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uthor:
- Da Rong Cheng
bibliography:
- 'compactness.bib'
title: 'A Compactness Result for Energy-minimizing Harmonic Maps with Rough Domain Metric'
---
---
abstract: 'Determining the properties of starbursts requires spectral diagnostics of their ultraviolet radiation fields, to test whether very massive stars are present. We test several such diagnostics, using new models of line ratio behavior combining Cloudy, Starburst99 and up-to-date spectral atlases [@pauldrach01; @hillmill]. For six galaxies we obtain new measurements of [ $1.7$ ]{}/, a difficult to measure but physically simple (and therefore reliable) diagnostic. We obtain new measurements of [ $2.06$ ]{}/ in five galaxies. We find that [ $2.06$ ]{}/ and \[\]/ are generally unreliable diagnostics in starbursts. The heteronuclear and homonuclear mid–infrared line ratios (notably \[\] $15.6$ / \[\] $12.8$ ) consistently agree with each other and with [ $1.7$ ]{}/; this argues that the mid–infrared line ratios are reliable diagnostics of spectral hardness. In a sample of $27$ starbursts, \[\]/\[\] is significantly lower than model predictions for a Salpeter IMF extending to $100$ . Plausible model alterations strengthen this conclusion. By contrast, the low–mass and low–metallicity galaxies II Zw 40 and NGC 5253 show relatively high neon line ratios, compatible with a Salpeter slope extending to at least $\sim40$–$60$ . One solution for the low neon line ratios in the high–metallicity starbursts would be that they are deficient in $\ga 40$ stars compared to a Salpeter IMF. An alternative explanation, which we prefer, is that massive stars in high–metallicity starbursts spend much of their lives embedded within ultra–compact regions that prevent the near– and mid–infrared nebular lines from forming and escaping. This hypothesis has important consequences for starburst modelling and interpretation.'
author:
- 'J. R. Rigby and G. H. Rieke'
nocite:
- '[@mrr]'
- '[@pauldrach01]'
- '[@kbfm; @ho3; @doherty95]'
-
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-1}=R/P_i$ and let $P\in
\Supp({\mathcal F})$. Since there are no non-trivial inclusions between the prime ideals in $\Supp({\mathcal F})$ it follows that $M_P$ has a filtration $(0)=(M_0)_P\subset (M_1)_P\subset \cdots \subset (M_r)_P=M_P$ such that $$(M_i)_P/(M_{i-1})_P = \left\{ \begin{array}{lll} R_P/PR_P, & \text{if} & P=P_i,\\
0, & \mbox{if} & P\neq P_i. \end{array} \right.$$ Hence we see that $\Ass_{R_P}(M_P)=\{PR_P\}$, and so $P\in \Ass(M)$. It follows that $\Supp({\mathcal F})=\Ass(M)$. Applying again assumption (b), we conclude that $\Ass(M)=\Min(M)$.
\[induced\] Let $0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a clean filtration of $M$. Then for all $i=0,\ldots,r$ $$0=M_i/M_i\subset M_{i+1}/M_i\subset\ldots \subset M_{r-1}/M_i\subset M_r/M_i,$$ and $$0=M_0\subset M_1\subset \ldots\subset M_{i-1}\subset M_i$$ are clean filtrations. In particular, $M_i$ and $M/M_i$ are clean.
A weakening of condition (b) of Lemma \[character\] leads to
\[pretty\]
*A prime filtration ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ of $M$ with $M_i/M_{i-1}=R/P_i$ is called [*pretty clean*]{}, if for all $i<j$ for which $P_i\subset P_j$ it follows that $P_i=P_j$.*
In other words, a proper inclusion $P_i\subset P_j$ is only possible if $i>j$. The module $M$ is called [*pretty clean*]{}, if it has a pretty clean filtration. A ring is called pretty clean if it is a pretty clean module, viewed as a module over itself.
[*Let ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a pretty clean filtration of $M$. It follows immediately from the definition that for all $i$ the filtrations $$0=M_i/M_i\subset M_{i+1}/M_i\subset\ldots \subset M_{r-1}/M_i\subset M_r/M_i,$$ and $$0=M_0\subset M_1\subset \ldots\subset M_{i-1}\subset M_i$$ are pretty clean.* ]{}
\[important\] Let ${\mathcal F}\: 0=M_0\subset M_1\subset \ldots \subset M_{r-1}\subset M_r=M$ be a pretty clean filtration of $M$. Then $P_i\in
\Ass(M_i)$ for all $i$.
We use the same argument as
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a primary field $\phi$. This computation relies on the prescription of appendix \[compositeOPEs\], and on the current-current and current-primary OPEs and .Since we computed these OPEs up to order $f^2$, we will also obtain the stress-tensor OPE up to order $f^2$. $$\begin{aligned}
\label{phiT}
\phi(z) 2 c_1 T(w)
&= \lim_{:x \to w:}\phi(z) j^a_{L,z}(x) j^b_{L,z}(w) \kappa_{ba} \cr
&= \kappa_{ab} \lim_{:x \to w:} \left[ \left(
-\frac{c_+}{c_++c_-} \frac{t^a \phi(z)}{x-z} + :j^a_{L,z} \phi:(x) \right. \right. \cr
& \left. + {A^a}_c \log |z-x|^2 :j^c_{L,z}\phi:(x) + {B^a}_c\frac{\bar z - \bar x}{z-x}:j^c_{L,\bar z}\phi:(x) + ... %\mathcal{O}(f^4)
\right) j^b_{L,z}(w) \cr
& + \left. j^a_{L,z}(x) \left( - \frac{c_+}{c_++c_-} \frac{t^b \phi(w)}{w-z} + \mathcal{O}\left( (z-w)^0 \right)
\right) \right]\end{aligned}$$ Let us first consider the first term in the previous expression. According to the prescription given in appendix \[compositeOPEs\], we have to evaluate the operator $\phi(z)$ at the point $x$ before we take the OPE with the remaining current $j^b_{L,z}(w)$. So we rewrite this term as: $$\begin{aligned}
\kappa_{ab} & \lim_{:x \to w:} \left( -\frac{c_+}{c_++c_-} \frac{t^a}{x-z} \sum_{n,\bar n=0}^{\infty} \frac{(z-x)^n}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p^n \bar \p^{\bar n} \phi(x) \right) j^b_{L,z}(w) \cr
%
& = \kappa_{ab} t^a \frac{c_+}{c_++c_-} \lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p_x^n \bar \p_x^{\bar n} \left(
-\frac{c_+}{c_++c_-} \frac{t^b \phi(x)}{w-x} - :j^b_{L,z} \phi:(w) + ...
\right) \cr
%
& = -\kappa_{ab} t^a t^b \left(\frac{c_+}{c_++c_-} \right)^2
\lim_{:x \to w:}\sum_{n,\bar n=0}^{\infty}\frac{(z-x)^{n-1}}{n!}\frac{(\bar z-\bar x)^{\bar n}}{\bar n!} \p_x^n \bar \p_x^{\bar n} \cr
& \qquad
\left(\frac{1}{w-x} \sum_{m,\bar m=0}^{\infty}\frac{(x-w)^{m}}{m!}\frac{(\bar x-\bar w)^{\bar m}}{\bar m!} \p^m \bar \p^{\bar m}\phi(w)
\right) - \kappa_{ab} t^a \frac{c_+}{c_++c_-} \frac{:j^b_{L,z} \phi:(w)}{w-z} + ... \nonumb
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_0^z \frac{d z'}{(1 + z')\,\sqrt{\Omega_\Lambda + \Omega_m (1 + z')^3}}\right)~.$$ It is clear that, in order to for $n_{\rm short}(z)$ to be reduced significantly, e.g., by two orders of magnitude, at the epoch $z \simeq 0.9$ of the GRB 090510 [@grb090510], while keeping $n^\star ={\cal O}(1)$, one must consider the magnitude of $v$ as well as the string scale $\ell_s$. For instance, for string energy scales of the order of TeV, i.e., string time scales $\ell_s/c = 10^{-27}~s$, one must consider a brane velocity $v \le \sqrt{10} \times 10^{-11} \, c$, which is not implausible for a slowly moving D-brane at a late era of the Universe [^5]. This is compatible with the constraint on $v$ obtained from inflation in [@emninfl], namely $v^2 \le 1.48 \times 10^{-5}\,g_s^{-1}$, where $g_s < 1$ for the weak string coupling we assume here. On the other hand, if we assume a 9-volume $V_9 = (K \ell_s)^9$: $K \sim 10^3$ and $\ell_s \sim 10^{-17}$/GeV, then the brane velocity $v \le 10^{-4} \, c$. In our model, due to the friction induced on the D-brane by the bulk D-particles, one would expect that the late-epoch brane velocity should be much smaller than during the inflationary era immediately following a D-brane collision.
Towards D3-Foam Phenomenology
=============================
The above discussion was in the context of type-IIA string theories, but may easily be extended to compactified type-IIB theories of D-particle foam [@li], which may be of interest for low-energy Standard Model phenomenology. In this case, as suggested in [@li], one may construct ‘effective’ D-particle defects by compactifying D3-branes on three-cycles. In the foam model considered in [@li], the rôle of the brane Universe was played by D7-branes compactified appropriately on four-cycles. The foam was provided by compactified D3-brane ‘D-particles’, in such a way that there is on average a D-particle in each three-dimensional volume, $V_{A3}$, in the large Minkowski space dimensions of the D7-brane. We recall that, although in the
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ger $N$ a curve $\cD=\cD_1\cup...\cup\cD_r$ can be found in $\PP^2_w$ satisfying the following extra conditions:
1. \[prop:3b\] $\cD\cap \operatorname{Sing}\PP^2_w \subseteq\{O\}$ and $\operatorname{Sing}\cD=\{O\}\cup \cN$, where $\cN$ is a set of nodal points of $\cD$ (at smooth points of $\PP^2_w$),
2. \[prop:4\] $\deg \cC$ divides $\deg_w \cD > N$,
3. \[prop:5\] $\deg \cD_j\equiv \deg \cC_j \mod (w_2)$, and
4. \[prop:6\] $q \equiv 1 \mod (w_2)$, where $q:=\frac{\deg_w \cD}{\deg_w \cC}\in \ZZ_{> 0}$.
Similarly as in the case of germs on a smooth surface, there is a polynomial representative $f(x,y)$ of $(\cC,P)$. In this case $f(x,y)=\sum_{(i,j)\in \Gamma(f)}a_{i,j}x^iy^j$ where $$\label{eq:ij}
\bar w_0 i + \bar w_1 j \equiv w_0 i + w_1 j\equiv \lambda \mod (w_2)$$ $(i,j)\in \Gamma(f)$ and the set $\Gamma(f)=\{(i,j)\in \ZZ^2\mid a_{i,j}\neq 0\}$ is finite and $\lambda$ is a fixed integer $0\leq \lambda<w_2$. Adding monomials of high enough $(w_0,w_1)$-degree satisfying , one can assure that the topological type of $\{g=f+F=0\}$ at $P=(0,0)$ coincides with that of $(\cC,P)$. The next step will be to homogenize $g$ with respect to $w=(w_0,w_1,w_2)$. Denote by $M:=\max\{w_0 i+w_1 j \mid (i,j)\in \Gamma(g)\}$. Given any $(i,j)\in \Gamma(g)$ note that $M-(w_0 i+w_1 j) = d_{i,j}w_2\geq 0$. Hence, $$G(X,Y,Z)=\sum_{(i,j)\in \Gamma(g)} a_{i,j} X^iY^jZ^{d_{i,j}}$$ is a $w$-homogeneous polynomial of degree $M$. For a generic choice of monomials in $F$, the global curve $\cD=\{G=0\}$ satisfies the required conditions \[prop:1\]-\[prop:3\].
For the second part, we proceed as before for each irreducible component $\cC_j$ obtaining $\cD=\cD_1\cup ...\cup\cD_r$. By the generic choice of monomials for each irreducible component, condition \[prop:3b\] is immediately satisfied and it will avoid the other singular points of $\PP^2_w$ if $M_j=\deg_w \cD_j$ is a multiple of $w_0w_1$. Also, note that $M_j=\deg_w \cD_j$ from the construction above and $M=\sum_j n_jM_j=\deg_w \cD$. Consider $\Delta\in \ZZ_{>0}$ such that $\
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root vectors. In Sect. \[sec:Verma\] we start to study Verma modules and special maps between them. Prop. \[pr:VTMiso\] gives a criterion for bijectivity of such maps, and Prop. \[pr:M=L\] identifies irreducible Verma modules. In Sect. \[sec:shapdet\] we study Shapovalov determinants following the approach in [@b-Joseph]. Here our main result is Thm \[th:Shapdet\], which gives a formula for the Shapovalov determinant of $U(\chi )$, where all values of $\chi $ are roots of $1$, and the root system of $\chi $ is finite. Then we pass to more general bicharacters: Thm. \[th:Shapdet2\] states a similar result for bicharacters with finite root system. We conclude the paper with the adaptation of our formulas to quantized enveloping algebras and Lusztig’s small quantum groups in Sect. \[sec:Uqg\], and with some commutative algebra in the Appendix.
Preliminaries {#sec:prelims}
=============
Let ${\Bbbk }$ be a field and ${{\Bbbk }^\times }={\Bbbk }\setminus \{0\}$. For all $n\in {\mathbb{N}}_0$ and $q\in {{\Bbbk }^\times }$ let $${(n)_{q}}=\sum _{j=0}^{n-1}q^j,\qquad {(n)^!_{q}}=\prod _{j=1}^n{(j)_{q}},$$ where ${(0)^!_{q}}=1$. For any finite set $I$ let $\{{\alpha }_i\,|\,i\in I\}$ be the standard basis of the free ${\mathbb{Z}}$-module ${\mathbb{Z}}^I$.
Cartan schemes, Weyl groupoids, and root systems {#ssec:CS}
------------------------------------------------
The combinatorics of a Drinfel’d double of a Nichols algebra of diagonal type is controlled to a large extent by its Weyl groupoid. We use the language developed in [@p-CH08]. Substantial part of the theory was obtained first in [@a-HeckYam08]. We recall the most important definitions and facts.
Let $I$ be a non-empty finite set. By [@b-Kac90 §1.1] a generalized Cartan matrix $C=(c_{ij})_{i,j\in I}$ is a matrix in ${\mathbb{Z}}^{I\times I}$ such that
1. $c_{ii}=2$ and $c_{jk}\le 0$ for all $i,j,k\in I$ with $j\not=k$,
2. if $i,j\in I$ and $c_{ij}=0$, then $c_{ji}=0$.
\[de:CS\] Let $I$ be a non-empty finite set, $A$ a non-empty set, ${r}_i :
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zero structure Fig.2(a).
We note that a back-bending dispersion was already observed in an early QMC study,[@ph97] where antiferromagnetic fluctuations were proposed as the origin of the pseudogap. Although our numerical data do not exclude the antiferromagnetic fluctuations from the possible mechanisms, the less-$\Vec{k}$-dependent pseudogap as well as the asymmetric location of the hole pocket (see Sec. \[ssec:arc\]) seems to oppose the mechanism; instead it rather supports that the pseudogap is a direct consequence of the proximity to the Mott insulator.
Energy-distribution curve {#ssec:edc}
-------------------------
Zeros of $G$ are not directly seen in spectra. Their footprints may, however, be detected through a sudden suppression of spectra due to a large Im$\S$ around the zeros. Figure \[fig:edc\](a) shows energy-distribution curves (EDC) of the spectral function along momentum cuts $(0,0)\text{-}(\pi,0)\text{-}(\pi,\pi)$, calculated by the CDMFT+ED with $\eta=0.1t$ for $t'=-0.2t$, $U=12t$, and $n=0.93$. The results exhibit a coherent peak around $(\pi,0)$ just below the Fermi level, its shift to lower energy from $(\pi,0)$ to $(\frac{\pi}{2},0)$, and the incoherent feature around $(0,0)$ and $(\pi,\pi)$. All these features are consistent with ARPES data \[Fig. 5 in Ref. , reproduced in Fig. \[fig:edc\](b)\]. This agreement supports our zero mechanism and shows that the incoherent feature can be interpreted as the effect of zeros of $G$: Since the zero surface exists just above the band around $(\pi,\pi)$ \[Fig. \[fig:fig2\](a)\] and since another one is located just below the band around $(0,0)$ \[not shown in Fig. \[fig:fig2\](a), but can refer to Fig. \[fig:rkw\_tp\](b) below\], the spectrum associated with the band is smeared around these momenta due to the large Im$\S$ around the zeros. While the suppression around $(\pi,\pi)$ is relevant to the emergence of Fermi arc,[@sk06; @sm09; @sm09-2] that around $(0,0)$ is related to the waterfall behavior discussed in the next subsection.
Waterfal
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at $p={p_\text{c}}$.
It thus remains to show the bounds in [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$. These bounds are proved by adapting the model-independent bootstrapping argument in [@hhs03] (see the proof of [@hhs03 Proposition 2.2] for self-avoiding walk and percolation), together with the fact that $G(x)$ decays exponentially as $|x|\uparrow\infty$ for every $p<{p_\text{c}}$ [@l80; @s80] so that $\sup_xG(x)$ is continuous in $p<{p_\text{c}}$ [@s05]. We complete the proof.
Strategy for the nearest-neighbor model
---------------------------------------
Since $\sigma^2=O(1)$ for short-range models, we cannot expect that $\theta_0$ in [(\[eq:IR-xbd\])]{} is small, or that Proposition \[prp:GimpliesPix\] is applicable to bound the expansion coefficients in this setting.
Under this circumstance, we follow the strategy in [@h05]. The following is the key proposition, whose proof will be explained in Section \[ss:proof-nn\]:
\[prp:GimpliesPik\] Let $J_{o,x}$ be the nearest-neighbor or spread-out interaction, and suppose that $$\begin{aligned}
{\label{eq:IR-kbd}}
\tau-1\leq\theta_0,&& \sup_x(D*G^{*2})(x)\leq\theta_0,&&
\sup_{\substack{x\equiv(x_1,\dots,x_d)\ne o\\ l=1,\dots,d}}
\bigg(\frac{x_l^2}{\sigma^2}\vee1\bigg)G(x)\leq\theta_0\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$. Then, for sufficiently small $\theta_0$ and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned}
{\label{eq:pi-sumbd}}
\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq
\begin{cases}
1+O(\theta_0^2)&(i=0),\\ O(\theta_0)^i&(i\ge1),
\end{cases}&&
\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq d\sigma^2(i+1)^2O(\theta_0)^{i
\vee2}.\end{aligned}$$ Furthermore, in addition to [(\[eq:IR-kbd\])]{} with $\theta_0\ll1$, if $$\begin{aligned}
{\label{eq:IR-xbdNN}}
G(x)\leq\lambda_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ holds for some $\lambda_0\in[1,\infty)$ and $q\in(0,d)$, then we have for $i\ge0$ $$\begin{aligned}
{\label{eq:pi-kbd}}
\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq O(\theta_0)^i\delta_{o,x}+\frac{\
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{RM}_2(3,5))=32-17=15$ in accordance with Example 6.12 in [@HP].
Theorem \[thm:main\] is somewhat similar in spirit as Theorem 6.8 from [@HP] in the sense that in both theorems a certain representation in terms of dimensions of Reed–Muller codes is used to give an expression for $d_r({\mathrm{RM}_q}(d,m))$. Where we studied decompositions of $\rho_q(d,m)-r$, in [@HP] the focus was on $r$ itself. This suggest there may exist a duality between the two approaches, but the similarities seem to stop there. The representation in [@HP] is not the Macaulay representation with respect to $q$ that we have used here. For us it is for example very important that each degree $i$ between $1$ and $d$ occurs once in Theorem \[thm:genrepMac\] (implying that the greedy algorithm terminates after at most $d$ iterations), while this is not the case in Theorem 6.8 [@HP]. It could be interesting future work to determine if a deeper lying relationship between the two approaches exists.
[22]{} E.F. Assmus Jr. and J.D. Key, Designs and their Codes, Cambridge University Press, 1992.
P. Beelen and M. Datta, Generalized Hamming weights of affine Cartesian codes, Finite Fields and Applications 51, 130–145, 2018.
M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, In: Algebraic Curves and Projective Geometry, Lecture Notes in Mathematics 1389, 76–86, 2006.
P. Heijnen and R. Pellikaan, Generalized [H]{}amming weights of [$q$]{}-ary [R]{}eed-[M]{}uller codes, IEEE Trans. Inform. Theory 44(1), 181–196, 1998.
[^1]: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800, Kongens Lyngby, Denmark, pabe@dtu.dk
---
abstract: 'While the analysis of airborne laser scanning (ALS) data often provides reliable estimates for certain forest stand attributes – such as total volume or basal area – there is still room for improvement, especially in estimating species-specific attributes. Moreover, while information on the estimate uncertainty woul
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ve Hamiltonian which has new Wilson coefficients and analyze the contributions coming from these coefficients within this model.\
The organization of this work as follows. In Section II, starting from the most general effective Hamiltonian, we compute the differential decay width of the $B \rar \pi\ell^+\ell^-$ decay,and the forward-backward asymmetry for this decay. In Section III, we carry out the numerical analysis to study the dependence of the asymmetry on the new Wilson coefficients and finally we conclude in Section IV.
Decay Width and Forward-Backward\
Asymmetry
=================================
The semileptonic decay $b \rar d\ell^+\ell^-$ is described,by effective Hamiltonian,at the quark level as: $$\begin{aligned}
{\cal
H}_{eff}=\,\frac{4G_F}{\sqrt{2}}\,V_{tb}V_{td}^*\,\sum_{i=1}^{10}C_i(\mu){\cal
O}_i(\mu),\end{aligned}$$ where the full set of the operators ${\cal O}_i(\mu)$ in the SM are given in [@Grinstein]. The effective coefficient of the operator ${\cal O}_9(\mu)$ can be defined as: $$\begin{aligned}
C_9^{eff}(\hat{s})&=&\,C_9+g(z,
\hat{s})\,(3C_1+C_2+3C_3+C_4+3C_5+C_6)\nonumber\\&-&\,\frac{1}{2}\,g(1,
\hat{s})\,(4C_3+4C_4+3C_5+C_6)-\,\frac{1}{2}\,g(0,
\hat{s})\,(C_3+3C_4)\nonumber\\&+&\,\frac{1}{2}\,(3C_3+C_4+3C_5+C_6).\end{aligned}$$ Here, $\hat{s}=q^2/m^2_B$ where q is the momentum transfer and $z=m_c/m_b$. The functions $g(z, \hat{s}), g(1, \hat{s})$ and $g(0, \hat{s})$ can be found in [@Kim].\
Neglecting the d quark mass, the above Hamiltonian leads to the following matrix element for the $b \rar d\ell^+\ell^-$ decay: $$\begin{aligned}
{\cal
M}&=&\,\frac{G_F\alpha}{\sqrt{2}\pi}\,V_{tb}V^*_{td}\Bigg[(C^{eff}_9-C_{10})\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_L\gamma^{\mu}\ell_L\nonumber\\&+&(C^{eff}_9+C_{10})\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_R\gamma^{\mu}\ell_R\nonumber\\&-&2C^{eff}_7\hat{m}_b\bar{d}i
\sigma_{\mu\nu}\frac{\hat{q}^{\nu}}{\hat{s}}Rb\,\hat{\ell}\gamma^{\mu}\ell\Bigg],\end{aligned}$$ Where, $L/R=(1\mp \gamma^5)/2$, $\hat{m}_b=m_b/m_B$ and $b(d)_{L,R}=[(1\mp \gamma^5)/2]b(d)$.\
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tegers; thus every sequence is a net but the family[^5] indexed, for example, by $\mathbb{Z}$, the set of *all* integers, is a net and not a sequence) with a sequence and provide the necessary background and motivation for the concept of graphical convergence.
***Begin Tutorial2: Convergence of Functions***
This Tutorial reviews the inadequacy of the usual notions of convergence of functions either to limit functions or to distributions and suggests the motivation and need for introduction of the notion of graphical convergence of functions to multifunctions. Here, we follow closely the exposition of @Korevaar1968, and use the notation $(f_{k})_{k=1}^{\infty}$ to denote real or complex valued functions on a bounded or unbounded interval $J$.
A sequence of piecewise continuous functions $(f_{k})_{k=1}^{\infty}$ is said to converge to the function $f$, notation $f_{k}\rightarrow f$, on a bounded or unbounded interval $J$[^6]
\(1) *Pointwise* if$$f_{k}(x)\longrightarrow f(x)\qquad\textrm{for all }x\in J,$$
that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$ that may depend on $x$, such that $|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$.
\(2) *Uniformly* if $$\sup_{x\in J}|f(x)-f_{k}(x)|\longrightarrow0\qquad\textrm{as }k\longrightarrow\infty,$$
that is: Given any arbitrary real number $\varepsilon>0$ there exists a $K\in\mathbb{N}$, such that $\sup_{x\in J}|f_{k}(x)-f(x)|<\varepsilon$ for all $k\geq K$.
\(3) *In the mean of order $p\geq1$* if $|f(x)-f_{k}(x)|^{p}$ is integrable over $J$ for each $k$ $$\int_{J}|f(x)-f_{k}(x)|^{p}\longrightarrow0\qquad\textrm{as }k\rightarrow\infty.$$
For $p=1$, this is the simple case of *convergence in the mean.*
\(4) *In the mean $m$-integrally* if it is possible to select indefinite integrals $$f_{k}^{(-m)}(x)=\pi_{k}(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f_{k}(x_{m})$$
and
$$f^{(-m)}(x)=\pi(x)+\int_{c}^{x}dx_{1}\int_{c}^{x_{1}}dx_{2}\cdots\int_{c}^{x_{m-1}}dx_{m}f(x_{m})$$
such that for some ar
| 3,911
| 4,256
| 4,141
| 3,423
| 2,798
| 0.776868
|
github_plus_top10pct_by_avg
|
Reference
AG 151 (15.7) 135 (10.0) Codominant 1.57 (1.20-2.05) \< 0.001
GG 11 (1.1) 3 (0.2) 6.18 (1.59-23.98)
AG + GG 162 (16.9) 138 (10.2) Dominant 1.66 (1.28-2.16) \< 0.001
Recessive 5.80 (1.50-22.48) 0.005
Overdominant 1.55 (1.18-2.03) 0.001
Log-additive 1.68 (1.31-2.15) \< 0.001
Abbreviations: OR, odds ratio; CI, confidence interval;
The ORs and *P* values were adjusted for age, sex, and underlying medical conditions.
{#F1}
PDGF-BB was persistently reduced in SFTSV-infected C57BL/6J mice {#s2_3}
--------------------------------------------------------
| 3,912
| 365
| 3,670
| 3,871
| null | null |
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|
i\ \bu_i,\bv_i \in \Z_m^k$. Then $\cF$ is called an $S$-matching vector family of size $n$ and dimension $k$ if $\forall\ i,j$, $$\begin{aligned}
{\langle \bu_i,\bv_j \rangle}\begin{cases}
= 0 & \mbox{if } i=j\\
\in S & \mbox{if } i\ne j
\end{cases}\end{aligned}$$ If $S$ is omitted, it implies that $S=\Z_m\setminus{\{0\}}$.
\[Grolmusz\] Let $m=p_1p_2\cdots p_r$ where $p_1,p_2\cdots, p_r$ are distinct primes with $r\ge 2$, then there exists an explicitly constructible $S$-matching vector family $\cF$ in $\Z_m^k$ of size $n\ge {\exp\left(\Omega\left(\frac{(\log k)^r}{(\log\log k)^{r-1}}\right)\right)}$ where $S={\{a\in \Z_m: a\mod p_i \in {\{0,1\}}\ \forall\ i \in[r]\}}\setminus {\{0\}}$.
\[CRT\] The size of $S$ in the above theorem is $2^r-1$ by the Chinese Remainder Theorem. Thus, there are matching vector families of size super-polynomial in the dimension of the space with inner products restricted to a set of size $2^r = |S \cup \{0\}|$.
In the special case when $p_1=2,p_2=3$, we have $m=6$ and the following corollary:
\[Grolmuszmod6\] There is an explicitly constructible $S$-matching vector family $\cF$ in $\Z_6^k$ of size $n\ge {\exp\left(\Omega\left(\frac{(\log k)^2}{\log\log k}\right)\right)}$ where $S={\{1,3,4\}}\subset \Z_6$
A number theoretic lemma
------------------------
We will need the following simple lemma. Recall that the [*order*]{} of an element $a$ in a finite multiplicative group $G$ is the smallest integer $w \geq 1$ so that $a^w=1$.
\[lem-order\] Let $\F_p$ be a field of prime order $p$ and let $k \geq 1$ be an integer co-prime to $p$. Then, the algebraic closure of $\F_p$ contains an element $\zeta$ of order $k$.
Since $k,p$ are co-prime, $p\in \Z_k^*$ which is the multiplicative group of invertible elements in $\Z_k$. Let $w \geq 1$ be the order of $p$ in the group $\Z_k^{*}$, so $k$ divides $p^w-1$. Consider the extension field $\F_{p^w}$, which is a sub field of the algebraic closure of $\F_p$. The multiplicative group $\F_{p^w}^*$ of this field is a cyclic group of size $p^w
| 3,913
| 4,279
| 3,909
| 3,495
| null | null |
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|
tarrows X$ is a [*finite*]{} equivalence relation if the maps $\sigma_1,\sigma_2$ are finite. In this case, $\sigma:R\to X\times_SX$ is also finite, hence a closed embedding (\[monom.defn\]).
\[setth.eq.rel.defn\] Let $X$ and $R$ be reduced $S$-schemes. We say that a morphism $\sigma:R\to X\times_SX$ is a [*set theoretic equivalence relation*]{} on $X$ if, for every geometric point ${\operatorname{Spec}}K\to S$, we get an equivalence relation on $K$-points $$\sigma(K):{\operatorname{Mor}}_S({\operatorname{Spec}}K,R)\into
{\operatorname{Mor}}_S({\operatorname{Spec}}K,X)\times {\operatorname{Mor}}_S({\operatorname{Spec}}K,X).$$ Equivalently,
1. $\sigma$ is geometrically injective.
2. (reflexive) $R$ contains the diagonal $\Delta_X$.
3. (symmetric) There is an involution $\tau_R$ on $ R$ such that $\tau_{X\times X}\circ\sigma\circ\tau_R=\sigma$ where $\tau_{X\times X}$ denotes the involution which interchanges the two factors of $X\times X$.
4. (transitive) For $1\leq i<j\leq 3$ set $X_i:=X$ and let $R_{ij}:=R$ when it maps to $X_i\times_SX_j$. Then the coordinate projection of ${\operatorname{red}}\bigl(R_{12}\times_{X_2}R_{23}\bigr)$ to $X_1\times_SX_3$ factors through $R_{13}$: $${\operatorname{red}}\bigl(R_{12}\times_{X_2}R_{23}\bigr)\to R_{13}
\stackrel{\pi_{13}}{\longrightarrow}
X_1\times_SX_3.$$
Note that the fiber product need not be reduced, and taking the reduced structure above is essential, as shown by (\[nonred.noneq.exmp\]).
It is sometimes convenient to consider finite morphisms $p:R\to X\times_SX$ such that the injection $i:p(R)\into X\times_SX$ is a set theoretic equivalence relation. Such a $p:R\to X\times_SX$ is called a [*set theoretic pre-equivalence relation.*]{}
\[nonred.noneq.exmp\] On $X:={{\mathbb C}}^2$ consider the ${{\mathbb Z}}/2$-action $(x,y)\mapsto (-x,-y)$. This can be given by a set theoretic equivalence relation $R\subset X_{x_1,y_1}\times X_{x_2,y_2}$ defined by the ideal $$(x_1-x_2,y_1-y_2)\cap (x_1+x_2,y_1+y_2)=
(x_1^2-x_2^2, y_1^2-y_2^2, x_1y_1-x_2y_2, x_
| 3,914
| 3,955
| 3,261
| 3,499
| 1,724
| 0.786386
|
github_plus_top10pct_by_avg
|
\quad
\overline{\nabla}_i\overline{\nabla}_{\alpha}v=0.$$ where $i,j=1,2,\cdots,n$, and $\alpha,\beta=n+1,n+2,\cdots,m$, moreover, $$\label{warpedLaplacian}
\Delta_{\overline{M}}=\Delta_f+e^{-\frac{2f}{q}}\Delta_N.$$ Thus, if $u:M\to[0,\infty)$ is a positive solution to , then $u$ satisfies the following equation $$\frac{\partial u}{\partial t}=\bar{\Delta}(u^{\gamma}).$$ Moreover, in term of , the weighted entropy functional is indeed the functional on $(\overline{M},g_{\overline{M}})$ $$\overline{\mathcal{W}}_K(v,t)=\sigma_K\beta_K\int_{\overline{M}}\left[\gamma\frac{|\nabla v|^2}{v}-\left(\frac{1}{\beta_K}+\frac{\dot{\sigma}_K}{\sigma_K}\right)\right]vu\,dV_{\overline{M}},$$ Applying the entropy formula on $(\overline{M},g_{\overline{M}})$ in Theorem \[KPMEentropy\], we have
$$\begin{aligned}
\label{KPMEnt}
&\frac{d}{dt}\overline{\mathcal{W}}_K(t)\notag\\
\le&
-2(\gamma-1)\bar{\sigma}_K\bar{\beta}_K\int_{\overline{M}}\left(\left| \bar{\nabla}_i\bar{\nabla}_jv+\frac{\bar{\eta}_K}{m(\gamma-1)}\bar{g}_{ij}\right|_{\bar{g}}^2
+(\overline{{\rm Ric}}+K\bar{g})(\bar{\nabla} v,\bar{\nabla} v )\right)vu\,dV_M dV_N\notag\\
&-2\bar{\sigma}_K\bar{\beta}_K\int_{\overline{M}}\left((\gamma-1)\bar{\Delta} v+\eta_K\right)^2vu\,dV_M dV_N.\end{aligned}$$
where $$\bar{\eta}_K=\frac{2\bar{a}{\kappa}}{1-e^{-2{\kappa}t}},\quad\bar{\beta}_K=\frac{\sinh(2{\kappa}t)}{2{\kappa}},\quad
\bar{\sigma}_K=\left(\frac{e^{2{\kappa}t}-1}{2\kappa}\right)^{{\bar{a}}},\quad \bar{a}=\frac{m(\gamma-1)}{m(\gamma-1)+2}.$$ An analogous calculation in [@LiLi] gives,
$$\begin{aligned}
\label{WHessian}
\Big|\overline{\nabla}_i\overline{\nabla}_jv+\frac{\bar{\eta}_K}{m(\gamma-1)}\bar{g}_{ij}\Big|^2
=&\Big|\nabla_i\nabla_jv+\frac{\bar{\eta}_K}{m(\gamma-1)}g_{ij}\Big|^2+\Big|\overline{\nabla}_{\alpha}\overline{\nabla}_{\beta}v+\frac{\bar{\eta}_K}{m(\gamma-1)}g_{\alpha\beta}\Big|^2\\
=&\Big|\nabla_i\nabla_jv+\frac{\bar{\eta}_K}{m(\gamma-1)}g_{ij}\Big|^2+\frac{1}{m-n}\Big|\nabla v\cdot\nabla f-\frac{\bar{\eta}_K(m-n)}{m(\gamma-1)}\Big|^2\end{ali
| 3,915
| 2,944
| 2,089
| 3,430
| null | null |
github_plus_top10pct_by_avg
|
de K}{\partial E}}}(E)\phi$ is provided by Lemma \[le:K\_C1\]. By assumptions , , we thus see that $h_\phi$ is in $C^1(I,L^2(G\times S))$.
By Theorem \[evoth\] there exists a unique solution $\phi\in C(I,W^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$ of (\[ecsd7\]). Then $\psi(x,\omega,E):=
e^{C(E_m-E)}\phi(x,\omega,E_m-E)$ is the required solution of the problem , , for $g=0$.
B. Suppose that more generally $g\in C^2(I,T^2(\Gamma_-))$ and that the compatibility condition (\[cc\]) holds. By [@tervo14 Lemma 5.10] there exists a lift $Lg\in C^2(I,\tilde W^2(G\times S))$ for which $\gamma_-(Lg)=g$, and $(Lg)(\cdot,\cdot,E_m)=0$ (follows from (\[cc\])), and furthermore $\omega\cdot\nabla_x(Lg)=0$. Substituting into the problem (\[se1\]), (\[se2\]), (\[se3\]) the function $u:=\psi-Lg$ for $\psi$ we obtain the following problem for $u$, $$\begin{gathered}
-{{\frac{\partial (S_0u)}{\partial E}}}+\omega\cdot\nabla_x u+\Sigma u
- Ku = \tilde{f},\nonumber\\
u_{|\Gamma_-}=0,\nonumber\\
u(\cdot,\cdot,E_m)=0,\label{ecsd8}\end{gathered}$$ where $$\tilde{f}:=f-\Big(-{{\frac{\partial (S_0(Lg))}{\partial E}}}+\Sigma (Lg)-K(Lg)\Big),$$ and we have $\tilde{F}\in C^1(I,L^2(G\times S))$ under the assumption , , .
By Part A of the proof, the problem , , (for $g=0$) has a unique solution $u\in C(I,\tilde{W}^2_{-,0}(G\times S))\cap C^1(I,L^2(G\times S))$. As argued above, $\psi:=u+Lg$ is then then desired unique solution for the problem , , , for the given, arbitrary $g\in C^2(I,T^2(\Gamma_-))$.
We claim that the solution $\psi$ belongs to $W^2_1(G\times S\times I)$. Indeed, since $\psi\in C^1(I,L^2(G\times S))$, we have $\psi\in L^2(G\times S\times I)$ and ${{\frac{\partial \psi}{\partial E}}}\in L^2(G\times S\times I)$. On the other hand, since $\psi$ solves , these imply, together with the assumptions made, that $\omega\cdot\nabla_x\psi\in L^2(G\times S\times I)$, which confirms the claim.
Hence, under the additional assumptions (\[ass2b\]), the estimate (\[evoest\]) follows from Corollary \[csdaco1\] (part (iii)). This comple
| 3,916
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| 1,698
| 3,849
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|
he same conditions we have imposed in section \[sec:energy-denominator\] the first one in (\[expansion-parameters\]) is $\simeq 7.6 \times 10^{-4}$ for $\Delta m^2_{J i} = 0.1$ eV$^2$ and $\rho E = 10 \text{ (g/cm}^3) \text{GeV}$, while the second and the third, which are comparable at around the first oscillation maximum, are estimated as $2.3 \times 10^{-2}$ for the same condition. Therefore, the smallness of the expansion parameter is ensured unless $\rho E \gg 10 \text{ (g/cm}^3) \text{GeV}$. In fact, a close examination of the order $W^4$ terms in the oscillation probability (see appendix \[sec:expression-probability-4th\]) shows that all the formally $W^4$ terms are actually further suppressed. The largest term in the fourth-order oscillation probabilities is of the one suppressed by a factor $\left\vert \left( \frac{A W }{ \Delta_{J} - h_{i} } \right) \left( A L W \right) W^2 \right\vert \lsim 1.7 \times 10^{-7}$, which is as small as $\sim10^{-4}$ even in the case $|W|=0.5$. Therefore, we expect that the formula for the oscillation probability in (\[P-beta-alpha-final\]) works under much relaxed conditions than the one in (\[suppression-cond\]).
On Uniqueness theorem and matter-dependent dynamical phase {#sec:U-theorem}
------------------------------------------------------------
We have shown in sections \[sec:probability-2nd\] and \[sec:probability-4th\] that there is no surviving matter dependent correction term in the oscillation probability up to order $W^4$ after averaging out fast oscillations and using the suppression by energy denominators. Should we expect that this feature is stable against higher order corrections beyond order $W^4$, as postulated by the Uniqueness theorem, in our perturbative framework? We argue that the answer is [*Yes*]{}.
We first note that higher-order corrections in terms of $W$ are computed by using $\Omega [1]$ as the kernel, as indicated in eq. (\[Omega-expand\]). Notice also that all the elements of $\Omega [1]$, except for $\Omega [1]_{JJ}$, carry the energ
| 3,917
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|
-P_{\sigma}}{2}
\left[ 1- \left(\dfrac{\varrho^{\mathrm{IS}}(\boldsymbol{r})}{\varrho_{0}}\right)^{\gamma} \right]
\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}). \label{eq:res_pp}$$ with $V_{0}=-390$ MeV $\cdot$fm$^{2}$ and $\varrho_{0}=0.16$ fm$^{-3}$, $\gamma=1$. Here, $\varrho^{\mathrm{IS}}(\boldsymbol{r})$ denotes the isoscalar density and $P_{\sigma}$ the spin exchange operator. The pairing strength $V_{0}$ is determined so as to approximately reproduce the experimental pairing gap of 1.25 MeV in $^{28}$Ne obtained by the three-point formula [@sat98]. Because the time-reversal symmetry and reflection symmetry with respect to the $x-y$ plane are assumed, we have only to solve for positive $\Omega$ and positive $z$. We use the lattice mesh size $\Delta\rho=\Delta z=0.6$ fm and the box boundary condition at $\rho_{\mathrm{max}}=9.9$ fm and $z_{\mathrm{max}}=9.6$ fm. The quasiparticle energy is cut off at 60 MeV and the quasiparticle states up to $\Omega^{\pi}=13/2^{\pm}$ are included.
Using the quasiparticle basis obtained by solving the HFB equation (\[eq:HFB1\]), we solve the QRPA equation in the matrix formulation [@row70] $$\sum_{\gamma \delta}
\begin{pmatrix}
A_{\alpha \beta \gamma \delta} & B_{\alpha \beta \gamma \delta} \\
B_{\alpha \beta \gamma \delta} & A_{\alpha \beta \gamma \delta}
\end{pmatrix}
\begin{pmatrix}
X_{\gamma \delta}^{\lambda} \\ Y_{\gamma \delta}^{\lambda}
\end{pmatrix}
=\hbar \omega_{\lambda}
\begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}
\begin{pmatrix}
X_{\alpha \beta}^{\lambda} \\ Y_{\alpha \beta}^{\lambda}
\end{pmatrix} \label{eq:AB1}.$$ The residual interaction in the particle-particle (p-p) channel appearing in the QRPA matrices $A$ and $B$ is the density-dependent contact interaction (\[eq:res\_pp\]). On the other hand, for the residual interaction in the particle-hole (p-h) channel, we employ the Landau-Migdal (LM) approximation [@bac75] applied to the density-dependent Skyrme forces [@gia81; @gia98], $$\begin{aligned}
v_{ph}(\boldsymbol{r},\boldsymbol{r}^{\prime})=&
N_{0}
| 3,918
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|
tum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M1_2004 "fig:"){width="\textwidth"}\
![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M1_2008 "fig:"){width="\textwidth"}
![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M0_2004 "fig:"){width="\textwidth"}\
![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M0_2008 "fig:"){width="\textwidth"}
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the German BMBF, the Maier-Leibnitz-Labor der LMU und TU München, the DFG Cluster of Excellence *Origin and Structure of the Universe*, and CERN-RFBR grant 08-02-91009.
[0]{} P. Abbon [*et al.*]{}, *Nucl. Instrum. Meth.* *
| 3,919
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|
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|
\Delta_{J} - h_{i} )^2 e^{- i h_{k} x}
+ ( h_{i} - h_{k} )( h_{i} + h_{k} - 2 \Delta_{J} ) e^{- i \Delta_{J} x}
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{J k}
\left\{ (UX)^{\dagger} A W \right\}_{k J}
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\nonumber \\
&+&
\sum_{K \neq J}
\biggl[
- \frac{ (ix) e^{- i h_{i} x} }{ (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) }
+ \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) (\Delta_{J} - h_{i})^2 ( \Delta_{K} - h_{i} )^2 }
\nonumber \\
&\times&
\biggl\{
( \Delta_{K} - h_{i} )^2 e^{- i \Delta_{J} x}
- (\Delta_{J} - h_{i})^2 e^{- i \Delta_{K} x}
+ e^{- i h_{i} x} (\Delta_{J} - \Delta_{K}) (\Delta_{J} + \Delta_{K} - 2 h_{i} )
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{J i}
\left\{ (UX)^{\dagger} A W \right\}_{i K}
\left\{ W^{\dagger} A (UX) \right\}_{K i}
\nonumber \\
&+&
\sum_{K \neq J}
\sum_{k \neq i}
\frac{ 1 }{ (\Delta_{J} - \Delta_{K}) ( h_{k} - h_{i} ) (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) (\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) }
\nonumber \\
&\times&
\biggl[
( h_{k} - h_{i} )
\biggl\{ (\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) e^{- i \Delta_{J} x}
- (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) e^{- i \Delta_{K} x}
\biggr\}
\nonumber \\
&-&
(\Delta_{J} - \Delta_{K})
\biggl\{ (\Delta_{J} - h_{k}) (\Delta_{K} - h_{k}) e^{- i h_{i} x}
- (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) e^{- i h_{k} x}
\biggr\}
\biggr]
\nonumber \\
&\times&
\left\{ W^{\dagger} A (UX) \right\}_{J k}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W^{\dagger} A (UX) \right\}_{K i}.
\label{hatS-Ji-W3-H2+3}\end{aligned}$$ Using (\[hatS-iJ-W3-H2+3\]) and (\[hatS-Ji-W3-H2+3\]), one can easily confirm generalized T invariance $\hat{S}_{i J}^{(3)} (U, W, \text{etc}) = \hat{S}_{J i}^{(3)} (U^*, W^*, \text{etc})$.
Order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j} [3]$ and $\hat{S}_{i j}^{(4)} \vert_{i \neq j} [4]$ {#sec:hatSij-3nd-4th}
-------------------------------------------------------
| 3,920
| 2,353
| 2,449
| 3,687
| null | null |
github_plus_top10pct_by_avg
|
ism $(P_{F_i}+(f_i))/P_{F_i}\iso (f_i)/(f_i)P_{F_i}$ results from the fact that $(f_i)\sect P_{F_i}=(f_i)P_{F_i}$ since the set of variables dividing $f_i$ and the set of variables generating $P_{F_i}$ have no element in common. Thus we have shown
\[shelling numbers\] Let $\Delta$ be a shellable simplicial complex with shelling $F_1,\ldots, F_r$ and shelling numbers $a_1,\ldots, a_r$. Then $(0)=M_0\subset M_1\subset\cdots\cdots M_{r-1}\subset M_r=K[\Delta]$ with $$M_i=\Sect_{j=1}^{r-i}P_{F_j}\quad \text{and}\quad M_i/M_{i-1}\iso S/P_{F_{r-i+1}}(-a_{r-i+1})$$ is a clean filtration of $S/I_{\Delta}$.
Multicomplexes
==============
The aim of this and the next section is to extend the result of Dress to multicomplexes. Stanley [@St] calls a subset $\Gamma\subset \NN^n$ a multicomplex if for all $a\in \Gamma$ and all $b\in\NN^n$ with $b\leq a$, it follows that $b\in\Gamma$. The elements of $\Gamma$ are called [*faces*]{}.
What are the facets of $\Gamma$? We define on $\NN^n$ the partial order given by $$(a(1),\ldots, a(n))\leq (b(1),\ldots,b(n))\quad \text{if}\quad a(i)\leq
b(i)\quad \text{for all}\quad i.$$ An element $m\in \Gamma$ is called maximal if there exists no $a\in \Gamma$ with $a> m$. We denote by ${\mathcal M}(\Gamma)$ the set of maximal elements of $\Gamma$. One would expect that ${\mathcal M}(\Gamma)$ is the set of facets of $\Gamma$. However ${\mathcal M}(\Gamma)$ may be the empty set, for example for $\Gamma=\NN^n$. To remedy this defect we will consider “closed" subsets $\Gamma$ in $\NN^n_\infty$, where $\NN_\infty=\NN\union \{\infty\}$.
Let $a\in\Gamma$. Then $$\ip a=\{i\: a(i)=\infty\}$$ is called the [*infinite part*]{} of $a$. We first notice that
\[finitem\] Let $\Gamma\subset \NN^n_\infty$. Then ${\mathcal M}(\Gamma)$ is finite.
Let $F\subset [n]$, and set $\Gamma_F=\{a\in \Gamma\: \ip a =F\}.$ It is clear that if $a\in \Gamma_F$ is maximal in $\Gamma$ then $a$ is maximal in $\Gamma_F$. Since there are only finitely many subsets $F$ of $[n]$, it suffices to show that $\Gamma_F$ has onl
| 3,921
| 2,363
| 3,176
| 3,525
| null | null |
github_plus_top10pct_by_avg
|
odel with many inflating branes along the fifth compact direction, thus extending on the recent models proposed by Randall-Sundrum and Oda \[1,2,3\]. Each brane is taken as a point in the fifth dimension (negligible thickness) $S^1$. They act as localized cosmological constants or alternatively as gravitational point sources.The whole 5D setup should be static in order to preserve Poincare invariance. That means that the 5D cosmological constant $\Lambda_5$ should cancel out the collected contribution of the 4D cosmological constants $\Lambda_{4i}$ of each brane (weighted by the appropriate conformal factor as shown below in Sect.2). Note that each $\Lambda_{4i}$ depends on the position of the i-th brane in $S^1$.
The existence of solution to the Einstein equation in the many branes scenario does not allow for a singular orbifold geometry $S^1/Z_2$ but instead, it requires a smooth manifold $S^1$ \[1\].
The 5D metric ansatz considered here is: $$ds^2 = g_{MN} dx^M dx^N = u(z)^2 dt^2 - a(z,t)^2 dx^2 - b(z)^2 dz^2$$ with $z$ ranging between $0$ and $2 L$.
The 5D Einstein-Hilbert action is: $$S = \frac{1}{2\kappa^2_{5}} \int d^4 x \int^{2L}_0 dz \sqrt{-g} (R+2\Lambda_5)
+ \sum^n_{i=1} \int_{z=L_i} d^4 x \sqrt{-g_i}({\cal L}_i + V_i)$$ where the 4D metric $g_{i\mu\nu}$ is obtained from the 5D one as follows: $g_{i\mu\nu}({\bf x},t) = g_{MN}({\bf x},t,z = L_i)$ and ${\cal L}_i, V_i$ are the lagrangian of the matter fields and the vacuum energy respectively, in the i-th brane.The 5D gravitational coupling constant $\kappa_{5}$ is related to the 4D Newton constant $G_N$ through $\kappa_5 =16 G_{N}\pi \int dz\sqrt{-g_{55}} =16 G_{N}\pi L_{phys} $.\
Many Inflating Branes Scenario. Solution
========================================
We are interested in constructing a solution for the inflating branes. Thus in Eqn. (1) we take: $$\begin{array}{ccl}
a(z,t) & = & f(z) v(t)\\
b(z) & = & f(z)\\
u(z) & = & f(z)
\end{array}$$ The dynamics of the boundaries is based on the slow roll assumption.The energy density of each bra
| 3,922
| 3,465
| 3,789
| 3,456
| 3,306
| 0.773317
|
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|
a(y)\approx e^{-\sqrt{-\Lambda/6}\;y}\left( 1 - {1\over5}\left({128\pi
\rho_0 \over -\Lambda M^3}\right)^{2/5} e^{2\sqrt{-\Lambda/6}\;y} +\dots
\right).$$
Conclusions and discussion
==========================
We have shown that in brane-world scenarios with a warped extra dimension, it is in principle possible to stabilize the radion $\phi$ through the Casimir force induced by bulk fields. Specifically, conformally invariant fields induce an effective potential of the form (\[confveff\]) as measured from the positive tension brane. From the point of view of the negative tension brane, this corresponds to an energy density per unit physical volume of the order $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}^{-}\sim m_{pl}^4
\left[{A\lambda^4 \over (1-\lambda)^4}+\alpha+\beta\lambda^4\right],$$ where $A$ is a calculable number (of order $10^{-3}$ per degree of freedom), and $\lambda \sim \phi/(M^3 \ell)^{1/2}$ is the dimensionless radion. Here $M$ is the higher-dimensional Planck mass, and $\ell$ is the AdS radius, which are both assumed to be of the same order, whereas $m_{pl}$ is the lower-dimensional Planck mass. In the absence of any fine-tuning, the potential will have an extremum at $\lambda \sim 1$, where the radion may be stabilized (at a mass of order $m_{pl}$). However, this stabilization scenario without fine-tuning would not explain the hierarchy between $m_{pl}$ and the $TeV$.
A hierarchy can be generated by adjusting $\beta$ according to (\[consts\]), with $\lambda_{obs}\sim (TeV/m_{pl}) \sim
10^{-16}$ (of course one must also adjust $\alpha$ in order to have vanishing four-dimensional cosmological constant). But with these adjustement, the mass of the radion would be very small, of order $$m^{2\ (-)}_{\phi} \sim\lambda_{obs}\ M^{-3} \ell^{-5}
\sim \lambda_{obs} (TeV)^2.
\label{smallmass}$$ Therefore, in order to make the model compatible with observations, an alternative mechanism must be invoked in order to stabilize the radion, giving it a mass of order $TeV$.
Goldberger and Wise [@gw
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nt family of closed, countably compact subspaces each with unbounded range.
Note that the paracompact subspace is the topological sum of $\leq \aleph_1$ $\sigma$-compact subspaces.
An early version of [@DT1] used the axioms ${\mathbf{\mathop{\pmb{\sum}}}}^-$ (defined in Section 5), ${\mathbf{PPI}}$, and the $\aleph_1$-collectionwise Hausdorffness of first countable normal spaces, as well as \[thm39\] to obtain “countably compact, hereditarily normal manifolds of dimension $> 1$ are metrizable" without the $\mathbf{P}_{22}$ axiom used in [@DT1] to get the stronger assertion in which “countably compact" is omitted.
Both of the conditions for paracompactness in \[thm38\] are necessary:
$\omega_1$ is locally compact, normal, first countable, its separable subspaces are countable, but it is not paracompact.
Van Douwen’s “honest example” [@vD] is locally compact, normal, first countable, separable, does not include a perfect pre-image of $\omega_1$ (because it has a $G_\delta$-diagonal), but is not paracompact.
Strengthenings of [PFA]{}$(S)[S]$
=================================
In addition to “front-loading” a PFA$(S)[S]$ model in order to get full collectionwise Hausdorffness, it has also been useful to employ strengthenings of PFA$(S)$ so as to obtain more reflection. E.g. in [@LT2] and [@T], **** is employed.
$C\subseteq[X]^{<\kappa}$ is **tight** if whenever $\{C_\alpha:\alpha<\delta\}$ is an increasing sequence from $C$ and $\omega<\text{\normalfont cf}(\delta)<\kappa$, $\bigcup\{C_\alpha:\alpha<\beta\}\in C$.
**Axiom R**
If $\mathcal{S}\subseteq[X]^{<\omega_1}$ is stationary and $C\subseteq[X]^{<\omega_2}$ is tight and unbounded, then there is a $Y\in C$ such that $\mathcal{P}(Y)\cap\mathcal{S}$ is stationary in $[Y] ^{<\omega_1}$.
**** (due to Fleissner [@Fle]) was obtained by using what is called *PFA$^{++}(S)$* in [@LT2], before forcing with $S$ [@LT2]. PFA$^{++}(S)$ holds if PFA$(S)$ is forced in the usual Laver-diamond way. Here we shall use a conceptually simple principle, MM$(S)$,
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8].
For $x\in X$, let $G_x\subset G$ denote the stabilizer. Let $x\in U_x\subset X$ be a $G_x$-invariant affine open subset. By shrinking $U_x$ we may assume that $G_{x'}\subset G_x$ for every $x'\in U_x$.
In the affine case, quotients by finite groups are easy to get (\[inv.of.fin.gps\]); this is where the conditions (1–3) are used. Thus $U_x/G_x$ exists and it is easy to see that the $U_x/G_x$ give étale charts for $X/G$.
In the general case, it is enough to construct the quotient when $X$ is irreducible. Let $m$ be the separable degree of the projections $\sigma_i:R\to X$.
Consider the $m$-fold product $X\times\cdots\times X$ with coordinate projections $\pi_i$. Let $R_{ij}$ (resp. $\Delta_{ij}$) denote the preimage of $R$ (resp. of the diagonal) under $(\pi_i,\pi_j)$. A geometric point of $\cap_{ij} R_{ij}$ is a sequence of geometric points $(x_1,\dots, x_m)$ such that any 2 are $R$-equivalent and a geometric point of $\cap_{ij} R_{ij}\setminus \cup_{ij}\Delta_{ij}$ is a sequence $(x_1,\dots, x_m)$ that constitutes a whole $R$-equivalence class. Let $X'$ be the normalization of the closure of $\cap_{ij} R_{ij}\setminus \cup_{ij}\Delta_{ij}$. Note that every $\pi_{\ell}:\cap_{ij} R_{ij}\to X$ is finite, hence the projections $\pi'_{\ell}:X'\to X$ are finite.
The symmetric group $S_m$ acts on $X\times\cdots\times X$ by permuting the factors and this lifts to an $S_m$-action on $X'$. Over a dense open subset of $X$, the $S_m$-orbits on the geometric points of $X'$ are exactly the $R$-equivalence classes.
Let $X^*\subset X'/S_m\times X$ be the image of $X'$ under the diagonal map.
By construction, $X^*\to X$ is finite and one-to-one on geometric points over an open set. Since $X$ is normal, $X^*\cong X$ in characteristic 0 and $X^*\to X$ is purely inseparable in positive characteristic.
In characteristic 0, we thus have a morphism $X\to X'/S_m$ whose geometric fibers are exactly the $R$-equivalence classes. Thus $X'/S_m=X/R$.
Essentially the same works in positive characteristic, see Section \[pos.c
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6 75.58 79.84 81.62 1.00 0.85 0.86 0.88
Sox2 1.82 4.23 4.59 4.65 98.37 90.34 90.57 92.93 1.00 0.96 0.98 0.98
Stat3 1.60 8.49 4.80 8.33 97.09 80.99 81.70 84.64 1.00 0.97 0.98 0.98
Tcfcp2l1 1.04 1.73 1.54 1.85 89.05 90.80 91.13 92.73 0.99 0.98 0.99 0.99
Zfx 2.62 3.81 3.94 4.06 94.89 87.67 87.61 88.42 1.00 0.96 0.97 0.97
Fold change is the ratio of newly detected peak number over the original peak number. Overlaps of the original peaks to the newly detected peaks were investigated with 200-bp window. The overlapped peaks were used to calculate the correlation of peak intensity.
{#F1}
The reason why the numbers vary is twofold. First, algorithmic differences in alignment tools cause the different numbers, particularly due to the gapped or ungapped alignment and random indel for mismatches. Second, thresholds for the peak intensity to distinguish experimental noise are different (Table S4 in Additional file [1](#S1){ref-type="supplementary-material"}). That is, Chen et al. used qPCR refinement with small number of peaks, whereas we used Monte Carlo simulation on each chromosome.
Remapped peaks improve the prediction of gene expression
--------------------------------------------------------
To assess the importance of TF bindings, Ouyang et al. \[[@B21]\] successfully applied a regression m
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small scale effects in the inversion procedure laid out in eq. (\[iteration\]) (see also eq. \[\[split\]\]). We “pretend” that we do not know the actual full distortion kernel (eq. \[\[Wfull\]\]), but instead assume only knowledge of the large scale distortion kernel (eq. \[\[Wlinear\]\]) when carrying out the inversion. Moreover, for method II, we have not been very careful in selecting the value of the constants $C_1$ and $C_2$: we simply set them to zero.
To estimate the effect of nonlinear distortions on our inversion procedure, we decrease $k^s_{\parallel}$ to $0.028
({\rm km/s})^{-1}$, and show the outputs of method I and II in Fig. \[pinvcomp\_ks\]. Here we are taking advantage of the fact that the factor of ${\rm exp} [- (k_\parallel / k^s_\parallel)^2]$ in eq. (\[Wfull\]) is commonly used to model nonlinear distortions in the case of galaxy surveys (see e.g. ). By raising the scale of nonlinear distortion by a factor of about $4$, we hope to gain an idea of how the as yet poorly understood nonlinear distortions on small scales might affect the inversion of the power spectrum on large scales. However, it remains to be checked using simulations how realistic this choice of scale, or this particular parametrization of nonlinear distortions, is. The agreement on large scales for method II is not as good as before, but is still within about $7 \%$, and is better than that of method I. It is possible to improve the agreement by playing with the input parameters $C_1$, $C_2$ or $k_{\star\star}$. We will not pursue that here.
We also show in Fig. \[pinvp\_ks\] what the inverted and input power spectra look like in a log-log plot. The subtle differences in the shapes of the power spectra are still possible, but harder, to discern in such a plot.
Lastly, we have assumed in all tests above that the input $\beta_f$ (eq. \[\[Wlinear\]\]) is known when performing the inversion. In practice, there is an uncertainty due to the lack of knowledge of the precise values of $\gamma$ and $f_\Omega$. In Fig. \[pinvcomp\
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ymmetry
-----------------------------
As $\MM{\pi}$ and $\MM{l}$ are still continuous in the discretised equations, the multisymplectic integrator will have a discrete particle-relabelling symmetry analogous to the one given in Section \[inverse map EPDiff\], with the only difference being the discretisation of the cotangent lift. Following the variational integrator programme described in Lew *et al.* (2003), the discrete form of Noether’s theorem will give rise to discrete conservation laws for the multisymplectic method.
Remapping labels
----------------
If this approach is to be applied to numerical solutions with intense vorticity then one needs to address the problem that eventually the numerical discretisation of the labels $\MM{l}$ will become very poor due to tangling, and hence the approximation to the momentum $$\label{mom} \MM{m} = -(\nabla\MM{l})^T\MM{\pi} = - \pi_k\nabla l_k
\,,$$ will degrade with time. One possible approach would be to apply discrete particle-relabelling, mapping the labels back to the Eulerian grid in such a way that the momentum (\[mom\]) stays fixed. This transformation is exactly the relabelling given in Section \[inverse map EPDiff\]. Numerically, one could construct a transformation (using a generating function for example) which satisfies $$\MM{l} \mapsto \MM{X} + \mathcal{O}(\Delta x^p,\Delta t^p),
\qquad \MM{\pi}(\nabla\MM{l})\mapsto \MM{\pi}(\nabla\MM{l})
+ \mathcal{O}(\Delta x^p,\Delta t^p),$$ where $p$ is the order of the method. For instance, one might use a variational discretisation of the relabelling transformation, which is generated by a symplectic vector field whose Hamiltonian is $\MM{\pi}\cdot\MM{\xi}(\MM{l})=\pi_k\xi_k(\MM{l})$. In this way, one may still retain some of the conservative properties of the method.
Summary and Outlook {#summary}
===================
Summary
-------
This paper describes a multisymplectic formulation of Euler-Poincaré equations (which are, in essence, fluid dynamical equations with a particle-relabelling symmetry). We have
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ere inscribed in $D$ that is centred at $\rho_{n-1}$. Calling this sphere $S_n$, we have that $S_n = \{y\in \mathbb{R}^d \colon |y-\rho_{n-1}| =r_n\}$. We now select $\rho_n$ to be a point that is uniformly positioned on $S_n$. Once again, we note that if $\rho_{n-1}\in D$ almost surely, then the uniform distribution of both $\rho_{n-1}$ and $\rho_{n}$ ensures that $\mathbb{P}(\rho_{n}\in \partial D) = 0$. Consequently, the sequence $\rho_n$ continues for all $n\geq1$. In the case that $\rho_n$ approaches the boundary, the sequence of spheres $S_n$ become arbitrarily small in size.
Thanks to the strong Markov property and the stationary and independent increments of Brownian motion, it is straightforward to prove the following result.
Fix $x\in D$ and define $\rho'_1 = W_{\tau_{S'_1}}$, where $\tau_{S'_1}=\inf\{t>0 \colon
W_t \in S'_1 \}$ and $S'_1$ is the largest sphere, centred at $x$, inscribed in $D$. For $n\geq 2$, given $\rho'_{n-1}\in D$, let $\rho'_n = W_{\tau_{S'_n}}$, where $\tau_{S'_n}=\inf\{t>0 \colon W_t \in S'_n \}$ and $S'_n$ is the largest sphere, centred at $\rho'_{n-1}$. Then the sequences $(\rho_n, n\geq 0)$ and $(\rho'_n, n\geq 0)$ have the same law.
As an immediate consequence, $\lim_{n\to\infty}\rho_n$ almost surely exists and, moreover, it it equal in distribution to $W_{\tau_D}$. The sequence $\rho\coloneqq (\rho_n, n\geq 0)$ may now replace the role of $(W_t, t\leq \tau_D)$ in , and hence in (\[FKMC\]), albeit that one must stop the sequence $\rho$ at some finite $N$. By picking a threshold $\varepsilon>0$, we can choose $N(\varepsilon)$ as a cutoff for the sequence $\rho$ such that $N(\varepsilon) = \min\{n\geq 0: \inf_{z\in\partial D}|\rho_n -z|\leq \varepsilon\}$. Intuitively, one is inwardly ‘thickening’ the boundary $\partial D$ with an ‘$\varepsilon$-skin’ and stopping once the walk-on-spheres hits the $\varepsilon$-skin. As the sequence $\rho$ is random, $N(\varepsilon)$ is also random. Starting with Theorem 6.6 of [@Mu] and the classical computations in [@Mo], it i
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\hat\Theta_{B}}=0$$ for $t=1,3$. Finally, we have ${\psi_{1,2}}\circ{\hat\Theta_{B}}=0$ if $v\equiv1\ppmod 4$, and ${\psi_{1,2}}\circ{\hat\Theta_{A}}=0$ if $a-v\equiv3\ppmod4$ or $v=b+3$ (where we apply Lemma \[lemma7\] in the latter case), and ${\psi_{1,2}}\circ{\hat\Theta_{A}}={\psi_{1,2}}\circ{\hat\Theta_{B}}$ if $a\equiv0\ppmod4$.
\[muladim\] $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la})=
\begin{cases}
2&(\text{if }a\equiv0\ppmod4,\ v\equiv1\ppmod4\text{ and }v\ls b+1)\\
1&(\text{otherwise}).
\end{cases}$$
By Lemma \[abnz\] and Proposition \[abhoms\] the dimension of the homomorphism space is at least that claimed. Now we show the reverse inequality, by considering linear combinations of semistandard homomorphisms.
Throughout this proof, we’ll write ${\calt[i]}$ for the set of semistandard $\mu$-tableaux of type $\la$ having exactly $i$ $2$s in the first row, for $i=0,1,2,3$, and let $\tau_i=\sum_{T\in{\calt[i]}}{\hat\Theta_{T}}$.
Suppose we have a linear combination $\theta$ of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to M^\la$ such that ${\psi_{d,t}}\circ\theta=0$ for all applicable $d,t$.
First consider ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ for $T\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$ and $d\gs3$. By Lemma \[lemma5\], ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a semistandard homomorphism (according to whether the $d$ and the $d+1$ in $T$ occur in the same row). If it is non-zero, then there is exactly one other $T'\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$ such that ${\psi_{d,1}}\circ{\hat\Theta_{T}}={\psi_{d,1}}\circ{\hat\Theta_{T'}}$, namely the tableau obtained by interchanging the $d$ and the $d+1$ in $T$. Hence ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ must occur with the same coefficient in $\theta$. Applying this for all $d\gs 3$, we find that for a fixed $i\in\{0,1,2,3\}$, all the homomorphisms ${\hat\Theta_{T}}$ for $T\in{\calt[i]}$ occur with the same coefficient in $\theta$. In other words, $\theta$ is a linear combination of $\tau_0,\tau_1,\tau_2
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r\Omega^{-2}(L\phi)^2|{\mathrm{d}}\ub\lesssim\delta\Omega_0^{-2}|u|^{-1}\mathscr{F}^2\mathcal{A}^2,\end{aligned}$$ which is the desired estimate . Using , using and , we have $$\begin{aligned}
|{\underline{h}}+1|\lesssim\int_0^\delta\left|\frac{\Omega^2(1+h{\underline{h}})}{r}\right|{\mathrm{d}}\ub\lesssim\delta(1+\mathscr{F}\mathcal{A})\cdot|u|^{-1}\lesssim\delta|u|^{-1}\mathscr{F}\mathcal{A},\end{aligned}$$ which is the desired estimate .
For $\omega$, we use the equation . The right hand side of can be estimated by, using , , and , $$\begin{aligned}
\left|\frac{\Omega^2(1+h{\underline{h}})}{r^2}-L\phi\Lb\phi\right|\lesssim|u|^{-2}\mathscr{F}\mathcal{A}+|u|^{-2}\mathscr{F}\mathcal{A}(|\psi|+\delta|u|^{-1}\mathscr{F}\mathcal{A}).\end{aligned}$$ Integrating and using , we then have $$\begin{aligned}
|\omega|\lesssim|u|^{-1}\mathscr{F}\mathscr{W}^{\frac{1}{2}}\mathcal{A},\end{aligned}$$ which is the desired estimate .
Finally, we find estimates - we have proved improve the bootstrap assumptions - if $C_0$ is sufficiently large. Therefore the estimates - hold without assuming -. Using these estimates to repeat the derivation as in , with the constant $C$ dropped, we know that also holds.
Instability theorems
====================
We then turn to the proof of the instability theorems. We divide the proof in two cases according to the behavior of $\varphi(u)$ as $u\to0^-$. The first case is the following.
\[instability1\] If $\varphi(u)$ is unbounded as $u\to0^-$, then there exists two sequences $\delta_n\to0^+$ and $u_n\to0^-$ such that holds for $\ub=\delta_n,u=u_n$.
Because $\varphi(u)$ is unbounded, we can find a sequence $u_n\to0^-$ such that $$\begin{aligned}
\varphi_n=\varphi(u_n)=\sup_{u_0\le u\le u_n}|\varphi(u)|\to\infty\ \text{as}\ n\to\infty.\end{aligned}$$ Define $\delta_n$ in terms of $u_n=-r_n$ by $$\begin{aligned}
\label{deltan}
\varphi_n^2=2^8c_1\Omega_n^4\log\frac{r_n}{4\Omega_n^2\delta_n}\end{aligned}$$ where $\Omega_n=\Omega_0(u_n)$. It is obvious that $\delta_n\to0^+$ because $\Omega_n\to0$.
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al V}_{i,j}, i=1,2,\ldots, \rho, j=0,1,\ldots, m-1\}\subseteq \mathbb{F}_q^l$ where each ${\mathcal V}_{i,j}$ is the vector corresponding to the coefficients $\widetilde{v}_{i,j} \in \mathbb{F}_{q^l}$ with respect to a $\mathbb{F}_q$-basis $\{1, \xi, \ldots, \xi^{l-1}\}$.* $\Box$
Define the Euclidean inner product of $u, v\in \mathbb{F}_q^{lm}$ by $$u\cdot v=\sum_{i=0}^{m-1}\sum_{j=0}^{l-1}u_{i,j}v_{i,j}.$$ Let $C$ be a skew QC code of length $lm$ with index $l$, $u\in C$ and $v\in C^\perp$. Since $\sigma^m=1$, we have $u\cdot T_{\sigma,l}(v)=\sum_{i=0}^{m-1}u_i\cdot \sigma(v_{i+m-1})=\sum_{i=0}^{m-1}\sigma(\sigma^{m-1}(u_i)\cdot v_{i+m-1})=\sigma(T_{\sigma,l}^{m-1}(u)\cdot v)=\sigma(0)=0$, where $i+m-1$ is taken modulo $m$. Hence $T_{\sigma, l}(v)\in C^\perp$, which implies that the dual code of skew QC code $C$ is also a skew QC code of the same index.
We define a conjugation map $^-$ on $R$ such that $\overline{ax^i}=\sigma^{-i}x^{m-i}$, for $ax^i\in R$. On $R^l$, we define the Hermitian inner product of $a(x)=(a_0(x), a_1(x), \ldots, a_{l-1}(x))$ and $b(x)=(b_0(x), b_1(x), \ldots, b_{l-1}(x))\in R^l$ by $$\langle a(x), b(x)\rangle=\sum_{i=0}^{l-1}a(x)\cdot \overline{b_i(x)}.$$ By generalizing Proposition 3.2 of [@Ling1], we get
[**Proposition 5.2**]{} *Let $u, v \in \mathbb{F}_q^{lm}$ and $u(x)$ and $v(x)$ be their polynomial representations in $R^l$, respectively. Then $T_{\sigma,l}^k(u)\cdot v=0$ for all $0\leq k \leq m-1$ if and only if $\langle u(x), v(x)\rangle=0$.* $\Box$
Let $C$ be a skew QC code of length $lm$ with index $l$ over $\mathbb{F}_q$. Then, by Theorem 5.1, $$C^\perp=\{v(x)\in R^l\mid \langle c(x), v(x)\rangle=0,~\forall c(x)\in C\}.$$ Furthermore, by Corollary 3.4 (iii), we have $C^\perp=\bigoplus_{i=1}^sC_i^\perp$.
In [@Ling3], some results for $\rho$-generator QC codes and their duals over finite fields are given. These results can also be generalized to skew $\rho$-generator QC codes over finite fields. By generalizing Corollary 6.3, Corollary 6.4 in [@Ling3] and Theorem 3.5 in this
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e broken vertical line)[]{data-label="fig2"}](f51_DOSall_Fermi_0.eps){width="\linewidth"}
![The (black curve) electronic density of states (DOS) and (orange drop lines) Inverse Participation Ratio (IPR) of the insulating model (a) and the metallized model (b). Energy axis for all datasets is shifted to have Fermi level at 0 eV (highlighted by the broken vertical line)[]{data-label="fig3"}](DOSnIPR25.eps){width="\linewidth"}
The metallized models, by construction, show a large density of states around Fermi energy (Fig. \[fig2\]) whereas the insulating models display small but well defined PBE gap of 0.41 eV and 0.54 eV for $x$=0.15 and 0.25 respectively. For disordered materials, a high DOS at $\epsilon_F$ [*alone*]{} may not produce conducting behaviour since these states can be localized (example: amorphous graphene, [@pablo]). We gauge the localization of these states by computing inverse participation ratio (IPR, [@ziman])(plotted for $x$=0.25 system in Figure \[fig3\]) and show that these states [*are*]{} indeed extended. We compute the electronic conductivity \[$\sigma(\omega)$\] using Kubo-Greenwood formula (KGF) in the following form: $$\label{eq_KGF}
\begin{aligned}
{\sigma}_{k}(\omega) = \frac{ 2 \pi e^{2} \hslash^{2}}{3 m^{2} \omega \Omega} \sum \limits_{j=1}^{N} \sum \limits_{i=1}^{N} \sum \limits_{\alpha=1}^{3}[F(\epsilon_{i},k)-F(\epsilon_{j},k)] \\
|\langle \psi_{j,k}|\bigtriangledown_{\alpha} |\psi_{i,k} \rangle|^{2} \delta(\epsilon_{j,k}-\epsilon_{i,k}-\hslash \omega)
\end{aligned}$$ It has been used with reasonable success to predict conductivity [@abtew2007; @*galli1990; @*allen1987]. Our calculations used 4 k-points to sample the Brillouin zone. To compensate for the sparseness in the DOS due to the size of the supercell, a Gaussian broadening () for the $\delta$-function is used. We note that the choice of between 0.01 eV and 0.1 eV does not significantly alter the computed values of DC conductivity \[$\sigma(\omega=0)$\] (Figure \[fig4\]). For the choice of =0.05 eV (which is small
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’\^2 + 2’\^2 = 0.
We are interested in the behavior of this extended solution in the sector $u < 0$, $v > 0$, i.e. $x < 0$. In this sector, Eqs. (\[einx1\]), (\[phix\]) and (\[einx2\]) can be integrated to (-x)\^[3/2]{}’ & = & \_x\^0 (-x)\^[1/2]{}e\^[2]{}dx,\
(-x)\^[5/4]{}’ & = & 14 \_x\^0 (-x)\^[1/4]{}’ dx,\
-x’ & = & 12 \_x\^0 ( e\^[2]{} + ’(12 - x’))dx . As long as $\rho > 0$, Eq. (\[sum1\]) (with $x < 0$, $\L < 0$) implies $\rho' < 0$, so that $\rho(x)$ decreases to 1 when $x$ increases to 0. It then follows from (\[sum2\]) that $\psi' <0$. Also, (\[sum2\]) can be integrated by parts to x’ = 14 - \_x\^0 (-x)\^[-3/4]{}dx, showing that $x\psi' < 1/4$. It then follows from (\[sum3\]) that $\s' < 0$. So, as $x$ decreases, the functions $\rho$ and $e^{2\s}$ increase and possibly go to infinity for a finite value $x = x_1$. If this is the case, the behavior of these functions near $x_1$ must be & = & \_1(1[|[x]{}]{} + 1[4x\_1]{} - + ... )\
e\^[2]{} & = & (1 + + ...)\
& = & \_1 + - (|[x]{}) + ... ($\bar{x} = x - x_1$).
These expectations are borne out by the actual numerical solution of the system x” + ’ &=& - e\^[2]{},\
-”+4’’ &=& ’\^2 +\^2’\^2, \[sist\] (this last equation comes from (\[einx4\]) where $\psi'$ is given by derivation of (\[rhospsi\]) ) where we have set $\L=-2$, with the boundary counditions $\rho(0)=1$, $\rho'(0)=-2/3$ (see eqs. (\[cmm\]) and (\[lmk\]) ), $\sigma(0)=0$. The plots of the functions $ \rho (x)$ , $\sigma (x)$ and $\psi' (x)$ are given in Figs. (4,5,6,). The value of $x_1$ is found to be approximately $-1.94$ (i.e. $\L x_1 = +3.88$).
The coordinate transformation[^3] u = Ł\^[-1]{}e\^[-|[U]{}]{}, v = e\^[|[V]{}]{} (|[U]{} = |[T]{} - |[R]{}, |[V]{} = |[T]{}+|[R]{}) leads to $x = \L^{-1} e^{2\bar{R}}$ and, on account of (\[anext\]) and (\[rhospsi\]), to the form of the metric ds\^2 = -Ł\^[-1]{}e\^[2((|[R]{})+|[R]{})]{}(d|[U]{}d|[V]{} - e\^[2(|[R]{}) -|[V]{}]{}d\^2). Near the spacelike boundary $\bar{R} = \bar{R}_1$ of the spacetime, the collapsing metr
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$$
Moreover, it is obvious by (2.2) that $\widehat{F}$ is a triangle. Thus, it has a unique inverse $\widehat{F}^{-1}$ which is also a triangle and the entries of $\widehat{F}^{-1}$ are given by$$\widehat{f}_{nk}^{-1}=\left \{
\begin{array}
[c]{cc}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}} & (0\leq k\leq n)\\
0 & (k>n)
\end{array}
\right. \tag{2.5}$$ for all $n,k\in\mathbb{N}
$. Therefore, we have by (2.4) that$$x_{n}=\sum_{k=0}^{n}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}}y_{k}\text{ ; }(n\in\mathbb{N}
). \tag{2.6}$$
In \[1\], the $\beta$-duals of the sequence spaces $\ell_{p}(\widehat{F})$ $(1\leq p<\infty)$ and $\ell_{\infty}(\widehat{F})$ have been determined and some related matrix classes characterized. Now, by taking into account that the inverse of $\widehat{F}$ is given by (2.5), we have the following lemma which is immediate by \[1, Theorem 4.6\].
Let $1\leq p\leq \infty$. If $a=(a_{k})\in \{ \ell_{p}(\widehat{F})\}^{\beta}$, then $\bar{a}=(\bar{a}_{k})\in \ell_{q}$ and we have $$\sum \limits_{k}a_{k}x_{k}=\sum \limits_{k}\bar{a}_{k}y_{k} \tag{2.7}$$ for all $x=(x_{k})\in \ell_{p}(\widehat{F})$ with $y=\widehat{F}x$, where $$\bar{a}_{k}={\displaystyle \sum \limits_{j=k}^{\infty}}
\frac{f_{j+1}^{2}}{f_{k}f_{k+1}}a_{j}\text{ ; }(k\in\mathbb{N}
). \tag{2.8}$$
Now, we prove the following results which will be needed in the sequel.
Let $1<p<\infty,$ $q=p/(p-1)$ and $\bar{a}=(\bar{a}_{k})$ be the sequence defined by (2.8) Then, we have
\(a) If $a=(a_{k})\in \{ \ell_{\infty}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{\infty}(\widehat{F})}^{\ast}=\sum_{k}\left \vert \bar
{a}_{k}\right \vert <\infty.$
\(b) If $a=(a_{k})\in \{ \ell_{1}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{1}(\widehat{F})}^{\ast}=\sup_{k}\left \vert \bar{a}_{k}\right \vert <\infty.$
\(c) If $a=(a_{k})\in \{ \ell_{p}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{p}(\widehat{F})}^{\ast}=\left( \sum_{k}\left \vert
\bar{a}_{k}\right \vert ^{q}\right) ^{1/q}<\infty.$
\(a) Let $a=(a_{k})\in \{
| 3,935
| 2,381
| 2,803
| 3,566
| null | null |
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|
\int d^3\xi \; f_{\epsilon} ({\bm \xi}) \, \phi(x(\tau, {\bm \xi}))
\ ,
\label{smearedoperator}$$ where ${\bm \xi} = (\xi^1, \xi^2, \xi^3)$ stands for the spatial coordinates associated with the local Fermi-Walker transported frame and $x(\tau, {\bm \xi})$ is a spacetime point written in terms of the Fermi-Walker coordinates. The smearing profile function $f_{\epsilon} ({\bm \xi})$ specifies the spatial size and shape of the detector in its instantaneous rest frame. In linear order perturbation theory, the detector’s transition probability is then proportional to the response function, $${\cal F}(\omega) = \int_{-\infty}^{\infty} du \, \chi(u) \int_{-\infty}^{\infty} ds \, \chi(u -s) e^{- i \omega s} \, W(u,u-s)
\ ,
\label{transprobability}$$ where $\omega$ is the transition energy, $W(\tau,\tau^\prime) = \langle \Psi | \phi(\tau) \phi(\tau^\prime) | \Psi \rangle$ and $|\Psi \rangle$ is the initial state of the scalar field. The choice for the smearing profile function $f_{\epsilon}$ made in [@schlicht] was the three-dimensional isotropic Lorentz-function, $$f_{\epsilon}({\bm \xi})= \frac{1}{\pi^2} \frac{\epsilon}{{(\xi^{2}+\epsilon^{2})}^2}
\ ,
\label{schlichtprofile}$$ where the positive parameter $\epsilon$ of dimension length characterises the effective size.
The selling point of the profile function is that it allows the switch-on and switch-off to be made instantaneous; for a strictly pointlike detector, by contrast, instantaneous switchings would produce infinities and ambiguities [@schlicht]. In particular, for a detector that is switched off at proper time $\tau$, the derivative of ${\cal F}$ with respect to $\tau$ can be understood as a transition rate, in the ‘ensemble of ensembles’ sense discussed in [@Langlois:2005if; @Louko:2007mu]. If the switch-on takes place in the infinite past, the transition rate formula becomes $${\dot {\cal F}}(\omega) = 2 \operatorname{Re}\int_0^\infty ds \,e^{-i \omega s} \, W(\tau,\tau - s)
\ .
\label{eq:transrate-schlicht}$$
When the trajectory is the R
| 3,936
| 4,063
| 3,718
| 3,502
| 3,384
| 0.772737
|
github_plus_top10pct_by_avg
|
)=\delta_{o,u}\delta_{o,v}+(1-\delta_{
o,u}\delta_{o,v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)\leq\delta_{o,u}
\delta_{o,v}+\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q{\vbu-v{|\!|\!|}}^q}.\end{aligned}$$ In addition, instead of using [(\[eq:bb1-bd\])]{}, we use $$\begin{aligned}
{\label{eq:bb1-dec}}
\sum_{b:{\underline{b}}=u}\tau_b\,Q'_{\Lambda;v}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)}
{{\vbz-u{|\!|\!|}}^q}\bigg(\delta_{z,v}\delta_{z,x}+(1-\delta_{z,x}\delta_{z,
v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z,x)+\sum_{j\ge1}P_{\Lambda;v}^{
\prime{{\scriptscriptstyle}(j)}}(z,x)\bigg){\nonumber}\\
&\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\sum_z\frac{O(
\theta_0^3)}{{\vbz-u{|\!|\!|}}^q{\vbx-z{|\!|\!|}}^{2q}{\vbv-z{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}{\nonumber}\\
&\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\frac{O(\theta_0^3)}
{{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}},\end{aligned}$$ due to [(\[eq:tildeG-bd\])]{}, [(\[eq:P’j-bd\])]{} and [(\[eq:P’0-dec\])]{}. Applying [(\[eq:P’0-dec\])]{}–[(\[eq:bb1-dec\])]{} to [(\[eq:piNbd\])]{} for $j=1$ and then using Proposition \[prp:conv-star\](ii), we end up with $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)&\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)}
{{\vbx{|\!|\!|}}^{3q}}+\sum_{u,v}\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q
{\vbu-v{|\!|\!|}}^q}\bigg(\frac{O(\theta_0)\,\delta_{v,x}}{{\vbx-u{|\!|\!|}}^q}+\frac{
O(\theta_0^3)}{{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$
To complete the proof of Proposition \[prp:GimpliesPix\], it thus remains to show [(\[eq:P’j-bd\])]{}–[(\[eq:P”j-bd\])]{}. The inequality [(\[eq:P’j-bd\])]{} for $j=1$ immediately follows from the definition [(\[eq:P’1-def\])]{} of $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (see also Figure \[fig:P-def\]) and the bound [(\[eq:psi-bd\])]{} on $\psi_\Lambda-
| 3,937
| 1,993
| 3,088
| 3,840
| null | null |
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|
We consider a Markov semigroup $P_{t}$ on ${\mathcal{S}({\mathbb{R}}^{d})}$ with infinitesimal operator $L$ and a sequence $P_{t}^{n},n\in {\mathbb{N}}$ of Markov semigroups on ${\mathcal{S}({\mathbb{R}}^{d})}$ with infinitesimal operator $L_{n}.$ We suppose that ${\mathcal{S}({\mathbb{R%
}}^{d})}$ is included in the domain of $L$ and of $L_{n}$ and we suppose that for $%
f\in {\mathcal{S}({\mathbb{R}}^{d})}$ we have $Lf\in {\mathcal{S}({\mathbb{R}}^{d})}$ and $L_{n}f\in {\mathcal{S}({\mathbb{R}}^{d})}$. We denote $\Delta _{n}=L-L_{n}.$ Moreover, we denote by $P_{t}^{\ast ,n}$ the formal adjoint of $P_{t}^{n}$ and by $\Delta _{n}^{\ast }$ the formal adjoint of $\Delta _{n}$ that is$$\left\langle P_{t}^{\ast ,n}f,g\right\rangle =\left\langle
f,P_{t}^{n}g\right\rangle \quad \mbox{and}\quad \left\langle \Delta
_{n}^{\ast }f,g\right\rangle =\left\langle f,\Delta _{n}g\right\rangle ,
\label{TR1}$$$\left\langle \cdot ,\cdot \right\rangle $ being the scalar product in $%
L^{2}({\mathbb{R}}^{d},dx).$
We present now our hypotheses. The first one concerns the speed of convergence of $L_{n}\rightarrow L.$
\[A1A\*1\] Let $a\in {\mathbb{N}}$, and let $(\varepsilon _{n})_{n\in {%
\mathbb{N}}}$ be a decreasing sequence such that $\lim_{n}\varepsilon
_{n}=0.$We assume that for every $q\in {\mathbb{N}},\kappa \geq 0$ and $p>1$ there exists $C>0$ such that for every $n$ and $f$, $$\begin{aligned}
(A_{1})& \qquad \left\Vert \Delta _{n}f\right\Vert _{q,-\kappa ,\infty }\leq
C\varepsilon _{n}\left\Vert f\right\Vert _{q+a,-\kappa ,\infty },
\label{TR3} \\
(A_{1}^{\ast })& \qquad \left\Vert \Delta _{n}^{\ast }f\right\Vert
_{q,\kappa ,p}\leq C\varepsilon _{n}\left\Vert f\right\Vert _{q+a,\kappa ,p}.
\label{TR3'}\end{aligned}$$
Our second hypothesis concerns the “propagation of regularity” for the semigroups $P_{t}^{n}$.
\[A2A\*2\] Let $\Lambda _{n}\geq 1,n\in {\mathbb{N}}$ be an increasing sequence such that $\Lambda _{n+1}\leq \gamma \Lambda _{n}$ for some $\gamma
\geq 1.$ For every $q\in {\mathbb{N}}$ and $\kappa \geq 0,p>1$
| 3,938
| 2,548
| 2,660
| 3,620
| null | null |
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|
ideals $\{J_1,\ldots, J_r\}$ is uniquely determined. In fact, this set corresponds bijectively to the set of facets of the multicomplex associated with $I$.
The statements (a) and (b) are obviously equivalent, while the existence of of the irreducible ideals $J_i$ is just the multigraded version of Proposition \[primary\].
Now we assume that the prime filtration $\mathcal F$ is pretty clean. Since $J_i$ is an irreducible monomial ideal, it follows that $J_i=\Gamma(a_i)$ for some $a_i\in\NN^n_\infty$, see Lemma \[irred\]. We claim that ${\mathcal A}=\{a_1,\ldots, a_r\}$ is the set of facets of the unique multicomplex $\Gamma$ with $I=I(\Gamma)$.
We first show that all $a_j$ are facets of $\Gamma$. Note that ${\mathcal M}(\Gamma)\subset \mathcal A$. Indeed, by Proposition \[irrdec\] we have that $$I(\Gamma)=\Sect _{a\in {\mathcal M}(\Gamma)}I(\Gamma(a))$$ is the unique irredundant decomposition of $I(\Gamma)$ into irreducible ideals. Since from any redundant such decomposition, like the decomposition $I=\Sect_{j=1}^rJ_j$, we obtain an irredundant by omitting redundant components we obtain the desired inclusion.
We also see that for each $J_j$ there exists a maximal facet $a$ of $\Gamma$ such that $I(\Gamma(a))\subset J_j$, that is, for each $a_j\in \mathcal A$ there exists a maximal facet $a$ of $\Gamma$ such that $a_j\leq a$. We claim that $\ip a_j=\ip a$, in other words, that $P_a= P_j$. In fact, since $a\in \mathcal A$ as we have just seen, there exists an integer $i$ such that $a=a_i$, and hence $I(\Gamma(a))=J_i$ is $P_i$-primary, and $P_i\subset P_j$. Suppose that $P_i\neq
P_j$. Then, since $\mathcal F$ is pretty clean, we conclude that $i>j$. It follows that $\Sect_{t>j}J_t=\Sect_{t\geq j}J_t$, contradicting (b).
Thus we have shown that all elements of $\mathcal A$ are facets of $\Gamma$. Next we prove that $r=|{\mathcal F}(\Gamma)|$. This then implies that ${\mathcal
A}={\mathcal F}(\Gamma)$, and that the elements of $\mathcal A$ are pairwise distinct.
We know from Corollary \[independent\] that $r$ e
| 3,939
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| 2,594
| 3,580
| null | null |
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|
`atacaattttgt` cation efflux family protein
`ataaaagacttt`
`gi 87160156 ref` `825` `SDSDSDSDSDSD`
`SDSDSDSDSDSD`
`SDSDSDSDSDSD`
`SRR022865_28556` `-37` `agtgtgagtgtg`
861340 SYN A:9 C:136 C:49 `gacacagacaca` clumping factor A
`ctatatctatac`
`gi 87160605 ref` `4` `FTQLSDRIKKAI`
`FTQLSDRIKK I`
`FTQLSDRIKK
| 3,940
| 3,958
| 4,407
| 3,742
| null | null |
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|
irness condition finally reads as: for any given $\varepsilon$, $$\! \bar{P}^{\cal{B}}_{ch}(m) \! = \! \int p(\alpha) P_{bind}^{\cal{B}}(m,\alpha)[1-P_{bind}^{\cal{A}}(m,\alpha)]{\mathbf d\alpha} < \varepsilon.$$ The coefficient $\alpha$ is sampled randomly by Trent to achieve stronger security requirements. This assures symmetric position of honest and cheating participant even before Trent is contacted during the Binding phase: if agents are temporarily unable to contact Trent, cheaters should not profit from this in a significant way. A client, say Bob, may be willing to take the risk and stop the protocol prematurely during the Exchange phase, provided such a situation can assure him some reasonable position. Consider a contract where Alice buys orange juice from Bob for $X$ units per litter. According to the market expectation, with probability $p$ the price should increase and with probability $(1-p)$ decrease. When the price goes up to $X'>X$, Alice wants to enforce the contract, since otherwise she should buy juice for higher price. Bob wants the contract to be canceled to sell the juice for higher price. In case the price drops, the situation is symmetric.
Bob may be willing to take the risk parameterized by $\delta$ in the following sense. The joint probability that the price drops and he will be able to enforce the contract is at least $\delta$ as well as the joint probability that price increases and Alice won’t be able to enforce the contract. The latter gives him protection from financial loses, while the former allows him to spare some money. This is formalized as $$(\exists\, 0 \! \le \! p \! \le \! 1\ \! ) \left[p(1-P_{bind}^{\cal{A}})\geq\delta\,\wedge\ (1-p)P_{bind}^{\cal{B}}\geq\delta\right].$$ Thus, to prevent reasonability of Bob’s cheating we require that $$(\forall\, 0 \!\! \le \! p \! \le \!\! 1\ \!\! ) \left[p(1-P_{bind}^{\cal{A}})\le\delta\,\vee\ (1-p)P_{bind}^{\cal{B}}\le\delta\right].$$ Let us denote $Y\stackrel{def}{=}P_{bind}^{\cal{B}}(m,\alpha)[1-P_{bind}^{\cal{A}}(m,\alpha)]
| 3,941
| 1,966
| 3,229
| 3,555
| null | null |
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|
\PSL(3,\Bbb{R})$.
Let $\Gamma$ be a subgroup of $Mob(\hat{\Bbb{R}})$ and $\gamma\in \Gamma$. Then $$\gamma=\left [\left [
\begin{array}{ll}
i & 0\\
0 & -i\\
\end{array}
\right ]\right ]
\left [\left [
\begin{array}{ll}
a & b\\
c & d\\
\end{array}
\right ]\right ]$$ where $a,b,c,d\in \Bbb{R}$ and $ad-bc=1$. A straightforward calculation shows that $$\iota\gamma=
\left [\left [
\begin{array}{lll}
-1 & 0 &0\\
0 & 1&\\
0 &0 &-1
\end{array}
\right ]\right ]
\left [
\left [
\begin{array}{lll}
a^2 & ab & b^2\\
2ac & ad+bc &2bd\\
c^2 & cd& d^2\\
\end{array}
\right ]\right ],$$ therefore $\iota\Gamma\subset\PSL(3,\Bbb{R})$.\
Let us assume that there is a real projective space $\Bbb{P}$ which is $\iota\Gamma$-invariant. Thus, as in Lemma \[l:semialg\], we conclude that $$Aut(PV)=\iota\PSL(2,\Bbb{C})\cap\{g\in \PSL(3,\Bbb{C})\vert g\Bbb{P}=\Bbb{P}\}$$ is a semi-algebraic group. Since $\Gamma\subset\iota^{-1}Aut(PV)$, we conclude that $\iota^{-1}Aut(PV)$ is a Lie group with positive dimension. From the classification of Lie subgroups of $\PSL(2,\Bbb{C})$ (see [@CS1]), we deduce that $\iota^{-1}Aut(PV)$ is either conjugate to $Mob(\hat{\Bbb{C}})$ or a subgroup of $Mob(\hat{\Bbb{R}})$. In order to conclude the proof, observe that the group $\iota^{-1}Aut(PV)$ can not be conjugated to $Mob(\hat{\Bbb{C}})$. In fact, assume on the contrary that $\iota^{-1}Aut(PV)=Mob(\hat{ \Bbb{C}})$. Since $Mob(\hat{ \Bbb{C}})$ acts transitively on $\hat{\Bbb{C}}$ we deduce that $Aut(PV)$ acts transitively on $Ver$. Finally, since $\psi(\Lambda(\Gamma))\subset Ver\cap\Bbb{P}$, we deduce $Ver\subset\Bbb{P}$, which is a contradiction.
Examples of Kleinian Groups with Infinite Lines in General Position {#s:rep}
===================================================================
Let us introduce the following projection, see [@goldman]. For each $z\in\Bbb{C}^3$ let $\eta$ be the function satisfying $\eta(z)^2=-<z,z>$ and consider the projection $\Pi:\Bbb{H}^2_{\Bbb{C}}\rightarrow\Bbb{H}^2_{\Bbb{R}}$ given by $$\Pi([z_1,z_2,z_3])=[\overline{\eta(
| 3,942
| 3,280
| 1,828
| 3,746
| null | null |
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acy classes of subgroups of $G$ such that $O^{p}(K)=_{G}J$.
Let $G$ be a finite group and $J$ be a $p$-perfect subgroup of $G$. If $p\mid \frac{|N_{G}(J)|}{|J|}$ and $p^{2}\nmid \frac{|N_{G}(J)|}{|J|}$, then there are exactly two conjugacy classes of subgroups $L$ of $G$ such that $O^{p}(L)=J$.
Let $S_{J}\leqslant N_{G}(J)$ such that $S_{J}/J$ is a Sylow $p$-subgroup of $N_{G}(J)/J$, then $O^{p}(S_{J})=J$. Conversely if $H$ is a subgroup of $G$ such $O^{p}(H)$ is conjugate to $J$, then changing $H$ by one of its conjugate one can assume that $O^{p}(H)=J$ and $H\leqslant N_{G}(J)$. Now $H/J$ is a $p$-subgroup of $N_{G}(J)/J$, so there are two possibilities: either $H=J$ or $H/J$ is a Sylow $p$-subgroup of $N_{G}(J)/J$, i-e $H$ is conjugate to $S_{J}$.
\[basis1\] Let $J$ be a $p$-perfect group. Let us denote by $\mathcal{S}_{J}$ a set of representatives of conjugacy classes of subgroups $L$ of $G$ such that $O^{p}(L)=J$. Then the set of $G/I f_{J}^{G}$ where $I\in \mathcal{S}_{J}$ and $J\in [s(G)]_{perf}$ is a basis of $RB(G)$.
Here, the proof of Deiml does not work for a general ring $R$, since there is a dimension argument. However by Lemma 5 ([@deiml]), we know that the family $\big(G/I f_{J}^{G}\big)$ is a free family, so we just need to check that it is a generating family. Let $K$ be a subgroup of $G$. It is enough to check that $G/K$ is a $R$-linear combination of elements of the form $G/I f_{J}^{G}$ where $O^{p}(I)=J$. If $|K|=1$, then $G/1 = G/1 f_{1}^{G}$. By induction on $|K|$, in $RB(G)$, we have: $$\begin{aligned}
G/K = G/K \times 1 &= \sum_{J\in [s(G)]_{perf}} G/K \times f_{J}^{G} \\
&= G/K f_{O^{p}(K)}^{G} + \sum_{O^{p}(K)\neq J\in [s(G)]_{perf}} G/K\times f_{J}^{G}.\end{aligned}$$ Now $G/K f_{J}^{G}$ is zero unless $J$ is conjugate to a subgroup of $K$. If it is the case, we have: $$\begin{aligned}
G/K f_{J}^{G} = \sum_{L\in [s(G)]\ ;\ O^{p}(L)=_{G}J} |G/K^{L}|e_{L}^{G}.\end{aligned}$$ Now $|G/K^{L}|$ is zero unless $L$ is conjugate to a subgroup of $K$. Moreover, since $O^{p}(L) = K \neq O^{p}(
| 3,943
| 2,113
| 2,785
| 3,546
| null | null |
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|
ion range to represent this pattern.
**OR nodes:** Both the top semantic-part node and latent-pattern nodes in the third layer are OR nodes. The parsing process assigns each OR node $u$ with an image region $\Lambda_{u}$ and an inference score $S_{u}$. $S_{u}$ measures the fitness between the parsed region $\Lambda_{u}$ and the sub-AOG under $u$. The computation of $\Lambda_{u}$ and $S_{u}$ for all OR nodes shares the same paradigm. $$S_{u}=\max_{v\in Child(u)}S_{v},\qquad\Lambda_{u}=\Lambda_{\hat{v}}$$ where let $u$ have $m$ children nodes $Child(u)=\{v_{1},v_{2},\ldots,v_{m}\}$. $S_{v}$ denotes the inference score of the child $v$, and $\Lambda_{v}$ is referred to as the image region assigned to $v$. The OR node selects the child with the highest score $\hat{v}={\arg\!\max}_{v\in Child(u)}S_{v}$ as the true parsing configuration. Node $\hat{v}$ propagates its image region to the parent $u$.
More specifically, we introduce detailed settings for different OR nodes.
- The OR node of the top semantic part contains a list of alternative part templates. We use $top$ to denote the top node of the semantic part. The semantic part chooses a part template to describe each input image $I$.
- The OR node of each latent pattern $u$ in the third layer naturally corresponds to a square deformation range within the feature map of a convolutional filter of a conv-layer. All neural units within the square are used as deformation candidates of the latent pattern. For simplification, we set a constant deformation range (with a center $\overline{{\bf p}}_{u}$ and a scale of $\frac{h}{3}\times\frac{w}{3}$ in the feature map where $h$ and $w$ ($h=w$) denote the height and width of the feature map) for each latent pattern. $\overline{{\bf p}}_{u}$ is a parameter that needs to be learned. Deformation ranges of different patterns in the same feature map may overlap. Given parsing configurations of children neural units as input, the latent pattern selects the child with the highest inference score as the true deformation configu
| 3,944
| 2,434
| 3,873
| 3,754
| 1,137
| 0.793782
|
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|
phonon coupling, we expect $\Im N_\sigma(-i0^+)\propto \lambda_c$, as confirmed by NRG calculations. Furthermore, the change in real part of correlation function $F_\sigma(z)$ must also depend quadratically on $\lambda_c$, because it scales with the polaron energy. Hence Eq. predicts an analytic form $1/(1+\alpha \lambda^2_c)$ for $\Gamma_{\rm eff}/\Gamma_0$, which agrees well with the data presented in Fig. \[fig:TK-Lc-b\]. Moreover, $\Gamma_{\rm eff}/\Gamma_0$ should decrease linearly with increasing $U$ for small $U$, as an expansion of Eq. in powers of $U$ shows. The inset in Fig. \[fig:TK-Lc-b\] confirms this prediction.
![Symmetric single impurity Anderson model plus unusual Holstein coupling $\lambda_c$ for a single vibrational mode $\omega_0$. $T_K$ plotted against the inverse of the renormalized hybridization. Different Coulomb interactions are indicated by different colors and different vibrational frequencies $\w_0$ by different points. For comparison we added the textbook expression for $T_K$ of the SIAM as the dashed line.[]{data-label="fig:TK-Lc-c"}](fig16-TK-Geff){width="50.00000%"}
Finally, we address the question whether the change of $T_{K}$ could also be understood by using an effective SIAM *without* explicitly including the phonons, whose effect would then be accounted for summarily by a renormalized $\Gamma_{\rm eff}$. As we will show below, the answer is no, one also needs a renormalization $U\to U_{\rm eff}$ *in the molecular orbital*. Since it is possible to reproduce any $T_{\rm K}$ with an appropriate combination of $U$ and $\Gamma$, not much understanding would be gained if both parameters were left free. Therefore, we demand that $U_{\rm eff}=Uf(x)$ depends only via a universal function $f(x)$ on the ratio $x=\Gamma_0/\Gamma_\text{eff}$. Assuming the validity of the standard expression for the Kondo temperature [@KrishWilWilson80a], the ratio of the Kondo temperatures for fixed band widths but different hybridization strengths $\Gamma_\text{eff}$ is given by $$\frac{T_K^{\rm S
| 3,945
| 1,975
| 2,806
| 3,690
| null | null |
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|
$ the vortex ring shrinks with respect to the domain $\Omega$ (cf. figure \[fig:ScalingLaws\_fixE\](d)) and the field ${\widetilde{\mathbf{u}}_{\E_0}}$ ultimately becomes axisymmetric (i.e., in this limit boundary effects vanish). At the same time, it is known that the 3D Navier-Stokes problem on an unbounded domain and with axisymmetric initial data is globally well posed [@k03], a results which is a consequence of the celebrated theorem due to @ckn83.
{width="90.00000%"}
. \[fig:dEdtE\]
![Vortex lines inside the region with the strongest vorticity in the extreme vortex state ${\widetilde{\mathbf{u}}_{\E_0}}$ with $\E_0 = 100$. The colour coding of the vortex lines is for identification purposes only.[]{data-label="fig:ring"}](Figs2/dnsNS3D_K00_E47_Instant_plane_vorLns){width="60.00000%"}
We close this section by comparing the different power laws characterizing the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ and the corresponding flow evolutions with the results obtained in analogous studies of extreme behaviour in 1D and 2D (see also Table \[tab:estimates\]). First, we note that the finite-time growth of enstrophy $\delta\E$ in 3D, cf. figure \[fig:Emax\_vsE0\_fixE\](a), exhibits the same dependence on the enstrophy $\E_0$ of the instantaneously optimal initial data as in 1D, i.e., is directly proportional to $\E_0$ in both cases [@ap11a]. This is also analogous to the maximum growth of palinstrophy $\P$ in 2D which was found by [@ap13a] to scale with the palinstrophy $\P_0$ of the initial data, when the instantaneously optimal initial condition was computed subject to [ *one*]{} constraint only (on $\P_0$). When the instantaneously optimal initial data was determined subject to [*two*]{} constraints, on $\K_0$ a
| 3,946
| 1,916
| 1,780
| 3,777
| 2,110
| 0.782673
|
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|
of the solar–metallicity models, and defer discussion of the low–mass, low–metallicity galaxies to § \[sec:lowZspecsynth\].
In figure \[fig:models\], as decreases, the line ratios decrease during the main sequence phase (because the ionizing spectrum softens), and the gap widens between the two phases of high line ratios (because fewer stars become Wolf–Rayets.) We now consider these models in light of the measured neon ratios of @thornley, which are overplotted in figure \[fig:ne-newplot\].
In the $=100$ model, for $46\%$ of the first $5$ Myr, the predicted \[\]/\[\] exceeds the highest line ratio measured by @thornley for a high–mass, $\sim$solar–metallicity galaxy; thus, this model poorly fits the data. A much better fit is the $Z=Z_{\sun}$, $=40$ model. For only $6\%$ of the first $5$ Myr does this model predict \[\]/\[\]$>1$; for $65\%$ of that time, it predicts neon line ratios within the range of the Thornley detections. The $=40$ model fits markedly better than the $=50$ and $30$ models. Because one–quarter of the Thornley datapoints are upper limits (excluding the three low–mass, low–metallicity galaxies), the M$_{up}=40$ model is a better fit to the Thornley data than the above percentages indicate.
One draws the same conclusion from continuous star formation models, as shown in figure \[fig:continuous\]. Such models with $=100$ and $75$ predict a constant neon ratio above $1$, while the neon ratio for the $=30$ model falls below the Thornley range. The $=40$ and $50$ models predict neon line ratios within the Thornley range; the $=40$ model comes closer to the median.
These results are consistent with those of § \[sec:midir-test\], which found that the heteronuclear and homonuclear mid–infrared line ratios within four regions of M82 required $<50$ to $<65$ (depending on the region), that He 2–10 required $<65$ , and that NGC 253 required $<100$ . Thus, \[\]/\[\] in the high–mass, solar–metallicity @thornley galaxies, and a concordance o
| 3,947
| 3,113
| 4,084
| 3,810
| null | null |
github_plus_top10pct_by_avg
|
}{n_b!\;m'_b!\;m''_b!}=\bigg(\sum_{{\partial}{{\bf n}}=\{
o,x\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\bigg)^3.\end{aligned}$$
It remains to show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. To do so, we use the following lemma, in which we denote by $\Omega_{z\to z'}^{{{\bf N}}}$ the set of paths on ${{\mathbb G}}_{{\bf N}}$ from $z$ to $z'$ and write $\omega\cap\omega'={\varnothing}$ to mean that $\omega$ and $\omega'$ are *edge*-disjoint (not necessarily *bond*-disjoint).
\[lmm:GHS-BK\] Given a current configuration ${{\bf N}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$, $k\ge1$, ${{\cal V}}\subset\Lambda$ and $z_i\ne z'_i\in\Lambda$ for $i=1,\dots,k$, we let $$\begin{aligned}
{\label{eq:fS-gen}}
{\mathfrak{S}}=\left\{({{\mathbb S}}_0,{{\mathbb S}}_1,\dots,{{\mathbb S}}_k):
\begin{array}{r}
{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{\raisebox{-3pt}{$\scriptstyle k$}}
{{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0={{\cal V}},\;{\partial}{{\mathbb S}}_i={\varnothing}~(i=1,\dots,k),\;\\
{{}^\exists}\omega_i\in\Omega^{{\bf N}}_{z_i\to z'_i}~(i=1,\dots,k)~
\text{\rm such that }\omega_i\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_i\\
\text{\rm and }\omega_i\cap\omega_j={\varnothing}~(i\ne j)
\end{array}\right\},\end{aligned}$$ and define ${\mathfrak{S}}'$ to be the right-hand side of [(\[eq:fS-gen\])]{} with “${\partial}{{\mathbb S}}_0={{\cal V}}$, ${\partial}{{\mathbb S}}_i={\varnothing}$” being replaced by “${\partial}{{\mathbb S}}_0={{\cal V}}{\,\triangle\,}\{z_1,z'_1\}{\,\triangle\,}\cdots{\,\triangle\,}\{z_k,z'_k\}$, ${\partial}{{\mathbb S}}_i=\{z_i,z'_i\}$”. Then, $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$.
We will prove this lemma at the end of this subsection.
Now we use Lemma \[lmm:GHS-BK\] with $k=2$ and ${{\cal V}}=\{z_1,z'_1\}=\{z_2,z'_2\}=\{o,x\}$. Note that ${\mathfrak{S}}_0$ in [(\[eq:Ssubset\])]{} is a subset of ${\mathfrak{S}}$, since ${\mathfrak{S}}$ includes partitions $({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2)$ in which there does not exist two *bond*-disjoi
| 3,948
| 2,365
| 2,753
| 3,763
| null | null |
github_plus_top10pct_by_avg
|
modular invariance. To see more clearly why this is the case we consider the decomposition of the $\bf{16}$ representation in the combinatorial notation of ref. [@xmap] $$\begin{aligned}
{\bf 16}
& \equiv &
\left[ \binom{5}{0} + \binom{5}{2} + \binom{5}{4} \right] \label{so1016}\\
& \equiv &
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{0} + \binom{2}{2} \right]
~+~
\left[ \binom{3}{1} \right]
\left[ \binom{2}{1} \right] \label{so64}\\
& \equiv &
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{0} \right]
~+~
\left[ \binom{3}{0} + \binom{3}{2} \right]
\left[ \binom{2}{2} \right]
~+~
\left[ \binom{3}{1} \right]
\left[ \binom{2}{1} \right]~~~~~
\label{su421decomposition}\end{aligned}$$ where the combinatorial factor counts the number of periodic fermions in the $\vert -\rangle$ state. The second line in eq. (\[so64\]) corresponds to the decomposition of the ${\bf16}$ under the Pati–Salam subgroup, whereas eq. (\[su421decomposition\]) shows its decomposition under the SU421 subgroup. The key point here, as seen from eq. (\[su421decomposition\]), is the even number of fermions in the $\vert -\rangle$ vacuum of the $Q_R$ states, resulting in $\pm1$ projections on the left–hand side of eq. (\[gso\]), whereas the right–hand side is fixed by the product $\beta\cdot B_j^{pqrs}= -1$ to be $\pm i$. Thus, the exclusion arises because the $\beta$ projection fixes the chirality of the vacuum of the world–sheet fermions ${\overline\psi}^{4,5}$ that generate the $SU(2)_L\times U(1)_R$ symmetry. We note that the situation here is different from the case of the SU421 models of ref. [@su421]. The reason is that our classification method only allows for symmetric boundary conditions for the set of internal fermions $\{y,\omega\vert{\overline y},{\overline\omega}\}^{1,\cdots,6}$, whereas the models of ref. [@su421] introduce additional freedom by allowing asymmetric boundary conditions. Thus, while the NAHE–based models of ref. [@su421] did not yield any model with three complete generation
| 3,949
| 2,678
| 1,086
| 3,842
| null | null |
github_plus_top10pct_by_avg
|
s of $\Bbb{P}_\Bbb{C}^2\setminus T_{\psi(x)}Ver$, thus $\psi x\in \partial B$ and $T_{\psi(x)}Ver$ is tangent to $\partial(B)$ at $x$. This concludes the proof.\
Now let us prove part (\[l:3\]). Since $\Gamma$ preserves the ball $B$, there is a Hermitian matrix $A=(a_{ij})$ with signature $(2,1)$ such that $B=\{[x]\in\Bbb{P}^2_{\Bbb{C}}:\overline{x}^t Ax<0\}$. Without loss of generality, we may assume that $[0,0,1]\notin C=\partial(B)\cap Ver$. Thus for each $x\in C$, there is a unique $z\in \Bbb{C}$ such that $x=[1,2z,z^2]=\psi [1,z]$ and $(1,2\bar z,\bar{z}^2)^tA(1, 2z,z^2)=0$. A straightforward calculation shows that $(1,2\bar z,\bar{z}^2)^tA(1, 2z,z^2)=0$ is equivalent to $$\label{e:cuadrica}
a_{11}+4Re(a_{12}z)+2Re(a_{13}z^2)+a_{33}\vert z\vert^4+4\vert z\vert^2 Re(a_{23}z)+4\vert z \vert^2a_{22}=0.$$ Taking $z=x+iy$ and $a_{ij}=b_{ij}+ic_{ij}$, Equation (\[e:cuadrica\]) can be written as $$\begin{array}{l}
a_{11}+4(b_{12}x-c_{12}y)+2(b_{13}(x^2-y^2)-2c_{13}xy)+a_{33}(x^2+y^2)^2+\\
+4(x^2+y^2)( b_{23}x-c_{23}y)+4(x^2+y^2)a_{22}=0,
\end{array}$$ which proves the assertion.\
Let us prove part (\[l:4\]). Since $\iota^{-1}Aut(BV)$ is a Lie group with positive dimension containing a non-elementary discrete subgroup, we deduce that (see [@CS1]) $\iota^{-1}Aut(BV)$ can be conjugated either to $\PSL(2,\Bbb{C})$ or a subgroup of $Mob(\hat{\Bbb{R}})$. On the other hand, we know that $\PSL(2,\Bbb{C})$ acts transitively on the Riemann sphere, but $\iota^{-1}Aut(BV)$ leaves an algebraic curve invariant, plus a point, therefore $\iota^{-1}Aut(BV)$ is conjugate to a subgroup of $Mob(\hat{\Bbb{R}})$, which concludes the proof.\
Let us prove part (\[l:5\]). We know that $C$ is $Aut(BV)$-invariant and by part (\[l:3\]) of the present lemma $\psi^{-1}C$ is an algebraic curve. Thus by Montel’s Lemma we conclude that $\Lambda_{Gr}\iota^{-1}Aut(BV)\subset \psi^{-1}C$, where $\Lambda_{Gr}\iota^{-1}Aut(BV)$ is the Greenberg limit set of $\iota^{-1}Aut(BV)$, see [@CS1]. Finally by part (\[l:3\]), we know that $ \iota^{-1}Aut(BV)$
| 3,950
| 1,822
| 2,196
| 3,648
| 2,185
| 0.781949
|
github_plus_top10pct_by_avg
|
C}}_{{\bf n}}^b(o){^{\rm c}}}=0$. As a result, $$\begin{aligned}
{\label{eq:0th-summand3}}
{(\ref{eq:0th-summand2})}~=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$
By [(\[eq:pre-1st-exp\])]{} and [(\[eq:0th-summand3\])]{}, we arrive at $$\begin{aligned}
{\label{eq:1st-exp}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in
{{\mathbb B}}_\Lambda}\pi_\Lambda^{{\scriptscriptstyle}(0)}({\underline{b}})\,\tau_b\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_\Lambda-R_\Lambda^{{\scriptscriptstyle}(1)}(x),\end{aligned}$$ where $$\begin{aligned}
{\label{eq:R1-def}}
R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_{b\in{{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}
\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\Big(
{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{
{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}\Big).\end{aligned}$$ This completes the proof of [(\[eq:Ising-lace\])]{} for $j=0$, with $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ being defined in [(\[eq:pi0-def\])]{} and [(\[eq:R1-def\])]{}, respectively.
### The second stage of the expansion {#sss:2ndexp}
In the next stage of the lace expansion, we further expand $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ in [(\[eq:1st-exp\])]{}. To do so, we investigate the difference ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}
\varphi_x \r
| 3,951
| 1,836
| 2,877
| 3,853
| null | null |
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|
tails see Ref. [@bruno_doublegama]) $$\begin{aligned}
\sigma \left( h_1 h_2 \rightarrow h_1 \otimes V_1V_2 \otimes h_2 ;s \right)
&=& \int \hat{\sigma}\left(\gamma \gamma \rightarrow V_1V_2 ;
W_{\gamma \gamma} \right ) N\left(\omega_{1},{\mathbf b_{1}} \right )
N\left(\omega_{2},{\mathbf b_{2}} \right ) S^2_{abs}({\mathbf b})
\frac{W_{\gamma \gamma}}{2} \mbox{d}^{2} {\mathbf b_{1}}
\mbox{d}^{2} {\mathbf b_{2}}
\mbox{d}W_{\gamma \gamma}
\mbox{d}Y \,\,\, .
\label{cross-sec-2}\end{aligned}$$ where $\omega_1$ and $\omega_2$ are the photon energies, $W_{\gamma \gamma} = \sqrt{4 \omega_1 \omega_2}$ is the invariant mass of the $\gamma \gamma$ system and $Y$ is the rapidity of the outgoing double meson system. Moreover, $S^2_{abs}({\mathbf b})$ is the absorption factor, given in what follows by $$\begin{aligned}
S^2_{abs}({\mathbf b}) = \Theta\left(
\left|{\mathbf b}\right| - R_{h_1} - R_{h_2}
\right ) =
\Theta\left(
\left|{\mathbf b_{1}} - {\mathbf b_{2}} \right| - R_{h_1} - R_{h_2}
\right ) \,\,,
\label{abs}\end{aligned}$$ where $R_{h_i}$ is the radius of the hadron $h_i$ ($i = 1,2$). In the dipole picture, the $\gamma \gamma \rightarrow V_1 V_2$ cross section can be expressed as follows $$\begin{aligned}
\sigma\, (\gamma \gamma \rightarrow V_1 \, V_2) = \frac{[{\cal I}m \, {\cal A}(W_{\gamma \gamma}^2,\,t=0)]^2}{16\pi\,B_{V_1 \,V_2}} \;,
\label{totalcs}\end{aligned}$$ where we have approximated the $t$-dependence of the differential cross section by an exponential with $B_{V_1 \, V_2}$ being the slope parameter. The imaginary part of the amplitude at zero momentum transfer ${\cal A}(W_{\gamma \gamma}^2,\,t=0)$ reads as $$\begin{aligned}
{\cal I}m \, {\cal A}\, (\gamma \gamma \rightarrow V_1 \, V_2) & = &
\int dz_1\, d^2{\mbox{\boldmath $r$}}_1 \,[\Psi^\gamma(z_1,\,{\mbox{\boldmath $r$}}_1)\,\, \Psi^{V_1*}(z_1,\,{\mbox{\boldmath $r$}}_1)]_T \nonumber \\
&\times & \int dz_2\, d^2{\mbox{\boldmath $r$}}_2 \,[\Psi^\gamma(z_2,\,{\mbox{\boldmath $r$}}_2)\,\, \Psi^{V_2 *}(z_2,\,{\mbox{\boldmath $r$}}_2)
| 3,952
| 3,840
| 3,497
| 3,522
| null | null |
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|
The matrix element for the $b \rar d\ell^+\ell^-$ decay coming from the most general effective Hamiltonian reads as [@Aliev4] $$\begin{aligned}
{\cal
M}&=&\,\frac{G_F\alpha}{\sqrt{2}\pi}\,V_{tb}V^*_{td}\Bigg[C_{LL}\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_L\gamma^{\mu}\ell_L+C_{LR}\bar{d}_L\gamma_{\mu}b_L
\bar{\ell}_R\gamma^{\mu}\ell_R\nonumber\\&+&C_{RL}\bar{d}_R\gamma_{\mu}b_R
\bar{\ell}_L\gamma^{\mu}\ell_L+C_{RR}\bar{d}_R\gamma_{\mu}b_R
\bar{\ell}_R\gamma^{\mu}\ell_R\nonumber\\&+&C_{LRLR}\bar{d}_Lb_R
\bar{\ell}_L\ell_R+C_{RLLR}\bar{d}_Rb_L
\bar{\ell}_L\ell_R\nonumber\\&+&C_{LRRL}\bar{d}_Lb_R
\bar{\ell}_R\ell_L+C_{RLRL}\bar{d}_Rb_L
\bar{\ell}_R\ell_L\nonumber\\&+&C_T\bar{d}\sigma_{\mu\nu}b\bar{\ell}
\sigma^{\mu\nu}\ell+iC_{TE}\bar{d}\sigma_{\mu\nu}b\bar{\ell}
\sigma_{\alpha\beta}\ell\epsilon^{\mu\nu\alpha\beta}\Bigg].\end{aligned}$$ The matrix element of the effective Hamiltonian over $\pi$ and B meson states in the $B \rar \pi\ell^+\ell^-$ decay are parametrized in terms of the form factors,and to calculate the amplitude of the $B \rar \pi\ell^+\ell^-$ decay,we need $$\begin{aligned}
\left< \pi\left|\bar{d}\gamma_{\mu}b\right|
B\right>&=&(P_{\mu}-\frac{1-\hat{m}^2_\pi}{\hat{s}}\,
q_{\mu})f_{+}+(\frac{1-\hat{m}^2_\pi}{\hat{s}})\, q_{\mu}f_{0}\,,\end{aligned}$$ with $f_+(0)=f_0(0)$, $$\begin{aligned}
\left< \pi\left|\bar{d}\sigma_{\mu\nu}b\right|
B\right>=-i(P_{\mu}q_{\nu}-P_{\nu}q_{\mu})\,\frac{f_T}{m_B+m_{\pi}}\,,\end{aligned}$$ $$\begin{aligned}
\left< \pi\left|\bar{d}i\sigma_{\mu\nu}q^{\nu}b\right|
B\right>=\Big[P_{\mu}q^2-(m^2_B-m^2_{\pi})q_{\mu}\Big]\,\frac{f_T}{m_B+m_{\pi}}\,,\end{aligned}$$ $$\begin{aligned}
\left< \pi\left|\bar{d}b\right|
B\right>=\,\frac{m_B(1-\hat{m}^2_\pi)}{\hat{m}_b}\,f_0\,,\end{aligned}$$ where $P=p_1+p_2$, $p_1$ and $p_2$ are the four momenta of the B and $\pi$ mesons, respectively.\
Using the matrix elements (5)-(8), we obtain the amplitude for $B
\rar \pi\ell^+\ell^-$ decay as: $$\begin{aligned}
{\cal M}&=&\,\frac{G_F\alpha}{\sqrt{2}\pi}\,V_{tb}V^*_{td}
\Bigg[A^{'}P_{\mu}(\bar{\ell}\
| 3,953
| 2,500
| 2,607
| 3,666
| null | null |
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|
(1,0.1)&(\ 0,\ 0)&36.0& 15.2 & 24.0 & 28.2 & 9.6 & 17.7 & 6.8 \\
&(24,\ 0) & & 23.6 & 28.5 & 30.9 & 20.1 & 24.5 & 18.7 \\
&(24,24) & & 32.7 & 33.2 & 33.5 & 31.0 & 31.4 & 32.4 \\
\hline
\end{array}
$
The simulated results for risk of $\ph_{GB}(Y|X)$ are given in Table \[tab:1\]. When the pair of eigenvalues of $\Th\Th^\top$ is $(0,\, 0)$, our simulations suggest that the risk of $\ph_{GB}(Y|X)$ decreases as $a$ increases under which $b$ is fixed or under which $a+b$ is fixed and also that the risk of $\ph_{GB}(Y|X)$ increases as $b$ increases under which $a$ is fixed. It is observed that $\ph_{GB}(Y|X)$ with $(a, b)=(1, 3)$ is superior to others.
When the pair of eigenvalues of $\Th\Th^\top$ is $(24,\, 24)$, $\ph_{GB}(Y|X)$ with $(a, b)=(-5, 3)$ or $(-5, 9)$ is best, but the improvement over $\ph_U(Y|X)$ is little. When the pair of eigenvalues of $\Th\Th^\top$ is $(24,\, 0)$, $\ph_{GB}(Y|X)$ with $(a, b)=(1, 3)$ is best.
Next, we investigate the risk of Bayesian predictive densities based on superharmonic priors when $r=2$ and $q=15$. If $\pi_s(\Th)$ is a superharmonic prior, then the Bayesian predictive density (\[eqn:BPD\]) can be expressed as $$\ph_{\pi_s}(Y|X)=\frac{\Er^{\Th|W}[\pi_s(\Th)]}{\Er^{\Th|X}[\pi_s(\Th)]}\ph_U(Y|X),$$ where $\Er^{\Th|W}$ and $\Er^{\Th|X}$ stand, respectively, for expectations with respect to $\Th|W\sim\Nc_{r\times q}(W, v_w I_r\otimes I_q)$ and $\Th|X \sim\Nc_{r\times q}(X, v_x I_r\otimes I_q)$. In our simulations, $\ph_{\pi_s}(Y|X)$ was estimated by means of $$\ph_{\pi_s}(Y|X)\approx \frac{\sum_{i=1}^{i_0}\pi_s(\Th_i)}{\sum_{i=1}^{i_0}\pi_s(\Th_i)}\ph_U(Y|X),$$ where $i_0=100,000$ and the $\Th_i$ and the $\Th_j$ are, respectively, independent replications from $\Nc_{r\times q}(W, v_w I_r\otimes I_q)$ and $\Nc_{r\times q}(X, v_x I_r\otimes I_q)$.
$
\begin{array}{cccc@{\hspace{20pt}}ccccc}
\hline
v_x & v_y & {\rm Eigenvalues} &{\rm Minimax}& $GB$ & $JS$\ & $EM$\, & $MS1$ & $MS2$ \\
&& {\rm of}\ \Th\Th^\top &{\rm risk}&&&&&\\
\hline
0.1& 1 &(\ 0,\ 0)&1.
| 3,954
| 1,811
| 1,397
| 4,015
| 3,469
| 0.77213
|
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|
(X)$ is generated by $$s_{i}=e_{i}(x_{1}+x_{1}^{-1},\dots,x_{n}+x_{n}^{-1}),\, i=1,\dots,n-1$$ and $$\Delta_{n}^{\pm}=\frac{1}{2}\bigg(\, \prod_{i=1}^{n}(x_{i}+\dfrac{1}{x_{i}})
\pm
\prod_{i=1}^{n}(x_{i}-\frac{1}{x_{i}})\,\bigg).$$ Moreover, $\Delta_{n}^{-}\in \c[s_{1},\dots,s_{p-1},\Delta^{+}]_P$, where $P$ is some polynomial in\
$\c[s_{1},\dots,s_{n-1},\Delta_{n}^{+}]$. In particular, $X/D_n$ is isomorphic to a principal open subset of $n$-dimensional affine space.
\(ii) Let $Z\subset \mathbb{T}^n$ be the variety defined by equation $\Delta=0$ and $U=X\setminus Z$. Then $U$ is an affine $D_n$-invariant subvariety of $X$ and the action of $D_n$ on $U$ is free. In particular, the projection $\pi:U \rightarrow U/D_n$ is etale.
\[proof-of-the-proposition-action-in-even-case\] Proof of $(i)$ is similar to the proof of $(i)$ in Proposition \[proposition-action-in-odd-case\]. Order the Laurent monomials lexicographically. Let $\pi=(k_{1}, \dots, k_{n})$ be a sequence of integers such that $k_{1}\geq k_{2}\geq\dots\geq |k_{n}|\geq 0$. Note that $k_{n}$ can be negative. Set $$\lambda_{\pi}=|\{g\in B_n\mid g\cdot (x_{1}^{k_{1}}\dots x_{n}^{k_{n}})
=x_{1}^{k_{1}}\dots x_{n}^{k_{n}}\}|, \, m_{\pi}=\lambda_{k}^{-1}
\sum_{g\in B_n}{}g\cdot (x_{1}^{k_{1}}\dots x_{n}^{k_{n}}).$$ Then polynomials $m_{\pi}$ form a basis of the space of $D_n$-invariant Laurent polynomials. The leading monomial in $m_{\pi}$ is $x_{1}^{k_{1}}\dots x_{n}^{k_{n}}$ and the same leading monomial has the element $$M_{k}:=s_{1}^{k_{1}-k_{2}}\dots s_{n-1}^{k_{n-1}-k_{n}}(\Delta_{n}^{sign(k_{n})})^{|k_{n}|}.$$ Then $ m_{k}-M_{k}$ has a smaller leading monomial and we can proceed by induction. Next we show how to choose $P$. Note that both $s_n=\Delta^{+}_{n}+\Delta^{-}_{n}$ and $D=\Delta^{+}_{n}\Delta^{-}_{n}$ are $D_n$-invariant and $D$ can be expressed as a polynomial in $s_{1},\dots, s_{n}$. The leading monomial in $D$ has the degree $(2,2,\dots,2,0)$ and, hence, $s_{n}$ can not enter in the expression for $D$ in the degree greater than $1$, as the de
| 3,955
| 3,098
| 2,573
| 3,653
| null | null |
github_plus_top10pct_by_avg
|
potential in presence of magnetic field, calculate the second-order QNS of deconfined QCD matter in this two scale hierarchies.
The paper is organized as follows: in Sec. \[setup\] we present the setup to calculate second-order QNS. In Subsec. \[quark\_f\], one-loop HTL free-energy of quark in presence of strong magnetic field at finite temperature and chemical potential is calculated. The gauge boson free-energy in presence of strong magnetic field is obtained in Subsec. \[gauge\_boson\]. We discuss in Subsec. \[pressure\] the anisotropic pressure and second-order QNS of QCD matter in a strong field approximation. Considering one-loop HTL pressure quark-gluon plasma in weak field approximation [@Bandyopadhyay:2017cle], we also calculate and discuss the second-order QNS in the presence of weak magnetic field in Sec. \[wfa\]. We conclude in Sec. \[conclusion\].
Setup
=====
Here we consider the deconfined QCD matter as grand canonical ensemble. The free-energy of the system can be written as F(T,V,)&=&u-Ts-n where $\mu$ is the quark chemical potential, $n$ number density and $s$ is the entropy density. The pressure of the system is given as P=-F. However, we consider the system to be anisotropic in presence of strong magnetic field and the free-energy of the system is defined in Eq. .
The second-order QNS is defined as =-\_[=0]{}=\_[=0]{}=\_[=0]{}\[chi\_def\] which is the measure of the variance or the fluctuation of the net quark number. One can find out the covariance of two conserved quantities when the quark flavors have different chemical potential. Alternatively, one can work with other basis according to the system [*[e.g.]{}*]{}, net baryon number $\mathcal B$, net charge $\mathcal Q$ and strangeness number $\mathcal S$ or $\mathcal B$, $\mathcal Q$ and third component of isospin $\mathcal I_3$. In our case we take strangeness and charge chemical potential to be zero. Moreover, we have considered same chemical potential for all flavors which results in zero off-diagonal quark number susceptibili
| 3,956
| 1,852
| 2,431
| 3,756
| null | null |
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|
($+,+,-$) ($+,-$) 1/6 2/3
($+,+,-$) ($-,+$) 1/6 -1/3
$\left( \, \textbf{4} \, , \textbf{2} , \, 0 \, \right)$ $(-,-,-)$ ($+,-$) -1/2 0
$(-,-,-)$ ($-,+$) -1/2 -1
($+,-,-$) $(-,-)$ -2/3 -2/3
$\left( \, \overline{\textbf{4}} \, , \textbf{1} , \, -1 \, \right)$ $(+,+,+)$ $(-,-)$ 0 0
($+,-,-$) $(+,+)$ 1/3 1/3
$\left( \, \overline{\textbf{4}} \, , \textbf{1} , \, +1 \, \right)$ $(+,+,+)$ $(+,+)$ 1 1
($+,-,-$) ($+,-$) -1/6 -2/3
($+,-,-$) ($-,+$) -1/6 1/3
$\left( \, \overline{\textbf{4}} \, , \textbf{2} , \, 0 \, \right)$ $(+,+,+)$ ($+,-$) 1/2 0
$(+,+,+)$ ($-,+$) 1/2 1
($+,+,-$) $(+,+)$ 2/3 2/3
$\left( \, \textbf{4} \, , \textbf{1} , \, -1 \, \right)$ $(-,-,-)$ $(+,+)$ 0 0
($+,+,-$)
| 3,957
| 6,109
| 695
| 3,162
| null | null |
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|
\gamma_\alpha(f) \\
& \qquad + \frac{1}{{\left\vert G \right\vert}}\sum_{\chi
\neq \overline{\chi}} {\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}} \sum_{a
\in G}
\chi(a) Y_a\right) - (1-p_2) \gamma_{\mathbb{C}}(f) \Biggr\vert \\
& \le p_2 \delta_n + (1-p_2) \delta_n = \delta_n,
\end{split}$$ where as above the subscripts $(n)$ are omitted. Since $p_2^{(n)}
\to p$, it follows that $\nu^{(n)} \Rightarrow (1-p) \gamma_{\mathbb{C}}+ p
\gamma_\alpha$, and so ${\mathbb{E}}\mu^{(n)} \Rightarrow (1-p) \gamma_{\mathbb{C}}+ p
\gamma_\alpha$.
Next observe that $$\label{E:var-int}
{\mathbb{E}}\bigl(\mu(f) \bigr)^2 =
\frac{1}{{\left\vert G \right\vert}^2} \sum_{\chi_1, \chi_2 \in \widehat{G}}
{\mathbb{E}}f(\lambda_{\chi_1}) f (\lambda_{\chi_2})
= \frac{1}{{\left\vert G \right\vert}^2} \sum_{\chi_1, \chi_2 \in \widehat{G}}
{\mathbb{E}}F\bigl( (\lambda_{\chi_1}, \lambda_{\chi_2}) \bigr),$$ where $F:{\mathbb{C}}^2 \to {\mathbb{R}}$ is defined by $F(w,z) = f(w)f(z)$, so that ${\left\Vert F \right\Vert}_{BL} \le 2$. We now consider $(\lambda_{\chi_1},\lambda_{\chi_2})$ as a sum of independent random vectors in ${\mathbb{R}}^4$. The upper-left and lower-right $2\times 2$ blocks of $\operatorname{Cov}\bigl((\lambda_{\chi_1}, \lambda_{\chi_2}) \bigr)$ are of course just $\operatorname{Cov}(\lambda_{\chi_1})$ and $\operatorname{Cov}(\lambda_{\chi_2})$, computed above. For the off-diagonal blocks, we use $w = \chi_1(a)
Y_a$ and $z = \chi_2(a) Y_a$ in to obtain for example $$\begin{aligned}
\sum_{a \in G} {\mathbb{E}}({\operatorname{Re}}\chi_1(a) Y_a) ({\operatorname{Re}}\chi_2(a) Y_a) &=
\sum_{a \in G} \left[\frac{1}{2}{\operatorname{Re}}{\mathbb{E}}\left( \chi_1(a) \chi_2(a)
Y_a^2 + \overline{\chi_1(a)} \chi_2(a)
{\left\vert Y_a \right\vert}^2 \right) \right] \\
&= \frac{1}{2} \left( \alpha \sum_{a \in G} \chi_1(a) \chi_2(a)
+ \sum_{a \in G} \overline{\chi_1(a)} \chi_2(a)\right) \\
&= \frac{{\left\v
| 3,958
| 5,103
| 1,976
| 3,313
| null | null |
github_plus_top10pct_by_avg
|
nitiated \[[@B9],[@B23]\], using a high protein high fibre weight loss diet (Table [1](#T1){ref-type="table"}), and fed according to manufacturer's instructions. The initial food allocation for weight loss was determined by first estimating maintenance energy requirement (MER = 440 kJ \[105 Kcal\] × body weight \[kg\]^0.75^/day) \[[@B26]\] using the estimated target weight. The exact level of restriction for each dog was then individualised based upon gender and other factors (i.e. presence of associated diseases such as osteoarthritis and other orthopaedic disorders), and was typically between 50-60% of MER at target weight \[[@B23]\]. Owners also implemented lifestyle and activity alterations to assist in weight loss. Dogs were reweighed every 7--21 days and changes were made to the dietary plan if necessary \[[@B9],[@B23]\].
######
Average analysis of the diet used for weight loss in the study dogs
**Nutrient** **Per Mcal** **Per 100 g as fed**
------------------------------ -------------- ----------------------
Kcal/kg Metabolizable Energy 2900^\*^ \-\--
Crude protein (g) 104.0 30.2
Arginine (g) 5.4 1.6
Histidine (g) 2.0 0.6
Isoleucine (g) 3.8 1.1
Met and Cys (g) 3.6 1.0
Leucine (g) 7.7 2.2
Lysine (g) 4.1 1.2
Phe and Tyr (g) 9.6 2.8
Threonine (g) 3.3 1.0
Tryptophan (g) 0.9 0.3
Valine (g) 4.4 1.3
Total fat (g) 33.0 9.6
Linoleic acid (g) 7.3 2.1
Calcium (g) 3.1 0.9
Phosphorus (g) 2.4 0.7
| 3,959
| 4,430
| 4,168
| 3,721
| null | null |
github_plus_top10pct_by_avg
|
ects and consider a single impurity in a two-dimensional BEC.
We rewrite the dGPE in dimensionless units by using the characteristic units of space and time in terms of the long-wavelength speed of sound $c=\sqrt{\mu/m}$ in the homogeneous condensate and the coherence length $\xi=\hbar/(m
c)=\hbar/\sqrt{m\mu}$. Space is rescaled as ${\boldsymbol{r}}\rightarrow
\tilde{{\boldsymbol{r}}} \xi$ and time as $t\rightarrow \tilde t \xi/c$. In addition, the wavefunction is also rescaled $\psi
\rightarrow \tilde \psi\sqrt{\mu/g}$, where $g/\mu$ is the equilibrium particle-number density corresponding to the solution with constant phase if $V_{ext},\mathcal U_p=0$. The external potential, $V_{ext}=\mu \tilde V_{ext}$, and the interaction potential, $g_p \mathcal U_p = \mu \tilde g_p
\tilde{\mathcal U_p}$, are measured in units of the chemical potential $\mu$ with $\tilde{\mathcal U_p} = 1/(2\pi a^2)
e^{-({{\boldsymbol{\tilde}} r}-{{\boldsymbol{\tilde}} r}_p)^2/(2a^2)}$, and $a =
\sigma/\xi$, $\tilde{g}_p = g_p/(\xi^2\mu)$. Henceforth, the dimensionless form of the dGPE reads as $$\begin{aligned}
\label{eq:GPe_dimless}
\tilde \partial_t \tilde\psi =
(i+\gamma)\left(\frac{1}{2}\tilde\nabla^2+1-\tilde V_{ext}-\tilde g_p\tilde{\mathcal U_p}-|\tilde\psi|^2\right)\tilde\psi .\end{aligned}$$ We use these dimensionless units and express the force (\[eq:forcegradrho\]) exerted on an impurity as ${\boldsymbol{F}}_p=
(\mu^2\xi/g) \tilde{{\boldsymbol{F}}}_p$, where $$\begin{aligned}
\tilde{{\boldsymbol{F}}}_p(t) =
-\tilde g_p\int d^2\tilde{{\boldsymbol{r}}} \tilde{\mathcal U_p}({\boldsymbol{r}}-{\boldsymbol{r}}_p)\tilde\nabla |\tilde\psi(\tilde{{\boldsymbol{r}}},t)|^2. \label{eq:fp2}\end{aligned}$$ For the rest of the paper, we will now omit the tildes over the dimensionless quantities.
In the limit of a point-like particle, $\mathcal U_p=
\delta({\boldsymbol{r}}-{\boldsymbol{r}}_p)$, the force from Eq. (\[eq:fp2\]) becomes $$\begin{aligned}
{\boldsymbol{F}}_p(t) = -g_p\nabla|\psi({\boldsymbol{r}},t)|^2 |_{{\boldsymbol{r}}={\boldsym
| 3,960
| 2,631
| 3,790
| 3,643
| 3,481
| 0.772049
|
github_plus_top10pct_by_avg
|
.5$\pm$1.8 11.9$\pm$0.8 4.9$\pm$0.3 0.8$\pm$0.1
21 05 41 40.44 -69 48 36.23 55 0.5$\pm$0.1 20.0$\pm$2.0 30.6$\pm$2.3 30.4$\pm$1.5 15.5$\pm$1.1 7.1$\pm$0.5 2.9$\pm$0.2 0.4$\pm$0.1
22 05 38 00.17 -69 42 32.98 55 0.7$\pm$0.1 19.8$\pm$2.0 31.2$\pm$1.9 32.4$\pm$1.6 16.8$\pm$1.2 7.9$\pm$0.6 3.2$\pm$0.2 0.4$\pm$0.1
23 05 40 13.59 -69 53 12.56 55 0.4$\pm$0.1 8.34$\pm$0.8 22.9$\pm$1.6 26.7$\pm$1.3 15.5$\pm$1.1 7.3$\pm$0.5 2.9$\pm$0.2 0.3$\pm$0.1
24 05 41 54.74 -69 45 30.83 55 0.4$\pm$0.1 7.3$\pm$0.7 16.1$\pm$1.8 17.9$\pm$0.9 10.5$\pm$0.8 5.0$\pm$0.4 2.1$\pm$0.2 0.5$\pm$0.1
------------ ------------- -------------- ---------- ---------------- ------------------ ------------------ ------------------ ---------------- ---------------- ---------------- -------------- --
: MIPS, PACS, SPIRE and LABOCA flux densities[]{data-label="Regions_Fluxes"}
Radio contamination is not subtracted from those values.
------------ --------- ----------------------------------------------------- ----------- ---------- ----------- ----------------------- ----------
Region T$_{c}$ M$_{dust}$ f$_{PAH}$ $\alpha$ U$_{min}$ U$_{max}$ H[i]{}/D
(K) ([$M_\odot$]{})
| 3,961
| 4,656
| 3,313
| 3,701
| null | null |
github_plus_top10pct_by_avg
|
A_{\alpha}$, it follows that $\bigcup A_{\alpha}$ is an upper bound of $\mathcal{A}$; this is also be the smallest of all such bounds because if $U$ is any other upper bound then every $A_{\alpha}$ must precede $U$ by Eq. (\[Eqn: upper bound\]) and therefore so must $\bigcup A_{\alpha}$ (because the union of a class of subsets of a set is the smallest that contain each member of the class: $A_{\alpha}\subseteq U\Rightarrow\bigcup A_{\alpha}\subseteq U$ for subsets $(A_{\alpha})$ and $U$ of $X$). Analogously, since $\bigcap A_{\alpha}$ is $\subseteq$-less than each $A_{\alpha}$ it is a lower bound of $\mathcal{A}$; that it is the greatest of all the lower bounds $L$ in $\mathcal{X}$ follows because the intersection of a class of subsets is the largest that is contained in each of the subsets: $L\subseteq A_{\alpha}\Rightarrow L\subseteq\bigcap A_{\alpha}$ for subsets $L$ and $(A_{\alpha})$ of $X$. Hence the supremum and infimum of $\mathcal{A}$ in $(\mathcal{X},\subseteq)$ given by $$A_{\leftarrow}=\sup_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcup_{A\in\mathcal{A}}A\qquad\textrm{and}\qquad_{\rightarrow}A=\inf_{(\mathcal{X},\subseteq)}\mathcal{A}=\bigcap_{A\in\mathcal{A}}A\label{Eqn: supinf3}$$ are both elements of $(\mathcal{X},\subseteq)$. Intuitively, an upper (respectively, lower) bound of $\mathcal{A}$ in $\mathcal{X}$ is any subset of $\mathcal{X}$ that contains (respectively, is contained in) every member of $\mathcal{A}$.$\qquad\blacksquare$
The statement of Zorn’s lemma and its proof can now be completed in three stages as follows. For Theorem 4.1 below that constitutes the most significant technical first stage, let $g$ be a function on $(X,\preceq)$ that assigns to every $x\in X$ an *immediate successor* $y\in X$ such that $${\textstyle \mathscr{M}(x)=\{\textrm{ }y\succ x\!:\not\exists\textrm{ }x_{*}\in X\textrm{ satisfying }x\prec x_{*}\prec y\}}$$ are all the successors of $x$ in $X$ with no element of $X$ lying strictly between $x$ and $y$. Select a representative of $\mathscr{M}(x)$ by a choice fu
| 3,962
| 4,981
| 4,080
| 3,616
| 2,350
| 0.780656
|
github_plus_top10pct_by_avg
|
tic moment in the region near the thresholds, and for fields $B \lesssim B_c$. The eigenvalues of the modes can be written approximately [@Hugo2] as $$\pi_{n,n^{\prime}}^{(i)}\approx-2\pi\phi_{n,n^{\prime}}^{(i)}/\vert\Lambda\vert
\label{eg5}$$ with $\vert\Lambda\vert=((k_\perp^{\prime 2 }-k_\perp^{\prime
\prime 2})(k_\perp^{\prime 2}-\omega^2+k_\parallel^2))^{1/2}$ with $k_\perp^{\prime\prime
2}=m_0^2[(1+2nB/B_c)^{1/2}-(1+2n^{\prime}B/B_c)^{1/2}]^2$, is the squared threshold energy for excitation between Landau levels $n,n^{\prime}$ of an electron or positron. The functions $\phi_{n,n^{\prime}}^{(i)}$ are expressed in terms of Laguerre functions of the variable $k_\perp^2/2e B$.
In the vicinity of the first resonance $n=n^{\prime}=0$ and considering $k_\perp\neq0$ and $k_\parallel\neq0$, according to [@shabad1; @shabad2] the physical eigenwaves are described by the second and third modes, but only the second mode has a singular behavior near the threshold and the function $\phi^{(2)}_{n n'}$ has the structure $$\phi_{0,0}^{(2)}\simeq-\frac{2\alpha e B m_0^2}{\pi}\textrm{exp}\left(-\frac{k_\perp^2}{2e B}\right)$$ In this case $k_{\perp}^{\prime\prime 2}=0$ and $k_{\perp}^{\prime
2}=4m_0^2$ is the threshold energy.
By using the approximation given by (\[eg5\]) the dispersion equation (\[egg\]) is turned into a cubic equation in the variable $z_1$ that can be solved by applying the Cardano formula. We will refer in the following to (\[eg2\]) as the real solution of this equation.
We should define the functions $
m_n=(k_\perp^{\prime}+k_\perp^{\prime\prime })/2$, $
m_{n^{\prime}}=(k_\perp^{\prime }-k_\perp^{\prime\prime })/2$ and $ \Lambda^{*}=4m_nm_{n^\prime} (k_\perp^{\prime 2}-k_\perp^2)$ to simplify the form of the solutions (\[eg2\]) of the equation ([\[egg\]]{}). The functions $f_{i}$ are dependent on $k_{\perp}^{2},k_{\perp}^{\prime 2},k_{\perp}^{\prime\prime 2},
B$, and are $$f_i^{(1)}=\frac{1}{3}\left[2k_\perp^{2}+k_\perp^{\prime
2}+\frac{\Lambda^{* 2}}{(k_\perp^{\prime\prime 2}-k_\perp^{\prime
2})\mathc
| 3,963
| 1,274
| 2,886
| 3,779
| null | null |
github_plus_top10pct_by_avg
|
set{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}
\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{i\ne i'}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\,\cap\,
{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)={\varnothing}\}$}}},{\label{eq:Theta''-2ndindbd3}}\end{gathered}$$ where, by conditioning on ${{\cal S}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox
{-2pt}{$\scriptstyle j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last two lines are (see below [(\[eq:lace-edges\])]{}) $$\begin{gathered}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\sum_{h=1}^j
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}
\bigg)\bigg(\prod_{i\ne i'}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}
(z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf m}}''={\partial}{{\bf k}}''={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,
{{\cal S}}_{{{\bf m}}+{{\bf
| 3,964
| 2,567
| 2,448
| 3,603
| null | null |
github_plus_top10pct_by_avg
|
erity; *p* \< 0.05). Men with CSB scored higher on the DRRI measure of relationship quality during deployment (*p* \< 0.05), as well as on the DRRI post-deployment stressors scale (*p* \< 0.05). Both findings are indicative of more negative experiences (i.e., poor relationship quality and more stressors).
######
Baseline descriptive comparison of those with and without compulsive sexual behavior
Compulsive sexual behavior
----------------------------------------------- ---------------------------- ------------- -------- -----------
*Sociodemographics*
Age 33.3 (8.2) 37.2 (15.0) --0.30 **0.016**
Race/ethnicity
White/Caucasian, non-Hispanic 150 (70.4) 21 (50.0) Ref.
Black/African-American, non-Hispanic 14 (7.0) 8 (19.1) 4.08 **0.005**
Other, non-Hispanic 15 (7.0) 4 (9.5) 1.90 0.290
Hispanic 33 (15.5) 9 (21.4) 1.95 0.132
Education level
≤high-school 52 (24.4) 12 (27.9) Ref.
\>high-school 160 (75.6) 31 (72.1) 0.84 0.641
Income (1--6) 3.1 (1.6) 3.3 (1.6) --0.13 0.321
Marital status
Married 104 (49.1) 25 (58.1) Ref.
Div
| 3,965
| 6,389
| 2,423
| 1,817
| null | null |
github_plus_top10pct_by_avg
|
$L$ and let $P_{t}^{n},n\in {%
\mathbb{N}}$, be a sequence of Markov semigroups on ${\mathcal{S}({\mathbb{R}%
}^{d})}$ with infinitesimal operators $L_{n}$, $n\in {\mathbb{N}}$. Classical results (Trotter Kato theorem, see e.g. [@EK]) assert that, if $L_{n}\rightarrow L$ then $P_{t}^{n}\rightarrow P_{t}.$ The problem that we address in this paper is the following. We suppose that $P_{t}^{n}$ has the regularity (density) property $P_{t}^{n}(x,dy)=p_{t}^{n}(x,y)dy$ with $%
p_{t}^{n}\in C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d})$ and we ask under which hypotheses this property is inherited by the limit semigroup $P_{t}.$ If we know that $p_{t}^{n}$ converges to some $p_{t}$ in a sufficiently strong sense, of course we obtain $P_{t}(x,dy)=p_{t}(x,y)dy.$ But in our framework $p_{t}^{n}$ does not converge: here, $p_{t}^{n}$ can even blow up as $n\rightarrow \infty $. However, if we may find a good equilibrium between the blow up and the speed of convergence, then we are able to conclude that $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ has some regularity properties. This is an interpolation type result.
Roughly speaking our main result is as follows. We assume that the speed of convergence is controlled in the following sense: there exists some $a\in {%
\mathbb{N}}$ such that for every $q\in {\mathbb{N}}$ $$\left\Vert (L-L_{n})f\right\Vert _{q,\infty }\leq \varepsilon _{n}\left\Vert
f\right\Vert _{q+a,\infty } \label{i1}$$Here $\left\Vert f\right\Vert _{q,\infty }$ is the norm in the standard Sobolev space $W^{q,\infty }.$ In fact we will work with weighted Sobolev spaces, and this is an important point. And also, we will assume a similar hypothesis for the adjoint $(L-L_{n})^{\ast }$ (see Assumption \[A1A\*1\] for a precise statement).
Moreover we assume a “propagation of regularity” property: there exist $%
b\in {\mathbb{N}}$ and $\Lambda _{n}\geq 1$ such that for every $q\in {%
\mathbb{N}}$$$\left\Vert P_{t}^{n}f\right\Vert _{q,\infty }\leq \Lambda _{n}\left\Vert
f\right\Vert _{q+b,\infty } \label{i2}$
| 3,966
| 2,855
| 1,183
| 3,846
| null | null |
github_plus_top10pct_by_avg
|
the $\alpha=-3$ branes, whose corresponding mixed-symmetry potentials are [@branesandwrappingrules] $$\begin{aligned}
& E_8 \ \ E_{8,2} \ \ E_{8,4} \ \ E_{9,2,1} \ \ E_{8,6} \ \ E_{9,4,1} \ \ E_{10,2,2} \ \ E_{10,4,2} \ \ E_{10,6,2} \ \ \ \ ({\rm IIB}) \nonumber \\
& E_{8,1} \ \ E_{8,3} \ \ E_{9,1,1} \ \ E_{8,5} \ \ E_{9,3,1}\ \ E_{9,5,1}\ \ E_{10,3,2} \ \ E_{10,5,2} \ \ \quad \quad \quad ({\rm IIA}) \quad , \label{allEpotentials}\end{aligned}$$ and one can recognise in the IIB list the field $E_8$ which is the S-dual of the RR field $C_8$. We find that the T-duality rule for these potentials is the following: if the potential of one theory has no indices along $a$, then $T_a$ maps it to a potential of the other theory with $a$ added on three sets of indices, while if the potential of one theory has an index $a$ only along the first set of indices, it is mapped to a potential of the other theory with $a$ added on the first and the second set of indices. Therefore, for instance the IIB potential $E_8$ with no indices along $a$ is mapped to the IIA potential $E_{9,1,1}$ where the index $a$ is present on all three sets of indices. If instead $E_8$ has one index $a$, then it is mapped to the IIA potential $E_{8,1}$ where $a$ appears on both sets of indices. The reader can appreciate that by performing repeated T-dualities one can map the S-dual of the D7-brane (which we denote as the $7_3$-brane) to all the exotic branes corresponding to the mixed-symmetry potentials in eq. .
The rule we find actually generalises to all the other branes with more-negative $\alpha$ that occur in string theory. Given a brane with $\alpha=-n$ such that in the corresponding potential the $a$ index occurs $p$ times (in $p$ different sets of antisymmetric indices), that this is mapped by T-duality along $a$ to the brane associated to the potential in which the $a$ index occurs $n-p$ times. Schematically, we write $$\alpha=-n \ : \qquad \quad \underbrace{a,a,...,a}_p \ \overset{T_a}{\longleftrightarrow} \ \underbrace{a,a, ....,a }_{
| 3,967
| 2,597
| 3,878
| 3,667
| 3,659
| 0.770865
|
github_plus_top10pct_by_avg
|
at{W}}\right) \right).
\end{array}$$ If we substitute (\[eq35\]) into the derivative of the Lyapunov function (\[eq38\]), it yields $$\begin{aligned}
\label{eq39}
{{\dot{V}}_{2}}=&-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{{s}}^{T}}{{c}_{2}}sign\left( {s} \right)-{{{s}}^{T}}{{c}_{3}}{s}+{{{s}}^{T}}{\varepsilon}\\ \nonumber
&+\varsigma \left\| {{s}} \right\|tr\left( {{{\tilde{W}}}^{T}}\left( W-\tilde{W} \right) \right).\end{aligned}$$ By the use of Cauchy-Schwarz inequality $$\label{eq40}
tr\left[ {{{\tilde{W}}}^{T}}\left( W-\tilde{W} \right) \right]\le {{\left\| {\tilde{W}} \right\|}_{F}}{{\left\| W \right\|}_{F}}-{{\left\| {\tilde{W}} \right\|}_{F}}^{2},$$ we can easily compute the inequality of the derivative of the Lyapunov function as follows, $$\begin{aligned}
\label{eq41}
{{\dot{V}}_{2}}\le &-{{{z}}_{1}}^{T}{{c}_{1}}{{{z}}_{1}}-{{{s}}^{T}}{{c}_{2}}sign\left( {s} \right)-{{{s}}^{T}}{{c}_{3}}{s}+{{{s}}^{T}}{\varepsilon}\\ \nonumber
&+\varsigma \left\| {{s}} \right\|\left( {{\left\| {\tilde{W}} \right\|}_{F}}{{\left\| W \right\|}_{F}}-{{\left\| {\tilde{W}} \right\|}_{F}}^{2} \right).\end{aligned}$$ Rearranging (\[eq41\]) by utilizing the attractor (\[eq36\]), it yields $$\label{eq42}
\begin{array}{r@{}l@{\qquad}l}
{{\dot{V}}_{2}}\le &-{{{s}}^{T}}{{c}_{2}}sign\left( {{s}} \right)-\varsigma \left\| {{s}} \right\|{{\left( {{\left\| {\tilde{W}} \right\|}_{F}}-\frac{1}{2}{{\left\| W \right\|}_{F}} \right)}^{2}}\\
&-\left\| {{s}} \right\|{{c}_{3\min }}\left\| {{s}} \right\|+{{\varepsilon }_{N}}\left\| {{s}} \right\|+\left\| {{s}} \right\|\varsigma \left( \frac{{{\left\| W \right\|}_{F}}^{2}}{4} \right)
\end{array}$$ In other words, if the sliding surface is outside the attractor, which is $$\left\| {s} \right\|>\frac{{{\varepsilon }_{N}}+\varsigma \frac{{{\left\| W \right\|}_{F}}^{2}}{4}}{{{c}_{3\min }}},$$ we then have ${{\dot{V}}_{2}}\le 0$. Therefore, the sliding surface ${s}$ is input-to-state stable.
Simulation Results {#sec_4}
==================
To demonstrate effectiveness of the proposed control law, we cond
| 3,968
| 924
| 1,979
| 4,049
| null | null |
github_plus_top10pct_by_avg
|
s frames are given by $$\widehat e_+ = \left(\begin{array}{c} \frac{hu}{a^4} ( a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}( a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad
\widehat e_- = \left(\begin{array}{c} \frac{hu}{a^4} (- a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}(- a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad
h = \frac{a^4}{1+a^4 u^4} \ .$$
The Lorentz rotation of induces a spinorial action according to given by $$\Omega = \sqrt{\frac{h}{a^4}} \left( \mathbb{I} - a^2 u^2 \Gamma^{23} \right) \ .$$ Now let us consider the duality transformation of the five-form RR flux supporting the $AdS_5 \times S^5$ geometry . The self-dual five-form flux can be written as $F_5 = (1+\star) f_5$, where $$f_5 = \frac{4}{g_0}u^3 du \wedge dt \wedge d x_1\wedge dx_2 \wedge dx_3 \equiv \frac{4}{g_0} e^u \wedge e^0 \wedge e^1 \wedge e^2 \wedge e^3 \ .$$ The corresponding poly-form of eq. is then given by $${\cal P} = 4 \Gamma^{u 0 1 2 3} - 4 \Gamma^{56789} \ .$$ The transformation of the poly-form under T-duality is given by $$\widehat{{\cal P}}= {\cal P} \cdot \Omega^{-1} = 4\sqrt{\frac{h}{a^4}} \Gamma^{u 0 1 2 3} - 4\sqrt{\frac{h}{a^4}}a^2 u^2 \Gamma^{u 0 1 } + \text{duals} \ .$$ Extracting the dual background from the above data we find $$\label{eq:mrback2}
\begin{aligned}
\widehat{ds}^2 &= \frac{du}{u^2} + u^2 \big( -dt^2 + dx_1^2 + \frac{h}{a^4} ( dx_2^2 + dx_3^2 ) \big) + d\Omega_5^2 \ , \\
\widehat{B}&= -a^{-2}\frac{h}{a^4} dx_2 \wedge dx_3 \ , \quad
\exp(2\widehat{\Phi}) = (g_0a^2)^2 \frac{h}{a^4} \ , \\
\widehat{F}_3&= -\frac{4}{g_0 a^2} a^2 u^3 du \wedge dt \wedge dx_1 \ , \quad
\widehat{F}_5 = \frac{4}{g_0 a^2} \frac{h}{a^4} u^3 (1+\star)\, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ . \\
\end{aligned}$$ Noting that $\tilde h = a^{-4}h$, we then immediately see that this is precisely the background up to the constant shift of the dilaton $g_0 \to g_0 a^{-2}$. A small subtlety is that while there is precise agreement between $H = dB$ in and , the $B$-field itself differs by a gauge: $$\widehat{B} =- a^{-2} \fr
| 3,969
| 2,927
| 3,410
| 3,444
| null | null |
github_plus_top10pct_by_avg
|
of $G$, and the class of all $p$-closed groups is a saturated formation, we deduce that ${{\operatorname}{\Phi}(G)}=1$ and that $G$ has a unique minimal normal subgroup, say $N$. Since $G/N$ has a normal Sylow $p$-subgroup, then ${{\operatorname}{O}_{p}(G)}=1$, and so $N$ is not a $p$-group. Also this implies that $PN \unlhd G$ for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$, and that $G/N$ is $p$-decomposable, by Lemma \[pclos\], as claimed. The last assertion follows from Frattini’s argument.
From now on $N$ is the unique minimal normal subgroup of $G$.
\[1\] $G=APN=BPN$, for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ .
Let $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$. Since $PN\unlhd G$, take for instance $T:=APN$. Let us suppose that $T<G$. Note that $T=A(T \cap B)$. If we take any $p$-regular element of prime power order $x\in A\cup (T\cap B)$, since $G=NN_{G}(P)$, then, by our hypotheses, there exists some $n\in N\leq T$ such that $P^n\leq {{\operatorname}{C}_{G}(x)}$, where $P\in{{\operatorname}{Syl}_{p}\left(T\right)}$. Whence $T$ satisfies the hypotheses and, by minimality, we deduce that $P \unlhd T$. But this means, by Lemma \[pclos\], that $T$, and so $N$, is $p$-decomposable. Since $N$ is not a $p$-group, we deduce that $N={{\operatorname}{O}_{p'}(N)}\leq {{\operatorname}{O}_{p'}(T)}\leq {{\operatorname}{C}_{G}(P)}$. But then $P$ is normal in $G=NN_{G}(P)$, a contradiction. Therefore, $G=APN$ and, analogously, $G=BPN$.
\[2\] Either $p\in\pi(A)$ or $p\in\pi(B)$. Moreover, if $X\in\{A, B\}$ and $p\in\pi(XN)$, then $G=XN$.
The first assertion is clear since $G=AB$ and $p \in \pi(G)$.
Without loss of generality, let us assume that $p\in\pi(AN)$. Consider $1\neq P_0\in{{\operatorname}{Syl}_{p}\left(AN\right)}$ and take $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$ with $P_0\leq P$, that is, $P_0=P\cap AN$.
Set $H:=AN$ and observe that $H=A(H \cap B)$. Note that for each $p$-regular element of prime power order $x\in A \cup(H\cap B)$, there exists some $n\in N$ wit
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cr
& \left. \qquad
+ :j^a_{L,z}(z) j^b_{L,z}(w): + ...
\right) j^c_{L,\bar z}(x) \cr
%
& = \frac{c_1 \kappa^{ab}j^c_{L,\bar z}(x)}{(z-w)^2} + \frac{c_3(c_2-g) {f^{abc}} (\bar z - \bar w)}{(z-w)^2(\bar w - \bar x)^2} + :j^a_{L,z}(z) j^b_{L,z}(w):j^c_{L,\bar z}(x) + ...\end{aligned}$$ The OPE involving the composite operator does not produces any term of order $f^{-2}$, thus we obtain : j\^c\_[L,|z]{}(x) = + + (f\^0) +... Now let us perform the same computation taking first the OPE between one $z$-component and one $\bar z$-component of the current: $$\begin{aligned}
j^a_{L,z}(z) & [j^b_{L,z}(w) j^c_{L,\bar z}(x)] \cr & =
j^a_{L,z}(z)
\left( \frac{(c_4-g) {f^{bc}}_d j^d_{L,z}(x)}{\bar w-\bar x}+ \frac{(c_2-g) {f^{bc}}_d j^d_{L,\bar z}(x)}{(w-x)}+ :j^b_{L,z}(w) j^c_{L,\bar z}(x): +... \right) \cr
%
& = \frac{c_1(c_4-g) {f^{abc}}}{(z-w)^2(\bar w - \bar x)} + j^a_{L,z}(z):j^b_{L,z}(w) j^c_{L,\bar z}(x): +... \cr
%
& = \frac{c_1(c_4-g) {f^{abc}}}{(z-w)^2(\bar w - \bar x)} + \frac{c_1 \kappa^{ab}j^c_{L,\bar z}(x)}{(z-w)^2} +...\end{aligned}$$ Thanks to the relations between the coefficients of the current algebra : c\_1(c\_4-g) = c\_3(c\_2-g) we find that the current algebra is indeed associative at the order at which we performed the computation. The coordinate dependence does not match exactly since we did not take into account the terms containing derivatives of the currents that appear in the current algebra as subleading terms. It is interesting to pursue the full proof of associativity.
The holomorphy of the stress-tensor
-----------------------------------
In this appendix we address the issue of the holomorphy of the stress-tensor[^7]: T(z) = \_[ab]{} :j\_[L,z]{}\^b j\_[L,z]{}\^a:(z). Since the $z$-component of the left-current is not holomorphic away from the WZW point, it is not obvious that the stress-tensor will be holomorphic in the quantum theory. The anti-holomorphic derivative of the stress-tensor reads: |T(z) = \_[ab]{} ( :|j\_[L,z]{}\^b j\_[L,z]{}\^a:(z) + :j\_[L,z]{}\^b |j\_[L,z]{}\^a:(z) )
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2k$ and $r(M^*) \ge r(M)$, then $M$ has a minor isomorphic to one of $M(K_n)$, $B(K_n)$ or $U_{n,2n}$.
The following conjecture, which is essentially posed in \[\[highlyconnected\]\], states that any highly vertically connected matroid omitting a given uniform minor is close to having one of three specific structures that preclude such a minor. Here a *frame matroid* is one of the form $M \del B$, where $M$ is a matroid framed by $B$; the matroid $U_{4,8}$ is not frame.
For all $s \ge 4$ there is an integer $k$ so that, if $M$ is a vertically $k$-connected matroid with no $U_{s,2s}$-minor, then there is a distance-$k$ perturbation $N$ of $M$ such that either
- $N$ or $N^*$ is a frame matroid, or
- $N$ is $\bF$-representable for some field $\bF$ over which $U_{s,2s}$ is not representable.
We assume familiarity with matroid theory, using the notation of Oxley \[\[oxley\]\].
Covering Number
===============
Let $a \ge 1$ be an integer. We write $\tau_a(M)$ for the *$a$-covering number* of a matroid $M$, defined to be the minimum number of sets of rank at most $a$ in $M$ required to cover $E(M)$. For $a \ge 2$, the parameter $\tau_{a-1}$ is a useful measure of density when excluding a rank-$a$ uniform minor; the following lemma, a strengthening of one proved in \[\[gkep\]\], finds such a minor when $\tau_{a-1}$ is large enough compared to the rank.
\[udensity\] Let $a,b$ be integers with $1 \le a < b$. If $B$ is a basis of a matroid $M$ satisfying $r(M) > a$ and $\tau_{a}(M) \ge \binom{b}{a}^{r(M)-a}$, then $M$ has a $U_{a+1,b}$-minor $U$ in which $E(U) \cap B$ is a basis.
If $r(M) = a+1$, then note that $M|B \cong U_{a+1,a+1}$; let $X \subseteq E(M)$ be maximal so that $B \subseteq X$ and $M|X$ is a rank-$(a+1)$ uniform matroid. We may assume that $|X| < b$. By maximality, every $x \in E(M) - X$ is spanned by some $a$-element subset of $X$; this also holds for every $x \in X$, so $\tau_{a}(M) \le \binom{|X|}{a} < \binom{b}{a}$, a contradiction.
Let $r(M) = r > a+1$ and suppose that the lemma holds for
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h these frames depend on $\nu_i$ the overall metric remains the round $S^5$ independent of $\nu_i$. The advantage of this basis is that the T-dualisation acts only on the $e_1$ and $e_2$ directions. We non-abelian T-dualise with respect to the central extension of $\tilde{h}_1$ and $\tilde{h}_2$ making the gauge fixing choice $$\hat{g} = e^{\frac{1}{2} \tilde{\phi}_3 \tilde{h}_3 } e^{-\frac{\xi}{2} \gamma^{13}} e^{\frac{i}{2} \alpha \gamma^1}$$ and by parametrising the Lagrange multiplier parameters as $$v_1 = - \frac{2 }{\nu_{3}} \tilde{\phi}_2 \ , \quad v_2 =\frac{2 }{\nu_{3}} \tilde{\phi}_1 \ , \quad v_3 = \frac{4}{\nu_3} \ , \quad dv_3= 0 \ .$$
After some work one finds the dual metric is exactly that of eq. with a $B$-field matching up to a gauge transformation.[^5] The dual dilaton is given by $$e^{\widehat{\Phi} - \phi_0} = \frac{\nu_3}{4 \sqrt{\lambda}} \ .$$
The frame fields produced by dualisation, using eq. , are $$\begin{aligned}
\widehat{e}^{\,\alpha} &= e^\alpha \ , \quad \widehat{e}^{\,\xi} = e^\xi \ , \quad \widehat{e}^{\,3} = e^3 \ , \\
\widehat{e}^{\,1} &\equiv \widehat{e}^{\,1}_{+} = \frac{1}{ \lambda \varphi \sqrt{\lambda-1} } \left( r_1^2 \varphi^2 d\phi_1 - r_2^2 (r_3^2 \nu_1 \nu_2 + (\lambda-1)\nu_3 ) d\phi_2 - r_3^2 (r_2^2 \nu_1 \nu_3 - (\lambda-1)\nu_2 d\phi_3 \right) \ , \\
\widehat{e}^{\,2} &\equiv \widehat{e}^{\,2}_{+} = \frac{1}{ \lambda \varphi } \left( r_1^2 \varphi^2 d\phi_1 + r_2^2 ( \nu_3 - \nu_1 \nu_2 r_3^2 ) d\phi_2 - r_3^2( \nu_2 + \nu_1 \nu_3 r_2^2 ) d\phi_3 \right) \ .
\end{aligned}$$ Following the dualisation procedure the Lorentz transformation in eq. is given by $$\Lambda = \frac{1}{\lambda} \left(\begin{array}{cc} 2-\lambda & - 2 \sqrt{\lambda - 1} \\ 2 \sqrt{\lambda -1} & 2- \lambda \end{array} \right) \ ,$$ for which the corresponding action on spinors is simply $$\Omega =\frac{1}{\sqrt{\lambda}} \mathbb{I} - \frac{\sqrt{\lambda -1 }}{\sqrt{\lambda} }\Gamma^{12} \ .$$ Then acting on the poly-form we ascertain the T-dual fluxes $$\begin{aligned}
\widehat{F}_3&= -
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[@SGA]. A [*section*]{} of this projection is a functor $\sigma:\Lambda \to {{\mathcal C}}_\#$ such that $\tau
\circ \sigma = {\operatorname{\sf id}}$ (since $\Lambda$ is small, there is no harm in requiring that two functors from $\Lambda$ to itself are equal, not just isomorphic). These sections obviously form a category which we denote by ${\operatorname{\sf Sec}}({{\mathcal C}}_\#)$. Explicitly, an object $M_\# \in
{\operatorname{\sf Sec}}({{\mathcal C}}_\#)$ is given by the following:
1. a collection of objects $M_n = M_\#([n]) \in {{\mathcal C}}^n$, and
2. a collection of transition maps $\iota_f:f_!M_{n'} \to M_n$ for any $n$, $n'$, and $f \in \Lambda([n'],[n])$,
subject to natural compatibility conditions.
A section $\sigma:\Lambda \to {{\mathcal C}}_\#$ is called cocartesian if $\sigma(f)$ is a cocartesian map for any $[n],[n'] \in \Lambda$ and $f:[n'] \to [n]$ – equivalently, a section is cocartesian if all the transition maps $\iota_f$ are isomorphisms. Cocartesian sections form a full subcategory ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$
\[cycl.str\] The category ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$ of cocartesian objects $M_\# \in
{\operatorname{\sf Sec}}({{\mathcal C}}_\#)$ is equivalent to the category of the following data:
1. an object $M = M_\#([1]) \in {{\mathcal C}}$, and
2. an isomorphism $\tau: I \times M \to M \times I$ in the category ${{\mathcal C}}^2 = {{\mathcal C}}\times {{\mathcal C}}$,
such that, if we denote by $\tau_{ij}$ the endomorphism of $I \times
I \times M \in {{\mathcal C}}^3$ obtained by applying $\tau$ to the $i$-th and $j$-th multiple, we have $\tau_{31} \circ \tau_{12} \circ \tau_{23}
= {\operatorname{\sf id}}$.
[[*Proof.*]{}]{} Straghtforward and left to the reader.
Thus the natural forgetfull functor ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#) \to {{\mathcal C}}$, $M_\# \mapsto M_\#([1])$ is faithful: an object in ${\operatorname{\sf Sec}}_{cart}({{\mathcal C}}_\#)$ is given by $M_\#([1])$ plus some extra structure on
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|
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ucted experiments in simulation environment. To simulate the DAR protocol, the two robot arms were first to track the desired trajectories to reach the object. The reference trajectories in the first 2 seconds are mathematically specified by $$\label{eq45}
\begin{array}{r@{}l@{\qquad}l}
{{x}_{a1}}(t)&={{x}_{f1}}+({{x}_{i1}}-{{x}_{f1}}){{e}^{-10{{t}^{2}}}}, \\
{{y}_{a1}}(t)&={{y}_{f1}}+({{y}_{i1}}-{{y}_{f1}}){{e}^{-10{{t}^{2}}}}, \\
{{x}_{a2}}(t)&={{x}_{f2}}+({{x}_{i2}}-{{x}_{f2}}){{e}^{-10{{t}^{2}}}}, \\
{{y}_{a2}}(t)&={{y}_{f2}}+({{y}_{i2}}-{{y}_{f2}}){{e}^{-10{{t}^{2}}}},
\end{array}$$ where $x_{a1},y_{a1},x_{a2},y_{a2}$ are the trajectories of the robot arms. $\left( {{x}_{i1}},{{y}_{i1}},{{x}_{i2}},{{y}_{i2}} \right)$ and $\left( {{x}_{f1}},{{y}_{f1}},{{x}_{f2}},{{y}_{f2}} \right)$ are the initial and final positions of the manipulators, respectively. After firmly holding the payload, the robot transports the object along the half of a circle so that it can avoid collision with an obstacle. The center of the object is expected to travel on a curve as follows, $$\begin{array}{r@{}l@{\qquad}l}
{{x}_{mr}}(t)&={{x}_{0}}+{{r}_{m}}\cos(\phi t), \\
{{y}_{mr}}(t)&={{y}_{0}}+{{r}_{m}}sin(\phi t),
\end{array}$$ where $\left( {{x}_{0}},{{y}_{0}} \right)$ is the position of the obstacle, which is also the center of the circle on which the center of the object travels. ${{r}_{m}}$ is the radius of the circle, while $\phi $ is a polar angle that varies from $-\pi$ to $0$. It is noted that the joint angles between the link and the base or its preceding link at the beginning $t=0$ were known, ${{q}_{1}}(0)=\frac{\pi }{6},\,\,{{q}_{2}}(0)=\frac{\pi }{2},\,\,{{q}_{3}}(0)=\pi$ and ${{q}_{4}}(0)=\frac{-2\pi }{3}$.
In the synthetic experiments, the physical model parameters of the DAR system were given. Furthermore, the parameters of the DSC controller were known. Those information are summarized in Table \[tab1\]. It was supposed that there is no prior knowledge of the r
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{\delta _{1},...,\delta
_{m}}(x,z)=Q_{2}^{\ast }p_{1}^{\beta ,x}(z)=\prod_{i=1}^{m-j}(S_{\delta
_{m-i+1}}^{\ast }U_{m-i}^{\ast })S_{\frac{1}{2}\delta _{j}}^{\ast
}p_{1}^{\beta ,x}(z).$$We will use (\[h’\]) $m-j$ times first and (\[B1\]) then. We denote $$q_{1}^{\prime }=q_{1}+(m-j)(a+b)$$and we write $$\begin{array}{rl}
\Vert \partial _{x}^{\beta }\tilde{p}_{\delta _{1},...,\delta _{m}}(x,\cdot
)\Vert _{q_{1},\kappa ,p} & \leq C_{q_{1}^{\prime },\kappa
,p}^{m-j}(U,S)\Vert S_{\frac{1}{2}\delta _{j}}^{\ast }p_{1}^{\beta ,x}\Vert
_{q_{1}^{\prime },\kappa ,p}\smallskip \\
& \displaystyle\leq C_{q_{1}^{\prime },\kappa ,p}^{m-j}(U,S)\,C\Big(\frac{2m%
}{\lambda t}\Big)^{\theta _{0}(q_{1}^{\prime }+\theta _{1})}\Vert
p_{1}^{\beta ,x}\Vert _{0,\nu ,1}%
\end{array}
\label{h9}$$with$$\nu =\pi (q_{1}^{\prime },\kappa +d)+\kappa +d.$$**Step 3.** We denote $g_{z}(u)=\prod_{l=1}^{d}1_{(0,\infty
)}(u_{l}-z_{l})$, so that $\delta _{0}(u-z)=\partial _{u}^{\rho }g_{z}(u)$ with $\rho =(1,2,\ldots ,d).$ We take $\mu =\nu +d+1$ and we formally write $$p_{1}(x,z)=\frac{1}{\psi _{\mu }(z)}Q_{1}(\psi _{\mu }\partial ^{\rho
}g_{z})(x).$$This formal equality can be rigorously written by using the regularization by convolution of the Dirac function.
We denote$$q_{2}^{\prime }=q_{2}+(j-1)(a+b),\quad \eta =\pi (d+q_{2}^{\prime },\mu
+d+1)+\mu$$and we write$$|p_{1}^{\beta ,x}(z)|=|\partial _{x}^{\beta }p_{1}(x,z)|\leq \frac{\psi
_{\eta }(x)}{\psi _{\mu }(z)}\Big\Vert\frac{1}{\psi _{\eta }}\partial
^{\beta }Q_{1}(\psi _{\mu }\partial ^{\rho }g_{z})\Big\Vert_{\infty }.$$Since $\mu =\nu +d+1$, $\int \psi _{\nu }\times \frac{1}{\psi _{\mu }}%
<\infty $, so using (\[NOT3c\]), we obtain (recall that $\left\vert \beta
\right\vert \leq q_{2})$ $$\begin{aligned}
\Vert p_{1}^{\beta ,x}\Vert _{0,\nu ,1}& \leq C\psi _{\eta }(x)\sup_{z\in {%
\mathbb{R}}^{d}}\big\Vert\frac{1}{\psi _{\eta }}\partial ^{\beta }Q_{1}(\psi
_{\mu }\partial ^{\rho }g_{z})\big\Vert_{\infty }\leq C\psi _{\eta
}(x)\sup_{z\in {\mathbb{R}}^{d}}\big\Vert\frac{1}{\psi _{\eta }}Q_{1}(
| 3,976
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|
x_2(x_1x_4))\\
-(x_1x_2)(x_3x_4)-(x_1x_3)(x_2x_4)-(x_3x_2)(x_1x_4)\end{gathered}$$ is the Jordan identity in a multilinear form [@Zhevl:78].
Hence using the general definition of a variety of dialgebras [@Kol:08] we obtain that the class of Jordan dialgebras is defined by two 0-identities (\[eq:0-DialgebraDef\]) and the following identities $$\label{eq:IdOfJordanDialgebras}
\begin{gathered}
x_1{\mathbin\vdash}x_2=x_2{\mathbin\dashv}x_1, \\
J(\dot{x}_1,x_2,x_3,x_4)=0,\quad J(x_1,\dot{x}_2,x_3,x_4)=0, \\
J(x_1,x_2,\dot{x}_3,x_4)=0,\quad J(x_1,x_2,x_3,\dot{x}_4)=0.
\end{gathered}$$
The variety of Jordan dialgebras is denoted ${\mathrm{Di}}{\mathrm{Jord}}$. We can express both operations in a Jordan dialgebra through one operation: $a{\mathbin\vdash}b=ab$, $a{\mathbin\dashv}b=ba$. Then an ordinary algebra arises that is a noncommutative analogue of a Jordan algebra. The corresponding variety is defined by the system of identities $$[x_1 x_2]x_3= 0, \quad (x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3), \quad
x_1(x_1^2 x_2)=x_1^2(x_1 x_2),$$ that is equivalent to identities (\[eq:IdOfJordanDialgebras\]).
Such algebras are investigated in [@Br:08; @Br:09; @GubKol:09].
Conformal algebras
------------------
The notion of a conformal algebra over a field of zero characteristic was introduced by V. G. Kac [@Kac:96] as a tool of the conformal field theory in mathematical physics. Over a field of an arbitrary characteristic, it is reasonable to use the following equivalent definition [@Kol:08]: a *conformal algebra* is a linear space $C$ endowed with a linear mapping $T\colon C\to C$ and a set of bilinear operations ($n$-products) $(\cdot{\mathbin{{}_{(n)}}}\cdot)\colon C\times C\to C$. For all $a,b\in C$ there exist just a finite number of elements $n\in \mathbb{Z}^+$ such that $a{\mathbin{{}_{(n)}}} b\not=0$ (locality property). In addition, these operations satisfy the following properties: $$\begin{gathered}
Ta{\mathbin{{}_{(n)}}}b=a{\mathbin{{}_{(n-1)}}}b,\ n\ge 1,\quad Ta{\mathbin{{}_{(0)}}}b=0,\\
T(a{\mathbin{{}_{(n)}}}b)=a{
| 3,977
| 1,355
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|
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|
symmetric KK theories. As usual we assume that the classical bubble solutions provide useful approximations to solutions in theories with moduli stabilized by additional fluxes or other objects. Ref. [@Dine:2004uw], for example, found that neutral bubble solutions persist after adding simple stabilizing potentials.
[^3]: The matter Lagrangian is normalized as $-\frac{1}{12}\int d^6x\, \sqrt{-g_6}C^2$.
[^4]: We thank Gary Shiu for bringing these solutions to our attention.
---
abstract: 'In this paper, we propose a new statistical inference method for massive data sets, which is very simple and efficient by combining divide-and-conquer method and empirical likelihood. Compared with two popular methods (the bag of little bootstrap and the subsampled double bootstrap), we make full use of data sets, and reduce the computation burden. Extensive numerical studies and real data analysis demonstrate the effectiveness and flexibility of our proposed method. Furthermore, the asymptotic property of our method is derived.'
author:
- ' Xuejun MA [^1] Shaochen WANG [^2] Wang ZHOU [^3]'
title: ' **Statistical inference in massive datasets by empirical likelihood** '
---
> [*Keywords*]{}: Bootstrap; divide-and-conquer; hypothesis test; empirical likelihood.
> [*MSC2010 subject classifications*]{}: Primary 62G10; secondary 62G05.
Introduction
============
With the rapid development of science and technologies, massive data can be collected at a large speed, especially in internet and financial fields. It is generally recognized that two major challenges in large-scale learning are estimation and inference due to large amount of computation.
For statistical inference on massive data sets, [@Kleiner2014] proposed the bag of little bootstrap (BLB) to assess the quality of estimators. However, they used only a small number of random subsets, and partial observations from each subset. This implies less efficiency in application. So, [@Sengupta2016] developed the subsampled double bootstrap (SDB) method which no
| 3,978
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the bar chart.
Pizza_bar <- ggplot(Pizza_Data_Research_Rockstar, aes(Number_of_times_eaten_pizza))
Times_eaten_pizza_7_days_bar <- Pizza_bar + geom_bar()
Times_eaten_pizza_7_days_bar
Don't know how to automatically pick scale for object of type tbl_df/tbl/data.frame. Defaulting to continuous.
The challenge becomes labeling the count of the different categories. Since I am newbie at R, I went searching for code examples but keep getting error messages.
ggplot(Pizza_Data_Research_Rockstar, aes(x= Number_of_times_eaten_pizza, y = count))+
geom_bar(stat = "identity", fill = "steelblue") +
geom_text(aes(label=count), vjust=-0.3, size=3.5) +
theme_minimal()
Don't know how to automatically pick scale for object of type
tbl_df/tbl/data.frame. Defaulting to continuous. Don't know how to
automatically pick scale for object of type function. Defaulting to
continuous. Error in (function (..., row.names = NULL, check.rows =
FALSE, check.names = TRUE, : arguments imply differing number of
rows: 46, 0
That is the closest that I have gotten to adding labels.
Could someone please help? Thank you.
Luis
A:
I really could't understand your data, so I made up my own:
Days = c("Monday", "Tuesday", "Thursday", "Friday")
Pizzas = c(40, 50, 10, 25)
Pizzadf = as.data.frame(cbind(Days, Pizzas))
Pizzadf$Pizzas = as.numeric(as.character(Pizzadf$Pizzas))
Pizzadf
ggplot(data = Pizzadf, aes(x = Days, y = Pizzas))+
geom_bar(stat = "identity", fill = "steelblue") +
geom_text(aes(label=Pizzas), vjust=-0.3, size=3.5) +
theme_minimal()
Try to modify and adjust.
Q:
Using Javascript to pull the url of a hyperlink with a class attribute
There will be only one link of a particular class on the page.
I'm trying to write a javascript snippet that will find this:
<a href="http://www.stackoverflow.com" class="linkclass"> ... </a>
and return a string consisting of this:
http://www.stackoverflow.com
Thanks in advance everyone.
A:
Try the following
var url = (function() {
var all = document.getElementsByTagName("a"
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| 0.829179
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, for some $i\in\{1,2\}$, we have that $|\cA(i)|\le |\cA(1,2)|$ then $(\cI\cup\{A\setminus \{i\}\mid A\in\cA(1,2)\}\cup\{\{1,2\}\setminus\{i\}\})\setminus\cA(i)$ is a star-subfamily of $\cH$ of size larger than $\cI$, which is a contradiction. Thus we know that $|\cA(1,2)|<\min(|\cA(1)|,|\cA(2)|)$.\
If $|\cA(1)|=|\cA(2)|=1$, then $\cA(1,2)=\emptyset$, and $(\cI\cup\{\{1,j\}:j\in C(1)\}\cup\{\{1\}\})\setminus\cA(2)$ is a star subfamily of size larger that $\cI$, a contradiction., so we may assume without loss of generality that $|\cA(1)|\ge 2$.\
For $i\in\{1,2\}$, set $\cA^\pr(i)=\{A\setminus\{i\}\mid A\in \cA(i)\}$; then $|\cA^\pr(i)|=|\cA(i)|$. Clearly $\cA^\pr(1)$ and $\cA^\pr(2)$ cross-intersect. If, for some $i\in\{1,2\}$, $\cA^\pr(i)$ is an intersecting family then $\cI\cup\cA^\pr(i)\setminus\cA(1,2)$ is an intersecting subfamily of $\cH$ that is larger that $\cI$, a contradiction, so we have that neither $\cA^\pr(1)$ nor $\cA^\pr(2)$ is intersecting. Since $\cA^\pr(1)$ is not intersecting and $|\cA^\pr(1)|\ge 2$, we may assume (by relabeling, if necessary) that $\{\{3,4\},\{5,6\}\}\subseteq\cA^\pr(1)$. Because $\cA^\pr(2)$ cross-intersects $\cA^\pr(1)$ we have $\cA^\pr(2)\subseteq\{\{3,5\},\{3,6\},\{4,5\},\{4,6\}\}$. In particular, $|\cA(2)|=|\cA^\pr(2)|\le 4$ and, for each $x\in\{3,4,5,6\}$, $\{1,x\}$ is a subset of some set in $\cA(1)$. But then $(\cI\setminus\cA(2)) \cup \{\{1,x\}\mid x\in\{3,4,5,6\}\}\cup\{\{1\}\}$ is an intersecting subfamily of $\cH$ that is larger than $\cI$, a contradiction.\
$|\cIt|=0$
----------
Here $\cIt=\emptyset$ and $\cI$ is an intersecting family of $3$-sets such that no 2-subset of $[n]$ is contained in every element of $\cI$ (otherwise that $2$-subset could be added to $\cI$).\
Let $\cS$ be the largest star in $\cI$ (clearly $|\cS|\ge 2$), and let $D$ be the head of $\cS$. If $\cS=\cI$ then we are done, so define $\cR=\cI\setminus\cS$ and assume that $\cR\ne\emptyset$. In particular, for every $R\in\cR$ we must have that $R\cap D=\emptyset$; otherwise $R$ could be added
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the correlator. By Lorentz invariance the expansion takes the schematic form ${1 \over (x-y. \bar n)^4 (x -y. n)^4}$ which scales like $\lambda^{-4}$ under rescaling of $n$. The additional factor of $\lambda$ comes from the measure $\int_{- \infty}^{\infty} d (x .n)$.
[c)]{} Momentum rescaling This follows from the fact that each detector measures energy. Equivalently, it could be seen using dilatation symmetry.
[d)]{} Permutation symmetry $n_{i} \leftrightarrow n_j$. Detectors are space-like separated and therefore commute.
[e)]{} Reality. Reality of energy correlators follows from Hermiticity of the stress-energy tensor.
More details and concrete examples of energy correlator computations can be found in . It is clear that we can think about energy correlators as correlation functions of scalar primary operators in 2d CFT with a vacuum that breaks conformal symmetry . If we choose the state to be given by a scalar operator instead of a tensor then kinematics of energy correlators is identical to that of BCFT where the role of the boundary is played by the momentum of the operator. This analogy could be useful when thinking about the small angle OPE for detector operators since instead of thinking about the light-cone OPE of light-ray operators we are dealing with the usual OPE in 2d Euclidean CFT even though the structure of spectrum in this case is different.
The one-point energy correlator originates from the three-point function of the stress tensor which itself depends on three real numbers as was reviewed above. This fact together with $\int d \Omega_{n} \la {\cal E}(n)\ra_{T.\eps.\eps(q)} = \la H \ra_{T.\eps.\eps(q)} = q^0$ where $H$ stands for the Hamiltonian operator fix the energy correlator up to two numbers The relation between $t_{2,4}$ and $n_{b,f,v}$ defined in section 2.1 can be found in appendix C of .
The two-point energy correlator is built out of the four-point function of the stress tensor. Its most general form is very complicated . Fortunately, the two-point energy correlator built
| 3,981
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n$. Then this (maximal) zero is achieved for some $T(f)$, where $f$ has at most one distinct zero in $(a,b)$.
Moreover, if the maximal zero above is achieved for some $T(f)$, where $f \in {\mathcal{M}}_d$ is $(a,b)$–rooted, then the maximal zero is also achieved for $T ((t-\epsilon/d)^d)$.
Let ${\mathcal{A}}={\mathcal{A}}(a,b,d,\epsilon)$ be the set of all $[a,b]$–rooted polynomials $f \in {\mathcal{M}}_d$ with $\tr(f)=\epsilon$. Note that continuity, compactness and Hurwitz’ theorem the maximum zero (say $\rho$) is achieved for some $T(f)$, where $f \in {\mathcal{A}}$. We argue that we may move zeros of $f$ to the boundary of $[a,b]$, while retaining $\tr(f)$ and the maximal zero of $T(f)$ as long as $f$ has at least two distinct zeros in $(a,b)$.
Suppose $a<\alpha<\beta<b$ are two zeros of $f\in {\mathcal{A}}$ and that the maximal zero is realized for $T(f)$. For $0<|s| \leq \min(b-\beta, \alpha-a, \beta-\alpha)$, let $$f_s(x) := \frac {(x-\alpha-s)(x-\beta+s)}{(x-\alpha)(x-\beta)} f(x),$$ and note that $f_s \in {\mathcal{A}}$ and $$f= (1-\theta)f_{s}+ \theta f_{-s}, \quad \mbox{ where } \quad \theta = \frac 1 2 \left(1- \frac s {\beta-\alpha}\right)\in [0,1].$$ By assumption $T(f_s)(\rho) \geq 0$. Since $0=T(f)(\rho)= (1-\theta)T(f_{s})(\rho)+ \theta T(f_{-s})(\rho)$, we conclude that $T(f_s)(\rho) = T(f_{-s})(\rho)=0$. Hence the maximal zero $\rho$ is realized also for $T(f_s)$ where $s= -\min(b-\beta, \alpha-a, \beta-\alpha)$. By possible iterating this process a few times we will have moved at least one interior zero to the boundary. We can continue until there is at most one distinct zero in $(a,b)$.
Suppose the maximal zero $\rho$ above is achieved for some $f \in {\mathcal{M}}_d$ which is $(a,b)$–rooted. Then $\rho$ is also attained for the same problem when we replace $[a,b]$ by $[r,s]$ where $a<r<s<b$ and $r-a$ and $b-s$ are sufficiently small. Hence, by what we have just proved, for each such $r,s$ there are nonnegative integers $i,j$ with $i+j \leq d$ such that $$\label{abs}
T\left( (t-r)
| 3,982
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tandard pairs of $I$ as defined by Sturmfels, Trung and Vogel in [@STV]. However the main justification of the definition is given by Proposition \[multiprimary\] where we show that a pretty clean filtration of $S/I$ determines uniquely the facets of $\Gamma$. This result finally leads us to the definition of shellable multicomplexes. In Proposition \[extend\] we show that our definition of shellable multicomplexes extends the corresponding notion known for simplicial complexes. However the main result of the final section is Theorem \[multi2\] which asserts that for a monomial ideal $I$ the ring $S/I$ is multigraded pretty clean if and only if the corresponding multicomplex is shellable.
The dimension filtration
========================
Let $M$ be an $R$-module of dimension $d$. In [@Sc] Schenzel introduced the [*dimension filtration*]{} $${\mathcal F}\: 0\subset D_0(M)\subset D_1(M)\subset \cdots \subset D_{d-1}(M)\subset D_d(M)=M$$ of $M$, which is defined by the property that $D_i(M)$ is the largest submodule of $M$ with $\dim D_i(M)\leq i$ for $i=0,\ldots,d$. It is convenient to set $D_{-1}(M)=(0)$.
For all $i$ we set $\Ass^i(M)=\{P\in\Ass(M)\: \dim R/P=i\}$. The following characterization of a dimension filtration will be useful for us:
\[characterization\] Let ${\mathcal F}\: 0\subset M_0\subset M_1\subset \cdots \subset M_{d-1}\subset M_d=M$ be a filtration of $M$. The following conditions are equivalent:
1. $\Ass(M_i/M_{i-1})=\Ass^{i}(M)$ for all $i$;
2. ${\mathcal F}$ is the dimension filtration of $M$.
That the dimension filtration satisfies condition (a) has been shown by Schenzel in [@Sc Corollary 2.3 (c)].
For the converse we show that if ${\mathcal F}$ satisfies condition (a), then it is uniquely determined. Since the dimension filtration satisfies this condition, it follows then that ${\mathcal F}$ must be the dimension filtration of $M$.
The integers $i$ for which $M_i=M_{i-1}$ are exactly those for which $\Ass^{i}(M)=\emptyset$, and hence this set is uniquely determined.
Thus it rema
| 3,983
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| 1,880
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M_1 \oplus (\bigoplus_{i\geq 2} M_i)$$ and $C(L^{j})$ is spanned by $$(\langle \pi e_i\rangle, e) ~(resp.~ (\langle \pi e_i\rangle, \pi a, e)) \textit{~and~} M_1 \oplus (\bigoplus_{i\geq 2} M_i).$$\
We now construct a morphism $\psi_j : \tilde{G} \rightarrow \mathbb{Z}/2\mathbb{Z}$ as follows. (There are 3 cases.)\
(1) Firstly, we assume that $M_0$ is *of type* $\textit{I}^e$. We choose a Jordan splitting for the hermitian lattice $(C(L^j), \xi^{-m}h)$ as follows: $$C(L^j)=\bigoplus_{i \geq 1} M_i^{\prime},$$ where $$M_1^{\prime}=(\pi)a\oplus Be\oplus M_1, ~~~ M_2^{\prime}=(\oplus_i(\pi)e_i)\oplus M_2, ~~~ \mathrm{and}~ M_k^{\prime}=M_k \mathrm{~if~} k\geq 3.$$ Here, $M_i^{\prime}$ is $\pi^i$-modular and $(\pi)$ is the ideal of $B$ generated by a uniformizer $\pi$. Notice that $M_2^{\prime}$ is *of type II*, since both $\oplus_i(\pi)e_i$ and $M_2$ are *of type II*, and that both $M_3^{\prime}$ and $M_4^{\prime}$ are *of type II* as well. The lattice $M_1^{\prime}$ is *of type I*, since the Gram matrix associated to $(\pi)a\oplus Be$ is $\begin{pmatrix} -2\delta&\pi\\-\pi&2b\end{pmatrix}$ with $\delta (\in A) \equiv 1 \mathrm{~mod~}2$. Thus $M_1^{\prime}$ is *free of type I* since the adjacent two lattices $M_0^{\prime}$ (which is empty) and $M_2^{\prime}$ are *of type II*.
Then consider the sublattice $Y(C(L^j))$ of $C(L^j)$ and choose a Jordan splitting for the hermitian lattice $(Y(C(L^j)), \xi^{-(m+1)}h)$ as follows: $$Y(C(L^j))=\bigoplus_{i \geq 0} M_i^{\prime\prime},$$ where $M_i^{\prime\prime}$ is $\pi^i$-modular. We explain the above Jordan splitting precisely. Since $C(L^j)=\bigoplus_{i \geq 1} M_i^{\prime}$ and $M_1'$ is *free of type I*, $Y(C(L^j))=Y(M_1')\oplus \bigoplus_{i \geq 2} M_i^{\prime}$.
1. If $b\in (2)$, then $Y(M_1')= (2)a\oplus Be\oplus \pi M_1$. The lattice $(2)a\oplus Be$ is $\pi^2$-modular *of type II* and $\pi M_1$ is $\pi^3$-modular *of type II*. Since we rescale $Y(C(L^j))$ by $\xi^{-1}$, we have that $$M_0''=\left((2)a\oplus Be\right)\oplus M_2'=\left((2)a\oplus Be\right)\oplus
| 3,984
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)\bar{\tau}^{(1)}_\sigma
+\bar{\tau}^{(2)}_\sigma
\nonumber\end{aligned}$$ The purely fermionic Green’s function $G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)$ picks up the factor $(1+\lambda^{\rm tip} x_0)^2$ which therefore appears as a prefactor in the elastic density of states and in the corresponding tunnel current. Remembering that our theory is accurate to quadratic order in $\lambda^{\rm tip}$, we may add corrections of order $O([\lambda^{\rm tip}]^3)$ and higher to the right hand side of Eq. . Since $\bar{\tau}^{1}_\sigma$ and $\bar{\tau}^{(2)}_\sigma$ are of orders $(\lambda^{\rm tip})$and $(\lambda^{\rm tip})^2$, respectively, we can thus write $$\begin{aligned}
\label{basis change density of states 2}
\tau^{(0)}_\sigma + \tau^{(1)}_\sigma + \tau^{(2)}_\sigma
&\simeq&(1 + \lambda^{\rm tip} x_0)^2 (\bar{\tau}^{(0)}_\sigma + \bar{\tau}^{(1)}_\sigma +\bar{\tau}^{(2)}_\sigma)\end{aligned}$$ after adding the corresponding higher-order correction terms to the prefactors of $\bar{\tau}^{(1)}_\sigma $ and $\bar{\tau}^{(2)}_\sigma$. Therefore, a finite displacement $x_0$ generates an overall prefactor $(1+\lambda x_0)^2$ in the total tunnel current. This can be absorbed into the tunneling matrix element $t^2_{\mu \sigma}\to \bar t^2_{\mu \sigma}= t^2_{\mu \sigma}( 1+\lambda x_0)^2 \approx [t_{\mu \sigma}\exp(\lambda x_0)]^2$, leading to an identical total tunnel current for the two bases $\hat{X}$ and $\hat{\bar{X}}$, up to $O([\lambda^{\rm tip}]^3)$ corrections.
However, while the total current is invariant under the basis change of the harmonic oscillator, the attribution of elastic and inelastic contributions remains basis-dependent, which becomes immediately obvious from the Eqs. and : the inelastic current in the original oscillator basis contains an elastic part with respect to the shifted oscillator basis.
We adopt the following strategy in order to ensure that all properties are discussed in the framework of a harmonic oscillator basis with vanishing displacements in the presence of the
| 3,985
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els, where spacetime gauge symmetries typically appear as worldsheet global symmetries visible in the UV, but is not contradicted by any physics we know. In any event, spectrum computations at Landau-Ginzburg points in these theories have not proven insightful.
Class III: Twisted bundles {#sect:type3:twisted}
==========================
The third fundamental class of examples we shall discuss involve cases in which the trivially-acting part of the gauge group acts nontrivially on the bundle, but is not one of the special cases discussed in section \[sect:het-gsomods\] in which the effect is merely to recreate part of the ten-dimensional left-moving GSO projection.
One reason for interest is that examples of this form have the potential to define new heterotic string compactifications. Other reasons also exist, revolving around making sense of heterotic orbifolds with invariant non-equivariant bundles. We review such motivations in subsection \[sect:class3-motivations\].
In subsection \[sect:otherexs-good\], we describe some (indirect) constructions of (0,2) SCFT’s of this form, via dimensional reduction of consistent four-dimensional theories, and via anomaly-free (0,2) GLSM’s.
Unfortunately, although there seem to exist consistent (0,2) SCFT’s, they do not seem to yield consistent perturbative heterotic string compactifications. The essential problem is that any finite group that acts only on left-movers, locally looks like a modification of the ten-dimensional left-moving GSO projection, and as the consistent ten-dimensional GSO projections are already known, if it is not one of them, the results cannot be well-behaved. We will give several examples of six-dimensional compactifications of this form, in which the six dimensional theory has anomalies and cannot be consistent. We outline in detail some examples in which heterotic string compactifications on these (0,2) SCFT’s break down in subsections \[sect:class3-caution1\], \[sect:class3-caution2\], and \[sect:class3-caution4\].
That said, it m
| 3,986
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$\overline{\Sigma}_L^{\mu\nu}\ :\ \
{\left(\overline{\Sigma}_L^{\mu\nu}\right)^{\dot{\alpha}}}_{\dot{\beta}}=
\frac{1}{4}i\left[
\overline{\sigma}_L^{\mu\dot{\alpha}\gamma}\,
{\sigma^\nu}_{\gamma\dot{\beta}}\,-\,
\overline{\sigma}^{\nu\dot{\alpha}\gamma}\,
{\sigma^\mu}_{\gamma\dot{\beta}}\right]\ .$$
Given these different considerations, it should not come as a surprise that once a free quantum field theory dynamics is constructed, it turns out that such fundamental spinor representations of the Lorentz group describe quanta which are massive or massless particles whose spin or helicity is 1/2.
Extending the above considerations to an arbitrary representation of the Dirac-Clifford algebra, any Dirac spinor may be decomposed into its chiral components, $$\psi=\psi_L+\psi_R\ \ ,\ \
\psi_L=P_L\,\psi=\frac{1}{2}\left(1-\gamma_5\right)\psi\ \ ,\ \
\psi_R=P_R\,\psi=\frac{1}{2}\left(1+\gamma_5\right)\psi\ .$$ The SL(2,$\mathbb C$) invariant tensors that enable the raising and lowering of dotted and undotted indices provide for a transformation which, given a Dirac spinor $\psi$ and its complex conjugate, constructs another Dirac spinor also transforming according to the correct rules under Lorentz transformations. This operation, known as charge conjugation since it exchanges the roles played by particles and their antiparticles, is represented through a matrix $C$ such that $$C\gamma^\mu C^{-1}=-{\gamma^\mu}^{\rm T}\ \ ,\ \
C=i\gamma^2\gamma^0\ \ ,\ \
C^\dagger=C^{\rm T}=-C\ \ ,\ \ C^2=-\one\ ,$$ where, except for the very first identity, the last series of properties is valid, for instance, in the Dirac and chiral representations of the $\gamma^\mu$ matrices, but not necessarily in just any other representation of the Dirac-Clifford algebra. The charge conjugate Dirac spinor $\psi_C$ associated to a given Dirac spinor $\psi$ is given by, $$\psi_C=C\overline{\psi}^{\rm T}\ \ ,\ \
\overline{\psi}=\psi^\dagger\,\gamma^0\ ,$$ up to an arbitrary phase factor. Consequently, a Majorana spinor $\psi$ obeys the Majoran
| 3,987
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==================
In this section, we consider a class of hierarchical priors inspired by Faith (1978) and derive a sufficient condition for minimaxity of the resulting Bayesian predictive density. Also, a proper Bayes and minimax predictive density is provided.
A class of hierarchical prior distributions
-------------------------------------------
Let $\Sc_r$ be the set of $r\times r$ symmetric matrices. For $A$ and $B\in\Sc_r$, write $A\prec(\preceq) B$ or $B\succ(\succeq) A$ if $B-A$ is a positive (semi-)definite matrix. The set $\Rc_r$ is defined as $$\Rc_r=\{ \La\in\Sc_r \mid 0_{r\times r}\prec \La \prec I_r\},$$ where $0_{r\times r}$ is the $r\times r$ zero matrix. Denote the boundary of $\Rc_r$ by $\partial\Rc_r$. It is noted that if $\Om\in\partial\Rc_r$ then $0_{r\times r}\preceq \Om \preceq I_r$ and also then $|\Om|=0$ or $|I_r-\Om|=0$.
Consider a proper/improper hierarchical prior $$\pi_H(\Th)=\int_{\Rc_r}\pi_1(\Th|\Om)\pi_2(\Om)\dd\Om.$$ The priors $\pi_1(\Th|\Om)$ and $\pi_2(\Om)$ are specified as follows: Assume that a prior distribution of $\Th$ given $\Om$ is $\Nc_{r\times q}(0_{r\times q},v_0\Om^{-1}(I_r-\Om)\otimes I_q)$, where $v_0$ is a known constant satisfying $$v_0\geq v_x.$$ Then the first-stage prior density $\pi_1(\Th|\Om)$ can be written as $$\label{eqn:pr_Th}
\pi_1(\Th|\Om)=(2\pi v_0)^{-qr/2}|\Om(I_r-\Om)^{-1}|^{q/2}\exp\Big[-\frac{1}{2v_0}\tr\{\Om(I_r-\Om)^{-1}\Th\Th^\top\}\Big].$$ Assume also that $\pi_2(\Om)$, a second-stage prior density for $\Om$, is a differentiable function on $\Rc_r$.
Denote by $\ph_H=\ph_H(Y|X)$ the resulting Bayesian predictive density with respect to the hierarchical prior $\pi_H(\Th)$. Assume that a marginal density of $W$ with respect to $\pi_H(\Th)$ is finite when $v=v_x$. The marginal density is given by $$\begin{aligned}
\label{eqn:m(W)}
m(W)&=\int_{\Re^{r\times q}} p(W|\Th)\pi_H(\Th)\dd\Th \non\\
&=\int_{\Rc_r}\int_{\Re^{r\times q}} \pi(\Th|\Om,W) \dd\Th \pi_2(\Om)\dd\Om,\end{aligned}$$ where $\pi(\Th|\Om,W)=p(W|\Th)\pi_1(\Th|\Om)$ is a posterior
| 3,988
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1\in\tilde{{\cal D}}$ and ${{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$, this chain of bubbles starts and ends with ${{\bf m}}$-connected clusters (possibly with a single ${{\bf m}}$-connected cluster), not with ${{\bf k}}$-connected clusters. Therefore, by following the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, we can easily show $$\begin{aligned}
{\label{eq:Theta'-2ndindbd3:j=1bd}}
{(\ref{eq:Theta'-2ndindbd3:j=1})}\leq\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z_1,z'_1).\end{aligned}$$
For $j\ge2$, since ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for $i=1,\dots,j$ are mutually-disjoint due to the last product of the indicators in [(\[eq:Theta’-2ndindbd3\])]{}, we can treat each bypath separately by the conditioning-on-clusters argument. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last line in the rightmost expression of [(\[eq:Theta’-2ndindbd3\])]{} equals $$\begin{gathered}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=
2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,l\ge
2\\ i\ne l}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}}
\bigg){\nonumber}\\
\times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+
{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,\frac{
w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}
| 3,989
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\delta$. To prove [(\[eq:P”j-bd\])]{} for $j=1$, we first recall the definition [(\[eq:P”1-def\])]{} of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (and Figure \[fig:P-def\]). Note that, by [(\[eq:GGpsi-bd\])]{}, $\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)$ obeys the same bound on $\sum_{v'}G(v'-y)\,G(z-v')$ (with a different $O(1)$ term). That is, the effect of an additional $\psi_\Lambda$ is not significant. Therefore, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ is identical, with a possible modification of the $O(1)$ multiple, to the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (or $P_{\Lambda;v}^{\prime{\scriptscriptstyle}(1)}$) with $v$ (resp., $u$) “being embedded” in one of the bubbles consisting of $\psi_\Lambda-\delta$. By [(\[eq:psi-bd\])]{}, $\psi_\Lambda(y,x)-\delta_{y,x}$ with $v$ being embedded in one of its bubbles is bounded as $$\begin{aligned}
{\label{eq:psipsi-bd}}
&\sum_{k=1}^\infty\sum_{l=1}^k\sum_{y',x'}\big(\tilde G_\Lambda^2
\big)^{*(l-1)}(y,y')\,\tilde G_\Lambda(y',x')\Big({{\langle \varphi_{y'}
\varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde G_\Lambda(y',x')\,
\delta_{v,x'}\Big)\big(\tilde G_\Lambda^2\big)^{*(k-l)}
(x',x){\nonumber}\\
&=\sum_{y',x'}\psi_\Lambda(y,y')\,\tilde G_\Lambda(y',x')\Big(
{{\langle \varphi_{y'}\varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde
G_\Lambda(y',x')\,\delta_{v,x'}\Big)\psi_\Lambda(x',x){\nonumber}\\
&\leq\sum_{y',x'}\frac{O(1)}{{\vby'-y{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)}
{{\vbx'-y'{|\!|\!|}}^q}\,\frac{O(\theta_0)}{{\vbv-y'{|\!|\!|}}^q{\vbx'-v{|\!|\!|}}^q}\,
\frac{O(1)}{{\vbx-x'{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)}
{{\vbx-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ By this observation and using [(\[eq:IR-xbd\])]{} to bound the remaining two two-point functions consisting of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (recall [(\[eq:P”1-def\])]{}), we obtain [(\[eq:P”j-bd\])]{} for $j=1$.
For [(\[eq:P’j-bd\])]{}–[(\[eq:P�
| 3,990
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-------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------
![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box1 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box2 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box3 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box4 "fig:")
(f) (g) (h) (i)
------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------
The technical details of the evaluation of the Feynman diagrams for the ball, triangle and box diagrams are given in the App. \[sec:balls\], \[sec:triangles\], and \[sec:boxs\] respectively. The main technique used
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times \R^{d}$$$\left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq \frac{C}{t^{\theta _{0}(1+\frac{a+b}{\delta }%
)(\left\vert \alpha \right\vert +\left\vert \beta \right\vert
+2d+\varepsilon )}}\times \frac{(1+\left\vert x\right\vert
^{2})^{\pi(\kappa) }}{(1+\left\vert x-y\right\vert
^{2})^{\kappa }}. \label{i3'}$$This is the transfer of regularity that we mention in the title and which is stated in Theorem \[TransferBIS-new\]. The proof is based on a criterion of regularity for probability measures given in [@[BC]], which is close to interpolation spaces techniques.
A second result concerns a perturbation of the semigroup $P_{t}$ by adding a compound Poisson process: we prove that if $P_{t}$ verifies (\[i2\]) and (\[i3\]) then the perturbed semigroup still verifies (\[i3\]) – see Theorem \[J\]. A similar perturbation problem is discussed in [@[Z]] (but the arguments there are quite different).
The regularity criterion presented in this paper is tailored in order to handle the following example (which will be treated in a forthcoming paper). We consider the integro-differential operator $$Lf(x)=\<b(x),\nabla f(x)\>+\int_{E}\big(f(x+c(z,x))-f(x)-\<c(z,x),\nabla f(x)\>%
\big)d\mu (z) \label{i5}$$where $\mu $ is an infinite measure on the normed space $(E,\left\vert \circ
\right\vert _{E})$ such that $\int_{E}1\wedge \left\vert c(z,x)\right\vert
^{2}d\mu (z)<\infty .$ Moreover, we consider a sequence $\varepsilon
_{n}\downarrow \emptyset ,$ we denote$$A_{n}^{i,j}(x)=\int_{\{\left\vert z\right\vert _{E}\leq \varepsilon
_{n}\}}c^{i}(z,x)c^{j}(z,x)d\mu (z)$$and we define$$\begin{array}{rl}
L_{n}f(x)= & \displaystyle\<b(x),\nabla f(x)\>+\int_{\{\left\vert
z\right\vert _{E}\geq \varepsilon _{n}\}}(f(x+c(z,x))-f(x)-\<c(z,x),\nabla
f(x)\>)d\mu (z)\smallskip \\
& \displaystyle+\frac{1}{2}\mathrm{tr}(A_{n}(x)\nabla ^{2}f(x)).%
\end{array}
\label{i6}$$By Taylor’s formula, $$\left\Vert Lf-L_{n}f\right\Vert _{\infty }\leq \left\Vert f\right\Vert
_{3,\infty }\varepsilon _{n}\quad \mbox{w
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can define a non-strict $S$-set $X$, $(s,x)\mapsto s\cdot x$, where defined, to be [*connected*]{} if for any $x\in X$ and any $e,f\in E$ such that $e\cdot x$ and $f\cdot x$ are defined, there is a sequence of idempotents $e=e_1,e_2,\dots, e_k=f$, called a [*connecting sequence over*]{} $x$, such that $e_i\cdot x$ is defined and $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$.
[*If $X$ is an $S$-set (that is, given by a homomomorphism), it is connected with $e,ef,f$ being a connecting sequence between $e$ and $f$ over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.* ]{}
[*If $S$ is a monoid, any non-strict $S$-set is connected with $e,1,f$ being a connecting sequence between $e$ and $f$, again over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.*]{}
It is not true that every non-strict $S$-set is connected, as the following example shows.
\[ex:ce\]
*Let $S=\{e,f,g\}$ be a three-element semilattice, given by the following Hasse diagram:*
\(e) [$e$]{}; (aux) \[node distance=0.6cm, right of=e\] ; (f) \[node distance=1.2cm, right of=e\] [$f$]{}; (g) \[node distance=1cm, below of=aux\] [$g$]{}; (e) edge node\[above\] (g) (f) edge node\[above\] (g);
Let $X=\{1,2\}$ and define the domains of action of $e$ and $f$ to be equal $\{1,2\}$, and the domain of action of $g$ to be equal $\{1\}$ (that is, $e$ and $f$ act by the identity map on $\{1,2\}$, and $g$ by the identity map on $\{1\}$). Thus $X$ becomes a non-strict $S$-set. It is however not connected, as both $e\cdot 2$ and $f\cdot 2$ are defined but there is no connecting sequence between $e$ and $f$ over $2$ as $g\cdot 2$ is undefined.
In view of Lemma \[lem:lem3\], it follows that the non-strict $S$-set from Example \[ex:ce\] can not be equal $\Psi(F)$ for any torsion-free functor $F$ on $L(S)$.
We now describe the correspondence between morphisms of non-strict $S$-sets and natural transformations of torsion-free functors on $L(S)$. Assume we are given non strict $S$-sets $(X,\mu)$, $(s,x)\mapsto s\cdot x$, where defined, an
| 3,993
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}}
\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}},~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\text{ (in }{{\cal B}}{^{\rm c}})\}$}}}{\nonumber}\\
&\leq\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\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}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\,
\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)},\end{aligned}$$ where $$\begin{aligned}
{\label{eq:Psi-def}}
\Psi_{y,z,v;{{\cal A}},{{\cal B}}}=\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=
y{\vartriangle}z}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ We note that, by ignoring the indicator in [(\[eq:Psi-def\])]{}, we have $0\leq\Psi_{y,z,v;{{\cal A}},{{\cal B}}}\leq{{\langle \varphi_y\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}$, which is zero whenever $z\in{{\cal B}}$. Therefore, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b\text{ even}\}}}$ in [(\[eq:EE’E”decpre3\])]{} to obtain $$\begin{aligned}
{\label{eq:EE'E''dec3}}
\Theta_{y,x;{{\cal
| 3,994
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| null | null |
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psi_{|\Gamma}$, which belongs to $T^2(\Gamma)=T^2(\Gamma_{e})$. Let $\tilde\Psi\in \tilde W^2(G_e\times S\times I)$ given in Remark \[wla\], Part C. Define ${{{\mathcal{}}}E}\psi$ by \[cg5\] [E]{}:=
& [on]{} GSI\
\_()=\_[e,]{}() & [on]{} \_\
& [on]{} G\_eSI
Then ${{{\mathcal{}}}E}\psi$ is in $W^2({\mathbb{R}}^3\times S\times I)$. This follows from the Green’s formula (\[green\]) since for all $v\in C_0^\infty({\mathbb{R}}^3\times S\times I^\circ)$ &\_[\^3SI]{}([E]{}) (\_x v) dx ddE\
=& \_[GSI]{}([E]{}) (\_x v) dx ddE+ \_[G\_eSI]{}([E]{}) (\_x v) dx ddE\
=& -\_[GSI]{} (\_x ) v dx ddE + \_[GSI]{}() (v) ()dddE\
& - \_[G\_eSI]{} (\_x ) v dx ddE + \_[G\_eSI]{}\_e() \_e(v) (\_e)dddE\
=& -\_[GSI]{} (\_x ) v dx ddE -\_[G\_eSI]{} (\_x ) v dx ddE where we used the facts that $\partial G=\partial G_e$ and $\nu_e=-\nu$ and so $\gamma_{\pm}(\psi)=:g_{\pm}=g_{e,\mp}:=\gamma_{e,\mp}(\tilde\Psi)$. Hence $\omega\cdot\nabla_x ({{{\mathcal{}}}E}\psi)\in L^2({\mathbb{R}}^3\times S\times I)$, as desired.
Finally, we find that by (\[trpr15\]) (recall Remark \[wla\], Part C.) &\_[W\^2(\^3SI)]{} =\_[W\^2(GSI)]{}+ \_[W\^2(G\_eSI)]{}\
& \_[W\^2(GSI)]{}+[L\_[e,-]{}(g\_[e,-]{})]{}\_[W\^2(G\_eSI)]{}+ [L\_[e,+]{}(g\_[e,+]{})]{}\_[W\^2(G\_eSI)]{}\
=& \_[W\^2(GSI)]{}+ ([\_+()]{}\_[T\^2(\_[e,-]{})]{}+ [\_-()]{}\_[T\^2(\_[e,+]{})]{}) which implies the boundedness of ${{{\mathcal{}}}E}$. This completes the proof.
Let $\tilde {\bf W}^2(G\times S\times I)$ be the completion of $C^1(\ol G\times S\times I)$ with respect to ${\left\Vert \cdot\right\Vert}_{\tilde { W}^2(G\times S\times I)}$-norm. For a convex set $G\subset{\mathbb{R}}^3$ have the following density result.
\[convg\] Suppose that $G\subset{\mathbb{R}}^3$ is as above and that it is *convex*. Then \[cg1\] \^2(GSI)=\^2(GSI), and \[cg2\] H\_2= \^2(GSI)W\_1\^2(GSI). (Definitions of the spaces $W_1^2$, $\tilde{W}^2$ and $H_2$ were given in , and , respectively.)
At first, we deal with the claim (\[cg1\]). The inclusion $"\subset"$ is clear and then it suffices to prove only
| 3,995
| 1,191
| 3,093
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follow the usual conventions of nonstandard analysis. ${I \kern -4.5pt N}= \{0,1,2,\ldots\}$ is the set of natural numbers, and $^{*}\! {I \kern -4.5pt N}$ is the set of hypernaturals. The standard hypernaturals are (i.e., can be identified with) the natural numbers. Also, $\langle a_{n}\rangle$ or $\langle a_{n}\!: n\in{I \kern -4.5pt N}\rangle$ or $\langle a_{0},a_{1},
a_{2},\ldots\rangle$ denotes a sequence whose elements can be members of any set, such as the set $X$ of nodes in a conventional graph $G=\{X,B\}$, where $B$ is the set of branches, a branch being a two-element set of nodes. On the other hand, $[a_{n}]$ denotes an equivalence class of sequences, where two sequences $\langle a_{n}\rangle$ and $\langle b_{n}\rangle$ are taken to be equivalent if $\{n\!:a_{n}=b_{n}\}\in {\cal F}$, where ${\cal F}$ is any chosen and fixed free ultrafilter.[^1] The $a_{n}$ appearing in $[a_{n}]$ are understood to be the elements of any one of the sequences in the equivalence class. At times, we will use the more specific notation $[\langle a_{0},a_{1},a_{2},\ldots\rangle]$. More generally, we adhere to the notations and terminology appearing in [@go].
The ordinals are denoted in the usual way: $\omega$ is the first transfinite ordinal. With $\tau\in{I \kern -4.5pt N}$, the product $\omega\cdot \tau$ is the sum of $\tau$ summands, each being $\omega$.
The Nonstandard Enlargement of a Graph
======================================
Throughout Sections 2 to 4, we assume that the conventionally infinite graph $G$ is connected and has infinitely many nodes. The definition of a nonstandard graph that we use herein is given in [@gn Section 8.1], a special case of which is the “enlargement” of a graph $G$.
Let us define the [*enlargement*]{} $^{*}\!G$ of $G$ here as well in order to remove any need for referring to [@gn]. $G=\{X,B\}$ is now taken to be a conventional connected graph having an infinite set $X$ of nodes and therefore an infinite set of branches as well, each branch being a two-element set of nodes. Thus,
| 3,996
| 5,228
| 3,784
| 3,702
| 1,922
| 0.784347
|
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{L}})$ given by thinking of an automorphism of $\oplus_n P$ as an automorphism of $\oplus_{n+1} P$ by taking the direct sum with the identity map on the last factor.
Now for any principal bundle $G \to P \to M$, consider the Atiyah-Bott equivalence [@atiyahbott] mentioned in the proof of Theorem \[main\]: $$\beta : B{\mathcal{G}}(P) \simeq Map_P(M, BG).$$ This equivalence is given by the observation that the equivariant mapping space $Map^G(P, EG)$ is a contractible space with a free action of the gauge group ${\mathcal{G}}(P) = Aut^G_M(P)$ given by precomposition. The orbit space of this action is then $Map_P (M, BG)$. The examples relevant here are the principal bundles $GL_n(R) \to \oplus_n P_{\mathcal{L}}\to M$. In this case we have an equivalence $$\beta_n: B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) {\xrightarrow}{\simeq} Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR).$$ With respect to these equivalences, the inclusions $j_n : B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \to B{\mathcal{G}}(\oplus_{n+1} P_{\mathcal{L}})$ are given up to homotopy by maps $$q_n : Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR) \to Map_{\oplus_{n+1}{\mathcal{L}}}(M, BGL_{n+1}R)$$ defined by sending $\phi : M \to BGL_n(R)$ to the composition $$q_n(\phi) : M {\xrightarrow}{\Delta} M \times M {\xrightarrow}{\phi \times \gamma_{\mathcal{L}}} BGL_n(R) \times BGL_1(R) {\xrightarrow}{\mu} BGL_{n+1}(R)$$ where $\gamma_{\mathcal{L}}$ is the fixed basepoint in $Map_{\mathcal{L}}(M, BGL_1(R))$, and $\mu$ is the usual block pairing.
If we denote the homotopy colimit of these maps $hocolim_n Map_{\oplus_n{\mathcal{L}}}(M, BGL_nR)$ by $Map_{\mathcal{L}}(M, BGL(R))$, we then have a homology equivalence $${\mathbb{Z}}\times Map_{\mathcal{L}}(M, BGL(R)) {\xrightarrow}{\simeq} {\mathbb{Z}}\times {\operatorname{hocolim}}_n B{\mathcal{G}}(\oplus_n P_{\mathcal{L}}) \simeq {\mathbb{Z}}\times BGL (End({\mathcal{L}})) \to {\mathbb{Z}}\times \Omega_0^\infty K_{conn}(End ({\mathcal{L}})),$$ which is the statement of Theorem \[ktheory\].
We now proceed to prove Coro
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ntro\](2). This is derived from an analogous result about the associated graded module of $B_{k0}\otimes_{U_c}eH_c$ that also implies Corollary \[cohh-intro\]. Section \[sect7\] then gives a reinterpretation of Theorem \[mainthm-intro\] in terms of a tensor product filtration of $B_{ij}$. In Appendix \[app-a\] we prove the following result that may be of independent interest: [*Suppose that $R=\bigoplus_{i\geq 0}R_i$ is an ${\mathbb{N}}$-graded algebra over a field $k$, with $R_0=k$. If $P$ is a right $R$-module that is both graded and projective, then $P$ is graded-free in the sense that $P$ has a free basis of homogeneous elements.*]{} This is a graded analogue of a classic result from [@Kap] for which we do not know a reference.
Acknowledgement
---------------
We would like to thank Victor Ginzburg for bringing his conjecture to our attention, since it really formed the starting point for this work. We would also like to thank Tom Nevins and Catharina Stroppel for suggesting many improvements to us.
Rational Cherednik algebras {#sect-rationalchered}
===========================
{#rcadef}
In this section we define the rational Cherednik algebras (which will always be of type $A$ in this paper) and give some of the basic properties that will be needed in the body of the paper.
Let ${{W}}=\mathfrak{S}_n$ \[symmetric-defn\] be the [*symmetric group*]{} on $n$ letters, regarded as the Weyl group of type $A_{n-1}$ acting on its $(n-1)$-dimensional representation ${\mathfrak{h}}\subset {\mathbb{C}}^n$ by permutations. Let $\mathcal{S}=\{s=(i,j)\ \text{with}\ i<j\}\subset {{W}}$ \[involution-defn\] denote the reflections, with reflecting hyperplanes $\alpha_s=0$. We make similar definitions for ${\mathfrak{h}}^*$ and normalise $\alpha_s^{\vee} \in {\mathfrak{h}}$ so that $\alpha_s(\alpha_s^{\vee}) = 2$.
Given $c\in {\mathbb{C}}$, [*the rational Cherednik algebra of type $A_{n-1}$*]{} is the ${\mathbb{C}}$-algebra $H_c$\[hc-defn\] generated by the vector spaces ${\mathfrak{h}}$ and ${\mathfrak{h}}^*$ and the
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bel{A.bar}
\begin{CD}
0 @>>> j_!M^\Delta_\# @>>> \overline{A_\#} @>>> A_\# @>>> 0
\end{CD}$$ of cyclic $k$-vector spaces.
Now assume in addition that $M$ is equipped with a structure of a cyclic $A$-bimodule $M_\#$, so that $M^\Delta_\# \cong
j^*M_\#$, and we have the structure map $\tau_\#:j_!M^\Delta_\# \to
M_\#$. Then we can compose the extension with the map $\tau_\#$, to obtain a commutative diagram $$\label{A.hat}
\begin{CD}
0 @>>> j_!M^\Delta_\# @>>> \overline{A_\#} @>>> A_\# @>>> 0\\
@. @V{\tau_\#}VV @VVV @|\\
0 @>>> M_\# @>>> {\widehat}{A_\#} @>>> A_\# @>>> 0
\end{CD}$$ of short exact sequences in ${\operatorname{Fun}}(\Lambda,k)$, with cartesian left square. It is easy to check that when ${\widetilde}{A} = A_R$ for some square-zero $R$, so that $M = A \otimes {{\mathfrak m}}$, and we take the cyclic $A$-bimodule structure on $M$ induced by the tautological structure on $A$, then ${\widehat}{A_\#}$ coincides precisely with the relative cyclic object $A_{R\#}$ (which we consider as a $k$-vector space, forgetting the $R$-module structure).
We believe that this is the proper generality for the Getzler connection; in this setting, the main result reads as follows.
\[spl\] Assume given a square-zero extension ${\widetilde}{A}$ of an associative algebra $A$ by an $A$-bimodule $M$, and assume that $M$ is equipped with a structure of a cyclic $A$-bimodule. Then the long exact sequence $$\begin{CD}
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A,M) @>>> HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#}) @>>> HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) @>>>
\end{CD}$$ of periodic cyclic homology induced by the second row in admits a canonical splitting $HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A) \to
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}({\widehat}{A_\#})$.
[[*Proof.*]{}]{} By definition, we have two natural maps $$\label{comp}
\begin{aligned}
HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(\overline{A_\#}) &\to HP_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(A_\#) = HP_{{\:\raisebox
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space, for which the matter sector is described by string coordinates $X^\mu(z)$ and their partners $\psi^\mu(z)$ $(\mu=0,1,\cdots,9)$. The reparametrization ghost sector and superconformal ghost sector are described by a fermion pair $(b(z),c(z))$ and a boson pair $(\beta(z),\gamma(z))$, respectively. The superconformal ghost sector has another description by a fermion pair ($\xi(z)$, $\eta(z)$) and a chiral boson $\phi(z)$ [@Friedan:1985ge]. The two descriptions are related through the bosonization relation: $$\beta(z)\ =\ \partial\xi(z) e^{-\phi(z)}\,,\qquad
\gamma(z)\ =\ e^{\phi(z)} \eta(z)\,.$$ The Hilbert space for the $\beta\gamma$ system is called the small Hilbert space and that for the $\xi\eta\phi$ system is called the large Hilbert space.
The theory has two sectors depending on the boundary condition on the world-sheet fermions $\psi^\mu$, $\beta$, and $\gamma$. The sector in which the world-sheet fermion obeys an antiperiodic boundary condition is known as the Neveu-Schwarz (NS) sector, and describes the space-time bosons. The other sector in which the world-sheet fermion obeys a periodic boundary condition is known as the Ramond (R) sector, and describes the space-time fermions. We can obtain the space-time supersymmetric theory by suitably combining two sectors[@Gliozzi:1976qd].
String fields and constraints
-----------------------------
In the WZW-like open superstring field theory, we use the string field $\Phi$ in the large Hilbert space for the NS sector. It is Grassmann even, and has ghost number 0 and picture number 0. Here we further impose the BRST-invariant GSO projection[^4] $$\Phi\ =\ \frac{1}{2}(1+(-1)^{G_{NS}})\, \Phi\,,
$$ where $G_{NS}$ is defined by $$\begin{aligned}
G_{NS}\ =&\ \sum_{r>0}(\psi^\mu_{-r}\psi_{r\mu}-\gamma_{-r}\beta_r+\beta_{-r}\gamma_r) - 1
\nonumber\\
\equiv&\ \sum_{r>0}\psi^\mu_{-r}\psi_{r\mu} + p_\phi\qquad (\textrm{mod}\ 2)\,,\end{aligned}$$ with $p_\phi=-\oint\frac{dz}{2\pi i}\partial\phi(z)$. This is necessary to remove the tachyon and makes the spectrum s
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