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.     

Explicit and exact special solutions for BBM-like B(m, n) equations with fully nonlinear dispersion

In this paper, the BBM-like equations with fully nonlinear dispersion, B(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxt = 0 which exhibit solutions with compact support and with solitary patterns, are studied. The exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of the equations are found by a new method. The special cases, B(2, 2) and B(3, 3), are used to illustrate the concrete scheme of our approach presented by this paper in B(m, n) equations. The nonlinear equations B(m, n) are addressed for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of B(m, n) equations are established.     

New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

The R(m,n) equations u_tt+a(uⁿ)_xx+b(u^m)_xxtt=0 (a, b const) are investigated by using some ansatze. As a result, new exact solitary patterns solutions and solitary wave solutions are obtained. These obtained solutions show that not only the nonlinearly dispersive R(m,m) equations (m≠1) but also the linearly dispersive R(1,n) equations (m=1) possess solitary patterns solutions, which has infinite slopes or cusps and solitary wave solutions.     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

A class of solutions to the gossip problem, part III

We complete the study of NOHO-graphs, begun in Parts I and II of this paper. NOHO- graphs correspond to solutions to the gossip problem where No One Hears his Own information. These are graphs with a linear ordering on their edges such that an increasing path exists from each vertex to every other, but from no vertex to itself. We discard the two such graphs with no 2-valent vertices. In Part I, we translated these graphs into quadruples of integer sequences. In Part II, we characterized and enumerated the realizable quadruples and various subclasses of them. In Part III, we eliminate the overcounting of isomorphic graphs and obtain recurrence relations and generating functions to enumerate the non-isomorphic NOHO-graphs. If u_m=(1,1,2,…) satisfies u_m=3u_m-1–u_m-3, then the number of non-isomorphic NOHO- graphs on 2m+2 vertices is (u_m + u_[m/2]+1 + u_[m/2]+1 - u_[m/2]). We also examine some re lated questions.     

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that (Q(x))(1/2) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q▽u) = b, we find sufficient conditions in order that Q(1/2)▽u has some first order smoothness. Specifically, if Ω is a bounded open set in R~n, we study when the components of Q(1/2)▽u belong to the first order Sobolev space W_Q~(1,2)(Ω)defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X_iu has some first order smoothness if u is a weak solution in Ω of ∑_(i=1)~n X′_iX_(iu) + b = 0.We do not assume that {X_i} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.     

Sperner''s theorem with constraints

A set X of subsets of an n-element set S is called an anti-chain if no two elements of X are related by set-wise inclusion. Sperner showed [8] that max |X|=(ⁿ_[n/2]), where |X| denotes the number of elements in X and the maximum is taken over all anti-chains of subsets of S.

Let non-negative integers i_o<n and m_io≠0, m_io+1,…m_n be given. In this paper we give an algorithm for calculating max |X| where the maximum is taken only over anti-chains containing exactly m_i i-element subsets of S for i_o i n.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets, and the Prohorov property

For a Hausdorff space X, let F be the hyperspace of all closed subsets of X and H a sublattice of F. Following Nogura and Shakhmatov, X is said to be H-trivial if the upper Kuratowski topology and the co-compact topology coincide on H. F-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with K-trivial spaces and Fin-trivial space, where K and Fin are respectively the lattices of compact and of finite subsets of X. It is proved that if C_k(X) is a Baire space or more generally if X has ‘the moving off property’ of Gruenhage and Ma, then X is K-trivial. If X is countable, then C_p(X) is Baire if and only if X is Fin-trivial and all compact subsets of X are finite. As for consonant spaces, it turns out that every regular K-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of K-trivial non-consonant spaces, of Fin-trivial K-nontrivial spaces and of countably compact Prohorov Fin-nontrivial spaces, are given. In particular, we show that all (generalized) Fréchet–Urysohn fans are K-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping f :X→Y, where X is Prohorov and Y is not Prohorov, answering a long-standing question by Topsøe.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

Regular expression searching on compressed text

We present a solution to the problem of regular expression searching on compressed text. The format we choose is the Ziv–Lempel family, specifically the LZ78 and LZW variants. Given a text of length u compressed into length n, and a pattern of length m, we report all the R occurrences of the pattern in the text in O(2^m+mn+Rmlogm) worst case time. On average this drops to O(m²+(n+Rm)logm) or O(m²+n+Ru/n) for most regular expressions. This is the first nontrivial result for this problem. The experimental results show that our compressed search algorithm needs half the time necessary for decompression plus searching, which is currently the only alternative.     

Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s,tX, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m²n²), where m is the number of vertices of P and n is the number of points in X.     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

Integral approach to compacton solutions of Boussinesq-like B(m,n) equation with fully nonlinear dispersion

There exists much good work in the area of usual solitons, but there appears little in the field of compacton solutions. Only a few mathematical tools were employed so far. Recently, Yan [Chaos, Solitons & Fractals 14 (2002) 1151] extended the decomposition method to seek compacton solutions of B(m,n) equation u_tt=(uⁿ)_xx+(u^m)_xxxx. In this paper we present a different approach, integral approach, to investigate the compacton solutions of the B(m,n) equation. Not only Yan’s results but also many new compacton solutions of the B(m,n) equation are obtained. Our approach is simple and also suitable for studying compacton solutions of some other equations.     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).     

Some explicit formulas for the polynomial decomposition of the matrix exponential and applications

This paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix A. More precisely, we suppose that the minimal polynomial M_A(X) of A is known and has degree m. Therefore, e^tA is given in terms of P₀(A),…,P_m−1(A), where the P_j(A) are polynomials in A of degree less than m, and some explicit analytic functions. Examples and applications are given. In particular, the two cases m=5 and m=6 are considered.     

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.     

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 Ding Projective Complexes

In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each X_m is a Ding projective R-module for all m ∈ Z.     

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.