Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

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.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

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.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

Nonlinear equations involving m-accretive operators

Let X be a Banach space and T an m-accretive operator defined on a subset D(T) of X and taking values in 2^x. For the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self-mappings, it is shown here that two boundary conditions which imply existence of zeroes for T, appear to be equivalent. This fact is then used to prove that if there exists x₀?D(T) and a bounded open neighborhood U of x₀, such that $\text{¦T(x}{}_0\text{)¦ < r ? ¦T(x)¦}$ for all x??U ∩ D(T), then the open ball B(0; r) is contained in the range of T.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

Geometric conditions for the existence of positive eigenvalues of matrices

Finite-dimensional theorems of Perron-Frobenius type are proved. For A∈$\text{C}$ⁿⁿ and a nonnegative integer k, we let w_k (A) be the cone generated by A^k, A^k+1,…in $\text{C}$ⁿⁿ. We show that A satisfies the Perron-Schaefer condition if and only if the closure $\text{W}$_k(A) of w_k(A) is a pointed cone. This theorem is closely related to several known results. If k?v₀(A), the index of the eigenvalue 0 in spec A, we prove that A has a positive eigenvalue if and only if w_k(A) is a pointed nonzero cone or, equivalently $\text{W}$_k(A) is not a real subspace of $\text{C}$ⁿⁿ. Our proofs are elementary and based on a method of Birkhoff's. We discuss the relation of this method to Pringsheim's theorem.     

The additive and logical complexities of linear and bilinear arithmetic algorithms

Let C_A(±) be the additive complexity of a (bi)linear algorithm A for a given problem; D(A) and $\text{D}\text{(A)}$ are two acyclic diagraphs that represent A, each of them is obtained from another one by reversing directions of all edges; ir(D) and do(D) are two numbers that are introduced to measure the structural deficiencies of an acyclic digraph D. K and Q are the numbers of outputs and input-variables. $\text{do(}\text{D}\text{(A))}$, do(D(A)), and ir(D(A)) characterize the logical complexity of A. It is shown that C_A(±) + do(D(A)) + ir(D(A)) = ω(K + Q)log(K + Q) and $\text{C}{}_{\mathrm{A}}\text{(±) + do(}\text{D}\text{(A)) = ω(K + Q)}\text{log}\text{(K + Q)}$ in the cases of DFT, vector convolution, and matrix multiplication. Also lower bounds on C_A(±) + do(D(A)) and on C_A(±) are expressed in terms of algebraic quantities such as the ranks of matrices and of multidimensional tensors associated with the problems.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

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 an inequality of Tosio Kato for degenerate-elliptic operators

Let Ω be a domain in Rⁿ and T = ∑_{j,k = 1}ⁿ(?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)), where the a_jk and the b_j are real valued functions in $\text{C}{}^1\text{(Ω)}$, and the matrix (a_jk(x)) is symmetric and positive definite for every $\text{x ? Ω}$. If T₀ is the same as T but with b_j = 0, j = 1,…, n, and if u and Tu are in $\text{L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, then T. Kato has established the distributional inequality $\text{T}{}_0\text{¦ u ¦ ?}\text{Re[(sign}$ ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rⁿ. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rⁿ.     

Densities for infinitely divisible random processes

Let {ξ_j(t), t ∈ [0, T]} j = 1, 2 be infinitely divisible processes with distinct Poisson components and no Gaussian components. Let X be the set of all real-valued functions on [0, T] which are not identically zero, and $\text{B}$ be the σ-ring generated by the cylinder sets of ξ_j(t), j = 1, 2. Let μ_j be the measure on $\text{B}$ induced by ξ_j(t).Necessary and sufficient conditions on the projective limits of the Levy-Khinchine spectral measures of the processes are found to make μ₂ ? μ₁, and a representation for the density $\text{dμ}{}_2\text{dμ}{}_1$ is obtained.     

Sharp nonlogarithmic Kneser theorems for fourth-order elliptic equations

Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim $\text{inf}\text{¦f(z) ¦ ? 1}$ on an arc A of ?Δ with length $\text{¦A ¦? ?}$. It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = sup_z?Δ$\text{(}\text{1 ? ¦z¦}{}^2\text{)¦f′(z)¦}\text{(1 + ¦f(z)¦}{}^2\text{) ? C}{}_1\text{sin}\text{(π ? (}\text{?}\text{2}\text{))/ (π ? (}\text{g9}\text{2}\text{))}$, where C₁ = lim_n→∞$\text{∥ z}{}^{\mathrm{n}}\text{∥}\text{and}\text{1}\text{2}\text{< C}{}_1\text{<}\text{2}\text{e}$. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that $\text{¦tf(e}{}^{\mathrm{i0}}\text{)¦ = 1}$ almost everywhere. It is proved that inf_f?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C₁. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥f^p∥ ? p∥f∥, for any positive integer p.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.