亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

The logics $\mathsf{CS4}$ and $\mathsf{IS4}$ are the two leading intuitionistic variants of the modal logic $\mathsf{S4}$. Whether the finite model property holds for each of these logics have been long-standing open problems. It was recently shown that $\mathsf{IS4}$ has the finite frame property and thus the finite model property. In this paper, we prove that $\mathsf{CS4}$ also enjoys the finite frame property. Additionally, we investigate the following three logics closely related to $\mathsf{IS4}$. The logic $\mathsf{GS4}$ is obtained by adding the G\"odel--Dummett axiom to $\mathsf{IS4}$; it is both a superintuitionistic and a fuzzy logic and has previously been given a real-valued semantics. We provide an alternative birelational semantics and prove strong completeness with respect to this semantics. The extension $\mathsf{GS4^c}$ of $\mathsf{GS4}$ corresponds to requiring a crisp accessibility relation on the real-valued semantics. We give a birelational semantics corresponding to an extra confluence condition on the $\mathsf{GS4}$ birelational semantics and prove strong completeness. Neither of these two logics have the finite model property with respect to their real-valued semantics, but we prove that they have the finite frame property for their birelational semantics. Establishing the finite birelational frame property immediately establishes decidability, which was previously open for these two logics. Our proofs yield NEXPTIME upper bounds. The logic $\mathsf{S4I}$ is obtained from $\mathsf{IS4}$ by reversing the roles of the modal and intuitionistic relations in the birelational semantics. We also prove the finite frame property, and thereby decidability, for $\mathsf{S4I}$

We give a corrected proof that if PP $\subseteq$ BQP/qpoly, then the Counting Hierarchy collapses, as originally claimed by Aaronson CCC'06 arXiv:cs/0504048. This recovers the related unconditional claim that PP does not have circuits of any fixed size $n^k$ even with quantum advice. We do so by proving that YQP*, an oblivious version of (QMA $\cap$ coQMA), is contained in APP, and so is PP-low.

A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $G^{epex}$ the class of graphs that are at most one edge away from being in $\mathcal{G}$. We note that $G^{epex}$ is hereditary and prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $G^{epex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. For the class of edge-apex cographs our main result bounds the order of such forbidden induced subgraphs by 8 and finds all of them by computer search.

We study empirical variants of the halfspace (Tukey) depth of a probability measure $\mu$, which are obtained by replacing $\mu$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of $\mu$ being the uniform distribution on a convex body $K$, the depth trimmed regions are convex floating bodies of $K$, and we obtain strong limit theorems for their empirical estimators.

We consider learning an unknown target function $f_*$ using kernel ridge regression (KRR) given i.i.d. data $(u_i,y_i)$, $i\leq n$, where $u_i \in U$ is a covariate vector and $y_i = f_* (u_i) +\varepsilon_i \in \mathbb{R}$. A recent string of work has empirically shown that the test error of KRR can be well approximated by a closed-form estimate derived from an `equivalent' sequence model that only depends on the spectrum of the kernel operator. However, a theoretical justification for this equivalence has so far relied either on restrictive assumptions -- such as subgaussian independent eigenfunctions -- , or asymptotic derivations for specific kernels in high dimensions. In this paper, we prove that this equivalence holds for a general class of problems satisfying some spectral and concentration properties on the kernel eigendecomposition. Specifically, we establish in this setting a non-asymptotic deterministic approximation for the test error of KRR -- with explicit non-asymptotic bounds -- that only depends on the eigenvalues and the target function alignment to the eigenvectors of the kernel. Our proofs rely on a careful derivation of deterministic equivalents for random matrix functionals in the dimension free regime pioneered by Cheng and Montanari (2022). We apply this setting to several classical examples and show an excellent agreement between theoretical predictions and numerical simulations. These results rely on having access to the eigendecomposition of the kernel operator. Alternatively, we prove that, under this same setting, the generalized cross-validation (GCV) estimator concentrates on the test error uniformly over a range of ridge regularization parameter that includes zero (the interpolating solution). As a consequence, the GCV estimator can be used to estimate from data the test error and optimal regularization parameter for KRR.

For any positive integer $q\geq 2$ and any real number $\delta\in(0,1)$, let $\alpha_q(n,\delta n)$ denote the maximum size of a subset of $\mathbb{Z}_q^n$ with minimum Hamming distance at least $\delta n$, where $\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$ and $n\in\mathbb{N}$. The asymptotic rate function is defined by $ R_q(\delta) = \limsup_{n\rightarrow\infty}\frac{1}{n}\log_q\alpha_q(n,\delta n). $ The famous $q$-ary asymptotic Gilbert-Varshamov bound, obtained in the 1950s, states that \[ R_q(\delta) \geq 1 - \delta\log_q(q-1)-\delta\log_q\frac{1}{\delta}-(1-\delta)\log_q\frac{1}{1-\delta} \stackrel{\mathrm{def}}{=}R_\mathrm{GV}(\delta,q) \] for all positive integers $q\geq 2$ and $0<\delta<1-q^{-1}$. In the case that $q$ is an even power of a prime with $q\geq 49$, the $q$-ary Gilbert-Varshamov bound was firstly improved by using algebraic geometry codes in the works of Tsfasman, Vladut, and Zink and of Ihara in the 1980s. The further investigation in algebraic geometry codes has shown that the $q$-ary Gilbert-Varshamov bound can also be improved in the case that $q$ is an odd power of a prime but not a prime with $q > 125$. However, it remains a long-standing open problem whether the $q$-ary Gilbert-Varshamov bound would be tight for those infinitely many integers $q$ which is a prime, except for Fermat primes not less than 257, and which is a generic positive integer not being a prime power. In this paper, we prove that the $q$-ary Gilbert-Varshamov bound can be improved for all but finitely many positive integers $q\geq 2$. It is shown that $ R_q(1/2) > R_\mathrm{GV}(1/2,q) $ for all integers $q > \exp(29)$. Furthermore, we show that the growth of the rate function $R_q(\delta)$ for $\delta\in(0,1)$ fixed and $q$ growing large has a nontrivial lower bound. These new lower bounds are achieved by using codes from geometry of numbers introduced by Lenstra in the 1980s.

We give a semidefinite programming characterization of the Crawford number. We show that the computation of the Crawford number within $\varepsilon$ precision is computable in polynomial time in the data and $|\log \varepsilon |$.

Given an undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Two optimization problems can be defined for quasi-cliques: the Maximum Quasi-Clique (MQC) Problem, which finds a quasi-clique with maximum vertex cardinality, and the Densest $k$-Subgraph (DKS) Problem, which finds the densest subgraph given a fixed cardinality constraint. Most existing approaches to solve both problems often disregard the requirement of connectedness, which may lead to solutions containing isolated components that are meaningless for many real-life applications. To address this issue, we propose two flow-based connectedness constraints to be integrated into known Mixed-Integer Linear Programming (MILP) formulations for either MQC or DKS problems. We compare the performance of MILP formulations enhanced with our connectedness constraints in terms of both running time and number of solved instances against existing approaches that ensure quasi-clique connectedness. Experimental results demonstrate that our constraints are quite competitive, making them valuable for practical applications requiring connectedness.

Backtracking linesearch is the de facto approach for minimizing continuously differentiable functions with locally Lipschitz gradient. In recent years, it has been shown that in the convex setting it is possible to avoid linesearch altogether, and to allow the stepsize to adapt based on a local smoothness estimate without any backtracks or evaluations of the function value. In this work we propose an adaptive proximal gradient method, adaPG, that uses novel estimates of the local smoothness modulus which leads to less conservative stepsize updates and that can additionally cope with nonsmooth terms. This idea is extended to the primal-dual setting where an adaptive three-term primal-dual algorithm, adaPD, is proposed which can be viewed as an extension of the PDHG method. Moreover, in this setting the "essentially" fully adaptive variant adaPD$^+$ is proposed that avoids evaluating the linear operator norm by invoking a backtracking procedure, that, remarkably, does not require extra gradient evaluations. Numerical simulations demonstrate the effectiveness of the proposed algorithms compared to the state of the art.

A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a dominating set of $G$. The secure domination number of $G$, denoted by $\gamma_{s}(G)$, is the cardinality of a smallest secure dominating sets of $G$. In this paper, we prove that for any outerplanar graph with $n \geq 4$ vertices, $\gamma_{s}(G) \geq (n+4)/5$ and the bound is tight.