All posts under category "Mathematics / Math.GM"
A trivial formalization is given for the informal reasonings presented in a series of papers by Ya.D.Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.
We introduce Hausdorff-Colombeau measure in respect with negative fractal dimensions. Axiomatic quantum field theory in spacetime with negative fractal dimensions is proposed.Spacetime is modelled as a multifractal subset of $R^{4}$ with positive and negative fractal dimensions.The cosmological cons
About global and local algebraic integrability of ovals. A contribution to clarify Newton results and relative comments on his work done by Arnol'd and Pourciau. A possibile application to air damper sections computation is offered, as example of unexpected link between pure mathematics and industri
New recursion relations for the Riemann zeta function are introduced. Their derivation started from the standard functional equation. The new functional equations have both real and imaginary increment versions and can be applied over the whole complex plane. We have developed various versions of th
In this paper, we investigate special Smarandache curves according to Bishop frame in Euclidean 3-space and we give some differential geometric properties of Smarandache curves. Also we find the centers of the osculating spheres and curvature spheres of Smarandache curves.
In this paper I present a counter-example to the conjecture: The Fiedler vector for the graph Laplacian of a tree has its most extreme values at the verticies which are the farthest apart. This counter-example looks roughly like a flower and so I have named it the 'Fiedler rose'.
We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $zeta(s)$.
Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partiti
We generalize the concept of disjunction.
Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms? In this paper we propose two answers: a) at the most elem
In this article, we first give a proof on the Arnold chord conjecture which states that every Reeb flow has at least as many Reeb chords as a smooth function on the Legendre submanifold has critical points on contact manifold. Second, we prove that every Reeb flow has at least as many close Reeb orb
The singularities of the $Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$ lim_{zrightarrow 0} frac{Gamma(nz)}{Gamma(z)} = frac 1 n; hspace{0.4in} li
In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers which are the leglengths of a Pythagorean triangle. Using th
In this paper, we introduce cone normed linear space, study the cone convergence with respect to cone norm. Finally, we prove the completeness of a finite dimensional cone normed linear space.
In this paper, to solve the invariant subspace problem, contraction operators are classified into three classes ; (Case 1) completely non-unitary contractions with a non-trivial algebraic element, (Case 2) completely non-unitary contractions without a non-trivial algebraic element, or (Case 3) contr
Intuitionistic Fuzzy derivative, Intuitionistic Fuzzy Gateaux derivative, Intuitionistic Fuzzy Fr'{e}chet derivative are defined and a few of their properties are studied. The relation between Intuitionistic Fuzzy Gateaux derivative and Intuitionistic Fuzzy Fr'{e}chet derivative are emphasized.
In this note, we generalize an ancient Greek inequality about the sequence of primes to the cases of arithmetic progressions even multivariable polynomials with integral coefficients. We also refine Bouniakowsky's conjecture [16] and Conjecture 2 in [22]. Moreover, we give two remarks on conjectures
We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.
We solve some forms of non homogeneous differential equations in one and two dimensions. By expanding the solution into whell-posed closed form-Eisenstein series the solution itself is quite simple and elementary. Also we consider Fourier series solutions of linear differential operator equations. I
In this article we present evaluations of continued fractions studied by Ramanujan. More precisely we give the complete polynomial equations of Rogers-Ramanujan and other continued fractions, using tools from the elementary theory of the Elliptic functions. We see that all these fractions are roots
In this article we give, for the fist time the solution of the general difference equation of 2-degree. We also give as application the expansion of a continued fraction into series, which was first proved, found in the past by the author.
The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be present. Such a clarification of the basic structures from wher
The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is presented by employing the Pl$ddot{u}$cker relations.
Suppose that the Riemann hypothesis for the Riemann $zeta$-function is false. There have yet been no workable results that the falsity of the Riemann hypothesis implies. The result of this article may be considered as more workable for the sake of deducing other new results on the Riemann hypothesis
In this note, we give some explicit upper and lower bounds for the summation $sum_{0<gammaleq T}frac{1}{gamma}$, where $gamma$ is the imaginary part of nontrivial zeros $rho=beta+igamma$ of $zeta(s)$.
The material of this work is aimed at mathematics educators, as well as math specialists with a keen interest in progressions. In this paper, we study the subject of arithmetic, geometric, mixed, and harmonic progressions or sequences. Some of the material found in Sections 2,3,4,and 5, can also be
Number of results in number theory have been developed using a new method. The Goldbach binary conjecture in strengthened formulation have been among them.
Following the definition of intuitionistic fuzzy n-norm [ 3 ], we have introduced the definition of intuitionistic fuzzy norm (in short IFN) over a linear space and there after a few results on intuitionistic fuzzy normed linear space and finite dimensional intuitionistic fuzzy normed linear space.
The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relati
Summing the values of the real portion of the logarithmic integral of n^rho, where rho is one of a consecutive series of zeros of the Riemann zeta function, reveals an unexpected periodicity to the sum. This is the cycle problem.
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always finite.
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 optimal data base of values d to be tested.
As we known, the {it Seifert-Van Kampen theorem} handles fundamental groups of those topological spaces $X=Ucup V$ for open subsets $U, Vsubset X$ such that $Ucap V$ is arcwise connected. In this paper, this theorem is generalized to such a case of maybe not arcwise-connected, i.e., there are $C_1$,
The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $U$, the points of $U$ different from the $z_i$, and such that $Arg(P(z))=theta$, form a regular $n$-gon, where $n$ is the deg
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the study of Jacobi elliptic theta functions theory.
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in implementation of the reductio method of proof and by identifying er
Using the delta correction to the standard free energy cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experime
Based on the Liouville-Weyl definition of the fractional derivative, a new direct fractional generalization of higher order derivatives is presented. It is shown, that the Riesz and Feller derivatives are special cases of this approach.
In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
We show that two tensor permutation matrices permutate tensor product of rectangle matrices. Some examples, in the particular case of tensor commutation matrices, for studying some linear matrix equations are given.
In the beginning of this paper, we present the general solution to the trigonometric equation asinx+bcosx=c. After that, we focus on the case when a^2+b^2=c^2. In this case, the general solution is expressed in terms of the acute angle theta which satisfies tan(theta)=a/b+c .From the right trianle w
The present study further strengthens the use of the Keedwell CIPQ against attack on a system by the use of the Smarandache Keedwell CIPQ for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell. This is done as follows. By constructing two S-isotopic S-quas
In this article using the theory of Eisenstein series, we give rise to the complete evaluation of two Gauss hypergeometric functions. Moreover we evaluate the modulus of each of these functions and the values of the functions in terms of the complete elliptic integral of the first kind. As applicati
A reproducing kernel Hilbert space (RKHS) has four well-known easily derived properties. Since these properties are usually not emphasized as a simple means of gaining insight into RKHS structure, they are singled out and proved in this brief article.
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 urbanization by migrants varies from district to district. In
Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of t
In this book, the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set n-codes are defined. Several examples are given to enable the reader to understand the concept. These new cl
Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is Aleph0. Cantor proved in 1891 with the diagonal argument that the set of real numbers is uncounta
The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method for nonlinear PDEs. In this regard, we consider an appropriate space ML(X) of normal lower semi-continuous functions. The space ML(X)= appears in the ring theory o
The main purpose of this paper is to calculate exact quasiclassical asymptotic of the quantum averages without any reference to the corresponding quasiclassical asimptotic of the Schr'odinger wave function {Psi}(x,t)given via Maslov canonical operator. We suggest a new asymptotic representation for
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