综合数学
In this paper, concept of fuzzy continuous operator, fuzzy bounded linear operator are introduced in fuzzy strong $\phi$-b-normed linear spaces and their relations are studied. Idea of operator fuzzy norm is developed and completeness of…
Let $f$ be a nonnegative function of class $C^k$ ($k \geq 2$) such that $f^{(k)}$ is H\''older continuous with exponent $\alpha$ in $(0,1]$. If $f'(x) = \cdots = f^{(k)}(x) = 0$ when $f(x) = 0$, we show that $f^{\mu}$ is differentiable for…
In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $a\in R$, $a<0$, then the equation has infinitely many roots.…
In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy…
Lie pseudogroups are groups of transformations solutions of systems of ordinary (OD) or partial differential (PD) equations. The purpose of this paper is to present an elementary summary of a few recent results obtained through the…
Despite many applications regarding fractional calculus have been reported in literature, it is still unknown how to model some practical process. One major challenge in solving such a problem is that, the nonlocal property is needed while…
Fermat's Last theorem (FLT) famously states that the equation $x^n+y^n=z^n$ has no solution in positive integers $x, y, z$ for any integer exponent $n>2$. But does this theorem have a quantitative version? Upon initial investigation we…
Being a pair of dual concepts, the normalized distance and similarity measures are very important tools for decision-making and pattern recognition under intuitionistic fuzzy sets framework. To be more effective for decision-making and…
Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the…
The Watson-Harkins sum involving the product of the cosine and cosecant functions is extended to derive the finite sum of generalized Hurwitz-Lerch Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A transformation…
In this paper we shall show amazing behavior of some discrete maps using Collatze like problems and some advanced theories in analytic number theory and dynamical system,we have investigated the driven cubic-quintic Duffing equation such…
Expressions of type $(p^q-1)/(p-1)$ and $a^2+ab+b^2$, where $a, b$ are natural and $p, q$ are prime numbers, are studied.
In this paper, we intend the concept of rough cubic Pythagorean fuzzy ideals in the semigroup. By using this notion, we discuss lower approximation and upper approximation of cubic Pythagorean fuzzy left (right) ideals, bi-ideals, interior…
We define the asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions in the paper. Classes of additive and multiplicative arithmetic functions are singled out for which the asymptotics coincides "almost…
In this paper, contractive mappings of \'Ciri\'c-Matkowski type in fuzzy metric spaces are studied. A class $\Psi_1$ of gauge functions $\psi:(0,1]\to(0,1]$ such that, for any $r\in(0,1)$, there exists $\rho\in(r,1)$ such that $1-r> \tau…
The division between two vectors belonging to the same vector space is obtained by elementary procedures of vector algebra and is defined by a matrix. This representation is obtained for two and three dimensional vector spaces. A new vector…
In this paper, we investigate the nature of Einstein solitons, whether it is steady, shrinking or expanding on almost $\alpha$-cosymplectic $3$-manifolds. We also prove that a simply connected homogeneous almost $\alpha$-cosymplectic…
In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and Euler's…
We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector…
Let $\alpha=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability…