Related papers: An improvement to a result about graph isomorphism…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of…
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…
We propose a generalization of the factorization method to the case when $\mathcal{G}$ is a finite dimensional Lie algebra such that $\mathcal{G}=\mathcal{G}_0\oplus M \oplus N$ (direct sum of vector spaces), where $\mathcal{G}_0$ is a…
In this work we produce a framework for constructing universal function approximators on graph isomorphism classes. We prove how this framework comes with a collection of theoretically desirable properties and enables novel analysis. We…
We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…
The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant "approximate" prime factorization, is the development of local approaches that…
An analytic proof is proposed of Wiener's theorem on factorization of positive definite matrix-functions.
To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…
Directed graphs occur throughout statistical modeling of networks, and exchangeability is a natural assumption when the ordering of vertices does not matter. There is a deep structural theory for exchangeable undirected graphs, which…
The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the…
In this short paper, following the most recent advances in complex network theory, a new approach to number theory with potential applications to other fields is proposed. The model by Garcia-Perez, Serrano and Boguna, introduces an…
The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit and compact determination of the isomorphism types of the groups…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…
In this note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be…
A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present…
It is well known [Lov\'asz, 67] that up to isomorphism a graph~$G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Thus, in principle, we can answer…
We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg…
The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly…