Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.     

Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x₁,…,xk) with xi∈G, there exists a non-empty subsequence (xj₁,…,xjl) and a₁,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x₁,…,xt) with xi∈G, there exist indices j₁,…,jn∈N,1?j₁<?<jn?t, and ?₁,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

On the zeros of exponential polynomials

Suppose r = (r₁, …, r_M), r_j ? 0, γ_kj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γ_k · r = ∑_jγ_kjr_j. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑_{j = 1}^Na_je^{?λγ_j · r}, where each a_j is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑_{k = 1}^MA_kx(t ? r_k) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.     

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

Two equivalent measures on weighted hypergraphs

Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2^N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x₀,x₁,…,x_n-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex x_i and define the area A_P(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size c_H(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:

•

A_P(H)?A_P(H^′) for all convex n-gons P.

•

c_H(i,j,k)?c_H^′(i,j,k) for all convex three-cuts C(i,j,k).

From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H^′ satisfy “A_P(H)?A_P(H^′) for all convex n-gons P” is immediately obtained.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Normal Forms for Nonautonomous Differential Equations

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λ_j−∑ⁿ_i=1 ?_iλ_i≠0 for eigenvalues is generalized to the new nonresonance condition λ_j∩∑ⁿ_i=1 ?_iλ_i=∅ for spectral intervals.     

Turán-Type Results for Complete <Emphasis Type="Italic">h</Emphasis>-Partite Graphs in Comparability and Incomparability Graphs

We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <₁,...,< _r be partial orders on P. Let G(P, <₁,...,< _r ) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x< _i y or y< _i x holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <₁, ... , < _r ) is strictly larger than 1 ? 1/(2h ? 2) ^r , then P contains h disjoint sets A ₁, ... , A _h such that A ₁ < _j ... < _j A _h holds for some 1 ≤ j ≤ r, and |A ₁| = ... = |A _h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 ? 1/(3h ? 3), then P contains h disjoint sets A ₁, ... , A _h such that the elements of A _i are incomparable with the elements of A _j for 1 ≤ i < j ≤ h, and |A ₁| = ... = |A _h | = |P|^1?o(1). Finally, we prove that if the edge density of the complement of G(P, <₁, <₂) is α, then there are disjoint sets A, B ? P such that any element of A is incomparable with any element of B in both <₁ and <₂, and |A| = |B| > n ^1?γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Inclusion relations involving k-numerical ranges

Let W_k(A) denote the k-numerical range of an n × n matrix A. It is known that W_i(A) ? W_j(A) for 1 ? j? i? n. It this paper we derive more general inclusion relations of the form Σⁿ_iλ_iW_i(A) ? Σⁿ_iμ_iW_i(A), where λ_i, μ_i are real coefficients.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.