Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately $3n^2$ logical variables for the binary field $\mathbb{F}_{2^n}$. In our estimations, we assume the existence of an optimal normal base of II type in the given fields. Such a QUBO instance can then be solved using quantum annealing.
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The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.
To address the sensitivity of parameters and limited precision for physics-informed extreme learning machines (PIELM) with common activation functions, such as sigmoid, tangent, and Gaussian, in solving high-order partial differential equations (PDEs) relevant to scientific computation and engineering applications, this work develops a Fourier-induced PIELM (FPIELM) method. This approach aims to approximate solutions for a class of fourth-order biharmonic equations with two boundary conditions on both unitized and non-unitized domains. By carefully calculating the differential and boundary operators of the biharmonic equation on discretized collections, the solution for this high-order equation is reformulated as a linear least squares minimization problem. We further evaluate the FPIELM with varying hidden nodes and scaling factors for uniform distribution initialization, and then determine the optimal range for these two hyperparameters. Numerical experiments and comparative analyses demonstrate that the proposed FPIELM method is more stable, robust, precise, and efficient than other PIELM approaches in solving biharmonic equations across both regular and irregular domains.
The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.
We consider random matrix ensembles on the set of Hermitian matrices that are heavy tailed, in particular not all moments exist, and that are invariant under the conjugate action of the unitary group. The latter property entails that the eigenvectors are Haar distributed and, therefore, factorise from the eigenvalue statistics. We prove a classification for stable matrix ensembles of this kind of matrices represented in terms of matrices, their eigenvalues and their diagonal entries with the help of the classification of the multivariate stable distributions and the harmonic analysis on symmetric matrix spaces. Moreover, we identify sufficient and necessary conditions for their domains of attraction. To illustrate our findings we discuss for instance elliptical invariant random matrix ensembles and P\'olya ensembles, the latter playing a particular role in matrix convolutions. As a byproduct we generalise the derivative principle on the Hermitian matrices to general tempered distributions. This principle relates the joint probability density of the eigenvalues and the diagonal entries of the random matrix.
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of $O(h^4)$ and $O(h^6)$ accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.
In this paper, we first study hypergraph rewriting in categorical terms in an attempt to define the notion of events and develop foundations of causality in graph rewriting. We introduce novel concepts within the framework of double-pushout rewriting in adhesive categories. Secondly, we will study the notion of events in $\lambda-$calculus, wherein we construct an algorithm to determine causal relations between events following the evaluation of a $\lambda-$expression satisfying certain conditions. Lastly, we attempt to extend this definition to arbitrary $\lambda-$expressions.
Aperiodic autocorrelation is an important indicator of performance of sequences used in communications, remote sensing, and scientific instrumentation. Knowing a sequence's autocorrelation function, which reports the autocorrelation at every possible translation, is equivalent to knowing the magnitude of the sequence's Fourier transform. The phase problem is the difficulty in resolving this lack of phase information. We say that two sequences are equicorrelational to mean that they have the same aperiodic autocorrelation function. Sequences used in technological applications often have restrictions on their terms: they are not arbitrary complex numbers, but come from a more restricted alphabet. For example, binary sequences involve terms equal to only $+1$ and $-1$. We investigate the necessary and sufficient conditions for two sequences to be equicorrelational, where we take their alphabet into consideration. There are trivial forms of equicorrelationality arising from modifications that predictably preserve the autocorrelation, for example, negating a binary sequence or reversing the order of its terms. By a search of binary sequences up to length $44$, we find that nontrivial equicorrelationality among binary sequences does occur, but is rare. An integer $n$ is said to be equivocal when there are binary sequences of length $n$ that are nontrivially equicorrelational; otherwise $n$ is unequivocal. For $n \leq 44$, we found that the unequivocal lengths are $1$--$8$, $10$, $11$, $13$, $14$, $19$, $22$, $23$, $26$, $29$, $37$, and $38$. We pose open questions about the finitude of unequivocal numbers and the probability of nontrivial equicorrelationality occurring among binary sequences.
We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.
Consider a finite directed graph without cycles in which the arrows are weighted. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. Thus, although our algorithm is based on arguments similar to those at work for the Bellman-Ford and Dijkstra methods, it is in fact essentially different. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
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