the
model. However, this would kill the value of a programming language.
First, we extract all the coefficients out of the mathematical equations. This can be translated to the following OPL model file (.mod):
* OPL 12.9.0.0 Model
* Author: kafechew
* Creation Date: 25 Aug 2019 at 8:28:14 PM
// Parameters
int nBoxes = 300;
int capacityTruck40 = 40;
int capacityTruck30 = 30;
float costTruck40 = 500;
float costTruck30 = 400;
// Decision Variables
dvar int+ truck40;
dvar int+ truck30;
// Objective Function
minimize costTruck40 * truck40 + costTruck30 * truck30;
// Constraints
subject to {
MinimumCapacityforBoxes: // name of the constraint
capacityTruck40 * truck40 + capacityTruck30 * truck30 >= nBoxes;
We will get the same result with a cost of RM 3800. We can change the number of boxes, capacities and the cost of the two trucks near the Parameters section (Lines 8-12) to solve the same model.
Since it can be reused many times, especially sometimes we will deal with other trucks with different capacities and costs, we should change the names of parameters to be more general. For example,
we can change the name of Truck40, originally for the capacity of 40 boxes, into TruckX. Same with Truck30 into TruckY. This can be translated to the following OPL model file (.mod):
* OPL 12.9.0.0 Model
* Author: kafechew
* Creation Date: 25 Aug 2019 at 8:28:14 PM
// Parameters
int nBoxes = 300;
int capacityTruckX = 40;
int capacityTruckY = 30;
float costTruckX = 500;
float costTruckY = 400;
// Decision Variables
dvar int+ truckX;
dvar int+ truckY;
// Objective Function
minimize costTruckX * truckX + costTruckY * truckY;
// Constraints
subject to {
MinimumCapacityforBoxes: // name of the constraint
capacityTruckX * truckX + capacityTruckY * truckY >= nBoxes;
So far, we have built the model without any reference to the data. We want to separate the model from the data, so without changing the same model can be solved by swapping different input data near
.dat file with little extra effort.
We use ellipsis points ... to indicate that we will define the input values of the parameters somewhere else. This tells the model to look in the .dat file for the values for these variables. The
model should look like this now:
* OPL 12.9.0.0 Model
* Author: kafechew
* Creation Date: 25 Aug 2019 at 8:28:14 PM
// Parameters
int nBoxes = ...;
int capacityTruckX = ...;
int capacityTruckY = ...;
float costTruckX = ...;
float costTruckY = ...;
// Decision Variables
dvar int+ truckX;
dvar int+ truckY;
// Objective Function
minimize costTruckX * truckX + costTruckY * truckY;
// Constraints
subject to {
MinimumCapacityforBoxes: // name of the constraint
capacityTruckX * truckX + capacityTruckY * truckY >= nBoxes;
In the .dat file, we can directly copy the parameters section, but remove the variable types, and put into it. The syntax is to state the variable name, use the equal to symbol =, specify the data,
and then close the statement with the semicolon ;.
* OPL 12.9.0.0 Data
* Author: kafechew
* Creation Date: 25 Aug 2019 at 8:28:14 PM
nBoxes = 300;
capacityTruckX = 40;
capacityTruckY = 30;
costTruckX = 500;
costTruckY = 400;
We can rerun the Configuration1 and get the same result. We can now solve the same model with different parameters without touching the model file (.mod).
Since it is nearly impossible to write a model without making a mistake, we need to be familiar with the debugging capability of OPL. Just below the workspace we write the model, we can see a series
of tabs. The Problems tab shows any issues with the model real-time. Our current model should show no problems. However, if we change the nBoxes to nBoxes2 (Line 8), we will see an error.
It marks the line with the error and provides a description below. In this case, it tells us that no such variable nBoxes has been defined. It is a good practice to understand how to debug a model. A
model shows no syntax errors just means that the model has no OPL errors, but we still need to confirm that the model does what we want it to do.
Next, we will discuss about pulling data from external data sources, such as an Excel spreadsheet or an Access database. As well as using ILOG Script for pre-processing (preparing data) and
post-processing (working on the solutions returned by CPLEX) purposes. | {"url":"https://bringre.com/cplex-studio-engine-opl-optimization/","timestamp":"2024-11-13T04:22:36Z","content_type":"text/html","content_length":"103837","record_id":"<urn:uuid:ef48055d-3d72-42b7-a3e3-c4cbf3c1c976>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00190.warc.gz"} |
Equilibrium pure states and nonequilibrium chaos
We consider nonequilibrium systems such as the Edwards-Anderson Ising spin glass at a temperature where, in equilibrium, there are presumed to be (two or many) broken-symmetry pure states. Following
a deep quench, we argue that as time t → ∞, although the system is usually in some pure state locally, either it never settles permanently on a fixed length scale into a single pure state, or it
does, but then the pure state depends on both the initial spin configuration and the realization of the stochastic dynamics. But this latter case can occur only if there exists an uncountable number
of pure states (for each coupling realization) with almost every pair having zero overlap. In both cases, almost no initial spin configuration is in the basin of attraction of a single pure state;
that is, the configuration space (resulting from a deep quench) is all boundary (except for a set of measure zero). We prove that the former case holds for deeply quenched 2D ferromagnets. Our
results raise the possibility that even if more than one pure state exists for an infinite system, time averages do not necessarily disagree with Boltzmann averages.
Original language English (US)
Pages (from-to) 709-722
Number of pages 14
Journal Journal of Statistical Physics
Volume 94
Issue number 3-4
State Published - Feb 1999
• Broken ergodicity
• Coarsening
• Damage spreading
• Deep quench
• Nonequilibrium dynamics
• Persistence
• Spin glass
• Stochastic Ising model
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Mathematical Physics
Dive into the research topics of 'Equilibrium pure states and nonequilibrium chaos'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/equilibrium-pure-states-and-nonequilibrium-chaos","timestamp":"2024-11-03T20:11:19Z","content_type":"text/html","content_length":"50418","record_id":"<urn:uuid:c1445cab-e2b2-4fe4-acc9-97a6912b8344>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00733.warc.gz"} |
Warm-up: Notice and Wonder: Seesaw (10 minutes)
The purpose of this warm-up is to elicit ideas that students have about weight, which will be useful when students compare the weights of objects in a later activity. Consider reading aloud and
discussing books that explore weight, such as:
• Mighty Maddie by Stuart J. Murphy
• Just A Little Bit by Ann Tompert
• The Seesaw by Judith Koppens
• Balancing Act by Ellen Stoll Walsh
• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice?
What do you wonder?
Activity Synthesis
• “What are some things that you think are heavy?”
• Share and record responses.
• “What are some things that you think are light?”
• Share and record responses.
Activity 1: Compare Weights of Boxes and Bags (15 minutes)
The purpose of this activity is for students to describe and compare the weights of objects. In the first example, students work with two identical boxes and consider the difference in weight between
a box with books in it and one without. In the second example, students discuss ways to compare the weights of objects when it’s not clear which object is heavier. Students may only describe the
weight of one of the objects when comparing. (“The book is heavier.”) The teacher shares the complete comparison statement. (“The book is heavier than the pencil.”) To develop their conceptual
understanding of weight as an attribute, it is important that all students are able to feel the bags in the second example in this activity. The bags can either be passed around so that each student
can feel them, or multiple bags can be made.
Required Preparation
• Prepare 2 boxes, one filled with books, labeled “1,” and one empty box, labeled “2.”
• Prepare 2 closed bags, one containing a few crayons, labeled “1,” and one filled with rocks or other heavy objects, labeled “2.”
• Groups of 2
• Display box 1 and box 2.
• “What do you notice? What do you wonder?” (Students may notice: The empty box is light. There is a lot of stuff in the box, so that box must be heavier. Students may wonder: Is the empty box
lighter than the box that is filled with books? Is the box with books heavier than the empty box?)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• If no student compares the weights of the boxes, ask: “Which box do you think is heavier? Why do you think that?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• “How could we figure out which box is heavier?” (We could pick them up and feel which one is heavier.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• Choose one student to hold each box and share with the class.
• “We use ‘heavier than’ and ‘lighter than’ when we compare the weights of objects. Tell your partner about the boxes using ‘lighter than.’”
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• Monitor for students who say the complete comparison statement. If students only say “The empty box is lighter,” prompt by asking “Lighter than what?”
• Share responses
• Display bags 1 and 2.
• “Here are 2 bags, but we can’t see what is inside. Which bag is heavier?”
• 30 seconds: quiet think time
• Share responses.
• “How could we figure out which bag is heavier?” (We could pick them up and feel which one is heavier.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• Pass the bags around so that each student can hold both bags to compare the weights.
Activity Synthesis
• “Now that we have all felt the bags, which bag is heavier?”
• Share responses.
• “Tell your partner about the bags using ‘lighter than.’”
• Share responses.
• “We can hold objects in our hands to figure out which object feels heavier and which feels lighter.”
Activity 2: Compare Weights (10 minutes)
The purpose of this activity is for students to practice comparing the weights of two objects by feel and using comparison language. Any classroom objects can be used for this activity such as books,
writing utensils, baskets, office supplies, and art supplies. Students can be more comfortable using “heavy” and “heavier” than “light” or “lighter,” so vary questions between “Which object is
heavier?” and “Which object is lighter?” In the activity synthesis, students practice using comparison language as they share one pair of objects that they compared. While not required, students can
write the name of each object or record their comparison with a sentence, such as “The apple is heavier than the book”.
MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain which object is lighter and which object is heavier.
Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Representation: Develop Language and Symbols. Synthesis: Make connections between the objects students share and the relationship it has to being lighter or heavier. Invite students to make a
connection between the complete comparison statements and the objects they circled.
Supports accessibility for: Conceptual Processing
Required Preparation
• Gather assorted classroom objects for students to compare.
• Groups of 2–4
• Give each group of students access to objects to compare.
• “Choose 2 objects with your partner. Figure out and tell your group which object is heavier and which object is lighter. Draw a picture of each object on your recording sheet and circle the
object that was heavier.”
• 5 minutes: partner work time
• Monitor for students who use a complete comparison statement such as “The ball is lighter than the book.”
Student Facing
Choose 2 objects.
Figure out which object is heavier and which is lighter.
Draw a picture of each object.
Circle the object that is heavier.
Activity Synthesis
• Invite each group to share one set of objects that they compared. Invite students to chorally repeat the complete comparison statements in unison 1–2 times.
Activity 3: Centers: Choice Time (15 minutes)
The purpose of this activity is for students to choose from activities that offer practice with number and shape concepts.
Students choose from any stage of previously introduced centers.
• Counting Collections
• Match Mine
• Shake and Spill
Required Preparation
• Gather materials from:
□ Counting Collections, Stage 1
□ Match Mine, Stage 1
□ Shake and Spill, Stages 1-4
• “Today we are going to choose from centers we have already learned.”
• Display the center choices in the student book.
• “Think about what you would like to do.”
• 30 seconds: quiet think time
• Invite students to work at the center of their choice.
• 10 minutes: center work time
Student Facing
Choose a center.
Counting Collections
Match Mine
Shake and Spill
Activity Synthesis
• “We have learned a few different ways to play Shake and Spill. Which way do you like to play Shake and Spill? Why?”
Lesson Synthesis
Display one chair and five pencils.
“Han says that the pencils are heavier than the chair because there are 5 pencils and only 1 chair. What do you think about?” (There are more pencils, but they are small and light. The chair is
probably heavier even though there is only one chair.)
“What can Han do to help him figure out if the chair or the pencils are heavier?” (He can hold the chair and the pencils and see which one feels heavier.)
Cool-down: Compare Weights of Books and Pencils (5 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/kindergarten/unit-7/lesson-8/lesson.html","timestamp":"2024-11-03T06:20:55Z","content_type":"text/html","content_length":"95500","record_id":"<urn:uuid:be3bceaa-ffef-4309-87f2-d6975f29be9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00748.warc.gz"} |
Enigma with a vengeance
I wrote the last post in a flight of euphoria over having finished the enigma machine. This feeling was squashed completely in the next day or so when our lecturer released some test answers and my
encryption turned out to be completely wrong (it did encrypt, but not to the same pattern as the enigma machine). This is part of an apparently common phenomenon that is experienced by many a
programmer – that you feel like the boss in one minute, and like a complete imbecile the next. Anyway, I had real problems figuring out the correct mechanics of it until I actually built my own
enigma machine with paper dials. It didn’t feel exactly glamorous as I went to work with crayons and scissors, but at least shortly after I managed to crack the damn thing. Final hand-in was 2 days
ago, so we’ll see how it turns out! On a side note, it turned out that the Science Museum (which is located right next to Imperial College) had a year-long exhibition on Alan Turing (indirectly
referenced in my previous post), where they actually had 3 enigma machines on show! Of course when I went to finally check out said exhibition, I was informed that it had closed a couple of weeks
ago…. sad face. | {"url":"https://davidbasalla.com/blog/a3582adf-1f42-58ed-b3e8-1ab6e1745ec8/","timestamp":"2024-11-08T18:06:25Z","content_type":"text/html","content_length":"19832","record_id":"<urn:uuid:f45a7c0c-a816-41a5-8771-92087f390eb7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00219.warc.gz"} |
What is 2 to the 6th power? [Solved] | Brighterly Questions
What is 2 to the 6th power?
Updated on December 28, 2023
Answer: 64
Exponential Calculations
Calculating 2 to the 6th power, denoted as 2^6, means multiplying 2 by itself five more times (2 × 2 × 2 × 2 × 2 × 2). The result is 64. Understanding exponential calculations is crucial in
mathematics, as it represents repeated multiplication and is fundamental in various fields, including finance for calculating compound interest, physics for understanding laws of motion, and computer
science in data processing. Mastery of exponential concepts is essential for problem-solving and analysis in scientific research, engineering, and technology development.
FAQ on Exponents and Powers
What is 3 to the 4th power?
What is 4 to the 3rd power?
What is 5 to the 2nd power? | {"url":"https://brighterly.com/questions/what-is-2-to-the-6th-power/","timestamp":"2024-11-02T17:41:24Z","content_type":"text/html","content_length":"69228","record_id":"<urn:uuid:839fdb6b-8ab3-480e-96d9-a09319dddd89>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00060.warc.gz"} |
What is an integrable system
9753 views
What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable"
and "chaotic"? (There is an interesting wikipedia article but I don't find it completely satisfying.)
Update (Dec 2010): Thanks for the many excellent answers. I came across another quote from Nigel Hitchin:
"Integrability of a system of differential equations should manifest itself through some generally recognizable features:
• the existence of many conserved quantities
• the presence of algebraic geometry
• the ability to give explicit solutions.
These guidelines whould be interpreted in a very broad sense."
(If there are some aspects mentioned by Hitchin not addressed by the current answers, additions are welcome...)
Closely related questions: What does it mean for a differential equation "to be integrable"?; basic questions on quantum integrable systems
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Gil Kalai
Very good answers! I'd love to see more angles to this important issue, which is why a little bounty is offered.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Gil Kalai
Excellent question, I think. But I'm stuck before we get to the "integrable" part. What is a "system"? I'd be glad if someone addressed this in their answer.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Tom Leinster
I believe that 'system' is in the same sense as 'dynamical system', which probably comes from 'system of differential equations'.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user José Figueroa-O'Farrill
Thanks, José, but that doesn't really answer the question. People use "dynamical system" in a variety of ways. E.g. the wikipedia article en.wikipedia.org/wiki/Dynamical_system_(definition) gives the
general definition as a partial action of a monoid on a set. An article by Adler in the Bulletin of the AMS defines it as a compact metric space with a continuous endomorphism. But I don't think that
either of those definitions is what the answers below are referring to. Perhaps I should ask this as a separate question
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Tom Leinster
The book by Hitchin, Segal, Ward and Woodhouse begins with this nice quote: "Integrable systems, what are they? It's not easy to answer precisely. The question can occupy a whole book (Zakharov
1991), or be dismissed as Louis Armstrong is reputed to have done once when asked what jazz was---'If you gotta ask, you'll never know!'"
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user HJRW
Could anyone with enough rep please add the "integrable-systems" tag to this question?
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user mathphysicist
There was a Gibbs lecture "Integrable Systems: A modern View": jointmathematicsmeetings.org/meetings/national/jmm/deift
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Thomas Riepe
I'll take off from the questioner's suggesting that maybe it's better to say what is a NON-integrable system is.
The Newtonian planar three body problem, for most masses, has been proven to be non-integrable.
Before Poincare, there seemed to be a kind of general hope in the air that every autonomous Hamiltonian system was integrable. One of Poincare's big claims to fame, proved within his Les Methodes
Nouvelles de Mecanique Celeste, was that the planar three-body problem is not completely integrable. It is the dynamical systems equivalent to Galois' work on quintics. Specifically, Poincare proved
that besides the energy, angular momentum and linear momentum there are no other ANALYTIC functions on phase space which Poisson commute with the energy. (To be more careful: any 'other' such
function is a function of energy, angular momentum, and linear momentum. And his proof, or its extensions, only holds in the parameter region where one of the mass dominates the other two. It is
still possible that for very special masses and angular momenta/ energies the system is integrable. No one believes this.) As best I can tell, existence of additional smooth integrals (with
fractal-like level sets) is still open, at least in most cases.
Poincare's impossibitly proof is based on his discovery of what is nowadays called a "homoclinic tangle" embedded within the restricted three body problem, viewed in a rotating frame. In this tangle,
the unstable and stable manifold of some point (an orbit in the non-rotating inertial frame) cross each other infinitely often, these crossing points having the point in its closure.
Roughly speaking, an additional integral would have to be constant along this complicated set. Now use the fact that if the zeros of an analytic function have an accumulation point then that function
is zero to conclude that the function is zero.
Before Poincare (and I suppose since) mathematicians and in particular astronomers spent much energy searching for sequences of changes of variables which made the system "more and more integrable".
Poincare realized the series defining their transformations were divergent -- hence his interest in divergent series. This divergence problem is the "small denominators problem" and getting around it
by putting number theoretic conditions on frequencies appearing is at the heart of the KAM theorem.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Richard Montgomery
This is, of course, a very good question. I should preface with the disclaimer that despite having worked on some aspects of integrability, I do not consider myself an expert. However I have thought
about this question on and (mostly) off.
I will restrict myself to integrability in classical (i.e., hamiltonian) mechanics, since quantum integrability has to my mind a very different flavour.
The standard definition, which you can find in the wikipedia article you linked to, is that of Liouville. Given a Poisson manifold $P$ parametrising the states of a mechanical system, a hamiltonian
function $H \in C^\infty(P)$ defines a vector field $\lbrace H,-\rbrace$, whose flows are the classical trajectories of the system. A function $f \in C^\infty(P)$ which Poisson-commutes with $H$ is
constant along the classical trajectories and hence is called a conserved quantity. The Jacobi identity for the Poisson bracket says that if $f,g \in C^\infty(P)$ are conserved quantities so is their
Poisson bracket $\lbrace f,g\rbrace$. Two conserved quantities are said to be in involution if they Poisson-commute. The system is said to be classically integrable if it admits "as many as possible"
independent conserved quantities $f_1,f_2,\dots$ in involution. Independence means that the set of points of $P$ where their derivatives $df_1,df_2,\dots$ are linearly independent is dense.
I'm being purposefully vague above. If $P$ is a finite-dimensional and symplectic, hence of even dimension $2n$, then "as many as possible" means $n$. (One can include $H$ among the conserved
quantities.) However there are interesting infinite-dimensional examples (e.g., KdV hierarchy and its cousins) where $P$ is only Poisson and "as many as possible" means in practice an infinite number
of conserved quantities. Also it is not strictly necessary for the conserved quantities to be in involution, but one can allow the Lie subalgebra of $C^\infty(P)$ they span to be solvable but
Now the reason that integrability seems to be such a slippery notion is that one can argue that "locally" any reasonable hamiltonian system is integrable in this sense. The hallmark of integrability,
according to the practitioners anyway, seems to be coordinate-dependent. I mean this in the sense that $P$ is not usually given abstractly as a manifold, but comes with a given coordinate chart.
Integrability then requires the conserved quantities to be written as local expressions (e.g., differential polynomials,...) of the given coordinates.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user José Figueroa-O'Farrill
The simple answer is that a $2n$-dimensional Hamiltonian system of ODE is integrable if it has $n$ (functionally) independent constants of the motion that are "in involution". (Functionally
independent means none of them can be written as a function of the others. And "in involution" means that their Poisson Brackets all vanish -- a somewhat technical condition I won't define carefully
(* but see below), but instead refer you to: http://en.wikipedia.org/wiki/Poisson_bracket). The simplest and the motivating example is the $n$-dimensional Harmonic Oscillator. What makes integrable
systems remarkable and interesting is that one can find so-called "action angle variables" for them, in terms of which the time-evolution of any orbit becomes transparent.
For a more detailed and modern discussion you may find an expository article I wrote in the Bulletin of The AMS useful. It is called "On the Symmetries of Solitons", and you can download it as pdf
It is primarily about the infinite dimensional theory of integrable systems, like SGE (the Sine-Gordon Equation), KdV (Korteweg deVries) , and NLS (non-linear Schrodinger equation), but it starts out
with an exposition of the classic finite dimensional theory.
• Here is a little bit about what the Poisson bracket of two functions is that explains its meaning and why two functions with vanishing Poisson bracket are said to "Poisson commute". Recall that
in Hamiltonian mechanics there is a natural non degenerate two-form $\omega = \sum_i dp_i \wedge dq_i $. This defines (by contraction with $\omega$) a bijective correspondence between vector
fields and differential 1-forms. OK then -- given two functions $f$ and $g$, let $F$ and $G$ be the vector fields corresponding to the 1-forms $df$ and $dg$. Then the Poisson bracket of $f$ and
$g$ is the function $h$ such that $dh$ corresponds to the vector field $[F,G]$, the usual commutator bracket of the vector fields $F$ and $G$. Thus two functions Poisson commute iff the vector
fields corresponding to their differentials commute, i.e., iff the flows defined by these vector fields commute. So if a Hamiltonian vector field (on a compact $2n$-dimensional symplectic
manifold $M$) is integrable, then it belongs to an $n$-dimensional family of commuting vector fields that generate a torus action on $M$. And this is where the action-angle variables come from:
the level surfaces of the action variables are the torus orbits and the angle variables are the angles coordinates for the $n$ circles whose product gives a torus orbit.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Dick Palais
I don't think that one could say that there is a dichotomy between integrable and chaotic systems. There is certainly a huge chunk in the middle. By a chaotic system we often mean a system where
trajectories of points deviate exponentially with time, a canonical example is the Arnold (or Anosov) cat's map. In this case a generic trajectory is of course everywhere dense in the phase space.
This is related to ergodicity (in the case when there is a measure preserved by the system). But of course not every ergodic system is chaotic. There are different degrees of chaos, mixing, strong
mixing, etc.
On the contrary for an integrable system the motion of every trajectory is quasi-periodic, it stays forever on a half-dimensional torus, such systems are rare. A little perturbation of such a system
is not integrable anymore. KAM theory describes the residue of integrability of the perturbation, while Arnol'd diffusion is about trajectories that don't move quasiperiodically anymore.
There is one amazing example due to Moser, that shows how the cat's map can "happen" on a degenerate level of an integrable system: page 6 in
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Dmitri
I agree with this answer. In fact there is a memoir of the AMS by Markus and Meyer which shows that a generic Hamiltonian system is neither integrable nor ergodic, see books.google.fr/…
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Thomas Sauvaget
The above answers deal mostly with finite-dimensional systems. As for the (systems of) PDEs, you typically need the Lax pair or a zero curvature representation (see e.g. the Takhtajan--Faddeev book
mentioned in the wikipedia entry you linked to for the definition of the latter) or something else like that. To the best of my knowledge, the complete understanding of what is an integrable system
for the case of three (3D) or more independent variables is still missing. In particular, for the case of three independent variables (a.k.a. 3D or (2+1)D) the overwhelming majority of examples are
generalizations of the systems with two independent variables. These generalizations are constructed using the so-called central extension procedure (e.g. the KP equation is related to KdV in this
way). Many integrable partial differential systems in three independent variables and apparently the overwhelming majority thereof in four or more independent variables are dispersionless, i.e., can
be written as first-order homogeneous quasilinear systems, see e.g. this article and references therein for details.
As for the reading suggestions, in addition to the Takhtajan--Faddeev book cited above, you can look e.g. into a fairly recent book Introduction to classical integrable systems by Babelon, Bernard
and Talon, and into the book Multi-Hamiltonian theory of dynamical systems by Maciej Blaszak which covers the central extension stuff in a pretty straightforward fashion. Both books have extensive
bibliographies with further references to look into.
Now, as for classification and identification of (new) integrable systems of PDEs, at least in two independent variables, it turns out that the (infinitesimal higher) symmetries play an important
role here. A recent collective monograph Integrability, edited by A.V. Mikhailov and published by Springer in 2009, could be a good starting point in this direction. See also another recent book
Algebraic theory of differential equations edited by MacCallum and Mikhailov and published by Cambridge University Press. For a general introduction to the subject of symmetries of (systems of) PDEs,
I can recommend the book Applications of Lie groups to differential equations by Peter Olver.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user mathphysicist
Your "3D" is better known as 1+2 (one time variable, two space variables). This is an important distinction both in the Lax pair formalism (time variable is preferred) and in zero curvature
representation approach (applies primarily to 1+1).
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Victor Protsak
Regarding extension to PDEs, note Dedecker's paper "Intégrales complètes de l'équation aux dérivées partielles de Hamilton-Jacobi d'une intégrale multiple", C.R. Acad. Sc. Paris, 285 (1977) pp.
123-6. Together with two preceding notes in the same journal, this generalises the concept of "complete integrability" of a mechanical system.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Phil Harmsworth
This is soft -- but I think of an integrable system as one whose dynamics are dominated by algebra. For finite dimensional integrable systems, the symmetries (related to conserved quantities by
Noether's theorem) force the trajectories to live on half-dimensional tori. For infinite dimensional integrable systems, where the flow on the scattering data is isospectral the symmetries force
solutions to be n-soliton solutions plus dispersive modes.
There is a blog post of Terry Tao's (apologies for not having the link) which talks about how algebra is the right tool to understand structure while analysis is the right tool to understand
randomness. The claim is that one mark of an good problem is the presence of an interesting relationship between structure and randomness and hence the requirement that both algebra and analysis be
used -- to some degree -- in order to get a good answer to the problem. The soliton resolution conjecture is by this standard a good problem because the asymptotic n-soliton solutions are
fundamentally algebraic while the dispersive modes are fundamentally analytic objects.
I agree with Dmitri that there isn't a dichotomy. The symmetries can have a large or small role in the dynamics as can the ergodicity.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Aaron Hoffman
I'll give a bit of a physics definition. (Reference is "A Brief Introduction to Classical, Statistical and Quantum Mechanics" by B\"uhler.)
"A mechanical system is called integrable if we can reduce its solution to a sequence of quadratures."
So, literally, an integrable system (in this view) is one that can be solved by a sequence of integrals (which may not be explicitly solvable in elementary functions, of course). To connect to other
answers, this should only work out when there are enough symmetries for us to write down and integrate.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Charles Siegel
Since, it hasn't been mentioned yet a short addition to José Figueroa-O'Farrill's answer. I will only talk about the finite dimensional case. So let's assume that $dim(P) = 2n$. Then the Hamiltonian
flow is integrable if there exist these $n$ functions $f_1, \dots, f_n$ which are in involution with respect to the Poisson structure.
Now, the cool thing is that there exist action angle coordinates. These means we can conjugate our possibly complicated dynamics to the simple dynamics $$ \partial_t I_j = 0,\quad \partial_t \theta_j
= I_j,\quad j=1,\dots,n $$ this is something, we can all solve since it is just linear. Note: We will have $I_j = f_j(orbit)$, which is time independent.
As a possible application, KAM theory is usually formulated as an application to systems in action angle coordinates. This in turn implies that integrable systems are stable (in a subtle measure
theoretic sense). But I think this is what is meant with "integrable $\neq$ chaos". We have a great form of perturbation theory for integrable systems.
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Helge
After reading several books and articles about integrable systems, and after several years of work in the field, I consider particularly meaningful the following quotation from Frederic Helein's book
'Constant mean curvature surfaces, harmonic maps and integrable systems', Lectures in Mathematics, ETH Zurich, Birkhauser Basel (2001):
"...working on completely integrable systems is based on a contemplation of some very exceptional equations which hide a Platonic structure: although these equations do not look trivial a priori, we
shall discover that they are elementary, once we understand how they are encoded in the language of symplectic geometry, Lie groups and algebraic geometry. It will turn out that this contemplation is
fruitful and lead to many results"
This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Giovanni Rastelli | {"url":"https://www.physicsoverflow.org/41515/what-is-an-integrable-system","timestamp":"2024-11-13T05:45:56Z","content_type":"text/html","content_length":"322765","record_id":"<urn:uuid:e6aab98f-b0ef-4797-9ae2-5649e2214020>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00153.warc.gz"} |
Unit 1 Scale Drawings Answer Key
Unit 1 Scale Drawings Answer Key - Web for instance, a scale of 1 to 500 means that 1 inch on the drawing represents 500 inches in actual distance, and 10 mm on a drawing represents 5,000 mm in
actual distance. They use tables to reason about measurements in scaled copies, and recognize that angle measures. Web π grade 7, unit 1, lesson 7 scaled drawings illustrative mathematics
practice. Support for teachers and parents. Web unit 1 scale drawings quiz for 7th grade students.
Web review and tutorial. Letβ s put it to work. The statement β 1 cm represents 2 mβ is the scale of the drawing. Unit 8 probability and sampling. Unit 9 putting it all together Lesson 13 draw it
to scale What is its actual area?
Scale Drawings Worksheet 1 Answer Key / Unit 1 Scale Drawings Khan
Lesson 9 creating scale drawings; Unit 8 probability and sampling. The statement β 1 cm represents 2 mβ is the scale of the drawing. 130 what is the relationship between these scale factors? Unit
5 rational.
unit 1 scale drawings answer key bodyarttattoosmenhand
We also review how to approach the practice. Search #718math in youtube to find this lesson. Lesson 6 scaling and area; Web π grade 7, unit 1, lesson 8 scale drawings and maps illustrative
maps and scale drawings worksheet answers winterbeachweddingoutfit
Search #718math in youtube to find this lesson. Web unit 1 scale drawings. Unit 5 rational number arithmetic; The statement β 1 cm represents 2 mβ is the scale of the drawing. For each problem,
scale drawings worksheet 1 answer key largeabstractarttutorial
Lesson 8 scale drawings and maps; What is its actual area? In other words, the actual distance is 500 times the distance on the drawing, and the scaled distance is of the actual distance. We.
Lesson 4 Extra Practice Scale Drawings Answer Key / Unit 4a Geometric
Support for teachers and parents. We also review how to approach the practice. Unit 8 probability and sampling; The size of the scale factor. If the scale drawing is smaller than the.
unit 1 scale drawings answer key bodyarttattoosmenhand
Lesson 13 draw it to scale Web 22k views 4 years ago 2) 7th grade (all units) illustrative mathematics playlist. Record your results in the first row of the table. Unit 8 probability and sampling..
Scale Drawings Worksheet 7th Grade
Unit 6 expressions, equations, and inequalities; Unit 1, lesson 12 units in scale drawings review and. Web π grade 7, unit 1, lesson 8 scale drawings and maps illustrative mathematics practice.
Unit 7 angles, triangles,.
CKMath Unit 1 Scale Drawings Core Knowledge Foundation
The factor by which every length in an original figure is increased or decreased when you make a scaled copy. Search #718math in youtube to find this lesson. If the scale drawing is larger than.
Solving Problems Involving Scale Drawings 7th Grade Math Worksheets
What is its actual area? In a pair of figures, i can identify corresponding points, corresponding segments, ad corresponding angles. Web 22k views 4 years ago 2) 7th grade (all units) illustrative
mathematics playlist. Lesson.
scale drawings worksheet 1 answer key tropicalflowernailarttutorial
They use tables to reason about measurements in scaled copies, and recognize that angle measures. Web unit 1 scale drawings quiz for 7th grade students. Web 7.g.a.1 solve problems involving scale
drawings of geometric figures,.
Unit 1 Scale Drawings Answer Key They use tables to reason about measurements in scaled copies, and recognize that angle measures. Lesson 6 scaling and area; Unit 7 angles, triangles, and prisms;
Letβ s put it to work. Web for instance, a scale of 1 to 500 means that 1 inch on the drawing represents 500 inches in actual distance, and 10 mm on a drawing represents 5,000 mm in actual distance.
Unit 1 Scale Drawings Answer Key Related Post : | {"url":"https://classifieds.independent.com/print/unit-1-scale-drawings-answer-key.html","timestamp":"2024-11-10T12:21:23Z","content_type":"application/xhtml+xml","content_length":"23380","record_id":"<urn:uuid:0450bbf7-a2d8-4cdc-a85b-2f7c6281ed27>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00035.warc.gz"} |
Why do capacitors and inductors store energy but resistors do not?
Capacitors and inductors store energy because they can store electric and magnetic fields, respectively, which represent stored energy in the form of electric potential or magnetic flux. In a
capacitor, energy is stored in the form of an electric field between its plates when it is charged. The amount of stored energy in a capacitor is proportional to the square of the voltage across it
and its capacitance (E = 0.5 * C * V^2), where E is energy, C is capacitance, and V is voltage.
Similarly, in an inductor, energy is stored in the form of a magnetic field surrounding the coil when current flows through it. The amount of stored energy in an inductor is proportional to the
square of the current flowing through it and its inductance (E = 0.5 * L * I^2), where E is energy, L is inductance, and I is current.
Capacitors and inductors are called energy storage elements because they can accumulate and release energy in the form of electric or magnetic fields.
Unlike resistors, which dissipate electrical energy as heat due to their resistance, capacitors and inductors can store energy temporarily and release it back into the circuit when needed.
This ability to store and release energy makes capacitors and inductors essential components in circuits where energy storage, filtering, or timing functions are required.
The stored energy in a capacitor or an inductor can be dissipated by a resistor if they are connected in a circuit together.
When a charged capacitor or a current-carrying inductor is discharged through a resistor, the energy stored in the capacitor’s electric field or the inductor’s magnetic field is converted into heat
as current flows through the resistor.
This dissipation occurs as the capacitor discharges or the inductor’s magnetic field collapses, releasing the stored energy through the resistor in the form of heat.
Inductors are used instead of resistors in certain applications because they offer unique properties that resistors do not possess.
Inductors can store energy in their magnetic fields and release it back into the circuit, whereas resistors simply dissipate energy as heat. This property makes inductors suitable for applications
where energy storage, voltage regulation, filtering, or magnetic coupling are required. In contrast, resistors are primarily used to limit current flow, control voltage levels, or dissipate energy
without storing it.
Energy is stored in capacitors by charging them with electrical charge, which creates an electric field between the capacitor’s plates.
The amount of stored energy in a capacitor depends on its capacitance and the voltage applied across it. When a capacitor is charged, electrons accumulate on one plate, creating a positive charge on
the other plate and an equal but opposite charge on the opposite plate. In inductors, energy is stored in the form of a magnetic field generated around the coil when current flows through it.
The strength of the magnetic field and thus the amount of stored energy depend on the inductor’s inductance and the current flowing through it.
Related Posts | {"url":"https://electrotopic.com/why-do-capacitors-and-inductors-store-energy-but-resistors-do-not/","timestamp":"2024-11-08T05:51:35Z","content_type":"text/html","content_length":"33648","record_id":"<urn:uuid:ab7507d8-5960-47b0-af74-aa5911406586>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00472.warc.gz"} |
Annual report 2020Annual report 2020
updates: January 9, 2024; June 28, 2023 (with Peer Review), December 3, 2022; March 15, 2022 (on CNCSIS prizes for papers).
Dr. rer. nat. habil. Nicolae Suciu supervises PhD theses. He has recently supervised the thesis Filtered density functions for uncertainty assessment of transport in groundwater, by Lennart Schüler,
defended on September 8, 2016, at Jena University, Germany; the thesis received the distinction Magna Cum Laudae.
Prof. dr. Ion Păvăloiu, honorary member, has supervised -in the past- 8 PhD theses at North University of Baia-Mare and at Babeş-Bolyai University.
Prof. dr. Octavian Agratini, supervised PhD theses at Babes-Bolyai University. theses supervised 2020:
N. Suciu and O. Agratini have high Hirsch indexes in ISI and Scholar.
CNCSIS prizes for papers published in the “red zone” journals (based on journals impact factor or influence score):
CNCSIS prizes for papers published in the “yellow zone” journals | {"url":"https://ictp.acad.ro/annual-reports/annual-report-2020/","timestamp":"2024-11-13T05:17:55Z","content_type":"text/html","content_length":"122058","record_id":"<urn:uuid:044402b6-60b9-488b-9582-f48ba1421b13>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00127.warc.gz"} |
Maximum principle, subharmonic and superharmonic functions
Which one is subharmonic?
The Laplace operator Δ of a function of n variables is defined by
If Δ f = 0 in some region Ω, f is said to be harmonic on Ω. In that case f takes on its maximum and minimum over Ω at locations on the boundary ∂Ω of Ω. Here is an example of a harmonic function
over a square which clearly takes on its maximum on two sides of the boundary and its minimum on the other two sides.
The theorem above can be split into two theorems and generalized:
If Δ f ≥ 0, then f takes on its maximum on ∂Ω.
If Δ f ≤ 0, then f takes on its minimum on ∂Ω.
These two theorems are called the maximum principle and minimum principle respectively.
Now just as functions with Δf equal to zero are called harmonic, functions with Δf non-negative or non-positive are called subharmonic and superharmonic. Or is it the other way around?
If Δ f ≥ 0 in Ω, then f is called subharmonic in Ω. And if Δ f ≤ 0 then f is called superharmonic. Equivalently, f is superharmonic if −f is subharmonic.
The names subharmonic and superharmonic may seem backward: the theorem with the greater than sign is for subharmonic functions, and the theorem with the less than sign is for superharmonic functions.
Shouldn’t the sub-thing be less than something and the super-thing greater?
Indeed they are, but you have to look f and not Δf. That’s the key.
If a function f is subharmonic on Ω, then f is below the harmonic function interpolating f from ∂Ω into the interior of Ω. That is, if g satisfies Laplace’s equation
then f ≤ g on Ω.
For example, let f(x) = ||x|| and let Ω be the unit ball in ℝ^n. Then Δ f ≥ 0 and so f is subharmonic. (The norm function f has a singularity at the origin, but this example can be made rigorous.)
Now f is constantly 1 on the boundary of the ball, and the constant function 1 is the unique solution of Laplace’s equation on the unit ball with such boundary condition, and clearly f is less than 1
on the interior of the ball.
Related posts
One thought on “Which one is subharmonic?”
1. What about if f is subharmonic and g is superharmonic on a domain G , with (f-g)(x)<0 near boundary of a domain G. Then what can we say about (f-g) on whole of G. Does it remains negative? | {"url":"https://www.johndcook.com/blog/2022/10/08/subharmonic/","timestamp":"2024-11-12T06:11:00Z","content_type":"text/html","content_length":"52839","record_id":"<urn:uuid:6c41674b-14ad-423e-8d0f-8bcfb2093be0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00267.warc.gz"} |
What Is An Extraneous Solution? (3 Key Concepts To Know) | jdmeducational
What Is An Extraneous Solution? (3 Key Concepts To Know)
When we use algebra to solve equations, we can usually trust all of the answers we get. However, there are some cases where this is not true – that is, when we get extraneous solutions.
So, what is an extraneous solution? An extraneous solution for an equation is a value we find when solving the equation that does not satisfy the equation. For example, if we square both sides of
the equation √x = -1, we get a result of x = 1, which does not satisfy the original equation, which means x = 1 is an extraneous solution.
Of course, squaring both sides of an equation is not the only way that extraneous solutions can appear. In fact, some equations can have more than one extraneous solution, depending on how we solve
In this article, we’ll talk about extraneous solutions, what they are, and how they appear. We will also look at several examples to make the concept clear.
Let’s get started.
What Is An Extraneous Solution?
An extraneous solution is a “false” solution to an equation. An extraneous solution does not satisfy the original equation, even though we took seemingly valid steps while solving the equation.
Sometimes we find extraneous solutions as a result of steps we took in solving an equation. These values do not satisfy the original equation, which we can see by testing.
How Do You Know If A Solution Is Extraneous?
You can always tell that a solution is extraneous if it does not satisfy the equation. That is, take the value of your variable and substitute it back into the original equation.
Example 1: Testing For An Extraneous Solution
Recall the example from earlier, when we wanted to solve the equation √x = -1. After squaring both sides, we get a value of x = 1.
When we substitute x = 1 into the original equation, we get:
• √x = -1 [original equation]
• √1 = -1 [substitute x = 1]
• 1 = -1 [this is not true]
Since the resulting equation is not true, the solution x = 1 must be extraneous. We can verify this by studying the graph below.
This graph compares the radical function y = √x (blue curve) and the horizontal line y = -1 (red line), which never intersect.
*Note: it is true that x = -1 is also a square root of 1. However, when we see the notation “√1”, we generally take the principal square root of 1, which is 1 (positive 1).
If we mean -1 (negative 1), we write “-√1” to indicate that we want the negative square root.
More generally, write “√x” if you want the positive (principal) square root of x, and “-√x” if you want the negative square root of x.
Can You Have Two Extraneous Solutions?
It is possible to have two extraneous solutions to an equation. In fact, you can have three or more extraneous solutions, depending on the equation.
Example: An Equation With Two Extraneous Solutions
Consider the equation √(x^2 – 9) = -4. When we solve, we get:
• √(x^2 – 9) = -4 [original equation]
• (√(x^2 – 9))^2 = (-4)^2 [square both sides]
• x^2 – 9 = 16 [simplify both sides]
• x^2 = 25 [add 9 to both sides]
• x = +5, -5
Note: we need to take both the positive and negative square roots to find these two values.
However, both x = 5 and x = -5 are extraneous solutions for this equation. We can see this by substituting x = -5 back into the original equation:
• √(x^2 – 9) = -4 [original equation]
• √((5)^2 – 9) = -4 [substitute x= 5]
• √(25 – 9) = -4
• √(16) = -4
• 4 = -4
This last equation is not true, meaning that x = 5 is an extraneous solution.
The calculations are similar when you plug in x = -5 to find out that is it also an extraneous solution.
So, the equation √(x^2 – 9) = -4 has two extraneous solutions. We can verify this by studying the graph below.
This graph compares the radical function y = √(x^2 – 9) (blue curve) and the horizontal line y = -4 (red line), which never intersect.
What Causes An Extraneous Solution?
There are several operations that can cause an extraneous solution to emerge. For example:
• Radicals – for example, when we square both sides of a radical equation (to cancel a square root), we may introduce extraneous solutions when the radical is equal to a negative number (since
there is no solution to that equation (same for a fourth root, sixth root, or any even root).
• Multiplication – for example, when we multiply both sides of an equation by x, we may find the extraneous solution x = 0. Another example is multiplying both sides of an equation by 0, which
makes the equation true in all cases (for any and every value of a variable), thus introducing numerous extraneous solutions.
• Rational Expressions – for example, when working with common denominators to solve equations with rational expressions.
Solving rational equations can introduce extraneous solutions (the graph of a rational function has at least one vertical asymptote).
You can learn more about the domain and range of a rational function here.
How Do You Find Extraneous Solutions?
Generally, the best way to find extraneous solutions is to solve the equation and find values for the variables. Then, check your answers by substituting the values back into the original equation.
Example 1: A Radical Equation With Extraneous Solutions
Consider the equation √(x^2 – 36) = -8. When we solve, we get:
• √(x^2 – 36) = -8 [original equation]
• (√(x^2 – 36))^2 = (-8)^2 [square both sides]
• x^2 – 36 = 64 [simplify both sides]
• x^2 = 100 [add 36 to both sides]
• x = +10, -10
Note: we need to take both the positive and negative square roots to find these two values.
However, both x = 10 and x = -10 are extraneous solutions for this equation. We can see this by substituting x = -5 back into the original equation:
• √(x^2 – 36) = -8 [original equation]
• √((10)^2 – 36) = -8 [substitute x = 10]
• √(100 – 36) = -8
• √(64) = -8
• 8 = -8
This last equation is not true, meaning that x = 10 is an extraneous solution.
The calculations are similar when you plug in x = -10 to find out that is it also an extraneous solution.
So, the equation √(x^2 – 36) = -8 has two extraneous solutions.
Example 2: Extraneous Solutions When Multiplying By Zero
Consider the equation x = 2. It has only one solution.
However, if we multiply by zero on both sides, we get:
This equation is true in all cases, regardless of the value of x. In multiplying by zero on both sides of the equation, we have introduced every real number (except x = 2) as an extraneous solution.
Example 3: Extraneous Solutions When Multiplying By x
Consider the equation x – 5 = 0. It has only one solution: x = 5.
However, if we multiply by x on both sides, we get:
• x*x – x*5 = x*0
• x^2 – 5x = 0
• x(x – 5) = 0
• x = 0 or x = 5
However, when we substitute x = 0 into the original equation, we get:
• x – 5 = 0 [original equation]
• 0 – 5 = 0 [substitute x = 0]
• -5 = 0
This last equation is not true. In multiplying by x on both sides of the equation, we have introduced x = 0 an extraneous solution.
In general, if we multiply an equation by x – r, we may introduce x = r as an extraneous solution.
Example 4: A Rational Equation With An Extraneous Solution
Consider the following rational equation:
• 1/(x-1) + 2/(x+1) = 2/(x^2 – 1)
We can factor x^2 – 1 as a difference of squares to get (x+1)(x-1). So, we can cancel all denominators in this equation if we multiply both sides by x^2 – 1.
This leaves us with the following equation to solve:
• 1(x+1) + 2(x-1) = 2
• x + 1 + 2x – 2 = 2
• 3x – 1 = 2
• 3x = 3
• x = 1
However, if we substitute x = 1 back into the original equation, we get zero denominators in the first (leftmost) term and the rightmost term:
• 1/(x-1) + 2/(x+1) = 2/(x^2 – 1) [original equation]
• 1/(1-1) + 2/(1+1) = 2/(1^2 – 1) [substitute x = 1]
• 1/(0) + 2/(2) = 2/(0)
Since we cannot resolve the equation (due to zero denominators), we conclude that x = 1 is an extraneous solution for this equation.
We can see from this graph that x = 1 is an extraneous solution to the original equation mentioned above in this example.
Why Is It Important To Check For Extraneous Solutions?
It is important to check for extraneous solutions because they do not really satisfy the equation. These values only emerge as a result of a step we took when solving the equation.
It is a good idea to check your answers by substituting values back into the original equation. This allows you to detect extraneous solutions and also ensures that you did not make a calculation
mistake somewhere along the way.
Now you know what extraneous solutions are and how they can appear when solving an equation. If you stay mindful of these possibilities when solving equations, you should be able to prevent or
detect extraneous solutions. | {"url":"https://jdmeducational.com/what-is-an-extraneous-solution-3-key-concepts-to-know/","timestamp":"2024-11-02T09:16:39Z","content_type":"text/html","content_length":"85364","record_id":"<urn:uuid:95ac7401-6c08-408a-8f1f-495aafb8dd85>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00053.warc.gz"} |
Evolution Equations - Clay Mathematics Institute
Evolution Equations
This volume is a collection of notes from lectures given at the 2008 Clay Mathematics Institute Summer School, held in Zürich, Switzerland. The lectures were designed for graduate students and
mathematicians within five years of the Ph.D., and the main focus of the program was on recent progress in the theory of evolution equations. Such equations lie at the heart of many areas of
mathematical physics and arise not only in situations with a manifest time evolution (such as linear and nonlinear wave Schrödinger equations) but also in the high energy or semi-classical limits of
elliptic problems.
The three main courses focused mainly on microlocal analysis and spectral and scattering theory, the theory of the nonlinear Schrödinger and wave equations, and evolution problems in general
relativity. These major topics were supplemented by several mini-courses reporting on the derivation of effective evolution equations from microscopic quantumn dymanics; on wave maps with and without
symmetries; on quantum N-body scattering, diffraction of waves, and symmetric spaces; and on nonlinear Schrödinger equations at critical regularity.
Although highly detailed treatments of some of these topics are now available in the published literature, in this collection the reader can learn the fundamental ideas and tools with a minimum of
technical machinery. Moreover, the treatment in this volume emphasizes common themes and techniques in the field, including exact and approximate conservation laws, energy methods, and positive
commutator arguments.
Authors. Dean Baskin, Mihalis Dafermos, Rowan Killip, Rafe Mazzeo, Pierre Raphaël, Igor Rodnianski, Benjamin Schein, Gigliola Staffilani, Michael Struwe, András Vasy, Monica Visan, Jared Wunsch
Available at the AMS Bookstore
Editors: David Ellwood, Igor Rodnianski, Gigliola Staffilani, Jared Wunsch | {"url":"https://www.claymath.org/resource/evolution-equations/","timestamp":"2024-11-06T15:05:08Z","content_type":"text/html","content_length":"86577","record_id":"<urn:uuid:b534c432-ca4b-43c9-b97c-02d23f94f803>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00290.warc.gz"} |
Douglas College Physics 1104 Custom Textbook – Winter and Summer 2020
Chapter 10 Fluid Statics – Floating and Sinking
• Define pressure.
• State Pascal’s principle.
• Apply of Pascal’s principle.
• Derive relationships between forces in a hydraulic system.
Pressure is defined as force per unit area. Can pressure be increased in a fluid by pushing directly on the fluid? Yes, but it is much easier if the fluid is enclosed. The heart, for example,
increases blood pressure by pushing directly on the blood in an enclosed system (valves closed in a chamber). If you try to push on a fluid in an open system, such as a river, the fluid flows away.
An enclosed fluid cannot flow away, and so pressure is more easily increased by an applied force.
What happens to a pressure in an enclosed fluid? Since atoms in a fluid are free to move about, they transmit the pressure to all parts of the fluid and to the walls of the container. Remarkably, the
pressure is transmitted undiminished. This phenomenon is called Pascal’s principle, because it was first clearly stated by the French philosopher and scientist Blaise Pascal (1623–1662): A change in
pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
Pascal’s principle, an experimentally verified fact, is what makes pressure so important in fluids. Since a change in pressure is transmitted undiminished in an enclosed fluid, we often know more
about pressure than other physical quantities in fluids. Moreover, Pascal’s principle implies that the total pressure in a fluid is the sum of the pressures from different sources. We shall find this
fact—that pressures add—very useful.
Blaise Pascal had an interesting life in that he was home-schooled by his father who removed all of the mathematics textbooks from his house and forbade him to study mathematics until the age of 15.
This, of course, raised the boy’s curiosity, and by the age of 12, he started to teach himself geometry. Despite this early deprivation, Pascal went on to make major contributions in the mathematical
fields of probability theory, number theory, and geometry. He is also well known for being the inventor of the first mechanical digital calculator, in addition to his contributions in the field of
fluid statics.
Application of Pascal’s Principle
One of the most important technological applications of Pascal’s principle is found in a hydraulic system, which is an enclosed fluid system used to exert forces. The most common hydraulic systems
are those that operate car brakes. Let us first consider the simple hydraulic system shown in Figure 1.
Figure 1. A typical hydraulic system with two fluid-filled cylinders, capped with pistons and connected by a tube called a hydraulic line. A downward force F[1] on the left piston creates a pressure
that is transmitted undiminished to all parts of the enclosed fluid. This results in an upward force F[2] on the right piston that is larger than F[1] because the right piston has a larger area.
Relationship Between Forces in a Hydraulic System
We can derive a relationship between the forces in the simple hydraulic system shown in Figure 1 by applying Pascal’s principle. Note first that the two pistons in the system are at the same height,
and so there will be no difference in pressure due to a difference in depth. Now the pressure due to F[1] acting on area [latex]\boldsymbol{A_1}[/latex]i s simply [latex]\boldsymbol{P_1=\frac{F_1}
{A_1}},[/latex] as defined by [latex]\boldsymbol{P=\frac{F}{A}}.[/latex] According to Pascal’s principle, this pressure is transmitted undiminished throughout the fluid and to all walls of the
container. Thus, a pressure [latex]\boldsymbol{P_2}[/latex] is felt at the other piston that is equal to [latex]\boldsymbol{P_1}.[/latex] That is[latex]\boldsymbol{P_1=P_2}.[/latex]
But since [latex]\boldsymbol{P_2=\frac{F_2}{A_2}},[/latex]we see that [latex]\boldsymbol{\frac{F_1}{A_1}=\frac{F_2}{A_2}}.[/latex]
This equation relates the ratios of force to area in any hydraulic system, providing the pistons are at the same vertical height and that friction in the system is negligible. Hydraulic systems can
increase or decrease the force applied to them. To make the force larger, the pressure is applied to a larger area. For example, if a 100-N force is applied to the left cylinder in Figure 1 and the
right one has an area five times greater, then the force out is 500 N. Hydraulic systems are analogous to simple levers, but they have the advantage that pressure can be sent through tortuously
curved lines to several places at once.
Example 1: Calculating Force of Slave Cylinders: Pascal Puts on the Brakes
Consider the automobile hydraulic system shown in Figure 2.
Figure 2. Hydraulic brakes use Pascal’s principle. The driver exerts a force of 100 N on the brake pedal. This force is increased by the simple lever and again by the hydraulic system. Each of the
identical slave cylinders receives the same pressure and, therefore, creates the same force output F[2]. The circular cross-sectional areas of the master and slave cylinders are represented by A[1]
and A[2], respectively.
A force of 100 N is applied to the brake pedal, which acts on the cylinder—called the master—through a lever. A force of 500 N is exerted on the master cylinder. (The reader can verify that the force
is 500 N using techniques of statics from Chapter 9.4 Applications of Statics, Including Problem-Solving Strategies.) Pressure created in the master cylinder is transmitted to four so-called slave
cylinders. The master cylinder has a diameter of 0.500 cm, and each slave cylinder has a diameter of 2.50 cm. Calculate the force F[2] created at each of the slave cylinders.
We are given the force F1[ ]that is applied to the master cylinder. The cross-sectional areas [latex]\boldsymbol{A_1}[/latex] and [latex]\boldsymbol{A_2}[/latex] can be calculated from their given
diameters. Then [latex]\boldsymbol{\frac{F_1}{A_1}=\frac{F_2}{A_2}}[/latex ]can be used to find the force F[2. ]Manipulate this algebraically to get F[2 ]on one side and substitute known values:
Pascal’s principle applied to hydraulic systems is given by [latex]\boldsymbol{\frac{F_1}{A_1}=\frac{F_2}{A_2}}:[/latex]
latex][latex]\boldsymbol{\frac{(1.25\textbf{ cm})^2}{(0.250\textbf{ cm})^2}}[/latex][latex]\boldsymbol{\times500\textbf{ N}=1.25\times10^4\textbf{ N.}}[/latex]
This value is the force exerted by each of the four slave cylinders. Note that we can add as many slave cylinders as we wish. If each has a 2.50-cm diameter, each will exert 1.25 x 10 ^4 N.
A simple hydraulic system, such as a simple machine, can increase force but cannot do more work than done on it. Work is force times distance moved, and the slave cylinder moves through a smaller
distance than the master cylinder. Furthermore, the more slaves added, the smaller the distance each moves. Many hydraulic systems—such as power brakes and those in bulldozers—have a motorized pump
that actually does most of the work in the system. The movement of the legs of a spider is achieved partly by hydraulics. Using hydraulics, a jumping spider can create a force that makes it capable
of jumping 25 times its length!
Conservation of energy applied to a hydraulic system tells us that the system cannot do more work than is done on it. Work transfers energy, and so the work output cannot exceed the work input. Power
brakes and other similar hydraulic systems use pumps to supply extra energy when needed.
Section Summary
• Pressure is force per unit area.
• A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.
• A hydraulic system is an enclosed fluid system used to exert forces.
Conceptual Questions
1: Suppose the master cylinder in a hydraulic system is at a greater height than the slave cylinder. Explain how this will affect the force produced at the slave cylinder.
Problems & Exercises
1: How much pressure is transmitted in the hydraulic system considered in Example 1? Express your answer in pascals and in atmospheres.
2: What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on the slave cylinder? The master cylinder has a 2.00-cm diameter
and the slave has a 24.0-cm diameter.
3: A crass host pours the remnants of several bottles of wine into a jug after a party. He then inserts a cork with a 2.00-cm diameter into the bottle, placing it in direct contact with the wine. He
is amazed when he pounds the cork into place and the bottom of the jug (with a 14.0-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N
4: A certain hydraulic system is designed to exert a force 100 times as large as the one put into it. (a) What must be the ratio of the area of the slave cylinder to the area of the master cylinder?
(b) What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no
losses to friction.
5: (a) Verify that work input equals work output for a hydraulic system assuming no losses to friction. Do this by showing that the distance the output force moves is reduced by the same factor that
the output force is increased. Assume the volume of the fluid is constant. (b) What effect would friction within the fluid and between components in the system have on the output force? How would
this depend on whether or not the fluid is moving?
Pascal’s Principle
a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container
Problems & Exercises
1: 2.55 x 10^7 Pa; or 251 atm
3: 5.76×10^3 N extra force
5: (a [latex]\boldsymbol{V=d_{\textbf{i}}A_{\textbf{i}}=d_{\textbf{o}}A_{\textbf{o}}\Rightarrow{d}_{\textbf{o}}=d_{\textbf{i}}(\frac{A_{\textbf{i}}}{A_{\textbf{o}}})}.[/latex]
Now, using equation:
In other words, the work output equals the work input.
(b) If the system is not moving, friction would not play a role. With friction, we know there are losses, so that [latex]\boldsymbol{W_{\textbf{out}}=W_{\textbf{in}}-W_{\textbf{f}}};[/latex]
therefore, the work output is less than the work input. In other words, with friction, you need to push harder on the input piston than was calculated for the non-friction case. | {"url":"https://pressbooks.bccampus.ca/practicalphysicsphys1104/chapter/11-5-pascals-principle/","timestamp":"2024-11-02T18:13:13Z","content_type":"text/html","content_length":"164683","record_id":"<urn:uuid:3dbc319b-1b70-4f22-9b1a-9ae2f2de2f15>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00051.warc.gz"} |
Meiosis in context of specific growth rate
31 Aug 2024
Title: The Role of Meiosis in Shaping Specific Growth Rates: A Theoretical Framework
Abstract: Meiosis, a specialized type of cell division, plays a crucial role in the reproduction and evolution of eukaryotic organisms. In this article, we explore the relationship between meiosis
and specific growth rates, with a focus on the theoretical framework that underlies this connection.
Introduction: Specific growth rate (SGR) is a measure of the rate at which a population grows or declines over time. It is defined as the ratio of the instantaneous rate of change of the population
size to the current population size:
SGR = dN/dt / N
where N is the population size and t is time.
Meiosis, on the other hand, is a process that occurs in reproductive cells (gametes) and involves the reduction of the chromosome number by half. This process is essential for the production of
genetically diverse offspring.
Theoretical Framework: We propose that meiosis influences specific growth rates through two main mechanisms:
1. Genetic Variation: Meiosis introduces genetic variation into a population, which can lead to increased fitness and adaptability in certain environments. As a result, populations with higher
levels of genetic variation may exhibit faster growth rates.
2. Reproductive Success: Meiosis allows for the production of genetically diverse offspring, which can increase reproductive success in certain contexts. For example, if a population is faced with
environmental stressors that require specific adaptations, individuals with the “right” combination of genes may be more likely to survive and reproduce.
Mathematical Representation: We can represent these mechanisms mathematically using the following equations:
SGR = f(GV) * RV
where GV represents genetic variation, RV represents reproductive success, and f is a function that describes the relationship between GV and SGR.
GV = μ * (1 - e^(-λt))
where μ is the mutation rate, λ is the selection coefficient, and t is time.
RV = β * (1 + γ * GV)
where β represents reproductive success in the absence of genetic variation, and γ is a parameter that describes the effect of genetic variation on reproductive success.
Conclusion: In conclusion, meiosis plays a crucial role in shaping specific growth rates through the introduction of genetic variation and increased reproductive success. Our theoretical framework
provides a mathematical representation of these mechanisms and highlights the importance of considering meiosis when studying population dynamics. Further research is needed to fully understand the
complex relationships between meiosis, genetic variation, and specific growth rates.
Related articles for ‘specific growth rate’ :
• Reading: Meiosis in context of specific growth rate
Calculators for ‘specific growth rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/0a4ac425f14331f7bda550da1f420bdb/JSON_TO_ARTCL_Meiosis_in_context_of_specific_growth_rate.html","timestamp":"2024-11-04T21:46:12Z","content_type":"text/html","content_length":"15824","record_id":"<urn:uuid:f05a205b-8eb9-4f0f-bbbb-a0f89ab37d7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00718.warc.gz"} |
Python Logical Operators
The logical operators are used to combine the conditional statements. Let us check out this to learn more about the Python logical operators in Python.
What is the Logical Operators in Python?
In Python, Logical operators will function on the conditional statements. They will work on Logical AND, Logical OR, and Logical NOT operations.
What is Truth Table for Logical Operators in Python?
X Y X and Z X or Y not (X) not (Y)
T T T T F F
T F F F F T
F T T F T F
F F F F T T
What is the Logical AND operator in Python?
The Logical and AND operator will return True if both the operands are True otherwise, it will give the result as False.
# Python program to demonstrate
# logical and operator
a = 10
b = 10
c = -10
if a > 0 and b > 0:
print("The numbers are greater than 0")
if a > 0 and b > 0 and c > 0:
print("The numbers are greater than 0")
print("Atleast one number is not greater than 0")
The numbers are greater than 0
Atleast one number is not greater than 0
Example 2
# Python program to demonstrate
# logical and operator
a = 10
b = 12
c = 0
if a and b and c:
print("All the numbers have boolean value as True")
print("Atleast one number has boolean value as False")
Atleast one number has boolean value as False
What is the Logical OR operator in Python?
The logical OR operator will return a True value if the operands come true.
# Python program to demonstrate
# logical or operator
a = 10
b = -10
c = 0
if a > 0 or b > 0:
print("Either of the number is greater than 0")
print("No number is greater than 0")
if b > 0 or c > 0:
print("Either of the number is greater than 0")
print("No number is greater than 0")
Either of the number is greater than 0
No number is greater than 0
What is the Logical NOT operator in Python?
The logical not operator will function with a single boolean value. Hence, if the boolean value will come True then it will return as False and Vice versa.
# Python program to demonstrate
# logical not operator
a = 10
if not a:
print("Boolean value of a is True")
if not (a%3 == 0 or a%5 == 0):
print("10 is not divisible by either 3 or 5")
print("10 is divisible by either 3 or 5")
10 is divisible by either 3 or 5
What is Order of Precedence of Logical Operators?
Python will calculate the expression from left to right in multiple operators.
# Python program to demonstrate
# order of evaluation of logical
# operators
def order(x):
print("Method called for value:", x)
return True if x > 0 else False
a = order
b = order
c = order
if a(-1) or b(5) or c(10):
print("Atleast one of the number is positive")
Method called for value: -1
Method called for value: 5
Atleast one of the number is positive
To conclude, these Python logical operators provide a straightforward way to make decisions in your code. “AND” ensures both conditions must be true for the overall expression to be true, while “or”
requires at least one condition to be true.
On the other hand, “not” negates the truth value of a condition. Several examples are illustrated in this for more clarity on Logical And, Logical, Logical Not operators.
Python Logical Operators- FAQs
Q1. What are logic operators?
Ans. Logical operators work to compare the logical expressions that show results either as true or false.
Q2. What is called a logical AND operator?
Ans. This AND logical operator will check if both the operands are true.
Q3. What is the syntax of logical operators?
Ans.Logical OR operator: ||
Hello, I’m Hridhya Manoj. I’m passionate about technology and its ever-evolving landscape. With a deep love for writing and a curious mind, I enjoy translating complex concepts into understandable,
engaging content. Let’s explore the world of tech together
Leave a Comment | {"url":"https://www.skillvertex.com/blog/python-logical-operators/","timestamp":"2024-11-09T16:19:47Z","content_type":"text/html","content_length":"96171","record_id":"<urn:uuid:5a9f9f9c-b1e5-45b8-a362-d0ea700fe6be>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00098.warc.gz"} |
Determine which of the following polynomials has (x + 1) a factor: x³ + x² + x + 1
You must login to ask question.
NCERT Solutions for Class 9 Maths Chapter 2
Important NCERT Questions
NCERT Books for Session 2022-2023
CBSE Board and UP Board Others state Board
EXERCISE 2.4
Page No:43
Questions No:1 part:1 | {"url":"https://discussion.tiwariacademy.com/question/determine-which-of-the-following-polynomials-has-x-1-a-factor-x%C2%B3-x%C2%B2-x-1/?show=oldest","timestamp":"2024-11-14T04:02:32Z","content_type":"text/html","content_length":"88421","record_id":"<urn:uuid:304de8dc-1690-4d36-94e7-d15b6cb939aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00291.warc.gz"} |
Syk Inhibitors
Fig. Fig.5b5b shows the resulting bifurcation diagram when r=1. We have Z-shaped curve of screening libraries fixed points. For larger values of ��, there are three fixed points; the lower fixed
point is stable, the middle is a saddle, and the upper is unstable. As �� decreases, lower stable and middle saddle fixed points merge at a saddle-node bifurcation (labeled SN). There is also a
subcritical Hopf bifurcation point on the upper branch and fixed points become stable once passed this point (thick black). A branch of unstable periodic orbits (thin gray), which turn to stable
orbits (thick black), emanates from the Hopf bifurcation point, and becomes a saddle-node homoclinic orbit when ��=��SN. In fact, this bifurcation structure persists for each r on [0, 1].
We trace the saddle-node bifurcation point (SN) in the bifurcation diagram as r varies to get a two dimensional bifurcation diagram, which is shown in Fig. Fig.6a.6a. We call the resulting curve
��-curve (the curve in the (��, r) plane at Fig. Fig.6a).6a). The fast subsystem shows sustained spiking in the region left to �� (spiking region) and quiescence in the region right �� (silent
region). Note that if r is sufficiently small, then, we cannot get an oscillatory solution. Fig. Fig.6a6a also shows frequency curves (dependence of frequency of spikes on the total synaptic input ��
for different values of r) in the spiking region. Fig. Fig.6b6b provides another view of these curves. There is a band-like region of lower frequency along ��, visible in the frequency curve when r=
This band is more prominent along the lower part of �� and this will play an important role in the generation of overlapped spiking. Figure 6 The frequency of firing in dependence on the slow
variables �� and r. (a) ��-curve (gray line in the (��, r) plane) divides the space of the slow variables (��, r) into silent and sustained spiking regions. Over the sustained … Regular out-of-phase
bursting solutions in the phase plane of slow variables and linear stability under constant calcium level Fig. Fig.77 shows the two parameter bifurcation diagram with the projection of regular
2-spike out-of-phase bursting solution when gsyn=0.86. Without loss of generality, let��s assume that active cell is cell 2 and silent cell is cell 1. We will follow trajectories of both cells from
the moment when cell 2 fires its second spike.
Upper filled circle in Fig. Fig.77 denotes (��1, r1) of cell 1 and lower filled circle denotes (��2, r2) of cell 2 at this moment. Figure 7 Two-parameter bifurcation diagram with projection
Anacetrapib of 2-spike out-of-phase bursting solution. The close-to-vertical curve in the middle of the figure is the ��-curve shown in Fig. Fig.66 when [Ca]=0.7. The moment when active … First note
that synaptic variable s of a cell rises once membrane potential rises, passes certain threshold (��g), and stays above it; s decreases otherwise (Eq. 4).
This is in contrast to the standard notion of essentiality, which
This is in contrast to the standard notion of essentiality, which is assigned to a gene or reaction whose single knockout abolishes a phenotype. k-essential links between genes/reactions and STI 571
systems-level functions arise from synergistic epistasis between parallel pathways in the network. Complex MCSs found using our method yield many k-essential reactions. To quantify novel k-essential
links between reactions and objectives, we compared the numbers of k-essential reactions to the number of 1-essential reactions obtained from a brute-force single knockout analysis of the human
metabolic network. Figure Figure44 shows how many reactions were deemed k-essential for each objective, with the numbers of reactions shown to be 1-essential for the objective shown in parentheses
next to the metabolite label.
We found that for most objectives we were able to associate many more k-essential reactions with the production of a given metabolite than were able to be found using a single knockout analysis. In
many cases, this difference was profound, such as for sphingomyelin, whose producibility we were able to epistatically link to 235 reactions in the metabolic network. Figure 4 Histogram showing
number of k-essential reactions discovered for each biosynthetic objective tested in our study. A reaction is k-essential for an objective if it contributes to at least one MCS for that objective.
The number of reactions found to be … MCSs span multiple compartments and metabolic subsystems MCSs discovered by our analysis span a breadth of cellular compartments.
However, the actual distributions of compartment span vary distinctly between specific metabolite classes (Fig. (Fig.5).5). In particular, amino acid-targeting MCSs discovered by our method employ
the fewest number of compartments, drawing from cytoplasmic fluxes alone or a combination of cytoplasmic and mitochondrial reactions. MCSs targeting core metabolites span between two and three
compartments, consisting of primarily cytoplasmic and mitochondrial reactions, however often also employing peroxisomal fluxes. Nucleotide-targeting MCSs sometimes employ cytoplasmic reactions only,
however more often pull combinations of reactions from two or three of the following compartments: cytoplasm, mitochondria, lysosome, and nucleus.
Across all metabolite classes studied, membrane-lipid-targeting MCSs are the most diverse: they harness up to five compartment combinations that employ reactions Dacomitinib from the cytoplasm,
endoplasmic reticulum, Golgi apparatus, nucleus, and peroxisome. Figure 5 Histogram showing number of compartments spanned by MCSs targeting the four metabolite classes. Frequencies are calibrated
separately for each metabolite class. There are also metabolite class differences in the subsystem span of discovered MCSs (Fig. (Fig.6).6). Nucleotide and amino acid-targeting MCSs span between one
and five subsystems.
Discrepancies of this type generally become more prevalent for sh
Discrepancies of this type generally become more prevalent for shorter loop lengths, where the attractor periods are short enough that nodes do not have time to rise to their saturation Nutlin-3a
mechanism values. Previous studies have emphasized the need for long time delays in regulatory oscillators. In the Elowitz-Leibler model of the repressilator (which is a frustration oscillator),
protein creation and degradation equations were added to the system in order to capture the oscillatory dynamics.2 From our present perspective, the protein dynamics simply serves to lengthen the
delay time for propagation of a pulse around the loop enough to allow elements to vary with sufficient amplitude. The explicit representation of protein variables is not necessary if the loop is made
longer. Norrell et al.
studied a different mechanism for lengthening the loop propagation times: inserting explicit delays into the differential equations.11 Using a slightly different form for fA and fR, they studied
frustration oscillations and pulse transmission oscillations, but did not address the distinct possibility of dip transmission oscillations. Finally, it is worth emphasizing that the distinction
between pulse transmission and dip transmission is not simply a matter of symmetry; that is, the dip transmission oscillations are not just pulse transmission oscillations with the on and off states
exchanged. If that were the case, we would have a dip that grows in width as it traverses the positive loop, but Figure Figure55 clearly shows that it is pulses (not dips) that grow in the dip
transmission oscillator.
The on-off symmetry is broken by the Hill function forms for fA and fR, but this is merely a quantitative effect that determines the parameter domains where oscillation is possible. The more
important symmetry breaking in the figure-8 system is the logic function for the two-input element A. If the default state (with both inputs off) were taken to yield A=1 and the activating input were
dominant, we could obtain oscillations in cases where dips grow rather than pulses. The language becomes a bit cumbersome: it might be best to refer to these cases as ��anti-pulse transmission�� and
��anti-dip transmission�� oscillations. Figure Figure88 shows an anti-pulse transmission oscillator, where the ODE system is the same as above except that Eq.
7 is replaced by A�B=(1?fr(Bn;?KBn)fa(Cm;?KCm))?A,? (12) and parameter values are given in Figure Figure88. Figure 8 An attractor showing anti-pulse transmission oscillations. Drug_discovery The
parameter values are n=9,?m=2,?��=5,?KBn=0.55,?KCm=0.5,KAB=0.52,?KAC=0.55. Top: The thick line shows A; the thin line Bn; and the dashed line … CONCLUSIONS This study serves to illustrate a sense in
which ABN modeling can be used to identify distinct classes of oscillatory solutions of ODE systems of a type often used to model activating and repressing regulatory interactions.
Using a right common femoral artery approach a diagnostic flush a
Using a right common femoral artery approach a diagnostic flush aortogram was performed to exclude extrarenal feeders Ganetespib Phase 3 to the tumor. A selective catheterization of the upper and
lower pole left renal artery revealed that the upper renal artery was exclusively supplying the renal parenchyma not affected by the AML with no significant feeding of the tumor (Fig. 3) whereas the
lower renal artery solely supplied the giant AML (Fig. 4). The diameter of the lower left artery was 6.5 mm. Embolization of the tumor-feeding lower left renal artery was performed with an 8-mm
Amplatzer Vascular Plug (AVP; AGA Medical, Golden Valley, MN, USA). The AVP was deployed through a long 6-F envoy-guiding catheter (Codman & Shurtleff, Raynham, MA, USA) with 0.070�� ID (1.8 mm).
An instant and complete occlusion of the lower left renal artery was achieved (Fig. 5). Fig. 3 Selective angiogram of the left upper renal artery supplying approximately two-thirds of the regular
renal parenchyma. There are no significant feeders to the angiomyolipoma Fig. 4 Selective angiogram of the left lower renal artery which is exclusively supplying the angiomyolipoma tumor mass Fig. 5
Implantation of an Amplatzer Vascular Plug Type II in the left lower renal artery. There is an abrupt and complete occlusion of the AML supplying vessel Immediately after embolization the patient
complained of left-sided abdominal pain, which was treated with a single dose of 50 mg pethidine i.v. As a consequence of tumor devascularization the patient developed post-embolization syndrome
characterized by acute pain, malaise, nausea, severe night sweats, and temperatures of up to 39��C 10 days following the procedure.
A follow-up CT scan showed necrosis of AML with signs of abscess formation (Fig. 6) 14 days post embolization. A nephron-sparing surgical resection of the residual AML was performed, preserving the
healthy upper pole of the left kidney, which was supplied by the separate upper renal artery. The patient was discharged from hospital 4 days later. Fig. 6 Coronal view of the CT demonstrates an
extended necrosis (large white arrows) of the angiomyolipoma tumor mass 10 days after the selective arterial embolization. The air bubbles are indicative for an abscess formation (small white arrows)
Discussion Predictive factors for bleeding complications in patients with renal AML are tumor size (10), presence of symptoms (11), and presence of tuberous sclerosis (4).
Different Brefeldin_A embolization techniques for the treatment of AML have been described. The ultimate goal of every SAE is to achieve complete tumor devascularization and to preserve healthy renal
parenchyma. Ramon et al. utilized a mixture of 20 mL ethanol and 1 mL (one bottle) of 45�C150 ��m PVA particles for SAE (10). Lee et al. describe a superselective approach using a coaxial
microcatheter: First, the targeted tumor vessel was tapped with microcoils (12).
Mean power of the propulsive phase was assessed for each load (cf
Mean power of the propulsive phase was assessed for each load (cf. figure 1) and maximum value obtained was registered for each test: squat (MPPsq); bench press (MPPbp) and lat pull down back
(MPPlpd). Figure 1 Load-power selleck chem inhibitor relationships for one representative subject, for each test. Statistical analysis Standard statistical methods were used for the calculation of
means and standard deviations (SD) from all dependent variables. The Shapiro-Wilk test was applied to determine the nature of the data distribution. Since the reduce sample size (N < 30) and the
rejection of the null hypothesis in the normality assessment, non-parametric procedures were adopted. Spearman correlation coefficients (��) were calculated between in water and dry land parameters
assessed. Significance was accepted at the p<0.
05 level. Results The mean �� SD value for the 50 m sprint test was 1.69 �� 0.04 m.s?1. The mean �� SD values of mean force production in tethered swimming tests were 95.16 �� 11.66 N for whole body;
80.33 �� 11.58 N for arms only; and 33.63 �� 7.53 N for legs only. The height assessed in the CMJ was 0.37 �� 0.05 m, being calculated the correspondent work of 219.30 �� 33.16 J. The maximum mean
propulsive power in the squat, bench press and lat pull down back were 381.76 �� 49.70 W; 221.77 �� 58.57; and 271.30 �� 47.60 W, respectively. The Table 1 presents the correlation coefficients (��)
between swimming velocities and average force in tethered tests with dry land variables assessed. It was found significant associations between in water and dry land tests.
Concerning the CMJ, work during the jump revealed to be more associated with in water variables, than the height. Both tests that involve the lower limbs musculature (CMJ and squat) presented
significant relationship with force production in water with the whole body and legs only, but not with swimming velocity. In bench press and lat pull down back, significant correlations were
observed with force production in water with the whole body and arms only, and with swimming velocity for the lat pull down back. Added to that, in the tethered swimming tests, arms only presented a
moderate correlation with swimming performance (�� = 0.68, p = 0.03). Table 1 Correlation coefficients (��) between in water and dry land tests variables Discussion The aim of this study was to
analyze the associations between dry land and in water tests.
The mean power of the propulsive phase in the lat pull down back was the only parameter that correlated significantly with swimming performance. Additionally, there were significant associations
between dry land tests and force exerted in water through tethered swimming. Concerning in water tests, velocity and mean force in tethered swimming seem to present descriptive data similar to other
papers in the literature for the same age and gender (Rohrs and Stager, 1991; GSK-3 Taylor et al., 2003b).
A training program is the expression of an ordered
A training program is the expression of an ordered leave a message sequence or series of efforts that have a dependency relationship to each other. Since we have used the term ��effort�� we must move
ahead to define it. The meaning of this term must be understood in the sense of the actual degree of demand in relation to the current possibilities of a given subject. We call this ��level of effort
(LE)�� (Gonz��lez Badillo and Gorostiaga, 1993, 1995). Therefore, when we talk about strength or resistance training, the nature of the effort will be best defined by the number of repetitions
actually performed in each exercise set with respect to the maximum possible number of repetitions that can be completed against a given absolute load. It thus seems reasonable that the degree or
level of effort is substantially different when performing, e.
g., eight out of twelve possible repetitions with a given load [8(12)] compared to performing all repetitions [12(12)]. Configuration of the exercise stimulus in resistance training mainly depends on
the manipulation of three variables: type of exercise, volume and intensity. Once the exercises have been selected, the training load will be defined by the manipulation of volume and intensity. Of
these two, the latter is the most important since it is the intensity which determines the amount of volume (number of repetitions) that can be performed. Furthermore, exercise intensity is generally
acknowledged as the most important stimulus related to changes in strength levels. It is for these reasons that we will focus on the study of training intensity in the following paragraphs.
Exercise intensity during resistance training has been commonly identified with relative load (percentage of one-repetition maximum, 1RM) or with performing a given maximal number of repetitions in
each set (XRM: 5RM, 10RM, 15 RM, etc.). However, for several reasons, none of these methods is entirely appropriate for precisely monitoring the real effort the athlete is performing in each training
session. The first approach requires coaches to individually assess the 1RM value for each athlete. It is true that expressing intensity as a percentage of the maximum repetition has the advantage
that it can be used to program resistance training for many diferent athletes at the same time, the loads being later transformed in absolute values (kg) for each person.
Another advantage is that this expression of the intensity can clearly reflect the dynamics of the evolution of the training load if we understand the percentage of 1RM as an effort, Cilengitide and
not as a simple arithmetic calculus. This would yield valuable information about the type of training being prescribed. Direct assessment of 1RM, however, has some potential disadvantages worth
noting. It may be associated with injury when performed incorrectly or by novice subjects and it is time-consuming and impractical for large groups. | {"url":"https://sykinhibitors.com/index.php/2016/04/","timestamp":"2024-11-14T04:43:08Z","content_type":"text/html","content_length":"57025","record_id":"<urn:uuid:788764e9-8e18-4071-b7ec-e7187f4c394d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00812.warc.gz"} |
Return on Invested Capital ROIC What’s It Worth?
Companies that rely on the wrong benchmark can overlook good investments or pursue bad ones. In contrast, ROCE is calculated using operating income generated prior to interest and tax payments. ROIC
generally is a bit more complicated to calculate compared to ROCE as there are several ways to calculate invested capital.
1. Generally, the higher the return on invested capital (ROIC), the more likely the company is to achieve sustainable long-term value creation.
2. The formula for ROI is the profit from the investment divided by the cost of the investment.
3. The reason the ROIC concept tends to be prioritized by value investors is that most investors purchase shares under the mindset of a long-term holding period.
4. For example, a 12% ROIC tells you that for every dollar you might put in a company, you would receive 12 cents in income.
ROIC measures the efficiency of total capital invested, while ROCE measures the efficiency of business operations. They are much suited for companies in capital-intensive industries such as
telecommunication, energy, and automotive. This tries to get to the operating results and growth of the core business based on reinvested capital. Return on Capital Employed relates a company’s net
operating income to its capital employed and can also provide insight into return on investment by showing the company’s overall financial health. In general, the higher the return percentage, the
more profitable the investment since the relatively high percentage shows that the company’s use of the invested capital can generate dividends for the investors.
There’s a few other reasons that a company’s ROI formulas, particularly ROCE, can depart from the real relationship between a company’s growth and return on reinvestments. If you were to measure the
difference in earnings from year-to-year, you’d see that 6% growth rate which corresponds with the 6% ROI/ROCE/ROIC. Remember this is a simplistic example, and it depends on a company reinvesting all
of its profits moving forward.
What Is Return on Capital Employed (ROCE)?
Another important use of the ROIC is to compare it to the same company’s weighted average cost of capital (WACC) — a weighted measure of the cost of capital provided by shareholders and debtholders.
The return on invested capital (ROIC) lets the company and other stakeholders estimate how much profit the company is creating for every dollar of invested capital. It tells us how well a company
uses its capital and whether it is creating value with its investments.
Return on Invested Capital Formula (ROIC)
When ROCE is below the cost of capital or the ROIC is negative, it shows that the company has not used invested capital effectively. Finally, we found that the median or mean returns of general,
broadly defined industry groups can be downright misleading. ROCE can be used to track a company’s capital efficiency over time as well as in comparison with other firms, either in its own industry
roce and roic or across industries. Keep in mind, however, that a high ROCE in one industry might be considered low in another. The tracking of ROIC over time through both rates allows the company to
better understand how it can “tweak” its operations to make more efficient use of its capital. A good ROIC is typically one that exceeds the company’s weighted average cost of capital (WACC) by at
least 2%.
How to Interpret ROIC?
The differences between the ROCE and ROA ratios are not many, but they are significant. In contrast, ROCE considers all funding sources for capital both debt and equity financing. ROCE also focuses
on earnings before interest and taxes, rather than after-tax profits.
ROIC Calculator
The ratios can also help in the comparison between different ventures to determine the venture with the highest returns possible. ROE is the percentage expression of a company’s net income, as it is
returned as value to shareholders. ROCE is the amount of profit a company generates for each dollar of capital employed in the business.
This indicates better financial performance for those companies which have significant debt. The trend of ROCE over the years is also an important indicator of a company’s performance. Investors
trust those companies which have stable and growing ROCE over those companies whose ROCE is volatile. Let’s understand ROCE in a better way with the help of the following example
Suppose there are two companies, X and Y, X has a profit margin of 20% and Y has a profit margin of 25%.
Suppose we’re tasked with calculating the return on invested capital (ROIC) of a company with the following financial profile as of Year 0. The complications with invested capital (IC) arise for
intangible-intensive industries, in which the intangible assets belonging to the companies that operate in the industry are not recognized yet. The alternative, simpler method to calculate the
invested capital is to add the net debt (i.e. subtract cash and cash equivalents from the gross debt amount) and equity values from the balance sheet. Since the return metric is presented in the form
of a percentage, the metric can be used to assess a company’s profitability as well as make comparisons to peer companies. Company Y pays an annual interest of Rs. 20 on its debt, and the tax rate is
ROIC vs ROCE: When to Use One Over the Other [Pros & Cons]
Andrew has always believed that average investors have so much potential to build wealth, through the power of patience, a long-term mindset, and compound interest. But, using this shortcut for ROIC
does have its potential weaknesses, as we saw with certain long term liabilities such as pension liabilities. You can see how it does boost UPS’s ROIC, even though these pension liabilities may not
really have had any impact whatsoever to the capital reinvested by the company lately. From a business perspective, ROCE could offer a more accurate view of the company’s overall health as it focuses
on profitability. Because cash on hand can reflect a company’s overall security, this amount is included in a company’s employed capital.
This can help neutralize financial performance analysis for companies with significant debt. ROCE, ROIC, ROA, and ROE are all return-based financial ratios that https://1investing.in/ measure a
company’s profitability and efficiency. They can be compared with bank rates and inflation to determine whether a company is performing well or not.
Investors can use these ratios to gain insights into a company’s ability to generate profits and how well it is utilizing its assets and capital to do so. They can analyze these ratios along with
other financial and non-financial metrics to gain a holistic view of a company’s performance and future prospects. Regarding this equation, net income is comprised of what is earned throughout a
year, minus all costs and expenses. It includes payouts made to preferred stockholders but not dividends paid to common stockholders (and the shareholders’ overall equity value excludes preferred
stock shares).
Both ratios inform investors how a company is performing, and how much of the net reported profits are returned to investors as dividends. The ratios also inform the investors how the company uses
its invested capital, as well as its ability to generate additional revenues in the future. Return on capital employed (ROCE) and return on assets (ROA) are two similar profitability ratios investors
and analysts use to evaluate companies. The ROCE ratio is a metric that evaluates how efficiently a company’s available capital is utilized. ROIC provides the necessary context for other metrics such
as the price-to-earnings (P/E) ratio.
Leave a Comment | {"url":"https://microntqm.ro/2023/11/20/return-on-invested-capital-roic-what-s-it-worth-2/","timestamp":"2024-11-04T02:09:45Z","content_type":"text/html","content_length":"176429","record_id":"<urn:uuid:4b77bb1d-0f78-4444-ac88-5d41d84ea3de>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00872.warc.gz"} |
11-785 Introduction to Deep Learning - Spring 2020
“Deep Learning” systems, typified by deep neural networks, are increasingly taking over all AI tasks, ranging from language understanding, and speech and image recognition, to machine translation,
planning, and even game playing and autonomous driving. As a result, expertise in deep learning is fast changing from an esoteric desirable to a mandatory prerequisite in many advanced academic
settings, and a large advantage in the industrial job market.
In this course we will learn about the basics of deep neural networks, and their applications to various AI tasks. By the end of the course, it is expected that students will have significant
familiarity with the subject, and be able to apply Deep Learning to a variety of tasks. They will also be positioned to understand much of the current literature on the topic and extend their
knowledge through further study.
If you are only interested in the lectures, you can watch them on the YouTube channel listed below.
Course description from student point of view
The course is well rounded in terms of concepts. It helps us understand the fundamentals of Deep Learning. The course starts off gradually with MLPs and it progresses into the more complicated
concepts such as attention and sequence-to-sequence models. We get a complete hands on with PyTorch which is very important to implement Deep Learning models. As a student, you will learn the tools
required for building Deep Learning models. The homeworks usually have 2 components which is Autolab and Kaggle. The Kaggle components allow us to explore multiple architectures and understand how to
fine-tune and continuously improve models. The task for all the homeworks were similar and it was interesting to learn how the same task can be solved using multiple Deep Learning approaches.
Overall, at the end of this course you will be confident enough to build and tune Deep Learning models.
Course description from student point of view
The course is well rounded in terms of concepts. It helps us understand the fundamentals of Deep Learning. The course starts off gradually with MLPs and it progresses into the more complicated
concepts such as attention and sequence-to-sequence models. We get a complete hands on with PyTorch which is very important to implement Deep Learning models. As a student, you will learn the tools
required for building Deep Learning models. The homeworks usually have 2 components which is Autolab and Kaggle. The Kaggle components allow us to explore multiple architectures and understand how to
fine-tune and continuously improve models. The task for all the homeworks were similar and it was interesting to learn how the same task can be solved using multiple Deep Learning approaches.
Overall, at the end of this course you will be confident enough to build and tune Deep Learning models. | {"url":"https://cogak.com/course/22","timestamp":"2024-11-02T17:24:29Z","content_type":"text/html","content_length":"180700","record_id":"<urn:uuid:0f157939-4105-4350-a11f-eed06f236af2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00053.warc.gz"} |