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Avraham Goldstein

Publications and source records attributed to Avraham Goldstein.

8 recordsLinked to original sources
arXiv2021

Generalization of Menger's Edge Theorem to Four Vertices

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there exist $k$ pairwise summand-disjoint formal sums of edges with coefficients in $\mathbb{F}_2$, each one of which is mapped by the boundary map to the sum of vertices $A$ and $B$, if and only if after a deletion of any $k-1$ or less edges there still exist a formal sum of edges with coefficients in $\mathbb{F}_2$ which is mapped by the boundary map to $A+B$. We extend this result to four vertices $A,B,C,D$. We prove that in an undirected graph, in which all the vertices different from $A,B,C,D$ have even degrees, the following two statements are equivalent: There exist $k$ pairwise summand-disjoint formal sums of edges with coefficients in $\mathbb{F}_2$, each one of which is mapped by the boundary map to $A+B+C+D$; After a deletion of any $k-1$ or less edges there still exists a formal sum of edges with coefficients in $\mathbb{F}_2$ which is mapped by the boundary map to $A+B+C+D$. Equivalently, if after a deletion of any $k-1$ or less edges, the four vertices $A,B,C,D$ can be split into two pairs of vertices, and the two vertices in each pair then can be connected by a path so that these two paths are edge-disjoint, then the four vertices $A,B,C,D$ can be split $k$ times into two pairs of vertices and the two vertices in each one of these $2k$ pairs can then be connected by a path in such a way that all these $2k$ paths are pairwise edge-disjoint.

math.CO↗
arXiv2015

Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

The Plaque Inverse Limit of a branched covering self-map of a Riemann surface was introduced and studied in \cite{CCG}. A point $x$ of P.I.L. was called regular if P.I.L. has the natural Riemann Surface structure at $x$ and was called irregular otherwise. The notion of the signature $sign(x,c)$ of $x$ with respect to a critical point $c$, which was shown to be a local invariant of P.I.L. was introduced and developed. It was shown that $sign(x,c)$ is nontrivial for some critical points $c$ if and only if $x$ is an irregular point. It was shown that the local topology of P.I.L. at an irregular point $x$ has a property, that removing $x$ from any its neighborhood breaks some path-connected component of that neighborhood into an uncountable number of path-connected components. Finally, various signatures, including signatures of the invariant lifts of super-attracting and attracting cycles and certain signatures of the invariant lift of a parabolic cycle, were computed. All these signatures had a maximal element. In this work we show that the local topology of P.I.L. at irregular points with different types of signatures is different. Namely, we prove that the local topology at an irregular point $x$ has a property, that for any neighborhood $V$ of $x$ and for some point $y\ne x$ in $V$, the open set $V-\{y\}$ consists of uncountable number of path-connected components, if and only if the signature $sign(x,c)$, for some critical point $c$, has no maximal element. Next, for a polynomial functions, we compute the signature of the invariant lift of a parabolic cycle with respect to a certain recurrent critical point. This signature, unlike the cases studied in \cite{CCG}, has no maximal element. We show that all other irregular points, except the invariant lifts of super-attracting, attracting, and parabolic cycles, have no maximal element with respect to some recurrent critical point.

math.DS↗
arXiv2014

Hultman Numbers and Generalized Commuting Probability in Finite Groups

Let $G$ be a finite group and $π$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}} \] when $π$ varies over all the permutations in $S_{n}$. The probability \[ Pr_π(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}}) \] is identical to $Pr_{1}^ω(G)$, with \[ ω=a_{1}a_{2}...a_{n-1}a_{n}a_{π_{1}}^{-1}a_{π_{2}}^{-1}\cdots a_{π_{n-1}}^{-1}a_{π_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativity degree, or the probability of a permutation equality $a_{1}a_{2}=a_{2}a_{1}$, for which $n=2$ and $π=\langle2\;\;1\rangle$, was introduced and assessed by P. Erdös and P. Turan in \cite{ET} in 1968 and by W. H. Gustafson in \cite{G} in 1973. In \cite{G} Gustafson establishes a relation between the probability of $a_{1},a_{2}\in G$ commuting and the number of conjugacy classes in $G$. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of $G$, and compute the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation $π$, depends only on the number $c(Gr(π))$ of the alternating cycles in the cycle graph $Gr(π)$ of $π$. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in \cite{BP}. We describe the spectrum of the probabilities of permutation equalities in a finite group as $π$ varies over all the elements of $S_{n}$. This spectrum turns-out to be closely related to the partition of $n!$ into a sum of the corresponding Hultman numbers.

math.GR↗
arXiv2014

On the topology of the inverse limit of a branched covering over a Riemann surface

We introduce the Plaque Topology on the inverse limit of a branched covering self-map of a Riemann surface of a finite degree greater than one. We present the notions of regular and irregular points in the setting of this Plaque Inverse Limit and study its local topological properties at the irregular points. We construct certain Boolean Algebra and certain sigma-lattice, derived from it, and use them to compute local topological invariants of the Plaque Inverse Limit. Finally, we obtain several results interrelating the dynamics of the forward iterations of the self-map and the topology of the Plaque Inverse Limit.

math.DS↗
arXiv2013

Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.

math.CO↗
arXiv2013

Balanced Abelian group valued functions on directed graphs

We discuss functions from the edges and vertices of a directed graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced and form an Abelian group. We study this group in two cases: when we allowed to walk against the direction of an edge taking the opposite value of the function and when we are not allowed to walk against the direction.

math.CO↗
arXiv2012

Generalization of the Menger's Theorem to Simplicial Complexes and Certain Invariants of the Underlying Topological Spaces

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by $k$ pairwise edge-disjoint paths if, and only if, after a deletion of any $k-1$ edges from the graph, there will still will exist a path connecting these two vertices. We introduce the notion of $k$-boundance of $(n-1)$-dimensional cycles in an $n$-dimensional simplicial complex over $\mathbb{F}_2$, which is a generalization of the classical notion of $k$-edge-connectivity in an undirected graph. For the case $n=1$, $k$-boundance of $0$-dimensional cycles in an undirected graph is just an extension of the classical notion of $k$-edge-connectivity of pairs of vertices, stated in the language of cycles and boundaries. Using the notion of $k$-boundance, we prove that a non-trivial $(n-1)$-dimensional cycle in an $n$-dimensional simplicial complex over $\mathbb{F}_2$ is a boundary of $k$ pairwise disjoint $n$-dimensional chains if, and only if, after a deletion of any $k-1$ $n$-dimensional simplices from that complex, there still remains some $n$-dimensional chain in it, for which this $(n-1)$-dimensional cycle is a boundary. In our last section we restate both the original Menger's Theorem and our generalization to $k$-boundance in $n$ dimensions, in terms of the underlying topological space. Thus, $k$-edge-connectivity of a pair of points in an undirected graph is really a topological property of the corresponding pair of points in the topological space, underlying that graph. Similarly, $k$-boundance of an $(n-1)$-dimensional cycle is a topological property of the topological subspace, underlying that $(n-1)$-dimensional cycle, in the topological space, underlying the $n$-dimensional simplicial complex.

math.GT↗