We introduce a numerical scheme that approximates solutions to linear PDE's by minimizing a residual in the $W^{-1,p'}(\Omega)$ norm with exponents $p> 2$. The resulting problem is solved by regularized Kacanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents $p\gg 2$. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.
相關內容
In the literature, there are many results about permutation polynomials over finite fields. However, very few permutations of vector spaces are constructed although it has been shown that permutations of vector spaces have many applications in cryptography, especially in constructing permutations with low differential and boomerang uniformities. In this paper, motivated by the butterfly structure \cite{perrin2016cryptanalysis} and the work of Qu and Li \cite{qu2023}, we investigate rotatable permutations from $\gf_{2^m}^3$ to itself with $d$-homogenous functions. Based on the theory of equations of low degree, the resultant of polynomials, and some skills of exponential sums, we construct five infinite classes of $3$-homogeneous rotatable permutations from $\gf_{2^m}^3$ to itself, where $m$ is odd. Moreover, we demonstrate that the corresponding permutation polynomials of $\gf_{2^{3m}}$ of our newly constructed permutations of $\gf_{2^m}^3$ are QM-inequivalent to the known ones.
In this paper, for any fixed integer $q>2$, we construct $q$-ary codes correcting a burst of at most $t$ deletions with redundancy $\log n+8\log\log n+o(\log\log n)+\gamma_{q,t}$ bits and near-linear encoding/decoding complexity, where $n$ is the message length and $\gamma_{q,t}$ is a constant that only depends on $q$ and $t$. In previous works there are constructions of such codes with redundancy $\log n+O(\log q\log\log n)$ bits or $\log n+O(t^2\log\log n)+O(t\log q)$. The redundancy of our new construction is independent of $q$ and $t$ in the second term.
This paper presents new upper bounds on the rate of linear $k$-hash codes in $\mathbb{F}_q^n$, $q\geq k$, that is, codes with the property that any $k$ distinct codewords are all simultaneously distinct in at least one coordinate.
Concurrency is an important aspect of (Petri) nets to describe and simulate the behavior of complex systems. Knowing which places and transitions could be executed in parallel helps to understand nets and enables analysis techniques and the computation of other properties, such as causality, exclusivity, etc.. All techniques based on concurrency detection depend on the efficiency of this detection methodology. Kovalyov and Esparza have developed algorithms that compute all concurrent places in $O\big((P+T)TP^2\big)$ for live nets (where $P$ and $T$ are the numbers of places and transitions) and in $O\big(P(P+T)^2\big)$ for live free-choice nets. Although these algorithms have a reasonably good computational complexity, large numbers of concurrent pairs of nodes may still lead to long computation times. Furthermore, both algorithms cannot be parallelized without additional effort. This paper complements the palette of concurrency detection algorithms with the Concurrent Paths (CP) algorithm for safe, live, free-choice nets. The algorithm allows parallelization and has a worst-case computational complexity of $O\big((P+T)^2\big)$ for acyclic nets and of $O\big(P^3+PT^2\big)$ for cyclic nets. Although the computational complexity of cyclic nets has not improved, the evaluation shows the benefits of CP, especially, if the net contains many nodes in concurrency relation.
Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels -- a PDE structure unaddressed thus far with DeepONet. In this note, we open the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$ coupled system in whose control a $2\times 2$ kernel PDE systems in Goursat form arises. Applications include oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. In this paper we establish the continuity of the mapping from (a total of five) plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and establish that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$\emph{-Globally-exponentially} stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures \emph{semi-global practical exponential stability (SG-PES).} The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.
We prove the following type of discrete entropy monotonicity for isotropic log-concave sums of independent identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ are independent and identically distributed uniform random vectors on $[0,1]^d$ and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. However, in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, its barycenter and its covariance matrix are close to their discrete counterparts. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which generalises a result of Bobkov, Marsiglietti and Melbourne (2022) and may be of independent interest.
We propose an original approach to investigate the linearity of $\mathbb{Z}_{2^L}$-linear codes, i.e., codes obtained as the image of the generalized Gray map applied to $\mathbb{Z}_{2^L}$-additive codes. To accomplish that, we define two related binary codes: the associated and concatenated, where one could perform a straightforward analysis of the Schur product between their codewords and determine the linearity of the respective $\mathbb{Z}_{2^L}$-linear code. This work expands on previous contributions from the literature, where the linearity was established with respect to the kernel of a code and/or operations on $\mathbb{Z}_{2^L}$. The $\mathbb{Z}_{2^L}$-additive codes we apply the Gray map and check the linearity are the well-known Hadamard, simplex, and MacDonald codes. We also present families of Reed-Muller and cyclic codes that yield to linear $\mathbb{Z}_{2^L}$-linear codes and perform a computational verification of our proposed method applied to other $\mathbb{Z}_{2^L}$-additive codes.
This paper proposes an $\alpha$-leakage measure for $\alpha\in[0,\infty)$ by a cross entropy interpretation of R{\'{e}}nyi entropy. While R\'{e}nyi entropy was originally defined as an $f$-mean for $f(t) = \exp((1-\alpha)t)$, we reveal that it is also a $\tilde{f}$-mean cross entropy measure for $\tilde{f}(t) = \exp(\frac{1-\alpha}{\alpha}t)$. Minimizing this R\'{e}nyi cross-entropy gives R\'{e}nyi entropy, by which the prior and posterior uncertainty measures are defined corresponding to the adversary's knowledge gain on sensitive attribute before and after data release, respectively. The $\alpha$-leakage is proposed as the difference between $\tilde{f}$-mean prior and posterior uncertainty measures, which is exactly the Arimoto mutual information. This not only extends the existing $\alpha$-leakage from $\alpha \in [1,\infty)$ to the overall R{\'{e}}nyi order range $\alpha \in [0,\infty)$ in a well-founded way with $\alpha=0$ referring to nonstochastic leakage, but also reveals that the existing maximal leakage is a $\tilde{f}$-mean of an elementary $\alpha$-leakage for all $\alpha \in [0,\infty)$, which generalizes the existing pointwise maximal leakage.
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(\alpha_1,\alpha_2, \alpha_3;t_1,t_2, t_3)$ with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, $\alpha_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, and $\alpha_3 \neq 0$. First, we show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for $3 \leq t \leq 11$, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$.
A stabilizer free weak Galerkin method with implicit $θ$-schemes for fourth order parabolic problems
In this paper, we combine the stabilizer free weak Galerkin (SFWG) method and the implicit $\theta$-schemes in time for $\theta\in [\frac{1}{2},1]$ to solve the fourth-order parabolic problem. In particular, when $\theta =1$, the full-discrete scheme is first-order backward Euler and the scheme is second-order Crank Nicolson scheme if $\theta =\frac{1}{2}$. Next, we analyze the well-posedness of the schemes and deduce the optimal convergence orders of the error in the $H^2$ and $L^2$ norms. Finally, numerical examples confirm the theoretical results.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191