Using the Brzdek fixed point theorem, we establish the Hyers?Ulam stability problem of Drygas functional equations begin{equation} delta(x+y-z)+delta(x-y)+delta(-y-z)+delta(y)=delta(x-y-z)+delta(y-z)+delta(x+y)+delta(-y)nonumber end{equation} for all $x,y,zin A$.
{"title":"The application of Brzdek's fixed point theorem in the stability problem of the Drygas functional equation","authors":"MEHDI DEHGHANIAN, YAMIN SAYYARI","doi":"10.55730/1300-0098.3462","DOIUrl":"https://doi.org/10.55730/1300-0098.3462","url":null,"abstract":"Using the Brzdek fixed point theorem, we establish the Hyers?Ulam stability problem of Drygas functional equations begin{equation} delta(x+y-z)+delta(x-y)+delta(-y-z)+delta(y)=delta(x-y-z)+delta(y-z)+delta(x+y)+delta(-y)nonumber end{equation} for all $x,y,zin A$.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with the one-dimensional porous medium system with sources begin{align*} begin{cases}u_t-( u^m)_{xx} =a_{11}u^{p}+a_{12} u^rv^{r+m}, (x,t)in Jtimes Isubsetmathbb{R}times mathbb{R} v_t-(v^m)_{xx} =a_{21} u^{r+m}v^{r}+a_{22}v^{p},;(x,t)in Jtimes Isubset mathbb{R}times mathbb{R}, end{cases} end{align*} where $p=2r+m$, $m>1$, $r>0$. Under the conditions $a_{12}geq 0, a_{21}geq 0$, $a_{11}>0$, and $a_{22}>0$, we prove that the system does not possess any nontrivial nonnegative weak solution.
{"title":"Liouville-type theorem for one-dimensional porous medium systems with sources","authors":"ANH TUAN DUONG","doi":"10.55730/1300-0098.3459","DOIUrl":"https://doi.org/10.55730/1300-0098.3459","url":null,"abstract":"In this paper, we are concerned with the one-dimensional porous medium system with sources begin{align*} begin{cases}u_t-( u^m)_{xx} =a_{11}u^{p}+a_{12} u^rv^{r+m}, (x,t)in Jtimes Isubsetmathbb{R}times mathbb{R} v_t-(v^m)_{xx} =a_{21} u^{r+m}v^{r}+a_{22}v^{p},;(x,t)in Jtimes Isubset mathbb{R}times mathbb{R}, end{cases} end{align*} where $p=2r+m$, $m>1$, $r>0$. Under the conditions $a_{12}geq 0, a_{21}geq 0$, $a_{11}>0$, and $a_{22}>0$, we prove that the system does not possess any nontrivial nonnegative weak solution.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we consider the monoid $DPC_n$ of all partial isometries of an $n$-cycle graph $C_n$. We show that $DPC_n$ is the submonoid of the monoid of all oriented partial permutations on an $n$-chain whose elements are precisely all restrictions of a dihedral group of order $2n$. Our main aim is to exhibit a presentation of $DPC_n$. We also describe Green's relations of $DPC_n$ and calculate its cardinality and rank.
{"title":"On the monoid of partial isometries of a cycle graph","authors":"VITOR H. FERNANDES, TANIA PAULISTA","doi":"10.55730/1300-0098.3460","DOIUrl":"https://doi.org/10.55730/1300-0098.3460","url":null,"abstract":"In this paper we consider the monoid $DPC_n$ of all partial isometries of an $n$-cycle graph $C_n$. We show that $DPC_n$ is the submonoid of the monoid of all oriented partial permutations on an $n$-chain whose elements are precisely all restrictions of a dihedral group of order $2n$. Our main aim is to exhibit a presentation of $DPC_n$. We also describe Green's relations of $DPC_n$ and calculate its cardinality and rank.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135768916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: The onset of magnetoconvection (known as Rayleigh-Bénard-Chandrasekhar convection) in two relaxation time viscoelastic liquids is studied here without seeking explicit recourse to a normal stress formulation as is usually done in these studies. Magnetoconvection refers to the flow of fluid in the presence of both thermal gradients (Rayleigh-Bénard convection) and a magnetic field. When these two effects are combined, they can lead to interesting and complex patterns of fluid motion. Understanding magnetoconvection in viscoelastic liquids is crucial for various industrial and scientific applications. The hyperbolic-type of linear momentum equation is decomposed into two first-order equations in time by cleverly separating the viscoelastic effect from the other effects in a clever manner as reported in a recent paper. The results of Maxwell, Rivlin-Ericksen, Walters’ liquid B, and Newtonian liquids are obtained as limiting cases of the present study. This research contributes to the understanding of magnetoconvection in viscoelastic liquids by using a novel approach that decouples the viscoelastic effect from other influences. The results obtained shed light on the behaviour of various types of viscoelastic materials and provide valuable insights for practical applications in fields such as materials science, engineering, and geophysics.
{"title":"A short note on a new approach to Rayleigh-Bénard-Chandrasekhar convection in weakly electrically conducting viscoelastic liquids","authors":"HATİCE MUTİ","doi":"10.55730/1300-0098.3466","DOIUrl":"https://doi.org/10.55730/1300-0098.3466","url":null,"abstract":": The onset of magnetoconvection (known as Rayleigh-Bénard-Chandrasekhar convection) in two relaxation time viscoelastic liquids is studied here without seeking explicit recourse to a normal stress formulation as is usually done in these studies. Magnetoconvection refers to the flow of fluid in the presence of both thermal gradients (Rayleigh-Bénard convection) and a magnetic field. When these two effects are combined, they can lead to interesting and complex patterns of fluid motion. Understanding magnetoconvection in viscoelastic liquids is crucial for various industrial and scientific applications. The hyperbolic-type of linear momentum equation is decomposed into two first-order equations in time by cleverly separating the viscoelastic effect from the other effects in a clever manner as reported in a recent paper. The results of Maxwell, Rivlin-Ericksen, Walters’ liquid B, and Newtonian liquids are obtained as limiting cases of the present study. This research contributes to the understanding of magnetoconvection in viscoelastic liquids by using a novel approach that decouples the viscoelastic effect from other influences. The results obtained shed light on the behaviour of various types of viscoelastic materials and provide valuable insights for practical applications in fields such as materials science, engineering, and geophysics.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
From the algebraic solution of $x^{m}-x+t=0$ for $m=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some definite integrals in terms of elementary functions.
{"title":"Identities involving special functions from hypergeometric solution of algebraic equations","authors":"JUAN LUIS GONZALEZ-SANTANDER","doi":"10.55730/1300-0098.3464","DOIUrl":"https://doi.org/10.55730/1300-0098.3464","url":null,"abstract":"From the algebraic solution of $x^{m}-x+t=0$ for $m=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some definite integrals in terms of elementary functions.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
As a continuation of the paper "Adjunction Identity to Hypersemigroup" in Turk J Math 2022; 46 (7): 2834--2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup $S$ an ordered hypersemigroup $V$ having a greatest element ($poe$-hypersemigroup) can be constructed in such a way that there exists a pseudoideal $T$ of $S$ such that $S$ is isomorphic to $T$. If $S$ does not have a greatest element, then this can be regarded as the embedding of an ordered hypersemigroup in an ordered semigroup with greatest element.
作为Turk J Math 2022论文“超移民群的附加恒等式”的延续;46(7): 2834—2853，本文证明了最大元与有序超群的共轭实际上是一个嵌入问题。引入了伪理想的概念，并证明了对于每一个有序超移民群$S$，一个具有最大元($poe$-超移民群)的有序超移民群$V$可以构造成$S$的伪理想$T$，使得$S$与$T$同构。如果$S$没有最大元，则这可以看作是在一个有最大元的有序半群中嵌入了一个有序超群。
{"title":"Adjunction greatest element to ordered hypersemigroups","authors":"NIOVI KEHAYOPULU","doi":"10.55730/1300-0098.3452","DOIUrl":"https://doi.org/10.55730/1300-0098.3452","url":null,"abstract":"As a continuation of the paper \"Adjunction Identity to Hypersemigroup\" in Turk J Math 2022; 46 (7): 2834--2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup $S$ an ordered hypersemigroup $V$ having a greatest element ($poe$-hypersemigroup) can be constructed in such a way that there exists a pseudoideal $T$ of $S$ such that $S$ is isomorphic to $T$. If $S$ does not have a greatest element, then this can be regarded as the embedding of an ordered hypersemigroup in an ordered semigroup with greatest element.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135866262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an extended calculus over the function algebra on $h$-deformed superplane. For this, we extend the $(h_1,h_2)$-deformed differential calculus on the $h$-deformed superplane by adding inner derivations. We reformulate the results with an $R$-matrix and present the tensor product realization of the wedge product. We also discuss Cartan calculus via a contraction.
{"title":"Extended calculus on ${cal O}({mathbb C}_{h}^{1vert1})$","authors":"SALİH ÇELİK","doi":"10.55730/1300-0098.3456","DOIUrl":"https://doi.org/10.55730/1300-0098.3456","url":null,"abstract":"We give an extended calculus over the function algebra on $h$-deformed superplane. For this, we extend the $(h_1,h_2)$-deformed differential calculus on the $h$-deformed superplane by adding inner derivations. We reformulate the results with an $R$-matrix and present the tensor product realization of the wedge product. We also discuss Cartan calculus via a contraction.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the following critical fractional semilinear Neumann problem begin{equation*} begin{cases} (-Delta)^{1/2}u+lambda u=u^{frac{n+1}{n-1}},~u>0quad&, mathrm{in} Omega, partial_nu{u}=0 &mathrm{on} partialOmega, end{cases} end{equation*} where $Omegasubsetmathbb{R}^n~(ngeq5)$ is a smooth bounded domain, $lambda>0$ and $nu$ is the outward unit normal to $partialOmega$. We prove that there exists a constant $lambda_0>0$ such that the above problem admits a minimal energy solution for $lambda
{"title":"Fractional semilinear Neumann problem with critical nonlinearity","authors":"ZHENFENG JIN, HONGRUI SUN","doi":"10.55730/1300-0098.3458","DOIUrl":"https://doi.org/10.55730/1300-0098.3458","url":null,"abstract":"In this paper, we consider the following critical fractional semilinear Neumann problem begin{equation*} begin{cases} (-Delta)^{1/2}u+lambda u=u^{frac{n+1}{n-1}},~u>0quad&, mathrm{in} Omega, partial_nu{u}=0 &mathrm{on} partialOmega, end{cases} end{equation*} where $Omegasubsetmathbb{R}^n~(ngeq5)$ is a smooth bounded domain, $lambda>0$ and $nu$ is the outward unit normal to $partialOmega$. We prove that there exists a constant $lambda_0>0$ such that the above problem admits a minimal energy solution for $lambda<lambda_0$. Moreover, if $Omega$ is convex, we show that this solution is constant for sufficiently small $lambda$.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The higher topological complexity of a space $X$, $text{TC}_r(X)$, $r=2,3,ldots$, and the topological complexity of a map $f$, $text{TC}(f)$, have been introduced by Rudyak and Pavesiç, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map $f$, $text{TC}_{r,s}(f)$, for $1leq sleq rgeq2$, which simultaneously extends Rudyak's and Pavesiç notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $text{TC}_{r,s}(fcolon Xto Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $text{TC}_{r,s}$ in the case of the nontrivial double covering over real projective spaces, as well as for their complex counterparts.
{"title":"Higher topological complexity of a map","authors":"CESAR AUGUSTO IPANAQUE ZAPATA, JESÚS GONZÁLEZ","doi":"10.55730/1300-0098.3453","DOIUrl":"https://doi.org/10.55730/1300-0098.3453","url":null,"abstract":"The higher topological complexity of a space $X$, $text{TC}_r(X)$, $r=2,3,ldots$, and the topological complexity of a map $f$, $text{TC}(f)$, have been introduced by Rudyak and Pavesiç, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map $f$, $text{TC}_{r,s}(f)$, for $1leq sleq rgeq2$, which simultaneously extends Rudyak's and Pavesiç notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $text{TC}_{r,s}(fcolon Xto Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $text{TC}_{r,s}$ in the case of the nontrivial double covering over real projective spaces, as well as for their complex counterparts.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135866248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we consider a velocity-vorticity formulation for the $g$-Navier-Stokes equations. The system is constructed by combining the velocity-pressure system which is included by using the rotational formulation of the nonlinearity and the vorticity equation for the $g$ -Navier-Stokes equations. We prove the existence and uniqueness of weak and strong solutions of this system with the periodic boundary conditions.
{"title":"On the weak and strong solutions of the velocity-vorticity model of the $g$-Navier-Stokes equations","authors":"ÖZGE KAZAR, MERYEM KAYA","doi":"10.55730/1300-0098.3457","DOIUrl":"https://doi.org/10.55730/1300-0098.3457","url":null,"abstract":"In this work, we consider a velocity-vorticity formulation for the $g$-Navier-Stokes equations. The system is constructed by combining the velocity-pressure system which is included by using the rotational formulation of the nonlinearity and the vorticity equation for the $g$ -Navier-Stokes equations. We prove the existence and uniqueness of weak and strong solutions of this system with the periodic boundary conditions.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135865214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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