rss_2.0Communications in Mathematics FeedSciendo RSS Feed for Communications in Mathematics in Mathematics 's Cover integral transform and its application in the propagation of Lorentz-Gaussian beams<abstract><title style='display:none'>Abstract</title><p>The aim of the present note is to derive an integral transform <disp-formula><alternatives><graphic xmlns:xlink="" xlink:href="graphic/j_cm-2021-0030_eq_001.png"/><mml:math xmlns:mml="" display="block"><mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mo>∞</mml:mo></mml:msubsup><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow/><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>γ</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ζ</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>J</mml:mi><mml:mi>μ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>χ</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>I = \int_0^\infty {{x^{s + 1}}{e^{ - \beta x}}^{2 + \gamma x}{M_{k,v}}} \left( {2\zeta {x^2}} \right)J\mu \left( {\chi x} \right)dx,</tex-math></alternatives></disp-formula> involving the product of the Whittaker function <italic>M<sub>k,</sub></italic><sub>ν</sub> and the Bessel function of the first kind <italic>J<sub>µ </sub></italic>of order <italic>µ</italic>. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters <italic>k</italic> and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]).</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00On the completeness of total spaces of horizontally conformal submersions<abstract><title style='display:none'>Abstract</title><p>In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of <italic>g-</italic>natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00G-tridiagonal majorization on M<abstract><title style='display:none'>Abstract</title><p>For <italic>X, Y</italic> ∈ M<italic><sub>n,m</sub></italic>, it is said that <italic>X</italic> is <italic>g-tridiagonal</italic> majorized by <italic>Y</italic> (and it is denoted by <italic>X</italic> ≺<italic><sub>gt </sub>Y</italic>) if there exists a tridiagonal g-doubly stochastic matrix <italic>A</italic> such that <italic>X</italic> = <italic>AY</italic>. In this paper, the linear preservers and strong linear preservers of ≺<italic><sub>gt </sub></italic>are characterized on M<italic><sub>n,m</sub></italic>.</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00A non-linear discrete-time dynamical system related to epidemic SISI model<abstract><title style='display:none'>Abstract</title><p>We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00Some type of semisymmetry on two classes of almost Kenmotsu manifolds<abstract><title style='display:none'>Abstract</title><p>The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (<italic>k, µ</italic>)-almost Kenmotsu manifold satisfying the curvature condition <italic>Q</italic> · <italic>R</italic> = 0 is locally isometric to the hyperbolic space ℍ<sup>2</sup><italic><sup>n</sup></italic><sup>+1</sup>(−1). Also in (<italic>k, µ</italic>)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇<italic>R</italic> = 0), (2) semisymmetry (<italic>R</italic>·<italic>R</italic> = 0), (3) <italic>Q</italic>(<italic>S, R</italic>) = 0, (4) <italic>R</italic>·<italic>R</italic> = <italic>Q</italic>(<italic>S, R</italic>), (5) locally isometric to the hyperbolic space ℍ<sup>2</sup><italic><sup>n</sup></italic><sup>+1</sup>(−1) are equivalent. Further, it is proved that a (<italic>k, µ</italic>)′ -almost Kenmotsu manifold satisfying <italic>Q</italic> · <italic>R</italic> = 0 is locally isometric to ℍ<italic><sup>n</sup></italic><sup>+1</sup>(−4) × ℝ<italic><sup>n</sup></italic> and a (<italic>k, µ</italic>)′ -almost Kenmotsu manifold satisfying any one of the curvature conditions <italic>Q</italic>(<italic>S, R</italic>) = 0 or <italic>R</italic> · <italic>R</italic> = <italic>Q</italic>(<italic>S, R</italic>) is either an Einstein manifold or locally isometric to ℍ<italic><sup>n</sup></italic><sup>+1</sup>(−4) × ℝ<italic><sup>n</sup></italic>. Finally, an illustrative example is presented.</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00A note on the solvability of homogeneous Riemann boundary problem with infinity index<abstract><title style='display:none'>Abstract</title><p>In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.</p></abstract>ARTICLE2021-10-01T00:00:00.000+00:00Existentially closed Leibniz algebras and an embedding theorem<abstract> <title style='display:none'>Abstract</title> <p>In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Rota-type operators on 3-dimensional nilpotent associative algebras<abstract> <title style='display:none'>Abstract</title> <p>We give the description of Rota–Baxter operators, Reynolds operators, Nijenhuis operators and average operators on 3-dimensional nilpotent associative algebras over ℂ.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00A generalisation of Amitsur’s A-polynomials<abstract> <title style='display:none'>Abstract</title> <p>We find examples of polynomials <italic>f</italic> ∈ <italic>D</italic> [<italic>t</italic>; <italic>σ, δ</italic>] whose eigenring <italic>ℰ</italic>(<italic>f</italic>) is a central simple algebra over the field <italic>F</italic> = <italic>C</italic> ∩ Fix(<italic>σ</italic>) ∩ Const(<italic>δ</italic>).</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Conservative algebras of 2-dimensional algebras, III<abstract> <title style='display:none'>Abstract</title> <p>In the present paper we prove that every local and 2-local derivation on conservative algebras of 2-dimensional algebras are derivations. Also, we prove that every local and 2-local automorphism on conservative algebras of 2-dimensional algebras are automorphisms.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Actions of the additive group on certain noncommutative deformations of the plane<abstract> <title style='display:none'>Abstract</title> <p>We connect the theorems of <xref ref-type="bibr" rid="j_cm-2021-0024_ref_018">Rentschler [18]</xref> and <xref ref-type="bibr" rid="j_cm-2021-0024_ref_010">Dixmier [10]</xref> on locally nilpotent derivations and automorphisms of the polynomial ring <italic>A</italic><sub>0</sub> and of the Weyl algebra <italic>A</italic><sub>1</sub>, both over a field of characteristic zero, by establishing the same type of results for the family of algebras <disp-formula id="j_cm-2021-0024_eq_001"> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_cm-2021-0024_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>|</mml:mo><mml:mi>y</mml:mi><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>〉</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math> <tex-math>{A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle ,</tex-math> </alternatives> </disp-formula> , where <italic>h</italic> is an arbitrary polynomial in <italic>x</italic>. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[<italic>t</italic>]-comodule algebra structures on <italic>A<sub>h</sub></italic>. We also compute the Makar-Limanov invariant of absolute constants of <italic>A<sub>h</sub></italic> over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of <italic>A<sub>h</sub></italic>.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Weak polynomial identities and their applications<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>R</italic> be an associative algebra over a field <italic>K</italic> generated by a vector subspace <italic>V</italic>. The polynomial <italic>f</italic>(<italic>x</italic><sub>1</sub>, . . . , <italic>x<sub>n</sub></italic>) of the free associative algebra <italic>K</italic>〈<italic>x</italic><sub>1</sub>, <italic>x</italic><sub>2</sub>, . . .〉 is a weak polynomial identity for the pair (<italic>R, V</italic>) if it vanishes in <italic>R</italic> when evaluated on <italic>V</italic>. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Gradedness of the set of rook placements in<abstract> <title style='display:none'>Abstract</title> <p>A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in <italic>A<sub>n−</sub></italic><sub>1</sub> with respect to a slightly different order and prove that this poset is graded.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras<abstract> <title style='display:none'>Abstract</title> <p>In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz algebras for the first and second families and determine those algebras in the first two classes of filiform Leibniz algebras that are non-strongly nilpotent.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices<abstract> <title style='display:none'>Abstract</title> <p>We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Certain partitions on a set and their applications to different classes of graded algebras<abstract> <title style='display:none'>Abstract</title> <p>Let (𝔘, <italic>ɛ<sub>u</sub></italic>) and (𝔅, <italic>ɛ<sub>b</sub></italic>) be two pointed sets. Given a family of three maps <italic>ℱ</italic> = {<italic>f</italic><sub>1</sub> : 𝔘 → 𝔘; <italic>f</italic><sub>2</sub> : 𝔘 × 𝔘 <italic>→</italic> 𝔘; <italic>f</italic><sub>3</sub> : 𝔘 × 𝔘 <italic>→</italic> 𝔅}, this family provides an adequate decomposition of 𝔘 \ {<italic>ɛ<sub>u</sub></italic>} as the orthogonal disjoint union of well-described <italic>ℱ</italic>-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak <italic>H</italic><sup>*</sup>-algebras.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Unified computational approach to nilpotent algebra classification problems<abstract> <title style='display:none'>Abstract</title> <p>In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.</p> </abstract>ARTICLE2021-07-15T00:00:00.000+00:00Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1<abstract> <title style='display:none'>Abstract</title> <p>It is known that a real symmetric circulant matrix with diagonal entries <italic>d</italic> ≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2<italic>d</italic> + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries <italic>d</italic> ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with <italic>d</italic> different from an odd integer is <italic>n</italic> = 2<italic>d</italic> + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ<italic><sub>m</sub></italic>. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00Jets and the variational calculus<abstract> <title style='display:none'>Abstract</title> <p>We review the approach to the calculus of variations using Ehresmann’s theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00On the Mathematical Theory of Records<abstract> <title style='display:none'>Abstract</title> <p>In the present work, we briefly analyze the development of the mathematical theory of records. We first consider applications associated with records. We then view distributional and limit results for record values and times. We further present methods of generation of continuous records. In the end of this work, we discuss some tests based on records.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00en-us-1