text
large_stringlengths 384
2.05k
| rank_avg
float64 1
4.19k
⌀ | rank_max
float64 1
8.21k
⌀ | rank_min
float64 1
5.03k
⌀ | rank_median
float64 1
4.21k
⌀ | rank_by_avgsim
float64 1
4.19k
⌀ | avgsim_to_github
float32 0.77
0.85
⌀ | dataset
large_stringclasses 1
value |
|---|---|---|---|---|---|---|---|
0 1
Marital status^[e](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.16 0.37 0 1
Education 7.21 2.18 0 11
Years in the Netherlands 14.31 4.46 0 25
Housing^[f](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.87 0.33 0 1
Number of friends in the Netherlands 31.47 39.21 0 294
Number of family members in the Netherlands 4.52 11.08 0 98
Dutch proficiency 1.36 1.05 0 4
Child \< 8 years^[g](#table-fn1-0192513X17710773){ref-type="table-fn"}^ 0.61 0.49 0 1
Job instability 2.00 1.97 0 9
Job absenteeism 2.41 4.01 0 24
*Note*. Superscripts indicate reference categories that include (a) nontransnational parent; (b) unhappy; (c) no work-to-family conflict (d) male; (e) married/in a relationship; (f) room, student housing, institution, other; and (g) no children \< 8 years of age.
*Source*. TCRAf-Eu Angolan parent survey, The Netherlands 2010-2011.
Mediation Analysis {#section10-0192513X17710773}
------------------
To test the hypotheses, we turn to regression analyses. Following the four steps as proposed by [@bibr5-0192513X17710773], it was first tested whether transnational parenting is associated with job absenteeism and job instability. These results are displayed in [Table 2](#table2-0192513X17710773){ref-type="table"}. Model 1 shows that transnational parents do not significantly differ from nontransnational parents in terms of the days they were absent from their jobs in the past 3 months. Model 2, however, indicates that, ceteris paribus, transnational
| 1,401
| 192
| 1,699
| 1,691
| 939
| 0.797256
|
github_plus_top10pct_by_avg
|
$\alpha\in L$ and that $\bar{s}$ is on a level above $\delta$. Similarly we may assume that for all $\xi,\rho<\delta$, $\bar{s}$ has decided the statement $$\dot{U}(\eta,0)\cap\dot{Z}(\xi,\rho)\neq\emptyset\quad
\text{ for all }\xi,\rho<\delta.$$ Now choose any $\alpha\in L$ (e.g.the least one), and then choose an infinite sequence $\{\beta_l:l\in\omega\}\subseteq
L\setminus(\alpha+1)$ so that $s_{\beta_l}\upharpoonright \delta^+({C_{\gamma_\alpha}})$ are all distinct. For each $l$, let $e(s_{\beta_l}\upharpoonright\delta^+({C_{\gamma_\alpha}})~)=n_l$. **Main Claim:** $\quad\bar{s}\Vdash
(\forall
l\in\omega)\left(\dot{W}(\gamma_\alpha,n_l)\cap\dot{U}(\eta,0)\neq
0\right).$ Once this claim is proven we are done, because we then have that $\bar{s}$ forces that $\dot{U}(\eta,0)$ cannot have compact closure, because it meets infinitely many members of the discrete family $\{\dot{W}(\gamma_\alpha,n):n\in\omega\}$.
To prove the claim, first note that there is a tail of $C_\zeta(\gamma_{\beta_l})\cap\delta$ included in $C_{\zeta(\gamma_{\alpha})}$. To see this, recall $C_{\zeta(\gamma_\alpha)}\setminus{\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is countable, so some tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is included in $C_{\zeta(\gamma_\alpha)}$. By elementarity, since $\gamma_\alpha$ and $\gamma_\beta$ are in $M$, a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap M$ is included in $C_{\zeta(\gamma_\alpha)}\cap M$, so a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap\delta$ is included in $C_{\zeta(\gamma_\alpha)}$.
Since there is a tail of $C_{\zeta(\gamma_{\beta_l})}\cap \delta$ included in $C_{\zeta(\gamma_\alpha)}$, $\dot{Z}(\xi,
C_{\zeta(\gamma_{\beta_l})})\subseteq\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)})$ for each $\xi<\delta$ (at least on a tail — which is all that matters for limits above $\delta$). Then $s_{\beta_l}$ forces that $\eta $ is a limit of the sequence $$\langle\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)}):
\xi\in\delta\text{ and }\dot{f}_{\gamma_\alpha}(\xi)= n_l\rangle.$$
| 1,402
| 1,775
| 1,678
| 1,266
| null | null |
github_plus_top10pct_by_avg
|
oint union $$\begin{aligned}
{\label{eq:fin-ind}}
&{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T}{\mathop{\Dot{\bigcup}}}_{\{s_it_i\}_{i=1}^j\in
{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap\bigcap_{i=1}^j
\big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\}\bigg\}\\
&={\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\Bigg\{{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T}
{\mathop{\Dot{\bigcup}}}_{\substack{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}\\ b_{t_i
+1}=e_{i+1}\;{{}^\forall}i=0,\dots,j-1}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap
\bigcap_{i=1}^j\big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\}
\bigg\}\Bigg\},{\nonumber}\end{aligned}$$ where $t_0=0$ by convention. On the left-hand side of [(\[eq:fin-ind\])]{}, the first two unions identify the number and location of the pivotal bonds for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$, and the third union identifies the indices of double connections associated with the bypaths between $z_i$ and $z'_i$, for every $i=1,\dots,j$. The union over $e_1,\dots,e_j$ on the right-hand side identifies some of the pivotal bonds $b_1,\dots,b_T$ that are essential to decompose the chain of double connections $H_{{{\bf n}};\vec b_T}(y,x)$ into the following building blocks (see Figure \[fig:I-def\]):
$$\begin{gathered}
I_1(y,z,x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I1}}\hspace{7pc}
I_2(y,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I2}}\\[1pc]
I_3(y,z,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I31}}\quad~
\cup\quad~\raisebox{-12pt}{\includegraphics[scale=0.12]{I32}}\end{gathered}$$
$$\begin{gathered}
I_1(y,z,x)=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}z\},\qquad
I_2(y,z',x)
| 1,403
| 485
| 1,430
| 1,494
| 3,856
| 0.769678
|
github_plus_top10pct_by_avg
|
K \sqrt{p}$ and $\mathbb{E}\left[ \| X_i \|^2
\right] \leq U p$ for all $i = 1,\ldots,n$, Proposition 1.2 in [@hsu12] yields that $$\label{eq:vector.bernstein}
\mathbb{P}\left( \frac{1}{n} \left\| \sum_{i=1}^n X_i \right\| \leq \sqrt{\frac{U p}{n}} + \sqrt{ 8 \frac{U p}{n} \log n} +
\frac{4 K \sqrt{p}}{3 n} \log n \right) \geq 1 - \frac{1}{n}.$$ Equation follows by bounding $\mathbb{E}\left[ \| X_i \|^2 \right]$ with $K^2 p$ instead of $Up$.
Next, we prove . We let $\preceq$ denote the positive semi-definite ordering, whereby, for any $p$-dimensional symmetric matrices $A$ and $B$, $A \preceq B$ if and only if $B-A$ is positive semi-definite. For each $i =1,\ldots,n$, the triangle inequality and the assumptions in the statement yield the bound $$\left\| X_i X_i^\top - \Sigma\right\|_{\mathrm{op}} \leq \| X_i \|^2 +
\lambda_{\max}(\Sigma) \leq K^2 p + U.$$ Similarly, $\| \mathbb{E}\left[ (X_i X_i^\top)^2 \right] - \Sigma\|_{\mathrm{op}} \leq
K^2p U$ for each $i = 1,\ldots, n$, since $$\mathbb{E}\left[ (X_i X_i^\top)^2 \right] - \Sigma^2 \preceq
\mathbb{E}\left[ \| X_i \|^2 X_i X_i^\top \right] \preceq K^2 p \Sigma
\preceq K^2 p U I_{p}.$$ with $I_p$ the $p $-dimensional identity matrix. Thus, applying the Matrix Bernstein inequality [see Theorem 1.4 in @Tropp2012], we obtain that $$\label{eq:matrix.bernstein}
\mathbb{P}\left( \| \widehat{\Sigma} - \Sigma \|_{\mathrm{op}} \leq \sqrt{ 2 K^2 p U \frac{\log p + \log 2n }{n}} +
\frac{2}{3} (K^2 p + U) \frac{\log p + \log 2n }{n}\right)\geq 1 -
\frac{1}{n}.$$ The bound follows from choosing $C$ large enough, depending on $\eta$, and using the fact that $p \leq n$. $\Box$
[**Remark.**]{} From , by using the looser bounds $$\left\| X_i X_i^\top - \Sigma\right\|_{\mathrm{op}} \leq 2 K^2 p \quad
\text{and} \quad \mathbb{E}\left[ (X_i X_i^\top)^2 \right] - \Sigma^2
\preceq K^4 p^2 I_p,$$ one can obtain directly that $$\label{eq:matrix.bernstein.simple}
\mathbb{P}\left( \| \widehat{\Sigma} - \Sigma \|_{\mathrm{op}} \leq C K^2
| 1,404
| 3,841
| 1,371
| 1,074
| null | null |
github_plus_top10pct_by_avg
|
and $U'\subset Y$ the “same” open subset of $Y$ then $U'$ is also affine by Chevalley’s theorem and so $Y$ has the Chevalley-Kleiman property.
Let $E$ be an elliptic curve and set $S:=E\times {{\mathbb P}}^1$. Pick a general $p\in E$ and $g:E\times \{0,1\}\to E$ be the identity on $E_0:=E\times \{0\}$ and translation by $-p$ on $E_1:=E\times \{1\}$. Where are the affine charts on the quotient $Y$?
If $P_i\subset E_i$ are 0-cycles then there is an ample divisor $H$ on $S$ such that $(H\cdot E_i)=P_i$ iff ${{\mathcal O}}_{E_0}(P_0)={{\mathcal O}}_{E_1}(P_1)$ under the identity map $E_0\cong E_1$.
Pick any $a,b\in E_0$ and let $a+p,b+p\in E_1$ be obtained by translation by $p$. Assume next that $2a+b=a+p+2(b+p)$, or, equivalently, that $3p=a-b$. Let $H(a,b)$ be an ample divisor on $S$ such that $H(a,b)\cap E_0=\{a,b\}$ and $H(a,b)\cap E_1=\{a+p,b+p\}$. Then $U(a,b):=S\setminus H(a,b)$ is affine and $g$ maps $E_i\cap U(a,b)$ isomorphically onto $E\setminus\{a,b\}$ for $i=0,1$. As we vary $a,b$ (subject to $3p=a-b$) we get an affine covering of $Y$.
Note however that the curves $H(a,b)$ do not give Cartier divisors on $Y$. In fact, for non-torsion $p\in E$, every line bundle on $Y$ pull back from the nodal curve obtained from the ${{\mathbb P}}^1$ factor by gluing the points $0$ and $1$ together.
Appendix by Claudiu Raicu {#appendix-by-claudiu-raicu .unnumbered}
=========================
\[raicu\] Let $A$ be a noetherian commutative ring and $X={{\mathbb A}}_S^n$ the $n$-dimensional affine space over $S={\operatorname{Spec}}A$. Then ${{\mathcal O}}_X\simeq A[{\textit{\textbf{x}}}]$, where ${\textit{\textbf{x}}}=(x_1,\dots,x_n)$. To give a finite equivalence relation $R\subset X\times_S X$ is equivalent to giving an ideal $I({\textit{\textbf{x}}},{\textit{\textbf{y}}})\subset A[{\textit{\textbf{x}}},{\textit{\textbf{y}}}]$ which satisfies the following properties:
1. (reflexivity) $I({\textit{\textbf{x}}},{\textit{\textbf{y}}})\subset (x_1-y_1,\dots,x_n-y_n)$.
2. (symmetry) $I({\textit{\textbf{x}}},{\te
| 1,405
| 627
| 1,295
| 1,358
| 4,147
| 0.767764
|
github_plus_top10pct_by_avg
|
lity, it suffices to show this for contraction. If $M \con X = N \con X$ then $M \dcon X = N \dcon X$, so $\dist(M,N) \le \dist(M,M \dcon X) + \dist(N \dcon X,N) \le 2|X|.$
We now consider an operation that turns various elements of a matroid into loops, and moves other elements into parallel with existing elements. Let $M$ be a matroid and let $\phi \notin E(M)$. Let $X \subseteq E(M)$ and let $\psi \colon X \to E(M \del X) \cup \{\phi\}$ be a function. We let $\psi(M)$ denote the matroid with ground set $E(M)$ so that
- $\psi(M) \del X = M \del X$,
- each $x \in X$ for which $\psi(x) = \phi$ is a loop of $\psi(M)$, and
- each $x \in X$ for which $\psi(x) \ne \phi$ is parallel to $\psi(x)$ in $\psi(M)$.
Note that $\si(\psi(M)) \cong \si(\psi(M) \del X) = \si(M \del X)$.
Suppose that there exists $C \subseteq E(M)$ such that $\psi(x) = \phi$ for all $x \in X \cap \cl_M(C)$, and for each $x \in X - \cl_M(C)$, the elements $x$ and $\psi(x)$ are parallel in $M \con C$. In this case, we say that the function $\psi$ is a *$C$-shift*, and that $\psi(M)$ is a *$C$-shift* of $M$.
If $\wh{M}$ is a $C$-shift of $M$, then $\dist(M,\wh{M}) \le 4|C|$.
Let $\psi\colon X \to E(M \del X) \cup \{\phi\}$ be a $C$-shift so that $\wh{M} = \psi(M)$. Let $\psi'$ be the restriction of $\psi$ to the domain $X-C$. Now $\psi'(M)$ is a $C$-shift of $M$. By the definition of a $C$-shift we have $\psi'(M) \con C = M \con C$, and clearly $\psi(M) \del C = \psi'(M) \del C$. Thus $\dist(\psi'(M),M) \le 2|C|$ and $\dist(\psi(M), \psi'(M)) \le 2|C|$, and the lemma follows.
The next theorem shows that small perturbations can not introduce arbitarily large projective geometries or balanced uniform matroids.
\[perturbthm\] Let $M$ and $N$ be matroids such that $\dist(M,N) = 1$.
- If $s \ge 2$ and $M$ has a $U_{s2^{4s},2s2^{4s}}$-minor, then $N$ has a $U_{s,2s}$-minor.
- If $n \ge 3$ and $M$ has a $\PG(n-1,q)$-minor, then $N$ has a $\PG(n-3,q)$-minor.
Let $\wh{M}$ be a matroid so that $\{\wh{M} \con e,\wh{M} \del e\} = \{M,N\}
| 1,406
| 867
| 905
| 1,378
| null | null |
github_plus_top10pct_by_avg
|
$-module via the ${\mathbb{K}}$-module structure of the second tensor factor.
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. Triangular decomposition of $U (\chi )$ gives the following standard fact.
The map $U ^-(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}\to M^\chi (\Lambda )$, $u{\otimes }x\mapsto uxv_\Lambda =u{\otimes }x1_\Lambda $, is an isomorphism of vector spaces over ${\mathbb{K}}$. \[le:MLiso\]
The isomorphism in Lemma \[le:MLiso\] and the ${\mathbb{Z}}^I$-grading of $U^-(\chi )$ induce a unique ${\mathbb{Z}}^I$-grading on $M^\chi (\Lambda )$ such that $$\begin{aligned}
M^\chi (\Lambda )_{\alpha }=U^-(\chi )_{\alpha }{\otimes }_{\Bbbk }{\mathbb{K}}_\Lambda
\quad \text{for all ${\alpha }\in {\mathbb{Z}}^I$.}
\label{eq:MLgrading}\end{aligned}$$ Then $$\begin{aligned}
M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda ,\,\,
U(\chi )_{\alpha }M^\chi (\Lambda )_\beta \subset M^\chi (\Lambda )_{{\alpha }+\beta }
\,\,\text{for all ${\alpha },\beta \in {\mathbb{Z}}^I$.}
\label{eq:MLgrading2}\end{aligned}$$ Moreover, $M^\chi (\Lambda )_{\alpha }\not=0$ implies that $-{\alpha }\in {\mathbb{N}}_0^I$.
The group algebra ${{\mathcal{U}}^0}$ acts on $M^\chi (\Lambda )$ via left multiplication. This action is given by characters: $$\begin{aligned}
uv=(\Lambda +{\zeta ^{\chi}} ({\alpha }))(u)v \quad
\text{for all $u\in {{\mathcal{U}}^0}$, ${\alpha }\in {\mathbb{Z}}^I$,
$v\in M^\chi (\Lambda )_{\alpha }$,}
\label{eq:U0M}\end{aligned}$$ see Eqs. , , , and .
Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$. The family of those $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodules of $M^\chi (\Lambda )$, which are contained in $\oplus _{{\alpha }\not=0}M^\chi (\Lambda )_{\alpha }$, have a unique maximal element $I^\chi (\Lambda )$. Let $$\begin{aligned}
L^\chi (\Lambda )=M^\chi (\Lambda )/I^\chi (\Lambda )
\label{eq:LLambda}\end{aligned}$$ be the quotient $U(\chi )$-module. The maximality of $I^\chi (\Lambda )$ implies that $$I^\chi (\Lambda )=\big(I^\
| 1,407
| 1,357
| 1,254
| 1,318
| null | null |
github_plus_top10pct_by_avg
|
t( \det {\cal E}_0^{\alpha *} \right) \otimes \left( \det T_0^{\alpha }
\right) \otimes {\cal L},$$ which, by symmetry, should also be the same as ${\cal L}^*$. In other words, $$\left( \det {\cal E}_0^{\alpha *} \right) \otimes \left( \det T_0^{\alpha }
\right) \otimes {\cal L}
\: \cong \: {\cal L}^*$$ or more simply $${\cal L}^2 \: \cong \:
\left( \det {\cal E}_0^{\alpha} \right) \otimes \left( \det
T_0^{\alpha *} \right) \: = \:
K_{\alpha} \otimes \det {\cal E}_0^{\alpha},$$ from which our claim is derived. In particular, taking ${\cal L} = {\cal O}$ will not, in general, be consistent.
In passing, note that the set of all Fock vacua in sector $\alpha$ form a vector bundle $$\left( \wedge^{\bullet} {\cal E}_0^{\alpha *}
\right) \otimes \left(
\wedge^{\bullet} T_0^{\alpha} \right) \otimes
\sqrt{ K_{\alpha} \otimes \det {\cal E}^{\alpha}_0 }
\otimes \otimes_{n > 0}
\left( \left( \det {\cal E}^{\alpha}_n \right) \left(
\det T^{\alpha}_n \right)^{-1}\right)^{- \frac{n}{t_{\alpha}} \: - \: \left[
- \frac{n}{t_{\alpha}} \right] \: - \: \frac{1}{2} }$$ over $I_{\mathfrak{X}}|_{\alpha}$, taking into account contributions from all boundary conditions.
The phenomenon of Fock vacua coupling to nontrivial bundles has also been noted in this context in [@manion-toappear], [@ando-s]\[section 2.1\]. However, aside from those two sources, we are not aware of many discussions of Fock vacua as sections of line bundles over target spaces[^20] in the literature, so it is perhaps useful to elaborate on this point. As we shall see in the present case and also in [@manion-toappear], it plays a crucial role in closing the spectrum under Serre duality of the sheaf cohomology groups, a basic symmetry of the spectra discussed in [@dist-greene]. The same behavior also arises elsewhere. For example, in open string theories, the Fock vacuum also transforms as a section of a line bundle, a square root of the canonical bundle of the D-brane worldvolume $B$ (assumed Spin), if the D-brane worldvolume is not Calabi-Yau. This can be understo
| 1,408
| 868
| 1,288
| 1,326
| null | null |
github_plus_top10pct_by_avg
|
%
\mathfrak{m}+q+2d/p_{\ast }}}+\overline{\Phi }_{n}^{\mathfrak{m}}(\delta
_{\ast })\Big)^{1+\varepsilon _{\ast }} \label{TR6''}\end{aligned}$$
**C**. Let $p>2d.$ Set $\bar{%
\mathfrak{m}}=1+\frac{q+1+2d/p_{\ast }}{\delta _{\ast }}$. There exist $%
C\geq 1,\eta \geq 0$ (depending on $q,p,\varepsilon _{\ast },\delta _{\ast
},\kappa $) such that for every $t>0$, $n\in {\mathbb{N}}$ and for every multi-indexes $\alpha ,\beta $ such that $\left\vert \alpha \right\vert
+\left\vert \beta \right\vert \leq q$, $$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq C\times Q_{n}(q+1,\bar{\mathfrak{m}})\times
t^{-\theta _{0}((a+b)\bar{\mathfrak{m}}+q+1+2d/p_{\ast })(1+\varepsilon _{\ast
})})\times \frac{\psi _{\eta +\kappa }(x)}{\psi _{\kappa }(x-y)}
\label{TR6d}$$for every $t\in (0,1]$ and $x,y\in {\mathbb{R}}^{d}.$
We stress that in hypothesis (\[TR6a\]) the order of derivation $q$ does not appear. However the conclusions (\[TR6’\]) and (\[TR6d\]) hold for every $q.$ The motivation of this is given by the following heuristics. The hypothesis (\[TR5\]) says that the semi-group $P_{t}^{n}$ has a regularization effect controlled by $1/(\lambda _{n}t)^{\theta _{0}}.$ If we want to decouple this effect $m_{0}$ times we write $%
P_{t}^{n}=P_{t/m_{0}}^{n}....P_{t/m_{0}}^{n}$ and then each of the $m_{0}$ operators $P_{t/m_{0}}^{n}$ acts with a regularization effect of order $%
(\lambda _{n}\times t/m_{0})^{\theta _{0}}.$ But this heuristics does not work directly: in order use it, we have to use a development in Taylor series coupled with the interpolation type criterion given in the following section.
The proof of Theorem \[Transfer\] is developed in Section [sect:proofTransfer]{}. We give now a variant of the estimate (\[TR6d\]), whose proof can be found in Section \[sect:proofTransferBIS\].
\[TransferBIS-new\] Suppose that Assumption \[A1A\*1\], \[A2A\*2\], [A3]{} and \[A5\] hold. Suppose also that (\[TR6\]) holds for some $\delta >0$ and that for every $\kappa>0$ there exist $\ba
| 1,409
| 533
| 1,383
| 1,414
| null | null |
github_plus_top10pct_by_avg
|
gg > = \alpha{\pi \over 2}
\sum_{m=1}^{\bar{K}}\sum_{n=1}^{K} d_{\bar{K}m} d_{Kn}\ n \
\Delta_{\bar{\nu}-\nu} (\Delta_{m-n+1}-\Delta_{n-m+1}) \eqno(A9)$$
$$\bigg < \bar{K}^- \bar{\nu} \bigg | {\alpha^2 \over
F^2}{\partial^2 \over \partial^2 \phi} \bigg | K^- \nu \bigg > =
{\alpha^2 \pi \over \sqrt{1-\alpha^2}}
\sum_{m=1}^{\bar{K}}\sum_{n=1}^{K} d_{\bar{K}m} d_{Kn}\ (\nu^2) \
\Delta_{\bar{\nu}-\nu} [f^{n+m}(\alpha)-f^{|n-m|}(\alpha)]
\eqno(A10)$$
$$\bigg < \bar{K}^- \bar{\nu} \bigg | i \tau_0 \alpha^2{\partial
\over
\partial \phi} \bigg | K^- \nu \bigg > =
\pi \tau_0 \alpha^2 \sum_{m=1}^{\bar{K}}\sum_{n=1}^{K}
d_{\bar{K}m} d_{Kn} (-\nu) \Delta_{\bar{\nu}-\nu}\big[
\Delta_{m-n} +$$ $${\alpha \over 2}
(\Delta_{m+n+1}+\Delta_{m-n+1}+\Delta_{m+n-1}+\Delta_{m-n-1})\big]
\eqno(A11)$$
$$\bigg < \bar{K}^- \bar{\nu} \bigg | -{\tau_0^2 \alpha^2 \over 4}
F^2\bigg | K^- \nu \bigg
> = - {\pi \tau_0^2 \alpha^2 \over 4}\sum_{m=1}^{\bar{K}}\sum_{n=1}^{K}
d_{\bar{K}m} d_{Kn} \Delta_{\bar{\nu}-\nu}$$ $$\big[ P_1 \Delta_{m-n}+{1\over 2}(P_2( \Delta_{m-n-1}+
\Delta_{n-m-1}- \Delta_{1-m-n}) +$$ $$P_3( \Delta_{m-n-1}+ \Delta_{n-m-2}- \Delta_{2-m-n})$$ $$+ P_4( \Delta_{m-n-1}+ \Delta_{n-m-2}- \Delta_{3-n-m}) ) \big]
\eqno(A12)$$
$$\bigg < \bar{K}^- \bar{\nu} \bigg | -{\tau_1^2 \alpha^2 \over 4}
F^2 {\rm sin^2\phi} \bigg | K^- \nu \bigg
> =- {\pi \tau_1^2 \alpha^2 \over 4}\sum_{m=1}^{\bar{K}}\sum_{n=1}^{K}
d_{\bar{K}m}d_{Kn} ({1\over 2}\Delta_{\bar{\nu}-\nu}-
{1\over 4} \Delta_{\nu -\bar{\nu}+2} - {1\over 4}
\Delta_{\nu-\bar{\nu}-2})$$ $$\big[ P_1 \Delta_{m-n}+{1\over 2}(P_2( \Delta_{m-n-1}+
\Delta_{n-m-1}- \Delta_{1-m-n}) +$$ $$P_3( \Delta_{m-n-2}+ \Delta_{n-m-2}- \Delta_{2-m-n})$$ $$+ P_4( \Delta_{m-n-3}- \Delta_{n-m-3}+ \Delta_{3-m-n}) ) \big]
\eqno(A13)$$
$$\bigg < \bar{K}^- \bar{\nu} \bigg | -{\tau_1^2 \alpha^4 \over 4}
{\rm sin^2\theta} \bigg | K^- \nu \bigg
> =- {\pi \tau_1^2 \alpha^4 \over 8}\sum_{m=1}^{\bar{K}}\sum_{n=1}^{K}
d_{\bar{K}m} d_{Kn} \Delta_{\bar{\nu}-\nu}
\big[ \Delta_{m-n}+$$ $${\alpha \over 4
| 1,410
| 2,823
| 925
| 1,382
| null | null |
github_plus_top10pct_by_avg
|
ch is different from the result of Ref. [@konno] for $DY_\hbar(sl_2)_k$. The reason for this difference is that, first, the Yangian double $DY_\hbar(sl_2)_k$ of Ref. [@konno] is realized in an asymmetric way which differs from the symmetric one which we are using by a shift of parameter [@KL]; Second, as we have remarked in Remarks \[rem1\] and \[rem2\], for the same realization of Yangian double, there still exist infinite many choices for the bosonization formulas. Therefore the difference of our result (\[hppp\]-\[ennn\]) from that of Ref. [@konno] is reasonable.
Problems in obtaining Vertex operators and screening currents
=============================================================
After successfully obtained the bosonization formulas for the Yangian double $DY_\hbar(sl_N)$ using the correspondence rules (Observations \[ob1\] and \[ob2\]), one naturally expects that the Vertex operators and screening currents for $DY_\hbar(sl_N)$ could also be obtained in the same way. In this section we briefly give why this is difficult.
Following Observations \[ob1\] and \[ob2\] and Ref.[@sln], we expects that the screening currents for $DY_\hbar(sl_N)$ might be written in the following form,
$$\begin{aligned}
S^i(u) = :\mbox{exp}\left( X^i [a](u) \right): \tilde{S}^i(u),\end{aligned}$$
where $\tilde{S}^i(u)$ is nothing but $E^+_{N-i}(u)$ with the replacement $\hat{b}^{ij}_\pm \rightarrow - \hat{b}^{N+1-j,N+1-i}_\mp$, $(b+c)^{ij} \rightarrow (b+c)^{N+1-j,N+1-i}$, and $X^i [a](u)$ is some field depending only on $a^i$ but not on $b^{ij}$ and $c^{ij}$. In the $q$-affine case, $X^i [a](u)$ is just the field $-\left( \frac{1}{k+g} a^i \right)(z; \frac{k+g}{2})$ [@sln]. At present, we expect that $:\mbox{exp}\left( X^i [a](u) \right):$ have the following OPE relations with $\mbox{exp}\left( \hat{a}^i_+(u) \right)$ and $\mbox{exp}\left( \hat{a}^i_-(u) \right)$ (see equations (C.17), (C.18) of Ref. [@sln])
$$\begin{aligned}
& & \mbox{exp} ( \hat{a}^i_+(u) )
:\mbox{exp} ( X^j [a](v) ): \nonumber \\
& &~~~~~~= \frac{u-v+(\f
| 1,411
| 613
| 1,129
| 1,430
| 1,201
| 0.792829
|
github_plus_top10pct_by_avg
|
FloorCondition2\] With Lemma \[lemma:FloorCondition\], the inequality condition $f\left(\floor{{{\bar{\a}}^\dagger}_k}_\ell\right) < f\left(\ceil{{{\bar{\a}}^\dagger}_k}_\ell\right)$ in line \[line:FloorCondition\] of Algorithm \[agorithm:SuccessiveQuantization\] can be simplified as $$\begin{aligned}
2\floor{{{\bar{\a}}^\dagger}_k(\ell)} - 2\left(\left(\floor{{{\bar{\a}}^\dagger}_k}_\ell\right)^T\u\right)\u(\ell) + 1 - \u(\ell)^2 < 0,\end{aligned}$$ where $\u$ is the normalized channel vector as defined in .
The proof is straightforward by writing $\bsG$ in terms of $\u$, and thus is omitted here.
After the quantization, a suboptimal coefficient vector ${{\bar{\a}}^\diamond}$ for ${\bar{\h}}$ is obtained with $$\begin{aligned}
\label{equation:aBarDiamond}
{{\bar{\a}}^\diamond}= \arg \min_{ \a \in \{ {{\bar{\a}}^\diamond}_k \} } \a^T \bsG \a.
\end{aligned}$$
Finally, a suboptimal coefficient vector ${\a^\diamond}$ for the original channel vector $\h$ is recovered from ${{\bar{\a}}^\diamond}$ according to Remark \[remark:Transformation\].
We summarize our proposed *QP relaxation* method in Algorithm \[agrm:QPRApproachOutline\]. The pseudocode is shown in Algorithm \[algorithm:QPRApproachCode\], where the function $[\bar{\w},\p]={\rm sort}(\w)$ sorts the elements in $\w$ in ascending order, returns the sorted vector $\bar{\w}$, and stores the original indices of the elements as vector $\p$, the function ${\rm floor}(\w)$ applies the floor operation to each element of $\w$ and returns the resulted integer vector.
1. \[item:outline:Preprocess\] Preprocess $\h$ to the nonnegative ordered ${\bar{\h}}$ with Remark \[remark:Transformation\].
2. \[item:outline:CalculateaBarDagger1\] Calculate ${{\bar{\a}}^\dagger}_1$, with Theorem \[theorem:aBarDagger1\] and Lemma \[lemma:r\].
3. \[item:outline:DetermineK\] Determine $K$ with .
4. \[item:outline:CalculateaBarDaggerk\] Calculate $\{ {{\bar{\a}}^\dagger}_k \}$, i.e., the real-valued approximations of the optimal coefficient vector for ${\bar{\h}}$, usin
| 1,412
| 432
| 720
| 1,462
| 2,827
| 0.776743
|
github_plus_top10pct_by_avg
|
rsity of Zurich
*Colin C. Venters*, University of Huddersfield
*Coral Calero*, University of Castilla-La Mancha
*Sedef Akinli Kocak*, Ryerson University
*Stefanie Betz*, Karlsruhe Institute of Technology\
---
abstract: |
Multi-dimensional data classification is an important and challenging problem in many astro-particle experiments. Neural networks have proved to be versatile and robust in multi-dimensional data classification. In this article we shall study the classification of gamma from the hadrons for the MAGIC Experiment. Two neural networks have been used for the classification task. One is Multi-Layer Perceptron based on supervised learning and other is Self-Organising Map (SOM), which is based on unsupervised learning technique. The results have been shown and the possible ways of combining these networks have been proposed to yield better and faster classification results.\
\
[*Keywords:*]{} Neural Networks, Multidimensional data classification, Self-Organising Maps, Multi-layer Perceptrons.\
author:
- |
F. Barbarino, P. Boinee, A. De Angelis\
\
[ ]{}
title: Multidimensional data classification with artificial neural networks
---
Introduction
============
Many high-energy gamma ray experiments have to deal with the problem of separating gammas from hadrons [@p1]. The experiments usually generate large data sets with many attributes in them. This multi-dimensional data classification problem offers a daunting challenge of extracting small number of interesting events (gammas) from an overwhelming sea of background (hadrons) . Many techniques are in active research for addressing this problem. The list includes classical statistical techniques to more sophisticated techniques like neural networks, classification trees and kernel functions.
The class of neural networks provides an automated technique for the classification of the data set into given number of classes [@kk]. It is in active research in both artificial intelligence and machine learning communities. Severa
| 1,413
| 144
| 1,654
| 1,411
| null | null |
github_plus_top10pct_by_avg
|
ar2005].
In our case, firstly, we took the improved parameters, such as the radial velocity curve amplitudes of the F hotter ($K_{1}$) and the K cooler ($K_{2}$) components, the inclination of the orbital axis ($i$), the conjunction time ($T_{0}$; the hotter component is behind) and the orbital period ($P$), from the paper by @eaton2007, who used both of the photometric and spectroscopic data; we did not change these values in the following steps. Then we used $\chi^{2}$ minimization method to determine the projected rotational velocity () of each component, from the combined data set of 2006 November and December; and the systematic radial velocity of the binary ($\gamma$), from each data set. We list the final adopted values for imaging SZ Psc in Table \[tab:par\], except for the values of $\gamma$ for five observing runs, which are given in the third column of Table \[tab:velocity\] for comparison. The rotational velocity of the cooler component we derived is obviously lower than that (80 ) determined by @eaton2007 but very close to the value (70 ) derived by @str1993 and @gla2008.
Parameter Value Ref.
----------------------- ---------------- ------
$q=M_{2}/M_{1}$ 1.40 a
$K_{1}$ (km s$^{-1}$) 103.98 a
$K_{2}$ (km s$^{-1}$) 74.2 a
$i$ ([$\degr$]{}) 69.75 a
$T_{0}$ (HJD) 2449284.4483 a
$P$ (d) 3.96566356 a
$_{1}$ (km s$^{-1}$) $3.0 \pm 0.6$ DoTS
$_{2}$ (km s$^{-1}$) $67.7 \pm 1.0$ DoTS
$T_{eff,1}$ (K) 6090 b
$T_{eff,2}$ (K) 4910 b
albedo$_{1}$ 1.0 b
albedo$_{2}$ 0.3 b
: Adopted stellar parameters of SZ Psc for Doppler imaging. The F hotter component is defined as the primary and the K cooler star is the secondary.[]{data-label="tab:par"}
\
References: a. @eaton2007; b. @lan2001.\
Results
-------
{width="95.00000%"}
| 1,414
| 2,019
| 2,947
| 1,651
| 1,263
| 0.791921
|
github_plus_top10pct_by_avg
|
r reasoning establishes that the other restriction $\phi$ is a member of ${\prod{\Phi}}$. The unique $\psi \in {\prod{\Psi}}$ and $\phi \in {\prod{\Psi}}$ such that $\upsilon = \psi\phi$ are expressed by the restrictions $\psi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Psi}}}}}$ and $\phi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$.
Persistent-volatile partition {#S:STATE_EVENT_PRTN}
-----------------------------
Definition \[D:BASIS\] asserts that basis $\langle \Psi, \Phi \rangle$ is comprised of two ensembles satisfying $\Phi \subseteq \Psi$. This allows partitioning terms of $\Psi$ into two sets: those terms that are members of both $\Psi$ and $\Phi$, and those terms that are members of $\Psi$ but not $\Phi$. The ensemble difference terminology of definition \[D:ENSEMBLE\_DIFFERENCE\] poses the minuend, subtrahend, and remainder of this basis as respectively $\Psi$, $\Phi$, and $\Psi \setminus \Phi$.
This partition is important when interpreted as systems theory. The minuend $\Psi$ generates a choice space ${\prod{\Psi}}$ called the *stimulus* space. The subtrahend $\Phi$ generates the *persistent* (alternatively *response*) space ${\prod{\Phi}}$. The remainder $\Psi \setminus \Phi$ generates the *event* space ${\prod{(\Psi \setminus \Phi)}}$.
\[T:STATE\_EVENT\_SPACES\] Let $\langle \Psi, \Phi \rangle$ be a basis. The persistent $(\Phi)$ and volatile $(\Upsilon = \Psi \setminus \Phi)$ generating ensembles are disjoint and complementary with respect to the generating ensemble $\Psi$ of the stimulus space. Expressed in dyadic product, $$\Psi = \Phi\Upsilon$$
Since $\langle \Psi, \Phi \rangle$ is a basis, then $\Phi \subseteq \Psi$ by definition \[D:BASIS\]. With that result and by lemma \[L:DISJOINT\_AND\_COMPLEMENTARY\], the subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are disjoint and complementary with respect to $\Psi$.
The dyadic product recapitulates these results. Since $\Psi \setminus \Phi$ and $\Phi$ are disjoint the dyadic product $[\Ps
| 1,415
| 222
| 2,177
| 1,547
| 2,102
| 0.782717
|
github_plus_top10pct_by_avg
|
\]) with $s=0$. By using Theorem \[th:1\], we obtain the following propositions. It should be noted that $T(0,a,b \,; -y,x-y) = T(0,a,b \,; y, y-x)$ when $(x,y) = (1,1)$, $(1,1/2)$, $(1/2,1)$ or $(1/2, 1/2)$. In this case, the next proposition coincides with [@NakamuraT Proposition 2.8] or [@Zhou Proposition 1].
For any admissible index, we have $$\begin{split}
&T(0,a,b \,; -y,x-y) -(-1)^{a+b} T(0,a,b \,; y, y-x)
= \\ &(-1)^b \zeta (a \,; x) \zeta (b \,; y) - (-1)^b K (a,b \,; x,y)
+ \zeta (a \,; -y) \zeta (b \,; x) - \zeta (a+b \,; x-y)
\label{eq:tasite}
\end{split}$$ \[pro:1\]
Let $\phi$, $\chi$ and $\psi$ are Dirichlet characters of conductor $h$, $k$, and $q$, respectively. For any admissible index, we define $L (0,a,b \,; \phi, \chi ,\psi)$ by $$L (0,a,b \,; \phi, \chi ,\psi) := \lim_{R \to \infty} \sum_{m,n=1}^{m+n=R}
\frac{\phi (m) \chi (n) \psi (m+n)}{n^a (m+n)^b}.
\label{eq:defL}$$
Terhune [@Terhune] showed that if and $\chi \psi (-1) = (-1)^{a+b+1}$ then $L (0,a,b \,; 1, \chi ,\psi)$ can be expressed as a polynomial in the values of polylogarithms at certain roots of unity, with coefficients in a cyclotomic field. In [@NakamuraT Proposition 4.5], the author obtained explicit evaluation formulas for $L (0,a,b \,; 1, \chi ,\psi)$ when $\chi \psi (-1) = (-1)^{a+b+1}$. Proposition \[pro:1\] and the following proposition give simpler ones. Denote the Gauss sum by $\tau (\chi) := \sum_{l=1}^{k-1} \chi (l) e^{2 \pi i l/k}$.
Define $2U(a,b \,; x,y):= T(0,a,b \,; x,y) -(-1)^{a+b} T(0,a,b \,; -x, -y)$. Let $\phi \chi \psi(-1) = (-1)^{a+b+1}$. For any admissible index, we have $$\begin{split}
&\tau (\overline{\phi}) \tau (\overline{\chi}) \tau (\overline{\psi}) L(0,a,b \,; \phi, \chi ,\psi) = \\
&\sum_{j=1}^{h-1} \sum_{l=1}^{k-1} \sum_{r=1}^{q-1} \overline{\phi}(j) \overline{\chi}(l) \overline{\psi}(r) U (a,b \,; j/h+r/q, l/k+r/q).
\label{eq:dl1}
\end{split}$$ \[pro:2\]
Proof of results
================
First, suppose $\Re (s) >1$. We define $S (a,b,s \,; x,y)$ by $$S (a,b,s \,; x,y) := T (a,b,s \,; x,y) +
| 1,416
| 630
| 1,001
| 1,450
| 3,644
| 0.770962
|
github_plus_top10pct_by_avg
|
\mathcal{F}_n^c \right).$$ Thus, $$\begin{aligned}
\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \hat{\gamma}_j, \forall j\right)
& \geq (1-\alpha)- \mathbb{P}\left( \mathcal{F}_n^c \right)+ \mathbb{P}\left(
|Z_{n,j}| \leq z_{\alpha/(2s)} (\gamma_j - \Xi_n), \forall
j \right) -\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right)\\
& \geq (1-\alpha) - \frac{1}{n} - \frac{ \Xi_n z_{\alpha/(2s)}}{\min_j
\gamma_j} \left(\sqrt{ 2 + \log(2s ) } + 2 \right),
\end{aligned}$$ since, by the Gaussian anti-concentration inequality of , $$\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} (\gamma_j - \Xi_n), \forall
j \right) -\mathbb{P}\left( |Z_{n,j}| \leq z_{\alpha/(2s)} \gamma_j,
\forall j\right) \geq - \frac{ \Xi_n z_{\alpha/(2s)}}{\min_j
\gamma_j} \left(\sqrt{ 2 + \log(2s ) } + 2 \right).$$
The result follows by combining all the above bounds and the fact that $\underline{\sigma}^2 = \min_{P \in \mathcal{P}_n} \min_j \Gamma(j,j)$. As usual, we have absorbed any lower order term (namely $\frac{1}{n}$) into higher order ones. $\Box$
[**Proof of .**]{} Let $Z_n \sim N(0,\Gamma)$ where $\Gamma = G V G^\top$ and $\hat Z_n \sim N(0,\hat\Gamma)$ where we recall that $\hat\Gamma = \hat G \hat V \hat G^\top$, $\hat G = G(\hat \psi)$ and $\hat V = n^{-1}\sum_{i=1}^n (W_i - \hat\psi)(W_i - \hat\psi)^\top$. Take $\mathcal{E}_n$ to be the event that $$\left\{ \max_{j,k} |\widehat{\Gamma} - \Gamma| \leq C \aleph_n \right\} \cap
\left\{ \| V - \hat{V} \|_{\mathrm{op}} \leq C \daleth_n
\right\},$$ where $C$ is the larger of the two constants in and in . Then, by and , $\mathbb{P}\left( \mathcal{E}_n \right) \geq 1-2/n$, uniformly over all the distributions in ${\cal P}_n$. By the triangle inequality, $$\label{eq:F.boot}
\mathbb{P}(\theta \in \hat{C}^*_n) =
\mathbb{P}(\sqrt{n}||\hat\theta - \theta||_\infty \leq \hat{t}^*_\alpha) \geq
\mathbb{P}( \sqrt{n}||\hat \theta^* - \hat{\theta}||_\infty \leq
\hat{t}^*_\alpha|(W_1,\ldots,W_n)) - (A_1 + A_2 + A_3),$$ where $$\begin{aligned}
A_1 & = \sup_{t >0
| 1,417
| 1,955
| 1,328
| 1,323
| null | null |
github_plus_top10pct_by_avg
|
same truncation must in fact be identical, hence they cannot differ at $y^C$, hence $C$ is not characteristic. Since the set of exponents of any branch is discrete, the first assertion follows.
The second assertion follows from Lemma \[quadconics\]: if $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit is a union of a multiple kernel line and conics with equation $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac
{\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C\quad:$$ these conics are different precisely when the coefficients $\gamma_C$ are different, and the statement follows.
Proposition \[typeV\] leads to the procedure giving components of type V explained in §\[details\] (also cf. [@MR2001h:14068], §2, Fact 5), concluding the proof of Theorem \[mainmain\].
Boundaries of orbits {#boundary}
====================
We have now completed the set-theoretic description of the PNC determined by an arbitrary plane curve ${{\mathscr C}}$. As we have argued in §\[prelim\], this yields in particular a description of the boundary of ${{\mathscr O_{{\mathscr C}}}}$. In this section we include a few remarks aimed at making this description more explicit.
If $\dim{{\mathscr O_{{\mathscr C}}}}=8$, then the boundary of ${{\mathscr O_{{\mathscr C}}}}$ consists of the image of the union of the PNC and of the proper transform $R$ of the complement of ${\text{\rm PGL}}(3)$ in ${{\mathbb{P}}}^8$ (cf. Remark \[eluding\]). Curves in the image of $R$ are stars (Lemma \[PNCtolimits\]). Curves in the image of the components of the PNC belong to the orbit closures of the limits of the marker germs listed in §\[germlist\]. We have proved that this list is exhaustive; therefore, the boundary of a given curve ${{\mathscr C}}$ may be determined (up to stars) by identifying the marker germs for ${{\mathscr C}}$, and taking the union of the orbit closures of the (finitely many) corresponding limits.
This reduces the determination of the curves in the boundary of the orbit of a given curve to the det
| 1,418
| 556
| 1,421
| 1,464
| 4,029
| 0.76857
|
github_plus_top10pct_by_avg
|
}={\left\langle}d\psi,\psi{\right\rangle}_{L^2(G\times S\times I)},$$ where $$d:=-\mathrm{div}_{\tilde{\omega}}(F),$$ and $\mathrm{div}_{\tilde{\omega}}$ is the divergence operator on $S$ with respect to its Riemannian metric.
The equation (\[add1\]) in velocity coordinates $(x,v)\in{\mathbb{R}}^3\times{\mathbb{R}}^3$ can be written as \[add2\] F\_v+v\_x+(-K)=f, which is known as the (linear) Vlasov-Boltzmann equation.
Similar observations concern the methods used in the next sections. Nevertheless, the $m$-dissipativity of the operator $\psi\mapsto {1\over a}( F\cdot\nabla_{\tilde\omega}\psi+\omega\cdot\nabla_x\psi+C\psi)$, for $C$ large enough (in relevant modified spaces), requires extra analysis.
Finally notice that no boundary condition is needed in with respect to $\omega$-variable because the unit sphere $S$ is a compact manifold without boundary.
Existence Results Based on $m$-dissipativity {#m-d}
--------------------------------------------
In this section we apply an alternative method based on the results of dissipative first-order partial differential operators. Let $$P(x,\omega,E,D)\phi := -{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi
+CS_0\phi$$ where $S_0\in C^1(\ol G\times S\times I)$ and $C$ is constant. Recall that the [*formal transpose (adjoint)* ]{} of $P(x,\omega,E,D)$ is P’(x,,E,D)v := S\_0-\_x v+CS\_0v. It should be pointed out that here the notation for $P(x,\omega,E,D)$ and $P'(x,\omega,E,D)$ differs from that used in section \[esols\] in that here the operators $P(x,\omega,E,D)$ and $P'(x,\omega,E,D)$ also include the term $CS_0$.
Define linear operators $P,P':L^2(G\times S\times I)\to L^2(G\times S\times I)$ with domains of definition $D(P)$, $D(P')$ by setting D(P):=D\^1(GSI),P:=P(x,,E,D), and $$D(P'):=C_0^\infty(G\times S\times I^\circ),\quad P'v:=P'(x,\omega,E,D)v.$$ Clearly both $P$ and $P'$ are densely defined.
Let $P'^*:L^2(G\times S\times I)\to L^2(G\times S\times I)$ be the adjoint operator of $P'$. Then $\phi\in L^2(G\times S\times I)$ is said to
| 1,419
| 496
| 1,411
| 1,472
| null | null |
github_plus_top10pct_by_avg
|
$, $U_c$ and $U_c^-$ are simple, Morita equivalent rings (see the introduction to [@BEG3] for the details). Since this also applies to $H_{c+1}$ the conditions are trivially satisfied and the result follows.
Thus we may assume that $c\in \mathcal{C}$. If $c\geq -1$, then necessarily $c\geq 0$ and so the result follows from Proposition \[morrat\]. Otherwise $c\leq -1$. In this case the discussion before [@De Remark 2.2] shows that there is an isomorphism $\chi: H_c\to H_{-c}$ satisfying $\chi(e_-)=e$. Thus, for any $c$, implies that $U_c\cong U^-_{-c} \cong eH_{-c-1}e=U_{-c-1}$. The result for $c\leq -1$ therefore follows from the cases already discussed.
Finally, let $k$ be an arbitrary subfield of ${\mathbb{C}}$ and consider $U(k)_c$. In order to prove, for example, that $U(k)_c$ is Morita equivalent to $U(k)_{c+1}$ we need to prove that $Q(k)P(k)=U(k)_{c+1}$ and $P(k)Q(k)=U(k)_c$. By construction, $Q({\mathbb{C}})=Q(k)\otimes_k{\mathbb{C}}$, and similarly for $P({\mathbb{C}})$. By the earlier part of the proof, $U({\mathbb{C}})_c/P({\mathbb{C}})Q({\mathbb{C}})=0$. The faithful flatness of $U({\mathbb{C}})_c=U(k)_c\otimes_k{\mathbb{C}}$ as a $U(k)_c$-module then ensures that $U(k)_c/P(k)Q(k)=0$, whence $PQ=0$. All the other steps in the proof follow in exactly the same way.
\(2) Using the identity $U_c\cong U^-_{-c}$, this follows from part (1).
Remarks {#morrat-cor-remark}
-------
\(1) The condition that $c\notin \frac{1}{2} + \mathbb{Z}$ is needed in Theorem \[morrat\] and Corollary \[morrat-cor\] in order to apply and may be unnecessary. This is the case when $n=2$ as $U_c$ is Morita equivalent to $U_{c+1} $ if and only if $c\not= -\frac{3}{2}, -\frac{1}{2}$ (see, for example, [@EG Proposition 8.2]). The point about the excluded cases is that $U_{-\frac{1}{2}}$ is simple but the two neighbouring algebras, $U_{\frac{1}{2}}$, $U_{-\frac{3}{2}}$ are not. Combining [@EG Proposition 8.2] with [@St Theorem B] shows that $U_{-\frac{1}{2}}$ has infinite global dimension, and so the next Corollary \[gldim
| 1,420
| 593
| 638
| 1,473
| 2,553
| 0.778921
|
github_plus_top10pct_by_avg
|
t 11/17/2001 at 1:00:00 PM CT thru Sat 11/17/2001 at 5:00:00 PM CT
Sat 11/17/2001 at 11:00:00 AM PT thru Sat 11/17/2001 at 3:00:00 PM PT
Sat 11/17/2001 at 7:00:00 PM London thru Sat 11/17/2001 at 11:00:00 PM London
Outage: Migrate VMS objects from Enpower test to production
Environments Impacted: Corp
Purpose: To allow production deployment of the VMS engine
Backout: None.
Contact(s): Charlene Fricker 713-345-3487
Impact: NAHOU-ORDB07P
Time: Sat 11/17/2001 at 9:00:00 AM thru Sat 11/17/2001 at 11:00:00 AM
Outage: NAHOU-ORDB07P Drive Failure.
Environments Impacted: Corp trying to access dbases on the server.
Purpose: To replace a bad hard drive.
Backout: Restore from tape and backed up DB.
Contact(s): David Devoll 713-345-8970
SITARA: No Scheduled Outages.
SUN/OSS SYSTEM: No Scheduled Outages.
TELEPHONY: No Scheduled Outages
TERMINAL SERVER: No Scheduled Outages.
UNIFY: No Scheduled Outages.
----------------------------------------------------------------------------------------------------------------------------
FOR ASSISTANCE
(713) 853-1411 Enron Resolution Center
Specific Help:
Information Risk Management (713) 853-5536
SAP/ISC (713) 345-4727
Unify On-Call (713) 284-3757 [Pager]
Sitara On-Call (713) 288-0101 [Pager]
RUS/GOPS/GeoTools/APRS (713) 639-9726 [Pager]
OSS/UA4/TARP (713) 285-3165 [Pager]
CPR (713) 284-4175 [Pager]
EDI Support (713) 327-3893 [Pager]
EES Help Desk (713)853-9797 OR (888)853-9797
TDS -Trader Decision Support On-Call (713) 327-6032 [Pager]
i am in, we had a great time.
-----Original Message-----
From: Thames, Davis
Sent: Tuesday, November 27, 2001 1:27 PM
To: Mckay, Brad
Subject: conf call
thursday 4:45p. let me know if you have a conflict, there will be a callin number.
hope your thanksgiving went well, we had a great trip to all the parent's farms. pretty ridiculous.
hdt
CALENDAR ENTRY: APPOINTME
| 1,421
| 504
| 1,262
| 1,961
| null | null |
github_plus_top10pct_by_avg
|
}$ with this highest weight. Given an arbitrary finite-dimensional (complex) $G$-representation $V$, we can always decompose it into irreducible sub-representations $
V \cong \bigoplus_{\lambda \in \Lambda^*_{G,+}} m_{G,V}(\lambda) \, V_{G,\lambda}
$. We shall call the function $m_{G,V}$ thus defined the *highest weight multiplicity function*.
If we restrict the representation to the maximal torus, we can similarly decompose into irreducible representations. Since $T_G$ is a compact Abelian group, we can always jointly diagonalize its action, and it follows that the irreducible representations are one-dimensional. The joint eigenvalues can be encoded as a weight $\beta \in \Lambda^*_G$, and we will denote the corresponding irreducible representation of $T_G$ by ${\mathbb C}_\beta$. The decomposition $
V \cong \bigoplus_{\beta \in \Lambda^*_G} m_{T_G,V}(\beta) \, {\mathbb C}_\beta
$ then defines the *weight multiplicity function* $m_{T_G,V}$. We also set $[k] = \{ 1,\ldots,k \}$, and write $f \sim g$ for the asymptotic equivalence $\lim_{k \rightarrow \infty} f(k)/g(k) = 1$.
An equivalent way of encoding weight multiplicities is in terms of the (formal) *character*, $$\operatorname{ch}V = \sum_\beta m_{T_G,V}(\beta) \, e^\beta,$$ which can be understood as the generating function of $m_{T_G,V}$. Formally, $\operatorname{ch}V$ is an element of the group ring ${\mathbb Z}[\Lambda^*_G]$, which consists of (finite) linear combinations of basis elements $e^\beta$ subject to the relation $e^\beta e^{\beta'} = e^{\beta + \beta'}$. The character of an irreducible representation $V_{G,\lambda}$ is given by the *Weyl character formula* [@knapp02 p. 319], $$\label{weyl character formula}
\operatorname{ch}V_{G,\lambda} =
\frac {\sum_{w \in W_G} \det(w) \, e^{w(\lambda + \rho)}} {e^\rho \prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)}.$$ Observe that we have $$\label{derivation kostant partition function}
\begin{aligned}
\frac 1 {\prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)}
&= \prod_{\m
| 1,422
| 4,206
| 1,720
| 1,131
| 3,536
| 0.771701
|
github_plus_top10pct_by_avg
|
ll-studied in the $O(1)$ model which is in the same universality class as $SU(2)$ Yang-Mills theory. Moreover, in Polyakov gauge the effective action $\Gamma[A_0]$ after integrating-out the spatial gauge field is close to that of an $O(1)$-model. Studies using the FRG in local potential approximation with an optimised cut-off for the $O(1)$ model yield $\nu = 0.65$, see [@Litim:2001hk]. The critical exponent is related to the screening mass of temporal gauge field by $$m^2(T) \propto |T-T_c|^{2 \nu},
\label{eq:critexp}$$ where $m^2 = V''(\varphi_{min,0}) / 2$. We have computed the temperature-dependence of the screening mass in the confined phase near the phase transition, and extracted the critical exponent $\nu$ from a linear fit to the data. This is shown in Fig. \[fig:critexp\]. The fit yields the anticipated value of $$\nu =
0.65^{+0.02}_{-0.01}\,,$$ for the critical exponent $\nu$. The critical exponent $\beta$ agrees within the errors with the Ising exponent $\beta=0.33$.
![Critical exponent $\nu$ from $m^2=V''(\varphi_{min,0}) / 2$[]{data-label="fig:critexp"}](critexpnu.eps "fig:"){width="8cm"}\
Finally we would like to compare the results obtained here with the results of [@Braun:2007bx]. There, the effective potential $V_{\rm
eff}[A_0]$ was computed from the flow [@jan; @Fischer:2008uz] of Landau gauge propagators [@Lerche:2002ep; @von; @Smekal:1997is; @Bonnet:2001uh] within a background field approach in Landau-DeWitt gauge. In this gauge the confining properties of the theory are encoded in the non-trivial momentum dependence of the gluon and ghost propagators. Indeed, in [@Braun:2007bx] the effective potential $V_k$ was computed solely from this momentum dependence and was not fed back into the flow. In $SU(2)$ Landau gauge Yang-Mills this is expected to be a good approximation with the exception of temperatures close to the phase transition, see [@Braun:2007bx]. The back-reaction of the effective potential is particularly important for the critical physics, and the value of the critical temp
| 1,423
| 490
| 1,080
| 1,516
| null | null |
github_plus_top10pct_by_avg
|
8,34)$$ 1 It is a simple exercise to show that $\la{^{\operatorname{reg}}}$ is a $2$-regular partition, and we have the following result.
[ ]{}\[jreg\] Suppose $\la$ and $\mu$ are partitions of $n$, with $\mu$ $2$-regular. Then $[S^\la:D^{\la{^{\operatorname{reg}}}}]=1$, while $[S^\la:D^\mu]=0$ if $\mu\ndom\la{^{\operatorname{reg}}}$.
In this paper we shall be concerned with the Specht modules labelled by partitions of the form $(a,3,1^b)$; so we compute the regularisations of these partitions.
\[reg\] Suppose $a\gs4$ and $b\gs2$. Then $$(a,3,1^b){^{\operatorname{reg}}}=
\begin{cases}
(a,b+1,2)&(a>b)\\
(b+2,a-1,2)&(a\ls b).
\end{cases}$$
Irreducible Specht modules
--------------------------
It will be very helpful to know the classification of irreducible Specht modules, which (in characteristic $2$) was discovered by James and Mathas [@jmp2]. If $k$ is a non-negative integer we let $l(k)$ denote the smallest positive integer such that $2^{l(k)}>k$.
[ ]{}\[irrspecht\] Suppose $\mu$ is a partition of $n$ and ${\operatorname{char}}(\bbf)=2$. Then $S^\mu$ is irreducible if and only if one of the following occurs:
- $\mu_i-\mu_{i+1}\equiv-1\ppmod{2^{l(\mu_{i+1}-\mu_{i+2})}}$ for each $i\gs1$;
- $\mu'_i-\mu'_{i+1}\equiv-1\ppmod{2^{l(\mu'_{i+1}-\mu'_{i+2})}}$ for each $i\gs1$;
- $\mu=(2^2)$.
Note that $\mu$ satisfies the first condition in the theorem if and only if $\mu'$ satisfies the second. In view of Lemma \[isospecht\] (and since we shall only be considering values of $n$ greater than $4$) we may assume that any irreducible Specht module is of the form $S^\mu$ where $\mu$ satisfies the first condition in the theorem.
The main results {#resultsec}
================
In this section, we describe the new family of decomposable Specht modules discussed in this paper, and the method we use to prove decomposability. *For the rest of this paper, we assume that $\bbf$ has characteristic $2$.*
Computer calculations show that the first few decomposable Specht modules which are not labelled by hook partit
| 1,424
| 606
| 443
| 1,398
| 882
| 0.798369
|
github_plus_top10pct_by_avg
|
$\bigcirc$ indicate the boundary of the chiral phases characterized by long-range ordered chirality-chirality correlations $\left<\kappa_i\kappa_j\right>$; $\square$ indicate the boundary of the SF-phases indicated by the critical Luttinger parameter $K=2$. Note that narrow CMI and CHI phases may occur as well. For $U,\, U_3 \geq 0.5$ keeping $n_{max}=4$ bosons per site in the DMRG-simulation can be shown to be sufficient.[]{data-label="fig:2"}](bh_u2_u3.eps){width="8.5cm"}
The $j>1/4$ case is best understood from bosonization in the $j\gg 1$ regime. We may then introduce two pairs of bosonic fields ($\theta_{1},\phi_{1}$) and ($\theta_{2},\phi_{2}$), describing, respectively, the subchains of even and odd sites. The effective model is governed by the Hamiltonian density $$\begin{aligned}
\label{effectivetwocomponent}
{\mathcal H}&=&\sum_{\alpha=\pm}\frac{ v_{\alpha}}{2}\left[ \frac{(\partial_x \phi_{\alpha})^2}{ K_{\alpha}}+ K_{\alpha}(\partial_x \theta_{\alpha})^2 \right] \\
&+& \lambda\partial_x \theta_+\sin \sqrt{2\pi}\theta_-- 2{\mathcal M} \cos\sqrt{2\pi} \phi_{+} \cos\sqrt{2\pi} \phi_{-}\nonumber\end{aligned}$$ where $\theta_\pm=(\theta_1\pm \theta_2)/\sqrt{2}$, $\phi_\pm=(\phi_1\pm \phi_2)/\sqrt{2}$, $v_{\pm}$, $K_{\pm}$, and ${\mathcal M}$ are phenomenological parameters (in the regimes displayed on Figure 2 (a)), $\lambda\sim j^{-1}$. Note that the chirality is given by $\kappa_i\to \sin \sqrt{2\pi}\theta_-(x)$. In weak-coupling, $Um'\ll1$, with $m'=(\partial^2 \epsilon(k)/\partial k^2)_{k_0}^{-1}=4j/t(16j^2-1)$, $v_{\pm}\sim \sqrt{\bar n U/m'\pi^2}$ and $K_{\pm}\sim \sqrt{\bar n\pi^2/ Um'}$. In this case only the term $\partial_x \theta_+\sin \sqrt{2\pi}\theta_-$ is relevant, resulting in $\langle \sin \sqrt{2\pi}\theta_-\rangle\neq 0$ [@Nersesyan]. Hence, a small $U$ is expected to favor a CSF for $j>1/4$, as our numerical results confirm (Fig. \[fig:2\](a)). The CSF phase is characterized by $G_{ij}\sim (-1)^{i-j}e^{-i\kappa(i-j)}{|i-j|^{-1/4 K_+}}$, where $\kappa\sim \langle \kappa_i\rangle
| 1,425
| 398
| 1,078
| 1,445
| null | null |
github_plus_top10pct_by_avg
|
Positive emotions 8.85 3.50 3--16 9.11 3.30 2--18 7.56 3.36 0--15 4.56 2.26 0--9
Quality of life 69.94 13.76 35--95 12.62 6.56 2--34 10.88 6.04 0--30 73.44 14.99 35--95
Family Functioning, Cancer Appraisal and Cancer-Related Emotions
----------------------------------------------------------------
The final models for the associations between family functioning, cancer appraisal and cancer-related emotions are shown in [Table 3](#T3){ref-type="table"}.
######
Final models for the associations between family functioning, cancer appraisal, and cancer-related emotions.
Loneliness (*N* = 220; 20 patients, 28 siblings, 99 mothers, 73 fathers)^1^ Uncertainty (*N* = 220; 20 patients, 28 siblings, 99 mothers, 73 fathers)^1^ Helplessness (*N* = 220; 20 patients, 28 siblings, 99 mothers, 73 fathers)^1^ Positive feelings (*N* = 220; 20 patients, 28 siblings, 99 mothers, 73 fathers)^1^
------------------------------------------------- ----------------------------------------------------------------------------- ------------------------------------------------------------------------------ ------------------------------------------------------------------------------- ------------------------------------------------------------------------------------ ------------------ --------------- ------- ------------------ --------------- -------- ------------------ ---------------
**Variables of interest**
| 1,426
| 407
| 1,187
| 1,426
| null | null |
github_plus_top10pct_by_avg
|
next case, we will describe $\psi_j|_{F_j} : F_j \rightarrow \mathbb{Z}/2\mathbb{Z}$ explicitly in terms of a formal matrix. To do that, we will first describe a morphism from $F_j$ to the special fiber of the smooth integral model associated to $L^j$. Then we will describe a morphism from $F_j$ to the even orthogonal group associated to $M_0''$, where $M_0''$ is a Jordan component of $Y(C(L^j))=\bigoplus_{i \geq 0} M_i''$, and compute the Dickson invariant of the image of an element of $F_j$ in this orthogonal group.
We write $M_0=N_0\oplus L_j$, where $N_0$ is unimodular with even rank. Thus $N_0$ is either *of type II* or *of type $I^e$*. First we assume that $N_0$ is *of type $I^e$*. Then we can write $N_0=(\oplus_{\lambda'}H_{\lambda'})\oplus A(1, 2b, 1)$ and $L_j=(\oplus_{\lambda''}H_{\lambda''})\oplus A(1, 2b', 1)$ by Theorem \[210\], where $H_{\lambda'}=H(0)=H_{\lambda''}$ and $b, b'\in A$. Thus we write $M_0=(\oplus_{\lambda}H_{\lambda})\oplus A(1, 2b, 1)\oplus A(1, 2b', 1)$, 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 $$\begin{pmatrix} id&0 &0\\ 0 &\begin{pmatrix} 1+\pi x_j & 2 z_j^{\ast}\\ 0 & 1 \end{pmatrix} &0 \\ 0& 0 &id \end{pmatrix}.$$ Here, $id$ in the $(1,1)$-block corresponds to the direct summand $(\oplus_{\lambda}H_{\lambda})\oplus A(1, 2b, 1)$ of $M_0$ and the diagonal block $\begin{pmatrix} 1+\pi x_j & 2 z_j^{\ast}\\ 0 & 1 \end{pmatrix} $ corresponds to the direct summand $A(1, 2b', 1)$ of $M_0$.
Let $(e_1, e_2, e_3, e_4)$ be a basis for the direct summand $A(1, 2b, 1)\oplus A(1, 2b', 1)$ of $M_0$. Since this is *unimodular of type $I^e$*, we can choose another basis based on Theorem \[210\]. With the basis $(-2be_1+e_2, (2b'-1)e_1+e_3-e_4, e_3, e_2+e_4)$, denoted by $(e_1', e_2', e_3', e_4')$, $A(1, 2b, 1)\oplus A(1, 2b', 1)$ becomes $A(2b(2b-1), 2b'(2b'-1), -(2b-1)(2b'-1))\oplus A(1, 2(b+b'), 1)$. Here, $A(2b(2b-1), 2b'(2b'-1
| 1,427
| 695
| 1,729
| 1,408
| null | null |
github_plus_top10pct_by_avg
|
tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}.\end{aligned}$$ So, modulo homomorphisms labelled by tableaux not dominated by $S$, we have $\sigma=\sum_i{\hat\Theta_{T'[i]}}+\sum_{i,j}{\hat\Theta_{T'[i,j]}}$. However, two applications of Lemma \[lemma7\] show that ${\hat\Theta_{T'[i,j]}}=0$ for all $i,j$, and Lemma \[lemma7\] also gives ${\hat\Theta_{T'[i]}}={\hat\Theta_{S}}$.
So the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is $b+2-v$, which is odd; so $\sigma\neq0$.
As before, we find that $\sigma$ is the only homomorphism from $S^\la$ to $S^{\mu'}$ up to scaling.
\[cd2homdim1\] With $\la,\mu$ as above, $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})=1.$$
The existence of the homomorphism $\sigma$ shows that the space of homomorphisms is non-zero. To show that it has dimension at most $1$, we again use the fact that $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})=\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la'}).$$ 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 $d,t$.
First of all, consider ${\psi_{2,1}}\circ\theta$. Given a semistandard tableau $T$, we can use Lemma \[lemma5\] to compute ${\psi_{2,1}}\circ{\hat\Theta_{T}}$, and then if necessary use Lemma \[lemma7\] (to move a $2$ from row $3$ to row $2$) to express this composition as a linear combination of semistandard homomorphisms. We find that if $T$ has two $2$s in its first row, then ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ involves a semistandard tableau which does not occur in any other ${\psi_{2,1}}\circ{\hat\Theta_{T'}}$; hence the coefficient of ${\hat\Theta_{T}}$ in $\theta$ must be zero.
Now we do the same thing with ${\psi_{2,2}}$: in this case we find that if $T$ is a semistandard tableau having two $3$s in its first row, then ${\psi_{2,2}}\circ{\hat\Theta_
| 1,428
| 1,510
| 1,209
| 1,461
| 2,837
| 0.776669
|
github_plus_top10pct_by_avg
|
of interest in this work.
{#f1-idr-11-1043}
{#f2-idr-11-1043}
######
Susceptibility testing of the isolated *Streptococcus pneumonia*
Antimicrobial agent MIC (µg/mL)
------------------------------- -------------
Penicillin 0.03
Ampicillin 0.06
Ceftriaxone 0.25
Meropenem 0.015
Erythromycin 4
Tosufloxacin ≤0.12
Clindamycin ≥8
Vancomycin 0.5
Sulfamethoxazole/trimethoprim ≤0.25
**Abbreviation:** MIC, minimal inhibitory concentration.
######
Clinical profiles of the patients with pneumococcal sacroiliitis
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Case number Age (yrs) Gender Risk factors HIV antibody Chief complaints Clinical diagnosis Clinical samples Bacteremia Drainage Antibiotic treatment Treatment duration (weeks) Outcome Year Reference
------------- ----------- -------- -------------- -------------- ------------------------------------ ------------------------------------------ -------------------------------------------- ------------ -------------- --------
| 1,429
| 600
| 935
| 1,945
| null | null |
github_plus_top10pct_by_avg
|
$C=0$) \[csda27a\] B\_0(,v)=& ,S\_0[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)]{} +,(\^\* -K\^\*)v\_[L\^2(GSI)]{}\
&+\_+(), \_+(v)\_[T\^2(\_+)]{} +(,,0),S\_0(,0) v(,,0)\_[L\^2(GS)]{}. The linear form $F_0:C^1(\ol G\times S\times I)\to{\mathbb{R}}$ is given by $$F_0(v)=
{\left\langle}{f},v{\right\rangle}_{L^2(G\times S\times I)}+
{\left\langle}g, \gamma_-(v){\right\rangle}_{T^2(\Gamma_-)},$$ and it admits a unique extension to a bounded linear form $F_0:H_1\to{\mathbb{R}}$. Note that the bilinear form (\[csda27a\]) is not necessarily coercive that is (\[csda30\]) does not necessarily hold, which justifies the need for the change of unknown $\phi=e^{CE}\psi$ performed above.
We have the following immediate corollary for the existence of solutions of the original CSDA-problem.
\[csdaco1\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]) (with $c>0$), (\[csda9\]), (\[csda9aa\]) and (\[csda9a\]) are valid. Let ${ f}\in L^2(G\times S\times I)$ and ${ g}\in T^2(\Gamma_-)$. Then the following assertions hold.
\(i) The variational equation \[vareq1\] B\_0(,v)=F\_0(v)vH\_2 has a solution $\tilde\psi=(\psi,q,p_0,p_m)\in H_1$.
Furthermore, $\psi\in{\mathcal{H}}_P(G\times S\times I)$ and it is a weak (distributional) solution of the equation (\[se1\]).
\(ii) Suppose that additionally the assumption ${\bf TC}$ holds. Then a solution $\psi$ of the equation obtained in part (i) is a solution of the problem , , .
In addition, we have $q_{|\Gamma_+}=\gamma_+(\psi)$ and $p_0=\psi(\cdot,\cdot,0)$, when $\tilde{\psi}=(\psi,q,p_0,p_m)$ is a solution in $H_1$ obtained in part (i).
\(iii) Under the assumptions imposed in part (ii) any solution $\psi\in {\mathcal{H}}_P(G\times S\times I^\circ)$ of the problem , , that further satisfies \[asscl-aa\] \_[|\_+]{}T\^2(\_+)(,,0)L\^2(GS), is unique and obeys the estimate \[csda40aaa\] \_[[H\_1]{}]{} (\_[L\^2(GSI)]{}+\_[T\^2(\_-)]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.)
Let ${ f}\in L^2(G\times S\times I)$ and ${ g}\in T^2(\Gamma_-)$
| 1,430
| 522
| 1,616
| 1,520
| null | null |
github_plus_top10pct_by_avg
|
}}^b (0) \sim \ \kappa^{ab} c_3 \frac{1}{\bar{z}^2}
+ {f^{ab}}_c \left[ \frac{c_4}{\bar{z}} j_{L,\bar{z}}^c(0) + \frac{(c_4-g)z}{\bar{z}^2} j_{L,z}^c(0)\right] \cr
& + {f^{ab}}_c \left[\frac{g}{4}\frac{z}{\bar z}(\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{c_4}{2} \partial_{\bar z} j_{L,\bar z}^c(0)+ \frac{c_4-g}{2} \frac{z^2}{\bar z^2} \partial_{z}j_{L,z}^c(0) \right] \cr
& + :j_{\bar z}^a j_{\bar z}^b:(0)
- {{A}^{ab}}_{cd} \log |z|^2 :j_{\bar z}^{ c} j_{\bar z}^{d }:(0) + {{B}^{ab}}_{cd}\frac{ z}{\bar z} :j_z^{ c} j_{\bar z}^{d }:(0) + {{C}^{ab}}_{cd} \frac{1}{2} \frac{z^2}{\bar z^2} :j^{ c}_z j_z^{d }:(0) \cr
& + ... \cr
%
j_{L,z}^a (z) &j_{L,\bar{z}}^b(0) \sim \ \tilde{c}\kappa^{ab} 2\pi \delta^{(2)}(z-w)
+ {f^{ab}}_c \left[ \frac{(c_4-g)}{\bar{z}} j_{L,z}^c(0) + \frac{(c_2-g) }{z} j_{L,\bar{z}}^c(0) \right] \cr
& + {f^{ab}}_c \left[ -\frac{g}{4} \log |z|^2 (\partial_z j_{\bar z}^c(0)-\partial_{\bar z}j_z^c(0)) + \frac{(c_4-g)z}{\bar{z}}\partial_z j_{L,z}^c(0) \right] \cr
& + :j_z^a j_{\bar z}^b:(0)
+ {{A}^{ab}}_{cd} \frac{\bar z}{z}:j_{\bar z}^{ c} j_{\bar z}^{d }:(0) - {{B}^{ab}}_{cd} \log |z|^2 :j_z^{ c} j_{\bar z}^{d }:(0)
+ {{C}^{ab}}_{cd} \frac{z}{\bar z} :j^{ c}_z j_z^{d }:(0) \cr
& + ... \cr\end{aligned}$$ Compared to [@Ashok:2009xx], we have added a few terms at order zero in the distance between the insertion points of the two current components[^2]. The ellipses refer to subleading terms in the expansion in the distance between the two insertion points (which includes logarithms). The right current components $j_{R,z}$ and $j_{R,\zbar}$ satisfy similar operator product expansions amongst themselves, with the holomorphic coordinates replaced by anti-holomorphic ones. This can be proven by using the $\mathbb{Z}_2$ symmetry that we noted before. Associativity of the current algebra is discussed in appendix \[associativity\].
For the supergroup non-linear sigma-model in equation , the coefficients of the second and first order poles in the conformal current algebra
| 1,431
| 1,387
| 1,627
| 1,304
| null | null |
github_plus_top10pct_by_avg
|
gives a suboptimal coefficient vector.
Proposed Method {#section:ProposedMethod}
===============
In this section, we will first derive our method for the real-valued channel model, and then extend the method for the complex-valued channel model.
Preliminaries
-------------
We start with investigating some properties of the problem, which is the basis of our new method.
(Signature Matrix) A signature matrix is a diagonal matrix whose diagonal elements are $\pm1$.
(Signed Permutation Matrix) A signed permutation matrix is a generalized permutation matrix whose nonzero entries are $\pm 1$.
After replacing $-1$’s with $1$’s, a signed permutation matrix becomes a permutation matrix. Obviously, signed permutation matrices are unimodular and orthogonal. Every signed permutation matrix can be expressed as $\S=\P\T$, where $\T$ is a signature matrix, and $\P$ is a permutation matrix.
\[theorem:ProblemTransformation\] If ${\a^\star}$ is the optimal coefficient vector for a channel vector $\h$ with power constraint $P$, then for any signed permutation matrix $\S\in\Zbb^{L\times L}$, $\S{\a^\star}$ is optimal for $\S\h$ with the same power constraint $P$, and $\bigR(\h,{\a^\star})=\bigR(\S\h,\S{\a^\star})$.
We first show $\bigR \left( \h, \a \right) = \bigR \left( \S\h, \S\a \right)$ for any $\h$ and $\a$ with the same power constraint $P$. $\S$ is unimodular, then $\S\a$ is an integer vector and can be applied as a coefficient vector. $\S$ is orthogonal, then $\S^T\S = \I$. $\norm{\S\h}^2 = \h^T\S^T\S\h = \h^T\h = \norm{\h}^2$, and similarly $\norm{\S\a}^2 = \norm{\a}^2$. $(\S\h)^T\S\a = \h^T\S^T\S\a = \h^T\a$. According to Theorem \[theorem:ComputationRate\], the computation rate $\bigR \left( \h, \a \right)$ is determined by $P$, $\norm{\h}^2$, $\norm{\a}^2$, and $\h^T\a$. Thus, $\bigR \left( \h, \a \right) = \bigR \left( \S\h, \S\a \right)$.
${\a^\star}$ is optimal for $\h$ means ${\a^\star}$ maximizes $\bigR \left( \h, \a \right)$. Then $\S{\a^\star}$ maximizes $\bigR \left( \S\h, \S\a \right)$ since $\bi
| 1,432
| 243
| 1,059
| 1,521
| null | null |
github_plus_top10pct_by_avg
|
as observed in [@Meckes].
\[T:C-Ginibre-eigenvalues\] Let $G$ be a finite abelian group and let $\{Y_a \mid a \in G\}$ be independent, standard complex Gaussian random variables. Then the eigenvalues $\bigl\{ \lambda_\chi \mid \chi \in \widehat{G} \bigr\}$ of $M$ given by are independent, standard complex Gaussian random variables.
The random matrix ensemble in Proposition \[T:C-Ginibre-eigenvalues\] is the $G$-circulant analogue of the complex Ginibre ensemble $X$, which consists of a square matrix with independent, standard complex Gaussian entries.
\[T:C-Ginibre-limit\] Suppose that for each $n$, $\bigl\{Y_a^{(n)} \mid a \in G^{(n)}
\bigr\}$ are independent, standard complex Gaussian random variables. Then ${\mathbb{E}}\mu^{(n)} = \gamma_{\mathbb{C}}$ for each $n$, and $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly in probability. Furthermore, if ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$ for some ${\varepsilon}> 0$, then $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly almost surely.
For each, say, Lipschitz $f: {\mathbb{C}}\to {\mathbb{R}}$, $$({\mathbb{E}}\mu)(f) := {\mathbb{E}}\bigl(\mu(f)\bigr)
= \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} {\mathbb{E}}f(\lambda_\chi),$$ where the $(n)$ superscripts are omitted for simplicity. By Proposition \[T:C-Ginibre-eigenvalues\], each $\lambda_\chi$ is distributed according to $\gamma_{\mathbb{C}}$, and so $ ({\mathbb{E}}\mu)(f) =
\gamma_{\mathbb{C}}(f) $. Thus ${\mathbb{E}}\mu = \gamma_{\mathbb{C}}$.
By the concentration properties of Gaussian measure (see [@Ledoux]), since the $\lambda_\chi$ are distributed as independent standard complex Gaussian random variables, if $f$ is $1$-Lipschitz, then $${\mathbb{P}}\bigl[{\left\vert \mu(f) - \gamma_{\mathbb{C}}(f) \right\vert} \ge t \bigr]
\le 2 e^{-{\left\vert G \right\vert} t^2}$$ for each $t > 0$. If ${\left\vert G^{(n)} \right\vert} = \Omega(n^{{\varepsilon}})$, then the Borel–Cantelli lemma implies that $\mu^{(n)}(f) \to \gamma_{\mathbb{C}}(f)$ almost surely. Applying
| 1,433
| 1,230
| 1,563
| 1,256
| null | null |
github_plus_top10pct_by_avg
|
eft\{ (UX)^{\dagger} A W \right\}_{m K}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\biggr\}.
\label{P-beta-alpha-W4-H3-Second} \end{aligned}$$
$$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{6th}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [2]
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
- \sum_{n}
\sum_{K} \sum_{k}
\frac{ 1 }{ \Delta_{K} - h_{k} }
\left[ (ix) e^{- i ( \Delta_{K} - h_{n} ) x}
+
\frac{e^{- i ( h_{k} - h_{n} ) x} - e^{- i ( \Delta_{K} - h_{n} ) x} }{ h_{k} - \Delta_{K} }
\right]
\nonumber \\
&\times&
(UX)^*_{\alpha n} (UX)_{\beta n}
W_{\alpha K} W^*_{\beta K}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\nonumber \\
&+&
\sum_{n}
\sum_{K \neq L}
\sum_{k}
\frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) }
\nonumber \\
&\times&
\biggl[
\Delta_{L} e^{- i ( \Delta_{K} - h_{n} ) x}
- \Delta_{K} e^{- i ( \Delta_{L} - h_{n} ) x}
+ \left( e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( \Delta_{K} - h_{n} ) x} \right) h_{k}
- ( \Delta_{L} - \Delta_{K} )
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)^*_{\alpha n} (UX)_{\beta n}
W_{\alpha K} W^*_{\beta L}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\biggr\}.
\label{P-beta-alpha-W4-H2} \end{aligned}$$
Neutrino evolution equation in flavor basis {#sec:flavor-basis-evolution}
===========================================
The Schrödinger equation takes the form with flavor basis Hamiltonian $H$ in (\[flavor-hamiltonian\]) $$\begin{aligned}
i \frac{d}{dx}
\left[
\begin{array}{c}
\nu_{a} \\
\nu_{s} \\
\end{array}
\right] =
\left[
\begin{array}{cc}
U {\bf \Delta_{a} } U^{\dagger} + W {\bf \Delta_{s} } W^{\dagger} + A &
U {\bf \Delta_{a} } Z^{\dagger} + W {\bf \Delta_{s} } V^{\dagger} \\
Z {\bf \Delta_{a} } U^{\dagger} + V {\bf \Delta_{s} } W^{\dagger} &
Z {\bf \Delta_{a} } Z^{\dagger} + V {\bf \Delta_{s} } V^{\dagger} \\
\end{array}
\rig
| 1,434
| 1,727
| 1,700
| 1,537
| null | null |
github_plus_top10pct_by_avg
|
tive map $\theta :
J^{k-1}\delta^ke\hookrightarrow M(k)$ of left $\mathbb C[{\mathfrak{h}}]$-modules such that:
1. $\theta$ is an ${\mathbf{E}}$-graded homomorphism and is a filtered homomorphism under the order filtration.
2. The associated graded map $\operatorname{{\textsf}{ogr}}\theta: J^{k-1}\delta^ke \to \operatorname{{\textsf}{ogr}}M(k)$ induced by $\theta$ is precisely $\operatorname{{\textsf}{ogr}}\theta = \Theta$.
3. In the notation of , the inclusion $ \theta[\delta^{-2}] :
(J^{k-1}\delta^k e)[\delta^{-2}] \to M(k)[\delta^{-2}] $ is an isomorphism. This map is ${\mathbf{E}}$-graded and is a filtered isomorphism under the order filtration.
(1,2) As in the proof of Lemma \[filter-injA\], one constructs $\theta$ by lifting a ${\mathbf{E}}$-homogeneous basis of the free ${\mathbb{C}}[{\mathfrak{h}}]$-module $\operatorname{{\textsf}{ogr}}^n(J^{k-1}\delta^k)e$ to a set of ${\mathbf{E}}$-homogeneous elements in $\operatorname{{\textsf}{ord}}^n M(k)$.
\(3) This is essentially the same as the proof of Lemma \[step-1\].
{#section-6}
By Lemma \[diaggrad\], $M(k)$ is graded under the adjoint ${\mathbf{h}}$-action and, as both copies of ${\mathbb{C}}$ are ${\mathbf{h}}$-graded modules, this grading restricts to one on $\overline{M(k)} $ and $
\underline{M(k)}$. In each case, we call this [*the ${\mathbf{h}}$-grading*]{}. For the reasons given in , this does not equal the ${\mathbf{E}}$-grading.
\[poincare-SC\] If $\overline{M(k)}$ is graded via the adjoint ${\mathbf{h}}$ action, then it has Poincaré series $$p(\overline{M(k)}, v) =
\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-(k-1)(n(\mu)
- n(\mu^t))}}{\prod_{i=2}^n
(1-v^{-i})}.$$
This is similar to the proof of Proposition \[poincare-SA\] except that we use the module $Y=H_ce\otimes_{R}{\mathbb{C}}$, where $R=e{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}e$, in place of $X=H_c\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]}{\mathbb{C}}$. As in that proposition, $Y$ is an object in $\widetilde{{\mathcal{O}}}_c$ and so we can write $ [Y] =
\sum_{\mu
| 1,435
| 902
| 557
| 1,447
| 4,089
| 0.76818
|
github_plus_top10pct_by_avg
|
n_i & \quad \textit{if $L_i$ is \textit{of type} $\textit{I}^o$ or \textit{free of type II}};\\
n_i+1 & \quad \textit{if $L_i$ is \textit{bound of type II}}.
\end{array} \right.$$ **
We next assume that $i=2m-1$ is odd. Recall that $Y_i$ is the sublattice $B_i$ such that $Y_i/\pi A_i$ is the radical of the alternating bilinear form $\frac{1}{\pi}\cdot\frac{1}{2^{m-1}} h$ mod $\pi$ on $B_i/\pi A_i$.
\[l42\] Let $i$ be odd. Then each element of $\underline{M}(R)$, for a flat $A$-algebra $R$, preserves $Y_i\otimes_A R$.
We claim that $Y_i=\pi^{i+1}B_i^{\perp}\cap B_i$. Then the lemma follows from this directly. The inclusion $Y_i \subseteq \pi^{i+1}B_i^{\perp}\cap B_i$ is clear by the definition of $Y_i$. For the other direction, we choose $a=\pi^{i+1}b^{\perp} \in \pi^{i+1}B_i^{\perp}\cap B_i$ where $b^{\perp}\in B_i^{\perp}$. Then $h(a, b')=\pi^{i+1}h(b^{\perp}, b')\in \pi^{i+1}B$ for any $b'\in B_i$. This completes the proof.
\[t44\] Assume that $i=2m-1$ is odd. Let $h_i$ denote the nonsingular alternating bilinear form $\frac{1}{\pi}\cdot\frac{1}{2^{m-1}} h$ mod $\pi$ on $B_i/Y_i$. Then there exists a unique morphism of algebraic groups $$\varphi_i:\tilde{G}\longrightarrow \mathrm{Sp}(B_i/Y_i, h_i)$$ defined over $\kappa$ such that for all étale local $A$-algebras $R$ with residue field $\kappa_R$ and every $\tilde{m} \in \underline{G}(R)$ with reduction $m\in \tilde{G}(\kappa_R)$, $\varphi_i(m)\in \mathrm{GL}(B_i\otimes_AR/Y_i\otimes_AR)$ is induced by the action of $\tilde{m}$ on $L\otimes_AR$ (which preserves $B_i\otimes_AR$ and $Y_i\otimes_AR$ by Lemma \[l42\]).
As in the above theorem, we only provide the image of an element $m$ of $\tilde{G}(\kappa_R)$ into $\mathrm{Sp}(B_i/Y_i, h_i)$, where $R$ is an étale local $A$-algebra with $\kappa_R$ as its residue field.
As in Theorem \[t43\], an element $m$ of $\tilde{G}(\kappa_R)$ may be written as, say, $(m_{i,j}, s_i \cdots w_i)$ and it has the following formal matrix description: $$m= \begin{pmatrix} \pi^{max\{0,j-i\}}m_{i,j} \end{pmatrix} \t
| 1,436
| 926
| 1,628
| 1,388
| null | null |
github_plus_top10pct_by_avg
|
e most dangerous area---the main tunnel, is mostly determined (besides smokiness level) by evacuees' familiarity with the situation, scenario and layout of the tunnel.
4.4 Movement speed {#sec011}
------------------
We calculated movement speed for experiments 1-3. Movement speed in the main tunnel and the evacuation tunnel was calculated separately, since the differences in conditions in these tunnels are significant---there is no smoke in the evacuation tunnel. Due to the size of the tunnel and the heavy smoke, determination of exact trajectories (and exact speed respectively) would be extremely hard. Thus, we calculated evacuees' speed in a simplified way, as the distance to travel divided by walking time. Fridolf et al. propose to call this method *modeling speed* \[[@pone.0201732.ref012]\]. However, this method is widely used in similar experiments \[[@pone.0201732.ref004], [@pone.0201732.ref031]\]. The calculated movement speeds for the main and the evacuation tunnel in consecutive experiments are shown in [Table 2](#pone.0201732.t002){ref-type="table"}.
10.1371/journal.pone.0201732.t002
###### Movement speed for the main and the evacuation tunnel for experiments 1-3.
First 9 persons, who stopped and discussed after leaving the bus during experiment 1 were excluded from the statistics.
{#pone.0201732.t002g}
Experiment section Minimum Maximum Mean Std. deviation
------------------------------------ --------- --------- ------- ----------------
experiment 1 the main tunnel 0.895 1.211 1.056 0.083
experiment 1 the evacuation tunnel 1.542 1.808 1.706 0.058
experiment 2 the main tunnel 0.917 2.422 1.321 0.375
experiment 2 the evacuation tunnel 1.489 1.953 1.635 0.081
experiment 3 the main tunnel 0.893 2.044 1.221 0.295
experiment 3 the evacuation tunnel 2.569 5.760 3.835 0.719
Movement speed in the evacuation tunnel ([Table 2](#pone.0201732.t002){ref-type
| 1,437
| 702
| 2,561
| 1,599
| null | null |
github_plus_top10pct_by_avg
|
ubscales) and quality of life (PedsQL scores and MMQ scores) per family member are presented in [Table 5](#T5){ref-type="table"}. Mean scores for mother, father, sibling and patients were compared.
######
Mean scores for cancer appraisal (PSS scores), family functioning (FES subscale scores), cancer related emotions (SSERQ subscale scores) and quality of life (standardized PedsQL and MMQ scores) for the different family members.
Patient *M (SD)* Mother *M (SD)* Father *M (SD)* Sibling *M (SD)*
-------------------------------- -------------------- ------------------ ----------------- ----------------- ------------------
Cancer appraisal 18.81 (5.31) 21.03 (6.55) 17.97 (6.28) 20.82 (6.19)
Family Functioning Cohesion 56.17 (5.32) 51.55 (7.66) 53.03 (7.21) 53.79 (6.65)
Expressiveness 52.52 (7.78) 53.06 (9.15) 51.37 (10.05) 52.73 (7.97)
Conflict 44.52 (11.92) 45.26 (9.47) 47.25 (10.11) 45.33 (10.25)
Organization 54.61 (6.97) 49.56 (8.35) 50.10 (10.24) 49.76 (8.87)
Control 51.78 (7.93) 49.44 (7.60) 48.18 (7.97) 51.76 (8.66)
Norms 53.09 (5.54) 48.88 (7.46) 50.48 (6.48) 52.91 (5.22)
Social orientation 48.35 (11.62) 48.64 (11.45) 48.38 (9.76) 51.18 (10.03)
Cancer-related emotion Loneliness 5.91 (3.63) 7.81 (6.81) 5.34 (5.13) 5.49 (4.70)
Uncertainty 5.65 (3.78) 8.88 (4.26) 7.40 (3.82) 7.29 (5.56)
Helplessness 12.87 (4.70) 13.36 (4.67) 11.23 (4.51) 13
| 1,438
| 3,486
| 1,743
| 1,097
| null | null |
github_plus_top10pct_by_avg
|
el obtained by forcing with a Souslin tree, if $X$ is locally compact normal, $D$ is a closed discrete subspace of $X$ of size $\aleph_1$ and $\{U_\alpha:\alpha\in\omega_1\}$ are open sets with compact closures, then for any countable $T\subseteq\omega_1$, $\stackrel{}{\overline{\bigcup\{U_\alpha:\alpha\in T\}}}\cap\; D$ is countable.
${\bigcup\{{\overline{U}_\alpha}:\alpha\in T\}}$ is dense in $\overline{\bigcup\{U_\alpha:\alpha\in T\}}$, which is locally compact normal.
Getting back to the proof of \[thm41\], let us assume we are in a model of ${\mathrm}{MM}(S)$ and that we have an $S$-name $\dot{X}$ for a locally compact normal space, with a closed discrete subspace labeled as $\omega_1$, with each of its points having character $\aleph_1$. Let us note that it follows from character reduction and Lemma \[LT1\] that if there is a discrete expansion of $\omega_1$ into compact $G_\delta$’s, then $\omega_1$ will have a separation. In fact, even more, it is shown in [@T3 Theorem 12] that if $\omega_1$ is forced to have an expansion by compact $G_\delta$’s that is $\sigma$-discrete, then $\omega_1$ will be separated. Since our proof is by contradiction, we will henceforth assume that it is forced (by the root of $S$) that there is no expansion of $\omega_1$ into a $\sigma$-discrete family of compact $G_\delta$’s.
For each $\xi,\alpha\in\omega_1$, let $\dot{U}(\xi,\alpha)$ be the name of the $\alpha$th neighbourhood from a local base for $\xi$ with $\dot U(\xi,0)$ forced to have compact closure. Corollary \[cor410\], and the fact that $S$ is ccc, ensure that for each $\delta\in
\omega_1$, every element of $S$ forces that $\omega_1 \cap \overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}$ is bounded by $\gamma$ for some $\gamma\in \omega_1$. Therefore there is a cub $C_0$ such that without loss of generality, we can assume that each of the following is forced by each element of $S$:
1. for each $\delta\in C_0$, $\omega_1\cap
\stackrel{}{\overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}}$ is included in $\delt
| 1,439
| 730
| 1,357
| 1,491
| null | null |
github_plus_top10pct_by_avg
|
omology concentrated in degree zero.
We now state our main theorem.
\[O\_hat\_Koszul\] If $\mathcal O$ is cyclic quadratic and Koszul, then $\widehat{\mathcal O}$ has a resolution, which for a given sequence $\vec X$ of colors, with $|\vec X|=n$, is given by the quasi-isomorphisms $$\begin{aligned}
\textbf{D}(\widehat{\mathcal O^!}) (\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})&\to& \widehat{
\mathcal O}(\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})=\mathcal O(n) \\
\textbf{D}(\widehat{\mathcal O^!}) (\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})&\to& \widehat{
\mathcal O}(\vec X;{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})=\mathcal O(n) \\
\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing) &\to&
\widehat{ \mathcal O}(\vec X;\varnothing)=\mathcal O(n-1)\end{aligned}$$
Koszulness of $\mathcal O$ means exactly that the first and second maps are quasi-isomorphisms. The proof that the third map is also a quasi-isomorphism will concern the rest of this section.
We need to show that the homology of $\textbf{D}(\widehat{\mathcal O^!}) (\vec X;
\varnothing)$ is concentrated in degree $0$: $$\begin{aligned}
\label{H0}
H_0 \left(\textbf{D}(\widehat{\mathcal O^!}) (\vec X;\varnothing)\right
)&=&\widehat{ \mathcal O}(\vec X;\varnothing)\\ \label{Hi<0} H_{i<0} \left(\textbf{D}(\widehat{\mathcal O^!})
(\vec X;\varnothing)\right)&=&\{0\}\end{aligned}$$ As mentioned in definition \[def\_O\_hat\], it is enough to restrict attention to the case where $\vec X = ({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},\ldots,{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth
| 1,440
| 1,012
| 1,093
| 1,321
| 519
| 0.807201
|
github_plus_top10pct_by_avg
|
ta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'}
\, G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.
address: 'Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania'
author:
- 'A. L. Agore'
- 'G. Militaru'
title: Deformations of a matched pair and Schreier type theorems for bicrossed product of groups
---
[^1]
Introduction {#introduction .unnumbered}
============
The aim of the paper is to bring back to attention and revitalize one of the most famous open problems of group theory formulated in the first half of the last century([@Douglas], [@Ore], [@Redei]). It can be seen as the dual of the more famous *extension problem* of O. L. Hölder and it is called the *factorization problem*. The statement is very simple and tempting:
*Let $H$ and $G$ be two given groups. Describe and classify up to an isomorphism all groups $E$ that factorize through $H$ and $G$: i.e. $E$ contains $H$ and $G$ as subgroups such that $E = H
G$ and $ H \cap G = 1$.*
Leaving aside the classification part introduced above, the first part of the problem was formulated in 1937 by O. Ore [@Ore] but it roots are much older and descend to E. Maillet’s 1900 paper [@Maillet]. Even though if the statement is very simple, as many famous problems in mathematics are, little progress has been made since then. We dare to say that this one is even more difficult than the more popular extension problem. In the case of two cyclic groups $H$ and $G$, not both finite, the problem was started by L. Rédei in [@Redei] and finished by P.M. Cohn in [@Cohn], without the classification part introduced above. To the best of our knowledge this seems to be the only case where the complete answer is known. If $H$ and $G$ are both finite cyclic groups the problem is more difficult and seems to be still an open question, even t
| 1,441
| 345
| 1,257
| 1,242
| null | null |
github_plus_top10pct_by_avg
|
on of the dimensionless detuning $\delta$ (full line). The dashed line corresponds to the asymptotic behavior $\epsilon_b \simeq -mg_1^2/4$ and the dotted line to the asymptotic behavior $\epsilon_b \simeq \nu$.](BFRMboundstate.eps "fig:"){height="6cm"} \[BFRMboundstate\]
The bound state energy $\epsilon_b$ is plotted as a function of the dimensionless detuning $\delta$ in Figure 1. The behavior of the bound state in the 1D BFRM is qualitatively similar to that of the confinement induced bound state found by Bergeman, Moore and Olshanii [@BMO] for two atoms trapped in a quasi-1D geometry (i.e., a waveguide with radial frequency $\omega_{\perp}/2\pi$). This fact reveals the connection, at the two-body level, between the 1D BFRM and the quasi-1D single channel model. In Figure 2 we have plotted the confinement induced (CI) bound state as a function of $\delta'$ (see below). In the quasi-1D case, the role of the dimensionless detuning $\delta$ is played by the parameter $\delta'\equiv a_{\perp}/a-A$, where $a_{\perp}\equiv (m\omega_{\perp})^{-1/2}$ is the radial oscillator length, $a$ is the 3D scattering length and $A\equiv
-\zeta(1/2,1)/\sqrt{2}\simeq 1.0326$ [@footnote2]. In the quasi-1D geometry, the 1D scattering amplitude shows a CI resonance [@Olshanii] and is given by $$\begin{aligned}
g_1'\equiv 2\omega_{\perp}a(1-Aa/a_{\perp})^{-1}=2/ma_{\perp}\delta'\end{aligned}$$ which is similar to $g_1=2/mr_{\star} \delta$, showing that $a_{\perp}$ plays the role of $r_{\star}$. The CI bound state energy $\epsilon_b'$ obeys the following equation $$\begin{aligned}
\sqrt{2}a_{\perp}/a+\zeta(1/2,\epsilon_b'/\epsilon_{\star}')=0\end{aligned}$$ where $\zeta(1/2,x)$ is a particular Hurwitz zeta function [@BMO] and $\epsilon_{\star}'\equiv \epsilon_b'(\delta'=0)=-2/ma_{\perp}^2$ is the CI bound state energy on resonance. When $\delta' \to
-\infty$, $\epsilon_b'\simeq -mg_1'^2/4$, in complete analogy with $\epsilon_b\simeq -mg_1^2/4$. On resonance $\delta'=0$, $\epsilon_{\star}'=-2/ma_{\perp}^2$, to be compared with $\epsil
| 1,442
| 275
| 1,701
| 1,502
| 2,354
| 0.780612
|
github_plus_top10pct_by_avg
|
sampled independently and are also independent of other iterations.
Intuitively, an $(s, \sigma, b_1, b_2)$-large-noise SGD should be considered as an SGD algorithm with step size $s$ and minibatch size $b_1$ and an additional noise term. The noise term computes the difference of two independent and unbiased estimates of the full gradient $\nabla U(w_k)$, each using a batch of $b_2$ data points. Using the definition of $\zeta$ in , we can verify that the update is equivalent to $$\begin{aligned}
\numberthis\label{e:sgd-noisy-2}
& w_{k+1} = w_k - s\nabla U(w_k) + s\zeta(w_k, \eta_k) \\
&\qquad + \sigma \sqrt{s}(\zeta(w_k, \eta_k'') - \zeta(w_k, \eta_k')),\end{aligned}$$ which is in the form of , with $$\begin{aligned}
\xi(w, \tilde{\eta}) = \sqrt{s} \zeta(w, \eta) + \sigma \lrp{\zeta(w, \eta'')-\zeta(w, \eta')},
\numberthis
\label{e:sgd-noisy-xi}\end{aligned}$$ where $\tilde{\eta} = (\eta, \eta', \eta'')$, and $|\eta| = b_1$, $|\eta'|=|\eta''|=b_2$. Further, the noise covariance matrix is $$\label{e:noise_cov_large_noise_sgd}
\E{\xi(w, \tilde{\eta}) \xi(w, \tilde{\eta})^T} = (\frac{s}{b_1} + \frac{2\sigma^2}{b_2}) H(w).$$ Therefore, if we have $$\label{e:var_matching}
\frac{s}{b_1} + \frac{2\sigma^2}{b_2} = \frac{\delta}{b},$$ then an $(s, \sigma, b_1, b_2)$-large-noise SGD should have the same noise covariance as a $(\delta, b)$-SGD *(but very different higher noise moments due to the injected noise)*, and based on our theory, the large-noise SGD should have similar test error to that of the SGD algorithm, even if the step size and batch size are different. In Section \[subsection:experiments\], we verify this experimentally. We stress that we are not proposing the large-noise SGD as a practical algorithm. The reason that this algorithm is interesting is that it gives us a family of $\lrp{w_k}_{k=1,2,\ldots}$ which converges to , and is implementable in practice. Thus this algorithm helps us uncover the importance of noise covariance (and the unimportance of higher noise moments) in Langevin
| 1,443
| 1,225
| 1,059
| 1,302
| 1,181
| 0.793075
|
github_plus_top10pct_by_avg
|
e boundaries fully cover the region of gravitational waves creation. The energy density of gravitons is given by $$d\rho_{\text{gw}} = 2 \cdot \hslash \omega \cdot \frac{4 \pi \omega^2 d\omega}{(2\pi c)^3} \cdot |B_{-}(k)|^2.$$ where we used definition (\[particles\]). The expression for the parameter $\Omega_{\text{gw}}$ defined by (\[omegaGW\]) takes now the form $$\Omega_{\text{gw}}(\nu) =\Omega_0 \cdot \nu^4 \cdot \bar{n}\left[ k = \nu \cdot 2\pi a_{\text{f}} \cdot
\left( \frac{a_\text{today}}{a_{\text{f}}} \right) \right]
\label{omegaGW1}$$ where $$\Omega_0 = \frac{\hslash c}{c^4}\frac{16\pi^2}{\rho_c} =
\frac{ 16 \pi^2 \cdot 197.3 \cdot 10^{-15}[\text{MeV} \cdot
\text{m}]}{3^4 \cdot 10^{32}[\text{m}^4/\text{s}^4] 1.05 \cdot 10^{-5} \cdot h^2_0
[\text{GeV}/\text{cm}^3] } = 3.66 \cdot h^{-2}_0 \cdot 10^{-49} \ [\text{Hz}^{-4}].$$ In the calculations we set present value of the Hubble factor for $h_{0}=0.7$. In Fig. \[spect1\] we show spectrum calculated with formula (\[omegaGW1\]). The obtained spectrum is extremely weak in the present epoch. The reason of this tiny amount of the background gravitons is the presence of the standard inflationary phase. The super-inflationary phase is placed before the inflation so the energy of gravitons decreases about $10^{27}$ times during this further phase.
![Spectrum of relic gravitons for the model with $j=100$ and $l=3/4$. Frequency scale in Hertz.[]{data-label="spect1"}](plot4.eps){width="7cm"}
To see better how presence on the inflation affect this spectrum we show in Fig. \[spect1\] the spectrum of relic gravitons in the model without the inflationary phase.
![Spectrum of relic gravitons for the model without inflation. Frequency scale in Hertz.[]{data-label="spect2"}](plot5.eps){width="7cm"}
It is clear that in such a model amount of relic gravitons would be extremely large. As we mentioned before it is possible that GUT energy scales cover partially with the inflation. In this situation the present amount of the relic gravitons would be high
| 1,444
| 3,621
| 2,243
| 1,488
| null | null |
github_plus_top10pct_by_avg
|
the general case. In the Coulomb gauge the vector potential ${\mathbf
A}(\theta,\phi) = {1 \over 2} \mathbf{B} \times \mathbf{r} $ expressed in surface variables reduces to $$\notag \mathbf {A}(\theta,\phi) = {1\over 2}\big [ B_1 (W
{\rm sin\phi \cos \theta} +
a \ {\rm sin^2\theta sin}\phi){\bm \theta} +
(B_0 W - B_1 a \ {\rm sin \theta cos\phi})]{\bm \phi}$$ $$+ B_1(F
{\rm sin\phi \sin\theta} - a \ {\rm cos\theta sin \theta \sin
\phi})\mathbf{n}.$$ with $\mathbf n = {\bm \phi} \ {\rm x} \ {\bm \theta}$. The normal component of $\mathbf A$ contributes a quadratic term to the Hamiltonian but leads to no differentiations in the coordinate normal to the surface as per Eq.(8). There is a wealth of literature concerning curvature effects when a particle is constrained to a two-dimensional surface in three-space [@burgsjens; @jenskoppe; @dacosta1; @dacosta2; @matsutani; @matsutani2; @duclosexner; @bindscatt; @popov; @ouyang; @midgwang; @ee1; @ee2; @lin; @goldjaffe; @exnerseba; @schujaff; @clarbrac], including some dealing with the torus specifically [@encmott], but the scope of this work will remain restricted to study of the Hamiltonian given by Eq. (9).
The Schrodinger equation (spin splitting will be neglected throughout this work) is more simply expressed by first defining $$\alpha = a/R$$ $$F = 1 + \rm \alpha \ cos\theta$$ $$\gamma_0 = B_0 \pi R^2$$ $$\gamma_1 = B_1 \pi R^2$$ $$\gamma_N = {\pi \hbar \over q}$$ $$\tau_0 = {\gamma_0 \over \gamma_N}$$ $$\tau_1 = {\gamma_1 \over \gamma_N}$$ $$\varepsilon = {2mEa^2 \over \hbar^2},$$ after which Eq. (9) may be written $$\bigg [ {\partial^2 \over \partial^2 \theta} -
{\alpha \ {\rm sin} \ \theta \over F}{\partial \over \partial
\theta} + {\alpha^2 \over F^2}{\partial^2 \over
\partial^2 \phi} +
i \bigg(\tau_0\alpha^2-{\tau_1\alpha^3 \over F}{\rm
sin\theta cos\phi} \bigg){\partial \over \partial \phi}$$ $$+ i\alpha\tau_1 {\rm sin \phi (\alpha+cos\theta)}{\partial
\over
\partial \theta}$$ $$\begin{aligned}
-{\tau_0^2 \alpha^2F^2 \over 4} - {\tau_1^2 \a
| 1,445
| 4,297
| 878
| 1,049
| null | null |
github_plus_top10pct_by_avg
|
}_g (c_2-g) {{B}^{gd}}_{xy} :j^{x}_{z} j^{y}_{\bar z}:
\nonumber \\ &
+ \mathcal{O}(f^2) %\mbox{higher order in $f^2$} \end{aligned}$$ We use current conservation and the Maurer-Cartan equation to write: $$\begin{aligned}
+i &(c_4-\frac{g}{2}) {f^{ac}}_g
{f^g}_{de} :j^e_z j^d_{\bar z}:
+c_- {B^{ac}}_{ed} : j_z^{e} j^{d}_{\bar z}:(w)
\nonumber \\ &
+c_+ {B^{ac}}_{ed} : j_z^{e} j^{d}_{\bar z}:(w)
\nonumber \\ &
+(c_4-g/2) ( i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} - i {f^c}_{dg} {f^{ag}}_e
) : j_z^e j_{\bar z}^d:(w)
\nonumber \\ &
+ g/2 (i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} + i {f^c}_{dg} {f^{ag}}_e
): j_z^e j_{\bar z}^d:(w)
\nonumber \\ &
- i {f^{c}}_{hx} {f^{ax}}_g (c_2-g) ){{B}^{gh}}_{ed} :j^{e}_{z} j^{d}_{\bar z}:
\nonumber \\ &
+ \mathcal{O}(f^2)\end{aligned}$$ where we have separated out (graded) symmetric and anti-symmetric terms. We now apply the super Jacobi identity to the first term in the third line and note that: $$\begin{aligned}
{f^{ce}}_{g} {f^{ag}}_d &=& {f^{ce}}_g {f^{ga}}_d (-1)^{1+a + ad}
\nonumber \\
&=& {f^{ec}}_g {f^{ga}}_d (-1)^{a + ad+ec}
\nonumber \\
&=& - (-1)^{a+ad+ec + cd}
( (-1)^{ac} { f^{ea}}_g {{f^g}_d}^c +
(-1)^{ad} {f^e}_{dg} f^{gca}),\end{aligned}$$ which leads to: $$\begin{aligned}
{f^c}_{eg} {f^{ag}}_d (-1)^{ed} - {f^c}_{dg} {f^{ag}}_e
&=&(-1)^{1+a+ad+ec+cd+ed+ac+e+g+g+1+cd}
{f^c}_{dg} {f^{ag}_e}
\nonumber \\
& & + (-1)^{1+a+ad+ec+cd+ad+g+ed+g+g+ca+ed} {f^{ac}}_g {f^g}_{de}
\nonumber \\
& &
- {f^c}_{dg} {f^{ag}}_e
\nonumber \\
&=& - {f^{ac}}_g {f^g}_{de}.\end{aligned}$$ Therefore, the third line cancels the first term in the first line and we are left with: $$\begin{aligned}
(c_- & {(B)^{ac}}_{ed}+ c_+
{(B)^{ac}}_{ed}) : j_z^{e} j^{d}_{\bar z}:(w)
\nonumber \\ &
+ g/2 (i {f^c}_{eg} {f^{ag}}_d (-1)^{ed} + i {f^c}_{dg} {f^{ag}}_e
)
: j_z^e j_{\bar z}^d:(w)
\nonumber \\ &
- i {f^{c}}_{hx} {f^{ax}}_g (c_2-g){{B}^{gh}}_{ed} :j^{e}_{z} j^{d}_{\bar z}: + \mathcal{O}(f^2).
\label{final}\end{aligned}$$ As expected the demand of the vanishing of this term gives the value
| 1,446
| 2,015
| 1,617
| 1,412
| null | null |
github_plus_top10pct_by_avg
|
e have: $$i(h, g) \cdot i(h', g') = \bigl( h (g \rhd h'), (g \lhd h')
g'\bigl)\, \stackrel{g \in {\rm Fix}(G)} {=} \bigl( h(g \rhd h'),
gg'\bigl) = i\bigl((h, g) (h',g')\bigl)$$
Thus the two semidirect products constructed above, $H
{}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G)$ and ${\rm Fix}(H)
\rtimes_{\psi_{\lhd}} G$, are subgroups of the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$. To conclude, we obtained a commutative diagram in the category of groups $${\label{eq:sfibr}}
\begin {CD}
Fix(H)\times Fix(G) @>\overline{j}>> Fix(H)\rtimes_{\psi} G\\
@VV\overline{i}V @VVjV\\
H {}_\varphi \ltimes Fix(G)@>i>> H {}_\alpha \bowtie_\beta G
\end{CD}$$
Using the construction of the pullback in the category of groups it follows that the pair $({\rm Fix}(H) \times {\rm Fix}(G),
(\overline{i}, \overline{j}))$ is a pullback of the morphisms $i:
H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) \hookrightarrow H\,
{}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $j: {\rm Fix}(H)
\rtimes_{\psi_{\lhd}} G \hookrightarrow H\, {}_{\alpha}\!\!
\bowtie_{\beta} \, G$.
[\[pr:pushout\]]{} Let $(H, G, \alpha, \beta)$ be a matched pair of groups and $\bigl(X, (\varphi, \psi)\bigl)$ be the pushout in the category of groups of the diagram $$\begin {CD}
Fix(H)\times Fix(G) @>\overline{j}>> Fix(H)\rtimes_{\psi} G\\
@VV\overline{i}V @VV\varphi V\\
H {}_\varphi \ltimes Fix(G)@>\psi >> X
\end{CD}$$ Then the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \,
G$ is isomorphic to a quotient group of $X$.
The diagram [(\[eq:sfibr\])]{} is commutative and $\bigl(X, (\varphi,
\psi)\bigl)$ is the pushout of the pair $(\overline{i}, \,
\overline{j})$: thus there exists an unique morphism of groups $\theta : X \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ such that $\theta \circ \psi = i$ and $\theta \circ \varphi = j$. Let $(h, g) \in H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$: as $(h,
1_G) \in H {}_{\varphi_{\rhd}}\!\! \ltimes Fix(G)$ and $(1_H, g)
\in Fix(H) \rtimes_{\psi_{\lhd}} G$ we obtain $$(h, g) = (h, 1_G)(1_H, g) = i
| 1,447
| 1,250
| 1,297
| 1,373
| null | null |
github_plus_top10pct_by_avg
|
\muhat^*},$$ where the sum is over types $\omhat$ as above. Comparison with Euler’s formula $$\Log\left(\sum_{n\geq 0}p(n)\,T^n\right)=\sum_{n\geq1}T^n,$$ shows that $L$ reduces to $\sum_{t\geq 1}
\,m_{t\muhat^*}$. Hence the coefficient of the lowest power of $q$ in $\H_\muhat\left(\sqrt{q},1/\sqrt{q}\right)$ is also $1$ in this case finishing the proof.
If $g\geq 1$, the dimension vector $\v$ is always in the fundamental set of imaginary roots of $\Gamma$. If $g=0$ the character variety if not empty if and only if $\v$ is a strict root of $\Gamma$ and if $\v$ is real then $\M_\muhat$ is a point [@crawley-par Theorem 8.3]. If $\v$ is imaginary then it can be taken by the Weyl group to some $\v'$ in the fundamental set and the two corresponding varieties $\M_\muhat$ and $\M_{\muhat'}$ are isomorphic for appropriate choices of conjugacy classes [@crawley-par Theorem 3.2, Lemma 4.3 (ii)], hence Theorem \[connectedness\].
Appendix by Gergely Harcos {#appendix}
==========================
Let $n,r$ be positive integers, and let $x_{ik}$ ($1\leq i\leq n$, $1\leq k\leq r$) be arbitrary nonnegative numbers. Let $c_i:=\sum_k x_{ik}$ and $c:=\max_i c_i$. Then we we have $$c\sum_k\biggl(\sum_i x_{ik}\biggr)^2-\biggl(\sum_i
c_i\biggr)\biggl(\sum_{i,k}x_{ik}^2\biggr)\leq c\biggl(\sum_i
c_i\biggr)^2-\biggl(\sum_i c_i\biggr)\biggl(\sum_i c_i^2\biggr).$$ Assuming $\min_i c_i>0$, equality holds if and only if we are in one of the following situations
\(i) $x_{ik}=x_{jk}$ for all $i,j,k$,
\(ii) there exists some $l$ such that $x_{ik}=0$ for all $i$ and all $k\neq l$. \[harcos\]
The assumption $\min_i c_i>0$ does not result in any loss of generality, because the values $i$ with $c_i=0$ can be omitted without altering any of the sums.
Without loss of generality we can assume $c=c_1\geq\dots\geq
c_n$, then the inequality can be rewritten as $$\biggl(\sum_i c_i\biggr)\biggl(\sum_j\sum_{k,l}x_{jk}x_{jl}-\sum_{j,k}x_{jk}^2\biggr)
\leq c\biggl(\sum_{i,j}\sum_{k,l}x_{ik}x_{jl}-\sum_{i,j}\sum_k
x_{ik}x_{jk}\biggr).$$ Here and lat
| 1,448
| 1,243
| 1,267
| 1,328
| 2,853
| 0.776532
|
github_plus_top10pct_by_avg
|
of . The Gaussian comparison yields that $A_2 \leq C \Delta_{n,3} + \frac{2}{n}$. To bound the term $A_3$, we repeat the arguments used in the first part of the proof of , applied to the larger class $\mathcal{P}_n^*$ and restricting to the event $\mathcal{E}_n$. As argued above, we will replace $\psi$ with $\hat\psi$ and $\hat\psi$ with $\hat\psi^*$ and, similarly, $\Gamma$ with $\hat{\Gamma}$ and $\hat{\Gamma}$ with $\hat{\Gamma}^*$. The assumption that $n$ is large enough guarantees that, with probability at least $1 - \frac{2}{n}$, both $v_n$ and $\sigma^2_n$ are positive. Thus, the right hand side of serves as an upper bound for the current term $A_3$ as well. The claimed bound then follows. $\Box$
Appendix 5: Proofs of Auxiliary Results {#appendix:auxilary}
=======================================
[**Proof of .**]{} Let $Z$ be the number of objects that are not selected. Then $\mathbb{E}[Z] = n \left( 1 - \frac{1}{n} \right)^n \leq \frac{n}{e}$. Next, by the bounded difference inequality, $$\mathbb{P}\left(| Z - \mathbb{E}[Z] | \geq t \right) \leq 2 e^{
-\frac{t^2}{2 n}},$$ which implies that $$\mathbb{P}\left( Z > n - d \right) \leq \exp\left\{ - \frac{(n -d
-n(1-1/n)^n)^2 }{2n} \right\}.$$ The claim follows immediately, since $n \geq \frac{d}{2}$ and $\left( 1 - \frac{1}{n} \right)^n \leq
e^{-1}$ for all $n=1,2,\ldots$.
$\Box$
[**Proof of Lemma \[lem:hyper\].**]{} Let $\psi$ be an arbitrary point in $\mathcal{S}_n$ and $G =
G(\psi) \in \mathbb{R}^{s \times b }$ be the corresponding Jacobian. Recall that, for $j=1,\ldots,s$ the $j^{\mathrm{th}}$ row of $G$ is the transpose of $G_j =
G_j(\psi)$, the gradient of $g_j$ at $\psi$. Let $\mathcal{V} =
\mathcal{V}(G) = \left\{ v_1,\ldots,v_{2s} \right\}$, where for $j=1,2,\ldots,s$, we define $v_{2j-1} =
\frac{G_j}{\| G_j \|}$ and $v_{2j} = -\frac{G_j}{\| G_j\|}$. For a given $t>0$ and for any Jacobian matrix $G = G(\psi)$, set $$\label{eq:polyhedron}
P(G,t) = \left\{ x \in \mathbb{R}^b \colon v_l^\top x \leq t_l , \forall v_l \in \mathcal{V}(G
| 1,449
| 961
| 978
| 1,375
| null | null |
github_plus_top10pct_by_avg
|
at{\pi}^{'} &=& i [ \hat{H}_{\text{t}}, \hat{\pi} ]. \label{Ham2}\end{aligned}$$
The Hamilton operator have the form $$\begin{aligned}
\hat{H}_{\text{t}} &=& \frac{1}{2}\int d^3 {\bf x} [ \hat{\pi}^{2}+D \delta^{ij} \partial_i \hat{u} \partial_j \hat{u}
+m^2_{\text{eff}}\hat{u}^2 ] \nonumber \\
&=& \frac{1}{2} \frac{1}{4(2\pi)^{3}} \int d^3 {\bf x} d^3 {\bf k} d^3 {\bf q}
\left[ \hat{\pi}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} +
\hat{\pi}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \left[ \hat{\pi}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} +
\hat{\pi}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\
&+& D \delta^{ij} i \left[ k_i\hat{u}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} -
k_i\hat{u}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] i\left[q_j \hat{u}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} -
q_j\hat{u}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\
&+& m^2_{\text{eff}} \left[ \hat{u}_{{\bf k}} e^{i{\bf k}\cdot {\bf x}} +
\hat{u}_{{\bf k}}^{\dagger} e^{-i{\bf k}\cdot {\bf x}} \right] \left[ \hat{u}_{{\bf q}} e^{i{\bf q}\cdot {\bf x}} +
\hat{u}_{{\bf q}}^{\dagger} e^{-i{\bf q}\cdot {\bf x}} \right] \nonumber \\
&=& \frac{1}{4} \int d^3 {\bf k}\left[ \hat{\pi}_{{\bf k}} \hat{\pi}_{{\bf k}}^{\dagger}+
\hat{\pi}_{{\bf k}}^{\dagger} \hat{\pi}_{{\bf k}} +\left(Dk^2 + m^2_{\text{eff} }\right)
\left( \hat{u}_{{\bf k}} \hat{u}_{{\bf k}}^{\dagger}+ \hat{u}_{{\bf k}}^{\dagger} \hat{u}_{{\bf k}} \right) \right]
\label{Ham}\end{aligned}$$ where we inserted decompositions (\[decomp1\]) and (\[decomp2\]). When we apply the Hamiltonian (\[Ham\]) and the decompositions (\[decomp1\]) and (\[decomp2\]), the Hamilton equations (\[Ham1\]) and (\[Ham2\]) take the forms $$\begin{aligned}
\hat{u}^{'}_{{\bf k}} &=& \hat{\pi} _{{\bf k}}, \label{Ham11} \\
\hat{\pi}^{'}_{{\bf k}} &=& -\left(Dk^2 + m^2_{\text{eff} }\right) \hat{u}_{{\bf k}}. \label{Ham22} \end{aligned}$$ The general solution of these equations has the form $$
| 1,450
| 4,724
| 289
| 1,144
| null | null |
github_plus_top10pct_by_avg
|
r \[item:pc4b\] implies statement \[item:pc3\]. Moreover, since the empty functor is a sieve and a cosieve, statement \[item:pc3\] implies \[item:pc2\]. By duality, it remains to show that \[item:pc1\] implies \[item:pc4a\]. Given a functor $u\colon A\to B$, the morphism $u_\ast\colon{\sD}^A\to{\sD}^B$ is a right adjoint and, as a pointed morphism of pointed derivators, $u_\ast$ preserves left Kan extensions along cosieves [@groth:can-can Cor. 8.2].
We now turn to the stable context. Let us recall that a category $A\in\cCat$ is **strictly homotopy finite** if it is finite, skeletal, and it has no non-trivial endomorphisms (equivalently the nerve $NA$ is a finite simplicial set). A category is **homotopy finite** if it is equivalent to a strictly homotopy finite category.
\[thm:stable-lim-I\] Homotopy finite colimits and homotopy finite limits commute in stable derivators.
Let be a stable derivator and let $A\in\cCat$. Denoting by $\pi_A\colon A\to\bbone$ the unique functor, there are defining adjunctions $$(\colim_A,\pi_A^\ast)\colon{\sD}^A\rightleftarrows{\sD}\qquad\text{and}\qquad (\pi_A^\ast,\mathrm{lim}_A)\colon{\sD}\rightleftarrows{\sD}^A,$$ and these exhibit $\colim_A,\mathrm{lim}_A\colon{\sD}^A\to{\sD}$ as exact morphisms of stable derivators [@groth:can-can Cor. 9.9]. Hence, by [@ps:linearity Thm. 7.1], $\colim_A$ preserves homotopy finite limits and $\lim_A$ preserves homotopy finite colimits.
For the converse to this theorem we collect the following lemma.
\[lem:lim-comm\] Let be a derivator such that homotopy finite colimits and homotopy finite limits commute in .
1. The derivator is pointed.
2. The morphisms ${\mathsf{cof}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$ and $C\colon{\sD}^{[1]}\to{\sD}$ preserve homotopy finite limits.
3. The morphism ${\mathsf{fib}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$ and $F\colon{\sD}^{[1]}\to{\sD}$ preserve homotopy finite colimits.
By assumption on , empty colimits and empty limits commute and this implies that is pointed (). Hence, by duality, it remains to take
| 1,451
| 996
| 1,395
| 1,353
| null | null |
github_plus_top10pct_by_avg
|
^\alpha}{\sum_j c_jk_j^\alpha},$$
where $c_i=\sum_j A_{ij}k_j^\alpha$,
Using Eqs. (\[eq:diff1\]) and (\[eq:diff2\]), we check that the transition matrix is reversible and then has $m$ real eigenvalues. From this stationary probability density, we can thus compute both the KSE and the second largest eigenvalue $\lambda(P)$ as a function of $\alpha$. The result is provided in (Fig. \[fig:KS5\]).
![KSE (top) and $\lambda(P)$ (bottom) function of $\alpha$ for a network of size $m=400$ with a proportion of $0$ in $A$ equal to $1/3$. []{data-label="fig:KS5"}](mixtimeandhksfuncalpham400rho1div34.jpg){width="10cm"}
We observe in (Fig. \[fig:KS5\]) that the KS entropy has a maximum at a value that we denote $\alpha_{KS}$, in agreement with the findings of [@gomez2008entropy]. Likewise, $\lambda(P)$ (i.e. the mixing time) presents a minimum for $\alpha=\alpha_{mix}$. Moreover, $\alpha_{KS}$ and $\alpha_{mix}$ are close. This means that the two optimal diffusion coefficients are close to each other. Furthermore, looking at the ends of the two curves, we can find two special Markov chains $P1$ and $P2$ such that $h_{KS}(P1) \leq h_{KS}(P2)$ and $t_1(\epsilon) \leq t_2(\epsilon)$, illustrating that the link between KSE and the minimum mixing time is only true in a general statistical sense.
We have thus shown that, for a given transition matrix $P$ (or equivalently for given jump rules) the greater the KSE, the smaller the mixing time. We now investigate whether a similar property holds for dynamics, i.e. whether transition rules that maximise KSE are close to the ones minimizing the mixing time. For a given network, i.e. for a fixed $A$, there is a well known procedure to compute the transition matrix $P_{KS}$ which maximizes the KSE with the constraints $A(i,j)=0 \Rightarrow P_{KS}(i,j)=0$ [@burda2009localization]. It proceeds as follow: let us note $\lambda$ the greatest eigenvalue of $A$ and $\Psi$ the normalized eigenvector associated i.e $A\Psi=\lambda \Psi$ and $ \sum_i \Psi^2_i=1$. We define $P_{KS}$ such that
| 1,452
| 2,193
| 2,594
| 1,572
| 2,833
| 0.776695
|
github_plus_top10pct_by_avg
|
weinberg].
Once Weinberg’s formalism is expressed by means of the symplectic form in Eq. (\[eq:wein\_eqofm\]), it can be generalized very easily in order to obtain a non-Hamiltonian quantum algebra. To this end, one can substitute the antisymmetric matrix $\mbox{\boldmath$\cal B$}$ with another antisymmetric matrix $\mbox{\boldmath$\Omega$}=\mbox{\boldmath$\Omega$}[\mbox{\boldmath$\zeta$}]$, whose elements might be functionals of $\mbox{\boldmath$\zeta$}\equiv(\vert\Psi\rangle,\langle\Psi\vert)$ obeying the homogeneity condition in Eq. (\[eq:homogeneity\]). By means of $\mbox{\boldmath$\Omega$}$ a non-Hamiltonian bracket $\left\{ \ldots ,\ldots \right\}_{\mbox{\tiny\boldmath$\Omega$}}$ can be defined as $$\begin{aligned}
\left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}}&=&
\sum_{\alpha=1}^2\frac{\partial a}{\partial\zeta{\alpha}}
{\Omega}_{\alpha\beta}[\zeta]\frac{\partial b}{\partial\zeta{\beta}}\;.
\label{eq:nhbracket}\end{aligned}$$ In general, the bracket in Eq. (\[eq:nhbracket\]) does no longer satisfy the Jacobi relation $${\cal J}=
\left\{ a,\left\{ b,c\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}}
+\left\{ c\left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}}
+\left\{ b,\left\{ c,a\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\neq 0\;.\label{eq:njacobi}$$ Thus, non-Hamiltonian equations of motion can be written as $$\frac{\partial\mbox{\boldmath$\zeta$}}{\partial t}=\frac{i}{\hbar}
\left\{ {\cal H},\mbox{\boldmath$\zeta$}\right\}_{\mbox{\tiny\boldmath$\Omega$}}
\;.
\label{eq:wein_nheqofm}$$ In principle, the non-Hamiltonian theory, specified by Eqs. (\[eq:nhbracket\]), (\[eq:njacobi\]), and (\[eq:wein\_nheqofm\]), can be used to address the problem of non-linear correction to quantum mechanics, as it was done in Refs. [@weinberg]. In the present paper, it has been shown that such a non-Hamiltonian and non-linear version of quantum mechanics is already implied when one formulates qua
| 1,453
| 1,537
| 1,594
| 1,474
| null | null |
github_plus_top10pct_by_avg
|
d will be denoted by $L^{n-4k}_z$. The structure group of the framing of this component is $\I_{2,z}$. Lemma 1 is proved.
### The last part of the proof of the Theorem 1 {#the-last-part-of-the-proof-of-the-theorem-1 .unnumbered}
Let us construct a pair of polyhedra $(P',Q') \subset \R^n$, $dim(P') =2s-n=n-2k-q-2$, $dim(Q') = dim(P')-1$. Obviously, $dim(P')< 2k -1$. Take a generic mapping $d': \RP^s \to \R^n$. Let us consider the submanifold with boundary $(\Delta'^{reg},
\partial \Delta'^{reg}) \subset \R^n$ (see the denotation in Lemma 1). Let $\eta_{\Delta'^{reg}}: (\Delta'^{reg},\partial
\Delta'^{reg}) \to (K(\D_4,1),K(\I_b,1))$ be the classifying mapping for the double point self-intersection manifold of $d$.
By a standard argument we may modify the mapping $d$ into $d'$ such that the mapping $\eta_{\Delta^{reg}}$ is a homotopy equivalence of pairs up to the dimension $q+1$. After this modification $d' \to d$ we define $(P,Q)=(\Delta^{reg},\partial \Delta^{reg}) \subset \R^n$ and the mapping $\eta_{\Delta^{reg}}$ is a $(q+1)$-homotopy equivalence.
The subpolyhedron $Q$ is equipped with two cohomology classes $\kappa_{Q,1}, \kappa_{Q,2} \in H^1(Q;\Z/2)$. Because $\Sigma$ is a submanifold in $\RP^s$, the restriction of the characteristic class $\kappa \in H^1(\RP^s;\Z/2)$ to $H^1(\Sigma;\Z/2)$ is well-defined. The inclusion $i_Q: Q \subset U_{\Sigma}$ determines the cohomology class $(i_Q)^{\ast}(\kappa) \in H^1(Q;\Z/2)$. The cohomology class $\kappa_{Q,1}$ is defined as the characteristic class of the canonical double points covering over $\Sigma$. The class $\kappa_{Q,2}$ is defined by the formula $\kappa_{Q,2} = (i_Q)^{\ast}(\kappa) + \kappa_{Q,1}$.
The immersed manifold (with boundary) $(N^{n-2k} \cap U_{\Sigma})
\looparrowright U_{\Sigma}$ is equipped with an $\I_b$-framing. Obviously the classes $\kappa_{Q,1}, \kappa_{Q,2} \in
H^1(U_{\Sigma};\Z/2)=H^1(Q;\Z/2)$ restricted to $H^1(g_2(N^{n-2k}_{ext});\Z/2)$ ( recall that $g_2(N^{n-2k}_{ext})=g_2(N^{n-2k}) \cap (\R^n \setminus
U_{\Delta})$) agree with t
| 1,454
| 683
| 1,256
| 1,408
| null | null |
github_plus_top10pct_by_avg
|
\alpha (t) \dot{\beta} (t) \dot{\gamma} (t)+\gamma (t)
\big(\dot{\alpha} (t) \dot{\beta} (t)+\alpha (t) \ddot{\beta} (t)\big)\bigg) +\beta (t)
\bigg(\alpha (t)^2 \dot{\beta} (t) \dot{\gamma} (t)^2
\\&
-\alpha (t)^2 \gamma (t) \big(\dot{\gamma} (t)
\ddot{\beta} (t)+\dot{\beta} (t) \ddot{\gamma} (t)\bigg) +\gamma (t)^2 \bigg(\dot{\alpha} (t)^2 \dot{\beta} (t)-\alpha (t) \dot{\alpha} (t) \ddot{\beta} (t)-\alpha (t) \big(\dot{\beta} (t) \ddot{\alpha} (t)+\alpha (t) \beta ^{(3)}(t)\big)\big)\bigg)
\\
&+\beta (t)^2 \Bigg[\alpha (t)
\dot{\gamma} (t) \bigg(\dot{\alpha} (t) \dot{\gamma} (t)+\alpha (t) \ddot{\gamma} (t)\bigg)+\gamma (t)^2
\bigg(\dot{\alpha} (t) \ddot{\alpha} (t)-\alpha (t) \alpha ^{(3)}(t)\bigg)+\gamma (t)
\bigg(\dot{\alpha} (t)^2 \dot{\gamma} (t)
\\
&-\alpha (t) \dot{\alpha} (t) \ddot{\gamma} (t)-\alpha (t)
\Big(\dot{\gamma} (t) \ddot{\alpha} (t)+\alpha (t) \gamma
^{(3)}(t)\Big)\bigg)\Bigg]\Bigg\}^2 \, \Bigg/ \Bigg\{ \alpha (t)^4 \beta (t)^4 \gamma (t)^4 \Bigg\}
\end{split} \end{aligned}$$ The associated equations of motion, very complicated, yet second order, for respectively $\alpha (t) $, $ \beta (t) $ and $\gamma (t)$ are : $$\begin{aligned}
\begin{split}
~&0 = -\alpha (t)^3 \gamma (t)^3 \dot{\beta} (t)^3+\alpha (t)^3 \beta (t) \gamma (t)^3 \dot{\beta} (t)
\ddot{\beta} (t)+\alpha (t) \beta (t)^2 \gamma (t)^2 \Bigg[2 \alpha (t) \dot{\alpha} (t) \dot{\beta}(t) \dot{\gamma} (t)
\\
&+\gamma (t) \Big(\dot{\alpha} (t)^2 \dot{\beta} (t)+\alpha (t) \dot{\beta} (t)
\ddot{\alpha} (t)+\alpha (t) \dot{\alpha} (t) \ddot{\beta} (t)\Big)\Bigg]+\beta (t)^3
\Bigg[-\alpha (t)^3 \dot{\gamma} (t)^3+\gamma (t)^3 \Big(-2 \dot{\alpha} (t)^3
\\
&+3 \alpha (t)
\dot{\alpha} (t) \ddot{\alpha} (t)\Big)+\alpha (t)^3 \gamma (t) \dot{\gamma} (t) \ddot{\gamma}(t)+\alpha (t) \gamma (t)^2 \Big(\dot{\alpha} (t)^2 \dot{\gamma} (t)+\alpha (t) \dot{\gamma} (t)
\ddot{\alpha} (t)+\alpha (t) \dot{\alpha} (t) \ddot{\gamma} (t)\Big)\Bigg]
\end{split} \end{aligned}$$ $$
| 1,455
| 2,195
| 1,666
| 1,468
| null | null |
github_plus_top10pct_by_avg
|
nBufferEnd(void * pBufferContext);
void OnBufferStart(void * pBufferContext);
void OnLoopEnd(void * pBufferContext);
void OnVoiceError(void * pBufferContext, HRESULT Error);
};
Everything is fine until I try to figure out how to call back from an instance of my native callback class to the parent SoundSample object. I was thinking I could pass an instance of the SoundSample class to the SoundCallback object, but it seems like it does not allow me to declare a ref class field in the native class:
SoundCallback.h(9): error C2143: syntax error : missing ';' before '^'
SoundCallback.h(9): error C4430: missing type specifier - int assumed. Note: C++ does not support default-int
SoundCallback.h(9): error C3699: '^' : cannot use this indirection on type 'int'
I looked back at implementing callbacks in native C++ and I could not find a reasonable solution so far. What is the best/easiest way to do this?
A:
Solved it (thanks to Jeremiah Morrill) - the problem is not with any barrier blocking the use of ref classes in basic classes. C4430 means that SoundSample is an unrecognized type, which was hidden by Intellisense - since that seemed to indicate that SoundSample is known.
What needs to be added is a declaration of the SoundSample type and this all starts working fine.
I just added
namespace MyNamespace { ref class SoundSample; }
before the SoundCallback class declaration and then SoundCallback class could declare:
MyNamespace::SoundSample^ sample;
Q:
Should I prefer namespaces or classes with static functions?
In TypeScript there are two possible ways to bundle and expose a group of functions. One is by exporting a class that contains nothing but public static functions. The other is by creating a namespace and and then exporting functions from within it. As near as I can tell this produces identical behavior within TypeScript (though probably generates different JavaScript.) Is there a method that is prefered, or is it pretty much down to personal preference.
namespace MyCollection {
export f
| 1,456
| 4,493
| 247
| 1,282
| 234
| 0.816969
|
github_plus_top10pct_by_avg
|
Proj}{\mathbb{S}}$, where ${\mathbb{S}}={\mathbb{C}}[{\mathbb{C}}^{2n}][t{\mathbb{J}}^1],$ is the blowup of ${\mathbb{C}}^{2n}$ at ${\mathbb{J}}^1$ [@hai3 Proposition 3.4.2].
{#subsec-5.5}
Observe that ${\mathbb{J}}^d$ is generated by its ${{W}}$-alternating or ${{W}}$-invariant elements, respectively, depending on whether $d$ is odd or even. Following Haiman we refer to these elements as having *correct parity*.
\[corpar\] [(1)]{} For any $d\geq 0$, ${\mathbb{A}}^d$ consists of the elements of ${\mathbb{J}}^d$ with the correct parity.
[(2)]{} If ${\mathbb{C}}^n$ denotes the first copy of that space in ${\mathbb{C}}^{2n}$, then ${\mathbb{J}}^d$ is a free module over both ${\mathbb{C}}[{\mathbb{C}}^n]$ and ${\mathbb{C}}[{\mathbb{C}}^n]^{{{W}}}$.
\(1) The statement is clearly true for $d=0,1$. Assume, by induction, that it is true for $d-1$. We will suppose that $d$ is even, the argument in the odd case being similar. Since ${\mathbb{A}}^1$ generates the ideal ${\mathbb{J}}^1$, any element $x\in {\mathbb{J}}^d$ can be decomposed as $x= \sum_i p_i q_i$ where $p_i \in {\mathbb{J}}^{d-1}$ and $q_i \in {\mathbb{A}}^1$. Since $q_ie = e_-q_i$ we have $(p_iq_i)e = (p_i e_-)q_i$ for all $i$. If $x$ has the correct parity then $x = xe = \sum_i (p_iq_i)e = \sum_i (p_ie_-q_i).$ But ${\mathbb{J}}^{d-1}e_{-}$ is the subset of ${{W}}$-alternating elements of ${\mathbb{J}}^{d-1}$ and so ${\mathbb{J}}^{d-1}e_{-}={\mathbb{A}}^{d-1}$ by induction. Thus $x\in {\mathbb{A}}^{d-1} {\mathbb{A}}^1
={\mathbb{A}}^d$.
\(2) By [@hai3 Proposition 4.11.1] ${\mathbb{J}}^d$ is a projective module over ${\mathbb{C}}[{\mathbb{C}}^n]$ and hence over ${\mathbb{C}}[{\mathbb{C}}^n]^{{W}}$. Since ${\mathbb{C}}[{\mathbb{C}}^n]$ and ${\mathbb{C}}[{\mathbb{C}}^n]^{{W}}$ are polynomial rings, any such projective module is free by the Quillen-Suslin Theorem.
Geometric interpretation {#geomint}
------------------------
There is a geometric description of both ${\mathbb{A}}^d$ and ${\mathbb{J}}^d$. Let ${\mathcal{B}}_1$\[tauto-defn\] be the [*ta
| 1,457
| 497
| 1,417
| 1,449
| null | null |
github_plus_top10pct_by_avg
|
\tilde{t}_1\right)^2 + \mathrm{Re}(\tilde{t}_2) &=
0.075\pm 0.018 \,. \label{eq:result-combi}\end{aligned}$$ Employing [@Tanabashi:2018oca] $$\begin{aligned}
\mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right) = (-6.3\pm 0.3)\cdot 10^{-4}\,,\end{aligned}$$ and inserting the measurement of $\Delta a_{CP}^{\mathrm{dir}}$ into Eq. (\[eq:penguinovertree\]), we obtain $$\begin{aligned}
\mathrm{Im}\,\tilde{p}_0 &= 0.65 \pm 0.12 \,. \label{eq:resultp0tilde} \end{aligned}$$ Using $\Sigma a_{CP}^{\mathrm{dir}} $ we get $$\begin{aligned}
2 \mathrm{Im}(\tilde{p}_0 ) \tilde{s}_1 + \mathrm{Im}(\tilde{p}_1) &= 1.7\pm 1.6\,. \end{aligned}$$
Few remarks are in order regarding the numerical values we obtained.
1. Among the five parameters defined in Eq. (\[eq:def-u-par\]), $\tilde{p}_1$ is the least constrained parameter as we have basically no information about it. In order to learn more about it we need measurements of $\Sigma a_{CP}^{\mathrm{dir}}$ as well as of the phases $\delta_{KK}$ and $\delta_{\pi\pi}$.
2. The higher order U-spin breaking parameters are consistently smaller than the first order ones, and the second order ones are even smaller. This is what we expect assuming the U-spin expansion.
3. Eqs. (\[eq:result-ret1tilde\])–(\[eq:result-combi\]) suggest that the SU(3)$_F$ breaking of the tree amplitude $\tilde{t}_1$ is smaller than the broken penguin contained in $\tilde{s}_1$.
4. Using Eqs. (\[eq:result-ret1tilde\])–(\[eq:result-combi\]) we can get a rough estimate for the $\mathcal{O}(\varepsilon^2)$ corrections that enter the expression for $\Delta a_{CP}^{\mathrm{dir}}$ in Eq. (\[eq:DeltaACPdirParameter\]). The results on the broken penguin suggest that these corrections do not exceed a level of $\sim 10\%$. We cannot, however, determine these corrections completely without further knowledge on $\tilde{p}_1$.
The $\Delta U=0$ rule \[sec:deltau0rule\]
==========================================
We now turn to discuss the implications of Eq. (\[eq:resultp0tilde\]). We rewrite
| 1,458
| 1,000
| 1,166
| 1,505
| 842
| 0.79905
|
github_plus_top10pct_by_avg
|
DI`
`SRR022865_49291` `-36` `tactagaaaaga`
1542366 NONSYN A:14 G:85 G:37 `tcatgagtaaat` hypothetical protein
`ttagttacaacc`
`gi 87160966 ref` `476` `RNDMVEFFGEKL` 5-
`RN+MVEFFGEKL`
`RNEMVEFFGEKL` Methyltetrahydropteroyltriglut amate\--homocysteine S- Methyltransferase
`SRR022865_53088` `1` `cagaggttggat`
`gaattattgaat`
408863 NONSYN T:6 G:160 G:37 `ttagtaccaaaa`
`gi 87161941 ref` `177` `VDVLDVYSDAY`
| 1,459
| 2,573
| 2,883
| 1,555
| null | null |
github_plus_top10pct_by_avg
|
the effective action. In turn, a local approximation such as requires $k_{0,\rm phys}=k_{\bot,\rm phys}$. In other word, a local approximation works best if the momentum transfer in the flow is minimised. More details about such a scale matching and its connection to optimisation [@Litim:2000ci; @Pawlowski:2005xe] can be found in [@Pawlowski:2005xe]. Note in this context that in the present case we also have to deal with the subtlety that $A_0$ only depends on spatial coordinates whereas $\vec A_\bot$ is space-time dependent. However, the requirement of minimal momentum transfer in the flow is a simple criterion which is technically accessible.
More specifically we restrict ourselves to regulators [@Litim:2006ag] $$R_{A,00} = Z_0 R_{{\rm opt},k}(\vec p^2)\,,\quad
R_{A,ij} = Z_i \Pi_{\bot,ij}(\vec p)R_{\rm opt,k_\bot }(\vec p^2)\,,
\label{eq:cutoffs}$$ where [@Litim:2000ci] $$\begin{aligned}
R_{{\rm opt},k}(\vec p^2)=(k^2-\vec p^2)\theta(k^2-\vec p^2)\,.
\label{eq:opt}\end{aligned}$$ The detailed scale-matching argument is deferred to Appendix \[app:match\], and results in a relation $k_{\bot}=k_\bot(k)$ depicted in Fig. \[fig:kbotk\] in the appendix. It is left to determine the effective cut-off scale $k_{\rm phys}$. This cut-off scale can be determined from the numerical comparison of the flows of appropriate observables: one computes the flow with the three-dimensional regulator $R_{{\rm opt},k_\bot }(\vec p^2)$ in , as well as with the four-dimensional regulator $R_{{\rm opt},k_{\rm phys}}(p^2)$. Then the respective physical scales are identified, i.e. $k_{\bot,\rm phys}(k_\bot)=k_{\rm phys}$. The results for this matching procedure are depicted in Fig. \[fig:kskphys\] in Appendix \[app:match\]. Another estimate comes from the flow related to the three-dimensional $A_0$-fluctuations, where we can directly identity $k_{\rm phys}=k$. We use the above choices as limiting cases for an estimate of the systematic error in our computation. Our explicit results are obtained for the best choice that works in
| 1,460
| 723
| 2,434
| 1,407
| 1,925
| 0.784346
|
github_plus_top10pct_by_avg
|
extit{ together with } z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}.$$ Here, if $i$ is odd and $L_i$ is *free of type I*, then $$m_{i,i}=\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},$$ where $s_i\in M_{(n_i-2)\times (n_i-2)}(B\otimes_A\kappa_R)$, etc. If $i$ is odd and $L_i$ is *of type II* or *bound of type I*, then $m_{i,i}\in M_{n_i\times n_i}(B\otimes_A\kappa_R)$.
We can write $m_{i, i}=(m_{i, i})_1+\pi\cdot (m_{i, i})_2$ when $L_i$ is *of type II* or *bound of type I* and for each block of $m_{i,i}$ when $L_i$ is *free of type I*, $s_i=(s_i)_1+\pi\cdot (s_i)_2$ and so on. Here, $(s_i)_1, (s_i)_2\in M_{(n_i-2)\times (n_i-2)}(\kappa_R) \subset M_{(n_i-2)\times (n_i-2)}(B\otimes_A\kappa_R)$ when $L_i$ is *free of type I* and so on, and $\pi$ stands for $\pi\otimes 1\in B\otimes_A\kappa_R$. Note that the description of the multiplication in $\tilde{M}(\kappa_R)$ given in Section \[m\] forces $(m_{i,i})_1$ (when $L_i$ is *of type II* or *bound of type I*) and $(s_i)_1$ to be invertible.
Then $m$ maps to $$\left\{
\begin{array}{l l}
(m_{i,i})_1 & \quad \textit{if $L_i$ is \textit{of type II} or \textit{bound of type I}};\\
(s_i)_1 & \quad \textit{if $L_i$ is \textit{free of type I}}.
\end{array} \right.$$
Note that the dimension of $B_i/Y_i$, as a $\kappa$-vector space, is as follows: $$\left\{
\begin{array}{l l}
n_i & \quad \textit{if $L_i$ is \textit{of type II} or \textit{bound of type I}};\\
n_i-2 & \quad \textit{if $L_i$ is \textit{free of type I}}.
\end{array} \right.$$
\[t45\] The morphism $\varphi$ defined by $$\varphi=\prod_i \varphi_i : \tilde{G} ~ \longrightarrow ~\prod_{i:even} \mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}\times \prod_{i:odd} \mathrm{Sp}(B_i/Y_i, h_i)$$ is surjective.
Let us first prove the theorem under the assumption that $$\label{e41}
\textit{dim $\tilde{G}$ = dim $\mathrm{Ker~}\varphi$ + $\sum_{i:\mathrm{even}}$ (dim $\mathrm{O}(B_i/Z_i, \bar{q}_i)_{\mathrm{red}}$)
+ $\sum_{i:\mathrm{odd}}$ (dim
| 1,461
| 2,846
| 1,453
| 1,261
| 1,893
| 0.784635
|
github_plus_top10pct_by_avg
|
| v
Stillwater & St. Paul | w
Winona & St. Peter | x
Winona, Mankato & New Ulm | y
=====================================================================
Road abbrev. | Termini. | Miles.
-----------------+-----------------------------------------+----------
a | From La Crescent to St. Paul | 128
b | " Hastings to Glencoe | 75
c | " St. Paul to Southern State line | 127
d | " Mendota to Minneapolis | 9
e | " Austin to Lyle | 12
f | " La Crescent to southern State Line| 25
g | " Mankato to Wells | 40
h | " St. Paul to Duluth | 156
i | " Minneapolis to White Bear | 15
j | " Minneapolis to Sioux City Junction| 27
k | " Duluth to Moorhead | 253-1/2
l | " St. Paul to St. James | 121-1/4
m | " St. James to southern State line | 66-1/4
n | " St. Anthony to Breckenridge | 207
o | " St. Paul to Sauk Rapids | 76
p | " Sauk Rapids to Melrose | 35
q | " Brainerd, 4-1/2 miles south | 4-1/2
| " a point 12 miles S. of Glyndon to |
r | a point 28 miles N. | 104
| of Crookston |
s | " St. Paul to Stillwater | 17-1/2
t | " Junction to Lake St. Croix | 3-1/4
u | " Stillwater to South Stillwater | 3
v | " Grand Crossing to Winnebago Cit
| 1,462
| 4,677
| 426
| 947
| null | null |
github_plus_top10pct_by_avg
|
27 (100) 154 (97) 54 (96) 235 (98) 302 (99) 697 (98)
Median; IQR 30; 20--48 60; 30--238 115;50--300 60; 30--230 200; 81--449 240; 60--600
**Centre status**
Single centre 25 (93) 46 (29) 37 (66) 108 (45) 70 (23) 191 (27)
Multi centre---national 1 (4) 45 (29) 13 (23) 59 (25) 67 (22) 92 (13)
Multi centre---international 1 (4) 66 (42) 6 (11) 73 (30) 155 (51) 404 (57)
Unclear/missing 0 1 (1) 0 1 (\<1) 14 (5) 24 (3)
**Labelled as pilot study** 14 (52) 34 (22) 19 (34) 67 (28) 36 (12) 47 (7)
**Sponsorship**
Industry[^2^](#t001fn002){ref-type="table-fn"} 9 (33) 99 (63) 10 (18) 118 (49) 182 (59) 455 (64)
Non-industry 18 (67) 59 (37) 46 (82) 123 (51) 124 (41) 256 (36)
**Medical field**
Oncology 3 (11) 40 (25)
| 1,463
| 4,781
| 652
| 840
| null | null |
github_plus_top10pct_by_avg
|
ment similar to the paragraph just before Equation (\[ea20\]) of Step (1), if we write the $(1, 2), (1,3), (2,3), (2,2)$-blocks of the $(i, i)$-block of the formal matrix product $\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$ as $\xi^{i/2}\cdot \mathcal{X}_{i,1,2}(\tilde{m})$, $\xi^{i/2}\cdot \pi\mathcal{X}_{i,1,3}(\tilde{m})$, $\xi^{i/2}\cdot (1+\pi\mathcal{X}_{i,2,3}(\tilde{m}))$, $\xi^{i/2}\cdot (1+2 \mathcal{X}_{i,2,2}(\tilde{m}))$, respectively, where $\mathcal{X}_{i,1,2}(\tilde{m}), \mathcal{X}_{i,1,3}(\tilde{m}) \in M_{(n_i-2)\times 1}(B\otimes_AR)$ and $\mathcal{X}_{i,2,3}(\tilde{m}), \mathcal{X}_{i,2,2}(\tilde{m}) \in B\otimes_AR$, then the images of $\mathcal{X}_{i,1,2}(\tilde{m}), \mathcal{X}_{i,1,3}(\tilde{m})$ in $M_{(n_i-2)\times 1}(B\otimes_AR)/(\pi\otimes 1)M_{(n_i-2)\times 1}(B\otimes_AR)$ and the images of $\mathcal{X}_{i,2,3}(\tilde{m}), \mathcal{X}_{i,2,2}(\tilde{m})$ in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$ are independent of the choice of the lift $\tilde{m}$ of $m$. Therefore, we may denote these images by $\mathcal{X}_{i,1,2}(m)$, $\mathcal{X}_{i,1,3}(m)$, $\mathcal{X}_{i,2,3}(m)$, and $\mathcal{X}_{i,2,2}(m)$ respectively. Note that $\mathcal{X}_{i,2,2}(\tilde{m})$ is indeed contained in $R$. Thus $\mathcal{X}_{i,2,2}(\tilde{m})$ is naturally identified with $\mathcal{X}_{i,2,2}(m)$.
As for Equation (\[ea20\]) of Step (1), we need to express $\mathcal{X}_{i,1,2}(m)$, $\mathcal{X}_{i,1,3}(m)$, $\mathcal{X}_{i,2,3}(m)$, and $\mathcal{X}_{i,2,2}(m)$ as matrices. Recall that $\pi^ih_i=\xi^{i/2} \begin{pmatrix} a_i&0&0\\ 0 &1&1 \\ 0 &1 &2\bar{\gamma}_i \end{pmatrix}
=\pi^i\cdot(-1)^{i/2}\begin{pmatrix} a_i&0&0\\ 0 &1&1 \\ 0 &1 &2\bar{\gamma}_i \end{pmatrix}$. We write $m_{i,i}$ as $\begin{pmatrix} id&r_i&\pi t_i\\ \pi y_i&1+\pi x_i&\pi z_i\\ v_i&u_i&1+\pi w_i \end{pmatrix}$ and $\tilde{m}_{i,i}$ as $\begin{pmatrix} \tilde{s}_i&\tilde{r}_i&\pi \tilde{t}_i\\ \pi \tilde{y}_i&1+\pi \tilde{x}_i&\pi \tilde{z}_i\\ \tilde{v}_i&\tilde{u}_i&1+\pi \tilde{w}_i \end{pmatrix}$ such that $\tild
| 1,464
| 2,030
| 1,665
| 1,479
| null | null |
github_plus_top10pct_by_avg
|
q:hess_posl_5}\end{aligned}$$ where follows from the definition of $D _{\max}$ in Equation and follows from the definition of $\beta$ in . Observe that from Equation , ${\|M^{(j)}\|} \leq 2\delta_{j,1} \leq 2\sqrt{\delta}$. Applying matrix Bernstein inequality, we have, $$\begin{aligned}
\mathbb{P}\Big[\big\|M - \E[M]\big\| \geq t\Big] \leq d \,\exp\Bigg(\frac{-t^2/2}{\frac{e^{6b}\eta\delta}{\beta \tau d} \sum_{j = 1}^n \tau_{j}\ell_j + 4\sqrt{\delta}t/3}\Bigg). \end{aligned}$$ Therefore, with probability at least $1 - d^{-3}$, we have, $$\begin{aligned}
\label{eq:hess_posl_6}
\big\|M - \E[M]\big\| \leq 4e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} +\frac{64 \sqrt{\delta}\log d}{3} \leq 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} \;,\end{aligned}$$ where the second inequality uses $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6} (\beta \tau /\eta)d\log d$ which follows from the assumption that $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6}e^{18b} \frac{\eta\delta}{\tau\gamma^2\alpha^2 \beta } d\log d$ and the fact that $\alpha, \beta \leq 1$, $\gamma \leq 1$, $\eta\geq 1$, and $\delta> \tau^2$.
### Proof of Lemma \[lem:posl\_lowerbound\]
Since providing a lower bound on $\P_{\theta}\big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \big] $ for arbitrary $\theta$ is challenging, we construct a new set of parameters $\{\ltheta_j\}_{j\in[d]}$ from the original $\theta$. These new parameters are constructed such that it is both easy to compute the probability and also provides a lower bound on the original distribution. We denote the sum of the weights by $W \equiv \sum_{j \in S} \exp(\theta_j)$. We define a new set of parameters $\{\ltheta_j\}_{j \in S}$: $$\begin{aligned}
\ltheta_j &=& \left\{ \begin{array}{rl}
\log(\widetilde{\alpha}_{i,i',\ell,\theta}/2) &\; \text{for} \; j = i \text{ or }\i\;, \\
0&\;\text{otherwise}\;. \end{array}\right. \end{aligned}$$ Similarly define $\widetilde{W} \equiv \sum_{j \in S} \exp(\ltheta_j) = \kappa
| 1,465
| 1,830
| 1,576
| 1,307
| null | null |
github_plus_top10pct_by_avg
|
000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mMSE 1.000 1.000 1.000 1.000 1.000 1.000 1.000
BLB($n^{0.6}$) 0.884 0.894 0.848 0.896 0.864 0.872 0.880
BLB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SDB($n^{0.6}$) 0.910 0.900 0.910 0.908 0.876 0.900 0.884
SDB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mMSE 1.000 1.000 1.000 1.000 1.000 1.000 1.000
BLB($n^{0.6}$) 0.464 0.472 0.428 0.458 0.480 0.478 0.488
BLB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SDB($n^{0.6}$) 0.006 0.010 0.006 0.008 0.010 0.004 0.008
SDB($n^{0.8}$) 1.000 1.000 1.000 1.000 1.000 1.000 1.000
| 1,466
| 5,945
| 533
| 714
| null | null |
github_plus_top10pct_by_avg
|
le}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}{\nonumber}\\
&~\leq\sum_{v_0}\big(\psi_\Lambda(v_0,v)-\delta_{v_0,v}\big)\sum_{u\in{{\cal A}}}
\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_0}.\end{aligned}$$ Following the same argument as in [(\[eq:Theta”-1stind1stcontr\])]{}–[(\[eq:Theta”-bd1stbd1\])]{}, we obtain $$\begin{aligned}
{\label{eq:Theta''-bd1stbd2}}
{(\ref{eq:Theta''-prebd1stbd2})}&\leq\sum_{u\in{{\cal A}},\;v_0}\big(\psi_\Lambda(v_0,
v)-\delta_{v_0,v}\big)\,{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_{v_0} \rangle}}_\Lambda{{\langle \varphi_{v_0}\varphi_x \rangle}}_\Lambda{\nonumber}\\
&\leq\sum_{u\in{{\cal A}}}\Big(P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)-
{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda
{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_v \rangle}}_\Lambda
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda\Big).\end{aligned}$$
Summarizing [(\[eq:Theta”-1stindbd\])]{}, [(\[eq:Theta”-bd1stbd1\])]{} and [(\[eq:Theta”-bd1stbd2\])]{}, we arrive at $$\begin{aligned}
{\label{eq:Theta''-0bdfin}}
{(\ref{eq:contr-(b)})}\leq\sum_{u\in{{\cal A}}}P_{\Lambda;u,v}^{\prime\prime
{{\scriptscriptstyle}(0)}}(y,x).\end{aligned}$$ This completes the bound
| 1,467
| 675
| 1,247
| 1,582
| null | null |
github_plus_top10pct_by_avg
|
s, E_{\tau^{-1}(s)})$. Then the following lemma holds.
\[lem:defect\] Let ${\mathbb M} = (M_1, \ldots, M_u)$ and ${\mathbb F} = (F_1, \ldots, F_v)$ be sequences of the upper and the lower (non-empty) parts of a seat-plan respectively. If $M_i$ \[resp. $F_i$\] is defective and $\sigma_i = (i,i+1)$, the $i$-th adjacent transposition, then the crank form expression ${\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ is moved to another crank form expression ${\cal C}(\sigma_i\mathbb{M}, \sigma, \mathbb{F})$ \[resp. ${\cal C}(\mathbb{M}, \sigma, \sigma_i\mathbb{F})$ \].
We consider the case $M_i$ is defective. In case $F_i$ is defective, the similar proof will hold. Let $P_{\mathbb{M},i}\in{\mathfrak S}_n$ be a permutation defined by $$P_{\mathbb{M},i}(x):=
\left\{
\begin{array}{ll}
x + |M_{i+1}|&
\mbox{if}\ \sum_{j=1}^{i-1}|M_{j}| <x \leq \sum_{j=1}^{i}|M_{j}|,\\
x - |M_{i}|&
\mbox{if}\ \sum_{j=1}^{i}|M_{j}| <x \leq \sum_{j=1}^{i+1}|M_{j}|,\\
x&
\mbox{otherwise}.
\end{array}
\right.$$ Then we find that $x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i}$ maps $j$ to the $j$-th coordinate of $\overline{\sigma_i\mathbb{M}}$. Hence we have $x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i}
= x_{\overline{\sigma_i\mathbb{M}}}$. (For the definition of $\overline{\mathbb{M}}$, see Section 3.2.)
On the other hand, since $M_i$ is defective, we have $P_{\mathbb{M},i}C[\mathbb{M}] = C[\sigma_i\mathbb{M}]$ by removing an excrescence of $M_i$ and iteratively using “defective part exchange” ($R17'$) in Figure \[fig:dpeE\] (if $M_{i+1}$ is defective) or iteratively using “defective part shift” ($R16'$) in Figure \[fig:dpsE\] (if $M_{i+1}$ is propagating), and then adding an excrescence to $M_i$ just moved. Thus we obtain $$\begin{aligned}
{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
&=& x_{\overline{\mathbb{M}}}
C[\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& (x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i})
(P_{\mathbb{M},i}C[\mathb
| 1,468
| 2,032
| 1,496
| 1,383
| null | null |
github_plus_top10pct_by_avg
|
-3\right)}{8 \left(u^2-1\right) \left(u^2+1\right)^3} \\
\mathcal{D}_{uu} & \frac{-\left(u^8+8 u^6+10 u^4-3\right) m^2+4 h \left(u^2-1\right) \left(u^2+1\right)^2+16 u^2 \left(u^2-1\right)}{8 \left(u^2-1\right)^2 \left(u^2+1\right)^3} & -\frac{(h+1) u}{u^4-1} \\
\mathcal{D}_{TR} & \frac{i m \left(u^4+6 u^2-3\right)}{4 \left(u^2+1\right)^3} & -\frac{i m u \left(u^2-3\right)}{\left(u^2+1\right)^2} \\
\mathcal{D}_{Tu} & -\frac{i m u \left(u^2-3\right)}{2 \left(u^2-1\right) \left(u^2+1\right)^2} & \frac{i m \left(u^4+6 u^2-3\right)}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi R} & \frac{i m \left(u^4+4 u^2-1\right)}{2 \left(u^2+1\right)^3} & -\frac{i m u \left(u^2-1\right)}{\left(u^2+1\right)^2} \\
\mathcal{D}_{\Phi u} & \frac{i m u}{2 \left(u^4-1\right)} & \frac{i m \left(u^4+6 u^2+h \left(u^2+1\right)^2-3\right)}{2 \left(u^2+1\right)^3} \\
\noalign{\bigskip}
\text{} & C_{uu}(u) & C_{TR}(u) \\
\noalign{\smallskip}
\hline \hline \noalign{\smallskip}
\mathcal{D}_{TT} & \frac{4 h^2 \left(u^6+5 u^4-9 u^2+3\right) \left(u^2+1\right)^2+m^2 \left(u^6+7 u^4+3 u^2-3\right)^2+8 \left(5 u^8+34 u^6-68 u^4+54 u^2-9\right)}{8 \left(u^2+1\right)^5} & \frac{i (2 h+3) m \left(u^4+6 u^2-3\right)}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{T\Phi} & \frac{\left(u^2-1\right) \left(\left(u^8+8 u^6+10 u^4-3\right) m^2+4 h^2 \left(u^2-1\right) \left(u^2+1\right)^2+2 h \left(u^2-1\right) \left(u^2+1\right)^2+8 \left(u^6+8 u^4-11 u^2+2\right)\right)}{2 \left(u^2+1\right)^5} & \frac{i m \left(2 \left(u^4+8 u^2-5\right)+h \left(u^4+10 u^2-7\right)\right)}{2 \left(u^2+1\right)^3} \\
\mathcal{D}_{\Phi \Phi } & \frac{2 \left(u^2-1\right)^2 \left(h^2 \left(u^2+1\right)^2+m^2 \left(u^2+1\right)^2+h \left(u^2+1\right)^2+2 \left(u^4+9 u^2-2\right)\right)}{\left(u^2+1\right)^5} & \frac{2 i (2 h+3) m \left(u^2-1\right)}{\left(u^2+1\right)^3} \\
\mathcal{D}_{RR} & -\frac{\left(u^8+8 u^6+10 u^4-3\right) m^2+4 h \left(u^2-1\right) \left(u^2+1\right)^2+8 \left(u^4+4 u^2-1\right)}{8 \left(u^2+1\right)^3} & -\frac{i m}{2 \left(u^2+1\right)} \
| 1,469
| 2,744
| 1,445
| 1,407
| null | null |
github_plus_top10pct_by_avg
|
his frame) for $V_p=1.6$ and $\gamma=0$. []{data-label="fig:BufferRegion"}](Figure_4.pdf){width="45.00000%"}
Numerical simulations of dGPE Eq. (\[eq:ComovingdGPE\]) are run for a system size of $128\times 256$ (in units of $\xi$) corresponding to the grid size $dx=0.25\xi$, and $dt=0.01\xi/c$. To simulate an infinite domain where the density variations emitted by the impurity do not recirculate under periodic boundary conditions, we use the fringe method from [@reeves2015identifying]. This means that we define buffer (fringe) regions around the outer rim of the computational domain (see Fig. \[fig:BufferRegion\]) where the thermal drag $\gamma$ is much larger than its value inside the domain, such that any density perturbation far from the impurity is quickly damped out and a steady inflow is maintained. The thermal drag becomes thus spatially-dependent and given by $\gamma({\boldsymbol{r}}) = \max[\gamma(x),\gamma(y)]$, where $$\begin{aligned}
&\gamma(x)= \frac{1}{2}\big(2 + \tanh{[(x-x_p-w_x)/d]}\nonumber\\
&-\tanh{[(x-x_p+w_x)/d]}\big) + \gamma_0,\end{aligned}$$ and similarly for $\gamma(y)$. Here ${\boldsymbol{r}}_p = (x_p,y_p)=
(128\xi,64\xi)$ is the position of the impurity and $\gamma_0$ is the constant thermal drag inside the buffer regions (bulk region). We set the fringe domain as $w_x=100\xi$, $w_y=50\xi$ and $d=7\xi$ as illustrated in Figure \[fig:BufferRegion\].
By separating the linear and non-linear terms in Eq. (\[eq:ComovingdGPE\]), we can write the dGPE formally as [@audunsthesis] $$\partial_t \psi = \hat\omega(-i\nabla)\psi + N({\boldsymbol{r}},t),$$ where $\hat\omega(-i\nabla) = i[\frac 1 2 \nabla^2+1]+{\boldsymbol{V}}_p\cdot \nabla$ is the linear differential operator and $N({\boldsymbol{r}},t)=-(i+\gamma)(\mathcal U_p+|\psi|^2)\psi + \gamma
\psi +\frac{1}{2}\gamma\nabla^2\psi$ is the nonlinear function including the spatially-dependent $\gamma$ and $\mathcal U_p$. Taking the Fourier transform, we obtain ordinary differential equations for Fourier coefficients $\psi({\boldsymbol{k}},t)$ as
| 1,470
| 1,111
| 2,362
| 1,609
| 1,185
| 0.792978
|
github_plus_top10pct_by_avg
|
ng the lines of proof of Theorem 1.8 in [@hayes2005large].
### Proof of Lemma \[lem:hessian\_positionl\]
The Hessian $H(\theta)$ is given in . For all $j\in [n]$, define $M^{(j)} \in \cS^d$ as $$\begin{aligned}
\label{eq:posl_M_j_def}
M^{(j)} &\equiv& \sum_{a=1}^{\ell_j} \lambda_{j,a} \sum_{i<\i \in S_j} \I_{\big\{(i,\i)\; \in \; G_{j,a}\big\}} (e_i - e_{\i})(e_i - e_{\i})^\top,\end{aligned}$$ and let $M \equiv \sum_{j=1}^n M^{(j)}$. Observe that $M$ is positive semi-definite and the smallest eigenvalue of $M$ is zero with the corresponding eigenvector given by the all-ones vector. If $|\theta_i| \leq b$, for all $i \in [d]$, $\frac{\exp(\theta_i + \theta_{\i})}{[\exp(\theta_i) + \exp(\theta_{\i})]^2} \geq \frac{e^{2b}}{(1+ e^{2b})^2}$. Recall the definition of $H(\theta)$ from Equation . It follows that $-H(\theta) \succeq \frac{e^{2b}}{(1+ e^{2b})^2} M$ for $\theta \in \Omega_b$. Since, $-H(\theta)$ and $M$ are symmetric matrices, from Weyl’s inequality we have, $\lambda_2(-H(\theta)) \geq \frac{e^{2b}}{(1+ e^{2b})^2} \lambda_2(M)$. Again from Weyl’s inequality, it follows that $$\begin{aligned}
\lambda_2(M) &\geq& \lambda_2(\E[M]) - {\|M-\E[M]\|} \;, \end{aligned}$$ where $\|\cdot\|$ denotes the spectral norm. We will show in that $\lambda_2(\E[M]) \geq 2 \gamma e^{-6b} (\alpha/(d-1)) \sum_{j = 1}^n \tau_j\ell_j$, and in that ${\|M-\E[M]\|}\leq 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} $.
$$\begin{aligned}
\label{eq:lambda2_M}
\lambda_2(M) &\geq& \frac{2e^{-6b} \alpha \gamma}{d-1} \sum_{j = 1}^n \tau_j\ell_j - 8e^{3b}\sqrt{\frac{\eta\delta\log d}{\beta \tau d}\sum_{j = 1}^n \tau_j \ell_j} \;\geq\; \frac{e^{-6b} \alpha \gamma}{d-1} \sum_{j = 1}^n \tau_j\ell_j \;, \end{aligned}$$
where the last inequality follows from the assumption that $\sum_{j = 1}^n \tau_j \ell_j \geq 2^{6}e^{18b} \frac{\eta\delta}{\alpha^2\beta \gamma^2\tau} d\log d$. This proves the desired claim. To prove the lower bound on $\lambda_2(\E[M])$, notice that $$\begin{aligned}
\label{e
| 1,471
| 1,136
| 1,132
| 1,364
| null | null |
github_plus_top10pct_by_avg
|
Moreover, for the groups $N$ listed in Table 2, there exist a prime $s \in \pi(N) \setminus (\pi({{\operatorname}{\textup{Out}}({N})}) \cup \{t\})$ and a Sylow $s$-subgroup of order $s$ which is self-centralising in $N$.
If $N=L_2(q)$, $C_N(x)$ is a $t$-group for each $t$-element $x \in N$.
If $N=L_3(q)$, there exists a maximal torus $T$ of order $(1/d) (q^2+q+1)$, $d=(3, q-1)$, such that each prime $r \in \pi(T)$ is a primitive prime divisor of $q^3-1$ (for $q\neq 4$), and $(|T|, 2t)=1$.
If $N=U_3(q)$, there exists a maximal torus $T$ of order $(1/d) (q^2-q+1)$, $d=(3, q+1)$, such that each prime $r \in \pi(T)$ is a primitive prime divisor of $q^6-1$, and $(|T|, 2t)=1$.
$$\begin{array}{c|c|c|c|c|c}
\hline
& & & & & \\
N& r &s& |T_1|& |T_2| & Remarks \\
& & & & & \\
\hline
& & & & & \\
L_n(q)& q_n &q_{n-1} & \frac{q^n-1}{(n, q-1)(q-1)}&\frac{q^{n-1}-1}{(n, q-1)}& (n, q) \neq (6, 2)\\
n \geq 4 & & & & & (n, q) \neq (4, 4), (7, 2)\\
&& s=7&&&(n,q)=(4,4)\\
& & & & & \\
\hline
& & & & & \\
U_{n}(q) & q_{n} & q_{2(n-1)} &\frac{q^{n}-1}{(n, q+1)(q+1)}& \frac{(q^{n-1}+1)}{(n, q+1)}& n \mbox{ even } \\
& & & & & (n, q) \neq (4, 2), (6, 2) \\
& & & & & \\
n\geq 4 & q_{2n} & q_{n-1} & \frac{q^{n}+1}{(n, q+1)(q+1)} & \frac{q^{n-1}-1}{(n, q+1)}& n \mbox{ odd} \\
& & & & & (n, q) \neq (7, 2) \\
& & & & & \\
\hline
& & & & & \\
PSp_{4}(q) & q_{4} & q_{2} & \frac{q^{2}+1}{(2, q-1)}& \frac{(q^{2}-1)}{(2, q-1)}& q\neq 8, l \quad (\star) \\
&& s=7&&& q=8\\
&& s \neq 2&&& q=l \\
& & & & & \\
\hline
& & & & & \\
PSp_{2n}(q) & q_{2n} & q_{2(n-1)} & \frac{q^{n}+1}{(2, q-1)}& \frac{(q^{n-1}+1)(q-1)}{(2, q-1)}& n \mbox{ even } \\
& && & & (n, q) \neq (4,2) \\
P\Omega_{2n+1}(q) & & & & & \\
& q_{2n} & q_{n} & \frac{q^{n}+1}{(2, q-1)} & \frac{(q^{n}-1)}{(2, q-1)}& n \mbox{ odd } \\
n\geq 3 & & & & & (n, q) \neq (3, 2) \\
& & & & & \\
\hline
& & & & & \\
P\Omega_{2n}^{-}(q) & q_{2n} & q_{2(n-1)} & \frac{q^{n}+1}{(4, q^n+1)}& \frac{(q^{n-1}+1)(q-1)}{(4, q^n+1)}& (n, q) \neq (4, 2) \\
n \geq
| 1,472
| 1,962
| 1,755
| 1,439
| null | null |
github_plus_top10pct_by_avg
|
as “T” here. This form of threshold can also be applied to other types of distributions even there is no intersection between these two.[]{data-label="fig:example-threshold"}](./ExampleThreshold.pdf){width="\linewidth"}
Moreover, as is verified in experiments, the meta-distribution of anomalous data collections lies in the right-hand-side to the normal one on the real line. As shown in Fig \[fig:example-threshold\], the black curve ($PDF_{normal}(x)$ or in short $PDF_n(x)$) displays the probability density function (PDF) fitting those divergences calculated from normal data collections; the blue curve ($PDF_{anomalous}(x)$ or in short $PDF_a(x)$) displays the PDF derived from anomalous data collections. Threshold is chosen to minimize total errors(both false negative and false positive).
Suppose: $$\begin{aligned}
PDF_n(x) &\approx \mathcal{N}(\mu_n, \sigma_n)\\
PDF_a(x) &\approx \mathcal{N}(\mu_a, \sigma_a)
\end{aligned}$$
Then the optimal threshold $T$ is calculated by E.q.(\[equ:equal-weight\]). The optimal threshold will minimize total errors and yield an optimal outcome. However, this is not accurate enough, since E.q.(\[equ:equal-weight\]) implicates an assumption that chances are the same for a new data collection to be either anomalous or not. If we can determine the probability for a new data collection to be anomalous in any segment of data sequence, the equation should be modified as E.q.(\[equ:linear-weight\]), where $\alpha$ is the anomaly probability.
$$\begin{aligned}
\label{equ:equal-weight}
T &= \mathop{\arg\min}_{T} \int_{0}^{T}PDF_{a}(x)dx +
\int_{T}^{\sup(D)}PDF_{n}(x)dx \nonumber\\
& \approx \mathop{\arg\min}_{T}
\int_{-\infty}^{T}
\frac{e^{-\frac{(x - \mu_a)^2}{2\sigma_a^2}}}{\sqrt{2\pi} \sigma_a}dx
+ \int_{T}^{+\infty}
\frac{e^{-\frac{(x - \mu_n)^2}{2\sigma_n^2}}}{\sqrt{2\pi} \sigma_n}dx \nonumber\\
| 1,473
| 6,009
| 1,759
| 624
| 2,716
| 0.777622
|
github_plus_top10pct_by_avg
|
pond to roots with zero and negative squared length respectively [@Kleinschmidt:2003mf]. In [@axel] it was shown that only the $E_{11}$ roots with positive squared length are associated to branes.
The constraint corresponding to $E_{9,3}$ is $$Q^{[ab}_e P^{c]e}_d+ P^{[ab}_e Q^{c]e}_d=0 \quad ,$$ and has already been proposed in [@Aldazabal:2006up]. In components one gets $$\begin{aligned}
& -b_{jj}\bar{g}_{jk}+\bar{b}_{kj}g_{kk}+h_j\bar{f}_k-b_{ij}\bar{g}_{ik}-g_{jj}\bar{b}_{jk}+f_j\bar{h}_k+\bar{g}_{kj}b_{kk}-g_{ij}\bar{b}_{ik}=0 \nonumber \\
& \bar{b}_{ij}g_{ik}-\bar{h}_jf_k-b_{kj}\bar{g}_{kk}+\bar{b}_{jj}g_{jk}+\bar{g}_{ij}b_{ik}-g_{kj}\bar{b}_{kk}-\bar{f}_jh_k+\bar{g}_{jj}b_{jk}=0\nonumber \\
& -b_{jj}g_{ik}+\bar{b}_{kj}f_k+h_j\bar{g}_{kk}-b_{ij}g_{jk}-g_{jj}b_{ik}+f_j\bar{b}_{kk}+\bar{g}_{kj}h_k-g_{ij}b_{jk}=0\nonumber \\
& \bar{b}_{ij}\bar{g}_{jk}-\bar{h}_jg_{kk}-b_{kj}\bar{f}_k+\bar{b}_{jj}\bar{g}_{ik}+\bar{g}_{ij}\bar{b}_{jk}-g_{kj}\bar{h}_k-\bar{f}_jb_{kk}+\bar{g}_{jj}\bar{b}_{ik}=0 \quad .\end{aligned}$$ From the multiplicity analysis of $E_{11}$ one can show that there are actually three independent $E_{10,2}$ potentials [@Kleinschmidt:2003mf]: with respect to the $SL(2,\mathbb{R})$ symmetry of the IIB theory, one belongs to the triplet that also contains $D_{10,2}$, while the other two are singlets. If only $P_1^2$ and $Q$ fluxes are turned on, the constraint arising from the triplet vanishes, while from the two singlets one gets $$Q^{e[a}_f P^{b]f}_e =0 \quad ,$$ which in components gives $$\begin{aligned}
& -b_{kk}\bar{g}_{kj}+h_k\bar{f}_j+\bar{b}_{ik}g_{ij}-b_{jk}\bar{g}_{jj}-b_{jj}\bar{g}_{jk}+h_j\bar{f}_k+\bar{b}_{ij}g_{ik}-b_{kj}\bar{g}_{kk} \nonumber \\
& + \bar{b}_{jk}g_{jj}-b_{ik}\bar{g}_{ij}-\bar{h}_kf_j+\bar{b}_{kk}g_{kj}+\bar{b}_{kj}g_{kk}-b_{ij}\bar{g}_{ik}-\bar{h}_jf_k+\bar{b}_{jj}g_{jk}=0 \quad .\end{aligned}$$
One can solve the whole set of equations that we have determined. In particular, in the isotropic case and without localised sources, a simple solution is $$g=\bar{g}=\gamma=\b
| 1,474
| 523
| 1,794
| 1,533
| null | null |
github_plus_top10pct_by_avg
|
-------- ----------------------- ---------
Discharged patients
D2P AED 51 (37, 68) 29 (21, 41) \<0.01
CED 58 (38, 77) 40 (26, 59) \<0.01
LOSD AED 289 (257, 320) 261 (238, 297) \<0.01
CED 241 (219, 264) 232.5 (209, 265) 0.01
Admitted patients
A2D AED 97.5 (84.5, 116) 111.0 (93, 144.5) \<0.01
CED 77 (66,92) 106 (80,140) \<0.01
*ED*, emergency department; *AED*, tertiary care academic emergency department; *CED*, community emergency department; *D2P*, arrival to being seen by physician; *LOSD*, total length of stay for discharged patients; *A2D*, admit request to departure for boarded patients awaiting hospital admission; *PIT*, physician in triage.
######
Quartile regression models examining the effect of boarding on median door-to-provider time and median discharged patient length of stay (in minutes).
Parameter AED CED
------------------------ ------------------------- ------ ---------------- ------------------------- ------ ----------------
D2P
Intercept 36.84 (33.18, 40.50) 1.87
| 1,475
| 354
| 1,218
| 1,825
| null | null |
github_plus_top10pct_by_avg
|
ion $$\begin{aligned}
\label{eq:w*functor}
(ww')^*\chi =w^*(w'{}^*\chi )\end{aligned}$$ holds for all $w,w'\in {\mathrm{Aut}}_{{\mathbb{Z}}}({\mathbb{Z}}^I)$ and all $\chi \in {\mathcal{X}}$.
\[de:Cartan\] Let $\chi \in {\mathcal{X}}$, $p\in I$, and $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. We say that $\chi $ is $p$-*finite*, if for all $j\in I$ there exists $m\in {\mathbb{N}}_0$ such that $\qnum{m+1}{q_{pp}}=0$ or $q_{pp}^m q_{pj}q_{jp}=1$.
Assume that $\chi $ is $p$-finite. Let $c_{p p}^\chi =2$, and for all $j\in I\setminus \{p\}$ let $$c_{pj}^\chi =-\min \{m\in {\mathbb{N}}_0 \,|\,
(m+1)_{q_{pp}}(q_{pp}^m q_{pj} q_{jp}-1)=0\}.$$ If $\chi $ is $i$-finite for all $i\in I$, then the matrix $C^\chi =(c_{ij}^\chi )_{i,j\in I}$ is called the *Cartan matrix* associated to $\chi $. It is a generalized Cartan matrix, see Sect. \[ssec:CS\].
For all $p\in I$ and $\chi \in {\mathcal{X}}$, where $\chi $ is $p$-finite, let ${\sigma }_p^\chi \in {\mathrm{Aut}}_{\mathbb{Z}}({\mathbb{Z}}^I)$, $$\begin{aligned}
{\sigma }_p^\chi ({\alpha }_j)={\alpha }_j-c_{pj}^\chi {\alpha }_p
\quad \text{for all $j\in I$.}\end{aligned}$$ Towards the definition of the Weyl groupoid of a bicharacter, we define bijections $r_p:{\mathcal{X}}\to {\mathcal{X}}$ for all $p\in I$. Namely, let $$\begin{aligned}
r_p: {\mathcal{X}}\to \cX,\quad
r_p(\chi )=
\begin{cases}
({\sigma }_p^\chi )^*\chi & \text{if $\chi $ is $p$-finite,}\\
\chi & \text{otherwise.}
\end{cases}\end{aligned}$$ Let $p\in I$, $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$. If $\chi $ is $p$-finite, then $$\begin{aligned}
r_p(\chi )({\alpha }_p,{\alpha }_p)=&q_{p p}, &
r_p(\chi )({\alpha }_p,{\alpha }_j)=&q_{p j}^{-1}q_{p p}^{c_{pj}^\chi },\\
r_p(\chi )({\alpha }_i,{\alpha }_p)=&q_{i p}^{-1}q_{p p}^{c_{pi}^\chi },&
r_p(\chi )({\alpha }_i,{\alpha }_j)=&q_{i j} q_{i p}^{-c_{p j}^\chi }
q_{p j}^{-c_{p i}^\chi } q_{p p}^{c_{pi}^\chi c_{p j}^\chi }
\end{aligned}
\label{eq:r
| 1,476
| 2,267
| 1,650
| 1,394
| null | null |
github_plus_top10pct_by_avg
|
representation in a probability space.
The rest of this paper is organized as follows. In Section 2, we present our results on the law of large numbers. Section 3 contains the results related to the CLT. A new representation of the $G$-normal distribution is derived in Section 4. Most of the proofs are deferred to Section 5.
Law of large numbers
====================
In this section, we first provide a rate of convergence for Peng’s law of large numbers, then discuss its implication on the statistical inference for uncertain distributions, and finally, we present a new law of large numbers with rates that may be of independent interest.
Rate of convergence
-------------------
Let $\{X_i\}_{i=1}^\infty$ be an i.i.d. sequence of random variables under a sublinear expectation $\E$ such that for a family of linear expectations $\{E_\theta: \theta\in \Theta\}$. Suppose both $\overline{\mu}=\E[X_1]\ \text{and} \ \underline{\mu}=- \E [-X_1]$ are finite. Define If $\overline{\sigma}^2$ is finite, then we can control the expected deviation of the sample mean $\overline{X}_n=\sum_{i=1}^n X_i/n$ from the interval $[\underline{\mu}, \overline{\mu}]$.
\[t7\] Under the above setting, we have
We can rewrite [1001]{} as where for $A\subset \mathbb{R}^d$ and $x\in \mathbb{R}^d$, $d_A(x):=\inf_{y\in A}|y-x|$. Clearly, for any interval $I$ larger than $[\underline{\mu}, \overline{\mu}]$, i.e., $[\underline{\mu}, \overline{\mu}]\subset I$, the conclusion of Theorem \[t7\] still holds for $d^2_{I}(\overline{X}_n)$. In fact, $[\underline{\mu}, \overline{\mu}]$ is the smallest interval satisfying Theorem \[t7\]. According to (\[001\]), if $[\underline{\nu},\overline{\nu}]\nsupseteq [\underline{\mu},\overline{\mu}]$, then $$\lim_{n\to \infty}\mathbb{E}\big[d_{[\underline{\nu},\overline{\nu}]}(\overline{X}_n)\big]=\sup_{x\in[\underline{\mu}, \overline{\mu}]}d_{[\underline{\nu},\overline{\nu}]}(x)>0.$$
[001]{} presents a law of large numbers under sublinear expectations where the convergence is in the distribution. In fact,
| 1,477
| 546
| 595
| 1,637
| null | null |
github_plus_top10pct_by_avg
|
\hat{\mathbf{\tau}}}^{2}$
-------------------------------------------------- -- ------------------------------------------------------- -------- -- -------- -------- -- --------- -------- -- --------- --------
Scenario [\*](#sim7930-note-0003){ref-type="fn"}
**Base Case** −100.00 −16.86 −41.50 −15.85 −100.00 −14.36 −56.17 −32.86
**A1** −100.00 −36.88 −80.33 −33.10 −100.00 −73.62 −100.00 −80.15
**A2** −100.00 −8.59 −20.86 −7.74 −100.00 −13.96 −39.78 −25.06
**B1** −49.64 −10.74 −22.94 −10.09 −100.00 −14.23 −28.91 −16.39
**B2** −56.93 −18.03 −35.11 −17.84 −72.90 −17.92 −35.95 −19.81
**B1‐A1** −77.28 −19.57 −42.62 −18.45 −100.00 −27.27 −54.23 −30.91
**B1‐A2** −40.64 −5.83 −13.08 −6.31 −88.22 −9.50 −19.59 −13.53
**B2‐A1** −72.73 −28.35 −56.23 −28.10 −100.00 −34.30 −64
| 1,478
| 4,929
| 1,745
| 911
| null | null |
github_plus_top10pct_by_avg
|
the direction of the root edge of the cubic graph. The cubic graph itself and its “simplification” are presented in the figure below. $$\begin{picture}(300,100) \linethickness{0.6mm} \put(0,30){\line(1,0){80}}
\put(60,30){\line(0,1){10}} \thinlines \put(20,30){\line(0,1){10}}
\put(20,30){\oval(40,40)[b]} \put(50,30){\oval(100,100)[t]}
\put(40,40){\oval(40,40)[t]} \put(60,30){\oval(40,20)[tr]}
\put(90,30){\oval(20,20)[b]} \put(78,23){$\downarrow$}
\put(140,50){$\Rightarrow$} \linethickness{0.6mm}
\put(200,20){\line(0,1){60}} \put(200,20){\line(1,0){80}}
\put(280,20){\line(0,1){30}} \qbezier(280,20)(310,50)(280,80)
\thinlines \qbezier(200,20)(170,50)(200,80)
\put(200,80){\line(1,0){80}} \put(200,50){\line(1,0){80}}
\put(280,50){\line(0,1){30}} \put(280,80){\vector(-1,0){15}}
\end{picture}$$
We can draw above mentioned arcs in such way, that they do not intersect in the exterior of $P$.
Let us connect midpoints of all sides in pairs by segments inside $P$. As the identification of sides in pairs generates a genus zero curve, then these segments do not intersect. The polygon $P$ is embedded into sphere, so we can interchange its interior and exterior domains.
Main statement
==============
The number of tree-rooted cubic maps with $2n$ vertices and a marked edge, that does not belong to the spanning tree, is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
Our theorem follows from two statements.
1. A convex $n$-gon with a marked side can be divided into triangles by non-intersecting diagonals in $C_{n-2}$ ways [@St].
2. There are $C_n$ ways to define a pairwise identification of sides of a convex $2n$-gon with a marked side to obtain a genus 0 curve [@LZ].
According to theorem, we have $C_4\cdot C_3=70$ tree-rooted cubic maps with $4$ vertices. In what follows a map with a marked spanning tree will be called *t-map*. The first cubic map in Figure 1 generates six t-maps. $$\begin{picture}(330,50) \qbezier[20](0,25)(0,45)(20,45)
\qbezier[20](0,25)(0,5)(20,5) \qbezier[20](60,25)(60,40)(75,40)
\
| 1,479
| 4,333
| 866
| 1,088
| 1,991
| 0.783777
|
github_plus_top10pct_by_avg
|
-------------------------------------------------------------------------
The third order in $H_{1}$ contribution to order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j}$ is given by $$\begin{aligned}
&& \hat{S}_{ij}^{(4)} \vert_{i \neq j} [3]
\nonumber \\
&=&
- \sum_{K}
(ix) e^{- i \Delta_{K} x}
\frac{ 1 }{ \Delta_{K} - h_{i} } \frac{ 1 }{ \Delta_{K} - h_{j} }
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A W \right\}_{K K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}
\nonumber \\
&+&
\sum_{K}
\frac{ 1 }{ ( h_{j} - h_{i} ) ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} )^2 }
\nonumber \\
&\times&
\biggl[
( h_{j} - h_{i} ) ( h_{j} + h_{i} - 2 \Delta_{K} ) e^{- i \Delta_{K} x} +
( \Delta_{K} - h_{i} )^2 e^{ - i h_{j} x} -
( \Delta_{K} - h_{j} )^2 e^{ - i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A W \right\}_{K K}
\left\{ W ^{\dagger} A (UX) \right\}_{K j}
\nonumber \\
&+&
\sum_{K \neq L}
\frac{ 1 }{ ( h_{j} - h_{i} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) }
\nonumber \\
&\times&
\biggl[
( h_{j} - h_{i} )
\biggl\{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) e^{- i \Delta_{L} x} -
( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) e^{- i \Delta_{K} x} \biggr\}
\nonumber \\
&+&
( \Delta_{L} - \Delta_{K} )
\biggl\{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) e^{- i h_{j} x} -
( \Delta_{K} - h_{j} ) ( \Delta_{L} - h_{j} ) e^{- i h_{i} x} \biggr\}
\biggl]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A W \right\}_{K L}
\left\{ W ^{\dagger} A (UX) \right\}_{L j}.
\label{hatS-3rd-order-ij-3}\end{aligned}$$ For bookkeeping purpose we decompose the fourth order in $H_{1}$ contribution to order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j}$ into the two terms $$\begin{aligned}
\hat{S}_{ij}^{(4)} [4] \vert_{i \neq j} =
\hat{S}_{ij}^{(4)} [4]
| 1,480
| 3,229
| 2,051
| 1,515
| null | null |
github_plus_top10pct_by_avg
|
}$ and $\bar{\beta}^i$ do not appear in the one-hypersurface IVP for $(\Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$.
Turning now to the conformal metrics in the IVP, we recall that two metrics $g_{i j}$ and $\bar{g}_{i j}$ are conformally equivalent if and only if there is a scalar $\psi > 0$ such that $\bar{g}_{i j} = \psi^4 g_{i j}$. The conformally invariant representative of the entire conformal equivalence class, in three dimensions, is the weight $(-2/3)$ unit-determinant “conformal metric” $\hat{g}_{i j}=\bar{g}^{-1/3} \bar{g}_{i j}=g^{-1/3} g_{i j}$ with $\bar{g}=\det(\bar{g}_{i j})$ and $g=(\det g_{i j})$. Note particularly that for any small perturbation, $\bar{g}^{i j} \delta \hat{g}_{i j}=0$. We will use the important relation $$\bar{g}^{i j} \partial_t \hat{g}_{i j} = g^{i j} \partial_t \hat{g}_{i j}
= \hat{g}^{i j} \partial_t \hat{g}_{i j} = 0\;.
\label{Eq:ggdot}$$
In the following, rather than use the mathematical apparatus associated with conformally weighted objects such as $\hat{g}_{i j}$, we find it simpler to use ordinary scalars and tensors to the same effect. Thus, let the role of $\hat{g}_{i j}$ on the first surface be played by a given metric $g_{i j}$ such that the physical metric that satisfies the constraints is $\bar{g}_{i j} = \psi^4 g_{i j}$ for some scalar $\psi > 0$. (This corresponds to “dressing” the initial unimodular conformal metric $\hat{g}_{i j}$ with the correct determinant factor $\bar{g}^{1/3} = \psi^4 g^{1/3}$. This process does not alter the conformal equivalence class of the metric.) The role of the conformal metric on the second surface is played by the metric $g^\prime_{i j} = g_{i j} + u_{i j} \delta t$, where, in keeping with (\[Eq:ggdot\]), the velocity tensor $u_{i j}= \dot{g}_{i j}$ is chosen such that $$g^{i j} u_{i j} = g^{i j} \dot{g}_{i j} = 0 \; .$$ Then, to first order in $\delta t$, $g^\prime_{i j}$ and $g_{i j}$ have equal determinants, as desired; but $g_{i j}$ and $g^\prime_{i j}$ are not in the same conformal equivalence class in
| 1,481
| 2,612
| 1,784
| 1,510
| null | null |
github_plus_top10pct_by_avg
|
in [@dress].
\[Dress\] A bivariant functor $M~=~(M^{*},M_{*})$ from $G$-$\rm{set}$ to $R$-$\rm{Mod}$ is a pair of functors from $G$-$\rm{set}\to$ $R$-$\rm{Mod}$ such that $M^{*}$ is a contravariant functor, and $M_{*}$ is a covariant functor. If $X$ is a $G$-set, then the image by the covariant and by the contravariant part coincide. We denote by $M(X)$ this image. A Mackey functor for $G$ over $R$ is a bivariant functor from $G$-set to $R$-Mod such that:
- Let $X$ and $Y$ be two finite $G$-sets, $i_{X}$ and $i_{Y}$ the canonical injection of $X$ (resp. $Y$) in $X\sqcup Y$, then $(M^{*}(i_X),M^{*}(i_Y))$ and $(M_{*}(i_X),M_{*}(i_Y))$ are inverse isomorphisms. $$M(X)\oplus M(Y)\cong M(X\sqcup Y).$$
- If $$\xymatrix{
X\ar[r]^{a}\ar[d]^{b}& Y\ar[d]^{c} \\
Z\ar[r]^{d} & T
}$$ is a pullback diagram of $G$-sets, then the diagram $$\xymatrix{
M(X)\ar[d]_{M_{*}(b)} & M(Y)\ar[l]_{M^{*}(a)}\ar[d]^{M_{*}(c)}\\
M(Z) & M(T)\ar[l]_{M^{*}(d)}
}$$ is commutative.
A morphism between two Mackey functors is a natural transformation of bivariant functors. Let us denote by $Mack_{R}(G)$ the category of Mackey functors for $G$ over $R$. Let us first recall an important example of Mackey functor:
[@bouc_green]\[burnside\] If $X$ is a finite $G$-set, then the category of $G$-sets over $X$ is the category with objects $(Y,\phi)$ where $Y$ is a finite $G$-set and $\phi$ is a morphism from $Y$ to $X$. A morphism $f$ from $(Y,\phi)$ to $(Z,\psi)$ is a morphism of $G$-sets $f:Y\to Z$ such that $\psi\circ f=\phi$. The Burnside functor at $X$ is the Grothendieck group of the category of $G$-sets over $X$, for relations given by disjoint union. This is a Mackey functor for $G$ over $R$ by extending scalars from $\mathbb{Z}$ to $R$. We denote by $RB$ the functor after scalar extension. If $X$ is a $G$-set, the Burnside module $RB(X^2)$ has an $R$-algebra structure. The product of (the isomorphism classes of) $(X\overset{\alpha}{\leftarrow} Y \overset{\beta}{\rightarrow} X)$ and $(X\overset{\gamma}{\leftarrow} Z
| 1,482
| 4,021
| 1,907
| 1,133
| null | null |
github_plus_top10pct_by_avg
|
0.001 (2) −0.001 (2)
C48 0.030 (3) 0.018 (2) 0.019 (3) −0.006 (2) −0.003 (2) 0.002 (2)
C49 0.027 (3) 0.030 (3) 0.017 (2) −0.006 (2) 0.007 (2) 0.009 (2)
C50 0.024 (3) 0.024 (2) 0.015 (2) 0.001 (2) 0.007 (2) 0.0054 (19)
C51 0.019 (2) 0.014 (2) 0.010 (2) 0.0035 (18) −0.0017 (18) 0.0002 (17)
C52 0.028 (3) 0.023 (3) 0.037 (3) 0.000 (2) 0.008 (3) −0.003 (2)
C53 0.044 (4) 0.015 (3) 0.051 (4) −0.004 (2) 0.008 (3) −0.012 (3)
C54 0.050 (4) 0.024 (3) 0.020 (3) 0.011 (3) 0.003 (3) −0.008 (2)
C55 0.034 (3) 0.027 (3) 0.029 (3) 0.013 (2) 0.011 (2) 0.004 (2)
C56 0.027 (3) 0.020 (2) 0.029 (3) 0.004 (2) 0.011 (2) 0.003 (2)
C57 0.014 (2) 0.013 (2) 0.013 (2) −0.0018 (17) 0.0026 (17) 0.0019 (17)
C58 0.015 (2) 0.026 (2) 0.014 (2) −0.0007 (19) 0.0050 (18) 0.0014 (19)
C59 0.019 (3) 0.025 (3) 0.013 (2) 0.001 (2) 0.0019 (19) 0.0066 (19)
C60 0.026 (3) 0.024 (2) 0.014 (2) −0.002 (2) 0.009 (2) 0.0015 (19)
C61 0.018 (2) 0.018 (2) 0.024 (3) −0.0007 (18) 0.013 (2) 0.0030 (19)
C62 0.014 (2) 0.015 (2) 0.022 (2) −0.0009 (17) 0.0066 (19) 0.0015 (18)
C63 0.008 (2) 0.016 (2) 0.012 (2) −0.0002 (16) 0.0050 (16) 0.0009 (17)
C64 0.017 (2) 0.0093 (19) 0.013 (2) −0.0019 (17) 0.0017 (18) −0.0012 (16)
C65 0.018 (3) 0.027 (3) 0.016 (2) 0.002 (2) 0.0018 (19) 0.001 (2)
C66 0.020 (3) 0.030 (3) 0.019 (2) −0.002 (2) −0.005 (2) −0.001 (2)
C67 0.035 (3) 0.036 (3) 0.011 (2) −0.002 (2) 0.001 (2) −0.005 (2)
C68 0.034 (3) 0.036 (3) 0.017 (3) −0.005 (2)
| 1,483
| 5,379
| 849
| 763
| null | null |
github_plus_top10pct_by_avg
|
of the leaves of the tree using the $S_2$-action on $E$, and the composition maps are given by attaching trees. This definition can readily be seen to define a colored operad.
An ideal $\mathcal I$ of a colored operad $\mathcal P$ is a collection of $S_n$-sub-modules $\mathcal I(\vec X;z)\subset
\mathcal P(\vec X;z)$ such that $f\circ_{i} g$ belongs to the ideal whenever $f$ or $g$ or both belong to the ideal. A colored operad $\mathcal P$ is said to be quadratic if $\mathcal P=\mathcal F(E)/(R)$ where $\mathcal F(E)$ is the free colored operad on some generators $E$, and $(R)$ is the ideal in $\mathcal F(E)$ generated by a subspace with 3 inputs, called the relations, $R\subset \bigoplus_{w,x,y,z\in C}
\mathcal F(E)(w,x,y;z)$.
We recall from [@GeK (3.2)], that an operad $\mathcal O$ is called cyclic quadratic if it is quadratic, with generators $E$ and relations $R$, so that the $S_2$-action on $E$ is naturally extended to a $S_3$-action via the sign-representation $sgn:S_3\to S_2$, and $R\subset\mathcal F(E)(3)$ is an $S_4$-invariant subspace. In this case, $\mathcal O$ becomes a cyclic operad, see [@GeK (3.2)].
The following lemma is straight forward to check.
\[O\_hat\_quadratic\] Let $\mathcal O$ be cyclic quadratic with generators $E$ and relations $R\subset\mathcal F(E)(3)$. Then $\widehat{\mathcal O}$ is generated by $\widehat{E}:=\widehat{E} ^{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}},{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}}}_{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}}}\oplus \widehat{E} ^{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt](0.1,0)(0.1,0.2)
\end{pspicture}},{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2)
\end{pspicture}}}_{{
\begin{pspicture}(0,0)(0.2,0.2)
\psline[linewidth=1pt, linestyle=dashed,dash=3pt 2pt](0.1,0)(0.1,0.2)
\end{pspicture}}} \oplus \widehat{E} ^{
| 1,484
| 892
| 1,392
| 1,421
| 948
| 0.79706
|
github_plus_top10pct_by_avg
|
ogarithmic factor larger than the number of parameters to be estimated. Such a logarithmic gap is also unavoidable and due to the fact that we require high probability bounds, where we want the tail probability to decrease at least polynomially in $d$. We discuss the role of the topology of the data in Section \[sec:role\]. The upper bound follows from an analysis of the convex program similar to those in [@NOS12; @HOX14; @SBB15]. However, unlike the traditional data collection scenarios, the main technical challenge is in analyzing the probability that a particular pair of items appear in the rank-breaking. We provide a proof in Section \[sec:proof\_main2\].
![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot1-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-169,50) (-115,-7)[number of separators ]{} (-52.5,87.5) (-82.5,83) (-52.5,78.5) ![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot6-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-96,-7)[sample size ]{} ![Simulation confirms $\|{\theta^* - \widehat{\theta}}\|_2^2 \propto 1/(\ell\,n)$, and smaller error is achieved for separators that are well spread out. []{data-label="fig:scaling_l_n"}](Plot2-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-162,50) (-90,-15) (-120,-3) (-25,-3)
In Figure \[fig:scaling\_l\_n\] , we verify the scaling of the resulting error via numerical simulations. We fix $d=1024$ and $\kappa_j=\kappa = 128$, and vary the number of separators $\ell_j=\ell$ for fixed $n = 128000$ (left), and vary the number of samples $n$ for fixed $\ell_j=\ell = 16$ (middle). Each point is average over $100$ instances. The plot confirms that the mean squared error scales as $1/(\ell \, n)$. Each sample is a partial ranking from a set of $\kappa$ alternatives chosen
| 1,485
| 425
| 307
| 1,587
| 235
| 0.816951
|
github_plus_top10pct_by_avg
|
us $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}^*=1$ and $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}= -1$. For an ${\mathbf{E}}$-graded module (or, indeed, any ${\mathbb{Z}}$-graded module) $M=\bigoplus_{i\in{\mathbb{Z}}}M_i$, we write the corresponding Poincaré series as $p(M,v)=\sum v^i\dim_{\mathbb{C}}M_i$. Set $$\label{factorial-defn}
[n]_v! = \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n} .$$
Under the ${\mathbf{E}}$-grading, the module $\overline{J^d} = {J^d}/{{{\mathbb{C}}[{\mathfrak{h}}]}^{{{W}}}_+J^d}$ has Poincaré series
$$\label{Z-grading0}
p(\overline{J^d}, v) = \frac{\sum_{\mu}
f_{\mu}(1)f_{\mu}(v^{-1}) v^{-d(n(\mu) - n(\mu^t))}[n]_v!}{\prod_{i=2}^n
(1-v^{-i})} .$$
Since ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}_+$ is ${\mathbf{E}}$-graded, so is $\overline{J^d}$, and so the result does make sense. By Lemma \[corpar\](2), the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ form an r-sequence in $J^d$ for any $d\geq 0$. Since these elements have degrees $2\leq r\leq n$, Corollary \[bigr\] implies that $\overline{J^d}$ has Poincaré series $$\label{Z-grading15}
p(\overline{J^d}, v) = \left( (1-t)\prod_{i=1}^n(1- s^i)\sum_{\mu}
P_{\mu}(s,t)\, \Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)}
\right)_{s=v, t=v^{-1}}$$ where $P_\mu$ and $\Omega(\mu)$ are defined in . Lemma \[babyverma\](2) implies that $$\label{Z-grading16}
\Big(\Omega(\mu)\Big)_{s=v,t=v^{-1}}
=f_{\mu}(v)^{-1}f_{\mu}(v^{-1})^{-1}
\prod_{i=1}^n(1-v^i)(1-v^{-i}).$$ This gives $$\label{Z-grading2}
p(\overline{J^d}, v) =
\frac{ \sum_{\mu} P_{\mu}(v,v^{-1})f_{\mu}(v)f_{\mu}(v^{-1})
v^{dn(\mu)}v^{-dn(\mu^t)} }
{\prod_{i=2}^n (1-v^{-i})}.$$ By Lemma \[babyverma\](3) the numerator of this expression can be described as $$\label{numerator}
\sum_{\mu} \left(\sum_{\lambda}
f_{\lambda}(v^{-1}) f_{\lambda}(1)\right) f_{\mu}(1) f_{\mu}(v)
v^{d(n(\mu)-n(\mu^t))}.$$ Applying Lemma \[babyverma\](1) and using the equality $f_{\mu}(1) = f_{\mu^t}(1)$ from we find that equals $$\label{rearrange} \sum_{\mu} \lef
| 1,486
| 3,085
| 1,536
| 1,345
| 3,710
| 0.77055
|
github_plus_top10pct_by_avg
|
\mathrm{Ker~}\tilde{\varphi}(R)$ be a lift of $m$. By using an argument similar to the paragraph just before Equation (\[ea20\]) of Step (1), if we write the $(1, 2), (1,3), (2,3), (2,2)$-blocks of the $(i, i)$-block of the formal matrix product $\sigma({}^t\tilde{m})\cdot h\cdot \tilde{m}$ as $\pi\cdot\xi^{(i-1)/2}\cdot \pi\mathcal{X}_{i,1,2}(\tilde{m})$, $\pi\cdot\xi^{(i-1)/2}\cdot \mathcal{X}_{i,1,3}(\tilde{m})$, $\pi\cdot\xi^{(i-1)/2}\cdot (1+\pi\mathcal{X}_{i,2,3}(\tilde{m}))$, $\pi\cdot\xi^{(i-1)/2}\cdot \pi^3\mathcal{X}_{i,2,2}(\tilde{m})$, respectively, where $\mathcal{X}_{i,1,2}(\tilde{m}), \mathcal{X}_{i,1,3}(\tilde{m}) \in M_{(n_i-2)\times 1}(B\otimes_AR)$ and $\mathcal{X}_{i,2,3}(\tilde{m}), \mathcal{X}_{i,2,2}(\tilde{m}) \in B\otimes_AR$, then the images of $\mathcal{X}_{i,1,2}(\tilde{m}), \mathcal{X}_{i,1,3}(\tilde{m})$ in $M_{(n_i-2)\times 1}(B\otimes_AR)/(\pi\otimes 1)M_{(n_i-2)\times 1}(B\otimes_AR)$ and the images of $\mathcal{X}_{i,2,3}(\tilde{m}), \mathcal{X}_{i,2,2}(\tilde{m})$ in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$ are independent of the choice of the lift $\tilde{m}$ of $m$. Therefore, we may denote these images by $\mathcal{X}_{i,1,2}(m)$, $\mathcal{X}_{i,1,3}(m)$, $\mathcal{X}_{i,2,3}(m)$, and $\mathcal{X}_{i,2,3}(m)$, respectively. Note that $\mathcal{X}_{i,2,2}(\tilde{m})$ is indeed contained in $R$. Thus $\mathcal{X}_{i,2,2}(\tilde{m})$ is naturally identified with $\mathcal{X}_{i,2,2}(m)$. As for Equation (\[ea20\]) of Step (1), we need to express $\mathcal{X}_{i,1,2}(m)$, $\mathcal{X}_{i,1,3}(m)$, and $\mathcal{X}_{i,2,3}(m)$ , and $\mathcal{X}_{i,2,2}(m)$ as matrices. Recall that $$\pi^ih_i=\xi^{(i-1)/2}\cdot\pi \begin{pmatrix} a_i&0&0\\ 0 &\pi^3\bar{\gamma}_i&1 \\ 0 &-1 &\pi \end{pmatrix}.$$ We write $$m_{i,i}=\begin{pmatrix} id&\pi r_i& t_i\\ y_i&1+\pi x_i&u_i\\ \pi v_i&\pi z_i&1+\pi w_i \end{pmatrix}
\mathrm{~and~}
\tilde{m}_{i,i}=\begin{pmatrix} \tilde{s}_i&\pi \tilde{r}_i& \tilde{t}_i\\ \tilde{y}_i&1+\pi \tilde{x}_i&\tilde{u}_i
\\ \pi \tilde{v}_i&\pi \tild
| 1,487
| 1,027
| 1,861
| 1,397
| 3,824
| 0.769859
|
github_plus_top10pct_by_avg
|
to compute high-probability bounds for $(\hat G - G) V G^\top$ and $G (\hat
V - V)G^\top$.
We first bound $(\hat G - G) V G^\top$. For any $j$ and $l$ in $\{1,\ldots,s\}$ and using the Cauchy-Schwartz inequality, we have that $$\label{eq::aaa}
\left| \left( \hat{G}_j - G_j \right) V G_l^\top \right| \leq \lambda_{\max}(V) \|
\hat{G}_j - G_j\| B \leq \overline{v} B \| \hat{G}_j - G_j\|,$$ by the definition of $B$ (see Equation \[eq:H.and.B\]), where we recall that $G_j$ denotes the $j^{\mathrm{th}}$ row of $G$.
It remains to bound the stochastic term $\max _j \|
\hat{G}_j - G_j\|$. Towards that end, we will show that, for some constant $C$ dependent on $A$ only, $$\label{eq:hatGjmGj}
\mathbb{P} \left( \max_j \|\hat G_j - G_j\|\leq C \overline{H} \sqrt{b
\frac{ \log
n}{ n}} \right) \geq 1 - 1/n.$$ Indeed, by a Taylor expansion, $$\begin{aligned}
\hat G_j - G_j =(\hat\psi -\psi)^\top \int_0^1 H_j((1-t)\psi +t\hat{\psi})dt
\quad \text{for all } j \in \{1,\ldots,s\},\end{aligned}$$ so that $$\max_j \|\hat G_j - G_j \| \leq \| \psi - \hat{\psi} \| \max_j \Big \| \int_0^1 H_j((1-t)\psi
+t\hat{\psi})dt \Big\|_{\mathrm{op}}.$$ Since the coordinates of $\hat{\psi}$ are bounded in absolute value by $A$, the bound implies that, for some positive constant $C$ dependent on $A$ only, $ \mathbb{P} \left( \|\hat\psi-\psi\| \leq C \sqrt{b (\log n)/n})
\right) \geq 1 - 1/n$, for all $P\in {\cal P}_n^{\mathrm{OLS}}$. We restrict to this event. By convexity of the operator norm $||\cdot ||_{\rm op}$ and our assumption, we have that $$\label{eq:here}
\max_j \Biggl|\Biggl|\int_0^1 H_j((1-t)\psi +t \hat{\psi})dt\Biggr|\Biggr|_{\mathrm{op}}\le \overline{H},$$ yielding the bound in . Combined with (\[eq::aaa\]), we conclude that on an event of probability at least $1 - 1/n$, $\max_{j,l} |\hat\Gamma(j,l) - \Gamma(j,l)|\preceq \aleph_n$. This bound holds uniformly over $P \in \mathcal{P}_n$.
As for the other term $G (\hat
V - V)G^\top$, we have that, by in , $$\max_{j,l} \left| G_j (\hat
V
| 1,488
| 3,557
| 1,874
| 1,237
| null | null |
github_plus_top10pct_by_avg
|
in LVAD patients, as shown in [Table 2](#t0002){ref-type="table"}. The pooled odds ratio (OR) of 30-day mortality was 3.66 (95% CI, 2.00--6.70, *I*^2^ = 71%, [Supplementary Figure S3](https://doi.org/10.1080/0886022X.2020.1768116)) and the pooled OR of 1 year mortality was 2.22 (95% CI, 1.62--3.04, *I*^2^ = 0%, [Supplementary Figure S4](https://doi.org/10.1080/0886022X.2020.1768116)) in LVAD patients with AKI, compared with no AKI. The pooled OR of 30-day mortality was 7.52 (95% CI, 4.58--12.33, *I*^2^ = 73%, [Supplementary Figure S5](https://doi.org/10.1080/0886022X.2020.1768116)) and the pooled OR of 1-year mortality was 5.41 (95% CI, 3.63--8.06, *I*^2^ = 0%, [Supplementary Figure S6](https://doi.org/10.1080/0886022X.2020.1768116)) in LVAD patients with severe AKI requiring RRT, compared with no RRT.
######
AKI associated Mortality in LVAD Patients.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Study Year Outcomes Confounder adjustment Quality assessment
----------------------------------- ------ ------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- --------------------
Kaltenmaier et al. \[[@CIT0040]\] 2000 30 days mortality\ None Selection: 4\
RRT: 2.54 (1.36--4.74)\
| 1,489
| 478
| 935
| 1,618
| null | null |
github_plus_top10pct_by_avg
|
x_{(k+1)\delta} - \by_{(k+1)\delta})}\\
=& \E{f(x_{\delta} - y_{\delta})}\\
\leq& e^{-\lambda \delta} \E{f(x_0 - y_0)} + 6\delta (L+\LN^2) \epsilon\\
=& e^{-\lambda \delta} \E{f(\bx_{k\delta} - \by_{k\delta})} + 6\delta (L+\LN^2) \epsilon
\end{aligned}$$ Applying the above recursively gives, for any $i$ $$\begin{aligned}
\E{f(\bx_{i\delta} - \by_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
[Proof of Theorem \[t:main\_gaussian\]]{}\[ss:proof:t:main\_gaussian\]
For ease of reference, we re-state Theorem \[t:main\_gaussian\] below as Theorem \[t:main\_gaussian:restated\] below. We make a minor notational change: using the letters $\bx_t$ and $\by_t$ in Theorem \[t:main\_gaussian:restated\], instead of the letters $x_t$ and $y_t$ in Theorem \[t:main\_gaussian\]. This is to avoid some notation conflicts in the proof.
\[Equivalent to Theorem \[t:main\_gaussian\]\] \[t:main\_gaussian:restated\] Let $\bx_t$ and $\by_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\by_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\twocase{\frac{\lambda^2 \hat{\epsilon}^2}{512 \beta^2\lrp{L^2 + \LN^4}\exp\lrp{\frac{14\aq\Rq^2}{3}}}}{\frac{2\lambda \hat{\epsilon}}{(L^2+\LN^4)\exp\lrp{\frac{7\aq\Rq^2}{3}}\sqrt{R^2 + \beta^2/m}}}.
\end{aligned}$$
If we assume that $\bx_0 = \by_0$, then there exists a coupling between $\bx_t$ and $\by_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{\bx_{k\delta} - \by_{k\delta}}_2} \leq \hat{\epsilon}
\end{aligned}$$
Alternatively, if we assume $k \geq \frac{ 3\aq\Rq^2}{ \delta} \log \frac{R^2 + \beta^2/m}{\hat{\ep
| 1,490
| 3,796
| 1,101
| 1,320
| null | null |
github_plus_top10pct_by_avg
|
ring of a particle in the spherically symmetric field with a barrier and with orbital quantum number $l=0$. In such case, we have: $$\begin{array}{ll}
W(r) =
\displaystyle\frac{2\beta - \alpha}{f(\bar{r})} -
\displaystyle\frac{\beta}{\bar{r}}, &
\mbox{при } 2\beta \ne \alpha,
\end{array}
\label{eq.3.1.1}$$ where $$f(\bar{r}) = C(2\beta - \alpha) \bar{r}^{2\beta / \alpha} +
\bar{r}.
\label{eq.3.1.2}$$ Here $\bar{r} = r+r_{0}$, $\beta$ and $C$ are arbitrary real positive constants, $r_{0}$ is a positive number close to zero, and a designation $\alpha = \displaystyle\frac{\hbar}{\sqrt{2m}}$ is introduced. This superpotential is defined on the positive semiaxis of $r$ (at $r > r_{0}$).
Let’s find potentials-partners for the superpotential (\[eq.3.1.1\]). In accordance with (\[eq.2.1.6\]), we obtain: $$\begin{array}{lcl}
V_{1,2}(r) & = &
\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{2\beta (2\beta - \alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac{\beta^{2}}{\bar{r}^{2}} \pm \\
& \pm &
\Biggl(
\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{2\beta (2\beta - \alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac{\alpha\beta}{\bar{r}^{2}}
\Biggr)
\end{array}
\label{eq.3.1.3}$$ or $$\begin{array}{lcl}
V_{1}(r) & = &
\displaystyle\frac{\beta (\beta - \alpha)} {\bar{r}^{2}}, \\
V_{2}(r) & = &
2\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{4\beta (2\beta-\alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac
{\beta (\beta+\alpha)} {\bar{r}^{2}}.
\end{array}
\label{eq.3.1.4}$$
From (\[eq.3.1.4\]) one can see that at $\beta = \alpha$ the first potential $V_{1}(r)$ obtains zero value and, therefore, it becomes reflectionless. Then, according to (\[eq.2.4.5\]), if the second potential $V_{2}(r)$ is finite in a whole region of its definition, then it should be reflectionless also. At $\beta = \alpha$ we obtain:
| 1,491
| 2,856
| 2,026
| 1,546
| null | null |
github_plus_top10pct_by_avg
|
p
\sqrt{\frac{ \log p + \log n}{n} } \right) \geq 1 - \frac{1}{n},$$ for some universal constant $C>0$. Clearly, the scaling in $p$ is worse.
[**Proof of Lemma \[lemma::horrible\].**]{} Throughout, we drop the dependence on ${\widehat{S}}$ in our notation and assume without loss of generality that ${\widehat{S}}= \{1,\ldots,k\}$. We refer the reader to [@magnus07] for a comprehensive treatment of matrix calculus techniques. Recall that $\psi = \left[ \begin{array}{c} \sigma \\ \alpha\\ \end{array} \right]$ and $\xi = \left[ \begin{array}{c} w\\ \alpha\\
\end{array} \right]$, where $\sigma =\mathrm{vec}(\Sigma)$ and $w = \mathrm{vec}(\Omega)$. The dimension of both $\psi$ and $\xi$ is $b = k^2 + k$. For $ 1 \leq j \leq n$, let $$\beta_j = g_j(\psi) = e^\top_j \Omega\alpha,$$ where $e_j$ is the $j^{\mathrm{th}}$ elements of the standard basis in $\mathbb{R}^n$. Then, we can write $$g_j(\psi) = g(f(\psi)),$$ with $f(\psi) = \xi \in \mathbb{R}^b$ and $g(\xi) = e^\top_j \Omega \alpha \in \mathbb{R}$.
Using the chain rule, the derivative of $g_j(\psi)$ is $$D g_j(\psi) = D g(\xi) D f(\psi) = e_j^\top \Big[\left( \alpha^\top \otimes I_k
\right) E + \Omega
F\Big]
\left[
\begin{array}{cc}
- \Omega \otimes \Omega & 0 \\
0 & I_k
\end{array}
\right],$$ where $$E = \Big[I_{k^2} \;\;\;\;\; 0_{k^2 \times k}\Big] = \frac{d w}{d \psi} \in
\mathbb{R}^{k^2 \times b} \quad \text{and} \quad
F =
\Big[0_{k \times k^2} \;\;\;\;\; I_k\Big] = \frac{d \alpha}{d \psi} \in
\mathbb{R}^{ k \times b}.$$ Carrying out the calculations, we have that $$\begin{aligned}
\left( \alpha^\top \otimes I_k
\right) E
\left[
\begin{array}{cc}
- \Omega \otimes \Omega & 0 \\
0 & I_k
\end{array}
\right] & =
\left( \alpha^\top \otimes I_k
\right) \Big[I_{k^2} \;\;\;\;\; 0_{k^2 \times k}\Big]
\left[
\begin{array}{cc}
- \Omega \otimes \Omega & 0 \\
0 & I_k
\end{array}
\right] \\
& = \Big[ - \left( \alpha^\top \otimes I_k
\right) (\Omega \otimes \Omega) \;\;\;\;\; 0_{k \times k}\Big
| 1,492
| 3,595
| 1,533
| 1,241
| null | null |
github_plus_top10pct_by_avg
|
Vascular invasion 0.472 0.492
Absent 72 33 39
Present 32 17 15
Cirrhosis 2.143 0.143
Yes 61 33 28
No 43 17 26
Recurrence 1.135 0.287
Yes 43 18 25
No 61 32 29
Expression of CLU and its association with clinical features of HCC patients {#sec3-2}
----------------------------------------------------------------------------
Relative expression of *CLU* mRNA in patients with HCC was determined via qRT-PCR. Accordingly, plasma *CLU* expression was higher in HCC cases than in normal controls (1.48 ± 0.22 vs. 0.22 ± 0.12, *P* \< 0.001, [Figure 1](#F1){ref-type="fig"}).
{#F1}
In addition, the expression of CLU protein in HCC tissues and non-malignant tissues was also estimated using IHC method. The results suggested that the expression of CLU protein in HCC tissues was significantly higher and the percentage of positively stained cells was as high as 94.2% (98/104); while CLU protein expression in non-malignant tissues was relatively weaker and the proportion of positively stained cells was only 14.4% (15/104). The difference between two sides was significant (*P* \< 0.001, [Figure 2](#F2){ref-type="fig"}).
{#F2}
| 1,493
| 478
| 1,408
| 1,604
| null | null |
github_plus_top10pct_by_avg
|
e desired net in $A$ by $${\textstyle \chi(N_{\beta},\beta)=x_{\beta}\in N_{\beta}\bigcap A}$$ where the family of nonempty decreasing subsets $N_{\beta}\bigcap A$ of $X$ constitute the filter-base in $A$ as required by the directed set $_{\mathbb{D}}N_{\beta}$. It now follows from Eq. (\[Eqn: DirectionIndexed\]) and the arguments in Example A1.3(3) that $x_{\beta}\rightarrow x$; compare the directed set of Eq. (\[Eqn: Closure\_Directed\]) for a more compact, yet essentially identical, argument. Carefully observe the dual roles of $\mathcal{N}_{x}$ as a neighbourhood filter base at $x$.
*Sufficiency.* Let $\chi$ be a net in $A$ that converges to $x\in X$. For any $N_{\alpha}\in\mathcal{N}_{x}$, there is a $\mathbb{R}_{\alpha}\in\textrm{Res}(\mathbb{D})$ of Eq. (\[Eqn: residual\]) such that $\chi(\mathbb{R}_{\alpha})\subseteq N_{\alpha}$. Hence the point $\chi(\alpha)=x_{\alpha}$ of $A$ belongs to $N_{\alpha}$ so that $A\bigcap N_{\alpha}\neq\emptyset$ which means, from Eq. (\[Eqn: Def: Closure\]), that $x\in\textrm{Cl}(A)$.$\qquad\blacksquare$
**Corollary.** Together with Eqs. (\[Eqn: Def: Closure\]) and (\[Eqn: Def: Derived\]), is follows that $$\textrm{Der}(A)=\{ x\in X\!:(\exists\textrm{ a net }\zeta\textrm{ in }A-\{ x\})(\zeta\rightarrow x)\}\qquad\square\label{Eqn: net derived}$$
The filter forms of Eqs. (\[Eqn: net closure\]) and (\[Eqn: net derived\]) $$\begin{aligned}
\textrm{Cl}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\label{Eqn: filter cls_der}\\
\textrm{Der}(A) & = & \{ x\in X\!:(\exists\textrm{ a filter }\mathcal{F}\textrm{ on }X)(A-\{ x\}\in\mathcal{F})(\mathcal{F}\rightarrow x)\}\nonumber \end{aligned}$$ then follows from Eq. (\[Eqn: Def: LimFilter\]) and the finite intersection property (F2) of $\mathcal{F}$ so that every neighbourhood of $x$ must intersect $A$ (respectively $A-\{ x\}$) in Eq. (\[Eqn: filter cls\_der\]) to produce the converging net needed in the proof of Theorem A1.3.
We end this discussion of conve
| 1,494
| 1,710
| 2,554
| 1,614
| 1,702
| 0.786649
|
github_plus_top10pct_by_avg
|
_R\^x (\_R\^x )\^ .
Using the standard definition of the Passarino-Veltman functions, B\_0(p, m\_1, m\_2) &=& 16 \^2\
p\_B\_1(p, m\_1, m\_2) &=& 16 \^2 \[passvelt\] ,we get B\_[LR]{}\^x &=& \_R\^x (\_L\^x )\^ B\_0 (p, , m\_[\_x]{})\
A\_[L]{}\^x &=& - \_L\^x (\_L\^x )\^ B\_1 (p, , m\_[\_x]{})\
A\_[R]{}\^x &=& - \_R\^x (\_R\^x )\^ B\_1 (p, , m\_[\_x]{}) .
Now we are ready to calculate the corrections to the bottom quark mass from the three diagrams as estimated in \[delmb\]:
m\_b\^ &=& \_[x=1,2]{} B\_[LR]{}\^x + (A\_L\^x + A\_R\^x)\
&=& \_[x=1,2]{} { \_R\^x (\_L\^x )\^ B\_0 (p, , m\_[\_x]{}) - B\_1 (p, , m\_[\_x]{})(\_L\^x (\_L\^x )\^ + \_R\^x (\_R\^x )\^) }.\
This is the exact expression for the one-loop threshold corrections to the bottom quark mass coming from the gluino-sbottom loops. In a full three family model, the $\Gamma_{L,R}$ are the $6 \times 3$ squark mixing matrices, and all the down-type squarks give rise to corrections to the bottom mass. Ignoring the off-diagonal elements that introduce the inter-generational mixing, we can consider a $2 \times 2$ block that mixes the two bottom squarks. The sbottom mixing matrix can be written as = (
[cc]{} \_L\^1 & \_R\^1\
\_L\^2 & \_R\^2
) = (
[cc]{} \_b & \_b\
-\_b & \_b
) ,such that (
[c]{} \_1\
\_2
) = (
[c]{} \_L\
\_R
) .Then, $\Delta m_b^{\tilde{g}}$ simplifies to m\_b\^ = .
Chargino-stop
=============
The charginos couple to the up-type squarks and down-type quarks proportional to the SU(2) coupling $g_2$ and the Yukawa couplings $\lam_{t,b}$ with strength depending upon their respective wino-higgsino composition. The corrections dominate when the squarks are from the third family due to CKM suppression of the contributions from the first two families of squarks. We calculate here the individual diagrams shown in \[charginocorrections\] considering the contributions from the two stop squarks. The calculation of the chargino-stop diagrams is similar to the calculation of the gluino-sbottom diagrams and yields
\
-i B\_[LR\_i]{}\^x &=&\
-i p |
| 1,495
| 241
| 1,836
| 1,597
| null | null |
github_plus_top10pct_by_avg
|
ight|
\approx \left|{\mathbf{I}}_{N_r}+\frac{\rho}{L_t} {\mathbf{H}_{\psi,V}}{\mathbf{H}_{\psi,V}}^H\right|\notag\\
&= \left|{\mathbf{I}}_{N_r}+\frac{\rho}{L_t} {\mathbf{H}}_{\psi}{\mathbf{H}}_{\psi}^H\right|.\end{aligned}$$ Based on , we find that a fast selection of reconfiguration state can be achieved by directly comparing their (full) physical channel matrices. Instead of finding the optimal beam selection of each reconfiguration state first, we can directly determine the optimal reconfiguration state by $$\label{eq:problemselect_2}
\widehat{\psi}=\arg\max_{\psi\in\left\{1,\cdots,\Psi\right\}} \left|{\mathbf{I}}_{N_r}+\frac{\rho}{L_t} {\mathbf{H}}_{\psi}{\mathbf{H}}_{\psi}^H\right|.$$ Based on , the near-optimal reconfiguration state can be selected by calculating and comparing the throughput among only $\Psi$ possible channel matrices. Note that ${\mathbf{H}}_{\psi}$ in is the full channel matrix rather than a low-dimensional virtual channel matrix associated with a particular selection of beams. In addition, with the fast reconfiguration state selection, we can select beams from only the beams that are associated with the selected reconfiguration state. In contrast, the exhaustive search needs to examine the performance of all beams associated with all reconfigurable states. Although the performance of this fast-selection method depends on the accuracy of the approximation in , we will show later by numerical results that usually near optimal performance can be achieved.
\[Alg:1\]
$\widehat{\psi}:=\arg\max_{\psi\in\left\{1,\cdots,\Psi\right\}} \left|{\mathbf{I}}_{N_r}+\frac{\rho}{L_t} {\mathbf{H}}_{\psi}{\mathbf{H}}_{\psi}^H\right|$;
$\mathcal{I}_r:=\left\{1, \cdots, N_r\right\}$; $\mathcal{I}_t:=\left\{1, \cdots, N_t\right\}$; ${\mathbf{h}}_j:=j$-th row of ${\mathbf{H}_{\widehat{\psi},V}}$, $\forall j\in\mathcal{I}_r$;
$J:=\arg\max_{j\in\mathcal{I}_r}{\mathbf{h}}_j{\mathbf{h}}_j^H$;
${\mathcal{M}_{r}}:=\left\{J\right\}$; ${\widetilde{\mathbf{H}}_{\widehat{\psi},V}}:=\left[{\mathbf{H}_{\wide
| 1,496
| 1,652
| 1,709
| 1,466
| 4,062
| 0.768381
|
github_plus_top10pct_by_avg
|
intermittent compound Poisson process.
#### Ordinary Poisson process {#S:INTERMITTENT_POISSON}
(Ordinary) Poisson processes are characterized simply by their rate or intensity:
- its fundamental rate $\lambda$, which is the expected number of arrivals per unit time.
Let $\lambda$ be the rate of a Poisson process. The probability of experiencing $k$ arrivals in a time interval $t$ units long is $$P_\lambda(k) = e^{-\lambda t}\frac{{(\lambda t)}^k}{k!}.$$
#### Compound Poisson process {#S:INTERMITTENT_POISSON}
A compound Poisson process is characterized by two rates:
- its fundamental rate $\lambda$ as before, and
- its rate of loss $L$, which is a random variable invoked once for each arrival.
Let $\lambda$ be the rate and $L$ be the loss random variable of a compound Poisson process. The expectation of the compound process for a time interval $t$ units long is $$\begin{aligned}
{1}
\E (\text{compound Poisson}) &= \lambda t \cdot \E (L)\\
&= \lambda t \cdot \mu_L.\end{aligned}$$
\[D:STATISTICAL\_RISK\] The statistical *risk*, written $h$, of a compound Poisson process is the time derivative of its expectation in a duration of length $t$; that is $$h = \frac{d}{dt} \E(\text{compound Poisson}) = \frac{d}{dt} (\lambda t \cdot \mu_L) = \lambda \mu_L,$$ which is the product of its rate $\lambda$ and its expected loss $\mu_L$.
#### Intermittent compound Poisson process {#S:INTERMITTENT_POISSON}
A variation of the CPP is the intermittent compound Poisson process, which is intermittently on or off with expected durations $\E(\text{on}) = \mu_\text{on}$ and $\E(\text{off}) = \mu_\text{off}$. An intermittent compound Poisson process (ICPP) is characterized by three rates:
- its fundamental rate $\lambda$ as before, and
- its rate of loss $L$, also as before,
- alternating durations of random lengths $\tau_\text{on}$ and $\tau_\text{off}$.
Random variables $\tau_\text{on}$ and $\tau_\text{off}$ converge to $\mu_\text{on}$ and $\mu_\text{off}$ in the limit. The *idle*
| 1,497
| 6,211
| 1,607
| 628
| 1,006
| 0.795872
|
github_plus_top10pct_by_avg
|
senting a *chaotic substate of* the system represented by the solution of the neutron transport equation — combine with the functional components $\phi(\pm\nu_{0},\mu)$ to produce the well-defined, non-chaotic, experimental end result of the neutron flux $\Phi(x,\mu)$.
The solution (\[Eqn: CaseSolution\_FR\]) is obtained by assuming $\Phi(x,\mu)=e^{-x/\nu}\phi(\mu,\nu)$ to get the equation for $\phi(\mu,\nu)$ to be $(\mu-\nu)\phi(\mu,\nu)=-c\nu/2$ with the normalization $\int_{-1}^{1}\phi(\mu,\nu)=1$. As $\mu_{\nu}^{-1}$ is not invertible in $\textrm{Multi}(V(\mu),\mathbb{R})$ and $\mu_{\nu\textrm{B}}\!:X_{\textrm{B}}\rightarrow f(X)$ does not exist, the alternate approach of regularization was adopted in [@Sengupta1988; @Sengupta1995] to rewrite $\mu_{\nu}\phi(\mu,\nu)=-c\nu/2$ as $\mu_{\nu\varepsilon}\phi_{\varepsilon}(\mu,\nu)=-c\nu/2$ with $\mu_{\nu\varepsilon}:=\mu-(\nu+i\varepsilon)$ being a net of bijective functions for $\varepsilon>0$; this is a consequence of the fact that for the multiplication operator every nonreal $\lambda$ belongs to the resolvent set of the operator. The family of solutions of the later equation is given by [@Sengupta1988; @Sengupta1995] $$\phi_{\varepsilon}(\nu,\mu)=\frac{c\nu}{2}\frac{\nu-\mu}{(\mu-\nu)^{2}+\varepsilon^{2}}+\frac{\lambda_{\varepsilon}(\nu)}{\pi_{\varepsilon}}\frac{\varepsilon}{(\mu-\nu)^{2}+\varepsilon^{2}}\label{Eqn: phieps}$$
where the required normalization $\int_{-1}^{1}\phi_{\varepsilon}(\nu,\mu)=1$ gives
$$\begin{array}{ccl}
{\displaystyle \lambda_{\varepsilon}(\nu)} & = & {\displaystyle \frac{\pi_{\varepsilon}}{\tan^{-1}(1+\nu)/\varepsilon+\tan^{-1}(1-\nu)/\varepsilon}\left(1-\frac{c\nu}{4}\ln\frac{(1+\nu)^{2}+\varepsilon^{2}}{(1-\nu)^{2}+\varepsilon^{2}}\right)}\\
& \overset{\varepsilon\rightarrow0}\longrightarrow & \pi\lambda(\nu)\end{array}$$
with $$\pi_{\varepsilon}=\varepsilon\int_{-1}^{1}\frac{d\mu}{\mu^{2}+\varepsilon^{2}}=2\tan^{-1}\left(\frac{1}{\varepsilon}\right)\overset{\varepsilon\rightarrow0}\longrightarrow\pi.$$
These discretized eq
| 1,498
| 2,450
| 1,498
| 1,444
| 3,705
| 0.770578
|
github_plus_top10pct_by_avg
|
and only if $D_u\cap D_t\neq \emptyset$ (that is, the $d$-choices of the $t$-th ball and the $u$-th ball contain a common bin).
We say a subgraph of ${\mathcal{C}}_m$ with vertex set $\{D_{t_1},\ldots, D_{t_k}\}$ is $c$-*loaded* if every bin in $D_{t_1}\cup D_{t_2}\cup \cdots \cup D_{t_k}$ has at least $c$ balls.
Our analysis will involve a useful combinatorial object, called an [*ordered tree*]{}. An ordered tree is a rooted tree, together with a specified ordering of the children of every vertex. Recall that $\frac{1}{k+1}\, \binom{2k}{k}$ is the $k$-th Catalan number, which counts numerous combinatorial objects, including the number of ways to form $k$ balanced parentheses. It is well known [@OEIS] that ordered trees with $k-1$ edges are counted by the $(k-1)$-th Catalan number, leading easily to the following proposition.
\[pro:ordered\] The number of $k$-vertex ordered trees is $\frac{1}{k}\,\binom{2k-2}{k-1}{\leqslant}4^{k-1}$.
More information regarding the enumeration of trees can be found in [@Knuth].
The following blue-red coloring will be very helpful in our analysis.
\[def:br\] Given $m\in \{1,2,\ldots, n\}$, suppose that $T\subset {\mathcal{C}}_m$ is a rooted and ordered $k$-vertex tree contained in ${\mathcal{C}}_m$. Let the vertex set of $T$ be $\{D_{t_1}\ldots,D_{t_k}\}$, where $D_{t_1}$ is the root. Perform depth-first search starting from the root, respecting the specified order of each vertex. [For $i=1,\ldots, k$, let $D(i)\in\{D_{t_1}\ldots,D_{t_k}\}$ be the vertex which is the $i$-th visited vertex in the depth-first search. Then $D(1)=D_{t_1}$ is the root.]{} for $j=1,\ldots, k$. We now define a blue-red coloring $col:\{D(2),\ldots, D(k)\}\rightarrow \{\text{blue, red}\}$ as follows. For $i= 2,\ldots, k$, $$col(D(i))= \begin{cases}
\text{blue} & \text{ if }\,\, |\, (\cup_{j=1}^{i-1}D(j)) \cap D(i)|= 1,\\
\text{red} & \text{ if }\,\, |\, (\cup_{j=1}^{i-1}D(j)) \cap D(i)|{\geqslant}2.
\end{cases}$$
The following key lemma presents a upper bound for the probability that
| 1,499
| 3,861
| 2,800
| 1,455
| 1,672
| 0.786935
|
github_plus_top10pct_by_avg
|
w)^2} :j^c_{L,z} \phi:(w) \cr
& \quad + \frac{\Delta_{\phi}:j^a_{L,z} \phi:(w)}{(z-w)^2} + \frac{:j^a_{L,z}\p \phi:(w)}{z-w} + \mathcal{O}(z-w)^0\end{aligned}$$ Using equation we obtain : $$\begin{aligned}
\label{TjphiStep1}
T(z) :j^a_{L,z}\phi:(w) &= -\frac{c_+}{c_++c_-}\frac{t^a \phi(w)}{(z-w)^3}
+\frac{(\Delta_\phi +1):j^a_{L,z}\phi:(w) + \frac{c_-}{(c_++c_-)^2}i{f^a}_{cb}t^b:j^c_{L,z}\phi:(w)}{(z-w)^2} \cr
&\quad +\frac{\partial :j^a_{L,z}\phi:(w)}{z-w} + \mathcal{O}(f^4)+ \mathcal{O}(z-w)^0.\end{aligned}$$ The matrices $t^a$ are the generators of the Lie algebra in the representation in which the operator $\phi$ transforms. Since one has a non-vanishing third-order pole, not all of the operators $:j^a_{L,z}\phi:$ are Virasoro primary. Indeed we know from equation that the operator $L_{-1}\phi = \partial \phi$, which is a Virasoro descendant, is a linear combination of these operators. However the remaining ones are all Virasoro primaries. In the case where the quadratic Casimir of the representation $\mathcal{R}$ associated to the representation of the operator $\phi$ is non-zero, it is straightforward to check that in the OPE between the stress-tensor and the operator $ c^{(2)}_\mathcal{R} :j^a_{L,z}\phi: - t^a t_b
:j^b_{L,z}\phi: $, the third-order pole vanishes. We adopt the notation $c^{(2)}_{\mathcal{R}}$ both for the (generalized) quadratic Casimir operator and for its eigenvalues in the irreducible representation or reducible indecomposable structure $\mathcal{R}$.
[From]{} the double pole in equation we can read off the action of the scaling operator $L_0$ on the operator $:j^a_{L,z}\phi:$: L\_0 :j\^a\_[L,z]{}: = (\_+1):j\^a\_[L,z]{}: + i[f\^a]{}\_[cb]{}t\^b:j\^c\_[L,z]{}:.The operators $:j^a_{L,z}\phi:$ do not diagonalize the scaling operator $L_0$. In order to extract the conformal dimensions of these operators we have to compute the eigenvalues of the following operator : \[offDiagL0\] [f\^a]{}\_[cb]{} [[(t\^b)]{}\_]{}\^[ ]{} where we wrote explicitly the indices $\alpha$, $\beta$ associated to
| 1,500
| 591
| 797
| 1,625
| null | null |
github_plus_top10pct_by_avg
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.