Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane

In this paper we study the monotonicity of positive (or non-negative) viscosity solutions to uniformly elliptic equations F(∇u,D²u)=f(u) in the half plane, where f is locally Lipschitz continuous (with f(0)?0) and zero Dirichlet boundary conditions are imposed. The result is obtained without assuming the u or |∇u| are bounded.     

Extended Deligne–Lusztig varieties for general and special linear groups

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group.Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL₂(Fq???/(?²)), with odd q, the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.     

Geometry of plane curves

In this paper we obtain, for a class of plane curves, extensions of the well-known relation of inflection points, double points and bitangencies established by Fabricius-Bjerre for closed curves.     

Explicit functorial correspondences for level zero representations of p-adic linear groups

Let F be a non-Archimedean local field and D a central F-division algebra of dimension n², n?1. We consider first the irreducible smooth representations of D^× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil-Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet-Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.     

Commuting graphs of matrices over semirings

We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.     

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v₁,v₂,…,v_m such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.     

A preconditioning approach to the pagerank computation problem

Some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of Euler-Richardson (ER) iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph. We choose the preconditioner from an algebra U of matrices, thereby obtaining an ER_U method, and we observe that ER_U can outperform ER in terms of rate of convergence. The proposed preconditioner can be updated at a very low cost whenever the graph changes, as is the case when it represents a generic set of information. The particular U utilized requires a surplus of operations per step and memory allocations, which make ER_U superior to ER for not too wide graphs. However, the observed high improvement in convergence rate obtained by preconditioning and the general theory developed, are a reason for investigating different choices of U, more efficient for huge graphs.     

On the sum of the Laplacian eigenvalues of a tree

Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10].     

Graphs whose normalized Laplacian has three eigenvalues

We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. Strongly regular graphs and complete bipartite graphs are examples of such graphs, but we also construct more exotic families of examples from conference graphs, projective planes, and certain quasi-symmetric designs.     

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.