Related papers: On a unified theory of numbers
In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
In this paper we present a new mathematical conception based on a new method for ordering the integers. The method relies on the assumption that negative numbers are beyond infinity, which goes back to Wallis and Euler. We also present a…
The present status of Unified Theories is summarized with special emphasis on their possible experimental tests. Outline: i) Unification of couplings; ii) Where can a positive signal come from? iii) HERA anomaly and Unification; iv) Recent…
In this paper, we introduced the theory of the sieve function transformation. Using the principle of sieve function transformation, we improved sieve method, and obtained the difference range of similar sieve function values. For this, we…
We develop large sample theory for merged data from multiple sources. Main statistical issues treated in this paper are (1) the same unit potentially appears in multiple datasets from overlapping data sources, (2) duplicated items are not…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
We present a set of principles and methodologies which may serve as foundations of a unifying theory of Mathematics. These principles are based on a new view of Grothendieck toposes as unifying spaces being able to act as `bridges' for…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing…
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
The physical models of a successful unified theory about the Universe must operate in different phase of matter evolution and different fields of physics. The attempts to build such wide range theory as a bunch of theories developed for…
We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.
Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement…