On the residuality of mixing by convolutions probabilities

A probability measureμ on a locally compactσ — compact amenable Hausdorff groupG is called mixing by convolutions if for every pair of probabilitiesν ₁,ν ₂ onG we have: $$\mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = \mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = 0.$$ . It is proved that the set of all mixing by convolutions probabilities is a norm (variation) dense subset of the setP(G) of all probabilities onG. IfG is additionally second countable the mixing measures are residual inP(G).     

Spanning forests and the golden ratio

For a graph G, let f_ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_ij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(f_ij)_n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ^-1 or to 1-φ^-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.     

Equality conditions for the singular values of 3 × 3 matrices with one-point spectrum

Let Γ_a be an upper triangular 3 × 3 matrix with diagonal entries equal to a complex scalar a. Necessary and su.cient conditions are found for two of the singular values of Γ_a to be equal. These conditions are much simpler than the equality discr ? = 0, where the expression in the left-hand side is the discriminant of the characteristic polynomial ? of the matrix _Ga = Γ_aΓ_a. Understanding the behavior of singular values of Γ_a is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix A.     

Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

In this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward.     

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

In this article we prove the Jacquet-Langlands local correspondence in non-zero characteristic. Let F be a local field of non-zero charactersitic and G′ an inner form of GL_n(F); then, following [17], we prove relations between the representation theory of G′ and the representation theory of an inner form of GL_n(L), where L is a local field of zero characteristic close to F. The proof of the Jacquet-Langlands correspondence between G′ and GL_n(F) is done using the above results and ideas from the proof by Deligne, Kazhdan and Vignéras [10] of the zero characteristic case. We also get the following, already known in zero characteristic: orthogonality relations for G′, inequality involving conductor and level for representations of G′ and finiteness for automorphic cuspidal representations with fixed component at almost every place for an inner form of GL_n over a global field of non-zero characteristic.     

Graphs determined by their generalized characteristic polynomials

For a given graph G with (0, 1)-adjacency matrix A_G, the generalized characteristic polynomial of G is defined to be ?_G=?_G(λ,t)=det(λI-(A_G-tD_G)), where I is the identity matrix and D_G is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ?_G. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ?_G. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching.     

Zero forcing parameters and minimum rank problems

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z₊(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented.     

Stay-in-a-set games

There exists a Nash equilibrium (ε-Nash equilibrium) for every n-person stochastic game with a finite (countable) state space and finite action sets for the players if the payoff to each player i is one when the process of states remains in a given set of states G _i and is zero otherwise. Received: December 2000     

Zero Divisor Graph of a Poset with Respect to an Ideal

In this paper, we introduce the zero divisor graph G _I (P) of a poset P (with 0) with respect to an ideal I. It is shown that G _I (P) is connected with its diameter ??3, and if G _I (P) contains a cycle, then the core K of G _I (P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of G _I (P) are shown to be equal. This proves a form of Beck??s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.     

Distance spectra and distance energy of integral circulant graphs

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G₁,G₂) of integral circulant graphs with equal distance energy - in the first family G₁ is subgraph of G₂, while in the second family the diameter of both graphs is three.     

Computational algorithm and parameter optimization for a multi-server system with unreliable servers and impatient customers

We consider an infinite capacity M/M/c queueing system with c unreliable servers, in which the customers may balk (do not enter) and renege (leave the queue after entering). The system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition of system equilibrium is obtained. System performance measures are explicitly derived in terms of computable forms. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of a matrix analytical approach. A cost model is derived to determine the optimal values of the number of servers, service rate and repair rate simultaneously at the minimal total expected cost per unit time. The parameter optimization is illustrated numerically by the Quasi-Newton method.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

An Algorithmic Approach for Sensitivity Analysis of Perturbed Quasi-Birth-and-Death Processes

In this paper, we present an algorithmic approach for sensitivity analysis of stationary and transient performance measures of a perturbed continuous-time level-dependent quasi-birth-and-death (QBD) process with infinitely-many levels. By developing a new LU-type RG-factorization using the censoring technique, we obtain the maximal negative inverse of the infinitesimal generator of the QBD process. The derivatives of the stationary performance measures of the QBD process can then be expressed and computed in terms of the maximal negative inverse, overcoming the computational difficulty arising from the use of group inverses of infinite size in the current literature (see Cao and Chen [11]). We also use a stochastic integral functional to study the transient performance measure of the QBD process and show how to use the algorithmic approach for its sensitivity analysis. As an example, a perturbed MAP/PH/1 queue is also analyzed.     

Von Neumann finite endomorphism rings

Let K be a field of characteristic zero, G a group acting on a nonempty set X and KX the permutation module induced by this action. By studying traces of idempotents, we prove that the endomorphism ring End_K[G](KX) is von Neumann finite under certain conditions for the action of G on X. This generalizes a classical result by Kaplansky for the group ring of G over K.     

A solution concept forn-person noncooperative games

The paper describes a solution concept forn-person noncooperative games, developed jointly by the author and Reinhard Selten. Its purpose is to select one specific perfect equilibrium points=s (G) as the solution of any given noncooperative gameG. The solution is constructed by an inductive procedure. In defining the solutions (G) of gameG, we use the solutionss (G ^*) of the component gamesG ^* (if any) ofG; and in defining the solutions (G^*) of any such component gameG ^*, we use the solutionss (G ^**) of its own component gamesG ^** (if any), etc. This inductive procedure is well-defined because it always comes to an end after a finite number of steps. At each level, the solution of a game (or of a component game) is defined in two steps. First, aprior subjectiveprobability distribution p _i is assigned to the pure strategies of each playeri, meant to represent the other players' initial expectations about playeri's likely strategy choice. Then, a mathematical procedure, called thetracing procedure, is used to define the solution on the basis of these prior probability distributionsp _i . The tracing procedure is meant to provide a mathematical representation for thesolution process by which rational players manage to coordinate their strategy plans and their expectations, and make them converge to one specific equilibrium point as solution for the game     

Distinguishing sceneries by observing the scenery along a random walk path

LetG be an infinite connected graph with vertex setV. Ascenery onG is a map ξ :V → 0, 1 (equivalently, an assignment of zeroes and ones to the vertices ofG). LetS _{n n≥0} be a simple random walk onG, starting at some distinguished vertex v₀. Now let ξ and η be twoknown sceneries and assume that we observe one of the two sequences ξ(S _n) _n≥0 or {η(S _n)} _n≥0 but we do not know which of the two sequences is observed. Can we decide, with a zero probability of error, which of the two sequences is observed? We show that ifG = Z orG = Z², then the answer is “yes” for each fixed ξ and “almost all” η. We also give some examples of graphsG for which almost all pairs (ξ, η) are not distinguishable, and discuss some variants of this problem.     

Tricyclic graphs with exactly two main eigenvalues

An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let G ₀ be the graph obtained from G by deleting all pendant vertices and δ(G) the minimum degree of vertices of G. In this paper, all connected tricyclic graphs G with δ(G ₀) ≥ 2 and exactly two main eigenvalues are determined.     

Semisimple strata for p-adic classical groups

Let F₀ be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F₀. In this paper, we prove some fundamental results towards the classification of the representations of G via types [8]. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].     

On Auslander-Reiten Components and Height Zero Lattices for Integral Group Rings

Let ?? G be the group ring of a finite group G over a complete discrete valuation ring ??. Then certain ?? G-lattices of height zero lie at the ends of their Auslander-Reiten components of tree class A _∞.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.