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t. Suppose -4/9*n**2 + g + 4/3*n = 0. Calculate n.
-2, 5
Let t = 9370 - 327938/35. Let q(w) be the first derivative of 23 + 0*w**2 + t*w**5 + 0*w + 2/7*w**3 + 1/14*w**6 + 15/28*w**4. Suppose q(b) = 0. What is b?
-2, -1, 0
Let p = 725 - 731. Let b be (200/35 + p)/(4/(-6)). Find d such that b*d**2 + 75/7 - 30/7*d = 0.
5
Let f(r) = -39*r**2 - 5494*r + 750746. Let p(a) = -16*a**2 - 2746*a + 375374. Let b(o) = 3*f(o) - 7*p(o). Find c such that b(c) = 0.
274
Let z(k) be the third derivative of 7*k**6/240 + 67*k**5/12 + 296*k**4/3 - 352*k**3/3 + k**2 - 20*k - 21. Factor z(b).
(b + 8)*(b + 88)*(7*b - 2)/2
Let p(j) be the second derivative of -4/5*j**3 - 88*j - 2 + 5/2*j**2 - 1/60*j**4. Factor p(m).
-(m - 1)*(m + 25)/5
Suppose 268/7*m**2 - 36/7*m**3 + 96/7 - 328/7*m = 0. What is m?
4/9, 1, 6
What is f in 0 + 34/9*f**2 - 34/9*f**4 + 140/9*f - 46/3*f**3 - 2/9*f**5 = 0?
-10, -7, -1, 0, 1
Suppose 0 - 10/3*f**2 + 0*f - 2/3*f**4 + 4*f**3 = 0. What is f?
0, 1, 5
What is y in -83*y**2 + y**3 - 56783*y + 56782*y - 48 + 131 = 0?
-1, 1, 83
Let a(q) = -29*q + 0*q**3 - q**3 - 227*q**2 + 214*q**2 + 12. Let d be a(-10). Factor -28*r**3 - 12*r**d - 37*r**3 + 66*r**3.
r**2*(r - 12)
Factor 22*a**2 - 68*a + 26*a**2 + 3*a**3 + 240 - 65*a - 95*a.
3*(a - 2)**2*(a + 20)
Let j(t) be the second derivative of -t**5/5 - t**4/6 + 47*t**2/2 + t - 111. Let o(z) be the first derivative of j(z). Factor o(y).
-4*y*(3*y + 1)
Let h = 598280/21 + -85448/3. Factor -h - 3/7*j**3 + 12/7*j**2 + 12/7*j.
-3*(j - 4)*(j - 2)*(j + 2)/7
Let y(r) be the second derivative of -r**5/5 + 26*r**4/3 - 80*r**3 + 269*r. Factor y(z).
-4*z*(z - 20)*(z - 6)
Let k(f) be the third derivative of f**6/120 + 2*f**5/5 + 11*f**4/24 - 46*f**3 - 4724*f**2. Solve k(c) = 0 for c.
-23, -4, 3
Let l(m) be the first derivative of -m**6/75 + m**4/5 - 8*m**3/15 + 3*m**2/5 - 54*m - 4. Let t(s) be the first derivative of l(s). Solve t(b) = 0 for b.
-3, 1
Let a(l) be the second derivative of -l**6/75 - 4*l**5/25 - l**4/10 + 32*l**3/15 + 28*l**2/5 - 9201*l. Solve a(n) = 0.
-7, -2, -1, 2
Let t(b) be
| 2,901
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s the proof.
Proof of Proposition \[pro:dy\] {#sub:dy}
===============================
In this subsection we will prove two lemmas and then combine them to establish the proposition. The lemmas and their proofs are inspired by [@KP06 Lemma 2.1 and 2.2]. Recall that a subgraph of ${\mathcal{C}}_n$ is $c$-loaded if every vertex (bin) in the subgraph has load at least $c$.
\[lem:tree\] [Let $k$ be a positive integer and let $c_1>0$.]{} The probability that conflict graph ${\mathcal{C}}_n$ contains a [$c_1$-loaded]{} connected component with $k$ vertices is at most $$n\cdot 8^{k}\cdot\left(\frac{2{\mathrm{e}}}{c_1}\right)^{c_1k}.$$ Moreover, by setting $c_1=12(c+1)$, we conclude that with probability at least $1-n^{-c}$, the conflict graph ${\mathcal{C}}_n$ does not contain a $c_1$-loaded tree with at least $\log n$ vertices.
A connected component in ${\mathcal{C}}_n$ with $k$ vertices contains a spanning tree with $k$ vertices. By Proposition \[pro:ordered\], there are at most $4^{k-1}$ ordered trees [with $k$ vertices]{}. For every ordered tree, we can choose its root in $n$ ways, as we have $n$ bins (vertices). Hence there are at most $n\cdot 4^{k-1}$ rooted and ordered trees. Let us fix an arbitrary ordered tree $T$ with a specified root. Also let $(t_1, \ldots, t_{k-1})$ denote an arbitrary sequence of rounds, where $t_i \in\{1,\ldots, n\}$ is the round when the $i$-th edge of the ordered tree $T$ is chosen. Notice that in an ordered tree with specified root, the $i$-th edge always connects the $i$-th child to its parent, and the parent is already known to us. Therefore, to build the tree, the $i$-th edge of the tree must be chosen from edges of $G^{(t_i)}$ that are adjacent to the known parent. This implies that the algorithm chooses the $i$-th edge [of $T$ in round $t_i$]{} with probability $\frac{\Delta_{t_i}}{n\Delta_{t_i}/2}=\frac{2}{n}$. Since balls are independent from each other, the tree $T$ is constructed at the given times $(t_1,\ldots,t_{k-1})$ with probability $$\begin{aligned}
\label{up:1}
| 2,902
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| 701
| 0.802132
|
github_plus_top10pct_by_avg
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ages implement a Cheddar program compiler and interpreter. Then scheduling simulation analysis is performed on AADL specifications with hierarchical schedulers.
- A two-level scheduler for RTSJ
The Real-Time Specification for Java (RTSJ) is a set of interfaces and behavioral specifications that allow for real-time computer programming in the Java programming language. It is modified to allow applications to implement two-level scheduling mechanism where the first level is the RTSJ priority scheduler and the second level is under application control [@Zerzelidis06b; @Zerzelidis10]. They also verify the two-level scheduler for RTSJ using Timed Automata in the UPPAAL tool [@Zerzelidis06]. The *Thread*, *BaseScheduler* (global scheduler), *EDFScheduler*(local scheduler) and other components are presented by timed automata. Five properties are verified on their model. Three of them are to check the correctness of their model: (1) a thread’s priority never takes an invalid value, (2) no thread can block due to locking after it starts, and (3) the system will always select a thread to run with higher absolute preemption level than the system ceiling, unless the selected thread is either currently locking a resource with higher ceiling than its apl or a thread that has just been released. The other two are liveness and deadlock free properties that state the system is livelock free and can never deadlock.
Summary {#sec:summary}
=======
Comparison of Related Work
--------------------------
We summarize the research work on formal specification and verification of separation kernels in [[[Table]{}]{}]{} \[tbl:comparison\_tab\]. In this table, “[[$\divideontimes$]{}]{}” means that the evidence for the data is not available and empty cells mean that the feature is not considered in the work. We compare seven features of them. The column “Target Kernel” is the object specified or verified in each work. The “Objective” shows the concerns of each work, in which *Specification* indicates that the work concen
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|
it may so happen that operators like $\widetilde{H}_N $, acting on a suitably chosen function subspace can preserve the space partially. In such cases we introduce the term quasi-solvability. A linear differential operator $H_N$ of several variables $\{z_j \vert j=1,\dots,N\}$, is said to be quasi-solvable if it preserves a finite dimensional function space $V_{\nu}$ whose basis admits an analytic expression in a closed form i.e., H\_N V\_ V\_, V\_ = n() < , 0.5cm 0.5cm V\_ = v\_1(z),…, v\_[n()]{}(z).
One of the advantages of quasi-solvability is that one can explicitly evaluate finite dimensional matrix elements $A_{kl}$ defined by H\_N v\_k = \_[l=1]{}\^[n()]{}A\_[kl]{}v\_l, (k = 1,…, n()). The finite dimensional submatrices $A_{kl}$ may be diagonalizable even when the entire $H_N$ is not. If the space $V_{\nu}$ is the subspace of a Hilbert space on which the operator $H_N$ is defined, the spectrum of $H_N$ can be computed algebraically, so as to obtain the exact eigenvectors of $H_N$ that belong entirely to $V_{\nu}$. This is the typical nature of quasi-solvability.
A quasi-solvable operator is said to be solvable if the quasi-solvability condition holds for an infinite number of sequences of finite dimensional proper subspaces each containing its previous descendant. V\_1 V\_2 …V\_ … Moreover, if the closure of $V_{\nu}$, as $\nu \rightarrow \infty $, is the Hilbert space on which $H_N$ acts, we call $H_N$ to be exactly solvable.
Now, we shall show that the Dunkl-type momentum operators obtained in Eq.(\[mom11\]) preserve the space $\mathcal{R}_n$ i.e., the space spanned by all monomials of the form $\prod_i z_i^{\ell_i}$, where $\ell_i \geq 0$ and $\sum_i \ell_i = n$, $n$ being a non negative integer.
It is easy to verify that the operator $(z_j\partial_j)$ preserves the space $\mathcal{R}_n$. (z\_j\_j) \_i z\_i\^[\_i]{} = \_j \_i z\_i\^[\_i]{}
We shall show that the second and third operators in Eq. (\[mom11\]) preserve $\mathcal{R}_n$, i.e., $(z_j+z_k)/(z_j-z_k)(1-\Lambda_{jk})\prod_i z_i^{
| 2,904
| 1,970
| 2,963
| 2,916
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github_plus_top10pct_by_avg
|
*m**2*s + m*s**2 + 5*s**2 wrt m.
-18*s**2 + 1770*s - 42
What is the second derivative of -4*c*m**5 + 34278*c*m**2 - 29*c*m - c - 2*m + 363 wrt m?
-80*c*m**3 + 68556*c
What is the third derivative of -822*b**6 - 2*b**5 + 8*b**4 + 72952*b**2?
-98640*b**3 - 120*b**2 + 192*b
What is the third derivative of 126472*w**6 + w**5 - 2*w**3 + 3*w**2 + 40117*w?
15176640*w**3 + 60*w**2 - 12
Find the third derivative of 1010*j**2*s**3 + 58*j**2*s**2 - 3*j**2 - 34*j*o**2*s**3 - 3*j*o - o**2*s**2 wrt s.
6060*j**2 - 204*j*o**2
Find the second derivative of -5103549*k**3 + 226399*k wrt k.
-30621294*k
What is the third derivative of 13*d*f**3*z**3 - 35*d*f**3*z**2 + 2*d*f**3*z + 13*d*f**2*z**2 - 2*d*z**3 + 5*d*z + 64009*z**3 wrt z?
78*d*f**3 - 12*d + 384054
What is the third derivative of 19221*d**4 - 2*d**3 - 174104*d**2?
461304*d - 12
What is the first derivative of 1024*j**2 + 15*j + 42258 wrt j?
2048*j + 15
What is the third derivative of 15*w**4 + 42820*w**3 + 41*w**2 - 18*w - 411?
360*w + 256920
What is the third derivative of -1668*l**2*n**4*x**2 + 20*l**2*n**2*x - l**2*n**2 + 54*l**2*n*x**2 + 686*l*n**4 - 13*n*x**2 wrt n?
-40032*l**2*n*x**2 + 16464*l*n
What is the second derivative of 57072*b*s**2 - b*s + 180*b - s + 10 wrt s?
114144*b
Differentiate 503*i**4*p**2 + 120332*p**2 wrt i.
2012*i**3*p**2
Differentiate -105*o**3 + 496*o**2 - 2*o + 280312 with respect to o.
-315*o**2 + 992*o - 2
Differentiate -82530*w**3 + 58517.
-247590*w**2
Find the third derivative of -2689*u**6 + u**4*x - 144*u**2*x + 24*u**2 + 2*u*x wrt u.
-322680*u**3 + 24*u*x
Find the first derivative of -j*l*v + 13306*j*v - 70*l*v + 358*l wrt j.
-l*v + 13306*v
What is the third derivative of -239670*l**3 + 91670*l**2 + 3 wrt l?
-1438020
What is the first derivative of o**2 - 6473*o + 119424?
2*o - 6473
Find the third derivative of 30*x**6 + 36*x**5 - 12*x**3 - 464650*x**2.
3600*x**3 + 2160*x**2 - 72
Find the third derivative of 1176*z**6 - 111*z**5 - 2*z**3 - 313084*z**2 wrt z.
141120*z**3 - 6660*z**2 - 12
Find the second derivative of 13831*d**4 - 2*d**3 -
| 2,905
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Assume that $i$ is even. A $\pi^i$-modular lattice $L$ is *of parity type I* if $n(L)=s(L)$, and *of parity type II* otherwise. The zero lattice is considered to be *of parity type II*. We caution that we do not assign a *parity type* to a $\pi^i$-modular lattice $L$ with $i$ odd.
\[r23\]
1. If $L$ is $\pi^i$-modular, then $\pi^j L$ is $\pi^{i+2j}$-modular for any integer $j$.
2. (Section 4 in [@J]) For a general lattice $L$, we have a Jordan splitting, namely $L=\bigoplus_i L_i$ such that $L_i$ is $\pi^{n(i)}$-modular and such that the sequence $\{n(i)\}_i$ increases. Two Jordan splittings $L=\bigoplus_{1\leqq i \leqq t} L_i$ and $K=\bigoplus_{1\leqq i \leqq T} K_i$ will be said to be of the same type if $t=T$ and, for $1\leqq i \leqq T$, the following conditions are satisfied: $s(L_i)=s(K_i)$, rank $L_i$ = rank $K_i$, and $n(L_i)=s(L_i)$ if and only if $n(K_i)=s(K_i)$. Jordan splitting is not unique but partially canonical in the sense that two Jordan splittings of isometric lattices are always of the same type.
3. If we allow some of the $L_i$’s to be zero, then we may assume that $n(i) = i$ for all $i$. In other words, for all $i\in \mathbb{N}\cup \{0\}$ we have $s(L_i)=(\pi^i)$, and, more precisely, $L_i$ is $\pi^i$-modular. Then we can rephrase part (b) above as follows. Let $L=\bigoplus_i L_i$ be a Jordan splitting with $s(L_i)=(\pi^i)$ for all $i\geq 0$. Then the scale, rank and parity type of $L_i$ depend only on $L$. We will deal exclusively with a Jordan splitting satisfying $s(L_i)=(\pi^i)$ from now on.
Lattices
--------
\[Section 2C in [@C2]\]\[lattices\]
In this subsection, we will define several lattices and associated notation. Fix a hermitian lattice $(L, h)$. We denote by $(\pi^l)$ the scale $s(L)$ of $L$.
- Define $A_i=\{x\in L \mid h(x,L) \in \pi^iB\}.$
- Define $X(L)$ to be the sublattice of $L$ such that $X(L)/\pi L$ is the radical of the symmetric bilinear form $\frac{1}{\pi^l}h$ mod $\pi$ on $L/\pi L$.
Let $l=2m$ or $l=2m-1$. We consider the function defined over $L$ b
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|
nt: method and leads Age (SD) No. of males (%) No. of Sp. type 1 patients (%) No. of SCN5a positive patients (%) Endpoints Comparisons No. of patients with adverse events /without adverse events/%/% per year Follow‐up duration (months) Quality score Ref.
------------------- ----------------- ---------------------------------------------- ---------- ------------------ -------------------------------- ------------------------------------ ----------------------- ------------------------------------------------ -------------------------------------------------------------------------- ----------------------------- --------------- -------------------------------------------
Morita 2017 471 Tangent method; V1, V2, V3, V5 47 (19) 447 (95) 118 (25) 27 (15) Syncope or VT/VF Syncope/VT/VF vs asymptomatic 145/326/31/4.09 91 7 [16](#joa312118-bib-0016){ref-type="ref"}
Mugnai 2017 448 End of the T‐wave; V1 to V6 45 (16) 273 (61) 96 (21) 55 (22) Spontaneous VF or SCD AT/SD vs asymptomatic 43/290/13/1.67 93 6 [7](#joa312118-bib-0007){ref-type="ref"}
Kawazoe 2016 143 Tangent method; V1 to V6 46 (12) 140 (98) 84 (59) -- VF VF vs no VF 35/108/24/1.9 105 7 [17](#joa312118-bib-0017){ref-type="ref"}
Zumhagen 2016 78
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&Education Assistance8
For additional information about Enron Corp's Organizational Development and
Training offerings, contact:
Suzanne Gruber, Senior Director
713/345-8314
Email: suzanne.gruber@enron.com
Nothing with or pertaining to ASpen to the best of my knowledge.
Kay C. Young
Legal Specialist
Enron North America Corp.
713-853-6794 Phone
713-646-3393 Fax
kay.young@enron.com
Tana Jones
04/17/2001 04:20 PM
To: Kay Young/HOU/ECT@ECT
cc:
Subject: Aspen Technologies NDA
Any conflicts
----- Forwarded by Tana Jones/HOU/ECT on 04/17/2001 04:20 PM -----
Bob Shults/ENRON@enronXgate
04/17/2001 03:27 PM
To: Tana Jones/HOU/ECT@ECT
cc:
Subject: Aspen Technologies NDA
Please email a two way NDA to Aspen Technologies
Wayne Bartel
wayne.bartel.petrovantage.com
Aspen Technologies
10 Canal Park
Cambridge, Mass 02141
617 949-1116 fax
617 949-1412
Steve:
(1) This is Carlos' email address:
c.j.ibarguen-bedoya@lse.ac.uk.
(2) His resume has been circulated to Mark Schroeder and Diane Bazelides, to
date.
(3) Attached is his resume for your review.
Lora
To: Diane Bazelides/HOU/AZURIX@AZURIX
cc: Joe Hillings/Corp/Enron@ENRON
Subject: Resume of Carlos Ibarguen
- Enron Fax.doc
- RESUME english.DOC
FYI
May be I was too assertive in the press criticizing the stupidity of revoking
Annex V and setting caps on spot. Mr. Mauro Arce got upset, but he realized
that his proposal to change the rules would face strong opposition, with
solid arguments.
In my simple view, he realized that not everyone would stay quiet.
Let's see.
LM
---------------------- Forwarded by Luiz Maurer/SA/Enron on 06/01/2001 09:48
AM ---------------------------
Joao Carlos Albuquerque
05/31/2001 07:20 PM
To: Luiz Maurer/SA/Enron@Enron
cc: Joe Kishkill/SA/Enron@Enron, Orlando Gonzalez/SA/Enron@Enron, Remi
Collonges/SA/Enron@Enron, Brett R Wiggs/SA/Enron@Enron, Jose
Bestard/ENRON_DEVELOPMENT@ENRON_DEVELOPMENT, Sergio Assad/SA/Enron@Enron,
David M Rosenberg/SA/Enron@Enron
| 2,908
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er98]. \[pageref:quasi-poly\] That is, there exists a decomposition of ${\mathbb Z}^{r_G+r_H}$ into polyhedral chambers such that on each chamber $C$ the function $n(y)$ is given by a single quasi-polynomial, i.e., there exists a sublattice $L \subseteq {\mathbb Z}^{r_G+r_H}$ of finite index and polynomials $(p_z)$ with rational coefficients, labeled by the finitely many points $z \in {\mathbb Z}^{r_G+r_H} / L$, such that $n(y) = p_{[y]}(y)$ for all $y \in {\mathbb Z}^{r_G+r_H}$ (cf. [@verdoolaegeseghirbeylsetal07 §2.2]). We record the following immediate consequence:
\[abstract cor\] For any fixed group homomorphism $f \colon H \rightarrow G$, the multiplicities $m^\lambda_\mu$ are given by a piecewise quasi-polynomial function in $\lambda$ and $\mu$.
In particular, this implies that the *stretching function* $k \mapsto m^{k\lambda}_{k\mu}$ is a quasi-polynomial function for large $k$. This is in fact true for all $k$, as has been observed in [@mulmuley07] (cf. [@meinrenkensjamaar99] for more general quasi-polynomiality results on convex cones, and also [@baldonivergne10] for further discussion).
Polynomial-Time Algorithm for the Subgroup Restriction Problem {#section:algorithm}
==============================================================
In this section we will formulate our algorithm for the subgroup restriction problem, . Recall that, by , the computation of the multiplicities $m^\lambda_\mu$ effectively reduces to counting the number of integral points in certain rational convex polytopes of the form . We shall suppose that the highest weights $\lambda$ and $\mu$, which are the *input* to our algorithm, are given in terms of their coordinates with respect to the fundamental weight bases fixed in . Clearly, for each of the finitely many $\gamma \in \Gamma_H$, the description of the polytope $\Delta_{\mathcal A,\mathcal B}(\lambda,\mu+\gamma)$ (say, in terms of linear inequalities) is of polynomial size in the bitlength of the input. It follows that *Barvinok’s algorithm* can be used to compute the
| 2,909
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|
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u$ final state, while in charm decays it generates to a very good approximation the same amount of $d \bar d$ and $s \bar s$ states.
We write the amplitudes very generally and up to a normalization factor as $${\cal A} = 1 + r a e^{i(\phi+\delta)}\,, \label{eq:generic-ampl}$$ such that $r$ is real and depends on CKM matrix elements, $a$ is real and corresponds to the ratio of the respective hadronic matrix elements, $\phi$ is a weak phase and $\delta$ is a strong phase. For kaons $a$ is the ratio of matrix elements of the operators $Q^{\Delta I=1/2}$ over $Q^{\Delta I=3/2}$, while for charm it is the ratio of matrix elements of the operators $Q^{\Delta U=0}$ over $Q^{\Delta U=1}$.
We first consider the case where we neglect the third generation. In that limit for kaons we have the decomposition $${\cal A}_K = V_{us}V_{ud}^* (A_{1/2} + r_{CG} A_{3/2})\,,$$ where $r_{CG}$ is the CG coefficient that can be read from Eq. (\[eq:kaondata\]). For charm we have $${\cal A}_D = V_{cs}V_{us}^* A_1.$$ That means that in the two-generational limit for kaons we have $r=1$ and in charm $r=0$. If we switch on the third generation we get small corrections to these values in each case: $r\ll 1$ for charm and $|r-1|\ll 1$ for kaons. These effects come from the non-unitarity of the $2 \times 2$ CKM. For the kaon case there is an extra effect that stems from SD penguins that come with $V_{ts}V_{td}^*$. In both cases we have $\delta \sim
\mathcal{O}(1)$ from non-perturbative rescattering, as well as $\phi \sim \mathcal{O}(1)$.
The general formula for direct CP asymmetry is given as [@Tanabashi:2018oca] $$\begin{aligned}
A_{CP} &= -\frac{2 r a \sin(\delta) \sin(\phi)
}{
1 + (ra)^2+ 2 ra \cos(\delta) \cos(\phi)
}
\approx
\begin{cases}
2 r a \sin(\delta) \sin(\phi)&{\mbox{for $ra \ll 1$}},\\
2 (r a)^{-1} \sin(\delta) \sin(\phi)&{\mbox{for $ra \gg 1$}}.\\
\end{cases} \label{eq:CPasym-general-formula} \end{aligned}$$ Non-perturbative effects enhance $a$ in both kaon and charm decays. This means the effect which is visible in the
| 2,910
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| 2,803
| 2,656
| 0.778067
|
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|
ig){\:\Dot{\cup}\:}{{\mathbb S}}_j\equiv F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j).\end{aligned}$$ Therefore, $\vec F_{\vec\omega_k}$ is a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$. This completes the proof of Lemma \[lmm:GHS-BK\].
Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ {#ss:pijbd}
----------------------------------------------------------------
First we prove [(\[eq:piNbd\])]{} for $j\ge1$ assuming the following two lemmas, in which we recall [(\[eq:Theta-def\])]{} and use $$\begin{aligned}
&E'_{{\bf N}}(z,x;{{\cal A}})=\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\Longleftrightarrow}}}x\},&
&E''_{{\bf N}}(z,x,v;{{\cal A}})=E'_{{\bf N}}(z,x;{{\cal A}})\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}v\},{\label{eq:E'E''-def}}\\
&\Theta'_{z,x;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\!
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}(z,x;{{\cal A}})$}}},\quad&
&\Theta''_{z,x,v;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\!
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}(z,x,v;{{\cal A}})$}}}.{\label{eq:Theta'Theta''-def}}\end{aligned}$$
\[lmm:Thetabds\] For the ferromagnetic Ising model, we have $$\begin{aligned}
\Theta_{y,x;{{\cal A}}}&\leq\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big)
\,\Theta'_{z,x;{{\cal A}}},{\label{eq:Theta[1]-bd}}\\[5pt]
\Theta_{y,x;{{\cal A}}
| 2,911
| 1,235
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aph topology effects the accuracy. When $\theta^*$ is chosen uniformly at random, the accuracy does not change with $d$ (left), and the accuracy is better for those graphs with larger spectral gap. However, for a certain worst-case $\theta^*$, the error increases with $d$ for the chain graph and the barbell-like graph, as predicted by the above analysis of the spectral gap. We use $\ell = 4$, $\kappa = 17$ and vary $d$ from $129$ to $2049$. $\kappa$ is kept small to make the resulting graphs more like the above discussed graphs. Figure on left shows accuracy when $\theta^*$ is chosen i.i.d. uniformly over $[-b,b]$ with $b=2$. Error in this case is roughly same for each of the graph topologies with chain graph being the worst. However, when $\theta^*$ is chosen carefully error for chain graph and barbell-like graph increases with $d$ as shown in the figure right. We chose $\theta^*$ such that all the items of a set have same weight, either $\theta_i = 0$ or $\theta_i = b$ for chain graph and barbell-like graph. We divide all the sets equally between the two types for chain graph. For barbell-like graph, we keep the two types of sets on the two different sides of the connector set and equally divide items of the connector set into two types. Number of samples $n$ is $100(d-1)/(\kappa-1)$ and each point is averaged over $100$ instances. Normalization constant $C$ is $n\ell/d^2$.
![For randomly chosen $\theta^*$ the error does not change with $d$ (left). However, for particular worst-case $\theta^*$ the error increases with $d$ for the Chain graph and the Barbell-like graph as predicted by the analysis of the spectral gap (right).[]{data-label="fig:topology"}](Plot11_rand-eps-converted-to.pdf "fig:"){width=".3\textwidth"} (-180,50) (-95,-5) (-90,100)[Random $\theta^*$]{} ![For randomly chosen $\theta^*$ the error does not change with $d$ (left). However, for particular worst-case $\theta^*$ the error increases with $d$ for the Chain graph and the Barbell-like graph as predicted by the analysis of the spectral gap (r
| 2,912
| 1,793
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$x$, $y\in D$. Let ${\mathrm{Var}}$ be a variety of ordinary algebras. In the paper [@Pozh:09] it is shown that $D\in{\mathrm{Di}}{\mathrm{Var}}$ if and only if $\bar D\in{\mathrm{Var}}$ and $D$ is a ${\mathrm{Var}}$-bimodule over $\bar D$ in the sense of Eilenberg, i. e., the split null extension $\widehat D=\bar D\oplus D$ belongs to the variety ${\mathrm{Var}}$.
In this way we can define a variety of dialgebras ${\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$ by a variety ${\mathcal{H}}{\mathrm{SJ}}$.
Let ${\mathrm{Di}}{\mathrm{SJ}}$ be the class of special Jordan dialgebras. Consider the closure ${\mathcal{H}}({\mathrm{Di}}{\mathrm{SJ}})$ of this class relative to the operator ${\mathcal{H}}$. The variety obtained we denote by ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$.
The purpose of this section is to show that ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}={\mathrm{Di}}{\mathcal{H}}{\mathrm{SJ}}$.
\[lemma:FreeSpecJordDialgebra\] ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle$ is a free algebra in the variety ${\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$.
Let $J'\in{\mathcal{H}}{\mathrm{Di}}{\mathrm{SJ}}$ be a homomorphic image of $J\in{\mathrm{Di}}{\mathrm{SJ}}$, $D$ be an associative dialgebra such that $J\hookrightarrow D^{(+)}$. We have the following commutative diagram $$\begin{CD}
J' @<\text{на}<< J @>\subseteq>> D \\
@AAA @AAA @AAA \\
X @>\subseteq>> {\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle @>\subseteq>>
{\mathrm{Di}}{\mathrm{As}}\langle X\rangle
\end{CD}$$
We have $X\subseteq J'$. Consider some preimages of elements of the set $X$ with respect to the mapping $J\to J'$. Since $J\subseteq D$, we obtain the embedding of $X$ into $D$. By the universal property of ${\mathrm{Di}}{\mathrm{As}}\langle X\rangle$ there exists an unique homomorphism ${\mathrm{Di}}{\mathrm{As}}\langle X\rangle\to D$ such that its restriction to ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle$ is the homomorphism ${\mathrm{Di}}{\mathrm{SJ}}\langle X\rangle\to J$. The last homomorphism in a composi
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isc\]). At this spectral resolution, H$_\alpha$ is blended with the \[N[II]{}\] doublet, and a fit to the three lines must be obtained simultaneously in order to measure their line strengths. With the exception of close doublets (i.e. \[O[II]{}\], \[S[II]{}\]) the other lines in the spectrum are all comparitively unblended, and all lines are consistent with being essentially unresolved at the instrumental resolution.
{width="1.97\columnwidth"}
Line Host Neighbour A B
-------------- ----------------- ---------------- ----------------
O[II]{}3726 61.9 $\pm$ 8.0
O[II]{}3729 31.0 $\pm$ 4.0
H$\gamma$ 6.5 $\pm$ 0.4
H$\beta$ 16.8 $\pm$ 1.0 3.2 $\pm$ 1.3 22 $\pm$ 9
O[III]{}4959 5.8 $\pm$ 0.3 4.1 $\pm$ 1.7 27 $\pm$ 11
O[III]{}5007 17.1 $\pm$ 1.0 12.4 $\pm$ 5.0 77 $\pm$ 32
He I 5875 2.8 $\pm$ 0.1
O[I]{}6300 6.1 $\pm$ 0.2
N[II]{}6548 10.3 $\pm$ 0.2 1.6 $\pm$ 0.1 4.1 $\pm$ 0.4
H$\alpha$ 103.1 $\pm$ 2.4 31.8 $\pm$ 1.5 116 $\pm$ 12
N[II]{}6583 31.5 $\pm$ 0.7 4.9 $\pm$ 0.2 12.5 $\pm$ 1.2
S[II]{}6716 13.3 $\pm$ 0.5
S[II]{}6730 13.0 $\pm$ 0.5
: Line strengths measured for the target objects. All measures are given as observed-frame equivalent widths in Angstroms. Measurement of weak lines is not attempted in the fainter neighbour, and it is impossible to isolate the two components in the \[O[II]{}\] line.\[tab:lines\]
{width="1.97\columnwidth"} {width="1.97\columnwidth"}
![The spectral region containing H$\alpha$ and the \[N[II]{}\] doublet. All three lines are consistent with the being unresolved at the instrumental FWHM. The relative strength of the doublet lines is consistent with that predicted from the electron transition probabilities. \[fig
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Phi$ and the Killing vector $I$ as in eq. of appendix \[app:sugra\].
To show that the dual background solves the modified supergravity equations we follow the derivation in [@Hoare:2016wsk]. After splitting the Lagrange multiplier as $v_a = u_a + y n_a$, it transpires that shifting $y$ is a symmetry of the dual background and T-dualising $y \to \tilde y$ gives a conformal $\sigma$-model with a dilaton linear in $\tilde y$. From the results of [@Arutyunov:2015mqj] this then implies that, in our conventions, the dual model solves the modified supergravity equations with $I^y = - 1$.
The classical bosonic string Lagrangian in conformal gauge, $$\label{eq:cbsacg}
\mathcal{L} = \partial_+ x^m (G_{mn} + B_{mn}) \partial_- x^n \ ,$$ has the property that when we replace $\partial_- x^m \to I^m$ it equals $W_n \partial_+ x^n$ where the one-form $W$, defined in eq. , is given by $$W_n = I^m (G_{mn} - B_{mn}) \ .$$ Following this procedure in the dual model with $I^y = - 1$ and the remaining components vanishing, we find that $$\label{eq:Ampush}
W_n \partial_+ x^n = - A_+^a n_a \ ,$$ with $A_+$ evaluated on the gauge field equation of motion . To summarise; if the T-duality is anomalous then the background solves the modified supergravity equations with the one-form $W$, which can be used to define the modification, given by the push forward of the $A_+$ component of the gauge field evaluated on its equations of motion.
Centrally-extended duality {#sec:centralext}
==========================
Let us now consider non-abelian T-dualities with respect to centrally-extended algebras. In particular we consider the setup considered in [@Hoare:2016wsk; @Borsato:2016pas] in which case the dualities are equivalent to Yang-Baxter deformations for homogeneous $r$-matrices. The aim of this section is to extend this to the RR fluxes using the technology outlined in section \[sec:natd\]. We start by recalling that for a homogeneous $r$-matrix for a Lie algebra $\mathfrak{f}$ $$\label{eq:rmatans}
r= \sum_{j} \eta_j \, \big( \sum_{i=1}^
| 2,915
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\tilde{{\bf k}}| = - [k_b u^b(\tau)]$. Substituting the above expression in Eq.(\[gwhitmann\]) and upon performing the straightforward ${\bf k}$ integral, we can write a compact expression for $W_{\Theta_0}$ of the following form $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) &=& \frac{-1}{16} \frac{\partial}{\partial \epsilon^\prime} \frac{\partial}{\partial \epsilon^{\prime \prime}} \Biggl( \frac{4\pi\epsilon^\prime \epsilon^{\prime \prime} }{ \left[ T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) \right]^2 - \left[ X \left( \epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) \right]^2 } \Biggr)_{\epsilon^\prime = \epsilon, \epsilon^{\prime \prime} = \epsilon}
\label{wfinalcompact}\end{aligned}$$ where the functions $T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ and $X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right)$ are found to be $$\begin{aligned}
T\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) &=& \left(t(\tau) -t(\tau^\prime) \right) - i |\cos\Theta_{0}| \left( \epsilon^{\prime \prime} {\dot{t}}(\tau) + \epsilon^{\prime} {\dot{t}}(\tau^\prime) \right) \label{Tdef} \\
X\left(\epsilon^\prime, \epsilon^{\prime \prime}, \tau, \tau^\prime \right) &=& \left({\bf x}(\tau) -{\bf x}(\tau^\prime) \right) - i |\cos\Theta_{0}| \left( \epsilon^{\prime \prime} {\dot{{\bf x}}}(\tau) + \epsilon^{\prime} {\dot{{\bf x}}}(\tau^\prime) \right) \label{Xdef}\end{aligned}$$ The overdot refers to the derivative with respect to the proper time $\tau$ or $\tau^\prime$. Expanding the above expression, $W_{\Theta_0}(\tau,\tau^\prime) $ can also be written as $$\begin{aligned}
W_{\Theta_0}(\tau,\tau^\prime) = \frac{1}{16} & \bigg\{& \frac{4 \pi}{-T^2 + X^2} + \frac{i 8 \pi \epsilon^{\prime \prime} |\cos\Theta_{0}| \left[ -T\dot{t} + X \dot{{\bf x}} \right] }{\left( -T^2+X^2 \right)^2} \nonumber \\
&& + \frac{i 8 \pi \epsilon^{\prime} |\cos\Theta_{0}| \left[-T \dot{t}^{\prime} + X \dot{{\
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here uses $\sum_{a = 1}^p \frac{1}{\kappa -a+1} \leq \log\big(\frac{\kappa}{\kappa-p}\big)$ and $C_3 \geq 0$. Equation follows from the fact that for any $x>0$, $\log(1+x) \leq x$. To prove , we have the first order partial derivative of $\P(\theta)$ given by $$\begin{aligned}
\label{eq:cr2}
\nabla_i \P(\theta) &=& \I_{\{\Omega^{-1}(i) \leq p \}}\P(\theta) - \sum_{\sigma \in \Omega} \Bigg(\frac{\exp\big(\sum_{m = 1}^{p} \theta_{\sigma(m)} \big)}{\prod_{a=1}^{p} \Big(\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big) \Big)} \Bigg( \sum_{a = 1}^p \frac{\I_{\{\sigma^{-1}(i) \geq a \}}\exp(\theta_i)}{\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big)} \Bigg) \Bigg) \,.\end{aligned}$$ Define constants $A_1$, $A_2$ and $A_3$ such that $$\begin{aligned}
A_1 & \equiv & \P(\theta) \big|_{\{\theta = {\boldsymbol{0}}\}} = \frac{(p-1)!}{\kappa(\kappa-1)\cdots(\kappa-p+1)}, \label{eq:crA1}\\
A_2 & \equiv & \Bigg( \sum_{a = 1}^p \frac{\exp(\theta_i)}{\sum_{m'=a}^{\kappa} \exp\big(\theta_{\sigma(m')}\big)} \Bigg)\Bigg|_{\{\theta = {\boldsymbol{0}}\}} = \Bigg(\frac{1}{\kappa} + \frac{1}{\kappa-1}+\cdots + \frac{1}{\kappa-p+1}\Bigg), \label{eq:crA2} \\
A_3 & \equiv & \Bigg(\frac{(p-1)(p-2)!}{(p-1)!(\kappa)} + \frac{(p-2)(p-2)!}{(p-1)!(\kappa-1)} + \cdots + \frac{(p-2)!}{(p-1)!(\kappa-p+2)} \Bigg) \,. \label{eq:crA3}\end{aligned}$$ Observe that, for all $i \in [d]$, $$\begin{aligned}
\label{eq:cr4}
\nabla_i \P(\theta) \big|_{\{\theta = {\boldsymbol{0}}\}} = A_1 \Big( \I_{\{\Omega_j^{-1}(i) = p\}}(1 - A_2) + \I_{\{\Omega_j^{-1}(i) < p\}}(1 - A_3) - \I_{\{\Omega_j^{-1}(i) > p\}}A_2 \Big) \;\; \,.\end{aligned}$$ Further define constants $B_1$, $B_2$, $B_3$ and $B_4$ such that $$\begin{aligned}
B_1 &\equiv & \Bigg(\frac{1}{\kappa^2} + \frac{1}{(\kappa-1)^2} + \cdots + \frac{1}{(\kappa - p+1)^2}\Bigg), \label{eq:crB1}\\
B_2 & \equiv & \Bigg(\frac{p-1}{(p-1)\kappa^2} + \frac{p-2}{(p-1)(\kappa-1)^2} + \cdots + \frac{1}{(p-1)(\kappa-p+2)^2} \Bigg), \label{eq:crB2} \\
B_3 & \equiv & \Bigg( \frac{(p-1)(p-2)(p-3)!}{(p-1)!\
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uld include another suitable criteria in general case which is still an active area of research.
The a priori estimates
----------------------
We are going to derive the a priori estimates for the geometric quantities. We fix a small constant $\delta>0$ and a constant $u_1\in(u_0,0)$ and denote $$\begin{aligned}
\mathscr{F}=\mathscr{F}(u_0,u_1)=\max\{1,\sup_{u_0\le u\le u_1}|\varphi(u)|\}.\end{aligned}$$ Without loss of generality, we also assume that $\Omega(u_0)\le1$. By the monotonicity of $\Omega_0$, we have $\Omega_0(u)\le1$ for all $u\in[u_0,0)$. Then we are going to prove
\[estimate\] There exists a universal large constant $C_0\ge1$ such that the following statement is true. Suppose that $$\begin{aligned}
\mathcal{A}=\mathcal{A}(\delta,u_0,u_1)=\max\{1,\sup_{0\le\ub\le\delta}\mathscr{F}^{-1}(|rL\phi(\ub,u_0)|+|u_0||\omega(\ub,u_0)|)\}<+\infty,\end{aligned}$$ and for some $C\ge C_0$ we have $$\begin{aligned}
\label{smallness}
C^2\delta|u_1|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}\le1,\ \text{where}\ \mathscr{W}=\mathscr{W}(u_0,u_1)=\max\left\{1,\left|\log\frac{\Omega_0(u_1)}{\Omega_0(u_0)}\right|\right\}.\end{aligned}$$ Then we have the following estimates for $0\le\ub\le\delta$, $u_0\le u\le u_1$:[^3] $$\begin{aligned}
\label{geometryestimate}\frac{1}{2}\Omega_0\le\Omega\le 2\Omega_0,&\ \frac{1}{2}|u|\le r\le 2|u|,\\
\label{estimate-Lphi}|rL\phi|\lesssim&\mathscr{F}\mathcal{A},\\
\label{estimate-Lbphi}|r\Lb\phi-\psi|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\
\label{estimate-h}|h-h_0|\lesssim&\delta|u|^{-1}\Omega_0^{-2}\mathscr{F}^2\mathcal{A}^2,\\
\label{estimate-hb}|{\underline{h}}+1|\lesssim&\delta|u|^{-1}\mathscr{F}\mathcal{A},\\
\label{estimate-omega}|u||\omega|\lesssim&\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A}.\end{aligned}$$ Moreover, we have the improved estimate $$\begin{aligned}
\label{estimate-Lphiimp}
|rL\phi(\ub,u)-\varphi(u)|\lesssim|rL\phi(\ub,u_0)-\varphi(u_0)|+\delta|u|^{-1}\mathscr{F}^2\mathscr{W}^{\frac{1}{2}}\mathcal{A}^2.\end{aligned}$$
We begin the proof by
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m $v=cz+d$ with the integers $(c,d)$ co-prime.
Now given the positive lattice basis $v=cz+d$ and $v'=az+b$, form the integer matrix $\g=\begin{pmatrix} a&b\\ c&d\end{pmatrix}$ , which has $\det(\g)=+1$ since $\{v,v'\}$ form a positive basis of the lattice. Thus we get a matrix in the modular group $\G=SL_2(\Z)$. Then with $\g$ applied as a Möbius transformation to $z$, the length of $v$ can be computed via $${\operatorname{Im}}(\g z) = \frac{{\operatorname{Im}}(z)}{|cz+d|^2}=\frac{{\operatorname{area}}(L)}{|v|^2}$$ The signed ratio between the lengths of $v$ and $v'$ (when $v'$ is chosen of minimal length) is $$\rho{(v)} = \pm |\gamma z| \;.$$ where the sign is $+$ if ${\operatorname{Re}}(\g z)>0$ and $-$ otherwise. Moreover, we have $${\operatorname{sk}}(v,v') = {\operatorname{Re}}(\g z)$$ Indeed, $${\operatorname{Re}}(\g z) = \frac{ac(x^2+y^2) +(ad+bc)x +bd}{|cz+d|^2}$$ which is ${\operatorname{sk}}(v,v')$ in view of . Consequently, the uniform distribution modulo one of ${\operatorname{sk}}(v)$ as $|v|\to\infty$ is then exactly the uniform distribution modulo one of ${\operatorname{Re}}(\g z)$ as $\g$ varies over $\GinfmodG$ with ${\operatorname{Im}}(\g z )\to 0$, that is Theorem \[equidistribution\].
A sketch of a proof of Good’s theorem {#sec:spectral}
=====================================
To prove Theorem \[equidistribution\], we use Weyl’s criterion to reduce it to showing that the corresponding “Weyl sums” satisfy $$\label{character asymptotics}
\sum_{\g \in(\GinfmodG)_{\varepsilon,z}}e(m{\operatorname{Re}}{\g
z})=\delta_{m=0}\frac{t_\G}{\vol{(\GmodH)}}\frac 1\varepsilon
+o(1/\varepsilon)$$ as $\varepsilon\to 0$. Here $t_\G$ equals $2$ if $-I\in \G$ and $1$ otherwise. In turn, will follow, by a more or less standard Tauberian theorem (see e.g. [@PetridisRisager:2004a p. 1035-1038]) from knowing the analytic properties of the series $$V_m(z,s):=\sum_{\g\in\GinfmodG}{\operatorname{Im}}(\g z)^se(m{\operatorname{Re}}(\g z)) \;.$$ studied also in [@Good:1981b; @Neunhoffer:1973a] Here $e(x)=\e
| 2,919
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Since any unmixed sequentially Cohen-Macaulay module is Cohen-Macaulay, all assertions are proved.
As a consequence of the previous theorem we get
\[interesting\] Let $M$ be an $R$-module. If the non-zero factors of the dimension filtration of $M$ are clean, then $M$ is pretty clean.
Conversely assume that $R$ is a local or standard graded CM ring with canonical module $\omega_R$, and that $M$ admits a pretty clean filtration ${\mathcal
F}$ such that $R/P$ is CM for all $P\in \Supp({\mathcal F})$. Furthermore assume that $M$ is graded if $R$ is graded. Then the non-zero factors of the dimension filtration of $M$ are clean.
Suppose all factors $D_i(M)/D_{i-1}(M)$ in the dimension filtration of $M$ are clean. Then it is obvious that the dimension filtration can be refined to yield a pretty clean filtration of $M$.
We prove the second statement of the corollary by induction on the length $r$ of the filtration $\mathcal F$. The claim is obvious if $r=1$. Now let $r>1$, and set $U=M_{r-1}$. We obtain the exact sequence $0\to U\to M\to R/P\to 0$ with $P\in \Spec(R)$. Let $d=\dim R/P$. Then, as we have seen in the proof of Theorem \[sequentially\], one has $ \Ext_R^{n-i}(M,\omega_R)\iso \Ext_R^{n-i}(U,\omega_R)$ for all $i\neq d$, as well as the exact sequence $$0\To\omega_{R/P}\to \Ext_R^{n-d}(M,\omega_R)\To \Ext_R^{n-d}(U,\omega_R)\To 0.$$ Since $M$ is sequentially CM by the previous theorem, these isomorphisms together with Proposition \[exti\](d) and Corollary \[comparison\] imply that $$D_i(M)/D_{i-1}(M)\iso D_i(U)/D_{i-1}(U)$$ for $i\neq d$. Hence, since the factors $D_i(U)/D_{i-1}(U)$ are clean by induction hypothesis, the same is true for the factors $D_i(M)/D_{i-1}(M)$ with $i\neq d$.
Applying the functor $\Ext_R^{n-d}(-,\omega_R)$ to the above exact sequence and using Proposition \[exti\](d) again we obtain the exact sequence $$0\To D_d(U)/D_{d-1}(U)\To D_d(M)/D_{d-1}(M)\To R/P\to 0.$$ Since all modules in this exact sequence are of dimension $d$, and since $D_d(U)/D_{d-1}(U)$ is clean, it follows that $
| 2,920
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chi (\Lambda )\cap (U^-(\chi ){\otimes }1)\big)\fiee$$ and that $L^\chi (\Lambda )$ is ${\mathbb{Z}}^I$-graded. For all ${\alpha }\in {\mathbb{Z}}^I$ let $$I^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }\cap
I^\chi (\Lambda ), \quad
L^\chi (\Lambda )_{\alpha }=M^\chi (\Lambda )_{\alpha }/I^\chi (\Lambda )_{\alpha }.$$ Since $M^\chi (\Lambda )_0={\mathbb{K}}v_\Lambda $, and any ${\mathbb{Z}}^I$-graded quotient of $M^\chi (\Lambda )$ by a $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-submodule containing $v_\Lambda $ is zero, $L^\chi (\Lambda )$ is the unique simple ${\mathbb{Z}}^I$-graded $U(\chi ){\otimes }_{\Bbbk }{\mathbb{K}}$-module quotient of $M^\chi (\Lambda )$.
\[de:fchar\] Let $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$ and $V$ a ${\mathbb{Z}}^I$-graded subquotient of $M^\chi (\Lambda )$. The *(formal) character of* $V$ is the sum $$\fch{V}=\sum _{{\alpha }\in {\mathbb{N}}_0^I} (\dim V_{-{\alpha }}) e^{-{\alpha }},$$ where $e$ is a formal variable.
Eq. implies that $$\begin{aligned}
\fch{M^\chi (\Lambda )}=\sum _{{\alpha }\in {\mathbb{N}}_0^I}\dim U^-(\chi )_{-{\alpha }}
e^{-{\alpha }}
\label{eq:chML}\end{aligned}$$ for all $\Lambda \in {{\mathrm{Hom}}({\mathcal{U}}^0,{{\mathbb{K}}^\times })}$.
\[re:fch\] For all ${\alpha },\beta \in {\mathbb{Z}}^I$ we let $e^{\alpha }e^\beta =e^{{\alpha }+\beta }$. Thus we can consider formal characters as elements of the ring $\cup _{{\alpha }\in {\mathbb{N}}_0^I}e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$, where $e^{\alpha }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]\subset
e^{{\alpha }+\beta }{\mathbb{Z}}[[e^{-{\alpha }_i}\,|\,i\in I]]$ for all ${\alpha },\beta \in {\mathbb{N}}_0^I$ in the natural way.
[*From now on let $\chi \in {\mathcal{X}}$, $p\in I$, and ${b}={b^{\chi}} ({\alpha }_p)={b^{r_p(\chi )}}({\alpha }_p)$. Assume that ${b}<\infty $.*]{} Then $\chi $ and $r_p(\chi )$ are $p$-finite. We deduce some phenomena which arise from the finiteness assumption on ${b}$.
For all $\Lambda \in {{\mathrm{Hom}}({\
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\ell_i}
\in \mathcal{R}_n $ and $(z_j-z_k)/(z_j+z_k)(1-\widetilde{\Lambda})_{jk}\prod_i z_i^{\ell_i}
\in \mathcal{R}_n $. They can be rewritten as \[ratio1\] (1-\_[jk]{})\_i z\_i\^[\_i]{} =(\_[i(j,k)]{} z\_i\^[\_i]{})(z\_j+z\_k)(z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r and \[ratio2\] (1-)\_[jk]{}\_i z\_i\^[\_i]{} =(\_[i ()]{} z\_i\^[\_i]{})(z\_j-z\_k)(z\_jz\_k) \^[(\_j,\_k)]{}(\_j,\_k) \_[r=0]{}\^[\_j-\_k-1]{}(-z\_j)\^[\_j -\_k-1-r]{}z\_k\^r where $\textrm{sign}(0) = 0$ and $\textrm{sign}(\alpha) = \alpha / \vert \alpha \vert$ for $\alpha \neq 0 $. And (,) = {
[cc]{} 1 &
. The right hand side of Eq.(\[ratio1\]) can be expressed as a sum of the following two terms, \[term1\] (\_[i]{} z\_i\^[\_i]{})z\_j (z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r and \[term2\] (\_[i]{} z\_i\^[\_i]{})z\_k(z\_jz\_k) \^[(\_j,\_k)]{}(\_j-\_k) \_[r=0]{}\^[\_j-\_k-1]{}z\_j\^[\_j -\_k-1-r]{}z\_k\^r . Let $p_r^j$ and $p_r^k$ denote the powers of $z_j$ and $z_k$ in the $r$-th summand of Eq. (\[term1\]). Then $p_r^j = \max(\ell_j, \ell_k) - r $ and $p_r^k = \min(\ell_j, \ell_k) + r$ . Thus, $p_r^j + p_r^k = \ell_j + \ell_k$. Therefore, the sum of powers of $z_j$ and $z_k$, in the $r$-th summand is $(\sum_{i\neq j, k} \ell_i ) + \ell_j+ \ell_k = \sum_i \ell_i$.
Hence, the expression (\[term1\]) is a member of $\mathcal{R}_n$. Similar calculation shows that the expression (\[term2\]) also belongs to a space spanned by monomials of degree $n$. Thus, the second operator in Eq. (\[mom11\]) preserves $\mathcal{R}_n$. In a similar manner it can be verified that the third operator in Eq. (\[mom11\]) also preserves $\mathcal{R}_n$. As the operators $\{D_j \vert j=1,\dots,N \}$ are linear and preserve the space $\mathcal{R}_n$, the Hamiltonian $\widetilde{H}_N(=\sum D_j^2)$ also preserves $\mathcal{R}_n$, and hence is quasi-solvable.
Conclusion
==========
In this article we have studied the behaviour of one dimensional chain of particles in a magnetic fi
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|
}_1,[\mathcal{O}_2,\cdots,
[\mathcal{O}_{i_0},\cdots,[\mathcal{O}_n,(p-X_0\tilde{p})]]]]\eta\Phi\
\nonumber\\
&\hspace{25mm}
+Q\eta(\xi_0(X_0)^p[\mathcal{O}_1,[\mathcal{O}_2,\cdots,
[\mathcal{O}_{i_0},\cdots,[\mathcal{O}_n,(p-X_0\tilde{p})]]]]\eta\Phi)\,.\end{aligned}$$ Using (\[p tilde p\]), it is easy to show that the transformations (\[general odd\]) or (\[general even\]) can further be written in the form of a gauge transformation as
$$\begin{aligned}
\delta_{[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l-1},\tilde{p}]]]}\eta\Phi
\cong& -(-1)^l Q\eta( (X_0)^{k+l-1}\xi_0[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l-1}, M]\cdots]]\Psi),\\
\delta_{[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l-1},\tilde{p}]]]}\,\Psi
\cong& -(-1)^l Q((X)^{k+l}\eta\xi_0[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l-1}, M]\cdots]]\,\eta\Phi)\,,\end{aligned}$$
or
$$\begin{aligned}
\delta_{[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l},\tilde{p}]]]}\,\eta\Phi\
\cong&\ (-1)^l Q \eta ((X_0)^{k+l}\xi_0[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l}, M]\cdots]]\,\eta\Phi)\,,\\
\delta_{[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l},\tilde{p}]]]}\,\Psi\
\cong&\ (-1)^l Q ( (X)^{k+l}\eta\xi_0[\mathcal{O}_1,[\mathcal{O}_2,\cdots,[\mathcal{O}_{2k+l}, M]\cdots]]\,\Psi)\,,\end{aligned}$$
respectively. Hence all the extra symmetries obtained as repeated commutators of $\delta_{\mathcal{S}}$’s and $\delta_{\tilde{p}}$’s act trivially on the on-shell physical states, and thus the physical S-matrix, defined by the asymptotic string fields.
[99]{}
N. Berkovits, “SuperPoincare invariant superstring field theory,” Nucl. Phys. B [**450**]{} (1995) 90 \[Erratum-ibid. B [**459**]{} (1996) 439\] \[hep-th/9503099\]. T. Erler, S. Konopka and I. Sachs, “Resolving Witten’s superstring field theory,” JHEP [**1404**]{}, 150 (2014) doi:10.1007/JHEP04(2014)150 \[arXiv:1312.2948 \[hep-th\]\]. H. Kunitomo and Y. Okawa, “Complete action for open superstring field th
| 2,923
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a)$ is expected to explode at affine parameter value $\lambda_e$.
[^4]: This map is defined such that $H: O\times [0,\lambda_0]\to M$ where $O$ is an open in $S$.
---
abstract: 'The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the [*same*]{} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal “thin sandwich” viewpoint coupled with the transformation properties of the lapse function.'
address: 'Department of Physics, North Carolina State University, Raleigh, NC 27695-8202'
author:
- 'James W. York, Jr.[@address]'
date: 'October 15, 1998'
title: 'Conformal “thin sandwich” data for the initial-value problem of general relativity[^1]'
---
In this paper I propose a new interpretation of the four Einstein vacuum initial-value constraints. (The presence of matter would add nothing new to the analysis.) Partly in the spirit of a “thin sandwich” viewpoint, I base this approach on prescribing the [*conformal*]{} metric [@York72] on each of two nearby spacelike hypersurfaces (“time slices” $t=t^\prime \mbox{ and } t=t^\prime + \delta t$) that make a “thin sandwich” (TS). Essential use is made of a new understanding of the role of the lapse function in general relativity [@AAJWY98; @YorkFest]. The new formulation could prove useful both conceptually, and in practice, as a way to construct initial data in which one has a hold on the input data different from that in the currently accepted approach. The new approach allows us to [*derive*]{} from its dynamical and metrical foundations the important scaling law $\bar{A}^{i j} = \psi^{-10} A^{i j}$ for the traceless part of the extrinsic curvature. This rule is simply postulated in the one-hypersurface approach.
The new formulation differs from the well-known TS conjecture of Baier
| 2,924
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rix} \right) }\in D({{{\mathcal{}}}A})$, we have $\psi(E)\in D(A)=\tilde W_{-,0}^2(G\times S)$, for any $E\geq 0$, and thus in particular, $\psi_{|\Gamma_-}=0$. We find by the first row of matrix equation (\[R7\]) that $\psi(0)=T(0)\phi+R(0)f=\phi$.
It might be worth attempting to generalize this method under less restrictive assumptions, especially for the case where $S_0$, $\Sigma$, $\sigma_1$ are allowed to be $E$-dependent.
Existence of Solutions for the Coupled System {#cosyst}
=============================================
In this section, we consider the coupled transport problem. For simplicity denote $\Sigma_j:=\Sigma_{j,r},\ S_j:=S_{j,r},\ \sigma_{jj,r}:=\sigma_{jj}$ for $j=2,3$. Let $f=(f_1,f_2,f_3)\in L^2(G\times S\times I)^3$ and $g=(g_1,g_2,g_3)\in T^2(\Gamma_-)^3$.
We deal with the following coupled system of integro-partial differential equations for $\psi=(\psi_1,\psi_2,\psi_3)$ on $G\times S\times I$, $$\begin{gathered}
\omega\cdot\nabla_x\psi_1+\Sigma_1\psi_1-K_{1}\psi=f_1, \label{csda1a}\\
-{{\frac{\partial (S_{j}\psi_j)}{\partial E}}}+\omega\cdot\nabla_x\psi_j+\Sigma_{j}\psi_j-K_{j}\psi=f_j,\quad j=2,3.\label{csda1b}\end{gathered}$$ In order to guarantee uniqueness of solutions, we moreover impose the inflow boundary condition on $\Gamma_-$, \[csda2\] [\_j]{}\_[|\_-]{}=g\_j,j=1,2,3, and initial value (or energy boundary) condition on $G\times S$, \[csda3\] \_j(,,E\_[m]{})=0,j=2,3, where $E_{\rm m}$ is the cut-off energy. As mentioned in the introduction the problem (\[csda1a\])-(\[csda3\]) is an approximation of the problem (\[intro1\]), (\[intro2\]).
We assume that the [*total (restricted) cross sections*]{} $\Sigma_j:G\times S\times I\to{\mathbb{R}}$, for $j=1,2,3$, are functions such that \[scateh\] \_jL\^(GSI),\_j0,j=1,2,3. Furthermore, we assume that the [*differential (restricted) cross sections*]{} $\sigma_{kj}:G\times S^2\times I^2\to{\mathbb{R}}$, $k,j=1,2,3$, are measurable functions such that $$\begin{aligned}
{3}\label{colleh}
&\sum_{k=1}^3\int_{S\times I} \sigma_{kj}(x,\omega
| 2,925
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| 2,204
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igma_D}) + \int_0^{ \sigma_D }{f}(X_s)\,{\rm d}s\right],
\end{aligned}$$ where in the final equality we have used that $\hat{u} = {g}$ on $D^{\rm c}$. Uniqueness now follows. $\square$
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Mateusz Kwaśniki for pointing out a number of references to us and Alexander Freudenberg for a close reading of an earlier version of this manuscript.
[^1]: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
[^2]: Supported by EPSRC grant EP/L002442/1.
[^3]: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK. Email: `a.kyprianou@bath.ac.uk`, `anaosojnik@gmail.com`, `t.shardlow@bath.ac.uk`.
---
abstract: 'It has been known that epidemic outbreaks in the SIR model on networks are described by phase transitions. Despite the similarity with percolation transitions, whether an epidemic outbreak occurs or not cannot be predicted with probability one in the thermodynamic limit. We elucidate its mechanism by deriving a simple Langevin equation that captures an essential aspect of the phenomenon. We also calculate the probability of epidemic outbreaks near the transition point.'
author:
- 'Junya Iwai${}^1$ and Shin-ichi Sasa${}^2$'
title: Intrinsic Unpredictability of Epidemic Outbreaks on Networks
---
Introduction
============
We start with the following question: How can it be determined whether an epidemic outbreak has occurred. Obviously, this is hard to answer, because an accurate model of epidemic spread in real societies, which include complicated and heterogeneous human-to-human contact, cannot be constructed. Then, is it possible to predict the outbreak for a simple mathematical model? Even in this case, the manner of the early spread of disease may significantly influence states that manifest after a sufficiently long time. For example, it seems reasonable to conjecture that whether a single infected individual with a very high infection rate causes an o
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e believe one should) and, moreover, we are required to use empirical and U-process theory. We adhere to their notation as much as possible.
The first step is to notice that, if we define $\delta_n(t)$ by the equation $$\label{delta}
\delta_n(t)=\frac{\hat f^{1/2}(t;h_{1,n})-f^{1/2}(t)}{f^{1/2}(t)}=\frac{\hat f(t;h_{1,n})-f(t)}{(\hat f^{1/2}(t;h_{1,n})+f^{1/2}(t))f^{1/2}(t)},$$ so that $\hat f^{1/2}(t;h_{1,n})=f^{1/2}(t)(1+\delta_n(t))$, then we have $$\label{zero}
\sup_{t\in D_r^\varepsilon}\delta_n(t)=o_{\rm a.s.}(1)\ \ {\rm uniformly\ in}\ f\ {\rm such\ that}\ \|f\|_\infty\le C,$$ where $D_r^\varepsilon$ denotes the $\varepsilon$-neighborhood of $D_r$ for $\varepsilon$ such that $f(t)>r/2$ in $D_r^\varepsilon$ ($f$ is uniformly continuous). We drop the subindex $n$ from $\delta$ from now on. Set $$D(t;h_{1,n})=\hat f(t;h_{1,n})-E\hat f(t;h_{1,n})\ \ {\rm and}\ \ b(t;h_{1,n})=E\hat f(t;h_{1,n})-f(t)$$ and note that $$\label{classic1}
\|D(\cdot;h_{1,n})\|_\infty=O_{a.s.}\left(\sqrt{\frac{\log h_{1,n}^{-1}}{nh_{1,n}}}\right)\ \ {\rm uniformly\ in}\ f\ {\rm such\ that}\ \|f\|_\infty\le C$$ for all $0<C<\infty$ by a result in Deheuvels (2000) and in Giné and Guillou (2002), and that $$\label{classic2}
\|b(\cdot;h_{1,n})\|_\infty\le \left(\int K(u)u^2du\right)\|f''\|_\infty h_{1,n}^2$$ by the classical bias computation for symmetric kernels. Since the numerator in the expression at the right hand side (\[delta\]) is just $D(t)+b(t)$ and the denominator is not smaller than $f(t)$ which is in turn larger than $r/2$, (\[zero\]) follows from (\[classic1\]) and (\[classic2\]). Define $$L_1(z)=zK'(z)\ \ {\rm and}\ \ L(z)=K(z)+zK'(z),\ \ z\in\mathbb R.$$ We then have $$\begin{aligned}
K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right)
&=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)+
\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i)\right)\\
&=&K\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)\\
&&~~~+K^\prime\left(\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\right)
\frac{t-X_i}{h_{2,n}}f^{1/2}(X_i)\delta (X_i)+\delta_2(t,X_i)\\
&
| 2,927
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| 2,788
| 3,693
| 0.770624
|
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|
a-1)^2},$$ where the quantity $P_{\pm}$ corresponds to the value calculated for $r_{\pm}$.
From equation (\[s\_grav\_nonrot\_chrg\]) we find that the gravitational entropy is proportional to the area of the event horizon of the black hole, just as in the case of the Bekenstein-Hawking entropy. We can further check the validity of our result by setting $\alpha=0$ in (\[s\_grav\_nonrot\_chrg\]), to see whether it leads us to the desired expression for the entropy of the Reissner–Nordstrom (RN) black hole. This exercise yields the result $$S_{grav}^{RN}=k_{s}(4\pi r_{\pm}^{2})\int_{\theta}P^{RN}_{\pm}(\theta)\dfrac{\sin\theta}{2} d\theta.$$ We can easily see that $P^{RN}_{\pm}(\theta)=P_{\pm}(\alpha=0)=\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}} , $ and therefore the gravitational entropy for the RN black hole is $$S_{grav}^{RN}=k_{s}(4\pi r_{\pm}^{2})\sqrt{\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}}}\int_{\theta}\dfrac{\sin\theta}{2}d\theta=k_{s}(4\pi r_{\pm}^{2})\sqrt{\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}}}.$$ This result matches with the expression of gravitational entropy for the RN black hole derived in [@entropy2] by Romero et al. The entropy density for the non-rotating charged black hole is obtained as $$\begin{aligned}
\label{s_nonrot_chrg}
\left.s\right. & = \frac{16\sqrt {6}k_{s}\sqrt { \left( -{\alpha}^{2}{r}^{2}+1 \right)
\left( {e}^{2}-2mr+{r}^{2} \right) }}{{{r}^{2} \left( 7{e}^{4}{\alpha}^{2} \cos^{2}\theta{r}^{2} + 10 \left( {e}^{2}-\dfrac{6mr}{5} \right) r\alpha{e}^{2}\cos\theta + 7{e}^{4}-12{e}^{2}mr+6{m}^{2}{r}^{2} \right) ^{3/2}}} \nonumber \\
& \times \left[ \cos^{3}\theta{\alpha}^{3}{e}^{6}{r}^{3} + {\frac {15{e}^{4}{\alpha}^{2} \cos^{2}\theta{r}^{2}}{8} \left( {e}^{2} - {\frac{13mr}{10}} \right) } + \dfrac{9 r\alpha{e}^{2} \cos\theta }{4} \left( {e}^{4}-{\frac {11{e}^{2}mr}{6}}+{m}^{2}{r}^{2} \right) \right. \nonumber \\
& \qquad\qquad\qquad + \left. {\frac {7{e}^{6}}{8}} - \frac{3mr}{4} \left({ \frac {13{e}^{4}
| 2,928
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ier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.'
address: 'Pembroke College, Cambridge, CB2 1RF, United Kingdom'
author:
- 'Matthew C. H. Tointon'
title: Polylogarithmic bounds in the nilpotent Freiman theorem
---
Introduction
============
This paper concerns *sets of small doubling* and *approximate groups* in non-abelian groups. This topic has been extensively covered in the recent mathematical literature; the reader may consult the author’s forthcoming book [@book] or the surveys [@bgt.survey; @app.grps; @ben.icm; @helf.survey; @sand.survey; @raconte-moi] for detailed background to the topic and examples of some of its many applications.
Given sets $A$ and $B$ in a group $G$ we define the *product set* $AB$ by $AB=\{ab:a\in A,b\in B\}$, and define $A^n$ recursively for $n\in{\mathbb{N}}$ by setting $A^1=A$ and $A^{n+1}=A^nA$. We also write $A^{-1}=\{a^{-1}:a\in A\}$ and $A^{-n}=(A^{-1})^n$. If $G$ is abelian we often use additive notation instead, for example writing $A+B$ or $nA$ in place of $AB$ or $A^n$, respectively.
By the *doubing* of a finite set $A$ we mean the ratio $|A^2|/|A|$, and when we say that a set has ‘small’ or ‘bounded’ doubling we mean that there is some constant $K\ge1$ such that $|A^2|\le K|A|$. Of course, this always holds for $K=|A|$, so $K$ should be thought of as being substantially smaller than $|A|$ in order for this to be meaningful.
One of the central aims in the theory of sets of small doubling is to describe the algebraic structure of such sets. The first result in this direction was Freiman’s theorem [@freiman], which describes sets of small doubling in terms of objects called *progressions*. Given elements $x_1,\ldots,x_r$ in an abelian group $G$ and reals $L_1,\ldots,L_r\ge0$, the *progression* $P(x;L)$ is defined via $P(x;L)=\{
| 2,929
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gned}
{\dot {\cal F}}_{\Theta_0}(\omega) &=& \frac{\pi g^{2}}{32 {\left(1+b^2 \epsilon^2 \right)}^3} \; \; \frac { e^{\frac{2\omega }{g} \tan^{-1}\left( g|\cos\Theta_{0}|\epsilon \right)}}{\left( e^{\frac{2 \pi \omega}{g}} -1\right) } \nonumber \\
&& \times \bigg\{ \; \; \frac{16 \pi}{3} \left(3 b^2 \epsilon^2 + b^4 \epsilon^4 \right) \frac{\omega}{g} \left(4+\frac{\omega^2}{g^2} \right) \nonumber \\
&& \; \; \; \; \; \; + 16 \pi \left(1-b^2\epsilon^2 \right) \frac{\omega}{g} + 32 \pi b \epsilon \frac{\omega^2}{g^2} \; \; \bigg\}\end{aligned}$$ Thus ${\dot {\cal F}}_{\Theta_0}(\omega)$ is not KMS thermal, in general, except when $\Theta_0 = \pi/2$. In the case $\Theta_0 = \pi/2$, $b$ vanishes and one recovers the usual Unruh temperature. Interestingly, even though the regularization in Eq.(\[Wfinal\]) does not hold in the $\Theta_0 = \pi/2$ case, we find that the final expression is indeed finite for the case.
For $\Theta_0 \neq \pi/2$, the quadratic term in the polynomial of $(\omega/g)$ breaks the thermality of the whole expression by just a sign. One can check that the polynomial in the braces does not possess a real root and hence is positive for all real values of $\omega$. Thus the transition rate ${\dot {\cal F}}_{\Theta_0}(\omega)$ is always positive as expected.
In the low frequency regime $|\omega/g| \ll 1$, the terms linear in $(\omega/g)$ dominate compared to the quadratic and cubic terms. Whereas, in the high frequency regime $|\omega/g| \gg 1$, the term cubic in $(\omega/g)$ dominate compared to the linear and quadratic terms. Hence, in both these limits, ${\dot {\cal F}}_{\Theta_0}(\omega)$ is KMS with the inverse of the temperature being equal to $2 \pi/ g - (4/g)\tan^{-1} \left( |\cos\Theta_{0}|g\epsilon \right)$, that is one observes a angle dependent temperature. In the $\Theta_0 = \pi/2$ direction, the temperature is same as the usual Unruh temperature while it increases as $\Theta_0$ decreases in the domain $0 \leq \Theta_0 \leq \pi/2$. Along the direction of acceleration, it is
| 2,930
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|
$ for some $e \in E(\wh{M})$, so $N$ is an elementary projection of $M$ if and only if $M = \wh{M} \del e$ and $N = \wh{M} \con e$.
Suppose that $M$ has a $U_{s2^{4s},2s 2^{4s}}$-minor. If $N$ is an elementary projection of $M$, then note that $M$ has a $U_{s+1,s2^{4s}}$-minor $M \con C \del D$. Let $M_0 = \wh{M} \con C \del D$. Note that $M_0 \con e$ is a minor of $N$ and that $s \le r(M_0 \con e) \le r(M_0 \del e) = s+1$. But we also have $M_0 \del e \cong U_{s+1,s2^{4s}}$, so $$\tau_{s-1}(M_0 \con e) \ge \tau_{s}(M_0) \ge s^{-1}(s2^{4s}) > \tbinom{2s}{s-1}^2 \ge \tbinom{2s}{s-1}^{r(M_0 \con e)-s},$$ and so $M_0 \con e$ has a $U_{s,2s}$-minor by Lemma \[udensity\]. Thus, $N$ has a $U_{s,2s}$-minor. If $N$ is an elementary lift of $M$, then $N$ has a $U_{s,2s}$-minor by duality.
Suppose that $M$ has a $\PG(n-1,q)$-minor $G = M \con C \del D$, and as before let $M_0 = \wh{M} \con C \del D$. If $N$ is an elementary projection of $M$ then $M_0 \del e = G \cong \PG(n-1,q)$, so there is a $\PG(n-2,q)$-restriction $R$ of $M_0 \del e$ that does not span $e$ in $M_0$. Thus $(M_0 \con e)|E(R) \cong \PG(n-2,q)$ and so $N$ has a $\PG(n-2,q)$-minor. Suppose that $N$ is an elementary lift of $M$. If there is some line $L$ of $G$ so that $e \in \cl_{M_0}(L)$, then $M_0 \del \{e\} \con L = M_0 \con (L \cup \{e\}) = G \con L$ and $\si(G \con L) \cong \PG(n-3,q)$, so $N $ has a $\PG(n-3,q)$-minor. If every line $L$ of $G$ is skew to $\{e\}$ in $M_0$, then let $B$ be a basis of $G$. Given $x_1,x_2 \in \cl_{M_0}(B)$, the line $\cl_G(\{x_1,x_2\})$ is skew to $\{e\}$ in $M_0$, so $\cl_G(\{x_1,x_2\}) \subseteq \cl_{M_0}(\{x_1,x_2\})$. Thus, $\cl_{M_0}(B)$ contains $B$ and is closed under taking lines of $G$ through two points; since $G \cong \PG(n-1,q)$ it follows that $\cl_{M_0}(B) \supseteq E(G)$. Moreover, $B$ is skew to $\{e\}$ in $M_0$ and so $M_0|\cl_{M_0}(B) = (M_0 \con e)|\cl_{M_0}(B) = G$; thus, $N$ has a $\PG(n-1,q)$-minor.
Spanning Cliques
================
In this section we prove Theorem \[main1\]. First we establi
| 2,931
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e the imaginary part (see Fig. \[fig.3\] (c)). Here, one can see also that at such choice of the real and imaginary parts of the partial components of the S-matrix the wave function leaves from its zero value at $r=0$.
![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f02.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f03.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$,
| 2,932
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|
be written as $$\rho_{t+1} = (1-p) U \rho_t U^{\dagger} + p \sum_{i} \mathbf{P_i}
U \rho_t U^{\dagger} \mathbf{P_i}$$ where $U$ is the unitary operator of the walk, $i$ runs over the dimensions where the decoherence occurs, and the $\mathbf{P_i}$ project in the usual “computational” basis [@KT03].
In the continuous setting, the unitary operator that governs the non-decohering walk is $U_t = e^{-iHt}$, where $H$ is the normalized adjacency matrix of the hypercube times an energy constant. To extend the above decoherence model to this setting, recall that the superoperator $U_t \otimes U_t^\dagger$ associated with these dynamics has the property that $$\frac{d\, U_t \otimes U_t^\dagger}{dt} = i\left(e^{-iHt} \otimes e^{iHt} \right) \left[
\identity \otimes H - H \otimes
\identity\right]\enspace;$$ wishing to augment these dynamics with measurement occurring at some prescribed rate $p$, we desire a superoperator $S_t$ that satisfies $$S_{t+dt} = S_t[e^{-iHdt} \otimes e^{iHdt}][(1 - p\,dt) \identity +
pdt(\mathbf{P})]$$ where $\mathbf{P}$ is the operator associated with the decohering measurement. Intuitively, the unitary evolution of the system is punctuated by measurements taking place with rate $p$, analogous to the discrete case.
Letting $e^{-iHdt} = \identity - iHdt$, we can expand and simplify: $$\begin{aligned}
S_{t+dt} & = S_t[e^{-iHdt} \otimes e^{iHdt}][(1 - pdt) \identity + pdt(\mathbf{P})] \\
& = S_t[(\identity - iHdt) \otimes (\identity + iHdt)][(1 - pdt) \identity + pdt(\mathbf{P})] \\
& = S_t[\identity \otimes \identity + idt(\identity \otimes H - H \otimes \identity) - pdt \identity \otimes \identity + pdt(\mathbf{P})]\enspace.\end{aligned}$$ In terms of a differential equation, $$\begin{aligned}
\frac{dS_t}{dt} & = \frac{S_{t+dt} - S_t}{dt}\\
& = \frac{S_t[\identity \otimes \identity + idt(\identity \otimes H - H \otimes \identity) - pdt \left(\identity \otimes \identity + \mathbf{P}\right)] - S_t}{dt}\\
& = S_t[i(\identity \otimes H - H \otimes \identity) - p\id
| 2,933
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| 2,081
| 1,979
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a _{2}^{2}$, the adsorption energy between colloids and droplets is much larger than the total repulsive energy. Therefore, we observe mostly closed structures \[Fig. \[fig:hist2\](c)\].
At a larger size asymmetry of $\sigma_{1}=2.0\sigma_{2}$, but at the same interfacial tension $\gamma=40, 100\, k_{\textrm{B}}T/\sigma _{2}^{2}$ \[Fig. \[fig:hist2\](b),(f) and \[fig:hist2\](c),(f)\] we observe a decrease of the number of large clusters, while the yield of smaller clusters increases.
-------------------------------- -------------------------------- --------------------------------
{width="5.9cm"} {width="5.9cm"} {width="5.9cm"}
{width="5.9cm"} {width="5.9cm"} {width="5.9cm"}
-------------------------------- -------------------------------- --------------------------------
Summary and Conclusions {#s:conc}
=======================
We investigated the cluster formation process of a mixture of colloidal dumbbells and droplets via emulsion droplet evaporation using Metropolis-based kinetic Monte Carlo simulations. The short-ranged attraction between colloids has a potential well depth of $9k_{\textrm{B}}T$ in order to ensure that neither dumbbells nor clusters are likely to break apart due to thermal fluctuations. In addition, the height of the repulsive barrier between colloids is about $9k_{\textrm{B}}T$, which is a large enough value to avoid spontaneous formation of clusters. The droplet-droplet interaction is a hard-sphere repulsion with an effective hard-sphere diameter chosen so that any two droplets cannot merge. The adsorption interaction between colloids and droplets has a minimum at the droplet surface to model the Pickering effect. In experiments, this energy has values up to millions of $k_{\textrm{B}}T$, depending on the contact angle, interfacial tension and particle size [@Aveyard2003]. In our simulations, however, we limited the colloid-droplet adsorption energy below $100k_{\textrm{B}}T
| 2,934
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rch Foundation of Korea. The work of ECO on this project is supported by grant 510940 from the Simons Foundation. The computation of this work was supported by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2018-C3-0015) and the PICSciE TIGRESS High Performance Computing Center at Princeton University.
SECOND-ORDER FINITE-DIFFERENCE IN LOGARITHMIC CYLINDRICAL COORDINATES {#s:fd}
=====================================================================
Here we derive a second-order finite-difference approximation to the radial part of the Laplace operator in logarithmic cylindrical coordinates. With the change of the variables $u\equiv \ln R$, the radial part of the Laplacian becomes $$\frac{1}{R} \frac{\partial}{\partial R} \left( R \frac{\partial\Phi}{\partial R} \right) = \frac{1}{R^2} \frac{\partial^2\Phi}{\partial u^2}.$$ Since the logarithmic grid in $R$ ($R_i = R_0 f^i$) corresponds to a uniform grid in $u$ ($u_i = u_0 + i\ln f$), we can apply a centered difference scheme in the $u$-space to obtain $$\label{eq:radiald_alt}
\frac{1}{R^2} \frac{\partial^2\Phi}{\partial u^2} = \frac{1}{R_i^2} \frac{\Phi_{i-1} - 2\Phi_i + \Phi_{i+1}}{(\delta u)^2} + {\cal O}((\delta u)^2).$$ Noting that $\delta u = \ln f = N_R^{-1}\ln (R_{\rm max}/R_{\rm min})$, the above expression can be expressed in the $R$-space as $$\label{eq:radiald2}
\frac{1}{R} \frac{\partial}{\partial R} \left( R \frac{\partial\Phi}{\partial R} \right) = \frac{\Phi_{i-1}-2\Phi_i + \Phi_{i+1}}{(R_i\ln f)^2} + {\cal O}\left( \frac{1}{N_R^2} \left(\ln\frac{R_{\rm max}}{R_{\rm min}}\right)^2 \right).$$ It is evident that the remainder decreases at a second-order rate with increasing $N_R$. We adopt Equation as our discrete Laplace operator in the logarithmic cylindrical grid (see Equation ).
COMPUTATION OF THE DISCRETE GREEN’S FUNCTION {#s:calc_dgf}
============================================
The DGF is needed in order to calculate the surface potenti
| 2,935
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| 3,507
| 2,860
| 2,503
| 0.779284
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_0 \epsilon _0/\epsilon _1$, and so the relation becomes explicitly, for all value of $(\alpha,\beta)$ : $$\begin{aligned}
\begin{split}
~& \sqrt{- \Bigg[ \Big(6\mathcal{L}_4 -12\mathcal{L}_6-\mathcal{L}_7 \Big)\alpha \big( 5\alpha +18\beta \big) +6 \big( \alpha +3 \beta \big) \Big( \alpha \curv{L}_5 + \beta \curv{L}_8 \Big) \Bigg] } \\ &- \frac{3\beta \epsilon_0 \sqrt{2}}{\epsilon_1} \sqrt{-\curv{L}_8 } -
\frac{\epsilon_0 \alpha}{\epsilon_2}
\sqrt{- \Big( 6 \curv{L}_5+5 \big( 6 \mathcal{L}_4 - 12\mathcal{L}_6 -\mathcal{L}_7\big) \Big) } =0.
\end{split} \end{aligned}$$ Therefore, this general formula for perfect squares depends only on $\sqrt{-J_{3,1}}$ and $\sqrt{-J_{3,2}}$ as we said.\
We note here that an interesting property coming from the existence of an infinite number $J_3\big(\alpha,\beta\big)$ of perfect squares for which the square-root can be decomposed in a small basis is that it gives some non-linear algebraic relations between the scalars of the FKWC-Basis, that reduce it in a non-trivial way, and allow to have a very small number of independent corrections. Indeed, solving the previous equation for $\mathcal{L}_4$, we find : $$\begin{aligned}
\begin{split}
\mathcal{L}_4 = \frac{1}{54} \Big( -9 \curv{L}_5 + 2 \curv{L}_8 + 108 \mathcal{L}_6 + 9 \mathcal{L}_7 - \sqrt{ \curv{L}_8 \big( 18 \curv{L}_5 -5 \curv{L}_8 \big) } \epsilon_0 \epsilon_1 \Big).
\end{split} \end{aligned}$$ We can now calculate the equations of motion for the 3 Lagrangian densities $\sqrt{-J_{2}}$, $\sqrt{-J_{3,1}}$ and $\sqrt{-J_{3,2}}$. First, one can check that the last one is in fact a topological term that does not bring any contribution to the equation of motion. Moreover, the first two lagrangian densities give the same equation of motion $54 \, \dot{a}(t) \ddot{a}(t)=0$ for the first one, and $18 \sqrt{2} \, \dot{a}(t) \ddot{a}(t)=0$ for the second one. It means that they are equal up to an invariant scalar $T$ for which $\sqrt{-g T}$ is a total derivative, $$\begin{aligned}
\sqrt{- \Big(2 \mathcal{L}_4+12 \mathc
| 2,936
| 3,710
| 2,831
| 2,554
| 3,763
| 0.770256
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e response in the $A$ units, especially for training on faces generated with half of the original principal component standard deviation. The lower standard deviation had a similar effect in the $R$ units, although training on the original rotating faces actually led to a smaller caricature response than the initial weights. The siamese VGG network used for the same/diff face identification task was constructed by taking the squared element-wise difference between the flattened features at the last convolutional layer for the two inputs, followed by a fully-connected, softmax classification layer.
![Responses of PredNet $A$ and $R$ units to varying levels of caricaturized faces, trained in different settings. Faces - Rotating synthetic faces. RandInit - Random initial weights. Static Faces - Same collection of images used in Rotating Faces except presented statically. Faces 0.5 SD - Rotating faces except each face generated from a distribution with half of the original standard deviation. []{data-label="norm_faces_AR"}](norm_faces_summary-relu_AR_v2.pdf){width="80.00000%"}
Illusory Contours
-----------------
Fig. \[illusory\_contours\_AR\] contains the illusory contour response plots for the $A$ and $R$ layers. The stimuli sequences consisted of $10$ time steps of the “four circles” image (see main text) followed by a test image for $10$ time steps. The response to the illusory stimuli begins one time step after the response to the line square for all unit types in the first layer.
![Illusory contours responses for $A$ and $R$ units.[]{data-label="illusory_contours_AR"}](illusory_contours-kitti_relu-AR.pdf){width="100.00000%"}
To quantify illusory responsiveness, we follow Lee et al. [@Lee_2001] in calculating the following two measures: $IC_a = \frac{R_i - R_a}{R_i + R_a}$ and $IC_r = \frac{R_i - R_r}{R_i + R_r}$, where $R_i$ is the response to the illusory contour (sum over stimulus duration), $R_a$ is the response to amodal stimuli, and $R_r$ is the response to the rotated J image. These indices were
| 2,937
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i^T\mathbf{X}_i\mathbf{e}},$$ and notice that $\mathbf{b}_j^T\mathbf{e}=0$ (because $\mathbf{b}_j$ and $\mathbf{e}$ are eigenvectors corresponding to different eigenvalues of $\mathbf{B}$) to produce $$\mathbf{B}_i=\mathbf{B}-\sum_{j=1}^{i-1}\beta_j\mathbf{b}_j\mathbf{b}_j^T=\mathbf{B}_{i-1}-\beta_{i-1}\mathbf{b}_{i-1}\mathbf{b}_{i-1}^T.$$ So by Brauer’s theorem again, the eigenpairs of $\mathbf{B}_i$ are the ones of $\mathbf{B}_{i-1}$ with $(\beta_{i-1}, \mathbf{b}_{i-1})$ replaced by an eigenpair with zero eigenvalue. So $\beta_i$ is the largest eigenvalue of $\mathbf{B}_i$ and $\mathbf{b}_i$ is the eigenvector of $\mathbf{B}_i$ corresponding to $\beta_i$.
Theorem \[thm7\] says when we build the new data matrix $\mathbf{X}_i$ from $\mathbf{X}$, $\mathbf{d}_i$ and $m_i$ change. Also $\mathbf{B}_i$ is different from $\mathbf{B}$, but the eigenpairs of $\mathbf{B}$ are retained by $\mathbf{B}_i$ except for the first $i-1$ pairs. The conclusion is that the first modularity component has the largest modularity of the data $\mathbf{X}$. Each succeeding modularity component has the largest modularity with the constraint that it is orthogonal to all previous modularity components.
Conclusion
==========
In this paper, the concept of modularity components is defined, and some important properties of modularity components are proven. The concept of modularity components can be used to explain why using more than one eigenvectors of the modularity matrix to do data clustering is reasonable. The combination of modularity clustering and modularity components gives a modularity component analysis that has some nice properties similar to the well known principal component analysis.
Appendices {#appendices .unnumbered}
==========
Proof of Lemma \[thm3\] {#app1}
=======================
The lemma is based on a theorem from [@bunch1978rank] about the interlacing property of a diagonal matrix and its rank-one modification and how to calculate the eigenvectors of a diagonal plus rank one (DPR1) matrix [@meyer2000matrix]. The
| 2,938
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. The complexity is $O(L\sqrt{P\norm{\h}^2})$ for a given channel vector $\h\in\Rbb^L$ and signal power constraint $P$, and is of average value $O(P^{0.5}L^{1.5})$ for i.i.d. standard Gaussian channel entries.
- For the complex-valued channels, we demonstrate how to apply our method in an efficient way to find the complex-valued coefficient vector.
- Extensive simulation results are presented to compare the effectiveness and efficiency of our method with the existing methods.
Part of this work has been presented in [@Zhou2014]. One main improvement here is the complexity order for i.i.d. Gaussian channel entries is further reduced from 3 to 1.5.
In the following, we will first introduce the system model of AWGN networks as well as the CF network coding design problem in Section \[section:ProblemStatement\]. Then in Section \[section:ProposedMethod\], we will present our proposed method in detail. Numerical results will be shown in Section \[section:NumericalResults\]. Finally, we will conclude our work in Section \[section:Conclusions\].
[*Notation.*]{} Let $\Rbb$ be the real field, $\Cbb$ be the complex field, and $\Zbb$ be the ring of integers. Boldface lowercase letters denote column vectors, and boldface uppercase letters denote matrices, e.g., $\w\in\Rbb^L$ and $\W\in\Rbb^{M\times L}$. $\norm{\w}$ denotes the $\ell^2$-norm of $\w$, and $\w^T$ denotes the transpose of $\w$. For a vector $\w$, let $\w(\ell)$ be the element with index $\ell$, and $\w(i\!:\!j)$ be the vector composed of elements with indices from $i$ to $j$. For a matrix $\W$, let be the submatrix containing elements with row indices from $i$ to $j$ and column indices from $k$ to $\ell$, be the submatrix containing elements with row indices from $i$ to $j$ and column index $k$, $\W(i,k\!:\!\ell)$ be the submatrix containing elements with row index $i$ and column indices from $k$ to $\ell$, and $\W(i,j)$ be the element with row index $i$ and column index $j$. Let $\floor{x}$ and $\ceil{x}$, i.e., the corresponding floor and ceiling f
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rrow\F, x\mapsto x^q$ is the Frobenius endomorphism.
We have $$\log\left(\sum_\muhat G_\muhat(q)m_\muhat\right)=\sum_{d=1}^\infty \phi_d(q)\cdot\log\left(\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)\right)$$where $\phi_n(q)=\frac{1}{n}\sum_{d|n}\mu(d)(q^{n/d}-1)$ is the number of $\langle f\rangle$-orbits of $\F^{\times}:=\F-\{0\}$ of size $n$. \[sumM\]
If $X$ is a finite set on which a finite group $H$ acts, recall Burnside’s formula which says that
$$\#X/H=\frac{1}{|H|}\sum_{h\in H}\#\{x\in X\,|\, h\cdot x=x\}.$$
Denote by ${\bm C}_n$ the set of conjugacy classes of $\GL_n(\F_q)$. Applying Burnside’s formula to ${\mathfrak G}_\muhat(\F_q)$, with $\muhat\in(\calP_n)^k$, we find that
$$\begin{aligned}
G_\muhat(q)&=|\GL_n(\F_q)|^{-1}\sum_{g\in \GL_n(\F_q)}\Lambda(g)\prod_{i=1}^k\#\{X\in\calF_{\mu^i}\,|\, g\cdot X=X\}\\
&=|\GL_n(\F_q)|^{-1}\sum_{g\in \GL_n(\F_q)}\Lambda(g)\prod_{i=1}^kR_{L_{\mu^i}}^G(1)(g)\\
&=\sum_{\calO\in {\bm C}_n}\frac{\Lambda(\calO)}{|Z_\calO|}\prod_{i=1}^kR_{L_{\mu^i}}^G(1)(\calO)\end{aligned}$$
For a conjugacy class $\calO$ of $\GL_n(\F_q)$, let $\omega(\calO)$ denotes its type. By Formula (\[alambda\]), we have
$$\frac{\Lambda(\calO)}{|Z_\calO|}=\calH_{\omega(\calO)}(0,\sqrt{q}).$$By Corollary \[R\], we deduce that
$$\sum_\muhat G_\muhat(q)m_\muhat=\sum_{\calO\in {\bm C}}\calH_{\omega(\calO)}(0,\sqrt{q})\prod_{i=1}^k\tilde{H}_{\omega(\calO)}(\x_i,q)$$where ${\bm C}:=\bigcup_{n\geq 1}{\bm C}_n$.
We denote by ${\bf F}^{\times}$ the set of $\langle f\rangle$-orbits of $\F^{\times}$. There is a natural bijection from the set ${\bm C}$ to the set of all maps ${\bf F}^{\times}\rightarrow\calP$ with finite support [@macdonald IV, 2]. If $C\in{\bm C}$ corresponds to $\alpha:{\bf F}^{\times}\rightarrow\calP$, then we may enumerate the elements of $\{s\in{\bf F}^{\times}\,|\, \alpha(s)\neq 0\}$ as $c_1,\dots,c_r$ such that $\omega(\alpha):=(d(c_1),\alpha(c_1))\cdots(d(c_r),\alpha(c_r))$, where $d(c)$ denotes the size of $c$, is the type $\omega(C)$.
We have
$$\begin{aligned}
\sum
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ta(1,2),\dots,\theta(m-1,m),\theta(m,1))$ of asymptotic directions of the interfaces, given that there are $m$ unbounded trees and assuming that the latter are labeled by following the trigonometric sense.
For this purpose, it is equivalent to study the distribution of the sectors $(\phi(i+1):=\theta(i+1,i+2)-\theta(i,i+1),\, i\in \{1,\dots,m\})$ (with the convention that $\theta(m,m+1)=\theta(m,1)$ and $\theta(m+1,m+2)=\theta(1,2)$), which characterize the asymptotic width of the unbounded trees.
\[lemm:conj2\] Conditionally on having $m$ unbounded trees, the angles between two interfaces are identically distributed with expectation $2\pi/m$.
Notice first that this rules out the possibility that the asymptotic directions $\theta(i,j)$’s are independent uniform r.v. on $[0,2\pi)$. Else, the distributions of the sectors would be Beta distributions $\mathbf{B}(1,m)$ which expectation is $2\pi/(m+1)$. There is thus interaction between the $\theta(i,j)$’s.\
Our conjecture is as follows:
\[conj1\]Conditionally on $m\in \{2,3,4,5\}$, the vector $(\phi(1),\dots,\phi(m))$ has a distribution close to a symmetric Dirichlet distribution of order $m$ on $[0,2\pi)$ with parameter $\alpha\not= 1$, $\mbox{Dir}(m,[0,2\pi),\alpha)$.
Symmetric Dirichlet distributions of order $m$ and parameter $\alpha>0$ on $[0,2\pi)$ are probability distributions on ${{\mathbb R}}^m$ with a support in $\Lambda=\{\eta=(\eta_1,\dots,\eta_m)\in {{\mathbb R}}^m,\, \sum_{i=1}^m \eta_i=2\pi\}$ and with the following density with respect to the Lebesgue measure on $\Lambda$: $$f(\eta_1,\dots,\eta_m ; \alpha)=\frac{1}{\mathbf{B}(\alpha)}\prod_{i=1}^{m}\Big(\frac{\eta_i}{2\pi}\Big)^{\alpha-1},\qquad \mbox{ where }\mathbf{B}(\alpha)=\frac{\Big(\int_0^{+\infty}t^{\alpha-1}e^{-t}dt\Big)^m}{\int_0^{+\infty}t^{m\alpha-1}e^{-t}dt}$$ is the Beta function. If we had a Dirichlet distribution conditionally on $m$, the marginal distribution of the exchangeable sectors would be a Beta distribution $\mathbf{B}(\alpha,(m-1)\alpha)$ on $[0,2\pi)$ with expectat
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The fact that these filtrations are induced from that of $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$ ensures that the associated graded object $$\operatorname{{\textsf}{ogr}}B = \bigoplus_{i\geq j\geq 0}\operatorname{{\textsf}{ogr}}B_{ij}$$ is also a ${\mathbb{Z}}$-algebra. Similarly, recall from the ${\mathbb{N}}$-graded algebra $A=\bigoplus_{i\geq 0} A^{i}$ associated to $\operatorname{Hilb(n)}$. In this section it is more convenient to use the isomorphic algebra $A=\bigoplus_{i\geq 0} A^{i}\delta^i$ to which we canonically associate the ${\mathbb{Z}}$-algebra \[Aij-defn\] $\widehat{A} = \bigoplus_{i\geq j \geq 0} A^{i-j}\delta^{i-j},$ in the notation of .
\[main\] Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and define $B$ and $\widehat{A}$ as above. Then:
1. There is an equivalence of categories ${U}_c{\text{-}{\textsf}{mod}}\ \xrightarrow{\sim}\ B{\text{-}{\textsf}{qgr}}$.
2. There is an equality $\operatorname{{\textsf}{ogr}}B = e \widehat{A}e$ and hence a graded ${\mathbb{Z}}$-algebra isomorphism $\operatorname{{\textsf}{ogr}}B \cong \widehat{A}$.
3. $\operatorname{{\textsf}{ogr}}B{\text{-}{\textsf}{qgr}}\simeq\operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$.
Combining Theorem \[main\] with Corollary \[morrat-cor\] and the isomorphism $U_c\cong U_{-c-1}$ from the proof of that result gives:
\[main-cor\] [(1)]{} Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\]. Then there exists a ${\mathbb{Z}}$-algebra $B'$ such that $U_c{\text{-}{\textsf}{mod}}\simeq B'{\text{-}{\textsf}{qgr}}$ and $\operatorname{{\textsf}{ogr}}B\cong \widehat{A}$. Thus $\operatorname{{\textsf}{ogr}}B'{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$.
[(2)]{} If $c\in {\mathbb{C}}$ with $c\not\in \frac{1}{2}+{\mathbb{Z}}$, then $H_c{\text{-}{\textsf}{mod}}\simeq B''{\text{-}{\textsf}{qgr}}$ and $\operatorname{{\textsf}{ogr}}B''{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ for some ${\mathbb{Z}}$-algebra $
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hi(\bar{G'})-k$ “pointed” colors (each given to the vertices corresponding to a set of lines passing through a common point).
Consider now $G$, the disjointness graph of the segments. Let $G_0$ denote the subgraph of $G$ induced by the set of segments whose supporting lines received one of the $k$ planar colors in the above coloring of $\bar{G}'$. These segments lie in at most $k$ planes. Therefore, applying Theorem 3.1 to $G_0$, we obtain $$\chi(G_0)\le (k-1)\omega(G_0)+(\omega(G_0))^4
\le (k-1)\omega(G)+(\omega(G))^4.$$
For $i, 1\le i\le \chi(\bar{G}')-k,$ let $G_i$ denote the subgraph of $G$ induced by the set of segments whose supporting lines are colored by the $i$th pointed color. It is easy to see that $G_i$ is the complement of a chordal graph. That is, the complement of $G_i$ contains no induced cycle of length larger than $3$. According to a theorem of Hajnal and Surányi [@HS58], any graph with this property is perfect, so that $$\chi(G_i)=\omega(G_i)\le\omega(G).$$
Putting these bounds together, we obtain that $$\chi(G)\le \chi(G_0)+\sum_{i=1}^{\chi(\bar{G}')-k}\chi(G_i)
\le(k-1)\omega(G)+(\omega(G))^4+\sum_{i=1}^{\chi(\bar{G}')-k}\omega(G)$$ $$\le((\omega(\bar{G}'))^2-1)\omega(G)+(\omega(G))^4<(\omega(G))^3
+(\omega(G))^4.$$
To prove the algorithmic claim in Theorem 1, we first apply the algorithm of Theorem 2 to the disjointness graph $\bar G'$. We distinguish between planar and pointed color classes and find the subgraphs $G_i$. We output a coloring of $G$, where for each $G_i, i>0$ we use the smallest possible number of colors ($G_i$ is perfect, so its optimal coloring can be found in polynomial time), and we color $G_0$ by the algorithm described in Theorem 3.1. The subgraphs $G_i$ are colored using pairwise disjoint sets of colors. We output the largest clique $K$ that we can find. This may belong to a subgraph $G_i$ with $i>0$, or may be found in $G_0$ or in $\bar G'$ by the algorithms given by Theorem 3.1 or Theorem 2, respectively. (In the last case, we need to turn a clique in
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.$$ The state in argument is obtained as a partial inner product of its index state and the maximally entangled state $|\Psi^+\rangle$. The mapping creating the index state from the original state, $$\label{eq:Lpsip}
L_{|\Psi^+\rangle}: {\cal H}_A\rightarrow {\cal H}_B,\quad
L_{|\Psi^+\rangle}|\Psi\rangle_A = |\Psi ^*\rangle_B$$ is antilinear, and in fact $\sqrt{N} L_{|\Psi^+\rangle}$ is antiunitary. Indeed, expanding an arbitrary $|\Psi\rangle_A$ on the computational basis, $$\label{eq:Lpsip_bas}
L_{|\Psi^+\rangle}|\Psi\rangle_A=L_{|\Psi^+\rangle}\sum_i C_i|i\rangle_A=
\frac{1}{\sqrt{N}} \sum_i C_i^*|i\rangle_B,$$ from which the above properties follow.
The introduction of $L$ via $|\Psi^+\rangle$ is also useful in describing channels $\$ _A$. Let us have the compound system in the state $|\Psi ^+\rangle_{AB}$, and send subsystem A through the channel $\$ _A$ while doing nothing with subsystem B. The effect of the channel on any pure state $|\Psi\rangle_A$ of system A is then obtained by the partial inner product with the corresponding index state: $$\label{eq:Relrep_chann}
\$_A\left(|\Psi\rangle_A\, _A\langle \Psi |\right)=
N\, _B\langle \Psi ^*|(\$ _A\otimes I_B)
\left (|\Psi^+\rangle _{AB} \,_{AB} \langle \Psi^+|\right)
|\Psi ^* \rangle_B,$$ where $|\Psi^*\rangle_B=L_{|\Psi^+\rangle}|\Psi\rangle_A$, and $I_B$ stands for the identity operator. This is the so called relative state representation of channels, which is widely used to describe them. But even more can be stated [@pra60_1888]. An affine isomorphism between the set of all $\$ _A$ channels on ${\cal S}_A$, and the set of bipartite states $\varrho _{AB}\in {\cal S}_{{\cal H}_A\otimes{\cal H}_B}$ with maximally mixed partial trace, i.e. with the property $$\label{eq:parctrac}
\mathop{\mbox{tr}}\nolimits _A \varrho_{AB}= \frac{1}{N} I_B,$$ can be found similarly to Eq. (\[eq:Relrep\_chann\]). The bipartite state corresponding to a channel can be obtained from $|\Psi^+\rangle$ by applying the channel on system A and doing nothing with sys
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omega,E',E) d\omega' dE'
\geq c, \label{csda4aa}\end{aligned}$$ hold for a.e. $(x,\omega,E)\in G\times S\times I$.
Note that if $\sigma_{kj}$ were (cf. Remark \[cosdare1\]) of the form $\sigma_{kj}(x,\omega',\omega,E',E)=\tilde\sigma_{kj}(x,\omega',\omega,E)\delta(E-E')$ then $$\int_{S\times I}\sigma_{kj,C}(x,\omega',\omega,E',E) d\omega' dE'=
\int_S\tilde\sigma_{kj}(x,\omega',\omega,E) d\omega',$$ for any $C$, and hence the conditions (\[csda3aa\]), (\[csda4aa\]) would be independent of $C$. All the considerations below, after obvious adaptations, are valid for this simplified case.
At first, we apply the variational formulations to deduce existence of solutions. Recall that the inner product in $L^2(G\times S\times I)^3$ is given by $${\left\langle}\phi,v{\right\rangle}_{L^2(G\times S\times I)^3}
=\sum_{j=1}^3 {\left\langle}\phi_j,v_j{\right\rangle}_{L^2(G\times S\times I)},$$ and analogously in other products of inner product spaces. Integrating by parts and applying the Green’s formula (\[green\]) we find (similarly as in section \[esols\]) that the bilinear form ${\bf B}(\cdot,\cdot):C^1(\ol G\times S\times I)^3\times C^1(\ol G\times S\times I)^3\to{\mathbb{R}}$ and the linear form ${\bf F}:
C^1(\ol G\times S\times I)^3 \to{\mathbb{R}}$ corresponding to the problem (\[cosyst1\])- (\[cosyst4\]) are \[cosyst5\] [**B**]{}(,v)=& \_[j=2,3]{}\_j,S\_j[E]{}\_[L\^2(GSI)]{} -,\_x v\_[L\^2(GSI)\^3]{}\
& +\_[j=2,3]{} C\_j,S\_jv\_j\_[L\^2(GSI)]{} +,(\^\*-K\_C\^\*) v\_[L\^2(GSI)\^3]{}\
& +\_+(), \_+(v) \_[T\^2(\_+)\^3]{} +\_[j=2,3]{} \_j(,,0),S\_j(,0) v\_j(,,0)\_[L\^2(GS)]{} , and \[cosyst6\] [**F**]{}(v) = [**f**]{},v\_[L\^2(GSI)\^3]{} +[**g**]{},\_-(v)\_[T\^2(\_-)\^3]{}.
For $j=2,3$, let $$q_j:={1\over 2}{\mathop{\mathrm{ess\hspace{1mm}sup}}}_{(x,E)\in G\times I}{{\frac{\partial S_j}{\partial E}}}(x,E).$$ Moreover, define $$\begin{aligned}
\label{cosyst6a}
&C_j:={{\max\{q_j,0\}}\over\kappa_j},\quad j=2,3, \nonumber \\
&C:=\max\{C_1,C_2\}. \end{aligned}$$
The appropriate Hilbert spaces are defined as &[H]{}:=HH\_1H
| 2,945
| 548
| 2,246
| 2,800
| null | null |
github_plus_top10pct_by_avg
|
$3^9, 27$ $9, 29$
$2,2,2,4,6$ $1^8, 5^2$ $9, 13$
$2,2,2,2,6$ $1^9, 9$ $3, 15$
$2,2,2,4,4$ $1^9, 15$ $5,19$
: This table lists combinatorial data for anomaly-free (0,2) GLSM’s describing rank 8 bundles over ${\mathbb Z}_2$ gerbes on Calabi-Yau’s.[]{data-label="table:DK-duality-exs"}
For example, the first entry in table \[table:DK-duality-exs\] describes a rank 8 bundle given as a kernel $$0 \: \longrightarrow \: {\cal E} \: \longrightarrow \:
\oplus_1^9 {\cal O}(1) \: \longrightarrow \: {\cal O}(9) \:
\longrightarrow \: 0$$ over the stack ${\mathbb P}^3_{[2,2,2,4]}[10]$, a ${\mathbb Z}_2$ gerbe over ${\mathbb P}^3_{[1,1,1,2]}[5]$.
We list a few rank 9 examples over ${\mathbb Z}_3$ gerbes in section \[sect:otherexs-good\]. These rank 8 examples are listed in this section because we are enumerating rank 8 bundles over ${\mathbb Z}_2$ gerbes, and the rank 9 examples are not candidates for the dualities discussed here.
Curiously, we were unable to find solutions of the combinatorial consistency conditions for GLSM’s for bundles of rank less than 8. We do not know whether this reflects a fundamental limitation of GLSM’s, or merely the inadequacy of our parameter space search.
Given a Distler-Kachru model with a phase describing a Landau-Ginzburg model over an orbifold of a vector space, methods exist to compute the massless spectrum in that Landau-Ginzburg phase [@kw; @dk1]. When these methods are applied to, for example, heterotic ${\rm Spin}(32)/{\mathbb Z}_2$ compactifications on typical examples from the table above, we find a large number of single vectors and matter representations which likely combine to form representations of a larger nonabelian gauge symmetry, but unfortunately the corresponding worldsheet global symmetry does not seem to be visible in the UV. We conclude that in these examples, much of the needed worldsheet global symmetry appears in the IR, where we have no direct access. This is atypical of Distler-Kachru mod
| 2,946
| 1,570
| 2,680
| 2,621
| null | null |
github_plus_top10pct_by_avg
|
perature distribution $\theta_{0}(x)$, the inverse problem calculates $\theta_{0}(x)$ from the integral equation $$\theta_{T}(x)=\frac{2}{L}\int_{0}^{a}k(x,x^{\prime})\theta_{0}(x^{\prime})dx^{\prime},\qquad0\leq x\leq L,$$
when this final temperature $\theta_{T}$ is known, and $$k(x,x^{\prime})=\sum_{n=1}^{\infty}\sin\left(\frac{n\pi}{L}x\right)\sin\left(\frac{n\pi}{L}x^{\prime}\right)e^{-\lambda_{n}^{2}T}$$ is the kernel of the integral equation. In terms of the final temperature the distribution becomes $$\theta_{T}(x)=\sum_{n=1}^{\infty}B_{n}\sin\left(\frac{n\pi}{L}x\right)e^{-\lambda_{n}^{2}(t-T)}\label{Eqn: heat2}$$
with Fourier coefficients $$B_{n}=\frac{2}{L}\int_{0}^{a}\theta_{T}(x^{\prime})\sin\left(\frac{n\pi}{L}x^{\prime}\right)dx^{\prime}.$$
In $L^{2}[0,a]$, Eqs. (\[Eqn: heat1\]) and (\[Eqn: heat2\]) at $t=T$ and $t=0$ yield respectively $$\Vert\theta_{T}(x)\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}A_{n}^{2}e^{-2\lambda_{n}^{2}T}\leq e^{-2\lambda_{1}^{2}T}\Vert\theta_{0}\Vert^{2}\label{Eqn: heat3}$$ $$\Vert\theta_{0}\Vert^{2}=\frac{L}{2}\sum_{n=1}^{\infty}B_{n}^{2}e^{2\lambda_{n}^{2}T}.\label{Eqn: heat4}$$
The last two equations differ from each other in the significant respect that whereas Eq. (\[Eqn: heat3\]) shows that the direct problem is well-posed according to (IP3), Eq. (\[Eqn: heat4\]) means that in the absence of similar bounds the inverse problem is ill-posed.[^10]$\qquad\blacksquare$
**Example 2.2.** Consider the ****Volterra integral equation of the first kind $$y(x)=\int_{a}^{x}r(x^{\prime})dx^{\prime}=Kr$$
where $y,r\in C[a,b]$ and $K\!:C[0,1]\rightarrow C[0,1]$ is the corresponding integral operator. Since the differential operator $D=d/dx$ under the sup-norm $\Vert r\Vert=\sup_{0\leq x\leq1}|r(x)|$ is unbounded, the inverse problem $r=Dy$ for a differentiable function $y$ on $[a,b]$ is ill-posed, see Example 6.1. However, $y=Kr$ becomes well-posed if $y$ is considered to be in $C^{1}[0,1]$ with norm $\Vert y\Vert=\sup_{0\leq x\leq1}|Dy|$. This illustrates the importance of the to
| 2,947
| 4,046
| 3,979
| 2,833
| null | null |
github_plus_top10pct_by_avg
|
ly the role of a proper choice of the index set $\mathbb{D}$ in the description of convergence.
**Example A1.3.** (1) Let $\gamma\in\mathbb{D}$. The eventually constant net $\chi(\delta)=x$ for $\delta\succeq\gamma$ converges to $x$.
\(2) Let $\mathcal{N}_{x}$ be a neighbourhood system at a point $x$ in $X$ and suppose that the net $(\chi(N))_{N\in\mathcal{N}_{x}}$ is defined by $$\chi(M)\overset{\textrm{def}}=s\in M;\label{Eqn: Def: Net1}$$ here the directed index set $_{\mathbb{D}}N$ is ordered by the natural direction (\[Eqn: Direction1\]) of $\mathcal{N}_{x}$. Then $\chi(N)\rightarrow x$ because given any $x$-neighbourhood $M\in\!\:_{\mathbb{D}}N$, it follows from $$M\preceq N\in\,{}_{\mathbb{D}}N\Longrightarrow\chi(N)=t\in N\subseteq M\label{Eqn: DirectedNet1}$$
that a point in any subset of $M$ is also in $M$; $\chi(N)$ is therefore eventually in every neighbourhood of $x$.
\(3) This slightly more general form of the previous example provides a link between the complimentary concepts of nets and filters that is considered below. For a point $x\in X$, and $M,N\in\mathcal{N}_{x}$ with the corresponding directed set $_{\mathbb{D}}M_{s}$ of Eq. (\[Eqn: Directed\]) ordered by its natural order (\[Eqn: Direction2\]), the net $$\chi(M,s)\overset{\textrm{def}}=s\label{Eqn: Def: Net2}$$ converges to $x$ because, as in the previous example, for any given $(M,s)\in\:\!_{\mathbb{D}}N_{s}$, it follows from $$(M,s)\preceq(N,t)\in\!\:_{\mathbb{D}}M_{s}\Longrightarrow\chi(N,t)=t\in N\subseteq M\label{Eqn: DirectedNet2}$$ that $\chi(N,t)$ is eventually in every neighbourhood $M$ of $x$. The significance of the directed set $_{\mathbb{D}}N_{t}$ of Eq. (\[Eqn: Directed\]), as compared to $_{\mathbb{D}}N$, is evident from the net that it induces *without using the Axiom of Choice*: For a subset $A$ of $X$, the net $\chi(N,t)=t\in A$ indexed by the directed set $${\textstyle _{\mathbb{D}}N_{t}=\{(N,t)\!:(N\in\mathcal{N}_{x})(t\in N\bigcap A)\}}\label{Eqn: Closure_Directed}$$ under the direction of Eq. (\[Eqn: Direction2\]),
| 2,948
| 4,402
| 3,369
| 2,792
| 2,161
| 0.782136
|
github_plus_top10pct_by_avg
|
a)$, where $f: M^{n-k}
\looparrowright \R^n$ is an immersion with the prescribed isomorphism $\Xi: \nu(g) \cong k \kappa$, called a skew-framing, $\nu(f)$ is the normal bundle of $f$, $\kappa$ is the given line bundle over $M^{m-k}$ with the characteristic class $w_1(\kappa)
\in H^1(M^{m-k};\Z/2)$. The cobordism relation of triples is standard.
The generalization of the group $Imm^{\D_4}(n-2,2)$ is following. Let us define the cobordism groups $Imm^{\D_4}(n-2k,2k)$. This group $Imm^{\D_4}(n-2k,2k)$ is represented by triples $(g,\Xi,\eta)$, where $g: N^{n-2k} \looparrowright \R^n$ is an immersion, $\Xi$ is a dihedral $k$-framing, i.e. the prescribed isomorphism $\Xi: \nu_g \cong k \eta$, where $\eta$ is a 2-dimensional bundle over $N^{n-2k}$. The characteristic mapping of the bundle $\eta$ is denoted also by $\eta: N^{n-2k} \to K(\D_4,1)$. The mapping $\eta$ is the characteristic mapping for the bundle $\nu_g$, because $\nu_g \cong k \eta$.
Obviously, the Kervaire homomorphism (1) is defined as the composition of the homomorphism (2) with a homomorphism $$\Theta_{\D_4} : Imm^{\D_4}(n-2,2) \to \Z/2. \eqno(3)$$ The homomorphism (3) is called the Kervaire invariant for $\D_4$-framed immersed manifolds.
The Kervaire homomorphisms are defined in a more general situation by a straightforward generalization of the homomorphisms (1) and (3): $$\Theta^k: Imm^{sf}(n-k,k) \to \Z/2, \eqno(4a)$$ $$\Theta^k_{\D_4} : Imm^{\D_4}(n-2k,2k) \to \Z/2, \eqno(4b)$$ (for $k=1$ the new homomorphism coincides with the homomorphism (3) defined above) and the following diagram $$\begin{array}{ccccc}
Imm^{sf}(n-1,1) & \stackrel {\delta}{\longrightarrow} &
Imm^{\D_4}(n-2,2) & \stackrel{\Theta_{\D_4}}{\longrightarrow} & \Z/2 \\
\downarrow J^k & & \downarrow J^k_{\D_4} & & \vert \vert \\
Imm^{sf}(n-k,k) & \stackrel{\delta^k}{\longrightarrow} & Imm^{\D_4}(n-2k,2k) &
\stackrel{\Theta_{\D_4}^k}{\longrightarrow} & \Z/2 \\
\end{array} \eqno(5)$$ is commutative. The homomorphism $J^k$ ($J^k_{\D_4}$) is determined by the regular cobordism clas
| 2,949
| 2,560
| 2,412
| 2,538
| null | null |
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|
},h_{2,n})-\bar f(t;h_{2,n})-T(t;h_{1;n},h_{2,n})\right|=o_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$ and in particular, $$\sup_{t\in D_r}\left|\hat f(t;h_{1,n},h_{2,n})-\bar f(t;h_{2,n})\right|=O_{\rm a.s.}\left(\left(\frac{\log n}{n}\right)^{4/9}\right)\ \ {\rm uniformly\ in}\ \ f\in{\cal P}_C.$$
[ a) We should remark that if we undersmooth the preliminary estimator a little more, by taking $h_{1,n}=n^{-(2+\eta)/9}$ with $0<\eta< 2$, then the three lemmas above are true and moreover we have $\sup_{t\in D_r^\varepsilon}|\varepsilon_i(t,h_{1,n},h_{2,n})|=o_{\rm a.s.}(n^{-4/9})$ in Lemma \[lemma1\]. So, for such $h_{1,n}$ the order of the first term in Proposition \[real-ideal\] is actually $o_{\rm a.s.}\left(n^{-4/9}\right)$. This is at odds with condition (9) in Hall, Hu and Marron (1995), as their condition does not necessarily imply undersmoothing of the preliminary estimator. b) It is worth mentioning that Proposition \[real-ideal\] does require that the indicators $I(|t-X_i|\le h_{2,n}B)$ be part of the definition of (\[ideal0\]) and (\[realest0\]): in fact none of the three lemmas in its proof seem to go through without it. This condition is required as well for the bias of the ideal estimator, but it is not necessary for its variance part. ]{}
Now we can complete the proof of the main theorems \[main0\] and \[mainu\]. Only the stronger Theorem \[mainu\] requires proof:
[**Proof of Theorem \[mainu\].**]{} Proposition \[real-ideal\] and Theorem \[unifidealthm\] together give (\[main1’\]). The limit (\[main2’\]) can be easily derived from (\[main1’\]), as follows. By (\[classic1\]) and (\[classic2\]), the preliminary estimator satisfies $$\label{prelimunif}
\sup_{t\in D_r}|\hat f(t;h_{1,n})-f(t)|=O_{\rm a.s. }\left(\frac{\sqrt{\log n}}{n^{7/18}} \right)\ \
{\rm uniformity\ in}\ {\cal D}_{C,z}$$ for all $C<\infty$, $z$ and $r$. Now, for all $n$ large enough, on the event $$\left\{\sup_{n\ge k}\frac{n^{7/18}}{\sqrt{\log n}}||\hat f(t;h_{1,n}
| 2,950
| 1,231
| 1,982
| 2,940
| null | null |
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|
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\nonumber \\
&-&
\sum_{m}
\sum_{k \neq l} \sum_{K}
\frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{- i ( h_{l} - h_{m} ) x}
- \left( \Delta_{K} - h_{l} \right) e^{- i ( h_{k} - h_{m} ) x}
- ( h_{l} - h_{k} ) e^{- i ( \Delta_{K} - h_{m} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K l}
\nonumber \\
&+&
\sum_{m}
\sum_{k, K}
\frac{e^{- i (\Delta_{K} - h_{m} ) x} - e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) }
\biggl[
(UX)_{\alpha k} W^*_{\beta K}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\nonumber \\
&+&
W_{\alpha K} (UX)^*_{\beta k}
(UX)^*_{\alpha m} (UX)_{\beta m}
\left\{ W ^{\dagger} A (UX) \right\}_{K k}
\biggr]
\nonumber \\
&+&
\sum_{m}
\sum_{K}
e^{- i (\Delta_{K} - h_{m} ) x}
W_{\alpha K} W^*_{\beta K}
(UX)^*_{\alpha m} (UX)_{\beta m}
\biggr\}.
\label{P-beta-alpha-0th+2nd}\end{aligned}$$ Notice that there is no matter dependent terms without suppression either by high-frequency oscillations $\propto \cos (\Delta_{K} - h_{m}) x$ (or $\sin$), or by large energy denominators $\propto \frac{ 1 }{ \Delta_{K} - h_{k} }$.
We take averaging over fast oscillations due to active-sterile and sterile-sterile mass squared differences which leads to $$\begin{aligned}
\left\langle \sin \Delta_{J i} x \right\rangle
&\approx&
\left\langle \sin \Delta_{J K} x \right\rangle \approx 0,
\nonumber \\
\left\langle \cos \Delta_{J i} x
\right\rangle &\approx&
\left\langle \sin \Delta_{J K} x
\right\rangle \approx 0,
\label{average-out}\end{aligned}$$ where $\langle ... \rangle$ stands for averaging over neutrino energy within the uncertainty of energy resolution, as well as averaging over uncertainty of distance between production and detection points of neutrinos.[^12] The second approx
| 2,951
| 4,153
| 2,954
| 2,815
| null | null |
github_plus_top10pct_by_avg
|
8.39 ± 0.91 8.37 ± 0.85 0.846
Post transfusion hematocrit - mean (SD) 26.54 ± 2.96 26.34 ± 2.93 1.000
Post transfusion RBC count - mean (SD) 3.24 ± 0.41 3.09 ± 0.54 0.571
Hemoglobin recheck time after transfusion-mean hours (range) 9 (3-17) 8.47 (4-23) 0.387
Mean corpuscular volume (MCV) 90.01 ± 9.2 88.11 ± 10.26 0.177
Red blood cell distribution width (RDW) 18.36 ± 4.05 19.82 ± 5.32 0.034
Mean corpuscular hemoglobin concentration (MCHC) 31.71 ± 2.7 31.5 ± 1.8 0.556
Coronary artery disease n (%) 22 (22.2) 18 (18.2) 0.479
Chronic kidney disease n (%) 23 (23.2) 16 (16.2) 0.211
Cancer n (%) 36 (36.4) 48 (48.5) 0.084
Bone marrow suppression n (%) 7 (7.1) 11 (11.1) 0.322
Chronic inflammatory state n (%) 9 (9.1) 7 (7.1) 0.602
Infection n (%) 5 (5.1) 6 (6.1) 0.756
Number of transfusions in the last six months - mean (SD) 2.06 ± 3.95 3.33 ± 4.5 0.058
-------------------------------------------------------------- ------------------ ------------------ ---------
The mean pre-transfusion hemoglobin in the group that received old blood was 7.41 ± 0.85 gm/dl as compared to 7.39 ± 0.74 gm/dl with no statistically significant difference (p-value 0.844). Similarly, there was no significant difference in the mean pre-transfusion hematocrit (23.18 ± 2.79 % vs 23.09 ± 2.66 % - p-value 0.833) and RBC count (2.62 ± 0.44 X10^6^/µl vs
| 2,952
| 6,136
| 733
| 1,197
| null | null |
github_plus_top10pct_by_avg
|
full virtual channel matrix. The low-dimensional virtual channel matrix is defined by $$\label{eq:sHv1}
{\widetilde{\mathbf{H}}_{\psi,V}}=\left[{\mathbf{H}_{\psi,V}}(i,j)\right]_{i\in{\mathcal{M}_{\psi,r}},j\in\mathcal{M}_{\psi,t}},$$ where $\mathcal{M}_{\psi,r}=\left\{i:(i,j)\in\mathcal{M}_{\psi}\right\}$, $\mathcal{M}_{\psi,t}=\left\{j:(i,j)\in\mathcal{M}_{\psi}\right\}$, and $\mathcal{M}_{\psi}$ is the beam selection mask. The beam selection mask $\mathcal{M}$ is related to the criterion of beam selection. For example, a common beam selection is based on the criterion of maximum magnitude, and the corresponding beam selection mask is defined as [@Brady_13_BeamspaceSAMAM] $$\label{eq:Mmagnit}
\mathcal{M}_{\psi}=\left\{(i,j):\left|{\mathbf{H}_{\psi,V}}(i,j)\right|^2\ge\gamma_{\psi}\max_{(i,j)}\left|{\mathbf{H}_{\psi,V}}(i,j)\right|^2\right\},$$ where $0<\gamma_{\psi}<1$ is a threshold parameter used to ensure that ${\widetilde{\mathbf{H}}_{\psi,V}}$ has the dimension of $L_r\times L_t$. Using , the resulting ${\widetilde{\mathbf{H}}_{\psi,V}}$ captures a fraction $\gamma_{\psi}$ of the power of ${\mathbf{H}_{\psi,V}}$.
Although the VCR and the beamspace hybrid beamforming for mmWave systems have been widely adopted and studied in the literature [@Health_16_OverviewSPTmmMIMO; @Gao8284058; @Brady_13_BeamspaceSAMAM; @Amadori_15_LowRDBStion; @Wang7974749; @Mo7094595], it is worth mentioning that there are some potential limitations on the utility of VCR in practical mmWave systems. For sub-6 GHz systems, motivated by the limitations of the S-V model and the statistical model, the intermediate VCR was introduced to keep the essence of S-V model without its complexity and to provide a tractable channel characterization [@Sayeed_02_Deconstuctingmfc; @Raghavan4487419; @Huang5074791]. The VCR for sub-6 GHz systems offers a simple and transparent interpretation of the effects of scattering and array characteristics [@Sayeed_02_Deconstuctingmfc]. However, the interpretation of VCR is based on the assumption of (ap
| 2,953
| 1,975
| 2,668
| 2,779
| 2,482
| 0.779405
|
github_plus_top10pct_by_avg
|
[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}$$ for $v+1\ls i\ls b+2$. Consider applying Lemma \[lemma7\] to $T[i]$, to move the two $1$s from row $2$ to row $1$. Of the tableaux appearing in that lemma with non-zero coefficient, the only ones dominated by $S$ are those having no more than four entries less than $4$ in the first row; these are the tableaux $T'[i]$ and $T'[i,j]$ for $4\ls j\ls v$, where $$\begin{aligned}
T'[i]&=
{\text{\footnotesize$\gyoungx(1.2,;1;1;1;2;4_{3.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3.5*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;2;3;i,;2,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}},
\\
T'[i,j]&=
{\text{\footnotesize$\gyoungx(1.2,;1;1;1;2;4;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!3};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};u,;2;3;j,;2,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{
| 2,954
| 1,457
| 1,412
| 2,943
| 186
| 0.819898
|
github_plus_top10pct_by_avg
|
limits_{n\in N}}
A_{n}\right \Vert _{X}^{\ast}\right) .$$
Now, we prove the following result:
Let $1<p<\infty$ and $q=p/(p-1)$. If $A\in(\ell_{p}(\widehat{F}),\ell_{1})$, then$$\lim_{m}\left \Vert A\right \Vert _{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq4.\lim_{m}\left \Vert A\right \Vert
_{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)} \tag{3.11}$$ and$$L_{A}\text{ is compact if and only if }\lim_{m}\left \Vert A\right \Vert
_{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}=0, \tag{3.12}$$ where $$\left \Vert A\right \Vert _{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}=\sup_{N\in\mathcal{F}_{m}}\left(
{\displaystyle \sum \limits_{k}}
\left \vert
{\displaystyle \sum \limits_{n\in N}}
\bar{a}_{nk}\right \vert ^{q}\right) ^{1/q};\text{ \ }(m\in\mathbb{N}
).$$
It is obvious that (3.11) is obtained by combining Lemmas 2.2(c), 2.3 and 3.6. Also, by using (1.5), we get (3.12) from (3.11).
Let $1\leq p<\infty$. If $A\in(\ell_{1}(\widehat{F}),\ell_{p})$, then$$\left \Vert L_{A}\right \Vert _{\chi}=\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) \tag{3.13}$$ and$$L_{A}\text{ is compact if and only if }\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) =0, \tag{3.14}$$
Let us remark that the limit in (3.13) exists by Lemma 2.4.
Now, we write $S=S_{\ell_{1}(\widehat{F})}$. Then, we have by Lemma 1.2 that $L_{A}(S)=AS\in M_{\ell_{p}}$. Thus, it follows from (1.4) and Lemma 1.4 that$$\left \Vert L_{A}\right \Vert _{\chi}=\chi(AS)=\lim_{m}\left( \sup_{x\in
S}\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}}\right) , \tag{3.15}$$ where $P_{m}:\ell_{p}\rightarrow \ell_{p}$ $(m\in\mathbb{N}
)$ is the operator defined by $P_{m}(x)=(x_{0},x_{1},...,x_{m},0,0,...)$ for all $x=(x_{k})\in \ell_{p}$ and $I$ is the identity operator on $\ell_{p}$.
On the other hand, let $x\in \ell_{1}(\widehat{F})$ be given. Then $y\in
\ell_
| 2,955
| 1,860
| 2,315
| 2,595
| null | null |
github_plus_top10pct_by_avg
|
lassical Rabi frequency, $\hat a$ the annihilation operator for the CM motion, $\hat \sigma_z = | e \rangle \! \langle e | - | g \rangle \! \langle g |$, $\hat\sigma_+ = \hat\sigma^\dag_- = | e \rangle \! \langle g | $, and $\eta$ the Lamb-Dicke parameter defined as $$\begin{aligned}
\label{etatrue}
\eta=\frac{\omega_L}{c}\sqrt\frac{\hbar}{2M\nu}\cos\phi,\end{aligned}$$ with $M$ being the mass of the trapped ion, $c$ the speed of light, and $\phi$ the angle between the laser wave vector and the trap axis (one dimensional motion).
Depending on the detuning $\omega_0 - \omega_L$, the laser will cause the coupling of different vibrational levels with electronic part, each case representing a different quantum-optical process [@wineland] with its own effective Hamiltonian. The procedure to reveal each of those Hamiltonians is very well described in the literature, e.g., [@leibfried; @orszag]. Basically, after setting $\omega_0 - \omega_L=\pm m\nu$, with $m=0,1,2,\ldots$, one applies a rotating wave approximation (RWA) to Hamiltonian (\[hamtot\]) in order to obtain $$\label{hamrwa}
\hat{\mathcal{H}}^{(m)}_{\pm} = \hat{\mathcal{H}}_{0} +
\hbar \left( \text{e}^{-i\omega_{L} t }\hat{\Omega}_{m}^\pm \hat{\sigma}_{+} +
\text{e}^{ i\omega_{L} t }\hat{\Omega}_{m}^\mp \hat{\sigma}_{-} \right),$$ where $$\label{Omegaux1}
\hat{\Omega}_{m}^{+} = \hat{\Omega}_{m}^{- \dag} =
\frac{\Omega}{2} \text{e}^{-{\eta^{2}\!}/{2}}
\sum_{l=0}^{\infty}\left(i\eta\right)^{2l+m}
\frac{\hat{a}^{\dagger l}\hat{a}^{l}}{l!(l+m)!} \hat a^m.$$ For consistency, one must notice that $\eta$ in Eq. (\[etatrue\]), besides being a function of $\phi$ and $\nu$, is also a function of $\omega_0$. This is so because $\omega_L$ is now fixed by the sideband choice (value of $m$).
The Hamiltonian $\hat{\mathcal{H}}^{(m)}_{+} $ is obtained with $\omega_0 - \omega_L=
m \nu $, and it describes a $m$-phonon process for the vibrational part accompanied with transitions in the at
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long the direction $\alpha$.
Our focus here on the renormalized value $u_r$ of the Josephson coupling $u$ in the SP regime. If the periodic boundary conditions are also imposed perpendicular to the layers (along $z$-direction), the inter-layer response $u_r$ is given by windings $W_z$ along $z$-direction: u\_r=W\^2\_z, W\_z = \_[i]{} J\_i, \[WU\] where the summation $\sum_i$ of the currents $J_i$ (oriented along $z$-direction) is performed over all sites of all layers. Similarly to the cases (\[KR\]) and (\[stif22\]), Eq.(\[WU\]) represents the full linear response at zero momentum – that is, the renormalized value $u_r$ of the Josephson coupling $u$.
At this point, we should comment on how to interpret the PBC for two layers, $N_z=2$. While in the case $N_z\geq 3$ it is a natural procedure to link the $z=N_z$th layer to the first one, $z=1$, by the Josephson term, the case $N_z=2$ needs an auxiliary construction because the layers 1 and 2 are coupled already directly. The formal procedure, then, consists of adding a third layer, $z=3$, with no rigidity along $x,y$ directions and coupled by the Josephson term to both layers, $z=1,2$. If the coupling $u_{13}$ between the layers 1 and 3 and the coupling $u_{23}$ between the layers 2 and 3 add up as $1/u_{13} + 1/u_{23}=1/u_V$, in the dual action (\[H\_J\]) the sum in the last term can be extended to the layers $z=1,2,3$, while the first term is still confined to the layers $z=1,2$. The key to this procedure is the Kirchhoff’s rule: the J-current from a site $(x,y)$ along $z$-direction from the layer 2 to the layer 3 must be exactly the same as the current from the site $(x,y)$ in the layer 3 to the layer 1. Then, in the form (\[H\_J\]) the same value $u_V$ can be used for the currents from the layer 1 to the layer 2 directly or through the layer 3.
Asymptotic expression for $u_r$ {#sec:AS}
-------------------------------
As mentioned above, the dual representation allows obtaining analytically the asymptotic values for $u_r$ . Let’s begin with the trivial
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}^q{\vbu_1-v_1{|\!|\!|}}^q}&\bigg(
\prod_{i=1}^{j-1}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i+1}-
v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\times\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\qquad(j\ge2).\end{aligned}$$ First, we consider the sum over $u_j$ and $v_j$. By successive applications of Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we obtain (see Figure \[fig:star\](b)) $$\begin{aligned}
{\label{eq:succ-appl}}
&\sum_{v_j}\sum_{u_j}\frac{O(\theta_0)}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbu_j
-v_j{|\!|\!|}}^q{\vbu_j-v_{j-1}{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q
{\vbx-v_j{|\!|\!|}}^{2q}}\\
&\leq\sum_{v_j}\frac{O(\theta_0)^2}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbv_{j-1}
-v_j{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^2}
{{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}},{\nonumber}\end{aligned}$$ and thus $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{\substack{u_1,\dots,u_{j-1}\\ v_1,
\dots,v_{j-1}}}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}
&\bigg(\prod_{i=1}^{j-2}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i
+1}-v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\times\frac{O(\theta_0)^2}{{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}}.\end{aligned}$$ Repeating the application of Proposition \[prp:conv-star\](ii) as in [(\[eq:succ-appl\])]{}, we end up with $$\begin{aligned}
{\label{eq:piNgeq2-bd}}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&\leq\sum_{u_1,v_1}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2
q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}\,\frac{O(\theta_0)^j}{{\vbx-u_1{|\!|\!|}}^q
{\vbx-v_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^j}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$
For the bound on $\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)$, we use the following bound, instead of [(\[eq:P’0-bd\])]{}: $$\begin{aligned}
{\label{eq:P'0-dec}}
P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u
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@SugiyamaDIE06; @SugiyamaADE07]. For instance, an iteration lemma due to Kowalczyk [@Kowalczyk05] may be extended easily to the case ${\mathbb R}^d$, $d \geq 2$ and to include the ${\nabla}\cdot (\eta \theta)$ term in [@CalvezCarrillo06].
Fix $p > d$. Then by Lemma \[lem:finite\_p\_bounded\], for sufficiently small $M$ and ${\|\theta_0\|}_{\overline{q}}$, ${\|\theta(\tau)\|}_p \in L_\tau^\infty({\mathbb R}^+)$. Therefore by Lemma \[lem:CZ\_rescale\] in the appendix, $${\|{\nabla}\vec{v}\|}_p = {\|e^{(1-\alpha-\beta)\beta^{-1}\tau}{\nabla}\left(e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau} \cdot)\ast\theta\right)\|}_p \lesssim e^{(1-\alpha)\beta^{-1}\tau}{\|\theta\|}_p \lesssim e^{(1-\alpha)\beta^{-1}\tau}.$$ Moreover, by ${\nabla}{\mathcal{K}}\in L^1({\mathbb R}^d)$, $$\begin{aligned}
{\|\vec{v}\|}_p & \leq e^{(1 - \alpha - \beta)\beta^{-1}\tau}{\|\theta\|}_p \lesssim e^{(1 - \alpha - \beta)\beta^{-1}\tau}. \end{aligned}$$ Since $1 - \alpha \leq 0$, Morrey’s inequality implies $\vec{v} \in L^\infty_{\tau,\eta}({\mathbb R}^+\times{\mathbb R}^d)$ and the lemma follows.
By Lemma \[lem:rescaled\_inftybdd\] and the definition of $\tau$, $${\|u(t)\|}_{L_x^\infty({\mathbb R}^d)} = e^{-d\tau}{\|\theta\|}_{L_\eta^\infty({\mathbb R}^d)} \lesssim (1+t)^{-d\beta},$$ establishing . A similar argument using Lemma \[lem:finite\_p\_bounded\_unifint\] in place of Lemma \[lem:finite\_p\_bounded\] implies
Theorem \[thm:Decay\] holds if ${\nabla}{\mathcal{K}}\in L^1$.
Now we turn to Theorem \[thm:IA\].
(Theorem \[thm:IA\]: **Intermediate Asymptotics I**) Now that the requisite decay estimate has been established, we proceed by estimating the decay of the relative entropy . By Young’s inequality, ${\nabla}{\mathcal{K}}\in L^1({\mathbb R}^d)$ and \[ineq:LinftyDecay\], $${\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast\theta\|}_\infty \leq {\|{\nabla}{\mathcal{K}}\|}_1{\|\theta\|}_\infty \lesssim 1. \label{ineq:velocity_bounded}$$
We first settle the case $m > 1$. By a standard computation, and Cauchy-Schwarz, for all $\delta
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ties to bind the contract in the next section. Here, we only note that we do not discuss cases when after step $m$ both clients want the same: if the protocol is fair for the cases when clients’ wishes are opposite (a conservative assumption), it will be when they wish the same.
Averaging over all possible Bob’s strategies $X^{\cal{B}}$ (Alice is honest, so her strategy is known), we obtain $$P_{bind}^{\cal{A}} (m, \alpha) = \sum_{X^{\cal{B}}} p(X^{\cal{B}}) P_{bind}^{\cal{A}} (m, \alpha, X^{\cal{A}}, X^{\cal{B}}),$$ and analogously for $P^{\cal{B}}_{bind} (m, \alpha)$. Here, $p(X^{\cal{B}})$ is the probability that Bob chooses the particular strategy $X^{\cal{B}} = X^{\cal{B}} (m, \delta m)$, and is given by the probability $p_w(\delta m) = 1-(3/4)^{\delta m}$ to obtain a wrong result, when measuring the Reject observable $\delta m$ times (obviously, wrong results are, in the case of ideal measurements with no errors, possible only on qubits from the Accept basis).
For our protocol to be fair, we require that at each step $m$ of the Exchange phase, the difference between the agents’ (Alice and Bob) probabilities to bind the contract can be made arbitrarily small: for any given $\varepsilon$, $$|P^{\cal{B}}_{bind}(m, \alpha) - P^{\cal{A}}_{bind}(m, \alpha) | < \varepsilon.$$ In order to make our protocol even more symmetric, we introduce [*the probability to cheat*]{} of a dishonest client (Bob). It is the product of Bob’s probability to bind, and the probability that Alice will not bind the contract: $$\begin{aligned}
P_{ch}^{\cal{B}} (m, \alpha, X^{\cal{A}}, X^{\cal{B}}) &=& P^{\cal{B}}_{bind} (m, \alpha, X^{\cal{A}}, X^{\cal{B}}) \\
&& \times [1 - P^{\cal{A}}_{bind} (m, \alpha, X^{\cal{A}}, X^{\cal{B}})]. \nonumber\end{aligned}$$
After the averaging over Bob’s strategies $X^{\cal{B}} (m)$ probability to cheat of a dishonest client Bob is: $$P_{ch}^{\cal{B}} (m, \alpha) = \sum_{X^{\cal{B}}} p(X^{\cal{B}}) P_{ch}^{\cal{B}} (m, \alpha, X^{\cal{A}}, X^{\cal{B}}).$$
Our second fairness requirement is
| 2,960
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| 3,080
| 0.774985
|
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|
}$ and $\{\dot xxyz\}$.
Homomorphic images of special Jordan dialgebras
-----------------------------------------------
In this section we construct the example of an exceptional two-generated Jordan dialgebra which is a homomorphic image of a special Jordan dialgebra.
Denote by $\widehat I$ the ideal of ${\mathrm{Di}}{\mathrm{As}}\,\langle x,y \rangle$ generated by the set $I$.
\[lemma:CriterionOfQuotientSpeciality\] Let $I$ be an ideal of ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X \rangle$. Then ${\mathrm{Di}}{\mathrm{SJ}}\,\langle X \rangle/I$ is special iff $\widehat I \cap {\mathrm{Di}}{\mathrm{SJ}}\,\langle X\rangle \subseteq I$.
The proof of this lemma is completely analogous to the proof of Theorem 2.2 [@Cohn:54].
\[prop:CriterionOfQuotientSpecialityTwoGenerated\] Let $I$ be an ideal of ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle$ is generated by elements $u_i$. Then ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle/I$ is special iff $\{u_i \dot xxy\} \in I$ and $\{u_i \dot yyx\} \in I$ for all $i$.
By Theorem \[thm:CohnForDialgebra\], ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle = {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$. Lemma \[lemma:CriterionOfQuotientSpeciality\] implies that ${\mathrm{Di}}{\mathrm{SJ}}\,\langle x,y \rangle/I$ is special iff $\widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle \subseteq I$.
“$\Rightarrow$”. It is clear that $\{u_i \dot xxy\} \in \widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$, hence the condition of proposition is necessary.
“$\Leftarrow$”. Suppose that $\{u_i \dot xxy\} \in I$ and $\{u_i \dot yyx\} \in I$ for all $i$ and let $w \in \widehat I \cap {\mathrm{Di}}{\mathrm{H}}\,\langle x,y \rangle$. It is clear (as in Lemma 3.2 [@Cohn:54]) that $w$ can be written as a symmetric polynomial $f=f^*$ in the $u$’s and $x$, $y$ which is linear homogenious in the $u$’s. We now regard $x$, $y$, $u_i$ as independent. Because $f\in{\mathrm{Di}}{\mathrm{H}}\,\langle x,y,u_i \rangle$, it can by Theorem \[thm:CohnForDialgebr
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nciple inquires both into mechanics of transduction, and how transducible values come into being. The automaton of the software mechanism answers the latter question. If software hazard is to be evaluated starting at points of transduction and proceeding backwards through internal logic, then the automaton must support reverse inference – meaning reversed in computational order, from final conclusion to possible premise (see §\[S:INTRO\_CONE\]).
CHOICE FORK {#S:CHOICE_FORK}
-----------
Chapter 2 (Discrete Systems Theory) details relationships between systems theory and automata. From a mathematical standpoint the material is necessary, but there are readers for whom this chapter would duplicate existing knowledge. After verifying their understanding of operational profiles, section \[S:OPERATIONAL\_PROFILE\_SECTION\], they are invited to skip forward to Chapter 3.
Chapter 2 is summarized here to decide whether to skip it. Software is described using a triad of structures: the process, the procedure, and the path; not all are independent. Rudiments underlying these structures consist of ensembles and Cartesian products. Walks, the actuated automaton, converse automata, reverse walks, and cones follow.
Those desiring detailed introduction to fundamentals may access Appendix \[Ch:GROUNDWORK\], which reviews groundwork and notation used here. Its highlights include that an ensemble is a mapping from a set of stimuli into a set of responses. Ensembles are denoted by uppercase Greek letters such as $\Psi$. The general Cartesian product of an ensemble, called a choice space, is denoted ${\prod{\Psi}}$.
Discrete systems theory
=======================
Discrete systems theory (software) is identified with the actuated automaton.
Process
-------
Chains of stimulus and response characterize reactive discrete systems. In this chain, successive links are not independent: the response effected in one link feeds forward into the stimulus of the following link. For instance, in a system of cog-wheels and escapements, g
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the following formula: $$\Theta_{\Z/2 \int \D_4}^k(h,\Psi,\zeta) = <w_2(\bar \eta)^{\frac{n-4k}{2}};[L^{n-4k}]>.$$ $$$$
This new invariant is a homomorphism $\Theta_{\Z/2 \int \D_4}^k:
Imm^{\Z/2 \int \D_4}(n,n-4k) \to \Z/2$ included into the following commutative diagram:
$$\begin{array}{ccc}
Imm^{\D_4}(n-2k,2k) & \stackrel{\Theta_{\D_4}}{\longrightarrow} & \Z/2 \\
\downarrow \delta_{\D_4}^k & & \vert \vert \\
Imm^{\Z/2 \int \D_4}(n-4k,4k) &
\stackrel{\Theta_{\Z/2 \int \D_4}^k}{\longrightarrow} & \Z/2. \\
\end{array} \eqno(7)$$
Let us formulate the first main results of the paper. In section 2 the notion of $\Z/2 \oplus \Z/2$-control ($\I_b$–control) on self-intersection of a skew-framed immersion is considered. Theorem 1 (for the proof see section 3) shows that under a natural restriction of dimensions the property of $\I_b$-control holds for an immersion in the regular cobordism class modulo odd torsion.
In section 4 we formulate a notion of $\Z/2 \oplus
\Z/4$–structure (or an $\I_4$–structure, or a cyclic structure) of a $\D_4$-framed immersion. In section 5 we prove Theorem 2. We prove under a natural restriction of dimension that an arbitrary $\D_4$-framed $\I_b$-controlled immersion admits in the regular homotopy class an immersion with a cyclic structure. For such an immersion Kervaire invariant is expressed in terms of $\Z/2 \oplus
\Z/4$–characteristic numbers of the self-intersection manifold. The proof (based on the two theorems from \[A2\] (in Russian)) of the Kewrvaire Invariant One Problem is in section 6.
The author is grateful to Prof. M.Mahowald (2005) and Prof. R.Cohen (2007) for discussions, to Prof. Peter Landweber for the help with the English translation, and to Prof. A.A.Voronov for the invitation to Minnesota University in (2005).
This paper was started in 1998 at the Postnikov Seminar. This paper is dedicated to the memory of Prof. Yu.P.Soloviev.
Geometric Control of self-intersection manifolds of skew-framed immersions
================================================
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nteractions. A common approach for the QCD dynamics in $\gamma \gamma$ and $\gamma h$ interactions is important to minimize the theoretical uncertainty and to perform a realistic comparison between the predictions of the two different mechanisms for the double vector production. In order to describe the vector meson production in $\gamma A$ interactions we need to specify the forward dipole - nucleus scattering amplitude, $\mathcal{N}_A(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_A)$. Following [@bruno1] we will use in our calculations the model proposed in Ref. [@armesto], which describes the current experimental data on the nuclear structure function as well as includes the impact parameter dependence in the dipole nucleus cross section. In this model the forward dipole-nucleus amplitude is given by $$\begin{aligned}
{\cal{N}}_A(x,{\mbox{\boldmath $r$}},{\mbox{\boldmath $b$}}_A) = 1 - \exp \left[-\frac{1}{2} \, \sigma_{dp}(x,{\mbox{\boldmath $r$}}^2)
\,T_A({\mbox{\boldmath $b$}}_A)\right] \,\,,
\label{enenuc}\end{aligned}$$ where $\sigma_{dp}$ is the dipole-proton cross section given by $$\begin{aligned}
\sigma_{dp} (x,{\mbox{\boldmath $r$}}^2) = 2 \int d^2{\mbox{\boldmath $b$}}_p \,\,\mathcal{N}_p(x,{\mbox{\boldmath $r$}},{{\mbox{\boldmath $b$}}_p}) \end{aligned}$$ and $T_A({\mbox{\boldmath $b$}}_A)$ is the nuclear profile function, which is obtained from a 3-parameter Fermi distribution for the nuclear density normalized to $A$. The above equation sums up all the multiple elastic rescattering diagrams of the $q \overline{q}$ pair and is justified for large coherence length, where the transverse separation ${\mbox{\boldmath $r$}}$ of partons in the multiparton Fock state of the photon becomes a conserved quantity, [*i.e.*]{} the size of the pair ${\mbox{\boldmath $r$}}$ becomes eigenvalue of the scattering matrix.
In the case of the double vector meson production in $\gamma \gamma$ interactions at hadronic colliders, represented in Fig. \[dia1\], we have that the total cross section is given by (For de
| 2,964
| 1,490
| 3,240
| 2,797
| 4,044
| 0.768481
|
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11.0 (4.0--17.0) 5.0 (5.0--6.0) 16.0 (16.0--17.0)
Glucose mmol/L 191 5.8 (2.5--12.9) 3.7 (3.1--4.1) 9.1 (8.3--10.2)
Aspartate aminotransferase U/L 191 161.0 (50.0--390.0) 84.4 (76.0--93.0) 304.8 (269.0--362.0)
Calcium mmol/L 191 1.9 (1.3--2.9) 1.4 (1.3--1.5) 2.5 (2.4--2.6)
Phosphorus mmol/L 191 2.2 (1.2--3.6) 1.4 (1.3--1.6) 3.3 (3.1--3.6)
Calcium/phosphorus ratio mmol/L/mmol/L 191 0.8 (0.4--1.8) 0.5 (0.4--0.6) 1.4 (1.3--1.6)
Sodium mmol/L 191 156.0 (145.0--168.0) 150.0 (145.0--150.0) 163.0 (162.0--165.0)
Potassium mmol/L 191 4.2 (2.5--6.1) 2.9 (2.5--3.1) 5.2 (5.1--5.7)
Chloride mmol/L 191 115.0 (101.0--129.0) 103.0 (102.0--107.0) 125.0 (123.0--125.0)
Globulin g/L 191 24.0 (13.0--46.0) 14.0 (14.0--16.0) 37.0 (33.0--38.0)
Creatine phosphokinase U/L 191 1,034.0 (153.0--13,310.0) 249.4 (165.0--297.0) 4409.0 (3782.0--5123.0)
Uric acid μmol/L 191 47.6 (5.9--166.5)
| 2,965
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| 2,034
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47 NONSYN A:5 C:164 C:37 `Tgagatgataat`
`gi 87160275 ref` `122` `NCLPVYKILLEK`
`NCLPVYKILL+K`
`NCLPVYKILLKK`
`SRR022865_59666` `-36` `attcgtaattaa` Lactose phosphotransferase system repressor phosphotransferase system, glucose-specific IIABC component
`agtctaatttaa`
2333470 NONSYN T:5 C:164 C:37 `ttgggtatgaaa`
`Query: 36 LV*IAPWLKNDI` `1`
`LV IAPWLKNDI`
2674216 TRUNC A:5 C:170 C:37 `Sbjct: 41 LVEIAPWLKNDI` `52`
| 2,966
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ak and the step-like feature indicate the same energy scale – we can thus conclude that the step is also a manifestation of the Kondo effect that leads to the peak recorded at the CH edge. Table \[tab:fwhm\_and\_q\] also reveals that $q$ is significantly smaller in the center of the molecule, indicating that there the probability to tunnel directly from the tip into the substrate ($t_1$) is larger than at the CH edge. The reason for the larger tunneling probability directly into the substate when the tip is located in the center of the molecule is a direct consequence of the spatial distribution of the LUMO wave function, which has a node in the center of the molecule and a pronounced lobe at the CH edge (Fig. \[fig:exp\_fig3\]b). This is reflected in Fig. \[fig:exp\_fig3\]a, which displays the LDOS of the LUMO $4$Å above the gas-phase NTCDA molecule.
![Local density of states (LDOS) of the LUMO of NTCDA calculated $4$Å above the gas-phase molecule (left panel). A graphical representation of the gas-phase NTCDA molecule has been overlaid for clarity. The right panel shows the top view of the LUMO of the gas-phase NTCDA molecule. The different colors indicate the positive ($\Psi (r) > 0$) and negative ($\Psi (r) < 0$) contributions of the wave function.[]{data-label="fig:exp_fig3"}](fig8){width="85mm"}
We note that the fits displayed in Fig. \[fig:exp\_fig2\] and the derived parameters in Table \[tab:fwhm\_and\_q\] are merely heuristic and should only be used to ascertain that there are at least two tunneling paths present, and that the center of the molecule is more transparent to the tunneling current than the CH edge. More elaborate fits to the spectra, based on a more solid theoretical foundation, will be presented in section section \[sec:application-to-NTCDA\].
molecule location $\delta$ $q$
---------- ---------- -------------------- ----------------
CH edge $ (28.5\pm 2.3)$mV $ 15.5\pm 6.9$
center $ (29.4\pm 6.8)$mV $ 1.2\pm 0.2$
| 2,967
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| 3,089
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- 1$. Since $k$ divides this size, there must be an element in $\F_{p^w}$ of order $k$.
Review of $O(n^{1/3})$ cost 2-server PIR {#oldconstruction}
========================================
There are several known constructions of 2-server PIR with $O(n^{1/3})$ communication cost. We will recall here in detail a particular construction due to [@WoodruffY05] which uses polynomial interpolation using derivatives (over a field). In the next section we will replace the field with a ring and see how to use matching vector families to reduce the communication cost.
Let $\ba=({a_1,\cdots,a_n})$ be the database, choose $k$ to be smallest integer such that $n \le \binom{k}{3} $. Let $\F_q$ be a finite field with $q > 3$ elements and suppose for simplicity that $q$ is prime (so that partial derivatives behave nicely for polynomials of degree at most $3$). Let $\phi:[n]\mapsto {\{0,1\}}^k\subset \F_q^k$ be an embedding of the $n$ coordinates into points in ${\{0,1\}}^k$ of Hamming weight 3. Such an embedding exists since $n\le \binom{k}{3}$.
Define $F({x_1,\cdots,x_k})=F(\bx) \in\F_q[x_1,\cdots,x_k]$ as $$F(\bx)=\sum_{i=1}^n a_i\left( \prod_{j: \phi(i)_j=1}x_j\right)$$ Note that $F(\bx)$ is a degree 3 polynomial satisfying $F(\phi(i))=a_i\ \forall\ i\in [n]$. Fix any two nonzero field elements $t_1 \neq t_2 \in \F_q \setminus \{0\}$.
Suppose the user $\cU$ wants to recover the bit $a_\tau$. The protocol is as follows: The user picks a uniformly random element $\bz\in \F_q^k$ and sends $\phi(\tau)+t_1\bz$ to $\cS_1$ and $\phi(\tau)+t_2\bz$ to $\cS_2$. Each server $S_i$ then replies with the value of $F$ at the point received $F(\phi(\tau)+t_i\bz)$ as well as the values of the $k$ partial derivatives of $F$ at the same point $$\nabla F(\phi(\tau)+t_i\bz)=\left(\frac{\partial F}{\partial z_1}(\phi(\tau)+t_i\bz),\cdots,\frac{\partial F}{\partial z_k}(\phi(\tau)+t_i\bz)\right)$$ The partial derivatives here are defined in the same way as for polynomials over the real numbers.
[align\*]{} &: \_q\^k\
\_i &: ()+t\_i\
\_i &: F(
| 2,968
| 2,658
| 3,286
| 2,918
| 1,371
| 0.790368
|
github_plus_top10pct_by_avg
|
he parameters $c_m$ are also selected randomly from a Gaussian distribution with zero mean and a variance we will specify. Under standard manipulations (Born, Markov and secular approximations), we find a master equation diagonal in the basis of eigenstates $\dot{P}_i = (\mathbb{W})_{ij} P_j$, where $P_i$ is the occupation probability of the eigenstate ${|i\rangle}$. The master operator $\mathbb{W}$ has elements $(\mathbb{W})_{ij}$ given by $$(\mathbb{W})_{ij} = W_{j\rightarrow i} - r_i\delta_{i,j}
\label{eq:W}$$ where the transition rates $W_{j\rightarrow i}$ are given by $$W_{j\rightarrow i} = J(\omega_{ji})\,\, |{\langle j|}S{|i\rangle}|^2\,.
\label{eq:Wij}$$ and $J(\omega_{ji}) = 2\pi \sum_k |h_k|^2 \delta(\omega_k - \omega_{ji})$ is the spectral density of the bath with $\omega_{ji} = E_j - E_i$. We will study the case of a bath with temperature $T=\infty$ such that the rates satisfy $W_{i\rightarrow j} = W_{j\rightarrow i}$. In this work, we consider an Ohmic bath with $J(\omega) = \omega$; this choice fixes the variance of the parameters $c_m$.
At long enough times, we anticipate that all knowledge of the initial location of the exciton will be lost and the probability of finding the exciton anywhere in the lattice will be uniform in accordance with the $T=\infty$ distribution. To ascertain how long the exciton has spent in different regions of the lattice we integrate the eigenstate occupation probabilities $P_i(t)$ over time and define $O_i(t) = \int_0^t dt' P_i(t')$. We express these occupation times in the local basis as $O_m(t) = \int_0^t dt' \sum_i |{\langle m | i \rangle}|^2 P_i(t')$
![(Colour online.) Histograms of the number of jumps $k$ in time intervals $t=300/J$ for simulations (as in Fig. \[fig1\]) with $10^8$ jumps in total. Plotted is the number of time intervals in which $k$ jumps occur, $N_\text{total}(k)$, for different strengths of disorder $d$ (labelled). Shown (dashed line) is a fit to the $d=1$ points assuming a Poisson distribution.[]{data-label="fig2"}](graph_dist.png){width="
| 2,969
| 2,183
| 3,228
| 2,908
| 2,204
| 0.781852
|
github_plus_top10pct_by_avg
|
ed volatility for these portfolios, on the other hand, need not (and in typical cases will not) diverge at all.
As stated earlier, the efficient frontier in the presence of a riskless asset has a simple allocation rule which requires that each principal portfolio be included in inverse proportion to its variance. For the current case, this rule clearly excludes the $N-2$ portfolios described above from the efficient frontier, leaving the first two principal portfolios and the riskless asset as the only constituents. Thus for the special case of constant residual variance, a knowledge of the two distinguished principal portfolios determined above is all that is needed to specify the efficient frontier when a riskless asset is present. For this reason, we will not continue with the explicit construction of the remaining $N-2$ eigenvectors.
At this point we can determine the expected value and the variance of the two principal portfolios determined above according to the definitions and formulae given in §2. Straightforward algebra leads to $${R}^{crv}_{N}={{\sum}_{i=1}^{N}{\hat{{\beta}}}_{i}({\bar{{\alpha}_{i}}}
+{\beta}_{i}{\bar{\rho}}_{mkt}) \over {N}^{{1 \over 2}} \cos
(\theta)}, \;\; {({V}_{N}^{crv})}^{2}={ {\bar{{\alpha}^{2}}} +
{\bm{\beta}} \cdot {\bm{\beta}} {\bar{{\rho}^{2}}}_{mkt} \over N
{\cos}^{2} (\theta)}, \label{448}$$ for the market-aligned portfolio, and $${R}^{crv}_{1}={{r}^{av} \over {\sin}^{2} (\theta)} -{\cot}^{2}
(\theta){R}^{crv}_{N}
, \;\; {({V}_{1}^{crv})}^{2}={ {\bar{{\alpha}^{2}}} \over N {\sin}^{2}
(\theta)}, \label{449}$$ for the market-orthogonal, minimum volatility principal portfolio. In order to facilitate comparison with the perturbative results of §2 for the general single-index model, we also record here the return-adjusted volatilities of these portfolios; $${\check{V}}^{crv}_{N} ={{({\bar{{\alpha}^{2}}} + {\bm{\beta}}
\cdot {\bm{\beta}} {\bar{{\rho}^{2}}}_{mkt})}^{1 \over 2} \over
{\sum}_{i=1}^{N}{\hat{{\beta}}}_{i}
({\bar{{\alpha}_{i}}}+{\beta}_{i}{\bar{\rho}}_{mkt})}, \lab
| 2,970
| 1,318
| 3,223
| 2,799
| null | null |
github_plus_top10pct_by_avg
|
m more accurate starburst diagnostics than a forbidden/recombination pair like \[\]/. To reduce reddening effects, we select lines close in wavelength to H lines. Unfortunately, the helium lines are weak: [ $6678$]{} saturates at $0.014$ of the strength of , and [ $4471$]{} saturates at $0.05$ of . As such, in the spectral atlas of @ho3, [ $6678$]{} was detected in only $108$ of $418$ galactic nuclei, and [ $4471$]{} in only $16$ nuclei. The small sample indicates that [ $4471$]{} is only marginally detected, and we do not consider it further.
Figures \[fig:optical\] and \[fig:o3\_Ne\] have already implicated \[\]/ as an unreliable indicator for \[\]/$\ga0.5$. This makes it hard to gauge the reliability of [ $6678$]{}/ in figure \[fig:optical\]. Also, the sample sizes are too small to compare the optical recombination line ratios to \[\]/\[\], [ $1.7$ ]{}/, or [ $2.06$ ]{}/individually. Instead, we use the latter three indicators together to test how well the optical recombination line ratios correlate with . In table \[tab-opt\], we list galaxies with measurements of at least two different indicators, in order of increasing , as determined from \[\]/\[\], [ $1.7$ ]{}/, and [ $2.06$ ]{}/ (when $\le 0.2$), as available. Due to measurement error and uncertainty in the relative calibrations of the diagnostics, the ordering is somewhat uncertain. The published plots of the @ho3 spectra lack the dynamic range to assign upper limits to the undetected optical recombination lines. These are marked as “non det” in table \[tab-opt\].
In general, table \[tab-opt\] shows some correlation between [ $6678$]{}/ and , though with considerable scatter. Using Kendall’s $\tau$ rank correlation test on the eight galaxies with measured [ $6678$]{}/, there is only a $5\%$ chance that and [ $6678$]{}/ are uncorrelated.
Testing the Mid–Infrared Fine Structure Line Ratios {#sec:midir-test}
---------------------------------------------------
In the mid–infrared, ratios of the fin
| 2,971
| 3,337
| 2,874
| 2,695
| null | null |
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|
\delta_a \mid a \in G\}$ is an orthonormal basis of $\ell^2(G)$. Then $\widehat{\delta_a}(\chi) = \chi(a)$ for each $\chi \in \widehat{G}$. By Lemma \[T:FT-isometry\](\[I:isometry\]), the number of $a \in G$ such that $a^2 = 1$ is equal to $$\begin{aligned}
\sum_{a \in G} {\left\langle \delta_a, \delta_{a^{-1}} \right\rangle}
&= \frac{1}{{\left\vert G \right\vert}} \sum_{a \in G}
{\left\langle \widehat{\delta_a}, \widehat{\delta_{a^{-1}}} \right\rangle}
= \frac{1}{{\left\vert G \right\vert}} \sum_{a \in G} \sum_{\chi \in \widehat{G}}
\chi(a) \overline{\chi}(a^{-1}) \\
&= \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} \sum_{a \in G}
\chi(a)^2
= \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}}
{\left\langle \chi, \overline{\chi} \right\rangle},
\end{aligned}$$ which by Lemma \[T:FT-isometry\](\[I:onb\]) is equal to the number of $\chi \in \widehat{G}$ such that $\chi = \overline{\chi}$.
Lemma \[T:p2\] says that $G$ and $\widehat{G}$ have equal numbers of elements of order $2$. A much stronger fact is also true: $G$ and $\widehat{G}$ are isomorphic groups. However, this isomorphism is noncanonical, depends on the classification of finite abelian groups, and in any case is not useful here.
\[T:extensions\] Let $H$ be a subgroup of a finite abelian group $H$. Then each character on $H$ extends to a character on $G$ in precisely ${\left\vert G \right\vert}/{\left\vert H \right\vert}$ distinct ways.
It is easy to check that restriction to $H$ defines a homomorphism $\widehat{G} \to \widehat{H}$. Since each coset of this homomorphism’s kernel has the same size, it suffices to prove that that it is surjective, or equivalently that each character on $H$ extends to a character on $G$ at all. For a proof of this fact see, e.g., [@Apostol p. 134].
From and Lemma \[T:FT-isometry\](\[I:convolution\]) it follows that the Fourier transform diagonalizes $G$-circulant matrices. In particular, if $M =
[f(ab^{-1})]_{a,b \in G}$ for $f \in \ell^2(G)$,
| 2,972
| 5,199
| 1,504
| 2,195
| null | null |
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|
74.0 ± 1.2 69.0 ± 1.3 100.8 ± 2.2 108.7 ± 1.1
Genotype 0.105 0.476 \<0.001 0.812 \<0.001 0.963
Date \<0.001 \<0.001 \<0.001 \<0.001 \<0.001 0.002
Geno × Date 0.052 0.12 0.1 0.973 0.239 0.767
John Wiley & Sons, Ltd
The % digestibility of cell wall glucose significantly declined between the two time points for both sets of plants whereas the % digestibility of xylose showed a nonsignificant change between July and October. The average difference in % digestibility of glucose was −16% of July levels in October for the mixed population and −7% of July levels in October for the mapping family. However, as biomass increased by 70% between July and October, yields of digestible sugars in October will still greatly exceed yields in July.
Nutrient remobilization {#gcbb12419-sec-0024}
-----------------------
The nitrogen (N), phosphorous (P) and potassium (K) of the total above‐ground material was analysed at six time points over 2 years: July (2011 & 2012), November (2011), December (2012) and January (2011 & 2012; Fig. [7](#gcbb12419-fig-0007){ref-type="fig"}). The climate data are shown in Fig. [1](#gcbb12419-fig-0001){ref-type="fig"}. Significant differences were observed between the harvest dates for all nutrients in both years (*P* =\< 0.01; Fig. [7](#gcbb12419-fig-0007){ref-type="fig"}). In July 2011 and 2012, N concentration was 13--21 g kg^−1^ but by January this had declined three‐ to fourfold to be only 5 g kg^−1^. A similar fourfold decline was also seen in P
| 2,973
| 1,191
| 2,998
| 3,070
| null | null |
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|
he quantum-classical dynamics of operators is transformed into a theory for phase space dependent wave fields evolving in time. Such a theory for wave fields is also expressed by means of suitable non-Hamiltonian brackets: in this way a link is found with the generalization of Weinberg’s non-linear formalism given in Appendix \[app:weinberg\]. More specifically, in Appendix \[app:weinberg\], Weinberg’s formalism is briefly reviewed and its symplectic structure is unveiled. Then, this structure is generalized by means of non-Hamiltonian brackets. Therefore, one can appreciate how the generalized Weinberg’s formalism establishes a more comprehensive mathematical framework for non-linear equations of motion, comprising phase space dependent wave fields as a special case. In Section \[sec:qcwdab\] the abstract non-linear equations of motion for quantum-classical fields are represented in the adiabatic basis and some considerations, which pertain to the numerical implementation, are made. By making an equilibrium *ansatz*, in Section \[sec:sb\] the non-linear equations of motion are put into a linear form and the theory is applied to the spin-boson model. Section \[sec:conclusions\] is devoted to conclusions and perspectives.
Non-Hamiltonian Mechanics of Quantum-Classical Operators {#sec:bracket}
========================================================
A quantum-classical system is composed of both quantum $\hat{\chi}$ and classical $X$ degrees of freedom, where $X=(R,P)$ is the phase space point, with $R$ and $P$ coordinates and momenta, respectively. Within the operator formalism of Refs. [@qc-bracket; @kcmqc; @b3; @bsilurante], the quantum variables depend from the classical point, $X$, of phase space. The energy of the system is defined in terms of a Hamiltonian operator $\hat{H}=\hat{H}(X)$, which couples quantum and classical variables, by $E={\rm Tr}'\int dX \hat{H}(X)$. The dynamical evolution of a quantum-classical operator $\hat{\chi}(X)$ is given by [@qc-bracket; @kcmqc] $$\begin{aligned}
\fr
| 2,974
| 1,918
| 1,497
| 2,623
| null | null |
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|
d we find that the discharge rate is fast if the WGC is satisfied by a light particle of mass $m\ll1/M$. For sufficiently large black holes $M\gtrsim {\rm max}(e^{|\Delta\phi|},1/m)$ the rate is slow.
We conclude with a loose conjecture: quantum gravity in asymptotically flat space requires a general bound on large localized moduli space excursions of the form $ |\Delta\phi|\lesssim |\log(R\Lambda)|$, where $R$ is the size of the minimal region enclosing the excitation and $\Lambda^{-1}$ is a short-distance cutoff. Both neutral and charged KK bubbles have finite $R$ and infinite excursions, but are strongly unstable. Dilatonic black holes in a controlled EFT also satisfy the bound. KK monopoles provide another example: they are stable and sample an infinite distance in moduli space, but only at a single point, so the visible excursion is limited by the short-distance cutoff.
Charged KK Bubbles
==================
We begin by discussing a representative class of Kaluza-Klein bubbles perturbatively stabilized by flux. We then add matter of mass $m$ and charge $q$ to the system and demonstrate the existence of a rapid pair-production instability for $q/m\gtrsim 1$.
Classical Solutions
-------------------
A number of static charged bubble solutions were obtained in [@Gibbons:1994vm; @Horowitz:2005vp]. In [@Horowitz:2005vp], a 6D bubble stabilized by electric and magnetic 3-form flux was constructed from a family of 5D zero-momentum initial data characterizing bubbles of different sizes. This method is particularly convenient for assessing bubble stability against radial perturbations. We review this construction here, simplifying to case of purely electric 3-form flux.
A family of five-dimensional spatial metrics is given by $$\begin{aligned}
ds^2_{spatial}=U(\rho)d\chi^2+\frac{d\rho^2}{U(\rho) h(\rho)}+\rho^2d\Omega_3
\label{eq:spatial}\end{aligned}$$ with $$\begin{aligned}
U(\rho)\equiv 1-\frac{\rho_0^2}{\rho^2}.
\label{eq:U}\end{aligned}$$ The function $h$ will be determined by the Hamiltonian constraint, a
| 2,975
| 1,587
| 1,762
| 2,949
| null | null |
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|
to be 0.089 events/Mton$\cdot$yr; we take double this value (0.18 $\pm$ 0.18 events/Mton$\cdot$yr) as a conservative estimate of the background rate for this decay mode. Similarly, we extrapolate for all of the dinucleon decay modes, finding background rates of 0.008 ($NN \rightarrow ee$), 0.033 ($NN \rightarrow e \mu$), and 0.006 ($NN \rightarrow \mu \mu$) events/Mton$\cdot$yr. We conservatively take the largest of these and double it as our estimate of expected background for all of the dinucleon decay modes: $0.07\pm 0.07$ events/Mton$\cdot$yr.
[l l @ cccc @cccc]{} & &\
& & SK-I & SK-II & SK-III & SK-IV & SK-I & SK-II & SK-III & SK-IV\
& High $P_{\text{tot}}$ & $51.0\pm0.2$ & $49.5 \pm 0.2$ & $50.8 \pm 0.2$ & $50.6 \pm 0.2$ & $0.01 \pm 0.01$ & $0.02 \pm 0.02$ & $< 0.01$ & $0.07 \pm 0.07$\
& Low $P_{\text{tot}}$ & $27.6 \pm 0.1$ & $26.1 \pm 0.1$ & $27.6 \pm 0.1$ & $27.5 \pm 0.1$ & $0.02 \pm 0.02 $ & $0.01 \pm 0.01$ & $0.01 \pm 0.01$ & $0.04 \pm 0.04$\
& High $P_{\text{tot}}$ & $50.2\pm 0.2$ & $49.7\pm0.2$ & $51.0 \pm 0.2$ & $48.1 \pm 0.2$ & $0.22 \pm 0.14$ & $0.14 \pm 0.11$ & $0.07 \pm 0.07$ & $0.23 \pm 0.14$\
& Low $P_{\text{tot}}$ & $29.1 \pm 0.1$ & $28.3 \pm 0.1$ & $29.0 \pm 0.1$ & $29.4 \pm 0.1$ & $0.02 \pm 0.02$ & $0.01 \pm 0.01$ & $<0.01$ & $0.02 \pm 0.02$\
$NN \rightarrow ee$ & & $80.9 \pm 0.1$ & $77.2 \pm 0.1$ & $79.5 \pm 0.1$ & $78.6 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\
$NN \rightarrow e\mu$ & & $84.1 \pm 0.1$ & $83.7 \pm 0.1$ & $83.4 \pm 0.1$ & $81.7 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\
$NN \rightarrow \mu\mu$ & & $86.3 \pm 0.1$ & $85.9 \pm 0.1$ & $86.0 \pm 0.1$ & $82.8 \pm 0.1$ & $0.01 \pm 0.01$ & $<0.01$ & $<0.01$ & $0.01 \pm 0.01$\
We find zero candidate events for the eight dinucleon decay modes. For the nucleon decay mode $p \rightarrow e^+ \gamma$, we also find zero candidate events. We observe two candidate events during the SK-IV period for the $p \rightarrow \mu^+ \gamma$ decay mode in the “High $P_{tot}$" signal box when $0.23\pm0
| 2,976
| 360
| 3,633
| 2,989
| null | null |
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|
rom \[prop:partition\], \[prop:ysmall\], and \[prop:ylarge\]) $|\cR\setminus\cR^*|\le 2$ and (from \[prop:bi\]) $|\cR^*|\le 2$. But this means that $|\cR|\le 4< 5+|Y|$, a contradiction. Therefore we know that if $|Y_x|\ge 2$ then $Y_{\hat{x}}=\emptyset$.\
If, for some $x\in X^*$, we have $|Y_x|=|Y_{\hat{x}}|=1$ (we may assume by relabeling, if necessary, that $x=2$, so $\hat{x}=3$), then (from \[prop:partition\] and \[prop:ysmall\]) $|\cR\setminus\cR^*|\le 4$ and (from \[prop:bi\]) $|\cR^*|\le 2$. Therefore, from $Y\ne\emptyset$, we get $|\cR|\le 6\le 5+|Y|$, therefore $|\cR|=5+|Y|$ and $|Y|=1$. Without loss of generality $Y_2=Y_3=Y=\{6\}$. Also, $|\cR^*|=2$, and $\cR^*=\{\{2,3,5\},\{2,3,4\}\}$ and (from \[prop:bi\]) $Y_4=Y_5=\emptyset$; consequently $\cS(Y)=\{\{1,2,6\},\{1,3,6\}\}$. Moreover (using \[prop:ysmall\]), from $|\cR\setminus\cR^*|=4$ we get that, for each $i\in\{2,3\}$ and $j\in\{4,5\}$, we have $\cR(i,j)=\{\{i,j,6\}\}$. Thus (from \[prop:sstar\]) $\cS^*=\emptyset$. But this yields $|\cI_2|=5>4=|\cS|$, a contradiction.\
Therefore we can now assume, for all $x\in X^*$, that $\min(|Y_x|,|Y_{\hat{x}}|)=0$. Set $L=\{x\in X^*\mid Y_x\ne\emptyset\}$. Then we have that $|L|\le 1$ or $L=\{i,j\}$ for some $i\in\{2,3\}$ and $j\in\{4,5\}$. For each $x\in X^*$ we have that $$\Bigl(|\cF_x|+|Y_x|+|\cS_x^*|\Bigr) + \Bigl(\sum_{j\in C_x}|\cR(x,j)|+|\cR^*_x|\Bigr)\ \le\ |\cI_x|\ \le\ |\cS|\ ,
\label{eq:individual}$$ where we have counted the sets containing 1 before those not containing 1. Of course, $|\cF_x|=1$ and $|S^*_x|\le 2$. By summing over $X^*$, we obtain $$4+\sum_{x\in X^*} |Y_x|+2|\cS^*|+2\left(\sum_{i\in\{2,3\}}\sum_{j\in\{4,5\}} |{\cal R}(i,j)|\right)+3|{\cal R}^*|\le 4|\cS|\ ,$$ which simplifies to $$4+\sum_{x\in X^*} |Y_x|+2|\cS^*|+2|\cR|+|\cR^*|\le 4|\cS|\ .$$ In particular, $$|\cR|\le 2|\cS|-2-\frac{1}{2}\sum_{x\in X^*}|Y_x|-|\cS^*|-\frac{1}{2}|\cR^*|\ .
\label{eq:general}$$ Now we consider the following three subcases, based on the size of $L$.\
[**Case**]{} $L=\emptyset$\
Then each $Y_x=Y=\emptyset$
| 2,977
| 1,470
| 1,659
| 2,924
| null | null |
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|
sum_{i,j}(\mu^i_j)^2+2$.
We have
$$\Log\left(\sum_\muhat q^{-\frac{1}{2}(d_\muhat-2)}V_\muhat(q)m_\muhat\right)=\frac{q}{q-1}\sum_\muhat A_\muhat(q)m_\muhat.$$ \[theohua\]
By Lemma \[moz\] and Formula (\[ExpA\]) we are reduced to prove the following.
We have $$\log\,\left(\sum_\muhat q^{-\frac{1}{2}(d_\muhat-2)}V_\muhat(q)m_\muhat\right)=\sum_{d=1}^\infty\varphi_d(q)\cdot\log\,\left(\Omega\left(\x_1^d,\dots,\x_k^d;0,q^{d/2}\right)\right)$$where $\varphi_n(q)=\frac{1}{n}\sum_{d|n}\mu(d)q^{n/d}$ is the number of $\langle f\rangle$-orbits of $\F$ of size $n$.
By Proposition \[inner\], we have
$$V_\muhat(q)=\frac{q^{-n^2+\frac{1}{2}(kn^2-\sum_{i,j}(\mu^i_j)^2)}}{|G|}\sum_{x\in\mathfrak{g}}\Lambda^\sim(x)R_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}(1)(x)\cdots R_{\mathfrak{l}_{\mu^1}}^\mathfrak{g}(1)(x).$$By Remark \[Rl=RL\] and Corollary \[R\], we see that $R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1)(x)=\left\langle\tilde{H}_\omega(\x;q),h_\lambda(\x)\right\rangle$ when the $G$-orbit of $x$ is of type $\omega$.
We now proceed exactly as in the proof of Proposition \[sumM\] to prove our formula.
Applications to the character theory of finite general linear groups {#applichar}
--------------------------------------------------------------------
The following theorem (which is a consequence of Theorem \[purity\] and Theorem \[multi\]) expresses certain fusion rules in the character ring of $\GL_n(\F_q)$ in terms of absolutely indecomposable representations of comet shaped quivers.
\[mult-thm\] We have $$\langle \Lambda\otimes R_\muhat,1\rangle=A_\muhat(q).$$ \[multi=A\]
From Theorem \[multi=A\] and Theorem \[kactheo\] we have the following result.
$\langle \Lambda\otimes R_\muhat,1\rangle\neq 0$ if and only if $\v_\muhat\in\Phi(\Gamma_\muhat)^+$. Moreover $\langle
\Lambda\otimes R_\muhat,1\rangle=1$ if and only if $\v_\muhat$ is a real root.
We will see in §\[delta-non-neg\] that $\v_\muhat$ is always an imaginary root when $g\geq 1$, hence the second assertion concerns only the case $g=0$ (i.e. $\Lambda=1$).
A
| 2,978
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| 2,148
| 2,614
| null | null |
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|
); // returns nothing
}
}
}
What am I doing wrong?
A:
I found the answer in the Raven Google group. It turns out I have to query with the same data type as the data. So, in this case since "Price" is a decimal, I have to pass in 60M to the where clause:
var gt = session.Advanced.LuceneQuery<Asset, AssetDataSearch>().WhereGreaterThan("Price", 60M).ToArray();
Q:
HTML5 Video in Amazon FireTV Apps
Pretty basic question.
I am using PhoneGap/Cordova (http://www.phonegap.com) to build an app for Amazon FireTV OS, which is built on Android. I am using HTML, Javascript, CSS and everything is working fine. However, when I try to use the < video > tag to pull in an .MP4 video, all I am getting is a gray background and film strip logo (picture: http://imgur.com/MXLvwy7). What does this mean? The documentation says is supported and I am 100% positive of the path and any other silly pitfalls. Do I need a cordova plugin of some sort? Perhaps the manifest file needs to be tweaked? Thank you for your assistance.
A:
As you didn't provide the code, I will try to form how I have done it before:
Make sure your code does not have video.load(); for android
You must call video.play();
example:
video.addEventListener('click',function(){
video.play();
},false);
If you trigger play please call video.pause(); before you call video.play();
Q:
Improving VBA Macro in Word to automatically create file relating to document name and location
I am currently using a macro to add an incrementing document number to the footer of a document. The problem is it uses a global "settings.txt" file.
I want to apply this macro to many documents and each much have its own record of the incrementing number.
I am not a pro a VBA/Macro writing so I do need some pretty clear instructions or copy and pasteable code.
Sub SerialNumber()
'
' SerialNumber Macro
'
'
Dim Message As String, Title As String, Default As String, NumCopies As Long
Dim Rng1 As Range
' Set prompt.
Message = "Enter the number of copies that you
| 2,979
| 2,867
| 125
| 2,021
| 2,452
| 0.779602
|
github_plus_top10pct_by_avg
|
a_{o,x}|\leq\theta\delta_{o,x}+\frac{\lambda(1
-\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ for any $p\leq{p_\text{c}}$ and any $x\in{{\mathbb Z}^d}$, where $(f*g)(x)=\sum_{y\in{{\mathbb Z}^d}}f(y)\,g(x-y)$. We note that the identity in [(\[eq:Ising-lace-Zdlim\])]{} is similar to the recursion equation for the random-walk Green’s function: $$\begin{aligned}
S_r(x)\equiv\sum_{i=0}^\infty r^iD^{*i}(x)=\delta_{o,x}+(rD*S_r)(x)
\qquad(|r|<1),\end{aligned}$$ where $f^{*i}(x)=(f^{*(i-1)}*f)(x)$, with $f^{*0}(x)=\delta_{o,x}$ by convention. The leading asymptotics of $S_1(x)$ for $d>2$ is known as $\frac{a_d}{\sigma^2}|x|^{-(d-2)}$, where $a_d=\frac{d}2\pi^{-d/2}\Gamma(\frac{d}2-1)$ (e.g., [@h05; @hhs03]). Following the model-independent analysis of the lace expansion in [@h05; @hhs03], we obtain the following asymptotics of the critical two-point function:
\[thm:x-asy\] Let $\rho=2(d-4)>0$ and fix any small ${\epsilon}>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, we have that, for $x\ne o$, $$\begin{aligned}
{\label{eq:thm-asy}}
G_{{p_\text{c}}}(x)=\frac{A}{\tau({p_\text{c}})}\,\frac{a_d}{\sigma^2|x|^{d-2}}
\times\begin{cases}
\big(1+O(|x|^{-\frac{(\rho-{\epsilon})\wedge2}d})\big)&(\text{NN model}),\\
\big(1+O(|x|^{-\rho\wedge2+{\epsilon}})\big)
&(\text{SO model}),
\end{cases}\end{aligned}$$ where constants in the error terms may vary depending on ${\epsilon}$, and $$\begin{aligned}
{\label{eq:constants}}
\tau({p_\text{c}})=\bigg(\sum_x\Pi_{{p_\text{c}}}(x)\bigg)^{-1},&&
A=\bigg(1+\frac{\tau({p_\text{c}})}{\sigma^2}\sum_x|x|^2\Pi_{{p_\text{c}}}(x)\bigg)^{-1}.\end{aligned}$$ Consequently, [(\[eq:MFbehavior\])]{} holds and $\eta=0$.
In this paper, we restrict ourselves to the nearest-neighbor model for $d\gg4$ and to the spread-out model for $d>4$ with $L\gg1$. However, it is strongly expected that our method can show the same asymptotics of the critical two-point function for *any* translation-invariant, ${{\mathbb Z}^d}$-symmetric finite-range model above fou
| 2,980
| 1,924
| 2,500
| 2,725
| 2,604
| 0.778511
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github_plus_top10pct_by_avg
|
as ΔCt mean ± SEM are shown in [Table 3](#pone.0214536.t003){ref-type="table"} (raw data are shown in [S2](#pone.0214536.s002){ref-type="supplementary-material"} and [S3](#pone.0214536.s003){ref-type="supplementary-material"} Tables). The results showed that the expression levels of the Hbl and Nhe toxin genes were considerably higher in all *B*. *cereus* reference strains compared to *B*. *toyonensis* BCT-7112^T^. The expression of *hblC* was absent in *B*. *toyonensis* BCT-7112^T^.
10.1371/journal.pone.0214536.t003
###### The quantification analysis data (ΔCt) of the Hbl and Nhe toxin genes expression after normalisation with the *udp* reference gene.
The toxin gene expression of *hblC* was absent in *B*. *toyonensis* BCT-7112^T^ (n = 3).
{#pone.0214536.t003g}
*Bacillus* Strain Name ΔCt mean (±SEM[^a^](#t003fn001){ref-type="table-fn"})
------------------------------------------------- ------------------------------------------------------- ----------------- ----------------- ----------------- ------------------- -----------------
***B*. *toyonensis* BCT-7112**^**T**^ **0.31** (0.18) \- **0.13** (0.02) **0.07** (0.00) **2.03** (0.74) **0.11** (0.02)
***B*. *cereus* 1230** **1.35** (0.80) **2.64** (0.58) **0.28** (0.12) **0.43** (0.04) **46.48** (9.73) **0.32** (0.19)
***B*. *cereus* DSM-4384** **1.41** (1.06) **1.94** (0.70) **0.46** (0.26) **2.26** (0.56) **74.49** (34.26) **3.07** (2.61)
***B*. *cereus* DSM-31** **0.44** (0.17) **4.09** (3.25) **0.21** (0.08) **0.63** (0.34) **66.13** (55.81) **0.62** (0.16)
***B*. *subtilis* subsp. *spizizenii* DSM-347** \-
| 2,981
| 439
| 1,643
| 2,934
| null | null |
github_plus_top10pct_by_avg
|
ligned}
&H_0\, = T\,\partial_T - R\,\partial_R, \\ \nonumber
&H_+ = \partial_T, \\ \nonumber
&H_- = (T^2 + \frac{1}{R^2})\,\partial_T - 2\,TR\,\partial_R
- \frac{2}{R}\,\partial_\Phi, \\ \nonumber
&Q_0\,\, = \partial_\Phi.\end{aligned}$$ $H_0$ is the infinitesimal generator of *dilation*, which leaves the metric invariant under $R \rightarrow cR$ and $T \rightarrow T/c$ for some constant $c\in(0,+\infty)$. $Q_0$ is the generator of the rotation along $\Phi$ which generates the $U(1)$ group. $H_+$ is the time translation generator inherited from Kerr. The four generators form a *representation* $\rho_{P}$ of the Lie algebra $\mathfrak{g} \equiv {\ensuremath{\mathfrak{sl}(2,\mathbb{R})\times \mathfrak{u}(1)}}$, $$\begin{aligned}
\label{eq:Lie-algebra-poincare}
[H_0 \,, H_\pm] &= \mp H_\pm \,, \\ \nonumber
[H_+ \,, H_-] &= 2\,H_0 \,, \\ \nonumber
[H_s \,, Q_0] &= 0 \,. \qquad (s=0,\pm)\end{aligned}$$
In global coordinates, we can similarly obtain four (different) generators that are KVFs of the NHEK spacetime, $$\begin{aligned}
L_\pm &= i e^{\pm i \tau} \sin\psi (-\cot\psi\partial_\tau \mp
i\partial_\psi + \partial_\varphi), \\ \nonumber
L_0 &= i \partial_\tau, \\ \nonumber
W_0 &= -i \partial_{\varphi}.\end{aligned}$$ This is a different representation, $\rho_{g}$. But since it is still a Lie algebra representation, they satisfy the same commutation relations as in Eq. with all $H$’s replaced by $L$’s, and $Q_0$ replaced $W_0$.
We say that the group $G$ acts on the manifold $\mathcal{M}$ by translation, $G \circlearrowleft \mathcal{M}$. That is, every element $g\in G$ determines an isomorphism $\phi_{g}: \mathcal{M} \to
\mathcal{M}$, and these isomorphisms, under composition, form a representation of the group $G$. There is an induced action on the space of functions/vector fields/forms/tensors/etc. living on $\mathcal{M}$ by pullback under the map $\phi_{g}$ [@MR2954043]. We call the pullback $\phi^{*}_{g}$, overloading this symbol to mean the pul
| 2,982
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|
.41)
Number of friends in the Netherlands (log) −0.06 (0.04) 0.08 (0.08)
Number of family members in the Netherlands (log) 0.03 (0.06) −0.05 (0.12)
Dutch proficiency 0.06 (0.07) −0.18 (0.12)
Child \< 8 years of age^[e](#table-fn3-0192513X17710773){ref-type="table-fn"}^ 0.48 (0.16)[\*\*](#table-fn6-0192513X17710773){ref-type="table-fn"} 0.33 (0.31)
*R* ^2^ .24 .13
*Note*. Superscripts indicate reference categories that include (a) nontransnational parent; (b) male; (c) married/in a relationship; (d) room, student housing, institution, other; (e) No children \< 8 years of age.
Standard errors in parentheses
*Source*. TCRAf-Eu Angolan parent survey, The Netherlands 2010-2011.
*p* \< .05. \*\**p* \< .01. \*\*\**p* \< .001 (one-tailed test).
[Figure 2](#fig2-0192513X17710773){ref-type="fig"} presents the results of the mediation analysis graphically and presents us with the relevant coefficients for each step of the mediation analysis. [Table 3](#table3-0192513X17710773){ref-type="table"} displays the indirect, direct, and total effects and the proportion of the total effect mediated with bias-corrected confidence intervals after bootstrapping. Although not presented, the model includes the same control variables as in [Table 2](#table2-0192513X17710773){ref-type="table"}. The second step of mediation requires the independent variable to be related to the mediating variable. Path a1 represents the association between transnational parenting and happiness and a2 between transnational parenting and family-to-work conflict. As graphically evidenced in
| 2,983
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|
a broad resonance. Specifically, the realization using a CI resonance in a tight waveguide requires a sufficiently dilute gas with $na_{\perp}\ll 1$. Taking typical values of order 50 nm for the transverse oscillator length which have been realized very recently in bosonic 1D gases [@Esslinger; @BlochKinoshita], this requires densities in the range of much less than 20 atoms per micron. In the case of photo-association, i.e. an optically induced resonance, the requirement is, that the effective 1D Rabi frequency $g\sqrt{n}$ is much larger than the Fermi energy. Using estimates for the Rabi-frequency taken over from photassociation of $^{87}$Rb in 3D [@GrimmRb], a rough estimate shows that the condition of a broad resonance can also be reached here. In particular the fact that the Franck-Condon overlap is enhanced in a 1D situation helps realizing this limit.
We acknowledge useful discussions with Walter Rantner, Stefano Cerrito and Andrea Micheli. Laboratoire de Physique des Solides is a mixed research unit (UMR 8502) of the CNRS and the Université Paris-Sud in Orsay.
Appendix A {#appendix-a .unnumbered}
==========
The estimates of $\mu$ used in the present article come from identifying $\delta \mu\equiv \mu -\epsilon_b/2$ (when in the broad resonance limit) with the chemical potential in the modified Gaudin-Yang model [@FRZ]. The chemical potential obtained from [@FRZ] gives the following estimate for $\delta \mu$: $$\delta \mu/\epsilon_F \simeq \left\{ \begin{array}{ll} 1 & \text{ when
} 1/\gamma \to -\infty \text{ BCS limit}\\ 1/4 & \text{ when }
1/\gamma \to 0 \text{ on resonance}\\ \gamma/4\pi^2 & \text{ when }
1/\gamma \to +\infty \text{ BEC limit}
\end{array}\right.$$
Appendix B {#appendix-b .unnumbered}
==========
In this appendix, we discuss the behavior of $1/\gamma$ as a function of $\nu$. Before resonance $\gamma=mg_1/n=-mg^2/n\nu$, which implies: $$\begin{aligned}
\frac{1}{\gamma}=-\frac{nr_{\star}}{2^{3/2}}\frac{\nu}{|\epsilon_{\star}|}.\end{aligned}$$ In the BCS limit $\nu\to +\infty$, $1/\ga
| 2,984
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|
(s+\lambda^{\varphi,n}_\epsilon(s)\frac{X_{1}}{\sqrt{n}})]\ge\mathbb{E}[\varphi(s+\frac{\xi_{1}}{\sqrt{n}})]-\epsilon.$$ For any $\sigma\in\Sigma^{\mathbb{N}}_G$ with $\sigma_{i+1}(s)=\lambda^{\varphi,n}_\epsilon(s)$, we have $$E[\varphi(W^{\sigma}_{i+1,n})]=E[E[\varphi(s+\sigma_{i+1}(s)\frac{X_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}]\ge E[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W^{\sigma}_{i,n}}]-\epsilon.$$ Therefore, $$\mathbb{E}[\varphi(W_{i+1,n})]\ge\mathbb{E}[\mathbb{E}[\varphi(s+\frac{\xi_{i+1}}{\sqrt{n}})]\big|_{s=W_{i,n}}].$$ Combining the above arguments, we prove equality (\[se2\]).
Let $\tilde{\xi}_1,\cdots, \tilde{\xi}_n$ be i.i.d random variables under a sublinear expectation $\tilde{\mathbb{E}}$ with $\tilde{\xi}\sim \mathcal{N}$, the distribution of $\xi_1$. On the basis of (\[se2\]), we have, for any $\varphi\in lip(\mathbb{R})$, $$\mathbb{E}[\varphi(W_n)]=\tilde{\mathbb{E}}[\varphi(\frac{\tilde{\xi}_1+\cdots+\tilde{\xi}_n}{\sqrt{n}})].$$ Therefore, by using Theorem 4.5 of [@So17], we obtain the desired estimate.
\[Proof of Theorem \[t10\]\] Without loss of generality, we shall only consider $\varphi$ that vanishes at infinity. Let $u$ be the solution to the $G$-heat equation with initial value $\varphi$. Set $\sigma_\varphi(t,x)=2G(\textmd{sgn}[\partial_{xx}^2u(1-t,x)])$, $(t,x)\in [0,1)\times\mathbb{R}$, where $$\textmd{sgn}[a]=
\begin{cases}
1, & \mbox{if }a\ge0; \\
-1, & \mbox{if }a<0.
\end{cases}$$ Then, $u$ satisfies $$\begin{aligned}
\partial_t u-\frac{1}{2}\widetilde{\sigma}_\varphi^2\partial^2_{xx} u&=&0, \ (t,x)\in (0,1]\times\mathbb{R},\\
u(0,x)&=& \varphi (x).\end{aligned}$$ By the mollification procedure, we can find $\{\sigma_n\}\subset\Sigma_G$ such that $\|\sigma_n-\sigma_\varphi\|_{L^2([0,1]\times \mathbf{B}(R))}\rightarrow0$ as $n\rightarrow\infty$ for any $R<\infty$. Next, set $v_n(t,x):=E[\varphi(W^{\sigma_n,x}_t)]$. Then, $v_n$ is the solution to the following equation: $$\begin{aligned}
\partial_t v_n-\frac{1}{2}\widetilde{\s
| 2,985
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Csiszar-Kullback inequality [@Csiszar67; @Kullback67] relates the relative entropy to the $L^1$ norm.
\[thm:CK\] Let $f\in L_+^1({\mathbb R}^d)$ with ${\|f\|}_1 = M$ and let $\theta_M$ be the ground state Barenblatt solution with mass $M$. Then, $${\|f - \theta_M\|}_1 \lesssim H(f|\theta_M)^{\min\left(\frac{1}{2},\frac{1}{m}\right)}.$$
Note that since we are interested in $1 \leq m \leq 2-2/d$, we will only apply the inequality with exponent $1/2$.
To prove Theorems \[thm:IA\] and \[thm:IA2\], the purpose of proving $\theta(\tau,\eta) \in L^\infty_{\tau,\eta}({\mathbb R}^+ \times {\mathbb R}^d)$ is to control the growth of ${\|e^{d\tau}{\nabla}{\mathcal{K}}(e^{\tau}\cdot)\ast \theta\|}_\infty$, which depends on the long-range effects of the kernel. Ultimately, this provides a bound essentially of the form, $$\frac{d}{d\tau}H(\theta(\tau)) \leq -I(\theta(\tau)) + C(M,{\|\theta\|}_{L_{\tau,\eta}^\infty({\mathbb R}^+ \times {\mathbb R}^d)})e^{-\gamma \tau},$$ for some $\gamma > 0$ (in reality, it is not quite as clean). Theorem then implies, $$\frac{d}{d\tau}H(\theta(\tau)|\theta_M) \leq -2H(\theta(\tau)|\theta_M) + C(M,{\|\theta\|}_{L_{\tau,\eta}^\infty({\mathbb R}^+ \times {\mathbb R}^d)})e^{-\gamma \tau}.$$ Integrating this and applying Theorem \[thm:CK\] implies, $${\|\theta - \theta_M\|}_1 \lesssim e^{-\frac{\tau}{2}\min\left(2,\gamma\right)},$$ which after rescaling and interpolation against the decay estimate , will prove Theorems \[thm:IA\] and \[thm:IA2\].
Preliminary Decay Estimates
===========================
Let $\overline{q} = (2-m)d/2$ and let $\eta, \tau$ and $\theta(\tau,\eta)$ be as defined in §\[sec:outline\]. As detailed above, we establish that $\theta(\tau,\eta) \in L^\infty_{\tau,\eta}({\mathbb R}^+\times{\mathbb R}^d)$ using Alikakos iteration [@Alikakos] (see also [@JagerLuckhaus92; @Kowalczyk05; @BRB10; @Blanchet09; @SugiyamaADE07; @SugiyamaDIE07; @SugiyamaDIE06]). The first step is to prove the following lemma which allows control over $L^p$ norms with $p < \infty$. In what follows w
| 2,986
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| 833
| 3,053
| 2,813
| 0.776801
|
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|
D\right\}=0=\left\{D^\dagger,D^\dagger\right\}\ \ ,\ \
\left\{D,D^\dagger\right\}=\left(-\frac{2}{\sqrt{\hbar\omega}}\right)^2\,
\left(i\hbar\partial_t\right)\ ,$$ as well as the required properties $$\left\{Q,D\right\}=0\ ,\
\left\{Q,D^\dagger\right\}=0\ ,\
\left\{Q^\dagger,D\right\}=0\ ,\
\left\{Q^\dagger,D^\dagger\right\}=0\ .$$
Consider now an arbitrary Grassmann even superfield on superspace, namely a function $X(t,\eta,\eta^\dagger)$. Without loss of generality (by distinguishing its real and imaginary parts), it is always possible to assume that such a superfield obeys a reality condition, $$X^\dagger(t,\eta,\eta^\dagger)=X(t,\eta,\eta^\dagger)\ .$$ On account of the Grassmann odd character of the coordinate $\eta$, namely the fact that $\eta^2=0={\eta^\dagger}^2$, the general form of such a real superfield is given by $$X(t,\eta,\eta^\dagger)=x(t)+i\eta\theta(t)+i\eta^\dagger\theta^\dagger(t)+
\eta^\dagger\eta\,f(t)\ ,$$ where $x(t)$ and $f(t)$ are real bosonic degrees of freedom, whereas $\theta(t)$ and $\theta^\dagger(t)$ are complex valued fermionic ones, complex conjugates of one another. Indeed, it will turn out that $x(t)$ and $\theta(t)$ correspond to the degrees of freedom considered above, while $f(t)$ will be seen to be simply an auxiliary degree of freedom without dynamics, whose equation of motion is purely algebraic and such that upon its reduction the system described in (\[eq:SUSYL\]) is recovered. This is a generic feature of superfields in supersymmetric field theories: they include auxiliary fields which are reduced through their algebraic equations of motion. However, in the superspace formulation, there are required for a supersymmetric covariant superspace calculus.
These choices having been specified, it is now straightforward to establish how the different components $(x,\theta,\theta^\dagger,f)$ (namely, the components of the terms in $1$, $i\eta$, $i\eta^\dagger$ and $\eta^\dagger\eta$ in the $\eta$-expansion of superfields) of real superfields transform under supersymmetry tra
| 2,987
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|
60*b**3/3 - 110*b**2 - 858*b. Suppose v(l) = 0. Calculate l.
-22, 1, 2
Suppose -73*t = -4*m - 68*t - 17, -2*m = t - 9. Let y(k) be the first derivative of -1/8*k**4 + 0*k - 1/6*k**3 + 0*k**m + 15. Solve y(d) = 0.
-1, 0
Let x(w) = 20*w - 240. Let z be ((-24)/(-10))/(14/(-280)*-4). Let r be x(z). Factor 0*m + 2/9*m**3 + 2/9*m**4 + r*m**2 + 0.
2*m**3*(m + 1)/9
Let g be 4/(-42) + (-4571)/(-147). Suppose 2*b**2 - 3*b + b + 3*b + 2*b + 30*b**3 - g*b**3 = 0. Calculate b.
-1, 0, 3
Let j(f) be the second derivative of f**6/30 - 11*f**5/15 + 13*f**4/3 - 32*f**3/3 + 14*f**2 + 2*f - 3. Let n(l) be the first derivative of j(l). Factor n(u).
4*(u - 8)*(u - 2)*(u - 1)
Let r(f) = -6*f**3 + 578*f**2 - 15120*f + 132214. Let g(p) = -3*p**3 + 291*p**2 - 7560*p + 66108. Let d(z) = -11*g(z) + 6*r(z). Factor d(s).
-3*(s - 36)**2*(s - 17)
Let w(l) be the first derivative of -5/3*l**3 + 1/5*l**5 - 1/4*l**4 + 0*l - 3/2*l**2 + 88. Determine u, given that w(u) = 0.
-1, 0, 3
Let i(d) be the first derivative of -d**5/25 - 9*d**4/10 + 139*d**3/5 + 217*d**2/5 + 5310. Find c, given that i(c) = 0.
-31, -1, 0, 14
Let q be 40/1*((-3)/(-28) + 0/3). Factor 2/7*c**3 + 18/7*c**2 - 50/7 + q*c.
2*(c - 1)*(c + 5)**2/7
Let z(f) be the third derivative of -f**5/160 + 9*f**4/8 + 225*f**3/16 - 41*f**2 + 13. Factor z(n).
-3*(n - 75)*(n + 3)/8
Let a(l) be the third derivative of -l**5/20 + 779*l**4/4 - 606841*l**3/2 - 22*l**2 + 2*l + 3. Find n such that a(n) = 0.
779
Let r(i) be the second derivative of 1/5*i**5 - 1/3*i**4 + 0*i**2 + 11*i - 1 + 0*i**3 - 1/30*i**6. Factor r(n).
-n**2*(n - 2)**2
Let f = 64 + -58. Suppose -3*q + 5*v + 37 = 0, f*q + 21 = 10*q - v. Factor -16*y**3 - 20*y**2 + 3*y**4 - 2*y**q - 2*y**4 - 3*y**4 - 8*y.
-4*y*(y + 1)**2*(y + 2)
Let l = 199 + -202. Let p(o) = -o**2 + 7*o - 6. Suppose -q - 12 = 9. Let u(b) = -b + 1. Let i(t) = l*p(t) + q*u(t). Factor i(w).
3*(w - 1)*(w + 1)
Determine v, given that -26/3*v**5 - 8/3*v - 36*v**4 - 146/3*v**3 - 24*v**2 + 0 = 0.
-2, -1, -2/13, 0
Let g = 357 + -355. Factor 2*h + 6*h**2 + 32*h**2 - 4 - 36*h**g.
2
| 2,988
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| 2,609
| 2,745
| null | null |
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|
\_linear\]). Since the force formulae require to obtain the condensate density in a neighborhood of the particle position, it is convenient to move to a coordinate frame with center always at the (possibly moving) particle location ${\boldsymbol{r}}={\boldsymbol{r}}_p(t)$. Thus we change variables from $({\boldsymbol{r}}, t)$ to $({\boldsymbol{z}}, t)$, with ${\boldsymbol{z}}={\boldsymbol{r}} -{\boldsymbol{r}}_p(t)$, and the velocity field will be now referred to the particle velocity ${\boldsymbol{V}}_p(t)={\boldsymbol{\dot r}}_p(t)$: $\delta {\boldsymbol{w}}^{(0)}({\boldsymbol{z}},t)=\delta{\boldsymbol{v}}^{(0)}({\boldsymbol{r}},t)- {\boldsymbol{V}}_p(t)$. Equation (\[eq:drhodv0\]) becomes: $$\begin{aligned}
\left(\nabla_z^2 - 4\right)&&\nabla_z\delta\rho_0 = \nonumber \\
&4& \left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_z-\frac{\gamma}{2}\nabla_z^2\right)\delta{\boldsymbol{w}}^{(0)} + {\boldsymbol{\dot V}}_p(t),
\label{eq:drhodv}\end{aligned}$$ which has the corresponding equation for its Green’s function given by $$\left(\nabla_z^2 - 4\right)G({\boldsymbol{z}}) = \delta({\boldsymbol{z}})$$ with the boundary condition $G(|{\boldsymbol{z}}|\rightarrow\infty)\rightarrow 0$ (corresponding to vanishing $\nabla_z \delta\rho_0({\boldsymbol{r}})$ at $|{\boldsymbol{r}}|=\infty$). The solution is given by the zeroth order modified Bessel function $G({\boldsymbol{z}})=-K_0(2|{\boldsymbol{z}}|)/(2\pi)$. Hence, the gradient of the density perturbation can be written as the convolution with the Green’s function:
$$\begin{aligned}
\nabla_z\delta\rho_0({\boldsymbol{z}},t)=
-\frac{2}{\pi}\int d{\boldsymbol{z}}' K_0(2|{\boldsymbol{z}}-{\boldsymbol{z}}'|)
\left[\left(\partial_t-{\boldsymbol{V}}_p\cdot\nabla_{{\boldsymbol{z}}'}-\frac{\gamma}{2}\nabla^2_{{\boldsymbol{z}}'}\right)\delta{\boldsymbol{w}}^{(0)}({\boldsymbol{z}}',t) + {\boldsymbol{\dot V}}_p(t) \right] ,
\label{eq:nablarho}\end{aligned}$$
and the expression for the force (\[eq:fp\_unperturb\]), using the comoving variables $({\boldsymbol{z}},t)$, becomes: $$\be
| 2,989
| 2,111
| 2,706
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| null | null |
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|
$and given $h\in {\mathbb{N}}$ we denote $$\rho _{h}=\frac{(a+b)m_{0}+q+2d/p_{\ast }}{2h}. \label{H5'}$$Notice that this is equal to the constant $\rho _{h}$ defined in (\[reg5\]) corresponding to $k=(a+b)m_{0}$ and $q$ and to $2d$ (instead of $d).$
**Step 1: a Lindeberg-type method to decompose $P_t-P^n_t$.** We fix (once for all) $t\in(0,1]$ and we write$$P_{t}f-P_{t}^{n}f=\int_{0}^{t}\partial
_{s}(P_{t-s}^{n}P_{s})fds=\int_{0}^{t}P^n
_{t-s}(L-L_{n})P_{s}fds=\int_{0}^{t}P^n_{t-s}\Delta _{n}P_{s}fds$$We iterate this formula $m_{0}$ times (with $m_{0}$ chosen in (\[H4\])) and we obtain$$P_{t}f(x)-P_{t}^{n}f(x)=\sum_{m=1}^{m_{0}-1}I_{n}^{m}f(x)+R_{n}^{m_{0}}f(x)
\label{R2}$$with (we put $t_{0}=t)$$$\begin{aligned}
I_{n}^{m}f(x)&=\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}...%
\int_{0}^{t_{m-1}}dt_{m}\prod_{i=0}^{m-1}(P_{t_{i}-t_{i+1}}^{n}\Delta
_{n})P_{t_{m}}^{n}f(x),\quad 1\leq m\leq m_0-1, \\
R_{n}^{m_{0}}f(x)&=\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}...%
\int_{0}^{t_{m_{0}-1}}dt_{m_{0}}\prod_{i=0}^{m_{0}-1}(P_{t_{i}-t_{i+1}}^{n}%
\Delta _{n})P_{t_{m_{0}}}f(x).\end{aligned}$$
In order to analyze $I_{n}^{m}f$ we use Lemma \[Reg\] for the semigroup $%
S_{t}=P_{t}^{n}$ and for the operators $U_{i}=\Delta _{n}=L-L_{n}$ (the same for each $i$), with $\delta _{i}=t_{i}-t_{i+1}$, $i=0,\ldots ,m$ (with $%
t_{m+1}=0$). So the hypotheses (\[h1\]) and (\[h1’\]) in Assumption \[H1H\*1\] coincide with the requests (\[TR3\]) and (\[TR3’\]) in Assumption \[A1A\*1\]. And we have $%
C_{q,\kappa ,\infty }(U)=C_{q,\kappa ,p}(U)=C\varepsilon _{n}.$ Moreover the hypotheses (\[h2\]) and (\[h2’\]) in Assumption \[H2H\*2\] coincide with the hypotheses (\[TR2\]) and (\[TR2’\]) in Assumption \[A2A\*2\]. And we have $%
C_{q,\kappa ,\infty }(P^{n})=C_{q,\kappa ,p}(P^{n})=\Lambda _{n}$. Hence, $$C_{q,\kappa ,\infty ,p}(\Delta _{n},P^{n})=C\,\varepsilon _{n}\times \Lambda
_{n}, \label{app1}$$Finally, the hypothesis (\[h3\]) in Assumption \[HH3\] coincides with (\[TR5\]) in Assumption \[A3\]. So, we can apply Lemma \[Reg\]: by using
| 2,990
| 1,025
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| null | null |
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|
rem \[evoth1\] are valid, and $\tilde{\sigma}\geq 0$. Let $f\in H^2(I,L^2(G\times S))$ and $g\in H^3(I,T^2(\Gamma_-))$ which satisfies the compatibility condition g(E\_m)=0. Then the problem (\[se1\]), (\[se2\]), (\[se3\]) has a unique solution $\psi\in C(I,\tilde W^2(G\times S))\cap C^1(I,L^2(G\times S))$.
If in addition the assumptions (with $c>0$) are also valid, the estimate (\[evoest\]) holds.
By the Sobolev Embedding Theorem $$H^m(I,X)\subset C^{j}(I,X)\ {\rm for}\ m>j+{1\over 2}$$ and then the assertion follows from Theorem \[coupthev\].
The evolution equation based approach given above can be generalized for $L^p$-theory when $1\leq p<\infty$. The approach based on the Lions-Lax-Milgram Theorem (section \[esols\]) is limited to the Hilbert space structure, and can therefore be only applied for $p=2$. However, some (recent) generalizations for reflexive Banach spaces of Lions-Lax-Milgram theory might allow methods of section \[esols\] to be generalized also for $1<p<\infty$.
On the Existence of Solutions for Volterra Type Collision Operators
-------------------------------------------------------------------
In the previous section, we assumed that the collision operator $K$ is of the form $(K\psi)(x,\omega,E)=\int_S \sigma(x,\omega',\omega,E)\psi(x,\omega',E) d\omega'$ in order to avoid integration over $I$ with respect to $E'$. Considerations were founded on the fact that $K\psi$ had a representation $(K\psi)(E)=K(E)\psi(E)$. However, for some collision operators of special type, also integration with respect to $E'$ is possible in evolution operator based approaches. In this section, we give a short and *formal* description of such a technique for Volterra type collision operators.
Consider the problem (\[se1\]), (\[se2\]), (\[se3\]) with $g=0$, $$\begin{gathered}
-{{\frac{\partial \psi}{\partial E}}}+{1\over{S_0(E)}}\omega\cdot\nabla_x\psi+{1\over{S_0(E)}}\Sigma(E)\psi-{1\over{S_0(E)}}{{\frac{\partial S_0}{\partial E}}}(E)\psi
-{1\over{S_0(E)}} K\psi= {1\over{S_0(E)}}f,\nonumber\\
\psi_{\Gamma_-
| 2,991
| 1,205
| 1,778
| 2,941
| null | null |
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|
,\tau_3$.
We can apply a similar argument in which we consider ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ for $T\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$. Again ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ is either zero or a semistandard homomorphism; and if it is non-zero, then the only other $T'$ having ${\psi_{3,1}}\circ{\hat\Theta_{T'}}={\psi_{3,1}}\circ{\hat\Theta_{T}}$ is the tableau obtained by exchanging the $3$ in $T$ with a $2$ in the other row. ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ occur with the same coefficient in $\theta$, and we deduce that $\theta$ must be a linear combination of $\tau_0+\tau_1$ and $\tau_2+\tau_3$.
Finally, we consider ${\psi_{1,2}}\circ\theta$. Each $\mu$-tableau $T$ of type $(a+2,1^{b+1})$ contains a single $2$; let $\phi$ denote the sum of ${\hat\Theta_{T}}$ for all those $T$ having the $2$ in row $1$, and $\chi$ the sum of all ${\hat\Theta_{T}}$ for $T$ having the $2$ in row $2$. Using Lemma \[lemma5\] and Lemma \[lemma7\] (and recalling that $a$ is even and $v$ is odd), we have $$\begin{aligned}
{\psi_{1,2}}\circ\tau_0&=\mbinom{v-1}2\chi,\\
{\psi_{1,2}}\circ\tau_1&=\mbinom v2\phi,\\
{\psi_{1,2}}\circ\tau_2&=\left(\mbinom{a+2}2+1\right)\phi+\chi,\\
{\psi_{1,2}}\circ\tau_3&=\mbinom{a+2}2\phi.\end{aligned}$$ So if $v\equiv 3\ppmod4$, then ${\psi_{1,2}}\circ(\tau_0+\tau_1)\neq0$, so $\theta$ cannot equal $\tau_0+\tau_1$. If $a\equiv2\ppmod4$, then ${\psi_{1,2}}\circ(\tau_2+\tau_3)\neq0$, so $\theta$ cannot be $\tau_2+\tau_3$. Hence $\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)\ls1$ in these cases. We also have $\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)\ls1$ in the case where $b=d-3$, since in this case ${\calt[2]}$ and ${\calt[3]}$ are empty, so $\tau_2+\tau_3=0$.
Composing the homomorphisms
---------------------------
Now we complete the analysis of when $S^\mu$ is a summand of $S^\la$, by composing the homomorphisms from the preceding subsections. This will be straightforward, using Proposition \[tabcomp\].
Recall that the space of homomor
| 2,992
| 1,888
| 1,040
| 3,000
| 3,158
| 0.774429
|
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|
m{2s}{s-1} \ge 4^{-s}n + s$ (this last inequality follows from $n \ge 4s$ and $\binom{2s}{s-1} \le \tfrac{1}{2}4^s$). Let $Y$ be a basis for $B''$ in $M \con A'$ and let $B_1 = B''-Y$, so $|B_1| \ge 4^{-s}n \ge 4^{s(t-1)}$. Let $M' = (M \con Y)|(A \cup B_0 \cup B_1).$ Note that, since they are bases for $M$, both $A$ and $A' \cup B_0$ are spanning in $M'$, and $r(M') = n - |Y|$. Moreover, $A'$ is independent in $M'$ and $B_1 \subseteq \cl_{M'}(A')$. Thus, $$\begin{aligned}
\sqcap_{(M')^*}(A_0,B_0) &= r(M' \del A_0) + r(M' \del B_0) - r(M' \del (A_0 \cup B_0)) - r(M') \\
&\ge r_{M'}(A' \cup B_0) + r_{M'}(A) - r_{M'}(A' \cup B_1) - r(M')\\
&= r(M') - r_{M'}(A')\\
&= (n-|Y|) - (n-s) > 0,
\end{aligned}$$ so $M'$ has a cocircuit $K \subseteq A_0 \cup B_0$ that intersects both $A_0$ and $B_0$. Let $a \in K \cap A_0$ and $b \in K \cap B_0$. Note that $B_1$ is independent in $M'$ and is spanned by $A'$; let $A_1 \subseteq A'$ be such that $|A_1| = |B_1|$ and $B_1$ spans $A_1$ in $M' \con (A'-A_1)$. Let $$M'' = M' \con (A'-A_1) \con (A_0 \cup B_0 - \{a,b\}).$$ Now $\{a,b\} \in A_0 \times B_0$ is a series pair of $M''$, and $A_1 \subseteq A'$ and $B_1 \subseteq B- B_0$ are bases of $M'' \con a \del b$. Since $r(M'' \con a \del b) = |B''| \ge 4^{s(t-1)}$, by the inductive hypothesis there is a rank-$(t-1)$ minor $N_0 = M'' \con (\{a\} \cup C) \del (\{b\} \cup D)$ of $M'' \con a \del b$ having a lower-triangular pair $(\bar{A_0}, \bar{B_0})$ with $\bar{A}_0 \subseteq A$ and $\bar{B}_0 \subseteq B$. Now $N = M'' \con C \del D$ has $\{a,b\}$ as a series pair and $N \con a \del b = N_0$. It follows that $((a,\bar{A}_0),(b,\bar{B}_0))$ is a lower-triangular pair in the rank-$t$ matroid $N$, giving the result.
\[triangulartwo\] Let $s \ge 2$ and $t \ge 0$ be integers. If $M$ is a matroid with $r(M) \ge (s4^s)^t$, and $(\bar{A},\bar{B})$ is an upper-triangular pair of $M$, then either
- $M$ has a $U_{s,2s}$-minor $U$ in which $E(U) \cap \bar{A}$ and $E(U) \cap \bar{B}$ are bases, or
- $M
| 2,993
| 2,081
| 2,389
| 2,691
| null | null |
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|
002ny] the value of this parameter is quantised according to $l_k=1-(2k)^{-1} \geq 1/2 , \ k \in \mathbb{N} $. For the further investigations we choose the representative value $l=3/4$. In the semi-classical region $ a_* \gg a \gg a_i$ expression (\[correction\]) simplify to the form $$D=D_*a^n$$ where $$D_* = \left( \frac{3}{1+l} \right)^{3/(2-2l)} a_*^{-3(2-l)/(1-l)} \ \ \text{and} \ \ n = 3(2-l)/(1-l) \ .$$
Now, due to the Hamilton equations we can derive the Friedmann and Raychaudhuri equations for the flat FRW universe filled with a homogeneous scalar field $$\begin{aligned}
H^2 &=& \frac{8\pi G}{3} \left[ \frac{\dot{\phi}^2}{2D} +V(\phi) \right] \ , \label{Fried1} \\
\frac{\ddot{a}}{a} &=& -\frac{8\pi G}{3} \left[ \frac{\dot{\phi}^2}{D} \left( 1-\frac{\dot{D}}{4HD} \right) -V(\phi) \right].
\label{Raych1}\end{aligned}$$ The equation of motion for the scalar field with quantum corrections has the form $$\ddot{\phi}+\left(3H - \frac{\dot{D}}{D} \right)\dot{\phi} + D\frac{dV}{d\phi} = 0.
\label{eom}$$ As we mentioned before, for the further investigations we simplify equations (\[Fried1\]), (\[Raych1\]) and (\[eom\]) assuming $V(\phi) = 0$.
The expression for the quantum correction $D$ is complicated and it is impossible to find an analytical solution for the equations of motion. In fact we even do not need it for the future investigations. To calculate the spectrum of gravitons we need to know analytical solutions only for the inner and outer states. We choose the $| \text{in} \rangle$ and $| \text{out} \rangle$ states respectively in the quantum and classical regimes. The expression for the quantum correction (\[correction\]) simplifies to the form $D=D_*a^n$ for the $a_i < a \ll a_*$ and $D=1$ for $a \gg a_* $. In these limits we can find the analytical solutions for the equations of motion (\[Fried1\]), (\[Raych1\]) and (\[eom\]). It is useful to introduce the conformal time $d\tau = dt/a $ to solve equations and for the further investigations. In the next step we must to fit obtain
| 2,994
| 3,365
| 3,138
| 2,863
| null | null |
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|
at(q)=q^{-\frac{d_\muhat}{2}}PH_c(\M_\muhat;q),$$where $PH_c(\M_\muhat;q):=\sum_ih_c^{i,i;2i}(\M_\muhat)q^i$ is the *pure part* of $H_c(\M_\muhat;q,t)$. \[purconj\]
Conjecture \[purconj\] implies Kac’s conjecture [@kacconj] for comet shaped quivers, namely, $A_\muhat(q)$ is a polynomial in $q$ with non-negative coefficients (see §\[genquiv\] for more details).
Characters of general linear groups over finite fields
------------------------------------------------------
Given two irreducible complex characters $\calX_1,\calX_2$ of $\GL_n(\F_q)$ it is a natural and difficult question to understand the decomposition of the tensor product $\calX_1\otimes\calX_2$ as a sum of irreducible characters. Note that the character table of $\GL_n(\F_q)$ is known (Green, 1955) and so we can compute in theory the multiplicity $\langle\calX_1\otimes\calX_2,\calX\rangle$ of any irreducible character $\calX$ of $\GL_n(\F_q)$ in $\calX_1\otimes\calX_2$ using the scalar product formula $$\langle\calX_1\otimes\calX_2,\calX\rangle=\frac{1}{|\GL_n(\F_q)|}\sum_{g\in\GL_n(\F_q)}\calX_1(g)\calX_2(g)\overline{\calX(g)}.\label{scalprod}$$However it is very difficult to extract any interesting information from this formula. In his thesis Mattig uses this formula to compute (with the help of a computer) the multiplicities $\langle
\calX_1\otimes\calX_2,\calX\rangle$ when $\calX_1,\calX_2,\calX$ are *unipotent characters* and when $n\leq 8$ (see [@Hiss]), and he noticed that $\langle \calX_1\otimes\calX_2,\calX\rangle$ is a polynomial in $q$ with positive integer coefficients.
In [@hausel-letellier-villegas] we define the notion of *generic* tuple $(\calX_1,\dots,\calX_k)$ of irreducible characters of $\GL_n(\F_q)$. We also consider the character $\Lambda:\GL_n(\F_q)\rightarrow \C$, $x\mapsto q^{g\cdot {\rm dim}\,
C_{\GL_n}(x)}$ where $C_{\GL_n}(x)$ denotes the centralizer of $x$ in $\GL_n(\overline{\F}_q)$ and where $g$ is a non-negative integer. If $g=1$, this is the character of the conjugation action of $\GL_n(\F_q)$ on the group al
| 2,995
| 1,560
| 2,438
| 2,582
| 3,012
| 0.775448
|
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|
than the other parameters. In particular, we find it necessary to impose uniform bounds on the largest and smallest eigenvalues of the covariance matrices of all $k$ marginals of the $d$ covariates, as well as bounds on the higher moments of $X$ and on the mixed moments of $X$ and $Y$. We will further assume, in most cases, that the distribution of the pair $(X,Y)$ in $[-A,A]^{d+1}$, for some fixed $A>0$. Such compactness assumptions are stronger than necessary but allow us to keep the statement of the results and their proofs simpler. In particular, they may be replaced with appropriate tail or moment bounds and not much will change in our analysis and results.
Although we have formulated the guarantees of honest validity, accuracy and concentration in asymptotic terms, all of our results are in fact obtained as finite sample bounds. This allow us to derive consistency rates in $n$ with all the relevant quantities, such as the dimension $d$, the size of the selected model $k$, and the variance and eigenvalue bounds needed for the projection parameters accounted for in the constants (with the exception of $A$, which we keep fixed). As a result, our results remain valid and are in fact most interesting when all these quantities are allowed to change with $n$.
Related Work {#sec:related}
------------
The problem of inference after model selection has received much attention lately. Much of the work falls broadly into three categories: inference uniformly over selection procedure, inference with regard to a particular debiased or desparsified model, and inference conditional on model selection. A summary of some of the various methods is in Table \[table::compare\]. We discuss these approaches in more detail in Section \[section::comments\].
The uniform approach includes POSI [@berk2013valid], which constructs valid inferential procedures regardless of the model selection procedure by maximizing over all possible model selections. This method assumes Normality and a fixed, known variance, as well as being compu
| 2,996
| 1,066
| 2,406
| 2,668
| null | null |
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|
a generic Weil divisor on $(X,P)$ of class $k$, that is, a generic germ in $\cO_{X,\zeta^k}$. In [@jiJM-correction] it is shown that $$A_{X,P}(D)=\delta^{\operatorname{top}}(D)-\kappa_P(D).$$ When applied this formula for generic germs one obtains a combinatorial way to calculate $A_{X,P}(D)$, that is, $$A_{X,P}(D)=\delta^{\operatorname{top}}(D)-r_P(D)+1.$$
Cyclic coverings of smooth algebraic surfaces d’après Esnault-Viehweg
---------------------------------------------------------------------
Esnault-Viehweg’s theory for cyclic covers in the smooth case can be presented as follows. Consider $X$ a projective smooth surface and let $D$ be a divisor which is linearly equivalent to $nH$ where $H$ is another divisor. Then $\mathcal{O}_X(D)$ is isomorphic to $\mathcal{O}_X(H)^{\otimes n}$. Then, given a meromorphic section $t:X\dashrightarrow\mathcal{O}_X(D)$, such that $\operatorname{div}(t)=D$, we can consider $$\hat{X}:=\overline{\{(x,v)\in\mathcal{O}_X(H)\mid v^{\otimes n}=t(x)\}}$$ and a suitable smooth model $\pi:\tilde{X}\to \hat X$. Recall that the first Betti number of $\tilde{X}$ equals $2\dim H^1(\tilde{X},\mathcal{O}_{\tilde{X}})$ (its irregularity) and it is a birational invariant.
Let us assume now that $D$ is a simple normal crossing divisor. We can assume that $D$ is effective and $D=\sum_{j=1}^r n_i D_i$ is its decomposition in irreducible components and $0\leq n_i<n$ (note that $n_i=0$ means that the covering is not ramified along $D_i$, but we allow this for technical reasons). Using the eigen-decomposition of $\pi_*(\mathcal{O}_{\tilde{X}})$ induced by that of $\mathcal{O}_{\tilde{X}}$ one can describe the irregularity of $\tilde X$ in cohomological terms using line bundles on $X$.
Under the previous conditions, the irregularity of the covering $\tilde{X}$ equals $2\sum_{k=0}^{n-1}\dim H^1(X,\mathcal{L}^{(k)})$, where $$\label{eq:Lkliso}
\mathcal{L}^{(k)}=
\mathcal{O}_X
\left(
-k H+
\sum_{i=0}^r
\left\lfloor\frac{k n_i}{n}\right\rfloor
D_i
\right).$$
One of the goals of this paper
| 2,997
| 2,360
| 1,989
| 2,898
| 2,614
| 0.778426
|
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|
can compute the tadpole conditions for the $6_3^1$ and $4_3^{3,2}$-branes in the IIA/O6 theory.
As an interesting application of our results, we now consider the IIB/O3 theory for the particular case in which $P^{1,4}=0$, and look at all the constraints related to $P_1^2 \cdot Q$ in the presence of exotic branes. From eq. one can see that the potential $E_{10,4,2}$ does not couple to $P_1^2$, and therefore we only have to consider, apart from $E_8$ (giving the constraint ), the potentials $E_{8,4}$ and $E_{9,2,1}$. The generalised Chern-Simons term for $E_{8,4}$ is $$\frac{1}{4!}\int E_{8,4} \wedge (P_1^2 \cdot Q)^4_2 \quad .\label{tadpoleE84}$$ with $(P_1^2 \cdot Q)^{abcd}_{ef}=12 P^{[ab}_{[e}Q^{cd]}_{f]}$. We denote with $\bigcirc abcd$ the isometry directions. We find the constraints $$\begin{aligned}
& N_{3_3^4}(\bigcirc x^jy^jx^k y^k)+\tfrac{1}{2}[g_{ii}\bar{b}_{ii}-\bar{g}_{ki}b_{ki}+\bar{f}_ih_i-\bar{g}_{ji}b_{ji}-f_i\bar{h}_i+g_{ji}\bar{b}_{ji}-\bar{g}_{ii}b_{ii}+g_{ki}\bar{b}_{ki}]=0 \nonumber \\
& N_{3_3^4}(\bigcirc y^iy^jx^ky^k)-\tfrac{1}{2}[\bar{g}_{ki}\bar{h}_j-\bar{f}_i\bar{b}_{kj}+\bar{g}_{kj}\bar{h}_i-\bar{f}_j\bar{b}_{ki}]=0 \nonumber \\
& N_{3_3^4}(\bigcirc y^ix^jx^ky^k)-\tfrac{1}{2}[-g_{ii}\bar{b}_{jj}+\bar{g}_{ji}b_{ij}+\bar{g}_{jj}b_{ii}-g_{ij}\bar{b}_{ji}]=0 \nonumber \\
& N_{3_3^4}(\bigcirc x^ix^jx^ky^k)+\tfrac{1}{2}[f_ib_{kj}-g_{ki}h_j-g_{kj}h_i+f_jb_{ki}]=0 \quad .\end{aligned}$$ As we have already discussed in the previous subsection for the NS-NS fluxes, eq. gives quadratic constraints also for the components that do not correspond to branes, [*i.e.*]{} components in which some on the downstairs indices are equal to some of the upstairs ones. These constraints are $$\begin{aligned}
& g_{ii}\bar{b}_{ij}-\bar{g}_{ji}b_{jj}-\bar{g}_{ij}b_{ii}+g_{jj}\bar{b}_{ji}=0 \nonumber \\
& -\bar{g}_{ki}\bar{b}_{ij}+\bar{f}_ib_{jj}+\bar{g}_{ij}\bar{b}_{ki}-g_{jj}\bar{h}_i=0 \nonumber \\
& g_{ii}b_{kj}-\bar{g}_{ji}h_j-g_{kj}b_{ii}+f_j\bar{b}_{ji}=0 \nonumber \\
& -\bar{g}_{ki}b_{kj}+\bar{f}_ih_j+g_{kj}
| 2,998
| 1,286
| 1,653
| 2,987
| null | null |
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|
[@gopi.volume]. That work finds that the cumulative distribution function of traded volume for time windows of $\Delta t = 15$ minutes decays as a power-law with a tail exponent $\lambda = 1.7 \pm 0.1$ for a wide range of stocks. This is the so called *inverse half cube law*, and it can be written as $${\mathbb P}_{\Delta t}(f) \propto f^{-(\lambda + 1)},
\label{eq:pl}$$ where $\mathbb P_{\Delta t}$ is the probability density function of the same quantity.
The estimation of tail exponents is often difficult due to poor statistics of rare events, large stock-to-stock variations and the presence of correlations. For the same $1994-1995$ period of data and the same $15$ minute time window certain stocks have $\lambda$ values significantly higher than $1.7$ \[see Fig. \[fig:distrib\](left)\]. The tails of these distributions can be fitted by a power law over an order of magnitude, for the top $3-10\%$ of the events. The exponent $\lambda$ is around $2.8$ for these examples. The question arises: Which value (if any) is correct?
In order to address this question we carried out a systematic investigation comprising the $1000$ stocks with the highest total traded value in the TAQ database. We used variants of Hill’s method [@hill; @alves] to estimate the typical tail exponent, see Ref. [@eisler.sizematters] for details. The results of this Section are summarized in Table \[tab:DETRlambda94-95\]. Note that in all cases the $U$-shaped intraday pattern of trading activity was removed.
Most descendants of Hill’s method, including the ones applied here, contain a free parameter, namely the fraction $p$ of top events to be considered to belong to the tail of the distribution (see Ref. [@alves] and refs. therein). According to Fig. \[fig:distrib\](left) this should be set around $p\approx 3-10\%$.
First, let us follow the methodology of Ref. [@gopi.volume]. In that paper, the authors first they deduct the mean from the time series by taking $f_i(t)-\ev{f_i}$, where $\ev{\cdot}$ denotes time averaging. Then this series i
| 2,999
| 672
| 3,146
| 2,857
| null | null |
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|
then the eigenvalues of $M$ are precisely the values $\bigl\{\widehat{f}(\chi) \mid \chi
\in \widehat{G}\bigr\}$ of the Fourier transform of $f$, and the characters of $G$ are eigenvectors of $M$. (For generalizations of these facts for nonabelian $G$, see [@Diaconis-book; @Diaconis-matrices].) Observe that every $G$-circulant matrix is normal, but that $M$ is Hermitian if and only if $f(a^{-1}) = \overline{f(a)}$ for each $a \in G$.
Given a family of random variables $\{ Y_a \mid a \in G\}$, define the random function $f \in \ell^2(G)$ by $f(a) = \frac{1}{\sqrt{{\left\vert G \right\vert}}}
Y_a$. (We are avoiding using $X$ to name random variables because of its typographical similarity to $\chi$.) The corresponding $G$-circulant matrix is the random matrix $M = \bigl[ Y_{ab^{-1}}
\bigr]_{a,b \in G}$. Its eigenvalues, indexed by $\chi \in
\widehat{G}$, are given by $$\label{E:eigenvalue-formula}
\lambda_\chi = \widehat{f}(\chi) = \frac{1}{\sqrt{{\left\vert G \right\vert}}}
\sum_{a \in G} Y_a \chi(a),$$ and the empirical spectral distribution of $M$ is $$\mu = \frac{1}{\bigl\vert \widehat{G} \bigr\vert} \sum_{\chi \in \widehat{G}}
\delta_{\lambda_\chi}
= \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}}
\delta_{\lambda_\chi},$$ where $\delta_z$ here denotes the point mass at $z \in {\mathbb{C}}$.
The Fourier transform $\widehat{f}$ is a random trigonometric polynomial on $G$, of the kind studied extensively by Marcus and Pisier [@MaPi]. From it follows in particular that ${\left\Vert M \right\Vert} = \bigl\Vert \widehat{f} \bigr\Vert_\infty$, where the former norm is the spectral norm of $M$. The following result is thus a special case of [@MaPi Theorem 1.4], which also applies to infinite compact abelian groups.
\[T:norm\] Suppose that $\{Y_a \mid a \in G\}$ are independent (except possibly for a constraint $Y_{a^{-1}} = \overline{Y_a}$ for each $a \in G$) and mean $0$ with finite second moments. Then $$c \left(\min_{a \in G} {\mathbb{E}}{\left\vert Y_a \right\vert}\right) \le
\frac{{\ma
| 3,000
| 3,140
| 2,694
| 2,639
| null | null |
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|
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