scholarly journals Channels, Billiards, and Perfect Matching 2-Divisibility

10.37236/9151 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Grant T. Barkley ◽  
Ricky Ini Liu
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Adjacency Matrix ◽  
Fast Algorithm ◽  
Perfect Matching ◽  
Combinatorial Proof ◽  
Domino Tilings ◽  
Local Graph ◽  
Simply Connected ◽  
Vertex Sets

Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.

scholarly journals Specific and Complete Local Integration of Patterns in Bayesian Networks

2017 ◽  
Author(s):  
Martin Biehl ◽  
Takashi Ikegami ◽  
Daniel Polani
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Bayesian Networks ◽  
Upper Bound ◽  
Formal Analysis ◽  
Special Role ◽  
Local Integration

We present a first formal analysis of specific and complete local integration. Complete local integration was previously proposed as a criterion for detecting entities or wholes in distributed dynamical systems. Such entities in turn were conceived to form the basis of a theory of emergence of agents within dynamical systems. Here, we give a more thorough account of the underlying formal measures. The main contribution is the disintegration theorem which reveals a special role of completely locally integrated patterns (what we call ι-entities) within the trajectories they occur in. Apart from proving this theorem we introduce the disintegration hierarchy and its refinement-free version as a way to structure the patterns in a trajectory. Furthermore we construct the least upper bound and provide a candidate for the greatest lower bound of specific local integration. Finally, we calculate the i-entities in small example systems as a first sanity check and find that ι-entities largely fulfil simple expectations.

On The Kakeya Constant

1965 ◽  
Vol 17 ◽  
pp. 946-956 ◽  
Author(s):  
F. Cunningham ◽  
I. J. Schoenberg
Keyword(s):  
Lower Bound ◽  
Small Area ◽  
Original Position ◽  
Connected Sets ◽  
Unit Segment ◽  
Simply Connected

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

Entropy of dynamical systems and perturbations of operators

1991 ◽  
Vol 11 (4) ◽  
pp. 779-786 ◽  
Author(s):  
Dan Voiculescu
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Hausdorff Measure ◽  
Spectral Measure ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
Hilbert Space Operators ◽  
Neumann Class ◽  
Adjoint Operators

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

scholarly journals Crossings in Grid Drawings

10.37236/3025 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vida Dujmović ◽  
Pat Morin ◽  
Adam Sheffer
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Geometric Graphs ◽  
Vertex Sets

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a $d$-dimensional grid of volume $N$ and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\Omega((m^2/N)\log(m/N))$ crossings. In $d$-dimensions, with $d\ge 4$, this bound becomes $\Omega(m^2/N)$. We provide matching upper bounds for all $d$. Finally, for $d\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.

scholarly journals Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

2021 ◽  
Vol 37 ◽  
pp. 295-315
Author(s):  
Derek Young
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Work Group ◽  
Zero Forcing ◽  
Vertex Connectivity ◽  
Aztec Diamond ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Graph Parameters

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

scholarly journals Tau invariants for balanced spatial graphs

2020 ◽  
Vol 29 (09) ◽  
pp. 2050066
Author(s):  
Katherine Vance
Keyword(s):  
Lower Bound ◽  
Floer Homology ◽  
Chain Complex ◽  
Homology Theory ◽  
Heegaard Floer Homology ◽  
Combinatorial Proof ◽  
Knot Concordance ◽  
Spatial Graphs ◽  
Heegaard Floer ◽  
Definition Of

In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

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