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At least 19 recordsLinked to original sources
arXiv2004

Some remarks on autoequivalences of categories

Prof. Boris I. Plotkin [arXiv:math.GM/0210194, arXiv:math.GM/0204245] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite important for universal algebraical geometry and concerns mainly the categories $Θ\sp 0 (X) $ of free universal algebras of some variety $Θ$ free generated by finite subsets of $X$. In the paper, a complete answer to the Plotkin's question is given: there are no proper autoequivalences of the category $Θ\sp 0 (X) $. Also some connected problems are discussed.

math.GM↗
arXiv2001

On the cardinality of the set of the real numbers

It is shown that any denumerable list L to which Cantor's diagonal method was applied is incomplete. However, this doesn't allow us to affirm that the cardinality of the real numbers of the interval [0, 1] is greater than the cardinality of the finite natural numbers. Paper withdrawn (its essential part is included in the version 3 of math.GM/0108119).

math.GM↗
arXiv2004

The second part of on duality triads' paper

The duality triads were defined in the preceding paper.(ArXiv: math.GM/0402260 v 1 Feb. 2004). Notation, enumeration of formulas and references is therefore to be continued hereby. In this paper Fibonomial triangle and further Pascal-like triangles including q-Gaussian one are given explicit interpretation as discrete time dynamical systems as it is the case with all duality triads.

math.GM↗
arXiv2019

A note on a recent attempt to prove Sendov's conjecture

Recently GM Sofi & SA Shabir [arXive: 1903.01850v2 [math.GM] 6 Mar 2019] made an attempt to prove the Sendov's conjecture. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

math.CV↗
arXiv2022

A New Operator of Primal Topological Spaces

Recently, Acharjee et al. [S. Acharjee, M. Özkoç and F. Y. Issaka, Primal topological spaces, arXiv:2209.12676[math.GM]] introduced a new structure in topology named primal. Primal is the duel structure of grill. The main purpose of this paper is to introduce an operator using primal and obtain some of its fundamental properties. Also, we define the notion of topology suitable for a primal. We not only obtain some characterizations of this new notion, but also investigate many properties.

math.GN↗
arXiv2012

Dynamical Sieve of Eratosthenes

In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers N, where each number n represents a time t. In the same way as the Eratosthenes' sieve, which iteratively mark as composite the multiples of each prime, starting at each prime. This dynamical prime number function P(s) zero-cross all composite numbers departing from primes, following a linear progression over time.

math.GM↗
arXiv2007

The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit

In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved: $gcd(R_a, R_b) = R_{gcd(a,b)}$; $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\geq1$, $b\geq1$ are integers. Dividers of numbers repunit, are researched by a degree of prime number.

math.GM↗
arXiv2007

Decartes' Perfect Lens

We give a new, elementary, purely analytical development of \textsc{Descartes}' theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry.

math.GM↗
arXiv2009

L'indice de Maslov dans les $JB^*$-triples

We construct a homotopy invariant index for pathes in the set of invertible tripotents in a JB*-triple that satisfy a Fredholm type condition with respect to a fixed invertible tripotent. That index generalizes the Maslov index in the Fredholm-Lagrangian of a symplectic Hilbert space.

math.GM↗
arXiv2007

On Carmichael's Conjecture

In this article we prove that equation $ϕ(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product of a very large number of primes (we conjecture that the number of such primes is infinite).

math.GM↗
arXiv2007

Combinatorics Of RNA Structures With Pseudoknots

In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all $k$-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore we enumerate for arbitrary $k$ all $k$-noncrossing, restricted RNA structures i.e. $k$-noncrossing RNA structures without 2-arcs i.e. arcs of the form $(i,i+2)$, for $1\le i\le n-2$.

math.CO↗
arXiv2007

Constructing a quadrilateral inside another one

Connect each vertex of a convex quadrilateral Q to the midpoint of the next (proceeding counterclockwise) side. The four connecting lines create an interior quadrilateral I. We study the ratio area(I)/area(Q). We also determine what happens to area(I)/area(Q) when the four midpoints are replaced by points which divide the sides in the ratio of rho to (1-rho) proceeding clockwise. Here rho is any fixed number satisfying 0 < rho < 1.

math.GM↗
arXiv2007

Remarks to Glazek's results on n-ary groups

It is a survey of the results obtained by K. Glazek's and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.

math.HO↗