text
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k} \to K(\Z/4,1)$ such that the characteristic mapping $\zeta: L^{n-4k} \to K(\Z/2 \int \D/4,1)$ of the $\Z/2
\int \D_4$-framing of the normal bundle over $L^{n-4k}$ is reduced to a mapping with the target $K(\I_b,1)$ such that the following equation holds: $$\zeta = i(\kappa_a \oplus \mu_a),$$ where $i: \Z/2
\oplus \Z/4 \to \I_4$ is the prescribed isomorphism.
The following Proposition is proved by a straightforward calculation.
### Proposition 2 {#proposition-2 .unnumbered}
Let $(g,\Psi_N, \eta)$ be a $\D_4$–framed immersion, that is a cyclic immersion. Then the Kervaire invariant, appearing as the top line of the diagram (7), can be calculated by following formula: $$\Theta_a=<\kappa_a^{\frac{n-4k}{2}}\mu_a^{\ast}(\tau)^{\frac{n-4k-2}{4}}
\mu_a^{\ast}(\rho);[L]>, \eqno(8)$$ where $\tau \in
H^2(\Z/4;\Z/2)$, $\rho \in H^1(\Z/4;\Z/2)$ are the generators. $$$$
### Proof of Proposition 2 {#proof-of-proposition-2 .unnumbered}
Let us consider the subgroup of index 2, $\I_b \subset \I_4$. This subgroup is the kernel of the epimorphism $\chi':
\I_4 \to \Z/2$, that is the restriction of the characteristic class $\chi: \Z/2 \int \D_4 \to \Z/2$ of the canonical double cover $\bar L \to L$ to the subgroup $\I_4 \subset \Z/2 \int
\D_4$. Obviously, the characteristic number (8) is calculated by the formula $$\Theta_a=<\hat \kappa_a^{\frac{n-4k}{2}} \hat
\rho_a^{\frac{n-4k}{2}};\bar L>, \eqno(9)$$ where the characteristic class $\hat \kappa_a \in H^1(\bar L;\Z/2)$ is induced from the class $\kappa_a \in H^1(L;\Z/2)$ by the canonical cover $\bar L \to L$, and the class $\hat \rho_a \in H^1(\bar
L;\Z/2)$ is obtained by the transfer of the class $\rho \in
H^1(L;\Z/4)$.
Note that $\hat \kappa_a = \tau_1$, $\hat \rho_a = \tau_2$, where $\tau_1$, $\tau_2$ are the two generating $\I_b$–characteristic classes. Therefore $\hat \kappa_a \hat \rho_a = \tau_1 \tau_2 =
w_2(\eta)$, where $\eta$ is the two-dimensional bundle that determines the $\D_4$–framing (over the submanifold $\bar
L^{n-4k} \subset N^{n-2k}$ this framing ad
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\lVert x-y\rVert_2.
$
2. $U$ has a stationary point at zero: $\nabla U(0) = 0.$
3. There exists a constant $m>0, \LR, R$ such that for all $\lrn{x-y}_2 \geq R$, $$\label{e:convexity_outside_ball}
\lin{ \nabla U(x) - \nabla U(y),x-y} \geq m \lrn{x-y}_2^2.$$ and for all $\lrn{x-y}_2 \leq R$, $
\lVert \nabla U(x) - \nabla U(y) \rVert_2 \le \LR \lVert x-y\rVert_2.
$
The assumption in roughly states that “$U(x)$ is convex outside a ball of radius $R$”. This assumption, and minor variants, is common in the nonconvex sampling literature [@eberle2011reflection; @eberle2016reflection; @cheng2018sharp; @ma2018sampling; @erdogdu2018global; @gorham2016measuring].
\[ass:xi\_properties\] We make the following assumptions on $\xi$ and $M$:
1. For all $x$, $\E{\xi(x,\eta)}=0$.
2. For all $x$, $\lrn{\xi(x,\eta)}_2 \leq \beta$ almost surely.
3. For all $x,y$, $\lrn{\xi(x,\eta) - \xi(y,\eta)}_2\leq L_{\xi} \lrn{x-y}_2$ almost surely.
4. There is a positive constant $\cm$ such that for all $x$, $2 \cm I \prec M(x)$.
We discuss these assumptions in a specific setting in Section \[ss:example\_ass\].
For convenience we define a matrix-valued function $N(\cdot) : \Re^d \to \Re^{d\times d}$: $$\begin{aligned}
N(x) := \sqrt{M(x)^2 - \cm^2 I}.
\numberthis
\label{d:N}
\end{aligned}$$ Under Assumption \[ass:U\_properties\], we can prove that $N(x)$ and $M(x)$ are bounded and Lipschitz (see Lemma \[l:M\_is\_regular\] and \[l:N\_is\_regular\] in Appendix \[ss:mnregularity\]). These properties will be crucial in ensuring convergence. Given an arbitrary sample space $\Omega$ and any two distribution $p\in \Pspace\lrp{\Omega}$ and $q\in \Pspace\lrp{\Omega}$, a joint distribution $\zeta \in \Pspace\lrp{\Omega \times \Omega}$ is a *coupling* between $p$ and $q$ if its marginals are equal to $p$ and $q$ respectively.
For a matrix, we use $\lrn{G}_2$ to denote the operator norm: $
\lrn{G}_2 = \sup_{v\in \Re
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|
her process such as mantle cooling may be necessary for Earth-like plate tectonics. Understanding the type of tectonics that might take place on a planet before plate tectonics, how much land and weatherable rock can be created through non-plate-tectonic volcanism, and the factors that then allow for Earth-like plate tectonics to develop, will all be crucial for determining how likely planets are to follow an evolutionary trajectory similar to Earth’s.
Acknowledgements
================
BF acknowledges funding from the NASA Astrobiology Institute under cooperative agreement NNA09DA81A. We thank Norm Sleep for a thorough and constructive review that helped to significantly improve the manuscript. No new data was used in producing this paper; all information shown is either obtained by solving the equations presented in the paper or is included in papers cited and listed in the references.
[245]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , & (). . , [ ** ]{}, . , , , & (). . , [ ** ]{}, . , , , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [**]{}. , , , , , , , , , et al. (). . , [**]{}, . (). . , [ ** ]{}, . (). . , [ ** ]{}, . (). . , [ ** ]{}, . , & (). . In , & (Eds.), [**]{} chapter . (pp. ). : volume . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [**]{}, . , , & (). . In , & (Eds.), [**]{} (pp. ). : volume . (). . . (). . , [ ** ]{}, . (). . , [ ** ]{}, . , & (). . , [**]{}, . , , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , , , , & (). . In [**]{} (pp. ). volume . , & (). . In , , & (Eds.), [**]{} (pp. ). : volume . , & (). . In (Ed.), [**]{} (pp. ). : volume . (). . , [ ** ]{}, . , , & (). . , [ ** ]{}, . (). , [ ** ]{}, . , , & (). . , [**]{}, . , & (). . , [ ** ]{}, . (). . , [**]{}, . (). . , [ ** ]{}, . , , & (). . , [**]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . , & (). . , [ ** ]{}, . (). . , [**]{},
| 503
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|
letting $d_i'=d_i=0$, we have $$\label{ea16}
d_i'=\pi(x_i'+ z_i'+w_i')=0.$$ This is an equation in $B\otimes_AR$. Thus there is exactly one independent linear equation among $x_i', z_i', w_i'$.
5. The $(2\times 2)$-block is $$1+2 f_i'=1+2 f_i+2\pi(u_i'+\pi x_i').$$ By letting $f_i'=f_i=0$, we have $$\label{ea17}
f_i'=f_i+\pi(u_i'+\pi x_i')=0. $$ This is an equation in $R$. Thus this equation is trivial.
6. We postpone to consider the $(3, 3)$-block later in Step (vi).\
By combining all five cases (a)-(e), there are exactly $((n_i-2)^2-(n_i-2))/2+2(n_i-2)+1=(n_i^2-n_i)/2$ independent linear equations. Thus $n_i^2-1-(n_i^2-n_i)/2=n_i(n_i+1)/2-1$ entries of $m_{i,i}'$ determine all entries of $m_{i,i}'$ except for $z_i'$ and $(z_i^{\ast})'$.\
6. Let $i$ be even and $L_i$ be *of type I*. Finally, we consider the $(2, 2)$-block of Equation (\[ea12\]) if $L_i$ is *of type $I^o$* or the $(3, 3)$-block of Equation (\[ea13\]) if $L_i$ is *of type $I^e$* given below: $$\label{ea18}
2\bar{\gamma}_i+4c_i'=2\bar{\gamma}_i+4c_i+ 2\pi^2z_i'+\pi^4(z_i')^2-\pi^4\delta_{i-2}k_{i-2, i}'-\pi^4\delta_{i+2}k_{i+2, i}'.$$ Here, $k_{i-2, i}'$ and $k_{i+2, i}'$ are as explained in the condition (5) given at the paragraph following (\[ea2\]). Note that the condition (5) yields Equation (\[52\]), $z_i'+\delta_{i-2}k_{i-2, i}'+\delta_{i+2}k_{i+2, i}'=0$, in $R$. Thus, by letting $c_i'=c_i=0$, we have $$\label{ea19}
0=c_i'=c_i+z_i'=z_i'$$ as an equation in $R$.
Note that if both $L_i$ and $L_{i+2}$ are *of type I*, then $k_{i+2, i}'=k_{i,i+2}'$ by Equation (\[ea4\]). We now choose an even integer $j$ such that $L_j$ is *of type I* and $L_{j+2}$ is *of type II*. For such $j$, there is a nonnegative integer $m_j$ such that $L_{j-2l}$ is *of type I* for every $l$ with $0\leq l \leq m_j$ and $L_{j-2(m_j+1)}$ is *of type II*. As mentioned in the above paragraph, the condition (5) yields the following equation in $R$: $$\mathcal{Z}_{j-2l}' : z_{j-2l}'+\delta_{j-2l-2}k_{j-2l-2, j-2l}'+\delta_{j-2l+
| 504
| 771
| 558
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| 2,272
| 0.78125
|
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|
H38B 0.6446 0.0288 0.4212 0.016\*
C39 0.6064 (2) 0.1611 (3) 0.49084 (16) 0.0119 (8)
C40 0.5981 (2) 0.2566 (3) 0.50130 (18) 0.0187 (10)
H40 0.6324 0.2970 0.5143 0.022\*
C41 0.5393 (2) 0.2920 (4) 0.4926 (2) 0.0233 (11)
H41 0.5335 0.3568 0.4998 0.028\*
C42 0.4894 (2) 0.2340 (4) 0.4736 (2) 0.0248 (11)
H42 0.4494 0.2594 0.4672 0.030\*
C43 0.4970 (2) 0.1386 (4) 0.46372 (18) 0.0193 (10)
H43 0.4624 0.0985 0.4513 0.023\*
C44 0.5557 (2) 0.1020 (3) 0.47208 (17) 0.0159 (9)
H44 0.5612 0.0370 0.4650 0.019\*
C45 0.6863 (2) 0.0014 (3) 0.52730 (17) 0.0144 (9)
C46 0.7188 (2) −0.0738 (3) 0.51465 (18) 0.0198 (10)
H46 0.7371 −0.0662 0.4885 0.024\*
C47 0.7243 (3) −0.1603 (4) 0.54060 (19) 0.0254 (11)
H47 0.7460 −0.2117 0.5318 0.030\*
C48 0.6979 (2) −0.1708 (4) 0.57924 (19) 0.0245 (11)
H48 0.7016 −0.2295 0.5968 0.029\*
C49 0.6666 (2) −0.0966 (4) 0.59207 (19) 0.0246 (11)
H49 0.6486 −0.1044 0.6185 0.030\*
C50 0.6608 (2) −0.0102 (4) 0.56679 (18) 0.0204 (10)
H50 0.6395 0.0410 0.5763 0.024\*
C51 0.9045 (2) 0.0612 (3) 0.58746 (16) 0.0156 (9)
C52 0.8728 (3) −0.0241 (4) 0.5823 (2) 0.0297 (12)
H52 0.8345 −0.0265 0.5890
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|
cup H_x\subset H$. From the classification of Lie subgroups of $\PSL(2,\Bbb{C})$, we deduce that $H$ is conjugate to a subgroup of $Rot_\infty$. Hence $H_y=H_x$ and so $x=y$.
\[t:rf\] Let $\Gamma\subset \PSL(2,\Bbb{C})$ be a discrete group. Then $\Gamma$ is conjugate to a subgroup $\Sigma$ of $\PSL(2,\Bbb{R})$ such that $ \Bbb{H}/\Sigma$ is a compact Riemann surface if and only if $\iota\Gamma$ is conjugate to a discrete compact surface group of $\PO^+(2,1)$.
Let $\Gamma\subset \PSL(2,\Bbb{R})$ be a subgroup acting properly, discontinuously, freely, and with compact quotient on $\Bbb{H}^+$. Let $R$ be a fundamental region for the action of $\Gamma$ on $\Bbb{H}^+$. We may assume without loss of generality that $\psi(R)\subset\Bbb{H}^2_{\Bbb{C}}$. Thus $\Pi\psi\overline{R} $ is a compact subset of $\Bbb{H}^2_\Bbb{R}$ satisfying $\iota\Gamma\Pi\psi\overline{R}=\Bbb{H}^2_{\Bbb{R}}$ which shows that $\iota\Gamma$ is a discrete compact surface group of $\PO^+(2,1)$.\
Now let us assume that $\iota\Gamma$ is a discrete compact surface group of $\PO^+(2,1)$. Then $\Gamma\subset\PSL(2,\Bbb{R})$. Thus $\iota\Gamma\subset\PO^+(2,1)$ and $\Bbb{H}^2_{\Bbb{R}}/\iota\Gamma$ is a compact surface, see [@tengren]. Now, consider the following commutative diagram
$$\xymatrix{
\Bbb{H}_\Bbb{R}^2 \ar[r]^{\Pi^{-1}} \ar
[d]^{q_1} & Ver\cap \Bbb{H}^2_\Bbb{C} \ar[d]^{q_2} \ar[r]^{\psi^{-1}} & \Bbb{H}^+\ar[d]^{q_3}\\
\Bbb{H}^2_{\Bbb{R}}/\iota
\Gamma
\ar[r]^{\widetilde {\Pi}} &
(Ver\cap \Bbb{H}^2_\Bbb{C})/ \iota\Gamma\ar[r]^{\widetilde \psi} &
\Bbb{H}^+/\Gamma
}$$
where $q_1,q_2,q_3$ are the quotient maps, $\widetilde {\Pi}(x)=q_2\Pi^{-1}q_1^{-1}x$, and $\widetilde\psi(x)=q_3\psi^{-1}q_2^{-1}(x)$. By Lemma \[l:prv\], we conclude that $\Bbb{H}^2_{\Bbb{R}}/\iota\Gamma,(Ver\cap \Bbb{H}^2_\Bbb{C})/\iota\Gamma,\Bbb{H}^+/\Gamma$ are homeomorphic compact surfaces, which concludes the proof.
Proof of theorem \[t:main2\] {#proof-of-theorem-tmain2 .unnumbered}
============================
If $\Gamma\subset\PO^+(2,1)$ is a discrete compa
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|
ddots}\pushoutcorner&0\ar@{}[dr]|{\ddots}\pushoutcorner&\\
&&&&&&&&
}
}$$
While $\pi^\ast$ points at the constant morphism in the middle of , for every $n$ the remaining $2n$-th adjoints to $\pi^\ast$ classify suitable iterated rotations of this morphism.
[^1]: In a weak sense; see below.
---
abstract: 'It is shown how initial conditions can be appropriately defined for the integration of Lorentz-Dirac equations of motion. The integration is performed [*forward*]{} in time. The theory is applied to the case of the motion of an electron in an intense laser pulse, relevant to nonlinear Compton scattering.'
address:
- |
$^1$Instituto de Física, Universidade de São Paulo,\
C.P.66318, 05315-970, São Paulo, SP, Brazil
- |
$^2$Institute for Theoretical Physics, University of California,\
Santa Barbara, 93106-4030, USA
- |
$^3$Institute for Theoretical Atomic and Molecular Physics,\
Harvard-Smithsonian Center for Astrophysics,\
Cambridge, Massachusetts, 02138, USA
author:
- |
M.S. Hussein$^{1}$, M.P. Pato$^{1,2}$, and J.C. Wells$^{2,3\thanks{Present address: Center for Computional Sciences, Oak Ridge National
Laboratory, Oak Ridge, Tenessee, 37831-6373, U.S.A.}}$
title: Causal Classical Theory of Radiation Damping
---
-20mm
The advent of a new generation of extremely high power lasers that uses chirped pulse amplification has put into focus the classical description of the dynamics of relativistic electrons. Under the action of high intensity electromagnetic field, a major ingredient of the dynamics is the electron self-interaction which implies in the damping of the movement caused by the interaction of the charge with its own field. The derivation of the damping force has been reviewed recently[@Luhm] revealing its relativistic origin asociated to the asymmetry introduced by the Doppler effect in the forward and backward emission of radiation. The inclusion of this force in the equation of motion leads to the nonlinear covariant Lorentz-Dirac(LD) equation[@Dirac] for a poi
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|
)}{q(x)}dx$$
Jensen-Shannon Divergence
-------------------------
Let $P,Q$ be discrete probability distributions, *Jensen-Shannon Divergence* between $P$ and $Q$ is defined to be:
$$JSD(P||Q) = \frac{1}{2}KLD(P||M) + \frac{1}{2}KLD(Q||M)$$
where $\displaystyle M = \frac{1}{2}(P+Q)$.
A more generalized form is defined to be:
$$JSD(P_1, \dots, P_n) = H\Big(\sum_{i=1}^n\pi_i P_i\Big) - \sum_{i=1}^n\pi_iH(P_i)$$
where $H$ is Shannon Entropy, $\displaystyle M = \sum_{i=1}^{n}\pi_iP_i$ and $\displaystyle \sum_{i=1}^{n}\pi_i = 1$.
Especially, if $\displaystyle \pi_i = \frac{1}{n}$, then: $$JSD(P_1, \dots, P_n) = \frac{1}{n}\sum_{i=1}^{n}KLD(P_i||M)$$
Jensen-Shannon divergence has some fine properties:
1. $JSD(P||Q) = JSD(Q||P), \forall P, Q\in \mathbb{S}$.
2. $0 \le JSD(P_1, \dots, P_n) \le log_k(n)$. If a $k$ based algorithm is adopted.
3. To calculate $JSD(P||Q)$, it need not necessarily to be true that $Q(x)=0$ implies $P(x)=0$.
Bhattacharyya Distance
----------------------
Let $P,Q$ be discrete probability distributions over same domain $X$, *Bhattacharyya Distance* between $P$ and $Q$ is defined to be:
$$BD(P||Q) = -ln\Big(\sum_{x\in X}\sqrt{P(x)Q(x)}\Big)$$
Hellinger Distance
------------------
Let $P,Q$ be discrete probability distributions, *Hellinger Distance* between $P$ and $Q$ is defined to be:
$$HD(P||Q) = \frac{1}{\sqrt{2}}\sqrt{\sum_x\bigg(\sqrt{P(x)} - \sqrt{Q(x)}\bigg)^2}$$
Kolmogorov-Smirnov Statistic
----------------------------
Let $P,Q$ be discrete one-dimensional probability distributions, $CDF_P$ and $CDF_Q$ are their cumulative probability functions respectively, *Kolmogorov-Smirnov Statistic* between $P$ and $Q$ is defined to be:
$$KSS(P||Q) = \sup_x | CDF_P(x) - CDF_Q(x) |$$
Statistical Detection {#sec:algorithm-details}
=====================
Diverse data sets in the real world show certain structures caused by hidden patterns or relationships among records. For example, traffic volume in the highway and the business transaction records, they may show a relatively stab
| 508
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uirement.
onTouch gives you Motion Event. Thus, you can do a lot of fancy things as it help you separate state of movement. Just to name a few:
ACTION_UP
ACTION_DOWN
ACTION_MOVE
Those are common actions we usually implement to get desire result such as dragging view on screen.
On the other hand, onClick doesn't give you much except which view user interacts. onClick is a complete event comprising of focusing,pressing and releasing. So, you have little control over it. One side up is it is very simple to implement.
do we need to implement both?
It is not necessary unless you want to mess up with your user. If you just want simple click event, go for onClick. If you want more than click, go for onTouch. Doing both will complicate the process.
From User point of view, it is unnoticeable if you implement onTouch carefully to look like onClick.
A:
A "Touch" event is when someone puts their finger on the screen. It gets called throughout the movement of the finger, down, drag, and up. A "Click" need not even come from the screen. It could be someone pressing the enter key.
Use OnTouchListener when you want to receive events from someone's finger on the screen.
Use OnClickListener when you want to detect clicks.
A:
onClickListener is used whenever a click event for any view is raised, say for example: click event for Button, ImageButton.
onTouchListener is used whenever you want to implement Touch kind of functionality, say for example if you want to get co-ordinates of screen where you touch exactly.
from official doc, definition for both are:
onClickListner: Interface definition for a callback to be invoked when a view is clicked.
onTouchListener: Interface definition for a callback to be invoked when a touch event is dispatched to this view. The callback will be invoked before the touch event is given to the view.
Q:
How to replace function in postgress?
I want to change code of function.
If i use
CREATE FUNCTION take_proxy (VARCHAR(255)) ....
I get function "take_proxy" already exists with same
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|
$\frac{d}{dx} y_{3}(a, x)$ $+\infty$ $+$ $+$ $+$ $+$ $+$ $0$ $-$ $0$
$y_{3}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $+$ $+$ $0$
$\frac{d}{dx} y_{2}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $+$ $+$ $0$
$y_{2}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $-$ $-$ $0$
$\frac{d}{dx} y_{1}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $-$ $-$ $0$
$y_{1}(a, x)$ $0$ $+$ $+$ $+$ $+$ $+$ $+$ $+$ $0$
: Case of $0 < a < 1$
$x$ $\;0\;$ $\cdots$ $\;p_{2}(a)\;$ $\cdots$ $\;p_{3}(a)\;$ $\cdots$ $\;+\infty\;$
---------------------------- --------- ---------- ---------------- ---------- ---------------- ---------- ------------------------------------------------------------------------
$\frac{d}{dx} y_{3}(a, x)$ $0$ $+$ $+$ $+$ $+$ $+$ $\begin{matrix}0\;(a < 2)\\ +\;(a = 2)\\ +\infty\;(a > 2)\end{matrix}$
$y_{3}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $0$
$\frac{d}{dx} y_{2}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $0$
$y_{2}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $0$
$\frac{d}{dx} y_{1}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $0$
$y_{1}(a, x)$ $0$
| 510
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|
atinkumarg.github.io/personal-portfolio, run:
npm run deploy
> personal-portfolio@0.1.0 deploy C:\react-projects\personal-portfolio
> gh-pages -d build
Cloning into 'node_modules\gh-pages\.cache\git@github.com!jatinkumarg!personal-portfolio.git'...
git@github.com: Permission denied (publickey).
fatal: Could not read from remote repository.
Please make sure you have the correct access rights
and the repository exists.
npm ERR! code ELIFECYCLE
npm ERR! errno 1
npm ERR! personal-portfolio@0.1.0 deploy: `gh-pages -d build`
npm ERR! Exit status 1
npm ERR!
npm ERR! Failed at the personal-portfolio@0.1.0 deploy script.
npm ERR! This is probably not a problem with npm. There is likely additional logging output above.
npm ERR! A complete log of this run can be found in:
npm ERR! C:\Users\jatin\AppData\Roaming\npm-cache\_logs\2019-07-23T04_40_54_788Z-debug.log
I have already set-up and tested my ssh key, it works fine.
There is only one remote url i.e. origin(SSH)
This is my package.json
{
"name": "personal-portfolio",
"version": "0.1.0",
"homepage": "https://jatinkumarg.github.io/personal-portfolio",
"dependencies": {
"jquery": "^3.4.1",
"json-loader": "^0.5.7",
"react": "^15.5.4",
"react-dom": "^15.5.4"
},
"devDependencies": {
"gh-pages": "^1.0.0",
"react-scripts": "1.0.1"
},
"scripts": {
"predeploy": "npm run build",
"deploy": "gh-pages -d build",
"start": "react-scripts start",
"build": "react-scripts build",
"test": "react-scripts test --env=jsdom",
"eject": "react-scripts eject"
}
}
At this point, I have no idea what's wrong. Can anyone please help me with this issue?
A:
You need the PRO plan to use GitHub pages on a private repository.
You can try adding a SSH identity in ~/.ssh/config
Host github.com
HostName github.com
User git
IdentityFile /Users/myusername/.ssh/my_github_ssh_private_key_registered_on_github
Q:
How do I end a behavior tree's action early without evaluating the entire active branch?
I'm reading about beh
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| 0.826824
|
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|
--------------------------------------------------------------------------------------------------------------------------
State Count
-------------------------------------------------------------------- ------------------------------------------------------------------------
$\overline{\partial} X^{3-4}_{-1/2} \otimes 8 spacetime scalars
| \pm \pm \rangle$
$\left( \lambda^{9-16}_{-1/4} \right)^2 \otimes | \pm \pm \rangle$ 2 spacetime scalars valued in $\wedge^2 {\bf 8} = {\bf 28}$ of $su(8)$
---------------------------------------------------------------------------------------------------------------------------------------------
There are no massless states in (R,NS) in this sector, as $E_{\rm left} =
+ 1/2$.
Copies of the states in the $k=1$ sector occur at each of the sixteen fixed points, hence the total state count should be obtained by multiplying the totals for this sector by sixteen.
Next, consider the twisted sector $k=2$.
In the (NS,NS) sector, fields have the following boundary conditions: $$\begin{aligned}
X^{1-4}(\sigma + 2\pi) & = & + X^{1-4}(\sigma), \\
\psi^{1-4}(\sigma + 2 \pi) & = & - \psi^{1-4}(\sigma), \\
\lambda^{1-8}(\sigma + 2 \pi) & = & - \lambda^{1-8}(\sigma), \\
\lambda^{9-16}(\sigma + 2 \pi) & = & + \lambda^{9-16}(\sigma).\end{aligned}$$ It is straightforward to compute $E_{\rm left} = 0$, $E_{\rm right} = -1/2$. The available field modes are $$\psi^{\mu}_{-1/2}, \overline{\psi}^{\mu}_{-1/2}, \: \: \:
\lambda^{1-8}_{-1/2}, \overline{\lambda}^{1-8}_{-1/2}.$$ There is a multiplicity of left Fock vacua, arising from $\lambda^{9-16}$. Let $|m,n\rangle$ denote a vacuum with $m$ +’s and $n$ -’s (note $m+n=8$), then under the action of the generator of ${\mathbb Z}_4$, it is straightforward to check that $|m=0,4,8\rangle$ are invariant, $|m=2,6\rangle$ get a sign flip, and
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N OPTIONS {#sim7930-sec-0002}
===================================================================
Consider that IPD have been obtained from *i* = 1 to *K* related randomized trials, each investigating a treatment effect based on a continuous outcome *Y* (say, blood pressure); that is, the mean difference in outcome value between a treatment and a control group. Suppose that there are *n* ~*i*~ participants in trial *i*. Let *Y* ~*Fi j*~ be the end‐of‐trial (F used to denote final) continuous outcome value, for participant *j* in trial *i*, and *Y* ~*Bi j*~ (B to denote baseline) be the pre‐treatment outcome value. Let *treat* ~*i j*~ take the value 1 or 0 for participants in the treatment or control group, respectively.
Given such IPD, there are several ways in which researchers can use a one‐stage meta‐analysis to model the summary treatment effect across trials. We focus initially on presenting one‐stage analysis of covariance (ANCOVA) mixed models, which either use a stratified intercept or a random intercept to account for clustering of participants within trials. We also assume a random treatment effect since heterogeneity is usually expected.
2.1. Model (1): stratified intercept {#sim7930-sec-0003}
------------------------------------
With the following approach, a stratified intercept is used to account for within‐trial clustering. $$\begin{matrix}
Y_{\mathit{Fij}} & {= \beta_{i} + \lambda_{i}\left( {Y_{\mathit{Bij}} - {\bar{Y}}_{\mathit{Bi}}} \right) + \left( {\theta + u_{i}} \right)\textit{treat}_{\mathit{ij}} + e_{\mathit{ij}}} \\
& {\mspace{99mu} u_{i} \sim N\left( {0,\tau^{2}} \right)} \\
& {\mspace{94mu} e_{\mathit{ij}} \sim N\left( {0,\sigma_{i}^{2}} \right)} \\
\end{matrix}$$ Here, *β* ~*i*~ denotes the intercept term for trial *i* (expected final outcome value for participants in the control group in trial *i* who have the mean baseline outcome value), and the distinct intercept for each trial is used to account for within trial clustering. The term *λ* ~*i*~ denotes
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) that $g(t_{\leftarrow})$ be an element of $_{\rightarrow}T$, unless of course
$$g(t_{\leftarrow})=t_{\leftarrow},\label{Eqn: fixed point}$$
which because of (\[Eqn: ChainedTower\]) may also be expressed equivalently as $g(c_{\leftarrow})=c_{\leftarrow}\in C_{\textrm{T}}$. As the sequential totally ordered set $_{\rightarrow}T$ is a subset of $X$, Eq. (48) implies that $t_{\leftarrow}$ is a maximal element of $X$ which allows (ST3) to be replaced by the remarkable inverse criterion that
$(\textrm{ST}3^{\prime})$ If $x\in X$ and $w$ precedes $x,$ $w\prec x$, then $w\in X$
that is obviously false for a general tower $T$. In fact, it follows directly from Eq. (\[Eqn: maximal\]) that under $(\textrm{ST}3^{\prime})$ *any $x_{+}\in X$ is a maximal element of $X$ iff it is a fixed point of $g$* as given by Eq. (\[Eqn: fixed point\]). This proves the theorem and also demonstrates how, starting from a minimum element of a partially ordered set $X$, (ST3) can be used to generate inductively a totally ordered sequential subset of $X$ *leading to a maximal $x_{+}=c_{\leftarrow}\in(X,\preceq)$ that is a fixed point of the generating function $g$* *whenever the supremum* $t_{\leftarrow}$ *of the chain $_{\rightarrow}T$ is in* $X$.$\qquad\blacksquare$
**Remarks.** The proof of this theorem, despite its apparent length and technically involved character, carries the highly significant underlying message that
[0.1in]{} *Any inductive sequential $g$-construction of an infinite chained tower* $C_{\textrm{T}}$ *starting with a smallest element $x_{0}\in(X,\preceq)$ such that a supremum $c_{\leftarrow}$ of the $g$-generated sequential chain* $C_{\textrm{T}}$ *in its own tower is contained in itself, must necessarily terminate with a fixed point relation of the type* (\[Eqn: fixed point\]) *with respect to the supremum. Note from Eqs. (\[Eqn: sup chain\]) and (\[Eqn: fixed point\]) that the role of* (ST2) *applied to a fully ordered tower is the identification of the maximal of the tower — which depends only the tower and has
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$\psi_j$ is similar to and simpler than that of the above cases when $N_0$ is *of type $I^e$* and so we skip it.\
2. Assume that $M_0$ is *of type $I^e$* and $L_j$ is *of type $I^o$*. We write $M_0=N_0\oplus L_j$, where $N_0$ is unimodular with odd rank so that it is *of type $I^o$*. Then we can write $N_0=(\oplus_{\lambda'}H_{\lambda'})\oplus (a)$ and $L_j=(\oplus_{\lambda''}H_{\lambda''})\oplus (a')$ by Theorem \[210\], where $H_{\lambda'}=H(0)=H_{\lambda''}$ and $a, a'\in A$ such that $a, a' \equiv 1$ mod 2. Thus we write $M_0=(\oplus_{\lambda}H_{\lambda})\oplus (a)\oplus (a')$, where $H_{\lambda}=H(0)$. For this choice of a basis of $L^j=\bigoplus_{i \geq 0} M_i$, the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $L^j$ is $$\begin{pmatrix} id&0 &0\\ 0 &\begin{pmatrix} 1+ 2 z_j^{\ast} \end{pmatrix} &0 \\ 0& 0 &id \end{pmatrix}.$$ Here, $id$ in the $(1,1)$-block corresponds to the direct summand $(\oplus_{\lambda}H_{\lambda})\oplus (a)$ of $M_0$ and the diagonal block $\begin{pmatrix} 1+ 2 z_j^{\ast} \end{pmatrix} $ corresponds to the direct summand $(a')$ of $M_0$.
Let $(e_1, e_2)$ be a basis for the direct summand $(a)\oplus (a')$ of $M_0$. Since this is *unimodular of type $I^e$*, we can choose another basis $(e_1, e_1+e_2)$ such that the associated Gram matrix is $A(a, a+a', a)$, where $a+a'\in (2)$, so that $$M_0=(\oplus_{\lambda}H_{\lambda})\oplus A(a, a+a', a).$$
For this basis, the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $L^j$ is $$\label{e4.3}
\begin{pmatrix} id&0 &0\\ 0 &\begin{pmatrix} 1 & -2 z_j^{\ast}\\ 0 & 1+2 z_j^{\ast} \end{pmatrix} &0 \\ 0& 0 &id \end{pmatrix}.$$ Here, the diagonal block $\begin{pmatrix} 1 & -2 z_j^{\ast}\\ 0 & 1+2 z_j^{\ast} \end{pmatrix} $ corresponds to $A(a, a+a', a)$ with a basis $(e_1, e_1+e_2)$ and $id$ in the $(1,1)$-block corresponds to the direct summand $\oplus_{\lambda}H_{\lambda}$ of $M_0$.
We now describe the image of a fixed eleme
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s monotonically with frequency. Thus $\lambda_{\rm oe}$ and $\lambda_{\rm hw}$ are oscillating in counter phase, see Eq. (\[eq:s14\]). This induces a corresponding oscillation in the strength of the fidelity decay as seen in Fig. \[fig:03\]. For a more quantitative discussion we compare the experimentally determined fidelity decay (solid lines) to the theoretical curves (dotted and dashed lines). First of all, one sees that fitting (dashed lines) the experimental results again works well for all cases. Focusing on the 50$\Omega$ load case (black lines) also the theoretical results without free parameter (dotted lines) show good agreement with the experiment for the frequency ranges plotted in Figs. \[fig:03\] (ii) and (iii). Only the plot in Fig. \[fig:03\] (i) shows significant deviation between the theoretical result without free parameter and the experimental curve. We want to stress that the case shown here is the worst among all the investigated frequency ranges. Here the experimental fidelity amplitude $f_{50\Omega}$ shows a significant imaginary part. Thus the imaginary part of $\lambda_{50\Omega}$ is not zero, i.e. the 50$\Omega$ terminator does not correspond to perfect absorption, and Eq. (\[eq:s14\]) does not hold. This might be due to an antenna resonance, leading to an increased reflection from the channel $c$.
$\nu$/GHz $\lambda^{\rm exp}_{\rm 50\Omega}$ $\lambda^{\rm fit}_{\rm 50\Omega}$ $\lambda_{\rm oe}$ $\lambda_{\rm hw}$ $\lambda_W$ $T_a$
------- ----------- ------------------------------------ ------------------------------------ -------------------- -------------------- ------------- --------
(i) $7.2-7.7$ $0.19$ $0.37$ $0.65\,\imath$ $-0.04\,\imath$ $0.16$ $0.19$
(ii) $8.0-8.5$ $0.21$ $0.20$ $0.19\,\imath$ $-0.23\,\imath$ $0.21$ $0.22$
(iii) $8.7-9.2$ $0.24$
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(r)[2-8]{} Base Arrangement \[0, 0, 1\] \[0, 1, 0\] \[0, 1, 1\] \[1, 0, 0\] \[1, 0, 1\] \[1, 1, 0\] \[1, 1, 1\]
[\[1, 1, 1\]]{} 0.219 **0.190** 0.249 0.648 0.773 0.728 0.949
----------------------------- ------------- ------------- ------------- ------------- ------------- ------------- -------------
: Cosine similarities of LeNet probability output vector gradients w.r.t. input images.[]{data-label="table:cosine_similarities_lenet"}
--------------------- ------- ------- -------
(r)[2-4]{} Cor Type FGS IGS CW2
Pearson 0.990 0.997 0.997
Spearman 0.829 0.943 0.986
--------------------- ------- ------- -------
: Correlations of cosine similarities between different defense arrangements (using \[1, 1, 1\] as the baseline defense) and the transferability between the attacks on the \[1, 1, 1\] defense to other defense arrangements.[]{data-label="table:pearson_correlations"}
Training for Gradient Orthogonality
-----------------------------------
In order to test the claim that explicitly training for gradient orthogonality will result in lower transferability of adversarial examples, we focus on a simple scenario. We trained 16 pairs of LeNet models, 8 of which were trained to have orthogonal input-output Jacobians, and 8 of which were trained to have parallel input-output Jacobians. As can be seen in Figure \[fig:parallel\_vs\_perpendicular\], the differences in transfer rates and relevant transfer rates are quite vast between the two approaches. The median relevant transfer attack success rate for the parallel approach is approx. 92%, whereas it is only approx. 17% for the perpendicular approach. This result further illustrates the importance of the input-output Jacobian cosine similarity between models when it comes to transferability.
!
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general structure known as a *family*. We employ nomenclature abridged from Halmos [@pH74 p. 34] as follows:
Let $I$ and $X$ be non-empty sets, and $\varphi \colon I \to X$ be a mapping. Each element $i \in I$ is an *index*, while $I$ itself is an *index set*. The mapping $\varphi$ is a *family*; its co-domain $X$ is an *indexed set*. An ordered pair $(i,x)$ belonging to the family is a *term*, whose value $x = \varphi(i)\in X$ is often denoted $\varphi_i$.
The family $\varphi$ itself is routinely but abusively denoted $\{\varphi_n\}$. This notation is a compound idiom.[^9] Especially in the case of sequences over a set $G$, the symbol $\{g_n\}$ signifies the mapping $\{ 1\mapsto g_1, \; 2\mapsto g_2, \; \ldots \;\}$.
\[D:ENSEMBLE\] An *ensemble* is a non-empty family.
\[D:CONSTANT\_VARIABLE\] Let $\Psi$ be an ensemble, with ${{\operatorname{ran}{\Psi}}}$ its range. If ${\lvert{{{\operatorname{ran}{\Psi}}}}\rvert} = 1$, the ensemble is *constant*; otherwise it is *variable*.
Since physical systems possess only a finite number of attributes, the scope of practical interest is limited to ensembles having finite-dimensional index sets.
A constant ensemble (definition \[D:CONSTANT\_VARIABLE\]) is also referenced under the historically colorful name *Hobson’s*[^10] choice. Using the word *choice* in its everyday sense, a Hobson’s choice is oxymoronic: a free choice in which only one option is offered, with gist [Only one choice is no choice.]{}
Let $\Psi$ be an ensemble. For term $(i,P) \in \Psi$, we denote $P = \Psi_i$.
Ensemble arithmetic {#S:ENSEMBLE_ARITHMETIC}
-------------------
\[D:DISJOINT\_ENSEMBLES\] Ensembles $\Psi$ and $\Phi$ are *disjoint* if ${{\operatorname{dom}{\Psi}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Phi}}} = \varnothing$ (that is, if their index sets have no member in common).
\[D:COMPLEMENTARY\_ENSEMBLES\] Ensembles $\Psi$ and $\Phi$ are *complementary* with respect to a third ensemble $\Upsilon$ if they are disjoint and $\Psi \cup \Phi = \Upsilon$.
Regarding ensembles $
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this section, we present experimental results that validate the importance of noise covariance in predicting the test error of Langevin MCMC-like algorithms. In all experiments, we use two different neural network architectures on the CIFAR-10 dataset [@krizhevsky2009learning] with the standard test-train split. The first architecture is a simple convolutional neural network, which we call CNN in the following,[^1] and the other is the VGG19 network [@simonyan2014very]. To make our experiments consistent with the setting of SGD, we do not use batch normalization or dropout, and use constant step size. In all of our experiments, we run SGD algorithm $2000$ epochs such that the algorithm converges sufficiently. Since in most of our experiments, the accuracies on the training dataset are almost $100\%$, we use the test accuracy to measure the generalization performance.
Recall that according to and , for both SGD and large-noise SGD, the noise covariance is a scalar multiple of $H(w)$. For simplicity, in the following, we will slightly abuse our terminology and call this scalar the *noise covariance*; more specifically, for $(\delta, b)$-SGD, the noise covariance is $\delta / b$, and for an $(s, \sigma, b_1, b_2)$-large-noise SGD, the noise covariance is $\frac{s}{b_1} + \frac{2\sigma^2}{b_2}$.
[Accuracy vs Noise Covariance]{} \[ss:acc\_rel\_var\]
In our first experiment, we focus on the SGD algorithm, and show that there is a positive correlation between the noise covariance and the final test accuracy of the trained model. One major purpose of this experiment is to establish baselines for our experiments on large-noise SGD.
We choose constant step size $\delta$ from $$\{0.001, 0.002, 0.004, 0.008, 0.016, 0.032, 0.064, 0.128\}$$ and minibatch size $b$ from $\{32, 64, 128, 256, 512\}$. For each (step size, batch size) pair, we plot its final test accuracy against its noise covariance in Figure \[fig:const\_lr\_acc\_vs\_var\]. From the plot, we can see that higher noise covariance leads to better final tes
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relate the entropy $H(x)$ to the compression rate $R(x)$, we obtain the following inequalities: $$H(x)\leq{R(x)}\leq{H(x)+1}.$$ Let $\Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{t}\}$ be an alphabet and $c:\Sigma\times\Sigma^{\leq{1}}\rightarrow\{0,1\}^{+}$ an adaptive code of order one constructed as shown in section 3. Also, consider $w=w_{1}w_{2}\ldots{w_{s}}\in{\Sigma^{+}}$, $w_{i}\in\Sigma$, $1\leq{i}\leq{s}$, and $p$ the number of symbols occurring in $w$. We denote by $R_{A}(w)$ the compression rate obtained when encoding the string $w$ by $\overline{c}$ and by $H_{A}(w)$ the entropy of $w$. It is useful to consider the following notations:
1. ${\it EH}(w)=\{i$ $\mid$ $2\leq{i}\leq{s}$ and $w_{i}\neq{w_{i-1}}\}$,
2. ${\it LNotHuffman}(w)=|c(w_{1},\lambda)|+\sum_{i\in{{\it Pairs}(w)}}|c(w_{i+1},w_{i})|$,
3. ${\it LHuffman}(w)$ is the entropy of $w_{j_{1}}w_{j_{2}}\ldots{w_{j_{r}}}$, $j_{k}\in{{\it EH}(w)}$, $1\leq{k}\leq{r}$,
4. $H_{A}(w)={\it LNotHuffman}(w)+{\it LHuffman}(w)$.
It is easy to verify that ${\it LNotHuffman}(w)={\it NRPairs}(w)+|c(w_{1},\lambda)|$. Using the notation above, we get that $${\it LHuffman}(w)=\sum_{i\in{{\it EH}(w)}}\{ \frac{1}{N(w_{i})}\sum_{q\in{{\it Prev}(w_{i})}}[F_{q}(w_{i})(1+\log_{2}\frac{N(w_{i})}{F_{q}(w_{i})})]\},$$ where
- $N(w_{i})=|\{j$ $\mid$ $j\in{{\it EH}(w)}$ and $w_{j}={w_{i}}\}|$,
- ${\it Prev}(w_{i})=\{j$ $\mid$ $j+1\in{{\it EH}(w)}$ and $w_{j+1}=w_{i}\}$,
- $F_{q}(w_{i})=|\{j$ $\mid$ $j\in{{\it EH}(w)}$ and $w_{j}=w_{i}$ and $w_{j-1}=w_{q}\}|$, $q\in{{\it Prev}(w_{i})}$.
Finally, we can relate the entropy $H_{A}(w)$ to the compression rate $R_{A}(w)$ by the following inequalities: $$H_{A}(w)\leq{R_{A}(w)}\leq{H_{A}(w)}+1,$$ where $R_{A}(w)$ is given by $$R_{A}(w)=\frac{|c(w_{1},\lambda)c(w_{2},w_{1})\ldots c(w_{s},w_{s-1})|}{s}.$$
GA Codes
========
In this section, we introduce a natural generalization of adaptive codes (of any order), called *GA codes* (**G**eneralized **A**daptive codes). Theorem 5.1 proves that adaptive codes are part
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e breakdown of the assumption in the analysis.
Discussion
==========
Our model showed a power law $v\sim\mu^{-1/2}$ under the assumptions that $r\ll1/\lambda_+$ and $\ell\gg1/\left|\lambda_-\right|$. In this section, we compare experimental results with the numerical results in Eq. (\[eq:v\]) in order to check whether our model is reasonable. Equation (\[eq:v\]) has several parameters such as $\Gamma$, $w$, $f_0$, $r$, $K$, and $\mu$. Since similar camphor boats were used, $w$, $r$, and $K$ were constant values in our experiments. We investigated the dependence of the other parameters, i.e., $\Gamma$, $f_0$, and $\mu$, on the glycerol concentration $p$ in Appendix A. Equation (\[eq:surface\]) showed $\Gamma=(\gamma_0 - \gamma)/c$. As $(\gamma_0 - \gamma)$ was independent of $p$ in our measurements, we considered that $\Gamma$ was constant. The supply rate $f_0$ corresponds to $\Delta M$, which is a loss of a camphor disk per unit time in our experiments, and we found that $\Delta M$ decreased with an increase in $p$. The viscosity $\mu$ of the base solution increased with $p$. Thus, $f_0$ and $\mu$ in Eq. (\[eq:v\]) are functions of $p$. In addition, the angular velocity is proportional to the camphor boat velocity in our experiments.
From the above discussion, Eq. leads to $$\begin{aligned}
\overline{\omega}(p) \propto\sqrt{\frac{\Delta M(p)}{\mu(p)}}.
\label{eq:v2}
\end{aligned}$$ Figure \[power\] shows a relationship between $\Delta M/\mu$ and $\overline{\omega}$ obtained from our experiments. The result almost agrees with the solid line in Eq. (\[eq:v2\]) [@Delta_M].
 Relationship between $\Delta M/\mu$ and $\overline{\omega}$, where $\Delta M$, $\mu$, and $\overline{\omega}$ are a weight loss of a camphor disk per one second, the viscosity of the base solution, and the angular velocity of the camphor boat, respectively. The solid line shows the numerical result; $\overline{\omega}\sim\sqrt{\Delta M/\mu}$ in Eq. .](power.eps){width="7cm"}
The power l
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:=f^{-}f(U)\subseteq X$ be the saturation of an open set $U$ of $X$ and $\textrm{comp}(V):=ff^{-}(V)=V\bigcap f(X)\in Y$ be the component of an open set $V$ of $Y$ on the range $f(X)$ of $f$. Let $\mathcal{U}_{\textrm{sat}}$, $\mathcal{V}_{\textrm{comp}}$ denote respectively the saturations $U_{\textrm{sat}}=\{\textrm{sat}(U)\!:U\in\mathcal{U}\}$ of the open sets of $X$ and the components $V_{\textrm{comp}}=\{\textrm{comp}(V)\!:V\in\mathcal{V}\}$ of the open sets of $Y$ whenever these are also open in $X$ and $Y$ respectively. Plainly, $\mathcal{U}_{\textrm{sat}}\subseteq\mathcal{U}$ and $\mathcal{V}_{\textrm{comp}}\subseteq\mathcal{V}$.
**Definition A2.1.** *For a function* $e\!:X\rightarrow(Y,\mathcal{V})$, *the* *preimage* *or* *initial topology of* $X$ *based on (generated by)* **$e$** *and $\mathcal{V}$* *is* $$\textrm{IT}\{ e;\mathcal{V}\}\overset{\textrm{def}}=\{ U\subseteq X\!:U=e^{-}(V)\textrm{ if }V\in\mathcal{V}_{\textrm{comp}}\},\label{Eqn: IT}$$
*while for $q\!:(X,\mathcal{U})\rightarrow Y$, the* *image* *or* *final topology of* $Y$ *based on (generated by) $\mathcal{U}$ and* **$q$** *is* $$\textrm{FT}\{\mathcal{U};q\}\overset{\textrm{def}}=\{ V\subseteq Y\!:q^{-}(V)=U\textrm{ if }U\in\mathcal{U}_{\textrm{sat}}\}.\qquad\square\label{Eqn: FT'}$$
Thus, the topology of $(X,\textrm{IT}\{ e;\mathcal{V}\})$ consists of, and only of, the $e$-saturations of all the open sets of $e(X)$, while the open sets of $(Y,\textrm{FT}\{\mathcal{U};q\})$ are the $q$-images *in* $Y$ (and not just in $q(X)$) of all the $q$-saturated open sets of $X$.[^32] The need for defining (\[Eqn: IT\]) in terms of $\mathcal{V}_{\textrm{comp}}$ rather than $\mathcal{V}$ will become clear in the following. The subspace topology $\textrm{IT}\{ i;\mathcal{U}\}$ of a subset $A\subseteq(X,\mathcal{U})$ is a basic example of the initial topology **by the inclusion map $i\!:X\supseteq A\rightarrow(X,\mathcal{U})$, and we take its generalization $e\!:(A,\textrm{IT}\{ e;\mathcal{V}\})\rightarrow(Y,\mathcal{V})$ that embeds a subset $A$ of $
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t $\phi = \gamma \mid {{\operatorname{dom}{\Phi}}}$. For each $i \in {{\operatorname{dom}{\Phi}}}$, the non-empty sets $\Phi(i) = \Theta\Phi(i)$ by definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\]. Since by construction $\phi(i) \in \Theta\Phi(i)$, and $\Phi(i) = \Theta\Phi(i)$, then $\phi(i) \in \Phi(i)$ for $i \in {{\operatorname{dom}{\Phi}}}$ – that is, $\phi$ is a choice in $\prod\Phi$.
It is thus established that for any $\gamma \in \prod\Theta\Phi$, there exists $\theta \in \prod\Theta$ and $\phi \in \prod\Phi$ such that $\gamma = \theta\phi$. Therefore the dyadic choice product is surjective.
Since the dyadic choice product is both injective and surjective, then it is a bijection $\prod\Theta \times \prod\Phi \leftrightarrow \prod\Theta\Phi$.
### Choice subspaces {#S:CHOICE_SUBSPACE}
Any mapping, including a choice mapping, may be restricted to subsets of its domain.
\[D:SUBCHOICE\] Let $\Psi$ be an ensemble, and let $R \subseteq {{\operatorname{dom}{\Psi}}}$ be a subset of its index set. Suppose $\chi \in \prod\Psi$ is a choice. A *subchoice* $\chi \mid R$ is the ordinary mapping restriction of $\chi$ to its domain subset $R$.
In the above, degenerate case $R = \varnothing$ yields $\chi \mid R = \varnothing$.
\[D:SUBSPACE\] Let $\Psi$ be an ensemble. For each $R \subseteq {{\operatorname{dom}{\Psi}}}$, the *subspace* $(\thinspace\prod\Psi) \mid R$ is the set of subchoices $\lbrace\thinspace\chi \mid R \medspace \colon \chi \in \prod\Psi\thinspace\rbrace$.
\[T:CHC\_RSTR\_EQ\_RSTR\_CHC\] Let ensemble $\Psi$ generate choice space $\prod \Psi$, and let $R \subseteq {{\operatorname{dom}{\Psi}}}$ be a subset of its index set. The restriction of the choice space equals the choice space of the restriction: $$(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}}).$$
Suppose $\xi \in (\thinspace\prod\Psi) \mid R$. By definition \[D:SUBCHOICE\], there exists $\chi \in \prod\Psi$ such that $\xi = \chi \mid R$. By definition of Cartesian product, for each $i \in {{\operatorname{dom}{\Psi}
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case of multiple options, one is chosen at random. Otherwise, the particle tries to diffuse to the neighbor $i'$ with a probability given by [@Leal_JPCM] $$\label{prob}
P_{\delta h}(i,i')=
\left\{
\begin{array}{cl}
1, & \textrm{if } ~ |\delta h|<2\\
\frac {1}{|\delta h|}, & \textrm{if } ~ |\delta h|\geq 2
\end{array}
\right.$$ where $\delta h = h_i-h_{i'}$. With probability $1 - P_{\delta h}(i,i')$ the particle remains at the site $i$. It is important to mention that Eq. is obtained [ assuming that the adatom first moves to top kink of the terrace and then start a unbiased one-dimensional random-walk normally to the initial substrate, stopping the movement if it either arrives at the bottom or return to top of the terrace. The result is the]{} solution of a non-directed one-dimensional random walk with absorbing boundaries separated by a distance $|\delta h|$ [@Shehawey]; see Fig. 1 of Ref. [@Leal_JPCM] for further details of this diffusion rule. This diffusion attempt is successively applied $N_s$ times (representing a $N_s$ diffusive steps) departing from the last position of the adatom. A unit time is defined as the deposition of $L^d$ particles.
The implementation of the DT model with kinetic barrier is similar. The difference is that diffusion to the nearest-neighbors are performed only if the adatom does not have lateral bounds and any neighbor with a number of bonds higher than 1 can be chosen with equal chance as the target site.
{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}\
{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}
![\[hh\_corr\_time\]Main panels: Height-height correlation function for the WV model at distinct times indicated in the legends for (a) one- and (b) two-dimensional substrates. The number of steps is $N_s=1$. The averages were computed over 100 independent runs. I
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Please help I am at my wits end.
A:
Can you try mapping your class like so? I have a hunch that your PropertyNamingStrategy is not correctly reflecting the JSON property name.
import com.fasterxml.jackson.annotation.JsonProperty;
public class Drivers {
@JsonProperty("userID");
private String userID = "";
@JsonProperty("userMemID");
private String userMemID = "";
@JsonProperty("userFirstName");
private String userFirstName = "";
@JsonProperty("userLastName");
private String userLastName = "";
@JsonProperty("userSkillLevel");
private String userSkillLevel = "";
@JsonProperty("userProviderID");
private String userProviderID = "";
public String getUserID() {
return userID;
}
public void setUserID(String userID) {
this.userID = userID;
}
public String getUserMemID() {
return userMemID;
}
public void setUserMemID(String userMemID) {
this.userMemID = userMemID;
}
public String getUserFirstName() {
return userFirstName;
}
public void setUserFirstName(String userFirstName) {
this.userFirstName = userFirstName;
}
public String getUserLastName() {
return userLastName;
}
public void setUserLastName(String userLastName) {
this.userLastName = userLastName;
}
public String getUserSkillLevel() {
return userSkillLevel;
}
public void setUserSkillLevel(String userSkillLevel) {
this.userSkillLevel = userSkillLevel;
}
public String getUserProviderID() {
return userProviderID;
}
public void setUserProviderID(String userProviderID) {
this.userProviderID = userProviderID;
}
}
Q:
let every item in ListVIew jump to
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WMTF response level rather than describing a distinct MU.
In light of these concerns, the marginal likelihood estimates are adjusted according to Section \[sec:ML\] with a conservative lower bound of ${\mu_{\min}}=15$mN to guard against small MUs that, when firing, are indistinguishable from other combinations. The corrected posterior mass function places 89.3% of the mass on the correct, seven-MU model, with 10.7% mass on the eight-MU model. The estimates of expected MUTF in Table \[tab:D1muEst\] for the seven-MU model are similar to those prior to the adjustment and are still close to the true values from which the data was generated. However, the prior adjustment for the eight-MU hypothesis has a significant effect on the penultimate MU and, so as to preserve the overall maximum WMTF, a small reduction in the estimated $\mu$s for its neighboring MUs.
--------------------------- ------------------- ------------------- ------------------- -------------------
Parameter $\mu_6$ $\mu_7$ $\mu_8$ $\nu^{-1}$
True 40.2mN 87.9mN – 4.54 mN${}^{2}$
$u=7$ 40.5 (37.7, 43.5) 91.2 (86.6, 95.9) – 3.85 (3.13, 4.81)
$u=8$ 36.3 (32.4, 40.2) 9.6 (4.7, 15.7) 86.7 (80.3, 92.7) 3.22 (2.57, 4.18)
$u=7$ & ${\mu_{\min}}=15$ 40.5 (37.8, 40.2) 91.3 (86.8, 95.7) – 3.90 (3.14, 4.92)
$u=8$ & ${\mu_{\min}}=15$ 35.7 (31.1, 40.0) 15.7 (15.0, 20.7) 80.4 (71.4, 86.6) 3.23 (2.60, 4.09)
--------------------------- ------------------- ------------------- ------------------- -------------------
: Expected MUTF median and 95% credible interval estimates for MUs with high excitation threshold from the true ($u^*=7$) and MAP ($\hat{u}=8$) models, with and without post-process truncation (${\mu_{\min}}=15$mN) on data set D1.[]{data-label="tab:D1muEst"}
![Predictive density (thick line) at st
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uality follows from the assumption that $n\ell \geq 2^{12}d\log d$.
### Proof of Lemma \[lem:prob\_bottomlbound\]
Without loss of generality, assume that $\i < i$, i.e., $\ltheta^*_{\i} \leq \ltheta^*_i$. Define $\Omega$ such that $\Omega = \{j: j\in S, j \neq i,\i\}$. For any $\beta_1 \in [0,(\ell-2)/\ell]$, define event $E_{\beta_1}$ that occurs if in the randomly chosen set $S$ there are at most ${\left \lfloor{\ell\beta_1} \right \rfloor}$ items that have preference scores less than $\ltheta^*_i$, i.e., $$\begin{aligned}
\label{eq:bl_prob_7}
E_{\beta_1} \; \equiv \; \Big\{\textstyle\sum_{j \in \Omega} \I_{\{\ltheta^*_i > \ltheta^*_j\}} \leq {\left \lfloor{\ell\beta_1} \right \rfloor} \Big\} \;.\end{aligned}$$ We have, $$\begin{aligned}
\label{eq:bl_prob_1}
&&\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) \;>\; \kappa - \ell \;\Big|\; i,\i \in S\Big] \nonumber\\
&>& \P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;\Big|\; i,\i \in S; E_{\beta_1} \Big] \P\Big[E_{\beta_1} \;\Big|\; i,\i \in S\Big]\end{aligned}$$ The following lemma provides a lower bound on $\P[\sigma^{-1}(i),\sigma^{-1}(\i) > \kappa - \ell \;|\; i,\i \in S; E_{\beta_1}]$.
\[lem:bl\_prob1\] Under the hypotheses of Lemma \[lem:prob\_bottomlbound\], $$\begin{aligned}
\label{eq:bl_prob2}
\P\Big[\sigma^{-1}(i),\sigma^{-1}(\i) \; > \; \kappa - \ell \;\Big|\; i,\i \in S; E_{\beta_1} \Big] \;\geq\; \frac{e^{-4b}(1-{\left \lfloor{\ell\beta_1} \right \rfloor}/\ell)^2}{2}\frac{\ell^2}{\kappa^2}\;.\end{aligned}$$
Next, we provide a lower bound on $\P[E_{\beta_1} \;|\;i,\i \in S]$. Fix $i,\i$ such that $i,\i \in S$. Selecting a set uniformly at random is probabilistically equivalent to selecting items one at a time uniformly at random without replacement. Without loss of generality, assume that $i,\i$ are the $1$st and $2$nd pick. Define Bernoulli random variables $Y_{\j}$ for $ 3 \leq \j \leq \kappa$ corresponding to the outcome of the $\j$-th random pick from the set of $(d-\j-1)$ items to generate the set $\Omega$ such that $Y_{\j} = 1$ if an
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, in this limit, the dynamics becomes reversible regardless of $m$. For low temperatures, one can write $\cosh \beta x \approx \tfrac{1}{2}{\rm e}^{\beta |x|}$ to find $$\label{limZ-T3}
\!\!\!\frac{\mathcal Z_{\pm}(\lambda_f)}{\mathcal Z(\lambda_i)} \!\to \!
\lim_{\beta\to\infty} \!\!
\left[\!(1\!-\!\delta_{m0}) {\rm e}^{-\tfrac{ \beta \hbar \omega_0 (1 \mp 1)}{2} }
\!\!+\!\! \sum_{n = 0}^{\infty} {\rm e}^{- \frac{\beta \hbar}{2} \Phi_n^m} \!\right]\!,$$ where we have defined $$\label{funcphi}
\Phi_n^m := \nu(2 n + m) + \omega_0 - \sqrt{(\omega_{0} \! \mp \! m\nu)^{2} \! +\!
\Omega^{2}\left|f_n^m\right|^{2}} \, .$$ From this, we can analyze individually the AJC and JC cases. For the AJC and the carrier $m = 0$, it is easy to see that $\Phi_0^m < 0, \, \forall m$. As a result, $$\label{limZ-T1}
\lim_{\beta \to \infty}\frac{\mathcal Z_{-}(\lambda_f)}{\mathcal Z(\lambda_i)} =
\infty, \,\,\, \forall m,$$ showing that for $\hat{\mathcal{H}}^{(m)}_{-}$ in Eq. (\[hamrwa\]) and $\hat{\mathcal{H}}^{(0)}$ in Eq. (\[hamct\]) the NL Eq. (\[nlfinal\]) always diverges when $\beta \to \infty$. For the JC case, we must give a closer look at the function $ \Phi_{n}^{m}$. From Eq. (\[limZ-T3\]), and remembering that the case $m=0$ was already analyzed in Eq. (\[limZ-T1\]), $$\label{limZ-T}
\frac{\mathcal Z_{+}(\lambda_f)}{\mathcal Z(\lambda_i)} \to
1 +
\lim_{\beta\to\infty} \sum_{n = 0}^{\infty} {\rm e}^{- \frac{\beta \hbar}{2} \Phi_n^m}.$$ If, for a given $m$, at least one of the $\Phi_n^m$ appearing in Eq. (\[limZ-T\]) is negative, the above limit diverges and $\mathcal L \to \infty$. On the other hand, provided $\Phi_n^m \ge 0$ for all $n$, then $$\label{limZ-T2}
\frac{\mathcal Z_{+}(\lambda_f)}{\mathcal Z(\lambda_i)} \to k + 1,$$ where $k$ is the number of times $\Phi_n^m$ equals zero. Consequently, $\mathcal L \to {\rm ln}(1 + k)$ for the JC case. For the parameters chosen in Fig. \[fig3L3\], the JC case corresponds to $\Phi_n^m \ge 0$ and the limit in Eq. (\[limZ-T2\]) holds with $k = 0$, [i.e.]{}, no dive
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athclap{\alpha \in R_{G,+}}} \left( 1 + e^{-\alpha} + e^{-2\alpha} + \ldots \right) \\
&= \sum_{\mathclap{\beta \in \Lambda^*_G}} \phi_{R_{G,+}}(\beta) e^{-\beta},
\end{aligned}$$ where $\phi_{R_{G,+}}$ is the *Kostant partition function* given by the formula $$\label{kostant partition function}
\phi_{R_{G,+}}(\beta) = \# \{ (x_j) \in {\mathbb Z}^{{\lvertR_{G,+}\rvert}}_{\geq 0} : \sum_j x_j \alpha_j = \beta \}.$$ That is, $\phi_{R_{G,+}}$ counts the number of ways that a weight can be written as a sum of positive roots (this number is always finite since the positive roots span a proper cone). It follows directly from and and that $$\begin{aligned}
\operatorname{ch}V_{G,\lambda}
= &\sum_{w \in W_G} \det(w) \sum_{\beta \in \Lambda^*_G} \phi_{R_{G,+}}(\beta) e^{w(\lambda + \rho) - \rho - \beta} \\
= &\sum_{\beta \in \Lambda^*_G} \sum_{w \in W_G} \det(w) \, \phi_{R_{G,+}}(w(\lambda + \rho) - \rho - \beta) e^\beta.\end{aligned}$$ In other words, the multiplicity of a weight $\beta$ in an irreducible representation $V_{G,\lambda}$ is given by the well-known *Kostant multiplicity formula* [@kostant59], $$\label{kostant multiplicity formula}
m_{T_G,V_{G,\lambda}}(\beta) = \sum_{w \in W_G} \det(w) \, \phi_{R_{G,+}}(w(\lambda + \rho) - \rho - \beta).$$
For any fixed group $G$, the Kostant partition function can be evaluated efficiently by using Barvinok’s algorithm [@barvinok94], since it amounts to counting points in a convex polytope in an ambient space of fixed dimension. Therefore, weight multiplicities for fixed groups $G$ can be computed efficiently. This idea has been implemented by Cochet [@cochet05] to compute weight multiplicities for the classical Lie algebras (using the method presented in [@welledabaldonibeckcochetetal06] instead of Barvinok’s algorithm). We remark that the problem of computing weight multiplicities is of course the special case of where $H$ is the maximal torus $T_G \subseteq G$.
Weight Multiplicities as a Single Partition Function {#weight-multiplicities-as-a-single-par
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or all $i$ and $Z_i\otimes_A R$ for all even integer $i$.
- $m $ maps $A_i\otimes_A R$ into $B_i\otimes_A R$ for all $i$.
- $m $ maps $W_i\otimes_A R$ into $(X_i\cap Z_i)\otimes_A R$ for all even integer $i$.
- $m $ maps $B_i^{\perp}\otimes_A R$ into $A_i^{\perp}\otimes_A R$ for all odd integer $i$.
Then, by Lemma 3.1 of [@C1], $\underline{M}^{\prime}$ is represented by a unique flat $A$-algebra $A(\underline{M'})$ which is a polynomial ring over $A$ of $2n^2$ variables. Moreover, it is easy to see that $\underline{M}'$ has the structure of a scheme of rings since $\underline{M}'(R)$ is closed under addition and multiplication.
Then the functor $\underline{M}$ is equivalent to the functor $1+\underline{M}^{\prime}$ as subfunctors of $\mathrm{Res}_{B/A}\mathrm{End}_B(L)$, where $(1+\underline{M}^{\prime})(R)=\{1+m : m \in \underline{M}^{\prime}(R) \}$ for a commutative flat $A$-algebra $R$. For a detailed explanation about this, we refer to Remark 3.1 of [@C2]. This fact induces that the functor $\underline{M}$ is also represented by a unique flat $A$-algebra $A[\underline{M}]$ which is a polynomial ring over $A$ of $2n^2$ variables. Moreover, it is easy to see that $\underline{M}$ has the structure of a scheme of monoids since $\underline{M}(R)$ is closed under multiplication.
We can therefore now talk of $\underline{M}(R)$ for any (not necessarily flat) $A$-algebra $R$. Before describing an element of $\underline{M}(R)$ for a non-flat $A$-algebra $R$, we explicitly, in terms of matrices, describe an element of $\underline{M}(R)$ for a flat $A$-algebra $R$. By using our chosen basis of $L$, each element of $\underline{M}(R)$ is written as $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \mathrm{~together~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$ Here,
1. When $i\neq j$, $m_{i,j}$ is an $(n_i \times n_j)$-matrix with entries in $B\otimes_AR$.
2. When $i=j$, $m_{i,i}$ is an $(n_i \times n_i)$-matrix with entries in $B\otimes_AR$ such that $$m_{i,i}=\left\{
\begin{arra
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bda\}-5.$$
(-1,.5) node\[above left\] [$\mathcal{C}_1$]{} to\[out=-60,in=180\] (0,0) to\[out=180,in=60\] (-1,-.5); (-1,0)–(.5,0) node\[right\] [$\mathcal{C}_2$]{};
(2.5,0)–(5.5,0) node\[right\] [$E_\nu$]{}; (3,0) circle \[radius=.1\] node\[below left\] [$\frac{1}{2}(1,1)$]{}; (5,0) circle \[radius=.1\] node\[below\] [$\frac{1}{3}(1,1)$]{}; (3,-1)–(3,1) node\[right\] [$\mathcal{C}_2$]{}; (4,-1)–(4,1) node\[right\] [$\mathcal{C}_1$]{};
The jumping values for $\lambda\in [0,1)$ are provided below $$\cM(\cC^\lambda)=
\begin{cases}
\CC\{x,y\} & \text{ for } \lambda\in [0,\frac{5}{12}),\\
\mathfrak{m}=(x,y) & \text{ for } \lambda\in [\frac{5}{12},\frac{1}{2}),\\
\CC\{x,y\} & \text{ for } \lambda\in [\frac{1}{2},\frac{2}{3}),\\
\mathfrak{m} & \text{ for } \lambda\in [\frac{2}{3},\frac{5}{6}),\\
(x^2,y) & \text{ for } \lambda\in [\frac{5}{6},\frac{11}{12}),\\
\mathfrak{m}^2 & \text{ for } \lambda\in [\frac{11}{12},1).\\
\end{cases}$$ As mentioned in Remark \[rem:independenciak\] the sequence $\cM(C^\lambda)$ is decreasing for $\lambda\in [0,\frac{1}{2})$. However this is not true for $\lambda\in [0,1)$.
Global study: irregularity of cyclic coverings {#sec:global}
==============================================
Let $w_0,w_1,w_2$ be three pairwise coprime positive integers and let $X:=\PP_w^2$ be the weighted projective plane with $w=(w_0,w_1,w_2)$. The section is split into three parts. First, an interpretation of $H^0(Y,\cO_Y(D'))$ for a blow-up $Y$ of $X$ and $\QQ$-divisors in terms of quasi-polynomials. Then the second cohomology groups of the cover are studied using Serre’s duality. Finally a formula for the irregularity is described with the help of the Euler characteristic and the Riemann-Roch formula on singular normal surfaces.
Global sections and weighted blow-ups
-------------------------------------
Consider $\pi: Y \to X$ a composition of $s$ weighted blowing-ups with exceptional divisors $E_1,\ldots,E_s$.
\[lemma:h0-weighted\] Given $D \in \operatorname{Weil}_{\QQ}(X)$, the space $H^0 (X,\!\
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] T\_[C,0]{}u=,u\_[\_-]{}=0,u(,,E\_[m]{})=0. where $$\tilde{\bf f} := {\bf f}-T_C(L({\bf g})).$$
By the Trotter’s formula, the semigroup $G(t)$ generated by $T_{C,0}$ is given by \[calceq2\] G(t) = \_[n]{}(T\_[B\_0]{}(t/n)T\_[A\_0]{}(t/n)T\_[-(+C I)]{}(t/n)T\_[K\_C]{}(t/n))\^n, where the convergence is uniform on compact $t$-intervals $[0,T]$. Note that the individual semi-groups $T_{B_0}(t)$, $T_{A_0}(t)$, $T_{-(\Sigma+C I)}(t)$ and $T_{K_C}(t)$ contributing to this expression can be computed explicitly (see section \[possol\]). Hence we get \[calceq4\] =&\_0\^G(t) dt\_0\^T G(t) dt\
=&\_0\^T\_[n]{} (T\_[B\_0]{}(t/n)T\_[A\_0]{}(t/n)T\_[-(+C I)]{}(t/n)T\_[K\_C]{}(t/n))\^ndt\
& \_0\^T(T\_[B\_0]{}(t/n\_0)T\_[A\_0]{}(t/n\_0)T\_[-(+C I)]{}(t/n\_0)T\_[K\_C]{}(t/n\_0))\^[n\_0]{}dt, for large enought $T$ and $n_0$. On the other hand, the semi-group $T_{K_C}$ generated by the bounded operator $K_C$ can be approximately computed from $$T_{K_C}(t)
\approx
\sum_{k=0}^{N_0} \frac{1}{k!}(tK_C)^k,$$ for large enough $N_0$.
We point out that this semi-group theory -based approach, unlike the one given in the previous section, does not require extra assumptions on cross-sections, like the ones imposed by the condition .
\[evocomp\]
Assume that $K$ is of the form $$(K\psi)(x,\omega,E)=\int_S\sigma(x,\omega',\omega,E)\psi(x,\omega',E)d\omega'.$$ Furthermore, as in Example \[desolex1\] we assume that $S_0=S_0(E)$ (independent of $x$) and we define $R(E):=\int_0^E{1\over{S_0(\tau)}}d \tau$, $\eta:=R(E)$, $r_m:=R(E_m)$ and $\tilde{I}:=R(I)=[0,r_{\rm m}]$. Let $$\begin{gathered}
v(x,\omega,\eta):=S_0(R^{-1}(\eta))\psi(x,\omega,R^{-1}(\eta)), \\[2mm]
\tilde\Sigma(x,\omega,\eta):=\Sigma(x,\omega,R^{-1}(\eta)),\\[2mm]
\tilde\sigma(x,\omega',\omega,\eta):=\sigma(x,\omega',\omega,R^{-1}(\eta)), \\
(\tilde K v)(x,\omega,\eta):=\int_S\tilde\sigma(x,\omega',\omega,\eta)
v(x,\omega',\eta)d\omega', \\
\tilde f(x,\omega,\eta):=S_0(R^{-1}(\eta))f(x,\omega,R^{-1}(\eta)), \\[2mm]
\tilde g(x,\omega,\eta):=S_0(R^{-1}(\eta))g(y,\omega,R^{-1}(\eta
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), -(2b-1)(2b'-1))$ is *unimodular of type II*. Thus we can write $$M_0=(\oplus_{\lambda}H_{\lambda})\oplus \left( Be_1'\oplus Be_2' \right)
\oplus \left(Be_3'\oplus Be_4'\right).$$ For this basis, the image of a fixed element of $F_j$ in the special fiber of the smooth integral model associated to $L^j$ is $$\begin{pmatrix} id&0 &0\\ 0 &\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&\pi x_j-2z_j^{\ast} &1 +\pi x_j & 2 z_j^{\ast}\\0&0& 0 & 1 \end{pmatrix} &0 \\ 0& 0 &id \end{pmatrix}.$$ Here, $id$ in the $(1,1)$-block corresponds to the direct summand $(\oplus_{\lambda}H_{\lambda})$ of $M_0$ and the diagonal block $\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&\pi x_j-2z_j^{\ast} &1 +\pi x_j & 2 z_j^{\ast}\\0&0& 0 & 1 \end{pmatrix}$ corresponds to $A(2b(2b-1), 2b'(2b'-1), -(2b-1)(2b'-1))\oplus A(1, 2(b+b'), 1)$ with a basis $(e_1', e_2', e_3', e_4')$.
We now describe the image of a fixed element of $F_j$ in the even orthogonal group associated to $M_0''$, where $M_0''$ is a Jordan component of $Y(C(L^j))=\bigoplus_{i \geq 0} M_i''$. There are 3 cases depending on whether $b+b'$ is a unit or not, and whether $M_1=\oplus H(1)$ (possibly empty) or $M_1=A(4b'', 2\delta, \pi) \oplus (\oplus H(1))$ with $b''\in A$. We will see in Step (iii) below that the case $M_1=A(4b'', 2\delta, \pi) \oplus (\oplus H(1))$ with $b''\in A$ is reduced to the case $M_1=\oplus H(1)$ (possibly empty).\
1. Assume that $M_1=\oplus H(1)$ and $b+b'\in (2)$. Then $$M_0''= \left( (\pi) e_1'\oplus (\pi) e_2' \right)\oplus
\left((2)e_3'\oplus Be_4'\right) \oplus (\oplus_{\lambda}\pi H_{\lambda})\oplus M_2,$$ as explained in the argument (i) of Step (1) in the construction of $\psi_j$. For this basis, the image of a fixed element of $F_j$ in the orthogonal group associated to $M_0''/\pi M_0''$ is $$T_1=\begin{pmatrix} \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0& (x_j)_1 &1& (z_j^{\ast})_1\\0&0& 0 & 1 \end{pmatrix} &0 \\ 0 & id \end{pmatrix}.$$ Here, $(z_j^{\ast})_1$ (resp. $(x_j)_1$) is in $R$ such that $z_j^{\ast}=(z_j^{\ast})_1+\pi
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$k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f04.eps "fig:"){width="50mm"}
In Fig. \[fig.4\] an evident picture of behavior of the imaginary part of the wave function close to point $r=0$ with continuous change of the wave vector $k$ and the parameter $C$ is shown.
![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f06.eps "fig:"){width="80mm"} ![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f07.eps "fig:"){width="80mm"}
Note, that according to (\[eq.2.4.9\]), the condition (\[eq.3.2.1.5\]) can bring to not zero values of the radial wave function $\chi_{l=0}^{(1)}(k,r)$ at $r \to 0$ and can give discontinuity of the total wave function. However, a variation of the phase of $S_{l=0}^{(1)}$ does not change the form of the potential $V_{1}(r)$, which remains zero and reflectionless. In other words, the reflectionless potential $V_{1}(r)$ allows an arbitrariness in a choice of boundary conditions for the wave function at point $r=0$, and the chosen boundary conditions define the shape of the total wave function and a process proceeding in the field of the potential $V_{1}(r)$. There is a similar situation for the potential $V_{2}(r)$, which remains reflectionless with the variation of the S-matrix phase.
Now let’s analyze the form of the wave function (\[eq.3.2.1.2\]) in asymptotic region. According to (\[eq.3.1.1\]), $W(r) \to 0$ at $r \to +\infty$ and we obtain: $$\chi_{l=0}^{(2)}(k,r) =
\bar{N}_{2} \biggl(e^{-ikr} - S_{l=0}^{(2)} e^{ikr}\biggr).
\label{eq.
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node comprised of 40 $2.4\,{\rm GHz}$ intel Skylake processors. We measure the wall clock time using the [MPI\_Wtime]{} function, after a call to [MPI\_Barrier]{} to synchronize all processors. We run the job with [SLURM –exclusive]{} option to make sure that other jobs do not interfere with ours.
For the weak scaling test, we divide the whole computational domain into $N_{\rm core}$ subdomains consisting of $64^3$ cells each. In the [Athena++]{} terminology, the whole domain and subdomain are referred to as [Mesh]{} and [MeshBlock]{}, respectively. We assign each [MeshBlock]{} to a single processor, while varying $N_{\rm core}$ from $1$ to $4096$. The corresponding size of [Mesh]{} varies from $64^3$ to $1024^3$. In each run, we call our Poisson solver, together with the MHD solver for comparison, 100 times, and then measure the wall clock time per cycle $t_{\rm wall}$ for various steps. We repeat the calculations five times in order to avoid unusual runs due to stale nodes.
Figure \[fig:scaling\] plots the mean values $\left\langle t_{\rm wall} \right\rangle$ of the wall clock times per cycle as functions of $N_{\rm core}$ for the Cartesian (left) and cylindrical (right) grids. As noted earlier, our Poisson solver requires to run the interior solver twice and the boundary solver once. The total time taken by the Poisson solver (triangles) is dominated by the interior solver (squares) rather than the boundary solver (circles). While the time taken by the MHD solver is comparable between Cartesian and cylindrical grids, the Poisson solver is more efficient in the cylindrical grid. This is because Cartesian coordinates has no inherent periodic direction and thus requires more operations to implement the open boundary conditions. In addition, the Cartesian grid has two more boundaries than the cylindrical grid and thus needs more boundary-to-boundary interactions in the boundary solver. Notwithstanding these differences, the Poisson solver in both Cartesian and cylindrical grids takes less time than the MHD so
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c{\ell^2}{\kappa^2}.\end{aligned}$$ Since, the above inequality is true for any fixed $i,\i$ and $j \in \Omega$ such that event $E$ holds, it is true for random indices $i,\i$ and $j \in \Omega$ such that event $E$ holds, hence the claim is proved.
### Proof of Lemma \[lem:bl\_prob2\]
Let $\widehat{\sigma}$ denote a ranking over the items of the set $S$ and $\P[\widehat{\sigma}]$ be the probability of observing $\widehat{\sigma}$. Let $$\begin{aligned}
\widehat{\Omega}_1 = \Big\{ \widehat{\sigma}: \widehat{\sigma}^{-1}(i_{\min_1}) = i_1, \widehat{\sigma}^{-1}(i_{\min_2}) = i_2 \Big\} \;\; \text{and} \;\; \widehat{\Omega}_2 = \Big\{ \widehat{\sigma} : \sigma^{-1}(i_{\min_1}) = 1, \sigma^{-1}(i_{\min_2}) = 2 \Big\}.\end{aligned}$$ Now, take any ranking $\widehat{\sigma} \in \widehat{\Omega}_1$ and construct another ranking $\widetilde{\sigma}$ from $\widehat{\sigma}$ as following. If $i_1 =2, i_2 = 1$, then swap the items at $i_1$-th and $i_2$-th position in ranking $\widehat{\sigma}$ to get $\widetilde{\sigma}$. Else, if $i_1 < i_2$, then first: swap items at $i_1$-th position and $1$st position, and second: swap items at $i_2$-th position and $2$nd position, to get $\widetilde{\sigma}$; if $i_2 < i_1$, then first: swap items at $i_2$-th position and $2$nd position, and second: swap items at $i_1$-th position and $1$st position, to get $\widetilde{\sigma}$.
Observe that $\P[\widetilde{\sigma}] \leq \P[\widehat{\sigma}]$ and $\widetilde{\sigma}_1^\ell \in \widehat{\Omega}_2$. Moreover, such a construction gives a bijective mapping between $\widehat{\Omega}_1$ and $\widehat{\Omega}_2$. Hence, the claim is proved.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the anonymous reviewers for their constructive feedback. This work was partially supported by National Science Foundation Grants MES-1450848, CNS-1527754, and CCF-1553452.
---
abstract: 'Notwithstanding the big efforts devoted to the investigation of the mechanisms responsible for the high-energy ($E>100$ MeV) $\gamm
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| 0.792382
|
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percolation. , 29:577–623, 2001.
J. Jacod and A.N. Shiryaev. . Springer-Verlag, Berlin, 1987.
J. Norris and A.G. Turner. Hastings-levitov aggregation in the small-particle limit. 2011. submitted.
L.P.R. Pimentel. Multitype shape theorems for first passage percolation models. , 39(1):53–76, 2007. submitted.
M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. , (81):73–205, 1995.
---
abstract: |
For fixed compact connected Lie groups H $\subseteq$ G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok’s algorithm for counting integral points in polytopes.
The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be \#P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.
author:
-
-
-
bibliography:
- 'multiplicities.bib'
title: |
Computing Multiplicities\
of Lie Group Representations
---
Introduction {#section:introduction}
============
The decomposition of Lie group representations into irreducible sub-representations is a fundamental problem of mathematics with a variety of applications to the sciences
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lic codes in Theorem 2.1.
[**Theorem 2.1** ]{}[@Boucher1; @Boucher2] *Let $C$ be a skew cyclic code ($\sigma$-cyclic code) of length $n$ over $\mathbb{F}_q$ generated by a right divisor $g(x)=\sum_{i=0}^{n-k-1}g_ix^i+x^{n-k}$ of $x^n-1$. Then\
* *(i) The generator matrix of $C$ is given by*
$$\left(
\begin{array}{ccccccc}
g_0 & \cdots & g_{n-k-1}& 1 & 0 & \cdots & 0\\
0 & \sigma(g_0) & \cdots & \sigma(g_{n-k-1}) & 1 & \cdots & 0\\
0 & \ddots & \ddots& & & \ddots& \vdots\\
\vdots & & \ddots &\ddots &\cdots &\ddots &0\\
0& \cdots & 0& \sigma^{k-1}(g_0) & \cdots & \sigma^{k-1}(g_{n-k-1})& 1\\
\end{array}
\right)$$
*and $\mid C\mid=q^{n-{\rm deg}(g(x))}$.\
*
*(ii) Let $x^n-1=h(x)g(x)$ and $h(x)=\sum_{i=0}^{k-1}h_ix^i$. Then $C^\perp$ is also a skew cyclic code of length $n$ generated by $\widetilde{h}(x)=x^{{\rm deg}(h(x))}\varphi(h(x))=1+\sigma(h_{k-1})x+\cdots+\sigma^k(h_0)x^k$, where $\varphi$ is an anti-automorphism of $\sigma$ defined as $\varphi(\sum_{i=0}^ta_ix^t)=\sum_{i=0}^tx^{-i}a_i$, where $ \sum_{i=0}^ta_ix^i\in R$. The generator matrix of $ C^\perp$ is given by* $$\left(
\begin{array}{ccccccc}
1 & \sigma(h_{k-1}) & \cdots& \sigma^k(h_0) & 0 & \cdots & 0\\
0 & 1 & \sigma^2(h_{k-1}) & \cdots & \sigma^{k+1}(h_0) & \cdots & 0\\
0 & 0 & \ddots& & & \ddots& 0\\
\vdots & & \ddots &\ddots &\cdots &\ddots &\vdots\\
0& \cdots & 0& 1 & \sigma^{n-k}(h_{k-1}) & \cdots& \sigma^{n-1}(h_0)\\
\end{array}
\right)$$\
*and $\mid C^\perp\mid=q^k$.*\
*(iii) For $c(x)\in R$, $c(x)\in C$ if and only if $c(x)h(x)=0$ in $R$.\
*
*(iv) $C$ is a cyclic code of length $n$ over $\mathbb{F}_q$ if and only if the generator polynomial $g(x)\in \mathbb{F}_{p^d}[x]/(x^n-1)$.*
The monic polynomials $g(x)$ and $h(x)$ in Theorem 2.1 are called the *generator polynomial* and the *parity-check polynomial* of the skew cyclic code $C$, respectively.
[**Theorem 2.2** ]{} *Let $C$ be a skew cyclic code with the generator polynomial $g(x)$ and the check polynomial $h(x)$. Then
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iterated adjoints {#sec:fun}
===============================
In particular, \[thm:stab-op\] implies that we can characterize $\Phi$-stability in terms of iterated adjoints to constant morphism morphisms. In this section we describe what this looks like more concretely in the pointed and stable cases.
\[prop:char-ptd\] The following are equivalent for a derivator .
1. The derivator is pointed.\[item:p1\]
2. The morphism $\emptyset_!\colon{\sD}^\emptyset\to{\sD}$ is a right adjoint.\[item:p2\]
3. The left Kan extension morphism $1_!\colon{\sD}\to{\sD}^{[1]}$ along the universal cosieve $1\colon\bbone\to[1]$ is a right adjoint.\[item:p3\]
4. For every cosieve $u\colon A\to B$ the left Kan extension morphism $u_!\colon{\sD}^A\to{\sD}^B$ is a right adjoint.\[item:p4\]
5. The morphism $\emptyset_\ast\colon{\sD}^\emptyset\to{\sD}$ is a left adjoint.\[item:p5\]
6. The right Kan extension morphism $0_\ast\colon{\sD}\to{\sD}^{[1]}$ along the universal sieve $0\colon\bbone\to[1]$ is a left adjoint.\[item:p6\]
7. For every sieve $u\colon A\to B$ the right Kan extension morphism $u_\ast\colon{\sD}^A\to{\sD}^B$ is a left adjoint.\[item:p7\]
By duality it suffices to show the equivalence of the first four statements. The implication \[item:p1\] implies \[item:p4\] is [@groth:ptstab Cor. 3.8]. Since the empty functor $\emptyset\colon\emptyset\to\bbone$ is a cosieve it remains to show that \[item:p2\] or \[item:p3\] imply \[item:p1\]. The case of \[item:p2\] is taken care of by the proof of [@groth:ptstab Cor. 3.5]. In the remaining case, if $1_!$ is a right adjoint it preserves all limits and hence terminal objects. Since the terminal object in ${\sD}([1])$ looks like $(\ast\to\ast)$, this has by [@groth:ptstab Prop. 1.23] to be isomorphic to $1_!(\ast)\cong(\emptyset\to\ast)$. Evaluating this isomorphism at $0$ shows that is pointed.
These additional adjoint functors are sometimes referred to as **(co)exceptional inverse image functors** (see [@groth:ptstab §3]).
\[rmk:C-inverse\] In [@groth:ptstab
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dels?
P.S. I use laravel 5.4
Brands Table:
id | name
-----------
1 | Adidas
2 | Nike
3 | Puma
Categories Table:
id | name
-----------
1 | Clothes
2 | Luxury
3 | Sport Wear
User_Interests Table:
id | user_id | type | reference_id
-----------------------------------------
1 | 113 | 'brand' | 2
1 | 113 | 'brand' | 3
2 | 113 |'category'| 3
3 | 224 | 'brand' | 1
A:
Yes - read the docs on polymorphic relations.
It works similar to what you have, but you might have columns called interestable_id and interestable_type (you can configure this to something else if you prefer). The interestable_type will literally be the string representation of the referenced class name, like App\Brand or App\Category and interestable_id will be the primary key of that model.
Best part is as it's all done through Eloquent it's already good to go with associations, eager-loading etc.
Edit: I should add that your 3 table setup is still the best way to represent this structure, just making suggestions on the column names you use and how to hook it in with Eloquent.
Q:
How do I translate the first two lines of Befriending the Learned? 入國而不存其士，見賢而不急
It's Mozi, first two lines of Befriending the Learned:
入國而不存其士，
見賢而不急，
I would translate it:
if one enters the country and doesn't have his soldiers(?), meets the valuable and doesn't have to worry.
It doesn't make sense? I agree, on ctext.org there is a different translation, Can someone explain me, possibly word by word?
A:
The original text is:
入國而不存其士，則亡國矣。見賢而不急，則緩其君矣。
could be translated word by word as:
If one governs the country but doesn't preserve his able and virtuous persons,
the country will perish.
If one sees (discovers) the able and virtuous persons but doesn't hurry up (to appoint them),
they will neglect their monarch.
治国而不优待贤士，国家就会灭亡。见到贤士而不急于任用，他们就�
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ed asymptotic solutions using a global numerical solution. The solution for the evolution of the scale factor in the quantum limit has the form $$a=\xi(-\tau+\beta)^p
\label{solution1}$$ where $p=2/(4-n)$. The solution in the classical limit we obtain putting simply $l=2$ what gives $D=1$ and $p=1/2$. The constants of integration $\xi$ and $\beta$ we fix with the use of a numerical solution applying formula $$\begin{aligned}
\xi = a|_{\tau} \left[-\frac{a'|_{\tau}}{p \ a|_{\tau} } \right]^p \
\ \text{and} \ \ \beta = -\tau-p\frac{a|_{\tau}}{a'|_{\tau}}.
\label{fixing}\end{aligned}$$ The value of the conformal time $\tau$ must be chosen in a proper way for the given regions. We will discuss this question in more details later.
Our point of reference is the numerical solution. To make this description complete we must choose the proper boundary conditions for the numerical solution. We use here the condition for the Hubble radius which must be larger than the limiting value $a_i$ [@Lidsey:2004ef], what gives us $$k \simeq |H|a < \frac{a}{a_i} \ \ \Rightarrow \ \ |H|a_i < 1.
\label{condition1}$$ The next condition requires that the scale factor must be greater than $a_i$ at the bounce. It is fulfilled taking $a|_{\tau_0} = a_{*}$ for some value of the conformal time $\tau_{0}$. In fact the conformal time is unphysical variable and their value can be chosen arbitrary. The physical outcomes do not depend on coordinates because the theory is invariant under local diffeomorphisms. So as an example we can choose $$\begin{aligned}
a|_{\tau_0=-4} &=& a_{*} \label{init1} \\
a'|_{\tau_0=-4} &=& l_{\text{Pl}} \label{init2}\end{aligned}$$ The chosen value of $a'|_{\tau_0=-4}$ holds the condition (\[condition1\]). Namely for $j=100$ we have $|H_*|a_i = 0.084 < 1$. The numerical solution is shown in Fig. \[fig:solution1\] as a black line.
![Numerically calculated evolution of the scale factor (top black line) and the Hubble parameter (bottom black line) for the model with $j=100$, $l=3/4$ and with initial conditio
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|
$+$ $OS$
int4 $+$ $+$ $+$ $OS$
int5 $+$ $+$ $+$ $OS$
int6 $+$ $+$ $+$ $OS$
int7 $+$ $+$ $OS$ $OS$
int8 $+$ $+$ $OS$ $OS$
int9 $-$ $+$ $OS$ $OS$
int10 $-$ $-$ $+$ $OS$
int11 $-$ $+$ $-$ $OS$
int12 $-$ $+$ $-$ $OS$
int13 $+$ $+$ $+$ $+$
int14 $+$ $+$ $+$ $OS$
: An overview of the experiments[]{data-label="Table:evaluation"}
We experimented with different bit-sizes in the translation to SAT and different classes of functions for the $prem$ functions in Proposition \[prop:rem\_imp\]. As $conc$ functions, the identity function was used. Table \[Table:evaluation\] shows the results for the considered settings, $+$ denotes that non-termination is proven successfully, $-$ denotes that non-termination could not be proven and $OS$ denotes that the computation went out of stack. The considered settings are 3 and 4 as bit-sizes and $linear$ and $max2$ as forms for the symbolic $prem$-functions. The $linear$ class is a weighted sum of each argument. The $max2$ class contains a weighted term for each multiplication of two arguments. The analysis time is between $1$ and $20$ seconds for all programs and settings.
Table \[Table:evaluation\] shows non-termination can be proven for any program of the benchmark when choosing the righ
| 542
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| 0.772707
|
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|
l} \; \mbox{if and only if}\; i = k\; \mbox{and}\; j=l.$$
In the array, the rows are the $\mathcal{R}$-classes of $\mathcal{B}$, the columns are the $\mathcal{L}$-classes and the $\mathcal{H}$-classes are points. There is only one $\mathcal{D}$-class; that is, $\mathcal{B}$ is a bisimple monoid (hence simple).
Following [@ruskuc], we start by introducing some basic subsets of $\mathcal{B}$, $$\begin{array}{rcl}D &= &\{a^{i}b^{i} : i \geq 0\} \:............................................ \mbox{the \emph{diagonal}}.\\
L^{p} &=& \{a^{i}b^{j} : 0 \leq j \leq p, i \geq 0\}\; \mbox{for} \; p \geq 0\: ............. \mbox{the \emph{left strip} (determined by $p$)}. \end{array}$$ For $0 \leq q \leq p$ we define the *triangle* $$T_{q,p} = \{a^{i}b^{j} : q \leq i \leq j < p\}.$$ For $i,m \geq 0$ and $d > 0$ we define the rows $$\Lambda_{i} = \{a^ib^j : j \geq 0\},\ \Lambda_{i,m,d} = \{a^{i}b^{j} : d|j- i, j \geq m\}$$ and in general for $I\subseteq \{0, \ldots ,m -1\}$, $$\Lambda_{I,m,d} =\bigcup_{i \in I}\Lambda_{i,m,d}=\{a^{i}b^{j}:i\in I, d|j-i , j\geq m\}.$$ For $p \geq 0, d > 0, r \in [d] = \{0,\ldots, d-1\}$ and $P \subseteq[d]$ we define the *squares* $$\Sigma_{p} = \{a^{i}b^{j}:i,j\geq p\}, \ \Sigma_{p,d,r} = \{a^{p+r+ud}b^{p+r+vd} : u, v\geq 0\},$$ $$\Sigma_{p,d,P}=\bigcup_{r\in P}\Sigma_{p,d,r}=\{a^{p+r+ud}b^{p+r+vd} : r \in P, u, v \geq 0\}.$$ It is worth pointing out that in [@ruskuc] it was shown that a subsemigroup of $\mathcal{B}$ is inverse if and only if it has the form $F_D \cup \Sigma_{p,d,P}$ where $F_D$ is a finite subset of the diagonal (which may be empty). The function $\widehat{} :\mathcal{B}\longrightarrow \mathcal{B}$ defined by $a^{i}b^{j} \rightarrow \widehat{a^{i}b^{j}}= a^{j}b^{i}$ is an anti-isomorphism. Geometrically it is the reflection with respect to the main diagonal.
{height="2.2in"} ![The subsets $\Lambda_{\{1,4,7\},7,3}, \widehat{\Lambda}_{\{1,4,6\},7,3},\
| 543
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| 0.778102
|
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|
72---C71 120.2 (4)
C10---C9---H9 120.3 C73---C72---H72 119.9
C9---C10---C11 120.8 (5) C71---C72---H72 119.9
C9---C10---H10 119.6 C72---C73---C74 120.2 (4)
C11---C10---H10 119.6 C72---C73---H73 119.9
C12---C11---C10 120.2 (5) C74---C73---H73 119.9
C12---C11---H11 119.9 C75---C74---C73 119.6 (5)
C10---C11---H11 119.9 C75---C74---H74 120.2
C11---C12---C7 119.7 (5) C73---C74---H74 120.2
C11---C12---H12 120.2 C74---C75---C70 120.4 (5)
C7---C12---H12 120.2 C74---C75---H75 119.8
P1---C13---P6 114.1 (2) C70---C75---H75 119.8
P1---C13---H13A 108.7 C76---N1---C77 121.0 (5)
P6---C13---H13A 108.7 C76---N1---C78 121.9 (4)
P1---C13---H13B 108.7 C77---N1---C78 117.1 (4)
P6---C13---H13B 108.7 O1---C76---N1 125.7 (5)
H13A---C13---H13B 107.6 O1---C76---H76 117.1
C19---C14---C15 118.8 (4) N1---C76---H76 117.1
C19---C14---P6 122.1 (3) N1---C77---H77A 109.5
C15---C14---P6 119.1 (4) N1---C77---H77B 109.5
C16---C15---C14 120.5 (4) H77A---C77---H77B 109.5
C16---C15---H15 119.7 N1---C77---H77C 109.5
C14---C15---H15 119.7 H77A---C77---H77C 109.5
C17---C16---C15 120.6 (5) H77B---C77---H77C 109.5
C17---C16---H16 119.7 N1---C78---H78A 109.5
C15---C16---H16 119.7 N1---C78---H78B 109.5
C16---C17---C18 119.7 (5) H78A---C78---H78B 109.5
C16---C17---H17 120.2 N1---C78---H78C 109.5
C18---C17---H17 120.2 H78A---C78---H78C 109.5
C19---C18---C17 120.0 (5) H78B---C78---H78C 109.5
C19---C18---H18 120.0 C79---N2---C81 125.7 (5)
C17---C18---H18 120.0 C79---N2---C80 120.2 (5)
C14---C19---C18 120.4 (4) C81---N2-
| 544
| 4,363
| 1,344
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|
$\underline{G}'$ and $G^{\circ}$ is the identity component of $G$. In our case, $G$ is the unitary group $\mathrm{U}(V, h)$, where $V=L\otimes_AF$. Since $\mathrm{U}(V, h)$ is connected, $G^{\circ}$ is the same as $G$ so that $[G:G^{\circ}]=1$.
Then based on Lemma 3.4 and Section 3.9 of [@GY], we finally have the following local density formula.
\[t52\] Let $f$ be the cardinality of $\kappa$. The local density of ($L,h$) is
$$\beta_L=f^N \cdot f^{-\mathrm{dim~} \mathrm{U}(V, h)} \#\tilde{G}(\kappa),$$
where $$N=N_H-N_M=\sum_{i<j}i\cdot n_i\cdot n_j+\sum_{\textit{i:even}} \frac{i+2}{2} \cdot n_i
+\sum_{\textit{i:odd}} \frac{i+3}{2} \cdot n_i+\sum_i d_i-\sum_{L_i:\textit{type I}}n_i.$$Here, $\#\tilde{G}(\kappa)$ can be computed explicitly based on Remark 5.3.(1) of [@C2] and Theorem \[t412\].
\[r53\]
As in Remark 7.4 of [@GY], although we have assumed that $n_i=0$ for $i<0$, it is easy to check that the formula in the preceding theorem remains true without this assumption.
The proof of Lemma \[l46\] {#App:AppendixA}
==========================
The proof of Lemma 4.5 is based on Proposition 6.3.1 in [@GY] and Appendix A in [@C2].\
We first state a theorem of Lazard which is repeatedly used in this paper. Let $U$ be a group scheme of finite type over $\kappa$ which is isomorphic to an affine space as an algebraic variety. Then $U$ is connected smooth unipotent group (cf. IV, $\S$ 4, Theorem 4.1 and IV, $\S$ 2, Corollary 3.9 in [@DG]).
For preparation, we state several lemmas.
\[la1\] (Lemma 6.3.3. in [@GY])
Let $1 \rightarrow X\rightarrow Y\rightarrow Z\rightarrow 1$ be an exact sequence of group schemes that are locally of finite type over $\kappa$, where $\kappa$ is a perfect field. Suppose that $X$ is smooth, connected, and unipotent. Then $1 \rightarrow X(R)\rightarrow Y(R)\rightarrow Z(R)\rightarrow 1$ is exact for any $\kappa$-algebra $R$.
Let $\tilde{M}$ be the special fiber of $\underline{M}^{\ast}$ and let $R$ be a $\kappa$-algebra. Recall that we have described an element and the multiplicatio
| 545
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| 2,550
| 0.778964
|
github_plus_top10pct_by_avg
|
1--3 times/week 846 0.85 0.48 −0.09, 1.78 843 0.003 0.003 −0.003, 0.009 845 0.60 0.34 −0.06, 1.26
4--6 times/week 157 −0.47 0.79 −2.01, 1.07 157 −0.003 0.005 −0.012, 0.007 158 −0.47 0.56 −1.56, 0.62
Daily 90 0.44 1.01 −1.55, 2.42 90 −0.001 0.006 −0.013, 0.011 91 0.50 0.71 −0.90, 1.90
2 \<weekly 578 Ref 576 Ref 578 Ref
1--3 times/week 846 **0.97** **0.48** **0.03, 1.91** 843 0.003 0.003 −0.002, 0.009 845 0.60 0.34 −0.07, 1.27
4--6 times/week 157 −0.44 0.79 −1.99, 1.10 157 −0.003 0.005 −0.012, 0.007 158 −0.54 0.56 −1.63, 0.56
Daily 90 0.37 1.02 −1.63, 2.37 90 −0.002 0.006 −0.014, 0.011 91 0.33 0.72 −1.08, 1.74
3 \<weekly 578 Ref 576 Ref 578 Ref
1--3 times/week 846 **1.02** **0.48** **0.08, 1.96** 843 0.004 0.003 −0.002, 0.010 845 0.63 0.34 −0.03, 1.30
4--6 times/week 157 −0.39 0.79 −1.93, 1.15 157 −0.002 0.005 −0.012, 0.007 158 −0.50 0.56 −1.59, 0.59
Daily 90 0.45 1.02 −1.54, 2.44 90 −0.001 0.006 −0.014, 0.011 91 0.39 0.71 −1.01, 1.79
4 \<weekly 578 Ref 576 Ref 578 Ref
1--3 times/week 846 0.60 0.38 −0.15, 1.35 843 0.002 0.003 −0.004, 0.007 845 0.40 0.26 −0.12, 0.91
4--6 times/week 157 −0.46 0.62 −1.68, 0.76 157 −0.003 0.005 −0.012, 0.006 158
| 546
| 3,681
| 159
| 399
| null | null |
github_plus_top10pct_by_avg
|
leotide polymorphisms stratified by gender in HCV-1 and HCV-2 infected patients receiving PEG-IFNα-RBV therapy with and without a RVR in a Chinese population in Taiwan**
**HCV-1** **HCV-2**
-------------------- ------------------ ------------ -------- -------------------- ----------- ------------ ------------ ---------- --------------------
**rs10907185** **rs10907185**
**Males** **Males**
A/A 6 (10.9) 2 (2.8) 4.32 (0.81, 23.17) A/A 8 (10.7) 0 (0.0) \-
A/G 24 (43.6) 36 (48.6) 0.96 (0.46, 1.98) A/G 27 (36.0) 6 (46.2) 0.79 (0.24, 2.60)
G/G 25 (45.5) 36 (48.6) 0.1600 1 G/G 40 (53.3) 7 (53.8) 0.4292 1
A/A + A/G 30 (54.5) 38 (51.4) 0.7193 1.14 (0.56, 2.29) A/A + A/G 35 (46.7) 6 (46.2) 0.9727 1.02 (0.31, 3.33)
**Females** **Females**
A/A 4 (7.6) 3 (3.6) 2.19 (0.46, 10.52) A/A 5 (6.0) 3 (12.5) 0.66 (0.14, 3.07)
A/G 21 (39.6) 34 (41.0) 1.01 (0.49, 2.08) A/G 35 (42.2) 4 (16.7) 3.46 (1.07, 11.22)
G/G 28 (52.8) 46 (55.4) 0.5986 1 G/G 43 (51.8) 17 (70.8) 0.0618 1
A/A + A/G 25 (47.2) 37 (44.6) 0.7673 1.11 (0.56, 2.22) A/A + A/G 40 (48.2) 7 (29.2)
| 547
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| 328
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| null | null |
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|
ne\phi^{*3,4}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_B \times U(1)_B$.
- $\{\overline\phi^{5,6}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*5,6}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SU(2)_C \times U(1)_C$.
- $\{\overline\phi^{7,8}\}|R\rangle_{pqrs}^{(4,5,6)}$ or $\{\overline\phi^{*7,8}\}|R\rangle_{pqrs}^{(4,5,6)}$. These states transform as a vector–like representations of $SO(4)$.
Similarly, for the matrix equations given above in eq. (\[matrixequations\]), we can write algebraic equations for the sectors in eq. (\[lighthiggssectors\]) given as follows:
$$\begin{aligned}
\begin{pmatrix} (e_1|e_3)&(e_1|e_4)&(e_1|e_5)&(e_1|e_6)\\
(e_2|e_3)&(e_2|e_4)&(e_2|e_5)&(e_2|e_6)\\
(z_1|e_3)&(z_1|e_4)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_3)&(z_2|e_4)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_1|b_1 + x)\\
(e_2|b_1 + x)\\
(z_1|b_1 + x)\\
(z_2|b_1 + x)
\end{pmatrix},\nonumber
\\[0.3cm]
\begin{pmatrix} (e_3|e_1)&(e_3|e_2)&(e_3|e_5)&(e_3|e_6)\\
(e_4|e_1)&(e_4|e_2)&(e_4|e_5)&(e_4|e_6)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_5)&(z_1|e_6)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_5)&(z_2|e_6) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (e_3|b_2 + x)\\
(e_4|b_2 + x)\\
(z_1|b_2 + x)\\
(z_2|b_2 + x)
\end{pmatrix},
\\[0.3cm]
\begin{pmatrix} (e_5|e_1)&(e_5|e_2)&(e_5|e_3)&(e_5|e_4)\\
(e_6|e_1)&(e_6|e_2)&(e_6|e_3)&(e_6|e_4)\\
(z_1|e_1)&(z_1|e_2)&(z_1|e_3)&(z_1|e_4)\\
(z_2|e_1)&(z_2|e_2)&(z_2|e_3)&(z_2|e_4) \end{pmatrix}
\begin{pmatrix} p\\q\\r\\s\end{pmatrix} &=
\begin{pmatrix} (es_5|b_3 + x)\\
(e_6|b_3 + x)\\
(z_1|b_3 + x)\\
(z_2|b_3 + x)
\end{pmatrix}.\nonumber\end{aligned}$$
The Observable Matter Spectrum {#observable}
==============================
The basis vectors $\alpha$ and $\beta$ given in eq. (\[421\]) break the $SO(10)$ symmetry to $SU(4) \times SU(2)_L \times U(1)_R$. Following the $\alpha$ and $\beta$ GGSO projections, the decomposition of the s
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function $\overline{c_{F}}$ is given by:
- $\overline{c_{F}}(\lambda)=\lambda$,
- $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=$ $c_{F}(\sigma_{1},\lambda)$ $c_{F}(\sigma_{2},\sigma_{1})$ $\ldots$ $c_{F}(\sigma_{n-1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-2}})$ $c_{F}(\sigma_{n},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-1}})$ $c_{F}(\sigma_{n+1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n}})$ $c_{F}(\sigma_{n+2},\sigma_{2}\sigma_{3}\ldots\sigma_{n+1})$ $c_{F}(\sigma_{n+3},\sigma_{3}\sigma_{4}\ldots\sigma_{n+2})\ldots$ $c_{F}(\sigma_{m},\sigma_{m-n}\sigma_{m-n+1}\ldots\sigma_{m-1})$,
for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$. It is easy to remark that
- $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=c_{F}(\sigma_{1},F(1,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))\ldots c_{F}(\sigma_{m},F(m,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))$,
for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, which proves the theorem. $\diamondsuit$
Conclusions and Future Work
===========================
We introduced a new class of non-standard variable-length codes, called adaptive codes, which associate a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. The main results of this paper are presented in Section 3, where we have shown that if an input data string $x$ has a significant number of pairs, then a good compression rate is achieved when encoding $x$ by adaptive codes of order one. In a further paper devoted to adaptive codes, we intend to extend the algorithm **Builder** to adaptive codes of any order.
---
abstract: 'It is well known that some quantum and statistical fluctuations of a quantum field may be recovered by adding suitable stochastic sources to the mean field equations derived from the Schwinger-Keldysh (Closed-time-path) effective action. In this note we show that this method can be extended to higher correlations and higher (n-particle irreducible) effective actions. As an example, we investigate three
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nsely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}},\end{aligned}$$]{} and $i-3$ more applications of Lemma \[lemma7\] show that ${\hat\Theta_{T''[i]}}={\hat\Theta_{S}}$.
Now consider applying Lemma \[lemma7\] to $T'[i,j]$, to move the $2$ from row $3$ to row $2$. If $j=2$, then neither of the two tableaux obtained is dominated by $S$. If $j>2$, then two of the three tableaux obtained are not dominated by $S$; the third has rows $3$ and $j+1$ both equal to $\young(j)$, so the resulting homomorphism is zero by Lemma \[lemma7\].
So we conclude that ${\hat\Theta_{T[i]}}$ equals ${\hat\Theta_{S}}$ plus a linear combination of homomorphisms indexed by tableaux not dominated by $S$.\
Now we consider applying Lemma \[lemma7\] to $U[i]$, moving the $1$ up from row $2$. The tableaux obtained that are dominated by $S$ are $T'[i]$ and the tableaux [$$\begin{aligned}
U'[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;j;i;{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls j\ls v$ with $i\neq j$.}\\
\intertext{\parbox{\linewidth}{(Note that if $i<j$ then $i$ and $j$ in the second row should be written the other way round; the case $i=j$ does not occur because the accompanying coefficient would be $\binom21=0$.)\\\hspace*{17pt}If $i=2$, then $U'[i,j]$ is a semistandard tableau diffe
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e rest of the data $\mathbf{y}_{-j}$. These likelihoods are used to monitor the predictive performance of the model. This performance is used to estimate the generalization error, and it can be used to carry out model selection [@kohavi1995study; @Rasmussen2006; @vehtari2017practical].
The Bayesian CV estimate of the predictive fit with given parameters ${\boldsymbol{\varphi}}$ is $$\mbox{CV} = \sum_{j=1}^n
\log p(\mathbf{y}_j \mid \mathbf{y}_{-j},{\boldsymbol{\varphi}}),$$ where $p(\mathbf{y}_j \mid \mathbf{y}_{-j},{\boldsymbol{\varphi}})$ is the predictive likelihood of the data $\mathbf{y}_j$ given the rest of the data. The best parameter values with respect to CV can be computed by enumerating the possible parameter values and selecting the one which gives the best fit in terms of CV.
Experimental results {#sec:expResults}
====================
In this section, we present numerical results using the GP model for limited x-ray tomography problems. All the computations were implemented in <span style="font-variant:small-caps;">Matlab</span> 9.4 (R2018a) and performed on an Intel Core i5 at 2.3 GHz and CPU 8GB 2133MHz LPDDR3 memory.
For both simulated data (see Section \[SimulatedData\]) and real data (see Section \[RealData\]) we use $m=10^4$ basis functions in . The measurements are obtained from the line integral of each x-ray over the attenuation coefficient of the measured objects. The measurements are taken for each direction (angle of view), and later they will be referred to as projections. The same number of rays in each direction is used. The computation of the hyperparameters is carried out using the Metropolis–Hastings algorithms with $5\,000$ samples, and the first $1\,000$ samples are thrown away ([*burn-in*]{} period). The reconstruction is computed by taking the conditional mean of the object estimate.
Simulated data: 2D Chest phantom {#SimulatedData}
--------------------------------
As for the simulated data, we use one slice of <span style="font-variant:small-caps;">Matlab</span>�
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|
, we should emphasize that the flow changes considerably when the parameters are varied and we have the freedom to choose the walls of the tool among the flow lines for certain fixed parameters $c_i(t)$. Moreover, the velocity and the orientation of the feeding (extraction) can vary somewhat for a given shape of the tool. Consequently, many types of extrusion dies can be drawn.
![Extrusion die corresponding to the solution (\[eq:4.21\]).[]{data-label="fig:2"}](outil_cas_ii.jpg){width="3.5in"}
Simple wave solutions of inhomogeneous quasilinear system. {#sec:ms}
==========================================================
\[sec:5\] Consider an inhogeneous first-order system of $q$ quasilinear PDEs in $p$ independent variables and $q$ unknowns of the form
\[eq:SW:1\] \^[i]{}\_(u) u\^\_i=b\^(u),,=1,…,q,i=1,…,p.
Let us underline that the system can be either hyperbolic or elliptic. We are looking for real solutions describing the propagation of a simple wave which can be realized by the system (\[eq:SW:1\]). We postulate a form of the solution $u$ in terms of a Riemann invariant $r$, [*i.e.* ]{}
\[eq:SW:2\] u=f(r),r=\_i(u) x\^i, i=1,…,p,
where $\lambda(u)={\left( \lambda_1(u),\ldots,\lambda_p(u) \right)}$ is a real-valued wave vector. We evaluate the Jacobian matrix $u^\beta_i$ by applying the chain rule: $$u^\beta_i=\frac{{\partial}f^\beta}{{\partial}r}{\left( r_{x^i}+r_{u^\alpha}u^\alpha_i \right)}=\frac{{\partial}f^\beta}{{\partial}r}{\left( \lambda_i+\lambda_{i,u^\alpha}x^iu^\alpha_i \right)}.$$We assume that the matrix
\[eq:SW:4\] =[( I\_q- )]{}\^[qq]{},
is invertible, where we have used the following notation ${\partial}f/{\partial}r={\left( {\partial}f^1/{\partial}r,\ldots, {\partial}f^q/{\partial}r \right)}^T$, ${\partial}r/{\partial}u ={\left( {\partial}r/{\partial}u^1,\ldots,{\partial}r/{\partial}u^q \right)}$. The Jacobian matrix ${\partial}u$ takes the form
\[eq:SW:5\] u=\^[-1]{}\^[qq]{}.
Replacing the Jacobian matrix (\[eq:SW:5\]) into the original system (\[eq:SW:1\]), we get
\[eq:SW:7\]
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)(60,40)
\qbezier[20](120,60)(100,60)(100,40)
\qbezier[20](120,20)(100,20)(100,40)
\qbezier[20](120,60)(140,60)(140,40)
\qbezier[20](120,20)(140,20)(140,40) \put(40,20){\circle*{3}}
\put(40,60){\circle*{3}} \put(120,20){\circle*{3}}
\put(120,60){\circle*{3}}
\put(170,37){\small 2)} \qbezier[20](210,60)(230,60)(230,40)
\qbezier[20](210,20)(230,20)(230,40)
\qbezier[20](290,60)(270,60)(270,40)
\qbezier[20](290,20)(270,20)(270,40) \put(210,20){\circle*{3}}
\put(210,60){\circle*{3}} \put(290,20){\circle*{3}}
\put(290,60){\circle*{3}} \qbezier[60](210,60)(250,90)(290,60)
\linethickness{0.5mm} \qbezier(40,60)(20,60)(20,40)
\qbezier(40,20)(20,20)(20,40) \qbezier(40,20)(80,-10)(120,20)
\qbezier(40,60)(80,90)(120,60) \qbezier(210,60)(190,60)(190,40)
\qbezier(210,20)(190,20)(190,40) \qbezier(210,20)(250,-10)(290,20)
\qbezier(290,60)(310,60)(310,40) \qbezier(290,20)(310,20)(310,40)
\end{picture}$$
$$\begin{picture}(310,80) \put(0,37){\small 3)} \qbezier[20](40,60)(60,60)(60,40)
\qbezier[20](40,20)(60,20)(60,40)
\qbezier[20](120,60)(140,60)(140,40)
\qbezier[20](120,20)(140,20)(140,40) \put(40,20){\circle*{3}}
\put(40,60){\circle*{3}} \put(120,20){\circle*{3}}
\put(120,60){\circle*{3}} \qbezier[60](40,60)(80,90)(120,60)
\put(170,37){\small 4)} \qbezier[20](210,60)(190,60)(190,40)
\qbezier[20](210,20)(190,20)(190,40)
\qbezier[20](290,60)(270,60)(270,40)
\qbezier[20](290,20)(270,20)(270,40) \put(210,20){\circle*{3}}
\put(210,60){\circle*{3}} \put(290,20){\circle*{3}}
\put(290,60){\circle*{3}} \qbezier[60](210,60)(250,90)(290,60)
\linethickness{0.5mm} \qbezier(40,60)(20,60)(20,40)
\qbezier(40,20)(20,20)(20,40) \qbezier(40,20)(80,-10)(120,20)
\qbezier(120,60)(100,60)(100,40) \qbezier(120,20)(100,20)(100,40)
\qbezier(210,60)(230,60)(230,40) \qbezier(210,20)(230,20)(230,40)
\qbezier(210,20)(250,-10)(290,20) \qbezier(290,60)(310,60)(310,40)
\qbezier(290,20)(310,20)(310,40)
\end{picture}$$ The first two t-maps have trivial groups of automorphisms. Thus, they generate six tree-rooted cubic maps each. But the group of automorphi
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|
rangle-\0{1}{\sqrt{1- \langle L\rangle ^2} }
\langle \delta L\rangle +
O\left(\langle \delta L^2\rangle\right)\,.$$ In the center-symmetric phase $\langle L\rangle =0$, c.f. . Under center transformations $L$ transforms according to (\[eq:centertrafo\]) $L\to Z\, L$ with $Z=\pm 1$ and hence $ \delta L\to Z\, \delta L$. It follows that $\langle \delta
L^{2n+1}\rangle = Z\langle \delta L^{2n+1}\rangle =0$, and all odd powers in vanish. The even powers vanish since $\arccos$ is an odd function and hence has vanishing even Taylor coefficients $\arccos^{(2n)}(0)$. Thus, in the center-symmetric phase we have $$\label{eq:expandA_0centersym} \012 g \beta \langle
A_0\rangle =\arccos \langle L\rangle=\0{\pi}{2}\,.$$ In summary we have shown $$\begin{aligned}
\nonumber
T<T_c: &\di\quad L[\langle {A_0} \rangle]=0
\quad \Leftrightarrow \quad\012 g \beta \langle A_0(\vec x)\rangle
= \0\pi2\,,
\\
T>T_c: &\di\quad L[\langle {A_0} \rangle]\neq 0 \quad \Leftrightarrow
\quad
\012 g \beta \langle A_0(\vec x)\rangle
<\0\pi2\,.
\label{eq:A0orderdisorder}\end{aligned}$$ We conclude that $\langle A_0\rangle $ in Polyakov gauge serves as an order parameter for the confinement-deconfinement (order-disorder) phase transition, as does $L[\langle A_0\rangle]$. Thus, we only have to compute the effective potential $V_{\rm eff}[\langle A_0\rangle]$ in order to extract the critical temperature, and e.g. critical exponents. This potential is more easily accessed than that for the Polyakov loop. It is here were the specific gauge comes to our aid as it allows the direct physical interpretation of a component of the gauge field. This property has been already exploited in the literature, where it has been shown that $\langle{A}_0\rangle$ in Polyakov gauge is sensitive to topological defects related to the confinement mechanism [@Reinhardt:1997rm; @Ford:1998bt].
Quantisation {#sec:quant}
============
We proceed by discussing the generating functional of Polyakov gauge Yang-Mills theory. For its derivation we use the
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along with the increasing of difficulty level), besides, the blocks that participants should complete would also increase along with the increasing of difficulty level, and the error times allowed in one level would also be changed based on difficulty level. This kind of design could ensure that the treatment of WM training and control group was almost the same (including the improvement of achieved difficulty level based on participants' performance which is known as "adaptive task"), but the control group was always conducting 1-back task. The last achieved difficulty level would be recorded after participants had finished one training session (i.e., 15 blocks), and then participants would start the next training session from this level.
######
The difficulty levels for WM training group and control group.
Level WM training group Control group
------- ------------------- --------------- --- ------ --- -------- ---- --- ------ ----
1 1-back 7 1 2500 2 1-back 25 1 1000 2
2 1-back 9 1 2500 2 1-back 25 2 1000 4
3 2-back 9 1 2500 2 1-back 25 2 700 4
4 2-back 11 1 2500 2 1-back 25 3 700 6
5 3-back 11 1 2500 2 1-back 25 3 600 6
6 3-back 13 1 2500 2 1-back 25 4 600 8
7 4-back 13 1 2500 2 1-back 25 4 550 8
8 4-back 15 1 2500 2 1-back 25 5 550 10
9 5-back 15 1 2500 2 1-back 25 5 500 10
10 5-back 17 1 2500 2 1-back 25 5 500 5
11 6-back 17 1 2500 1 1-back 25 5 400 5
12 6-back 19 1 2500 1 1-back 25 5
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kly increasing along the first row and down the first column. Let $\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}$.
\[sigmahom2\] With the notation and assumptions above, we have ${\psi_{d,t}}\circ\sigma=0$ for all $d,t$.
For $d\gs v$ and $t=1$, we use the same argument as that used in several proofs above: for $T\in{\calu}$ either ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$, or there is a unique other $T'\in{\calu}$ with ${\psi_{d,1}}\circ{\hat\Theta_{T'}}={\psi_{d,1}}\circ{\hat\Theta_{T}}$.
The cases where $2\ls d\ls v$ are easier: in this case Lemma \[lemma5\] and Lemma \[lemma7\] imply that we have ${\psi_{d,t}}\circ{\hat\Theta_{T}}=0$ for all $T\in{\calu}$.
So we are left with the cases where $d=1$ and $t\in\{1,2,3\}$. For $T\in {\calu}$ we have ${\psi_{1,3}}\circ{\hat\Theta_{T}}$ immediately from Lemma \[lemma5\], while ${\psi_{1,2}}\circ{\hat\Theta_{T}}$ is a homomorphism labelled by a tableau of the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;1,;1,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;\star).$$ But this homomorphism is zero by Lemma \[lemma7\]. Finally, ${\psi_{1,1}}\circ{\hat\Theta_{T}}$ is the sum of the homomorphisms labelled by two tableaux $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;1,;2,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,dens
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|
\delta
_{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$.
Let us observe now that $\mathcal{L}_{QD}^{P}$ obviously inherits the notions of p-validity and order defined in $\mathcal{L}_{Q}^{P}$ (Sec. 3.4). Hence, we can illustrate the role of the connective $I_{Q}$ within $\mathcal{L}_{QD}^{P}$ by means of the following *pragmatic deduction lemma*.
PDL.* Let* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*. Then,* $\delta _{1}\prec $* *$\delta _{2}$* iff for every* $S\in S$*,* $\pi _{S}(\delta _{1}I_{Q}\delta
_{2})=J$* (equivalently, iff* $\delta _{1}I_{Q}\delta _{2}$*is p-valid).*
Proof. The following sequence of equivalences holds.
For every $S\in \mathcal{S}$, $\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff for every $S\in \mathcal{S}$, $S\in \mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$ iff $\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})=\mathcal{S}$ iff $\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}}=$ $\mathcal{S}_{\delta _{1}}$ iff $\mathcal{S}_{\delta
_{1}}\subset \mathcal{S}_{\delta _{2}}$ iff $\delta _{1}\prec $ $\delta _{2}$.$\blacksquare \smallskip $
PDL shows that the quantum pragmatic implication $I_{Q}$ plays within $\mathcal{L}_{QD}^{P}$ a role similar to the role of material implication in classical logic.
Interpreting QL onto $\mathcal{L}_{QD}^{P}$
-------------------------------------------
In order to show that the physical QL $(\mathcal{E},\prec )$ introduced in Sec. 2.2 can be interpreted into $\mathcal{L}_{QD}^{P}$, a further preliminary step is needed. To be precise, let us make reference to the preorder introduced on $\psi _{A}^{Q}$ in Sec. 3.4 and consider the pre-ordered set $(\phi _{AD}^{Q},\prec )$ of all afs of $\mathcal{L}_{QD}^{P} $. Furthermore, let us denote by $\approx $ (by abuse of language) the restriction of the equivalence relation introduced on $\psi _{A}^{Q}$ in Sec. 3.4 to $\phi _{AD}^{Q}$, and let us denote by $\prec $ (
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commute in . Such a derivator is pointed and for every $X\in{\sD}^\square$ the canonical morphism $$\label{eq:stable-lim-II}
C(F_2X)\toiso F(C_1X)$$ is an isomorphism (). For every $x\in{\sD}$ we consider the square $$X=X(x)=(i_\lrcorner)_!\pi_\lrcorner^\ast x\in{\sD}^\square.$$ The morphism $\pi_\lrcorner^\ast\colon{\sD}\to{\sD}^\lrcorner$ forms constant cospans. Since $i_\lrcorner\colon\lrcorner\to \square$ is a cosieve, $(i_\lrcorner)_!$ is left extension by zero [@groth:ptstab Prop. 3.6] and the diagram $X\in{\sD}^\square$ looks like $$\xymatrix{
0\ar[r]\ar[d]&x\ar[d]^-\id\\
x\ar[r]_-\id&x.
}$$ We calculate $CF_2(X)\cong C(\Omega x\to 0)\cong \Sigma\Omega x$ and $FC_1(X)\cong F(x\to 0)\cong x$, showing that the canonical isomorphism induces a natural isomorphism $\Sigma\Omega\toiso\id$. Using constant spans instead one also constructs a natural isomorphism $\id\toiso\Omega\Sigma$, showing that $\Sigma,\Omega\colon{\sD}\to{\sD}$ are equivalences. It follows from that is stable.
It is now straightforward to obtain the following variant of this theorem. We recall from [@groth:can-can §9] that **left homotopy finite left Kan extensions** are left Kan extensions along functors $u\colon A\to B$ such that the slice categories $(u/b),$ $b\in B,$ admit a homotopy final functor $C_b\to(u/b)$ from a homotopy finite category $C_b$. The point of this notion is that right exact morphisms of derivators preserve left homotopy finite left Kan extensions [@groth:can-can Thm. 9.14].
\[thm:stable-lim-III\] The following are equivalent for a derivator .
1. The derivator is stable.\[item:sl1\]
2. Homotopy finite colimits and homotopy finite limits commute in .\[item:sl2\] Left homotopy finite left Kan extensions and arbitrary right Kan extensions commute in .\[item:sl3a\]
Arbitrary left Kan extensions and right homotopy finite right Kan extensions commute in .\[item:sl3b\] Every left exact morphism ${\sD}^A\to{\sD}^B,A,B\in\cCat,$ preserves left homotopy finite left Kan extensions.\[item:sl4a\]
Every right exac
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|
icular cases of GA codes.
Let $\Sigma$ and $\Delta$ be two alphabets and $F:N^{*}\times\Sigma^{+}\rightarrow\Sigma^{*}$ a function, where $N$ is the set of natural numbers, and $N^{*}=N-\{0\}$. A function $c_{F}:\Sigma\times\Sigma^{*}\rightarrow\Delta^{+}$ is called a if its unique homomorphic extension $\overline{c_{F}}:\Sigma^{*}\rightarrow\Delta^{*}$ given by
- $\overline{c_{F}}(\lambda)=\lambda$,
- $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=c_{F}(\sigma_{1},F(1,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))\ldots c_{F}(\sigma_{m},F(m,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))$,
for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, is injective.
The function $F$ in Definition 5.1 is called the of the GA code $c_{F}$. Clearly, a GA code $c_{F}$ can be constructed if its adaptive function $F$ is already constructed.
Let $\Sigma$ and $\Delta$ be two alphabets. We denote by $GAC(\Sigma,\Delta)$ the set $\{c_{F}:\Sigma\times\Sigma^{*}\rightarrow\Delta^{+}$ $\mid$ $c_{F}$ is a GA code$\}$.
Let $\Sigma$ and $\Delta$ be alphabets. Then, $AC(\Sigma,\Delta,n)\subset{GAC(\Sigma,\Delta)}$, for all $n\geq{1}$.
**Proof** Let $c_{F}\in{AC(\Sigma,\Delta,n)}$ be an adaptive code of order $n$, $n\geq{1}$, and $F:N^{*}\times\Sigma^{+}\rightarrow\Sigma^{*}$ a function given by: $$F(i,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=
\left\{
\begin{array}{ll}
\lambda
& \textrm{if $i=1$ or $i>m$,} \\
{\sigma_{1}\sigma_{2}\ldots\sigma_{i-1}}
& \textrm{if $2\leq{i}\leq{m}$ and $2\leq{i}\leq{n+1}$,} \\
\sigma_{i-n}\sigma_{i-n+1}\ldots\sigma_{i-1}
& \textrm{if $2\leq{i}\leq{m}$ and $i>n+1$,}
\end{array}
\right.$$ for all $i\geq{1}$ and ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in{\Sigma^{+}}$. One can verify that $|F(i,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}})|\leq{n}$, for all $i\geq{1}$ and ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in{\Sigma^{+}}$. According to Definition 2.1, the
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|
0.058 0.058 0.058 0.048 0.062 0.068 0.062
K=100 0.056 0.062 0.058 0.054 0.052 0.054 0.044
K=150 0.046 0.062 0.058 0.054 0.058 0.054 0.060
mVC 0.078 0.072 0.070 0.080 0.054 0.070 0.066
mMSE 0.070 0.080 0.058 0.068 0.060 0.066 0.068
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.002 0.002 0.000
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TB 0.070 0.082 0.062 0.074 0.064 0.060 0.084
5 K=50 0.066 0.062 0.046 0.074 0.046 0.058 0.048
K=100 0.056 0.058 0.054 0.070 0.048 0.066 0.050
K=150 0.060 0.060 0.066 0.084 0.044 0.068 0.052
mVC 0.060 0.082 0.066 0.090 0.070 0.068 0.066
mMSE 0.074 0.074 0.048 0.068 0.052 0.070 0.064
BLB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
BLB($n^{0.8}$) 0.000 0.000 0.000 0.002 0.000 0.000 0.002
SDB($n^{0.6}$) 0.000 0.000
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he good elements at the beginning of the list.
Currently, my function to generate the random indices looks essentially as follows:
def pick():
p = 0.2
for i in itertools.count():
if random.random() < p:
break
return i
It does a good job, but I wonder:
What's the name of the generated random distribution?
Is there a built-in function in Python for that distribution?
A:
What you are describing sounds a lot like the exponential distribution. It already exists in the random module.
Here is some code that takes just the integer part of sampling from an exponential distribution with a rate parameter of 100.
import random
import matplotlib.pyplot as plt
d = [int(random.expovariate(1/100)) for i in range(10000)]
h,b = np.histogram(d, bins=np.arange(0,max(d)))
plt.bar(left=b[:-1], height=h, ec='none', width=1))
plt.show()
A:
You could simulate it via exponential, but this is like making square peg fit round hole. As Mark said, it is geometric distribution - discrete, shifted by 1. And it is right here in the numpy:
import numpy as np
import random
import itertools
import matplotlib.pyplot as plt
p = 0.2
def pick():
for i in itertools.count():
if random.random() < p:
break
return i
q = np.random.geometric(p, size = 100000) - 1
z = [pick() for i in range(100000)]
bins = np.linspace(-0.5, 30.5, 32)
plt.hist(q, bins, alpha=0.2, label='geom')
plt.hist(z, bins, alpha=0.2, label='pick')
plt.legend(loc='upper right')
plt.show()
Output:
Q:
Two divs with 50% width while content inside container
How to accomplish the look like it is in the image?
It is just regular Bootstrap code for columns, but first two columns should be on white background inside .container, while the other two columns on black background and that black background is going right all the way to the end.
CodePen
<div class="container">
<div class="col-md-3">
<p>First column</p>
</div>
<div class="col-md-3">
<p>Second column</p>
</div>
<div class="col-md-3">
<p>Third colum
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that the right-hand side is nonzero only when $x$ and $y$ are connected by a chain of bonds with *odd* currents (see Figure \[fig:RCrepr\]).
![\[fig:RCrepr\]A current configuration with sources at $x$ and $y$. The thick-solid segments represent bonds with odd currents, while the thin-solid segments represent bonds with positive even currents, which cannot be seen in the high-temperature expansion.](RCrepr)
We will exploit this peculiar underlying percolation picture to derive the lace expansion for the two-point function.
Derivation of the lace expansion {#ss:derivation}
--------------------------------
In this subsection, we derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ using the random-current representation. In Section \[sss:1stexp\], we introduce some definitions and perform the first stage of the expansion, namely [(\[eq:Ising-lace\])]{} for $j=0$, simply using inclusion-exclusion. In Section \[sss:2ndexp\], we perform the second stage of the expansion, where the source-switching lemma (Lemma \[lmm:switching\]) plays a significant role to carry on the expansion indefinitely. Finally, in Section \[sss:complexp\], we complete the proof of Proposition \[prp:Ising-lace\].
### The first stage of the expansion {#sss:1stexp}
As mentioned in Section \[ss:RCrepr\], the underlying picture in the random-current representation is quite similar to percolation. We exploit this similarity to obtain the lace expansion.
First, we introduce some notions and notation.
\[defn:perc\]
(i) Given ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ and ${{\cal A}}\subset\Lambda$, we say that $x$ is ${{\bf n}}$-connected to $y$ in (the graph) ${{\cal A}}$, and simply write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y$ *in* ${{\cal A}}$, if either $x=y\in{{\cal A}}$ or there is a self-avoiding path (or we simply call it a path) from $x$ to $y$ consisting of bonds $b\in{{\mathbb B}}_{{\cal A}}$ with $n_b>0$. If ${{\bf n}}\in{{\
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| 1,503
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|
github_plus_top10pct_by_avg
|
t(\frac{1 + P^{(0)}_{m+1,n}}{P^{(2)}_{m+1,n}} - \frac{1+P^{(0)}_{m-1,n}}{P^{(2)}_{m-1,n}} \right),\end{gathered}$$
where
\[eq:N5-P-G\] $$\begin{gathered}
P^{(0)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m,n}, \\
P^{(1)}_{m,n} = \phi^{(0)}_{m-1,n} \big(\phi^{(0)}_{m,n}\big)^2 \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m+1,n},\\
P^{(2)}_{m,n} = 2 +2 P^{(0)}_{m,n} + P^{(1)}_{m,n}.\label{eq:G-5}\end{gathered}$$
The case $\boldsymbol{N=7}$, with level structure $\boldsymbol{(3,4;3,4)}$
--------------------------------------------------------------------------
=-1 The fully discrete system (\[eq:dLP-gen-sys-1-a\]) and its lower order symmetries (\[eq:phi-sys-sym\]) and master symmetries (\[eq:phi-sys-msym\]) can be easily adapted to our choices $N=7$ and $(k,\ell)=(3,4)$. In the same way the corresponding reduced system follows from (\[self-dual-equn\]) with $k=3$. Thus we omit all these systems here and present only the lowest order symmetry of the reduced system which takes the following form $$\begin{gathered}
\partial_{t_2} \phi^{(0)}_{m,n} = \phi^{(0)}_{m,n} \frac{ P^{(1)}_{m,n}}{P^{(3)}_{m,n}} \left(\frac{1}{P^{(3)}_{m+1,n}} - \frac{1}{P^{(3)}_{m-1,n}}\right), \nonumber \\
\partial_{t_2} \phi^{(1)}_{m,n} = \phi^{(1)}_{m,n} \frac{P^{(0)}_{m,n}}{P^{(3)}_{m,n}} \left(\frac{1+ P^{(0)}_{m+1,n}}{P^{(3)}_{m+1,n}} - \frac{1+ P^{(0)}_{m-1,n}}{P^{(3)}_{m-1,n}}\right), \label{eq:N7-red-dd} \\
\partial_{t_2} \phi^{(2)}_{m,n}= \phi^{(2)}_{m,n} \frac{1}{P^{(3)}_{m,n}} \left(\frac{1+ P^{(0)}_{m+1,n}+ P^{(1)}_{m+1,n}}{P^{(3)}_{m+1,n}} - \frac{1+ P^{(0)}_{m-1,n}+ P^{(1)}_{m-1,n}}{P^{(3)}_{m-1,n}}\right), \nonumber\end{gathered}$$ where
\[eq:N7-P-G\] $$\begin{gathered}
P^{(0)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(2)}_{m,n}, \\
P^{(1)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \big(\phi^{(1)}_{m,n}\big)^2 \phi^{(1)}_{m+1,n} \phi^{(2)}_{m,n}, \\
P^{(2)}_{m,n} =\phi^{(0)}_{m-1,n} \big(\
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| 1,010
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github_plus_top10pct_by_avg
|
\delta_{\tilde{p}(v)}\,\Psi\ =&\ p(v)\Psi - X\eta F\Xi D_{\tilde{p}(v)}F\Psi\,,
\label{p tilde Psi}\end{aligned}$$
where the former is determined so as to induce $$\begin{aligned}
\delta_{\tilde{p}(v)}A_\eta\ =&\
D_\eta\Big( A_{p(v)}- f\xi_0\big(QA_{\tilde{p}(v)} + [F\Psi,\,F\Xi D_{\tilde{p}(v)}F\Psi]\big)\Big)
\nonumber\\
\cong&\ p(v)A_\eta - QA_{\tilde{p}(v)} - [F\Psi,\,F\Xi D_{\tilde{p}(v)}F\Psi]\,.\end{aligned}$$ This extra contribution can be absorbed into the gauge transformation, up to the equations of motion, at the linearized level as we will see shortly. Let us consider the transformation (\[p tilde\]) at the linearized level:
\[translation tilde\] $$\begin{aligned}
\delta_{\tilde{p}}^{(0)}\Phi\ =&\ p(v)\Phi
- \xi_0 Q\tilde{p}(v)\Phi\ =\ \big(p(v)- X_0\tilde{p}(v)\big)\Phi +Q(\xi_0\tilde{p}(v)\Phi)\,,
\label{trans tilde ns}\\
\delta_{\tilde{p}}^{(0)}\Psi\ =&\ \big(p(v)-X\tilde{p}(v)\big)\Psi\,.
\label{trans tilde ramond}\end{aligned}$$
Thanks to (\[p tilde p\]), the transformation of $\Phi$ (\[trans tilde ns\]) becomes the form of the gauge transformation up to the equation of motion at the linearized level: $$\delta_{\tilde{p}}^{(0)}\Phi\
=\
Q \big((M(v) + \xi_0\tilde{p}(v))\Phi\big)
+ \eta \big(\xi_0 M(v) Q\Phi\big)
+ \xi_0M(v)Q\eta\Phi\,.
\label{trans tilde ns 2}$$ We can similarly show that the transformation of $\Psi$ in (\[trans tilde ramond\]) can also be written as a gauge transformation up to the equation of motion at the linearized level as shown in Appendix \[app B\]. Here we assume that the asymptotic condition[@Lehmann:1954rq] holds for string field theory as well as the conventional (particle) field theory. Then, at least perturbatively, we can identify that the transformation (\[translation tilde\]), or (\[trans tilde ns 2\]) and (\[B4\]) can be interpreted, with appropriate (finite) renormalization, as that of asymptotic string fields. If we further assume asymptotic completeness, this implies that the extra transformation (\[translation tilde\]) acts trivially on th
| 564
| 2,184
| 1,015
| 653
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|
plicity, we take $\epsilon$ and $\tau$ to be fixed but we will keep explicit track of these quantities in the constants.
Since each coordinate of $\hat{\gamma}_{{\widehat{S}}}$ is an average of random variables that are bounded in absolute value by $2(A+\tau) + \epsilon$, and $\mathbb{E}\left[
\hat{\gamma}_{{\widehat{S}}} | \mathcal{D}_{1,n}\right] = \gamma_{{\widehat{S}}}$, a standard bound for the maxima of $k$ bounded (and, therefore, sub-Gaussian) random variables yields the following concentration result. As usual, the probability is with respect to the randomness in the full sample and in the splitting.
$$\sup_{w_n \in \mathcal{W}_n} \sup_{P \in \mathcal{P}_n^{\mathrm{LOCO}}}
\mathbb{P}\left( \| \hat{\gamma}_{{\widehat{S}}} -
\gamma_{{\widehat{S}}}\|_\infty \leq \left( 2(A+\tau) + \epsilon \right) \sqrt{ 2
\frac{\log k + \log n}{n} } \right) \geq 1 - \frac{1}{n}.$$
The bound on $\| \hat{\gamma}_{{\widehat{S}}} - \gamma_{{\widehat{S}}}\|_\infty $ holds with probability at least $1 - \frac{1}{n}$ conditionally on $\mathcal{D}_{1,n}$ and the outcome of data splitting, and uniformly over the choice of the procedure $w_n$ and of the distribution $P$. Thus, the uniform validity of the bound holds also unconditionally.
We now construct confidence sets for $\gamma_{{\widehat{S}}}$. Just like we did with the projection parameters, we consider two types of methods: one based on Normal approximations and the other on the bootstrap.
### Normal Approximation {#normal-approximation .unnumbered}
Obtaining high-dimensional Berry-Esseen bounds for $\hat\gamma_{{\widehat{S}}}$ is nearly straightforward since, conditionally on $\mathcal{D}_{1,n}$ and the splitting, $\hat\gamma_{{\widehat{S}}}$ is just a vector of averages of bounded and independent variables with non-vanishing variances. Thus, there is no need for a Taylor approximation and we can apply directly the results in [@cherno2]. In addition, we find that the accuracy of the confidence sets for this LOCO parameter is higher than for the project
| 565
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|
$ for some formal expansion $\tilde{z}_i''^{\ddag}$. Here, if $m$ is a formal matrix of size $a\times b$, then $m^{\dag}$ is the $(a, b)^{th}$-entry of $m$ (resp. $(a-1, b)^{th}$-entry of $m$) if $L_i$ is *of type* $\textit{I}^o$ (resp. $\textit{I}^e$).
Note that $$\left \{ \begin{array}{l}
z_i+\delta_{i-2}\cdot k_{i-2, i}+ \delta_{i+2}\cdot k_{i+2, i}=\pi z_i^{\ast};\\
z_i'+\delta_{i-2}\cdot k_{i-2, i}'+ \delta_{i+2}\cdot k_{i+2, i}'=\pi (z_i^{\ast})';\\
\textit{$(m_{i, i-1}m_{i-1, i}')^{\dag}+\delta_{i-2}\cdot((m_{i-2, i-1}m_{i-1, i}')^{\dag})=\pi (m_{i-1, i-1}^{\ast}\cdot m_{i-1, i}')^{\dag}$};\\
\textit{$(m_{i, i+1}m_{i+1, i}')^{\dag}+\delta_{i+2}\cdot((m_{i+2, i+1}m_{i+1, i}')^{\dag})=\pi (m_{i+1, i+1}^{\ast}\cdot m_{i+1, i}')^{\dag}$}.
\end{array} \right.$$ Therefore, $$\tilde{z}_i''+\delta_{i-2}\tilde{k}_{i-2, i}''+\delta_{i+2}\tilde{k}_{i+2, i}''=
\pi\left(z_i^{\ast}+(z_i^{\ast})'+(m_{i-1, i-1}^{\ast}\cdot m_{i-1, i}')^{\dag}+m_{i+1, i+1}^{\ast}\cdot m_{i+1, i}')^{\dag}+\tilde{z}_i''^{\ddag}\right)$$ for some formal expansion $\tilde{z}_i''^{\ddag}$. Then $$(z_i^{\ast})''=z_i^{\ast}+(z_i^{\ast})'+(m_{i-1, i-1}^{\ast}\cdot m_{i-1, i}')^{\dag}+m_{i+1, i+1}^{\ast}\cdot m_{i+1, i}')^{\dag}+\tilde{z}_i''^{\ddag}$$ as an equation in $B\otimes_AR$.\
6. Assume that $i$ is odd and $L_i$ is *bound of type I*. Then the following formal sum $$\delta_{i-1}v_{i-1}\cdot \tilde{m}_{i-1, i}''+\delta_{i+1}v_{i+1}\cdot \tilde{m}_{i+1, i}''$$ equals $$\delta_{i-1}v_{i-1}\cdot (m_{i-1, i}m_{i,i}'+m_{i-1, i-1}m_{i-1,i}')+\delta_{i+1}v_{i+1}\cdot (m_{i+1, i}m_{i,i}'+m_{i+1, i+1}m_{i+1,i}')+\pi \tilde{z}_i''^{\dag}=$$ $$\left(\delta_{i-1}v_{i-1}\cdot (m_{i-1, i}m_{i,i}')+\delta_{i+1}v_{i+1}\cdot (m_{i+1, i}m_{i,i}')\right)+$$ $$\left(\delta_{i-1}v_{i-1}\cdot (m_{i-1, i-1}m_{i-1,i}')+\delta_{i+1}v_{i+1}\cdot (m_{i+1, i+1}m_{i+1,i}')\right)
+\pi \tilde{z}_i''^{\dag}$$ for some formal expansion $\tilde{z}_i''^{\dag}$. Here, $\delta_{j}v_{j}$ is as explained in Step (d) of the above de
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|
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|
.00035
5000 .00014
10000 .00007
: Indifference proportion[]{data-label="Ta:INDIFFERENCE_PROPORTION"}
#### Indemnification formula {#S:UPPER_BOUND}
The indemnification formula provides a statistical upper bound on hazard intensity. Indemnification data may be expressed as an equivalent statistical upper bound on hazard intensity. This differs fundamentally from estimating the intensity of a hazard. By a statistically [guaranteed]{} hazard intensity, we mean an upper bound such that the true hazard intensity is likely to fall beneath this level with known confidence (probability).
Suppose we choose ${^1\!\!/\!_2}$ as the known confidence. The indifference proportion $\hat{\rho}_\text{\,I} = 1 - \sqrt[N]{{^1\!\!/\!_2}}$ then has a second interpretation as an upper bound with confidence ${^1\!\!/\!_2}$. For any $\rho \leq \hat{\rho}_\text{\,I}$, it is true that power function $P_{N,0}(\rho) \leq {^1\!\!/\!_2}$, so $\hat{\rho}_\text{\,I}$ is an upper bound of confidence ${^1\!\!/\!_2}$.
To convert from the size of the indemnification sample into its equivalent upper bound hazard intensity, find the indifference proportion $\hat{\rho}_{\,\text{I}} = 1 - \sqrt[N]{{^1\!\!/\!_2}}$.
Check the sample physics. This amounts to analysis of the originating cone ${\mathcal{C}}$, which is the point of exhibition of a hazard whenever safety constraints are not met. The cone’s edge has an absolute operational profile expressed as a rate. This quantity is the counting norm of the cone’s edge.
We have shown that the probable upper bound of the hazard intensity is proportional to the indifference proportion, with constant of proportionality furnished by the counting norm of the cone’s edge. The indemnification formula is: $$\hat{\lambda}_{\,\text{I}} = \hat{\rho}_{\,\text{I}} \cdot {\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}
= \hat{\rho}_{\,\text{I}} \cdot {\Vert{\lbrace{\mathit{s}}_{\text{crux}}\rbrace}\Vert}.$$
Epilogue
========
From previous discussion two stru
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| 0.791208
|
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|
convention that $\H^{[0]}(z,w):=1$. It is known by work of Göttsche and Soergel [@Gottsche-Soergel] that the mixed Hodge polynomial $H_c\left(X^{[n]};q,t\right)$ is given by $$H_c\left(X^{[n]};q,t\right)
=(qt^2)^n\H^{[n]}\left(-t\sqrt{q}, \frac{1}{\sqrt{q}}\right).$$
We have $$\H^{[n]}(z,w)=\H_{(n-1,1)}(z,w).$$ \[conjHS\]
This together with the conjectural formula (\[mainconj\]) implies that the Hilbert scheme $X^{[n]}$ and the character variety $\M_{(n-1,1)}$ should have the same mixed Hodge polynomial. Although this is believed to be true (in the analogous additive case this is well-known; see Theorem \[adpure\]) there is no complete proof in the literature. (The result follows from known facts modulo some missing arguments in the non-Abelian Hodge theory for punctured Riemann surfaces; see the comment after Conjecture \[conjCV=HS\].) We prove the following results which give evidence for Conjecture \[conjHS\].
We have $$\begin{aligned}
&\H^{[n]}(0,w)=\H_{(n-1,1)}\left(0,w\right),\\
&\H^{[n]}(w^{-1},w)=\H_{(n-1,1)}(w^{-1},w).\end{aligned}$$ \[theospe\]
The second identity means that the $E$-polynomials of $X^{[n]}$ and $\M_{(n-1,1)}$ agree. As a consequence of Theorem \[theospe\] we have the following relation between character varieties and quasi-modular forms.
We have $$1+\sum_{n\geq 1}\H_{(n-1,1)}\left(e^{u/2},e^{-u/2}\right)T^n
=\frac{1}{u}\left(e^{u/2}-e^{-u/2}\right)\exp\left(2\sum_{k\geq
2}G_k(T)\frac{u^k}{k!}\right),$$ where $$G_k(T)=\frac{-B_k}{2k}+\sum_{n\geq 1}\sum_{d\,|\, n}
d^{k-1}T^n$$(with $B_k$ is the $k$-th Bernoulli number) is the classical Eisenstein series for $SL_2(\Z)$.
In particular, the coefficient of any power of $u$ in the left hand side is in the ring of *quasi-modular* forms, generated by the $G_k$, $k\geq 2$, over $\Q$.
Relation between Hilbert schemes and modular forms was first investigated by Göttsche [@Gottsche].
Quiver representations {#quiver-repns}
----------------------
For a partition $\mu=\mu_1\geq\dots \geq \mu_r>0$ of $n$ we denote by $l(\mu)=r$ its lengt
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|
od as a marginalisation of ${\boldsymbol{\mu}}$ and substituting the definition produces the first equality by: $$\begin{aligned}
\tilde{f}(y_{1:T}|s_{1:T}) =\int \tilde{\pi}({\boldsymbol{\mu}}) f(y_{1:T}|{\boldsymbol{\mu}},s_{1:T}) d{\boldsymbol{\mu}}= \frac{\int_M \pi({\boldsymbol{\mu}})f(y_{1:T}|{\boldsymbol{\mu}},s_{1:T}) d{\boldsymbol{\mu}}}{\pi(M)}
= \frac{\pi(M|y_{1:T},s_{1:T})}{\pi(M)} f(y_{1:T}|s_{1:T}).\end{aligned}$$ The second equality in arises as the first $\tau-1$ observations relate exclusively to the baseline.
Evaluation of $\pi(M|y_{1:\tau-1},s_{1:\tau-1})$ is straightforward since it is an orthant probability for the multivariate Student-t distribution. In contrast, the posterior probability is estimated from the $N$-component-mixture approximation of the posterior distribution by the final particle set: $$\begin{aligned}
\hat{\pi}(M|y_{1:T},s_{1:T}) & = \frac{1}{N} \sum_{i=1}^{N} \pi(M|{\mathbf{x}}_{1:T}^{(i)}, y_{1:T}). $$
There is no theoretical argument against assuming from the outset. Indeed, the firing events sampled in the propagation step of Algorithm \[tab:Alg\] would then account for the restriction to the ${\boldsymbol{\mu}}$ parameter space and therefore directing particle samples to a more appropriate approximation for posterior parameter estimates. However, implementing this scheme requires at most $N2^u$ orthant evaluations of the multivariate Student’s t-distribution per time step in calculating the re-sampling weights. Standard procedures for evaluating these orthant probabilities [@Genz09] are expensive, so the computational time of the resulting SMC-MUNE algorithm would increase substantially.
Simulation study {#sec:SimStudy}
================
The performance of the SMC-MUNE algorithm is now assessed using $200$ simulated data sets, $20$ for each true number of MUs of $u^*=1,\ldots,10$. Each data set consists of $T=220$ measurements with $\tau=21$ so that the first $20$ observations correspond to the baseline, $s_t=0$V, and these are followed by the supramaxim
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|
ne-forms $\{\text{d}\tau,\text{d}\varphi,\text{d}\psi\}$ via $$\mathbf{V}^{(m\,h\,k)}= V_{i}^{(m\,h\,k)}\text{d}x^i,\quad x\in\{\tau,\varphi,\psi\}\,.$$ The covector components are given by $$\begin{aligned}
V_{j}^{(m\,h\,k)} \propto
\begin{bmatrix}
v^{(m\,h\,k)}_{\tau}(\sin \psi )^{-1} \\
v^{(m\,h\,k)}_{\varphi} (\sin \psi )^{+0} \\
v^{(m\,h\,k)}_{\psi} (\sin \psi )^{-1}
\end{bmatrix}
(\sin \psi )^{-h} e^{i [(h-k) \tau + m \varphi] +m \psi}
\,,\end{aligned}$$ where $$\begin{aligned}
v_{\tau}^{(m\,h\,0)} &= -\frac{1}{4 }\left(c_1 e^{- i \psi }+ 2c_1 e^{i \psi }-2 c_2 e^{ -i \psi } +4 c_3 e^{ i \psi }\right)\,, \\ {\nonumber}v_{\varphi}^{(m\,h\,0)} &= c_1 \,, \\ {\nonumber}v_{\psi}^{(m\,h\,0)} &= +\frac{1}{4 }\left(c_1 e^{- i \psi }+2 c_2 e^{ -i \psi } +4 c_3 e^{ i \psi }\right) \,,\end{aligned}$$ and $$\begin{aligned}
v_{\tau}^{(m\,h\,1)} =& -\frac{1}{4}\bigg\{
c_1 [2 (h+i m)e^{2 i \psi } +(3 h-i m-1)+(h-i m+1)e^{- 2i \psi }]-\\ {\nonumber}&-2c_2 [ (h+i m+1)+ (h-i m-1)e^{-2 i \psi }]
+4 c_3 [(h+i m-1)e^{2i \psi } +(h-i m+1)]
\bigg\}\,,\\ {\nonumber}v_{\varphi}^{(m\,h\,1)} =& -2 c_1(m \sin \psi -h \cos \psi )\,, \\ {\nonumber}v_{\psi}^{(m\,h\,1)} =& +\frac{1}{4} \bigg\{
c_1 [(h+i m+1)+(h-i m-1)e^{-2 i \psi } ]+
2 c_2 [(h+i m+1)+(h-i m-1)e^{-2 i \psi } ]\\{\nonumber}&+4 c_3 [(h+i m-1)e^{2 i \psi} +(h-i m+1)]
\bigg\}\,.\end{aligned}$$
### Symmetric tensor bases {#app:tensor-basis-global}
The symmetric tensor bases in global coordinates can be decomposed using the dual basis one-forms $\{\text{d}\tau,\text{d}\varphi,\text{d}\psi\}$ via $$\mathbf{W}^{(m\,h\,k)} = W^{(m\,h\,k)}_{ij}\,\text{d}x^i\otimes \text{d}x^j,\quad x\in\{\tau,\varphi,\psi\}\,.$$ The tensor components are given by $$\begin{aligned}
W_{ij}^{(m\,h\,k)} \propto
\begin{bmatrix}
w^{(m\,h\,k)}_{\tau\tau} (\sin \psi )^{-2} &
w^{(m\,h\,k)}_{\tau\varphi} (\sin \psi )^{-1}&
w^{(m\,h\,k)}_{\tau\psi} (\sin \psi )^{-2}\\
* &
w^{(m\,h\,k)}_{\varphi\varphi} (\sin \psi )^{+0}&
w^{(m\,h\,k)}_{\varphi\psi} (\s
| 570
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|
\,
\label{eq:sigma-full} \\
%
A_{l'l}=|A_{l'}| |A_{l}|\; , \;\;A_l= C_l
N_l(E)\,e^{i\delta_l(E)}\int_0^{r_0}dr j_l(q_lr)\; , \nonumber \\
%
q_l=\sqrt{2M(E-V_l)}\; , \;\;
\phi_{l'l}(E) = \phi^{(0)}_{l'l} + \delta_{l'}(E) - \delta_{l}(E)\, ,
\nonumber
%\end{aligned}$$ where $N_l$ is defined by matching condition on the well boundary for internal function $j_l(q_lr)$. The three coefficients $C_l$ give rise to the two phases $\phi^{(0)}_{10}$ and $\phi^{(0)}_{12}$.
Positions and widths of the states are fixed by the three parameters $V_l$. Their relative contributions to the inclusive energy spectrum (Fig. \[fig:distrib\]a–c) are fixed by the three parameters $|C_l|$. The phase $\phi^{(0)}_{10}$ is fixed by the angular distributions at low energy, where the contribution of the $d$-wave resonance is small. After that the phase $\phi^{(0)}_{12}$ was varied to fit the angular distributions at higher energies (Fig. \[fig:distrib\], $E>2.2$ MeV). So, the model does not have redundant parameters and the ambiguity of the theoretical interpretation is defined by the quality of the data.
Set $|C_0|^2$ $|C_1|^2$ $V_0$ $V_1$ $V_2$ $\phi^{(0)}_{10}$ $\phi^{(0)}_{12}$
----- ----------- ----------- ---------- --------- --------- ------------------- -------------------
1 0.26 0.35 $-4.0$ $-20.7$ $-43.4$ $0.80 \pi$ $-0.03 \pi$
2 0.03 0.52 $-4.0$ $-20.7$ $-43.4$ $0.85 \pi$ $-0.02 \pi$
3 0.12 0.43 $-5.817$ $-20.7$ $-43.4$ $1.00 \pi$ $-0.03 \pi$
: Parameters of theoretical model used in the work. $V_i$ values are in MeV, weight coefficients for different states are normalized to unity $\sum
|C_i|^2=1$.
\[tab:th\]
We have found that the weight and interaction strength for the $1/2^+$ state can be varied in a relatively broad range, still providing a good description of data. The results of MC simulations of the experiment with different $s$-wave contributions are shown in Fig.
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\{j,k\}$ such that $i_\nu =j$ if $\nu $ is even and $i_\nu =k$ if $\nu $ is odd. Let $\chi '=r_{i_0}r_{i_1}\cdots r_{i_{m-1}}(\chi )=
r_{i_1}r_{i_2}\cdots r_{i_m}(\chi )$ (by (R4)), and $$\Lambda '={t}_{i_0}{t}_{i_1}\cdots {t}_{i_{m-1}}^\chi (\Lambda ),\quad
\Lambda ''={t}_{i_1}{t}_{i_2}\cdots {t}_{i_m}^\chi (\Lambda ).$$ These definitions make sense, since ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Lemma \[le:VTMrel1\] implies that the claim of the lemma is equivalent to $\Lambda '=\Lambda ''$.
Let ${\hat{T}}'={\hat{T}}_{i_0}{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}}:
M^{\chi '}(\Lambda ')\to M^\chi (\Lambda )$ and ${\hat{T}}''={\hat{T}}_{i_1}{\hat{T}}_{i_2}\cdots {\hat{T}}_{i_m}:
M^{\chi '}(\Lambda '')\to M^\chi (\Lambda )$. For all $\nu \in \{1,2,\dots ,m\}$ let $$\beta '_\nu =1_\chi {\sigma }_{i_0}{\sigma }_{i_1}\cdots {\sigma }_{i_{\nu -2}}
({\alpha }_{i_{\nu -1}}), \quad
\beta ''_\nu =1_\chi {\sigma }_{i_1}{\sigma }_{i_2}\cdots {\sigma }_{i_{\nu -1}}
({\alpha }_{i_\nu }).$$ By definition of ${\hat{T}}_j$ and ${\hat{T}}_k$, $$\begin{aligned}
{\hat{T}}''(v_{\Lambda ''})=&{\hat{T}}_{i_1}\cdots {\hat{T}}_{i_{m-1}}
(F_{i_m}^{{b^{\chi '}}({\alpha }_{i_m})-1}v_{{t}_{i_m}^{\chi '}(\Lambda '')})
=\cdots
\\
=&F_{\beta ''_m}^{{b^{\chi}} (\beta ''_m)-1}\cdots
F_{\beta ''_2}^{{b^{\chi}} (\beta ''_2)-1}
F_{\beta ''_1}^{{b^{\chi}} (\beta ''_1)-1}v_\Lambda ,\\
{\hat{T}}'(v_{\Lambda '})=&F_{\beta '_m}^{{b^{\chi}} (\beta '_m)-1}\cdots
F_{\beta '_2}^{{b^{\chi}} (\beta '_2)-1}
F_{\beta '_1}^{{b^{\chi}} (\beta '_1)-1}v_\Lambda .
\end{aligned}$$ Both expressions are nonzero by Thm. \[th:PBWtau\]. Since $$\{\beta '_\nu \,|\,1\le \nu \le m\}=\{\beta ''_\nu \,|\,1\le \nu \le m\}
=R^\chi _+\cap ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k),$$ we obtain that
- ${\hat{T}}'({\mathbb{K}}v_{\Lambda '})$ and ${\hat{T}}''({\mathbb{K}}v_{\Lambda ''})$ are isomorphic ${{\mathcal{U}}^0}$-modu
| 572
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| 527
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| null | null |
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|
chosen top level category. Thus, in this new dataset we replaced each navigational step over a page with an appropriate Wikipedia category (topic) and the dataset contains paths of topics which users visited during navigation (see Figure \[fig:pathexample\]). Figure \[fig:histograms\] illustrates the distinct topics and their corresponding occurrence frequency (A).
#### Wikispeedia dataset
This dataset is based on a similar online game as the Wikigame dataset called *Wikispeedia*[^9]. Again, the players are presented with two randomly chosen Wikipedia pages and they are as well connected via the underlying link structure of Wikipedia. Furthermore, users can also select their own start and target page instead of getting randomly chosen ones. Contrary to the Wikigame, this game is no multiplayer game and you do not have a time limit. Again, we only look at navigational paths with at least two nodes in the path. The main difference to the Wikigame dataset is that Wikispeedia is played on a limited version of Wikipedia (Wikipedia for schools[^10]) with around 4,600 articles. Some main characteristics are presented in Table \[tab:datasetfacts\]. Conducted research and further explanations of the dataset can be found in [@west; @west2; @west3; @scaria2014last]. As we want to look at transitions between topics we determine a corresponding top level category (topic) for each page in the dataset. We do this in similar fashion as for our Wikigame dataset, but the Wikipedia version used for Wikispeedia has distinct top level categories compared to the full Wikipedia. Figure \[fig:histograms\] illustrates the distinct categories and their corresponding occurrence frequency (B).
![**Topic frequencies.** Frequency of categories (in percent) of all paths in (A) the Wikigame topic dataset (B) the Wikispeedia dataset and (C) the MSNBC dataset. The colors indicate the categories we will investigate in detail later and are representative for a single dataset – this means that the same color in the datasets does not represen
| 573
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| 1,420
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|
{kl}=\Lambda_{kl}\Lambda_{ij}$ , (v) $\Lambda_{jk} z_k \partial_k = z_j \partial_j$ .
The sign reversing operator $\Lambda_j$ may be defined by its action on the coordinates of the $j$-th particle as $\Lambda_j f(z_1,..,z_j,.., z_N)=
f(z_1,..,-z_j,.., z_N)$. This operator is (i) self-inverse $\Lambda_{j}^{-1} = \Lambda_{j}$, (ii) mutually commuting $[\Lambda_j, \Lambda_k] = 0$, and satisfies (iii) $[\Lambda_{ij},\Lambda_k]= 0$ , $i\neq j\neq k$, (iv) $\Lambda_{ij}\Lambda_j=\Lambda_i\Lambda_{ij}$.
In terms of $\Lambda_{jk}$ and $\Lambda_j$, we introduce operator $\widetilde{\Lambda}_{jk} = \Lambda_j \Lambda_k
\Lambda_{jk}$, which is (i) self-adjoint and (ii) unitary. In addition it satisfies (iii) $\widetilde{\Lambda}_{ij}\widetilde{\Lambda}_{jk}
=\widetilde{\Lambda}_{ik}\widetilde{\Lambda}_{ij}
=\widetilde{\Lambda}_{jk}\widetilde{\Lambda}_{ik}$, (iv) $\widetilde{\Lambda}_{ij}\widetilde{\Lambda}_{kl}
=\widetilde{\Lambda}_{kl}\widetilde{\Lambda}_{ij}$, (v) $\widetilde{\Lambda}_{jk} z_k \partial_k = z_j \partial_j $. In terms of the above mentioned operators, the Dunkl-type momentum operators $\{ D_j \vert j=1,\dots, N \}$ may be represented by the following equation, \[mom11\] D\_j = z\_j\_j +\_[k(j)]{} (1-\_[jk]{}) . +\_[k(j)]{} (1-\_[jk]{}) $ \widetilde{H}_N = \sum_j D_j^2 $. These Dunkl-type operators commute with the coordinate exchange operators and the operators $\widetilde{\Lambda}_{jk}$, i.e; $[D_j, \Lambda_{jk}] = 0$, $[D_j, \widetilde{\Lambda}_{jk}] = 0$. They also commute among themselves and because of the very nature of their construction, commute with the Hamiltonian; $[D_j, D_k] = 0$, $[D_j, \widetilde{H}_N ] = 0$. The existence of such an operator establishes the integrability of the system.
Quasi-solvability of the model Hamiltonian
==========================================
The integrability does not necessarily imply the existence of a function space involving the variables $\{z_j \vert j = 1,\dots, N \}$ such that $ \widetilde{H}_N $ can be represented in a diagonal form. However, sometimes
| 574
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| null | null |
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|
ve viewpoint if we endow $\mathcal{L}_{QD}^{P}$ with some further derived pragmatic connectives which can be made to correspond with connectives of physical QL. To this end, we introduce the following definitions.
D$_{1}$. *We call* quantum pragmatic disjunction *the connective* $A_{Q}$* defined as follows.*
*For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*,* $\delta _{1}A_{Q}\delta _{2}=N((N\delta _{1})K(N\delta
_{2}))$*.*
D$_{2}$. *We call* quantum pragmatic implication* the connective* $I_{Q}$* defined as follows.*
*For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*,* $\delta _{1}I_{Q}\delta _{2}=(N\delta _{1})A_{Q}(\delta
_{1}K\delta _{2})$*.*
Let us discuss the justification rules which hold for afs in which the new connectives $A_{Q}$ and $I_{Q}$ occur.
By using the function $f$ introduced in Sec. 3.2 we get (since the set-theoretical operation $\cap $ coincides with the lattice operation $\Cap
$ in $(\mathcal{L(S)},\subset )$, see Sec. 2.2),
$\mathcal{S}_{\delta _{1}A_{Q}\delta _{2}}=\mathcal{S}_{(N\delta
_{1})K(N\delta _{2})}^{\bot }=(\mathcal{S}_{N\delta _{1}}\cap \mathcal{S}_{N\delta _{2}})^{\bot }=(\mathcal{S}_{\delta _{1}}^{\bot }\Cap \mathcal{S}_{\delta _{2}}^{\bot })^{\bot }=(\mathcal{S}_{\delta _{1}}\Cup \mathcal{S}_{\delta _{2}})$.
Hence, for every $S\in \mathcal{S}$,
$\pi _{S}(\delta _{1}A_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{\delta
_{1}}\Cup \mathcal{S}_{\delta _{2}}$.
Let us come to the quantum pragmatic implication $I_{Q}$. By using the definition of $A_{Q}$, one gets
$\delta _{1}I_{Q}\delta _{2}=N((NN\delta _{1})K(N(\delta _{1}K\delta _{2}))$.
By using the function $f$ and the above result about $A_{Q}$, one then gets
$\mathcal{S}_{\delta _{1}I_{Q}\delta _{2}}=\mathcal{S}_{N\delta _{1}}\Cup
\mathcal{S}_{\delta _{1}K\delta _{2}}=\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$.
It follows that, for every $S\in \mathcal{S}$,
$\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{
| 575
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| null | null |
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|
tition-function .unnumbered}
----------------------------------------------------
If $G$ is semisimple, we can find $s, t \in {\mathbb Z}_{\geq 0}$ and group homomorphisms $A \colon {\mathbb Z}^s \rightarrow {\mathbb Z}^t$ and $B \colon \Lambda^*_G \oplus \Lambda^*_G \rightarrow {\mathbb Z}^t$ such that $$\label{bliem multiplicity formula}
m_{T_G,V_{G,\lambda}}(\beta) = \phi_A \left( B {\left(\begin{smallmatrix}\lambda \\ \beta\end{smallmatrix}\right)} \right)
\>\>
(\forall \lambda \in \Lambda^*_{G,+}, \beta \in \Lambda^*_G),$$ where $\phi_A$ is the *vector partition function* defined by $$\phi_A(y) = \# \{ x \in {\mathbb Z}^s_{\geq 0} : A x = y \}.$$ Note that this improves over the Kostant multiplicity formula , where weight multiplicities are expressed as an alternating sum over vector partition functions. In particular, is an evidently positive formula. It has been established by Billey, Guillemin, and Rassart for the Lie algebra ${\mathfrak{su}}(d)$ [@billeyguilleminrassart04], and was later extended to the general case by Bliem [@bliem08] by considering Littelmann patterns [@littelmann98] instead of Gelfand–Tsetlin patterns [@gelfandtsetlin88]. The assumption of semisimplicity for is not a restriction. Indeed, if $G$ is a general compact connected Lie group then its Lie algebra can always decomposed as $$\label{compact decomposition}
\mathfrak g = [\mathfrak g,\mathfrak g] \oplus \mathfrak z,$$ where the commutator $[\mathfrak g, \mathfrak g]$ is the Lie algebra of a compact connected semisimple Lie group $G_{\operatorname{ss}}$, and where $\mathfrak z$ the Lie algebra of the center $Z(G)$ of $G$ [@knapp02 Corollary 4.25]. Let us choose a maximal torus $T_{G_{\operatorname{ss}}}$ of $G_{\operatorname{ss}}$ that is contained in $T_G$. Consider now an irreducible $G$-representation $V_{G,\lambda}$ with highest weight $\lambda$. By Schur’s lemma, each element in $Z(G)$ acts by a scalar. Therefore, all weights $\beta$ that appear in the weight-space decomposition have the same restriction to $\math
| 576
| 172
| 714
| 711
| 1,743
| 0.786186
|
github_plus_top10pct_by_avg
|
termined by the least significant two bits of $a$, $b$ and $m$.
The new algorithm
-----------------
The algorithm works with two non-negative integers $a$ and $b$, where multiples of the smaller one is subtracted from the larger. To compute the Jacobi symbol we maintain these additional state variables: $$\begin{aligned}
e &\in {\mathbb{Z}}_2 && \text{Current sign is $(-1)^e$} \\
\alpha &\in {\mathbb{Z}}_4 && \text{Least significant bits of $a$} \\
\beta &\in {\mathbb{Z}}_4 && \text{Least significant bits of $b$} \\
d &\in {\mathbb{Z}}_2 && \text{Index of denominator}\end{aligned}$$
The value of $d$ is one if the most recent reduction subtracted $b$ from $a$, and zero if it subtracted $a$ from $b$. We collect these four variables as the state $S = (e, \alpha, \beta, d)$. The state is updated by the function , Algorithm \[alg:jupdate\].
In: $d' \in {\mathbb{Z}}_2$, $m \in {\mathbb{Z}}_4$, $S = (e, \alpha, \beta, d)$ $d \neq d'$ and both $\alpha$ and $\beta$ are odd $e {\leftarrow}e + (\alpha - 1)(\beta - 1)/4$ Reciprocity $d {\leftarrow}d'$ $d = 1$ $\beta = 2$ $e {\leftarrow}e + m (\alpha - 1)/2 + m (m-1)/2$ $\alpha {\leftarrow}\alpha - m \beta$ $\alpha = 2$ $e {\leftarrow}e + m (\beta - 1)/2 + m (m-1)/2$ $\beta {\leftarrow}\beta - m \alpha$ $S' = (e, \alpha, \beta, d)$
Since the inputs of this function are nine bits, and the outputs are six bits, it’s clear it can be implemented using a lookup table consisting of $2^9$ six-bit entries, which fits in 512 bytes if entries are padded to byte boundaries.[^4]
Algorithm \[alg:jacobi\] extends the generic left-to-right algorithm to compute the Jacobi symbol. The main loop of this algorithm differs from Algorithm \[alg:gcd\] only by the calls to for each reduction step.
In: $a, b > 0$, $b$ odd Out: The Jacobi symbol $(a | b)$ State: $S = (e, \alpha, \beta, d)$ \[li:jacobi-init\] $S {\leftarrow}(0, a \bmod 4, b \bmod 4, 1)$ $a \geq b$ \[li:update-a\] $a {\leftarrow}a - m b$, with $1 \leq m \leq {\lfloor a/b \rfloor}$ \[li:jacobi-update-a\] $S {\leftarrow}\proc
| 577
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| 266
| 588
| 978
| 0.79649
|
github_plus_top10pct_by_avg
|
t\in \{1,2,\ldots, n\} \mid \{i,j\}\in E_t\}|{\leqslant}2n^{1-\varepsilon}$$ for some constant $\varepsilon\in(0, 1)$. This property is called $\varepsilon$-visibility.
\[thm:s2c\] Let $(G^{(1)},\ldots, G^{(n)})$ be a regular dynamic graph which satisfies the $\varepsilon$-visibility condition, for some $\varepsilon\in (0,1)$. Suppose that the balanced allocation process on $(G^{(1)}, \ldots, G^{(n)})$ has allocated $n$ balls. Then the maximum load is at most $\log_2\log n+{\mathcal{O}}(1/\varepsilon)$, with high probability.
The proof, which can be found in Section \[sec:graph\], is again based on the witness tree technique. We remark that Theorem \[thm:s2c\] can be extended to the case where the dynamic graph is *almost regular*, meaning that the ratio of the minimum and maximum degree of $G^{(t)}$ is bounded above by an absolute constant for $t=1,\ldots, n$.
### Dynamic Graphs and Hypergraphs with Low Pair Visibility {#dynamic-graphs-and-hypergraphs-with-low-pair-visibility .unnumbered}
In order to show the ubiquity of the visibility condition, we will describe some dynamic graphs with low pair visibility. One can easily construct a dynamic hypergraph from a dynamic graph by considering the $r$-neighborhood of each vertex of the $t$-th graph as a hyperedge in the $t$-th hypergraph, for $t=1,\ldots, n$.
- - Suppose that $n$ is a prime number and fix $s=s(n)$ [such that $\log n{\leqslant}s{\leqslant}n^{1/5}$. (Here $n$ is large enough so that this range is non-empty.)]{} For $t=1,\ldots,n$, let $k_t=\lceil \sqrt{n} \rceil + \lceil \frac{t}{n^{3/4}} \rceil$ and for each $\alpha\in \mathbb{Z}_n$ define $$H_t(\alpha)=\{ \, \alpha + j k_t\, (\operatorname{mod}\, n) \mid j=0,1,\ldots,s-1 \, \}.$$ Then $H_t(\alpha)$ is a subset of $\mathbb{Z}_n$ of size $s$, as $n$ is prime. Now for each $t=1,\ldots,n$ we define the dynamic $s$-uniform hypergraph ${\mathcal{H}}^{(t)}=(\mathbb{Z}_n, \mathcal{E}_t)$, where $\mathcal{E}_t=\{H_t(\alpha) \mid \alpha\in \mathbb{Z}_n\}$. Then ${\mathcal{H}}^{(t)}$ is $s$-regular
| 578
| 1,124
| 1,032
| 585
| 1,675
| 0.786892
|
github_plus_top10pct_by_avg
|
-------------------------------------------------------------------------- ------------- -------------------- ----------------------------------------------- ------------- ------------- ------------ ---------- ---------- ----------
Age, year
Mean (SD) 60.7 (12.0) 59.8 (12.1) 62.7 (11.7) 59.8 (12.0) 67.5 (10.4) 47.8(19.2) \< 0.001 \< 0.001 \< 0.001
≤ 60, n. (%) 473 (46.4) 361 (49.5) 112 (38.6) 453 (49.7) 20 (18.5) 972 (71.8) \< 0.001 0.002 \< 0.001
Male, n (%) 428 (42.0) 280 (38.4) 148 (51.0) 373 (40.9) 55 (50.9) 694 (51.3) \< 0.001 \< 0.001 0.046
Underlying medical conditions[^a^](#tfn_001){ref-type="table-fn"}, n (%) 278 (27.3) 171 (23.4) 107 (36.9) 234 (25.7) 44 (40.6) 309 (22.8) 0.013 \< 0.001 0.001
Abbreviations: SD, standard deviation.
The underlying medical conditions were defined as patients presenting with one of the following: hypertension, diabetes, cancer, active hepatitis, cerebral infarction, et al.
*χ*^2^ test for categorical variables and the Mann Whitney U test for continuous variable
*PDGF-B* rs1800818 *G* allele conferred increased susceptibility to SFTS disease {#s2_2}
--------------------------------------------------------------------------------
The initial small-scale association study was performed in the first cohort of 250 SFTS patients and 250 controls. The genotyping
| 579
| 472
| 920
| 801
| null | null |
github_plus_top10pct_by_avg
|
CA \- 107.5 35.0 109.8 95.4 2158 19 0.75 382 130 9
GP16 CA \- 108.6 36.0 115.3 95.4 2939 26 1.03 677 240 43
GP01 CA \+ 117.5 39.0 104.0 95.7 6652 38 2.33 1105 187 10
GP07 CA \+ 106.5 34.9 121.9 95.4 2113 16 0.74 359 102 4
GP09 CA \+ 111.9 37.2 105.3 95.5 2907 15 1.02 511 165 17
GP17 CA \+ 111.3 36.6 112.2 95.6 3238 35 1.13 551 186 16
Mean 113.4 37.4 111.4 95.6 2821 22 1.00 561 230 17
Total 1587.3 523.8 1559.0 1337.9 39489 313 13.82 7856 3213 243
######
*PTEN* deletion status evaluated by FISH assay.
Race *PTEN* status *p*-Value
------------ --------------- ----------- -------
AA, N = 40 35 (88%) 6 (15%) 1E-06
CA, N = 59 22 (37%) 37 (63%)
######
*PTEN* deletion frequencies by worst Gleason sum.
Gleason Sum AA (N = 40) CA (N = 52) *p*-Value
------------- ------------- ------------- ----------- ---------- -------
6 or less 14 (93%) 1 (7%) 9 (47%) 10 (53%) 0.004
7 11 (73%) 4 (27%) 7 (33%) 14 (67%) 0.02
8 to 10 7 (70%) 3 (30%) 4 (33%) 8 (67%) 0.09
[^1]: Equal contributions t
| 580
| 4,396
| 320
| 235
| null | null |
github_plus_top10pct_by_avg
|
algebra $K(H)[X, X^{-1}; \s] = K(H)[Y, Y^{-1}; \s^{-1}]$ where $K(H)$ is the field of rational functions in the variable $H$ and the automorphism $\s$ of $K(H)$ is given by the rule $\s(H) = H-1$. By [@Bav-DixPr5 Proposition 2.1(1)], the centralizer $C_B(P_p)$ of the element $P_p$ in $B$ is a Laurent polynomial algebra $K[\alpha X^n, (\alpha X^n)^{-1}]$ for some nonzero element $\alpha\in K(H)$ and $n \geq 1$. In general, $\alpha \notin K[H]$. Similarly, $C_B(P_{-q}) = K[\beta Y^m, (\beta Y^m)^{-1}]$ for some nonzero element $\beta \in K(H)$ and $m \geq 1$.
Since $[P_p, Q_q]=0$, $Q_q \in C_B(P_p)$ and $$P_p = \lambda(P_p) (\alpha X^n)^i =
\lambda(P_p) \alpha \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) X^{ni} = \alpha_{n,i} X^p,$$ $$Q_q = \lambda(Q_q) (\alpha X^n)^j = \lambda(Q_q) \alpha \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) X^{nj} = \alpha'_{n,j} X^q,$$ for some nonzero scalars $\lambda(P_p), \lambda(Q_q) \in K^*$ and some $i \geq 1$ and $j\geq 1$ where $$\alpha_{n,i} = \lambda(P_p) \alpha \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) \in K[H], \; p=ni,$$ $$\alpha'_{n,j} = \lambda(Q_q) \alpha \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) \in K[H], \; q=nj.$$
Since $[P_{-p}, Q_{-p}]=0$, $Q_{-p} \in C_B(P_{-q})$ and $$P_{-q} = \lambda(P_{-q}) (\beta Y^m)^s =
\lambda(P_{-q}) \beta \s^{-m}(\beta)\cdots \s^{-m(s-1)}(\beta) Y^{ms} = \beta_{m,s} Y^p,$$ $$Q_{-p} = \lambda(Q_{-p}) (\beta Y^m)^t =
\lambda(Q_{-p}) \beta\s^{-m}(\beta)\cdots \s^{-m(t-1)}(\beta) Y^{mt} = \beta'_{m,t} Y^q,$$ for some nonzero scalars $\lambda(P_{-q}), \lambda(Q_{-p}) \in K^*$ and some $s \geq 1$ and $t\geq 1$ where $$\beta_{m,s} = \lambda(P_{-q}) \beta \s^{-m}(\beta) \cdots \s^{-m(s-1)}(\beta) \in K[H],\; p=ms,$$ $$\beta'_{m,t} = \lambda(Q_{-p})\beta \s^{-m}(\beta) \cdots \s^{-m(t-1)}(\beta) \in K[H], \; q=mt.$$
Now, $$1 = [P,Q] = [P_p, Q_{-p}] + [P_{-q}, Q_q]
= [\alpha_{n,i} X^p, \beta_{m,t}' Y^p ] + [\beta_{m,s} Y^q , \alpha_{n,j}' X^q]$$ $$= \alpha_{n,i} \s^p (\beta'_{m,t}) (p,-p) - \beta'_{m,t} \s^{-p}(\alpha_{n,i}) (-p,p)$$ $$+
\beta_{m,s} \
| 581
| 2,552
| 700
| 625
| null | null |
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|
act-sde}$ and $\eqref{e:discrete-clt}$ over an epoch which consists of an interval $[k\delta , (k+n)\delta)$ for some $k$. The coupling in consists of four processes $(x_t,y_t,v_t,w_t)$, where $y_t$ and $v_t$ are auxiliary processes used in defining the coupling. Notably, the process $(x_t,y_t)$ has the same distribution over the epoch as .
2. In Appendix \[ss:epoch\_nongaussian\], we prove Lemma \[l:non\_gaussian\_contraction\_stationary\] and Lemma \[l:non\_gaussian\_contraction\_anisotropic\], which, combined with Lemma \[l:gaussian\_contraction\] from Appendix \[ss:step\_gaussian\], show that under the coupling constructed in Step 1, a Lyapunov function $f(x_T - w_T)$ contracts exponentially with rate $\lambda$, plus a discretization error term. In Corollary \[c:main\_nongaussian:1\], we apply the results of Lemma \[l:gaussian\_contraction\], Lemma \[l:non\_gaussian\_contraction\_stationary\] and Lemma \[l:non\_gaussian\_contraction\_anisotropic\] recursively over multiple steps to give a bound on $f(x_{k\delta}-w_{k\delta})$ for all $k$, and for sufficiently small $\delta$.
3. Finally, in Appendix \[ss:proof:t:main\_nongaussian\], we prove Theorem \[t:main\_nongaussian\] by applying the results of Corollary \[c:main\_nongaussian:1\], together with the fact that $f(z)$ upper bounds $\lrn{z}_2$ up to a constant.
[Constructing a Coupling]{} \[ss:4\_coupling\] In this subsection, we construct a coupling between and , given arbitrary initialization $(x_0,w_0)$. We will consider a finite time $T=n\delta$, which we will refer to as an *epoch*.
1. Let $V_t$ and $W_t$ be two independent Brownian motion.
2. Using $V_t$ and $W_t$, define $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_x}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(w_0) dW_s
\end{aligned}$$
3. Using the same $V_t$ and $W_t$ in , we will define $y_t$ as $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_y}
y_t =& w_0
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n{aligned}
X \otimes_{[B]} \big((\id\times u\op)^\ast\lI_B\big) &\cong (\id\times u\op)^\ast X\\
Y \otimes_{[A]} \big((\id\times u\op)_!\lI_A\big) &\cong (\id\times u\op)_! Y\end{aligned}$$ In fact, these dual base change objects are actually isomorphic to the first two swapped: $$\begin{aligned}
(\id\times u\op)^\ast\lI_B &\cong (u\times\id)_!\lI_A\\
(\id\times u\op)_!\lI_A &\cong (u\times\id)^\ast\lI_B\end{aligned}$$ This all follows from the fact that $\cProf({\sV})$ is actually a “framed bicategory”; see [@shulman:frbi] and [@ps:linearity (15.2)].
Let be a monoidal left derivator and a -module. For $u\colon A\to B$ in $\cCat$ we obtain an isomorphism $$u_!\cong ((\id\times u\op)^\ast\lI_B)\otimes_{[A]}-\colon{\sD}^A\to{\sD}^B.$$ Specializing to $u=\pi_A\colon A\to\bbone$ we deduce that colimits are weighted colimits with constant weight $\pi_{A\op}^\ast\lS_\bbone$. More generally, the weight for $u_!$ has components $$((\id\times u\op)^\ast\lI_B)_{b,a}\cong\coprod_{\hom_B(ua,b)}\lS_\bbone,$$ and the isomorphism $u_!X\cong((\id\times u\op)^\ast\lI_B)\otimes_{[A]}X$ is hence a left derivator version of the usual coend formula for left Kan extensions in sufficiently cocomplete categories ([@maclane Thm. X.4.1]).
Stability via weighted colimits {#sec:stab-via-wcolim}
===============================
\[thm:wcolim\]\[item:wcl3\] and \[item:wcl4\] cry out for a generalization to $\Phi$-stability.
If $\Phi$ is a class of functors $u\colon A\to B$, we define a left derivator to be **right $\Phi$-stable** if it *has* right Kan extensions along each $u\in \Phi$ which moreover commute with arbitrary left Kan extensions.
By \[thm:ldh-ran\], if ${\sD}$ is right $\Phi$-stable, then so is ${\mathsf{END}\ccsub}({\sD})$.
\[thm:stable-dual\] Let be a monoidal left derivator and $u\colon A\to B$ a functor. The following are equivalent:
1. ${\sV}$ is right $u\op$-stable.\[item:sd0\]
2. The base change profunctor $(u\times\id)_!\lI_A\in \cProf({\sV})(B,A)$ has a right adjoint in $\cProf({\sV})$.\[item:sd1
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on of the 1-form $\omega\in T_u^\ast U$ with the vector field $\gamma\in T_uU$ and similarly for the vector field $\bar{\gamma}\in T_uU$. Multiplying the equations (\[eq:2.8\]) by the 1-form $\omega\in\Phi$ we get
\[eq:2.9\] |<>|=0.
We look for the compatibility condition for which the system (\[eq:2.9\]) does not provide any algebraic constraints on the coefficients $\xi$ and $\bar{\xi}$. This postulate means that the profile of the simple modes associated with $\gamma\otimes \lambda$ and $\bar{\gamma}\otimes
\bar{\lambda}$ can be chosen in an arbitrary way for the initial (or boundary) conditions. It follows from (\[eq:2.9\]) that the commutator of the vector fields $\gamma$ and $\bar{\gamma}$ is a linear combination of $\gamma$ and $\bar{\gamma}$, where the coefficients are not necessarily constant, [*i.e.* ]{}$${\left[ \gamma,\bar{\gamma} \right]}\in\operatorname{span}{\left\{ \gamma,\bar{\gamma} \right\}}.$$So, the Frobenius theorem is satisfied. That is, at every point $u_0$ of the space of dependent variables $U$, there exists a tangent surface $S$ spanned by the vector fields $\gamma$ and $\bar{\gamma}$ passing through the point $u_0\in U$. Moreover the above condition implies that there exists a complex-valued function $\alpha$ such that
\[eq:2.10\] =-||,
since $\overline{{\left[ \gamma,\bar{\gamma} \right]}}={\left[ \bar{\gamma},\gamma \right]}=-{\left[ \gamma,\bar{\gamma} \right]}$ holds. Next, from the vector fields $\gamma$ and $\bar{\gamma}$ defined on $U$-space, we can construct the coframe $\Psi$, that is the set of 1-forms $\sigma_1$ and $\sigma_2$ defined on $U$ satisfying the conditions
\[eq:2.11\] &<\_1>=1,&&<\_1|>=0,\_1,\_2,\
&<\_2>=0, &&<\_2|>=1.
Substituting (\[eq:2.6\]) and (\[eq:2.10\]) into the prolonged system (\[eq:2.8\]) and multiplying by $\sigma_1$ and $\sigma_2$ respectively, we get
\[eq:2.11b\] &(i)&& d+[( ||\_[,|]{}+|\_[,]{} )]{}+||=0,\
&(ii) &&d||+|[( |\_[,]{}+||\_[,|]{} )]{}+|||=0.
If $d\xi=d\bar{\xi}$ then equation (\[eq:2.11b\].i) is a c
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f [<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{} from the perspective of instance-wise optimality.'
author:
- |
Lijie Chen Jian Li\
Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University
bibliography:
- 'team.bib'
title: On the Optimal Sample Complexity for Best Arm Identification
---
Concluding Remarks
==================
The most interesting open problem from this paper is to obtain an almost instance optimal algorithm for [<span style="font-variant:small-caps;">Best-$1$-Arm</span>]{}, in particular to prove (or disprove) Conjecture \[conj:optimal\]. Note that for the clustered instances, and the instances where the gap entropy is $\Omega(\ln\ln n)$, we already have such an algorithm. Our techniques may be helpful for obtaining better bounds for the [<span style="font-variant:small-caps;">Best-$k$-Arm</span>]{} problem, or even the combinatorial pure exploration problem. In an ongoing work, we already have some partial results on applying some of the ideas in this paper to obtain improved upper and lower bounds for [<span style="font-variant:small-caps;">Best-$k$-Arm</span>]{}.
---
abstract: 'Earth’s climate, mantle, and core interact over geologic timescales. Climate influences whether plate tectonics can take place on a planet, with cool climates being favorable for plate tectonics because they enhance stresses in the lithosphere, suppress plate boundary annealing, and promote hydration and weakening of the lithosphere. Plate tectonics plays a vital role in the long-term carbon cycle, which helps to maintain a temperate climate. Plate tectonics provides long-term cooling of the core, which is vital for generating a magnetic field, and the magnetic field is capable of shielding atmospheric volatiles from the solar wind. Coupling between climate, mantle, and core can potentially explain the divergent evolution of Earth and Venus. As Venus lies too close to the sun for liquid water to exist, there is no long-term carbon cycle and thus an extremely hot cl
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s, and 60°C for 60 s. A melt curve was performed in order to verify single PCR products. The comparative threshold cycle (*C~T~*) method was used to quantify transcripts, with the ribosomal *rrs* gene serving as the endogenous control. Δ*C~T~* values were calculated by subtracting the *C~T~* value of the control gene from the *C~T~* value of the target gene, and the ΔΔ*C~T~* value was calculated by subtracting the wild-type Δ*C~T~* value from the mutant Δ*C~T~* value. Fold change represents 2^−ΔΔ*CT*^. Experiments included three technical replicates, and the data represent the qPCR results from five separate RNA isolation experiments. The oligonucleotides used are listed in [Table 2](#T2){ref-type="table"}.
######
Oligonucleotides used in this study
Primer purpose and gene or plasmid Direction and use[^*a*^](#T2F1){ref-type="table-fn"} Sequence Reference
------------------------------------ ------------------------------------------------------ ------------------------------------------------------------------------ -----------
qPCR
*hilA* Fwd ATAGCAAACTCCCGACGATG [@B79]
Rev ATTAAGGCGACAGAGCTGG
*hilD* Fwd GGTAGTTAACGTGACGCTTG [@B79]
Rev GATCTTCTGCGCTTTCTCTG
*rtsA* Fwd
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y denominator, as shown in (\[Omega-1st-order\]). Then, higher order correction terms are always accompanied by the energy denominators which are composed of some of the first three in (\[expansion-parameters\]), and therefore they are suppressed. The unique exception for it is the terms generated only by $\Omega [1]_{JJ}$ which lacks the energy denominator. Therefore, apart from this special case, we have shown that higher-order corrections in $W$ does not produce the surviving terms after averaging over fast oscillation and using the energy denominator suppression. It is consistent with what we saw in our explicit computation to order $W^4$. This concludes our justification of the Uniqueness theorem.
We need to clear up the issue of special type of perturbative correction terms which involve only $\Omega [1]_{JJ}$ as the kernel in (\[Omega-expand\]). It produces the unique form of $\hat{S}_{JJ}$ as $$\begin{aligned}
\hat{S}_{JJ} =
e^{ - i \Delta_{J} x }
\sum_{n}
\frac{ (-ix)^n }{n !} \left\{ (W^{\dagger} A W)_{JJ} \right\}^n,
\label{hatS-JJ-term}\end{aligned}$$ a collection of terms of matter-dependent higher order renormalization to $\sum_{J} W_{\alpha J} W_{\beta J}^*$, the probability leaking term at the amplitude level. However, it exponentiates and has contribution to the $S$ matrix element as[^22] $$\begin{aligned}
S_{\alpha \beta} =
\sum_{J}
W_{\alpha J} W_{\beta J}^*
\exp { \left[ -i \left\{ \Delta_{J} + (W^{\dagger} A W )_{JJ} \right\} x \right] }.
\label{S-alpha-beta-WW-term}\end{aligned}$$ The unique form of $S$ matrix, in principle, raises an interesting issue of dynamically generated phase produced jointly by unitarity violation and matter effect.[^23] In our setting, however, it either disappears from the amplitude squared, or has vanishing effect when the high frequency oscillation is averaged out.
Absence of enhancement due to small solar mass splitting denominator {#sec:no-enhancement}
----------------------------------------------------------------------
In perturbation theory one
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labeled as input. $\hfill \square$
As stated, we want to prove that every query in the denotation of the considered moded query is either non-terminating or terminates due to the evaluation of an integer condition. To achieve this, we need to guarantee that integer constructors are repeatedly evaluated with a free variable and an integer expression as arguments and that integer conditions are repeatedly evaluated with integer expressions as arguments. Proposition \[prop:mmg\] already implies that the first argument of all integer constructors are free variables in the subsequent iterations of the loop.
To prove the repeated behavior on integer constructors and integer expressions stated above, the *integer-similar to* relation is defined. Intuitively, given some loop in the computation, if an atom at the end of the loop is integer-similar to an atom at the start of the loop, then it will provide the required integer expressions to the first atom. First, we introduce positions to identify subterms and a function to obtain a subterm from a given position.
\[def:func\_subterm\] Let $L$ be a list of natural numbers, called a *position*, and $A$ a moded atom or term. The function *subterm(L,A)* returns the subterm obtained by:
- if $L = [I]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = A_I$
- else if $L=[I|T]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = subterm(T,A_I)$ $\hfill \square$
An atom $A$ is integer-similar to an atom $B$ if it has integer expressions on all positions corresponding to integer expressions in $B$.
Let $A$ and $B$ be moded atoms. $A$ is *integer-similar to* $B$ if for every integer expression $t_B$ of $B$, with $subterm(L,B) = t_B$, there exists an integer expression $t_A$ of $A$, with $subterm(L,A) = t_A$. $\hfill \square$
- $count(0,\underline{N},L)$ is integer-similar to $count(\underline{M},\underline{N},L)$
- $count(\underline{M},\underline{N},L)$ is integer-similar to $count(0,\underline{N},L)$
- $count(\underline{M} + 1,\under
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left( \wedge^{m_n} {\cal E}_n^{\alpha^{-1} *}
\otimes \wedge^{p_n} {\cal E}_n^{\alpha^{-1} }
\otimes \wedge^{\ell_n} T_n^{\alpha^{-1} *}
\otimes \wedge^{k_n} T_n^{\alpha^{-1} } \right)
\\
& & \hspace*{3.5in} \left.
\otimes {\cal F}^{\alpha^{-1}}_+ \otimes
\sqrt{ K_{\alpha^{-1}} \otimes \det {\cal E}^{\alpha^{-1} }_0 }
\right)^*,\end{aligned}$$ which is of the same form as equation (\[eq:countstates1\]), as desired. Note that the Fock vacuum contribution is essential for the spectrum to close under Serre duality in this fashion: otherwise, Serre duality would generate a factor of $K_{\alpha}$ in the coefficients, unmatched by anything else, and which is nontrivial if the $\alpha$ component is not Calabi-Yau[^23]. Our computations so far have focused on the (R,R) sector, but one should note that identical considerations hold in the (NS,R) sector as well.
In the special case that the stack $\mathfrak{X}$ is a smooth Calabi-Yau manifold $X$, these computational methods reduce to those of [@dist-greene]. In this case, the inertia stack $I_{\mathfrak{X}}$ has no nontrivial components: $I_{\mathfrak{X}} = X$. Furthermore, we typically take $\det {\cal E}$ to be trivial, so the Fock vacuum is a section of a trivial line bundle.
In the special case that the stack $\mathfrak{X}$ is a toroidal orbifold, again these methods reduce to known results. In this case, all of the bundles involved are trivial, so sheaf cohomology is nontrivial only in degree zero, and sheaf cohomology on a stack just takes group invariants of the coefficients.
A less trivial example is discussed in section \[sect:class3-caution1\]. Further examples and computational techniques will appear in [@manion-toappear].
Just as in [@dist-greene], in principle the number of generations can be computed as an index based on the spectrum. We shall not work through details here, but appendix \[app:chern-reps\] contains general results on index theory computations on stacks.
A/2 model spectra
-----------------
In this appendix we have focused on physica
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$v_{i+1}=(0,\cdots, 0, 1, 0)$) of size $1\times n_{i+1}$ if $L_{i+1}$ is *of type* $\textit{I}^o$ (resp. *of type* $\textit{I}^e$).
5. Assume that $i$ is odd and that $L_i$ is *bound of type I*. Then $$\delta_{i-1}v_{i-1}\cdot {}^tm_{i, i-1}+\delta_{i+1}v_{i+1}\cdot {}^tm_{i, i+1} = \pi m_{i,i}^{\ast\ast}$$ such that $ m_{i,i}^{\ast\ast} \in M_{1\times n_i}(B\otimes_AR)$. Here, $v_{i-1}$ (resp. $v_{i+1}$)$=(0,\cdots, 0, 1)$ of size $1\times n_{i-1}$ (resp. $1\times n_{i+1}$).\
In conclusion, each element of $\underline{M}(R)$ for a flat $A$-algebra $R$ is written as the following matrix $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \mathrm{~together~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$
**\
However, for a general $R$, the above description for $\underline{M}(R)$ will no longer be true. For such $R$, we use our chosen basis of $L$ to write each element of $\underline{M}(R)$ formally. We describe each element of $\underline{M}(R)$ as formal matrices $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \mathrm{~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}$$ such that
1. When $i\neq j$, $m_{i,j}$ is an $(n_i \times n_j)$-matrix with entries in $B\otimes_AR$.
2. When $i=j$, $m_{i,i}$ is an $(n_i \times n_i)$-formal matrix such that $$m_{i,i}=\left\{
\begin{array}{l l}
\begin{pmatrix} s_i&\pi y_i\\ \pi v_i&1+\pi z_i \end{pmatrix} & \quad \textit{if $i$ is even and $L_i$ is \textit{of type $I^o$}};\\
\begin{pmatrix} s_i&r_i&\pi t_i\\ \pi y_i&1+\pi x_i&\pi z_i\\ v_i&u_i&1+\pi w_i \end{pmatrix} & \quad \textit{if $i$ is even and $L_i$ is \textit{of type $I^e$}};\\
\begin{pmatrix} s_i&\pi r_i&t_i\\ y_i&1+\pi x_i& u_i\\\pi v_i&\pi z_i&1+\pi w_i \end{pmatrix}
& \quad \textit{if $i$ is odd and $L_i$ is \textit{free of type $I$}};\\
m_{i,i} & \quad \textit{otherwise}.
\end{array} \right.$$ Here, $s_i$ is an $(n_i-1 \times n_i-1)$-matrix (resp. $(n_i-2 \times n_i-2)$-matrix) with entries in $B\otimes_AR$ if $i$ is ev
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| 0.780068
|
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|
ran}{\Phi}}}$.
\[D:DYADIC\_SPACE\_PRODUCT\] Let $\Theta$ and $\Phi$ be disjoint ensembles. The dyadic space product of $\prod\Theta$ and $\prod\Phi$ is $$\prod \Theta \prod \Phi = \prod \Theta\Phi$$ (that is, with $\Upsilon = \Theta\Phi$, the set of all $\Upsilon$-choices).
\[D:DYADIC\_CHOICE\_PROD\] Let $\Theta$ and $\Phi$ be disjoint ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$. Suppose $\alpha \in \prod\Theta$ and $\beta \in \prod\Phi$ are choices. Their dyadic *choice* product $(\alpha,\beta) \mapsto \alpha\beta$ is $$\alpha\beta = \alpha \cup \beta.$$
\[T:DYADIC\_CHOICE\_PROD\] Let $\Theta$ and $\Phi$ be disjoint ensembles. The dyadic choice product is a bijection $$\prod\Theta \times \prod\Phi \leftrightarrow \prod\Theta\Phi.$$
Suppose $(\theta, \phi) \in \prod\Theta \times \prod\Phi$, whence it follows that choices $\theta \in \prod\Theta$ and $\phi \in \prod\Phi$. Consider the dyadic products $\theta\phi = \theta \cup \phi$ and $\Theta\Phi = \Theta \cup \Phi$. Since ${{\operatorname{dom}{\theta}}} = {{\operatorname{dom}{\Theta}}} = T$ and ${{\operatorname{dom}{\phi}}} = {{\operatorname{dom}{\Phi}}} = P$, then ${{\operatorname{dom}{\theta\phi}}} = T \cup P = {{\operatorname{dom}{\Theta\Phi}}}$. Let $i \in {{\operatorname{dom}{\theta\phi}}} = T \cup P$. Since $\Theta$ and $\Phi$ are disjoint, then exactly one of two cases holds: either $i \in T$ and $i \notin P$, or $i \notin T$ and $i \in P$.
In the first case, $i \in T$, $\theta\phi(i) = \theta(i)$ and $\Theta\Phi(i) = \Theta(i)$. Since $\theta$ is a $\Theta$-choice, then $\theta(i) \in \Theta(i)$. But since $\theta\phi(i) = \theta(i)$ and $\Theta\Phi(i) = \Theta(i)$, then $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in T$.
The second case is similar and leads to $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in P$. From these two cases we conclude that $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in T \cup P = {{\operatorname{dom}{\theta\phi}}}$, and that $\theta\phi \in \prod\Theta\Phi$.
The preceding establishes that the cho
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ceptible state. At a given time $t$, the network management system updates a module of a controller with software that contains a bug, as shown in Fig. \[fig:prop2\]. As a result, this node becomes infected and propagates the failure (e.g., the bug) to his neighbors, as observed in Fig. \[fig:prop3\]. The epidemic continues to spread while the CP of an infected node eventually affects its DP operation. In this case, the incapability to resolve the problem at the CP might necessitate a complete node replacement (e.g., shutdown), thus impacting the operation of the DP. Consequently, this node becomes disabled as shown in Fig. \[fig:prop4\].
Throughout the last decades, several failures have spread through communication networks. In the early 90s, a rapidly spreading malfunction collapsed the AT&Ts long distance network[^2], causing the loss of more than \$60 million in terms of unconnected calls. In 2002, a failure propagation in the IP layer of the Internet was caused by a vulnerability of the BGP protocol. More recently, a BGP update bug which propagated through Juniper routers caused a major Internet outage in 2011[^3]. In this latter case, routers were reseting and re-establishing its functioning state after five minutes.
Although there are no commercial references nor reports with respect to the occurrence of propagation of failures in BTNs, in the following sections we identify several failure scenarios that could be modelled as epidemic-like spreadings.
Failure Propagation Scenarios in Transport Networks\[vulnerability\]
====================================================================
This section describes BTNs and SDNTNs, showing that both contemporary and future networks might be predisposed to enduring epidemic-like failures.
Backbone Transport Networks
---------------------------
As explained in Section \[sec:introduction\], a BTN is divided into Data Plane (DP), Control Plane (CP) and Network Management Plane (NMP), which have the following functionalities:
- Data plane (or transport p
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solutions are $$t_{hit} = 2n \pi \left( \frac{2c + 1}{\beta} \right)$$ where $c$ ranges over the non-negative integers. At these points in time, the value of $P[1]$ is $$\frac{1}{2} + \frac{1}{2}e^{-(2c+1)\frac{p \pi}{\beta}}$$ which immediately yields Theorem 2.
The breakpoint case $p = 4k$
----------------------------
We first observe that $t_{mix} \to \infty$ as $p \to 4k$. Hence, we do not expect to see any mixing in this case. To analyze the probabilities exactly, we take the limit of $\gamma$ as $p \to 4k$. The solution is $$\label{pequals4k}
\lim_{p \to 4k} \gamma = \frac{1}{2}e^{-\frac{2kt}{n}}\left[1 + \frac{2kt}{n} \right]\enspace.$$ Indeed, since $k$, $t$ and $n$ are all positive, $\gamma$ is zero only in the limit as $t \to \infty$. The linear mixing and hitting behavior from the previous section has entirely disappeared. As in the critical damping of simple harmonic motion, a small decrease in the rate $p$ can result in drastically different behavior, in this case a return to linear mixing and hitting. We leave the limiting mixing analysis of this case for the next section, where we develop some relevant tools.
The case $p > 4k$ and the limit to the classical walk
-----------------------------------------------------
![The $p>4k$ case: a plot of $P[0]$ and $P[1]$ versus time, for $k=1$, $n=5$, $p = 9$[]{data-label="fig3"}](pgreater4k){width="3in"}
The goal of this section is to show two interesting consequences of the presence of substantial decoherence in the quantum walk on the hypercube. First, we will show that for a fixed $p \geq 4k$, the walk behaves much like the classical walk on the hypercube, mixing in time $\Theta(n \log n)$. Second, we show that as $p \to \infty$, the walk suffers from the quantum Zeno effect. Informally stated, the rate of decoherence is so large that the walk is continuously being reset to the initial wave function $|0\rangle^{\otimes n}$ by measurement.
### Recovering classical behavior
Consider a single qubit. Let $P$ be the distribution obtained by full measur
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dient Richardson number ($Ri_g = N^2/S^2$); where, $S$ is the magnitude of wind shear.
It is evident that the simulations with larger cooling rates result in smaller $\mathcal{L}$ as would be physically expected. In contrast, $\eta$ marginally increases with higher stability due to lower $\overline{\varepsilon}$. The ratio of $\mathcal{L}$ to $\eta$ decreases from about 100 to 20 as stability is increased from weakly stable condition to strongly stable condition.
{width="49.00000%"} {width="49.00000%"}
Next, we compute four outer length scales: Ozmidov ($L_{OZ}$), Corrsin ($L_C$), buoyancy ($L_b$), and Hunt ($L_H$). They are defined as [@corrsin58; @dougherty61; @ozmidov65; @brost78; @hunt88; @hunt89; @sorbjan08; @wyngaard10]:
$$L_{OZ} \equiv \left(\frac{\overline{\varepsilon}}{N^3}\right)^{1/2},$$
$$L_C \equiv \left(\frac{\overline{\varepsilon}}{S^3}\right)^{1/2},
\label{LC}$$
$$L_b \equiv \frac{\overline{e}^{1/2}}{N},$$
$$L_H \equiv \frac{\overline{e}^{1/2}}{S},$$
\[OLS\]
Please note that, in the literature, $L_b$ and $L_H$ have also been defined as $\sigma_w/N$ and $\sigma_w/S$, respectively. Both $L_{OZ}$ and $L_C$ are functions of $\overline{\varepsilon}$, a microscale variable. In contrast, $L_b$ and $L_H$ only depend on macroscale variables.
Both shear and buoyancy prefer to deform the larger eddies compared to the smaller ones [@itsweire93; @smyth00; @chung12; @mater13]. Eddies which are smaller than $L_C$ or $L_H$ are not affected by shear. Similarly, buoyancy does not influence the eddies of size less than $L_{OZ}$ or $L_b$. In other words, the eddies can be assumed to be isotropic if they are smaller than all these OLSs.
{width="49.00000%"} {width="49.00000%"}\
{width="49.00000%"} {width="49.00000%"}
Since $\mathcal{L}$ changes across the simulations, all the OLS values are normalized by corresponding $\mathcal{L}$ values and plotted as functions of $Ri_g$ in Fig. \[
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-v^{i})^{-1}$. On the other hand, the coinvariant ring $ {\mathbb{C}}[{\mathfrak{h}}]^{\text{co}{{W}}}$ has graded Poincaré polynomial $\sum_{\lambda} \sum_i [{\mathbb{C}}[{\mathfrak{h}}]_i^{\text{co} {{W}}} : \lambda] [\lambda]v^i$. By definition, this is just $\sum_{\lambda} f_{\lambda}(v)[\lambda]$. Combining these observations gives .
Completion of the proof of Theorem \[morrat\] {#subsec-4.5}
---------------------------------------------
We first prove that $H_de_-H_d=H_d$ (where $d=c+1$, as before). Since $\mu\cong \mu^*$ for symmetric groups, the sign representation is a direct summand of $\mu\otimes \nu$ if and only if $\nu =\mu^t$. Thus implies that $\operatorname{{\textsf}{sign}}$ first appears in $\Delta_d(\mu)$ in the weight space $$m+d(n(\mu)-n(\mu^t)) + n(\mu^t) = m+ dn({\mu}) - (d-1)n({\mu^t})
\qquad\text{where}\quad m=(n-1)/2.$$ If $\lambda \leq \mu$ then $n({\lambda}) \geq n(\mu)$ by [@Shi Theorem B and Proposition 1.6]. Moreover, as $\lambda^t\geq \mu^t$, we have $n(\lambda^t)\leq
n(\mu^t)$. Since $d\in \mathbb{R}_{\geq 1}$, $$m+dn({\lambda}) -(d-1)n({\lambda^t}) \geq m+dn({\mu}) -(d-1)n({\mu^t})$$ with equality if and only if $ \lambda=\mu$.
It follows that the copy of $\operatorname{{\textsf}{sign}}$ appearing in the lowest possible weight space of $\Delta_d(\mu)$ is never a weight of $\Delta_d(\lambda)$ for $\lambda
< \mu$. By Corollary \[poono\], this means that this copy of $\operatorname{{\textsf}{sign}}$ is a weight for $L_d(\mu)$ and hence that $e_-L_d(\mu)\not=0$. By this implies that $H_de_-H_d =H_d$, and so the first equality of is proven.
It remains to show that $H_ceH_c=H_c$ for $c\in {\mathbb{R}}_{\geq 0}$. The argument of shows that we need to prove that $e$ does not annihilate any simple module from $\mathcal{O}_c$. In this case $\operatorname{{\textsf}{triv}}$ appears in $\mu\otimes \nu$ if and only if $\nu=\mu$. Therefore, now implies that $\operatorname{{\textsf}{triv}}$ first appears in $\Delta_c(\mu)$ in degree $m+c(n(\mu) -n(\mu^t))+n(\mu).$ Let $\lambda\leq \mu$.
| 595
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od{\Phi}}$. This chain of implication concludes that $(\phi_i \in {\prod{\Phi}}) \Rightarrow (\phi_{i+1} \in {\prod{\Phi}})$.
Use $\lbrace \phi_n \rbrace$ and $\lbrace \xi_n \rbrace$ to define another sequence $\lbrace {\mathbf{f}}_n \rbrace$ by setting ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$. Since $\phi_i\xi_i \in {\prod{\Psi}}$ and $\phi_{i+1} \in {\prod{\Phi}}$, then $(\phi_i\xi_i, \phi_{i+1}) \in {\prod{\Psi}} \times {\prod{\Phi}}$ so $\lbrace {\mathbf{f}}_n \rbrace$ is a sequence of frames. Note that ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$ and ${\mathbf{f}}_{i+1} = (\phi_{i+1}\xi_{i+1}, \phi_{i+2})$. In this case, the sequence is successively conjoint (definitions \[D:CONJOINT\] and \[D:SUCCESIVELY\_CONJOINT\]) because ${{\phi_{i+1}\xi_{i+1}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi_{i+1}$. As a successively conjoint sequence of frames, $\lbrace {\mathbf{f}}_n \rbrace$ is a process (definition \[D:PROCESS\]).
Since ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$, and by construction $\phi_{i+1} = {\mathit{f}}_i(\phi_i\xi_i)$, then ${\mathbf{f}}_i = (\phi_i\xi_i, {\mathit{f}}_i(\phi_i\xi_i))$ and ${\mathbf{f}}_i \in {\mathit{f}}_i$. By definition \[D:COVERING\_PROCEDURE\], procedure $\lbrace {\mathit{f}}_n \rbrace$ covers process $\lbrace {\mathbf{f}}_n \rbrace$.
### Uncoverable process {#S:UNCOVERABLE_PROCESS}
Although any procedure does cover some process, some processes have no covering procedure. See §\[S:UNCOVERABLE\_PROCESS\_APPENDIX\].
Automata
--------
The algorithm is conceived as a method to solve problems using a network of mechanistic steps consisting of decisions and contingent actions. An automaton is a formal machine whose architecture of states and transitions concretizes some aspects of the algorithm. The deterministic finite automaton (DFA, see example in §\[S:DFA\]) is a simple structure describing transit-based behavior. However, the DFA leaves unexplained the working mechanism underlying transitions. The DFA’s definition can be modified to effect
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possibly zero, by our convention) if $j$ is even (resp. odd).
\[t411\] The morphism $$\psi=\prod_j \psi_j : \tilde{G} \longrightarrow (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is surjective.
Moreover, the morphism $$\varphi \times \psi : \tilde{G} \longrightarrow \prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}} \times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)\times (\mathbb{Z}/2\mathbb{Z})^{\beta}$$ is also surjective.
We first show that $\psi_j$ is surjective. Recall that for such an integer $j$, $L_j$ is *of type I* and $L_{j+2}, L_{j+3}, L_{j+4}$ (resp. $L_{j-1}, L_{j+1},$ $L_{j+2}, L_{j+3}$) are *of type II* if $j$ is even (resp. odd). We define the closed subgroup scheme $F_j$ of $\tilde{G}$ defined by the following equations:
- $m_{i,k}=0$ *if $i\neq k$*;
- $m_{i,i}=\mathrm{id}, z_i^{\ast}=0, m_{i,i}^{\ast}=0, m_{i,i}^{\ast\ast}=0$ *if $i\neq j$*;
- and for $m_{j,j}$, $$\left \{
\begin{array}{l l}
s_j=\mathrm{id~}, y_j=0, v_j=0, z_j=\pi z_j^{\ast} & \quad \textit{if $i$ is even and $L_i$ is \textit{of type} $\textit{I}^o$};\\
s_j=\mathrm{id~}, r_j=t_j=y_j=v_j=u_j=w_j=0, z_j=\pi z_j^{\ast} & \quad \textit{if $i$ is even and $L_i$ is \textit{of type} $\textit{I}^e$};\\
s_j=\mathrm{id~}, r_j=t_j=y_j=v_j=u_j=w_j=0 & \quad \textit{if $i$ is odd and $L_i$ is \textit{free of type I}}.\\
\end{array} \right.$$
A formal matrix form of an element of $F_j(R)$ for a $\kappa$-algebra $R$ is then $$\begin{pmatrix} id&0& & \ldots& & &0\\ 0&\ddots&& & & &\\ & &id& & & & \\ \vdots & & &m_{j,j} & & &\vdots
\\ & & & & id & & \\ & & & & &\ddots &0 \\ 0& & &\ldots & &0 &id \end{pmatrix}$$
such that $$m_{j,j}=\left\{
\begin{array}{l l}
\begin{pmatrix}id&0\\0&1+2 z_j^{\ast} \end{pmatrix} & \quad \textit{if $j$ is even and $L_j$ is of type $I^o$};\\
\begin{pmatrix}id&0&0\\0&1+\pi x_j&2 z_j^{\ast}\\0&0&1 \end{pmatrix} & \quad \textit{if $j$ is even and $L_j$ is of type $I^e$};\\
\begin{pmatrix}id&0&0\\0&1+\pi x_j&0\\0&\pi z_j&1 \end{pmatrix} & \quad \textit{if $j$ is odd and
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| null | null |
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a,3,1^b)}$ unless $(u,v)\dom(a,3,1^b){^{\operatorname{reg}}}$, which is the partition $(\max\{a,b+2\},\min\{a-1,b+1\},2)$. So we may assume that this is the case, which is the same as saying $v\ls\min\{a+1,b+3\}$. For easy reference, we set out notation and assumptions for this section.
**Assumptions and notation in force throughout Section \[uvsec\]:**
$\la=(a,3,1^b)$ and $\mu=(u,v)$, where $a,b,u,v$ are positive integers with $a,b,u$ even, $a\gs4$, $u>v$, $n=a+b+3=u+v$ and $v\ls\min\{a+1,b+3\}$.
Homomorphisms from $S^\la$ to $S^{\mu'}$ {#hlamu'1}
----------------------------------------
In this subsection we consider $\bbf{\mathfrak{S}_}n$-homomorphisms from $S^\la$ to $S^{\mu'}$. We begin by constructing such a homomorphism in the case where $3\ls v\ls a-1$.
Let ${\calu}$ be the set of $\la$-tableaux having the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;\star;\star,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star)$$ in which the $\star$s represent the numbers from $2$ to $u$, and in which
- the entries along each row are strictly increasing,
- the entries down each column are weakly increasing.
Now define $$\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}.$$
\[sigmahom\] With the assumptions and notation above, we have ${\psi_{d,t}}\circ\sigma=0$ for each $d,t$.
First take $v<d\ls u$ and $t=1$. If $T\in{\calu}$, then $T$ contains a single $d$ and a single $d+1$. If these occur in the same row or the same column of $T$, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$ by Lemma \[lemma5\] and Lemma \[lemma7\]. Otherwise, there is another tableau $T'\in{\calu}$ obtained by interchanging the $d$ and the $d+1$. By Lemma \[lemma5\] we have ${\psi_{d,1}}\circ({\hat\Theta_{T}}+{\hat\Theta_{T'}})=0$. Hence by summing ${\psi_{d,1}}
| 598
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| 0.79359
|
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. $Y \cap (S \times \{n\}) = \{x_1, x_2\} \times \{n\}$.
In Case 1, each $\mathbb{Z}_{k+1}^{2}$ layer contains exactly one point of $Y$. $T$ then tiles the rest of the layer by Proposition \[onepoint\].
In Case 2, some of the layers contain two points of $Y$, and some of the layers contain no points. Holes of size 0 and 2 do not exist, so we will need copies of $T$ in the third direction to fill some gaps (where $Y$ consists of copies of $T$ in the fourth direction). The following lemma provides us with a way to do this:
\[otherlemma\] Let $A \subset \mathbb{Z}^d$, $|S| = 3k$. Then there exists $B \subset S \times \mathbb{Z}$ such that $T$ tiles $B$, and $$|B \cap (S \times \{n\})| =
\begin{cases}
k+1 & \text{\emph{if} } n \equiv 1, \ldots, k \pmod{2k}\\
k-1 & \text{\emph{if} } n \equiv k+1, \ldots, 2k \pmod{2k}
\end{cases}$$
Let $A = \{a_1, \ldots, a_{3k}\}$. Then:\
For $i = 1, \ldots, k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i \pmod{6k}$.\
For $i = k+1, \ldots, 2k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i+k \pmod{6k}$.\
For $i = 2k+1, \ldots, 3k$, place a copy of $T$ beginning at $\{a_i\} \times \{n\}$ for every $n \equiv i+2k \pmod{6k}$.\
We now observe that the union $B$ of these tiles has the required property.\
For $n \equiv 1, \ldots, k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{2k+n}, \ldots, a_{3k}, a_1, \ldots, a_n\}$ (size $k+1$).\
For $n \equiv k+1, \ldots, 2k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_1, \ldots, a_k\}\setminus\{a_{n-k}\}$ (size $k-1$).\
For $n \equiv 2k+1, \ldots, 3k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{n-2k}, \ldots, a_{n-k}\}$ (size $k+1$).\
For $n \equiv 3k+1, \ldots, 4k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{k+1}, \ldots, a_{2k}\}\setminus\{a_{n-2k}\}$ (size $k-1$).\
For $n \equiv 4k+1, \ldots, 5k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{n-3k}, \ldots, a_{n-2k}\}$ (size $k+1$).\
For $n \equiv 5k+1, \ldots, 6k \pmod{6k}$, $B \cap (A \times \{n\}) = \{a_{2k+1}, \ldots,
| 599
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ta_{C}}\neq0$.
Showing the first statement is very easy, using Lemma \[lemma5\]. The only homomorphisms that occur in that lemma with non-zero coefficient are labelled by tableaux with more than $v$ $1$s in rows $2$ and $3$, and therefore by Lemma \[lemma7\] are zero.
Showing that ${\hat\Theta_{C}}\neq0$ is also straightforward using Lemma \[lemma7\]. We apply this lemma to move the $1$s from row $2$ to row $1$, and then again to move the $2$s from row $3$ to row $2$. The tableau $${\text{\footnotesize$\gyoungx(1.2,;1;1;1_{4.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(4.5*.25,0);\end{tikzpicture}}};1;{v\!\!+\!\!2}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;2;2;2;5_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{v\!\!+\!\!1},;3;4)$}}$$ (for example) labels a homomorphism occurring with non-zero coefficient.
\[cd2homdim2\] With $\la,\mu$ as above, $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)=1.$$
The existence of the homomorphism ${\hat\Theta_{C}}$ above shows that the space of homomorphisms is non-zero. So we just need to show the upper bound on the dimension. So suppose $\theta$ is a linear combination of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to S^\la$ such that ${\psi_{d,t}}\circ\theta=0$ for all $d,t$.
For $3\ls d\ls b+1$, say that a semistandard tableau $T$ is *$d$-bad* if the entries $d,d+1$ appear in the same column of $T$. Note that this must be column $1$ or $2$, because the assumption $v\ls a-1$ guarantees that any column of length greater than $1$ has a $1$ at the top.
We claim that if $T$ is $d$-bad, then ${\hat\Theta_{T}}$ cannot appear in $\theta$. To show this, we consider ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ for every semistandard $T$. If $T$ is not $d$-bad, then by Lemma \[lemma5\] ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a homomorphism labelled by a semistandard tableau
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