It is shown that any polynomial written as an infinite product with all positive real roots may be split in two steps into the product of four infinite polynomials: two with all imaginary and two with all real roots. Equations between such infinite products define adjoint infinite polynomials with roots on the adjoint roots (real and imaginary). It is shown that the shifting of the coordinates to a parallel line of one of the adjoint axes does not influence the rela-tive placement of the roots: they are shifted to the parallel line. General relations between original and adjoint polynomials are evaluated. These relations are generalized representations of the relations of Euler and Pythagoras in form of infinite polynomial products. They are inherent properties of split polynomial products. If the shifting of the coordinate system corresponds to the shifting of the imaginary axes to the critical line, then the relations of Euler take the form corresponding to their occurrence in the functional equation of the Riemann zeta function: the roots on the imaginary axes are all shifted to the critical line. Since it is known that the gamma and the zeta functions may be written as composed functions with exponential and trigonometric parts, this opens the possibility to prove the placement of the zeta function on the critical line.
{"title":"The Exponential Function as Split Infinite Product","authors":"P. Doroszlai, Horacio Keller","doi":"10.4236/apm.2022.124024","DOIUrl":"https://doi.org/10.4236/apm.2022.124024","url":null,"abstract":"It is shown that any polynomial written as an infinite product with all positive real roots may be split in two steps into the product of four infinite polynomials: two with all imaginary and two with all real roots. Equations between such infinite products define adjoint infinite polynomials with roots on the adjoint roots (real and imaginary). It is shown that the shifting of the coordinates to a parallel line of one of the adjoint axes does not influence the rela-tive placement of the roots: they are shifted to the parallel line. General relations between original and adjoint polynomials are evaluated. These relations are generalized representations of the relations of Euler and Pythagoras in form of infinite polynomial products. They are inherent properties of split polynomial products. If the shifting of the coordinate system corresponds to the shifting of the imaginary axes to the critical line, then the relations of Euler take the form corresponding to their occurrence in the functional equation of the Riemann zeta function: the roots on the imaginary axes are all shifted to the critical line. Since it is known that the gamma and the zeta functions may be written as composed functions with exponential and trigonometric parts, this opens the possibility to prove the placement of the zeta function on the critical line.","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"137 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86489816","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}
Pub Date : 2022-01-01DOI: 10.4236/apm.2022.1211046
S. Kanagawa, K. Tchizawa
{"title":"Structural Stability in 4-Dimensional Canards","authors":"S. Kanagawa, K. Tchizawa","doi":"10.4236/apm.2022.1211046","DOIUrl":"https://doi.org/10.4236/apm.2022.1211046","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"453 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79722804","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}
{"title":"Regular Decimations Result in Irregular Distribution of Primes","authors":"Xin Wang","doi":"10.4236/apm.2022.126032","DOIUrl":"https://doi.org/10.4236/apm.2022.126032","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"331 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74707148","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}
V. Gonçalves de Brito dos Santos, P.T. Gomes dos Anjos
{"title":"Finite Difference Method Applied in Two-Dimensional Heat Conduction Problem in the Permanent Regime in Rectangular Coordinates","authors":"V. Gonçalves de Brito dos Santos, P.T. Gomes dos Anjos","doi":"10.4236/apm.2022.129038","DOIUrl":"https://doi.org/10.4236/apm.2022.129038","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78731418","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}
Pub Date : 2022-01-01DOI: 10.21494/iste.op.2022.0888
S. Heidarkhani, Ahmad Ghobadi, G. Caristi
. In this paper, we study a discrete anisotropic Kirchhoff type problem using variational methods and critical point theory, and we discuss the existence of two solutions for the problem. A example is presented to demonstrate the application of our main results.
{"title":"Critical point approaches for discrete anisotropic Kirchhoff type problems","authors":"S. Heidarkhani, Ahmad Ghobadi, G. Caristi","doi":"10.21494/iste.op.2022.0888","DOIUrl":"https://doi.org/10.21494/iste.op.2022.0888","url":null,"abstract":". In this paper, we study a discrete anisotropic Kirchhoff type problem using variational methods and critical point theory, and we discuss the existence of two solutions for the problem. A example is presented to demonstrate the application of our main results.","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68703274","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}
In this paper, we study the traveling wave solutions of the fractional generalized reaction Duffing equation, which contains several nonlinear conformable time fractional wave equations. By the dynamic system method, the phase portraits of the fractional generalized reaction Duffing equation are given, and all possible exact traveling wave solutions of the equation are obtained.
{"title":"Phase Portraits and Traveling Wave Solutions of a Fractional Generalized Reaction Duffing Equation","authors":"Kelei Zhang, Zhenfeng Zhang, Tao Yuwen","doi":"10.4236/apm.2022.127035","DOIUrl":"https://doi.org/10.4236/apm.2022.127035","url":null,"abstract":"In this paper, we study the traveling wave solutions of the fractional generalized reaction Duffing equation, which contains several nonlinear conformable time fractional wave equations. By the dynamic system method, the phase portraits of the fractional generalized reaction Duffing equation are given, and all possible exact traveling wave solutions of the equation are obtained.","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87282748","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}
{"title":"On <i>M</i>-Asymmetric Irresolute Multifunctions in Bitopological Spaces","authors":"L. K. Matindih, P. J. Banda, D. Mukonda","doi":"10.4236/apm.2022.128037","DOIUrl":"https://doi.org/10.4236/apm.2022.128037","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"350 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75121278","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}
{"title":"A New Method to Prove Goldbach’s Conjecture","authors":"Zengyong Liang","doi":"10.4236/apm.2022.121001","DOIUrl":"https://doi.org/10.4236/apm.2022.121001","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"91 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85383332","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}
Pub Date : 2022-01-01DOI: 10.4236/apm.2022.1212058
P. Doroszlai, Horacio Keller
{"title":"The Gaps between Primes","authors":"P. Doroszlai, Horacio Keller","doi":"10.4236/apm.2022.1212058","DOIUrl":"https://doi.org/10.4236/apm.2022.1212058","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82216063","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}
{"title":"Photoconductivity under Pulsed Excitation","authors":"Ilhan M. Izmirli","doi":"10.4236/apm.2022.126031","DOIUrl":"https://doi.org/10.4236/apm.2022.126031","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87952950","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}
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