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This paper introduces a new generalized superfactorial function (referable to as $n^{th}$- degree superfactorial: $sf^{(n)}(x)$) and a generalized hyperfactorial function (referable to as $n^{th}$- degree hyperfactorial: $H^{(n)}(x)$), and…
We consider the problem of determining the Fourier integral in the Hilbert space of square integrable functions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the…
Multimodular functions, primarily used in the literature of queueing theory, discrete-event systems, and operations research, constitute a fundamental function class in discrete convex analysis. The objective of this paper is to clarify the…
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the…
Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show…
We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the…
In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook…
Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on…
An unconditional inequality of the totient function is contributed to the literature. This result is associated with various problems about the distribution of prime numbers.
We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using…
By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second…
Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.
It is quite well-known from Kurt Godel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are…
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…
The Ackermann function is a famous total recursive binary function on the natural numbers. It is the archetypal example of such a function that is not primitive recursive, in the sense of classical recursion theory. However, and in seeming…
Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in…
Using the concepts of Hyperbolic Classification of Natural Numbers, Essential Regions and Goldbach Conjecture Function we prove that the existence of a proof of the Goldbach Conjecture in First-Order Arithmetic would imply the existence of…
The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
We prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices. We apply these results to study dependence of the solution of the initial…