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The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of…

Computational Complexity · Computer Science 2020-09-22 Philipp Schepper

This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the…

History and Overview · Mathematics 2019-01-31 Daniel Reem

In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth…

Numerical Analysis · Mathematics 2019-01-30 Baoli Yin , Yang Liu , Hong Li , Siriguleng He

This work constructs codes that are efficiently decodable from a constant fraction of \emph{worst-case} insertion and deletion errors in three parameter settings: (i) Binary codes with rate approaching 1; (ii) Codes with constant rate for…

Information Theory · Computer Science 2016-05-17 Venkatesan Guruswami , Ray Li

Finding the solution to linear ordinary differential equations of the form $\partial_t u(t) = -A(t)u(t)$ has been a promising theoretical avenue for \textit{asymptotic} quantum speedups. However, despite the improvements to existing quantum…

Quantum Physics · Physics 2025-11-06 Matthew Pocrnic , Peter D. Johnson , Amara Katabarwa , Nathan Wiebe

The task of maximizing a monotone submodular function under a cardinality constraint is at the core of many machine learning and data mining applications, including data summarization, sparse regression and coverage problems. We study this…

Data Structures and Algorithms · Computer Science 2023-05-26 Silvio Lattanzi , Slobodan Mitrović , Ashkan Norouzi-Fard , Jakub Tarnawski , Morteza Zadimoghaddam

We present two algorithms, one quantum and one classical, for estimating partition functions of quantum spin Hamiltonians. The former is a DQC1 (Deterministic quantum computation with one clean qubit) algorithm, and the first such for…

Quantum Physics · Physics 2023-02-01 Andrew Jackson , Theodoros Kapourniotis , Animesh Datta

Partition functions of a canonical ensemble of non-interacting bound electrons are a key ingredient of the super-transition-array approach to the computation of radiative opacity. A few years ago, we published a robust and stable recursion…

Statistical Mechanics · Physics 2020-10-28 Jean-Christophe Pain , Franck Gilleron , Brian G. Wilson

Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial…

Numerical Analysis · Mathematics 2024-09-19 Fredrik Fryklund , Leslie Greengard , Shidong Jiang , Samuel Potter

There are quantum algorithms for finding a function $f$ satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when…

Quantum Physics · Physics 2023-06-02 Koichi Miyamoto , Hiroshi Ueda

We present an exact quantum algorithm for solving the Exact Satisfiability (XSAT) problem, which belongs to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts:…

Quantum Physics · Physics 2016-08-30 Salvatore Mandrà , Gian Giacomo Guerreschi , Alán Aspuru-Guzik

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…

Classical Analysis and ODEs · Mathematics 2026-04-30 Alexandre Benoit , Nicolas Brisebarre , Bruno Salvy

We present algorithms that substantially accelerate partition-based cross-validation for machine learning models that require matrix products $\mathbf{X}^\mathbf{T}\mathbf{X}$ and $\mathbf{X}^\mathbf{T}\mathbf{Y}$. Our algorithms have…

Machine Learning · Computer Science 2025-09-29 Ole-Christian Galbo Engstrøm , Martin Holm Jensen

New formulas for approximation of zeta-constants were derived on the basis of a number-theoretic approach constructed for the irrationality proof of certain classical constants. Using these formulas it's possible to approximate certain…

Number Theory · Mathematics 2018-05-08 Ekatherina A. Karatsuba

Efficient and accurate algorithm for partition function, free energy and thermal entropy calculations is of great significance in statistical physics and quantum many-body physics. Here we present an unbiased but low-technical-barrier…

Statistical Mechanics · Physics 2024-11-19 Yi-Ming Ding , Jun-Song Sun , Nvsen Ma , Gaopei Pan , Chen Cheng , Zheng Yan

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…

Computational Complexity · Computer Science 2016-08-15 Peter Franek , Stefan Ratschan , Piotr Zgliczynski

We present an algorithm for efficient computation of the constant term of a power of a multivariate Laurent polynomial. The algorithm is based on univariate interpolation, does not require the storage of intermediate data and can be easily…

Symbolic Computation · Computer Science 2012-11-19 Pavel Metelitsyn

The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem.…

Optimization and Control · Mathematics 2018-06-13 Valentina Ciccone , Augusto Ferrante , Mattia Zorzi

We consider an algorithm to approximate complex-valued periodic functions $f(e^{i\theta})$ as a matrix element of a product of $SU(2)$-valued functions, which underlies so-called quantum signal processing. We prove that the algorithm runs…

Quantum Physics · Physics 2020-05-07 Jeongwan Haah

In many applications, it is of interest to approximate data, given by mxn matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time…

Numerical Analysis · Mathematics 2007-05-23 Shmuel Friedland , Mostafa Kaveh , Amir Niknejad , Hossein Zare
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