Related papers: Fundamentals of Computing
Computability logic is a formal theory of computational tasks and resources. Its formulas represent interactive computational problems, logical operators stand for operations on computational problems, and validity of a formula is…
``The purpose of life is to obtain knowledge, use it to live with as much satisfaction as possible, and pass it on with improvements and modifications to the next generation.'' This may sound philosophical, and the interpretation of words…
Quantum computing comes with the potential to push computational boundaries in various domains including, e.g., cryptography, simulation, optimization, and machine learning. Exploiting the principles of quantum mechanics, new algorithms can…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…
We give a survey of the foundations of statistical queries and their many applications to other areas. We introduce the model, give the main definitions, and we explore the fundamental theory statistical queries and how how it connects to…
The goal of this article is to clarify the meaning of Computational Thinking. We differentiate logical from computational reasoning and discuss the importance of Computational Thinking in solving problems. The three pillars of Computational…
These lectures introduce the basic ideas and practices of statistical analysis for particle physicists, using a real-world example to illustrate how the abstractions on which statistics is based are translated into practical application.
We introduce a lightweight and accessible approach to computation over the real numbers, with the aim of clarifying both the underlying concepts and their relevance in modern research. The material is intended for a broad audience,…
Accounting for resources is the central issue in computational efficiency. We point out physical constraints implicit in information readout that have been overlooked in classical computing. The basic particle-counting mode of read-out sets…
An efficient numerical algorithm for the computation of linking number is presented. The algorithm keep tracks or rounding error so that it can ensure the correctness of the results.
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
Human societies continuously transform scattered information into collective judgments and coordinated action, whether through markets discovering prices, governments allocating resources, communities enforcing norms, or science converging…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
In this paper we give a self contained introduction to the conceptional and mathematical foundations of quantum information theory. In the first part we introduce the basic notions like entanglement, channels, teleportation etc. and their…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
An indispensable part of our lives, computing has also become essential to industries and governments. Steady improvements in computer hardware have been supported by periodic doubling of transistor densities in integrated circuits over the…
The recent development of quantum computing, which uses entanglement, superposition, and other quantum fundamental concepts, can provide substantial processing advantages over traditional computing. These quantum features help solve many…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…