Graphes connexes représentation des entiers et équirépartition

Let q be an integer ≥2 and Ω a suitable subset of {0,…,q ? 1}²; $\text{C}$(q; Ω) denotes the set of natural integers, the pairs of successive q-adic digits of which are in Ω. If P is an irrational polynomial, the sequence (P(n): n ∈ $\text{C}$(q; Ω)) is uniformly distributed modulo one.     

Finitely generated submodules of differentiable functions II

Let [$\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of [$\text{E}$(Ω)]^p are of the form im(F) where F: [$\text{E}$(Ω)]^q → [$\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in [$\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.     

Transformation group C1-algebras with continuous trace

We obtain several results characterizing when transformation group C¹-algebras have continuous trace. These results can be stated most succinctly when ($\text{G, Ω}$) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if the stability groups vary continuously on Ω and compact subsets of Ω are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if ($\text{G, Ω}$) is not second countable, that the natural maps of $\text{G}\text{S}{}_{\mathrm{x}}$ onto G · x are homeomorphisms for each x. Then $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if compact subsets of Ω are wandering and an additional C¹-algebra, constructed from the stability groups and Ω, has continuous trace.     

Subordination,rank, and determinism of multivariate stationary sequences

Necessary and sufficient conditions for an arbitrary q-variate stationary sequence x_t, t ∈ $\text{Z}$, to be deterministic are presented. A characterization of the rank r(x) of x_t, t ∈ $\text{Z}$, and a method to construct the Wold-Cramér decomposition for x_t, t ∈ $\text{Z}$, are given. Subordination of q-variate bounded orthogonally scattered vector measures is considered.     

On the compactness of certain integral operators

This paper develops practical methods for deciding whether a given kernel function induces a compact integral operator from certain spaces of functions, defined on a compact subset Ω of $\text{R}$ⁿ, into the space of continuous functions over Ω. Necessary and sufficient conditions for compactness are introduced, and several tests for deciding if these conditions are satisfied are developed. The paper concludes with an illustration of the practical use of the theory.     

Equivalence,linearization, and decomposition of holomorphic operator functions

This paper discusses the problem of classifying holomorphic operator functions up to equivalence. A survey is given in 41 of the main results about equivalence classes of holomorphic matrix functions and holomorphic Fredholm-operator functions. In 42, it is shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A(λ) (for each λ ? Ω) by an identity operator I_Z in such a way that the extended operator function A(·) ⊕ I_Z is equivalent on Ω to a linear function of λ, T ? λI. Other versions of this “linearization by extension” are described, including the cases of entire functions and polynomials (where Ω = $\text{C}$). As an application of these results, we consider the operator function equation $\text{A}{}_2\text{(λ) Z}{}_2\text{(λ) + Z}{}_1\text{(λ) A}{}_1\text{(λ) = C(λ), λ ? Ω}$, (1) and explicitly construct the solutions Z₁ and Z₂. The formulas for Z₁ and Z₂ seem to be new, even when A₁, A₂ and C are matrix polynomials. The existence of solutions of (¹) makes it possible to analyze an operator function A whose spectrum decomposes into pairwise disjoint compact subsets σ₁, …, σ_n of Ω. In this case, a suitable extension of A is equivalent on Ω to a direct sum of operator functions, A₁, …, A_n, such that the spectrum of A_i is σ_i (i = 1, …, n). In the final section of the paper, we discuss the relation between local and global equivalence on Ω, and show that there exist operator functions A and B which are locally equivalent on Ω, but admit no extensions (of the sort considered in this paper) which are globally equivalent on Ω.     

Weight enumerators of self-orthogonal codes

Canonical forms are given for (i) the weight enumerator of an $\text{|n,}\text{1}\text{2}\text{(n?1)|}$ self-orthogonal code, and (ii) the split weight enumerator (which classifies the codewords according to the weight of the left-and right-half words) of an $\text{|n,}\text{1}\text{2}\text{n|}$ self-dual code.     

Weight enumerators of reducible cyclic codes and their dual codes

Let ${\mathbb{F}}_q$ be a finite field and $n$ a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length $n$ over ${\mathbb{F}}_q$, which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.     

Approximation numbers of Sobolev imbeddings over unbounded domains

Let Ω ? $\text{R}$^N be an open set with $\text{dist}\text{(x, ?Ω) = O(¦ x ¦}{}^{\mathrm{?l}}\text{)}\text{for}\text{x ? Ω}$ and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers α_n of Sobolev imbeddings

over these quasibounded domains Ω. Here

denotes the Sobolev space obtained by completing $\text{C}{}_0{}^{\mathrm{staggered\infty }}\text{(Ω)}$ under the usual Sobolev norm. We prove $\text{α}{}_{\mathrm{n}}\text{(I}{}_{\mathrm{p,q}}{}^{\mathrm{m}}\text{)}\text{\$}\text{?}\text{n}{}^{\mathrm{?\gamma }}$, where

. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of I_p,q^m. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded domain of the above type.     

A multiplication of distributions

If Ω denotes an open subset of $\text{R}$ⁿ (n = 1, 2,…), we define an algebra $\text{g}$ (Ω) which contains the space $\text{D}$′(Ω) of all distributions on Ω and such that $\text{C}{}^{\mathrm{\infty }}\text{(Ω)}$ is a subalgebra of $\text{G}$ (Ω). The elements of $\text{G}$ (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in $\text{G}$(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and $\text{?∈O}{}_{\mathrm{M}}\text{(}\text{R}{}^{\mathrm{2q}}\text{)}$—a classical Schwartz notation—for any G₁,…,G_q∈G(σ), we define naturally an element $\text{?G}{}_1\text{,…,G}{}_{\mathrm{q}}\text{∈G(σ)}$. These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.     

A Good Method of Combining Codes

Let q be an odd prime power, and suppose q?1 (mod8), Let C(q) and C(q)¹ be the two extended binary quadratic residue codes (QR codes) of length q+1, and let

$\text{T(q)={(a+x;b+x;a+b+x):a,b∈C(q),x∈C(q)}{}^1\text{}}$

. We establish a square root bound on the minimum weight in T(q). Since the same type of bound applies to C(q) and C(q)¹, this is a good method of combining codes.     

Symmetry codes and their invariant subcodes

We define and study the invariant subcodes of the symmetry codes in order to be able to determine the algebraic properties of these codes. An infinite family of self-orthogonal rate $\text{1}\text{2}$ codes over GF(3), called symmetry codes, were constructed in [3]. A (2q + 2, q + 1) symmetry code, denoted by C(q), exists whenever q is an odd prime power ≡ ?1, (mod 3). The group of monomial transformations leaving a symmetry code invariant is denoted by G(q). In this paper we construct two subcodes of C(q) denoted by R_σ(q) and R_μ(q). Every vector in R_σ(q) is invariant under a monomial transformation τ in G(q) of odd order s where s divides (q + 1). Also R_μ(q) is invariant under τ but not vector-wise. The dimensions of R_σ(q) and R_μ(q) are determined and relations between these subcodes are given. An isomorphism is constructed between R_σ(q) and a subspace of $\text{W = V}{}_3{}^{\mathrm{<mtext>(2q+2)</mtext><mtext>s</mtext>}}$. It is shown that the image of R_σ(q) is a self-orthogonal subspace of W. The isomorphic images of R_σ(17) (under an order 3 monomial) and R_σ(29) (under an order 5 monomial) are both demonstrated to be equivalent to the (12, 6) Golay code.     

On the inverse problem for three-dimensional potential scattering

We consider a linear perturbation for the wave equation $\text{□u = 0}\text{in}\text{Ω = E}{}^3$ by “repulsive” smooth potentials q(y) that are small at infinity and suitably small (in magnitude). We use a time-dependent approach to prove that the scattering operator S(q) determines uniquenes uniquely the scatterer q (at least in this class). Energy inequalities will play a central role in our discussion.     

On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation

In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation

does not possess global solutions for arbitrary choices of $\text{u(x, 0). (x ? Ω ? R}{}^{\mathrm{n}}\text{, Ω}\text{bounded}\text{, Δ}{}_{\mathrm{n}}\text{= n}\text{dimensional Laplacian}$) when 0 < k ? 1 for a wide class of nonlinearities $\text{T}$, which includes all the even powers of u and the functions u^{2n + 1}, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy $\text{?u}\text{?t(x, 0)}\text{= 0, x ? Ω}$. The result is independent of the space dimension.     

Self-dual codes over GF(4)

This paper studies codes $\text{C}$ over GF(4) which have even weights and have the same weight distribution as the dual code C^⊥. Some of the results are as follows. All such codes satisfy $\text{C}{}^{\mathrm{\perp }}\text{=}\text{C}$ (If C^⊥= C, $\text{T}$ has a binary basis.) The number of such C's is determined, and those of length ?14 are completely classified. The weight enumerator of $\text{C}$ is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for $\text{C}$ to be extended cyclic. Two new 5-designs are constructed. A generator matrix for $\text{C}$ can be taken to have the form [I | B], where $\text{B}{}^{\mathrm{\perp }}\text{=}\text{B}$. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.     

Local duality theorem for q-ary 1-perfect codes

In this paper, we derive the relationship between local weight enumerator of q-ary 1-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened q-ary 1-perfect code.     

Solvability of quasilinear elliptic equations with nonlinear boundary conditions. II

It is shown that solvability of the second order quasilinear elliptic equation Qu = 0 in Ω with first order nonlinear boundary condition Nu = 0 on ?Ω can be inferred from appropriate estimates on solutions of the problem $\text{Qu = ?}$ in Ω, Nu = 0 on ?Ω as ? varies over a suitable function class. This result improves previous work of the author, where estimates were required on solutions of $\text{Qu = ?}$ in Ω, Nu = ? on ?Ω as (?, ?) varies over some function space. The value of this improvement is demonstrated by some examples.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Mappings of subspaces into subsets

Let V_n(q) denote the n-dimensional vector space over the finite field with q elements, and L_n(q) be the lattice of subspaces of V_n(q). Two rank- and order-preserving maps from L_n(q) onto the lattice of subsets of an n-set are constructed. Three equivalent formulations of these maps are given: an inductive procedure based on an elementary combinatorial interpretation of a well-known pair of difference equations satisfied by the Gaussian coefficients [$\text{n}\text{k}$], a direct set-theoretical definition, and, a direct definition involving a certain pair of modular chains in L_n(q). The direct set-theoretical definition of one of these maps has already been given by Knuth. Knuth's map, however, may be systematically discovered by means of the inductive procedure and the direct lattice-theoretic definition shows how it can be generalized. As a further application of the pair of difference equations satisfied by [$\text{n}\text{k}$], a direct-combinatorial proof of an identity of Carlitz that expands Gaussian coefficients in terms of binomial coefficients has been formulated.