Related papers: WZ Theory, Chapter II
Computing is a critical driving force in the development of human civilization. In recent years, we have witnessed the emergence of intelligent computing, a new computing paradigm that is reshaping traditional computing and promoting…
Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial…
According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true,…
In this paper, we examine the state art of quantum computing and analyze its potential effects in scientific computing and cybersecurity. Additionally, a non-technical description of the mechanics of the listed form of computing is provided…
We provide an historical overview of how advances in technology influenced high school and university mathematical competitions in the United States and at the International Mathematical Olympiad. While students are not allowed the usage of…
The laptops, cell phones, and internet applications commonplace in our daily lives are all rooted in the idea of zeros and ones - in bits. This foundational element originated from the combination of mathematics and Claude Shannon's Theory…
Computational imaging has been playing a vital role in the development of natural sciences. Advances in sensory, information, and computer technologies have further extended the scope of influence of imaging, making digital images an…
In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…
In this article we are investigating the computers development process in the past decades in order to identify the factors that influence it the most. We describe such factors and use them to predict the direction of further development.…
This paper reviews the Church-Turing Thesis (or rather, theses) with reference to their origin and application and considers some models of "hypercomputation", concentrating on perhaps the most straight-forward option: Zeno machines (Turing…
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
Quantum mechanics---the theory describing the fundamental workings of nature---is famously counterintuitive: it predicts that a particle can be in two places at the same time, and that two remote particles can be inextricably and…
We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result. The unusual feature of the use of computers here was that a computer generated but human…
Artificial intelligence has impacted many aspects of human life. This paper studies the impact of artificial intelligence on economic theory. In particular we study the impact of artificial intelligence on the theory of bounded rationality,…
Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely…
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
The effects of computer-assisted and distance learning of geometric modeling and computer aided geometric design are studied. It was shown that computer algebra systems and dynamic geometric environments can be considered as excellent tools…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
More than hundred years ago the 'classic physics' was it in its full power, with just a few unexplained phenomena; which however led to a revolution and the development of the 'modern physics'. Today the computing is in a similar position:…
Despite of an active work of many researchers in the theory of quantum computations, this area still saves some mysterious charm. It is already an almost common idea, that maybe many fashionable current projects will fade in future, but…