Pub Date : 2022-06-18DOI: 10.15673/tmgc.v15i1.2139
Тетяна Осіпчук
In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.
{"title":"On closed weakly m-convexsets","authors":"Тетяна Осіпчук","doi":"10.15673/tmgc.v15i1.2139","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.2139","url":null,"abstract":"In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80017478","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 : 2022-06-18DOI: 10.15673/tmgc.v15i1.2058
N. Konovenko
The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. In this paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [4], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants.
{"title":"On symplectic invariants of planar 3-webs","authors":"N. Konovenko","doi":"10.15673/tmgc.v15i1.2058","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.2058","url":null,"abstract":"The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. \u0000In this paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [4], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85008650","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 : 2022-06-18DOI: 10.15673/tmgc.v15i1.2084
A. Skryabina, P. Stegantseva, N. Bashova
The problem of counting non-homeomorphic topologies as well as all topologies on an n-elements set is still open. The topologies with the weight k>2n-1, where k is the number of the elements of the topology on an n-elements set, which are called close to the discrete topology have been studied completely. Moreover R.~Stanley in 1971, M.~Kolli in 2007 and in 2014 have been found the number of T0-topologies on an n-elements set with weights k≥7·2n-4, k ≥3·2n-3, and k≥5·2n-4 respectively. In the present paper we investigate T0-topologies using the topology vector, being an ordered set of the nonnegative integers that define the minimal neighborhoods of the elements of the given finite set, and also using the special form of 2-CNF of Boolean function. In 2021 the authors found the form of the vector of T0-topologies with k≥5·2n-4 and the values k∈[5·2n-4, 2n-1], for which there are no T0-topologies with the weight k. The method of describing of T0-topologies using the special form of 2-CNF of Boolean function is used for the identification of the mutually dual and self-dual T0-topologies, and the properties of such 2-CNF Boolean function are used for counting T0-topologies with the weight 25·2n-6.
{"title":"Properties of 2-CNF mutually dual and self-dual T_0 -topologies on a finite set and calculation of T_0-topologies of a certain weight","authors":"A. Skryabina, P. Stegantseva, N. Bashova","doi":"10.15673/tmgc.v15i1.2084","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.2084","url":null,"abstract":"The problem of counting non-homeomorphic topologies as well as all topologies on an n-elements set is still open. The topologies with the weight k>2n-1, where k is the number of the elements of the topology on an n-elements set, which are called close to the discrete topology have been studied completely. Moreover R.~Stanley in 1971, M.~Kolli in 2007 and in 2014 have been found the number of T0-topologies on an n-elements set with weights k≥7·2n-4, k ≥3·2n-3, and k≥5·2n-4 respectively. \u0000In the present paper we investigate T0-topologies using the topology vector, being an ordered set of the nonnegative integers that define the minimal neighborhoods of the elements of the given finite set, and also using the special form of 2-CNF of Boolean function. In 2021 the authors found the form of the vector of T0-topologies with k≥5·2n-4 and the values k∈[5·2n-4, 2n-1], for which there are no T0-topologies with the weight k. The method of describing of T0-topologies using the special form of 2-CNF of Boolean function is used for the identification of the mutually dual and self-dual T0-topologies, and the properties of such 2-CNF Boolean function are used for counting T0-topologies with the weight 25·2n-6.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75157062","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 : 2022-05-23DOI: 10.15673/tmgc.v15i1.2225
Luca Di Beo, A. Prishlyak
We study flows on the Boy's surface. The Boy's surface is the image of the projective plane under a certain immersion into the three-dimensional Euclidean space. It has a natural stratification consisting of one 0-dimensional stratum (central point), three 1-dimensional strata (loops starting at this point), and four 2-dimensional strata (three of them are disks lying on the same plane as the 1-dimensional strata, and having the loops as boundaries). We found all 342 optimal Morse-Smale flows and all 80 optimal projective Morse-Smale flows on the Boy's surface.
{"title":"Flows with minimal number of singularities in the Boy's surface","authors":"Luca Di Beo, A. Prishlyak","doi":"10.15673/tmgc.v15i1.2225","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.2225","url":null,"abstract":"We study flows on the Boy's surface. The Boy's surface is the image of the projective plane under a certain immersion into the three-dimensional Euclidean space. It has a natural stratification consisting of one 0-dimensional stratum (central point), three 1-dimensional strata (loops starting at this point), and four 2-dimensional strata (three of them are disks lying on the same plane as the 1-dimensional strata, and having the loops as boundaries). We found all 342 optimal Morse-Smale flows and all 80 optimal projective Morse-Smale flows on the Boy's surface.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90519376","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 : 2022-05-21DOI: 10.15673/tmgc.v15i1.1860
M. Shoptrajanov
In this paper we will discuss a dynamical approach to an open problem from shape theory. We will address the problem in compact metric spaces using the notion of Lebesgue number for a covering and the intrinsic approach to strong shape.
{"title":"A dynamical approach to shape","authors":"M. Shoptrajanov","doi":"10.15673/tmgc.v15i1.1860","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.1860","url":null,"abstract":"\u0000In this paper we will discuss a dynamical approach to an open problem from shape theory. \u0000We will address the problem in compact metric spaces using the notion of Lebesgue number for a covering and the intrinsic approach to strong shape. \u0000","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85392829","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 : 2022-05-10DOI: 10.15673/tmgc.v15i1.2196
A. Zaim
Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.
{"title":"Relative Gottlieb groups of mapping spaces and their rational cohomology","authors":"A. Zaim","doi":"10.15673/tmgc.v15i1.2196","DOIUrl":"https://doi.org/10.15673/tmgc.v15i1.2196","url":null,"abstract":"Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"4160 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86755033","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 : 2022-02-07DOI: 10.15673/pigc.v14i3.2205
Y. Drozd, N. Konovenko, S. Maksymenko, S. Plaksa, Olexander Prishlyak
On May 25-28, 2021 held an International online conference "Algebraic and geometric methods of analysis" dedicated to the memory of an outstanding mathematician, the Corresponding member of National Academy of Sciences of Ukraine Yuriy Yuriyovych Trokhymchuk.
{"title":"Yuriy Yuriyovych Trokhymchuk","authors":"Y. Drozd, N. Konovenko, S. Maksymenko, S. Plaksa, Olexander Prishlyak","doi":"10.15673/pigc.v14i3.2205","DOIUrl":"https://doi.org/10.15673/pigc.v14i3.2205","url":null,"abstract":"\u0000On May 25-28, 2021 held an International online conference \"Algebraic and geometric methods of analysis\" dedicated to the memory of an outstanding mathematician, the Corresponding member of National Academy of Sciences of Ukraine Yuriy Yuriyovych Trokhymchuk. \u0000 \u0000 \u0000 ","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"145 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80468233","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 : 2022-02-06DOI: 10.15673/tmgc.v14i4.2153
Анатолій Петрович Петравчук
abstract{ukrainian}{Нехай $mathbb K$ -- алгебраїчно замкнене поле харатеристики нуль,$A = mathbb K[x_1,dots,x_n]$ -- кільце многочленів і$R = mathbb K(x_1,dots,x_n)$ -- поле раціональних функцій від $n$ змінних. Позначимо через $W_n = W_n(mathbb K)$ алгебру Лі всіх$mathbb K$-диференціювань на $A$(у випадку $mathbb C$ це алгебра Лі всіх векторних полів на $ mathbb C^n$ з поліноміальними коефіцієнтами). Для заданого $D in W_n(mathbb K)$ будова централізатора$C_{W_n (mathbb K)}(D)$ залежить від поля констант$Ker D = {phi in R | D(phi)=0}$(тут ми природнім чином розширюємо кожне диференціювання $D$ на $A$ на поле $R$).Досліджено випадок, коли $tr.deg_{mathbb K} Ker D le 1$, охарактеризована будова підалгебри $C_{W_n(mathbb K)}(D)$, зокрема доведено, що якщо $Ker D$ не містить несталих многочленів, то$C_{W_n(mathbb K)}(D)$ скінченновимірний над $mathbb K$. Отримано деякі результати про централізатори лінійних диференціювань в $W_n(mathbb K).$}
{"title":"Centralizers of elements in Lie algebras of vector fields with polynomial coefficients","authors":"Анатолій Петрович Петравчук","doi":"10.15673/tmgc.v14i4.2153","DOIUrl":"https://doi.org/10.15673/tmgc.v14i4.2153","url":null,"abstract":"abstract{ukrainian}{Нехай $mathbb K$ -- алгебраїчно замкнене поле харатеристики нуль,$A = mathbb K[x_1,dots,x_n]$ -- кільце многочленів і$R = mathbb K(x_1,dots,x_n)$ -- поле раціональних функцій від $n$ змінних. Позначимо через $W_n = W_n(mathbb K)$ алгебру Лі всіх$mathbb K$-диференціювань на $A$(у випадку $mathbb C$ це алгебра Лі всіх векторних полів на $ mathbb C^n$ з поліноміальними коефіцієнтами). Для заданого $D in W_n(mathbb K)$ будова централізатора$C_{W_n (mathbb K)}(D)$ залежить від поля констант$Ker D = {phi in R | D(phi)=0}$(тут ми природнім чином розширюємо кожне диференціювання $D$ на $A$ на поле $R$).Досліджено випадок, коли $tr.deg_{mathbb K} Ker D le 1$, охарактеризована будова підалгебри $C_{W_n(mathbb K)}(D)$, зокрема доведено, що якщо $Ker D$ не містить несталих многочленів, то$C_{W_n(mathbb K)}(D)$ скінченновимірний над $mathbb K$. Отримано деякі результати про централізатори лінійних диференціювань в $W_n(mathbb K).$}","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77934330","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 : 2022-02-06DOI: 10.15673/tmgc.v14i4.2204
S. Maksymenko, Eugene Polulyakh Institute of Mathematics of Nas of Ukraine, Kyiv, Ukraine
Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R×(0,1) with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines R×t, t∊(0,1), and boundary intervals which gives a foliation Δ on all of Z. Denote by H(Z,Δ) the group of all homeomorphisms of Z that maps leaves of Δ onto leaves and by H(Z/Δ) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π0H(Z,Δ) with a group of automorphisms of a certain graph G with additional structure which encodes the combinatorics of gluing Z from strips. That graph is in a certain sense dual to the space of leaves Z/Δ. On the other hand, for every hinH(Z,Δ) the induced permutation k of leaves of Δ is in fact a homeomorphism of Z/Δ and the correspondence h→k is a homomorphism ψ:H(Δ)→H(Z/Δ). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ0:π0H(Z,Δ)→π0H(Z/Δ) which turns out to be either injective or having a kernel Z2. This gives a dual description of π0H(Z,Δ) in terms of the space of leaves.
设Z是一个非紧二维流形，它是由一组具有边界区间的开带rx(0,1)通过将这些开带沿着边界区间的某些对粘接而得到的。每条这样的条带都有一个自然的叶理，在所有Z上形成平行线R×t, t(0,1)和边界区间，这些边界区间给出了一个叶理Δ。用H(Z，Δ)表示将Δ的叶子映射到叶子上的Z的所有同胚群，用H(Z/Δ)表示具有相应紧开拓扑的叶子空间的同胚群。最近，作者发现了一类具有附加结构的图G的自同构群π0H(Z，Δ)，该自同构群编码了带胶合Z的组合。这个图在某种意义上是对叶空间Z/Δ的对偶。另一方面，对于每一个hin h (Z，Δ)，Δ的叶的诱导排列k实际上是Z/Δ的同态，对应h→k是ψ: h (Δ)→h (Z/Δ)的同态。本文的目的是证明ψ引申出相应的同构群ψ0:π0H(Z，Δ)→π0H(Z/Δ)的一个同态，这个同态要么是内射，要么有一个核Z2。给出了π0H(Z，Δ)在叶空间中的对偶描述。
{"title":"Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves","authors":"S. Maksymenko, Eugene Polulyakh Institute of Mathematics of Nas of Ukraine, Kyiv, Ukraine","doi":"10.15673/tmgc.v14i4.2204","DOIUrl":"https://doi.org/10.15673/tmgc.v14i4.2204","url":null,"abstract":"Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R×(0,1) with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines R×t, t∊(0,1), and boundary intervals which gives a foliation Δ on all of Z. Denote by H(Z,Δ) the group of all homeomorphisms of Z that maps leaves of Δ onto leaves and by H(Z/Δ) the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group π0H(Z,Δ) with a group of automorphisms of a certain graph G with additional structure which encodes the combinatorics of gluing Z from strips. That graph is in a certain sense dual to the space of leaves Z/Δ. \u0000On the other hand, for every hinH(Z,Δ) the induced permutation k of leaves of Δ is in fact a homeomorphism of Z/Δ and the correspondence h→k is a homomorphism ψ:H(Δ)→H(Z/Δ). The aim of the present paper is to show that ψ induces a homomorphism of the corresponding homeotopy groups ψ0:π0H(Z,Δ)→π0H(Z/Δ) which turns out to be either injective or having a kernel Z2. This gives a dual description of π0H(Z,Δ) in terms of the space of leaves.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76465985","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 : 2021-12-24DOI: 10.15673/tmgc.v14i3.2078
V. Fedorchuk, V. Fedorchuk
We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.
{"title":"On symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation","authors":"V. Fedorchuk, V. Fedorchuk","doi":"10.15673/tmgc.v14i3.2078","DOIUrl":"https://doi.org/10.15673/tmgc.v14i3.2078","url":null,"abstract":"We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"321 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80250709","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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