The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme

Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the irreducible $T$-modules occur in the standard module. We compute this multiplicity for those irreducible $T$-modules which have diameter at least $D-3$.Comment: 22 page

Higher Dimensional Lattice Chains and Delannoy Numbers

Fix nonnegative integers n1 , . . ., nd, and let L denote the lattice of points (a1 , . . ., ad) ∈ ℤd that satisfy 0 ≤ ai ≤ ni for 1 ≤ i ≤ d. Let L be partially ordered by the usual dominance ordering. In this paper we use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in L. Setting ni = n (for all i) in these expressions yields a new proof of a recent result of Duichi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension

Proof of the Kresch-Tamvakis Conjecture

In this paper we resolve a conjecture of Kresch and Tamvakis. Our result is the following. Theorem: For any positive integer $D$ and any integers $i,j$ $(0\leq i,j \leq D)$, the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left[ \begin{array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem

Data Analysis and Material Property Inferences from Passive Microscopic Probe Experiments in Heterogeneous Biological Systems

Passive particle-tracking microrheology (PPTM) exploits thermal fluctuations of passive microscale probes dispersed in soft matter. The probes are tracked using microscopy; then application of the generalized Stokes-Einstein relation determines the equilibrium dynamic moduli of the sample material. The methodology was designed and applied since the 1990s for presumed homogeneous soft matter materials. The standard PPTM data analysis calculates the ensemble-averaged mean-squared displacement (MSD) of the position time series of tracked particles to determine dynamic moduli of the source material, assuming a faithful random particle sampling of the material. In this dissertation, we confront the challenge of using PPTM to probe heterogeneous materials. Our motivation and the primary application of the tools we develop is the pathology of human airway mucus induced by cystic fibrosis (CF). Normal human bronchial epithelial (HBE) mucus is comprised of a mixture of two large molecular weight mucin polymers, MUC5B and MUC5AC. The mucin ratio of MUC5B:MUC5AC progressively drops during progression of CF, coincident with progressively more extreme heterogeneity in structure whereby the mucin polymers aggregate and phase separate into insoluble, dense flakes within otherwise dilute solutions. These structure properties and the associated heterogeneous rheology are the focus of our analysis. We develop a classifier method based on fractional Brownian motion applied to 200 nm and 1 micron diameter beads that: 1. coarse-grains the particle tracking data into beads within versus outside of flakes; 2. separates within-flake data into increment time series that are distinguishable or not from the noise floor; makes inferences subject to data availability of 3. the heterogeneity in flake structure, 4. the dynamic moduli of flakes and of the dilute solution as inferred from both probe diameters.Doctor of Philosoph

The Multiplicities of a Dual-thin Q-polynomial Association Scheme

Let Y=(X,{Ri}1≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E0,...,ED of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi (0≤i≤D) of Yis unimodal. Talking to Terwilliger, Stanton made the related conjecture that mi≤mi+1 and mi≤mD−i for i\u3cD/2. We prove that if Y is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true

Kernels of Directed Graph Laplacians

Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace--namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights

The girth, odd girth, distance function, and diameter of generalized Johnson graphs

For any non-negative integers $v > k > i$, the {\em generalized Johnson graph}, $J(v,k,i)$, is the undirected simple graph whose vertices are the $k$-subsets of a $v$-set, and where any two vertices $A$ and $B$ are adjacent whenever $|A \cap B| =i$. In this article, we derive formulas for the girth, odd girth, distance function, and diameter of $J(v,k,i)$

Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration

The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have been for a long time a central focus of complexity science, and physics. Here, we introduce group-theoretic concepts to identify and enumerate the symmetric inputs, which result in irreversible system behaviors with undesired effects on many computational tasks. The concept of so-called configuration shift-symmetry is applied on two-dimensional cellular automata as an ideal model of computation. The results show the universal insolvability of “non-symmetric” tasks regardless of the transition function. By using a compact enumeration formula and bounding the number of shift-symmetric configurations for a given lattice size, we efficiently calculate how likely a configuration randomly generated from a uniform or density-uniform distribution turns shift-symmetric. Further, we devise an algorithm detecting the presence of shift-symmetry in a configuration. The enumeration and probability formulas can directly help to lower the minimal expected error for many crucial (non-symmetric) distributed problems, such as leader election, edge detection, pattern recognition, convex hull/minimum bounding rectangle, and encryption. Besides cellular automata, the shift-symmetry analysis can be used to study the non-linear behavior in various synchronous rule-based systems that include inference engines, Boolean networks, neural networks, and systolic arrays

Counting lattice chains and Delannoy paths in higher dimensions

AbstractLattice chains and Delannoy paths represent two different ways to progress through a lattice. We use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in a lattice of arbitrary finite dimension. Specifically, fix nonnegative integers n1,…,nd, and let L denote the lattice of points (a1,…,ad)∈Zd that satisfy 0≤ai≤ni for 1≤i≤d. We prove that the number of chains in L is given by 2nd+1∑k=1kmax′∑i=1k(−1)i+kk−1i−1nd+k−1nd∏j=1d−1nj+i−1nj, where kmax′=n1+⋯+nd−1+1. We also show that the number of Delannoy paths in L equals ∑k=1kmax′∑i=1k(−1)i+k(k−1i−1)(nd+k−1nd)∏j=1d−1(nd+i−1nj). Setting ni=n (for all i) in these expressions yields a new proof of a recent result of Duchi and Sulanke [9] relating the total number of chains to the central Delannoy numbers. We also give a novel derivation of the generating functions for these numbers in arbitrary dimension