Related papers: A New Definition of Hypercomplex Analyticity
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In our thesis, we try to shed more light onto the complexity of quantum complexity classes by refining the related part of the hierarchy. First, we review the basic concepts of quantum computing in general. Then, inspired by BQP, we define…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions,…
Physicists study a wide variety of phenomena creating new interdisciplinary research fields by applying theories and methods originally developed in physics in order to solve problems in economics, social science, biology, medicine,…
It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The…
Quaternion, an extension of complex number, is the first discovered non-commutative division algebra by William Rowan Hamilton in 1843. In this article, we review the recent progress on building up the connection between the mathematical…
We are concerned with rigid analytic geometry in the general setting of Henselian fields $K$ with separated analytic structure, whose theory was developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and approaches of numerous…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a…
This paper is devoted to octonions that are the eight-dimensional hypercomplex numbers characterized by multiplicative non-associativity. The decomposition of the product of three octonions with the conjugated central factor into the sum of…
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…
Ultrafunctions are a particular class of functions defined on a Non Archimedean field E. They have been introduced and studied in some previous works. In this paper we develop the notion of fine ultrafunctions which improves the older…
In the algebra of complex quaternions $\mathbb{H(C)}$ we consider for the first time left- and right-$\psi$-hyperholomorphic functions. We justify the transition in left- and right-$\psi$-hyperholomorphic functions to a simpler basis i.e.…
Some examples are easier for humans to classify than others. The same should be true for deep neural networks (DNNs). We use the term example perplexity to refer to the level of difficulty of classifying an example. In this paper, we…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
Ultrafunctions are a particular class of functions defined on some non- Archimedean field. They provide generalized solutions to functional equa- tions which do not have any solutions among the real functions or the distributions. In this…
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb H$. In this work we deals with a…