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The kernel embedding algorithm is an important component for adapting kernel methods to large datasets. Since the algorithm consumes a major computation cost in the testing phase, we propose a novel teacher-learner framework of learning…
Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while…
Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware,…
We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…
Fast Fourier transform was included in the Top 10 Algorithms of 20th Century by Computing in Science & Engineering. In this paper, we provide a new simple derivation of both the discrete Fourier transform and fast Fourier transform by means…
The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the…
FaaS allows an application to be decomposed into functions that are executed on a FaaS platform. The FaaS platform is responsible for the resource provisioning of the functions. Recently, there is a growing trend towards the execution of…
We obtain two new algorithms for partial fraction decompositions; the first is over algebraically closed fields, and the second is over general fields. These algorithms takes $O(M^2)$ time, where $M$ is the degree of the denominator of the…
The algebraic degree is an important parameter of Boolean functions used in cryptography. When a function in a large number of variables is not given explicitly in algebraic normal form, it might not be feasible to compute its degree.…
Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions…
We present ensmallen, a fast and flexible C++ library for mathematical optimization of arbitrary user-supplied functions, which can be applied to many machine learning problems. Several types of optimizations are supported, including…
We propose a general framework for computing Retarded Green's Functions (RGFs) on quantum computers by recasting their evaluation as a problem of circuit differentiation. Our proposal is based on real-time evolution and specifically…
Classical Principal Component Analysis (PCA) approximates data in terms of projections on a small number of orthogonal vectors. There are simple procedures to efficiently compute various functions of the data from the PCA approximation. The…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
The security of the RSA cryptosystem is based on the intractability of computing Euler's totient function phi(n) for large integers n. Although deriving phi(n) deterministically remains computationally infeasible for cryptographically…
The non-equidistant fast Fourier transform (NFFT) is an extension of the famous fast Fourier transform (FFT), which can be applied to non-equidistantly sampled data in time/space or frequency domain. It is an approximative algorithm that…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…