Capture in linear functional differential games of pursuit

The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W₂⁽¹⁾. The target is either a subset of W₂⁽¹⁾ a point in W₂⁽¹⁾, a subset of the Euclidean space Eⁿ or a point of Eⁿ. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of E^m and Eⁱ respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ε E^mp′(t) ε E^r and B(t) is an n × r′-matrices and C(t) an n × m-matrix.     

On the non-existence of linear codes attaining the Griesmer bound

A lower bound on the size of a set K in PG(3, q) satisfying for any plane of PG(3, q), q4 is given. It induces the non-existence of linear [n,4,n + 1 – q ²]-codes over GF(q) attaining the Griesmer bound for .     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

An approximation for the complex normal probability integral

An analytic approximation to the complex normal probability integral, (x+iy)=(2)^–1/2 _–t ^x exp[–(t–iy)²/2]dt, is given together with a study of the error in the approximation.     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

On the periodic solutions of the second-order wave equations. V

It is established that the linear problemu _u –a ² u _xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g( –x,t) = –g(–x,t)} provided thataTq = (2p – 1) and (2p – 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.     

Normal Spreads

In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t – 1, q) into the Grassmannian of the subspaces of rank t of PG(2t – 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt – 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt – 1, q) by a cap V _{r, t} of PG(r ^t – 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r – 1), and that the collineation group of PG(r ^t – 1, q) stabilizing V _{r, t} acts 2-transitively on V _{r, t} . In particular, we prove that V _{3, 2} is the union of q ² – q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q ²) is represented by a hyperplane section U of V _{3, 2}. For q 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q ⁺ (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q ²). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino.     

A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay

In this note we consider a reaction-diffusion problem which describes a simple model chemical reaction scheme for quadratic autocatalysis with linear decay. We show that withk>1 (wherek is a parameter measuring the relative strength of the decay step to the autocatalytic step) the dimensionless unreacting state 1, 0 is globally asymptotically stable, with 1+0(t ^–1/2) and 0(t ^–1/2 e ^–(k–1)t) ast-. Here and are the concentrations of the reactant and the autocatalyst respectively, andt is time. The casek<1 has been considered in detail by Merkin et al. [1].     

On nonautonomous linear systems of differential and difference equations with -symmetric coefficient matrices

Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e^-2πi/k, d₀+d₁++d_k-1=n, and with . We say that is R-symmetric if R(t)A(t)R^-1(t)=A(t), , and we show that if A is R-symmetric then solving x^′=A(t)x or x^′=A(t)x+f(t) reduces to solving k independent d_ℓ×d_ℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations.     

On the number of solutions to the linear comple-mentarity problem

Given ann × n matrixA and ann-dimensional vectorq letN(A, q) be the cardinality of the set of solutions to the linear complementarity problem defined byA andq. It is shown that ifA is nondegenerate thenN(A, q) + N(A, –q) 2 ⁿ , which in turn impliesN(A, q) 2 ⁿ – 1 ifA is also aQ-matrix.It is then demonstrated that min _q0 N(A, q) 2 ^n–1 – 1, which concludes that the complementary cones cannot spanR ⁿ more than 2 ^n–1 – 1 times around. For anyn, an example of ann × n nondegenerateQ-matrix spanning allR ⁿ , but a subset of empty interior, 2^[n/3] times around is given.     

Optimal pseudo-cyclic codes and caps in PG (3,q)

Existence and uniqueness of pseudo-cyclic [q ²+1,q ²–3, 4]-codes over GF(q) are proved. Elliptic quadrics are characterized as those (q ²+1)-caps in PG(3,q) whose corresponding [q ²+1,q ²–3, 4]-codes are pseudo-cyclic.     

Approximation of the unit step function by a linear combination of exponential functions

We approximate the unit step function, which equals 1 if t ε [0, T] and equals 0 if t > T, by functions of the form ∑_{n = 1}^N Ax_n^N e^−λ_n^t/T, where each λ_n is a given positive constant. We find the coefficients A_n^(N) by minimizing the integrated square of the difference between the unit step function and the approximating function. We first solve the specialized case where each λ_n = n. The resulting sum can be shown to converge in the mean to the unit step function as N → ∞. The general case is then solved and some interesting properties of the numbers A_n^(N) are noted.     

Constructing codes identifying sets of vertices

For all t ≥ 2, we give an explicit construction of an infinite family of graphs such that G admits a code identifying sets of at most t vertices of G of cardinality O(t ² ln |V(G)|) for all members G of the family. This beats the best previously known explicit construction by a factor of t ².      

Parameter estimation in linear filtering

Suppose on a probability space (Ω, F, P), a partially observable random process (x_t, y_t), t ≥ 0; is given where only the second component (y_t) is observed. Furthermore assume that (x_t, y_t) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W₁(t)) and (W₂(t)): dx_t=−βx_tdt+dW₁(t), x₀=0, dy_t=αx_tdt+dW₂(t), y₀=0; α, β∞(a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter θ = (α, β). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and θ is a k-dimensional vector.     

Designs from subcode supports of linear codes

We present new constructions of t-designs by considering subcode supports of linear codes over finite fields. In particular, we prove an Assmus-Mattson type theorem for such subcodes, as well as an automorphism characterization. We derive new t-designs (t ≤ 5) from our constructions.      

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

On the Rate of Almost Everywhere Convergence of Certain Classical Integral Means

The main purpose of this article is to establish nearly optimal results concerning the rate of almost everywhere convergence of the Gauss–Weierstrass, Abel–Poisson, and Bochner–Riesz means of the one-dimensional Fourier integral. A typical result for these means is the following: If the function f belongs to the Besov spaceB^s_{p, p}, 1<p<∞, 0<s<1, thenT_mtf (x)−f(x)=o_x(t^s) a.e. ast→0+.