综合数学
We present a new one parameter family of second derivative discontinuous solutions to the simplest scale invariant linear ordinary differential equation. We also point out how the construction could be extended to generate families of…
The origin of complex structures, randomness, and irreversibility are analyzed in the scale free SL(2,R) analysis, which is an extension of the ordinary analysis based on the recently uncovered scale free $C^{2^n-1}$ solutions to linear…
The framework of a new scale invariant analysis on a Cantor set $C\subset $ $% I=[0,1] $, presented originally in {\it S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009)}, is clarified and extended further. For an arbitrarily small…
The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant…
It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear…
From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.
The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the…
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$ ($q$-Airy function), Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson…
From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…
Migration has various dimensions; urbanization due to migration is one of them. In Rajasthan State, district level analysis of urbanization due to migrants shows trend invariably for all districts of the state, though the contribution in…
We prove that for any partition of a set which contains an infinite arithmetic (respectively geometric) progression into two disjoint subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is…
Hilbert's machine is a supertask machine inspired by Hilbert's Hotel whose functioning leads to a contradictory result involving w-ordering.
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…
This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of w-order leads to the same absurdity of Zeno's Dichotomy II. It also derives a contradictory result from the first Zeno's…
In [1], we give Dickson's conjecture on $N^n$. In this paper, we further give Dickson's conjecture on $Z^n$ and obtain an equivalent form of Green-Tao's conjecture [2]. Based on our work, it is possible to establish a general theory that…
The one-sided power spectrum of the fluctuation of Chebyshev's weighted prime counting function is numerically estimated based on samples of the fluctuating function of different sizes. The power spectrum is also estimated analytically for…
In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not…
A recursive approach for shrinking coefficients of an atomic decomposition is proposed. The corresponding algorithm evolves so as to provide at each iteration a) the orthogonal projection of a signal onto a reduced subspace and b) the index…
A few of the algebraic and topological properties of int. fuzzy continuity and int. fuzzy uniform continuity are investigated. Also, the concept of int. fuzzy uniform convergence is introduced thereafter a few result on int. fuzzy uniform…