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":"128 1","pages":"0"},"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":"2014 1","pages":"0"},"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}
We prove the conjugacy of Sylow $2$-subgroups in pseudofinite $mathfrak{M}_c$ (in particular linear) groups under the assumption that there is at least one finite Sylow $2$-subgroup. We observe the importance of the pseudofiniteness assumption by analyzing an example of a linear group with non-conjugate finite Sylow $2$-subgroups which was constructed by Platonov.
{"title":"A note on the conjugacy problem for finite Sylow subgroups of linear pseudofinite groups","authors":"Pinar Uugurlu","doi":"10.3906/mat-1604-11","DOIUrl":"https://doi.org/10.3906/mat-1604-11","url":null,"abstract":"We prove the conjugacy of Sylow $2$-subgroups in pseudofinite $mathfrak{M}_c$ (in particular linear) groups under the assumption that there is at least one finite Sylow $2$-subgroup. We observe the importance of the pseudofiniteness assumption by analyzing an example of a linear group with non-conjugate finite Sylow $2$-subgroups which was constructed by Platonov.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":"41 1","pages":"1458-1466"},"PeriodicalIF":1.0,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3906/mat-1604-11","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41481582","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}
: Let M and N be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from M to N and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices M or N , these two classes are the same. By using this relation, we show that if µ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from M into N then µ − 1 : N → M is an orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator).
{"title":"On orthogonally additive band operators and orthogonally additive disjointness preserving operators","authors":"Bahri̇ Turan, Demet Tülü","doi":"10.55730/1300-0098.3425","DOIUrl":"https://doi.org/10.55730/1300-0098.3425","url":null,"abstract":": Let M and N be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from M to N and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices M or N , these two classes are the same. By using this relation, we show that if µ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from M into N then µ − 1 : N → M is an orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator).","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42837298","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 study, we have shown that numerical semigroups M = < 3 , C + 1 , C + 2 > and M = < 3 , C, C + 2 > have maximal or almost maximal length, with conductor C , where C ≡ 0(3) and C ≡ 2(3) , respectively. We also examined whether half of these numerical semigroups were of maximal or almost maximal length.
在本研究中，我们证明了数值半群M = < 3, C + 1, C + 2 >和M = < 3, C, C + 2 >具有极大或几乎极大长度，导体C，其中C分别≡0(3)和C≡2(3)。我们还检查了这些数值半群中是否有一半是最大或几乎最大的长度。
{"title":"A note on half of some MED semigroups of maximal or almost maximal length","authors":"A. Çeli̇k","doi":"10.55730/1300-0098.3445","DOIUrl":"https://doi.org/10.55730/1300-0098.3445","url":null,"abstract":": In this study, we have shown that numerical semigroups M = < 3 , C + 1 , C + 2 > and M = < 3 , C, C + 2 > have maximal or almost maximal length, with conductor C , where C ≡ 0(3) and C ≡ 2(3) , respectively. We also examined whether half of these numerical semigroups were of maximal or almost maximal length.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42943737","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 main objective of this paper is to study certain geometric properties (like univalence, starlikeness, convexity, close-to-convexity) for the normalized Miller-Ross function. The various results, which we have established in the present investigation, are believed to be new, and their importance is illustrated by several interesting consequences and examples. Furthermore, some of the main results improve the corresponding results available in the literature [15]
{"title":"The normalized Miller-Ross function and its geometric properties","authors":"K. Mehrez","doi":"10.55730/1300-0098.3388","DOIUrl":"https://doi.org/10.55730/1300-0098.3388","url":null,"abstract":": The main objective of this paper is to study certain geometric properties (like univalence, starlikeness, convexity, close-to-convexity) for the normalized Miller-Ross function. The various results, which we have established in the present investigation, are believed to be new, and their importance is illustrated by several interesting consequences and examples. Furthermore, some of the main results improve the corresponding results available in the literature [15]","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48296949","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, including αϵ p 0 , 1 q ; we examined the Dirac system in the frame which includes α order right and left Reimann-Liouville fractional integrals and derivatives with exponential kernels, and the Dirac system which includes α order right and left Caputo fractional integrals and derivatives with exponential kernels. Furthermore, we have given some definitions and properties for discrete exponential kernels and their associated fractional sums and fractional differences, and we have studied discrete fractional Dirac systems.
{"title":"Some fractional Dirac systems","authors":"Yüksel Yalçinkaya","doi":"10.55730/1300-0098.3349","DOIUrl":"https://doi.org/10.55730/1300-0098.3349","url":null,"abstract":": In this work, including αϵ p 0 , 1 q ; we examined the Dirac system in the frame which includes α order right and left Reimann-Liouville fractional integrals and derivatives with exponential kernels, and the Dirac system which includes α order right and left Caputo fractional integrals and derivatives with exponential kernels. Furthermore, we have given some definitions and properties for discrete exponential kernels and their associated fractional sums and fractional differences, and we have studied discrete fractional Dirac systems.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49191415","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}
: For any two non-empty (disjoint) chains X and Y , and for a fixed order-preserving transformation θ : Y → X , let GO ( X, Y ; θ ) be the generalized order-preserving transformation semigroup. Let O ( Z ) be the order-preserving transformation semigroup on the set Z = X ∪ Y with a defined order. In this paper, we show that GO ( X, Y ; θ ) can be embedded in O ( Z, Y ) = { α ∈ O ( Z ) : Zα ⊆ Y } , the semigroup of order-preserving transformations with restricted range. If θ ∈ GO ( Y, X ) is one-to-one, then we show that GO ( X, Y ; θ ) and O ( X, im ( θ )) are isomorphic semigroups. If we suppose that | X | = m , | Y | = n , and | im ( θ ) | = r where m, n, r ∈ N , then we find the rank of GO ( X, Y ; θ ) when θ is one-to-one but not onto. Moreover, we find lower bounds for rank ( GO ( X, Y ; θ )) when θ is neither one-to-one nor onto and when θ is onto but not one-to-one.
:对于任意两个非空(不相交)链X和Y，对于一个固定保序变换θ: Y→X，令GO (X, Y;θ)为广义保序变换半群。设O (Z)是集Z = X∪Y上具有定义阶的保序变换半群。本文证明了GO (X, Y;θ)可以嵌入到O (Z, Y) = {α∈O (Z): Zα≤Y}中，即保序变换的受限范围半群。如果θ∈GO (Y, X)是一对一的，则证明GO (X, Y;θ)和O (X, im (θ))是同构半群。假设| X | = m， | Y | = n， | im (θ) | = r，其中m, n, r∈n，则求出GO (X, Y;θ)， θ是一对一的，但不是映上的。此外，我们还找到了rank (GO (X, Y;θ既不是一对一的也不是映上的，θ是映上的但不是一对一的。
{"title":"On the rank of generalized order-preserving transformation semigroups","authors":"Haytham Darweesh Mustafa Abusarris, G. Ayık","doi":"10.55730/1300-0098.3420","DOIUrl":"https://doi.org/10.55730/1300-0098.3420","url":null,"abstract":": For any two non-empty (disjoint) chains X and Y , and for a fixed order-preserving transformation θ : Y → X , let GO ( X, Y ; θ ) be the generalized order-preserving transformation semigroup. Let O ( Z ) be the order-preserving transformation semigroup on the set Z = X ∪ Y with a defined order. In this paper, we show that GO ( X, Y ; θ ) can be embedded in O ( Z, Y ) = { α ∈ O ( Z ) : Zα ⊆ Y } , the semigroup of order-preserving transformations with restricted range. If θ ∈ GO ( Y, X ) is one-to-one, then we show that GO ( X, Y ; θ ) and O ( X, im ( θ )) are isomorphic semigroups. If we suppose that | X | = m , | Y | = n , and | im ( θ ) | = r where m, n, r ∈ N , then we find the rank of GO ( X, Y ; θ ) when θ is one-to-one but not onto. Moreover, we find lower bounds for rank ( GO ( X, Y ; θ )) when θ is neither one-to-one nor onto and when θ is onto but not one-to-one.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46238803","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}
: With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.
{"title":"Generalization of statistical limit-cluster points and the concepts of statistical limit inferior-superior on time scales by using regular integral transformations","authors":"Ceylan Yalçin","doi":"10.55730/1300-0098.3370","DOIUrl":"https://doi.org/10.55730/1300-0098.3370","url":null,"abstract":": With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.","PeriodicalId":51206,"journal":{"name":"Turkish Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49261709","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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