We present the first unquenched latticeQCD calculation of the form factors for the decay
We study the classical Hénon family
 NSFPAR ID:
 10432927
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Bulletin of the Brazilian Mathematical Society, New Series
 Volume:
 54
 Issue:
 3
 ISSN:
 16787544
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract at nonzero recoil. Our analysis includes 15 MILC ensembles with$$B\rightarrow D^*\ell \nu $$ $B\to {D}^{\ast}\ell \nu $ flavors of asqtad sea quarks, with a strange quark mass close to its physical mass. The lattice spacings range from$$N_f=2+1$$ ${N}_{f}=2+1$ fm down to 0.045 fm, while the ratio between the light and the strangequark masses ranges from 0.05 to 0.4. The valence$$a\approx 0.15$$ $a\approx 0.15$b andc quarks are treated using the Wilsonclover action with the Fermilab interpretation, whereas the light sector employs asqtad staggered fermions. We extrapolate our results to the physical point in the continuum limit using rooted staggered heavylight meson chiral perturbation theory. Then we apply a modelindependent parametrization to extend the form factors to the full kinematic range. With this parametrization we perform a joint latticeQCD/experiment fit using several experimental datasets to determine the CKM matrix element . We obtain$$V_{cb}$$ ${V}_{\mathrm{cb}}$ . The first error is theoretical, the second comes from experiment and the last one includes electromagnetic and electroweak uncertainties, with an overall$$\left V_{cb}\right = (38.40 \pm 0.68_{\text {th}} \pm 0.34_{\text {exp}} \pm 0.18_{\text {EM}})\times 10^{3}$$ $\left({V}_{\mathrm{cb}}\right)=(38.40\pm 0.{68}_{\text{th}}\pm 0.{34}_{\text{exp}}\pm 0.{18}_{\text{EM}})\times {10}^{3}$ , which illustrates the tensions between the experimental data sets, and between theory and experiment. This result is in agreement with previous exclusive determinations, but the tension with the inclusive determination remains. Finally, we integrate the differential decay rate obtained solely from lattice data to predict$$\chi ^2\text {/dof} = 126/84$$ ${\chi}^{2}\text{/dof}=126/84$ , which confirms the current tension between theory and experiment.$$R(D^*) = 0.265 \pm 0.013$$ $R\left({D}^{\ast}\right)=0.265\pm 0.013$ 
Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
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p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ ${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$ with coefficients$$j=1,\dots ,m$$ $j=1,\cdots ,m$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ ${a}_{j,i}\in {F}_{p}$ , that$$k\ge 3m$$ $k\ge 3m$ for$$a_{j,1}+\dots +a_{j,k}=0$$ ${a}_{j,1}+\cdots +{a}_{j,k}=0$ and that every$$j=1,\dots ,m$$ $j=1,\cdots ,m$ minor of the$$m\times m$$ $m\times m$ matrix$$m\times k$$ $m\times k$ is nonsingular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ ${\left({a}_{j,i}\right)}_{j,i}$n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq {F}_{p}^{n}$ contains a solution$$A> C\cdot \Gamma ^n$$ $\leftA\right>C\xb7{\Gamma}^{n}$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ ${x}_{1},\cdots ,{x}_{k}\in A$C and are constants only depending on$$\Gamma $$ $\Gamma $p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma $\Gamma <p$
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be a positive map from the$$\phi $$ $\varphi $ matrices$$n\times n$$ $n\times n$ to the$$\mathcal {M}_n$$ ${M}_{n}$ matrices$$m\times m$$ $m\times m$ . It is known that$$\mathcal {M}_m$$ ${M}_{m}$ is 2positive if and only if for all$$\phi $$ $\varphi $ and all strictly positive$$K\in \mathcal {M}_n$$ $K\in {M}_{n}$ ,$$X\in \mathcal {M}_n$$ $X\in {M}_{n}$ . This inequality is not generally true if$$\phi (K^*X^{1}K) \geqslant \phi (K)^*\phi (X)^{1}\phi (K)$$ $\varphi \left({K}^{\ast}{X}^{1}K\right)\u2a7e\varphi {\left(K\right)}^{\ast}\varphi {\left(X\right)}^{1}\varphi \left(K\right)$ is merely a Schwarz map. We show that the corresponding tracial inequality$$\phi $$ $\varphi $ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.$${{\,\textrm{Tr}\,}}[\phi (K^*X^{1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{1}\phi (K)]$$ $\phantom{\rule{0ex}{0ex}}\text{Tr}\phantom{\rule{0ex}{0ex}}\left[\varphi \left({K}^{\ast}{X}^{1}K\right)\right]\u2a7e\phantom{\rule{0ex}{0ex}}\text{Tr}\phantom{\rule{0ex}{0ex}}\left[\varphi {\left(K\right)}^{\ast}\varphi {\left(X\right)}^{1}\varphi \left(K\right)\right]$