General Mathematics
Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The result $ \sum_{n\leq x}\varphi([x/n])=(6/\pi^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log…
We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.
The concept of porous numbers is presented. A number $k$ which is not a multiple of 10 is called {\it porous} if every number $m$ with sum of digits = $k$ and $k$ a divisor of both $m$ and digit reversal of $m$ has a zero in its digits. It…
In this paper, we connect four different branches of Mathematics: Statistics, Probability, Algebra and Discrete Mathematics with the objective of introducing new results on Markov chains and evolution algebras obtained by following a…
We present properties and invariants of Hamiltonian circuits in rectangular grids. It is proved that all circuits on a $2n \times 2n$ chessboard have at least $4n$ turns and at least $2n$ straights if $n$ is even and $2n+2$ straights if $n$…
When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that…
Fuzzy implication functions have been widely investigated, both in theoretical and practical fields. The aim of this work is to continue previous works related to fuzzy implications constructed by means of non necessarily associative…
A random variable X that is base b Benford will not in general be base c Benford when c is not equal to b. This paper builds on two of my earlier papers and is an attempt to cast some light on the issue of base dependence. Following some…
Most if not all of today's revolutionary technologies have a common foundation, namely the intelligent use of information. It is clear that computers play a central role: but the contribution of mathematics, though less visible, is no less…
We investigate a class of locally conformal almost K\"ahler structures and prove that, under some conditions, this class is a subclass of almost K\"ahler structures. We show that a locally conformal almost K\"ahler manifold admits a…
This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows from their…
The famous (3n + 1) or Collatz conjecture has admitted some progress over the last several decades towards the conclusion that the conjecture is true (i.e. that all Collatz sequences will eventually reach a value of one), but has stubbornly…
In an article entitled The Counterexample of a theorem in Wiener index of a fuzzy graph and application to illegal immigration networks, we have shown a few examples of incorrect proof of a theorem in the Wiener index. Now, in this article,…
To demonstrate the ability in standard arithmetic operations to perform a variety of digit manipulation tasks, a closed-form representation of the Conway Base-13 Function over the integers is given.
In this paper, we present a new definition and generalization of higher order Daehee of the first and second kind. Some new results for these polynomials and numbers are derived. Furthermore, some interesting special cases of the new…
In this paper, we give explicit evaluation for some integrals involving polylogarithm functions of types $\int_{0}^{x}t^{m} Li_{p}(t)\mathrm{d}t$ and $\int_{0}^{x}\log^{m}(t) Li_{p}(t)\mathrm{d}t$. Some more integrals involving the…
In this paper, we obtain explicit solutions to the Navier-Stokes equation and the Euler equation. For any initial velocity u0 and the force vector f, exact solutions can be explicitly solved as series, where the coefficients are all known…
In this paper, a general class of mixture of some densities is proposed. The proposed class contains some of classical and weighted distributions as special cases. Formulas for each of cumulative distribution function, reliability function,…