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asymptotic quantum numbers $[Nn_{3}\Lambda]\Omega$. Only components with $X_{\alpha\beta}^{2}-Y_{\alpha\beta}^{2} > 0.001$ are listed. Two-quasiparticle excitation energies are given by $E_{\alpha}+E_{\beta}$ in MeV and two-quasiparticle transition matrix elements $Q_{10,\alpha\beta}$ in $e \cdot$fm. In the row (i), the label $\nu 1/2^{-}$ denotes a non-resonant discretized continuum state of neutron $\Omega^{\pi}=1/2^{-}$ level. []{data-label="26Ne_0-"}
----- --------------- --------------- ------------------------ -------------------------------------------- ----------------------
$E_{\alpha}+E_{\beta}$ $Q_{11,\alpha\beta}$
$\alpha$ $\beta$ (MeV) $X_{\alpha \beta}^{2}-Y_{\alpha\beta}^{2}$ ($e\cdot$ fm)
(a) $\nu[312]3/2$ $\nu[211]1/2$ 8.68 0.849 0.339
(b) $\nu[310]1/2$ $\nu[211]1/2$ 8.16 0.040 $-0.131$
(c) $\nu[301]1/2$ $\nu[211]1/2$ 9.32 0.010 0.294
(d) $\nu[321]3/2$ $\nu[220]1/2$ 12.0 0.007 0.250
(e) $\nu[303]7/2$ $\nu[202]5/2$ 12.1 0.006 0.414
(f) $\nu[330]1/2$ $\nu[220]1/2$ 11.4 0.004 $-0.127$
(g) $\nu[312]5/2$ $\nu[211]3/2$ 12.1 0.004 0.348
(h) $\nu[321]3/2$ $\nu[202]5/2$ 10.3 0.001 $-0.010$
(i) $\nu[321]3/2$ $\nu[202]5/2$ 11.8 0.003 $-0.214$
(j) $\nu[330]1/2$ $\nu[211]1/2$ 6.54
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ots,v\},\quad X^{12}=\{1,\dots,v\},\quad X^{i1}=T^i\text{ for }i\gs2.$$ Thus we have ${\hat\Theta_{T}}\circ{\hat\Theta_{A}}={\hat\Theta_{D}}$.
In the case $S=B$, let $y$ be the $(2,3)$-entry of $T$. Then $y>x$. $X^{22}$ must contain either $x$ or $y$, so if $x>v$ then we cannot possibly achieve ($\dagger$). So we get ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}=0$ in this case. If $x\ls v<y$, then the only way to achieve ($\dagger$) is to have $X^{22}=\{1,x\}$ and $X^{12}=\{2,\dots,\hat x,\dots,v\}$, and this yields ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}={\hat\Theta_{D}}$. Finally, if $y\ls v$, then there are three possible ways to achieve ($\dagger$); each of these gives a coefficient of $1$, and again we have ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}={\hat\Theta_{D}}$.
This result is very helpful: it tells us that the composition of $\sigma$ with a combination of ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ is a scalar multiple of ${\hat\Theta_{D}}$; hence this composition is non-zero if and only if this scalar is non-zero. In order to use this result, we need to find the number of tableaux in ${\calu}$, and also the number of tableaux in ${\calu}$ in which the $(2,2)$-entry is at most $v$. This is a straightforward count.
\[countu\]
- The number of tableaux in ${\calu}$ is $\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2$.
- The number of tableaux in ${\calu}$ whose $(2,2)$-entry is greater than $v$ is $\mbinom{u-v}{a-v}\mbinom{u-a}2$.
Suppose $S^\mu=S^{(u,v)}$ is irreducible, with $u+v=a+b+3$. Throughout this proof, all congruences are modulo $4$.
Suppose first that $u,v$ satisfy the given conditions, i.e. $v\equiv3$ and $\mbinom{u-v}{a-v}$ is odd. The second condition implies in particular that $0\ls a-v\ls u-v$, which gives $v\ls\min\{a-1,b+3\}$; so the assumptions in force in this section are satisfied. In addition, Theorem \[irrspecht\] gives $u\equiv2$.
We need to show that there are homomorphisms $S^\mu\stackrel\gamma\longrightarrow S^\la\stackrel\delta\longrightarrow S^{\mu'}$ such that $\delta\circ\gamma\neq0$. S
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} = {\bf 32_2} \oplus {\bf 32_0} \oplus {\bf 32_{-2}} \nonumber \\
& {\bf (220,2)} = {\bf 220_1} \oplus {\bf 220_{-1}} \\
& {\bf (12,2)} = {\bf 12_1} \oplus {\bf 12_{-1}} \quad ,\nonumber\end{aligned}$$ where the subscript denotes the $\mathbb{R}^+$ weight. The representation ${\bf 32_2}$ corresponds to the RR fluxes, the ${\bf 220_1}$ corresponds to the NS-NS fluxes and the $ {\bf 352_0}$ corresponds to the $P$ fluxes. This can be seen by decomposing each of the $SO(6,6)$ representations in terms of $GL(6,\mathbb{R})$, which we do in detail now.
It is straightforward to see how this decomposition works for the case of the RR fluxes. The ${\bf 32}$ is the spinorial representation $\theta_\alpha$ of $SO(6,6)$, and according to the convention that one chooses for the chirality of this spinor, one has the two possible decompositions $$\theta_\alpha \rightarrow \left\{ \begin{array}{ll} F_a \ \ F_{abc} \ \ F_{abcde} \ \ \ \ \ \ \ \ \ \ \ ({\rm IIB})
\\
\\
F \ \ F_{ab} \ \ F_{abcd} \ \ F_{abcdef} \ \ \ ({\rm IIA})\end{array} \right. \quad , \label{decompositionofRRfluxes}$$ corresponding to the RR fluxes of odd rank in the IIB theory and of even rank in the IIA theory. Obviously this representation contains only geometric fluxes. The T-duality rule given in eq. maps a given flux in one theory to a flux in the other theory. All the components in eq. are connected by chains of T-duality transformations.
The next representation is the ${{\bf 220 }}$, which is the representation $\theta_{MNP}$ of $SO(6,6)$ with three antisymmetrised vector indices. This corresponds to the NS-NS fluxes, and indeed the embedding tensor decomposes under $GL(6,\mathbb{R})$ as $$\theta_{MNP} \ \rightarrow \ H_{abc} \quad f_{ab}^c \quad Q_{a}^{bc} \quad R^{abc} \quad .\label{decompositionofNSfluxes}$$ The T-duality rule given in eq. connects the different components in the equation above, but in this case not all the components can be reached by chains of T-dualities starting for instance from a
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not quite so obvious as the following examples which are significantly useful in dealing with convergence questions in topological spaces, amply illustrate.
The neighbourhood system $$_{\mathbb{D}}N=\{ N\!:N\in\mathcal{N}_{x}\}$$ at a point $x\in X$, directed by the reverse inclusion direction $\preceq$ defined as $$M\preceq N\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x},\label{Eqn: Direction1}$$
is a fundamental example of a *natural direction of $\mathcal{N}_{x}$*. In fact while reflexivity and transitivity are clearly obvious, (c) follows because for any $M,N\in\mathcal{N}_{x}$, $M\preceq M\bigcap N$ and $N\preceq M\bigcap N$. Of course, this direction is not a total ordering on $\mathcal{N}_{x}$. A more naturally useful directed set in convergence theory is $$_{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N)\}\label{Eqn: Directed}$$
under its *natural direction* $$(M,s)\preceq(N,t)\Longleftrightarrow N\subseteq M\qquad\textrm{for }M,N\in\mathcal{N}_{x};\label{Eqn: Direction2}$$ **$_{\mathbb{D}}N_{t}$ is more useful than $_{\mathbb{D}}N$ because, unlike the later, $_{\mathbb{D}}N_{t}$ does not require a simultaneous choice of points from every $N\in\mathcal{N}_{x}$ that implicitly involves a simultaneous application of the Axiom of Choice; see Examples A1.2(2) and (3) below. The general indexed variation
$$_{\mathbb{D}}N_{\beta}=\{(N,\beta)\!:(N\in\mathcal{N}_{x})(\beta\in\mathbb{D})(x_{\beta}\in N)\}\label{Eqn: DirectedIndexed}$$
of Eq. (\[Eqn: Directed\]), with natural direction $$(M,\alpha)\leq(N,\beta)\Longleftrightarrow(\alpha\preceq\beta)\wedge(N\subseteq M),\label{Eqn: DirectionIndexed}$$
often proves useful in applications as will be clear from the proofs of Theorems A1.3 and A1.4.
**Definition A1.5.** ***Net.*** *Let $X$ be any set and $\mathbb{D}$ a directed set. A net $\chi\!:\mathbb{D}\rightarrow X$* *in $X$* *is a function* *on the directed set $\mathbb{D}$ with values in $X$.$\qquad\square$*
A net, to be denoted as $\chi(\alpha)$, $\alpha\in\mathbb{D}$,
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rt{r_1},
\qquad x\in D.$$ The result now follows by taking logarithms.
Exact simulation of stable paths {#stable_paths}
================================
The key ingredient to the walk-on-spheres in the Brownian setting is the knowledge that spheres are exited continuously and uniformly on the boundary of spheres. In the stable setting, the inclusion of path discontinuities means that the process will exit a sphere by a jump. The analogous key observation that makes our analysis possible is the following result, which gives the distribution of a stable process issued from the origin, when it first exits a unit sphere.
\[BGR\] Suppose that $B(0,1)$ is a unit ball centred at the origin and write $\sigma_{B(0,1)} = \inf\{t>0 : X_t \not\in B(0,1)\}$. Then, $$\mathbb{P}_0(X_{\sigma_{B(0,1)}}\in \mathrm{d}y) = \pi^{-(d/2+1)}\,\Gamma(d/2)\,\sin(\pi\alpha/2)\,\left|1-|y|^2\right|^{-\alpha/2}|y|^{-d}\,{\rm d}y, \qquad |y|>1.$$
This result provides a method of constructing precise sample paths of stable processes in phase space (i.e. exploring sample paths as ordered subsets of $\mathbb{R}^d$ rather than as functions $[0,\infty]\to \mathbb{R}^d$). Choose a tolerance $\epsilon$ and initial point $X_0 =x$. Denote by $E_1$ a sampling from the distribution given in Theorem \[BGR\]. This gives the exit from a ball of radius one when $X$ is issued from the origin. By the scaling property and the stationary and independent increments, $x +\epsilon\, E_1$ is distributed as the exit position from a ball of radius $\epsilon$ centred at $x$ when the process is issued from $x$. Hence, we define $X_1=x + \epsilon\, E_1$ and then, inductively for $n\geq 1$, generate $X_{n+1}$ as the exit point of the ball centred on $X_n$ with radius $\epsilon$ by noting this is equal in distribution to $X_n + \epsilon \, E_{n+1}$, where $E_{n+1}$ is an [*iid*]{}copy of $E_1$. It is important to remark for later that the value of $\epsilon$ in this algorithm does not need to be fixed and may vary with each step. Note, however, the method do
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${{X}^{crv}}_{\mu}=0
$ for $2 \leq \mu \leq N-1$, since the corresponding inverse variances ${{Z}^{crv}}_{\mu}$ all vanish. The three components of the efficient frontier are the riskless asset together with ${{\bf
e}^{crv}}^{1}$ and ${{\bf e}^{crv}}^{N}$. Furthermore, the latter portfolio will be strongly disfavored relative to the former for large $N$ since its variance grows in proportion to $N$ relative to that of the former; cf. Eq. (\[452\]) et seq. Indeed for reasonably large $N$, the efficient frontier is essentially a combination of ${{\bf e}^{crv}}^{1}$ and the riskless asset; $${X}^{crv}_{0} {\rightarrow} { {R}^{crv}_{1}-{\cal R} \over {R}^{crv}_{1}-
{R}_{0}}, \; {X}^{crv}_{1}{\rightarrow}{{\cal R}-{R}_{0} \over
{R}^{crv}_{1}-{R}_{0} }, \; \;{\rm as}\; {N \rightarrow \infty},
\label{4840}$$ while $${X}^{crv}_{N}{\rightarrow} 0, \; {V}_{eff}=\vert {{\cal R}-{R}_{0} \over
{R}_{1}^{crv}-{R}_{0} } \vert {[{ {\bar{{\alpha}^{2}}} \over N {\sin}^{2}
(\theta)}]}^{1 \over 2} {\rightarrow}0 \;{\rm as}\; N \rightarrow \infty.
\label{485}$$ This last property, i.e., the vanishing of the efficient portfolio volatility (i.e., the market as well as the specific risk) in proportion to ${N}^{-{1 \over 2}}$ in the limit as $N\rightarrow \infty$, also holds for the general single-index model, as can be discerned from the results of §2. As discussed earlier, this total vanishing of the portfolio volatility is a specific consequence of leveraging.
We close this section by summarizing the results established above.
[**Proposition 2.**]{} [*The principal portfolios of the constant variance single-index model consist of a market-aligned portfolio ${{\bf
e}^{crv}}^{N}$, a minimum-volatility, market-orthogonal portfolio ${{\bf
e}^{crv}}^{1}$, and $N-2$ critically leveraged market-orthogonal portfolios with infinite volatility and expected return, as given in Eqs.(\[446\])-(\[452\]). Furthermore, as $N \rightarrow \infty$, the efficient portfolio reduces to a combination of ${{\bf e}^{crv}}^{1}$ and the riskless asset with
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alent.
1. $f_1(x), \ldots, f_m(x)$ have a common interleaver.
2. for all $p_1, \ldots, p_m \geq 0$, $\sum_{i}p_i=1$, the polynomial $$p_1f_1(x)+ \cdots+ p_mf_m(x)$$ is real–rooted.
\[largestz\] Let $f_1,\ldots, f_m$ be real–rooted polynomials that have the same degree and positive leading coefficients, and suppose $p_1, \ldots, p_m \geq 0$ sum to one. If $\{f_1,\ldots, f_m\}$ is compatible, then for some $1 \leq i \leq m$ with $p_i >0$ the largest zero of $f_i$ is smaller or equal to the largest zero of the polynomial $$f=p_1f_1 + p_2f_2 + \cdots + p_mf_m.$$
If $\alpha$ is the largest zero of the common interleaver, then $f_i(\alpha) \leq 0$ for all $i$, so that the largest zero, $\beta$, of $f(x)$ is located in the interval $[\alpha, \infty)$, as are the largest zeros of $f_i$ for each $1\leq i \leq m$. Since $f(\beta)=0$, there is an index $i$ with $p_i >0$ such that $f_i(\beta) \geq 0$. Hence the largest zero of $f_i$ is at most $\beta$.
Suppose $S_1, \ldots, S_m$ are finite sets. A family of polynomials, $\{f({\mathbf{s}};t)\}_{{\mathbf{s}}\in S_1 \times \cdots \times S_m}$, for which all non-zero members are of the same degree and have the same signs of their leading coefficients is called *compatible* if for all choices of independent random variables ${\mathsf{X}}_1 \in S_1, \ldots, {\mathsf{X}}_m \in S_m$, the polynomial $
{\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_n;t)
$ is real–rooted.
The notion of compatible families of polynomials is less general than that of *interlacing families of polynomials* in [@MSS1; @MSS2]. However since all families appearing here (and in [@MSS1; @MSS2]) are compatible we find it more convenient to work with these. The following theorem is in essence from [@MSS1].
\[expfam\] Let $\{f({\mathbf{s}};t)\}_{{\mathbf{s}}\in S_1 \times \cdots \times S_m}$ be a compatible family, and let ${\mathsf{X}}_1 \in S_1, \ldots, {\mathsf{X}}_m \in S_m$ be independent random variables such that ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t) \not \equiv 0$. Th
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}({\mathcal{W}}(\chi ))$, then $R^\chi _+=R^{\chi '}_+$.
By Thm. \[th:rschi\], ${\mathcal{R}}(\chi )$ is a root system of type $\cC
(\chi )$.
\(i) Assume that $R^\chi _+=R^{\chi '}_+$. Then $C^{\chi }=C^{\chi '}$ by Lemma \[le:cm\]. Therefore ${\sigma }_i^\chi ={\sigma }_i^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$. Since $\chi ,\chi '\in {\mathcal{X}}_2$, by induction it follows that $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots
{\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ and $C^{(\s_{i_1}\cdots {\sigma }_{i_k}^\chi )^*\chi }=
C^{(\s_{i_1}\cdots {\sigma }_{i_k}^{\chi '})^*\chi '}$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots ,i_k\in I$. Hence $C^{w^*\chi }=C^{w^*\chi '}$ for all $w\in {\mathrm{Hom}}(\chi ,\underline{\,\,})
\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))$.
\(ii) Since $\chi \in {\mathcal{X}}_3$, $R^\chi =\{w^{-1}({\alpha }_i)\,|\,w\in \Hom
(\chi ,\underline{\,\,})\subset {\mathrm{Hom}}({\mathcal{W}}(\chi ))\}$ by [@p-CH08 Prop.2.12]. By assumption on the Cartan matrices, $\s_{i_1}\cdots {\sigma }_{i_k}^\chi ={\sigma }_{i_1}\cdots
{\sigma }_{i_k}^{\chi '}$ in ${\mathrm{Aut}}({\mathbb{Z}}^I)$ for all $k\in {\mathbb{N}}_0$ and $i_1,\dots,i_k\in I$. Hence $R^\chi =R^{\chi '}$, and the lemma holds by (R1).
For our study of Drinfel’d doubles we will use an analog of the sum of fundamental weights, commonly known as $\rho $. More precisely, we define a character version of the linear form $(2\rho ,\cdot )$, where $(\cdot ,\cdot )$ is the usual bilinear form on the weight lattice.
Let ${\widehat{{\mathbb{Z}}^I}}={\mathrm{Hom}}({\mathbb{Z}}^I,{{\Bbbk }^\times })$ denote the group of characters of ${\mathbb{Z}}^I$ with values in ${{\Bbbk }^\times }$.
\[de:rhomap\] Let $\chi \in {\mathcal{X}}$. Let ${\rho ^{\chi}} \in \widehat{{\mathbb{Z}}^I}$ such that $${\rho ^{\chi}} ({\alpha }_i)=\chi ({\alpha }_i,{\alpha }_i) \quad \text{for all $i\in I$}.$$
Let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)$. Assume that ${b}<\infty $. Then
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exes. Let $\Delta$ be a simplicial complex on the vertex set $[n]$. To each facet $F\in \Delta$ we associate the element $a_F\in\NN^n_\infty$ with $$a_F(i)=\left\{ \begin{array}{lll} \infty, & \text{if} & i\in F\,\\
0, & \mbox{if} & i\not\in F, \end{array} \right.$$ Then $\{a_F\: F\in \Delta\}$ is the set of facets of a multicomplex $\Gamma(\Delta)$, and $I(\Gamma(\Delta))=I_\Delta$, where $I_\Delta$ is the Stanley-Reisner ideal of $\Delta$. Moreover one has $\dim \Gamma=\dim \Delta(\Gamma)$.
For a multicomplex $\Gamma$ and $a\in \Gamma$ we let $P_a$ be the prime ideal generated by all $x_i$ with $i\not\in \ip a$. Thus $P_a$ is generated by all $x_i$ with $a(i)\in \NN$.
\[irred\] Let $\Gamma$ be a multicomplex. The following statements are equivalent:
1. $\Gamma$ has just one maximal facet $a$;
2. $I(\Gamma)$ is an irreducible ideal.
If the equivalent conditions hold, then $I(\Gamma)$ is generated by $\{x_i^{a(i)+1}\: i\in [n]\setminus \ip a\}$. In particular, $I(\Gamma)$ is a $P_a$-primary ideal.
If $a$ is the unique maximal facet of $\Gamma$ then $$I(\Gamma)=(x^b\: b\in\NN^n,\; b(i)>a(i)\; \text{for some $i$}) =(x_i^{a(i)+1}\: i\in [n]\setminus \ip a).$$ Conversely, if $I(\Gamma)$ is irreducible, then according to [@Vi Theorem 5.1.16] there exists a subset $A\subset \{1,\ldots,n\}$ and for each $i\in
A$ an integer $a_i>0$ such that $I(\Gamma)=(x_i^{a_i}:i\in A, a_i>0)$. Set $a(i)=a_i-1$ for $i\in A$ and $a(i)=\infty$ for $i\not \in A$. Then $a$ is the unique facet of $\Gamma$.
\[irrprime\] Let $\Gamma\subset \NN_{\infty}^n$ be a multicomplex with just one facet $a$. Then $I(\Gamma)=P_a$.
Suppose $a(i)\neq 0$ for some $i\not\in \ip a$. Then $a-e_i$ is a facet, different from $a$. Here $e_i$ is the canonical $i$th unique vector. Thus we see that $a(i)\in \{0,\infty\}$ for $i=1,\ldots,n$, so that $I(\Gamma)=I(\Gamma(a))=P_a$.
The next result describes how the maximal facets of a multicomplex $\Gamma$ are related to the irreducible components of $I(\Gamma)$.
\[irrdec\] Let $\Gamma\subset \NN_{\infty}^n
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X(j)$ (in both the model selection and estimation steps). Of course, it is possible to extend the definition of this parameter by leaving out several variables from ${\widehat{S}}$ at once without additional conceptual difficulties.\
The parameter $\gamma_{{\widehat{S}}}$ has several advantages over the projection parameter $\beta_{{\widehat{S}}}$: it is more interpretable since it refers directly to prediction error and we shall see that the accuracy of the Normal approximation and the bootstrap is much higher. Indeed, we believe that the widespread focus on $\beta_{{\widehat{S}}}$ is mainly due to the fact that statisticians are used to thinking in terms of cases where the linear model is assumed to be correct.\
The second type of LOCO parameters that we consider are the median LOCO parameters $\phi_{{\widehat{S}}} = (\phi_{{\widehat{S}}}(j):\ j\in {{\widehat{S}}})$ with $$\label{eq:median.LOCO}
\phi_{{\widehat{S}}}(j) = {\rm median}\Biggl[|Y-\hat\beta_{{\widehat{S}}(j)}^\top X_{{\widehat{S}}}|-
|Y-\hat\beta_{{\widehat{S}}}^\top X_{{\widehat{S}}}|\,\Biggr],$$ where the median is over the conditional distribution of $(X,Y)$ given $\mathcal{D}_n$. Though one may simply regard $\phi_{{\widehat{S}}}$ as a robust version of $\gamma_{{\widehat{S}}}$, we find that inference for $\phi_{{\widehat{S}}}$ will remain valid under weaker assumptions that the ones needed for $\gamma_{{\widehat{S}}}$. Of course, as with $\gamma_{{\widehat{S}}}$, we may leave out multiple covariate at the same time.
- [**The prediction parameter $\rho_{{\widehat{S}}}$**]{}. It is also of interest to obtain an omnibus parameter that measures how well the selected model will predict future observations. To this end, we define the future predictive error as $$\rho_{{\widehat{S}}} = \mathbb{E}_{X,Y}\Bigl[| Y - \hat\beta_{{\widehat{S}}}^\top X_{{\widehat{S}}}|\, \Bigr],$$ where $\widehat{\beta}_{{\widehat{S}}}$ is any estimator the projection parameters $\beta_{{\widehat{S}}}$.
[**Remarks.**]{}
1. The LOCO and predict
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probability, $r<c_2\cdot d$.
Now suppose that after allocating $m$ balls, there is a ball at height $\ell+c_1+c_2+1$. This implies that there is a $d$-choice, denoted by $R$, whose minimum load is at least $\ell+c_1+c_2+1$. Let us consider all balls placed in the bins contained in $R$ with height at least $\ell+c_1+1$. Recover the corresponding $d$-choices for these balls, say $D_1,D_2,\ldots, D_w$, then colour them blue-red with respect to the root $R$ and an arbitrary ordering of the children of each vertex. Since $w{\geqslant}c_2\cdot d$, w.h.p., there are $b{\geqslant}1$ blue vertices and $w-b$ red vertices. We now consider every blue vertex $D_t\in \{D_1,D_2,\ldots, D_w \}$ as a root and start the recursive construction of the witness graph. [Assuming that the number of red vertices is strictly less than]{} $c_2\cdot d< w$, it follows that at least one recursive construction (with root $D_i$) does not produce any red vertex. Moreover, the recursion from $D_i$ gives a $c_1$-loaded tree with at least [$k=d^\ell$]{} vertices. [We take $\ell = \log_d\log n$, so that $k=\log n$.]{} Another application of Lemma \[lem:col\] implies that a $c_1$-loaded $k$-vertex tree [with no red vertices]{} exists with probability at most [ $$\begin{aligned}
n^{c_0+3}\, \exp\{4k\log(2\beta d)-c_1(d-1)(k-1)\} &{\leqslant}\exp\left\{ \big(c_0 + 4 + 4\log(2\beta d) - c_1(d-1)\big)\log n \right\}\\
&{\leqslant}\exp\left\{ \big(c_0 + 4 + 4\log(4\beta) - c_1\big)\log n\right\},
\end{aligned}$$ using the fact that $2{\leqslant}d = o(\log n)$ and $k=\log n$. Setting $c_1$ to be a large enough positive constant,]{} we conclude that with high probability the maximum load is at most $${\log_d\log n+{\mathcal{O}}(1)+c_2} = \log_d\log n+{\mathcal{O}}(1/\varepsilon),$$ where $c_2={\mathcal{O}}(1/\varepsilon)$. This proves the first statement of Theorem \[thm:d-choice\]. [The proof of the second statement is presented in Appendix \[app:missmain\].]{}
Balanced Allocation on Dynamic Graphs {#sec:graph}
===================================
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----------+
| Capital investment costs | 10‐40 | Million USD | Production scale, grade and containment level of the facility | β |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
| Vial or container cost | 0.1‐0.6 | USD/dose | Number of doses per container or vial | [56](#amp210060-bib-0056){ref-type="ref"}, [61](#amp210060-bib-0061){ref-type="ref"} |
+------------------------------------------------------------------+------------------+-------------+---------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
*Note*: α---Assumed by the authors; β---calculated using SuperPro Designer V10 (Intelligen, Inc.).
Recycling or re‐use of the materials for producing multiple batches of the same product. For this, high cost raw material (eg, the 5′ cap analogue and enzymes) can be separated from the RNA product using TFF and fed back into the RNA synthesis bioreactor.
3. CHALLENGES AHEAD AND POTENTIAL SOLUTIONS {#amp210060-sec-0006}
===========================================
The biggest challenge for saRNA and mRNA vaccines currently is to demonstrate efficacy against target disease in clinical trials, especially the Phase II/III efficacy trials, where the highest pro
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|
now ready to put these observations together to prove the hard part of Proposition \[pre-cohh\].
\[grsameA\] Fix $k\geq 0$ and set ${\mathcal{J}}= eJ^k\delta^k$ and ${\mathcal{N}}=N(k)$. Then the map $\theta:{\mathcal{J}}\to{\mathcal{N}}$ is an isomorphism.
Set $\mathfrak{m}= {\mathbb{C}}[{\mathfrak{h}}]^{W}_+$ and note that ${\mathcal{N}}/\mathfrak{m}{\mathcal{N}}=\overline{N(k)}$. On the other hand, in the notation of Corollary \[gr\], ${\mathcal{J}}/\mathfrak{m}{\mathcal{J}}\cong \overline{J^{k}}[K]$ is the shift of $\overline{J^k}$ by $\deg \delta^{k} = K=kn(n-1)/2$. By Corollaries \[gr\] and \[poincare-S2A\], we therefore have an equality of Poincaré series under the ${\mathbf{E}}$-gradation: $$\label{eqpoiA}
p( {\mathcal{J}}/\mathfrak{m}{\mathcal{J}}, v) =
v^{K}\frac{\sum_{\mu}
f_{\mu}(1)f_{\mu}(v^{-1})v^{-k(n(\mu) - n(\mu^t))}[n]_v!}
{\prod_{i=2}^n (1-v^{-i})} = p( {{\mathcal{N}}/\mathfrak{m}{\mathcal{N}}}, v).$$
Keep the ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-bases of $\theta({\mathcal{J}})\cong {\mathcal{J}}$ and ${\mathcal{N}}$ described in Notation \[eqpoi-sect\]. We write ${a(g\ell m)} =
g$ whenever $a_{g\ell m}$ exists for that choice of $g,\ell,m$; thus $\sum_{g\ell m} v^{a(g\ell m)}$ denotes the sum $\sum v^g$, where one has one copy of $v^g$ for each $\ell,m$ for which $a_{g\ell m}$ exists. Define ${b(gu)}$ analogously. Since the bases $\{a_{g\ell m}\}$ and $\{b_{gu}\}$ induce ${\mathbb{C}}$-bases of ${\mathcal{J}}/\mathfrak{m}{\mathcal{J}}$, respectively $\overline{N(k)}$, can be reinterpreted as $$\begin{aligned}
\label{eqpoi2}
\sum_{g,\ell,m} v^{a(g\ell m)} &= &
v^{K} \frac{\sum_{\mu} f_{\mu}(v^{-1})f_{\mu}(v)v^{-k(n(\mu)-
n(\mu^t))}[n]_v!}
{\prod_{i=2}^n (1-v^{-i})} =
\sum_ {g,u} v^{b(gu)}.\end{aligned}$$
We note that has several consequences for the $a(g\ell m)$ and $b(gu)$.
1. For fixed $g$, there exist only finitely many elements $a_{g\ell m}$ and $b_{gu}$. This is because the middle expression in is a well-defined series.
2. There exists a universal upper bound $a(g\el
| 2,213
| 2,464
| 1,879
| 1,995
| 4,076
| 0.768248
|
github_plus_top10pct_by_avg
|
=H^{-1}\cdot h_0(t)$, and $K=k(0)$, this shows that $\alpha(t)$ is equivalent to $$H\cdot h_1(t) \cdot
\begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}
\cdot K$$ with $h_1(0)=I$, and $K$ constant and invertible. By Corollary \[MPIcorol\], this matrix is equivalent to $$\beta(t)=H\cdot \begin{pmatrix}
1 & 0 & 0 \\
q & 1 & 0 \\
r & s & 1 \end{pmatrix}\cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}
\cdot L\cdot K$$ with $L$ invertible, and $q$, $r$, $s$ polynomials satisfying the needed conditions. Letting $M=L\cdot K$ gives the statement in the case $b<c$.
If $b=c$, then the condition that $\deg s<c-b=0$ forces $s\equiv 0$. When $q$ and $r$ are not both $0$, the inequality $v(q)<v(r)$ may be obtained by conjugating with a constant matrix.
If $q(t)\not\equiv 0$ and $v(q)=a$, then we can extract its $a$-th root as a power series. It follows that there exists a unit $\nu(t)\in{{\mathbb{C}}}[[t]]$ such that $q(t\nu(t))=t^a$. Therefore, $$\beta(t\nu(t))=H\cdot \begin{pmatrix}
1 & 0 & 0 \\
t^a & 1 & 0 \\
r(t\nu(t)) & s(t\nu(t)) & 1 \end{pmatrix}\cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}
\cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & \nu(t)^b & 0 \\
0 & 0 & \nu(t)^c
\end{pmatrix}
\cdot M\quad.$$ Another application of Corollary \[MPIcorol\] allows us to truncate the power series $r(t\nu(t))$ and $s(t\nu(t))$ to obtain polynomials $\underline r$, $\underline s$ satisfying the same conditions as $r$, $s$, at the price of multiplying to the right of the 1-PS by a constant invertible matrix $\underline K$: that is, $\beta(t\nu(t))$ (and hence $\alpha(t)$) is equivalent to $$H\cdot \begin{pmatrix}
1 & 0 & 0 \\
t^a & 1 & 0 \\
\underline r & \underline s & 1 \end{pmatrix}\cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & t^b & 0 \\
0 & 0 & t^c
\end{pmatrix}
\cdot \left[ \underline K \cdot \begin{pmatrix}
1 & 0 & 0 \\
0 & \nu(0)^b & 0 \\
0 & 0 & \nu(0)^c
\end{pmatrix}
\cdot M \right]\quad.$$ Renaming $r=\underline r$, $s=\underline s$, and absorbing the factors
| 2,214
| 899
| 1,481
| 2,007
| null | null |
github_plus_top10pct_by_avg
|
ta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_1\to
z'_1}^{{\bf N}}$ earlier than $\omega_1$. Then, we define $\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ to be the set of paths $\zeta\in\Xi_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ such that there are $k-2$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf N}}}\setminus(\omega_1{\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=3,\dots,k$.
3. More generally, for $l<k$ and $\vec\omega_l=(\omega_1,
\dots,\omega_l)$ with $\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$, $\omega_2\in\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1},\dots$, $\omega_l\in\tilde\Omega_{z_l\to z'_l}^{{{\bf N}};\vec\omega_{l-1}}$, we define $\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Omega_{z_{l+1}\to z'_{l+1}}^{{\bf N}}$ on ${{\mathbb G}}_{{\bf N}}\setminus{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle l$}}
\omega_i$ such that $\zeta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ earlier than $\omega_i$, for every $i=1,\dots,l$. Then, we define $\tilde\Omega_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ such that there are $k-(l+1)$ edge-disjoint paths on ${{\mathbb G}}_{
{{\bf N}}}\setminus({\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle
l$}}\omega_i {\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=l+2,\dots,k$.
4. If $l=k-1$, then we simply define $\tilde\Omega_{z_k\to z'_k}^{{{\bf N}};\vec
\omega_{k-1}}=\Xi_{z_k\to z'_k}^{{{\bf N}};\vec\omega_{k-1}}$. We will also abuse the notation to denote $\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ by $\tilde\Omega_{z_1\to z'_1}^{{{\bf N}};\vec\omega_0}$.
Using the above notation, we can decompose ${\mathfrak{S}}^{(\prime)}$ disjointly as follows. For a collection $\omega_i\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ for $i=1,\dots,k$, we denote by ${\mathfrak{S}}_{\vec\omega_k}^{(\
| 2,215
| 2,102
| 1,814
| 2,053
| null | null |
github_plus_top10pct_by_avg
|
eroth-order terms are rewritten in a familiar way. However, the expression in (\[P-beta-alpha-2nd-averaged\]) does not mean a great simplification. That is, the matter dependent terms with sterile sector model-dependence remain.
Suppression by the energy denominator {#sec:energy-denominator}
--------------------------------------
We, therefore, have to examine the effect of suppression by the large energy denominator which characterizes transition between active-sterile sates, $1/ ( \Delta_{K} - h_{k} )$. We demand that the matter dependent terms in (\[P-beta-alpha-2nd-averaged\]) be smaller than the probability leaking and the normalization terms of order $\sim W^4$. It leads to $$\begin{aligned}
\biggl | \frac{ AA L }{ ( \Delta_{J} - h_{i} ) } \biggr |
< W^2,
\hspace{4mm}
\biggl | \frac{ AA }{ ( h_{k} - h_{j} ) ( \Delta_{J} - h_{i} ) } \biggr |
< W^2,
\hspace{4mm}
\text{and}
\hspace{4mm}
\biggl | \frac{ A }{ ( \Delta_{J} - h_{i} ) } \biggr |
< W^2,
\label{denominator-size}\end{aligned}$$ where $L$ is the baseline distance and $i$ and $J$ denote, respectively, generic indices for active and sterile states. For notational convenience, we define $\lambda_{i}$ $(i=1,2,3)$ to be the eigenvalues of $3 \times 3$ submatrix $2E \tilde{H}_0$ in (\[tilde-H0+H1\]) corresponding to the active neutrino mass squared in matter. Then, $h_{i} = \frac{ \lambda_{i} }{2E}$. As one can easily see[^13] the last one in (\[denominator-size\]) gives the severest constraint (taking the matter potential due to CC in $A$ and removing the factor $\frac{1}{2E}$) $$\begin{aligned}
\biggl | \frac{ a }{ m^2_{J} - \lambda_{i} } \biggr |
\approx
\biggl | \frac{ a }{ \Delta m^2_{J i} } \biggr |
< W^2.
\label{suppression-cond}\end{aligned}$$ Notice that, in order for the first inequality in (\[suppression-cond\]) to be valid, we have restricted the energy region for a given matter density such that $\lambda_{i}$ remain in the order of active neutrino masses. Roughly speaking, it corresponds to $- 50 \,\text{ (g/cm}^3) \text{GeV} \lsim Y
| 2,216
| 371
| 1,124
| 2,313
| 1,297
| 0.791431
|
github_plus_top10pct_by_avg
|
morphism is represented by an affine group scheme which is isomorphic to $\underline{G}_j$. Note that $T$ preserves the hermitian form attached to the lattice $M_0^{\prime}\oplus C(L^j)$.
We claim that $\begin{pmatrix} 1&0 \\ 0&m \end{pmatrix}$ is contained in $\underline{G}_j'(R)$. If this is true, then the above matrix description defines the morphism from $\underline{G}_j$ to $\underline{G}_j'$ we want to describe (cf. the last paragraph of page 489 in [@C2]).
We rewrite the hermitian lattice $M_0^{\prime}\oplus C(L^j)$ as $(M_0^{\prime}\oplus M_0^{\prime})\oplus (\bigoplus_{i \geq 1} M_i^{\prime})$. Let $(e_1, e_2)$ be a basis for $(M_0^{\prime}\oplus M_0^{\prime})$ so that the corresponding Gram matrix of $(M_0^{\prime}\oplus M_0^{\prime})$ is $\begin{pmatrix} a&0 \\ 0&a \end{pmatrix}$, where $a \equiv 1$ mod 2. Then the hermitian lattice $(M_0^{\prime}\oplus M_0^{\prime})$ has Gram matrix $\begin{pmatrix} a&a \\ a&2a \end{pmatrix}$ with respect to the basis $(e_1, e_1+e_2)$. The lattice $(M_0^{\prime}\oplus M_0^{\prime})$ is *unimodular of type $I^e$* with rank 2. With this basis, $T$ becomes
$$\tilde{T}=\begin{pmatrix} 1&-2 z_0^{\ast} &-m_1\\ 0&1+2 z_0^{\ast} &m_1\\0& m_2&m_3\end{pmatrix}.$$
On the other hand, an element of $\underline{G}_j'(R)$, with respect to a basis for $M_0^{\prime}\oplus C(L^j)$ obtained by putting together the basis $(e_1, e_1+e_2)$ for $(M_0^{\prime}\oplus M_0^{\prime})$ and a basis for $C(L^j)$, is given by an expression $$\begin{pmatrix} 1+\pi x_0'&-2 z_0'^{\ast} & m_1' \\ u_0'&1+\pi w_0' &m_1'' \\ m_2' &m_2'' & m_3''\end{pmatrix},$$ cf. Section \[mc\]. Then we can easily see that the congruence conditions on $m_1, m_2, m_3$ are the same as those of $m_1', m_2'', m_3''$, respectively, and that the congruence conditions on $m_1'$ are included in those of $m_1''$. We caution that the congruence conditions on $m_1'$ are not the same as those of $m_1''$ because of the condition (d) of the description of an element of $\underline{M}(R)$ mentioned in the argument following Remark
| 2,217
| 2,559
| 2,400
| 1,990
| null | null |
github_plus_top10pct_by_avg
|
tion whether we can improve the bootstrap rates. For example, the remainder term in the Taylor approximation of $\sqrt{n}(\hat\beta(j) - \beta(j))$ is $$\frac{1}{2n}\int
\int \delta^\top H_j((1-t)\psi + t \hat\psi) \delta\, dt$$ where $\delta=\sqrt{n}(\hat\psi - \psi)$. By approximating this quadratic term it might be possible to correct the bootstrap distribution. [@pouzo2015bootstrap] has results for bootstrapping quadratic forms that could be useful here. In Section \[section::improving\] we saw that a modified bootstrap, that we called the image bootstrap, has very good coverage accuracy even in high dimensions. Future work is needed to compute the resulting confidence set efficiently.
Finally, we remind the reader that we have taken a assumption-free perspective. If there are reasons to believe in some parametric model then of course the distribution-free, sample splitting approach used in this paper will be sub-optimal.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the AE and the reviewers for comments that led to substantial improvements on the paper and the discovery of a mistake in the original version of the manuscript. We also thank Lukas Steinberger, Peter Buhlmann and Iosif Pinelis for helpful suggestions and Ryan Tibshirani for comments on early drafts.
Appendix 1: Improving the Coverage Accuracy of the Bootstrap for the Projection Parameters {#section::improving}
==========================================================================================
Throughout, we treat $S$ as a fixed, non-empty subset of $\{1,\ldots,d\}$ of size $k$ and assume an i.i.d. sample $(Z_1,\ldots,Z_n$) where $Z_i = (X_i,Y_i)$ for all $i$, from a distribution from $\mathcal{P}_n^{\mathrm{OLS}}$.
The coverage accuracy for LOCO and prediction parameters is much higher than for the projection parameters and the inferences for $\beta_S$ are less accurate if $k$ is allowed to increase with $n$. Of course, one way to ensure accurate inferences is simply to focus on $\gam
| 2,218
| 759
| 993
| 2,256
| 2,747
| 0.777308
|
github_plus_top10pct_by_avg
|
proof for ${\hat{T}}^-_p$ is analogous.
Let $\chi '=r_p(\chi )$ and $\Lambda '={t}_p^\chi (\Lambda )$. By Lemma \[le:MLiso\] and Thm. \[th:PBWtau\], $$M^{\chi }(\Lambda )_{{\alpha }_j+a{\alpha }_p}=0=
M^{\chi }(\Lambda )_{-{b}{\alpha }_p} \quad
\text{for all $a\in {\mathbb{Z}}$.}$$ Thus, since $\deg {T}_p(E_j)={\sigma }_p^{\chi '}({\alpha }_j)$ for $E_j\in U(\chi ')$, $$\begin{aligned}
{T}_p(E_j)F_p^{{b}-1}v_\Lambda \in
M^{\chi }(\Lambda )_{{\alpha }_j+(1-{b}-c^{\chi '}_{pj}){\alpha }_p}=0
\quad \text{for all $j\in I$.}
\end{aligned}$$ Moreover, Eqs. , give that $$\begin{aligned}
K_{{\alpha }}L_{\beta }F_p^{{b}-1} v_{\Lambda }
=\chi ({\alpha },{\alpha }_p)^{1-{b}}\, \chi ({\alpha }_p,\beta )^{{b}-1}\,
\Lambda (K_{\alpha }L_\beta ) F_p^{{b}-1}v_{\Lambda }\qquad &\\
=\Lambda '(K_{{\sigma }_p^\chi ({\alpha })}
L_{{\sigma }_p^\chi (\beta )}) F_p^{{b}-1}v_\Lambda &
\end{aligned}$$ for all ${\alpha },\beta \in {\mathbb{Z}}^I$. Hence ${T}_p(u)F_p^{{b}-1}v_{\Lambda }={\hat{T}}_p(\Lambda '(u)v_{\Lambda '})$ for all $u\in {{\mathcal{U}}^0}$. Therefore ${\hat{T}}_p$ is well-defined.
The last claim of the lemma follows from the equations $$\begin{aligned}
{\hat{T}}_p(uv)={T}_p(u){\hat{T}}_p(v),\quad
{\hat{T}}^-_p(uv)={T}^-_p(u){\hat{T}}^-_p(v),
\label{eq:Tuv}
\end{aligned}$$ where $u\in U(r_p(\chi ))$ and $v\in M^{r_p(\chi )}({t}_p^\chi (\Lambda ))$, and from the surjectivity of the maps ${T}_p,{T}^-_p$.
\[le:VTMrel2\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $j,k\in I$ such that $j\not=k$. Assume that $m_{j,k}^\chi =|({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+|<\infty $ and that ${b^{\chi}} ({\alpha })<\infty $ for all ${\alpha }\in ({\mathbb{N}}_0{\alpha }_j+{\mathbb{N}}_0{\alpha }_k)\cap R^\chi _+$. Then $$\begin{aligned}
\label{eq:VTCox}
({t}_j{t}_k)^{m _{j,k}^\chi -1}{t}_j{t}_k^\chi (\Lambda )=\Lambda .
\end{aligned}$$
Let $m=m^\chi _{j,k}$, and let $i_0,i_1,\dots ,i_{m_{j,k}^\chi }\in
| 2,219
| 2,253
| 2,163
| 2,064
| null | null |
github_plus_top10pct_by_avg
|
) is true under the Euler specialization $(z,w)\mapsto\left(\sqrt{q},1/\sqrt{q}\right)$; namely, we have $$\H_{(n-1,1)}(z,z^{-1})
=\H^{[n]}(z,z^{-1}).
\label{CV=HS1}$$ Equivalently, the two varieties $\M_{(n-1,1)}$ and $X^{[n]}$ have the same $E$-polynomial.
Consider the generating function $$F:=(1-z)(1-w)\sum_\lambda\phi_\lambda(z,w)T^{|\lambda|}.$$
It is straightforward to see that for $\lambda\neq 0$ we have $$\begin{aligned}
(1-z)(1-w)\phi_\lambda(z,w)&=1+\sum_{i=1}^{l(\lambda)}(w^i-w^{i-1})z^{\lambda_i}
-w^{l(\lambda)}\\
&=1+\sum_{i\geq 1}(w^i-w^{i-1})z^{\lambda_i}.\end{aligned}$$ Interchanging summations we find $$F=\sum_{i\geq 1}(w^i-w^{i-1})\sum_{\lambda\neq
0}z^{\lambda_i}T^{|\lambda|}+\sum_{\lambda\neq 0}T^{|\lambda|}.$$ To compute the sum over $\lambda$ for a fixed $i$ we break the partitions as follows: $$\lambda_1\geq\lambda_2\geq\cdots\geq
\lambda_{i-1}\geq\underbrace{\lambda_i\geq\lambda_{i+1}\geq\cdots}_{\rho}$$ and we put $$\begin{aligned}
&\rho:=(\lambda_i,\lambda_{i+1},\dots)\\
&\mu:=(\lambda_1-\lambda_i,\lambda_2-\lambda_i,\dots,\lambda_{i-1}-\lambda_i)\end{aligned}$$ Notice that $\mu_1'=l(\mu)<i$, $\rho_1=l(\rho')=\lambda_i$ and $|\lambda|=|\mu|+|\rho|+l(\rho')(i-1)$.
We then have $$\sum_\lambda
z^{\lambda_i}T^{|\lambda|}=\sum_{\mu_1<i}T^{|\mu|}\sum_\rho
z^{l(\rho)}T^{|\rho|+(i-1)l(\rho)}$$(changing $\rho$ to $\rho'$ and $\mu$ to $\mu'$). Each sum can be written as an infinite product, namely $$\sum_\lambda
z^{\lambda_i}T^{|\lambda|}=\prod_{k=1}^{i-1}(1-T^k)^{-1}\prod_{n\geq
1}(1-zT^{n+i-1})^{-1}.$$
So $$\begin{aligned}
F&=\sum_{\lambda\neq 0}T^{|\lambda|}+\sum_{i\geq
1}(w^i-w^{i-1})\left(\prod_{k=1}^{i-1}(1-T^k)^{-1}\prod_{n\geq
1}(1-zT^{n+i-1})^{-1}-1\right)\\&=\sum_{\lambda\neq
0}T^{|\lambda|}+\prod_{n\geq 1}(1-zT^n)^{-1}\sum_{i\geq
1}(w^i-w^{i-1})\prod_{k=1}^{i-1}\frac{(1-zT^k)}{(1-T^k)}-\sum_{i\geq
1}(w^i-w^{i-1}).\end{aligned}$$ The last sum telescopes to $1$ and we find $$F=\sum_\lambda T^{|\lambda|}+\prod_{n\geq
1}(1-zT^n)^{-1}(w-1)\sum_{i\geq 1}w^{i-1}
| 2,220
| 3,658
| 1,617
| 1,832
| null | null |
github_plus_top10pct_by_avg
|
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3 K=50 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=100 1.000 1.000 1.000 1.000 1.000 1.000 1.000
K=150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
mVC 0.944 0.960 0.944 0.966 0.950 0.960 0.968
mMSE 0.976 0.976 0.978 0.976 0.952 0.986 0.962
BLB($n^{0.6}$) 0.030 0.070 0.042 0.066 0.042 0.044 0.032
BLB($n^{0.8}$) 0.994 0.988 0.998 0.998 0.994 0.996 0.994
SDB($n^{0.6}$) 0.000 0.000 0.000 0.000 0.000 0.000 0.000
SDB($n^{0.8}$) 0.998 1.000 1.000 1.000 0.998 1.000 0.998
TB 1.000 1.000 1.000 1.000 1.000 1.000 1.000
: Empirical powers comparison for Cases 1-3 in Example \[example2\].
\[table6\]
Case Method $\beta_{1}$ $\beta_{2}$ $\beta_{3}$ $\beta_{4}$ $\beta_{5}$ $\beta_{6}$ $\beta_7$
------ ---------------- ------------- ------------- ------------- ------------- ------------- ------------- -----------
4 K=50 0.976 0.982 0.972 0.962 0.972 0.962 0.964
K=100 0.972 0.970 0.964 0.956 0.978 0.960 0.950
K=150 0.974 0.976 0.956 0.956 0.962 0.956 0.966
mVC 0.354 0.388
| 2,221
| 5,515
| 694
| 1,575
| null | null |
github_plus_top10pct_by_avg
|
actly equal to the relative entropy between the thermal state used to evaluate $F(\lambda_f, \beta)$ and the postwork state [@daffner]. It is important to remark that the relative entropy is zero for identical states and it diverges for orthogonal states [@vedral].
Short Review on Trapped Ions Interacting with Classical Laser Fields {#iiwaef}
====================================================================
We now present the basic elements needed to work with trapped ions subjected to laser fields. More information can be found in the many reviews available in the literature, e.g., [@leibfried]. Usually, the laser-ion setup is described by a model consisting of a two-level system (electronic degrees of freedom) coupled to a harmonic oscillator (center of mass motion). The latter is the result of electromagnetic confinement achieved by the use of trapping technology, e.g., Paul traps [@ghosh], and the electronic-motion coupling occurs due to momentum exchange with the laser.
By considering the center of mass (CM) degree of freedom as an oscillator with natural frequency $\nu$, and the two levels $\{ | g \rangle$, $ | e \rangle \}$ with an energy separation of $\hbar\omega_0$, the system Hamiltonian reads [@blockley] $$\label{hamtot}
\hat{\mathcal{H}} = \hat{\mathcal{H}}_0 + \hat{\mathcal{H}}_{\rm I},$$ with $$\label{hamfree}
\hat{\mathcal{H}}_0 = \hbar\nu\hat{a}^{\dagger}\hat{a} +
\frac{ \hbar \omega_0 }{2} \hat{\sigma}_{z},$$ and $$\label{Hamint}
\!\!\!\!\hat{\mathcal{H}}_{\rm I} = \frac{\hbar \Omega}{2}
\left[ \hat{\sigma}_{+} \,
\text{e}^{ i \eta ( \hat{a} + \hat{a}^{\dagger} )
- i \omega_L t } +
\hat{\sigma}_{-} \,
\text{e}^{ - i \eta ( \hat{a} + \hat{a}^{\dagger} )
+ i \omega_L t }
\right],$$ where $\omega_L$ is the laser frequency, $\Omega$ the c
| 2,222
| 4,987
| 568
| 1,785
| null | null |
github_plus_top10pct_by_avg
|
G allele 77 (72.6) 126 (75.9) 0.5465 1 G allele 121 (72.9) 38 (79.2) 0.3809 1
**rs6603797** **rs6603797**
C allele 94 (88.7) 146 (88.0) 1.07 (0.50, 2.30) C allele 150 (90.4) 42 (87.5) 1.34 (0.49, 3.64)
T allele 12 (11.3) 20 (12.0) 0.8559 1 T allele 16 (9.6) 6 (12.5) 0.5653 1
**rs4648727**^**a**^ **rs4648727**
A allele 31 (29.2) 58 (35.4) 0.76 (0.45, 1.28) A allele 54 (32.5) 12 (25.0) 1.45 (0.70, 3.00)
C allele 75 (70.8) 106 (64.6) 0.2961 1 C allele 112 (67.5) 36 (75.0) 0.3198 1
**rs12126768**^**a**^ **rs12126768**
G allele 22 (20.8) 46 (28.0) 0.67 (0.38, 1.20) G allele 40 (24.1) 12 (25.0) 0.95 (0.45, 2.00)
T allele 84 (79.2) 118 (72.0) 0.1776 1 T allele 126 (75.9) 36 (75.0) 0.8977 1
^a^: Contains 1 missing data point in the RVR (−) group.
Abbreviations: SNP, single nucleotide polymorphism; RVR, rapid virological response; OR, odds ratio; CI, confidence interval.
Allele frequencies were determined by the *χ*^2^ test using 2 × 2 tables. Odds ratios and 95% CI per genotype were estimated by unconditional logistic regression. *P* values less than 0.05 were considered statistically significant.
For LD analysis, our results indicated the existence of a low degree of pairwise LD among these SNPs in HCV-1 and HCV-2 infected populations with or without
| 2,223
| 2,219
| 2,058
| 2,392
| null | null |
github_plus_top10pct_by_avg
|
Vert _{\chi}\leq \underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert$$ and$$L_{A}\text{ is compact if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert =0.$$
\(b) If $A\in(\ell_{\infty}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert =0.$$
\(c) If $A\in(\ell_{\infty}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \leq \left \Vert L_{A}\right \Vert
_{\chi}\leq \underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert =0,$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
Let $\bar{A}=(\bar{a}_{nk})$ be the associated matrix defined by (2.9). Then we have
\(a) If $A\in(\ell_{1}(\widehat{F}),\ell_{\infty})$, then$$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
\sup_{k}\left \vert \bar{a}_{nk}\right \vert \right)$$ and$$L_{A}\text{ is compact if }\underset{n}{\lim}\left( \sup_{k}\left \vert
\bar{a}_{nk}\right \vert \right) =0.$$
\(b) If $A\in(\ell_{1}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}\left( \sup
_{k}\left \vert \bar{a}_{nk}\right \vert \right)$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left( \sup
_{k}\left \vert \bar{a}_{nk}\right \vert \right) =0.$$
\(c) If $A\in(\ell_{1}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}\left( \sup_{k}\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \right) \leq \left \Vert L_{A}\right \V
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ct that the gauge algebra involves kinematic factors is also the reason why it is possible for higher-form gauge symmetries to be non-Abelian. Higher-form global symmetries are always Abelian [@Gaiotto:2014kfa]. Hence an ordinary procedure of “gauging” the global symmetry by introducing space-time dependence to the generators in a way analogous to eq.(\[factorizable\]) can never result in a non-Abelian gauge symmetry (unless the space-time coordinates are noncommutative). Conversely, for a non-Abelian higher-form gauge symmetry, when the transformation parameters are restricted to be constant, all kinematic factors become trivial, and the symmetry algebra becomes Abelian. The Nambu-Poisson gauge symmetry is clearly an example of this fact. The noncommutative $U(1)$ gauge symmetry is the lower-form analogue.
Another example of non-Abelian gauge symmetry with a 2-form gauge potential is the low energy effective theory for multiple M5-branes proposed in Ref.[@MM5; @Ho:2012nt]. The M5-branes are compactified on a circle, and the gauge transformation laws distinguish zero-modes from KK modes. The distinct treatment on zero-modes and KK modes can be viewed as a dependence on the kinematic factor (whether the momentum is zero or not), and so it is also an example of the generalized gauge theory for higher forms.
There are other examples of non-Abelian gauge symmetry with higher-form gauge potentials [@2form], in addition those mentioned above. It will be interesting to explore further how the idea promoted above on generalized gauge symmetry will help the construction of a mathematical framework for non-Abelian higher-form gauge theories.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author would like to thank Chong-Sun Chu, Kazuo Hosomichi, Yu-Tin Huang, Takeo Inami for their interest and discussions. This work is supported in part by the National Science Council, Taiwan, R.O.C. and by the National Taiwan University.
.8cm
[99]{}
P. M. Ho and S. Ramgoolam, “Higher dimensional geometr
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nt to specifying an initial joint prior on ${\bar{\nu}}$ and $\nu$). Letting ${\bar{\nu}}_{\tau-1}^{\mbox{\scriptsize med}}$ denote the posterior median of ${\bar{\nu}}$ at time $\tau-1$, tuning parameters $\epsilon<<1$ and $\delta<<1$ are chosen such that $\mathbb{P}(\nu>\epsilon{\bar{\nu}})\approx \delta$ is desired. Given that $$\begin{aligned}
1-\delta = \mathbb{P}(\nu\leq\epsilon{\bar{\nu}}) \approx \mathbb{P}(\nu\leq\epsilon{\bar{\nu}}_{\tau-1}^{\mbox{\scriptsize med}}) = \mbox{\textsf{Gam}}(b_{\tau-1}\epsilon{\bar{\nu}}_{\tau-1}^{\mbox{\scriptsize med}};~a_{\tau-1}),\label{eq:DefineNuPrior}\end{aligned}$$ where $\mbox{\textsf{Gam}}(x; \alpha)$ is the cumulative distribution function evaluated at $x$ of a gamma random variable with shape $\alpha$ and unit rate, a practical specification for the prior for $\nu$ is obtained by defining a small $a_{\tau-1}=a_0$ and then solving for $b_{\tau-1}$.
Improving the marginal likelihood estimate {#sec:ML}
------------------------------------------
The following post-processing development is motivated by the analysis of a particular simulated dataset where the point estimate for the number of MUs is one greater than the true number. The detailed analysis in Section \[sec:SimStudyOver\] shows that the extra MU has a very weak expected MUTF and that it, effectively, acts simply to increase the variability in the response. The problem arises because the $u$-vector, ${\boldsymbol{\mu}}$, of expected MUTF contributions has a Gaussian prior which, to allow reasonable uncertainty across the typical range of believable MUTF contributions, also places a non-negligible prior mass at low and even negative values. Negative expectations for an individual MU need not be prohibited by the data provided that MU is always inferred to fire alongside another MU with a positive expectation of similar or larger magnitude. The fact that the parameter suport permits this possibility potentially increases the marginal likelihood for a model which is larger than that necessary to explain
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t and right products. Then $\bar J\in{\mathcal{H}}{\mathrm{SJ}}$ and $\widehat J=\bar
J\oplus J=\bar J\oplus\bar J\in{\mathcal{H}}{\mathrm{SJ}}$, so $J\in{\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}={\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$. We obtain $J\vDash f$, hence $\bar
J\vDash \bar f$. Therefore, ${\mathcal{H}}{\mathrm{SJ}}\vDash\bar f=f(x,y,z)$, so by the classical Macdonald’s Theorem we have ${\mathrm{Jord}}\vDash\bar f$.
It remains to note that if $f= f(x,y,\dot z)$ is a multilinear dipolynomial such that ${\mathrm{Jord}}\vDash\bar f=f(x,y,z)$ then ${\mathrm{Di}}{\mathrm{Jord}}\vDash f$: It follows immediately from the definition [@Kol:08] of what is a variety of dialgebras. The polynomial $f(x,y,z)$ can be nonlinear in $x$ and $y$. Suppose $\deg_x f=n$, $\deg_y f=m$. Consider the full linearization $$g(x_1,\dots, x_n, y_1,\dots, y_m, z)= L_x^n L_y^m f(x,y,z)$$ of the identity $f(x,y,z)$ (notations from [@Zhevl:78 ch. 1]). Then ${\mathrm{Jord}}\vDash g(x_1,\dots, x_n, y_1,\dots, y_m, z)$ and so ${\mathrm{Di}}{\mathrm{Jord}}\vDash g(x_1,\dots, x_n,y_1,\dots, y_m, \dot z)$.
If we now identify variables, then $$g(x,\dots, x, y,\dots, y,\dot
z)=n!m!f(x,y,\dot z).$$ In this section the characteristic of the basic field is equal to zero, so we can divide by $n!m!$ and hence $f(x,y,\dot z)$ is an identity on ${\mathrm{Di}}{\mathrm{Jord}}$.
P. M. Cohn in [@Cohn:59] proposed an axiomatic characterization of Jordan algebras $J_1(A)=A^{(+)}$ and $J_2(A,*)=H(A,*)$, where $A$ is an associative algebra and $*$ is an involution on $A$, in terms of $n$-ary operations. This is an interesting task to generalize these results to Jordan dialgebras.
Acknowledgements {#acknowledgements .unnumbered}
----------------
In the end of paper the author thanks P. S. Kolesnikov, A. P. Pozhidaev and V. Yu. Gubarev for helpful discussions and valuable comments. The author is grateful to the referee for valuable comments that allowed to improve the manuscript. In particular, the statement about tetrads in Theorem \[thm:CohnFor
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qbezier[20](60,25)(60,10)(75,10)
\qbezier[20](75,40)(90,40)(90,25)
\qbezier[20](75,10)(90,10)(90,25) \qbezier[30](20,5)(20,25)(20,45)
\qbezier[20](180,25)(180,40)(195,40)
\qbezier[20](180,25)(180,10)(195,10)
\qbezier[20](195,40)(210,40)(210,25)
\qbezier[20](195,10)(210,10)(210,25)
\qbezier[20](120,25)(120,45)(140,45)
\qbezier[20](120,25)(120,5)(140,5)
\qbezier[20](140,5)(160,5)(160,25)
\qbezier[20](300,25)(300,40)(315,40)
\qbezier[20](300,25)(300,10)(315,10)
\qbezier[20](315,40)(330,40)(330,25)
\qbezier[20](315,10)(330,10)(330,25)
\qbezier[20](260,5)(280,5)(280,25)
\qbezier[30](260,5)(260,25)(260,45)
\linethickness{0.5mm} \qbezier(20,45)(40,45)(40,25)
\qbezier(20,5)(40,5)(40,25) \put(40,25){\line(1,0){20}}
\put(160,25){\line(1,0){20}} \put(280,25){\line(1,0){20}}
\qbezier(140,45)(160,45)(160,25) \put(140,5){\line(0,1){40}}
\qbezier(240,25)(240,45)(260,45) \qbezier(240,25)(240,5)(260,5)
\qbezier(260,45)(280,45)(280,25)
\end{picture}$$
$$\begin{picture}(210,50) \qbezier[20](0,25)(0,45)(20,45)
\qbezier[20](0,25)(0,5)(20,5) \qbezier[20](60,25)(60,40)(75,40)
\qbezier[20](60,25)(60,10)(75,10)
\qbezier[20](75,40)(90,40)(90,25)
\qbezier[20](75,10)(90,10)(90,25)
\qbezier[20](20,45)(40,45)(40,25)
\qbezier[20](180,25)(180,40)(195,40)
\qbezier[20](180,25)(180,10)(195,10)
\qbezier[20](195,40)(210,40)(210,25)
\qbezier[20](195,10)(210,10)(210,25)
\qbezier[20](140,45)(160,45)(160,25) \put(140,5){\line(0,1){40}}
\qbezier[20](140,5)(160,5)(160,25)
\qbezier[30](140,5)(140,25)(140,45)
\linethickness{0.5mm} \put(20,5){\line(0,1){40}}
\put(40,25){\line(1,0){20}} \qbezier(20,5)(40,5)(40,25)
\qbezier(120,25)(120,45)(140,45) \qbezier(120,25)(120,5)(140,5)
\put(160,25){\line(1,0){20}} \qbezier(140,5)(160,5)(160,25)
\end{picture}$$ In each case we have six ways to choose a marked edge, that does not belong to the tree. Thus, the first map generates $30$ tree-rooted maps.
The second cubic map in Figure 1 generates four t-maps. $$\begin{picture}(310,80) \put(0,37){\small 1)} \qbezier[20](40,60)(60,60)(60,40)
\qbezier[20](40,20)(60,20
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Superantigen-like protein 5
1633215 A:5 C:83 C:37 5 18.4 28 putative traG membrane protein
2114835 A:6 C:136 C:37 7 17.14 27 phiPVL ORF046-like protein
1859648 C:5 X:1 G:121 G:37 6 16 24 FtsK/SpoIIIE family protein
467549 A:89 T:23 A:58 23 15.56 40 Staphylococcal tandem lipoprotein
2123177 A:57 X:5 A:37 5 15.4 21 putative phage transcriptional regulator
36501
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_1(x)=h_2(x)=\cdots =h_l(x)$. Therefore, from the discussion above, we have if $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $d_{\rm H}(C)=\sum_{i=1}^ld_i\geq \sum_{i=1}^l(\delta_i+1)$. $\Box$
From Theorems 4.2 and 4.3, we have the following corollary immediately.
[**Corollary 4.4**]{} *Let $C$ be a $1$-generator skew QC code of length $ml$ generated by $c(x)=(c_1(x), c_2(x), \ldots, c_l(x))\in (R/(x^m-1))^l$. Suppose $h_i(x)=(x^m-1)/{\rm gcld}(c_i(x), x^m-1)$, $i=1,2,\ldots,l$, and $h(x)={\rm lclm}\{h_1(x),h_2(x),\ldots,h_l(x)\}$. Then\
(i) The dimension of $C$ is the degree of $h(x)$.\
(ii) Let $\delta_i$ denote the number of consecutive powers of a primitive $m_i$-th root of unity that among the right zeros of $(x^{m}-1)/h_i(x)$. Then $d_{\rm H}(C)\geq \sum_{{i}\not \in K}(\delta_i+1)$, where $K\subseteq \{1,2,\ldots,l\}$ is a set of maximum size such that ${\rm lclm}_{i\in K}h_i(x)\neq h(x)$.\
(iii) If $h_1(x)=h_2(x)=\cdots =h_l(x)$, then $\delta_i=\delta$ for each $i=1,2,\ldots,l$ and $d_{\rm H}(C)\geq l(\delta+1)$.* $\Box$
[**Example 4.5**]{} Let $R=\mathbb{F}_{3^2}[x,\sigma]$, where $\sigma$ is the Frobenius automorphism of $\mathbb{F}_{3^2}$ over $\mathbb{F}_3$. The polynomial $g(x)=x-\alpha^2$ is a right divisor of $x^4-1$, where $\alpha$ is a primitive element of $\mathbb{F}_{3^2}$. Consider the $1$-generator GQC code $C$ of block length $(4,8)$ and length $4+8=12$ generated by $c(x)=(g(x), g(x))$. Then, by Theorem 4.3, $h(x)=(x^8-1)/(x-\alpha^2)$ and $d_H(C)\geq 2$. A generator matrix for $C$ is given as follows $$G=\left(
\begin{array}{cccccccccccc}
-\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -\alpha^6 & 1 & 0 & 0 & -\alpha^6 & 1& 0 & 0 & 0& 0 & 0\\
0 & 0 & -\alpha^2 & 1 & 0 & 0 & -\alpha^2 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & -\alpha^6 & 0 & 0 & 0 & -\alpha^6 & 1 & 0 & 0 & 0\\
-\alpha^2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -\alpha^2 & 1 & 0 & 0\\
0 & -\alpha^6 & 1 & 0 & 0 & 0 & 0& 0 & 0 & -\alpha^6& 1 & 0\\
0 & 0 & -\alpha^2 & 1 & 0 & 0& 0 & 0 & 0 & 0 & -\al
| 2,230
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thbb{E}}{\left\Vert M \right\Vert}}{\sqrt{\log{\left\vert G \right\vert}}} \le
C \sqrt{\max_{a \in G} {\mathbb{E}}{\left\vert Y_a \right\vert}^2},$$ where $c, C > 0$ are constants, independent of $G$ and the distributions of the $Y_a$.
The rest of this paper deals mainly with infinite sequences of finite abelian groups $G^{(n)}$, always assumed to satisfy ${\left\vert G^{(n)} \right\vert} \to
\infty$. For each $n$ a family of random variables $\bigl\{Y_g^{(n)}
\mid g \in G^{(n)} \bigr\}$ will be used to construct a random $G^{(n)}$-circulant matrix $$M^{(n)}= \left[\frac{1}{\sqrt{{\left\vert G^{(n)} \right\vert}}} Y^{(n)}_{ab^{-1}}
\right]_{a,b \in G^{(n)}}$$ with empirical spectral measure $\mu^{(n)}$. As mentioned earlier, an important role will be played by the quantity $$p_2^{(n)} = \frac{{\left\vert \{ a \in G^{(n)} \mid a^2 = 1\} \right\vert}}{{\left\vert G^{(n)} \right\vert}}
=
\frac{\bigl\vert \bigl\{\chi \in \widehat{G} \mid \chi = \overline{\chi} \bigr
\}\bigr\vert }
{\bigl\vert\widehat{G}^{(n)}\bigr\vert}.$$
The standard real Gaussian measure is denoted $\gamma_{\mathbb{R}}$, and the standard complex Gaussian distribution, normalized such that ${\mathbb{E}}{\left\vert Z \right\vert}^2 = 1$ when $Z$ is a standard complex Gaussian random variable, is denoted $\gamma_{\mathbb{C}}$. For $\alpha \in [0,1]$, $\gamma_\alpha$ denotes the Gaussian measure on ${\mathbb{C}}\cong {\mathbb{R}}^2$ with covariance $\frac{1}{2} \bigl[ \begin{smallmatrix} 1+\alpha & 0 \\
0 & 1-\alpha \end{smallmatrix}\bigr]$, so that in particular $\gamma_0 = \gamma_{\mathbb{C}}$ and $\gamma_1 = \gamma_{\mathbb{R}}$.
The integral of a function $f$ with respect to a measure $\nu$ will be denoted by $\nu(f)$.
Gaussian matrix entries {#S:Gaussian}
=======================
The following is an immediate consequence of Lemma \[T:FT-isometry\](\[I:isometry\]) and the rotation-invariance of the standard Gaussian distribution. The special case of this result for classical circulant matrices (that is, when $G$ is a cyclic group) w
| 2,231
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[Baseline attack success rates and transfer success rates for an IGS attack with an epsilon of 1.0 on LeNet models trained on MNIST. 8 pairs of models were trained for the parallel Jacobian goal, and 8 pairs of models were trained for the perpendicular goal to obtain error bars around attack success rates.[]{data-label="fig:parallel_vs_perpendicular"}](figures/parallel_vs_perpendicular.pdf){width="90.00000%"}
Effect of Gradient Magnitude
----------------------------
When training the dozens of models used for the results of this paper, we noticed that attack success rates varied greatly between models, even though the difference in hyperparameters seemed negligible. We posited that a significant confounding factor that determines susceptibility to attacks such as FGS and IGS is the magnitude of the input-output Jacobian for the true class. Intuitively, if the magnitude of the input-output Jacobian is large, then the perturbation required to cause a model to misclassify an image is smaller than had the magnitude of the input-output Jacobian been small. This is seen in Figure \[fig:effect\_of\_grad\_magnitude\] where there is a clear increasing trend in attack success rate as the input-output Jacobian increases. This metric can be a significant confounding factor when analyzing robustness to adversarial examples, so it is important to measure it before concluding that differences in hyperparameters are the cause of varying levels of adversarial robustness.
![Relationship between input-output Jacobian L2 norm and susceptibility to IGS attack with epsilon of 0.3. 8 LeNet models were trained on MNIST for each magnitude with different random seeds. The ’mean’ and ’std’ reported underneath the goal note the actual input-output Jacobian L2 norms, whereas the ’goal’ is the target that was used during training to regularize the magnitude of the Jacobian.[]{data-label="fig:effect_of_grad_magnitude"}](figures/effect_of_grad_magnitude_igs.pdf){width="\textwidth"}
Detector Analysis
=================
The effect
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lace operator ($-k^2$).
The discrete analog of the Sturm-Liouville theory (e.g., @hil68 [chap. 1.10–1.16]; see also, @atk64) guarantees that the eigenfunctions given in Equation – satisfy discrete orthogonality relations, for example, $$\label{eq:orthogorel}
\frac{2}{N_x+1} \sum_{i=1}^{N_x}{\cal X}_i^l{\cal X}_i^{l'} = \delta_{ll'}\quad\text{and}\quad\frac{2}{N_x+1} \sum_{l=1}^{N_x} {\cal X}_i^l{\cal X}_{i'}^l = \delta_{ii'},$$ where the symbol $\delta$ denotes the Kronecker delta. These orthogonality relations allow us to expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ as $$\begin{aligned}
\widetilde{\Phi}_{i,j,k} = \frac{8}{(N_x+1)(N_y+1)(N_z+1)}\sum_{l=1}^{N_x}\sum_{m=1}^{N_y}\sum_{n=1}^{N_z}\widetilde{\Phi}^{lmn}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_Phi_forward}\\
\rho_{i,j,k} = \frac{8}{(N_x+1)(N_y+1)(N_z+1)}\sum_{l=1}^{N_x}\sum_{m=1}^{N_y}\sum_{n=1}^{N_z}\rho^{lmn}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_rho_forward}\end{aligned}$$ where the expansion coefficients $\widetilde{\Phi}^{lmn}$ and $\rho^{lmn}$ are given by $$\begin{aligned}
\widetilde{\Phi}^{lmn} &= \sum_{i=1}^{N_x}\sum_{j=1}^{N_y}\sum_{k=1}^{N_z}\widetilde{\Phi}_{i,j,k}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_Phi_backward}\\
\rho^{lmn} &= \sum_{i=1}^{N_x}\sum_{j=1}^{N_y}\sum_{k=1}^{N_z}\rho_{i,j,k}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n.\label{eq:car_rho_backward}\end{aligned}$$ Plugging Equations – in Equation , we obtain a simple algebraic relation $$\label{eq:car_Poisson_transform}
\widetilde{\Phi}^{lmn} = \frac{4\pi G \rho^{lmn}}{\lambda_x^l + \lambda_y^m + \lambda_z^n}.$$ Therefore, the Poisson equation in Cartesian coordinates can be solved by the following three steps:
1. Perform a forward transform $\rho_{i,j,k}\to\rho^{lmn}$ using Equation : ${\cal O}(N_xN_yN_z\log_2[N_xN_yN_z])$.
2. Convert $\rho^{lmn} \to \widetilde{\Phi}^{lmn}$ using the kernel in Equation : ${\cal O}(N_xN_yN_z)$.
3. Perform a backward transform $\widetilde{\Phi}^{lmn}\to\widetilde{\Phi}_{i,j,k}$ using Equa
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patterns to obtain a spare AOG. Besides, for each part template $v$, we estimated $n_{k}$ latent patterns in the $k$-th conv-layer. We assumed that scores of all latent patterns in the $k$-th conv-layer follow the distribution of [$Score(u)\sim\alpha\exp[-(\xi\cdot{rank})^{0.5}]+\gamma$]{}, where $rank$ denotes the score rank of [$u$]{}. We set [$n_{k}=\lceil0.5/\xi\rceil$]{}, which learned the best AOG.
Datasets
--------
Because evaluation of part localization requires ground-truth annotations of part positions, we used the following three benchmark datasets to test our method, *i.e.* the PASCAL VOC Part Dataset [@SemanticPart], the CUB200-2011 dataset [@CUB200], and the ILSVRC 2013 DET Animal-Part dataset [@CNNAoG]. Just like in [@SemanticPart; @CNNAoG], we selected animal categories, which prevalently contain non-rigid shape deformation, for testing. *I.e.* we selected six animal categories—*bird, cat, cow, dog, horse*, and *sheep*—from the PASCAL Part Dataset. The CUB200-2011 dataset contains 11.8K images of 200 bird species. We followed [@ActivePart; @CNNSemanticPart; @CNNAoG] and used all these images as a single bird category for learning. The ILSVRC 2013 DET Animal-Part dataset [@CNNAoG] contains part annotations of 30 animal categories among all the 200 categories in the ILSVRC 2013 DET dataset [@ImageNet].
{width="\linewidth"}
The 2nd column shows the number of part annotations for training. The 3rd column indicates whether the baseline used all object-box annotations in the category to pre-fine-tune a CNN before learning the part (*object-box annotations are more than part annotations*).
Baselines
---------
We used the following thirteen baselines for comparison. The first two baselines were based on the Fast-RCNN [@FastRCNN]. We fine-tuned the fast-RCNN with a loss of detecting a single class/part for a fair comparison. The first baseline, namely *Fast-RCNN (1 ft)*, fine-tuned the VGG-16 using part annotations to detect parts on well-cropped objects. To enable
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mrc; @Miranda:2016wdr; @Ge:2016xya; @Dutta:2016vcc; @Dutta:2016czj; @Rout:2017udo].
[^5]: To understand the formulas presented in this section, minimal explanation of definitions of the quantities may be helpful. Let the flavor mixing matrix $\bf{U}$ in $(3+N) \times (3+N)$ space that connect the flavor and mass eigenstates as $\nu_{\zeta} = {\bf U}_{\zeta z} \nu_{z}$, where $\zeta$ runs over active $\alpha = e,\mu,\tau$ and sterile flavor $s=s_1,\cdot \cdot \cdot,s_N$ indices, and $z$ over mostly active $i=1,2,3$ and mostly sterile mass eigenstate $J=4,5,\cdot \cdot \cdot, N+3$ indices. Let $U$ and $W$ be a part of $\bf{U}$ so that they connect the mass eigenstates to active neutrino flavor states, $\nu_{\alpha} = \sum_{i} (U)_{\alpha i} \nu_{i} + \sum_{J} (W)_{\alpha J} \nu_{J}$. The kinematical phase factors $\Delta_{j}$ and $\Delta_{J}$ are defined as $\Delta_{j} \equiv \frac{m^2_{j}}{2E}$ and $\Delta_{J} \equiv \frac{m^2_{J}}{2E}$, respectively. $m_{j}$ and $m_{J}$ denote the active and sterile neutrino masses, respectively.
[^6]: Speaking more precisely, we mean that all the $W$ matrix elements are assumed to be small, of the order of $\simeq 0.1$.
[^7]: Our result also means that taking care of all order matter effect does not change the feature obtained by first-order treatment in matter perturbation theory. That is, the same condition on sterile states masses derived by using first-order matter perturbation theory suffices to guarantee the absence of matter-dependent higher order correction terms in $W$. The reasons for this feature will be explained at the end of section \[sec:energy-denominator\].
[^8]: In matter, we have $\sum_{j=1}^{3} (UX)_{\alpha j} (UX)^{*}_{\beta j}
= \sum_{j,k,l=1}^{3} (U)_{\alpha k} (U)^{*}_{\beta l} X_{kj} X^*_{lj}
= \sum_{k=1}^{3} (U)_{\alpha k} (U)^{*}_{\beta k}$ where in the last step, we have used the unitarity relation $\sum_{j=1}^3 X_{kj} X^*_{lj} = \delta_{kl}$.
[^9]: Through unitarity (\[unitarity\]), $U$ matrix elements have some dependence on $W$ matri
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|
)(1.5,3)
\psline(0.4,2.6)(0.4,3)
\psline(0.7,2.3)(0.7,3)
\rput(1.4,1.3){\tiny $\alpha_1$}
\rput(0.2,2.4){\tiny $\alpha_2$}
\rput(0.6,2.1){\tiny $\alpha_3$}
\rput(1,2.3){\tiny $\alpha_4$}
\pscircle[linestyle=dotted](1,2.2){1.4}
\rput(5.2,1){$\psi$}
\psline(3.5,1.5)(3,3)
\psline(3.5,1.5)(4,3)
\psline(4.5,2.5)(4.2,3)
\psline(3.3,2.1)(3.3,3)
\psline(3.3,2.1)(3.6,3)
\rput(3.6,1.2){\tiny $\beta_1$}
\rput(3,2){\tiny $\beta_2$}
\rput(4.6,2.2){\tiny $\beta_3$}
\rput(-1.5,2){$\varphi * \psi =$}
\end{pspicture}$$ As for $\varphi\# \psi$, we assume, that $\varphi\in \textbf{D}(\mathcal O^!)(k)$ with $k\geq 2$. Then identify $\varphi$ with an element in $\textbf{D} (\widehat{
\mathcal O^!})({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},\ldots,{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}};\varnothing)$ by interpreting the lowest decoration $\alpha_1\in O^!(m)$ of $\varphi$, as an inner product $\alpha_1\in \widehat{\mathcal O^!}({
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},\ldots,{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt](0.1,0)(0.1,0.4)
\end{pspicture}},{
\begin{pspicture}(0,0.1)(0.2,0.4)
\psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
\end{pspicture}};\varnothing)=\mathcal O^!(m)$. $\varphi\# \psi$ is defined by attaching $\psi$ to the dashed input labeled by $\alpha_1$: $$\begin{pspicture}(-3,0)(6,4)
\psline[linestyle=dashed](2.5,0.5)(0,3)
\psline[linestyle=dashed](2.5,0.5)(5,3)
\rput(2.7,0.3){\tiny $\alpha_1$}
\pscircle[linestyle=dotted](4
| 2,236
| 1,057
| 1,458
| 2,181
| 489
| 0.808103
|
github_plus_top10pct_by_avg
|
e impurity Anderson model plus unusual Holstein coupling $\lambda_c$ for a single vibrational mode $\omega_0$. Renormalized hybridization $\Gamma_0\rightarrow\Gamma_{\rm eff}(\lambda_c)$ as function of $\lambda_c$ for two different values for $\omega_0$ at $U/\Gamma_0=10$. The inset depicts the same quantity for $\omega_0/\Gamma_0=0.2$, $\lambda_c/\Gamma_0=0.35$ as function of the Coulomb interaction.[]{data-label="fig:TK-Lc-b"}](fig15-Geff-U-Lc){width="50.00000%"}
Particle-hole symmetric demands $$\begin{aligned}
G_{d_{0\sigma},d^\dagger_{0\sigma}}(-i0^+)&=& \frac{i}{\Gamma_\text{eff}} .\end{aligned}$$ We substitute Eq. into Eq. with $\lambda_d=0$, $$\begin{aligned}
\Gamma_\text{eff} &=&\Gamma_0 \\
&& \nonumber - \Gamma_\text{eff} \left[
U \Re F_\sigma(-i0^+) -\frac{ \lambda_c }{V_0} \Gamma_0\Im N_\sigma(-i0^+)
\right] ,\end{aligned}$$ which we solve for the ratio $$\begin{aligned}
\label{eq:gamma-eff}
\frac{\Gamma_\text{eff}}{\Gamma_0} &=&
\frac{1}
{
1 + U \Re F(-i0^+)- \frac{ \lambda_c}{V_0}\Gamma_0 \Im N_\sigma(-i0^+)
}\end{aligned}$$ A negative $\Im N_\sigma(-i0^+)$ in combination with a positive $\Re F(-i0^+)$ leads to a reduction of $\Gamma_{\rm eff}$ which is quadratic in $\lambda_c$ for small $\lambda_c$, since then $\Im N_\sigma(-i0^+)$ is proportional to $\lambda_c$. Clearly, the reduction $\Gamma_0 \to \Gamma_{\rm eff}$ is not only effected by $\lambda_c$, but also depends on $U$.
Fig. \[fig:TK-Lc-b\] shows the dependence, as calculated by NRG, of the effective hybridization on the coupling $\lambda_c$ for two different vibrational frequencies $\omega_0$ and a fixed $U/\Gamma_0=10$. For a fixed coupling strength $\lambda_c$, the reduction of $\Gamma_{\rm eff}$ decreases with decreasing polaron energy $E_{p}$, as expected from the discussion in the context of Fig. \[fig:TK-Lc-a\], confirming the microscopic mechanism outlined above: the larger $E_{p}$, the more severe is the suppression of the hybridization and the stronger thus the reduction of the Kondo temperature. For weak electron-
| 2,237
| 1,722
| 2,977
| 2,379
| 3,326
| 0.773149
|
github_plus_top10pct_by_avg
|
the electric charge.[^17] In contradistinction, the quanta associated to a Dirac spinor may be distinguished in terms of particles and their antiparticles carrying opposite values of a conserved quantum number, such as for instance the electric charge (or baryon or lepton number), associated to a symmetry under arbitrary global phase transformations of the Dirac spinor. As the above construction clearly shows, in a 4-dimensional Minkowski spacetime, one cannot have both a Weyl and a Majorana condition imposed on a Dirac spinor. In such a case, one has either only Dirac spinors, Majorana spinors, or Weyl spinors of definite chirality, while the fundamental constructs of Lorentz covariant spinors are the two fundamental right- and left-handed Weyl spinors. In fact, it may be shown,[@PvN] using the properties of the Dirac-Clifford algebra, that Majorana-Weyl spinors exist only in a Minkowski spacetime of dimension $D=2$ (mod 8), which includes the dimension $D=10$ in which superstrings may be constructed, which is not an accident.
Given that the Dirac $\gamma^\mu$ matrices provide a representation space of the Lorentz group, it should be possible to display explicitly the associated generators. Indeed, it may be shown that the latter are obtained as $$\Sigma^{\mu\nu}=\frac{1}{2}i\gamma^{\mu\nu}\ \ \ ,\ \ \
\gamma^{\mu\nu}=\frac{1}{2}\left[\gamma^\mu,\gamma^\nu\right]\ ,$$ with $$\gamma^{\mu\nu}=\frac{1}{2}\left(\begin{array}{c c}
\sigma^\mu\overline{\sigma}^\nu-\sigma^\nu\overline{\sigma}^\mu & 0 \\
0 & \overline{\sigma}^\mu\sigma^\nu-\overline{\sigma}^\nu\sigma^\mu
\end{array}\right)\ .$$ Thus a right-handed spinor $\psi_\alpha$ transforms according to the generators, $$\Sigma_R^{\mu\nu}\ :\ \
{\left(\Sigma_R^{\mu\nu}\right)_\alpha}^\beta=\frac{1}{4}i\left[
{\sigma^\mu}_{\alpha\dot{\gamma}}\,
\overline{\sigma}^{\nu\dot{\gamma}\beta}\,-\,
{\sigma^\nu}_{\alpha\dot{\gamma}}\,
\overline{\sigma}^{\mu\dot{\gamma}\beta}\right]\ ,$$ while a left-handed Weyl spinor $\overline{\chi}^{\dot{\alpha}}$ according to $
| 2,238
| 4,550
| 2,200
| 1,883
| null | null |
github_plus_top10pct_by_avg
|
.Experience, ethics, planning and selection, ethic education, thought leaders.Protocols (4 mentions).Truthfulness, exclusion criteria, diagnostic criteria, treatment design, patient selection, patient follow-up.Sites (4 mentions).Certifications, requirements, screening by authorities, proceedings and operation.Declaration of Helsinki (4 mentions).Providing medication at the end of the study, giving patients information, benefits for individuals and the community.Others (3 mentions).Management and operation of health services \[*Authorities*\]. Intervention by specialists in philosophy of science \[*Justification in clinical research*\]. Data protection. \[*Ethics in a study evaluation, monitoring*\].
Q5 introduces the subject of "ethics dumping" with a brief definition. When asked whether this practice occurred in our country, nearly 50% of respondents agreed.
Q4 asked respondents to suggest areas that need improvement; the summarized answers are shown in Table [2](#Tab2){ref-type="table"}, which presents the answers to the first open ended question. Participants may contribute more than once. The categories with the greatest number of responses were the following: patients, committees, and consent.
General perception of ethics dumping in Mexico {#Sec5}
----------------------------------------------
Q6 to Q10 (Fig. [2](#Fig2){ref-type="fig"}) explore the general perception of the participants on ethics dumping in Mexico. Q6 elaborates on the topic of ethics dumping, asking whether this practice is common in our country. The answers are like those obtained in the previous question, with nearly 50% of respondents expressing agreement. Q7 asks about the severity and urgency of the situation. Again, the answers are similar: nearly 50% agree that the situation is serious and urgent; close to 20% do not believe this to be the case, and a third of respondents are unsure. Q8 elicits related examples, and in this case only 20% says they could provide an example of ethics dumping. When probed for knowledge of more tha
| 2,239
| 161
| 2,721
| 2,536
| null | null |
github_plus_top10pct_by_avg
|
har.sec\] for details.
\[quot.norm.lem\] Let $p_1,p_2:R\rightrightarrows X$ be a finite, set theoretic equivalence relation such that $(X/R)^{cat}$ exists.
1. If $X$ is normal and $X, R$ are pure dimensional then $(X/R)^{cat}$ is also normal.
2. If $X$ is seminormal then $(X/R)^{cat}$ is also seminormal.
Proof. In the first case, let $Z\to (X/R)^{cat}$ be a finite morphism which is an isomorphism at all generic points of $(X/R)^{cat}$. Since $X$ is normal, $\pi:X\to (X/R)^{cat}$ lifts to $\pi_Z:X\to Z$. By assumption, $\pi_Z\circ p_1$ equals $\pi_Z\circ p_2$ at all generic points of $R$ and $R$ is reduced. Thus $\pi_Z\circ p_1=\pi_Z\circ p_2$. The universal property of categorical quotients gives $(X/R)^{cat}\to Z$, hence $Z=(X/R)^{cat}$ and $(X/R)^{cat}$ is normal.
In the second case, let $Z\to (X/R)^{cat}$ be a finite morphism which is a universal homeomorphism (\[univ.homeo.defn\]). As before, we get liftings $\pi_Z\circ p_1,\pi_Z\circ p_2:R\rightrightarrows X\to Z$ which agree on closed points. Since $R$ is reduced, we conclude that $\pi_Z\circ p_1=\pi_Z\circ p_2$, thus $(X/R)^{cat}$ is seminormal.
The following result goes back at least to E. Noether.
\[inv.of.fin.gps\] Let $A$ be a Noetherian ring, $R$ a Noetherian $A$-algebra and $G$ a finite group of $A$-automorphisms of $R$. Let $R^G\subset R$ denote the subalgebra of $G$-invariant elements. Assume that one of the following holds:
1. $\frac1{|G|}\in A$,
2. $R$ is essentially of finite type over $A$, or
3. $R$ is finite over $A[R^p]$ for every prime $p$ that divides $|G|$.
Then $R^G$ is Noetherian and $R$ is finite over $R^G$.
Proof. Assume first that $R$ is a localization of a finitely generated $A$ algebra $A[r_1,\dots, r_m]\subset R$. We may assume that $G$ permutes the $r_j$. Let $\sigma_{ij}$ denote the $j$th elementary symmetric polynomial of the $\{g(r_i):g\in G\}$. Then $$A\bigl[\sigma_{ij}\bigr]\subset A[r_1,\dots, r_m]^G\subset R^G$$ and, with $n:=|G|$, each $r_i$ satisfies the equation $$r_i^n-\sigma_{i1}r_i^{n-1}+\sigma_{i2}r
| 2,240
| 3,937
| 2,483
| 2,072
| null | null |
github_plus_top10pct_by_avg
|
h{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid (i\in D_t) \text{~and~} \neg {\mathcal{F}}\right]}}\cdot{\ensuremath{\operatorname{\mathbf{Pr}}\left[\neg {\mathcal{F}}\right]}}\nonumber\\
&{\leqslant}{\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid (i\in D_t) \text{~and~} {\mathcal{F}}\right]}}+n^{-2}{\leqslant}{\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid (i\in D_t) \text{~and~} {\mathcal{F}}\right]}}+1/d.
\end{aligned}$$ Let $X$ be the random variable that counts the number of empty bins of a random $(d-1)$-element subset of $H_t\setminus\{i\}$, [conditioned on the event that “$(i\in D_t)$ and ${\mathcal{F}}$” holds. Then $X$]{} is a hypergeometric random variable with parameters $({s}-1, K, d-1)$, where $K$ is the number of empty bins contained in $H_t\setminus\{i\}$. Thus $${\ensuremath{\operatorname{\mathbf{E}}\left[X\right]}}=\frac{(d-1)K}{{s}-1} \text{~~and~~} {\ensuremath{\operatorname{\mathbf{Var}}\left[X\right]}}{\leqslant}\frac{(d-1)K}{{s}-1}{\leqslant}d.$$ Then ${\ensuremath{\operatorname{\mathbf{E}}\left[X\right]}} > d/3$, since $K{\geqslant}{s}/2-1$ when $i\in D_t$ and ${\mathcal{F}}$ holds (and using the size property $s = \Omega(\log n)$) [and the fact that $d{\geqslant}7$]{}). Therefore $$\begin{aligned}
& {\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid (i\in D_t) \text{~and~} {\mathcal{F}}\right]}}\\
&{\leqslant}{\ensuremath{\operatorname{\mathbf{Pr}}\left[X<d/6\right]}} {\leqslant}{\ensuremath{\operatorname{\mathbf{Pr}}\left[|X-{\ensuremath{\operatorname{\mathbf{E}}\left[X\right]}}|{\leqslant}{\ensuremath{\operatorname{\mathbf{E}}\left[X\right]}}/2\right]}} < \frac{4{\ensuremath{\operatorname{\mathbf{Var}}\left[X\right]}}}{{\ensuremath{\operatorname{\mathbf{E}}\left[X\right]}}^2}{\leqslant}\frac{36d}{d^2}=\frac{36}{d},
\end{aligned}$$ using Chebychev’s inequality. Substituting the above upper bound in Inequality (\[ineq:sec\]) establishes (\[u:cl\]) [with $\hat{c}=38$]{}, which complete
| 2,241
| 2,182
| 1,795
| 2,131
| null | null |
github_plus_top10pct_by_avg
|
ned}$$
\[c:clt\_sum\] Let $X_1...X_n$ be random vectors with mean 0, covariance $\Sigma$, and $\lrn{X_i}\leq \beta$ almost surely for each $i$. let $Y$ be a Gaussian with covariance $n\Sigma$. Then $$\begin{aligned}
W_2\lrp{\sum_i X_i, Y}\leq 6\sqrt{d}\beta \sqrt{\log n}\end{aligned}$$
This is simply taking the result of Theorem \[t:zhai\] and scaling the inequality by $\sqrt{n}$ on both sides.
The following Lemma is taken from [@cheng2019quantitative] and included here for completeness.
\[l:xlogxbound\] For any $c> 0$, $x> 3 \max\lrbb{\frac{1}{c} \log \frac{1}{c},0}$, the inequality $$\begin{aligned}
\frac{1}{c}\log(x) \leq x\end{aligned}$$ holds.
We will consider two cases:
**Case 1**: If $c\geq \frac{1}{e}$, then the inequality $$\log(x) \leq c x$$ is true for all $x$.
**Case 2**: $c \leq \frac{1}{e}$.
In this case, we consider the Lambert W function, defined as the inverse of $f(x) = x e^x$. We will particularly pay attention to $W_{-1}$ which is the lower branch of $W$. (See Wikipedia for a description of $W$ and $W_{-1}$).
We can lower bound $W_{-1}(-c)$ using Theorem 1 from [@chatzigeorgiou2013bounds]: $$\begin{aligned}
& \forall u>0,\quad W_{-1} (-e^{-u-1}) > -u-\sqrt{2u} -1\\
\text{equivalently}\quad &\forall c\in (0,1/e),\quad -W_{-1} (-c) < \log\lrp{\frac{1}{c}} + 1 + \sqrt{2\lrp{\log\lrp{\frac{1}{c}}-1}}-1 \\
&\qquad \qquad\qquad\qquad\qquad\ \ = \log\lrp{\frac{1}{c}} + \sqrt{2\lrp{\log\lrp{\frac{1}{c}}-1}}\\
&\qquad \qquad\qquad\qquad\qquad\ \ \leq 3\log \frac{1}{c}\end{aligned}$$
Thus by our assumption, $$\begin{aligned}
& x\geq 3\cdot \frac{1}{c}\log\lrp{\frac{1}{c}}\\
\Rightarrow & x\geq \frac{1}{c}\lrp{-W_{-1}(-c)}\end{aligned}$$
then $W_{-1}(-c)$ is defined, so $$\begin{aligned}
&x \geq \frac{1}{c}\max\lrbb{-W_{-1} (-c), 1}\\
\Rightarrow & (-cx) e^{-cx} \geq -c\\
\Rightarrow & x e^{-cx} \leq 1\\
\Rightarrow & \log(x) \leq cx\end{aligned}$$
The first implication is justified as follows: $W_{-1}^{-1}: [-\frac{1}{\epsilon}, \infty) \to (-\infty, -1)$ is monotonically decreasing. Thus it
| 2,242
| 2,356
| 2,280
| 2,119
| 3,516
| 0.771836
|
github_plus_top10pct_by_avg
|
erent entering end exiting vertices from $P_3$ chain) cross $M$ times and have $K$ pairs of sites belonging to different chains and neighboring the crossing sites. Functions $A_i^{(r)}$ and $B_i^{(r)}$ satisfy the following recursion relations $$\begin{aligned}
A'_i&=& \sum_{\cal{N}}
a_i({\cal{N}})\,
A^{N_A}B^{N_B}C^{N_C}
\prod_{j=1}^{4} A_{j}^{N_{A_j}}
\prod_{k=1}^2B_{k}^{N_{B_k}}\,,\quad i=1,2,3,4\>,
\label{eq:RGAi}\\
B'_i&=& \sum_{\cal{N}} b_i({\cal{N}})\,
A^{N_A}B^{N_B}C^{N_C}
\prod_{j=1}^{4} A_{j}^{N_{A_j}}\prod_{k=1}^2
B_{k}^{N_{B_k}}\,,\quad i=1,2\>, \label{eq:RGBi}\end{aligned}$$ where $\cal{N}$ denotes the set of numbers ${\cal{N}}=\{N_{A},N_{B},N_{C},N_{A_1},N_{A_2},N_{A_3,}N_{A_4},
N_{B_1},N_{B_2}\}$, and where we have used the prime symbol as a superscript for $(r+1)$-th restricted partition functions and no indices for the $r$-th order partition functions. The above set of relations (\[eq:RGAi\])–(\[eq:RGBi\]), together with the previously introduced relations (\[eq:RGA\])–(\[eq:RGC\]) for the functions $A$, $B$, and $C$, can be considered as the system of RG equations for the problem under study, with the initial conditions $$\begin{aligned}
&&A^{(0)}=x_3\,,\quad B^{(0)}=x_3^2u^4\,,\quad C^{(0)}=x_2\,,\nonumber\\
&& A_1^{(0)}=x_3x_2w^2\,,\quad
A_2^{(0)}=A_3^{(0)}=x_3x_2wt\,,\quad
A_4^{(0)}=x_3x_2\,, \label{pocuslovi}\\
&& B_1^{(0)}=B_2^{(0)}=x_3^2x_2w^2u^4\,,\nonumber\end{aligned}$$ corresponding to the unit tetrahedron. Because the number of all possible configurations is extremely large, we have been able to find explicit form of the RG equations (\[eq:RGAi\])–(\[eq:RGBi\]) only for $b=2$ and $b=3$ SG fractals (see \[app:CSAWsRG\]). For both cases numerical analysis shows that, for each considered value of $t$, there is a critical line $w_c(u,t)$ dividing the $u-w$ plane into regions where the two polymers are either segregated ($w<w_c(u,t)$) or entangled ($w\geq w_c(u,t)$). Depending on the value of the intra-chain interaction parameter $u$, the area $w\leq w_c(u,t)$ is furt
| 2,243
| 1,026
| 2,057
| 2,308
| null | null |
github_plus_top10pct_by_avg
|
mily in July and October. N = 3 ± SE. Statistics show two‐way [anova]{.smallcaps} with genotype and date as factors *P* = ≤ 0.05
Mixed Population Acid digestible carbohydrate Enzymatic carbohydrate release \% Digestibility
------------------ ------------------------------ -------------------------------- ------------------ -------------- -------------- -------------- -------------- -------------- ------------ ------------ ------------- -------------
Sin 1 432.9 ± 4.1 429.2 ± 17.8 275.2 ± 10.5 290.4 ± 9.7 286.2 ± 8.7 273.0 ± 3.6 257.6 ± 8.4 308.5 ± 1.7 66.1 ± 1.5 64.1 ± 3.7 93.7 ± 1.2 106.6 ± 3.1
Sin 2 429.7 ± 11.8 427.6 ± 9.5 259.0 ± 2.9 299.9 ± 5.6 315.7 ± 6.0 291.8 ± 13.5 256.0 ± 6.6 306.5 ± 13.9 73.5 ± 0.6 68.5 ± 4.4 98.8 ± 1.5 102.1 ± 3.5
Sin 3 410.0 ± 16.9 450.9 ± 8.5 268.7 ± 11.2 308.6 ± 5.3 311.1 ± 2.4 267.5 ± 8.5 250.5 ± 13.1 303.6 ± 2.4 76.3 ± 3.4 59.3 ± 1.0 93.1 ± 1.0 98.4 ± 1.0
Sin 4 422.0 ± 11.8 399.6 ± 11.0 278.7 ± 4.3 295.7 ± 6.7 287.9 ± 0.2 265.2 ± 4.1 252.0 ± 4.6 296.1 ± 5.3 68.4 ± 1.9 66.5 ± 1.7 90.5 ± 2.5 100.2 ± 0.7
Sin 5 397.1 ± 15.6 404.5 ± 12.2 279.2 ± 5.0 291.1 ± 3.5 308.2 ± 6.0 294.7 ± 11.7 262.3 ± 12.2 285.1 ± 8.7 78.0 ± 3.7 72.8 ± 1.3 93.9 ± 3.8 98.0 ± 3.0
Sin 6 410.7 ± 4.5 432.8 ± 20.9 233.8 ± 3.0 291.7
| 2,244
| 4,807
| 1,556
| 1,855
| null | null |
github_plus_top10pct_by_avg
|
\]](oke "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](contactw0 "fig:")
------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------
For completeness, we rewrite here the LO EFT already presented in Ref. [@PPJ11], and then build the NLO contributions in the next section.
At tree level, the transition potential $\Lambda N\to NN$ involves the LO contact terms, and $\pi$ and $K$ exchanges, as depicted in Fig. \[fig:loc\]. First, the contact interaction can be written as the most general Lorentz invariant potential with no derivatives. The four-fermion (4P) interaction in momentum space at leading order (in units of $G_F$) is $$\begin{aligned}
V_{4P} ({\vec q} \, ) &=&
C_0^0 + C_0^1 \; {\vec \sigma}_1
{\vec \sigma}_2 \,,\label{eq:vlo}\end{aligned}$$ where $C_0^0$ and $C_0^1$ are low energy constants which need to be fitted by direct comparison to experimental data. In Ref. [@PPJ11] we presented several sets of values which were to a large extent compatible with the scarce data on hypernuclear decay.
The potentials for the one pion and one kaon exchanges, as functions of transferred momentum ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, read, respectively [@PRB97] $$\begin{aligned}
{V_{\pi}} ({\vec q}\,) =&
\nonumber
-\frac{G_F m_\pi^2g_{NN\pi}}{2 M_N}
\left(
A_\pi - \frac{B_\pi}{2 {\overline{M}}}{\vec \sigma}_1 \, {\vec q} \,
\right)
\frac{{\vec \sigma}_2 \, {\vec q}\,}{-q_0^2+{\vec q}^{\; 2}+m_\pi^2} \,
\\&\times {\vec{\tau}_1}\cdot{\vec{\tau}_2}{\rm ,}
\label{eq:pion}\\
{V_{K}} ({\vec q}\,) =&
-\frac{G_F m_\pi^2g_{\Lambda N K}}{2{\overline{M}}}
\left(
\hat{A} - \frac{\hat{B}}{2 M_N}{\vec \sigma}_1 \, {\vec q} \,
\right)
\nonumber\\&\times\frac{{\vec \sigma}_2 \, {\vec q}\,} {-q_0^2+{\vec q}^{\; 2}+m_K^2} \,
{\rm ,}
\label{eq:kaon}\end{aligned}$$ where
| 2,245
| 978
| 557
| 2,426
| null | null |
github_plus_top10pct_by_avg
|
oung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;\star;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};\star,;2;3;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;\star;\star)
&\qquad
8.\quad&
\gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;3;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;3;\star)\end{aligned}$$ In each case, the $\star$s represent the numbers from $4$ to $a$. Note that in each of these tableaux, the entries $4,\dots,a$ must all occur in different columns (the assumption $v\ls b+1$ means that any column of length at least two has a $1$ at the top). So we can consider the homomorphisms ${\psi_{d,1}}$ for $d\gs3$, and repeat the argument used in the last paragraph of the proof of Proposition \[cdhomdim1\], to show that if $T,T'$ are two tableaux which have their $1$s, $2$s and $3$s in the same positions, then ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ occur with the same coefficient in $\theta$. Hence $\theta$ is a linear combination of the homomorphisms $\tau_1,\dots,\tau_8$, where $\tau_i$ is the sum of all homomorphisms ${\hat\Theta_{T}}$ for $T$ of type $i$.
Once more we can consider ${\psi_{2,1}}\circ\theta$: when we compute ${\psi_{2,1}}\circ\tau_5$, we obtain (in addition to some other semistandard tableaux) the sum of the semistandard tableaux of the form $$\gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;2;2;\star_2{{\begin{tikzpicture}[baseline=
| 2,246
| 1,829
| 814
| 2,347
| 308
| 0.814095
|
github_plus_top10pct_by_avg
|
he optical potential at 1 GeV nucleon energy has been constrained to 50 MeV. With the increase of the effective nucleon mass from DD-ME1 to D$^3$C and D$^3$C$^*$, we also note the corresponding decrease of the nuclear matter compression modulus $K_{\infty}$ . This correlation between $K_{\infty}$ and $m^*$ is also well known in non-relativistic Skyrme effective interactions [@Cha.97].
DD-ME1 D$^{3}$C D$^3$C
------------------------------------- -------- ---------- ----------
$\varrho_{\rm sat}$ \[fm${}^{-3}$\] 0.152 0.151 0.152
$a_{V}$ \[MeV\] -16.20 -15.98 -16.30
$a_{4}$ \[MeV\] 33.1 31.9 33.0
$K_{\infty}$ \[MeV\] 244.5 232.5 224.9
$m_{D}/m$ 0.58 0.54 0.57
$m^{*}/m$ 0.66 0.71 0.79
$\Gamma_{S}$ 0.0 -21.632 -146.089
$\Gamma_{V}$ 0.0 302.188 180.889
: \[TabA\] Properties of symmetric nuclear matter at saturation density calculated with the models DD-ME1, D$^{3}$C, and D$^3$C.
In Fig. \[Fig1\] we display the neutron and proton single-particle levels in $^{132}$Sn calculated in the relativistic mean-field model with the DD-ME1, D$^{3}$C, and D$^3$C effective interactions, in comparison with available data for the levels close to the Fermi surface [@Isa.02]. Compared to the DD-ME1 interaction, the enhancement of the effective mass in D$^{3}$C and D$^3$C results in the increase of the density of states around the Fermi surface, and the calculated spectra are in much better agreement with the empirical energy spacings.
![(Color online) Neutron (left panel) and proton (right panel) single-particle levels in $^{132}\textrm{Sn}$ calculated with the DD-ME1 (a), D$^{3}$C (b) and D$^3$C (c) interactions, compared to experimental levels (d) [@Isa.02].[]{data-label="Fig1"}](Fig1.eps)
In the next step the thr
| 2,247
| 1,331
| 2,656
| 2,466
| null | null |
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|
difference plots for bagging. Both subset selection methods improve when utilising multiple subset selection. In the case when class-balanced selection is used, as was observed for single nested dichotomies, the average ranks across all datasets closely correspond to the integer values, showing that increasing the number of subsets evaluated consistently improves performance. For random-pair selection, a more constrained subset selection method, each value of $\lambda > 1$ is statistically equivalent and superior to the single subset case.
The critical difference plots in Figure \[fig:cd\_boosting\] (top) show boosted nested dichotomies are significantly improved by increasing the number of subsets sufficiently when class-balanced nested dichotomies are used. Results are less consistent for random-pair selection, with few significant results in either direction. This is reflected in the critical differences plot (Fig. \[fig:cd\_boosting\], bottom), which shows single subset evaluation statistically equivalent to multiple subset selection for all values of $\lambda$, with $\lambda = 7$ performing markedly worse on average. As RMSE is based on probability estimates, this may be in part due to poor probability calibration, which is known to affect boosted ensembles [@niculescu2005predicting] and nested dichotomies [@leathart2018calibration].
\
Conclusion\[sec:conclusion\]
============================
Multiple subset selection in nested dichotomies can improve predictive performance while retaining the particular advantages of the subset selection method employed. We present an analysis of the effect of multiple subset selection on expected RMSE and show empirically in our experiments that adopting our technique can improve predictive performance, at the cost of a constant factor in training time.
The results of our experiments suggest that for class-balanced selection, performance can be consistently improved significantly by utilising multiple subset evaluation. For random-pair selection, $\lambda=3$ yiel
| 2,248
| 1,472
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| 2,151
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|
/g) \cosh (g\tau) $ and $y = z = 0$ in Eqs.(\[Tdef\]) and (\[Xdef\]), we have $$\begin{aligned}
-T^2+X^2
& = \frac{2}{g^2} \bigg\{ 1 - \sqrt{1-{c^{\prime}}^2} \sqrt{1-{c^{\prime\prime}}^2} \cosh \bigl[ g(\tau-\tau^{\prime})-i(\alpha_{c^{\prime\prime}}+\alpha_{c^{\prime}} ) \bigr]
\notag
\\
& \hspace{8ex}
- \left( \frac{{c^\prime}^2 + {c^{\prime\prime}}^2 }{2} \right)
\bigg\}
\ ,
\label{TXRindler}\end{aligned}$$ where $c^{\prime\prime}=i|\cos\Theta_{0}| g \epsilon^{\prime\prime}$, $c^{\prime}=i|\cos\Theta_{0}| g \epsilon^{\prime}$, $\cos \alpha_{c^{\prime\prime}} = 1/\sqrt{1-{c^{\prime\prime}}^2}$ and $\cos \alpha_{c^{\prime}} = 1/\sqrt{1-{c^{\prime}}^2}$. As is expected for a stationary trajectory, the above expression depends on the proper time $\tau$ and $\tau^\prime$ through their difference $\tau - \tau^\prime$ only. Further substituting Eq.(\[TXRindler\]) in Eqs.(\[wfinalcompact\]) and (\[angtransitionrate\]), we get $${\dot {\cal F}}_{\Theta_0}(\omega) =\frac{1}{16} \frac{\partial}{\partial \epsilon^{\prime}} \frac{\partial}{\partial \epsilon^{\prime\prime}} \bigg\{ 2 \operatorname{Re}\int_0^{\infty} ds \, e^{-i\omega s} \frac{4\pi\epsilon^{\prime\prime} \epsilon^{\prime}}{-T^{2}+X^{2}} \bigg\}_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{angFinter}$$ Identifying the symmetry in the integrand under the simultaneous exchange of $s \rightarrow -s$ and $i \rightarrow -i$ using Eq.(\[TXRindler\]), we can express the integral as a contour integral over the full real line $s \rightarrow (-\infty, \infty)$. Further, $(- T^2 + X^2)$ has the periodicity in $s \rightarrow s + 2\pi i /g$. One can then close the contour at $s + 2\pi i /g$ and evaluate the residue at the poles, to get $$\begin{aligned}
{\dot {\cal F}}_{\Theta_0}(\omega)
&=
\frac{\partial}{\partial \epsilon^{\prime}} \frac{\partial}{\partial \epsilon^{\prime\prime}} \bigg\{\frac{\pi^{2}g\epsilon^{\prime} \epsilon^{\prime\prime}e^{\frac{\omega}{g} \left[ \tan^{-1} \left( |\cos\Theta_{0}|g\epsilon^{\prime} \right)+ \tan^{
| 2,249
| 3,093
| 2,731
| 2,258
| null | null |
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|
messageProperty: 'message'
},
writer: {
type: 'json',
encode: true,
root: 'details'
},
fields: ['id','document', 'date'],
},
});`.
In both the cases i am calling the same action and getting the same JSON.
Can anyone help me out here that how can I declare another store object and get the value from the existing 'store' so that i should not have to load the same data for 2 times.
Regards : Dev
A:
You can make the first store have autoload set to false. Then in the second store use the load event to inject the records into the first using that stores active proxy data:
var store2 = Ext.create('Ext.data.Store', {
model: 'Writer.Document',
autoLoad: true,
proxy: {
type: 'ajax',
url : 'findPatientRecordAction',
reader: {
type: 'json',
successProperty: 'success',
root: 'details',
messageProperty: 'message'
},
writer: {
type: 'json',
encode: true,
root: 'details'
},
fields: ['id','document', 'date'],
},
listeners: {
load: function(store, records, successful, eOpts){
if(successful)
{
store1.loadData(store.proxy.reader.rawData.data);
}
}
}
});
Q:
Replace function not replacing
I followed some documentation to use the JavaScript replace function and it's not changing anything. No errors are thrown. Any idea what I'm doing wrong? The variable is retrieved from XML - maybe it needs to be cast as a string or something?
for (var i = 0, iln = projects.length; i < iln; i++){
var thumb = projects[i].get('thumb');
thumb.replace("200.jpg", "640.jpg");
console.log(thumb) //200.jpg not replaced
}
The full thumb value should look like this:
http://b.vimeocdn.com/ts/160/895/160895498_200.jpg
| 2,250
| 2,431
| 67
| 1,995
| 15
| 0.839742
|
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|
7 15.28 40 repressor
putative fibronectin/fibrinogen binding
\* 1206348 A:1 T:6 G:145 G:37 7 14.71 25 protein
A:153 (T:1)
\* 2262790 G:6 A:37 7 13.14 29 cation efflux family protein
II\. Heterogeneity sites that did not pass the SNPfilter but have an average per base Phred value greater than 13
| 2,251
| 6,189
| 318
| 816
| null | null |
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|
ms and by the indicator of $|X_i-t|\le h_{2,n}B$). The three terms from $\delta_3$ involve, instead of $\delta_4$, respectively $\delta_2$, $\delta^2$ and $\delta_2\delta$ (see (\[e1\])-(\[e3\])). We have from (\[delta\]), (\[classic1\]) and (\[classic2\]) that $$\sup_{t\in D_r^\varepsilon}\delta^2_n=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ that the same is true for $\delta_4$ by (\[delta4\]), and, moreover, by (\[delta2\]), $$|\delta_2(t,X_i)|I(|t-X_i|\le h_{2,n}B)\le \frac{\|K''\|_\infty}{2}B^2\|f\|_\infty\delta^2(X_i)=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ then, if we define $\tilde Q_i(t)$ by $$\varepsilon_3(t,h_{1,n},h_{2,n})=\frac{1}{nh_{2,n}}\sum_{i=1}^n\tilde Q_i(t),$$ we have $$\sup_{t\in D_r}|\tilde Q_i(t)|=O_{\rm a.s.}\left(n^{-7/9}\log n\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ and therefore, $$\sup_{t\in D_r}\left|\epsilon_3(t;h_{1,n}, h_{2,n})\right|=O_{\rm a.s.}( h_{2,n}^{-1}n^{-7/9}\log n)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C,$$ proving the lemma for $\varepsilon_3$ as $h_{2,n}^{-1}n^{-7/9}\log n<< n^{-4/9}$.
\[eps1-t\] Let $$\begin{aligned}
\label{tien}
&&T(t; h_{1,n}, h_{2,n})=\\
&&\frac{1}{nh_{1,n}h_{2,n}}\sum_{i=1}^{n}
E_{X}\Big[f^{-1/2}(X)\Big\{K\Big(\frac{X-X_i}{h_{1,n}}\Big)-E_YK\Big(\frac{X-Y}{h_{1,n}}\Big)\Big\}
L\Big(\frac{t-X}{h_{2,n}}f^{1/2}(X)\Big)I(|t-X|\le h_{2,n}B)\Big],\notag\end{aligned}$$ where $L(z)=K(z)+zK^\prime(z)$. Then, $$\sup_{t\in D_r}\left|\varepsilon_1(t,h_{1,n},h_{2,n})-T(t;h_{1,n},h_{2,n})\right|=o_{\rm a.s.}(n^{-4/9})\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$
Given a function $H$ of two variables, and two i.i.d. random variables $X$ and $Y$ such that $H(X,Y)$ is integrable, we recall that the second order Hoeffding projection of $H(X,Y)$ is $$\pi_2(H)(X,Y)=H(X,Y)-E_XH(X,Y)-E_YH(X,Y)+EH.$$ We also recall the $U$-statistic notation $$U_n(H)=\frac{1}{n(n-1)}\sum_{1\le i\ne j\le n}H(X_i,X_j),$$ where the variables $X_i$ are i.i.d. Set $$H_t(X,Y):=L\left(\frac{t-X}
| 2,252
| 920
| 2,033
| 2,197
| null | null |
github_plus_top10pct_by_avg
|
t $$(\bar{L} \bar{\beta})_{i j} = \psi^4 (L \beta)_{i j} \; ; \qquad
(\bar{L} \beta)^{i j} = \psi^{-4} (L \beta)^{i j} \; .$$
Next, we note, perhaps surprisingly, that the lapse function $\bar{N}$ has essential non-trivial conformal behavior. Furthermore, this is [*the*]{} new element in the IVP analysis. In [@Teitel; @Ashtekar; @AAJWY98; @YorkFest] the “slicing function” $\alpha(t,x) > 0$ replaces the lapse function $\bar{N}$, $$\bar{N} = \bar{g}^{1/2} \alpha \; ,
\label{Eq:slicingfunction}$$ with important improvements then appearing in Teitelboim’s path integral [@Teitel], in Ashtekar’s new variables program [@Ashtekar], in the canonical action principle [@AAJWY98; @YorkFest], and in making clear the role of the contracted Bianchi identities [@AAJWY98; @YorkFest]. The lapse is now a dynamical variable because of the $\bar{g}^{1/2}$ factor [@Ashtekar; @AAJWY98; @YorkFest]. Furthermore, in the construction of mathematically hyperbolic systems for the Einstein [*evolution*]{} equations with explicitly physical characteristics, and only such, (for example [@CBY97; @ACBY97; @CBYAnew]), it turns out to be $\alpha(t,x)$, not the usual lapse function $\bar{N}$, that can be freely specified. This use of $\bar{N} = \bar{g}^{1/2} \alpha$ is Choquet-Bruhat’s “algebraic gauge” [@CBRug; @CBY95] with, in general, a “gauge source” [@Friedrich]. Actually, $\bar{N} = \bar{g}^{1/2} \alpha$ should be seen as a change of variables in which one specifies freely $\alpha(t,x)>0$ rather than $N$. For these reasons, we conclude that $\alpha$ is not a dynamical variable, $\bar{\alpha} = \alpha$. For the lapse, we have from (\[Eq:slicingfunction\]), with $N$ given and positive, $$\bar{N} = \psi^6 N \; .$$
Finally, we recall from the standard initial value problem for $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$, that the separation of the extrinsic curvature into (its irreducible) trace and traceless parts is fundamental, as it is here, and that $\bar{K}=K$ [@York72]: the trace is not transformed even
| 2,253
| 3,309
| 3,391
| 1,997
| 1,495
| 0.788827
|
github_plus_top10pct_by_avg
|
dagger} A W \right\}_{k K}
\left\{ W^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m L}
\biggr\}.
\label{P-beta-alpha-W4-H3-First} \end{aligned}$$
$$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-2nd}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [3]_\text{Second}
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
- \sum_{n}
\sum_{L k}
\frac{ 1 }{ \Delta_{L} - h_{k} }
\left[
(ix) e^{- i ( \Delta_{L} - h_{n} ) x}
+
\frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) }
\right]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A W \right\}_{L L}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\nonumber \\
&+&
\sum_{n}
\sum_{L k}
\sum_{K \neq L}
\frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) ( \Delta_{L} - h_{k} ) ( \Delta_{K} - h_{k} ) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right)
e^{- i ( \Delta_{L} - h_{n} ) x}
- \left( \Delta_{L} - h_{k} \right)
e^{- i ( \Delta_{K} - h_{n} ) x}
- \left( \Delta_{K} - \Delta_{L} \right)
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A W \right\}_{L K}
\left\{ W^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&-&
\sum_{n}
\sum_{L k}
\frac{ 1 }{ (\Delta_{L} - h_{k})^2 }
\biggl[
(ix) \left( e^{- i ( h_{k} - h_{n} ) x} + e^{- i ( \Delta_{L} - h_{n} ) x} \right)
+ 2 \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) }
\biggr]
\nonumber \\
&\times&
W_{\alpha L} (UX)^*_{\beta k}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\nonumber \\
&+&
\sum_{n}
\sum_{L k}
\sum_{m \neq k}
\biggl[
- \frac{ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} )( \Delta_{L} - h_{m} ) }
+ \frac{ 1 }{ ( h_{
| 2,254
| 3,522
| 2,304
| 2,202
| null | null |
github_plus_top10pct_by_avg
|
0$ since the propagator is of order $f^0$. Now let us assume that the statement has been proven for $p <n+2$, and consider a Feynman diagram with $n+2$ external lines. We isolate $m$ of these external legs that are contracted on the same vertex with $m+1$ legs. This piece is of order $f^{m-1}$. The other piece of the Feynman diagram has $n+2-m+1$ external lines, and by induction is of order $f^{n-m+1}$. Thus the result is of order $f^{n}$, and the proof is completed. We deduce that: X\^a(z) X\^b(w) :X\^[a\_1]{}:X\^[a\_2]{}...:X\^[a\_[p-1]{}]{}X\^[a\_p]{}:...::(x) \_[connected]{} = (f\^p).Since two-point functions of (composites of) the fields $X^a$ behave at least as $\mathcal{O}(f^0)$, we can now combine the previous result with equation to evaluate the order of the term in the current OPE under consideration[^5]: = ... + (f\^p) ::...::...::(w) +...Given that $f c_+ = \mathcal{O}(f^{-1})$, we obtain: j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) = ... + (f\^[2p-2]{}) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_[p-1]{}]{}\_[L,z]{}j\^[a\_p]{}\_[L,z]{}:...::(w) +...This is a property we repeatedly confirm as well as use in the bulk of the paper.
The semi-classical behavior of the current-primary OPE {#the-semi-classical-behavior-of-the-current-primary-ope .unnumbered}
------------------------------------------------------
We can perform a similar analysis to determine the behavior of the terms in the current-primary OPE at large radius. Let us consider a primary field $\phi$. We assume that all the terms that appear in the OPE between a left current and this primary field are composite operators including an arbitrary number of left currents and one field $\phi$ only. This is the case at the WZW point. Then by continuously deforming the OPEs away from the WZW point, this is the case over the whole moduli space of the theory. Let us isolate one term in the OPE between the left current $j^a_{L,z}$ and the primary field $\phi$: j\^a\_[L,z]{}(z) (w) = ... + [B\^[a]{}]{}\_[a\_p a\_[p-1]{}... a\_1]{}(z-w, |z - |w) :j\^[a\
| 2,255
| 1,260
| 3,218
| 2,291
| 2,602
| 0.778526
|
github_plus_top10pct_by_avg
|
have closed structure and for $k=0.5$ a variety of intermediate structures can be found. These observations are in good agreement with our results for colloid-droplet radial distribution functions, as discussed above.
{width="5.9cm"} {width="5.9cm"} {width="5.9cm"}
Symmetric wetting properties and asymmetric sizes {#s:fluid2}
-------------------------------------------------
We next investigate the cluster formation of colloidal dumbbells built with spheres of different diameters, ${\sigma_{1}=1.5\sigma_{2}}$ and $\sigma_{1}=2.0\sigma_{2}$ but equal wetting properties, which are obtained by the interfacial tension $\gamma_1=\gamma_2\equiv\gamma$. We investigate the setting values $\gamma=$10, 40 and $100k_{\textrm{B}}T/\sigma _{2}^{2}$. We note that a size asymmetry between the colloids forming the dumbbells causes an asymmetry in colloid-droplet adsorption energies \[Eqs. (\[eqn:phicd1\]) and (\[eqn:phicd2\])\]. For this reason, the structures found in this case are the same as those shown in Fig. \[fig:cluster\] for asymmetric wetting properties.
We analyze the size distribution of the clusters. Figure \[fig:hist2\] shows stacked histograms of the number of clusters $N_{n_{c}}$ with $n_c$ colloids for different values of $\gamma$ and two different size ratios. For the case $\sigma_{1}=1.5\sigma_{2}$ and $\gamma=10k_{\textrm{B}}T/\sigma _{2}^{2}$ \[Fig. \[fig:hist2\](a) and Fig. \[fig:hist2\](d)\], all clusters have open structures with $n_b=3$. In addition, we do not find cluster with a high $n_c$ \[Fig. \[fig:hist2\](a) and (d)\] because the Yukawa repulsion and the thermal fluctuations dominate over the adsorption energy between colloids and droplets that keeps the colloids in a compact arrangement. On the other hand, in the case of $\gamma=40k_{\textrm{B}}T/\sigma _{2}^{2}$ \[Fig. \[fig:hist2\](b)\] we observe many clusters of bond numbers $n_b$ in the range $3-6$, corresponding to intermediate structures. Finally, when $\gamma=100k_{\textrm{B}}T/\sigm
| 2,256
| 1,210
| 2,976
| 2,502
| 2,116
| 0.782605
|
github_plus_top10pct_by_avg
|
^3 \det H_0.\end{aligned}$$ The adjugate of $H_{0}$ is defined as $\text{Adj} H_{0} \equiv (H_{0})^{-1} \text{det} H_{0}$. Notice that $T$, $A$ and $D$ are invariant under unitary transformation of $ H_0\to K H_0 K^{\dagger}$ with $K$ any unitary matrix and so are $\lambda_i$.
Following the notation in [@Kimura:2002wd] we define $p_{ij}$ and $q_{ij}$ as ($i,j=1,2,3$) $$\begin{aligned}
\frac{ p_{ij} }{ 2E } &\equiv& \left( H_{0} \right)_{ij},
\nonumber \\
\frac{ q_{ij} }{ (2E)^2 } &\equiv& \left( \text{Adj} H_{0} \right)_{ij}.
\label{pq-def}\end{aligned}$$ Notice that $p_{ij}$ and $q_{ij}$ are written only by the known (or given) quantities. Then, the equations $$\begin{aligned}
\left( H_{d} \right)_{ij} &=& \frac{ p_{ij} }{ 2E },
\nonumber \\
\left( \text{Adj} H_{d} \right)_{ij} &=& \frac{ q_{ij} }{ (2E)^2 },
\label{KTY-eq}\end{aligned}$$ together with unitarity of $X$, become the equations to determine $X X^{\dagger}$: $$\begin{aligned}
X_{i 1} X_{j 1}^* + X_{i 2} X_{j 2}^* + X_{i 3} X_{j 3}^* &=& \delta_{ij},
\nonumber \\
\lambda_{1} X_{i 1} X_{j 1}^* + \lambda_{2} X_{i 2} X_{j 2}^* + \lambda_{3}X_{i 3} X_{j 3}^* &=& p_{ij},
\nonumber \\
\lambda_{2} \lambda_{3} X_{i 1} X_{j 1}^* + \lambda_{3} \lambda_{1} X_{i 2} X_{j 2}^* + \lambda_{1} \lambda_{2} X_{i 3} X_{j 3}^* &=& q_{ij}.
\label{KTY-eq-explicit}\end{aligned}$$ They lead to the solution ($k=1,2,3$) $$\begin{aligned}
X_{i k} X_{j k}^* =
\frac{ q_{ij} + p_{ij} \lambda_{k} - \delta_{ij} \lambda_{k} ( \lambda_{l} + \lambda_{m} ) }{ (\lambda_{l} -\lambda_{k} ) (\lambda_{m} -\lambda_{k} ) },
\label{KTY-eq-solution}\end{aligned}$$ where $k,l,m$ is cyclic, and sum over $k$ is not implied in (\[KTY-eq-solution\]).
Therefore, to zeroth-order in $W$ expansion, the $S$ matrix elements are given by $$\begin{aligned}
S_{\alpha \beta}^{(0)} &=&
\sum_{k}
\left(
\sum_{i, j}
U_{\alpha i}
\left[ q_{ij} + p_{ij} \lambda_{k} - \delta_{ij} \lambda_{k} ( \lambda_{l} + \lambda_{m} ) \right]
U^*_{\beta j}
\right)
\frac{ e^{-i h_{k} x} }{ (\lambda_{l} -\lambda_{k} )
| 2,257
| 2,962
| 2,270
| 2,204
| 4,144
| 0.767793
|
github_plus_top10pct_by_avg
|
\overset{\delta}{\rightarrow} X)$ is given by (the isomorphism class of ) the pullback along $\beta$ and $\gamma$. $$\xymatrix{
& & P\ar@{..>}[dr]\ar@{..>}[dl] & & \\
& Y\ar[dl]_{\alpha}\ar[dr]^{\beta} & & Z\ar[dl]_{\gamma}\ar[dr]^{\delta} & \\
X & & X & &X
}$$ The identity of this $R$-algebra is (the isomorphism class of )$\xymatrix{&X\ar@{=}[rd]\ar@{=}[dl]&\\X& &X }$
The usual Burnside algebra of a finite group $H$, previously denoted by $RB(H)$ is isomorphic to the Burnside functor evaluated at the $H$-set $H/H$. In the Mackey functors’ langage the first notation correspond to Green’s notation and the second one correspond to Dress’ notation. In the rest of the paper the notation $RB(H)$ will always be used for the usual Burnside algebra of the group $H$. If we want to speak about the Burnside functor evaluated at the $H$-set $H/1$, we will write $RB(H/1)$.
Another definition of Mackey functors was given by Thévenaz and Webb in [@tw].
The Mackey algebra $\mu_{R}(G)$ for $G$ over $R$ is the unital associative algebra with generators $t_{H}^{K}$, $r^{K}_{H}$ and $c_{g,H}$ for $H\leqslant K\leqslant G$ and $g\in G$, with the following relations:
- $\sum_{H\leqslant G}t^{H}_{H}=1_{\mu_{R}(G)}$.
- $t^{H}_{H}=r^{H}_{H}=c_{h,H}$ for $H\leqslant G$ and $h\in H$.
- $t^{L}_{K}t_{H}^{K}=t^{L}_{H}$, $r^{K}_{H}r^{L}_{K}=r^{L}_{H}$ for $H\subseteq K\subseteq L$.
- $c_{g',{^{g}H}}c_{g,H}=c_{g'g,H}$, for $H\leqslant G$ and $g,g'\in G$.
- $t^{{^{g}K}}_{{^{g}H}}c_{g,H}=c_{g,K}t^{K}_{H}$ and $r^{{^{g}K}}_{{^{g}H}}c_{g,K}=c_{g,H}r^{K}_{H}$, $H\leqslant K$, $g\in G$.
- $r^{H}_{L}t^{H}_{K}=\sum_{h\in [L\backslash H / K]} t^{L}_{L\cap {^{h} K}} c_{h, L^{h} \cap H} r^{K}_{L^{h}\cap H}$ for $L\leqslant H \geqslant K$.
- All the other products of generators are zero.
\[basis\] The Mackey algebra is a free $R$-module, of finite rank independent of $R$. The set of elements $t^{H}_{K}xr^{L}_{K^{x}}$, where $H$ and $L$ are subgroups of $G$, where $x\in [H\backslash G/L]$, and $K$ is a subgroup of $H{\cap~{\
| 2,258
| 2,806
| 1,870
| 1,963
| null | null |
github_plus_top10pct_by_avg
|
epair protein MutS
`Sbjct: 754 ASAGKKSSISN` `764`
`gi 87161934 ref` `619` `QQANVELSPTSD`
`Q+ANVELSPTSD`
`QKANVELSPTSD`
`SRR022865_10478` `2` `cagaggtacatg`
950365 NONSYN A:5 C:113 C:37 `aacatatgccca`
`ggttcgataaat` exonuclease RexB
`Query: 1 FK*YFKQFEENY` `36`
`FK YFKQFEENY`
`Sbjct: 223 FKSYFKQFEENY` `234`
2512836 TRUNC C:7 G:115 G:37
| 2,259
| 4,668
| 2,271
| 1,681
| null | null |
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|
ulated periods fitted with our observed ones in Table \[table:g207params\]. We also list the $\sigma_\mathrm{{rms}}$ values of the models. The $T_{\rmn{eff}}=12\,000$K solutions are in agreement with the spectroscopic value. The $T_{\rmn{eff}}=12\,400$K model seems somewhat too hot comparing to the $\sim12\,100$K spectroscopic temperature, but considering that the uncertainties of both values are estimated to be around $200$K, this model still not contradicts to the observations. The $0.855-0.870\,M_*$ stellar masses are also close to the value derived by spectroscopy, considering its uncertainty. Summing it up, we can find models with stellar parameters and periods close to the observed values even if we assume that at least half of the modes is $l=1$, including the dominant mode.
{width="17.5cm"}
{width="17.5cm"}
[llrrrr]{} $T_{\rmn{eff}}$ (K) & & & & & Reference\
\
12078$\pm$200 & 0.84$\pm$0.03 & & & & @2011ApJ...743..138G\
& & & & & @2013AA...559A.104T\
\
12000 & 0.815 & 2.0 & 8.5 & 259.0, 292.0, 317.3, 557.3, 740.7, 787.5$^\star$ & @2009MNRAS.396.1709C\
11700 & 0.530 & 3.5 & 6.5 & 259.0, 292.0, 317.3, 557.3, 740.7, 787.5$^\star$ & @2009MNRAS.396.1709C\
12030 & 0.837 & 2.5 & 6–7 & 259.1 (1,4), 292.0 (2,10), 318.0 (1,5),& @2012MNRAS.420.1462R [@2013ApJ...779...58R]\
& & & & 557.4 (1,12), 740.4 (1,17) &\
& & & & &\
& & &\
& & & & &\
12000 (1.06s) & 0.870 & 2.0 & 4.0 & 291.0 (1,7), 595.5 (2,32), 195.8 (2,9), &\
& & & & 625.6 (2,34), 129.0 (2,5), 319.7 (2,16), &\
& & & & 305.4 (2,13), 558.6 (2,28) &\
12000 (1.61s) & 0.865 & 2.0 & 6.0 & 290.6 (1,5), 594.5 (1,14), 193.1 (2,6), &\
& & & & 623.9 (1,15), 130.4 (2,3), 316.6 (1,6), &\
& & & & 306.2 (2,12), 555.1 (2,24) &\
12400 (1.50s) & 0.855 & 2.0 & 4.6 & 290.5 (1,6), 594.7 (1,16), 194.0 (1,3), &\
& & & & 624.7 (1,17), 132.3 (2,4), 318.7 (2,14), &\
& & & & 304.6 (2,13), 557.0 (2,27) &\
\
\
Best-fitting models for LP 133-144
----------------------------------
The model with the lowest $\sigma_\mathrm{{rms}}$ ($0.46
| 2,260
| 1,132
| 1,788
| 2,218
| null | null |
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|
D. P., [Strand]{}, N. E., [Hall]{}, P. B., [Blomquist]{}, J. A., & [York]{}, D. G. 2008, , 678, 635
, A. D. [et al.]{} 2006, , 638, 622
, T., [Johansson]{}, P. H., & [Ostriker]{}, J. P. 2009, , 699, L178
, T., [Johansson]{}, P. H., [Ostriker]{}, J. P., & [Efstathiou]{}, G. 2007, , 658, 710
, T., [Khochfar]{}, S., & [Burkert]{}, A. 2006, , 636, L81
, P. B., [van den Bergh]{}, S., & [Abraham]{}, R. G. 2010, , 715, 606
, J. A. [et al.]{} 2012, in prep
, C., [Treu]{}, T., [Auger]{}, M. W., & [Bolton]{}, A. S. 2009, , 706, L86
, J. B. & [Gunn]{}, J. E. 1983, , 266, 713
, C. [et al.]{} 2006, , 640, 92
—. 2011, in prep
, C. Y., [Ho]{}, L. C., [Impey]{}, C. D., & [Rix]{}, H.-W. 2002, , 124, 266
—. 2010, , 139, 2097
, C., [Magliocchetti]{}, M., & [Norberg]{}, P. 2004, , 355, 1010
, W. H., [Flannery]{}, B. P., & [Teukolsky]{}, S. A. 1986, [Numerical recipes. The art of scientific computing]{}, ed. [Press, W. H., Flannery, B. P., & Teukolsky, S. A.]{}
, A. [et al.]{} 2011, ArXiv e-prints
, A. [et al.]{} 2010, , 709, 512
, K. H. R., [Prochaska]{}, J. X., [M[é]{}nard]{}, B., [Murray]{}, N., [Kasen]{}, D., [Koo]{}, D. C., & [Phillips]{}, A. C. 2011, , 728, 55
, K. H. R., [Weiner]{}, B. J., [Koo]{}, D. C., [Martin]{}, C. L., [Prochaska]{}, J. X., [Coil]{}, A. L., & [Newman]{}, J. A. 2010, , 719, 1503
, G. [et al.]{} 2009, , 700, 1559
, D. S., [Veilleux]{}, S., & [Sanders]{}, D. B. 2005, , 632, 751
, R. P. [et al.]{} 2006, , 651, L93
, W., [Bahcall]{}, N., [M[é]{}nard]{}, B., & [Richards]{}, G. 2006, , 643, 68
, J. L. 1968, [Atlas de galaxias australes]{}, ed. [Sersic, J. L.]{}
, J. D. [et al.]{} 2009, , 695, 171
, V., [Di Matteo]{}, T., & [Hernquist]{}, L. 2005, , 620, L79
, E. N., [Franx]{}, M., [Glazebrook]{}, K., [Brinchmann]{}, J., [van der Wel]{}, A., & [van Dokkum]{}, P. G. 2010, , 720, 723
, S., [Franx]{}, M., [van Dokkum]{}, P., [F[ö]{}rster Schreiber]{}, N. M., [Labbe]{}, I., [Wuyts]{}, S., & [Marchesini]{}, D. 2009, , 705, 255
| 2,261
| 767
| 3,720
| 2,461
| null | null |
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|
ga}-\xi_{k_2-k}) \nonumber \\ &\times&
(i\tilde{\omega}_1+i\tilde{\omega}_2-i\tilde{\omega}_3-i\tilde{\omega}-\xi_{k_1+k_2-k_3-k})
\nonumber \\ &\times&
(-i\tilde{\omega}_2+i\tilde{\omega}_3+i\tilde{\omega}-\xi_{-k_2+k_3+k})
\Big]^{-1}\end{aligned}$$ with $\xi_k\equiv k^2/2m - \mu$. Using again the condition characterizing a broad resonance, we see that the momentum and frequency dependance of the interaction is irrelevant. This implies that $$\begin{aligned}
g_B(1,2,3)&\simeq& g_B(k_j=0,\tilde{\omega}_j=0;j=1,2,3) \nonumber \\
&\simeq& g^4\int_{k,\tilde{\omega}} \Big[\tilde{\omega}^2+\xi_k^2
\Big]^{-2}=\frac{3g^4\sqrt{m}}{8|\epsilon_b|^{5/2}}
\label{gB}\end{aligned}$$ with $\mu\simeq \epsilon_b/2$.
**$l\geq 3$ term**: For all $l$, we obtain the following estimate for the corresponding term in the sum (\[suml\]): $$\begin{aligned}
t_l\equiv \frac{1}{2l}\text{Tr}\left[(G_0\Delta)^{2l}\right]\sim
\frac{g^{2l}}{|\epsilon_b|^{2l-1}r_b}\frac{k_F^{l-1}}{\epsilon_F}\sim
(nr_b)^{l-3}.\end{aligned}$$ For $l=1$ and $l=2$, in the broad resonance limit where $1\gg
nr_{\star} >nr_b$, we obtain that $t_1$ and $t_2\gg 1$. For $l\geq 3$, the ratio $t_2/t_l\gg 1$ in the broad resonance limit and the corresponding terms can therefore be neglected.
In the broad resonance limit, the effective action for bosons becomes $$\begin{aligned}
S_B^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg(\bar{\psi}_{B}
\Big[\partial_{\tau}-\frac{\partial_x^2}{4m}-(2\mu-\epsilon_b)\Big]\psi_{B}
\nonumber \\ &+& \frac{g_B}{2}
\bar{\psi}_{B}\bar{\psi}_{B}\psi_{B}\psi_{B} \bigg)\end{aligned}$$ where $g_B\equiv 3g^4\sqrt{m}/8|\epsilon_b|^{5/2}$ describes a repulsive interaction between the strongly bound dimers. From the partition function (\[Z3\]) and the preceding effective action, we can obtain the average total number of atoms: $$\begin{aligned}
\langle N \rangle = -T\frac{\partial \ln Z_F^0}{\partial \mu}
-T\frac{\partial \ln Z_B^{eff}}{\partial \mu}.
\label{N45}\end{aligned}$$ The first term is given by the usual expression for an ideal Fermi gas: $$\be
| 2,262
| 2,511
| 2,373
| 2,227
| null | null |
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|
d at the hadronic supercolliders and $e^+e^-$ colliders. The advantages of the processes in Fig. \[one\] are that the $W_LW_L$ scattering is no longer on loop level, and additional vector bosons in the final state can be tagged on to eliminate the large $W_T W_T$ and $Z_T Z_T$ backgrounds. In addition, both the $W_L^+
W_L^-$ and $W_L^\pm W_L^\pm$ scattering can be studied in $\gamma\gamma$ collision but only one of them can be studied in the $e^+e^-$ or $e^-e^-$ collisions. Also any $Z_LZ_L$ pair in the final state must come from the $W_LW_L$ fusion because photon will not couple to $Z$ on tree level. Totally, we can study four scattering processes, $W_L^\pm W_L^\pm \to W^\pm_L W^\pm_L$, $W^+_L W^-_L \to W^+_L W^-_L,\, Z_L Z_L$.
For simplification we will use the effective $W_L$ luminosity inside a photon in analogy to the effective $W$ approximation. This approximation will suffice for the purpose here for we will consider the kinematic region where the EWSBS will interact strongly, or in another words, in the large invariant mass region of the vector boson pair. The luminosity function of a $W_L$ inside a photon in the asymtotic energy limit is given by [@zerwas] $$\label{lum}
f_{W_L/\gamma}(x) = \frac{\alpha}{\pi} \left [ \frac{1-x}{x} +
\frac{x(1-x)}{2}\; \left ( \log \frac{s(1-x)^2}{m_W^2} - 2 \right ) \right
]\,,$$ which is in analogy to the luminosity function $f_{W_L/e}(x) = \frac{\alpha}{4\pi x_{\rm w}} \frac{1-x}{x}$ of $W_L$ inside an electron. The first term in Eq. (\[lum\]) is approximately equal to the luminosity of $W_L$ inside an electron, and the logarithm factor will enhance the luminosity at high energy. This is the reason why the signal rates can be achieved higher than those in the $e^+e^-$ colliders at the same energy.
Models & Predictions
====================
In this section, we will calculate the number of signal events predicted by some of the models that have been proposed for the EWSBS. In $\gamma\gamma$ collision we can study the following subprocesses $$\begin{aligned}
W_L^+ W_L
| 2,263
| 1,788
| 2,856
| 2,331
| 2,850
| 0.776537
|
github_plus_top10pct_by_avg
|
_{i}f(x)=\int_{{\mathbb{R}}^{d}}U_{i}^{\ast }s_{t}^{x}(y)f(y)dy.$$As a consequence, one gets the kernel in (\[h6\]): $$\tilde{p}_{\delta_{1},\ldots ,\delta_{m}}(x,y)=\int_{{\mathbb{R}}^{d\times
(m-1)}}U_{1}^{\ast }s_{\delta_{1}}^{x}(y_{1})\Big(\prod_{j=2}^{m-1}U_{j}^{%
\ast }s_{\delta_{j}}^{y_{j-1}}(y_{j})\Big)s_{\delta_{m}}(y_{m-1},y)dy_{1}%
\cdots dy_{m-1},$$and the regularity immediately follows.
**B.** We split the proof in several steps.
**Step 1: decomposition**. Since $\sum_{i=1}^{m}\delta_{i}=t$ we may find $j\in \{1,...,m\}$ such that $\delta_{j}\geq \frac{t}{m}.$ We fix this $j$ and we write$$\prod_{i=1}^{m-1}(S_{\delta_{i}}U_{i})S_{\delta_{m}}=Q_{1}Q_{2}$$with$$Q_{1}=\prod_{i=1}^{j-1}(S_{\delta_{i}}U_{i})S_{\frac{1}{2}\delta_{j}}\quad %
\mbox{and}\quad Q_{2}=S_{\frac{1}{2}\delta_{j}}U_{j}\prod_{i=j+1}^{m-1}(S_{%
\delta_{i}}U_{i})S_{\delta_{m}}=S_{\frac{1}{2}\delta_{j}}%
\prod_{i=j}^{m-1}(U_{i}S_{\delta_{i+1}}).$$Here we use the semi-group property $S_{\frac{1}{2}\delta_{j}}S_{\frac{1}{2}%
\delta_{j}}=S_{\delta_{j}}.$
We suppose that $j\leq m-1$. In the case $j=m$ the proof is analogous but simpler. We will use Lemma \[lemmaB\] in order to estimate the terms corresponding to each of these two operators. As already seen, both $Q_1$ and $Q_2$ are given by means of smooth kernels, that we call $p_1(x,y)$ and $%
p_2(x,y)$ respectively.
**Step 2**. We take $\beta $ with $\left\vert \beta \right\vert \leq
q_{2}$ and we denote $g^{\beta ,x}(y):=\partial _{x}^{\beta }g(x,y)$. For $%
h\in L^{1}$ we write$$\begin{aligned}
& \int_{{\mathbb{R}}^{d}}h(z)\partial _{x}^{\beta }\tilde{p}_{\delta
_{1},...,\delta _{m}}(x,z)dz=\int_{{\mathbb{R}}^{d}}h(z)\int_{{\mathbb{R}}%
^{d}}\partial _{x}^{\beta }p_{1}(x,y)p_{2}(y,z)dydz \\
& \qquad =\int_{{\mathbb{R}}^{d}}\partial _{x}^{\beta }p_{1}(x,y)\int
h(z)p_{2}(y,z)dzdy=\int_{{\mathbb{R}}^{d}}\partial _{x}^{\beta
}p_{1}(x,y)Q_{2}h(y)dy \\
& \qquad =\int_{{\mathbb{R}}^{d}}Q_{2}^{\ast }p_{1}^{\beta ,x}(y)h(y)dy.\end{aligned}$$It follows that $$\partial _{x}^{\beta }\tilde{p}_
| 2,264
| 1,123
| 928
| 2,375
| null | null |
github_plus_top10pct_by_avg
|
w $\varepsilon (\delta )=\frac{h\delta }{1+\delta }$ which gives $%
\frac{2h}{2(h-\varepsilon (\delta ))}=1+\delta .$ And we take $l(\delta
)\geq 1$ such that $2^{l\delta /(1+\delta )}\geq l$ for $l\geq l(\delta ).$ Since $h\geq 1$ it follows that $\varepsilon (\delta )\geq \frac{\delta }{%
1+\delta }$ so that, for $l\geq l(\delta )$ we also have $2^{l\varepsilon
(\delta )}\geq l.$ Now we check that $$2^{2(h-\varepsilon (\delta ))l_{\ast }}\leq 2^{2hl(\delta )}\theta (n_{\ast
}). \label{reg14}$$If $l_{\ast }\leq l(\delta )$ then the inequality is evident (recall that $%
\theta (n)\geq 1$ for every $n).$ And if $l_{\ast }>l(\delta )$ then $%
2^{l_{\ast }\varepsilon (\delta )}\geq l_{\ast }.$ By the very definition of $l_{\ast }$ we have $$\frac{2^{2h(l_{\ast }-1)}}{(l_{\ast }-1)^{2}}<\theta (n_{\ast })$$so that $$2^{2hl_{\ast }}\leq 2^{2h}(l_{\ast }-1)^{2}\theta (n_{\ast })\leq
2^{2h}\times 2^{2l_{\ast }\varepsilon (\delta )}\theta (n_{\ast })$$and this gives (\[reg14\]).
**Step 2**. We define$$\mbox{$\nu _{l} =0$ if $l<l_{\ast }$ and $\nu_l=\mu _{n(l)}$ if $l\geq
l_{\ast }$}$$and we estimate $\pi _{k,q,h,p}(\mu ,(\nu _{l})_{l}).$ First, by (\[reg9\]) and (\[reg13\]) $$\sum_{l=l_{\ast }}^{\infty }\frac{1}{2^{2hl}}\left\Vert f_{n(l)}\right\Vert
_{q+2h,2h,p}\leq \sum_{l=l_{\ast }}^{\infty }\frac{1}{2^{2hl}}\theta
(n(l))\leq \Theta \sum_{l=l_{\ast }}^{\infty }\frac{1}{l^{2}}\leq \Theta .$$Then we write$$\sum_{l=1}^{\infty }2^{(q+k+d/p_{\ast })l}d_{k}(\mu ,\nu _{l})=S_{1}+S_{2}$$with$$S_{1}=\sum_{l=1}^{l_{\ast }-1}2^{(q+k+d/p_{\ast })l}d_{k}(\mu ,0),\quad
S_{2}=\sum_{l=l_{\ast }}^{\infty }2^{(q+k+d/p_{\ast })l}d_{k}(\mu ,\mu
_{n(l)}).$$Since $d_{k}(\mu ,0)\leq d_{0}(\mu ,0)\leq \left\vert \mu \right\vert ({%
\mathbb{R}}^{d})$ we use (\[reg14\]) and we obtain $$\begin{aligned}
S_{1} &\leq &\left\vert \mu \right\vert ({\mathbb{R}}^{d})\times
2^{(q+k+d/p_{\ast })l_{\ast }}=\left\vert \mu \right\vert ({\mathbb{R}}%
^{d})\times (2^{2(h-\varepsilon (\delta ))l_{\ast }})^{(q+k+d/p_{\ast
})/2(h-\varepsilon (\delta ))} \\
&\
| 2,265
| 610
| 2,042
| 2,274
| null | null |
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|
site $\gamma > 0$ be sufficiently strong that its amplitudes in the left half of the lattice are negligible. One then has $$\widetilde{\left<\psi_1\right|}\left.{\psi}_0\right>\simeq
\frac1{\sqrt{2}}\widetilde{\left<\psi_0\right|}\left.{\psi}_0\right>
= \frac1{\sqrt{2}} \; ,$$ leading to $$\begin{aligned}
E_0-E_1&\simeq&
-\frac W{4}\sum_{\ell=1}^{\infty} \Big\{
\left(a_{\ell}\!+\!a_{-\ell}\right)a_{\ell+1}
+\left(a_{\ell}\!+\!a_{-\ell}\right)a_{\ell-1}\Big\}
\nonumber \\
& &+\frac W{4}\sum_{\ell=1}^{\infty} \Big\{
a_{\ell}\left(a_{\ell+1}\!+\!a_{-\ell-1}\right)
\nonumber \\ & & \qquad \qquad \qquad
+a_{\ell}\left(a_{\ell-1}\!+\!a_{-\ell+1}\right) \Big\}
\nonumber \\
&= &-\frac W{4}\sum_{\ell=1}^{\infty}
\Big\{ \big(a_{-\ell}a_{\ell+1}-a_{-\ell+1}a_{\ell} \big)
\nonumber \\ & & \qquad \qquad
+ \big(a_{-\ell}a_{\ell-1}-a_{-\ell-1}a_{\ell}\big) \Big\} \; .\end{aligned}$$ These telescope series can be summed immediately ($\sum_{n=1}^N\left(b_n-b_{n-1}\right)=b_N-b_0$ and $\lim_{\ell\rightarrow\infty}a_{\ell}=0$), resulting in $$E_0-E_1=\frac W{4} \left(a_1-a_{-1}\right)a_0 \; .$$ In the same manner, one also derives $$E_2-E_0=\frac W{4} \left(a_1-a_{-1}\right)a_0 \; .$$ Summing these two equations finally leads to a surprisingly simple expression for the splitting of the energies associated with the two defects: $$\label{eq:fin}
E_2-E_1=\frac W{2} \left(a_1-a_{-1}\right)a_0 \; .$$ Since $\left(a_1-a_{-1}\right)/2$ can be taken as the discrete derivative of the wave function at the site $\ell = 0$, the energy splitting is determined by the product of the wave function itself and its derivative halfway between the two defects, [*i.e.*]{}, by the current prevailing there. Thus, this Eq. (\[eq:fin\]) constitutes a discrete analog of Herring’s formula, which describes the tunneling splitting for wave functions in double well potentials [@LL; @Herring62; @Gutzwiller90]. In view of Eqs. (\[eq:al\]) and (\[eq:normerg\]), one then has $$\begin{aligned}
\Delta E &\eq
| 2,266
| 5,332
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| 1,595
| null | null |
github_plus_top10pct_by_avg
|
eta} \right|
\sim
\frac{ \lambda_{s\Phi\eta}^2 v_s^2 }
{ 64 \pi^2 ({\mathcal M}_0)_{\eta\eta}^2 }
\sim
10^{-3} \text{-} 10^{-7} .\end{aligned}$$ Thus, even if the value of $\mu_\eta$ is in the TeV scale, we can obtain $\mu \sim 0.1\,{{\text{MeV}}}$ although we need further suppression with $h_{\ell{{\ell^\prime}}} \lesssim 10^{-5}$ to have $m_\nu \lesssim 1\,{{\text{eV}}}$. If we use $h_{\ell{{\ell^\prime}}} \sim \lambda_{s\Phi\eta} \sim 10^{-3}$, we obtain $|\mu/\mu_\eta| h_{\ell{{\ell^\prime}}} \sim 10^{-12}$ which can connect the TeV scale $\mu_\eta$ to the eV scale $m_\nu$.
Collider {#subsec:col}
--------
![ The unique process in our model for ${\mathcal H}_1^0 \simeq \eta^0_r$. The bosonic decay of $H^{+}$ contains information of $\mu_\eta$ indicated by a red blob. []{data-label="fig:LHC"}](loop_muterm_LHC.eps)
The characteristic feature of our model is that $\mu_\eta$ is much larger than $\mu$. Let us consider possibility to probe the large $\mu_\eta$ in collider experiments.
A favorable process is shown in Fig. \[fig:LHC\] for ${\mathcal H}_1^0 \simeq \eta^0_r$. For simplicity, we take $\lambda_5=0$ which results in $m_{H^{\pm\pm}}^{} \simeq m_{H^\pm}^{}
\simeq m_{H^0,A^0}^{}$. Recently, it was found in Ref. [@Kanemura:2012rs] that the electroweak precision test prefers $\lambda_5 > 0$ in the HTM where the electroweak sector is described by four input parameters. However, results in Ref. [@Kanemura:2012rs] might not be applied directly to our model[^4] because the scalar sector is extended. Since $H^{\pm\pm} \to \ell^\pm {\ell^\prime}^\pm$ is the most interesting decay in the HTM, we assume $2 m_{{\mathcal H}^\pm}^{} > m_{H^{\pm\pm}}^{}$ in order to forbid $H^{\pm\pm} \to {\mathcal H}^\pm {\mathcal H}^\pm$. Even in this case, the DM ${\mathcal H}_1^0$ can be light enough ($m_{H^\pm}^{} > m_{{\mathcal H}^\pm}^{} + m_{{\mathcal H}_1^0}^{}$) so that $Z_2$-even charged scalar $H^\pm$ ($\simeq \Delta^\pm$) can decay into ${\mathcal H}^\pm {\mathcal H}_1^0$ via $\mu_\eta$-term which is indicated b
| 2,267
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| 1,333
| 2,397
| 2,768
| 0.777128
|
github_plus_top10pct_by_avg
|
-> android -> existing android code into workspace
Browse to the unzipped folder of ZXing 2.1
You will have two projects in the list. Select both and click OK.
They will be imported into your workspace and you will have errors.
For the first Project (CaptureActivity in my case), you have to add the core.jar file present in ZXing2.1\zxing2.1\core
For the second project (ZXingTestActivity in my case), you have to add the core.jar file present in ZXing2.1\zxing2.1\core and android-integration.jar present in ZXing2.1\zxing2.1\android-integration
That's it.. You are done...
Hope this helps...
I have an application where we use ZXing. I have to update the library to the latest one. So I downloaded the code from ZXing-2.1.zip from here. When I try to compile I get errors in the import statements.
import com.google.zxing.BarcodeFormat;
import com.google.zxing.DecodeHintType;
import com.google.zxing.Result;
I couldn't find these libraries anywhere in the zip that I have downloaded. Can someone please guide me on how this is done? What am I doing wrong?
If you have a step by step tutorial, please do link them.
A:
Extract zip
Right click on project-> properties-> Java Build Path-> Libraries-> Add External Jars-> select the jar file where it is in your Pc-> click ok.
Q:
Searching for names in a MySQL database that probably has typos
I'm currently writing a script tasked with going through tens of thousands of rows of account information and cleaning mistyped addresses, as well as printing out reports on how the address was cleaned. Currently the biggest source of unclean addresses is mistyped street-names (it's amazing how many ways you can spell a street-name). In any case, currently my script grabs the input street-name and performs a series of edits specific to the Norwegian language (v. becomes vegen, gt. becomes gata etc.) and searches for the street-name in a ~2 million row database of addresses. If it doesn't find a match it proceeds to split off the latter half of the street-name and replacing it with a
| 2,268
| 3,221
| 1,757
| 2,835
| null | null |
github_plus_top10pct_by_avg
|
+b$ for integers $a$ and $b$ satisfying $a \ge 0$ and $1 \le b \le q-1$. With this notation, we obtain that for any $0 \le j \le a-1$ and $0 \le \ell \le q-2$ we have that $$n_{d-j(q-1)-\ell} \le n_d-j \le m_d-j-1.$$
In particular choosing $j=a-1$ and $\ell=0$, this implies that $m_d \ge a+n_{q-1+b} \ge a+1+n_b \ge a$. Using these observations, we obtain from Equation that $$\label{eq:uniqueness2}
\rho_q(d,m_d) \le N = \sum_{i=e}^d \rho_q(i,n_i) \le \sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m_d-j-1) + \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1).$$ Applying the same technique as in the proof of Corollary \[cor:help\], we derive that $$\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m_d-j-1)=\rho_q(d,m_d)-\rho_q(b+e-1,m_d-a)$$ and Equation can be simplified to $$\label{eq:uniqueness3}
\rho_q(d,m_d) \le \rho_q(d,m_d)-\rho_q(b+e-1,m_d-a) + \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1).$$ For $m_d=a$ the right-hand side equals $\rho_q(d,m_d)-1$, leading to a contradiction. If $m_d>q$, Equation implies $$\begin{array}{rcl}
\rho_q(b+e-1,m_d-a) &\le & \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1)\\
\\
&= & \sum_{j=0}^{b-1}\rho_q(e+b-1-j,m_d-a-1)\\
\\
&< & \sum_{j=0}^{\min\{e+b-1,q-1\}}\rho_q(e+b-1-j,m_d-a-1)
\\
& = & \rho_q(b+e-1,m_d-a),
\end{array}$$ where in the last equality we used Lemma \[lem:help\]. Again we arrive at a contradiction. This completes the proof of uniqueness of the $d$-th Macaulay representation with respect to $q$.
Now we show existence. Let $d$, $N$ and $q$ be given. We will proceed with induction on $d$. For $d=1$, note that $\rho_q(1,m)=m+1$ for any $m \ge -1$. Therefore, for a given $N \ge 0$, we can write $N=\rho_q(1,N-1)$.
Now assume the theorem for $d-1$. There exists $m_d \ge -1$ such that $$\label{eq:existence}
\rho_q(d,m_d) \le N < \rho_q(d,m_d+1).$$ Applying the induction hypothesis on $N-\rho_q(d,m_d)$, we can find $m_{d-1},\dots,m_1$ satisfying the conditions of the theorem for $d-1$. In particular we have that
1. $N-\rho_q(d,m_d)=\sum_{i=1}^{d-1} \rho_q(i,m_i),$
2. $-1 \le m_1 \le \c
| 2,269
| 1,414
| 2,016
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ntary consequence is: If $\overline{m_1}$ has a well-defined critical exponent $\beta_1^{\text{dis}}$ in the sense that [@Sta71] $$\label{defce}
\beta_1^{\text{dis}}=\lim_{\tau\to 0^-}
\frac{\ln \overline{m_1}(\tau)}{\ln \tau}$$ exists, then we have $$\label{ordeq}
\beta_1^{\text{dis}}=\beta_1^{\text{ord}}\,.$$
Two further implications of (\[ineq\]) are worth mentioning. First, if a surface critical exponent $\tilde\beta_1^{\text{dis}}$ can be defined via the analog of (\[defce\]) for the most probable value of $m(i_s;.)$ [@comment], then it must have the same value $\beta_1^{\text{ord}}$. Second, the inequality (\[ineq\]) also rules out a limiting $\tau$ dependence of the form $\sim |\tau|^{\beta_1}\,|\ln|\tau||^\varphi$ (standard logarithmic corrections) for $\overline{m_1}$ and the most probable value of $m(i_s;.)$.
Consider next the case $r^>=r_c$. Let us again make the assumption that the limit (\[defce\]) or the analogous one defining $\tilde\beta_1^{\text{dis}}$ exist. Then the inequalities $$\label{betadisineq}
\beta_1^{\text{sp}}\le \beta_1^{\text{dis}}\le \beta_1^{\text{ord}}$$ and their analogs for $\tilde\beta_1^{\text{dis}}$ can be deduced from (\[ineq\]). (Cf. Lemma 3 of [@Sta71].)
The same reasoning applied in the case $r^>>r_c$ shows that $\beta_1^{\text{dis}}$ or $\tilde\beta_1^{\text{dis}}$ must obey the relations $$0\le \beta_1^{\text{dis}}\le \beta_1^{\text{ord}}$$ whenever the limits (\[defce\]) through which we defined them exist.
Likewise in the case $r^<=r_c$, the possible values of $\beta_1^{\text{dis}}$ or $\tilde\beta_1^{\text{dis}}$ are restricted by $$\label{bsp}
0\le \beta_1^{\text{dis}}\le \beta_1^{\text{sp}}$$ at transitions at which $\overline{m_1}$ or the most probable value of $m(i_s;.)$ [@comment] approach zero, respectively. On the other hand, it should be recalled that the surface critical exponent $\beta_1^{\text{ex}}$ of the pure system’s extraordinary transition requires a definition other than (\[defce\]): One must subtract a regular background contribution $m_1^{\
| 2,270
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| 2,036
| 0.783287
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g the decreasingly nested filter-base associated with $\textrm{Orb}(x_{0})$. The so-called *$\omega$-limit set of* $x_{0}$ given by $$\begin{array}{ccl}
\omega(x_{0}) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)\textrm{ }(f^{n_{k}}(x_{0})\rightarrow x)\}\\
& = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall f_{\mathbb{R}_{i}}\in\,_{\textrm{F}}\mathcal{B}_{x_{0}})\textrm{ }(f_{\mathbb{R}_{i}}(x_{0})\bigcap N\neq\emptyset)\}\end{array}\label{Eqn: Def: omega(x)}$$ is simply the adherence set $\textrm{adh}(f^{j}(x_{0}))$ of the sequence $(f^{j}(x_{0}))_{j\in\mathbb{N}}$, see Eq. (\[Eqn: net adh\]); hence Def. A1.11 of the filter-base associated with a sequence and Eqs. (\[Eqn: adh net2\]), (\[Eqn: adh filter\]), (\[Eqn: filter adh\*\]) and (\[Eqn: net-fil\]) allow us to express $\omega(x_{0})$ more meaningfully as $$\omega(x_{0})=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f_{\mathbb{R}_{i}}(x_{0})).\label{Eqn: adh_omega_x}$$ It is clear from the second of Eqs. (\[Eqn: Def: omega(x)\]) that for a continuous $f$ and any $x\in X$, $x\in\omega(x_{0})$ implies $f(x)\in\omega(x_{0})$ so that the entire orbit of $x$ lies in $\omega(x_{0})$ whenever $x$ does implying that the $\omega$-limit set is positively invariant; it is also closed because the adherent set is a closed set according to Theorem A1.3. Hence $x_{0}\in\omega(x_{0})\Rightarrow A\subseteq\omega(x_{0})$ reduces the $\omega$-limit set to the closure of $A$ without any isolated points, $A\subseteq\textrm{Der}(A)$. In terms of Eq. (\[Eqn: PrinFil\_Cl(A)\]) involving principal filters, Eq. (\[Eqn: adh\_omega\_x\]) in this case may be expressed in the more transparent form $\omega(x_{0})=\bigcap\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty}))$ where the principal filter $_{\textrm{F}}\mathcal{P}(\{ f^{j}(x_{0})\}_{j=0}^{\infty})$ at $A$ consists of all supersets of $A=\{ f^{j}(x_{0})\}_{j=0}^{\infty}$, and $\omega(x_{0})$ represents the adherence set of the principal filter at $A$, see the discus
| 2,271
| 508
| 1,826
| 2,383
| 1,345
| 0.790676
|
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|
{- i ( \Delta_{K} - h_{n} ) x}
- \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{k} ) (\Delta_{L} - h_{l} )^2 }
e^{- i ( \Delta_{L} - h_{n} ) x}
\nonumber \\
&+&
\frac{ 1 }{ (\Delta_{K} - h_{k} ) (\Delta_{L} - h_{k} ) ( h_{l} - h_{k} )^2 }
e^{- i ( h_{k} - h_{n} ) x}
\nonumber \\
&-&
\frac{ 1 }{ (\Delta_{K} - h_{l} )^2 (\Delta_{L} - h_{l} )^2 ( h_{l} - h_{k} )^2 }
\biggl\{
3 h_{l}^2 - 2 h_{k} h_{l} - \left( 2 h_{l} - h_{k} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L}
\biggr\}
e^{- i ( h_{l} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\left\{ (UX)^{\dagger} A W \right\}_{l L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\nonumber \\
&+&
\sum_{n}
\sum_{k \neq l }
\sum_{K \neq L} \sum_{m \neq k, l}
\biggl[
\frac{1}{ ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) ( h_{l} - h_{k} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) }
\nonumber \\
&+&
\biggl\{
( h_{m} - h_{l} ) (\Delta_{K} - h_{l} ) ( \Delta_{L} - h_{l} )
e^{- i ( h_{k} - h_{n} ) x}
-
( h_{m} - h_{k} ) (\Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} )
e^{- i ( h_{l} - h_{n} ) x}
\biggr\}
\nonumber \\
&+&
\frac{1}{ ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{m} ) }
e^{- i ( h_{m} - h_{n} ) x}
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) ( \Delta_{L} - h_{m} ) ( \Delta_{L} - \Delta_{K} ) }
\nonumber \\
&\times&
\biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} )
e^{- i ( \Delta_{L} - h_{n} ) x}
- ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) ( \Delta_{L} - h_{m} )
e^{- i ( \Delta_{K} - h_{n} ) x}
\biggr\}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha n} (UX)_{\beta n}
\nonumber \\
&\times&
\left\{ (UX)^{\dagger
| 2,272
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| 2,531
| 2,372
| null | null |
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|
)=\_[I’]{}\_[S’]{}(x,’,,E’,E)(x,’,E’)d’ dE’. The simplest case is where $\sigma=\sigma_0(x,\omega',\omega,E',E)$ is a measurable non-negative function $G\times S'\times S\times (I'\times I\setminus D)\to{\mathbb{R}}$, where $D=\{(E,E)\ |\ E\in I=I'\}$ is the diagonal of $I'\times I$, obeying for $E\neq E'$ the estimates $$\begin{aligned}
{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega)}\int_{S'}\sigma_0(x,\omega',\omega,E',E)d\omega'\leq {}& {C\over{|E-E'|^\kappa}},
\label{coll-3} \\
{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega')}\int_{S}\sigma_0(x,\omega',\omega,E',E)d\omega\leq {}& {C\over{|E-E'|^\kappa}},
\label{coll-3a}\end{aligned}$$ where $\kappa<1$, meaning that $\sigma_0(x,\omega',\omega,E',E)$ may have a so-called *weak singularity* with respect to energy. We see that $$\begin{gathered}
{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,\omega,E)\in G\times S\times I}\int_{I'}\int_{S'}\sigma_0(x,\omega'\omega,E',E)d\omega'dE'
\leq \sup_{E} C\int_{I'}{1\over{|E-E'|^\kappa}}dE' \\
={}
\sup_{E} C{1\over{1-\kappa}}[(E_{\rm m}-E)^{1-\kappa}+(E-E_0)^{1-\kappa}]\leq {{2CE_{\rm m}^{1-\kappa}}\over{1-\kappa}},
\label{coll-4a}\end{gathered}$$ and similarly for $\int_{I}\int_{S}\sigma_0(x,\omega'\omega,E',E)d\omega dE$. Hence we see that $\sigma_0(x,\omega',\omega,E',E)$ satisfies conditions below, and the corresponding collision operator (K\_0)(x,,E) = \_[I’]{}\_[S’]{}\_0(x,’,,E’,E)(x,’,E’)d’ dE’, is the usual partial Schur (singular) integral operator. It is bounded $L^2(G\times S\times I)\to L^2(G\times S\times I)$.
Nevertheless, the collision operator $K$ is not generally of the above form $K_0$. $(E',E)$-dependence in differential cross section $\sigma(x,\omega'\omega,E',E)$ may contain hyper-singularities of higher order, ${1\over{(E'-E)}^m}$, for $m=1,2$ for example; see Example \[moller\] below.
Moreover, the $(\omega',\omega)$-dependence in differential cross-sections typically contain Dirac’s $\delta$-distributions (on ${\mathbb{R}}$). More precisely, in $\sigma(x,\omega
| 2,273
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| 2,180
| 2,339
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|
lies that $x=y$.
The constructed functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$ is torsion-free.
This follows from the definition of $\sim$ since $$\Phi(X,\mu)(e',e)(x)=e'\cdot x=e\cdot x=x$$ for any $e'\geq e$ in $E$ and any $x\in X$ such that $e\cdot x$ is defined.
We have therefore assigned to $(X,\mu)$ a torsion-free functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$. We now describe the reverse direction. Assume that $F$ is a torsion-free functor $L(S)\to {\mathsf{Sets}}$. By $[e,x]$ we will denote the $\sim$-class of $(e,x)$. For $s\in S$ and $\alpha\in \Psi(F)$ we define $$\label{eq:action} s\circ\alpha=\left\lbrace\begin{array}{ll}[{\mathbf{r}}(s), F({\mathbf{r}}(s),s)(x)],& \text{ if } \alpha=[{\mathbf{d}}(s),x];\\ \text{undefined,} & \text{otherwise.} \end{array}\right.$$
If $\alpha\in \Psi(F)$ we define
$$\label{eq:connected}\pi_1(\alpha)=\{e\in E\colon \text{ there is some }(e,x)\in \alpha\}=\{e\in E\colon e\circ \alpha \text{ is defined}\}.$$
It follows that $s\circ \alpha$ is defined if and only if ${\mathbf{d}}(s)\in\pi_1(\alpha)$.
\[lem:lem3\]
1. The map $\alpha\mapsto s\circ\alpha$, given by , is injective on its domain.
2. The assignment defines on $\Psi(F)$ the structure of a non-strict $S$-set $(\Psi(F), \nu)$.
3. For any $\alpha\in \Psi(F)$ and $e,f\in \pi_1(\alpha)$, there are $$e=e_1,e_2,\dots, e_k=f$$ in $\pi_1(\alpha)$ such that $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$.
\(1) Follows from , since $F$ is torsion-free and thus all the translation maps are injective.
\(2) We use the fact that a map $\varphi\colon S\to T$ between inverse semigroups is a prehomomorphism if and only if $\varphi(st)=\varphi(s)\varphi(t)$ for any $s,t$ such that ${\mathbf{r}}(t)={\mathbf{d}}(s)$ and $\varphi(ef)\leq \varphi(e)\varphi(f)$ for any $e,f\in E(S)$. It is immediate from that both of these conditions hold for $\Psi(F)$.
\(3) Follows from the construction of $\Psi(F)$ and .
Since the set $\pi_1(\alpha)$ is expressable in terms of the action, as is given in , we
| 2,274
| 3,486
| 2,298
| 2,065
| null | null |
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|
}^k u_iv_i$. For a commutative ring $\cR$ we will denote by $\cR[{x_1,\cdots,x_k}]$ the ring of polynomials in formal variables $x_1,\ldots,x_k$ with coefficients in $\cR$. We will use the notation $\bx^\bz$ with $\bx=({x_1,\cdots,x_k}),\ \bz=({z_1,\cdots,z_k}) \in \Z^k$ to denote the monomial $\prod_{i=1}^k x_i^{z_i}$. So any polynomial $F(\bx)\in \cR[{x_1,\cdots,x_k}]$ can be written as $F(\bx)=\sum_{\bz} c_{\bz}\bx^{\bz}$.
$\Z_m=\Z/m\Z$ is the ring of integers modulo $m$. When $\bu\in \Z_m^k$, $\bx^\bu$ denotes $\bx^{\tilde{\bu}}$ where $\tilde{\bu}\in {\{0,1,\cdots,m-1\}}^k$ is the unique vector such that $\bu\equiv \tilde{\bu} \mod m$. $\F_q$ denotes the finite field of size $q$.
The rings $\cR_{m,r}$
---------------------
For our construction it will be convenient (although not absolutely necessary, see Section \[sec-overZm\]) to work over a ring which has characteristic $6$ and contains an element of order $6$. We now discuss how to construct such a ring in general.
Let $m>1$ be an integer and let $\gamma$ be a formal variable. We denote by $$\cR_{m,r} = \Z_m[\gamma]/(\gamma^r-1)$$ the ring of univariate polynomials $\Z_m[\gamma]$ in $\gamma$ with coefficients in $\Z_m$ modulo the identity $\gamma^r = 1$.[^4] More formally, each element $f \in \cR_{m,r}$ is represented by a degree $\leq r-1$ polynomial $f(\gamma) = \sum_{\ell=0}^{r-1}c_\ell \gamma^\ell$ with coefficients $c_i \in \Z_m$. Addition is done as in $\Z_m[\gamma]$ (coordinate wise modulo $m$) and multiplication is done over $\Z_m[\gamma]$ but using the identity $\gamma^r=1$ to reduce higher order monomials to degree $\leq r-1$. It is easy to see that this reduction is uniquely defined: to obtain the coefficient of $\gamma^\ell$ we sum all the coefficients of powers of $\gamma$ that are of the form $\ell + km$ for some integer $k \geq 0$. This implies the following lemma.
\[nonzerolemma\] Let $f = \sum_{\ell=0}^{r-1}c_\ell \gamma^\ell$ be an element in $\cR_{m,r}$. Then, $f=0$ in the ring $\cR_{m,r}$ iff $c_i=0$ (in $\Z_m$) for all $0 \leq i
| 2,275
| 3,162
| 2,285
| 2,040
| 4,009
| 0.768663
|
github_plus_top10pct_by_avg
|
various Freiman-type theorems in non-nilpotent groups. As in \[sec:details\], at various points we separate the trivial case $K<2$ from the meaningful case $K\ge2$ so as to avoid the need for multiplicative constants.
Our first corollary improves an earlier result of the author for residually nilpotent groups [@resid Corollary 1.4], and partially improves on \[cor:ruzsa\] for large values of $s$.
\[cor:resid\] Let $K\ge1$. Let $G$ be a residually nilpotent group, and suppose that $A\subset G$ is a finite $K$-approximate group. Then $A$ is contained in the union of at most $\exp(K^{O(1)})$ left translates of a coset nilprogression $P\subset A^{O_K(1)}$ of rank at most $\exp(O(K^{12}))$ and step at most $K^6$.
This compares with the bound of $\exp(\exp(K^{O(1)}))$ on the rank of $P$ obtained previously by the author using \[thm:old\].
It follows from [@resid Theorem 1.2] that there exist subgroups $H\lhd N<G$ such that $H\subset A^{O_K(1)}$, such that $N/H$ is nilpotent of step at most $K^6$, and such that $A$ is contained in a union of at most $\exp(K^{O(1))}$ left cosets of $N$. \[lem:fibre.pigeonhole\] then implies that $A$ is contained in a union of at most $\exp(K^{O(1))}$ left translates of $A^2\cap N$, which is a $K^3$-approximate group by \[lem:slicing\]. The desired result therefore follows from applying \[cor:chang.ag\] to the image of $A^2\cap N$ in $N/H$.
\[cor:resid\] gives a better rank bound than \[cor:ruzsa\] if the step of the ambient group is greater than $K^6$. It gives a better bound on number of translates of $P$ required to cover $A$ as soon as the step is logarithmic in $K$.
Our next corollary applies to linear groups over fields of prime order, and arises from combining \[cor:ruzsa\] with a result of Gill, Helfgott, Pyber and Szabó [@gill-helf Theorem 3].
\[cor:ghps\] Let $n\in{\mathbb{N}}$ and $K\ge1$, and let $p$ be a prime. Suppose that $A\subset GL_n({\mathbb{F}}_p)$ is a finite $K$-approximate group. Then there is a coset nilprogression $P\subset A^{K^{O_n(1)}}$ of rank at most
| 2,276
| 1,756
| 1,825
| 2,162
| 2,012
| 0.783544
|
github_plus_top10pct_by_avg
|
enever the null model is nested within the alternative model the likelihood ratio approximately follows a $\chi^2$ distribution with degrees of freedom specified by $(|S|^m-|S|^k)(|S|-1)$. If the p-value is below a specific significance level we can reject the null hypothesis and prefer the alternative model [@bartlett][^4].
[[ Likelihood ratios and corresponding tests have been shown to be a very understandable approach of specifying evidence [@perneger]. They also have the advantage of specifying a clear value (i.e., the likelihood ratio) with can give us intuitive meaning about the advantage of one model over the other. However, the likelihood-ratio test also has limitations like that it only works for nested models, which is fine for our approach but may be problematic for other use cases. It also requires us to use elements from frequentist approaches (i.e., the p-value) for deciding between two models which have been criticized in the past (e.g., [@morrison2006significance]). Furthermore, we are only able to compare two models with each other at a time. This makes it difficult to choose one single model as the most likely one as we may end up with several statistical significant improvements. Also, as we increase the number of hypothesis in our test, we as well increase the probability that we find at least one significant result (Type 1 error)[^5]. ]{}]{}
![**Log-likelihoods for random path dataset.** Simple log-likelihoods of varying Markov chain orders would suggest higher orders as the higher the order the higher the corresponding log-likelihoods are. This suggests that looking at these log-likelihoods is not enough for finding the appropriate Markov chain order as methods are necessary that balance the goodness-of-fit against the number of model parameters.[]{data-label="fig:randomloglikelihood"}](paths_wikigame_random_loglikelihoods){width="0.7\columnwidth"}
### Bayesian Method {#subsubsec:bayesianinference .unnumbered}
[[ Bayesian inference is a statistical method utilizing the Bayes’ rule –
| 2,277
| 5,498
| 2,438
| 1,741
| null | null |
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|
230 1 1 1 1 1 0 0 0
270 1 1 1 1 1 1 0 0 270 1 1 1 1 1 1 0 0
320 1 1 1 1 1 1 1 0 300 1 1 1 1 1 1 1 0
360 1 1 1 1 1 1 1 1 350 1 1 1 1 1 1 1 1
------------ --- ----- --- --- --- --- --- --- -- ------------ --- --- --- --- --- --- --- ---
: Most probable firing events (1=fire, 0=latent) for each level in the stimulus-response curve. The labeled MUs for R50 are re-ordered to demonstrate similarity between the two data sets. The response level around 70mN is not present in the R10 data set.
Discussion {#sec:Discussion}
==========
This paper presents a new sequential Bayesian procedure for motor unit number estimation (MUNE), the assessment of the number of the operating motor units (MUs) from an electromyography investigation into muscle function. The fully adpated sequential Monte Carlo (SMC) filter uses the approximate conditional conjugacy of the twitch process. The principal purpose of SMC-MUNE is to estimate the marginal likelihood for the neuromuscular model based on a fixed number of MUs. From this, motor unit number estimation (MUNE) is then performed by comparing the evidence between competing MU-number hypotheses. As is demonstrated in Sections \[sec:SimStudyOver\] and \[sec:SimStudy\_under\] SMC-MUNE also allows detailed scrutiny of the quality of each model fit.
SMC-MUNE performed well on simulated data, but two scenarios that may cause incorect estimation were identified. In the first scenario, one or more MUs were estimated to have a negligible or negative twitch force, allowing a model that was larger than the truth to fit the data and resulting in overestimation of the number of motor units (MUs). This lead to the development of a post-process correction that restricts the parameter space. By contrast, the second scenario resulted in under-estimation because of difficulty in estimating the underlying pr
| 2,278
| 2,218
| 3,341
| 2,378
| null | null |
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|
USD 339.00 PER NIGHT
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-----Original Message-----
From: Collins, Angie
Sent: Tuesday, January 15, 2002 4:53 PM
To: Holcombe, Tina ; Snow, Dina; Leigh, Lorie; Gonzales, Kathryn; Quezada, Daniel; Presas, Jessica
Subject: UPDATE ON CHURN
I just met with the move coordinator. We are now scheduled to move to the 4th floor, however, the information collected in the meeting will need to be approved. Needless to say the churn will not have to be submitted today.
The following issues were discussed in the meeting today:
Fax Machine/Lines TBD (I will process the churn for all fax lines)
Churn Due Date Thursday, January 17 (tentative)
Move Date Tuesday, January 21 (tentative)
Boxes Will be moved by move team (if staff is available)
Filing Space Please get with your group and determine what files can be achieved if any (space is being looked at)
Once move to locations have been determined an email will need to be sent to your group requesting that they g
| 2,279
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| 2,701
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|
r\\
&-\frac{13}{24}e^4)a^2 -\frac{1}{48}e^2r^4\alpha^2(e^2-2mr))\cos^2(\theta)+60(((\frac{1}{30}r^3\alpha^2-r)m+e^2)ma^2+\frac{1}{10}r(e^4-\frac{1}{3}e^2mr-\frac{1}{3}m^2r^2)) \times \nonumber\\
&\alpha r^5\cos(\theta) -4a^2\alpha^2m^2r^7+6e^4r^5-10e^2mr^6+4m^2r^7)\sin(\theta)+(a^2\cos^2(\theta)+r^2)(a^2(a^2\alpha m+\alpha e^2r-am)\cos^3(\theta)+ \nonumber\\
&(-3a^3mr\alpha+(e^2-3mr)a^2-2ae^2\alpha r^2)\cos^2(\theta)+(-3a^2mr^2\alpha+(-2e^2r+3mr^2)a-e^2\alpha r^3)\cos(\theta)+r^2(a\alpha mr-e^2 \nonumber\\
&+mr))(r\alpha \cos(\theta)-1)(a^2(a^2\alpha m+\alpha e^2 r+am)\cos^3(\theta)+(3a^3mr\alpha+(e^2-3mr)a^2+2ae^2\alpha r^2)\cos^2(\theta) \nonumber\\
&+(-3a^2mr^2\alpha+(2e^2r-3mr^2)a-e^2\alpha r^3)\cos(\theta)-r^3am\alpha-e^2r^2+r^3m)\cos(\theta)\Big)\bigg)\Bigg|\Bigg)\end{aligned}$$
Therefore from FIG. \[fig8\], we find that the entropy density measure diverges not only at the ring singularity but also at $ \theta=\pi $, which renders this measure inappropriate for determining the gravitational entropy in these cases. There is another singularity at $ \theta=0 $, though not visible in FIG. \[fig8\], but can be inferred from the mathematical analysis. This is in agreement with the observations in [@entropy2] for non-accelerating axisymmetric black holes. This is a disturbing feature of this method of analysis. For a possible resolution of this problem we want to point out that in the case of rotating black holes, the existence of stationary observer is not well defined because of the effect of frame dragging. Nevertheless, we have worked with the chosen definition of gravitational entropy density to get an overall idea of the way things work out. For such cases of axisymmetric space-times, it is not possible to determine the spatial metric $h_{ij} $ because of the presence of the metric coefficient $g_{t\phi}$ in the metric (\[ht\]) and in metric (\[htn\]). This is because the object is rotating and the spatial position of each event in the space-time depends on time. Therefore the covariant divergence
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mmetric in the indices $c,b$. Moreover the double contraction of the structure constant vanishes since the dual Coxeter number of the Lie super algebra vanishes. This concludes the proof of equation up to terms of order $f^4$. Let us mention that the same equation also guaranties the quantum integrability of the model up to this order, as discussed in section \[integrability\]. The same argument leads to the equality: f\_[abc]{} :j\_[L,z]{}\^a :j\^b\_[L,|z]{} j\^c\_[L,z]{} ::(z) = f\_[abc]{} :j\_[L,z]{}\^a j\^b\_[L,|z]{} j\^c\_[L,z]{} :(z). Thus we have: |T(z) = f\_[abc]{} ( : j\^c\_[L,z]{} j\^b\_[L,|z]{} j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^a j\^b\_[L,|z]{} j\^c\_[L,z]{} :(z) ) =0 which vanishes thanks to the (graded) anti-symmetry of the structure constants. It would be interesting to have a non-perturbative understanding of the consistency of the normal-ordering and the holomorphy of the energy-momentum tensor.
Details on primary operators {#AppPrimaries}
============================
Behavior of current primaries under perturbation of the kinetic term {#WZWaffine}
--------------------------------------------------------------------
In this section we will show that current primaries at a given point of moduli space remain current primaries after perturbation of the kinetic term. More precisely we will show that if an operator $\phi$ satisfies the OPEs at a given point of moduli space, then it also satisfies the same OPEs after exactly marginal deformation of the theory. This implies that it is consistent to think of a current primary as being the group element taken in a given representation, at any point of the moduli space. It also proves the claim in section \[primaries\] that the affine primary fields at the WZW points become current primaries after deformation of the theory.
For convenience let us recall the OPEs that define a primary operator $\phi$ : $$\begin{aligned}
\label{defPrimariesBis}
j^a_{L,z}(z) \phi(w) &= - \frac{c_+}{c_++c_-} t^a \frac{\phi(w)}{z-w} + \text{less singular} \cr
j^a_{L,\bar
| 2,281
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$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$, we have $$\begin{aligned}
{\label{eq:Theta''-1stindbd}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
v}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\,\Big({\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
v\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\\}$}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}\Big).\end{aligned}$$ We investigate the contributions from the two indicators in the parentheses separately.
We begin with the contribution from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$, which is independent of ${{\bf m}}$. Since $$\begin{aligned}
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}&\subset\{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\circ\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\
| 2,282
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q:posl_expec1a}
\E[M] &=& \sum_{j = 1}^n \sum_{a =1}^{\ell_j} \lambda_{j,a} \sum_{i<\i \in S_j} \P\Big[(i,\i) \in G_{j,a} \Big| (i,\i \in S_j) \Big] (e_i - e_{\i})(e_i - e_{\i})^\top\;.\end{aligned}$$ The following lemma provides a lower bound on $\P[(i,\i) \in G_{j,a} | (i,\i \in S_j)]$.
\[lem:posl\_lowerbound\] Consider a ranking $\sigma$ over a set $S \subseteq [d]$ such that $|S| = \kappa$. For any two items $i,\i \in S$, $\theta\in\Omega_b$, and $1 \leq \ell \leq \kappa-1$, $$\begin{aligned}
\label{eq:posl_lowerbound_eq}
\P_{\theta}\Big[\sigma^{-1}(i) = \ell, \sigma^{-1}(\i) > \ell \Big] \;\;\geq\;\; \frac{e^{-6b}(\kappa-\ell)}{\kappa(\kappa-1)} \bigg(1 - \frac{\ell}{\kappa}\bigg)^{\alpha_{i,i',\ell,\theta} -2} \;,\end{aligned}$$ where the probability $\prob_\theta$ is with respect to the sampled ranking resulting from PL weights $\theta\in\Omega_b$, and $\alpha_{i,i',\ell,\theta}$ is defined as $1 \leq \alpha_{i,i',\ell,\theta} = {\left \lceil{\widetilde{\alpha}_{i,i',\ell,\theta}} \right \rceil}$, and $\widetilde{\alpha}_{i,i',\ell,\theta}$ is, $$\begin{aligned}
\label{eq:posl_alpha}
\widetilde{\alpha}_{i,i',\ell,\theta} \;\; \equiv \;\;
\max_{\ell'\in[\ell]} \max_{\substack{\Omega \subseteq S\setminus\{i,\i\} \\ : |\Omega| = \kappa-\ell'}} \Bigg\{\frac{\exp(\theta_i)+\exp(\theta_{\i})}{\big(\sum_{j\in \Omega} \exp(\theta_j)\big)/|\Omega|} \Bigg \}\;.
\end{aligned}$$
Note that we do not need $\max_{\ell' \in[\ell]}$ in the above equation as the expression achieves its maxima at $\ell' = \ell$, but we keep the definition to avoid any confusion. In the worst case, $2e^{-2b} \leq \widetilde{\alpha}_{i,i',\ell,\theta} \leq 2e^{2b}$. Therefore, using definition of rank breaking graph $G_{j,a}$, and Equations and we have, $$\begin{aligned}
\E[M] &\succeq& \gamma e^{-6b} \sum_{j = 1}^n \sum_{a =1}^{\ell_j} \lambda_{j,a} \frac{2(\kappa_j-p_{j,a})}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top \nonumber\\
&\succeq& 2\gamma e^{-6b} \sum_{j = 1}^n \frac{1}{\
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{{\frac{\partial (S_0\phi)}{\partial E}}}+\omega\cdot\nabla_x\phi+CS_0\phi +\Sigma\phi -K_C\phi ={\bf f},
\nonumber\\
{\phi}_{|\Gamma_-}=0,\quad\ \phi(\cdot,\cdot,E_{\rm m})=0. \label{co3-d}\end{gathered}$$ Let (for clarity, we have included here the subscript $C$ into $P$) P\_C(x,,E,D):=& -[E]{}+\_x+CS\_0\
=& -S\_0[E]{}+\_x-[E]{}+CS\_0.
We shall seek a strong solution of with the homogeneous inflow boundary conditions. Using the above notations, the problem is equivalent to $$(\tilde{P}_{C,0}+\Sigma-K_C)\phi={\bf f},$$ where $\phi\in D(\tilde{P}_{C,0})$. The following arguments are analogous to those used in [@tervo14 Section 5.3].
\[coth2-d\] Suppose that the assumptions (\[ass1\]), (\[ass2\]), (\[ass3\]), (\[evo16\]), (\[evo8-a\]) and (\[evo9-a\]) are valid with $C={{\max\{q,0\}}\over{\kappa}}$ and $c>0$. Then for every ${\bf f}\in L^2(G\times S\times I)$ the problem has a unique strong solution $\phi\in D(\tilde P_{C,0})={{{\mathcal{}}}H}_{P_{C,0}}(G\times S\times I^\circ)$ with homogeneous inflow boundary conditions.
Recall that $q={1\over 2}\sup_{(x,E)\in G\times I}{{\frac{\partial S_0}{\partial E}}}(x,E)$ (see ). For $C={{\max\{q,0\}}\over{\kappa}}$ we have $${{\frac{\partial S_0}{\partial E}}}\leq 2q=2\kappa{q\over\kappa}\leq 2S_0{{\max\{q,0\}}\over\kappa}=2S_0C,$$ that is \[flp1-d\] -[E]{}+2CS\_00, and hence by Corollary \[d-cor\], the operator $-{\tilde P}_{C,0}:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative. On the other hand, due to Lemma \[csdale1a\] the bounded operator $-(\Sigma-K_C)+c I:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is dissipative. These facts allow us to conclude that $-{\tilde P}_{C,0}-(\Sigma-K_C)+cI :L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative ([@pazy83 Theorem 4.3 and Corollary 3.3], or [@tervo14 Theorem 4.4]). This implies, as $c>0$, that $R\big(cI-(-{\tilde P}_{C,0}-(\Sigma-K_C)+cI)\big)=R({\tilde P}_{C,0}+\Sigma-K_C)=L^2(G\times S\times I)$, and so the existence of solutions follows.
Because $c>0$ and because $-{\tilde
| 2,284
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| 2,317
| 4,027
| 0.768577
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s $k$ times, i.e. $${\xi}^{(m\,h\,k)} = (\mathcal{L}_{H_-})^k\,{\xi}^{(m\,h\,0)}.$$
#### Lifting to the whole manifold.
Finally, we promote the basis functions living on $\Sigma_u$ to functions living on the whole manifold $\mathcal{M}$ by sending all unknown constant coefficients $c_{\beta}$ (from the end of step b) to be unknown smooth functions $c_{\beta}(u)$. While lifting the vector and tensor bases from $\Sigma_u$ to $\mathcal{M}$, i.e. $ V_i\rightarrow V_a $ and $W_{ij}\rightarrow W_{ab}$, we also set all their projections on the $u$ direction to be zero, i.e. $V_u=0$, $W_{iu}=W_{ui}=W_{uu}=0$.
To obtain the basis functions in global coordinates, one just replaces $H_s$ by $L_s$, where $s=0,\pm$, and $Q_0$ by $iW_0$ in steps b and c. To construct the lowest-weight modules of NHEK’s isometry group, one should instead impose the lowest-weight condition ${\mathcal{L}}_{H_-}\,{\xi}^{(m\,h\,0)} = 0$, and the condition $\Omega\,{\xi}^{(m\,h\,k)} = h(h-1)\,{\xi}^{(m\,h\,k)}$, in step b. All descendant states will then be obtained by applying the raising operator ${\mathcal{L}}_{H_+}$ on the lowest-weight states. In Poincaré coordinates, we focus on the basis functions that form the highest-weight module because their expressions are simpler. In global coordinates, we show both representations explicitly in App. \[app:scalar-basis-highest-reps\] and \[app:scalar-basis-lowest-reps\]. Unless otherwise specified, our basis functions will refer to those obtained using the highest-weight method.
Let us remark on the allowed values of $m,\,h,\,k$. It is straightforward to see $k\in \mathbb{Z}^{+}$ by construction, and $m\in \mathbb{Z}$ due to the periodic boundary conditions for the azimuthal angle. In order to have a unitary representation of the isometry group, there are conditions on $h$ as well. For the scalar case, for instance, if we apply the raising operator on a scalar in the highest-weight module, we get $${\mathcal{L}}_{H_+}\,F^{(m\,h\,k)} = k(k-1-2h)\,F^{(m\,h\,k-1)}.$$ A nontrivial unitary represe
| 2,285
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C_{q}}\sum_{0\leq \left\vert \alpha \right\vert \leq q}\left\vert
\partial ^{\alpha }(\psi _{k}f)\right\vert \leq \sum_{0\leq \left\vert
\alpha \right\vert \leq q}\psi _{k}\left\vert \partial ^{\alpha
}f\right\vert \leq C_{q}\sum_{0\leq \left\vert \alpha \right\vert \leq
q}\left\vert \partial ^{\alpha }(\psi _{k}f)\right\vert . \label{n5}$$
**Proof**. One proves by recurrence that, if $\left\vert \alpha
\right\vert \geq 1$ then $\partial ^{\alpha }\psi
_{k}=\sum_{q=1}^{\left\vert \alpha \right\vert }\psi _{k-q}P_{q}$ with $%
P_{q} $ a polynomial of order $q.$ Since $1+\left\vert x\right\vert \leq
2(1+\left\vert x\right\vert ^{2})$ it follows that $\left\vert
P_{q}\right\vert \leq C\psi _{q}$ and (\[n4\]) follows. Now we write $$\psi _{k}\partial ^{\alpha }f=\partial ^{\alpha }(\psi _{k}f)-\sum
_{\substack{ (\beta ,\gamma )=\alpha \\ \left\vert \beta \right\vert \geq 1
}}c(\beta ,\gamma )\partial ^{\beta }\psi _{k}\partial ^{\gamma }f$$and the same arguments as in the proof of (\[n3\]) give (\[n5\]).
Semigroup estimates {#app:semi}
-------------------
We consider a semigroup $P_{t}$ on $C^{\infty }({\mathbb{R}}^{d})$ such that $P_{t}f(x)=\int f(y)P_{t}(x,dy)$ where $P_{t}(x,dy)$ is a probability transition kernel and we denote by $P_{t}^{\ast }$ its formal adjoint.
\[B1B2\] There exists $Q\geq 1$ such that for every $t\leq T$ and every $f\in
C^{\infty }({\mathbb{R}}^{d})$$$\left\Vert P_{t}f\right\Vert _{1}\leq Q\left\Vert f\right\Vert _{1}.
\label{A31}$$ Moreover, for every $k\in {\mathbb{N}}$ there exists $K_{k}\geq 1$ such that for every $x\in {\mathbb{R}}^{d}$ $$\left\vert P_{t}(\psi _{k})(x)\right\vert \leq K_{k}\psi _{k}(x).
\label{A32}$$
Under Assumption \[B1B2\], one has $$\left\Vert \psi _{k}P_{t}^{\ast }(f/\psi _{k})\right\Vert _{p}\leq
K_{kp}^{1/p}Q^{1/p_{\ast }}\left\Vert f\right\Vert _{p}. \label{A34}$$
**Proof**. Using Hölder’s inequality, the identity $\psi
_{k}^{p}=\psi _{kp},$ and (\[A32\])$$\left\vert P_{t}(\psi _{k}g)(x)\right\vert \leq \left\vert P_{t}(\psi
_{k}^{p})(x)\right\vert
| 2,286
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le $J$ over the special Jordan algebra $\bar J$ is special. The bimodule $J$ is embedded into $D$ and $D$ is a $\bar
J$-bimodule too. Consider mappings $\sigma_1,\,\sigma_2\colon\bar
J\to\mathrm{Hom}(D,D)$ defined by the rule $$\sigma_1(\bar a)\colon d\mapsto a{\mathbin\vdash}d\in D,\,
\sigma_2(\bar a)\colon d\mapsto d{\mathbin\dashv}a\in D,\quad d\in D,\,a\in
J\subseteq D.$$
These mappings are well-defined. We show that they are associative specializations. Indeed for every $\bar{a\vphantom b},\,\bar
b\in\bar J$, $d\in D$ $$\begin{gathered}
\sigma_1(\bar{a\vphantom b}\bar b)(d) =\sigma_1(\overline{a{\mathbin{{}_{(\vdash)}}}b})(d)=\frac{1}{2}(a{\mathbin\vdash}b+b{\mathbin\dashv}a){\mathbin\vdash}d=\\
\frac{1}{2}(b{\mathbin\vdash}a{\mathbin\vdash}d+a{\mathbin\vdash}b{\mathbin\vdash}d)=\frac{1}{2}(\sigma_1(\bar{a\vphantom b})\sigma_1(\bar
b)+\sigma_1(\bar b)\sigma_1(\bar{a\vphantom b}))(d).\end{gathered}$$
(We write a composition of mappings as $fg(x)=g(f(x))$.) Analogously, one may check that $\sigma_2$ is an associative specialization.
The relation (\[eq:SpecBiModCond\]) follows from the definition of the operation in our bimodule.
Moreover, $\sigma_1$ and $\sigma_2$ are commuting because $$[\sigma_1(\bar{a\vphantom b}),\sigma_2(\bar b)](d)
=(\sigma_1(\bar{a\vphantom b})\sigma_2(\bar b) +\sigma_2(\bar
b)\sigma_1(\bar{a\vphantom b}))(d)=(a{\mathbin\vdash}d){\mathbin\dashv}b-a{\mathbin\vdash}(d{\mathbin\dashv}b)=0.$$
So, $J$ is a special $\bar J$-bimodule and by Theorem \[thm:SpecSplitNullExtCrit\] we obtain that $\widehat J$ is special.
In papers [@Kol:08; @Kol:06] conformal algebras were investigated and the following fact was proved.
\[prop:VarCur0DiVar\] If an algebra $A$ belongs to a variety ${\mathrm{Var}}$ then a dialgebra $({\mathop{\mathrm{Cur}}\nolimits}A)^{(0)}$ belongs to a variety ${\mathrm{Di}}{\mathrm{Var}}$.
We first prove an auxiliary statement.
\[lemma:ifHomSJthenHomDiSJ\] If $\widehat J\in{\mathcal{H}}{\mathrm{SJ}}$ then $J\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$.
Use conformal algeb
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x_i^{a(i)}$ and $Z=\{x_i\: i\in\ip a\}$. Then $(u,Z)\in\mathcal A$ and $a=\log
u+c(Z)$. Since $a\in\Gamma$ it follows that $u\cdot K[Z]\sect I=\{0\}$. Suppose that $(u,Z)$ is not minimal with this property. Then there exists $(v,W)\in
\mathcal A$ with $v\cdot K[W]\sect I=\{\ 0\}$ and $(v,W)<(u,Z)$, and we have
1. $b=\log v+c(W)\in \Gamma$;
2. $v$ divides $u$;
3. $\supp(u/v)\union Z\subset W$.
The properties (2) and (3) imply that $a(i)=b(i)$ for all $i$ such that $b(i)<\infty$. Thus $a\leq b$, and $a=b$ if and only if $\ip a=\ip b$. However since $a\neq b$, we have $\ip a\neq \ip b$. By property (1) there exists $m\in {\mathcal M}(\Gamma)$ with $b\leq m$. Then $a\leq m$ and $\ip b\subset \ip m$. In particular, $\ip a \neq \ip m$. It follows that $a\not \in {\mathcal F}(\Gamma)$, a contradiction.
Pretty clean filtrations and shellable multicomplexes
=====================================================
In this section we introduce shellable multicomplexes and show how this concept is related to clean filtrations. Our concept of shellability is a translation of Corollary \[primary1\] into the language of multicomplexes. In that corollary we characterized pretty clean filtrations in terms of primary decompositions. Here we need a refined multigraded version of this result.
\[multiprimary\] Let $S=K[x_1,\ldots, x_n]$ be the polynomial ring, and $I\subset S$ a monomial ideal. The following conditions are equivalent:
1. $S/I$ admits a multigraded prime filtration ${\mathcal F}: (0)=M_0\subset M_1\subset \cdots \subset M_{r-1}\subset M_r=S/I$ such that $M_i/M_{i-1}\iso S/P_i(-a_i)$ for all $i$;
2. there exists a chain of monomial ideals $I=I_0\subset I_1\subset \cdots \subset I_r=S$ and monomials $u_i$ of multidegree $a_i$ such that $I_i=(I_{i-1},u_i)$ and $I_{i-1}:u_i=P_i$;
If the equivalent conditions hold, then there exist irreducible monomial ideals $J_1,\ldots J_r$ such that $I_i=\Sect_{j=i+1}^r J_j$ for $i=0,\ldots, r$. Moreover, if the prime filtration is pretty clean, then this set of irreducible
| 2,288
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+ R_{2} e^{ikr}), &
\chi_{tr}^{(2)}(k, r \to +\infty) \to
\bar{N}_{2} T_{2} e^{ikr},
\end{array}
\label{eq.2.4.2}$$ where the coefficients $\bar{N}_{1}$ and $\bar{N}_{2}$ can be found from the normalization conditions.
Using (\[eq.2.4.2\]) for the wave functions in asymptotic region, taking into account the interdependence (\[eq.2.3.6\]) between them and definitions (\[eq.2.1.1\]) for the operators $A_{1}$ and $A_{1}^{+}$, we obtain: $$\begin{array}{l}
\bar{N}_{1} \biggl( e^{-ikr} + R_{1} e^{ikr} \biggr) =
\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}}
\biggl(e^{-ikr} - R_{2} e^{ikr} \biggr), \\
\bar{N}_{1} T_{1} e^{ikr} =
-\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}} T_{2} e^{ikr}.
\end{array}
\label{eq.2.4.3}$$ These expressions are carried out only, if items with the same exponents are equal between themselves. We find: $$\bar{N}_{1} =
\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}}
\label{eq.2.4.4}$$ and $$\begin{array}{cc}
R_{1}(k) = - R_{2}(k), &
T_{1}(k) = - T_{2}(k).
\end{array}
\label{eq.2.4.5}$$
Exp. (\[eq.2.4.5\]) establish the interdependence between the amplitudes of the transmission $T_{1}(k)$, $T_{2}(k)$ and the amplitudes of the reflection $R_{1}(k)$, $R_{2}(k)$ for the particle relatively two potentials. Squares of modules of the transmitted and reflected amplitudes represent the resonant and potential components of the S-matrixes for two systems. We see, that all these values do not depend on the normalized coefficients $N_{1}$, $N_{2}$, $\bar{N}_{1}$, $\bar{N}_{2}$.
One can introduce the matrix of scattering $S_{l}(k)$ for $l$-partial wave: $$\chi_{nl}(r) \sim S_{l}(k) e^{ikr} - (-1)^{l} e^{-ikr}
\label{eq.2.4.6}$$ and determine a phase shift $\delta_{l}(k)$: $$e^{i\delta_{l}(k)} = S_{l}(k).
\label{eq.2.4.7}$$ Then with taking into account (\[eq.2.4.2\]), we find: $$S_{l}(k) = (-1)^{l+1} (R_{l}(k) + T_{l}(k)).
\label{eq.2.4.8}$$ One can see, how these part
| 2,289
| 4,318
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| 2,137
| 2,632
| 0.778248
|
github_plus_top10pct_by_avg
|
on}(\overline{D})$ enters. Note in particular that Theorem \[corr\] follows as a corollary.
We can develop the expression in in terms of the walk-on-spheres $(\rho_n, n\leq N)$, providing the basis for a Monte Carlo simulation. What will work to our advantage here is another explicit identity that appears in [@BGR]. Define $$\begin{aligned}
V_r(x,{\rm d}y) & \coloneqq \int_0^\infty \mathbb{P}_x(X_t\in {\rm d}y, \, t<{\sigma_{B(x,r)}} )\,{\rm d}t, \qquad x\in\mathbb{R}^d, \;|y|<1,\; r>0.
\end{aligned}$$
The expected occupation measure of the stable process prior to exiting a unit ball centred at the origin is given, for $|y|<1$, by $$\begin{aligned}
V_1(0,{\rm d}y)= 2^{-\alpha}\,\pi^{-d/2}\, \frac{\Gamma(d/2)}{\Gamma(\alpha/2)^{2}}\,
|y|^{\alpha -d}\, \pp{\int_0^{|y|^{-2}-1}(u+1)^{-d/2}u^{\alpha/2-1}{\rm d}u}\,{\rm d}y.\label{V}
\end{aligned}$$
Whilst the above identity is presented in a probabilistic context, it has a much older history in the analysis literature. Known as Boggio’s formula, the original derivation in the setting of potential theory dates back to [@Bog]. See the discussion in [@DR; @bucur].
In the next result, we will write as a slight abuse of notation $V_r(x,{f}(\cdot)) = \int_{|y-x|<r}{f}(y)\,V_r(x,{\rm d}y)$ for bounded measurable ${f}$.
\[integral\]For $x\in D$, ${g}\in L^1_\alpha(D^\mathrm{c})$ and ${f}\in C^{\alpha +\varepsilon}(\overline{D})$, we have the representation $$u(x) =\mathbb{E}_x[{g}(\rho_{N})] + \mathbb{E}_x\left[\sum_{n=0}^{N-1} r_n^{\alpha} V_{1}(0, {f}(\rho_n + r_n\cdot))\right].$$
Given the walk-on-spheres $(\rho_n, n\leq N)$ with $\rho_0 = x\in D$, define $\sigma_n$ jointly with $\rho_n$ so that, given $\rho_{n-1}$, $(\rho_n, \sigma_n)$ is equal in law to $(X_{\sigma_{B_n}}, \sigma_{B_n})$ under $\mathbb{P}_{\rho_{n-1}}$. We can now represent the second expectation on the right-hand side of (\[non\_homg\_FK\]) in the form $$\mathbb{E}_x\left[\sum_{n\geq 0} \mathbf{1}_{\{\rho_n\in D\}} \int_{0}^{\sigma_{n+1}
| 2,290
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|
ication map $\phi: E\to (A^1\delta)^{i-j}$ is surjective and its kernel is the largest torsion $A^0$-submodule of $(A^1\delta)^{i-j}$. On the other hand $\operatorname{{\textsf}{ogr}}B_{ij}\subseteq e{\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{W}}$ is a torsion-free $A^0$-module and so $\mathrm{ker}(\phi) \subseteq \mathrm{ker}(\chi)$. Thus $\operatorname{{\textsf}{ogr}}B_{ij} = E/\mathrm{ker}(\chi) $ is a homomorphic image of $ (A^1\delta)^{i-j} $. Since $ (A^1\delta)^{i-j} $ is a right ideal of the domain $A^0$, any proper factor of $ (A^1\delta)^{i-j} $ will be torsion. Thus $\mathrm{ker}(\phi)=
\mathrm{ker}(\chi)$ and $\operatorname{{\textsf}{ogr}}B_{ij} \cong (A^1\delta)^{i-j}$.
{#order-counter}
The observation in suggests that Lemma \[ord-tens\] will only hold for very special decompositions and this is indeed the case. In essence, Theorem \[main\] says that the identity $B_{ij}\cong B_{i,i-1}\otimes\cdots\otimes B_{j+1,j}$ is a filtered isomorphism. On the other hand, an identity like $H_c\cong H_ce\otimes_{U_c} eH_c$ from Theorem \[morrat\] is clearly not filtered; in writing the element $1$ as an element of $H_ce \otimes eH_c$ an easy computation shows that one needs to use commutators of elements from ${\mathbb{C}}[{\mathfrak{h}}]$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$ and so $1\notin \operatorname{{\textsf}{ten}}^0(H_c)$. However, $ge=ge\cdot 1 \in \operatorname{{\textsf}{ten}}^0(H_c)$ for any $0\not=g\in {\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ and so $\sigma(ge)\sigma(1)=0$ in $\operatorname{{\textsf}{tgr}}H_c$. On the other hand, as $1$ is a regular element of $\operatorname{{\textsf}{ogr}}H_c$, no such equation is possible $\operatorname{{\textsf}{ogr}}H_c$. Thus $\operatorname{{\textsf}{ten}}H_c\not\cong \operatorname{{\textsf}{ogr}}H_c$.
As a second example, it is easy to check that Lemma \[ord-tens\] will fail for $M(i)$ if one introduces one more tensor product, $M(i) \cong H_{c+i}e \otimes_{U_{c+i}} B_{i0}$. Indeed, Lemma \[thetainjC\] implies that $ \operatorname{{\textsf}{ogr}}
| 2,291
| 1,669
| 937
| 2,399
| 3,084
| 0.77498
|
github_plus_top10pct_by_avg
|
eq:split.hom\] for every $x,y\in G/N$ with $x,y,xy\in\pi(A)$ we have $\varphi(xy)\in\varphi(x)\varphi(y)\left(A^3\cap N\right)$.
This is essentially just an observation: by definition of $\varphi$ we have $a\varphi(\pi(a))^{-1}\in A^2\cap N$ and $\varphi(y)^{-1}\varphi(x)^{-1}\varphi(xy)\in A^3\cap N$.
\[lem:pullback.large\] Let $G$ be a group, let $N\lhd G$ be a normal subgroup, and let $\pi:G\to G/N$ be the quotient homomorphism. Let $A$ be a finite symmetric subset of $G$, and let $P\subset\pi(A^m)$. Suppose that $|P|\ge c|\pi(A)|$. Then $|\pi^{-1}(P)\cap A^{m+2}|\ge c|A|$.
\[lem:fibre.pigeonhole\] implies that $|N\cap A^2|\ge|A|/|\pi(A)|$, which in turn implies that $|\pi^{-1}(x)\cap A^{m+2}|\ge|A|/|\pi(A)|$ for every $x\in\pi(A^m)$. In particular, $|\pi^{-1}(P)\cap A^{m+2}|\ge|A||P|/|\pi(A)|\ge c|A|$, as desired.
Write $\pi:G\to G/[G,G]$ for the quotient homomorphism, and note that $\pi(A)$ is a finite $K$-approximate subgroup of the abelian group $G/[G,G]$. Theorem \[thm:sanders\] therefore implies that there exists a finite subgroup $H\subset G/[G,G]$, and a progression $P=\{x_1^{\ell_1}\cdots x_r^{\ell_r}:|\ell_i|\le L_i\}$ with $r\le\log^{O(1)}2K$ such that $HP\subset\pi(A^4)$ and $|HP|\ge\exp(-\log^{O(1)}2K)|\pi(A)|$. Lemma \[lem:pullback.large\] then implies that $$\label{eq:pullback.large}
|\pi^{-1}(HP)\cap A^6|\ge\exp(-\log^{O(1)}2K)|A|.$$ Now let $\varphi:\pi(A^6)\to A^6$ be an arbitrary map such that $\pi(\varphi(x))=x$ for every $x\in\pi(A^6)$. Suppose that $a\in\pi^{-1}(HP)\cap A^6$, so that there exist $h\in H$ and $\ell_1,\ldots,\ell_r\in{\mathbb{Z}}$ such that $\pi(a)=hx_1^{\ell_1}\cdots x_r^{\ell_r}$. It follows from \[lem:splitting\] \[eq:split.inv\] that $$\begin{aligned}
a&\in\left(A^{12}\cap[G,G]\right)\varphi(\pi(a))\\
&=\left(A^{12}\cap[G,G]\right)\varphi(hx_1^{\ell_1}\cdots x_r^{\ell_r}),\end{aligned}$$ and hence by repeated application of \[lem:splitting\] \[eq:split.hom\] that $$\begin{aligned}
a&\in\left(A^{12}\cap[G,G]\right)\varphi(h)\prod_{i=1}^r\varphi(x_i^{\ell_i})\left(A^{
| 2,292
| 1,655
| 1,234
| 2,187
| null | null |
github_plus_top10pct_by_avg
|
\Omega) } {d \Omega} = \Big[ (I_k \otimes \Omega) \otimes I_{k^2}
\Big] \frac{d (\Omega \otimes I_k)}{d \Omega} + \Big[ I_{k^2} \otimes(
\Omega \otimes I_k) \Big] \frac{d (I_k \otimes \Omega)}{d \Omega}.$$ Next, $$\frac{d (\Omega \otimes I_k)}{d \Omega} = \Big( I_k \otimes K_{k,k} \otimes
I_k \Big) \Big( I_{k^2} \otimes \mathrm{vec}(I_k) \Big) = \Big(
I_{k^2} \otimes K_{k,k} \Big) \Big(I_k \otimes \mathrm{vec}(I_k) \otimes I_k \Big)$$ and $$\frac{d (I_k \otimes \Omega )}{d \Omega} = \Big( I_k \otimes K_{k,k} \otimes
I_k \Big) \Big( \mathrm{vec}(I_k) \otimes I_{k^2} \Big) = \Big(
K_{k,k} \otimes I_{k^2} \Big) \Big(I_k \otimes \mathrm{vec}(I_k) \otimes I_k
\Big).$$ The formula for $J$ follows from the last three expressions. Notice that $J$ is matrix of size $k^4 \times k^2$. Finally, plugging the expressions for $\frac{d (\alpha^\top \otimes I_k) (\Omega \otimes \Omega) }{d \psi}$ and $\frac{ d \Omega }{d \psi}$ in we get that the Hessian $H g_j(\psi)$ is $$\label{eq::Hessian}
\frac{1}{2}\left( (I_b \otimes e^\top_j) H + H^\top (I_b \otimes e_j) \right)$$ where $$\label{eq:Halcazzo}
H = \left[
\begin{array}{c}
- \Big( (\Omega \otimes \Omega) \otimes I_k \Big) \Big[0_{k^3 \times
k^2} \;\;\;\;\; I_k \otimes \mathrm{vec}(I_k) \Big] +
\Big( I_{k^2} \otimes (\alpha^\top
\otimes I_k )\Big) J \Big[ \Omega \otimes \Omega \;\;\;\;\;
0_{k^2 \times k}\Big]\\
\;\\
\Big[ - \Omega \otimes \Omega \;\;\;\;\; 0_{k^2 \times k} \Big]
\end{array}
\right].$$
So far we have ignored the facts that $\Sigma$ is symmetric. Account for the symmetry, the Hessian of $g_j(\psi)$ is $$D_h^\top H g_j(\psi) D_h,$$ where $D_h$ is the modified duplication matrix such that $D \psi_h = \psi$, with $\psi_h$ the vector comprised by the sub-vector of $\psi$ not including the entries corresponding to the upper (or lower) diagonal entries of $\Sigma$.
We now prove the bounds and . We will use repeatedly the fact that $\sigma_1(A \otimes B) = \sigma_1(A)
\sigma_1(B)$ and, for a vector $x$, $\sigma_1(x) = \
| 2,293
| 4,376
| 1,674
| 1,661
| null | null |
github_plus_top10pct_by_avg
|
I}_{\space\text{n}}=-\int\frac{1}{\sqrt{1-\text{u}^2}}\space\text{d}\text{u}=\text{C}-\arcsin\left(\text{u}\right)=\text{C}-\arcsin\left(\frac{\cot\left(x\right)}{\sqrt{\text{n}}}\right)\tag3$$
Q:
PHP: Does password_hash() check if the hash generated is unique? (Understanding!)
Simple question because i did not find a really helping answer on google: Does the password_hash() function also check if there is already such a hash generated for instance in the userdata file?
I basically get what the function is doing, but i am fairly new to php, so i was not really able to see if the password is checked for uniqueness.
Please be gentle on this noob question right here. I simply want to understand what i am using right there, and not only do it because my exercise sheet at university tells me so.
A:
No.
Two reasons:
the function will not know anything about "other" passwords.
You don't want this. What will you do if you hit a duplicate? Tell the user? He will then know someone's password.
Q:
Create an alias for a AWS S3 bucket
Is it possible to setup a working bucket or to setup an alias for a bucket in AWS S3 CLI console? Because for every single command you have to enter the bucket path and I could't find a solution.
A:
There IS an alias capability for the AWS Command-Line Interface (CLI). See:
AWS re:Invent 2016: The Effective AWS CLI User (DEV402)
GitHub - awslabs/awscli-aliases: Repository for AWS CLI aliases.
It can be used to substitute whole commands, but I'm not sure if it can be used to substitute variables within a line. For that, just use your the capabilities of your OS, eg:
$ my_bucket=s3://my-bucket/
$ aws s3 ls $my_bucket
Q:
Getting and storing methods of a python class
First of all I am new to python and a bit rusty on .NET so bear with me if this sounds
too obvious.
Assuming the following python classes;
class foo1(object):
def bar1(self):
print "bar1 called."
def bar2(self):
print "bar2 called."
class foo2 ..... same as above
I want to store th
| 2,294
| 1,907
| 2,172
| 2,207
| 755
| 0.800755
|
github_plus_top10pct_by_avg
|
y such constraint one wished to impose on individual twisted sectors.
[^25]: For $m>0$. The total spaces of line bundles of positive degree over projective spaces do not seem to admit a GLSM description, even though they are toric varieties – they can be described as GIT quotients of open subsets of ${\mathbb C}^{n+2}$ by ${\mathbb C}^{\times}$, but not as a GIT quotient of the full complex vector space, and they naturally compactify to ${\mathbb P}^{n+1}_{[1,\cdots,1,m]}$. We would like to thank D. Skinner for asking a question that made this manifest.
[^26]: We would like to thank T. Pantev for explaining this example to us.
---
abstract: 'The super-inflationary phase is predicted by the Loop Quantum Cosmology. In this paper we study the creation of gravitational waves during this phase. We consider the inverse volume corrections to the equation for the tensor modes and calculate the spectrum of the produced gravitons. The amplitude of the obtained spectrum as well as maximal energy of gravitons strongly depend on the evolution of the Universe after the super-inflation. We show that a further standard inflationary phase is necessary to lower the amount of gravitons below the present bound. In case of the lack of the standard inflationary phase, the present intensity of gravitons would be extremely large. These considerations give us another motivation to introduce the standard phase of inflation.'
author:
- Jakub Mielczarek
- 'Marek Szyd[ł]{}owski'
title: 'Relic gravitons from super-inflation'
---
Introduction {#sec:intro}
============
The cosmological creation of the gravitational waves was proposed by Grishchuk [@Grishchuk:1974ny] in the mid-seventies. Since that time this phenomenon has been studied extensively, especially in the context of the inflation. The accelerating expansion phase gives the conditions for the abundant creation of the gravitational waves. Gravitons produced during the inflation fill the entire space in the form of a stochastic background. Together with the scalar modes, prod
| 2,295
| 1,590
| 792
| 2,127
| null | null |
github_plus_top10pct_by_avg
|
evel 1 list so maintaining parent child relationship. Normalized structure would be cumbersome for now but it would have long term benefits and flexibility
Step 2 : Drawing Chart
Method 1 : Jquery Charts
Fetch data from SharePoint List using Rest API or CSOM on client side.
CRUD-Operation-to-List-Using-SharePoint-Rest-API
Complete basic operations using JavaScript library code in SharePoint 2013
Create either JSON Structure or HTML Structure depending on Widget or Plugin for Org chart you using.
Few of the reference are below and you can find many more options online...
dabeng/OrgChart
caprica/jquery-orgchart
Google Org Chart
Highcharts Org Chart
Method 2 : SSRS
You can also use SSRS to achieve this but the output is not as elegant as Jquery plugins would provide you.
Recursive Hierarchy Group in SSRS 2008
SSRS Org Charts
Q:
Frobenius isomorphism for Hopf algebras
It is known that a finite dimensional Hopf algebra $H$ over a field $k$ is a Frobenius algebra. Thus there is an isomorphism $H \cong H^\ast$ of left $H$-modules.
Question: Is it possible to write down such an isomorphism entirely in terms of the Hopf algebra, i.e. by product, coproduct, unit, counit and involution ?
A:
This isn't an answer but a lengthy comment.
The proofs for the $H$-modul ismorphism $H \cong H^\ast$ I know of use (some variant of) the $H$-module isomorphism $I_L(H^\ast) \otimes_k H \cong H^\ast$ where
$$I_L(H^\ast) = \lbrace g \in H^\ast \mid \forall f\in H^\ast: f\ast g = f(1) \cdot g \rbrace$$ is the space of left integrals of $H^\ast$ ($f \ast g$ is the convolution). A dimension argument shows that $I_L(H^\ast)$ is one-dimensional. If $\lambda$ is a non-zero left integral then $I_L(H^\ast) = k\cdot \lambda \cong k$ yields $H \cong H^\ast$ as left $H$-modules. This is just the isomorphism described by darij.
The problem is that the proof doesn't yield a formula for such a $\lambda$. Choose a $k$-basis $\lbrace e_1=1,e_2,...,e_n\rbrace$ of $H$ and suppose $\Delta(e_k) = \sum_{i,j}d_{ij}^{(k)}\cdot e_i \otimes e_j$. Se
| 2,296
| 2,644
| 1,876
| 2,216
| 2,192
| 0.781896
|
github_plus_top10pct_by_avg
|
rt}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
We write the value of $c$ satisfying $\frac{d}{dc} \operatorname{{E}}[\Pe(Z + c)] = 0$ as $C$. Then, we find that $\operatorname{{E}}[\Pe(Z + c)]$ has a minimum value at $c = C$. Also, it follows from $$\begin{aligned}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
= \operatorname{{sgn}}(C) \frac{k_{2} - k_{1}}{k_{1} + k_{2}} \G(a) \end{aligned}$$ that $\operatorname{{sgn}}(C) = \operatorname{{sgn}}(k_{2} - k_{1})$, where $\operatorname{{sgn}}(c) := 1 \: (c \geq 0); -1 \: (c < 0)$, and $C = 0$ only when $k_{1} = k_{2}$. This equation implies that the ratio of $\G(a)$ and $\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)$ is $1 : \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$. That is, the vertical axis $t = \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}}$ divides the area between $t^{a - 1} e^{- t}$ and the $t$-axis into $\frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}} : 1- \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$.
Substituting $c = C$ in the equation $(1)$ of Lemma $\ref{lem:1.1}$, from the equation $(3)$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + C)]
= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ This is the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. From this and the $c = 0$ case of the equat
| 2,297
| 4,401
| 2,491
| 1,828
| null | null |
github_plus_top10pct_by_avg
|
})$ a root system of type ${\mathcal{C}}$. Let ${\mathcal{W}}$ be the abstract groupoid with ${\mathrm{Ob}}({\mathcal{W}})=A$ such that ${\mathrm{Hom}}({\mathcal{W}})$ is generated by abstract morphisms $s_i^a\in {\mathrm{Hom}}(a,{r}_i(a))$, where $i\in I$ and $a\in A$, satisfying the relations $$\begin{aligned}
s_i s_i 1_a=1_a,\quad (s_j s_k)^{m_{j,k}^a}1_a=1_a,
\qquad a\in A,\,i,j,k\in I,\, j\not=k,
\end{aligned}$$ see Conv. \[con:uind\]. Here $1_a$ is the identity of the object $a$, and $(s_j s_k)^\infty 1_a$ is understood to be $1_a$. The functor ${\mathcal{W}}\to {\mathcal{W}}({\mathcal{R}})$, which is the identity on the objects, and on the set of morphisms is given by $s _i^a\mapsto \s_i^a$ for all $i\in I$, $a\in A$, is an isomorphism of groupoids.
If ${\mathcal{C}}$ is a Cartan scheme, then the Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ admits a length function $\ell :{\mathcal{W}}(\cC
)\to {\mathbb{N}}_0$ such that $$\begin{aligned}
\ell (w)=\min \{k\in {\mathbb{N}}_0\,|\,\exists i_1,\dots ,i_k\in I,a\in A:
w={\sigma }_{i_1}\cdots {\sigma }_{i_k}1_a\}
\label{eq:ell}\end{aligned}$$ for all $w\in {\mathcal{W}}({\mathcal{C}})$. If there exists a root system of type ${\mathcal{C}}$, then $\ell $ has very similar properties to the well-known length function for Weyl groups, see [@a-HeckYam08].
Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{R}}$ a root system of type ${\mathcal{C}}$. Let $a\in A$. Then $-c^a_{ij}=\max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in
R^a_+\}$ for all $i,j\in I$ with $i\not=j$. \[le:cm\]
By (C2) and (R3), ${\sigma }_i^{r_i(a)}({\alpha }_j)={\alpha }_j-c^a_{ij}{\alpha }_i\in R^a_+$. Hence $-c^a_{ij}\le \max\{m\in {\mathbb{N}}_0\,|\,{\alpha }_j+m{\alpha }_i\in R^a_+\}$. On the other hand, if ${\alpha }_j+m{\alpha }_i\in R^a_+$, then ${\sigma }_i^a({\alpha }_j+m\al
_i)={\alpha }_j+(-c^a_{ij}-m){\alpha }_i\in R^{r_i(a)}_+$ by (R3) and (R1), and hence $m\le -c^a_{ij}$. This proves the lemma.
Let ${\mathcal{C}}$ be a Cartan scheme and ${\mathcal{
| 2,298
| 2,432
| 2,142
| 2,115
| null | null |
github_plus_top10pct_by_avg
|
ever we point out that this is *not the same as the mapping space to the group completion, which would be the zero space of the mapping spectrum $Map_0 (M, K(\Sigma^\infty (G_+))$. This spectrum calculates the ${K(\Sigma^\infty (G_+))}$-cohomology of $M$. However, as we will show below, we can define a homomorphism of $K$-theory groups,*
$$\label{kgcoho}
\gamma: K_{conn}^{-q}({\mathcal{S}}(P)) \to {K(\Sigma^\infty (G_+))}^{-q}(M)$$
which gives a partial geometric understanding of the ${K(\Sigma^\infty (G_+))}$-cohomology theory in terms of the algebraic $K$-theory of the string topology spectrum. The situation when $q=0$ was studied in detail by Lind in [@lind].
We conclude by observing two important applications of Theorem \[ktheory\].
\[application\] Let $M$ be a closed manifold. There are homology equivalences $$\begin{aligned}
Map_{\Sigma^\infty_M ({\mathcal{P}}_+)} (M, BGL(\Sigma^\infty (\Omega M_+)) &\to \Omega^\infty_0 K_{conn}(LM^{-TM}) \notag \\
Map_{{\mathbb{S}}} (M, BGL({\mathbb{S}})) &\to \Omega_0^\infty K_{conn}(DM) \notag\end{aligned}$$ where ${\mathbb{S}}$ is the sphere spectrum, and ${LM^{-TM}}$ is the Thom spectrum of the virtual bundle $-TM$ over $M$, pulled back over $LM$ via the map $e : LM \to M$ that evaluates a loop at the basepoint of the circle. $DM$ denotes the Spanier-Whitehead dual of the manifold $M$, which is an $E_\infty$-ring spectrum.
We point out that in these cases, the map $\gamma$ defined above (\[kgcoho\]) gives homomorphisms $$\gamma : K_{conn}^{-q}(LM^{-TM}) \to A(M)^{-q}(M) \quad \text{and} \quad \gamma : K_{conn}^{-q}(D(M)) \to A(point)^{-q}(M).$$
The algebraic $K$-theory of nonconnective spectra was defined in terms of Waldhausen categories of modules by Blumberg and Mandell in [@blumbergmandell]. When $X$ is simply connected, they related the Waldhausen category defining $K(D(X))$ to the Waldhausen category defining $A(X) = K(\Sigma^\infty (\Omega X_+))$. It would be interesting to relate Corollary \[application\] regarding the $K_{conn}$-theory
| 2,299
| 1,507
| 2,212
| 2,060
| null | null |
github_plus_top10pct_by_avg
|
eduling in the real-time specification for java. In: [*Proceedings of the 4th international workshop on Java technologies for real-time and embedded systems*]{}, 2006, 20–29.
Alves-Foss J. Multiple independent levels of security. In: [ *Encyclopedia of Cryptography and Security*]{}, Springer US, 2011, 815–818.
Clarkson M and Schneider F. Hyperproperties. , 2010, 18(6):1157–1210.
Clarkson M, Finkbeiner B, Koleini M, Micinski K, Rabe M, and S[á]{}nchez C. Temporal logics for hyperproperties. In: [*Proceedings of International Conference on Principles of Security and Trust*]{}, 2014, 265–284.
Abrial JR. Formal methods in industry: Achievements, problems, future. In: [*Proceedings of the 28th International Conference on Software Engineering*]{}, 2006, 761–768.
[^1]: http://www.euromils.eu/
[^2]: http://www.verisoftxt.de/StartPage.html
[^3]: http://research.microsoft.com/en-us/projects/vcc/
---
author:
- Masashi KOSUDA
title: |
**Partition Algebra,\
Its Characterization and Representations**
---
[**Abstract**]{}
In this note we give representations for the partition algebra $A_{3}(Q)$ in Young’s seminormal form. For this purpose, we also give characterizations of $A_{n}(Q)$ and $A_{n-\frac{1}{2}}(Q)$.
Introduction
============
Definition of the partition algebra
-----------------------------------
Let $M = \{1, 2, \ldots, n\}$ be a set of $n$ symbols and $F = \{1', \ldots, n'\}$ another set of $n$ symbols. We assume that the elements of $M$ and $F$ are ordered by $1<2<\cdots <n$ and $1'<2'<\cdots<n'$ respectively. Consider the following set of set partitions: $$\begin{aligned}
\Sigma_n^1
&=&
\{\{T_1, \ldots, T_s\}\ |\ s=1,2, \ldots \ ,\nonumber\\
& & \quad T_j(\neq\emptyset)\subset M\cup F\
(j = 1, 2, \ldots, s),\\
& & \quad \cup T_j = M\cup F,\quad T_i\cap T_j = \emptyset \mbox{ if }
i\neq j\nonumber\}.\end{aligned}$$ We call an element $w$ of $\Sigma_n^1$ [*a seat-plan*]{} and each element of $w$ a [*part*]{} of $w$. It is easy to see that the number of seat-plans is equa
| 2,300
| 662
| 2,192
| 2,319
| 616
| 0.803903
|
github_plus_top10pct_by_avg
|
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