Pub Date : 2021-11-10DOI: 10.5556/j.tkjm.54.2023.4085
A. Uçum, Ç. Camcı, K. Ilarslan
In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.
{"title":"A New Approach to Mannheim Curve in Euclidean 3-Space","authors":"A. Uçum, Ç. Camcı, K. Ilarslan","doi":"10.5556/j.tkjm.54.2023.4085","DOIUrl":"https://doi.org/10.5556/j.tkjm.54.2023.4085","url":null,"abstract":"In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79758774","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 the human respiratory tract, air breathed in is often contaminated with strange particles such as dust and chemical spray, which may cause people respiratory diseases. However, the human body has an innate immune system that helps to trap the debris by secreting mucus to catch the foreign particles, which are removed from the body by the movement of tiny hairs lining on the surface of the epithelial cells in the immune system. The layer containing the tiny hairs or cilia is called Periciliary Layer (PCL). In this research, we find the velocity of the fluid in the PCL moved by a ciliary beating by using the Navier-Stokes-Brinkman equations. We apply the Galerkin finite element method to determine numerical solutions. For the steady linear case of the equation, the numerical result is in good agreement with an exact solution. Including the time derivative and nonlinear terms, we show that the velocity of the liquid is affected by the velocity of the solid, which follows the physical meaning of the fluid flow. The result can be applied as a bottom boundary condition of the mucous layer to be able to find the velocity of mucus in the human lungs.
{"title":"The Flow in Periciliary Layer in Human Lungs with Navier-Stokes-Brinkman Equations","authors":"Kanognudge Wuttanachamsri, Nattapol Oangwatcharaparkan","doi":"10.5556/j.tkjm.54.2023.3738","DOIUrl":"https://doi.org/10.5556/j.tkjm.54.2023.3738","url":null,"abstract":"In the human respiratory tract, air breathed in is often contaminated with strange particles such as dust and chemical spray, which may cause people respiratory diseases. \u0000However, the human body has an innate immune system that helps to trap the debris by secreting mucus to catch the foreign particles, which are removed from the body by the movement of tiny hairs lining on the surface of the epithelial cells in the immune system. \u0000The layer containing the tiny hairs or cilia is called Periciliary Layer (PCL). In this research, we find the velocity of the fluid in the PCL moved by a ciliary beating by using the Navier-Stokes-Brinkman equations. \u0000We apply the Galerkin finite element method to determine numerical solutions. For the steady linear case of the equation, the numerical result is in good agreement with an exact solution. \u0000Including the time derivative and nonlinear terms, we show that the velocity of the liquid is affected by the velocity of the solid, which follows the physical meaning of the fluid flow. \u0000The result can be applied as a bottom boundary condition of the mucous layer to be able to find the velocity of mucus in the human lungs.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82175906","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 : 2021-11-10DOI: 10.5556/j.tkjm.54.2023.4069
M. Azis
The anisotropic-diffusion convection equation of spatiallyvariable coefficients which is relevant for functionally graded mediais discussed in this paper to find numerical solutions by using acombined Laplace transform and boundary element method. The variablecoefficients equation is transformed to a constant coefficients equation.The constant coefficients equation is then Laplace-transformed sothat the time variable vanishes. The Laplace-transformed equationis consequently written in a pure boundary integral equation whichinvolves a time-free fundamental solution. The boundary integral equationis therefore employed to find numerical solutions using a standardboundary element method. Finally the results obtained are inverselytransformed numerically using the Stehfest formula to get solutionsin the time variable. The combined Laplace transform and boundaryelement method is easy to be implemented, efficient and accurate forsolving unsteady problems of anisotropic functionally graded mediagoverned by the diffusion convection equation.
{"title":"Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow","authors":"M. Azis","doi":"10.5556/j.tkjm.54.2023.4069","DOIUrl":"https://doi.org/10.5556/j.tkjm.54.2023.4069","url":null,"abstract":"The anisotropic-diffusion convection equation of spatiallyvariable coefficients which is relevant for functionally graded mediais discussed in this paper to find numerical solutions by using acombined Laplace transform and boundary element method. The variablecoefficients equation is transformed to a constant coefficients equation.The constant coefficients equation is then Laplace-transformed sothat the time variable vanishes. The Laplace-transformed equationis consequently written in a pure boundary integral equation whichinvolves a time-free fundamental solution. The boundary integral equationis therefore employed to find numerical solutions using a standardboundary element method. Finally the results obtained are inverselytransformed numerically using the Stehfest formula to get solutionsin the time variable. The combined Laplace transform and boundaryelement method is easy to be implemented, efficient and accurate forsolving unsteady problems of anisotropic functionally graded mediagoverned by the diffusion convection equation.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82049402","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 : 2021-11-08DOI: 10.5556/j.tkjm.54.2023.4136
Tapatee Sahoo, B. Davvaz, Harikrishnan Panackal, B. S. Kedukodi, S. P. Kuncham
Let $G$ be an $N$-group where $N$ is a (right) nearring. We introduce the concept of relative essential ideal (or $N$-subgroup) as a generalization of the concept of essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or $N$-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of $N$-groups and obtain their properties under homomorphism.
{"title":"Relative Essential Ideals in N-groups","authors":"Tapatee Sahoo, B. Davvaz, Harikrishnan Panackal, B. S. Kedukodi, S. P. Kuncham","doi":"10.5556/j.tkjm.54.2023.4136","DOIUrl":"https://doi.org/10.5556/j.tkjm.54.2023.4136","url":null,"abstract":"Let $G$ be an $N$-group where $N$ is a (right) nearring. We introduce the concept of relative essential ideal (or $N$-subgroup) as a generalization of the concept of essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or $N$-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of $N$-groups and obtain their properties under homomorphism.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85059311","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 : 2021-11-07DOI: 10.5556/j.tkjm.54.2023.4120
S. Pirzada, Saleem Khan
Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $delta_{1}, ~delta_{1},~dots,delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.
{"title":"On the Sum of Distance Laplacian Eigenvalues of Graphs","authors":"S. Pirzada, Saleem Khan","doi":"10.5556/j.tkjm.54.2023.4120","DOIUrl":"https://doi.org/10.5556/j.tkjm.54.2023.4120","url":null,"abstract":"Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $delta_{1}, ~delta_{1},~dots,delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89536436","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 : 2021-10-02DOI: 10.5556/j.tkjm.53.2022.3639
B. Esmaeili, G. Haghighatdoost, F. Pashaie
It is well-known that some of minimal (or maximal) hypersurfaces are stable. However, there is growing recognition on unstable hypersurfaces by introducing the concept of index of stability for minimal ones. For instance, the index of stability for minimal hypersurefces in Euclidean n-sphere has been defined by J. Simons and followed by many people. Also, Barros and Sousa have studied a high order extention of index as the concept of r-index (i.e. index of r-stability) on r-minimal hypersurfaces of n-sphere. They gave low bonds for r-stability index of r-minimal hypersurfaces in Euclidean sphere. In this paper, we low bounds for the r-stability index of r-maximal closed spacelike hypersurfaces in the de Sitter space.
{"title":"r-Stablity Index of r-Maximal Closed Hypersurfaces in de Sitter Spaces","authors":"B. Esmaeili, G. Haghighatdoost, F. Pashaie","doi":"10.5556/j.tkjm.53.2022.3639","DOIUrl":"https://doi.org/10.5556/j.tkjm.53.2022.3639","url":null,"abstract":"It is well-known that some of minimal (or maximal) hypersurfaces are stable. However, there is growing recognition on unstable hypersurfaces by introducing the concept of index of stability for minimal ones. For instance, the index of stability for minimal hypersurefces in Euclidean n-sphere has been defined by J. Simons and followed by many people. Also, Barros and Sousa have studied a high order extention of index as the concept of r-index (i.e. index of r-stability) on r-minimal hypersurfaces of n-sphere. They gave low bonds for r-stability index of r-minimal hypersurfaces in Euclidean sphere. In this paper, we low bounds for the r-stability index of r-maximal closed spacelike hypersurfaces in the de Sitter space.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89543897","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 : 2021-06-15DOI: 10.5556/J.TKJM.23.1992.4522
Sharief Deshmukh
The normal bundle $bar nu$ of a totally real surface $M$ in $S^6$ splits as $barnu= JTMoplus barmu$ where $TM$ is the tangent bundle of $M$ and $barmu$ is subbundle of $barnu$ which is invariant under the almost complex structure $J$. We study the totally real surfaces M of constant Gaussian curvature K for which the second fundamental form $h(x, y) in JTM$, and we show that $K = 1$ (that is, $M$ is totally geodesic).
在$S^6$中，$M$的正常束$bar nu$分裂为$barnu= JTMoplus barmu$，其中$TM$是$M$的切线束，$barmu$是$barnu$的子束，$J$在几乎复杂的结构下是不变的。我们研究了具有恒定高斯曲率K的全实曲面M，其第二种基本形式为$h(x, y) in JTM$，并且我们证明了$K = 1$(即$M$是完全测地线)。
{"title":"TOTALLY REAL SURFACES IN $S^6$","authors":"Sharief Deshmukh","doi":"10.5556/J.TKJM.23.1992.4522","DOIUrl":"https://doi.org/10.5556/J.TKJM.23.1992.4522","url":null,"abstract":"\u0000 \u0000 \u0000The normal bundle $bar nu$ of a totally real surface $M$ in $S^6$ splits as $barnu= JTMoplus barmu$ where $TM$ is the tangent bundle of $M$ and $barmu$ is subbundle of $barnu$ which is invariant under the almost complex structure $J$. We study the totally real surfaces M of constant Gaussian curvature K for which the second fundamental form $h(x, y) in JTM$, and we show that $K = 1$ (that is, $M$ is totally geodesic). \u0000 \u0000 \u0000","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83724976","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 : 2021-06-06DOI: 10.5556/J.TKJM.25.1994.4423
C. Adiga
We obtain Jacobi's two-square and four-square theorems as an application of an identity of L.J. Rogers.
作为罗杰斯恒等式的一个应用，我们得到了雅可比二平方定理和四平方定理。
{"title":"JACOBI'S TWO-SQUARE AND FOUR-SQUARE THEOREMS VIA ROGER'S IDENTITY","authors":"C. Adiga","doi":"10.5556/J.TKJM.25.1994.4423","DOIUrl":"https://doi.org/10.5556/J.TKJM.25.1994.4423","url":null,"abstract":"We obtain Jacobi's two-square and four-square theorems as an application of an identity of L.J. Rogers.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74189260","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 : 2021-05-25DOI: 10.5556/J.TKJM.29.1998.4289
T. Chern
Let J be a function meromorphic in the finite complex plane C. We donate by T(r, J)(To(r, !)) the Nevanlinna(Ahlfors-Shmizu) characteristic function of J. A mero morphic function a(z) (including the case f(z) == c where c in Cu {oo}) is called small with respect to f if T(r, a(z)) = o(T(r, J)) as r -, +oo. Ve let n(兀 0. This paper deals with the existence of the Borel directions concerning small functions for mermorphic functions of finite positive order. Using Tsuji's method, we shall mainly prove Theorem 1 stated in the abstract. Theorem~extends a result of Chuang [2, p.127, Corollary 5.3], there a(z} are restricte
{"title":"ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS","authors":"T. Chern","doi":"10.5556/J.TKJM.29.1998.4289","DOIUrl":"https://doi.org/10.5556/J.TKJM.29.1998.4289","url":null,"abstract":"Let J be a function meromorphic in the finite complex plane C. We donate by T(r, J)(To(r, !)) the Nevanlinna(Ahlfors-Shmizu) characteristic function of J. A mero morphic function a(z) (including the case f(z) == c where c in Cu {oo}) is called small with respect to f if T(r, a(z)) = o(T(r, J)) as r -, +oo. Ve let n(兀 <p, a, J = a(z)) be the number of roots (multiple roots being counted with their multiplicities) of the equation j(z) = a(z) for z in the angular domain D(r,cp,a) = {z: largz 列< c..t, lzl < r} where 0 :::; cp < 21r, a > 0. This paper deals with the existence of the Borel directions concerning small functions for mermorphic functions of finite positive order. Using Tsuji's method, we shall mainly prove Theorem 1 stated in the abstract. Theorem~extends a result of Chuang [2, p.127, Corollary 5.3], there a(z} are restricte<l over all extended complex numbm·s. Chuang's method rs different from ours and is區ed on the existence of a sequence of filling disk with their roots in the works of Milloux [3] and Valiron [7].","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75671532","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 : 2021-05-16DOI: 10.5556/J.TKJM.28.1997.4247
L. Olszowy
In this paper we give an estimate of the modulus of near smoothness of the space $c_o(E_i)$. In the case of the space $c_o(l_{p_i})$ the exact formula for this modulus is derived. Moreover, we show that the properties of near uniform smoothness and local near uniform smoothness are hereditary with respect to the product space $c_o(E_i)$.
{"title":"LOCAL AND UNIFORM NEAR SMOOTHNESS OF SOME BANACH SPACES","authors":"L. Olszowy","doi":"10.5556/J.TKJM.28.1997.4247","DOIUrl":"https://doi.org/10.5556/J.TKJM.28.1997.4247","url":null,"abstract":"\u0000 \u0000 \u0000In this paper we give an estimate of the modulus of near smoothness of the space $c_o(E_i)$. In the case of the space $c_o(l_{p_i})$ the exact formula for this modulus is derived. Moreover, we show that the properties of near uniform smoothness and local near uniform smoothness are hereditary with respect to the product space $c_o(E_i)$. \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76879964","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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