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Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ𝑛), 𝑛 ≤ 1 (f)中Schwartz分布的连续小波变换，𝑛≤1
Philosophic research and analysis Pub Date : 2022-06-29 DOI: 10.1515/anly-2021-0002
J. Pandey
{"title":"Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ𝑛), 𝑛 ≤ 1","authors":"J. Pandey","doi":"10.1515/anly-2021-0002","DOIUrl":"https://doi.org/10.1515/anly-2021-0002","url":null,"abstract":"Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in D ′ ⁢ ( R n ) mathcal{D}^{prime}(mathbb{R}^{n}) , n ≥ 1 ngeq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element ψ ⁢ ( x ) psi(x) of D ⁢ ( R n ) mathcal{D}(mathbb{R}^{n}) , n ≥ 1 ngeq 1 , which, when integrated along each of the real axes X 1 , X 2 , X 3 , … , X n X_{1},X_{2},X_{3},ldots,X_{n} vanishes, but none of its moments ∫ R n ψ ⁢ ( x ) ⁢ x m ⁢ d x int_{mathbb{R}^{n}}psi(x)x^{m},dx is zero; here x m = x 1 m 1 ⁢ x 2 m 2 ⁢ … ⁢ x n m n x^{m}=x_{1}^{{m_{1}}},x_{2}^{{m_{2}}}ldots x_{n}^{{m_{n}}} , d ⁢ x = d ⁢ x 1 ⁢ d ⁢ x 2 ⁢ … ⁢ d ⁢ x n dx=dx_{1},dx_{2}ldots dx_{n} and m = ( m 1 , m 2 , … , m n ) m=(m_{1},m_{2},ldots,m_{n}) and each of m 1 , m 2 , … , m n m_{1},m_{2},ldots,m_{n} is at least 1. The set of such kernel will be denoted by D m ⁢ ( R n ) mathcal{D}_{m}(mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space D F ′ ⁢ ( R n ) mathcal{D}_{F}^{prime}(mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space D ′ ⁢ ( R n ) mathcal{D}^{prime}(mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for x 1 2 + x 2 2 + ⋯ ⁢ x n 2 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 = 1 - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 frac{x_{1}^{2}+x_{2}^{2}+cdots x_{n}^{2}}{1+x_{1}^{2}+x_{2}^{2}+cdots x_{n}^{2}}=1-frac{1}{1+x_{1}^{2}+x_{2}^{2}+cdots x_{n}^{2}} , 1 is deleted and - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 frac{-1}{1+x_{1}^{2}+x_{2}^{2}+cdots x_{n}^{2}} is retained.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"21 1","pages":"133 - 139"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80952362","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
Construction of partially degenerate Bell–Bernoulli polynomials of the first kind and their certain properties 第一类部分退化Bell-Bernoulli多项式的构造及其若干性质
Philosophic research and analysis Pub Date : 2022-06-29 DOI: 10.1515/anly-2021-1028
W. Khan, M. Kamarujjama, Daud
{"title":"Construction of partially degenerate Bell–Bernoulli polynomials of the first kind and their certain properties","authors":"W. Khan, M. Kamarujjama, Daud","doi":"10.1515/anly-2021-1028","DOIUrl":"https://doi.org/10.1515/anly-2021-1028","url":null,"abstract":"Abstract Various applications of degenerate polynomials in different areas call for the thoughtful study and research, and many extensions and variants can be found in the literature. In this paper, we introduce partially degenerate Bell–Bernoulli polynomials of the first kind and investigate their properties and identities. Furthermore, we introduce a generalized form of partially degenerate Bell–Bernoulli polynomials of the first kind and derive some interesting properties and identities. The results obtained are of general character and can be reduced to yield formulae and identities for relatively simple polynomials and numbers.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"37 1","pages":"171 - 184"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79824448","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: 2
Estimates for weighted Ostrowski–Grüss type inequalities with applications 加权ostrowski - gr<s:1>型不等式的估计及其应用
Philosophic research and analysis Pub Date : 2022-06-29 DOI: 10.1515/anly-2021-0044
Muhammad Awais Shaikh, Asif R Khan, F. Mehmood
{"title":"Estimates for weighted Ostrowski–Grüss type inequalities with applications","authors":"Muhammad Awais Shaikh, Asif R Khan, F. Mehmood","doi":"10.1515/anly-2021-0044","DOIUrl":"https://doi.org/10.1515/anly-2021-0044","url":null,"abstract":"Abstract In the present paper, some error bounds for weighted Ostrowski–Grüss type inequalities are found by using the weighted Čebyšev functional and Diaz–Metcalf inequality. These bounds give some better and new estimates. Applications are also obtained for numerical integration. Furthermore, this paper recaptures some previous established results as its special cases.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"1 1","pages":"159 - 169"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85409415","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: 1
Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator 一类具有𝑝(1)-类拉普拉斯算子的Neumann边值问题的弱解
Philosophic research and analysis Pub Date : 2022-06-29 DOI: 10.1515/anly-2022-1063
Mohamed El Ouaarabi, C. Allalou, S. Melliani
{"title":"Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator","authors":"Mohamed El Ouaarabi, C. Allalou, S. Melliani","doi":"10.1515/anly-2022-1063","DOIUrl":"https://doi.org/10.1515/anly-2022-1063","url":null,"abstract":"Abstract In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"29 1","pages":"271 - 280"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72611187","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: 6
Erratum to: Conglomerability, disintegrability, and the comparative principle 勘误:聚集性、可分解性和比较原则
Philosophic research and analysis Pub Date : 2022-05-05 DOI: 10.1093/analys/anac013
Rush T. Stewart, Michael Nielsen
{"title":"Erratum to: Conglomerability, disintegrability, and the comparative principle","authors":"Rush T. Stewart, Michael Nielsen","doi":"10.1093/analys/anac013","DOIUrl":"https://doi.org/10.1093/analys/anac013","url":null,"abstract":"","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80898015","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
Correction to: Be modest: you’re living on the edge 纠正:谦虚点:你生活在边缘
Philosophic research and analysis Pub Date : 2022-04-24 DOI: 10.1093/analys/anac024
Kevin Dorst
{"title":"Correction to: Be modest: you’re living on the edge","authors":"Kevin Dorst","doi":"10.1093/analys/anac024","DOIUrl":"https://doi.org/10.1093/analys/anac024","url":null,"abstract":"","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75427366","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
Growth properties of meromorphic solutions of some higher-order linear differential-difference equations 一类高阶线性微分-差分方程亚纯解的生长性质
Philosophic research and analysis Pub Date : 2022-04-08 DOI: 10.1515/anly-2021-1000
Rachid Bellaama, B. Belaïdi
{"title":"Growth properties of meromorphic solutions of some higher-order linear differential-difference equations","authors":"Rachid Bellaama, B. Belaïdi","doi":"10.1515/anly-2021-1000","DOIUrl":"https://doi.org/10.1515/anly-2021-1000","url":null,"abstract":"Abstract This paper is devoted to the study of the growth of meromorphic solutions of homogeneous and non-homogeneous linear differential-difference equations ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = 0 , displaystylesum_{i=0}^{n}sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=0, ∑ i = 0 n ∑ j = 0 m A i ⁢ j ⁢ f ( j ) ⁢ ( z + c i ) = F , displaystylesum_{i=0}^{n}sum_{j=0}^{m}A_{ij}f^{(j)}(z+c_{i})=F, where A i ⁢ j {A_{ij}} ( i = 0 , … , n {i=0,ldots,n} , j = 0 , … , m {j=0,ldots,m} ), F are meromorphic functions and c i {c_{i}} ( 0 , … , n {0,ldots,n} ) are non-zero distinct complex numbers. Under some conditions on the coefficients, we extend early results due to Zhou and Zheng.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"116 1","pages":"71 - 88"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79266861","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 λ-Changhee–Hermite polynomials 关于λ-Changhee-Hermite多项式
Philosophic research and analysis Pub Date : 2022-04-08 DOI: 10.1515/anly-2021-1012
Mushtaque Ahmed Pathan, W. Khan
{"title":"On λ-Changhee–Hermite polynomials","authors":"Mushtaque Ahmed Pathan, W. Khan","doi":"10.1515/anly-2021-1012","DOIUrl":"https://doi.org/10.1515/anly-2021-1012","url":null,"abstract":"Abstract In this paper, we introduce a new class of λ-analogues of the Changhee–Hermite polynomials and generalized Gould–Hopper–Appell type λ-Changhee polynomials, and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould–Hopper, as well as the 1D and 2D holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the Appell type λ-Changhee polynomials are also discussed.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"5 1","pages":"57 - 69"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74744252","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: 1
Weak solutions for fractional p(x,·)-Laplacian Dirichlet problems with weight 分数阶p(x，·)-带权拉普拉斯Dirichlet问题的弱解
Philosophic research and analysis Pub Date : 2022-04-08 DOI: 10.1515/anly-2021-1007
M. Ait Hammou
{"title":"Weak solutions for fractional p(x,·)-Laplacian Dirichlet problems with weight","authors":"M. Ait Hammou","doi":"10.1515/anly-2021-1007","DOIUrl":"https://doi.org/10.1515/anly-2021-1007","url":null,"abstract":"Abstract The main purpose of this paper is to show the existence of weak solutions for a problem involving the fractional p ⁢ ( x , ⋅ ) {p(x,cdot,)} -Laplacian operator of the following form: { ( - Δ p ⁢ ( x , ⋅ ) ) s ⁢ u ⁢ ( x ) + w ⁢ ( x ) ⁢ | u | p ¯ ⁢ ( x ) - 2 ⁢ u = λ ⁢ f ⁢ ( x , u ) in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω , left{begin{aligned} displaystyle(-Delta_{p(x,cdot,)})^{s}u(x)+w(x)% lvert urvert^{bar{p}(x)-2}u&displaystyle=lambda f(x,u)&&displaystyle% phantom{}text{in }Omega, displaystyle u&displaystyle=0&&displaystylephantom{}text{in }mathbb{R}^{% N}setminusOmega,end{aligned}right. The main tool used for this purpose is the Berkovits topological degree.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"110 1","pages":"121 - 132"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80169209","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 transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations 一类非线性广义时滞微分方程的超越全解和亚纯解
Philosophic research and analysis Pub Date : 2022-04-08 DOI: 10.1515/anly-2021-1011
A. Banerjee, T. Biswas
{"title":"On the transcendental entire and meromorphic solutions of certain non-linear generalized delay-differential equations","authors":"A. Banerjee, T. Biswas","doi":"10.1515/anly-2021-1011","DOIUrl":"https://doi.org/10.1515/anly-2021-1011","url":null,"abstract":"Abstract The prime intention of this paper is to study the conditions under which certain non-linear generalized delay-differential equations possess a solution. In this respect, by extending and improving recent results of [5, 15], we characterize the nature of solutions. By providing relevant examples in a remark, we also show that for the uniqueness of solutions the conditions on the Borel exceptional value can not be removed. Finally, to determine explicitly all forms of the solutions of a traditional generalized delay-differential equation, we deal with the situation under the aegis of generalized c-delay-differential equations. We exhibit some examples to show that the conclusions of the theorems actually occur.","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"24 1","pages":"89 - 109"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88333849","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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