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The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing…
It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to…
Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp-Massey…
Three-body recombination is a prime example of the fundamental interaction between three particles. Due to the complexity of this process it has resisted a comprehensive description. Experimental investigations have mainly focussed on the…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
Binary idempotent semirings govern classical path algebras. Their multiplicative structure is dyadic. We examine whether this restriction is structural or accidental. We define ternary idempotent $\Gamma$-semirings as higher-arity ordered…
This work proposes deep network models and learning algorithms for unsupervised and supervised binary hashing. Our novel network design constrains one hidden layer to directly output the binary codes. This addresses a challenging issue in…
Summary: Human alpha satellite and satellite 2/3 contribute to several percent of the human genome. However, identifying these sequences with traditional algorithms is computationally intensive. Here we develop dna-brnn, a recurrent neural…
While many papers have proposed implementations of ternary adders and ternary multipliers, no comparisons have generally been done with the corresponding binary ones. We compare the implementations of binary and ternary adders and…
The binary operation of usual addition is associative in all common matrices over R. However, here we define a binary operation of addition in matrices over Zn which present the concept of nonassociativity. These structures form Matrix…
We investigate universal behavior in the recombination rate of three bosons close to threshold. Using the He-He system as a reference, we solve the three-body Schr\"odinger equation above the dimer threshold for different potentials having…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
With an aim towards modeling cosmologies beyond the $\Lambda$CDM paradigm, we demonstrate the automatic construction of recombination history emulators while enforcing a prior of causal dynamics. These methods are particularly useful in the…
We survey the results of a long-term study of the process of radiative recombination. A rigorous theory of nonrelativistic electron radiative recombination with a hydrogen-like ion is used to calculate the total cross section of the…
In this work we set out to find a method to classify protein structures using a Deep Learning methodology. Our Artificial Intelligence has been trained to recognize complex biomolecule structures extrapolated from the Protein Data Bank…
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this…
Working in the multitape Turing model, we show how to reduce the problem of matrix transposition to the problem of integer multiplication. If transposing an $n \times n$ binary matrix requires $\Omega(n^2 \log n)$ steps on a Turing machine,…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
Inspired by biology's most sophisticated computer, the brain, neural networks constitute a profound reformulation of computational principles. Remarkably, analogous high-dimensional, highly-interconnected computational architectures also…
Biclustering, the process of simultaneously clustering the rows and columns of a data matrix, is a popular and effective tool for finding structure in a high-dimensional dataset. Many biclustering procedures appear to work well in practice,…