Minimum Rank Problems, 2010 Iowa State University

#### Minimum Rank Problems, Leslie Hogben

*Mathematics Publications*

A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal ...

Maximum Generic Nullity Of A Graph, 2010 Iowa State University

#### Maximum Generic Nullity Of A Graph, Leslie Hogben, Bryan Shader

*Mathematics Publications*

For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is introduced. Maximum generic nullity provides insight into the structure of the null-space of a matrix realizing maximum nullity of a graph. It is shown that ...

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, 2010 West Chester University of Pennsylvania

#### Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills

*Mathematics*

Lucy Slater used Bailey's 6*Ã*6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10*Ã*10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters ...

Recognizing Graph Theoretic Properties With Polynomial Ideals, 2010 University of California - Davis

#### Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar

*All HMC Faculty Publications and Research*

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term *polynomial method* to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

On The Equivalence Between Real Mutually Unbiased Bases And A Certain Class Of Association Schemes, 2010 Worcester Polytechnic Institute

#### On The Equivalence Between Real Mutually Unbiased Bases And A Certain Class Of Association Schemes, Nicholas Lecompte, William J. Martin, William Owens

*Mathematical Sciences Faculty Publications*

Mutually unbiased bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given dimension and their construction remain open. In an attempt to understand the complex case better, some authors have also considered real MUBs, mutually unbiased bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiased bases and 4-class cometric association schemes which are both Q-bipartite and Q-antipodal. We then explore the consequences of this equivalence, constructing new cometric ...

The Maximum Rectilinear Crossing Number Of The Petersen Graph, 2010 CUNY Kingsborough Community College

#### The Maximum Rectilinear Crossing Number Of The Petersen Graph, Elie Feder, Heiko Harborth, Steven Herzberg, Sheldon Klein

*Publications and Research*

We prove that the maximum rectilinear crossing number of the *Petersen graph *is 49. First, we illustrate a picture of the Petersen graph with 49 crossings to prove the lower bound. We then prove that this bound is sharp by carefully analyzing the ten Cs's which occur in the Petersen graph and their properties.

Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, 2010 Claremont McKenna College

#### Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger

*CMC Senior Theses*

Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.

Minimum Rank, Maximum Nullity And Zero Forcing Number For Selected Graph Families, 2010 University of Puerto Rico, Río Piedras Campus

#### Minimum Rank, Maximum Nullity And Zero Forcing Number For Selected Graph Families, Edgard Almodovar, Laura Deloss, Leslie Hogben, Kirsten Hogenson, Kaitlyn Murphy, Travis Peters, Camila A. Ramírez

*Mathematics Publications*

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ij-th entry (for i≠j) is nonzero whenever {i,j}{i,j} is an edge in G and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper defines the graph families ciclos and estrellas and ...

Linear Stochastic State Space Theory In The White Noise Space Setting, 2010 Chapman University

#### Linear Stochastic State Space Theory In The White Noise Space Setting, Daniel Alpay, David Levanony, Ariel Pinhas

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior.

On The Characteristics Of A Class Of Gaussian Processes Within The White Noise Space Setting, 2010 Chapman University

#### On The Characteristics Of A Class Of Gaussian Processes Within The White Noise Space Setting, Daniel Alpay, Haim Attia, David Levanony

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

Using the white noise space framework, we define a class of stochastic processes which include as a particular case the fractional Brownian motion and its derivative. The covariance functions of these processes are of a special form, studied by Schoenberg, von Neumann and Krein.

Linear Stochastic Systems: A White Noise Approach, 2010 Chapman University

#### Linear Stochastic Systems: A White Noise Approach, Daniel Alpay, David Levanony

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study ℓ1-ℓ2 stability in the discrete time case, and L2-L∞ stability in the continuous time case.

Krein Systems And Canonical Systems On A Finite Interval: Accelerants With A Jump Discontinuity At The Origin And Continuous Potentials, 2010 Chapman University

#### Krein Systems And Canonical Systems On A Finite Interval: Accelerants With A Jump Discontinuity At The Origin And Continuous Potentials, Daniel Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, A. Sakhnovich

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

This paper is devoted to connections between accelerants and potentials of Krein systems and of canonical systems of Dirac type, both on a finite interval. It is shown that a continuous potential is always generated by an accelerant, provided the latter is continuous with a possible jump discontinuity at the origin. Moreover, the generating accelerant is uniquely determined by the potential. The results are illustrated on pseudo-exponential potentials. The paper is a continuation of the earlier paper of the authors [1] dealing with the direct problem for Krein systems.

Discrete-Time Multi-Scale Systems, 2010 Chapman University

#### Discrete-Time Multi-Scale Systems, Daniel Alpay, Mamadou Mboup

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the poly-disc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier.

Exit Frequency Matrices For Finite Markov Chains, 2009 Macalester College

#### Exit Frequency Matrices For Finite Markov Chains, Andrew Beveridge, László Lovász

*Andrew Beveridge*

No abstract provided.