Some special vapnik-chervonenkis classes

For a class $\text{C}$ of subsets of a set X, let V($\text{C}$) be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ $\text{C}$. The condition V($\text{C}$) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈$\text{C}$, then $\text{V(}\text{C}\text{)=2}$ if and only if $\text{C}$ is linearly ordered by inclusion. If $\text{C}$ is of the form $\text{C}\text{={∩}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{C}{}_{\mathrm{i}}\text{:C}{}_{\mathrm{i}}\text{∈}\text{C}{}_{\mathrm{i}}$, i=1,2,…,n}, where each $\text{C}{}_{\mathrm{i}}$ is linearly ordered by inclusion, then $\text{V(}\text{C}\text{)?n+1}$. If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and $\text{C}$ is the collection of all sets {x: f(x)>0} for f in H, then $\text{V(}\text{C}\text{)=n}$.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

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.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

A characterization of first order nonlinear partial differential operators on Sobolev spaces

Nonlinear partial differential operators $\text{G: W}{}^{\mathrm{1,p}}\text{(Ω) → L}{}^{\mathrm{q}}\text{(Ω) (1 ? p, q ∞)}$ having the form G(u) = g(u, D₁u,…, D_Nu), with g?C(R × R_N), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, $\text{W}{}^{\mathrm{1,\infty }}\text{(Ω)}$, and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of $\text{W}{}^{\mathrm{1,p}}\text{(Ω)}\text{and}\text{L}{}^{\mathrm{q}}\text{(Ω)}$.     

On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth

We consider the equation −Δu+V(x)u=f(x,u) for $\text{x∈}\text{R}{}^2$ where $\text{V:}\text{R}{}^2\text{→}\text{R}$ is a positive potential bounded away from zero, and the nonlinearity $\text{f:}\text{R}{}^2\text{×}\text{R}\text{→}\text{R}$ behaves like exp(α|u|²) as |u|→∞. We also assume that the potential V(x) and the nonlinearity f(x,u) are asymptotically periodic at infinity. We prove the existence of at least one weak positive solution $\text{u∈H}{}^1\text{(}\text{R}{}^2\text{)}$ by combining the mountain-pass theorem with Trudinger–Moser inequality and a version of a result due to Lions for critical growth in $\text{R}{}^2$.     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

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.     

Estimées Ck pour l'équation ∂̄b sur un convexe de type fini de Cn

Let q=1,…,n?1 and D be a bounded convex domain in $\text{C}{}^{\mathrm{n}}$ of finite type m. We construct two integral operators T_q and $\text{T}{}_{\mathrm{q}}$ such that for all $\text{p∈}\text{N}\text{,}\text{T}{}_{\mathrm{q}}\text{,}\text{T}{}_{\mathrm{q}}\text{:C}{}^{\mathrm{p}}{}_{\mathrm{0,q}}\text{(bD)→C}{}^{\mathrm{p+1/m}}{}_{\mathrm{0,q?1}}\text{(bD)}$ are continuous, and for all (0,q)-forms h continuous on bD with $\text{?}{}_{\mathrm{b}}\text{h}$ continuous on bD too, with the additional hypothesis when q=n?1 that ∫_bDh∧φ=0 for all φ∈C^∞_n,0(bD) $\text{?}\text{?}{}_{\mathrm{b}}$-fermée, we show $\text{h=}\text{?}\text{?}{}_{\mathrm{b}}\text{(T}{}_{\mathrm{q}}\text{?}\text{T}{}_{\mathrm{q}}\text{)h+(T}{}_{\mathrm{q+1}}\text{?}\text{T}{}_{\mathrm{q+1}}\text{)}\text{?}\text{?}{}_{\mathrm{b}}\text{h}$. For this construction, we use the Diederich–Fornæss support function of Alexandre (Publ. IRMA Lille 54 (III) (2001)). To prove the continuity of T_q, we integrate by parts and take care of the tangential derivatives. The normal component in z of the kernel of $\text{T}{}_{\mathrm{q}}$ will have a bad behaviour, so, in order to find a good representative of its equivalence class, we isolate the tangential component of the kernel and then integrate by parts again. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

Nonstandard points on algebraic curves

Let Γ be an algebraic curve which is given by an equation f(x, y) = 0, f(x, y) ∈ k[x, y] where k is an algebraic number field and f(x, y) is irreducible. Suppose that there exists an ${}^1\text{Γ}$ a nonstandard point $\text{(ξ, η) ∈}{}^1\text{k ×}{}^1\text{k}$. Then k(ξ, η) is (isomorphic to) the algebraic function field of Γ and, at the same time, is a subfield of ${}^1\text{k}$. Correlating the divisors of the function field k(ξ, η) and of the number field ${}^1\text{k}$, we develop an analogue of the Artin-Whaples theory of the product formula. This leads to one of Siegel's basic inequalities for rational points on algebraic curves.     

Singular sets of Sobolev functionsEnsembles singuliers des fonctions de Sobolev

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.     

Estimées Ck pour les couronnes de convexes de type fini de Cn

C^k estimates for convex domains of finite type in $\text{C}{}^{\mathrm{n}}$ are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of $\text{C}{}^{\mathrm{n}}$, of finite type m and m′ such that $\text{D}\text{?D′}$, for all q=1,…,n?2, there exists a linear operator $\text{T}{}^1{}_{\mathrm{q}}$ from $\text{C}{}_{\mathrm{0,q}}\text{(}\text{D′?D}\text{)}$ to $\text{C}{}_{\mathrm{0,q?1}}\text{(}\text{D′?D}\text{)}$ such that for all $\text{k∈}\text{N}$ and all (0,q)-form f, $\text{?}\text{?}$-closed of regularity C^k up to the boundary, $\text{T}{}^1{}_{\mathrm{q}}\text{f}$ is of regularity C^{k+1/max(m,m′)} up to the boundary and $\text{?}\text{?}\text{T}{}_{\mathrm{q}}{}^1\text{f=f}$. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the $\text{?}\text{?}$ problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.