Related papers: Dividing by zero - how bad is it, really?
An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…
Field experiments are often difficult and expensive to make. To bypass these issues, industrial companies have developed computational codes. These codes intend to be representative of the physical system, but come with a certain amount of…
We explore the Weihrauch degree of the problems ``find a bad sequence in a non-well quasi order'' ($\mathsf{BS}$) and ``find a descending sequence in an ill-founded linear order'' ($\mathsf{DS}$). We prove that $\mathsf{DS}$ is strictly…
Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning…
In this article we call a sequence $(a_n)_n$ of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function $r:\mathbb{N}\to\mathbb{N}$ the sequence $(d(a_{r(n+1)},a_{r(n)}))_n$ converges…
We show that the nonlinear real arithmetic theory (NRA) as defined in the SMTLIB standard is undecidable. The undecidability arises from the treatment of division by zero as an uninterpreted function, which allows encoding integer…
Computational feasibility is a widespread concern that guides the framing and modeling of biological and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the…
In our previous work there was some indication that Partition Sort could be having a more robust average case O(nlogn) complexity than the popular Quick Sort. In our first study in this paper, we reconfirm this through computer experiments…
In this review we present hyper-dual numbers as a tool for the automatic differentiation of computer programs via operator overloading. We start with a motivational introduction into the ideas of algorithmic differentiation. Then we…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
The remarkable results for denoising in computer vision using diffusion models given in \cite{SDWMG,HJA,HHG} yield a robust mathematical justification for algorithms based on crucial properties of a sequence of Gaussian independent $N(0,1)$…
We study the computational content of various theorems with reverse mathematical strength around Arithmetical Transfinite Recursion ($\mathsf{ATR}_0$) from the point of view of computability-theoretic reducibilities, in particular Weihrauch…
Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known.…
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,…
Despite remarkable achievements in its practical tractability, the notorious class of NP-complete problems has been escaping all attempts to find a worst-case polynomial time-bound solution algorithms for any of them. The vast majority of…
A major hurdle in machine learning is scalability to massive datasets. One approach to overcoming this is to distribute the computational tasks among several workers. \textit{Gradient coding} has been recently proposed in distributed…
The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group…
The decision to incorporate cross-validation into validation processes of mathematical models raises an immediate question - how should one partition the data into calibration and validation sets? We answer this question systematically: we…
The Deustch-Jozsa problem is one of the most basic ways to demonstrate the power of quantum computation. Consider a Boolean function f : {0,1}^n to {0,1} and suppose we have a black-box to compute f. The Deutsch-Jozsa problem is to…
Given a large dataset and an estimation task, it is common to pre-process the data by reducing them to a set of sufficient statistics. This step is often regarded as straightforward and advantageous (in that it simplifies statistical…