Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

The initial and boundary value problem for the degenerate parabolic equation v_t = Δ(?(v)) + F(v) in the cylinder $\text{Ω × ¦0, ∞), Ω ? R}{}^{\mathrm{n}}$ bounded, for a certain class of point functions ? satisfying ?′(v) ? 0 (e.g., $\text{?(v) = ¦v¦}{}^{\mathrm{m}}\text{sign}\text{v}$) is considered. In the case that F(v) sign $\text{v ? C(1 + ¦?(v)¦}{}^{\mathrm{\alpha }}\text{), α < 1}$, the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that $\text{F(v)}\text{?(v)}$ is nondecreasing a condition is given on the initial state v(x, 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.     

One-sided continuous dependence of maximal solutions

We consider the maximal solution of $\text{u′(s) = ?(s, u(s))}$, where ? satisfies a one-sided variant of Carathéodory's conditions. A best-possible condition is proved for the dependence of u on ?. Also we show that a function v satisfies $\text{v(t) ? v(r) ? ?}{}_{\mathrm{r}}{}^{\mathrm{t}}\text{?(s, v(s))ds}$ if and only if v is dominated by the maximal solution u.     

Character formulas and spectra of compact nilmanifolds

The authors give a new method for calculating the spectrum and multiplicities of the irreducible unitary representations appearing in the quasi-regular representation U: N × L²(ΓβN) → L²(ΓβN) on a compact nilmanifold ΓβN. They proceed by decomposing the trace of U into traces of irreducible representations. The basic calculations in the paper deal with lattice subgroups (Λ = log Γ an additive lattice in the Lie algebra $\text{N}$), essentially using the Poisson summation formula. Let Ad′ be the contragredient adjoint action of N on $\text{N}$¹. If ?₀ ? $\text{N}$¹, the multiplicity of π(?₀) in U is zero unless the Ad′(N) orbit of ?₀ meets $\text{Λ}{}^{\mathrm{\perp }}\text{= {h ? N}{}^1\text{: <h, Λ> ?}\text{Z}\text{}}$. If ?₀ ? Λ^⊥, then the multiplicity is a sum over representatives of certain Ad′(Γ)-orbits in,

$\text{m(π(?}{}_0\text{),U) =}\text{∑}\text{Ad′(N)?}{}_0\text{∩Λ}{}^{\mathrm{\perp }}\text{Ad′(Γ)}\text{k(?)}$

.The constants $\text{k(?)}$ are given both algebraic and geometric interpretations that lead to simple and effective calculations. Similar formulas hold if Γ is not a lattice subgroup.     

A lower bound for v in a t − (v,k, λ) design

In this paper it is shown that if v ? k + 1 then $\text{v ? t ? 1 + (k ? t + 1)(k ? t + 2)}\text{λ}$, where v, k, λ and t are the characteristic parameters of a t ? (v, k, λ) design. We compare this bound with the known lower bounds on v.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

Unbounded derivations tangential to compact groups of automorphisms

We consider unbounded derivations in C¹-algebras commuting with compact groups of ¹-automorphisms. A closed ¹-derivation δ in a C¹-algebra $\text{U}$ is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈$\text{R}$→τ(t)? Aut($\text{U}$) such that $\text{δ =}\text{d}\text{dt}\text{τ(t)¦}{}_{\mathrm{t\; =\; 0}}$. If δ is known to commute with a compact abelian action α:G→Aut($\text{U}$), and if δ(a) = 0 for all a in the fixed point algebra $\text{U}$^α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that $\text{v(γ)}{}^1\text{= ?v(γ), v(γ)}$ is possibly unbounded but affiliated with the center of {$\text{U}$^α}″, and e^tδ(x) = xe^tv(γ) for all x in the Arveson spectral subspace $\text{U}$^α(γ). In particular, if $\text{U}$ is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group.     

Function algebra invariants and the conformal structure of planar domains

Let A(S) be the sup-normed Banach algebra of analytic functions with continuous boundary values on the compact bordered Riemann surface S.For (?) in $\text{A(S)}{}^{\mathrm{?1}}\text{exp}\text{(A(S))}$, the colength of (?) is defined by $\text{∥(?)∥ =}\text{1}\text{2}\text{log inf}\text{{∥ g ∥ ∥ g}{}^{\mathrm{?1}}\text{∥; g ? (?)}}$. Colength is shown to induce a norm on the cohomology group H¹(S,R) dual to the norm induced on the homology group H₁(S,R) by harmonic length, or, equivalently, dual to the norm on Re A(S)^⊥.The existence and uniqueness of extremal functions for the colength functional is demonstrated. The aforementioned norms are shown to determine the conformal structure of S (up to reflection) and to be related to the mapping properties of S.     

Neighbor designs

A neighbor design is an arrangement of r copies of each of v varieties into b circular blocks of size k > 1 such that neighboring objects in each block are distinct and every pair of distinct varieties appears as neighbors in the set of circular blocks exactly λ times. Necessary conditions for the existence of a neighbor design with these parameters v, k, λ, r, b are that $\text{r =}\text{λ(v ? 1)}\text{2}$, and $\text{b =}\text{λv(v ? 1)}\text{2k}$ be integers for k > 2 and v > 2; and for k = 2 or v = 2, it is also necessary that λ be even or k be even, respectively. In this paper we show that these necessary conditions are also sufficient by giving a method to construct a neighbor design for all values of the parameters satisfying the necessary conditions.     

A remark on Hamiltonian cycles

Every 2-connected graph G with $\text{δ ? (v + κ)}\text{3}$ is hamiltonian where v denotes the order, δ the minimum degree and κ the point connectivity of G.     

Minimal codes in abelian group algebras

In this paper we show that two minimal codes $\text{M}$₁ and $\text{M}$₂ in the group algebra $\text{F}$₂[G] have the same (Hamming) weight distribution if and only if there exists an automorphism θ of G whose linear extension to $\text{F}$₂[G] maps $\text{M}$₁ onto $\text{M}$₂. If θ(M₁) = M₂, then $\text{M}$₁ and $\text{M}$₂ are called equivalent. We also show that there are exactly τ(l) inequivalent minimal codes in $\text{F}$₂[G], where ? is the exponent of G, and τ(?) is the number of divisors of ?.     

Local properties of Boolean functions I: injectivity

This work inaugurates a cycle of papers based on the following common idea. Given a property $\text{P}\text{(f)}$ which is defined for Boolean functions $\text{?:}\text{B}{}^{\mathrm{n}}\text{→}\text{B}$, introduce and study a suitable “local” property $\text{p(?,}\text{X}\text{)}$ depending both on the function ? and on a point $\text{X}\text{∈}\text{B}{}^{\mathrm{n}}$ in such a way that for every Boolean function ?, property $\text{P(?)}$ is true if and only if $\text{p(?,}\text{X>}\text{)}$ holds for every $\text{X}\text{∈}\text{B}{}^{\mathrm{n}}$.The first article of the series deals with local injectivity, showing in particular that for n ? 1 local injectivity coincides with global injectivity, whereas for n > 1 there is no injective Boolean function (in the classical sense). Two forthcoming papers will study local isotony and applications to extremal solutions of Boolean equations.     

Graphical t-wise balanced designs

A pair (X, $\text{B}$) will be a t-wise balanced design (tBD) of type t?(v, K, λ) if $\text{B}\text{= (B}{}_{\mathrm{i}}\text{: i ? I)}$ is a family of subsets of X, called blocks, such that: (i) $\text{|X| = v ?}\text{N}$, where $\text{N}$ is the set of positive integers; (ii) $\text{1?t?|B}{}_{\mathrm{i}}\text{|?K?}\text{N}$, for every i ? I; and (iii) if T ? X, |T| = t, then there are λ ? $\text{N}$ indices i ? I where T ? B_i. Throughout this paper we make three restrictions on our tBD's: (1) there are no repeated blocks, i.e. $\text{B}$ will be a set of subsets of X; (2) t ? K or there are no blocks of size t; and (3) $\text{P}{}_{\mathrm{k}}\text{(X)?}\text{B}$ or $\text{B}$ does not contain all k-subsets of X for any t<k?v. Note then that X ? $\text{B}$. Also, if we give the parameters of a specific tBD, then we will choose a minimal K.We focus on the $\text{t?((}\text{p}\text{2}\text{), K, λ)}$ designs with the symmetric group S_p as automorphism group, i.e. X will be the set of $\text{v = (}\text{p}\text{2}\text{)}$ labelled edges of the undirected complete graph K_p and if B ? $\text{B}$ then all subgraphs of K_p isomorphic to B are also in $\text{B}$. Call such tBD's ‘graphical tBD's’. We determine all graphical tBD's with λ = 1 or 2 which will include one with parameters 4?(15,{5,7},1).     

Limit sets of dynamical systems on the space of probability measures

Let m be a dynamical system on the space of probability measures $\text{M}$₁(R^d), and let Λ + (?) be the positive limit set for ? ∈ $\text{M}$₁(R^d), where ? has compact support K ?R^d. The main result of this paper states that support of Λ⁺(?) ?

,support of Λ + (δ_x), where δ_x is the Dirac measure at point x.     

Probabilistic analysis of solving the assignment problem for the traveling salesman problem

This paper deals with probabilistic analysis of optimal solutions of the asymmetric traveling salesman problem. The exact distribution for the number of required next-best solutions of the assignment problem with random data in order to find an optimal tour is given. For every n-city asymmetric problem, there exists an algorithm such that (i) with probability 1 ? s, s?(0,1) the algorithm produces an optimal tour, (ii) it runs in time $\text{O}\text{(}\text{n}{}^4\text{3}\text{)}$, and (iii) it requires less than w((w + n ? 1)$\text{log}\text{(w + n ? 1) + w + 1) +}\text{1}\text{6}\text{w(n}{}^3\text{+ 3n}{}^2\text{+ 2n ? 6)}$ computational steps, where w = log(s)/log(1 ? E_n); E_n ?(0,1) is given by a simple mathematical formula. Additionally, the polynomial of (iii) gives the exact (deterministic) execution time to find w =1 ,2…. next-best solutions of the assignment problem.     

Zeros for accretive operators satisfying certain boundary conditions

Let X be a Banach space with the dual space $\text{X}{}^1$ to be uniformly convex, let D ? X be open, and let $\text{T:}\text{D}\text{?}\text{→ X}$ be strongly accretive (i.e., for some k < 1: (λ ? k)∥ u ? v∥ ? ∥(λ ? 1)(u ? v)+ T(u) ? T(v)∥ for all $\text{u, v ?}\text{D}\text{?}$ and λ > k). Suppose T is demicontinuous and strongly accretive and suppose there exists z?D satisfying: T(x) t(x ? z) for all x??D and t < 0. Then it is shown that T has a unique zero in $\text{D}\text{?}$. This result is then applied to the study of existence of zeros of accretive mappings under apparently different types of boundary conditions on T.     

A Euler-Poincaré characteristic for the open set lattices of finite topologies

Let $\text{T}$ be a finite topology. If P and Q are open sets of $\text{T}$ (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of $\text{T}$ such that Q ? R ? P. A subcollection $\text{D}$ of the open sets of $\text{T}$ is termed an i-discrete collection of $\text{T}$ provided $\text{D}$ contains every open O ∈ $\text{T}$ with the property that ? $\text{D}$ ? O ? ? $\text{D}$, $\text{D}$ contains exactly i minimal covers of ? $\text{D}$, and provided ?$\text{D}$ = ?{O | O ∈ $\text{D}$ and O is a minimal cover of ? $\text{D}$}. A single open set is a O-discrete collection. The number of distinct i-discrete collections of $\text{T}$ is denoted by p($\text{T}$, i). If there does not exist any i-discrete collection then p($\text{T}$,i) = 0, and this happens trivially for the case when i is greater than the number of points on which $\text{T}$ is defined. The object of this article is to establish the theorem: For any finite topology $\text{T}$, the quantity E($\text{T}$) = Σ_{i = 0}^∞ (?1)ⁱp($\text{T}$, i) = 1.     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

Some properties of Q-matrices

This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ $\text{R}$^n×1, the linear complementarity problem

$\text{Iw ? Mz = q}$

,

$\text{w ? 0, z ? 0, and w}{}^{\mathrm{T}}\text{z = 0}$

, has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q.     

Semi-crossed products of C1-algebras

Given a C¹-algebra $\text{U}$ and endomorphim α, there is an associated nonselfadjoint operator algebra $\text{Z}$⁺ X_α$\text{U}$, called the semi-crossed product of $\text{U}$ with α. If α is an automorphim, $\text{Z}$⁺ X_α$\text{U}$ can be identified with a subalgebra of the C¹-crossed product $\text{Z}$⁺ X_α$\text{U}$. If $\text{U}$ is commutative and α is an automorphim satisfying certain conditions, $\text{Z}$⁺ X_α$\text{U}$ is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C₀(S) given by α(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z⁺X_αC₀(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.