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Many computational problems can be formulated in terms of high-dimensional functions. Simple representations of such functions and resulting computations with them typically suffer from the "curse of dimensionality", an exponential cost…
The increasing difficulty in continued development of digital electronic logic has led to a renewed interest in alternative approaches. Oscillatory computing is one such approach that leverages alternative physical systems and computation…
High-dimensional distributed semantic spaces have proven useful and effective for aggregating and processing visual, auditory, and lexical information for many tasks related to human-generated data. Human language makes use of a large and…
Hyperdimensional Computing (HDC), also known as Vector-Symbolic Architectures (VSA), is a promising framework for the development of cognitive architectures and artificial intelligence systems, as well as for technical applications and…
We report the first steps in creating an optical computing system. This system may solve NP-Hard problems by utilizing a setup of exponential sized masks. This is exponential space complexity but the production of those masks is done with a…
Program synthesis is a class of regression problems where one seeks a solution, in the form of a source-code program, mapping the inputs to their corresponding outputs exactly. Due to its precise and combinatorial nature, program synthesis…
Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic…
In article the basic principles put in a basis of algorithmicallysoftware of hypercomplex number calculations, structure of a software, structure of functional subsystems are considered. The most important procedures included in subsystems…
Vector Symbolic Architectures (VSAs) are a powerful framework for representing compositional reasoning. They lend themselves to neural-network implementations, allowing us to create neural networks that can perform cognitive functions, like…
Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for…
Following up on a previous analysis of graph embeddings, we generalize and expand some results to the general setting of vector symbolic architectures (VSA) and hyperdimensional computing (HDC). Importantly, we explore the mathematical…
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The…
Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…
Synthesizing programs from examples requires searching over a vast, combinatorial space of possible programs. In this search process, a key challenge is representing the behavior of a partially written program before it can be executed, to…
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation…
This paper presents a high-accuracy higher-order multiscale method for solving multi-continuum problems in in highly heterogeneous media. First, microscopic unit cell functions are defined, leading to the derivation of macroscopic…
Hypercomplex signal processing (HSP) provides state-of-the-art tools to handle multidimensional signals by harnessing intrinsic correlation of the signal dimensions through Clifford algebra. Recently, the hypercomplex representation of the…
The method of obtaining the set of noncanonical hypercomplex number systems by conversion of infinite hypercomplex number system to finite hypercomplex number system depending on multiplication rules and factorization method is described.…
This monograph presents a detailed analysis of hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions. It continues with a detailed analysis of hypercomplex numbers in n…
Cyber-Physical Systems (CPS) pose new challenges to verification and validation that go beyond the proof of functional correctness based on high-level models. Particular challenges are, in particular for formal methods, its heterogeneity…