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Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of discrete Fourier transforms over finite fields, which have widespread applications in cryptography and error control codes. They are of great interest because of…
We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over…
Sinkhorn's alternative minimization algorithm applied to a positive $n\times n$ matrix converges to a doubly stochastic matrix. If the algorithm, applied to a $2\times 2$ matrix, converges in a finite number of iterations, then it converges…
It is shown that a piecewise linear function can be represented as a Max-Min polynomial of its linear components.
The suitable basis functions for approximating periodic function are periodic, trigonometric functions. When the function is not periodic, a viable alternative is to consider polynomials as basis functions. In this paper we will point out…
A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show…
Let G be a block matrix function with one diagonal block A being positive definite and the off diagonal blocks complex conjugates of each other. Conditions are obtained for G to be factorable (in particular, with zero partial indices) in…
We develop an elementary theory of partially additive rings as a foundation of ${\mathbb F}_1$-geometry. Our approach is so concrete that an analog of classical algebraic geometry is established very straightforwardly. As applications, (1)…
This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction.…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
Motivated by an application in computational biology, we consider low-rank matrix factorization with $\{0,1\}$-constraints on one of the factors and optionally convex constraints on the second one. In addition to the non-convexity shared…
This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication…
We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a…
Matrix completion is a classical problem in data science wherein one attempts to reconstruct a low-rank matrix while only observing some subset of the entries. Previous authors have phrased this problem as a nuclear norm minimization…
We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address the generic minimal rank problem.
We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity…
Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix multiplication, must be ordinary matrix…
We extend the technique of constructive expansions to compute the connected functions of matrix models in a uniform way as the size of the matrix increases. This provides the main missing ingredient for a non-perturbative construction of…