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Operator learning offers a powerful paradigm for solving parametric partial differential equations (PDEs), but scaling probabilistic neural operators such as the recently proposed Gaussian Processes Operators (GPOs) to high-dimensional,…
A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and…
Boolean algebra, the branch of mathematics where variables can assume only true or false value, is the theoretical basis of classical computation. The analogy between Boolean operations and electronic switching circuits, highlighted by…
Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…
A polynomial Turing compression (PTC) for a parameterized problem $L$ is a polynomial time Turing machine that has access to an oracle for a problem $L'$ such that a polynomial in the input parameter bounds each query. Meanwhile, a…
We investigate whether there are inherent limits of parallelization in the (randomized) massively parallel computation (MPC) model by comparing it with the (sequential) RAM model. As our main result, we show the existence of hard functions…
This paper serves as a bridge between quantum computing and analogical modeling (a general theory for predicting categories of behavior in varying contexts). Since its formulation in the early 1980s, analogical modeling has been…
Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high performance numerical simulation, on the other hand, transparent environments for distributed computing which follow…
We report the consequences of a destabilization process on a simulated General Purpose Analog Computer. This new technology overcomes problems linked with serial ambiguity, and provides an analog bias to encode algorithms whose complexity…
We introduce an error resilient distributed computing method based on an extension of the channel polarization phenomenon to distributed algorithms. The method leverages an algorithmic split operation that transforms two identical compute…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
A conjecture of Jozsa (arXiv:quant-ph/0508124) states that any polynomial-time quantum computation can be simulated by polylogarithmic-depth quantum computation interleaved with polynomial-depth classical computation. Separately, Aaronson…
A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the…
Conventional coded computing frameworks are predominantly tailored for structured computations, such as matrix multiplication and polynomial evaluation. Such tasks allow the reuse of tools and techniques from algebraic coding theory to…
Photonics is a promising platform for demonstrating a quantum computational advantage (QCA) by outperforming the most powerful classical supercomputers on a well-defined computational task. Despite this promise, existing proposals and…
A rational number can be naturally presented by an arithmetic computation (AC): a sequence of elementary arithmetic operations starting from a fixed constant, say 1. The asymptotic complexity issues of such a representation are studied e.g.…
Many domains, from deep learning to finance, require compounding real numbers over long sequences, often leading to catastrophic numerical underflow or overflow. We introduce generalized orders of magnitude (GOOMs), a principled extension…
Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…
Computer models are used as replacements for physical experiments in a large variety of applications. Nevertheless, direct use of the computer model for the ultimate scientific objective is often limited by the complexity and cost of the…
We motivate the use of differentiable probabilistic programming techniques in order to account for the large model-space inherent to astrophysical $\gamma$-ray analyses. Targeting the longstanding Galactic Center $\gamma$-ray Excess (GCE)…