rss_2.0Communications in Mathematics FeedSciendo RSS Feed for Communications in Mathematicshttps://sciendo.com/journal/CMhttps://www.sciendo.comCommunications in Mathematics 's Coverhttps://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/61c4b769f9200d2343fd48a9/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220128T003823Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKDOZOEZ7H%2F20220128%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b800a043ef9a1e3b6319d3ec59ca1360f7dda80582bccb24b603fa87715bb7e0200300A note on the solvability of homogeneous Riemann boundary problem with infinity indexhttps://sciendo.com/article/10.2478/cm-2021-0033<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>ARTICLE2021-12-23T00:00:00.000+00:00On the completeness of total spaces of horizontally conformal submersionshttps://sciendo.com/article/10.2478/cm-2021-0031<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>ARTICLE2021-12-23T00:00:00.000+00:00G-tridiagonal majorization on Mhttps://sciendo.com/article/10.2478/cm-2021-0027<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 g-tridiagonal majorized by <italic>Y</italic> (and it is denoted by <italic>X ≺<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>ARTICLE2021-12-23T00:00:00.000+00:00()-derivations on semiprime rings and Banach algebrashttps://sciendo.com/article/10.2478/cm-2021-0013<abstract><title style='display:none'>Abstract</title><p>Let <italic>ℛ</italic> be a semiprime ring with unity <italic>e</italic> and <italic>ϕ</italic>, <italic>φ</italic> be automorphisms of <italic>ℛ</italic>. In this paper it is shown that if <italic>ℛ</italic> satisfies <disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0013_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mn>2</mml:mn><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right)</tex-math></alternatives></disp-formula> for all <italic>x</italic> ∈ <italic>ℛ</italic> and some fixed integer <italic>n</italic> ≥ 2, then <italic>𝒟</italic> is an (<italic>ϕ</italic>, <italic>φ</italic>)-derivation. Moreover, this result makes it possible to prove that if <italic>ℛ</italic> admits an additive mappings <italic>𝒟, 𝒢</italic> : <italic>ℛ</italic> → <italic>ℛ</italic> satisfying the relations <disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0013_eq_002.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mn>2</mml:mn><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right)</tex-math></alternatives></disp-formula><disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0013_eq_003.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mn>2</mml:mn><mml:mi>𝒢</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>𝒢</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>𝒟</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right),</tex-math></alternatives></disp-formula> for all <italic>x</italic> ∈ <italic>ℛ</italic> and some fixed integer <italic>n</italic> ≥ 2, then <italic>𝒟</italic> and <italic>𝒢</italic> are (<italic>ϕ</italic>, <italic>φ</italic>)--derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00A non-linear discrete-time dynamical system related to epidemic SISI modelhttps://sciendo.com/article/10.2478/cm-2021-0032<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>ARTICLE2021-12-23T00:00:00.000+00:00Multiplicative Lie triple derivations on standard operator algebrashttps://sciendo.com/article/10.2478/cm-2021-0012<abstract><title style='display:none'>Abstract</title><p>Let <italic>χ</italic> be a Banach space of dimension <italic>n &gt;</italic> 1 and 𝔘 ⊂ <italic>𝔅</italic>(<italic>χ</italic>) be a standard operator algebra. In the present paper it is shown that if a mapping <italic>d</italic> : 𝔘 → 𝔘 (not necessarily linear) satisfies <disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0012_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>U</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>W</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>U</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>W</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>U</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>W</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>U</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>W</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right]</tex-math></alternatives></disp-formula> for all <italic>U, V, W</italic> ∈ 𝔘, then <italic>d</italic> =<italic>ψ</italic> + <italic>τ</italic>, where <italic>ψ</italic> is an additive derivation of 𝔘 and <italic>τ</italic> : 𝔘 → 𝔽<italic>I</italic> vanishes at second commutator [[<italic>U, V</italic> ], <italic>W</italic> ] for all <italic>U, V, W</italic> ∈ 𝔘. Moreover, if <italic>d</italic> is linear and satisfies the above relation, then there exists an operator <italic>S</italic> ∈ 𝔘 and a linear mapping <italic>τ</italic> from 𝔘 into 𝔽<italic>I</italic> satisfying <italic>τ</italic> ([[<italic>U, V</italic> ], <italic>W</italic> ]) = 0 for all <italic>U, V, W</italic> ∈ 𝔘, such that <italic>d</italic>(<italic>U</italic>) = <italic>SU − US</italic> + <italic>τ</italic> (<italic>U</italic>) for all <italic>U</italic> ∈ 𝔘.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00On Balancing and Lucas-balancing Quaternionshttps://sciendo.com/article/10.2478/cm-2021-0010<abstract><title style='display:none'>Abstract</title><p>The aim of this article is to investigate two new classes of quaternions, namely, balancing and Lucas-balancing quaternions that are based on balancing and Lucas-balancing numbers, respectively. Further, some identities including Binet’s formulas, summation formulas, Catalan’s identity, etc. concerning these quaternions are also established.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00A Weighted Eigenvalue Problems Driven by both (·)-Harmonic and (·)-Biharmonic Operatorshttps://sciendo.com/article/10.2478/cm-2020-0011<abstract><title style='display:none'>Abstract</title><p>The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both <italic>p</italic>(·)-Harmonic and <italic>p</italic>(·)-biharmonic operators <disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2020-0011_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>w</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mtext>in</mml:mtext><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mn>0</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><tex-math>\eqalign{&amp; \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left| u \right|^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr &amp; \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr}</tex-math></alternatives></disp-formula> is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces <italic>L<sup>p</sup></italic><sup>(·)</sup>(Ω) and <italic>W<sup>m,p</sup></italic><sup>(·)</sup>(Ω).</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00Sur la variation de certaines suites de parties fractionnaireshttps://sciendo.com/article/10.2478/cm-2020-0021<abstract><title style='display:none'>Abstract</title><p>Let <italic>b &gt; a &gt;</italic> 0. We prove the following asymptotic formula <disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2020-0021_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:munder><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>a</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>b</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mfrac><mml:mi>ζ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msqrt><mml:mrow><mml:mi>c</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msqrt><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mo>/</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>\sum\limits_{n \geqslant 0} {\left| {\left\{ {x/\left( {n + a} \right)} \right\} - \left\{ {x/\left( {n + b} \right)} \right\}} \right| = {2 \over \pi }\zeta \left( {3/2} \right)\sqrt {cx} + O\left( {{c^{2/9}}{x^{4/9}}} \right),}</tex-math></alternatives></disp-formula> with <italic>c</italic> = <italic>b</italic> − <italic>a</italic>, uniformly for <italic>x</italic> ⩾ 40<italic>c</italic><sup>−5</sup>(1 + <italic>b</italic>)<sup>27/2</sup>.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00Remarks on Ramanujan’s inequality concerning the prime counting functionhttps://sciendo.com/article/10.2478/cm-2021-0014<abstract><title style='display:none'>Abstract</title><p>In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0014_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>π</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><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:mo>&lt;</mml:mo><mml:mfrac><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>log</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac><mml:mi>π</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mi>e</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>\pi \left( {{x^2}} \right) &lt; {{ex} \over {\log x}}\pi \left( {{x \over e}} \right)</tex-math></alternatives></inline-formula> for <italic>x</italic> sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0014_eq_002.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mo>log</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math><tex-math>{x \over {\log x}}</tex-math></alternatives></inline-formula> on its right hand side by the factor <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0014_eq_003.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mo>log</mml:mo><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math><tex-math>{x \over {\log x - h}}</tex-math></alternatives></inline-formula> for a given <italic>h</italic>, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00An integral transform and its application in the propagation of Lorentz-Gaussian beamshttps://sciendo.com/article/10.2478/cm-2021-0030<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="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0030_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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,ν </sub></italic>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 <italic>ν</italic> 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>ARTICLE2021-12-23T00:00:00.000+00:00Some type of semisymmetry on two classes of almost Kenmotsu manifoldshttps://sciendo.com/article/10.2478/cm-2021-0029<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>(<italic>−</italic>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>)<italic><sup>′</sup></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>(<italic>−</italic>4) <italic>×</italic> ℝ<italic><sup>n </sup></italic>and a (<italic>k, µ</italic>)<italic><sup>′-</sup></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) <italic>×</italic> ℝ<italic><sup>n</sup></italic>. Finally, an illustrative example is presented.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00A note on the volume of ∇-Einstein manifolds with skew-torsionhttps://sciendo.com/article/10.2478/cm-2020-0009<abstract><title style='display:none'>Abstract</title><p>We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville [15] related with the first variation of the volume on a compact Einstein manifold.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00Symmetric identity for polynomial sequences satisfying ′() = ( + 1)()https://sciendo.com/article/10.2478/cm-2021-0011<abstract><title style='display:none'>Abstract</title><p>Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying <italic>A</italic>′<italic><sub>n</sub></italic><sub>+1</sub>(<italic>x</italic>) = (<italic>n</italic> + 1)<italic>A<sub>n</sub></italic>(<italic>x</italic>) with <italic>A</italic><sub>0</sub>(<italic>x</italic>) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol--Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00On the gaps between -binomial coefficientshttps://sciendo.com/article/10.2478/cm-2020-0010<abstract><title style='display:none'>Abstract</title><p>In this note, we estimate the distance between two <italic>q</italic>-nomial coefficients <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2020-0010_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow/><mml:mi>k</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>q</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow/><mml:msup><mml:mi>k</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>q</mml:mi></mml:msub></mml:mrow></mml:math><tex-math>{\left( {_k^n} \right)_q} - {\left( {_{k'}^{n'}} \right)_q}</tex-math></alternatives></inline-formula>, where (<italic>n, k</italic>) ≠ (<italic>n<sup>′</sup>, k<sup>′</sup></italic>) and <italic>q</italic> ≥ 2 is an integer.</p></abstract>ARTICLE2021-12-23T00:00:00.000+00:00Existentially closed Leibniz algebras and an embedding theoremhttps://sciendo.com/article/10.2478/cm-2021-0015<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 algebrashttps://sciendo.com/article/10.2478/cm-2021-0020<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-polynomialshttps://sciendo.com/article/10.2478/cm-2021-0025<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, IIIhttps://sciendo.com/article/10.2478/cm-2021-0023<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 planehttps://sciendo.com/article/10.2478/cm-2021-0024<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="http://www.w3.org/1999/xlink" xlink:href="graphic/j_cm-2021-0024_eq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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+00:00en-us-1