Pub Date : 2022-06-01DOI: 10.21494/iste.op.2022.0837
H. Marzougui
The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and V is a divergence-free C-vector field with finitely many singularities on M then every orbit L of V is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M∗ = M − Sing(V), (iv) a locally dense orbit, where Sing(V) denotes the set of singular points of V . On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M∗. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431–445] established when the divergence-free vector field V is regular that is all its singular points are non-degenerate. 2010 Mathematics Subject Classification. Primary: 37A, 37B20, 37C10, 37E35
{"title":"Limit sets and global dynamic for 2-D divergence-free vector fields","authors":"H. Marzougui","doi":"10.21494/iste.op.2022.0837","DOIUrl":"https://doi.org/10.21494/iste.op.2022.0837","url":null,"abstract":"The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and V is a divergence-free C-vector field with finitely many singularities on M then every orbit L of V is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M∗ = M − Sing(V), (iv) a locally dense orbit, where Sing(V) denotes the set of singular points of V . On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M∗. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431–445] established when the divergence-free vector field V is regular that is all its singular points are non-degenerate. 2010 Mathematics Subject Classification. Primary: 37A, 37B20, 37C10, 37E35","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48781940","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 : 2022-02-26DOI: 10.4236/apm.2022.1212054
N. Tsirivas
Let θ be a fixed positive number, θ ∈ (0, 1) and let λ = (λn)n∈N be a fixed sequence of non-zero complex numbers, so that λn→∞. We shall apply the functions gn : [0, θ]× C→C, defined as gn((t, z)) = z + λne for each (t, z) ∈ [0, θ]× C. We shall consider the space C([0, θ]× C) of continuous functions on [0, θ]× C, as endowed with the topology of uniform convergence on compacta and let ρ be the usual metric in C([0, θ]×C). For an entire function f ∈ H(C) we shall denote that f̄ : [0, θ]× C→C, f̄((t, z)) = f(z) for every (t, z) ∈ [0, θ]× C. We will prove that the equation: lim n→+∞ ρ((x ◦ gyn , f̄)) = 0 does not have any solution (x, yn) where x ∈ H(C) and yn is an strictly increasing subsequence of natural numbers and f ∈ H(C) is a given non-constant entire function. When f is a constant entire function, then the above equation has infinitely several solutions, according to a result provided by G. Costakis.
{"title":"Uniform Convergence of Translation Operators","authors":"N. Tsirivas","doi":"10.4236/apm.2022.1212054","DOIUrl":"https://doi.org/10.4236/apm.2022.1212054","url":null,"abstract":"Let θ be a fixed positive number, θ ∈ (0, 1) and let λ = (λn)n∈N be a fixed sequence of non-zero complex numbers, so that λn→∞. We shall apply the functions gn : [0, θ]× C→C, defined as gn((t, z)) = z + λne for each (t, z) ∈ [0, θ]× C. We shall consider the space C([0, θ]× C) of continuous functions on [0, θ]× C, as endowed with the topology of uniform convergence on compacta and let ρ be the usual metric in C([0, θ]×C). For an entire function f ∈ H(C) we shall denote that f̄ : [0, θ]× C→C, f̄((t, z)) = f(z) for every (t, z) ∈ [0, θ]× C. We will prove that the equation: lim n→+∞ ρ((x ◦ gyn , f̄)) = 0 does not have any solution (x, yn) where x ∈ H(C) and yn is an strictly increasing subsequence of natural numbers and f ∈ H(C) is a given non-constant entire function. When f is a constant entire function, then the above equation has infinitely several solutions, according to a result provided by G. Costakis.","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81390146","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.2022.1212055
G. Cirier
{"title":"A Probabilistic Approach of the Poincaré-Bendixon Problem in R<sup><i>d</i></sup>","authors":"G. Cirier","doi":"10.4236/apm.2022.1212055","DOIUrl":"https://doi.org/10.4236/apm.2022.1212055","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74457440","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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"A Method for the Squaring of a Circle","authors":"Lyndon O. Barton","doi":"10.4236/apm.2022.129041","DOIUrl":"https://doi.org/10.4236/apm.2022.129041","url":null,"abstract":"","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72845435","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}
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":null,"pages":null},"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":null,"pages":null},"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}
Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations m f mapping m onto . Even for the simplest entity, the pre-image by m f of a unique point, there is no way of visualization. Pre-images by m f of figures from are like ghost figures in m . This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most im-portant achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in m are well known and vector calculus in m is familiar, yet the pre-image by m f of a vector from is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in m . Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points
{"title":"Some Geometric Properties of the <i>m</i>-Möbius Transformations","authors":"D. Ghisa","doi":"10.4236/apm.2022.123013","DOIUrl":"https://doi.org/10.4236/apm.2022.123013","url":null,"abstract":"Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations m f mapping m onto . Even for the simplest entity, the pre-image by m f of a unique point, there is no way of visualization. Pre-images by m f of figures from are like ghost figures in m . This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most im-portant achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in m are well known and vector calculus in m is familiar, yet the pre-image by m f of a vector from is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in m . Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points","PeriodicalId":43512,"journal":{"name":"Advances in Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79578607","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":null,"pages":null},"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}
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