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One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot \log n$, and it is widely open how to prove a super-$3\log n$ depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering formulas with restriction on top, we can break the $3\log n$ barrier. Formally, they prove there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<\alpha<0.4$ and constant $0<\epsilon<\alpha/2$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{(\alpha-\epsilon)n}$ formulas of size at most $2^{(1-\alpha/2-\epsilon)n}$. This implies a modified version of Andreev function is not computable by any circuit of depth $(3.2-\epsilon)\log n$ with the restriction that top $0.4-\epsilon$ layers only consist of AND gates for any small constant $\epsilon>0$. They ask whether the parameter $\alpha$ can be push up to nearly $1$ thus implying a nearly-$3.5\log n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<\alpha<2-o(1)$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{\alpha n}$ formulas of size at most $2^{(1-\alpha/2-o(1))n}$. This implies a $(4-o(1))\log n$ depth lower bound with the restriction that top $2-o(1)$ layers only consist of AND gates. We prove it by observing that one crucial component in Mihajlin and Sofronova's work, called the well-mixed set of functions, can be significantly simplified thus improved. Then with this observation and a more careful analysis, we obtain these nearly tight results.

We study the asymptotic eigenvalue distribution of the Slepian spatiospectral concentration problem within subdomains of the $d$-dimensional unit ball $\mathbb{B}^d$. The clustering of the eigenvalues near zero and one is a well-known phenomenon. Here, we provide an analytical investigation of this phenomenon for two different notions of bandlimit: (a) multivariate polynomials, with the maximal polynomial degree determining the bandlimit, (b) basis functions that separate into radial and spherical contributions (expressed in terms of Jacobi polynomials and spherical harmonics, respectively), with separate maximal degrees for the radial and spherical contributions determining the bandlimit. In particular, we investigate the number of relevant non-zero eigenvalues (the so-called Shannon number) and obtain distinct asymptotic results for both notions of bandlimit, characterized by Jacobi weights $W_0$ and a modification $\widetilde{W_0}$, respectively. The analytic results are illustrated by numerical examples on the 3-d ball.

We derive eigenvalue bounds for the $t$-distance chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo, Du and Graham [Inf. Process. Lett., 2002], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [Discrete Appl. Math., 2011]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance $3$. In order to prove our results, we use a mix of spectral and number theory tools. Our results, which provide the first application of spectral methods to Lee codes, illustrate that such methods succeed to capture the nature of the Lee metric.

Kalai's $3^d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3^d$ faces. We give short proofs for two special cases: if $P$ is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if $P$ is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.

Circuit complexity, defined as the minimum circuit size required for implementing a particular Boolean computation, is a foundational concept in computer science. Determining circuit complexity is believed to be a hard computational problem [1]. Recently, in the context of black holes, circuit complexity has been promoted to a physical property, wherein the growth of complexity is reflected in the time evolution of the Einstein-Rosen bridge (``wormhole'') connecting the two sides of an AdS ``eternal'' black hole [2]. Here we explore another link between complexity and thermodynamics for circuits of given functionality, making the physics-inspired approach relevant to real computational problems, for which functionality is the key element of interest. In particular, our thermodynamic framework provides a new perspective on the obfuscation of programs of arbitrary length -- an important problem in cryptography -- as thermalization through recursive mixing of neighboring sections of a circuit, which can be viewed as the mixing of two containers with ``gases of gates''. This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits with same size and functionality that cannot be connected via local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of computational complexity theory to its first level.

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.

This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Pad\'e methods shown to be superior in performance to the approximants used in state-of-the-art software. The algorithm computes matrix--matrix products and also matrix inverses, but it can be implemented to avoid the computation of inverses, making it convenient for some problems. If the matrix A belongs to a Lie algebra, then exp(A) belongs to its associated Lie group, being a property which is preserved by diagonal Pad\'e approximants, and the algorithm has another option to use only these. Numerical experiments show the superior performance with respect to state-of-the-art implementations.

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is approximately solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, a preconditioned correction equation is derived that extracts useful information from current searching subspaces to construct effective preconditioners for the correction equation and is proved to retain the same convergence of outer iterations of JDSVD. The resulting method is called inner preconditioned JDSVD (IPJDSVD) method. Convergence results show that MINRES for the preconditioned correction equation can converge much faster when there is a cluster of singular values closest to a given target, so that IPJDSVD is more efficient than JDSVD. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets of a large matrix. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD.

The problem of enumerating connected subgraphs of a given size in a graph has been extensively studied in recent years. In this short communication, we propose an algorithm with a delay of $\mathcal{O}(k\Delta)$ for enumerating all connected induced subgraphs of size $k$ in an undirected graph $G=(V, E)$, where $k$ and $\Delta$ are respectively the size of subgraphs and the maximum degree of $G$. The proposed algorithm improves upon the current best delay bound $\mathcal{O}(k^2\Delta)$ for the connected induced subgraph enumeration problem in the literature.

We propose a novel statistical test to assess the mutual independence of multidimensional random vectors. Our approach is based on the $L_1$-distance between the joint density function and the product of the marginal densities associated with the presumed independent vectors. Under the null hypothesis, we employ Poissonization techniques to establish the asymptotic normal approximation of the corresponding test statistic, without imposing any regularity assumptions on the underlying Lebesgue density function, denoted as $f(\cdot)$. Remarkably, we observe that the limiting distribution of the $L_1$-based statistics remains unaffected by the specific form of $f(\cdot)$. This unexpected outcome contributes to the robustness and versatility of our method. Moreover, our tests exhibit nontrivial local power against a subset of local alternatives, which converge to the null hypothesis at a rate of {${\tiny n^{\tiny -1/2}h_n^{\tiny -{d/4}}}$}, $d\geq 2$, where $n$ represents the sample size and $h_n$ denotes the bandwidth. Finally, the theory is supported by a comprehensive simulation study to investigate the finite-sample performance of our proposed test. The results demonstrate that our testing procedure generally outperforms existing approaches across various examined scenarios.