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/608cb0de66e75927c4a7af40/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210624T102113Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKDOZOEZ7H%2F20210624%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=a5485eb8890c56d938a9d3d4329434b844ed9499bf9628f2425d9f41a30881b7200300Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1https://sciendo.com/article/10.2478/cm-2021-0005<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 calculushttps://sciendo.com/article/10.2478/cm-2021-0004<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 Recordshttps://sciendo.com/article/10.2478/cm-2021-0009<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+00:00Metric groups, unitary representations and continuous logichttps://sciendo.com/article/10.2478/cm-2021-0007<abstract> <title style='display:none'>Abstract</title> <p>We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find <italic>L<sub>ω</sub></italic><sub>1</sub><italic><sub>ω</sub></italic>-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00Rota-Baxter operators and Bernoulli polynomialshttps://sciendo.com/article/10.2478/cm-2021-0001<abstract> <title style='display:none'>Abstract</title> <p>We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00The inverse problem in the calculus of variations: new developmentshttps://sciendo.com/article/10.2478/cm-2021-0008<abstract> <title style='display:none'>Abstract</title> <p>We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of <italic>n</italic> second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for <italic>n</italic> = 2. We then examine a new class of solutions in arbitrary dimension <italic>n</italic> and give some non-trivial examples in dimension 3.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00A look on some results about Camassa–Holm type equationshttps://sciendo.com/article/10.2478/cm-2021-0006<abstract> <title style='display:none'>Abstract</title> <p>We present an overview of some contributions of the author regarding Camassa–Holm type equations. We show that an equation unifying both Camassa–Holm and Novikov equations can be derived using the invariance under certain suitable scaling, conservation of the Sobolev norm and existence of peakon solutions. Qualitative analysis of the two-peakon dynamics is given.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00Crystallographic actions on Lie groups and post-Lie algebra structureshttps://sciendo.com/article/10.2478/cm-2021-0003<abstract> <title style='display:none'>Abstract</title> <p>This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+00:00Entropy in Thermodynamics: from Foliation to Categorizationhttps://sciendo.com/article/10.2478/cm-2021-0002<abstract> <title style='display:none'>Abstract</title> <p>We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.</p> </abstract>ARTICLE2021-04-30T00:00:00.000+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</p><p><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></p><p>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>ARTICLE2020-08-10T00:00:00.000+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>ARTICLE2020-08-10T00:00:00.000+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 ⊂ ℬ(<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>ARTICLE2021-04-03T00:00:00.000+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</p><p><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></p><p>with <italic>c</italic> = <italic>b</italic> − <italic>a</italic>, uniformly for <italic>x</italic> ⩾ 40<italic>c <sup>−</sup></italic><sup>5</sup>(1 + b)<sup>27/2</sup>.</p></abstract>ARTICLE2020-09-13T00:00:00.000+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>ARTICLE2021-04-27T00:00:00.000+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:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><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 \right)^2} &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>ARTICLE2021-04-27T00:00:00.000+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 ℛ be a semiprime ring with unity <italic>e</italic> and <italic>ϕ</italic>, <italic>φ</italic> be automorphisms of ℛ. In this paper it is shown that if ℛ 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> ∈ ℛ and some fixed integer <italic>n</italic> ≥ 2, then 𝒟 is an (<italic>ϕ</italic>, <italic>φ</italic>)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, Gscr; : ℛ <italic>→</italic> ℛ 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>𝒢</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>𝒢</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>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:mo>,</mml:mo></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{G}\left( x \right) + \mathcal{G}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{G}\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> ∈ ℛ and some fixed integer <italic>n</italic> ≥ 2, then 𝒟 and 𝒢 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>ARTICLE2021-04-03T00:00:00.000+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>ARTICLE2021-04-27T00:00:00.000+00:00Upgrading Probability via Fractions of Eventshttps://sciendo.com/article/10.1515/cm-2016-0004<abstract><title style='display:none'>Abstract</title><p> The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.</p></abstract>ARTICLE2016-08-20T00:00:00.000+00:00Toeplitz Quantization for Non-commutating Symbol Spaces such as SU(2)https://sciendo.com/article/10.1515/cm-2016-0005<abstract><title style='display:none'>Abstract</title><p> Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SU<sub>q</sub>(2) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck’s constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck’s constant and a Hilbert space where natural, densely defined operators act. </p></abstract>ARTICLE2016-08-20T00:00:00.000+00:00A note on the pp conjecture for sheaves of spaces of orderingshttps://sciendo.com/article/10.1515/cm-2016-0001<abstract><title style='display:none'>Abstract</title><p> In this note we provide a direct and simple proof of a result previously obtained by Astier stating that the class of spaces of orderings for which the pp conjecture holds true is closed under sheaves over Boolean spaces.</p></abstract>ARTICLE2016-08-20T00:00:00.000+00:00en-us-1