Related papers: Finitely many implies infinitely many
It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.
We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new…
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…
"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…
Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…
The concept of infinity took centuries to achieve recognized status in the field of mathematics, despite the fact that it was implicitly present in nearly all mathematical endeavors. Here I explore the idea that a similar development might…
In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…
The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of…
The main result is the following: Let X be a finite set and D be a non empty family of choice functions for (X choose 2) closed under permutation of X. Then the following conditions are equivalent: (A) for any choice function c on (X choose…
Symmetries for wave equation with additional conditions are found. Some conditions yield infinite-dimensional symmetry algebra for the nonlinear equation. Ansatzes and solutions corresponding to the new symmetries were constructed.
We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…
The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that every model $M$ of the theory of finite arithmetic -- arithmetic with a…
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…