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 |
|---|---|---|---|---|---|---|---|
\right)$, which implies ${\a^\star}$ is not optimal. Thus, ${\a^\star}(i) - {\a^\star}(j) \geq -1$. Similarly, ${\a^\star}(j) - {\a^\star}(i) \geq -1$. Therefore, ${\a^\star}(i) = {\a^\star}(j)$ or $\fabs{{\a^\star}(i) - {\a^\star}(j)}=~1$.
\[remark:hijEqual\] In Lemma \[lemma:hijEqual\], for the case where $\h(i) = \h(j)$ with $\fabs{{\a^\star}(i) - {\a^\star}(j)} = 1$, $i < j$, we will always set ${\a^\star}(j) = {\a^\star}(i) + 1$ since setting ${\a^\star}(i) = {\a^\star}(j) + 1$ results the same computation rate. Then, as long as $\h(i) = \h(j)$, $i < j$, it holds that ${\a^\star}(i)\leq{\a^\star}(j)$.
\[theorem:aNonnegativeOrdered\] For a nonnegative ordered channel vector $\h$, the optimal coefficient vector ${\a^\star}$ is also nonnegative ordered.
According to Lemma \[lemma:aNonnegative\], all the elements in ${\a^\star}$ are nonnegative. Suppose ${\a^\star}$ is not nonnegative ordered, then there must exist $i, j $ ($1 \leq i < j \leq L $) such that ${\a^\star}(i) > {\a^\star}(j) \geq 0$. According to Lemma \[lemma:a0\], ${\a^\star}(i) > 0$ implies $\h(i) > 0$. According to Lemma \[lemma:hijEqual\] and Remark \[remark:hijEqual\], ${\a^\star}(i) > {\a^\star}(j)$ implies $\h(i) \neq \h(j)$ and thus $\h(j) > \h(i) > 0$. Then, $\h(i) {\a^\star}(j) + \h(j) {\a^\star}(i) > \h(i) {\a^\star}(i) + \h(j) {\a^\star}(j)$.
Define $\a'$ as: $\a'(i) = {\a^\star}(j)$, $\a'(j) = {\a^\star}(i)$, and $\a'(\ell) = {\a^\star}(\ell)$, $\forall \ell \notin \{i,j\}$. Obviously, $\norm{\a'} = \norm{{\a^\star}}$, and $\h^T \a'
= \sum_{\ell = 1}^L \h(\ell) \a'(\ell)
= \sum_{\ell \notin \{i,j\}} \h(\ell) \a'(\ell) + \h(i) \a'(i) + \h(j) \a'(j)
= \sum_{\ell \notin \{i,j\}} \h(\ell) {\a^\star}(\ell) + \h(i) {\a^\star}(j) + \h(j) {\a^\star}(i)
> \sum_{\ell \notin \{i,j\}} \h(\ell) {\a^\star}(\ell) + \h(i) {\a^\star}(i) + \h(j) {\a^\star}(j)
= \sum_{\ell = 1}^L \h(\ell) {\a^\star}(\ell)
= \h^T {\a^\star}\geq 0$. Then according to , $\mathcal{R} \left( \h, \a' \right) > \mathcal{R} \left( \h, {\a^\star}\right)$, which implies $
| 1,601
| 825
| 1,799
| 1,528
| null | null |
github_plus_top10pct_by_avg
|
} \left| \mathbb{P}\left( \sqrt{n} \| \hat{\theta} -
\theta \|_\infty
\leq t \right) - \mathbb{P}( \| Z_n \|_\infty \leq t ) \right|,\\
A_2 & = \sup_{t>0} \left| \mathbb{P}( \| Z_n \|_\infty \leq t ) - \mathbb{P}( \|
\hat{Z}_n \|_\infty \leq t ) \right|,\\
\text{and} & \\
A_3 & = \sup_{t >0} \left| \mathbb{P}( \| \hat{Z}_n \|_\infty \leq
t - \mathbb{P}\left( \sqrt{n} \| \hat{\theta}^* - \hat{\theta} \|_\infty \leq t
\Big| (W_1,\ldots,W_n) \right) \right|.\end{aligned}$$ Since, by definition, $\mathbb{P}( \sqrt{n}||\hat \theta^* - \hat{\theta}||_\infty \leq
\hat{t}^*_\alpha|(W_1,\ldots,W_n)) \geq 1 - \alpha$, it follows from that, in order to establish (\[eq::boot-cov\]) we will need to upper bound each of the terms $A_1$, $A_2$ and $A_3$ accordingly. The term $A_1$ has already been bounded by $C( \Delta_{1,n} + \Delta_{2,n} )$ in the earlier . For $A_2$ we use the Gaussian comparison as in the proof of restricted to the event $\mathcal{E}_n$ to conclude that $A_2 \leq C \Delta_{n,3} + \frac{2}{n}$. Finally, to bound $A_3$, one can apply the same arguments as in Theorem \[theorem::deltamethod\], but restricted to the event $\mathcal{E}_n$, to the larger class of probability distributions $\mathcal{P}^*_n$ differing from $\mathcal{P}_n$ only in the fact that $v$ is replaced by the smaller quantity $v_n > 0$ and $\overline{v}$ by the larger quantity $\overline{v}_n =\overline{v} + C \daleth_n$. In particular, the bootstrap distribution belongs to $\mathcal{P}^*_n$. In detail, one can replace $\psi$ and with $\hat\psi$, and $\hat\psi$ with $\hat\psi^*$ and, similarly, $\Gamma$ with $\hat{\Gamma}$ and $\hat{\Gamma}$ with $\hat{\Gamma}^* = G(\hat{\psi}^*)
\hat{V}^* G(\hat{\psi}^*)^\top$, where $\hat{V}^*$ is the empirical covariance matrix based on a sample of size $n$ from the bootstrap distribution. The assumption that $n$ is large enough so that $v_n$ and $\sigma^2_n$ are positive ensures that, on the event $\mathcal{E}_n$ of probability at least $1-2/n$, $\min_j
\sqrt{
| 1,602
| 2,769
| 1,566
| 1,460
| null | null |
github_plus_top10pct_by_avg
|
decode $\alpha\y$ as an integer linear combination, whose coefficients form $\a$, of the original codewords $\{\x_\ell\}$. The *computation rate* [@Nazer2011] is the maximum transmission rate from the associated sources to a relay such that the integer linear combinations at the relay can be decoded with arbitrarily small error probability. Assume the $\log$ function is with respect to base 2, and define $\log^+(w) \triangleq \max \left( \log(w),0 \right)$. The computation rate can be calculated with Theorem \[theorem:ComputationRate\] from [@Nazer2011].
*(Computation Rate in Real-Valued Channel Model)* \[theorem:ComputationRate\] For a relay with coefficient vector $\a$ in the real-valued channel model defined in Definition \[definition:RealChannelModel\], the following computation rate is achievable,
$$\begin{aligned}
\label{equation:ComputationRate}
\bigR \left( \h, \a \right) =
\frac{1}{2} \log^+
\left( \left(
\norm{\a}^2 -
\frac{P (\h^T\a)^2}{1 + P\norm{\h}^2}
\right)^{-1}\right).
\end{aligned}$$
With the computation rate being the metric, we define the optimal coefficient vector as follows.
*(The Optimal Coefficient Vector)* The optimal coefficient vector ${\a^\star}$ for a channel vector $\h$ is the one that maximizes the computation rate, $$\begin{aligned}
\label{equation:aOptimal}
{\a^\star}= \arg \max_{\a \in \Zbb^L \backslash\{\0\}} \bigR \left( \h, \a \right).
\end{aligned}$$
After a few simple manipulations, the optimization problem stated in can be written in the following quadratic form [@Wei2012], $$\label{equation:IntegerQP}
\begin{aligned}
{\a^\star}=
\arg\min_{\a\in\Zbb^L\backslash\{\0\}}
\a^T\bsG\a,
\end{aligned}$$ where $$\begin{aligned}
\label{equation:G}
\bsG \triangleq
\I - \frac{P}{1 + P\norm{\h}^2} \h\h^T.
\end{aligned}$$
If we take $\bsG$, which is positive definite, as the *Gram matrix* of a lattice $\Lambda$, then the problem turns out to be the SVP in the lattice $\Lambda$. In the next section, we will propose an efficient approximation method based on QP relaxation that
| 1,603
| 298
| 1,448
| 1,758
| null | null |
github_plus_top10pct_by_avg
|
] We have $$\sum_{n\geq
1}\H_{(n-1,1)}(z,w)T^n=(z^2-1)(1-w^2)\frac{A_1(z,w;T)}{A_0(z,w;T)}.$$
The coefficient of the monomial symmetric function $m_{(n-1,1)}(\x)$ in a symmetric function in $\Lambda(\x)$ of homogeneous degree $n$ is the coefficient of $u$ when specializing the variables $\x=\{x_1,x_2,\dots\}$ to $\{1,u,0,0\dots\}$. Hence, the generating series $\sum_{n\geq 1}\H_{(n-1,1)}(z,w)T^n$ is the coefficient of $u$ in $$(z^2-1)(1-w^2)\Log\left(\sum_\lambda\calH_\lambda(z,w)
\tilde{H}_\lambda(1,u,0,0,\dots;z^2,w^2)\,T^{|\lambda|}\right).$$ We know that $$\tilde{H}_\lambda(\x;z,w)
=\sum_\rho\tilde{K}_{\rho\lambda}(z,w)s_\rho(\x),$$ and $s_\rho(\x)=\sum_{\mu\unlhd\rho} K_{\rho\mu}m_\mu(\x)$ where $K_{\rho\mu}$ are the Kostka numbers. We have $$\begin{aligned}
&s_{(n)}(1,u,0,0,\dots)=1+u+O(u^2)\\
&s_{(n-1,1)}(1,u,0,0,\dots)=u+O(u^2)\end{aligned}$$ and $$s_\rho(1,u,0,0,\dots)=O(u^2)$$ for any other partition $\rho$. Hence, $$\tilde{H}_\lambda(1,u,0,0,\dots;z,w)
=\tilde{K}_{(n)\lambda}(z,w)(1+u)
+\tilde{K}_{(n-1,1)\lambda}(z,w)u+O(u^2).$$ From Macdonald [@macdonald p. 362] we obtain $\tilde{K}_{(n)\lambda}(a,b)=1$ and $\tilde{K}_{(n-1,1)\lambda}(a,b)=\phi_\lambda(a,b)-1$. Hence, finally, $$\tilde{H}_\lambda(1,u,0,0,\dots;z,w)=
1+\phi_\lambda(z,w)u+O(u^2).$$
It follows that $(z^2-1)^{-1}(1-w^2)^{-1}\sum_{n\geq
1}\H_{(n-1,1)}(z,w)T^n$ equals the coefficient of $u$ in $$\Log\left(\sum_\lambda\calH_\lambda(z,w)\left(1
+\phi_\lambda(z^2,w^2)u+O(u^2)\right)
T^{|\lambda|}\right)=\Log\left(A_0(T)+A_1(T)u+O(u^2)\right).$$ The claim follows from the general fact $$\Log\left(A_0(T)+A_1(T)u+O(u^2)\right)=\Log \,
A_0(T)+\frac{A_1(T)}{A_0(T)}u+O(u^2).$$
Combining Proposition \[Log-fmla\] with we obtain the following.
Conjecture \[conjCV=HS\] is equivalent to the following combinatorial identity $$1+(z^2-1)(1-w^2)\frac{A_1(z,w;T)}{A_0(z,w;T)}
=\prod_{n\geq 1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)}.
\label{comb}$$
The main result of this section is the following theorem.
\[euler-spec\] Formula (\[comb\]
| 1,604
| 3,315
| 2,019
| 1,498
| null | null |
github_plus_top10pct_by_avg
|
f symbols $x_1, \dots, x_n, x\in \{{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},\varnothing \}$, we have (differential graded) vector spaces $\mathcal P(x_1,\dots, x_n; x_0)$ over $k$, which label the operations with $n$ inputs with colors $x_1,\dots, x_n$, and one output with color $x$. The operad $\mathcal P$ comes with maps $\circ_i: \mathcal P(x_1,\dots,x_n;x) \otimes \mathcal P(y_1,\dots,y_m;x_i) \to \mathcal P(x_1,\dots,x_{i-1},y_1,\dots, y_m,x_{i+1},\dots, x_n;x)$ which label the composition in $\mathcal P$, and with an action of the symmetric group $S_n$, which, for $\sigma\in S_n$, maps $\mathcal P(x_1,\dots, x_n;x)\to \mathcal P(x_{\sigma(1)},\dots,x_{\sigma(n)};x)$. These maps have to satisfy the usual associativity and equivariance axioms of colored operads.
$\mathcal P$ is called a 0/1-operad if the color $\varnothing$ can appear only as an output, and the only nontrivial spaces with one input are $\mathcal P({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}};{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})=k$ and $\mathcal P({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}};{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})=k$. We assume furthermore, that there are fixed generators of the spaces $\mathcal P({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}};{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}})=k$ and $\mathcal P({
\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[
| 1,605
| 1,396
| 1,713
| 1,478
| 757
| 0.800722
|
github_plus_top10pct_by_avg
|
certain irreducible representation (irrep) of ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$ labeled by $(m,h,k)$. Then the ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$ structure factors straight through the differential operator $\mathcal{D}_{x}$, leaving a new differential operator $\mathcal{D}_{u}^{(m,h)}$ which only takes $u$ derivatives. This greatly simplifies computations, since the partial differential equations have been converted into ordinary differential equations (ODEs). Because of the ${\ensuremath{SL(2,\mathbb{R})\times U(1)}}$-invariance, notice that $\mathcal{D}_{u}^{(m,h)}$ only depends on $m$ and $h$, which label the irrep, and not on $k$, which labels the descendant number within the irrep.
Covariant differentiation preserves isometry group irrep labels {#sec:general-statement}
---------------------------------------------------------------
Let us first make a general statement about how the presence of a group of isometries acting on the manifold can be useful in separation of variables. The conclusions obtained in this subsection will also justify our motivations of finding group representations for NHEK’s isometry. Consider a manifold $\mathcal{M}$ with metric $g_{ab}$, metric-compatible connection ${\nabla}$, and an isometry Lie group $G$ acting on the manifold. Let $\alpha_{(i)}\in \mathfrak{g}$ be a basis for the Lie algebra, with representation $\{X_{(i)}\}$ on the manifold. Further, let $c^{(i)(j)}$ be the inverse of the Killing form of the Lie algebra in this basis [@Barut:1986dd]. Then we also have a quadratic Casimir element, which acts on any tensor $\mathbf{t}$ as $$\begin{aligned}
\Omega \cdot \mathbf{t} \equiv \sum_{i,j} c^{(i)(j)}
{\mathcal{L}}_{X_{(i)}} {\mathcal{L}}_{X_{(j)}} \mathbf{t}
\,.\end{aligned}$$ Irreps of $G$ will be labeled by eigenvalues $\lambda_{i}$ of *some* of the KVFs, and the eigenvalue $\omega$ of the Casimir $\Omega$.
First, we need a lemma on the commutation relation of manifold isometries and covariant derivatives, $$\left[ \mathcal{L}_{X_{(i)
| 1,606
| 2,681
| 2,004
| 1,591
| null | null |
github_plus_top10pct_by_avg
|
,t)$, but, for both $b=2$ and $b=3$, the function $w_c(u,t)$ for fixed $t$ is monotonically decreasing function (see figure \[fig:fdCSAWs\]). Dependence of $w_c(u,t)$ on $t$, when $u$ is fixed, is presented in figure \[fig:wcODt\], for several values of $u$.
![Critical value of the inter-chain interaction parameter $w_c(u,t)$, depicted as a function of $t$, for three different values of intra-chain interaction parameter $u$, in the cases of $b=2$ and $b=3$ 3D SG fractals. []{data-label="fig:wcODt"}](figure6.eps)
As one can see, the limiting values $t=0$ and $t=1$ are also included in this figure. However, in these cases different fixed points, from those obtained for $0<t<1$, can be reached, which is expounded in the following paragraphs.
First, we analyze the value $t=0$, which represents the limiting case, within the CSAWs model, when the energy $\varepsilon_t$ (corresponding to the repelling of two different chain monomers, placed at sites which are nearest neighbours to a crossing site) is infinitely large. Starting with the initial values (\[pocuslovi\]), it can be shown that, in the case of the $b=3$ fractal, the same fixed points of the RG equations (\[eq:RGAi\]) and (\[eq:RGBi\]), as for $0<t<1$ are reached. However, for the $b=2$ fractal, it can be seen, from the explicit form of the RG equations (\[eq:b2jednacinaA1\])–(\[eq:b2jednacinaB2\]), that $t=0$ leads to $A_2^{(r)}=A_3^{(r)}=0$, for every $r$, $x_2$, $x_3$, $u$ and $w$. This is due to the topology of this fractal, and a consequence is that the fixed point $(A_E,B_E,C^*,C^*,0,0,A_4^*,A_4^*,0)$ corresponds to the critical values $w=w_c(1\leq u< u_\theta,t=0)$. The coordinates of this fixed point $A_E=A^*$, $B_E=B^*$ and $C^*$ are given in the part “extended 3D chain ($u<u_\theta$)" of the table \[tab:CSAWs\], while $A_4^*=0.1164$, and the concomitant critical exponent is $\varphi=0.8439$. The remaining fixed points are the same as for $0<t<1$.
The second limiting value ($t=1$) corresponds to the case $\varepsilon_t=0$ (when there is no r
| 1,607
| 930
| 2,279
| 1,657
| 1,704
| 0.786621
|
github_plus_top10pct_by_avg
|
_{ki}\Big\|^4\\
&=O(K^{-1})\end{aligned}$$ as $K, m\to \infty$. On the other hand, from the Assumption \[assumption2\], we get $${\mathbb{P}}\Big(\max_{1\leq i\leq K} \|R_{km}\|>\epsilon K^{1/2}\Big)\to 0$$ as $K, m\to \infty$. So we can complete the proof.
\[lem-2\] Let $$S_K =\frac{1}{K}\sum_{k=1}^{K}(Y_{km}-\mu)(Y_{km}-\mu)^\top.$$ Under the conditions of Theorem \[theorem2\], we have $S_K \stackrel{p}{\longrightarrow} \Sigma $ as $K,m\to \infty$.
Note that $$\begin{aligned}
& (Y_{km}-\mu)(Y_{km}-\mu)^\top \\
&=\Big(\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}\Big)\Big(\frac{1}{\sqrt{m}}\sum_{i=1}^m\eta_{ki}+R_{km}\Big)^\top\\
&=\frac{1}{m}\Big(\sum_{i=1}^m\eta_{ki}\Big)\Big(\sum_{i=1}^m\eta_{ki}\Big)^\top
+2\frac{1}{\sqrt{m}}\Big(\sum_{i=1}^m\eta_{ki}\Big)R_{km}^\top + R_{km}R_{km}^\top.\end{aligned}$$ Now we consider the convergence of the $(j,l)$ element of $S_K$ for $1\leq j,l\leq p$. For any $\epsilon>0$, $$\begin{aligned}
& {\mathbb{P}}\Big(K^{-1}\Big|\frac{1}{\sqrt{m}}\sum_{k=1}^K (\sum_{i=1}^m\eta_{kij})R_{kml}\Big| > \epsilon\Big)\\
&\leq {\mathbb{P}}\Big(\max_{1\leq k\leq K}\Big| \Big(\sum_{i=1}^m\eta_{kij}\Big)R_{kml}\Big| > \sqrt{m}\epsilon\Big) \\
&\leq \sum_{k=1}^K {\mathbb{P}}\Big(| \sum_{i=1}^m\eta_{kij}| >C m^{1/2+\alpha} \Big) + {\mathbb{P}}\Big(\max_{1\leq k\leq K}|R_{kml}| >C^{-1}m^{-\alpha} \epsilon \Big),\end{aligned}$$ where $C$ is a constant which will go to infinity finally. Since $\eta_{kij}$, $k,i=1,2,\cdots$ are independent and identically distributed random variables with mean zero and finite fourth moment, $${\mathbb{P}}\Big(| \sum_{i=1}^m\eta_{kij}| > Cm^{1/2+\alpha}\Big)\leq C^{-4}m^{-2-4\alpha} {\mathbb{E}}| \sum_{i=1}^m\eta_{kij}|^4 =C^{-4}O(m^{-4\alpha}).$$ It follows from Assumption \[assumption2\] that $${\mathbb{P}}\Big(K^{-1}|\sum_{k=1}^K (\sum_{i=1}^m\eta_{kij})R_{kml}| > \epsilon\Big) \to 0,$$ if we let $K,m\to \infty$ as a first step, then let $C\to \infty$ as a second step. Similarly, $${\mathbb{P}}\Big(K^{-1}|\sum_{k=1}^K R_{kmj}R_{kml}|>\e
| 1,608
| 1,760
| 1,083
| 1,456
| null | null |
github_plus_top10pct_by_avg
|
his completes the proof, since that element $\bar s$ is above $s_0$ and forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary.
$(S)[S]$ implies ****.
We next need:
\[P. Larson\]\[larson\] Suppose
- ****, and
- for sufficiently large $\theta$ and stationary $E\subseteq\omega_1$, for any $X\in H(\theta)$, there is a Chang model $M$ with $M\cap\omega_1\in E, X\in M$ and $|M\cap\omega_2|=\aleph_1$.
Then if $\{A_\alpha:\alpha<\omega_2\}$ are stationary subsets of $\omega_1$, $M\cap\omega_1=\delta$ is in uncountably many $A_\alpha,\alpha\in M$.
It is well known that ${\mathrm}{NS}_{\omega_1}$ is $\aleph_1$-complete, since the diagonal union of $\aleph_1$ non-stationary subsets of $\omega_1$ is non-stationary. It follows that $\mathcal{P}(\omega_1)/{\mathrm}{NS}_{\omega_1}$ is a complete Boolean algebra, because (1) says it satisfies the $\aleph_2$-chain condition. Since it is complete, for each $\alpha<\omega_2$ there is a stationary $B_\alpha$ which is the sup of $\{A_\beta:\beta\in(\alpha,\omega_2)\}$. Let $E$ be the inf of the family of $B_\alpha$’s. By saturation, $E$ is really the inf of an $\aleph_1$-sized family, and so is itself stationary. Given any $\alpha\in\omega_2$, we can find an $\eta(\alpha)>\alpha$ such that the diagonal union of $\{A_\beta:\beta\in(\alpha,\eta(\alpha))\}$ includes $E$, mod ${\mathrm}{NS}_{\omega_1}$. It follows that there is a cub $C\subseteq \omega_2$ such that for each $\alpha\in C$, there is a subset of $\{A_\beta:\beta\in(\alpha,\alpha^+)\}$ of cardinality $\aleph_1$ with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$, where $\alpha^+$ denotes the next element of $C$ after $\alpha$.
Now let $M$ be an elementary submodel of a suitable $H(\theta)$, with $\langle A_\alpha:\alpha<\omega_2\rangle$, $E$, and $C\in M$ and $\delta=M\cap\omega_1\in E$, $|M\cap\omega_2|=\aleph_1$. We claim $\delta$ is an element of uncountably many $A_\alpha,\alpha\in M$.
Since the cub $C$ divides $\omega_2$ into $\aleph_2$ disjoint intervals, $C\hspace{.03cm}\cap M$ div
| 1,609
| 3,517
| 2,236
| 1,646
| null | null |
github_plus_top10pct_by_avg
|
rhd_{r} h := r(g)hr(g)^{-1}$, for all $g\in G$ and $h\in H$. Hence $\widehat{(\alpha_{0}, \beta_{0})} = \{(\rhd_{r}, \beta_{0})
~|~ r: G\to H ~~ {\rm ~is~ a~ morphism~ of~ groups}\}$. We restate this observation as follows: let $H$ and $G$ be two groups. Then there exists $(H, G, \alpha', \beta')$ a matched pair such that $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong H\times G$ (isomorphism of groups that fixes $H$) if and only if the action $\beta'$ is trivial and there exists a morphism of groups $r: G
\rightarrow H$ such that the action $\alpha'$ is given by $g \rhd'
h = r(g) h r(g)^{-1}$ for all $g \in G$, $h \in H$.
More generally, as a first application of [Theorem \[th:sch1\]]{}, we shall prove the following necessary and sufficient condition for a bicrossed product to be isomorphic to a left version of a semidirect product in the category $B_{1}(H,G)$:
[\[co:lake1\]]{} Let $H$, $G$ be two groups and $\alpha : G\times H \to H$ be an action as automorphisms of $G$ on $H$. The following statements are equivalent:
1. There exists $(H, G, \alpha', \beta')$ a matched pair of groups such that $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong
H {}_{\alpha}\ltimes G$ an isomorphism of groups that fixes $H$.
2. The action $\beta'$ is trivial and there exists a pair $(r,
v)$, where $v\in {{\rm Aut}\,}(G)$ is an automorphism of $G$, $r: G \to H$ is a map such that $${\label{eq:s3'}}
r(g_{1}g_{2}) = r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2}) \bigl)$$ for all $g_1$, $g_2 \in G$ and the action $\alpha '$ is given by $${\label{eq:s1'}}
g \rhd' h = r(g)(v(g) \rhd h) r(g)^{-1}$$ for all $g\in G$ and $h\in H$.
The isomorphism $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G
\to H {}_{\alpha}\ltimes G$ in $B_{1}(H,G)$ is given by $\psi (h,
g) = (h r(g), v(g))$, for all $h\in H$, $g\in G$.
We apply [Corollary \[co:cosch\]]{} in the case that $\beta$ is the trivial action. It this context, using the fact that $v$ is bijective, it follows from [(\[eq:s2\])]{} that the action $\beta'$ is trivial and [(\
| 1,610
| 3,022
| 1,901
| 1,608
| null | null |
github_plus_top10pct_by_avg
|
s then given by ${\mathbf{F}}={\mathbf{I}}_{N_s}$ with equal power allocation between the $N_s$ data streams, since the transmitter does not have the full CSI. At the receiver, the digital decoder is the joint ML decoder for maximizing the throughput.[^6] With the aforementioned transceiver architecture and CSI assumptions, the system throughput with a selected ${\widetilde{\mathbf{H}}_{\psi,V}}$ is given by [@Tse_05_Fundamentals] $$\label{eq:Coptimal}
R_{{\widetilde{\mathbf{H}}_{\psi,V}}}=\log_2\left|{\mathbf{I}}_{L_r}+\frac{\rho}{L_t}{\widetilde{\mathbf{H}}_{\psi,V}}{\widetilde{\mathbf{H}}_{\psi,V}}^H\right|,$$ where $\rho=P/\sigma^2_n$ denotes the transmit power to noise ratio.
Throughput Gain of Employing Reconfigurable Antennas {#sec:thrgainana}
====================================================
In this section, we analyze the performance gain of employing the reconfigurable antennas in terms of the throughput. With the optimal reconfiguration state selection, the instantaneous system throughput is given by $$\label{eq:defRsc}
R_{{\widehat{\psi}}}=\max_{\psi\in\left\{1,\cdots,\Psi\right\}}R_{\psi},$$ where $$\begin{aligned}
\label{eq:defRs}
R_{\psi}&=\log_2\left|{\mathbf{I}}_{L_r}+\frac{\rho}{L_t}{\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}{\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}^H\right|\notag\\
&=\max_{{\widetilde{\mathbf{H}}_{\psi,V}}\in\left\{\tilde{\mathcal{H}}_\psi\right\}}\log_2\left|{\mathbf{I}}_{L_r}+\frac{\rho}{L_t}{\widetilde{\mathbf{H}}_{\psi,V}}{\widetilde{\mathbf{H}}_{\psi,V}}^H\right|\end{aligned}$$ represents the maximum achievable throughput under the reconfiguration state $\psi$, ${\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}$ denotes the optimal low-dimensional virtual channel of ${\mathbf{H}_{\psi,V}}$, and $\tilde{\mathcal{H}}_\psi$ denotes the set of all possible $L_r\times L_t$ submatrices of ${\mathbf{H}_{\psi,V}}$. Here, the optimal reconfiguration state is the reconfiguration state that maximizes the throughput.
Average Throughput Gain {#sec:appAvethrougai}
-------------
| 1,611
| 575
| 1,993
| 1,552
| 3,021
| 0.775408
|
github_plus_top10pct_by_avg
|
:flutter_convertor/ItemBought.dart';
class task extends StatelessWidget{
@override
...
}
class taskScreen extends StatefulWidget{
@override
taskState createState() => new taskState();
}
class taskState extends State<taskScreen> {
bool isButtonEnabled = false;
//Callback function i want to call in order to change the state of my Button
formReady(){
setState(() {
isButtonEnabled = !isButtonEnabled ;
});
}
@override
Widget build(BuildContext context) {
Column taskScreen = Column(
children: <Widget>[...
ItemBought(), //ITEMBOUGHT WIDGET
...
Column(
children: <Widget>[
FlatButton(
onPressed: isButtonEnabled ? _completePage : null,
child: Text(
...
),
),
],
)]);
return Scaffold(
appBar: AppBar(
title: Text('Task Screen')),
body: taskScreen,
);
} }
ItemBought.dart
import 'package:flutter/material.dart';
import 'package:flutter_convertor/addImage.dart';
class ItemBought extends StatefulWidget{
@override
_ItemBoughtState createState() => _ItemBoughtState();
}
class _ItemBoughtState extends State<ItemBought> {
...
@override
Widget build(BuildContext context) {
return Container(
child: Column(
children: <Widget> [
...
addImage(),
...
]
)
);
}
}
addImage.dart
import 'package:flutter/material.dart';
import 'package:flutter_convertor/task.dart';
class addImage extends StatefulWidget{
const addImage({this.formReady});
final VoidCallback formReady;
@override
_addImageState createState() => _addImageState();
}
class _addImageState extends State<addImage> {
// const void({taskState.formReady});
// final VoidCallback formReady;
@override
Widget build(BuildContext context) {
return Container(
child: TextField(
onChanged: (text){
| 1,612
| 2,372
| 62
| 928
| 82
| 0.827561
|
github_plus_top10pct_by_avg
|
66] implied that there is no algorithm which decides whether copies of a given finite set of polyominoes tile $\mathbb{Z}^2$. It is unknown whether the same is true for tilings by a single polyomino. For tilings of $\mathbb{Z}$ by sets of general one-dimensional tiles, such an algorithm does exist, as demonstrated by Adler and Holroyd [@ah81]. Kisisel [@kisisel01] introduced an ingenious technique for proving that certain tiles do not tile $\mathbb{Z}^2$ without having to resort to case analysis.
A similar problem is to consider whether a tile $T$ tiles certain finite regions, such as cuboids. There is a significant body of research, sometimes involving computer searches, on tilings of rectangles in $\mathbb{Z}^2$ by polyominoes (see, for example, Conway and Lagarias [@cl90] and Dahlke [@dahlke]). Friedman [@friedman] has collected some results on tilings of rectangles by small one-dimensional tiles. More recently, Gruslys, Leader and Tomon [@gltomon16] and Tomon [@tomon16] considered the related problem of partitioning the Boolean lattice into copies of a poset, and similarly Gruslys [@gruslys16] and Gruslys and Letzter [@gl16] have worked on the problem of partitioning the hypercube into copies of a graph.
Preliminaries and the odd case
==============================
We begin with the case of $k$ odd. This is technically much simpler than the general case, and allows us to demonstrate some of the main ideas in the proof of Theorem \[mainthm\] in a less complicated setting.
\[kodd\] Let $T$ be the punctured interval $\underbrace{\texttt{\emph{XXXXX}}}_{k}\!\texttt{.}\!\underbrace{\texttt{\emph{XXXXX}}}_{k}$, with $k$ odd. Then $T$ tiles $\mathbb{Z}^3$.
Throughout this section, $T$ is fixed, and $k \geq 3$. We will not yet assume that $k$ is odd, because the tools that we are about to develop will be relevant to the general case too.
We start with an important definition from [@gltan16]: a *string* is a one-dimensional infinite line in $\mathbb{Z}^d$ with every $(k+1)$th point removed. Crucially, a string i
| 1,613
| 282
| 2,166
| 1,472
| 1,994
| 0.783758
|
github_plus_top10pct_by_avg
|
N$.
Then $\mathcal K$ admits the skew field of fractions $F(\mathcal K)$ and $F(\mathcal K)\simeq F_{n,N-n}.$
Since the action of $\mathcal M$ is trivial on $L(t_{n+1}, \ldots, t_N)$ then we have the $G$-equivariant embedding $$(L(t_1, \ldots, t_n)*\mathcal M)\otimes L(t_{n+1}, \ldots, t_N) \hookrightarrow L*\mathcal M$$ and $$((L(t_1, \ldots, t_n)\otimes L(t_{n+1}, \ldots, t_N))*\mathcal M)^G\hookrightarrow (L*\mathcal M)^G.$$ Moreover, both algebras have the same skew fields. But $$((L(t_1, \ldots, t_n)*\mathcal M)\otimes L(t_{n+1}, \ldots, t_N))^G\simeq ((L(t_1, \ldots, t_n)*\mathcal M)^G\otimes L(t_{n+1}, \ldots, t_N)^G.$$ Since $(L(t_1, \ldots, t_n)*\mathcal M\simeq R_n$, $R_n^G=R_n^{G_n}$, $F(R_n)\simeq F(A_n)$ and $L(t_{n+1}, \ldots, t_N)^G\simeq
L(z_1, \ldots z_{N-n})$ we obtain $$F(\mathcal K)\simeq F(A_n)^G\otimes L(z_1, \ldots z_{N-n}).$$ The result follows from Theorem \[main\].
\[theorem-main-theorem\] Let $U$ be a linear Galois algebra in $(L*\mathcal M)^G$ such that
- $L=\c(t_{ij}, i=1, \ldots, N; j=1, \ldots, n_i; z_1, \ldots z_m)$, for some integers $m$, $n_1$, $\ldots$, $n_N$;
- $G=G_1\times \ldots \times G_N$, where $G_s$ acts normally only on variables $t_{s1}$, $\ldots$, $t_{s, n_s}$, $s=1, \ldots, N$;
- $\mathcal M\simeq \mathbb Z^{n}$ acts by shifts on $t_{11},\dots, t_{N,n_N}$, $n=n_1+\ldots + n_N$.
Then $U$ admits the skew field of fractions $F(U)$ and $F(U)\simeq F_{n, m}.$
Follows from Theorem 3 and Lemma \[lem-skew-main\].
\[remark-case-of-s-n\] The action of the symmetric group $S_{N}$ on $\c^{N}$ by permutations of the coordinates is obviously linear and it normalizes the action of $\mathcal M=\mathbb Z^{N}$ on $\c^{N}$ by shifts. Recall that $U(gl_{n})$ and $U(sl_{n})$ are linear Galois algebras with respect to their Gelfand-Tsetlin subalgebras [@FO]. Then Theorem \[theorem-main-theorem\] implies immediately the Gelfand-Kirillov conjecture for $gl_{n}$ and $sl_{n}$. In a similar manner one obtains the Gelfand-Kirillov conjecture for restricted Yangians of type $A$
| 1,614
| 822
| 1,458
| 1,504
| null | null |
github_plus_top10pct_by_avg
|
C'\subset{\mathbb{R}}^3\backslash G$ implies that for all $n$ large enough, one has $x_n-\lambda \omega_n\in y_0+\lambda_0' C'\subset{\mathbb{R}}^3\backslash G$ for all $0<\lambda\leq \lambda_0'$, hence $t(x_n,\omega_n)\leq \lambda$, from which $\limsup_{n\to\infty} t(x_n,\omega_n)\leq \lambda$, and finally $\limsup_{n\to\infty} t(x_n,\omega_n)=0$.
We then suppose that $(x_n,\omega_n)\to (y_+,\omega_0)$. Since $y_0=y_+ - \tau_-(y_0,\omega_0)\omega_0$ and $y_0-\lambda_0'\omega_0\in y_0+\lambda_0'C'\subset{\mathbb{R}}^3\backslash G$, one has for all $n$ large enough and all $0<\lambda\leq \lambda_0'$, that $$x_n- (\tau_-(y_0,\omega_0)+\lambda)\omega_n \in y_0+\lambda_0'C'\subset{\mathbb{R}}^3\backslash G,$$ whence $t(x_n,\omega_n)\leq \tau_-(y_0,\omega_0)+\lambda$, which gives the upper limit $$\limsup_{n\to\infty} t(x_n,\omega_n)\leq \tau_-(y_0,\omega_0).$$
In order to obtain the corresponding lower limit, which then shows the existence and the correct value of the limit we were set out to show, notice that if $0<\sigma<\tau_-(y_0,\omega_0)$, then $y_0+\sigma\omega_0\in G$, hence for all $n$ large enough $x_n-(\tau_-(y_0,\omega_0)-\sigma)\omega_n
\in G$ (since $x_n-(\tau_-(y_0,\omega_0)-\sigma)\omega_n\to
x_0-(\tau_-(y_0,\omega_0)-\sigma)\omega_0=y_0+\sigma\omega_0$) , which implies that $t(x_n,\omega_n)\geq \tau_+(y_+,\omega_0)-\sigma$, and thus $\liminf_{n\to\infty} t(x_n,\omega_n)\geq \tau_-(y_0,\omega_0)-\sigma$. Letting $\sigma\to 0+$ allows us to conclude that $$\liminf_{n\to\infty} t(x_n,\omega_n)\geq \tau_-(y_0,\omega_0).$$
\[le:esccont:1\] Let $(x_0,\omega_0,E_0)\in (G\times S\times I)\cup \Gamma_+\cup \Gamma_-$. Then $$\lim_{(x,\omega,E)\to (x_0,\omega_0,E_0)} t(x,\omega)
=\begin{cases}
t(x_0,\omega_0),\ & \textrm{if}\ (x_0,\omega_0,E_0)\in G\times S\times I \\
& \textrm{and}\ \big(x_0-t(x_0,\omega_0)\omega_0,\omega_0,E_0\big)\in \Gamma_-, \\
\tau_+(x_0,\omega_0),\ & \textrm{if}\ (x_0,\omega_0,E_0)\in \Gamma_+ \\
& \textrm{and}\ (x_0-\tau_+(x_0,\omega_0)\omega_0,\omega_0,E_0)\in \Gamma_-,\\
0,\ & \te
| 1,615
| 1,697
| 1,683
| 1,564
| null | null |
github_plus_top10pct_by_avg
|
ad {\left|\psi_2\right>}={\left|x_1^4\right>}$\
$\lambda_{max}\simeq 7,40$
- Third orbit\
${\left|\varphi_3\right>}={\left|x_0^1\right>},\quad {\left|\psi_3\right>}={\left|x_1^8\right>}$\
$\lambda_{max}\simeq 6,63$
The maximal eigenvalue of $X$:\
$\lambda_{max}(X)\simeq 17,38$.\
The corresponding sums of probabilities appearing on the right hand side of eq. (\[a3\]) are written out explicitly in Appendix. Having computed the (maximal) quantum mechanical values of the relevant sums of probabilities one can study the corresponding Bell inequalities. To this end we compute the coefficients $c(\alpha)$ entering the inequalities (\[a4\]). There are 16 observables 8 for $"$Alice$"$ and 8 for $"$Bob$"$. Therefore, the assumed joint probability is defined for $3^{16}$ configurations. We used the computer to check, for three examples above, how many times any given configuration appears in 72 terms in $"$classical$"$ counterpart of the right hand side of eq. (\[a3\]). The result are summarized in Appendix. It follows that the relevant sums of probabilities have the upper bounds 16, 18 and 16 for the examples I, II and III, respectively. This implies that in all three examples the Bell inequalities are broken.
Interpretation in terms of game theory
======================================
As it has been described in Refs. [@ugur] and [@ugur1] the Bell inequalities can be discussed in terms of a nonlocal game. To this end we assume there are two players, Alice and Bob and an arbitrator who sends Alice a value $s$ and Bob a value $t$, $s=1,2,\ldots,8$, $t=1,2\ldots,8$; assume that all of 64 possible values of ${\left(s,t\right)}$ are equally likely. After receiving the numbers $s$ and $t$ from an arbitrator both Alice nad Bob transmit back the numbers $a$ and $b$, respectively, where $a=0,1,2$, $b=0,1,2$. They win iff the configuration ${\left(a_s=a,b_t=b\right)}$ appears in the sum of probabilities corresponding to the right hand side of eq. (\[a3\]). Let us consider for definiteness the example I. Using (\[
| 1,616
| 2,044
| 2,518
| 1,813
| 1,910
| 0.784466
|
github_plus_top10pct_by_avg
|
IIA/O6 result is concerned, our expression is valid for the specific orbifold model we are considering, but it would be interesting to investigate whether the same superpotential can be derived for more general orientifolds and whether it has a natural explanation in the context of generalised geometry.
We now want to derive the explicit form of the superpotential in terms of the $S,T,U$ moduli. We use for the fluxes the conventions defined in the first column of Tables \[TableRRfluxes\], \[TableNSfluxes\] and \[allPfluxes\]. Following [@Aldazabal:2006up], we consider the simpler case of three equivalent tori, [*i.e.*]{} the isotropic case. This greatly simplifies the explicit expressions for the superpotentials in both the IIB and IIA theory. We consistently remove the indices from all the fluxes $q_i$, $e_i$, $a_i$, $b_{ij}$, $h_i$, $f_i$, $g_{ij}$ and from the corresponding primed and/or barred fluxes in Tables \[TableRRfluxes\], \[TableNSfluxes\] and \[allPfluxes\]. We rename $b_{ii}$ and $g_{ii}$ as $\beta$ and $\gamma$, and similarly for the equivalent primed and/or barred fluxes [@Aldazabal:2006up]. The IIB/O3 superpotential in eq. leads to $$\begin{aligned}
W_{\rm IIB/O3} & = e_0 +3ieU -3qU^2 +imU^3\nonumber \\
& + S\big( ih_0 -3aU +3i\bar{a}U^2 - \bar{h}_0 U^3 \big) \nonumber \\
& +3T \Big( -i h - (2b+\beta ) U + i (2 \bar{b} + \bar{\beta} ) U^2 + \bar{h} U^3 \Big) \nonumber \\
& + 3ST \big( -f +i(2g +\gamma)U + (2\bar{g} + \bar{\gamma} )U^2 -i\bar{f}U^3 \big) \nonumber \\
& - 3T^2\big( f' +i(2g'+ \gamma' )U - (2\bar{g}' + \bar{\gamma}' )U^2 -i\bar{f}' U^3 \big) \quad , \label{isotropicsuperpotential}\end{aligned}$$ and it can be shown that the IIA/O6 superpotential in eq. gives the same expression with $U$ and $T$ interchanged.
T-duality rules and exotic branes
=================================
In the IIB $T^6/[\mathbb{Z}_2\times \mathbb{Z}_2 ]$ O3 orientifold setup the
| 1,617
| 2,644
| 1,972
| 1,607
| 1,212
| 0.792642
|
github_plus_top10pct_by_avg
|
correction from the approximate form. Because the approximate form of the gluino-sbottom correction is equal to the terms in the exact form proportional to the $B_0$ Passarino-Veltman functions the discrepancy must be due to the terms in the exact form proportional to the $B_1$ Passarino-Veltman functions.
![We plot the $B_0^{\widetilde{g}}$ term against the $B_1^{\widetilde{g}}$ term ($B_0^{\widetilde{g}}$ and $B_1^{\widetilde{g}}$ are defined in the text). Darker shades represent increasing sbottom masses. As the sbottom masses increase from 1 TeV to $\ge$4 TeV, the $B_0^{\widetilde{g}}$ terms becomes smaller and the two terms are nearly the same magnitude. Furthermore, the points along the vertical line have $(\mu\tanb -A_b)\simeq0$.[]{data-label="fig:gl-B0B1"}](new_PLOTS/gluino-B0vsB1.pdf){width="56.00000%"}
We refer to the term in \[Eq:fullgluino\] containing the $B_{0(1)}$ Passarino-Veltman functions and its prefactor as the $``B_{0(1)}^{\widetilde{g}}"$ term. In \[fig:gl-B0B1\], the $B_0$ term is plotted against the $B_1^{\widetilde{g}}$ term. The color gradient from light to dark represents increasing sbottom masses from 1 TeV to $\ge$4 TeV. As the sbottom masses get pushed toward more than a few TeV, the $B_0^{\widetilde{g}}$ term decreases while the $B_1^{\widetilde{g}}$ term slightly increases, and the two terms are nearly the same magnitude. The increase in the $B_1^{\widetilde{g}}$ term can be understood by considering \[Eq:pass-velt-B1\] in the limit of large sbottom masses. For a fixed gluino mass[^3] and in the limit of large $x$, one finds for the $B_1^{\widetilde{g}}$ term, . Thus the $B_1^{\widetilde{g}}$ term grows logarithmically with increasing sbottom masses, which explains why there appears to be a constant vertical shift of $\sim$8% from the diagonal line along which the approximation is equal to the exact expression in \[fig:gl-ex-app\]. In this regime, where the $B_0^{\widetilde{g}}$ term is small, it is therefore important that the $B_1^{\widetilde{g}}$ term not be ignored. Fina
| 1,618
| 696
| 819
| 1,313
| 1,505
| 0.788776
|
github_plus_top10pct_by_avg
|
ULTS {#s:tests}
============
We implement our Poisson solver in [Athena++]{} which is a state-of-art astrophysical magnetohydrodynamics (MHD) code with very flexible coordinate and grid options. Using Cartesian and uniform/logarithmic cylindrical grids, we test our solver on a few test problems to check its accuracy, convergence, and parallel performance. We also run time-dependent simulations of a gravitationally-unstable isothermal ring to check if the gravity module combines well with the MHD solver of [Athena++]{} to produce the expected results of ring fragmentation. For all tests presented below, we set the gravitational constant to $G = 1$.
Uniform Sphere Test {#s:staticPot}
-------------------
To test the accuracy of our Poisson solver, we consider a uniform sphere with radius $r_0$ and density $\rho_0$. The analytic gravitational potential of such a sphere is given by $$\Phi_a(r) =
\begin{dcases}
-2\pi G \rho_0 \left(r_0^2 - \tfrac{1}{3} r^2\right), & (r < r_0),\\
-\frac{4\pi G\rho_0 r_0^3}{3r}, & (r > r_0),
\end{dcases}$$ where $r$ denotes the distance from the center of the sphere. We take $r_0=0.2$ and $\rho_0=1$, and place it at an off-centered position $(x_0, y_0, z_0)$ in Cartesian coordinates and $(R_0, \phi_0, z_0)$ in cylindrical coordinates, and calculate the gravitational potential $\Phi$ numerically. Table \[tb:sphere\_test\] lists the grid dimension, resolution, and sphere position in each coordinate system adopted. As a measure of accuracy, we define the relative error between the numerical solution and the analytic solution as $${\epsilon} \equiv \left|\frac{\Phi-\Phi_a}{\Phi_a}\right|,$$ evaluated at the cell centers. Figures \[fig:car\_sphere\]–\[fig:cyl\_sphere\] plot the test results on the Cartesian, uniform cylindrical, and logarithmic cylindrical grid, respectively. Panels (a) and (b) plot the one-dimensional (1D) cut profiles of $\Phi$ and $\epsilon$ along the $x$- or $R$-direction, respectively, while panel (c) gives the 2D distribution of $\epsilon$ in the $z=0$
| 1,619
| 2,010
| 2,550
| 1,745
| 3,923
| 0.769283
|
github_plus_top10pct_by_avg
|
i-algebraic group.
Since $\iota(\PSL(2,\Bbb{C}))$ and $\PU(2,1)$ are simple Lie groups with trivial centers, we deduce that they are semi-algebraic groups (see [@semi]). Thus the sets $$\begin{array}{l}
\{(g,h,gh): g,h\in Aut(BV)\}\\
\{(g,g^{-1}): g\in Aut(BV)\}
\end{array}$$ are semi-algebraic sets. Therefore $Aut(BV)$ is a semi-algebraic group.
\[c:liedim\] Let $\Gamma\subset\PSL(2,\Bbb{C})$ be a discrete non-elementary group such that $\iota\Gamma$ leaves invariant a complex ball $B$. Then:
1. \[l:1\] The group $Aut(BV)$ is a Lie group of positive dimension.
2. \[l:2\] We have $\psi\Lambda(\Gamma)\subset Ver\cap\partial B$.
3. \[l:3\] Set $C=\partial B\cap Ver$. Then the set $\psi^{-1}(C)$ is an algebraic curve of degree at most four.
4. \[l:4\] The group $\iota^{-1}Aut(BV)$ can be conjugated to a subgroup of $Mob(\hat{\Bbb{R}})$, where $Mob(\hat{\Bbb{R}})=\{\gamma\in\PSL(2,\Bbb{C}):\gamma(\Bbb{R}\cup\{\infty\})=\Bbb{R}\cup\{\infty\}\}$.
5. \[l:5\] The set $\psi^{-1}(C)$ is a circle in the Riemann sphere.
6. \[l:6\] The set $C$ is an $\Bbb{R}$-circle, [*i.e.*]{} $C=\gamma(\partial\Bbb{H}^2_{\Bbb{C}}\cap\Bbb{P}^2_{\Bbb{R}})$, where $\gamma\in\PSL(3,\Bbb{C})$ is some element satisfying $\gamma(\Bbb{H}^2_{\Bbb{C}})=B$.
7. \[l:7\] The set $Ver\cap (\Bbb{P}^2_{\Bbb{C}}\setminus\overline{B})$ is non-empty.
8. \[l:8\] The set $Ver\cap B$ is non-empty.
Let us start by showing (\[l:1\]). Since $Aut(BV)$ is semi-algebraic, we deduce that it is a Lie group with a finite number of connected components (see [@semi]). On the other hand, since $Aut(BV)$ contains a discrete subgroup, we conclude $Aut(BV)$ has positive dimension.\
Now let us prove part (\[l:2\]). Let $x\in \Lambda(\Gamma)$. Then there is a sequence $(\gamma_n)\subset \Gamma$ of distinct elements such that $\gamma_n\xymatrix{\ar[r]_{m\rightarrow\infty}&}x$ uniformly on compact sets of $\widehat{\Bbb{C}}\setminus \{x\}$. From Lemma \[l:pseudo\] we know that $\iota\gamma_n\xymatrix{\ar[r]_{m\rightarrow\infty}&}\psi(x)$ uniformly on compact set
| 1,620
| 824
| 1,422
| 1,543
| null | null |
github_plus_top10pct_by_avg
|
We obtain: $$\begin{aligned}
\mbox{Term 3} &=& (-i) (-1)^{ea} {f^c}_{de} (
(c_3 \kappa^{ad} \frac{1}{(\bar z - \bar w})^2 j^e_z (w)
\nonumber \\
& &
+ {f^{ad}}_g (\frac{c_4}{\bar{z}-\bar{w}} :j^e_z j^g_{\bar z}: (w) +
\frac{(c_4-g)(z-w)}{(\bar z- \bar w)^2}
:j^e_z j^g_{z}: (w)
\nonumber \\
& & + \mbox{order zero in the separation.}\end{aligned}$$ There is also the more involved term where we contract first with $j^e_z(x)$, and then further with $j^d_{\bar z}(w)$: $$\begin{aligned}
\mbox{Term 4} &= & \lim_{:x \to w:} X^{ae}(z,x) (-i) {f^c}_{de} j^d_{\bar z}(w)\end{aligned}$$ where $$\begin{aligned}
X^{ae}(z,x)
& \sim & \tilde{c} \kappa^{ae} 2 \pi \delta(z-x)
\nonumber \\
& &
+ {f^{ae}}_g (\frac{c_4-g}{\bar{z}-\bar{x}} j^g_z (z) + \frac{(c_2-g)}{z-x} j_{\bar{z}}^g (z)
\nonumber \\
& &
+ \frac{g}{4} \log |z-x|^2 (\partial_z j_{\bar{z}}^g(z) - \partial_{\bar z} j^g_z(z)))
\nonumber \\
& &
+ (-1)^{ae} : j_z^e j_{\bar z}^a :(z)
\nonumber \\
& &
+ {A_{}^{ac}}_{gh} \frac{\bar z - \bar x}{z-x} :j^{g}_{\bar z}
j^{h}_{\bar z}: (z) - {B_{}^{ac}}_{gh} \log |z-x|^2 :j^{g}_{z}
j^{h}_{\bar z}: (z)\nonumber \\
& & + {C_{}^{ac}}_{gh} \frac{z-x}{\bar{z}-\bar{x}}
: j_z^{g} j_z^{h}:(z)
\nonumber \\
& & + \mbox{order 1 in the separation and higher order in the parameter $f^2$.} \nonumber\end{aligned}$$ Let’s sum these four terms and discuss the vanishing of the total operator product order by order. The contact terms and double pole terms were already treated in [@Ashok:2009xx]. We cancel them as follows:
1\. There are terms proportional to $\partial_{\bar w} 2 \pi \delta(z-w)$. These have coefficients: $$\begin{aligned}
c_- \tilde{c} \kappa^{ac}
+c_+ c_3 \kappa^{ac} \end{aligned}$$ which vanishes since the coefficients of the current algebra satisfy : $$\begin{aligned}
c_- \tilde{c} &=&- c_+ c_3\end{aligned}$$
2\. There are terms proportional to $2 \pi \delta(z-w)$ with coefficient: $$\begin{aligned}
- c_- & {f^{ac}}_g (c_2-g) j_{\bar z}^g (w)+ c_+ {f^{ac}}_g c_4 j_{\bar z}^g(w)
-i {f^c}_{de} \tilde{c} \kappa^{
| 1,621
| 1,929
| 1,640
| 1,665
| null | null |
github_plus_top10pct_by_avg
|
_j^\top(\hat\psi - \psi) +
\frac{1}{2n}\delta^\top \Lambda_j \delta, \quad \forall j \in \{1, \ldots s\}$$ where $\delta = \sqrt{n}(\hat\psi - \psi)$ and $\Lambda_j = \int_0^1 H_j( (1-t)\psi + t \hat\psi) dt \in \mathbb{R}^{b \times b}$. Hence, $$\label{eq::taylor}
\sqrt{n}(\hat\theta - \theta) = \sqrt{n}(\hat\nu - \nu) + R$$ where $\nu = G\psi$, $\hat\nu = G \hat\psi$ and $R$ is a random vector in $\mathbb{R}^s$ whose $j^{\mathrm{th}}$ coordinate is $$R_j = \frac{1}{2\sqrt{n}} \delta^\top
\left[ \int_0^1 H_j( (1-t)\psi + t \hat\psi) dt \right]
\delta.$$ By Lemma \[lem:hyper\] below, there exists a constant $C>0$, depending on $A$ only, such that $$\label{eq::CLT}
\sup_{P\in {\cal P}_n}
\sup_t
\Bigl|\mathbb{P}(\sqrt{n}||\hat\nu - \nu||_\infty \leq t) -
\mathbb{P}(||Z_n||_\infty \leq t)\Bigr| \leq C
\frac{1}{\sqrt{v}} \left( \frac{ \overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6},$$ where $Z_n \sim N_s(0,\Gamma)$.
Now we bound the effect of remainder $R$ in (\[eq::taylor\]). First, by assumption (see Equation \[eq:H.and.B\]), we have that, almost everywhere, $$\label{eq:bound.H}
\sup_{u \in [0,1]} \| H_j( (1-u)\psi + u \hat\psi) \|_{\mathrm{op}} \leq
\overline{H},$$ from which it is follows that $$\| R \|_\infty \leq \frac{\overline{H} ||\delta||^2}{2\sqrt{n}},$$ with the inequality holding uniformly in $\mathcal{P}_n$. Next, consider the event $\mathcal{E}_n = \Bigl\{ \frac{\overline{H} ||\delta||^2}{2\sqrt{n}} < \epsilon_n\Bigr\}$ where $$\label{eq:epsilon}
\epsilon_n = C \sqrt{\frac{b \overline{v} \overline{H}^2 (\log n)^2}{n}},$$ for a sufficiently large, positive constant $C$ to be specified later. Thus, since $\delta = \sqrt{n} (\hat{\psi}- \psi)$, we have that $$\begin{aligned}
\nonumber
\mathbb{P}(\mathcal{E}_n^c) &= \mathbb{P}\left( \frac{\overline{H}
||\delta||^2}{2\sqrt{n}} > \epsilon_n\right)\\
\nonumber
& = \mathbb{P}\left( ||\hat{\psi} - \psi || >
\sqrt{ \frac{2 \epsilon_n}{ \sqrt{n} \overline{H}}}\right)\\
\nonumber
& = \mathbb{P}\left( ||\hat{\psi} - \psi || >
C \sqrt{
| 1,622
| 2,035
| 1,156
| 1,541
| null | null |
github_plus_top10pct_by_avg
|
=========
In this appendix we gather various technical results related to the current algebra .
The current algebra at order $f^2$ {#jMCOPE}
----------------------------------
In [@Ashok:2009xx] the current algebra was computed at the order of the poles. The discussion of section \[bootstrap\] shows that we can compute the less-singular terms by demanding consistency with current conservation and the Maurer-Cartan equation. In this appendix we will give details of this computation, and derive in particular the value of the new coefficients in the current algebra .
In this particular calculation, we show how to restore various signs that are associated to the fact that we deal with a super Lie algebra. Since we use the special algebraic structure of supergroups with zero Killing form, these signs are crucial. To set up the problem, we establish conventions for the metric inverse and the contraction of indices: $$\begin{aligned}
\kappa_{ab} \kappa^{cb} &=& {\delta_a}^c
\nonumber \\
j_a &=& \kappa_{ab} j^b \nonumber \\
{[} t_a, t_b {]} &=& i t_c {f^c}_{ab}.\end{aligned}$$ We contract indices south-west north-east[^6].
As explained in section \[bootstrap\] current conservation implies that the tensors $A,B,C$ that appear in each one of the three OPEs are equal. To compute them we ask for the vanishing of the OPE between a current and the Maurer-Cartan operator : $$\begin{aligned}
c_- \partial_{\bar{z}} j_{L,z}^c
- c_+ \partial_z j^c_{L,\bar z} - i {f^c}_{de} :j_{L,z}^e j_{L,\bar z}^d:.\end{aligned}$$ Below we compute the OPE between the (left) current $j_{\bar z}^a$ and the Maurer-Cartan operator. For ease of writing, we will separate various terms in the calculation. We first calculate the operator product of the current with the first term: $$\begin{aligned}
\mbox{Term 1} &=& j_{\bar z}^a (z) \cdot c_- \partial_{\bar{w}} j_z^c (w)
\nonumber \\
& \sim & c_- \partial_{\bar w} ( \tilde{c} \kappa^{ac} 2 \pi \delta(z-w)
\nonumber \\
& &
+ {f^{ac}}_g (\frac{c_4-g}{\bar{z}-\bar{w}} j^g_z (z) + \frac{(c_2-g)}{z-w}
| 1,623
| 1,574
| 2,103
| 1,519
| 3,203
| 0.774067
|
github_plus_top10pct_by_avg
|
g or dwindling) by turning the two evidence sets as sliding windows and adopting certain update strategies such as *Least Recently Used*(LRU). Time complexity for this optimization is $O(n\cdot(|\mathbb{E}_N|+|\mathbb{E}_A|)\cdot T_D)$, where $T_D$ denotes time complexity of divergence calculation.
Threshold {#sec:alg-threshold}
---------
One important factor in algorithm SDD-E is the value of threshold. A lower threshold rejects more instances, improving the sensitivity of anomalous data while increasing the number of false alarms. A higher threshold provide higher true negative rates yet neglecting more possible threats.
A naive but prevalent approach is to set a fixed value as the threshold(As is shown in Algorithm \[alg:sdd-r\]). This approach is easy to implement and may give satisfying results in specific cases. However, a fixed threshold requires specific analysis in the certain scenario, manual observations and tuning of parameters, which involves lots of human labour. The rule of “$3 \sigma$” declares all instances outside $[\mu - 3\sigma, \mu + 3\sigma]$ to be anomalous. It can be used to automatically determine a threshold. But as a rigid metric, it is merely an estimation of a suitable boundary considering average situations, which is far from optimality when concrete data is provided. It would be either lower than the optimum if anomalous data lies far away from the normal cluster, or higher than the optimum if the anomalies sit close to the cluster centre.
Fortunately, applying divergence as the distance measurement among data collections provides a fine property. With a reference distribution, divergences of normal data collections form a quasi-Gaussian distribution as we have seen in section \[sec:alg-opt-reference\]. The same applies to those anomalous ones.
![Threshold can be determined either by a probability density value, or a radius from the centre. In the scenario shown in the figure, the threshold can be determined by the position between two centres of the distribution, denoted
| 1,624
| 1,581
| 2,701
| 1,480
| null | null |
github_plus_top10pct_by_avg
|
ly that $$\begin{aligned}
\E{f(x_T - v_T) - f(x_T - y_T)} \leq 2TL\epsilon
\end{aligned}$$
\[l:non\_gaussian\_contraction\_anisotropic\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon\leq \frac{\Rq}{\aq\Rq^2 + 1}$. Let $x_t$, $v_t$ and $w_t$ be as defined in , , . Let $n$ be an integer and $\delta$ be a step size, and let $T:= n\delta$.
If we assume that $\E{\lrn{x_0}_2^2}$, $\E{\lrn{v_0}_2^2}$, and $\E{\lrn{w_0}_2^2}$ are each upper bounded by $8 \lrp{R^2 + \beta^2/m}$ and that $T \leq \min\lrbb{ \frac{1}{16L}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$, then $$\begin{aligned}
& \E{f(x_T - w_T)} - \E{f(x_T - v_T)} \leq 4T(L+\LN^2)\epsilon
\end{aligned}$$
For sufficiently small $\epsilon$, our assumption on $T$ boils down to $T = o(\epsilon^4)$
First, we can verify using Taylor’s theorem that for any $x,y$, $$\begin{aligned}
f(y) =& f(x) + \lin{\nabla f(x), y-x} + \int_0^1\int_0^s \lin{\nabla^2 f(x + s(y-x)), (y-x) (y-x)^T} ds dt
\numberthis \label{e:taylor1}\\
\nabla f(y) =& \nabla f(x) + \lin{\nabla^2 f(x), y-x} + \int_0^1\int_0^s \lin{\nabla^3 f(x + s(y-x)), (y-x) (y-x)^T} ds dt
\numberthis \label{e:taylor2}
\end{aligned}$$
Thus $$\begin{aligned}
& \E{f(x_T - w_T)}\\
=& \E{f(x_T - v_T)+ \lin{\nabla f(x_T - v_T), v_T - w_T} + \int_0^1\int_0^s \lin{\nabla^2 f(x_T - v_T + s(v_T-w_T)), (v_T - w_T) (v_T - w_T)^T} ds dt}\\
=& \E{f(x_T - v_T)+ \underbrace{\lin{\nabla f(x_0 - v_0), v_T - w_T}}_{\circled{1}} + \underbrace{\lin{\nabla f(x_T - v_T) - \nabla f(x_0 - v_0), v_T - w_T}}_{\circled{2}}}\\
&\quad + \E{\underbrace{\int_0^1\int_0^s \lin{\nabla^2 f(x_T - v_T + s(v_T-w_T)), (v_T - w_T) (v_T - w_T)^T} ds dt}_{\circled{3}}}
\end{aligned}$$ Recall from and that $$\begin{aligned}
v_{n\delta} =& w_0 + \sum_{i=0}^{n-1} \delta \nabla U(w_0) + {\sqrt{\delta}} \sum_{i=0}^{
| 1,625
| 2,989
| 1,506
| 1,500
| null | null |
github_plus_top10pct_by_avg
|
$t \mapsto t^3$. $0 \in [-1,1]$ is the vertex of degree $2$. The other points are not in the vertex set. The map is regarded as a $D_2$-symmetric map onto $L_2$.\
\
Step 2 Around a vertex of degree $n \geq 3$.\
We consider a $C^{\infty}$ map on a $C^{\infty}$ manifold of dimension $m>1$ into the plane whose image is the bounded domain surrounded by the five arcs as FIGURE \[fig:2\]. We construct the map as a special generic map such that the restriction to the singular set is embedding and the singular value set is the disjoint union of two thick arcs. Note that inverse images of regular values are standard spheres ($m>2$) or two point sets ($m=2$). For more precise facts on special generic maps into the plane, see [@saeki2] for example.
![A special generic map into the plane.[]{data-label="fig:2"}](zahyoone.eps){width="30mm"}
Let $f$ be a $C^{\infty}$ function whose value is $1$ on the interval $x \leq 0$ and which is strictly increasing on the interval $x \geq 0$. We can do in the step before so that by composing a $C^{\infty}$ map $T$ defined as $$T(x,y):=
\begin{cases}
(x,y) & (y \leq 0) \\
(f(y)x,y) & (y \geq 0)
\end{cases}$$ the resulting image is as FIGURE \[fig:3\]: the arc connecting two thick arcs is defined as a part of a quadratic curve of a form $x^2-y^2=a$. We can determine $c(t)$ and we can naturally determine a $C^{\infty}$ function $c$ on $[0,t]$ based on quadratic curves of this form. We can extend this to the bounded domain except two arcs in the bottom as a $C^{\infty}$ map. However, we can extend to the arcs as a continuous map: we define the map so that the values on these two arcs are $(0,-1)$. Thus we can define a map $C$ on the bounded domain, Through the steps, we can define a $(C^r,C^0)$ function on the given $m$-dimensional manifold to a closed interval $[-1,c(t)]$ where we identify $x$ with $(0,x)$ for $-1 \leq x \leq c(t)$.
![A deformation of the special generic map into the plane of FIGURE \[fig:2\].[]{data-label="fig:3"}](zahyotwo.eps){width="30mm"}
We consider the map on an
| 1,626
| 273
| 963
| 1,609
| 2,556
| 0.778885
|
github_plus_top10pct_by_avg
|
etting. By Jensen’s inequality, we have $$\begin{aligned}
\sum_{i = 2}^d \frac{1}{\lambda_i(L)} \geq \frac{(d-1)^2}{\sum_{i = 2}^d \lambda_i(L)} = \frac{(d-1)^2}{\Tr(L)} = \frac{(d-1)^2}{n}.\end{aligned}$$
### Proof of Lemma \[lem:cr\_lem\]
Define $\L_j(\theta)$ for $j \in [n]$ such that $\L(\theta) = \sum_{j = 1}^n \L_j(\theta)$. Let $H^{(j)}(\theta) \in \mathcal{S}^d$ be the Hessian matrix such that $H^{(j)}_{i\i}(\theta) = \frac{\partial^2\L_j(\theta)}{\partial\theta_i \partial \theta_{\i}}$ for $i,\i \in S_j$. We prove that for all $j \in [n]$, $$\begin{aligned}
\label{eq:cr01}
\E_\theta[H^{(j)}({\boldsymbol{0}})] \;\; \succeq\;\; - \frac{2p\log(\kappa_j)^2}{\kappa_j(\kappa_j-1)} \sum_{\i<i \in S_j}(e_i - e_{\i})(e_i - e_{\i})^{\top} \,.\end{aligned}$$ In the following, we omit superscript/subscript $j$ for brevity. With a slight abuse of notation, we use $\I_{\{\Omega^{-1}(i) = a\}} = 1$ if item $i$ is ranked at the $a$-th position in all the orderings $\sigma \in \Omega$. Let $\P[\theta]$ be the likelihood of observing $\Omega^{-1}(p) = i^{(p)}$ and the set $\Lambda$ (the set of the items that are ranked before the $p$-th position). We have, $$\begin{aligned}
\label{eq:cr1}
\P(\theta) = \sum_{\sigma \in \Omega} \Bigg(\frac{\exp\big(\sum_{m = 1}^{p} \theta_{\sigma(m)} \big)}{\prod_{a=1}^{p} \Big(\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big) \Big)}\Bigg)\,.\end{aligned}$$ For $i,\i \in S_j $, we have $$\begin{aligned}
\label{eq:cr7}
H_{i\i}(\theta) = \frac{1}{\P(\theta)} \frac{\partial^2\P(\theta)}{\partial\theta_i \partial \theta_{\i}} - \frac{\nabla_i \P(\theta) \nabla_{\i} \P(\theta)}{\big(\P(\theta)\big)^2} \end{aligned}$$ We claim that at $\theta = {\boldsymbol{0}}$,
$$\begin{aligned}
-H_{i\i}({\boldsymbol{0}}) = \left\{ \begin{array}{rl}
C_1 & \;\; \text{if} \; i = \i, \; \big\{\Omega^{-1}(i) \geq p \big\} \label{eq:cr81} \\
C_2 + A_3^2 - C_3 & \;\; \text{if} \; i = \i,\; \big\{\Omega^{-1}(i) < p \big\} \label{eq:cr82}\\
-B_1 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) \g
| 1,627
| 1,396
| 1,601
| 1,517
| null | null |
github_plus_top10pct_by_avg
|
3}\pi^9 {\alpha '}^5} -n_{\rm short} \frac{\pi }{12} v(1 - v)~.
\label{vtotal2}$$ As above, $n_{\rm short}$ denotes the nine-dimensional bulk density of D-particles near the (moving) brane world.
As in the previous oversimplified example, the transition of the D8-brane world from a region densely populated with D-particles to a depleted D-void causes these negative contributions to the total energy density to diminish, leading potentially to an acceleration of the expansion of the Universe. The latter lasts as long as the energy density remains positive and overcomes matter contributions. In the particular example shown in Fig. \[fig:inhom\], $R_2(t)$ diminishes as time elapses and the D8-brane moves towards the right-hand stack of D-branes, so the net long-distance contribution to the energy density (\[total2\]) (the first term) increases, tending further to increase the acceleration.
It is the linear density of the D-particle defects $n(z)$ encountered by a propagating photon that determines the amount of refraction. The density of D-particles crossing the D-brane world cannot be determined from first principles, and so may be regarded as a parameter in phenomenological models. The flux of D-particles is proportional also to the velocity $v$ of the D8-brane in the bulk, if the relative motion of the population of D-particles is ignored.
Towards D-Foam Phenomenology
============================
In order to make some phenomenological headway, we adopt some simplifying assumptions. For example, we may assume that between a redshift $z < 1$ and today ($z=0$), the energy density (\[vtotal2\]) has remained approximately constant, as suggested by the available cosmological data. this assumption corresponds to [^4]: $$\label{zero}
0 \simeq \frac{d\mathcal{\rho}_8}{d t } = H(z) (1 + z) \frac{d\mathcal{\rho}_8}{d z} =
\frac{v^5}{2^{7}\pi^9 {\alpha '}^5} -\frac{d n_{\rm short}}{d t} \frac{\pi }{12} v(1 - v)~.$$ where $t$ denotes the Robertson-Walker time on the brane world, for which $d/dt = -H(z) (1 + z) d/d z$,
| 1,628
| 3,591
| 3,119
| 1,491
| null | null |
github_plus_top10pct_by_avg
|
use the same width was used for every target: $$TDI~ = ~{log}_{2}\left( {1 + D} \right),$$ where D is the target distance from the center in the GUI interface, and throughput was finally calculated by dividing completion time from TDI. Even though throughput contains the information of completion time, throughput and completion time were separately reported in this study because throughput cannot directly represent completion time. Among the seven performance metrics, user effort is introduced as a measure of effort to complete the target acquisition task. Since the RMS feature used in this study has been used for evaluating user effort in previous EMG studies \[[@pone.0186318.ref055]--[@pone.0186318.ref057]\], we also used the mean RMS value averaged over all recording channels and the whole experimental time to calculate the user effort for each experimental run.
10.1371/journal.pone.0186318.t001
###### Description of seven performance metrics.
{#pone.0186318.t001g}
Metric Description
------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------
Completion Rate (%) Ratio of successfully achieved targets to the total number of targets
Completion Time (s) Mean time taken achieving a target; the time limit of 10 s was counted for missed targets
Throughput (bit/s) Ratio of the task difficulty index (TDI) and the completion time; information transfer rate defined by Fitt's law \[[@pone.0186318.ref058]\]
Path Efficiency (%) Ratio of the shortest path between a target and the initial (center) position to the actual path travelled
Overshoot (%) Ratio of the number of times a target is reached but left before 1 s dwell time to the total number of targets
Stopping Path (travel length) Path length travelled in a target circle for 1 s dwell time; it is applied only for achieved
| 1,629
| 104
| 2,442
| 1,869
| null | null |
github_plus_top10pct_by_avg
|
(\chi ){\otimes }_{\Bbbk }{\mathbb{K}})v=M^\chi (\Lambda )$. Since $v$ is contained in any nonzero $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule of $M^\chi (\Lambda )$ by Thms. \[th:EErel\], \[th:PBWtau\], it follows that $I^\chi (\Lambda )=0$.
\[le:FFv\] Let $t\in \{1,2,\dots ,{b}-1\}$. Assume that $\Lambda (K_{i_1}L_{i_1}^{-1})=\chi ({\alpha }_{i_1},{\alpha }_{i_1})^{t-1}$. Then in $M^\chi (\Lambda )$ $$\begin{aligned}
F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1}
F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots
F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} F_{i_1}^t v_\Lambda =0
\end{aligned}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$.
By Eq. and Lemma \[le:hwvector\] it suffices to prove that $$F_{\beta _{n+\nu -\mu }} F_{\beta _{n+1-\mu }}^{{b^{\chi}} (\beta _{n+1-\mu })-1}
F_{\beta _{n+2-\mu }}^{{b^{\chi}} (\beta _{n+2-\mu })-1}\cdots
F_{\beta _n}^{{b^{\chi}} (\beta _n)-1} \in
\sum _{\kappa =1}^{\nu -\mu } U(\chi )F_{\beta _{n+\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. By replacing $\chi $ with an appropriate element in ${\mathcal{G}}(\chi )$ and applying isomorphisms ${T}_i^-$, it suffices to show that $$F_{\beta _\nu } F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}
F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}\cdots
F_{\beta _\mu }^{{b^{\chi}} (\beta _\mu )-1} \in
\sum _{\kappa =1}^{\nu -\mu } U^-(\chi )F_{\beta _{\mu +\kappa }}$$ for all $\mu \in \{0,1,\dots ,n-1\}$ and $\nu \in \{\mu +1,\mu +2,\dots ,n\}$. The latter follows from Thms. \[th:EErel\] and \[th:PBW\] by considering ${\mathbb{Z}}^I$-degrees.
\[th:hwv\] Assume that $\chi \in {\mathcal{X}}_4$. Let $\mu \in \{1,2,\dots ,n\}$, $t\in \{1,2,
\dots ,{b^{\chi}} (\beta _\mu )-1\}$, and $\chi _\mu =r_{i_{\mu -1}}\cdots
r_{i_2}r_{i_1}(\chi )$. Assume that $$\begin{aligned}
{\rho ^{\chi}} (\beta _\mu )\Lambda (K_{\beta _\mu }L_{\beta _\mu }^{-1})
={\rho ^{\chi _\mu }}({\alpha }_{
| 1,630
| 1,598
| 1,511
| 1,532
| null | null |
github_plus_top10pct_by_avg
|
c spinor and its Dirac equation, describing massive spin 1/2 particles and antiparticles invariant under parity, which is to be discussed hereafter.
An Interlude on SU(2) Representations {#Sec3.2}
-------------------------------------
Let us pause for a moment to recall a few well known facts concerning SU(2) representations, that will become relevant in the next section. The $su(2)$ Lie algebra is spanned by three generators $T^i$ $(i=1,2,3)$ with the Lie bracket algebra $$\left[T^i,T^j\right]=i\epsilon^{ijk}\,T^k\ \ ,\ \
\epsilon^{123}=+1\ .$$
As is the case for any SU(N) algebra, [*a priori*]{}, SU(2) possesses two fundamental representations of dimension two, complex conjugates of one another, namely the spinor representations of SU(2) or SO(3). There is the “covariant” 2-dimensional representation $\underline{\bf 2}$, a vector space spanned by covariant complex valued doublet vectors $a_\alpha$ $(\alpha=1,2)$ transforming under a SU(2) group element ${U_\alpha}^\beta$, with $U^\dagger=U^{-1}$ and ${\rm det}\,U=1$, as $${a'}_\alpha={U_\alpha}^\beta\,a_\beta\ .$$ This representation is also associated to the generators $$T^i=\frac{1}{2}\sigma_i\ \ \ ,\ \ \ i=1,2,3\ ,$$ the $\sigma_i$ being the usual Pauli matrices,[^14] $$\sigma_1=\left(\begin{array}{c c}
0 & 1 \\
1 & 0
\end{array}\right)\ \ ,\ \
\sigma_2=\left(\begin{array}{c c}
0 & -i \\
i & 0
\end{array}\right)\ \ ,\ \
\sigma_3=\left(\begin{array}{c c}
1 & 0 \\
0 & -1
\end{array}\right)\ .$$
Correspondingly, the “contravariant” complex conjugate 2-dimensional representation $\overline{\underline{\bf 2}}$, spanned by vectors $a^\alpha$ ($\alpha=1,2)$, consists of complex valued vectors transforming under SU(2) group elements as $${a'}^\alpha=a^\beta\,{U^\dagger_\beta}^\alpha=
a^\beta\,{U^{-1}_\beta}^\alpha\ ,$$ and associated to the generators $T^i=\sigma^*_i/2$.
Similar considerations apply to the SU(N) case. The fact that in this general case these are the two fundam
| 1,631
| 4,439
| 1,269
| 1,233
| null | null |
github_plus_top10pct_by_avg
|
h_{i} )^2 }
e^{- i h_{k} x}
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( h_{k} - h_{i} )^2 }
\left( \Delta_{K} + 2 h_{k} - 3 h_{i} \right)
e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\nonumber \\
&+&
\sum_{K \neq L}
\biggl[
- \frac{x^2}{2} e^{- i h_{i} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) }
\nonumber \\
&-& (ix) e^{- i h_{i} x}
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 }
\left( \Delta_{K} + \Delta_{L} - 2 h_{i} \right)
\nonumber \\
&-&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{L} - \Delta_{K} ) }
e^{- i \Delta_{K} x}
+ \frac{ 1 }{ ( \Delta_{L} - h_{i} )^3 ( \Delta_{L} - \Delta_{K} ) }
e^{- i \Delta_{L} x}
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{L} - h_{i} )^3 }
\biggl\{
\Delta_{L}^2 + \Delta_{L} \Delta_{K} + \Delta_{K}^2 - 3 h_{i} ( \Delta_{L} + \Delta_{K} ) + 3 h_{i}^2
\biggr\}
e^{- i h_{i} x}
\biggr]
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W ^{\dagger} A (UX) \right\}_{K i}
\left\{ (UX)^{\dagger} A W \right\}_{i L}
\left\{ W ^{\dagger} A (UX) \right\}_{L i}
\nonumber \\
&+&
\sum_{K \neq L} \sum_{k \neq i}
\biggl[
\frac{1}{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) ( h_{k} - h_{i} ) }
(ix) e^{- i h_{i} x}
\nonumber \\
&-&
\frac{1}{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 ( h_{k} - h_{i} )^2 }
\biggl\{
\Delta_{K} \Delta_{L} + ( h_{k} - 2 h_{i} ) ( \Delta_{K} + \Delta_{L} ) + 3
h_{i}^2 - 2 h_{k} h_{i}
\biggr\}
e^{- i h_{i} x}
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) ( h_{k} - h_{i} )^2 } e^{- i h_{k} x}
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) } e^{- i \Delta_{K} x}
- \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{L} - h_{i} )^2 ( \Delta_{L} - h_{k} ) } e^{- i \Delta_{L} x}
\biggr]
\
| 1,632
| 681
| 2,018
| 1,703
| null | null |
github_plus_top10pct_by_avg
|
een these scalars, coming from the fact that we are going to restrict our study to specific backgrounds, the Friedmann-Lemaître-Robertson-Walker (FLRW) metric and the static spherically symmetric one, both in four dimensions.
For both of them, there are between the scalars relations coming from the Lovelock theorem (that are not taken into account in Ref [@17]), and also for FLRW, relations coming from the fact that this is a conformally invariant flat metric. All these relations are written in the first Appendix, and we need to express them to define the Weyl tensor, as : $$\begin{aligned}
W_{\mu\nu\alpha\beta}=R_{\mu\nu\alpha\beta}-\frac{1}{2}(R_{\mu\alpha}g_{\nu\beta}-R_{\mu\beta}g_{\nu\alpha}+R_{\nu\beta}g_{\mu\alpha}-R_{\nu\alpha}g_{\mu\beta})+\frac{1}{6}(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha})R.\end{aligned}$$ And the following rank 2 tensor that is null in four dimension because of the Lovelock theorem : $$\begin{aligned}
L_{\mu\nu} = -\frac{1}{2}g_{\mu \nu} \mathcal{E}_4 + 2 Q_{\mu \nu} -4 P_{\mu \nu} +4 R_{\ \nu \mu \ }^{\alpha \ \ \gamma} R_{\alpha \gamma} + 2 R R_{\mu \nu} =0,\end{aligned}$$ where $Q_{\mu \nu} =R_{\mu\eta\alpha}^{\ \ \ \ \beta}R_{\nu \ \ \beta}^{\ \eta \alpha}$ and $P_{\mu \nu} = R_{\nu\gamma}R^{\gamma}_{\ \mu}$. Indeed, if we vary the Lagrangian associated with the Gauss-Bonnet invariant with respect to the metric field we find : $$\begin{aligned}
\begin{split}
\delta \big( \sqrt{-g} \mathcal{E}_4 \big) =&
\sqrt{-g} \; \delta g^{\mu \nu} L_{\mu\nu}.
\end{split}\end{aligned}$$ But as we saw in equation (3), this Lagrangian can be written as a total derivative in four dimensions, which means that its contribution to the equations of motion is identically zero.
Friedmann-Lemaître Space-time
=============================
Order 6
-------
We start with the flat Friedmann-Lemaître cosmological metric describing the dynamics of the universe at very large scale, in the simpliest manner : $$\begin{aligned}
ds^2=-dt^2 + a(t)^2 \big( dr^2 + r^2 d\Omega ^2 \big),\end{align
| 1,633
| 2,524
| 1,274
| 1,626
| null | null |
github_plus_top10pct_by_avg
|
f(y_t|y_{1:t-1},~s_{1:t})$ in as required; however, it is infeasible to track $\mathbb{P}\left({\mathbf{x}}_{1:t}|y_{1:t},s_{1:t}\right)$ as the dimension of the event space increases exponentially with time. Instead, combining , and gives the conditional mass function for the current firing vector given all previous firing vectors and all MUTFs to date, $$\begin{aligned}
\mathbb{P}\left({\mathbf{x}}_{t} |~ y_{1:t},~ {\mathbf{x}}_{1:t-1},~ s_{1:t}\right) & =
\frac{f\left(y_t|~{\mathbf{x}}_{1:t},~y_{1:t-1},~s_{1:t}\right) \mathbb{P}\left({\mathbf{x}}_t|~{\mathbf{x}}_{1:t-1},~s_{1:t}\right)}{f\left(y_t|~{\mathbf{x}}_{1:t-1},~y_{1:t-1},~s_{1:t}\right)}. \label{eq:Xupdate}\end{aligned}$$ Expressions in and together lead to a fully adaptive sequential Monte Carlo (SMC) sampler which approximates the historical firing event mass function by the particle set $\left\{{\mathbf{x}}_{1:t}^{(i)}\right\}_{i=1}^{N}$, for a suitably large $N$, recursively updating the set for $t=1,\dots,T$. Algorithm \[tab:Alg\] presents the auxiliary SMC sampler [@Pit99] which, given the set of samples drawn from $\mathbf{X}_{1:t-1}|~y_{1:t-1},~ s_{1:t-1}$, creates an unweighted sample from the filtering distribution $\mathbf{X}_{1:t}|y_{1:t},s_{1:t}$, and approximates via Monte Carlo so as to update the marginal likelihood estimate $\hat{f}(y_{1:t}|~s_{1:t})$.
$\omega_t^{(i)}=f(y_t|{\mathbf{x}}_{1:t-1}^{(i)},~y_{1:t},~s_{1:t})$ $\bar{\omega}_t^{(i)}=\omega_t^{(i)}/\sum_{k} \omega_t^{(k)}$. Sample auxiliary indices $\{\zeta^{(i)}\}_{i=1}^N$ with probabilities $\{\bar{\omega}_{t}^{(i)}\}_{i=1}^N$. Sample ${\mathbf{x}}_t^{(i)}$ with probability $\mathbb{P}\left({\mathbf{x}}_t|~y_t,~{\mathbf{x}}_{1:t-1}^{(\zeta_i)},~s_{1:t}\right)$. Set ${\mathbf{x}}_{1:t}^{(i)} = \left({\mathbf{x}}_{1:t-1}^{(\zeta_i)},~
{\mathbf{x}}_t^{(i)}\right)$. Set $\log \hat{f}\left(y_{1:t}|~s_{1:t},~u\right) =
\log \hat{f}\left(y_{1:t-1}|~s_{1:t-1},~u\right) - \log N + \log \sum_i
\omega^{(i)}_t$.
Although primary interest lies in the marginal-likelihood est
| 1,634
| 1,223
| 2,196
| 1,650
| 4,014
| 0.768645
|
github_plus_top10pct_by_avg
|
e PNC will arise depending on the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$, which we now determine.
\[quadconics\] If $C>\lambda_0$ and $B=\frac{C-\lambda_0}2+1$, then the limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ consists of a union of quadritangent conics, with distinguished tangent equal to the kernel line $x=0$, and of a multiple of the distinguished tangent line.
Both $\gamma_{\lambda_0}$ and $\gamma_{\frac{\lambda_0+C}2}$ are determined by the truncation $f_{(C)}$ (since $C>\lambda_0$); hence the equations of the conics $$z=\frac {\lambda_0(\lambda_0-1)}2\gamma_{\lambda_0}y^2+\frac
{\lambda_0+C}2\gamma_{\frac{\lambda_0+C}2}y+\gamma_C$$ contributed (according to Lemma \[dominant\]) by different branches with truncation $f_{(C)}$ may only differ in the coefficient $\gamma_C$.
It is immediately verified that all such conics are tangent to the kernel line $x=0$, at the point $(0:0:1)$, and that any two distinct such conics meet only at the point $(0:0:1)$; thus they are necessarily quadritangent.
Finally, the branches that do not truncate to $f_{(C)}(y)$ must contribute kernel lines, by Lemmas \[otherbranches\] and \[nottrunc\].
The degenerate case in which only one conic arises corresponds to germs not contributing components of the projective normal cone, by dimension considerations. A component is present as soon as there are two or more conics, that is, as soon as two branches contribute distinct conics to the limit.
This leads to the description given in §\[germlist\]. We say that a rational number $C$ is ‘characteristic’ for ${{\mathscr C}}$ (with respect to $z=0$) if at least two formal branches of ${{\mathscr C}}$ (tangent to $z=0$) have the same nonzero truncation, but different coefficients for $y^C$.
\[typeV\] The set of characteristic rationals is finite.
The limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ obtained in Lemma \[quadconics\] determines a component of the projective normal cone precisely when $C$ is characteristic.
If $C\gg 0$, then branches with the
| 1,635
| 1,640
| 2,112
| 1,638
| null | null |
github_plus_top10pct_by_avg
|
ents ensures that the $\theta(a_{jm})\in {\mathcal{N}}^m$ are a free basis for the module they generate. The other conclusions of the lemma follow automatically from the construction of $\theta$.
{#step-1}
As happens with many questions about ${{W}}$-invariants, it is easy to prove that $\Theta$ is surjective on ${\mathfrak{h}^{\text{reg}}}$. Given a left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module $M$, we will write $M[\delta^{-2}]$ for the localisation ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}[\delta^{-2}]\otimes_{{\mathbb{C}}[{\mathfrak{h}}]^{{W}}}M$. Clearly, when $M$ is a left ${\mathbb{C}}[{\mathfrak{h}}]$-module, $M[\delta^{-2}]$ is naturally isomorphic to ${\mathbb{C}}[{\mathfrak{h}}][\delta^{-1}]\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}M$.
\(1) The inclusion $ \Theta[\delta^{-2}] : {\mathcal{J}}[\theta^{-2}]
\hookrightarrow (\operatorname{{\textsf}{ogr}}{\mathcal{N}})[\delta^{-2}]$ is an equality.
\(2) The induced map $ \theta[\delta^{-2}] : {\mathcal{J}}[\theta^{-2}]
\to {\mathcal{N}}[\delta^{-2}]$ is an isomorphism. This map is graded under the ${\mathbf{E}}$-grading and is a filtered isomorphism under the order filtration, in the sense that $\theta[\delta^{-2}]$ maps $ \operatorname{{\textsf}{ord}}^n{\mathcal{J}}[\delta^{-2}] $ isomorphically to $\operatorname{{\textsf}{ord}}^n{\mathcal{N}}[\delta^{-2}]$ for each $n$.
\(1) By $B_{k,k-1}[\delta^{-2}] = eH_{c+k}\delta[\delta^{-2}]e= e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})e,$ for any $k\in{\mathbb{C}}$. Repeated application of this shows that $B_{ij}[\delta^{-2}] = e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})e$ and hence, by Corollary \[morrat-cor\], that ${\mathcal{N}}[\delta^{-2}] = e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})eH_c=e(D({\mathfrak{h}^{\text{reg}}})\ast {{W}}) .$ Since $\operatorname{{\textsf}{ord}}(\delta^2)=0$, we deduce that $(\operatorname{{\textsf}{ogr}}{\mathcal{N}})[\delta^{-2}] =
e({\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]\ast {{W}}).$ On the other hand, since $\delta^{2k}\in J^k\delta^k\subsete
| 1,636
| 1,197
| 983
| 1,615
| null | null |
github_plus_top10pct_by_avg
|
i y_i\\ \pi v_i&1+\pi z_i \end{pmatrix}$ and $\tilde{m}_{i,i}$ as $\begin{pmatrix} \tilde{s}_i&\pi \tilde{y}_i\\ \pi \tilde{v}_i&1+\pi \tilde{z}_i \end{pmatrix}$ such that $\tilde{s}_i=\mathrm{id}$ mod $\pi \otimes 1$. Then $$\label{ea25'}
\sigma({}^t\tilde{m}_{i,i})h_i\tilde{m}_{i,i}=(-1)^{i/2}\begin{pmatrix}\sigma({}^t\tilde{s}_i)&\sigma(\pi\cdot {}^t \tilde{v}_i)\\ \sigma(\pi\cdot {}^t \tilde{y}_i)&1+\sigma(\pi \tilde{z}_i) \end{pmatrix}
\begin{pmatrix} a_i&0\\ 0 &1 +2\bar{\gamma}_i \end{pmatrix} \begin{pmatrix} \tilde{s}_i&\pi \tilde{y}_i\\ \pi \tilde{v}_i&1+\pi \tilde{z}_i \end{pmatrix}.$$ The $(1,2)$-block of $\sigma({}^t\tilde{m}_{i,i})h_i\tilde{m}_{i,i}$ is $(-1)^{i/2}\pi(a_i\tilde{y}_i-\sigma({}^t\tilde{v}_i))+\pi^2(\ast)$ for a certain polynomial $(\ast)$. Therefore, by observing the $(1, 2)$-block of Equation (\[ea1\]), we have $$\mathcal{X}_{i,1,2}(m)=\bar{a}_iy_i+{}^tv_i+\mathcal{P}^i_{1, 2}.$$ Here, $\mathcal{P}^i_{1, 2}$ is a polynomial with variables in the entries of $m_{i-1, i}, m_{i+1, i}$. Note that this is an equation in $(B\otimes_AR)/(\pi\otimes 1)(B\otimes_AR)$. Since $m$ actually belongs to $\mathrm{Ker~}\varphi(R)/\tilde{G}^1(R)$, we have the following equation by the argument made at the beginning of this paragraph: $$\label{ea25}
\mathcal{X}_{i,1,2}(m)=\bar{a}_iy_i+{}^tv_i+\mathcal{P}^i_{1, 2}=\bar{b}_i=0.$$ Thus we get polynomials $\mathcal{X}_{i,1,2}$ on $\mathrm{Ker~}\tilde{\varphi}/\tilde{M}^1$, vanishing on the subscheme $\mathrm{Ker~}\varphi/\tilde{G}^1$.\
5. Assume that $i$ is even and that $L_i$ is *of type* $\textit{I}^e$. The argument used in this step is similar to that of the above Step (4). By Equations (\[ea14\]), (\[ea15\]), (\[ea16\]), and (\[ea17\]) which involve an element of $\tilde{M}^1(R)$, each entry of $b_i', e_i', d_i', f_i'$ has $\pi$ as a factor so that $b_i'\equiv b_i=0, e_i'\equiv e_i=0, d_i'\equiv d_i=0, f_i'\equiv f_i=0$ mod $(\pi\otimes 1)(B\otimes_AR)$. Let $\tilde{m}\in \mathrm{Ker~}\tilde{\varphi}(R)$ be a lift of $m$. By using an argu
| 1,637
| 1,297
| 1,774
| 1,567
| 3,456
| 0.772228
|
github_plus_top10pct_by_avg
|
adherent* *to* *$A$ is given by* *is the union* ***of $A$ with its boundary.*
*The* *interior of $A$* $$\textrm{Int}(A)\overset{\textrm{def}}=\{ x\in X\!:(\exists N\in\mathcal{N}_{x})\textrm{ }(N\subseteq A)\}\label{Eqn: Def: Interior}$$ *consisting of those points of $X$ that are in $A$ but not in its boundary,* $\textrm{Int}(A)=A-\textrm{Bdy}(A)$*, is the largest open subset of $X$ that is contained in $A$. Hence it follows that* $\textrm{Int}(\textrm{Bdy}(A))=\emptyset$, *the boundary of $A$ is the intersection of the closures of $A$ and $X-A$,* *and a subset $N$ of $X$ is a neighbourhood of $x$ iff* $x\in\textrm{Int}(N)$*.$\qquad\square$*
The three subsets $\textrm{Int}(A)$, $\textrm{Bdy}(A)$ and *exterior* of $A$ defined as $\textrm{Ext}(A):=\textrm{Int}(X-A)=X-\textrm{Cl}(A)$, are pairwise disjoint and have the full space $X$ as their union.
**Definition 2.3.** ***Derived and Isolated sets.*** *Let $A$ be a subset of $X$. A point $x\in X$ (which may or may not be a point of $A$) is a* *cluster point of* $A$ *if every neighbourhood $N\in\mathcal{N}_{x}$ contains atleast one point of $A$* ***different from*** *$\mathbf{x}$. The* *derived set of $A$* $${\textstyle \textrm{Der}(A)\overset{\textrm{def}}=\{ x\in X\!:(\forall N\in\mathcal{N}_{x})\textrm{ }(N\bigcap(A-\{ x\})\neq\emptyset)\}}\label{Eqn: Def: Derived}$$ *is the set of all cluster points of $A$. The complement of* $\textrm{Der}(A)$ in $A$ $$\textrm{Iso}(A)\overset{\textrm{def}}=A-\textrm{Der}(A)=\textrm{Cl}(A)-\textrm{Der}(A)\label{Eqn: Def: Isolated}$$ *are the* *isolated* *points* *of* $A$ *to which no proper sequence in $A$ converges, that is there exists a neighbourhood of any such point that contains no other point of $A$ so that* *the only sequence that converges to* $a\in\textrm{Iso}(A)$ *is the constant sequence $(a,a,a,\cdots)$.$\qquad\square$*
Clearly,
$$\begin{array}{ccl}
{\textstyle \textrm{Cl}(A)} & = & A\bigcup\textrm{Der}(A)=A\bigcup\textrm{Bdy}(A)\\
& = & \textrm{Iso}(A)\bigcup\textrm{Der}(A)=\textrm{Int}(A)\bigcup\textrm{Bdy}(A
| 1,638
| 2,669
| 1,907
| 1,569
| 2,107
| 0.782691
|
github_plus_top10pct_by_avg
|
_n$, given in , can be bounded, on an event of probability at least $1 - \frac{1}{n}$ and using again , by $$\label{eq:new.aleph}
C \frac{k^{5/2}}{u_n^3 u^2} \overline{v} \sqrt{ \frac{ \log n}{n}},$$ for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$ and some $C>0$ dependent on $A$ only. (In light of the bounds derived next in , the dominant term in the bound on $\aleph_n$ given in is $ \overline{H} B \overline{v} \sqrt{ b\frac{
\log n}{n}}$, from which follows. We omit the details).
Thus, for each $P \in \mathcal{P}_n^{\mathrm{OLS}}$, we may now apply Theorems \[thm::coverage\] and \[thm::bonf\] on event with probability no smaller than $ 1- \frac{1}{n}$, whereby the term $\overline{H}$ is replaced by $C \frac{k}{u^3_n}$ and the terms $B$ and $\overline{\sigma}$ are bounded as in .
\[lemma::horrible\] For any $j \in {\widehat{S}}$, let $\beta_{{\widehat{S}}}(j) = e_j^\top \beta_{{\widehat{S}}} = g_j(\psi)$ where $e_j$ is the $j^{\mathrm{th}}$ standard unit vector. Write $\alpha = \alpha_{{\widehat{S}}}$ and $\Omega = \Sigma^{-1}_{{\widehat{S}}}$ and assume that $k
\geq u^2$. The gradient and Hessian of $g_j$ are given by $$\label{eq:Gj}
G^\top_j = e^\top_j \Big( \left[ - \left( \alpha^\top \otimes I_k \right)
(\Omega \otimes \Omega) \;\;\;\;\; \Omega\right] \Big) D_h$$ and $$\label{eq:Hj}
H_j = D_h^\top A_j D_h,$$ respectively, where $$A_j =
\frac{1}{2}\left( (I_b \otimes e^\top_j) H + H^\top (I_b \otimes e_j)
\right),$$ and $$H = \left[
\begin{array}{c}
- \Big( ( \Omega \otimes \Omega) \otimes I_k \Big) \Big[0_{k^3 \times
k^2} \;\;\;\;\; ( I_k \otimes \mathrm{vec}(I_k)) \Big] +
\Big( I_{k^2} \otimes (\alpha^\top
\otimes I_k) \Big) G \Big[ (\Omega \otimes \Omega) \;\;\;\;\;
0_{k^2 \times k}\Big]\\
\;\\
\Big[ - (\Omega \otimes \Omega) \;\;\;\;\; 0_{k^2 \times k} \Big]
\end{array}
\right],$$ and $D_h$ is the modified duplication matrix defined by $D \psi_h = \psi$, with $\psi_h$ the vector consisting of the subset of $\psi$ not including entries that correspond to the upper diagonal
| 1,639
| 3,984
| 1,096
| 1,182
| null | null |
github_plus_top10pct_by_avg
|
psilon\Big)\to 0$$ as $K,m\to \infty$. It remains to consider $$\label{eqA2}
(Km)^{-1} \sum_{k=1}^K\Big(\sum_{i=1}^m \eta_{kij}\Big)\Big(\sum_{i=1}^m \eta_{kil}\Big)=(Km)^{-1} \sum_{k=1}^K\sum_{i=1}^m \eta_{kij}\eta_{kil}+(Km)^{-1} \sum_{k=1}^K\sum_{1\leq i_1\not=i_2\leq m}\eta_{ki_1j}\eta_{ki_2l}.$$ The second sum on the right hand side of equality in converges to zero in probability by Markov’s inequality as $K,m\to \infty$. The first sum on the right hand side of equality in converges to the $(j,l)$ element of $\Sigma$ in probability by law of large numbers. Combining all above completes the proof.
[**Proof Theorem \[theorem2\]**]{}. (\[eq24\]) can be re-expression as $$\label{eqA3}
f(\lambda)=\frac{1}{K}\sum_{k=1}^{K} \frac{Y_{km}- \mu}{1 + \lambda^\top(Y_{km}- \mu)}=0.$$ Let $\lambda=\| \lambda\|\theta$, where $\theta\in \Theta$ is a unit vector, and $\Theta$ denotes the set of unit vector in $\mathbb{R}^p$. In the following, we show $$\|\lambda\|= O_{p}(K^{-1/2}).$$
Let $$U_{km}= \lambda^\top(Y_{km}- \mu).$$ Using the representation $1/(1+ U_{km})=1 - U_{m,k}/ (1+U_{km})$, and $\theta^\top f(\lambda)=0$, we have $$\label{eqA4}
\theta^\top (\bar{Y}_{Km}- \mu) =\| \lambda \| \theta^\top \tilde{S} \theta,$$ where $$\tilde{S}=\frac{1}{K}\sum_{k=1}^{K}\frac{(Y_{km}-\mu)(Y_{km}-\mu)^\top}{1+ U_{km}}$$ and $$\bar{Y}_{Km} = \frac{1}{K} \sum_{k=1}^{K}Y_{km}.$$ Since $0<\omega_{k}<1$, we have $1+U_{m, k} >0$, hence $$\begin{aligned}
\|\lambda\|\theta^\top S_K \theta & \leq \|\lambda\|\theta^\top \tilde{S} \theta (1+ \max_{1\leq k\leq K} U_{km})\\
&\leq \|\lambda\|\theta^\top \tilde{S} \theta (1+ \|\lambda\| Z_{K})\\
&= \theta^\top (\bar{Y}_{Km}- \mu)(1+ \|\lambda\| Z_{K}).\end{aligned}$$ The last equality follows by (\[eqA4\]). Hence, $$\|\lambda\|[ \theta^\top S_K\theta - \theta^\top(\bar{Y}_{Km}-\mu) Z_{K}] \leq \theta^\top (\bar{Y}_{Km}-\mu).$$ By the central limit theorem, $\bar{Y}_{Km}-\mu=O_{p}(K^{-1/2})$. Lemma \[lem-1\] shows $Z_{K}=o_{p}(K^{1/2})$. By Lemma \[lem-2\], the smallest eigenvalue
| 1,640
| 1,561
| 1,830
| 1,674
| null | null |
github_plus_top10pct_by_avg
|
-547ins+275 *MX1* Interferon-induced GTP metabolizing enzyme, antiviral properties Indel 275 bp Promoter SSC13 \[[@B12-viruses-11-00706]\]
-1533G\>A *USP18* Ubiquitin-specific proteases, Downregulation of interferon responses G\>A Promoter SSC5 \[[@B16-viruses-11-00706]\]
rs325981825 *HDAC6* Epigenetic labeling of histones by acetylation/deacetylation G\>A Exon 3 SSCX This study
g.2360C\>T C\>T Exon 15 SSCX This study
^1^ SSC: *Sus scrofa* chromosome; PRRSV: porcine reproductive and respiratory syndrome virus.
viruses-11-00706-t002_Table 2
######
Polymorphisms found in the sequenced fragments of the *HDAC6* gene.
*HDAC6* Fragment Polymorphism Position from ATG \* Location Change Type
------------------ -------------- ---------------------- ---------------------- ----------------------
Fragment 1 C/T −1538 Exon 1 5′UTR
Fragment 2 G/A +35 Exon 3 Missense (Arg12Lys)
C/G +63 Exon 3 Synonym (His21)
Fragment 3 G/A +2180 Intron 13 \-
G/A +2222 Exon 14 Synonym (Gln337)
G/A +2340 Intron 14 \-
C/T +2360 Exon 15 Missense (Pro360Leu)
Fragment 4 C/A +3785 Exon 19 Missense (Pro503His)
Fragment 5 G/T +9813 Exon 25 Synonym (Gln799)
C/A
| 1,641
| 5,569
| 1,266
| 551
| null | null |
github_plus_top10pct_by_avg
|
linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}})=k$.
Graphically, we represent $\mathcal P(x_1,\dots,x_n;x)$ by a tree with $n$ inputs and one output of the given color. Since the color $\varnothing$ cannot appear as an input, we may use the following convention: we represent the output $\varnothing$ with a blank line, i.e., with no line, and we say that the operation “has no output”. $$\begin{pspicture}(0,.5)(4,4)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(1.2,3)
\psline[arrowsize=0.1, arrowinset=0](2,2)(1.6,3)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,3)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2.4,3)
\psline[arrowsize=0.1, arrowinset=0](2,2)(2.8,3)
\rput[b](2,.5){$\mathcal P ({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}};\varnothing)$}
\rput[b](2,3.2){$1\,\,\, 2\,\,\, 3\,\,\, 4\,\,\, 5$}
\end{pspicture}
\quad \quad \quad
\begin{pspicture}(0,0)(4,3.6)
\psline[arrowsize=0.1, arrowinset=0](2,2)(1.2,3)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(1.6,3)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,3)
\psline[arrowsize=0.1, arrowinset=0](2,2)(2.4,3)
\psline[arrowsize=0.1, arrowinset=0](2,2)(2.8,3)
\psline[linestyle=dashed, arrowsize=0.1, arrowinset=0](2,2)(2,1)
\rput[b](2,0){$\mathcal P ({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=
| 1,642
| 469
| 1,896
| 1,836
| 672
| 0.802639
|
github_plus_top10pct_by_avg
|
tars can populate (again assuming solar metallicity.) For maximum effective temperatures of $^{MS}=50,000$ K, $^{WN}=120,000$ K, and $^{WC}=150,000$ K, for [@pauldrach01] model O stars and @hillmill WR stars, we find the following:
- Main sequence O stars can only produce $[{\hbox{{\rm Ar}\kern 0.1em{\sc iii}}}]/[{\hbox{{\rm Ar}\kern 0.1em{\sc ii}}}] \le 18$, whereas WN and WC stars can reach ratios of $40$.
- MS O stars can only produce \[\]/\[\]$=0.5$, while WC stars can reach $1.2$, and WN can reach $3$.
- MS O stars can only produce \[\]/\[\]$=4$, whereas WC can reach $12$ and WN can reach $130$.
- MS O stars and WC stars can only produce \[\]/\[\]$=2.5$, whereas WN stars can reach $14$.
We have just seen that the conversion from mid–infrared line ratio to is very different for main sequence stars than for Wolf–Rayet stars. Thus, even a modest portion of WR stars within a hot stellar population can significantly affect the line ratios. We also conclude that in a solar–metallicity starburst, if the line flux ratios exceed the maximum that main sequence stars can produce, then WR stars dominate the ionizing flux.
Spectral Synthesis Models {#sec:specsynth}
-------------------------
Given the influence of WR stars, we must consider the more realistic scenario of an evolving stellar population as the ionizing source. We used the spectral synthesis code Starburst99 version 4.0 [@starburst99] to create instantaneous starbursts with an initial mass function of Salpeter–slope [@salpeter] and initial stellar masses between $1$ M$_{\sun}$ and a variable upper mass cutoff, “” ($=100$, $75$, $60$, $50$, $40$, and $30$ .) ( in this paper always refers to the IMF, not the present-day mass function.) As in our single–star models, this version of Starburst99 uses O star model spectra from @pauldrach01 and Wolf–Rayet model spectra from the code of @hillmill, as prepared by @snc.
We created two suites of models: the first set assumed solar metallicity in Starburst99 and Cloudy, an
| 1,643
| 2,206
| 2,750
| 1,938
| null | null |
github_plus_top10pct_by_avg
|
C}}$ are tangent to the line $y=0$, leaving to the reader the necessary adjustments in the presence of such branches. We write the generator $F$ for the ideal of ${{\mathscr C}}$ as a product of formal branches $F =\prod_{i=1}^m (z-f_i(y))$. We will focus on the formal branches that are tangent to the line $z=0$, which may be written explicitly as $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$.
Now we begin the proof of Proposition \[standardform\].
Reduction to $q\ne 0$
---------------------
The first remark is that, under the assumptions that $q$, $r$, and $s$ do not all vanish, we may in fact assume that $q(t)$ is not identically zero.
\[qnot0\] If $\alpha(t)$ is as in $(\dagger)$, and $q= 0$, then $\lim_{t\to 0}\mathcal
C\circ\alpha(t)$ is a rank-2 limit.
(Sketch.) Assume $q=0$, and study the action of $\alpha(t)$ on individual monomials $x^Ay^Bz^C$ in an equation for ${{\mathscr C}}$: $$m_{ABC}:=x^A y^B (r(t) x+s(t) t^b y+t^c z)^C t^{bB}\quad.$$ There are various possibilities for the vanishing of $r$ and $s$, but the dominant terms in $m_{ABC}$ are always kernel stars, which are rank-2 limits by Lemma \[rank2lemma\].
Reduction to $b<c$
------------------
By Lemma \[qnot0\] and the last part of Lemma \[faber\] we may replace $\alpha(t)$ with an equivalent germ $$\begin{pmatrix}
1 & 0 & 0 \\
t^a & t^b & 0 \\
r(t) & s(t)t^b & t^c\end{pmatrix}$$ with $a<b\le c$, and $r(t)$, $s(t)$ polynomials of degree $<c$, $<(c-b)$ respectively and vanishing at $t=0$.
Next, we have to show that if $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit then $b<c$ and $r(t)$, $s(t)$ are as stated in Proposition \[standardform\].
\[b=c\] Let $\alpha(t)$ be as above. If $b=c$, then $\lim_{t\to 0}
{{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit.
Decompose $F(1:y:z)$ in ${{\mathbb{C}}}[[y,z]]$: write $F(1:y:z)=G(y,z)\cdot H(y,z)$, where $G(y,z)$ collects the branches that are [*not*]{} tangent
| 1,644
| 1,268
| 1,251
| 1,470
| 3,072
| 0.775028
|
github_plus_top10pct_by_avg
|
e downsampled sinogram with $140$ rays and 15 projections from $360^\circ$ angle of view. In the computations, the size of the target is set to $120 \times 120$.
Figure \[CheeseRec\](c) shows the GP reconstruction (Matérn covariance function) of the cross section of the carved cheese slice using 15 projections (uniformly spaced) out of 360$^\circ$ angle of view. For comparison, the FBP reconstruction is shown in Figure \[CheeseRec\](b).
------------ ----------------- -------------- ---------------
Covariance $\sigma_f$ (SD) $l$ (SD) $\sigma$ (SD)
function
Matérn 0.012 (0.07) 11.00 (0.08) 0.02 (0.04)
------------ ----------------- -------------- ---------------
: Estimated GP parameters for the carved cheese using Matérn covariance function. The estimates are calculated using the conditional mean, and the standard deviation (SD) values are also reported in parentheses.[]{data-label="GP parameter cheese"}
The computation times for the carved cheese are reported in Table \[Computation time cheese\].
(100,200) (150,38)[![(a) FBP reconstruction (Ram–Lak filter) of the carved cheese using dense $360$ projections. (b) Filtered backprojection reconstruction from 15 projections. (c) GP reconstruction using Matérn covariance from 15 projections.[]{data-label="CheeseRec"}](Cheese15AngFBP "fig:"){width="6.21cm"}]{} (0,-31)[![(a) FBP reconstruction (Ram–Lak filter) of the carved cheese using dense $360$ projections. (b) Filtered backprojection reconstruction from 15 projections. (c) GP reconstruction using Matérn covariance from 15 projections.[]{data-label="CheeseRec"}](Cheese_slice_1120b.pdf "fig:"){width="6.8cm"}]{} (295,38)[![(a) FBP reconstruction (Ram–Lak filter) of the carved cheese using dense $360$ projections. (b) Filtered backprojection reconstruction from 15 projections. (c) GP reconstruction using Matérn covariance from 15 projections.[]{data-label="CheeseRec"}](Cheese15AngMatern "fig:"){width="6.21cm"}]{} (90,30)[(a)]
| 1,645
| 80
| 1,805
| 1,856
| 1,082
| 0.794633
|
github_plus_top10pct_by_avg
|
farming behaviours: Centralized and Equalized. Centralized click farming refers to the scenarios that transactions are randomly generated throughout the day. A significant feature of this approach is that the cheating transactions usually assemble together in a short period of time since most workers work at the same time. Equalized click farming refers to the circumstances that click farms are arranged by some well programmed applications or teams carefully managed and strictly commit transactions according to a timetable. Thus the transaction distribution may not vary too much with and without click farming.
A research performed in China showed that 81.9% of investigated people had heard of the behaviour of click farming, 51.2% are aware of click farm and 18.9% of them had experience of click farming themselves [@yan2015report]. American researchers reported in 2015 that over 11000 sellers on Taobao were detected to have click farmed records and only 2.2% of 4000 investigated dishonest sellers had been penalized because of the cheating attempts [@netease2015research].
Current detection techniques for click farming mainly focus on user behaviours, such as browsing frequencies and periods, most common purchasing time, favourite products, remarks and whether they communicate with sellers [@simpleDetection]. Those techniques require the platform to keep lots of records and user features. However, the detection can be easily bypassed by trained workers and some well programmed applications.
![Example cumulative distribution function of original and click farmed daily transaction data[]{data-label="fig:example-ecdf"}](./ExampleCDF.pdf){width="\linewidth"}
Although it is hard to classify users as honest or malicious, we can still find clues from the sellers’ aspect. For normal sellers, their customers are usually similar since choices of products are seldom changed. Therefore, the distribution of transactions in a fixed period of time, say one day, is relatively stable. No matter how much alike between hones
| 1,646
| 3,189
| 1,706
| 1,601
| null | null |
github_plus_top10pct_by_avg
|
heorem \[t1\] we are therefore motivated to look closer at the following problem.
\[central\] Let $h$ be a polynomial of degree $d$ which is hyperbolic with respect to ${\mathbf{e}}$, and let $\epsilon >0$ and $m \in {\mathbb{Z}}_+$ be given. Determine the largest possible maximal zero, $\rho=\rho(h,{\mathbf{e}},\epsilon,m)$, of mixed characteristic polynomials: $$\chi[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t):=h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+ {\mathbf{1}})$$ subject to the conditions
1. ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+$,
2. ${\mathbf{v}}_1 + \cdots + {\mathbf{v}}_m = {\mathbf{e}}$, and
3. $\tr({\mathbf{v}}_i) \leq \epsilon$ for all $1 \leq i \leq m$.
The following conjecture was made by Marcus *et al.* [@MSS2] in the case when $h = \det$, but we take the liberty to extend the conjecture to any hyperbolic polynomial.
\[maxmax\] The maximal zero in Problem \[central\] is achieved for $${\mathbf{v}}_1=\cdots ={\mathbf{v}}_k= \frac \epsilon d {\mathbf{e}}, {\mathbf{v}}_{k+1}= \left(1-\frac k d \epsilon\right){\mathbf{e}}, {\mathbf{v}}_{k+2}={\mathbf{v}}_{k+3}= \cdots= {\mathbf{v}}_{m}=0,$$ where $k= \lfloor d/\epsilon \rfloor$.
We will prove here that Conjecture \[maxmax\] is equivalent to the following seemingly weaker conjecture.
\[maxmax2\] The maximal zero in Problem \[central\] is achieved for some ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ where ${\mathbf{v}}_i \in \Lambda_{++}\cup \{0\}$ for each $i \in [m]$.
We start by proving that there is a solution to Problem \[central\] for which the ${\mathbf{v}}_i$’s have correct traces, i.e., as those in Conjecture \[maxmax\]. By a “solution" to Problem \[central\] we mean a list of vectors ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$, as in Problem \[central\], which realize the maximal zero. First a useful lemma.
\[nicein\] Suppose ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_+$. Then $$(D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}))^2 \geq D_{\mathbf{u}}^2 h({\mathbf{w}}) \cdot D_{\mathbf{v}}^2 h({\math
| 1,647
| 934
| 2,126
| 1,448
| null | null |
github_plus_top10pct_by_avg
|
{and}\ t\in F^{n-s}R,$ as required.
\(2) Here, $rF^{n}I \subseteq F^n(rI)$ whence $rF^{n}I = r^2F^{n}I \subseteq rF^n(rI)
\subseteq rF^nI$. Since $rF^n(rI) = F^n(rI) $ this implies that $rF^n(I)=F^n(rI)$.
[**Example**]{}. It is easy to check that some hypotheses are required for the lemma to hold. For example, filter the polynomial ring $R={\mathbb{C}}[x,y]$ by $x,xy\in F^0R$ but $y\in F^1R$. Then $x,xy\in F^0(xR)$, yet $xy\not\in \sigma(x)\operatorname{gr}_FR$.
{#section-2}
We now turn to the proof of Proposition \[pre-cohh\]. As was mentioned in the inclusion $J^k\delta^k e \subseteq \operatorname{{\textsf}{ogr}}N(k)$ is easy.
\[thetainjA\] [(1)]{} For $i\geq j\geq 0$ we have $e(A^{i-j}\delta^{i-j})e
\subseteq \operatorname{{\textsf}{ogr}}B_{ij}$.
[(2)]{} The inclusion of part (1) is an equality for $i=j$ and for $i=j+1$.
[(3)]{} For $k\geq 0$ there is an inclusion $eJ^{k}\delta^{k}
\subseteq \operatorname{{\textsf}{ogr}}N(k)$ of left $eA^0e$-modules. This is an equality for $k=0$.
\(2) By the PBW Theorem \[PBW\], $\operatorname{{\textsf}{ogr}}B_{ii} = e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast {{W}})e$ and so the claim holds for $i=j$. Similarly, since $ e,\delta\in\operatorname{{\textsf}{ord}}^0(D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}})$ and $\delta$ is regular in $\operatorname{{\textsf}{ogr}}(D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}})$, Lemma \[grade-elements\] implies that $$\operatorname{{\textsf}{ogr}}B_{j+1,j} = \operatorname{{\textsf}{ogr}}( eH_{c+j+1}e_- \delta) =
\operatorname{{\textsf}{ogr}}(eH_{c+j+1}e_-)\delta=
e(\operatorname{{\textsf}{ogr}}H_{c+j+1})e_-\delta = e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast{{W}})e_-\delta = eA^1\delta e.$$
\(1) Combining part (2) with Lemma \[abstract-products\](1) and induction shows that $$(eA^1\delta^1e)^{i-j} \ = \
\operatorname{{\textsf}{ogr}}B_{i,i-1}\operatorname{{\textsf}{ogr}}B_{i-1,i-2}\cdots \operatorname{{\textsf}{ogr}}B_{j+1,j}
\ \subseteq\ \operatorname{{\textsf}{ogr}}\left( B_{i,i-1} \cdots B_{j+
| 1,648
| 1,114
| 1,412
| 1,557
| 2,818
| 0.776773
|
github_plus_top10pct_by_avg
|
^d, \|v\|_2 =1} \lrn{Gv}_2.
$.
Finally, we define a few useful constants which will be used throughout the paper: $$\begin{aligned}
&\LN := \frac{4\beta L_\xi}{\cm}, \ \ \aq:=\frac{\LR+L_N^2}{2\cm^2}, \\ &\Rq:=\max\lrbb{R,{\frac{16\beta^2 L_N}{m\cdot \cm}}} \\
&\lambda :=\min\lrbb{\frac{m}{2}, \frac{2\cm^2}{32\Rq^2}}\exp\lrp{-\frac{7}{3}\aq\Rq^2}.
\numberthis \label{d:constants}
\end{aligned}$$
$\LN$ is the smoothness parameter of the matrix $N(x)$, and we show in Lemma \[l:N\_is\_regular\] that $\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \LN^2\lrn{x-y}_2^2$. The constants $\aq$ and $\Rq$ are used to define a Lyapunov function $q$ in Appendix \[ss:defining-q\]. A key step in our proof uses the fact that, under the dynamics , $q$ contracts at a rate of $e^{-\lambda}$, plus discretization error.
[Main Results]{}\[s:main\_results\] In this section, we present our main convergence results beginning with convergence under Gaussian noise and proceeding to the non-Gaussian case.
\[t:main\_gaussian\] Let $x_t$ and $y_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{x_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{y_0}_2^2}\leq R^2 + \beta^2/m$. 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 $x_0 = y_0$, then there exists a coupling between $x_t$ and $y_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{x_{k\delta} - y_{k\delta}}_2} \leq \hat{\epsilon}
\end{aligned}$$
Alternatively, if we assume $n \geq \frac{ 3\aq\Rq^2}{\delta} \log \frac{R^2 + \beta^2/m}{
| 1,649
| 954
| 1,455
| 1,574
| null | null |
github_plus_top10pct_by_avg
|
(\nu_\beta \rightarrow \nu_\alpha) &=&
\mathcal{C}_{\alpha \beta} +
\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2
\nonumber\\
&-&
2 \sum_{j \neq k}
\mbox{Re}
\left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right]
\sin^2 \frac{ ( h_{k} - h_{j} ) x }{ 2 }
\nonumber\\
&-&
\sum_{j \neq k} \mbox{Im}
\left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right]
\sin ( h_{k} - h_{j} ) x,
\label{P-beta-alpha-final}\end{aligned}$$ where $h_{i}$ $(i=1,2,3)$ denote the energy eigenvalues of zeroth-order states of active neutrinos in matter, and $X$ is the unitary matrix which diagonalize the zeroth-order Hamiltonian used to formulate our perturbation theory. $\mathcal{C}_{\alpha \beta}$ is given in (\[Cab\]). The expression is valid under the restriction on sterile neutrino masses $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$ for $\vert W \vert^4 \gsim 10^{-4}$. For the restriction needed on the sterile state masses for smaller $W$, see section \[sec:probability-2nd\].
Notice that the normalization term, the second term both in (\[P-beta-alpha-ave-vac\]) and (\[P-beta-alpha-final\]), comes out from the contribution of zeroth-order Hamiltonian which contains all orders effect of the matter potential. It is true despite the fact that it is a vacuum effect, which indeed exists in vacuum.[^8]
The expression (\[P-beta-alpha-final\]) is a very transparent result in the sense that (1) the vacuum non-unitary mixing matrix $U$ is “dressed” in a simple way by the matter effect represented by $X$, the unitary matrix which diagonalizes the zeroth-order Hamiltonian, and (2) the probability leaking term and the normalization term stay as they are in vacuum.
Enhanced higher-order $W$ corrections and nature of unitarity violation {#sec:higher-order-corrections}
------------------------------------------------------------------------
We will point out that, despite our above theorem, there are regions of neutrino energy and baseline that condition for
| 1,650
| 2,025
| 2,316
| 1,800
| null | null |
github_plus_top10pct_by_avg
|
irst attempt to control the value of that plateau the initiator approximation has been implemented for walkers that belong to replica 1. We allow the initiator threshold to be different in replica 0 and replica 1. As an illustration we studied the carbon dimer molecule with the cc-pVQZ basis set[@jr_gaussian_1989], with ${\cal H}_{0}$ corresponding to a CAS (8,8) which is the valence space of the molecule. We use a initiator threshold of 3 and a targeted number of walker of 50k for replica 0. In Fig.\[fig1:swalknum\] we present the evolution of the number of walker on replica 0 and on replica 1, the different calculations have been run with a initiator criterion of 3, 1 and no initiator approximation on replica 1.
![Number of walkers in replica 0 (black), and in replica 1 with an initiator approximation of 3 (green), 1 (red) and no initiator approximation (blue) for the $C_{2}$ molecule in the cc-pVQZ basis set. \[fig1:swalknum\]](fig1){width="70.00000%"}
Looking at Fig.\[fig1:swalknum\] it can be seen that with no initiator approximation on $\Ket{\Psi_{1}}$, the number of walkers grows to more than 60 times the number of walkers in the reference. When using the most moderate initiator criteria of 1, this number is already reduced by more than a factor of 2, it can be further decreased by increasing the initiator threshold. However going from a threshold of 1 to 3 only reduced the total number of walkers by roughly 30 %. This is still not satisfying since there is no way to know *a priori* what the number of walkers on $\Ket{\Psi_{1}}$ is going to be. The cost of the calculation cannot be known before running it; for this reason we describe hereafter a way to control the population on replica 1 independently from the initiator threshold.
In Eq.\[eq:ps1equ\] it can be seen that $\Ket{\Psi_{1}}$ scales linearly with the perturbation $\hat{V}$. and thus scale down the perturbation by a real prefactor $\alpha$, that is typically small. This is done in practice by multiplying the matrix elements $H_{ij}$ by this fa
| 1,651
| 1,284
| 964
| 1,324
| null | null |
github_plus_top10pct_by_avg
|
v_{\eta'} v_{\eta'}^T}}}
\end{aligned}$$
For any fixed $\eta$ and $\eta'$, let’s further simplify notation by letting $u,u',v,v'$ denote $u_\eta, u_{\eta'}, v_\eta, v_{\eta'}$. Thus $$\begin{aligned}
&\tr\lrp{\lrp{uu^T - vv^T} \lrp{u'u'^T - v'v'^T}}\\
=& \tr\lrp{ \lrp{(u-v)v^T + v(u-v)^T + (u-v)(u-v)^T} \lrp{(u'-v')v'^T + v'(u'-v')^T + (u'-v')(u'-v')^T} }\\
=& \tr\lrp{ (u-v) v^T (u'-v') v'^T} + \tr\lrp{ (u-v) v^T v'(u'-v')^T } + \tr\lrp{ (u-v) v^T (u'-v')(u'-v')^T }\\
&\quad + \tr\lrp{ v(u-v)^T (u'-v') v'^T} + \tr\lrp{ v(u-v)^T v'(u'-v')^T } + \tr\lrp{ v(u-v)^T (u'-v')(u'-v')^T }\\
&\quad + \tr\lrp{ (u-v)(u-v)^T (u'-v') v'^T} + \tr\lrp{ (u-v)(u-v)^T v'(u'-v')^T }\\
&\quad + \tr\lrp{ (u-v)(u-v)^T (u'-v')(u'-v')^T }\\
\leq& \min\lrbb{16 \beta^2 L_\xi^2 \lrn{x-y}_2^2, 32\beta^3 L_{\xi}\|x-y\|_2}
\end{aligned}$$
Where the last inequality uses Assumption \[ass:xi\_properties\].2 and \[ass:xi\_properties\].3; in particular, $\lrn{v}_2\leq \beta$ and $\lrn{u-v}_2 \leq \min\lrbb{2\beta, L_\xi \|x-y\|_2}$. This proves 2. and 3. of the Lemma statement.
\[l:N\_is\_regular\] Let $N(x)$ be as defined in and $\LN$ be as defined in . Then $$\begin{aligned}
1.\ &\tr\lrp{N(x)^2} \leq \beta^2\\
2.\ &\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \LN^2\lrn{x-y}_2^2\\
3.\ &\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \frac{8 \beta^2}{\cm} \cdot \LN \lrn{x-y}_2.
\end{aligned}$$
The first inequality holds because $N(x)^2 := M(x)^2 - \cm^2 I$, and then applying Lemma \[l:M\_is\_regular\].1, and the fact that $\tr\lrp{M(x)^2 - \cm^2 I } \leq \tr\lrp{M(x)^2}$ by Assumption \[ass:xi\_properties\].4.
The second inequality is a immediate consequence of Lemma \[l:eldan-matrix\], Lemma \[l:M\_is\_regular\].2, and the fact that $\lambda_{min} \lrp{N(x)^2} = \lambda_{min} \lrp{M(x)^2 - \cm^2} \geq \cm^2$ by Assumption \[ass:xi\_properties\].4.
The proof for the third inequality is similar to the second
| 1,652
| 2,943
| 1,555
| 1,529
| null | null |
github_plus_top10pct_by_avg
|
------------------------------------------------------------------------- --
Using strong and weak LO contact interactions and two baryonic propagators one can also build three diagrams that enter at NLO. These caramel-like diagrams are shown in Fig. \[fig:caramels\]. They only differ in the position of the strong and weak vertices and in the mass of upper-leg baryonic propagator. In order to write a general expression for the three caramel diagrams we label the mass of the upper-leg propagating baryon $M_\alpha$ ($M_a=M_N$, $M_b=M_\Lambda$ and $M_c=M_\Sigma$) and the corresponding strong and weak contact vertices $C_{S(s)}^\alpha+C_{T(s)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$ and $C_{S(w)}^\alpha+C_{T(w)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$, where $\alpha=a,b,c$ corresponds to the labels of Fig. \[fig:caramels\]. It is also convenient to define $M_\alpha=M_N+\Delta_\alpha$. In the heavy baryon formalism these diagrams only contribute with an imaginary part of the form $$\begin{aligned}
V_\alpha&=i\frac{G_Fm_\pi^2}{16\pi M_N}
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times\nonumber
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times\nonumber
\sqrt{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\,.\end{aligned}$$ Few more details are given in App. \[sec:caramels\].
One pion corrections to the LO contact interactions, shown in Fig. \[fig:contact.corrections\], also enter at NLO. The net contribution of these diagrams is to shift the coefficients of the LO contact terms with functions dependent on $m_\pi$, $M_\Lambda-M_N$ and $M_\Sigma-M_N$.
![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate shift of the coefficients of the LO contact terms.[]{data-label="fig:contact.corrections"}](contacttots "fig:")\
![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate sh
| 1,653
| 492
| 2,238
| 1,709
| 2,113
| 0.78264
|
github_plus_top10pct_by_avg
|
erms are for $J'\leqslant K \leqslant S_{J}$ and since $|S_{J}|/|J'|=p$ either $K=J'$ or $K=S_{J}$. If $K=S_{J}$, then $O^{p}(K)=J'$ is $H$-conjugate to $O^{p}(S_{J})=J$, that is $J'$ is $H$-conjugate to $J$. If $K=J'$ and $J\neq J'$, then there we have the following situation: $$\xymatrix{
& S_{J} &\\
J\ar@{=}[ur]^{p} & & J'\ar@{-}[ul]_{p}\\
& J\cap J'\ar@{-}[ul]\ar@{=}[ur] &
}$$ The two subgroups $J$ and $J'$ are of index $p$ in $S_{J}$. We have $JJ'=S_{J}$. Since $J$ is normal in $S_{J}$, the intersection $J\cap J'$ is normal in $J'$. Then by the second isomorphism theorem, we have $|J'|/|J'\cap J|=p$. This implies that $p^2/|S_{J}|$ which is not possible by hypothesis. So we have $H/S_{J}Res^{G}_{H}(f_{J}^{G})=H/S_{J} f_{J}^{H}$. Using the Frobenius identity (see Proposition $3.13$ [@bouc_burnside]), we have: $$\begin{aligned}
Ind_{H}^{G}(H/S_{j} f_{J}^{H})&=Ind_{H}^{G}(H/S_{J}Res^{G}_{H}f^{G}_{J})\\
&= G/S_{J}f^{G}_{J}. \end{aligned}$$
\[mod1\] Let $G$ be a finite group.
1. $\phi_{G}(G/1)=1$.
2. if $p\mid |G|$ and $p^2 \nmid |G|$, then the family $\big(\phi_{H}\big)_{H\leqslant G}$ is stable by induction.
<!-- -->
1. The first part is obvious since $G/1f_{1}^{G}=|G|e^{G}_{1}=G/1$.
2. The second part follow from Lemma \[indu\].
\[mod2\] Let $G$ be a finite group such that $p\mid |G|$ and $p^2\nmid |G|$. Then the Burnside algebra $RB(G)$ is a symmetric algebra.
In the basis of Lemma \[basis1\] the matrix of $b_{\phi_{G}}$ is a diagonal by block matrix. The blocks are indexed by the conjugacy classes of $p$-perfect subgroups of $G$. If $J$ is a $p$-perfect subgroup such that $p\nmid |N_{G}(J)/J|$, then there is only one conjugacy class of subgroup $L$ of $G$ such that $O^{p}(L)=J$, so the block indexed by $J$ is of size $1$. The entry in this block is : $$\begin{aligned}
b_{\phi_{G}}(G/Jf_{J}^{G},G/Jf_{J}^{G}) &= \phi(G_{J}f_{J}^{G}\times G/Jf_{J}^{G})\\
&= \sum_{g\in [J\backslash G/J]} \phi(G/J\cap J^{g}f_{J}^{G})\\
&=\sum_{g\in [N_{G}(J)/J]}\phi(G/Jf_{J}^{G})\\
&=\frac{|N_{G}(J)|}{|J|} \in R^{\tim
| 1,654
| 1,167
| 1,422
| 1,467
| null | null |
github_plus_top10pct_by_avg
|
some”.
Suppose that $t-1$ is injective. Note that the injectivity of $t-1$ is equivalent to $\p_*:H_2(M)\to H_1(M_\infty)$ being the zero map. Then, for [**any**]{} $[V_y]$ as above, $\p_*([V_y])=0$. But we claim that $\p_*([V_y])$ is represented by $[\tl c(x,y)]$, since $V_x$ is Poincaré Dual to the class $x$ defining $M_\infty$. For if $Y=M-{\operatorname{int}}(V_x\x[-1,1])$ then a copy of $Y$, denoted $\wt Y$, can be viewed as a fundamental domain in $M_\infty$, as shown in Figure \[cover\].
(177,84) (10,10)[![Fundamental Domain of $M_\infty$[]{data-label="cover"}](cover.eps "fig:")]{} (15,62)[$\wt c(x,y)$]{} (80,33)[$\wt V_y$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(x,y)$]{} (87,-5)[$\wt Y$]{}
Moreover if $\wt V_y$ denotes $p^{-1}(V_y)\cap \wt Y$ then $\wt V_y$ is a compact surface in $\wt M$ whose boundary is $t_*(\tl c(x,y))-\tl c(x,y)$. Thus $\wt V_y$ is a 2-chain in $M_\infty$ such that $\pi_\#(\wt V_y)$ gives the chain representing $[V_y]$. Since $\p\wt V_y$ is $(t-1)\tl c(x,y)$ in $C_*(M_\infty;\BQ)$, it follows from the explicit construction of $\p_*$ in the proof of the Zig-Zag Lemma \[Mu,Section 24\] that $\p_*([V_y])=[\tl
c(x,y)]$.
Conversely, if $\p_*([V_y])=0$ for [**some**]{} $[V_y]$ then $\p_*$ is the zero map (note that since $V_x$ lifts to $M_\infty$, $[V_x]$ lies in the image of $H_2(M_\infty)\lra H_2(M)$ so $\p_*([V_x])=0$ t).
Therefore the injectivity statement implies the “for each” statement which clearly implies the “for some” statement. Conversely, the “for some” statement implies the injectivity statement.
[**Step 4**]{}: The class $[\tl c(x,y)]$ from Step 3 is 0 if and only if it is divisible by $(t-1)^k$ for every positive $k$. In fact it suffices that it be divisible by $(t-1)^N$ where $N$ is the largest nonnegative integer such that $\La/\<(t-1)^N\>$ is a summand of $H_1(M_\infty,\BQ)$.
One implication is immediate, so assume that there exists a class $[V_y]$ as in Step 3 such that $\p_*([V_y])=[\tl c_{1}]=(t-1)^N\b$ for some $\b\in H_1
| 1,655
| 524
| 1,462
| 1,720
| 2,654
| 0.778107
|
github_plus_top10pct_by_avg
|
ith entries in $B\otimes_AR$. Here, $\dag$ is a polynomial of $m_{i-1,i-1}', m_{i-1,i}', m_{i,i-1}', m_{i,i}', m_{i,i+1}', m_{i+1,i}', m_{i+1,i+1}'$. Thus $$\label{ea3'}
(f_{i, i}^{\ast})'=\pi\left((m_{i,i}^{\ast})'-(m_{i,i}^{\ast\ast})'h_i+\dag\right).$$ Since this is an equation in $B\otimes_AR$, it is of the form $X+\pi Y=0$. By letting $(f_{i, i}^{\ast})'=f_{i, i}^{\ast}=0$, we obtain $$\label{ea4'}
(m_{i,i}^{\ast})'=(m_{i,i}^{\ast\ast})'\bar{h}_i+\bar{\dag},$$ where $\bar{h}_i$ (resp. $\bar{\dag}$) is obtained by letting each term in $h_i$ (resp. $\dag$) having $\pi$ as a factor be zero so that this equation is considered in $R$.\
On the other hand, we apply Equation (\[ea4\]) to the cases $(i-1, i)$ and $(i, i+1)$ and then we have $$\bar{h}_{i-1} m_{i-1,i}'+{}^tm_{i,i-1}'\bar{h}_i=0,$$ $$\bar{h}_i m_{i,i+1}'+{}^tm_{i+1,i}'\bar{h}_{i+1}=0.$$ When $i$ is odd and $L_i$ is *bound of type I*, the above two equations with the first equation of Equation (\[ea72\]) yield the second equation of Equation (\[ea72\]). Thus by combining Equations (\[ea4\]) and (\[ea4’\]) together with the first equation of Equation (\[ea72\]), we conclude that $$2\left(\sum_{\textit{$L_i$:bound of type I with i odd}}n_i\right)+\left(\sum_{i<j}n_in_j\right)-\textit{variables}$$ can be eliminated among $$2\left(\sum_{\textit{$L_i$:bound of type I with i odd}}n_i\right)+2\left(\sum_{i<j}n_in_j\right)-\textit{variables }
\textit{$\{m_{i,j}'\}_{i\neq j}$, $(m_{i,i}^{\ast})'$, $(m_{i,i}^{\ast\ast})'$}.$$ **\
Next, we put (1)-(7) into (\[ea1\]). Then we obtain $$\label{ea5}
\pi^i\left(\sigma(1+\pi\cdot {}^tm_{i,i}')h_i(1+\pi m_{i,i}')+\pi^3(\ast)\right).$$ Here, $(\ast)$ is a certain formal polynomial. We interpret this so as to obtain equations defining $\tilde{G}^1$. There are 6 cases, indexed by (i) - (vi), according to types of $L_i$.\
1. Assume that $i$ is odd and that $L_i$ is *of type II* or *bound of type I*. Then $\pi^ih_i=\xi^{(i-1)/2}\pi a_i$ as explained in Section \[h\] and thus we have $$a_i'=\sigma(1+\pi\cdot {}^tm_{i,i}')a_
| 1,656
| 560
| 1,173
| 1,688
| 2,706
| 0.777712
|
github_plus_top10pct_by_avg
|
\,c\ .$$
The chiral point
================
Asymptotic behavior
-------------------
Hereafter we only consider $\mu l=1$, since the case of $\mu l=-1$ just corresponds to the interchange $x^{+}\longleftrightarrow x^{-}$.
In the case of $\mu l=1$, the appropriate asymptotic behavior for $\Delta
g_{\mu\nu}$ reads $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+\cdot\cdot\cdot\\
\Delta g_{r+} & = & f_{r+}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{r-} & = & h_{r-}\ r^{-3}\ln\left( r\right) +f_{r-}r^{-3}%
+\cdot\cdot\cdot\\
\Delta g_{++} & = & f_{++}+\cdot\cdot\cdot\\
\Delta g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & h_{--}\;\ln\left( r\right) +f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric}%$$ where $f_{\mu\nu}$ and $h_{--}$ depend only on $x^{\pm}=\frac{t}{l}\pm\phi$. This behavior accommodates the known solutions with constant curvature at infinity [@DS; @OST; @Gaston], whose metric is given by $$ds^{2}=l^{2}\frac{dr^{2}}{r^{2}}-r^{2}dx^{+}dx^{-}+F(x^{-})\log(r)\left(
dx^{-}\right) ^{2}\ . \label{pp-wave chiral}%$$ with $F(x^{-})$ being an arbitrary function.
Asymptotic symmetry
-------------------
Just as for $\mu l\not =1$, the asymptotic conditions are invariant under diffeomorphisms that behave at infinity as in Eq. (\[Asympt KV\]), where the $\cdots$ terms are again of lowest order and do not contribute to the surface integrals. Hence, the boundary conditions are invariant under the conformal group in two dimensions, generated by $T^{+}(x^{+})$ and $T^{-}(x^{-})$.
Under the action of the Virasoro symmetry, one obtains$$\delta_{\eta}h_{--}=2h_{--}\partial_{-}T^{-}+T^{-}\partial_{-}h_{--}%
+T^{+}\partial_{+}h_{--}\ \label{deltah--chiral}%$$ and $$\delta_{\eta}f_{++} =2f_{++}\partial_{+}T^{+}+T^{-}\partial_{-}%
f_{++}+T^{+}\partial_{+}f_{++}
-l^{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right) \! /2 .
\label{deltaf--chiral}%$$
The field equations are easily verified to imply that$$\partial_{-}f_{++}=0\text{, and }\partial_{+}h_{--}=0\text{.}%$$ Note th
| 1,657
| 744
| 1,294
| 1,715
| null | null |
github_plus_top10pct_by_avg
|
a}_j = \sqrt{ \hat{\Gamma}_{j,j}}$ and $\hat{t}_j = z_{\alpha/(2s)} \hat{\gamma}_j$ We use the same arguments and notation as in the proofs of and . Thus, let $\mathcal{E}_n$ be the event that $ \frac{\overline{H} ||\delta||^2}{2\sqrt{n}} < \epsilon_n$, where $\frac{\overline{H} ||\delta||^2}{2\sqrt{n}}$ is an upper bound on $\|
R\|_\infty$, with $R$ the reminder in the Taylor series expansion and $\epsilon_n$ as in . Then, $\mathbb{P}\left( \mathcal{E}_n^c \right) \leq n^{-1}$ (see equation \[eq:Ac\]).
Next, for each $t \in \mathbb{R}^{2s}_+$ and any Jacobian matrix $G = G(\psi(P))$, with $P \in \mathcal{P}_n$, let $$\label{eq:polyhedron2}
P(G,t) = \left\{ x \in \mathbb{R}^b \colon v_l^\top x \leq t_l , \forall
v_l \in \mathcal{V}(G) \right\},$$ where $\mathcal{V}(G)$ is defined in the proof of . Then, for any positive numbers $(t'_1,\ldots,t'_s)$ $$| \sqrt{n}(\hat{\nu}_j - \nu_j ) | \leq t'_j, j=1,\ldots,s \quad
\text{if and only if } \quad
\sqrt{n} (\hat{\psi} - \psi) \in P(G,t),$$ where the coordinates of $t \in \mathbb{R}^{2s}$ are as follows: for $j=1,\ldots,s$, $t_{2l-1} = t_{2l} = \frac{t'_l}{\|G_j\|}$.
Consider now the class of subsets of $\mathbb{R}^b$ of the form specified in , where $t$ ranges over the positive vectors in $\mathbb{R}^{2s}$ and $G$ ranges in $ \{ G(\psi(P)), P \in
\mathcal{P}_n\}$. This is a class comprised by polytopes with at most $2s$ faces in $\mathbb{R}^b$. Thus, using the same arguments as in the proof of , we obtain that $$\label{eq:delta1n.bonf}
\sup_{t =(t_1,\ldots,t_s) \in \mathbb{R}^s_+} \left|
\mathbb{P}\left(\sqrt{n}|\hat\nu_j - \nu_j| \leq t_j, \forall j\right) -
\mathbb{P}\left(|Z_{n,j} | \leq t_j, \forall j\right) \right| \leq
C \frac{1}{\sqrt{v}} \left( \frac{\overline{v}^2 b (\log 2bn)^7}{n}
\right)^{1/6},$$ for some $C>0$ depending only on $A$, where $Z_n \sim N(0,\Gamma)$. Using the above display, and following the same arguments as in the proof of , we have that $$\begin{aligned}
\mathbb{P}( \sqrt{n} |\hat\theta_j - \thet
| 1,658
| 3,671
| 1,343
| 1,382
| null | null |
github_plus_top10pct_by_avg
|
e because the two set-theoretic defining requirements of $fGf=f$ and $GfG=G$ for the generalized inverse are satisfied, as Fig. \[Fig: GenInv\] shows, in the following forms $$jf_{\textrm{B}}Gf=f\qquad Gjf_{\textrm{B}}G=G.$$ In fact the commutativity embodied in these equalities is self evident from the fact that $e=if_{\textrm{B}}$ is a left inverse of $G$, that is $eG=\bold1_{Y}$. On putting back $X_{\textrm{B}}$ into $X$ by identifying each point of $X_{\textrm{B }}$ with the set it came from yields the required set-valued inverse $f^{-}$, and $G$ may be viewed as a functional selection of the multiinverse $f^{-}$.
An *injective branch* of a function $f$ in this work refers to the restrictions $f_{\mathrm{B}}$ and its associated inverse $f_{\mathrm{B}}^{-1}$.
The following example of an inverse ill-posed problem will be useful in fixing the notations introduced above. Let $f$ on $[0,1]$ be the function of \[Fig: gen-inv\].
Then $f(x)=y$ is well-posed for $[0,1/4)$, and ill-posed in [\[]{}1/4,1[\]]{}. There are two injective branches of $f$ in $\{[1/4,3/8)$$\bigcup$ $(5/8,1]\}$, and $f$ is constant ill-posed in $[3/8,5/8]$. Hence the basic component $f_{\textrm{B}}$ of $f$ can be taken to be $f_{\textrm{B}}(x)=2x$ for $x\in[0,3/8)$ having the inverse $f_{\textrm{B}}^{-1}(y)=x/2$ with $y\in[0,3/4]$. The generalized inverse is obtained by taking $[0,3/4]$ as a subspace of $[0,1]$, while the multiinverse $f^{-}$ follows by associating with every point of the basic domain $[0,1]_{\textrm{B}}=[0,3/8]$, the respective equivalent points $[3/8]_{f}=[3/8,5/8]$ and $[x]_{f}=\{ x,7/4-3x\}\textrm{ for }x\in[1/4,3/8)$. Thus the inverses $G$ and $f^{-}$ of $f$ are[^17]
$$G(y)=\left\{ \begin{array}{ccl}
y/2, & & y\in[0,3/4]\\
0, & & y\in(3/4,1]\end{array}\right.,\quad f^{-}(y)=\left\{ \begin{array}{ccl}
y/2, & & y\in[0,1/2)\\
\{ y/2,7/4-3y/2\}, & & y\in[1/2,3/4)\\
{}[3/8,5/8], & & y=3/4\\
0, & & y\in(3/4,1],\end{array}\right.$$
which shows that $f^{-}$ is multivalued. In order to avoid cumbersome notations, an inje
| 1,659
| 4,505
| 2,764
| 1,643
| 1,713
| 0.786503
|
github_plus_top10pct_by_avg
|
abel{eigval+}
\begin{aligned}
\mu_{+}^{(n,m)} & =
\hbar\nu\left(n+\frac{m}{2}\right)-
\frac{\hbar}{2}\sqrt{\omega_{L}^{2}+
\Omega^{2}\left|f_n^m\right|^{2}}, \\
\gamma_{+}^{(n,m)} & = \hbar\nu\left(n+\frac{m}{2}\right)+
\frac{\hbar}{2}\sqrt{\omega_{L}^{2}+\Omega^{2}\left|f_n^m\right|^{2}},
\end{aligned}$$ respectively, associated to the eigenvectors $$\label{eigevec+}
\begin{aligned}
\left|\mu_{+}^{(n,m)}\right\rangle & =
\tfrac{\text{e}^{-i\omega_{\!L}t}\left[\omega_{\!L}-\sqrt{\omega_{\!L}^{2}+
\Omega^{2}\left|f_n^m\right|^{2}}\right]}{\Omega f_n^{m\ast}\,\sqrt{1+\frac{\left|\omega_{\!L}-
\sqrt{\omega_{\!L}^{2}+\Omega^{2}\left|f_n^m\right|^{2}}\right|^{2}}{\Omega^{2}\left|f_n^m\right|^{2}}}}
\left|n,e\right\rangle \\
& + \tfrac{1}{\sqrt{1+\frac{\left|\omega_{\!L}-\sqrt{\omega_{\!L}^{2}+
\Omega^{2}\left|f_n^m\right|^{2}}\right|^{2}}{\Omega^{2}\left|f_n^m\right|^{2}}}}\left|n+m,g\right\rangle, \\
\left|\gamma_{+}^{(n,m)}\right\rangle & =\tfrac{\text{e}^{-i\omega_{\!L}t}\left[\omega_{\!L}
+\sqrt{\omega_{\!L}^{2}+\Omega^{2}\left|f_n^m\right|^{2}}\right]}
{\Omega f_n^{m\ast}\,\sqrt{1+\frac{\left|\omega_{\!L}+\sqrt{\omega_{\!L}^{2}+
\Omega^{2}\left|f_n^m\right|^{2}}\right|^{2}}{\Omega^{2}\left|f_n^m\right|^{2}}}}\left|n,e\right\rangle \\
& + \tfrac{1}{\sqrt{1+\frac{\left|\omega_{\!L}+\sqrt{\omega_{\!L}^{2}+\Omega^{2}
\left|f_n^m\right|^{2}}\right|^{2}}{\Omega^{2}\left|f_n^m\right|^{2}}}}\left|n+m,g\right\rangle ,
\end{aligned}$$ where in this regime $\omega_{L} = (\omega_0 - m\nu)$.
Diagonalization of {#do-}
-------------------
For the AJC like Hamiltonian, $\hat{\mathcal{H}}^{(m)}_{-}$, in Eq. (\[hamrwa\]), the invariant subspace is $\{ | n + m , e \rangle, | n , g \rangle \}$ for all $m,n$, while for $m > n$ it should be replaced by $\{ \left|n,e\right\rangle, | n + m , e \rangle, | n , g \rangle \}$. Taking the matrix elements of the Hamiltonian in these subspaces, and rearranging the basis as before, one finds $$\label{hblocks-0}
\hat{\mathcal{H}}^{(m)}_{-} = \hat{\mathcal{H}}^{[1]}_{-} \oplus
| 1,660
| 2,051
| 2,080
| 1,684
| null | null |
github_plus_top10pct_by_avg
|
^\dag
L_q)=1$. Then we obtain that the probability is $$p_q = \left[ {\mathrm{tr}}_B \left( (L^\dag L)^{-1} \right) \right]^{-1}.
\label{eq:p_q2}$$
For an arbitrary entangled shared pair described by invertible $L$, the set of measurement outcomes providing fidelity 1 conditional teleportation is given by the set $${\mathcal M}_L = \left\{\, L_q =
\frac{i_{AC}^{-1} U L^{-1}\strut^\dag}
{\sqrt{{{\mathrm{tr}}}_B \big( \textstyle L^{-1} L^{-1}\strut^\dag \big)}}
\, \Bigg| \, \mbox{$U$ is unitary} \,\right\}.
\label{eq:M_L}$$ Thus not every possible measurement outcome allows teleportation, only those described by ${\mathcal M}_L$. The measurement and the shared state should be “matched” to each other. This can be regarded as a generalization of “entanglement matching” introduced in Ref. [@pra61_034301].
It is worth to note that (\[eq:p\_q2\]) is the same for every outcome $q$ that matches the shared state in the above sense. The probability of a successful outcome depends only on the shared state. Another important result is that the set ${\mathcal M}_L$ of matching outcomes is spanned by local unitary transformations: if one finds a measurement outcome which enables probabilistic teleportation, then every matching outcome can be obtained from it by a local unitary transformation on system $A$.
We give an example for entanglement matching. Suppose that the antilinear operator $L$ describing the shared state $|\sigma\rangle_{BC}$ is given by the following matrix: $$L = \left( \begin{matrix} \alpha_1\cr & \ddots\cr &&\alpha_n\end{matrix} \right),
\qquad |\sigma\rangle_{AB}=\sum_i \alpha_i | i \rangle_B | i \rangle_C,$$ where all $\alpha_i$ are nonzero (consider a Schmidt decomposition for example). Taking that the unitary transformation $U_q$ is identity, we obtain from (\[eq:L\_q\]) that a matching measurement outcome is given by $$\begin{aligned}
L_1&=& \left( \sum_i \frac1{|\alpha_i|^2} \right)^{-1/2} \left(
\begin{matrix}1/\alpha_1^\ast \cr & \ddots\cr && 1/\alpha_n^\ast\end{matrix} \right
| 1,661
| 4,443
| 801
| 1,091
| 2,146
| 0.782285
|
github_plus_top10pct_by_avg
|
ong\_T\] for two different temperatures.
![Variation of the transverse part of the second-order QNS scaled with that of free field value in presence of strong magnetic field with temperature (left panel) and magnetic field (right panel) strength for $N_f=3$.[]{data-label="QNS_sfa_trans_T"}](chi2_sfa_trans.pdf "fig:") ![Variation of the transverse part of the second-order QNS scaled with that of free field value in presence of strong magnetic field with temperature (left panel) and magnetic field (right panel) strength for $N_f=3$.[]{data-label="QNS_sfa_trans_T"}](chi2_sfa_trans_eB.pdf "fig:")
In the left panel of Fig. \[QNS\_sfa\_trans\_T\] the variation of transverse second-order QNS with temperature is displayed for two values of magnetic field strength. It is found that the transverse second-order QNS decreases with temperature. This is an indication that the system may shrink in the transverse direction. For a given temperature the transverse second-order QNS is found to increase with the increase of the magnetic field strength as shown in the right panel of Fig. \[QNS\_sfa\_trans\_T\] for two different temperatures. This behaviour is in contrary to that of longitudinal one.
Weak magnetic field {#wfa}
===================
In this section we consider magnetic field strength to be the lowest among all the scales $T$, $m_{th}$ as $\sqrt{q_fB} < m_{th}\sim gT <T$. The HTL one-loop free energy for the deconfined QCD matter has been calculated upto $\mathcal O[g^4]$ in Ref. [@Bandyopadhyay:2017cle]. The total renormalized free-energy in presence of weak magnetic field is sum of renormalized quark and gluon free-energy and can be written [@Bandyopadhyay:2017cle] as F=F\_q\^r + F\_g\^r, where the renormalized quark free-energy is F\_q\^r &=& N\_c N\_f. \[Eq:Fqr\] $M_{B,f}$ is the thermomagnetic mass for quark flavor $f$ in presence of weak magnetic field and $M_B$ represents flavor summed thermomagnetic quark mass as M\_B\^2=\_f M\_[B,f]{}\^2 &=& \_f . \[mgmass\] $\aleph(z)$ in Eq. is abbreviated as (z) (z)+
| 1,662
| 1,342
| 362
| 1,442
| 3,201
| 0.774078
|
github_plus_top10pct_by_avg
|
}} & := & \{(\varepsilon,\varepsilon), (r_{12},r_{12}), (r_{13},r_{13}),(r_{12}r_{14},r_{12}r_{14}), (r_{13}r_{14},r_{13}r_{14})\}).\end{aligned}$$ One can check that ${\rm Id_{C,1}}$ is a prefix of the strategy, for the game with initial position $(C,C)$, $${\rm Id}_{C,\infty}:=\{(u,u) \mid u \in {\cal R}^*, C(L_1) {\stackrel{u}{\longrightarrow_{}}}\}.$$ The set ${\rm Id_{D,2}}$ (resp. ${\rm Id_{E,2}}$) is really a strategy for the game with initial position $(D,D)$ (resp. $(E,E)$) since no rule $r_i$ is applicable on $\bot$. For every $N \in \{C,D,E\}$, the symbol ${\rm Id_{N,i}}$ will denote a residual of length $i$ of the strategy ${\rm Id_{N,n}}$: $$\begin{aligned}
{\rm Id_{C,0}} & = & {\rm Id_{D,0}} = {\rm Id_{E,0}} = \{(\varepsilon,\varepsilon)\},\\
{\rm Id_{D,1}} & = & {\rm Id_{E,1}} = \{(\varepsilon,\varepsilon), (r_{14},r_{14})\}\end{aligned}$$
The Non-equivalence (meta-) proof
=================================
$A(\bot) \not{\!\!\sim} B(\bot)$ \[lem-nonequivalence\_proof\]
$$\forall u \in {\cal R}^*, ACT(u) = aaab \Rightarrow A(\bot) \not{\!\!{\stackrel{u}{\longrightarrow_{}}}}$$ while $$\exists u \in {\cal R}^*, ACT(u) = aaab \mbox { and } B(\bot) {\stackrel{u}{\longrightarrow_{}}}$$ hence $A(\bot) \not{\!\!\sim} B(\bot)$.
From section \[sec-equivalence\_proof\] and Lemma \[lem-nonequivalence\_proof\] we conclude
The family of formal systems $({\cal J}(T_0,T'_0,S_0,{\cal B}))$ is not sound.
Variations
==========
Let us describe variations around this example.\
#### Description of the proofs
$\;\;$\
We chosed to write the proofs with judgments of the form $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S)$ or $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}(T_1,T'_1,S_1)$ or $m {\:|\!\!\!=\!\!\!\!=\:}(T,T',S) \leadsto \alpha {\:|\!\!\!=\!\!\!\!=\:}{{\rm SUCC}}$, where, in the case of forms 2,3, the prefix $\alpha$ is given by its image under the map ${{\rm LAB}_{\cal T}}$ (its image is enough to determine $\alpha \in ({\cal R} \times {\cal R})^*$ just because the grammar is determinis
| 1,663
| 912
| 2,100
| 1,669
| 3,192
| 0.774128
|
github_plus_top10pct_by_avg
|
t $x_0$ be a basepoint in $X$ and $y_0 = f(x_0)$ be a basepoint in $Y$. Because $g\circ f$ is close to ${\text{id}}_X$, we can say that there is a $D$ such that ${\text{d}}(x,g\circ f(x))\leq D$ for all $x \in X$. Let $K$ be the integer provided by $X$ being $\sigma$-stable and $K'$ be the integer provided by $Y$ being $\sigma$-stable. As $f$ is bornologous, there is an $M$ such that $f$ sends $K$-sequences to $M$-sequences in $Y$. We can assume $ M\geq K'$. Similarly, because $g$ is bornologous, there is an $L$ such that $g$ sends $M$-sequences to $L$-sequences in $X$, choosing $L \geq D$.
Let $z_L:\sigma_L(X, x_0)\to \sigma_L(X, g\circ f(x_0))$ be the function that sends the equivalence class of $x_0, x_1, \ldots$ to the equivalence class of $g\circ f(x_0),x_0, x_1, \ldots$. We chose $L\geq D$, so we can say that this addition does not prevent $x_0, g\circ f(x_0), x_1, \ldots$ from being an $L$ sequence. By \[zlemma\] we know $z_L$ is a bijection.
Let $f_K$ be the function that sends an element$[s]_K\in\sigma_K(X,x_0)$ to the element $[f(s)]_M\in\sigma_M(Y,f(x_0))$ and let $g_M$ be the function that sends an element $[s]_M\in\sigma_M(Y,f(x_0))$ to the element $[g(s)]_L\in\sigma(X,g\circ f(x_0))$.
We show the following diagram commutes:
\_L(X, gf(x\_0)) & & & &\
\^[z\_L]{} &(4,4)\^[g\_M]{} & & &\
\_L(X, x\_0) & & & &\
\^[\_[KL]{}]{} & & & &\
\_K(X, x\_0) & & \_[f\_K]{} & & \_M(Y, f(x\_0))\
Let $(x_n)$ be a $K$-sequence in $X$. Then $g\circ f([(x_n)])$ is the equivalence class of the sequence $g\circ f(x_0), g\circ f(x_1), \ldots$ and $z_L\circ\phi_{KL}([x_n])$ is the equivalence class of the sequence $g\circ f(x_0),x_0,x_1,\ldots$. Consider the sequence $$g\circ f(x_0), x_0, x_1, g\circ f(x_1),g\circ f(x_2), x_2, x_3, g\circ f(x_3),g\circ f(x_4), \ldots$$ There are three distances to consider: the distance between successive elements of ${x_n}$, the distance between successive elements of ${g\circ f(x_n)}$, and the distance between any $x_i$, and its counterpart $g\circ f(x_i)$. Because $d(x_i, g\circ f(x_i
| 1,664
| 3,179
| 2,292
| 1,624
| 2,841
| 0.776626
|
github_plus_top10pct_by_avg
|
al{R}_s$, we change the coordinates such that $x_0$ is the origin and regard $\mathbb{R}^d$ as $s^\perp \bigotimes s$, where $s^\perp$ is the orthogonal space of $s$. Suppose that $s^\perp$ is $d_1$-dimensional. Then, under this new coordinate system and for $y\in \mathcal{R}_s$, we have Hence $D^2 \varphi(y)$ is a diagonal matrix with the first $d_1$ diagonal entries being $2$ and the rest being $0$, and $||D^2\varphi(y)||_{op}{\leqslant}2.$ Similar arguments and results apply to $x\in \mathcal{R}_e$ and to $x\in \mathcal{R}_0$. Recall that $y_0$ is a continuous function of $y$. We conclude that $D\varphi$ is continuous. Therefore, we have [42]{}.
We now prove [41]{}. Recall that we assumed that $v=0$. On one hand, as $0\in \mathcal{P}$, $$\sup_{\mu\in \mathcal{P}}\mu \cdot D\varphi(x){\geqslant}0.$$ On the other hand, by considering $x\in \mathcal{R}_0, \mathcal{R}_e, \mathcal{R}_s$ separately as above, as $\mu\in \mathcal{P}$ points “inwards" and $D\varphi(x)=2(x-x_0)$ points “outwards", it is clear that which proves [41]{}.
Let $\mathbf{B}_0(R)$ denote a disk of radius $R$ in a plane. For $m\in \mathbb{N}$, denote as $P_m$ a regular $m$-sided polygon with $\mathbf{B}_0(R)$ as the inscribed circle (see Figure 1 below).
(0,0) circle (0.866)\[thick\];
iin [0,1,2,3,4,5]{} [ (i 60:1) edge\[thick\] ([(i+ 1) \* 60]{}:1); ]{}
(120:1) edge \[left\] node [$r_m$]{} (0,0);
(0,0.866) edge \[right\] node [R]{} (0,0);
at (0, -1.3) [Figure 1: $\mathcal{P}$ & $P_m$]{};
Write $r_m$ as the radius of the regular $m$-sided polygon. Then, $r_m=\frac{R}{\cos\frac{\pi}{m}}$. We can easily check that $$r_m-R{\leqslant}\frac{7\pi^2R}{m^2} \ \emph{for} \ m\ge 3$$ and $$\lim_{m\rightarrow+\infty}m^2(r_m-R)= \frac{\pi^2R}{2}.$$ Now, we expand the set $\Theta$ as $\Theta_m$ such that $\{E_\theta [X_1]: \theta\in \Theta_m\}=P_m$ and $$\sup_{\theta\in \Theta_m} E_\theta[|X_1-E_\theta[X_1]|^2]=\sup_{\theta\in \Theta} E_\theta[|X_1-E_\theta[X_1]|^2]=:\bar{\sigma}^2.$$ Set $\mathbb{E}_m=\sup_{\theta\in \Theta_m}E_\theta$. Then, $$
| 1,665
| 2,565
| 1,771
| 1,649
| null | null |
github_plus_top10pct_by_avg
|
raints\] for a possible application of $c_1^{\rm rep}$).
For any stack $\mathfrak{X}$, let $V$ be a vector bundle over $\mathfrak{X}$, and $I_{\mathfrak{X}}$ the inertia stack of $\mathfrak{X}$. Let $q: I_{\mathfrak{X}} \rightarrow \mathfrak{X}$ denote the natural projection operator onto one component.
We define Chern classes of $V$ as follows. First, pullback $V$ to $I_{\mathfrak{X}}$ along $q$. Then, on each component $\alpha$ of $I_{\mathfrak{X}}$, $q^* V$ will decompose into eigenbundles of the action of the stabilizer for that component: $$q^* V|_{\alpha} \: = \: \oplus_{\chi} V_{\alpha, \chi}.$$ (When $\alpha$ is the identity, our conventions are that there is only one component, associated to the trivial character.) Define ${\rm ch}^{\rm rep}(V)$ over a component $\alpha$ to be $${\rm ch}^{\rm rep}(V)|_{\alpha} \: \equiv \:
\bigoplus_{\chi} {\rm ch}(V_{\alpha, \chi}) \otimes \chi,$$ where $\chi$ is the eigenvalue of that component of $q^* V$ under the stabilizer, and ${\rm ch}$ denotes the naive notion of Chern classes, living in equivariant cohomology pertinent to the stack itself. (These seem to be the same as the Chern classes in “delocalized cohomology” described in [*e.g.*]{} [@at-seg; @bbmp; @baum-fete], though our starting point is different.)
Intuitively, the idea is that on any component of the inertia stack determined by some generic automorphism, the bundle should decompose into eigenbundles, and $\chi$ is the eigenvalue associated with the action of that automorphism on the bundle. Slightly more generally, one can define a “diagonalization map” $$d: \: K^0(I_{\mathfrak{X}}) \otimes {\mathbb C} \: \longrightarrow \:
K^0(I_{\mathfrak{X}}) \otimes {\mathbb C},$$ which on a component $\alpha$ maps a sheaf ${\cal F}$ to its isotypic decomposition, weighted by characters: $$d( [{\cal F}] ) |_{\alpha}
\: = \: \sum_{\chi} {\cal F}_{\alpha, \chi} \otimes \chi.$$ In this language, $${\rm ch}^{\rm rep}(V) \: = \: {\rm ch}( d( q^* V) ).$$
To clarify these ideas, let us work through some examp
| 1,666
| 1,213
| 2,028
| 1,611
| 1,539
| 0.788516
|
github_plus_top10pct_by_avg
|
)\Xi\Psi)\,.\end{aligned}$$ Using the fact that ${\mathcal{S}}$ is BPZ odd, $$\langle {\mathcal{S}}A, B\rangle\ =\ -\langle A, {\mathcal{S}}B\rangle\,,
\label{BPZ S}$$ it is easy to see that the quadratic terms of the action (\[complete action\]), $$S^{(0)}\ =\ - \frac{1}{2} \langle\Phi, Q\eta\Phi\rangle
- \frac{1}{2} {\langle\!\langle}\Psi, YQ\Psi{\rangle\!\rangle}\,,
\label{kinetic}$$ are invariant under the transformation $$\delta_{\mathcal{S}}^{(0)}\Phi\ =\ {\mathcal{S}}\Xi\Psi\,,\qquad
\delta_{\mathcal{S}}^{(0)}\Psi\ =\ X{\mathcal{S}}\eta\Phi\,.
\label{linearized tf}$$ However, the action at the next order, $$S^{(1)}\ =\ -\frac{1}{6}\langle\Phi, Q[\Phi, \eta\Phi]\rangle
- \langle\Phi, \Psi^2\rangle\,,$$ is not invariant under $\delta^{(0)}_{\mathcal{S}}$ but is transformed as $$\begin{aligned}
\delta^{(0)}_{\mathcal{S}}S^{(1)}\ =\
\langle\left(\frac{1}{2}[\Phi, {\mathcal{S}}\Xi\Psi]
-{\mathcal{S}}\Xi[\Phi, \Psi]
+\{\Psi, \Xi{\mathcal{S}}\Phi\}\right),Q\eta\Phi\rangle
\nonumber\\
+\, {\langle\!\langle}\left(-\frac{1}{2}X\eta[\Phi,{\mathcal{S}}\Phi]
+X\eta[\Phi,\Xi{\mathcal{S}}\eta\Phi]\right), YQ\Psi{\rangle\!\rangle}\,.
\label{var one}\end{aligned}$$ We have thus to modify the transformation by adding $$\begin{aligned}
\delta^{(1)}_{\mathcal{S}}\Phi\ =&\ \frac{1}{2}[\Phi, {\mathcal{S}}\Xi\Psi]
-{\mathcal{S}}\Xi[\Phi,\Psi]+\{\Psi,\Xi {\mathcal{S}}\Phi\},\\
\delta^{(1)}_{\mathcal{S}}\Psi\ =&\ -\frac{1}{2}X\eta[\Phi,{\mathcal{S}}\Phi]
+X\eta[\Phi,\Xi {\mathcal{S}}\eta\Phi]\,,\end{aligned}$$ under which the kinetic terms (\[kinetic\]) are transformed so as to cancel the contribution (\[var one\]): $\delta_{\mathcal{S}}^{(1)}S^{(0)}+\delta_{\mathcal{S}}^{(0)}S^{(1)}=0$. Then at the next order we have two contributions, $\delta_{\mathcal{S}}^{(1)}S^{(1)}$ and $\delta_{\mathcal{S}}^{(0)}S^{(2)}$, which are again nonzero and require to add $$\begin{aligned}
\delta^{(2)}_{\mathcal{S}}\Phi\ =&\
\frac{1}{12}[\Phi,[\Phi,{\mathcal{S}}\Xi\Psi]]
+\frac{1}{2}\{[\Phi,\Psi],\Xi {\mathcal{S}}\Phi\}
+\frac{1}{2}[\Xi[\P
| 1,667
| 1,348
| 2,027
| 1,576
| null | null |
github_plus_top10pct_by_avg
|
{f\rho}_\ell k dk/2\pi$ (which is restricted to its upper, or lower depending on one’s convention, triangular entries by the limits of integration), treating $\Delta$ as a free parameter or ignoring it altogether.
Instead of doing so, we will rewrite eq. (\[split\]) into a form that is closer to the original analysis by Croft et al. [-@croft98], thereby making manifest the differences from our procedure suggested here.
By taking the derivative of eq. (\[split\]) with respect to $k_\parallel$, it can be shown that $$\begin{aligned}
\label{iteration}
A' {\tilde P^\rho} (k=k_i) =&& - {2 \pi \over (1+\beta_f)^2
k_i} \Biggl[ \left.{d P^f \over d
k_\parallel}\right|_{k_\parallel=k_i} - 4 \beta_f k_i \Biggl(
\int_{k_i}^{k_{\star\star}} A' {\tilde P^\rho} (k) k^{-1} {dk \over 2 \pi} +
C_1 \Biggr) \\ \nonumber && - 4 \beta_f^2 k_i^3
\Biggl(\int_{k_i}^{k_{\star\star}} A'
{\tilde P^\rho} (k) k^{-3} {dk \over 2 \pi} + C_2 \Biggr) \Biggr]\end{aligned}$$ where we have used the form of $W^{f\rho}_\ell$ in eq. (\[Wlinear\]). The value of $k_i$ for which we will perform the inversion would range from some maximum $k_{\star\star}$ to whatever small $k_i$ (large scale) one might wish. The constraint is that $k_{\star\star}$ has to be sufficiently smaller than $k_{\star}$ such that condition number 2 as set out for eq. (\[split\]) is satisfied.
The constants $C_1$ and $C_2$ should be $$C_1 = \int_{k_{\star\star}}^{k_{\star}} A' {\tilde P^\rho} (k) k^{-1} {dk
\over 2 \pi} \, \, , \, \, C_2 = \int_{k_{\star\star}}^{k_{\star}} A' {\tilde P^\rho} (k) k^{-3} {dk
\over 2 \pi}
\label{C}$$
Assuming some values for $C_1$, $C_2$ and the starting wavenumber $k_{\star\star}$, eq. (\[iteration\]) can be used to obtain $A' {\tilde P^{\rho}} (k=k_i)$ for successively smaller $k_i$’s (at, say, evenly spaced intervals). The method adopted by Croft et al. [-@croft98] is equivalent to keeping only the first term within the square brackets on the right hand side of eq. (\[iteration\]) i.e. a simple differentiation.
Because we are interest
| 1,668
| 2,919
| 2,287
| 1,675
| 1,681
| 0.786837
|
github_plus_top10pct_by_avg
|
e coupling variation in the channel $c$ amounts then to replacing in this expression $T_c$ with $T_c^{\mathrm{eff}}$. This suggests the following heuristic formula for the coupling fidelity $$\label{eq:app2}
F_{\mathrm{surm}}(t) = \left[\frac{(1+2T_c t/\beta)(1+2T'_c t/\beta) }{
|1+2T^{\rm eff}_c t/\beta|^2}\right]^{\beta/2}\,.$$ For the parameters of $\lambda$ and $\lambda^\prime$ found in the experiment only deviations on the % level were found while making fit to surmise (\[eq:app2\]). We stress, however, that there is no control on approximations involved to derive this expression. One should generally expect that surmise (\[eq:app2\]) coincides closely with the exact result at small times (when it is given by an exponential dependence), while the exact asymptotic behavior at large times is reproduced up to a factor of the order of unity (as was indeed confirmed numerically). Therefore, we have used the exact supersymmetry result for all the figures and analysis of the main text.
[42]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, eds., **, Landmarks in Mathematics and Physics (, , ).
, , , ****, ().
, , , ****, ().
, , , , ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ** (, , ).
, ** (, , ).
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ** (, , ).
We omit the overall factor equal to the length of the energy spectrum used to compute the Fourier transforms in Eq. (\[eq:ss\_ft\]), since it is finally canceled due to the normalization in definition (\[eq:f\_ab\]).
| 1,669
| 604
| 1,772
| 1,659
| null | null |
github_plus_top10pct_by_avg
|
ete gamma functions, respectively, defined by $$\begin{aligned}
\G(a, x) := \int_{x}^{+\infty} t^{a - 1} e^{-t} dt, \qquad
\g(a, x) := \int_{0}^{x} t^{a - 1} e^{-t} dt, \end{aligned}$$ where $\text{Re}(a) > 0$ and $x \geq 0$. These functions have the following properties:
\[lem:3.0\] For ${\rm Re}(a) > 0$ and $x \geq 0$,
&(1)(a, x) + (a, x) = (a);\
&(2)\_[x ]{} (a, x) = (a);\
&(3)(a, 0) = (a);\
&(4) (a, x) = x\^[a - 1]{} e\^[-x]{};\
&(5) (a, x) = - x\^[a - 1]{} e\^[-x]{}. &
Also, for $c \in \mathbb{R}$, let $\operatorname{{sgn}}(c) := 1 \: (c \geq 0); -1 \: (c < 0)$. Then, the expected value and the variance of $\Pe(Z + c)$ are as follows:
\[lem:3.1\] For any $c \in \mathbb{R}$, we have $$\begin{aligned}
(1)\quad
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right), \\
(2)\quad
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
See the last two sections for the proof of Lemma $\ref{lem:3.1}$.
From Lemma $\ref{
| 1,670
| 4,091
| 1,999
| 1,390
| null | null |
github_plus_top10pct_by_avg
|
eq p, \;\Omega^{-1}(\i) \geq p \big\} \label{eq:cr83} \\
-B_2 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) \geq p, \; \Omega^{-1}(\i) < p \big\} \label{eq:cr84} \\
-B_2 & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) < p, \; \Omega^{-1}(\i) \geq p\big\} \label{eq:cr85} \\
-(B_3 + B_4 - A_3^2) & \;\; \text{if} \; i \neq \i, \; \big\{\Omega^{-1}(i) < p,\; \Omega^{-1}(\i) < p\big\} \;. \label{eq:cr86}
\end{array}
\right.\end{aligned}$$
where constants $A_3, B_1, B_2, B_3, B_4, C_1, C_2$ and $C_3$ are defined in Equations , , , , , , and respectively. From this computation of the Hessian, note that we have $$\begin{aligned}
\label{eq:cr11}
H({\boldsymbol{0}}) = \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \Big(H_{i\i}({\boldsymbol{0}}) \Big) \;. \end{aligned}$$ which follows directly from the fact that the diagonal entries are summations of the off-diagonals, i.e. $C_1 = B_1(\kappa-p) + B_2(p-1)$ and $C_2 + A_3^2 - C_3 = B_2(\kappa-p+1) + (B_3 + B_4 - A_3^2)(p-2)$. The second equality follows from the fact that $C_2 = B_2(\kappa-p+1) + B_3(p-2)$ and $A_3^2(p-1) = B_4(p-2) + C_3$. Note that since $\theta = {\boldsymbol{0}}$, all items are exchangeable. Hence, $\E[H_{i\i}({\boldsymbol{0}})] = \E[H_{ii}({\boldsymbol{0}})]/(\kappa-1)$, and substituting this into and using Equations , we get $$\begin{aligned}
&& \E\Big[ H({\boldsymbol{0}})\Big] \nonumber\\
&=& -\frac{1}{\kappa-1}\bigg(\P\big[\Omega^{-1}(i) \geq p \big]C_1 + \P\big[\Omega^{-1}(i) < p \big](C_2 + A_3^2 - C_3)\bigg) \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \nonumber\\
&\succeq & - \frac{1}{\kappa(\kappa-1)} \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \nonumber\\
&& \Bigg((\kappa-p+1)\log\bigg(\frac{\kappa}{\kappa-p}\bigg) + (p-1)\bigg(\log\bigg(\frac{\kappa}{\kappa-p+1}\bigg) + \log\bigg(\frac{\kappa}{\kappa-p+1}\bigg)^2 \bigg)\Bigg) \nonumber\\\label{eq:cr12}\\
&\succeq & -\frac{2p\log(\kappa)^2}{\kappa(\kappa-1)} \sum_{\i<i \in S}(e_i - e_{\i})(e_i - e_{\i})^{\top} \;, \label{eq:cr13} \end{aligned}$$ w
| 1,671
| 1,306
| 1,876
| 1,679
| null | null |
github_plus_top10pct_by_avg
|
a (4 years, inclusive). Incidences include 95% CI
Junior Senior Combined
----------------------------- --------------------- ------------------------- ---------------------
Permanent (ND+Quad.+Fatal) 0.24 (0 to 0.65) **4.52 (0.74 to 8.30)** 1.04 (0.25 to 1.82)
Neurological deficit (ND) 0.14 (0 to 0.46) 2.05 (0 to 4.60) 0.50 (0 to 1.04)
Quadriplegics (Quad.) 0.09 (0 to 0.36) 1.85 (0 to 4.27) 0.42 (0 to 0.92)
Fatal 0 (--) 0.62 (0 to 2.01) 0.12 (0 to 0.38)
Non-permanent ('near miss') 0.57 (0 to 1.21) 0.62 (0 to 2.01) 0.58 (0 to 1.16)
Not provided\* 0.09 (0 to 0.36) 0.21 (0 to 1.01) 0.12 (0 to 0.38)
Total ASCIs 0.90 (0.09 to 1.70) **5.34 (1.24 to 9.45)** 1.73 (0.72 to 2.74)
Acute spinal cord injuries (ASCIs) are divided into outcomes: not provided, non-permanent ('near misses') and permanent (neurological deficit, quadriplegic and fatal). Incidences are shown for Junior, Senior and Combined (Junior+Senior).
\*Specific diagnosis not available/supplied, but confirmed as ASCI.
**Bold** text indicates value is significantly different from Junior level: this occurs either if the 95% CI do not overlap or if the p value is less than 0.05.
In Senior players, 85% (n=22 of 26) of all ASCIs had *permanent* outcomes (neurological deficit, quadriplegia or fatal) in comparison to 26% (n=5 of 19) in Junior players. When considering the different numbers for the populations at risk, *permanent* ASCIs occurred significantly more often in Senior players (4.52 per 100 000 players; 0.74--8.30) than Junior players (0.24 per 100 000 players; 0--0.65; p=0.04; combined: 1.04 per 100 000 players, 95% CI 0.25 to 1.82) between 2008 and 2011 ([table 2](#BMJOPEN2012002475TB2){ref-type="table"}).
Matches, as opposed to training, were associated with 88% (n=38 of 43) of all ASCIs (informatio
| 1,672
| 3,268
| 1,696
| 1,718
| null | null |
github_plus_top10pct_by_avg
|
e invariant subspaces in $\Bbb{P}^2_\Bbb{C}$.
Let us assume that there is a complex line $\ell$ invariant under $\iota(\Gamma)$. By Bézout’s theorem $Ver\cap \ell$ has either one or two points. From the following commutative diagram $$\xymatrix{
\Bbb{P}_\Bbb{C}^1 \ar[r]^{ \tau}\ar[d]^\psi & \Bbb{P}_\Bbb{C}^1 \ar[d]^\psi \\
Ver \ar[r]^{\iota\tau} & Ver
}$$ where $\tau\in\Gamma$, we deduce that $\Gamma$ leaves $\psi^{-1}(Ver\cap\ell)$ invariant. Therefore $\Gamma$ is an elementary group, which is a contradiction, thus $\iota\Gamma$ does not have invariant lines in $\Bbb{P}^2_\Bbb{C}$. Finally, if there is a point $p\in\Bbb{P}_\Bbb{C}^2$ fixed by $\iota\Gamma$, then by Lemmas \[l:3gen\], \[l:eq\] and \[l:pseudo\], there is a sequence of distinct elements $(\gamma_m)_{m\in\Bbb{N}}\subset\Gamma$ and a pseudo-projective transformation $\gamma\in QP(3,\Bbb{C})$ such that $\iota\gamma_m\xymatrix{\ar[r]_{m\rightarrow\infty}&}\gamma $ and $Ker(\gamma)$ is a complex line not containing $p$. Since $p$ is invariant and outside $Ker(\gamma)$ we conclude $\{p\}=Im(\gamma)$. On the other hand, by Lemma \[l:pseudo\] we deduce $p\in Ver$. Therefore $\Gamma$ is elementary, which is a contradiction.
The following theorem follows easily from the previous discussion.
\[l:eq\] Let $\Gamma$ be a discrete subgroup of $\PSL(2,\Bbb{C})$. Then $$\Bbb{P}_\Bbb{C}^2\setminus Eq(\iota(\Gamma))=\bigcup_{z\in\Lambda(\Gamma)}T_{\psi(z)}(\psi(\Bbb{P}_\Bbb{C}^)).$$ Moreover $\Omega_{\Kul}(\iota\Gamma)=Eq(\iota(\Gamma))$ is Kobayashi hyperbolic, pseudo-convex, and is the largest open set on which $\Gamma$ acts properly discontinuously.
Complex Hyperbolic Groups Leaving $Ver$ Invariant {#s:chvh}
=================================================
In this section we characterize the subgroups of $\PU(2,1)$ that leave invariant a projective translation of the Veronese curve $Ver$. We need some preliminary lemmas.
\[l:semialg\] Let $B$ be a complex ball. Then $$Aut(BV)=\{g\in\PSL(3,\Bbb{C})\vert g\in \iota\PSL(2,\Bbb{C}),gB=B\}$$ is a sem
| 1,673
| 1,192
| 790
| 1,756
| null | null |
github_plus_top10pct_by_avg
|
j})}\prod_j p_{ij}^{\alpha_{ij} - 1} d\theta_k\\
\nonumber &=& p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \int \prod_j p_{ij}^{n_{ij}} \prod_j p_{ij}^{\alpha_{ij} - 1} d\theta_k\\
\nonumber &=& p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \int \prod_j p_{ij}^{n_{ij}+\alpha_{ij} - 1} d\theta_k\end{aligned}$$
Please note, that: $$\begin{aligned}
\nonumber \int \frac{\Gamma(\sum_j \alpha_{j})}{\prod_j \Gamma(\alpha_{j})}\prod_j x_{j}^{\alpha_{j} - 1} dx &=& 1\\
\nonumber \frac{\Gamma(\sum_j \alpha_{j})}{\prod_j \Gamma(\alpha_{j})} \int \prod_j x_{j}^{\alpha_{j} - 1} dx &=& 1\\
\nonumber \int \prod_j x_{j}^{\alpha_{j} - 1} dx &=& \frac{\prod_j \Gamma(\alpha_{j})}{\Gamma(\sum_j \alpha_{j})}\end{aligned}$$
Thus, we have $$\int \prod_j p_{ij}^{n_{ij}+\alpha_{ij} - 1} d\theta_k = \frac{\prod_j \Gamma(n_{ij}+\alpha_{ij})}{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}$$
And thus, $$P(D | M_k) = p(x_1)\prod_i\frac{\Gamma(\sum_j \alpha_{ij})}{\prod_j \Gamma(\alpha_{ij})} \frac{\prod_j \Gamma(n_{ij}+\alpha_{ij})}{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}
\label{eq:evidence}$$
*Posterior.* For the posterior distribution over the parameters $\theta_k$ we obtain: $$\begin{aligned}
\nonumber P(\theta_k | D, M_k) &=& \prod_i \prod_j p_{ij}^{n_{ij}} \prod_i \prod_j p_{ij}^{\alpha_{ij} - 1} \frac{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}{\prod_j \Gamma(n_{ij}+\alpha_{ij})}\\
\nonumber &=& \prod_i \prod_j p_{ij}^{n_{ij} + \alpha_{ij} - 1} \frac{\Gamma(\sum_j (n_{ij}+\alpha_{ij}))}{\prod_j \Gamma(n_{ij}+\alpha_{ij})}\end{aligned}$$ This equation is the product of the Dirichlet distributions for each row with parameters $n_{j} + \alpha_{j}$: $$P(\theta_k | D, M_k) = \prod_i Dir(n_i + \alpha_{i})$$
The posterior distribution is a combination of our prior belief and the data that we have observed. In fact, the expectation and the variance of the posterior distribution are: $$E[p_{ij}] = \frac{n_{ij} + \alpha_{ij}}{\sum_j (n_{ij} + \alpha_{ij})}$$ $$Var[(p_{ij}] = \frac{(n_{ij} + \alpha
| 1,674
| 4,019
| 2,255
| 1,540
| null | null |
github_plus_top10pct_by_avg
|
is conjugate to a subgroup of $Mob(\hat{\Bbb{R}})$, therefore $ \Lambda_{Gr}\iota^{-1}Aut(BV)$ is a circle in the Riemann sphere and $\Lambda_{Gr}\iota^{-1}Aut(BV) = \psi^{-1}C$.\
In order to prove part (\[l:6\]), observe that after a projective change of coordinates we can assume that $\psi^{-1}C=\hat{\Bbb{R}}$. Thus $C=\psi \hat{\Bbb{R}}=\{[z^2,2zw,w^2]:z,w\in \Bbb{R}, \vert a\vert +\vert b\vert \neq 0\}$. The following claim concludes the proof.\
Claim. The sets $C$ and $\partial \Bbb{H}^1_{\Bbb{R}}=\{[x,y,z]\in\Bbb{P}^2_{\Bbb{R}}:x^2+y^2=z^2\}$ are projectively equivalent. Let $\gamma\in PSL(3,\Bbb{R})$, be the projective transformation induced by: $$\widetilde \gamma=
\begin{pmatrix}
1 & 0 & -1\\
0 & 1 &0\\
1 & 0 &1
\end{pmatrix}.$$ Given $[p]=[x^2,2xy, y^2]\in C$, we get $\gamma(p)=(x^2 - y^2, 2 xy, x^2 + y^2)$ and $$(x^2 - y^2)^2+ (2 xy)^2=( x^2 + y^2)^2.$$ Thus $\gamma C\subset \partial \Bbb{H}^1_{\Bbb{R}}$. Since $C$ is a compact, connected and contains more than two points we conclude that $\gamma$ is a projective equivalence between $C$ and $\partial \Bbb{H}^1_{\Bbb{R}}$.\
Now we prove part (\[l:7\]). Let $x\in B$. Then $x^{\bot}$ is a complex line in $\Bbb{P}_\Bbb{C}^2\setminus \bar{B}$; by Bézout’s theorem we know $Ver\cap x^\bot$ is non-empty, thus $Ver\cap(\Bbb{P}^2_\Bbb{C}\setminus\bar{B})\neq\emptyset$.\
Finally, let us prove part (\[l:8\]). After conjugating by an element in $\iota\PSL(3,\Bbb{C})$ we can assume that $[0,0,1]\notin\partial B$. Let $A=(a_{ij})$ be the Hermitian matrix introduced in part (\[l:3\]) of the present lemma. Clearly $a_{33}\neq 0$. Now let $F:\Bbb{R}^2\rightarrow \Bbb{R}$ be given by
$$F(x,y)=a_{11}+4(b_{12}x-c_{12}y)+2(b_{13}(x^2-y^2)-2c_{13}xy)+a_{33}(x^2+y^2)^2
+4(x^2+y^2) ( b_{23}x-c_{23}y+a_{22}).$$
Thus by part (\[l:5\]) of this lemma we know $\psi^{-1}C=F^{-1}0$ is a circle. Moreover $$\begin{array}{l}
\iota F^{-1}\Bbb{R}^+=Ver\cap\Bbb{P}_\Bbb{C}^2\setminus \bar{B}.\\
\iota F^{-1}\Bbb{R}^-=Ver\cap B.\\
\iota F^{-1}0=Ver\cap\partial B.\\
\end{array}$$ I
| 1,675
| 2,692
| 2,421
| 1,549
| 4,074
| 0.768254
|
github_plus_top10pct_by_avg
|
\leq e^{(1-\alpha-\beta)\beta^{-1}\tau}\left( {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{B_1(0)} \ast \theta\|}_q + {\| e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)} \ast \theta\|}_q \right) \\
& \leq e^{(1-\alpha-\beta)\beta^{-1}\tau}\left( {\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{B_1(0)}\|}_1{\|\theta\|}_q + {\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\mathbf{1}_{{\mathbb R}^d \setminus B_1(0)}\|}_qM\right). \end{aligned}$$ Since ${\|\theta(\tau)\|}_q \lesssim 1$, by and we have, $$\begin{aligned}
{\|\vec{v}\|}_q & \lesssim e^{(1-\alpha-\beta)\beta^{-1}\tau}\left(1 + e^{(d-\gamma)\tau} \right) \\
& \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau} + e^{(1-\beta - \gamma\beta)\beta^{-1}\tau}. \end{aligned}$$ Since $1 - \beta - \gamma\beta \leq 0$ and $1 - \alpha \leq 0$, by Morrey’s inequality we may conclude $\vec{v} \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$. Similarly if $\gamma = d$, then by the same reasoning as above, and imply, $$\begin{aligned}
{\|\vec{v}\|}_q & \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau}\left(1 + \tau + e^{(d-\gamma)\tau} \right) \\
& \lesssim e^{(1-\alpha - \beta)\beta^{-1}\tau}\left(1 + \tau\right) + e^{(1-\beta - \gamma\beta)\beta^{-1}\tau}. \end{aligned}$$ Since $1 - \alpha - \beta < 0$, we may conclude also in this case that $\vec{v} \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$. Therefore Lemma \[lem:rescaled\_inftybdd\] applies with the hypotheses of Theorem \[thm:IA2\]. Re-writing in terms of $x$ and $t$, this implies . A similar proof with Lemma \[lem:finite\_p\_bounded\_unifint\] in place of Lemma \[lem:finite\_p\_bounded\] also proves *(ii)*
We now prove Theorem \[thm:IA2\].
(Theorem \[thm:IA2\]: **Intermediate Asymptotics II**) To complete the proof of Theorem \[thm:IA2\], we estimate the decay of the relative entropy . The proof of Theorem \[thm:IA\] used the estimate . Here we use the bound ${\|\theta(\tau)\|}_\infty \lesssim 1$ and to imply, if $\ga
| 1,676
| 557
| 595
| 1,808
| null | null |
github_plus_top10pct_by_avg
|
$(f,\Xi_M,\kappa) \in Imm^{sf}(n-k,k)$ be an arbitrary element, where $f: M^{n-k} \looparrowright \R^n$ is an immersion of codimension $k$ with the characteristic class $\kappa \in H^1(M^{n-k};\Z/2)$ of the skew-framing $\Xi_M$. We say that the pair $(M^{n-k},\kappa)$ admits a retraction of order $q$, if the mapping $\kappa : M^{n-k} \to \RP^{\infty}$ is represented by the composition $\kappa = I \circ \bar \kappa : M^{n-k} \to
\RP^{n-k-q-1} \subset \RP^{\infty}$. The element $[(f, \Xi_M,
\kappa)]$ admits a retraction of order $q$, if in the cobordism class of this skew-framed immersion there exists a triple $(M'^{n-k}, \Xi_{M'}, \kappa')$ that admits a retraction of order $q$.
$$$$
### Theorem 1 {#theorem-1 .unnumbered}
Let $q = q(l)$ be a positive integer, $q=2(mod 4)$. Let us assume that an element $\alpha \in Imm^{sf}(\frac{3n+q}{4},\frac{n-q}{4})$ admits a retraction of the order $q$ and $3n-12k-4>0$. Then the element $\delta(\alpha) \in
Imm^{\D_4}(n-2k,2k)$, $k=\frac{n-q}{4}$, is represented by a $\D_4$-framed immersion $[(g,\Psi_N,\eta)]$ with $\I_b$-control. $$$$
Proof of Theorem 1
==================
Let us denote $n-k-q-1 = 3k-1$ by $s$. Let $d: \RP^{s} \to \R^n$ be a generic mapping. We denote the self-intersection points of $d$ (in the target space) by $\Delta(d)$ and the singular points of $d$ by $\Sigma(d)$.
Let us recall a classification of singular points of generic mappings $\RP^{s} \to \R^n$ in the case $4s < 3n$, for details see \[Sz\]. In this range generic mappings have no quadruple points. The singular values (in the target space) are of the following two types:
– a closed manifold $\Sigma^{1,1,0}$;
– a singular manifold $\Sigma^{1,0}$ (with singularities of the type $\Sigma^{1,1,0}$).
The multiple points are of the multiplicities 2 and 3. The set of triple points form a manifold with boundary and with corners on the boundary. These “corner” singular points on the boundary of the triple points manifold coincide with the manifold $\Sigma^{1,1,0}$. The regular part of boundary
| 1,677
| 662
| 366
| 1,781
| 3,432
| 0.77243
|
github_plus_top10pct_by_avg
|
=\bigcup_u\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\}\circ I_1(u,z',x)\big\},
{\label{eq:I12-def}}\\
I_3(y,z,z',x)=\bigcup_u\Big\{\{I_2(y,z,u)\circ I_2(u,z',x)\}\cup\big\{
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\}\circ\{I_1(u,z,x)\cap I_1(u,z',x)\}\big\}\Big\}.
{\label{eq:I3-def}}\end{gathered}$$
For example, since ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, we have $$\begin{aligned}
{\label{eq:fin-ind:=1}}
&({(\ref{eq:fin-ind})}\text{ for }j=1)={\mathop{\Dot{\bigcup}}}_{e_1}{\mathop{\Dot{\bigcup}}}_{T\ge1}\,
{\mathop{\Dot{\bigcup}}}_{\vec b_T:b_1=e_1}\Big\{H_{{{\bf n}};\vec b_T}(y,x)\cap\big\{
z_1\in{{\cal D}}_{{{\bf n}};0},\,z'_1\in{{\cal D}}_{{{\bf n}};T}\big\}\Big\}{\nonumber}\\
&\qquad\subset{\mathop{\Dot{\bigcup}}}_{e_1}\Big\{\big\{I_1(y,z_1,{\underline{e}}_1)\circ I_2
({\overline{e}}_1,z'_1,x)\big\}\cap\big\{n_{e_1}>0,~e_1\text{ is pivotal for }
y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\Big\}.\end{aligned}$$ It is not hard to see in general that $$\begin{aligned}
{\label{eq:fin-ind:geq2}}
&({(\ref{eq:fin-ind})}\text{ for }j\ge2){\nonumber}\\
&\quad\subset{\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\bigg\{\Big\{I_1(y,z_1,{\underline{e}}_1)
\circ I_3({\overline{e}}_1,z_2,z'_1,{\underline{e}}_2)\circ\cdots\circ I_3({\overline{e}}_{j-1},
z_j,z'_{j-1},{\underline{e}}_j)\circ I_2({\overline{e}}_j,z'_j,x)\Big\}{\nonumber}\\
&\hspace{5pc}\cap\bigcap_{i=1}^j\big\{n_{e_i}>0,~e_i
\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\bigg\}.\end{aligned}$$
To bound [(\[eq:Theta’-2ndindbd4\])]{} using Lemma \[lmm:GHS-BK\], we further consider an event that includes [(\[eq:fin-ind:=1\])]{}–[(\[eq:fin-ind:geq2\])]{} as subsets. Without losing generality, we can assume that $y\ne{\underline{e}}_1$, ${\overline{e}}_{i-1}\ne{\underline{e}}_i$
| 1,678
| 811
| 1,502
| 1,714
| null | null |
github_plus_top10pct_by_avg
|
W\^[,(AB)]{}= to distinguish it from
W\^[,AB]{}= The 2PI CTP EA is the full Legendre transform
=W-J\_A\^A-12K\_[AB]{}Therefore the mean field equations of motion are
\_[,A]{}=-J\_A-K\_[AB]{}\^B
\_[,(AB)]{}=-12K\_[AB]{}
One further variation yields the identities
\^[C,E]{}+\_[,A(CD)]{}G\^[CD,E]{}=-\_[A]{}\^E
\^[C,(EF)]{}+\_[,A(CD)]{}G\^[CD,(EF)]{}=-\_[(AB)]{}\^[(EF)]{}\^B
\_[,(AB)C]{}\^[C,E]{}+\_[,(AB)(CD)]{}G\^[CD,E]{}=0
\_[,(AB)C]{}\^[C,(EF)]{}+\_[,(AB)(CD)]{}G\^[CD,(EF)]{}=-12\_[(AB)]{}\^[(EF)]{} where $\delta_{\left(AB\right)}^{\left(EF\right)}$ stands for the symmetrized identity operator
\_[(AB)]{}\^[(EF)]{}=12{\_A\^E\_B\^F+\_A\^F\_B\^E}
We now write the derivatives of the mean fields in terms of correlations
\^[C,E]{}=W\^[,CE]{}=iG\^[CE]{}
\^[C,(EF)]{}=W\^[,C(EF)]{}=i2
G\^[CD,E]{}&=&2W\^[,(CD)E]{}-\^[C,E]{}\^D-\^[D,E]{}\^C&=&i
G\^[CD,(EF)]{}&=&2W\^[,(CD)(EF)]{}-\^[C,(EF)]{}\^D-\^[D,(EF)]{}\^C&=&i2{\_H\^C\_H\^D\_H\^E\_H\^F-(\^C\^D+G\^[CD]{})(\^E\^F+G\^[EF]{}).&-&\^D&-&.\^C} This last equation may be rewritten as
G\^[CD,(EF)]{}&=&i2{\_H\^C\_H\^D\_H\^E\_H\^F-\^D\_H\^C\_H\^E\_H\^F-\^C\_H\^D\_H\^E\_H\^F.&-&. (-\^C\^D+G\^[CD]{})(\^E\^F+G\^[EF]{})} and further
-2iG\^[CD,(EF)]{}&=&\_H\^C\_H\^D\_H\^E\_H\^F&-&\^C\^D\^E\^F&-&\^C\^DG\^[EF]{}-\^C\^EG\^[DF]{}-\^C\^FG\^[DE]{}&-&\^D\^EG\^[CF]{}-\^D\^FG\^[CE]{}-\^E\^FG\^[CD]{}&+&i\^CG\^[EF,D]{}+i\^DG\^[EF,C]{}
We also notice the identity
-2i\^[C,(EF)]{}=-iG\^[CE,F]{}+\^EG\^[CF]{}+\^FG\^[CE]{}
Early stochastic formulations
-----------------------------
As we have said in the Introduction, it has been known for a long time that the Boltzmann equation is just a mean field equation, and can be improved by upgrading it to a full Langevin type equation where particle number fluctuations are explicitly included. Since from the point of view of field theory the Boltzmann equation is just a particular limit of the Kadanoff-Baym equations, which are in turn equivalent to the Schwinger-Dyson equations [@CH08], it is natural to seek a corresponding stochastic gener
| 1,679
| 3,739
| 3,079
| 1,317
| null | null |
github_plus_top10pct_by_avg
|
/v) \;.$$
If we replace $v'$ by adding to it an integer multiple of $v$, then ${\operatorname{sk}}(v,v')$ changes by $${\operatorname{sk}}(v,v'+nv) = {\operatorname{sk}}(v,v') + n \;.$$ In particular, since $v'$ is unique up to addition of an integer multiple of $v$, looking at the fractional part, that is in $\R/\Z$, we get a quantity ${\operatorname{sk}}(v)\in (-1/2,1/2]$ depending only on $v$: $${\operatorname{sk}}(v) : ={\operatorname{sk}}(v,v') \mod 1 \;.$$ This is the least skewness of a fundamental domain for the lattice constructed from the primitive vector $v$.
The signed ratio $\rho(v) = \pm |v'|/|v|$ and the least skewness ${\operatorname{sk}}(v)$ are asymptotically equivalent: $$\rho(v) =
{\operatorname{sk}}(v)\left(1+O(\frac 1{|v|^2})\right) \;.
$$
In terms of the angle $0<\alpha<\pi$ between the vectors $v$ and $v'$, we have $$\label{relation between ecc and alpha}
{\operatorname{sk}}(v,v') = \frac{|v'|}{|v|}\cos\alpha \;.
$$ Our claim follows from this and the fact $\cos\alpha = \pm 1+O(1/|v|^2)$, which follows from the upper bound of Lemma \[lem:angle\].
Thus the sequences $\{\rho(v)\}$, $\{{\operatorname{sk}}(v)\}$ are asymptotically identical, hence uniform distribution of one implies that of the other. To prove Theorem \[unif dist of rho\] it suffices to show
\[unif dist of ecc\] As $v$ ranges over all primitive vectors in the lattice $L$, the least skewness ${\operatorname{sk}}(v)$ become uniformly distributed modulo one.
This result, for the standard lattice $\Z[\sqrt{-1}]$, was highlighted by Good in the introduction to [@Good:1983a]. Below we review the reduction of Theorem \[unif dist of ecc\] to Theorem \[equidistribution\].
Proof of Theorem \[unif dist of ecc\]
-------------------------------------
Our problems only depend on the lattice $L$ up to scaling. So we may assume that $L$ has a basis $L=\{1,z\}$ with $z=x+iy$ in the upper half-plane. The area of a fundamental domain for $L$ is ${\operatorname{area}}(L)={\operatorname{Im}}(z)$. Any primitive vector has the for
| 1,680
| 3,402
| 2,084
| 1,621
| 2,729
| 0.777455
|
github_plus_top10pct_by_avg
|
the dependence on the topology of the data, and $C_b'$ and $C_b$ are constants that only depend on $b$. Putting these together, we will show that there exists a $\theta\in\Omega_b$ such that $$\begin{aligned}
\|\widehat\theta -\theta^* \|_2 &\leq& \frac{2\|\nabla \cL_{\rm RB}(\theta^*)\|_2 }{-\lambda_2(H(\theta))} \;\, \leq \;\, C''_b \frac{\sqrt{\log d \, \sum_{j\in[n]} \ell_j}}{\gamma \,\lambda_2(L)} \;. \label{eq:intro_bound}\end{aligned}$$ Here $\lambda_2(H(\theta))$ denotes the second largest eigenvalue of a negative semi-definite Hessian matrix $H(\theta)$ of the objective function. The reason the second largest eigenvalue shows up is because the top eigenvector is always the all-ones vector which by the definition of $\Omega_b$ is infeasible. The accuracy depends on the topology of the collected data via the comparison graph of given data.
\[def:comparison\_graph1\] (Comparison graph $\H$). We define a graph $\H([d],E)$ where each alternative corresponds to a node, and we put an edge $(i,i')$ if there exists an agent $j$ whose offerings is a set $S_j$ such that $i, \i \in S_j$. Each edge $(i,\i) \in E$ has a weight $A_{i \i}$ defined as $$\begin{aligned}
A_{i\i} &=& \sum_{j\in[n] : i,\i \in S_j} \frac{\ell_j}{\kappa_j(\kappa_j-1)}\;,
\end{aligned}$$ where $\kappa_j = |S_j|$ is the size of each sampled set and $\ell_j$ is the number of separators in $S_j$ defined by rank-breaking in Section \[sec:intro\].
Define a diagonal matrix $D = {\rm diag}(A{\boldsymbol{1}})$, and the corresponding graph Laplacian $L = D - A$, such that $$\begin{aligned}
\label{eq:comparison1_L}
L &=& \sum_{j = 1}^n \frac{\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top.
\end{aligned}$$ Let $ 0 = \lambda_1(L) \leq \lambda_2(L) \leq \cdots \leq \lambda_d(L)$ denote the (sorted) eigenvalues of $L$. Of special interest is $\lambda_2(L)$, also called the spectral gap, which measured how well-connected the graph is. Intuitively, one can expect better accuracy when the sp
| 1,681
| 1,579
| 1,806
| 1,563
| 3,035
| 0.775313
|
github_plus_top10pct_by_avg
|
frac{1}{2}u_iu_i -\frac{\lambda^2}{2}W_{ij}W_{ij}\right) = -\left(\frac{1}{2}|\MM{u}|^2 -\frac{\lambda^2}{2}|W|^2\right).$$
One-form quasi-conservation law
-------------------------------
For our multisymplectic formulation of EPDiff($H^1$), the independent variables are $$q^j = x_j, \quad j=1,\ldots n, \qquad q^{n+1} = t,$$ and the dependent variables are $$\begin{aligned}
z^i = u_i, \qquad z^{n + k} =
l_k, \qquad z^{2n + k} = \pi_k,\qquad z^{(i+2)n+j}=W_{ij}\,,\end{aligned}$$ where $i,j$ and $k$ range from $1$ to $n$. Comparing (\[epmslag\]) with (\[mslag\]) gives the following non-zero components $L^{\alpha}_j$: $$L^j_i = \lambda^2W_{ij}, \qquad L_{n+k}^j = \pi_ku_j, \qquad
L_{n+k}^{n+1} = \pi_k, \qquad i,j,k=1,\ldots,n.$$ Therefore the one-form quasi-conservation law amounts to $$\label{epqcl}
\left(\pi_k{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2W_{ij}{\mathrm{d}}u_i+\pi_ku_j{\mathrm{d}}l_k\right)_{,j}={\mathrm{d}}L.$$ The exterior derivative of this expression yields the structural conservation law $$\label{epsympcl}
\left({\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2{\mathrm{d}}W_{ij}\wedge{\mathrm{d}}u_i+u_j{\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k+\pi_k{\mathrm{d}}u_j\wedge{\mathrm{d}}l_k\right)_{,j}=0.$$
Conservation of energy
----------------------
For EPDiff($H^1$), the ${\mathrm{d}}t$-component of the pullback of the one-form conservation law (\[epqcl\]) gives $$\left(\pi_kl_{k,t} - L\right)_{,t} +\left(
\lambda^2W_{ij}u_{i,t} + \pi_ku_jl_{k,t} \right)_{,j} = 0.$$ In terms of $\MM{u}$ and its derivatives, this amounts to $$\left(u_im_i-\frac{1}{2}u_iu_i-\frac{\lambda^2}{2}u_{i,j}u_{i,j}\right)_{,t}
+\left(\lambda^2u_{i,j}u_{i,t}+u_iu_jm_i\right)_{,j} =
0,$$ where $$m_i=u_i-\lambda^2u_{i,kk}.$$ This is the energy conservation law for EPDiff($H^1$).
Conservation of momentum
------------------------
Similarly, the conservation law that is associated with translations in the $x_i$-direction is $$\left(\pi_kl_{k,i}\right)_{,t} +\left( \lambda^2
W_{kj}u_{k,i} + \pi_ku_jl_{k,i} - \de
| 1,682
| 1,357
| 2,319
| 1,632
| 4,052
| 0.768438
|
github_plus_top10pct_by_avg
|
ration.
**AND nodes:** Each part template is an AND node, which uses its children (latent patterns) to represent its constituent or contextual regions. We use $v$ and $Child(v)=\{u_{1},u_{2},\ldots,u_{m}\}$ to denote the part template and its children latent patterns. We learn the average displacement from $\Lambda_{u}$ to $\Lambda_{v}$ among different image, denoted by $\Delta{\bf p}_{u}$, as a parameter of the AOG. Given parsing results of children latent patterns, we use the image region of each child node $\Lambda_{u}$ to infer the region for the parent $v$ based on its spatial relationships. Just like a deformable part model, the parsing of $v$ can be given as $$S_{v}\!=\!\!\!\!\!\!\!\sum_{u\in Child(v)}\!\!\!\!\!\!\!\big[S_{u}\!+\!S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})\big],\;\;\Lambda_{v}\!=\!f(\Lambda_{u_{1}},\ldots,\Lambda_{u_{m}})\!$$ where we use parsing results of children nodes to infer the parent part template $v$. $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$ denotes the spatial compatibility between $\Lambda_{u}$ and $\Lambda_{v}$ *w.r.t.* their average displacement $\Delta{\bf p}_{u}$. Please see the appendix for details of $S^{\textrm{inf}}(\Lambda_{u}|\Lambda_{v})$.
For the region parsing of the part template $v$, we need to estimate two terms, *i.e.* the center position ${\bf p}_{v}$ and the scale $scale_{v}$ of $\Lambda_{v}$. We learn a fixed scale for each part template, which will be introduced in Section \[sec:learnAOG\]. In this way, we can simply implement region parsing by computing the region position that maximizes the inference score ${\bf p}_{v}=f(\Lambda_{u_{1}},\Lambda_{u_{2}},\ldots,\Lambda_{u_{m}})={\arg\!\max}_{{\bf p}_{v}}S_{v}$.
**Terminal nodes (neural units):** Each terminal node under a latent pattern represents a deformation candidate of the latent pattern. The terminal node has a fixed image region, *i.e.* we propagate the neural unit’s receptive field back to the image plane as its image region. We compute a neural unit’s inference score based on both its neura
| 1,683
| 1,272
| 2,344
| 1,839
| 582
| 0.805145
|
github_plus_top10pct_by_avg
|
isor-Ck}
{\left \lfloor \pi_{*} D' \right \rfloor} = {\left \lfloor \pi_{*}(K_Y-L^{(k)}) \right \rfloor} = K_X + k H
- \sum_{j=1}^r {\left \lfloor \frac{k n_j}{d} \right \rfloor} \mathcal{C}_j,$$ which has $w$-degree $$\label{eq:degree-sk}
k - |w| - \sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} d_j = \sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} d_j - |w|
= s_k - |w|.$$ Note that if the curve $\mathcal{C}$ is reduced and thus all $n_i=1$, then $s_k = k$ and $e_{\v k} = 0$. Moreover, $$\pi^{*} {\left \lfloor \pi_{*}D' \right \rfloor} - D' = - K_\pi +
\sum_{P \in S} \sum_{\v \in \Gamma_P} \bigg( {\left \lfloor \frac{k(m_\v-d\b_\v)}{d} \right \rfloor} + k \b_\v - e_{\v k} \bigg) E_\v.$$
From Proposition \[prop:H0YD\]\[item2-lemma-h0\], $F \in H^0(Y,\cO_{Y}(K_Y-L^{(k)}))$ if and only if for all $\v \in \Gamma_P$ and $P \in \Si$ one has $$\label{eq:condition-F}
\operatorname{mult}_{E_\v} \pi^{*} F \geq
{\left \lfloor \frac{k m_\v}{d}-k\b_\v \right \rfloor} + k\b_\v - e_{\v k}
- (\nu_\v-1).$$ It remains to show that this condition is equivalent to the one given in the statement. If $F \in \CC[x,y,z]$ is $w$-homogeneous of degree $s_k-|w| = k - |w| - \sum_j {\left \lfloor \frac{kn_j}{d} \right \rfloor} d_j$, then the divisor $(F)-kH+K_X+\sum_{j=1}^r {\left \lfloor \frac{kn_j}{d} \right \rfloor} \mathcal{C}_j$ has degree $0$ and thus $\operatorname{mult}_{\v} \pi^{*} F - k\b_\v + \nu_\v-1 + e_{\v k}$ is an integer. Hence the condition can be rewritten as $${\left \lceil \operatorname{mult}_{E_\v} \pi^{*} F - \frac{km_\v}{d} + e_{\v k} + \nu_\v-1 \right \rceil} \geq 0$$ or equivalently $\operatorname{mult}_{E_\v} \pi^{*} F > \frac{km_\v}{d} - e_{\v k} - \nu_\v$. Since $m_\v = \sum_{j=1}^r n_j m_{\v j}$ the latter term equals $\sum_{j=1}^r {\left \{ \frac{kn_j}{d} \right \}} m_{\v j}$ and the proof is complete.
Irregularity of a cyclic cover of a weighted projective plane
-------------------------------------------------------------
Let $\rho_X:\tilde X\to X$ be a cyclic covering and $\pi:Y\to X$ a $
| 1,684
| 1,134
| 899
| 1,559
| null | null |
github_plus_top10pct_by_avg
|
Meropenem 8 (6) 1--10
Ampicillin-sulbactam 6 (4) 1--37
Ceftazidime 6 (4) 1--9
Ampicillin 4 (3) 2--7
Azithromycin 4 (3) 1--6
Fluconazole 4 (3) 1--37
Nafcillin 3 (2) 2--6
Clindamycin 3 (2) 2--6
Gentamycin 2 (1) 6--9
Daptomycin 2 (1) 5--7
Amoxicillin 2 (1) 3--6
Trimethoprim-\ 1 (\<1) 3
sulphamethoxazole\*
Ertapenem 1 (\<1) 2
Clarithromycin 1 (\<1) 3
Caspafungin 1 (\<1) 11
Micafungin 1 (\<1) 1
Aztreonam 1 (\<1) 2
Moxifloxacin 1 (\<1) 2
Linezolid 1 (\<1) 9
Cefotaxime 1 (\<1) 1
Cefazolin 1 (\<1) 3
-----------------------------------------------------------------
\[\[i\] \*Recorded as treatment. This analysis excluded patients continued on ciprofloxacin or trimethoprim-sulphamethoxazole for prophylaxis.\]
###### Spontaneous bacterial peritonitis outcome and ceftriaxone dosage data
Anonymized outcome data from medical records of patients treated with ceftriaxone for spontaneous bacterial peritonitis at the Beth Israel Deaconess Medical Center, Boston, USA, between January 2003 and December 2011. Exclusion criteria from dataset were: \<250 neutrophils in ascites, prior liver transplant, evidence of intra-abdominal source of infection (abscess, perforation, recent (within 2 weeks) intra-abdominal surgery), peritoneal dialysis and documentation of a secondary infection (urinary tract infection, pneumonia, blood stream infection,
| 1,685
| 4,577
| 862
| 1,014
| null | null |
github_plus_top10pct_by_avg
|
K)$, the group $L$ canot be equal to $K$. So $G/K f_{J}^{G}$ is a $R$-linear combination of transitive $G$-set $G/L'$ where $|L'|<|K|$. By induction, $G/K$ is a $R$-linear combination of elements of the form $G/I f_{J}^{G}$.
Following [@deiml], let us consider the linear form $\phi_{G}$ on $RB(G)$ defined on a basis element by: $$\phi\big(G/I f_{J}^{G}\big) = \left\{\begin{array}{c}1 \hbox{ if $I=J$,} \\0 \hbox{ if $I\neq J$.}\end{array}\right.$$
\[re1\]
- If $R=k$ is a field of characteristic $p$, and if $p\nmid |G|$, then the idempotents $f_{J}^{G}$ are the idempotents $e_{J}^{G}$ so it is easy to check that
$\phi_{G}(X) = \sum_{H\in [s(G)]}\frac{|H|}{|N_{G}(H)|} |X^{H}|$, for $X\in kB(G)$.
- If $p\mid |G|$, it seems rather difficult to compute the value of $\phi_{G}$ on a transitive $G$-set.
\[indu\] Let $H\leqslant G$ and $J$ be a $p$-perfect subgroup of $H$. Then:
1. $Ind_{H}^{G}(H/J f_{J}^{H})=G/Jf_{J}^{G}$.
2. Moreover if $p\mid |N_{H}(J)/J|$ and $p^2\nmid |N_{H}(J)/J|$, let $S_{J}$ be a subgroup of $H$ such that $J\subset S_{J}$ and $O^{p}(S_{J})=J$. Then: $$Ind_{H}^{G}(H/S_{J}f_{J}^{H})=G/S_{J} f_{J}^{G}.$$
Using Lemma \[form\], we have: $$\begin{aligned}
Ind_{H}^{G}\big(H/J f_{J}^{H}\big)&= \frac{|N_{H}(J)|}{|J|}Ind_{H}^{G}(e_{H}^{G})\\
&=\frac{|N_{H}(J)|}{|J|} \frac{|N_{G}(J)|}{|N_{H}(J)|} e_{J}^{G}\\
&=G/J f_{J}^{G}.\end{aligned}$$ For the second part, by Lemma $3.5$ of [@yoshida_idempotent], we have $Res^{G}_{H}(f^{G}_{J})=\sum_{J'} f^{H}_{J'}$ where $J'$ runs the subgroups of $H$ up to $H$-conjugacy such that $J'$ is conjugate to $J$ in $G$. So, we have: $$\begin{aligned}
H/S_{J} Res^{G}_{H}(f^{G}_{J})&=\sum_{J'} H/S_{J} f^{H}_{J'},\end{aligned}$$ but we have: $$\begin{aligned}
H/S_{J}f_{J'}^{H}= \sum_{\underset{O^{p}(K)=J'}{K\leqslant J\hbox{ {\footnotesize up to $H$-conjugacy}}}} |(H/S_{J})^K|e_{K}^{H}. \end{aligned}$$ But $|(H/S_{J})^{K}|=0$ unless $K$ is $H$-conjugate to a subgroup of $S_{J}$. Without lost of generality one can assume $K\subseteq S_{J}$. So the only non zero t
| 1,686
| 1,725
| 1,761
| 1,457
| null | null |
github_plus_top10pct_by_avg
|
V}}$. Then, we compare the obtained ${\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}$ among all reconfiguration states to complete the selection of optimal ${\widetilde{\mathbf{H}}_{\psi,V}}$, denoted by ${\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}$. Since there are $\Psi$ reconfiguration states and $\binom{N_t}{L_t}\binom{N_r}{L_r}$ possible submatrices for each state, the total number of possible selections to search is given by $$\label{}
N_{\mathrm{total}}=\Psi\binom{N_t}{L_t}\binom{N_r}{L_r}=\frac{\Psi N_r!N_t!}{L_r!L_t!\left(N_r-L_r\right)!\left(N_t-L_t\right)!}.$$ When $N_t\gg L_t$, $N_r\gg L_r$, and/or $\Psi\gg 1$, the total number to search, $N_{\mathrm{total}}$, would be too large for practical applications due to the high complexity. Thus, in what follows, we propose a low-complexity design to obtain ${\widehat{\widetilde{\mathbf{H}}}_{\widehat{\psi},V}}$ which achieves the near optimal throughput performance.
As discussed earlier, the mmWave MIMO channel has a sparse nature, and the number of non-vanishing rows and columns of the virtual channel matrix is relatively small in the clustered scattering environment. Now let us consider an extreme scenario such that all of the non-vanishing entries of ${\mathbf{H}_{\psi,V}}$ are contained in the low-dimensional submatrix, and ${\mathbf{H}_{\psi,V}}$ is approximated by $$\label{eq:ssHv}
{\mathbf{M}}\odot{\mathbf{H}_{\psi,V}},$$ where $$\label{eq:MaskM}
{\mathbf{M}}(i,j)=\left\{
\begin{array}{ll} 1\;, &\mbox{if}~(i,j)\in \widehat{{\mathcal{M}}}_{\psi},\\
0\;, &\mbox{otherwise,}
\end{array}
\right.$$ and $\widehat{{\mathcal{M}}}_{\psi}$ is the beam selection mask corresponding to ${\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}$. Note that a similar approximation was adopted in, e.g., [@Sayeed_07_maxMcsparseRAA] to approximate the sparse virtual MIMO channel. With , we have $$\begin{aligned}
\label{eq:appHtoHvl}
&\left|{\mathbf{I}}_{L_r}+\frac{\rho}{L_t} {\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}{\widehat{\widetilde{\mathbf{H}}}_{\psi,V}}^H\r
| 1,687
| 1,468
| 1,817
| 1,647
| 1,764
| 0.785887
|
github_plus_top10pct_by_avg
|
M_N}
\label{eq:vnlo}
+ \,C_1^1 \; \displaystyle\frac{{\vec \sigma}_2{\vec q}}{2 M_N}
+ {\im} \, C_1^2 \; \displaystyle
\frac{({\vec \sigma}_1 \times {\vec\sigma}_2)\;{\vec q}}{2 M_N}
\\
&+ C_2^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q}
\;{\vec\sigma}_2{\vec q}}{4 M_N^2}+
C_2^1 \; \displaystyle\frac{{\vec \sigma}_1
{\vec \sigma}_2 \; {\vec q}^{\; 2}}
{4 M_N^2}+
C_2^2 \; \displaystyle\frac{{\vec q}^{\,2}}{4 M_N^2} \,.
\nonumber\end{aligned}$$
----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- --
![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel1 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel2 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel3 "fig:")
(a) (b) (c)
----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------
| 1,688
| 359
| 936
| 1,956
| null | null |
github_plus_top10pct_by_avg
|
le cell. Even if both inputs are ‘1’ the Physarum cells have no space to avoid collision and therefore the merge and propagate into the output channel.
The gate [and]{} looks like distorted ‘H’: $
\begin{smallmatrix}
& & \, & \downarrow & & \downarrow\\ \hline
\downarrow & & \, & & \downarrow& \\
\end{smallmatrix}
$ When only one input is ‘1’ the Physarum propagates towards closes attractant and exits along right output channel. When both inputs are ‘1’ the Physarum from the right input channel propagates into the right output channel. The Physarum from the left input channel avoids merging with another Physarum and propagates towards left output channel. The left output channel realises [and]{}.
Ballistic logical gates {#ballistic}
-----------------------
In designs of ballistic gates [@adamatzky2010slimeballistic] we employ inertia of the Physarum growing zones. On a non-nutrient substrate the plasmodium propagates as a traveling localisation, as a compact wave-fragment of protoplasm. The plasmodium-localisation travels in its originally predetermined direction for a substantial period of time even when no gradient of chemo-attractants is present. We explore this feature of Physarum localisations to design a two-input two-output Boolean gates. The gate realising [and]{} on one output and [or]{} on another output look like horizontally flipped ‘K’: $
\begin{smallmatrix}
\searrow & \downarrow \\
\swarrow & \downarrow
\end{smallmatrix} .
$ When left input is ‘1’ the Physarum propagates inertially along the vertical (on the right) output channel. The same happens when right input is ‘1’. If both inputs are ‘1’ then the Physarum from the right input propagates along vertical output channel but the Physarum from the left input repels from the right-input-Physarum and moves into the left output channel. The left output channel realises [and]{} and the right output channel realises [or]{}.
The gate [not]{} is an asymmetric cross junction
| 1,689
| 3,453
| 2,493
| 1,702
| null | null |
github_plus_top10pct_by_avg
|
such ethical approval is not mandatory for experimental studies that do not involve any risk or discomfort for the participants as long as anonymity is preserved (Spanish Law 15/1999 for Personal Data Protection) and participants are fully informed about the procedures of the study and give written informed consent to participate. The current experiment is in line with this regulation and further complies with the international standards of experimental economics research. The participants did not learn the identity of the other participants they interacted with and the identity of the participants cannot be inferred from the data which is entirely anonymous. Finally, the experimental protocols were approved by the LINEEX (University of Valencia), the institution hosting the experiment (see their webpage for further details about their data protection policy).
Results {#sec005}
=======
Descriptive statistics {#sec006}
----------------------
We summarize the data in [Table 1](#pone.0204392.t001){ref-type="table"}. Panel A presents the decision of the 48 investors for each possible distribution of endowments. The data for allocators is summarized in Panel B. We note that allocators can only make a decision if they have received any transfer from investors, thus Panel B reports the behavior of those allocators who received a positive transfer from investors. This, in turn, implies that the number of observations may differ across distribution. We report the correlation between the investor's transfer (*X*) and the allocator's returned share (*y*) in Panel C.
10.1371/journal.pone.0204392.t001
###### Summary of the data.
{#pone.0204392.t001g}
*e*~*i*~ = 40 *e*~*i*~ = 10
---------------------------------------------------- --------------- --------------- -------------- --------------
**A. Investor's behavior**
Amount sent (*X*)
| 1,690
| 314
| 1,820
| 1,695
| null | null |
github_plus_top10pct_by_avg
|
unction doSomething(macguffin: any) {
//todo: implement doSomething
}
}
export class MyCollection {
public static doSomething(macguffin: any) {
//todo: implement doSomething
}
}
A:
It's probably best to use modules instead of namespaces or static class methods. From the TypeScript official documentation page on namespaces and modules:
Namespaces are simply named JavaScript objects in the global namespace. [...] Just like all global namespace pollution, it can be hard to identify component dependencies, especially in a large application.
[...]
Modules provide for better code reuse, stronger isolation and better tooling support for bundling.
[...]
Starting with ECMAScript 2015, modules are native part of the language, and should be supported by all compliant engine implementations. Thus, for new projects modules would be the recommended code organization mechanism.
Q:
C#: How does the static object.Equals check for equality?
Say you have two different classes where each have their own implementation of Equals; which one is used? What if only one of them have one? Or none of them? Are any of the following lines equivalent?
object .Equals( first, second )
first .Equals( second )
second .Equals( first )
I'm guessing that the first two might be equivalent, but I don't really have a clue.
What does it really do?
A:
Basically it does three things:
Check for reference equality (return true if so)
Check for reference nullity (return false if either value is null; by now the null == null case has been handled)
Check for value equality with first.Equals(second)
The ordering shouldn't matter if both values have well-behaved equality implementations, as equality should be implemented such that x.Equals(y) implies y.Equals(x). However, the offline documentation I've got installed does state that first.Equals(second) (or objA.equals(objB) to use the real parameter naming) is specified. The online documentation doesn't mention this, interestingly enough.
Just to make all of this concrete,
| 1,691
| 2,315
| 1,479
| 1,165
| 1,219
| 0.792505
|
github_plus_top10pct_by_avg
|
le unitarity violation poses highly nontrivial features such as non-Hermitian Hamiltonian [@Antusch:2006vwa], or the evolution equation $i \frac{d}{dx} \nu_{\alpha} = \sum_{j} \left[ U \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) U^{\dagger} \right]_{\alpha \beta} \nu_{\beta}$ [@Escrihuela:2016ube]. The latter is not equivalent to (\[Schroedinger-eq-0th\]) in the vacuum mass eigenstate basis due to non-unitarity of the $U$ matrix.
[^17]: One can apply our formulas of $S$ matrix obtained under the constant matter density approximation to semi-realistic calculation for earth crossing neutrinos by using them in each shell (core, mantle, and crust regions, etc.) with proper connecting conditions at the boundaries.
[^18]: It can be re-parametrized in terms of the “$\alpha$ matrix parametrization” defined in ref. [@Escrihuela:2015wra]. The resultant values of $\alpha$ parameters are given as follows: $\alpha_{11} = 0.990$, $\alpha_{21} = - 0.0141$, $\alpha_{22} = 0.995$, $\alpha_{31} = -0.0445$, $\alpha_{32} = -0.0316$, $\alpha_{33} = 0.949$.
[^19]: This feature must be obvious if one goes back to the derivation of bound on $\mathcal{C}_{\alpha \beta}$ in [@Fong:2016yyh].
[^20]: We are aware that the assumption of equal sterile neutrino masses is contradictory to the assumption of no accidental degeneracy in the sterile mass spectrum we made in section \[sec:probability-2nd\]. It was done not to complicate term by term evaluation of the perturbative series, and to avoid using degenerate perturbation theory. Fortunately, we can remove this assumption to second order in $W$ in which no purely sterile sector energy denominator is involved.
[^21]: Of course, there is an issues of how to separate effects of $W^2$ correction terms from unitarity violation through $U$ matrix in leading order.
[^22]: It might be easier to obtain the phase factor if we use a different decomposition of $\tilde{H}$ from (\[tilde-H0+H1\]) by absorbing $W^{\dagger} A W$ into $\tilde{H}_{0}$.
[^23]: The phase itself needs not be s
| 1,692
| 875
| 2,371
| 1,763
| null | null |
github_plus_top10pct_by_avg
|
sider $D_1, D_2 \in \operatorname{Weil}_\QQ(X)$. The *intersection number* $(D_1 \cdot D_2)_X$ is defined as $$(D_1 \cdot D_2)_X :=
\frac{1}{k_1 k_2} (k_1 D_1 \cdot k_2 D_2 )_X \in \QQ,$$ where $k_1, k_2 \in \ZZ$ are chosen so that $k_1 D_1\in \text{Weil}(X)$, $k_2 D_2\in \text{Cart}(X)$ and either the divisor $D_1$ is compact or $D_1 \cap D_2$ is finite [@Fulton-Intersection Ch.2]. The local intersection number at $P \in D_1 \cap D_2$ is defined analogously as long as the condition $D_1 \not\subset D_2$ is satisfied.
\[thm:intersection\] Let $F : Y \to X$ be a proper morphism between two irreducible $V$-surfaces, and $D_1, D_2 \in \operatorname{Weil}_\QQ(X)$.
1. The cardinal of $F^{-1}(P)$, $P \in X$ generic, is a finite constant. This number is denoted by $\deg(F)$.
2. If $(D_1 \cdot D_2)_X$ is defined, then so is the number $(F^*(D_1) \cdot F^*(D_2))_Y$. In that case $(F^*(D_1) \cdot F^*(D_2))_Y=\deg(F)(D_1 \cdot D_2)_X$.
3. If $(D_1 \cdot D_2)_P$ is defined for some $P \in X$, then so is $(F^*(D_1) \cdot F^*(D_2))_Q$, $\forall Q \in F^{-1}(P)$, and $\sum_{Q\in F^{-1}(P)}(F^*(D_1)\cdot F^*(D_2))_Q=\deg(F)(D_1\cdot D_2)_P$.
\[ex:QNC:intersection\] Following Example \[ex:QNC\], note that $3D_1, 3D_2\in \operatorname{Cart}(X_1)$ and hence, according to Definition \[def:multintVM\], one has $(3D_1\cdot 3D_2)_{X_1}=\dim_\CC\frac{\cO_{X_1}}{(x^3,y^3)}=\dim_\CC \langle 1,x^2y,xy^2\rangle_\CC=3$. Hence $(D_1\cdot D_2)_{X_1}=\frac{1}{3}$.
Analogously one can check $(3D_1\cdot 3D_2)_{X_2}=\dim_\CC \langle 1,xy,x^2y^2\rangle_\CC=3$, and thus also $(D_1\cdot D_2)_{X_2}=\frac{1}{3}$.
\[formula\_self-intersection\] Let $X$ be a cyclic $V$-surface. Let $\pi: \hat{X} \to X$ be the weighted blow-up at a point $P$ of type $\frac{1}{d}(p,q)$ with respect to $w=(a,b)$. Assume $(d,p)=(d,q)=(a,b)=1$ and write $e = \gcd(d,aq-bp)$.
Consider $C$ and $D$ two $\QQ$-divisors on $X$, denote by $E$ the exceptional divisor of $\pi$, and by $\widehat{C}$ (resp. $\widehat{D}$) the strict transform of $C$ (resp. $D$). Let $\nu
| 1,693
| 1,769
| 1,674
| 1,513
| 2,607
| 0.778487
|
github_plus_top10pct_by_avg
|
==
In this section we show an upper bound for maximum load attained by the balanced allocation on [regular]{} dynamic graphs (i.e., Theorem \[thm:s2c\]). Suppose that the balanced allocation process has allocated $n$ balls to the dynamic regular graph $(G^{(1)},\ldots, G^{(n)})$. Define the *conflict graph* ${\mathcal{C}}_n$ formed by the edges selected by the $n$ balls. The vertex set of ${\mathcal{C}}_n$ is the set $[n]$ of bins, and the loads of these bins are updated during the process.
[Given a tree $T$ which is a subgraph of ${\mathcal{C}}_n$, [and vertices $u$, $v$ of the tree]{}, if $\{u,v\}$ is an edge of ${\mathcal{C}}_n$ then we say it is a *cycle-producing edge* with respect to the tree $T$. The name arises as adding this edge to the tree would produce a cycle, which may be a 2-cycle if the edge $\{u,v\}$ is already present in $T$.]{} For a positive integer $c>0$, a subgraph of ${\mathcal{C}}_n$ is called $c$-*loaded* if each vertex (bin) contained in the subgraph has load at least $c$. The following proposition presents some properties of connected components of ${\mathcal{C}}_n$.
\[pro:dy\] Let $(G^{(1)}, \ldots, G^{(n)})$ be a regular dynamic graph on vertex set $[n]$ which is $\varepsilon$-visible. Let that ${\mathcal{C}}_n$ be the conflict graph obtained after allocating $n$ balls using the balanced allocation process. Then for every given constant $c>0$, with probability at least $1-n^{-c}$, every $12(c+1)$-loaded connected component of ${\mathcal{C}}_n$ contains [strictly fewer than]{} $\log n$ vertices. Moreover, the number of cycle-producing edges in the component is at most $2(c+1)/\varepsilon$.
We will prove the proposition in Appendix \[sub:dy\]. We now explain how to recursively build a witness graph, provided there exists a bin whose load is higher than a certain threshold.
#### Construction of the Witness Graph {#construction-of-the-witness-graph-1 .unnumbered}
[ Let us start with a bin, say $r$, with $\ell+c$ balls. Clearly, if a ball is in bin $r$ at height $h$ then the other
| 1,694
| 216
| 854
| 1,779
| 1,737
| 0.786219
|
github_plus_top10pct_by_avg
|
l area and volume. The Bayesian linear results show notable bias in $N$, $\mathit{BA}$, and $V$.
{width="\textwidth"}
The CI coverages of GPR and the reference Bayesian inference method are compared in Figure \[fig:ci\]. The numerical CI% value is shown above each bar; the ideal value is here 95%. The CI% for the Bayesian linear estimates fall short of the 95% target, that is, the prediction intervals that are too narrow. The GPR prediction intervals perform well on basal area and stem volume, with good coverage also on stem number. The GPR CI% for these stand attributes is consistently better than the Bayesian linear. The GPR prediction intervals for height and diameter are overconfident, especially in the deciduous variables, and the performance is roughly similar to the Bayesian linear estimates.
{width="140mm"}
Total attributes
----------------
Point estimates and credible intervals for the total stem number, basal area, and stem volume were calculated from the species-specific results. The point estimates were computed by summing up the corresponding species-specific estimates. The GPR prediction interval for the total attributes is acquired from the prediction covariance $\boldsymbol{\Gamma}_{\mathbf{y}_*}$, because summation is a linear transformation.
The results for the total attributes are shown in Figure \[fig:totvars\]. In RMSE%, GPR estimates show the best performance. Bayesian linear estimates have lower RMSE% than kNN in the total basal area and volume, while kNN is better in the stem number. In the relative bias, GPR has fairly low bias and performs worst in the total stem volume. kNN has consistent slight bias, while the Bayesian linear estimates show large bias in the stem number. In credible interval coverage, GPR produces too wide intervals for basal area and stem volume (CI% between 98-99%). The Bayesian linear intervals are, on the other hand, with CI% around roughly 80%.
{width="140mm"}
Effect of training set size
----
| 1,695
| 204
| 3,373
| 1,698
| null | null |
github_plus_top10pct_by_avg
|
{R}}%
^{d})\}.$$Then $\overline{\pi }_{k,q,h,p}$ is equivalent with the interpolation norm of order $\rho =\frac{k+q+d/p_{\ast }}{2h}$ between the spaces $W_{\ast
}^{k,\infty }$ (the dual of $W^{k,\infty })$ and $W^{2h+q,2h,p}=\{f:$ $%
\left\Vert f_{n}\right\Vert _{2h+q,2h,p}<\infty \}$. This is proved in [\[BC\]]{}, see Section 2.4 and Appendix B. So the inequality (\[reg4\]) below says that the Sobolev space $W^{q,p}$ is included in the above interpolation space. However we prefer to remain in an elementary framework and to derive directly the consequences of (\[reg4\]) - see Lemma \[REG\] below
The following result is the key point in our approach (this is Proposition 2.5 in [@[BC]]):
\[lemma-inter\] Let $k,q,h\in {\mathbb{N}}$ with $h\geq 1$, and $p>1$ be given. There exists a constant $C_*$ (depending on $k,q,h$ and $p$ only) such that the following holds. Let $\mu $ be a finite measure for which one may find a sequence $\mu _{n}(dx)=f_{n}(x)dx$, $n\in {\mathbb{N}}$ such that $\pi _{k,q,h,p}(\mu ,(\mu _{n})_{n})<\infty .$ Then $\mu (dx)=f(x)dx$ with $%
f\in W^{q,p}$ and moreover $$\left\Vert f\right\Vert _{q,p}\leq C_{\ast }\times \pi _{k,q,h,p}(\mu ,(\mu
_{n})_{n}). \label{reg4}$$
The proof of Lemma \[lemma-inter\] is given in [@[BC]], being a particular case (take $\mathbf{e}=\mathbf{e}_{p}$) of Proposition A.1 in Appendix A.
We give a first simple consequence.
\[reg\] Let $p_{t}\in C^{\infty }(\R^{d}),t>0$ be a family of non negative functions and let $\varphi=\varphi(x)\geq 0$ be such that $\int \varphi(x)p_{t}(x)dx\leq m<\infty$ for every $t<1$. We assume that for some $\theta _{0}>0$ and $\theta _{1}>0$ the following holds: for every $q\in \N$ and $%
p> 1$ there exists a constant $C=C(q,p)$ such that $$\left\Vert \varphi p_{t}\right\Vert _{q,p}\leq Ct^{-\theta
_{0}(q+\theta _{1})},\quad t<1. \label{a1}$$Let $\delta>0$. Then, there exists a constant $C_*=C_*(q,p,\delta) $ such that $$\left\Vert \varphi p_{t}\right\Vert _{q,p}\leq C_\ast t^{-\theta _{0}(q+%
\frac{d}{p_{\ast }}+\delta)} ,\quad t<1
| 1,696
| 1,078
| 852
| 1,668
| null | null |
github_plus_top10pct_by_avg
|
odate, for example, the following term. $$\begin{aligned}
&&\langle \pi|N^*(t_{\rm snk})|N\rangle
\langle N| \bar N^*(t_{\rm src})|\pi\rangle \nonumber \\
&\times&e^{-E_N(t_{\rm snk}-t_{\rm src})}\times
e^{-E_\pi (N_t-t_{\rm snk}+t_{\rm src})}.\end{aligned}$$ Here, $N_t$ denotes the temporal extent of a lattice. Such a term is quite problematic and mimic a fake plateau at $E_N-E_\pi$ in the effective mass plot because it behaves as $\sim e^{-(E_N-E_\pi)(t_{\rm snk}-t_{\rm src})}$. Although these contaminations disappear when one employ enough large-$N_t$ lattice, our lattices do not have so large $N_t$. In order to eliminate such contributions, we impose the Dirichlet condition on the temporal boundary for valence quarks, which prevents valence quarks from going over the boundary. Though the boundary is still transparent for the states with the same quantum numbers as vacuum, [*e.g.*]{} glueballs, such contributions will be suppressed by the factor of $e^{-E_GN_t}$ and we neglect them in this paper. (Wraparound effects can be found even in quenched calculations [@Takahashi:2005uk].)
[**Formulation.**]{} We here give a brief introduction to our formulation [@Takahashi:2005uk; @Burch:2006cc]. Let us assume that we have a set of $N$ independent operators, $O_{\rm snk}^I$ for sinks and $O_{\rm src}^{I\dagger}$ for sources. We can then construct an $N\times N$ correlation matrix ${\cal C}^{IJ}(T)
\equiv
\langle O_{\rm snk}^I(T)O_{\rm src}^{J\dagger}(0)\rangle
=C_{\rm snk}^\dagger\Lambda(T)C_{\rm src}
$. Here, $
(C^\dagger_{\rm snk})_{Ii}\equiv \langle 0 | O_{\rm snk}^I | i \rangle
$ and $
(C_{\rm src})_{jI}\equiv \langle j | O_{\rm src}^{J \dagger} | 0 \rangle
$ are general matrices, and $\Lambda(T)_{ij}$ is a diagonal matrix given by $
\Lambda(T)_{ij}\equiv \delta_{ij} e^{- E_iT}.
$ The optimal source and sink operators, ${\cal O}_{\rm src}^{i \dagger}$ and ${\cal O}_{\rm snk}^{i}$, which couple dominantly (solely in the ideal case) to $i$-th lowest state, are obtained as $
{\cal O}_{\rm src}^{i \dagger}=\sum_J
O^
| 1,697
| 697
| 1,826
| 1,696
| 1,403
| 0.790034
|
github_plus_top10pct_by_avg
|
ces {#sec:RR}
----------------------------------------
For a given $X$ a normal projective surface and $D\in \operatorname{Weil}(X)$, the following formula is a generalization of the classical Riemann-Roch formula from the smooth case.
\[thm:RR\] There is a rational map $R_{X,P}:\operatorname{Weil}(X,P)/\operatorname{Cart}(X,P)\to \QQ$ for each singular point $P\in \operatorname{Sing}(X)$ such that $$\chi(\cO_X(D))-\chi(\cO_X)=\frac{1}{2}D\cdot (D-K_X) + \sum_{P\in \operatorname{Sing}(X)} R_{X,P}(D).$$
A consequence of this result via Serre duality is that $R_{X,P}(D)=R_{X,P}(D-K_X)$. One major breakthrough accomplished by Blache in [@Blache95 Thm. 2.1] is to provide an interpretation for $R_{X,P}(D+K_{X})$ as follows. Consider $\sigma:(\tilde X,E)\to (X,P)$ a resolution of $X$ at $P$, where $E$ is the exceptional part of the resolution, $\sigma^* D=E_{D}+\hat D$, $\hat D$ is the strict transform of $D$, $E_{D}$ is its exceptional part, and $K_{\tilde X}$ (resp. $K_X$) is the canonical divisor of $\tilde{X}$ (resp. $X$). Then $$\label{eq:A_X}
A_{X,P}(D):=-R_{X,P}(D+K_{X})=\frac{1}{2}E_{D}(\hat D+K_{\tilde X})-h^0(\sigma_* \cO_{\hat D}/{\cO_{D}}),$$ where both summands depend on $\sigma$. Moreover, if $\sigma$ is a good resolution of $(X,D)$, then $$\label{eq:delta-top}
\delta^{\operatorname{top}}(D):=\frac{1}{2}E_{D}(\hat D+K_{\tilde X})=-\frac{1}{2}E_{D}(E_D+Z_{\tilde X}),$$ ($Z_{\tilde X}$ is a numerical canonical cycle) and $\delta(D):=h^0(\sigma_* \cO_{\hat D}/{\cO_{D}})$, the classical $\delta$-invariant, do not depend on $\sigma$. Denote by $K_{\sigma} := K_{\tilde{X}} - \sigma^{*} K_X$ the relative canonical divisor, then the canonical cycle $Z_{\tilde X}$ is numerically equivalent to $-K_{\sigma}$.
A second interpretation of this invariant is provided in [@jiJM-correction], where the second and third authors define a new invariant $\kappa_P(D)$, which in the cyclic $V$-manifold case has a geometric interpretation as $r_P(k)-1$, where $r_P(k)$ is defined as the number of irreducible components of
| 1,698
| 1,308
| 1,707
| 1,618
| null | null |
github_plus_top10pct_by_avg
|
tional problem (\[vareqcoad\]) and vice versa.
The method founded on the $m$-dissipativity (see sections \[m-d\] and \[mdiss-op\]) can be applied also to the adjoint problem. Define $$P_{1}^*(x,\omega,E,D)\psi_1^*:={}&-\omega\cdot\nabla_x\psi_1^*, \\[2mm]
P_{j}^*(x,\omega,E,D)\psi_j^*:={}&S_j{{\frac{\partial \psi_j^*}{\partial E}}}-\omega\cdot\nabla_x\psi_j^*,\quad j=2,3, \\[2mm]
{\bf P}^*(x,\omega,E,D)\psi^*:={}&\big(P_{1}^*(x,\omega,E,D)\psi_1^*, P_{2}^*(x,\omega,E,D)\psi_2^*,P_{3}^*(x,\omega,E,D)\psi_3^*\big),$$ and the space $$\begin{gathered}
{{{\mathcal{}}}H}_{{\bf P}^*}(G\times S\times I^\circ)
:=\{\psi^*\in L^2(G\times S\times I)^3\ | \\
{\bf P}^*(x,\omega,E,D)\psi^*\in
L^2(G\times S\times I)^3\ \textrm{in the weak sense}\}. \label{eq:H_bfP_star}\end{gathered}$$
The relevant operator here is the smallest closed extension (closure) $\tilde{\bf P}^*_0$ of the operator ${\bf P}^*_0$ defined by $$D({\bf P}_0^*):={}&\big\{
\psi^*\in \tilde{W}^2(G\times S\times I)\times \big(\tilde{W}^2(G\times S\times I)\cap H^1(I,L^2(G\times S)\big)^2
\ \big| \\[2mm]
&\hspace{2.5mm} \psi^*_{|\Gamma_+}=0,\ \psi^*_j(\cdot,\cdot,0)=0,\ j=2,3\big\} \\[2mm]
{\bf P}_0^*\phi:={}&{\bf P}^*(x,\omega,E,D)\phi.$$ When $g^*=0$, the problem - is equivalent, in the strong sense, to $$(\tilde{\bf P}_0^*+\Sigma^*-K^*)\psi^*=f^*,$$ where $\psi^*\in D(\tilde {\bf P}_0^*)$, and $\Sigma^*\psi^*=(\Sigma_1^*\psi_1^*, \Sigma_2^*\psi_2^*, \Sigma_3^*\psi_3^*)$, $K^*\psi^*=(K_1^*\psi^*, K_2^*\psi^*, K_3^*\psi^*)$.
The result analogous to Theorem \[m-d-j-co1\] is the following.
\[m-d-ad\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[csda3aa\]), (\[csda4aa\]) (with $c>0$) and , , are valid. Furthermore, suppose that ${f^*}\in L^2(G\times S\times I)^3$ and $g^*\in T^2(\Gamma_+)\times H^1(I,T^2(\Gamma_+'))^2$ is such that the compatibility condition \[comp-d-j-ad\] g\_j\^\*(,,E\_0)=0, j=2,3, holds. Then the problem - has a unique solution $\psi^*\in {{{\mathcal{}}}H}_{\bf P^*}(G\times S\times I^\circ)$. In addition, there exists a con
| 1,699
| 570
| 1,048
| 1,716
| null | null |
github_plus_top10pct_by_avg
|
pha\beta}W_{\mu\sigma\rho\beta}W_{\;\;\;\nu\alpha}^{\sigma\;\,\;\,\,\;\rho}\end{aligned}$$ are the two independent cubic contractions of the Weyl tensor in four dimensions.
From the relations of Ref [@25], we have found the following geometrical identities : $$\begin{aligned}
L_{\mu\nu}R^{\mu\nu}=-(2 \mathcal{L}_3 + \frac{1}{2}\mathcal{L}_4+4\mathcal{L}_5+4\mathcal{L}_6-4\mathcal{L}_7+\frac{1}{2}\mathcal{L}_8)\end{aligned}$$ $$\begin{aligned}
W_{\mu\nu\alpha\beta}\nabla^{\nu} \nabla^{\beta}R^{\mu\alpha}=\mathscr{L}_3-\frac{1}{2}\mathscr{L}_2+\frac{1}{12}\mathscr{L}_1+\frac{1}{2}\mathcal{L}_6-\frac{1}{2}\mathcal{L}_5\end{aligned}$$ $$\begin{aligned}
\nabla^\rho R^{\mu\nu\alpha\beta}\nabla_{\rho}W_{\mu\nu\alpha\beta}=\mathscr{L}_7-2\mathscr{L}_5+\frac{1}{3}\mathscr{L}_8\end{aligned}$$ $$\begin{aligned}
R^{\mu\alpha}\nabla^{\beta}\nabla^{\nu}W_{\mu\nu\alpha\beta}=\frac{1}{2}\curv{L}_2-\frac{1}{12}\curv{L}_1-\frac{1}{6}\curv{L}_4-\frac{1}{2}L_6+\frac{1}{2}L_5\end{aligned}$$ $$\begin{aligned}
\nabla^\mu W_{\mu\nu\alpha\beta}\nabla^{\alpha} R^{\nu\beta}=\frac{1}{2}\curv{L}_5-\frac{1}{2}\curv{L}_6-\frac{1}{24}\curv{L}_8\end{aligned}$$ Note that $\nabla^\mu \nabla^\nu L_{\mu\nu}=0$ identically, such that there are no new relations coming from this term.
Because in four dimensions, $\mathcal{E}_6=0$ and $L_{\mu\nu}=0$, leading to $L_{\mu\nu}R^{\mu\nu}=0$ for order 6 scalars, the FKWC basis, that does not take into account these relations, is then reduced.
As for conformally invariant space-times, their metrics verify $W_{\mu\nu\alpha\beta}=0$ which provides again new relations between the scalars coming from :
$R \, W^2 =0$, $W^3_1 =0$, $W^3_2 =0$ (note that because $\mathcal{E}_6$ is a linear combination of $W^3_1$ and $W^3_2$ [@24], the relation $\mathcal{E}_6=0$ becomes redundant), $W_{\mu\nu\alpha\beta}\nabla^{\nu} \nabla^{\beta}R^{\mu\alpha}=0$, $\nabla^\rho R^{\mu\nu\alpha\beta}\nabla_{\rho}W_{\mu\nu\alpha\beta}=0$, $R^{\mu\alpha}\nabla^{\beta}\nabla^{\nu}W_{\mu\nu\alpha\beta}=0$ and $\nabla^\mu W_{\mu\nu\alpha
| 1,700
| 1,303
| 1,931
| 1,566
| 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.