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 method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

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.     

Arithmétique d'une famille de corps cubiques

In this Note, we study the family of polynomials: P(X)=X³?nX²?n, with n=3^sp₁…p_t, where s=0 or 1 and where the p_i, for 1?i?t, are distinct prime numbers and all different from 3, and (4n²+27)/9^s is squarefree. For this family, we determine the arithmetic invariants of the number field $\text{K=}\text{Q}\text{(α),}$ where α is the only real root of the polynomial P(X), and we find the following results: $\text{O}{}_{\mathrm{K}}\text{=}\text{Z}\text{[α]}$ is the ring of integers of K, d_K=?n²(4n²+27) is the discriminant of K; ε=α²+1 is the fundamental unit of O_K and R_K=Log(α²+1) is the regulator of K. To cite this article: O. Lahlou, M. El Hassani Charkani, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A general approach to optimal control of a regression experiment

Let $\text{θ}\text{θ}\text{?}$ = ($\text{θ}\text{θ}\text{?}$₁, $\text{θ}\text{θ}\text{?}$₂, …, $\text{θ}\text{θ}\text{?}$_n)′ be the least-squares estimator of θ = (θ₁, θ₂, …, θ_n)′ by the realization of the process y(t) = Σ_{k = 1}ⁿθ_kf_k(t) + ξ(t) on the interval T = [a, b] with f = (f₁, f₂, …, f_n)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and D_s-optimality are used as criteria for choosing f in X. A-, D-, and D_s-optimal models can be constructed explicitly by means of r.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

On maximal antichains containing no set and its complement

Let 1?k₁?k₂?…?k_n be integers and let S denote the set of all vectors x = (x₁, …, x_n with integral coordinates satisfying 0?x_i?k_i, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of k_i elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities x_i?y_i, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k₁ + k₂ + …k_n. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k₁-x₁, k₂-x₂, …, k_n -x_n). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and$\text{|(}\text{1}\text{2}\text{K)X|}$ does not exceed the number of vectors in $\text{(}\text{1}\text{2}\text{K)S}$ with first coordinate different from k₁, then

$\text{∑}\text{i=0}\text{K}\text{i≠}\text{1}\text{2}\text{K}\text{|(i)X|}\text{|(i)S|}\text{+}\text{|(}\text{1}\text{2}\text{K)X|}\text{|(}\text{1}\text{2}\text{K-1)S|}\text{?1}$

.     

An interpolant defined by subdivision and the analysis of the error

Given a set of points x_i, i=0,…,n on [−1,1] and the corresponding values y_i, i=0,…,n of a 2-periodic function y(x), supplied in some way by interpolation or approximation, we describe a simple method that by doubling iteratively this original set, produces in the limit a smooth function. The analysis of the interpolation error is given.We show that if $\text{y∈}\text{C}{}^4$ then the error in the p-norm, $\text{p=1,}\text{2}$ and ∞ depends on the magnitude of the fourth derivative of the function y(x) and on a function α(x) which is even, concave and bounded on [−1,1].     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

On the duality operator of a convex cone

Let C be a convex set in Rⁿ. For each y?C, the cone of C at y, denoted by cone(y, C), is the cone {α(x ? y): α ? 0 and x?C}. If K is a cone in Rⁿ, we shall denote by $\text{K}{}^1$ its dual cone and by F(K) the lattice of faces of K. Then the duality operator of K is the mapping $\text{d}{}_{\mathrm{K}}\text{: F(K) → F(K}{}^1\text{)}$ given by $\text{d}{}_{\mathrm{K}}\text{(F) = (}\text{span}\text{F)}{}^{\mathrm{\perp }}\text{∩ K}{}^1$. Properties of the duality operator d_K of a closed, pointed, full cone K have been studied before. In this paper, we study d_K for a general cone K, especially in relation to d_{cone(y, K)}, where y?K. Our main result says that, for any closed cone K in Rⁿ, the duality operator d_K is injective (surjective) if and only if the duality operator d_{cone(y, K)} is injective (surjective) for each vector y?K ～ [K ∩ (? K)]. In the last part of the paper, we obtain some partial results on the problem of constructing a compact convex set C, which contains the zero vector, such that cone (0, C) is equal to a given cone.     

A version of Chevet's theorem for stable processes

Let $\text{X}\text{}\text{?}\text{bo}\text{Y}$ be the injective tensor product of the separable Banach spaces X and Y and let S_X, S_Y and $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$ be the unit spheres of these spaces. The tensor product of two symmetric finite measures η₁ on S_X and η₂ on S_Y, η₁?η₂, is defined in a natural way as a measure on $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$. It is shown that η₁? η₂ is the spectral measure of a p-stable random variable W on $\text{X}\text{}\text{?}\text{bo}\text{Y}$, 0 <p < 2, if and only if η₁ and η₂ are the spectral measures of p-stable random variables U and V on X and Y, respectively. Actually upper and lower bounds for $\text{(E∥ W∥}{}^{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}$ in terms of the random variables U and V are obtained. When X = C(S), Y = C(T) with S, T compact metric spaces, and η₁, and η₂ are discrete, our results imply that if θ_i, θ_ij are i.i.d. standard symmetric real valued p-stable random variables, 0 < p <2, x_i?C(S), and y_i?C(T), then the series ∑_ijθ_ijx_i(s) y_j(t) converges uniformly a.s. iff the series ∑_iθ_ix_i(s) and ∑_iθ_iy_i(t) both converge uniformly a.s. When p = 2 this follows from Chevet's theorem on Gaussian processes. Several examples are given. One of them requires an interesting upper bound on the probability distribution of the maximum of i.i.d. p-stable random variables taking values in a general Banach space.     

Corresponding residue systems in normal extensions

Let K and K′ be number fields with L = K · K′ and F = KφK′. Suppose that $\text{K}\text{F}$ and $\text{K′}\text{F}$ are normal extensions of degree n. Let $\text{B}$ be a prime ideal in L and suppose that $\text{B}$ is totally ramified in $\text{K}\text{F}$ and in $\text{K′}\text{F}$. Let π be a prime element for $\text{B}$_K = $\text{B}$ φ K, and let f(x) ∈ F[x] be the minimum polynomial for π over F. Suppose that $\text{B}$_K · $\text{D}$_L = ($\text{B}$^≠)^e. Then,

$\text{M(B\# : K, K′) =}\text{min}\text{{m, e(t + 1)}}$

, where $\text{t =}\text{min}\text{{t(}\text{K}\text{F}\text{), t(}\text{K′}\text{F}\text{)}}$ and m is the largest integer such that ($\text{B}$_K′)^nm/e φ f($\text{D}$_K′) ≠ {φ}.If we assume in addition to the above hypotheses that [K : F] = [K′: F] = pⁿ, a prime power, and that $\text{B}$ divides p and is totally ramified in $\text{L}\text{F}$, then

$\text{M(B\# : K, K′) ? p}{}^{\mathrm{n?1}}\text{[(p ? 1)(t + p]}$

, with t = t($\text{B}$ : L/F).     

The existence,uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system $\text{?x}\text{?t}\text{= Δx + F(x,y),}\text{?y}\text{?t}\text{= G(x,y)}$ is investigated, where x and y are scalar functions of time (t ? 0), and n space variables $\text{(ξ}{}_1\text{,…, ξ}{}_{\mathrm{n}}\text{), Δx ≡ ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{x}\text{?ξ}{}_{\mathrm{i}}{}^2$, and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution $\text{(x(r),y(r)),}\text{where}\text{r = (ξ}{}_1{}^2\text{+ … + ξ}{}_{\mathrm{n}}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which is bounded for r ? 0 and satisfies x(0) >x₀, y(0) > y₀, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.     

On the stabilization of iterative processes

Processes of the type A^tz, A being a symmetric matrix, are considered. Such a process is called strongly stabilizable iff given an arbitrary sequence of interference times (t_i), it admits a stabilization consisting of the addition of a constant vector y at ”times“ t_i. It is shown (Theorem 3) that A^t is strongly stabilizable iff no eigenvalue of A lies in the interval $\text{( ?}\text{1}\text{2}\text{(1 +}\text{5}\text{), ? 1)}$.     

Levy's stochastic area formula in higher dimensions

Let (W_t) = (W¹_t,W²_t,…,W^d_t), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from $\text{R}$⁺ into the space of d × d skew-symmetric matrices and x(t) such a function into $\text{R}$^d. A class of stochastic processes (L_t^A,x), a particular example of which is Levy's “stochastic area” $\text{L}{}_{\mathrm{t}}\text{=}\text{1}\text{2}\text{∝}{}_0{}^{\mathrm{?t}}\text{(W}{}^1{}_{\mathrm{s}}\text{,dW}{}^2{}_{\mathrm{s}}\text{? W}{}^2{}_{\mathrm{s}}\text{,dW}{}^1{}_{\mathrm{s}}\text{)}$, is dealt with.The joint characteristic function of W_t and L₁^A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (W_t,L_t^A,x) is given.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Oscillation of two-dimensional linear second-order differential systems

This article is concerned with the oscillatory behavior at infinity of the solution y:[a, ∞) → $\text{R}$² of a system of two second-order differential equations, y″(t) + Q(t) y(t) = 0, t?[a, ∞); Q is a continuous matrix-valued function on [a, ∞) whose values are real symmetric matrices of order 2. It is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix ∝_a^tQ(s) ds tends to infinity as t → ∞. This proves a conjecture of D. Hinton and R. T. Lewis (Rocky Mountain J. Math.10 (1980), 751–766) for the two-dimensional case. Furthermore, it is shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity.