General Mathematics
For coprime positive integers $a, b, c$, where $a+b=c$, $\gcd(a,b,c)=1$ and $1\leq a < b$, the famous $abc$ conjecture (Masser and Oesterl\`e, 1985) states that for $\varepsilon > 0$, only finitely many $abc$ triples satisfy $c >…
The question of a possible excitation and emergence of fractional type dynamics, as a more realistic framework for understanding emergence of complex systems, directly from a conventional integral order dynamics, in the form a continuous…
Let T=(V,E) be a tree with vertex set V and edge set E. A graceful labelling f of T is an injective function f from V into {0, 1, ..., |E|} such that if edge uv is assigned the label g(uv)=|f(u)-f(v)| then the function g from E into {1,…
The ultrapower $T^{\ast}$ of an arbitrary ordered set $T$ is introduced as an infinitesimal extension of $T$. It is obtained as the set of equivalence classes of the sequences in $T$, where the corresponding relation is generated by an…
In this article, we explore the Riemann zeta function with a perspective on primes and non-trivial zeros. We develop the Golomb's recurrence formula for the $n$th+1 prime, and assuming (RH), we propose an analytical recurrence formula for…
On this thesis we present the fuzzy sets, fuzzy numbers, the fractional derivative and also we discuss the solution of the first order of fuzzy hybrid equation.
In this paper we give a comprehensive presentation of the notions of filter base, filter and ultrafilter on single valued neutrosophic set and we investigate some of their properties and relationships. More precisely, we discuss properties…
We prove the Zabreiko's lemma in 2-Banach spaces. As an application we shall prove a version of the closed graph theorem and open mapping theorem.
A square, upper-triangular matrix U is a Cholesky root of a matrix M provided U*U=M, where * represents the conjugate transpose. Over finite fields, as well as over the reals, it suffices for U^TU=M. In this paper, we investigate the number…
In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind…
A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
Two articles published by Information Science discuss the derivatives of interval functions, in the sense of Svetoslav Markov. The authors of these articles tried to characterize for which functions and points such derivatives exist.…
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…
It is proposed that the validity, or not, of the Riemann Hypothesis might be established on the basis of the integral $$\int\frac{\xi(2s)}{\xi(s)}ds$$ where $$\xi(s)=(s-1)\pi^{-s/2}\Gamma(1+s/2)\zeta(s).$$
We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of…
Ova-angular rotations of a prime number are characterized, constructed using the Dirichlet theorem. The geometric properties arising from this theory are analyzed and some applications are presented, including Goldbach's conjecture, the…
Let $\lambda(m)$ be the $m$th coefficient of a modular form $f(z)=\sum_{m\geq 1} \lambda(m)q^m$ of weight $k\geq 4$, let $p^n$ be a prime power, and let $\varepsilon>0$ be a small number. An approximate of the Atkin-Serre conjecture on the…
What are simplest ways to construct a finite group from its atomic constituents? To understand part-whole relations between finite simple groups and the global structure of finite groups, we axiomatize complexity measures on finite groups.…
In this paper we generalize to associative superalgebras Gerstenhaber's work on cohomology structure of an associative algebra. We introduce two multiplications U and [-,-] on the cochain complex C^*(A;A) of an associative superalgebra A.…