The range of the p-Laplacian

We prove that for λ ≥ 0, p ≥ 3, there exists an open ball B L²(0,1) such that the problem

− (|u′|^p−2 u′)′ − λ|u|^p−2u = f, in (0,1)

, subject to certain separated boundary conditions on (0,1), has a solution for f B.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms

Significant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.

For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.

For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory.     

Further results on irregular, critical perfect systems of difference sets II: systems without splits

An (m, n; u, v; c)-system is a collection of components, m of valency u−1 and n of valency v−1, whose difference sets form a perfect system with threshold c. If there is an (m, n; 3, 6; c)-system, then m2c−1; and if there is a (2c−1, n; 3, 6; c)-system, then 2c−1n. For all sufficiently large c, there are (2c−1, n; 3, 6; c)-systems with a split at 3c+6n−1 at least when n=1, 5, 6 and 7, but such systems do not exist for n=2, 3 or 4.

We describe here a general method of construction for (2c−1, n; 3, 6; c)-systems and use it to show that there are such systems for 2n4 and certain values of c depending on n. We also discuss the limitations of this method.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).     

On ramsey numbers for large disjoint unions of graphs

Let be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r(G, H) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from . Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and mn,

r(mK_k, nK_l)=(k − 1)m+ln+r(K_k−1, K_l−1)−2.

     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Cubic theta functions

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show that

a′(q,z)³=b(q,z)³+c(q)²c(q,z).

This is a counterpart of the identity

a(q,z)³=b(q)²b(q,z³)+c(q,z)³,

which was found by Hirschhorn et al.

The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.