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An Alternative Equation for Generalized Polynomials of Degree Two 二阶广义多项式的一个可选方程
Annales Mathematicae Silesianae Pub Date : 2023-10-26 DOI: 10.2478/amsil-2023-0017
Zoltán Boros, Rayene Menzer
{"title":"An Alternative Equation for Generalized Polynomials of Degree Two","authors":"Zoltán Boros, Rayene Menzer","doi":"10.2478/amsil-2023-0017","DOIUrl":"https://doi.org/10.2478/amsil-2023-0017","url":null,"abstract":"Abstract In this paper we consider a generalized polynomial f : ℝ → ℝ of degree two that satisfies the additional equation f ( x ) f ( y ) = 0 for the pairs ( x, y ) ∈ D , where D ⊆ ℝ 2 is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mrow> <m:mi>D</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>|</m:mo> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>-</m:mo> <m:mi>m</m:mi> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> D = left{ {left( {x,y} right) in mathbb{R}{^2}|{x^2} - m{y^2} = 1} right}, we prove that f ( x ) = 0 for all x ∈ ℝ.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136376830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Zeros of Polynomials with Restricted Coefficients 关于限制系数多项式的零点
Annales Mathematicae Silesianae Pub Date : 2023-09-01 DOI: 10.2478/amsil-2023-0016
B. A. Zargar, M. H. Gulzar, M. Ali
{"title":"On the Zeros of Polynomials with Restricted Coefficients","authors":"B. A. Zargar, M. H. Gulzar, M. Ali","doi":"10.2478/amsil-2023-0016","DOIUrl":"https://doi.org/10.2478/amsil-2023-0016","url":null,"abstract":"Abstract Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"> <m:mrow> <m:mi>P</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msub> <m:msup> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mi>j</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> Pleft( z right) = sumnolimits_{j = 0}^n {{a_j}{z^j}} be a polynomial of degree n such that a n ≥ a n− 1 ≥ . . . ≥ a 1 ≥ a 0 ≥ 0. Then according to Eneström-Kakeya theorem all the zeros of P ( z ) lie in |z| ≤ 1. This result has been generalized in various ways (see [1, 3, 4, 6, 7]). In this paper we shall prove some generalizations of the results due to Aziz and Zargar [1, 2] and Nwaeze [7].","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"14 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135304928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups Drygas函数方程在Abel半群上的新氧化
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-08-29 DOI: 10.2478/amsil-2023-0015
Y. Aissi, D. Zeglami
{"title":"New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups","authors":"Y. Aissi, D. Zeglami","doi":"10.2478/amsil-2023-0015","DOIUrl":"https://doi.org/10.2478/amsil-2023-0015","url":null,"abstract":"Abstract Let (S, +) be an abelian semigroup, let (H, +) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f, h : S → H of each of the functional equations f(x+y)+f(x+ϕ(y))=h(x)+f(y)+f∘ϕ(y), x,y∈S,f(x+y)+f(x+ϕ(y))=h(x)+2f(y), x,y∈S, matrix{ {fleft( {x + y} right) + fleft( {x + varphi left( y right)} right) = hleft( x right) + fleft( y right) + f circ varphi left( y right),,x,y in S,} hfill cr {fleft( {x + y} right) + fleft( {x + varphi left( y right)} right) = hleft( x right) + 2fleft( y right),,x,y in S,} hfill cr } in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45680293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Generalized Version of the Lions-Type Lemma lions型引理的一个推广版本
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-08-28 DOI: 10.2478/amsil-2023-0014
M. Chmara
{"title":"A Generalized Version of the Lions-Type Lemma","authors":"M. Chmara","doi":"10.2478/amsil-2023-0014","DOIUrl":"https://doi.org/10.2478/amsil-2023-0014","url":null,"abstract":"Abstract In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48742821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces Hilbert空间中正算子的积分变换的梯度不等式
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0008
S. Dragomir
{"title":"Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces","authors":"S. Dragomir","doi":"10.2478/amsil-2023-0008","DOIUrl":"https://doi.org/10.2478/amsil-2023-0008","url":null,"abstract":"Abstract For a continuous and positive function w (λ) , λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform 𝒟(w,μ)(T):=∫0∞w(λ)(λ+T)-1dμ(λ), mathcal{D}left( {w,mu } right)left( T right): = int_0^infty {wleft( lambda right){{left( {lambda + T} right)}^{ - 1}}dmu left( lambda right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. Assume that A ≥ α > 0, δ ≥ B > 0 and 0 < m ≤ B − A ≤ M for some constants α, δ, m, M. Then 0≤-m𝒟′(w,μ)(δ)≤𝒟(w,μ)(A)-𝒟(w,μ)(B)≤-M𝒟′(w,μ)(α), 0 le - mmathcal{D}'left( {w,mu } right)left( delta right) le mathcal{D}left( {w,mu } right)left( A right) - mathcal{D}left( {w,mu } right)left( B right) le - Mmathcal{D}'left( {w,mu } right)left( alpha right), where D′(w, µ) (t) is the derivative of D(w, µ) (t) as a function of t > 0. If f : [0, ∞) → ℝ is operator monotone on [0, ∞) with f (0) = 0, then 0≤mδ2[ f(δ)-f′(δ)δ≤f(A)A-1-f(B)B-1 ]≤Mα2[ f(α)-f′(α)α ]. matrix{ {0 le {m over {{delta ^2}}}left[ {fleft( delta right) - f'left( delta right)delta le fleft( A right){A^{ - 1}} - f{{left( B right)}^{B - 1}}} right]} cr { le {M over {{alpha ^2}}}left[ {fleft( alpha right) - f'left( alpha right)alpha } right].} cr } Some examples for operator convex functions as well as for integral transforms D (·, ·) related to the exponential and logarithmic functions are also provided.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":"0 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41549626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Fourier Analysis Based New Look at Integration 基于傅立叶分析的积分新论
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0011
P. Imkeller, Nicolas Perkowski
{"title":"A Fourier Analysis Based New Look at Integration","authors":"P. Imkeller, Nicolas Perkowski","doi":"10.2478/amsil-2023-0011","DOIUrl":"https://doi.org/10.2478/amsil-2023-0011","url":null,"abstract":"Abstract We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48826509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers 由包含除数的类二项式公式产生的恒等式
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0013
K. Gryszka
{"title":"Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers","authors":"K. Gryszka","doi":"10.2478/amsil-2023-0013","DOIUrl":"https://doi.org/10.2478/amsil-2023-0013","url":null,"abstract":"Abstract In this article, we derive a great number of identities involving the ω function counting distinct prime divisors of a given number n. These identities also include Pochhammer symbols, Fibonacci and Lucas numbers and many more.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47062665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Probability on Submetric Spaces 子度量空间上的概率
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0012
Adam Jakubowski
{"title":"Probability on Submetric Spaces","authors":"Adam Jakubowski","doi":"10.2478/amsil-2023-0012","DOIUrl":"https://doi.org/10.2478/amsil-2023-0012","url":null,"abstract":"Abstract A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology). We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47902293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fuzzy Weak Filters of Sheffer Stroke Hilbert Algebras Sheffer Stroke Hilbert代数的模糊弱滤波器
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0010
R. Borzooei, G. Rezaei, Y. Jun
{"title":"Fuzzy Weak Filters of Sheffer Stroke Hilbert Algebras","authors":"R. Borzooei, G. Rezaei, Y. Jun","doi":"10.2478/amsil-2023-0010","DOIUrl":"https://doi.org/10.2478/amsil-2023-0010","url":null,"abstract":"Abstract The (fuzzy) filter of the Sheffer stroke Hilbert algebra was addressed by Oner, Katican and Borumand Saeid. The weak version of the filter was discussed by Jun and Oner. In this manuscipt, with the fuzzy version of the weak filter in mind, the notions of fuzzy weak filters and (∈, ∈ ∨q)-fuzzy weak filters are introduced, and their properties are explored. Conditions under which t-level set, Qt-set and t-∈ ∨q-set become weak filters are explored in relation to fuzzy weak filters and (∈, ∈ ∨q)-fuzzy weak filters. The relationship and characterization of the fuzzy weak filter and the (∈, ∈ ∨q)-fuzzy weak filter are investigated.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47139260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives 具有Caputo–Fabrizion和Riemann–Liouville分数阶导数的时空扩散偏微分方程的数值Fem解
IF 0.4
Annales Mathematicae Silesianae Pub Date : 2023-07-26 DOI: 10.2478/amsil-2023-0009
Malika Boutiba, Selma Encadreur Baghli-Bendimerad, Michal Feckan
{"title":"Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives","authors":"Malika Boutiba, Selma Encadreur Baghli-Bendimerad, Michal Feckan","doi":"10.2478/amsil-2023-0009","DOIUrl":"https://doi.org/10.2478/amsil-2023-0009","url":null,"abstract":"Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.","PeriodicalId":52359,"journal":{"name":"Annales Mathematicae Silesianae","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41720541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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