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In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups.…
A method for absolute calibration of a photon-number resolving detector producing analog signals as the output is developed using a twin beam. The method gives both analog-to-digital conversion parameters and quantum detection efficiency…
Several predictive algorithms are described. Highlighted are variants that make predictions by superposing fields associated to the training data instances. They operate seamlessly with categorical, continuous, and mixed data. Predictive…
We develop a new algorithm for fitting circles that does not have drawbacks commonly found in existing circle fits. Our fit achieves ultimate accuracy (to machine precision), avoids divergence, and is numerically stable even when fitting…
Evaluating performance across optimization algorithms on many problems presents a complex challenge due to the diversity of numerical scales involved. Traditional data processing methods, such as hypothesis testing and Bayesian inference,…
There is no, nor will there ever be, single best clustering algorithm. Nevertheless, we would still like to be able to distinguish between methods that work well on certain task types and those that systematically underperform. Clustering…
We present a systematic, algebraically based, design methodology for efficient implementation of computer programs optimized over multiple levels of the processor/memory and network hierarchy. Using a common formalism to describe the…
Ideal quantum random number generators (QRNGs) can produce algorithmically random and thus incomputable sequences, in contrast to pseudo-random number generators. However, the verification of the presence of algorithmic randomness and…
Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by…
This paper considers the problem of cardinality estimation in data stream applications. We present a statistical analysis of probabilistic counting algorithms, focusing on two techniques that use pseudo-random variates to form…
The main object of this paper is to construct new Durrmeyer type operators which have better features than the classical one. Some results concerning the rate of convergence and asymptotic formulas of the new operator are given. Finally,…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
Historically, the notion of effective algorithm is closely related to the Church-Turing thesis. But effectivity imposes no restriction on computation time or any other resource; in that sense, it is incompatible with engineering or physics.…
Chernoff approximations to strongly continuous one-parameter semigroups give solutions to a wide class of differential equations. This paper studies the rate of convergence of the Chernoff approximations. We provide simple natural examples…
Hartle and Srednicki have suggested that standard quantum theory does not favor our typicality. Here an alternative version is proposed in which typicality is likely, Eventual Quantum Mechanics. This version allows one to calculate…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was…
We propose a distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) to minimize the sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications…
We consider classical and quantum algorithms which have a duality property: roughly, either the algorithm provides some nontrivial improvement over random or there exist many solutions which are significantly worse than random. This enables…
In nearly every discipline, scientific computations are limited by the cost and speed of computation. For example, the best-known exact algorithms for the canonical Traveling Salesman Problem would take centuries to run on an instance of…