Maximization,under equality constraints,of a functional of a probability distribution

Let $\text{F}$ be a family of probability distributions. Let O, C₁…C_n be real functions on $\text{F}$. Let z₁…z_n be real numbers. Then we consider the problem of maximization of the object function O(F)(F?$\text{F}$) under the equality constraints C₁(F)=z₁(i=1,…,n) . The theory is developed in order to solve problems of the following kind: Find the maximal variance of a stop-loss reinsured risk under partial information on the risk such as its range and two first moments.     

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 generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

Splines with maximal zero sets

Consider a spline s(x) of degree n with L knots of specified multiplicities R₁, …, R_L, which satisfies r sign consistent mixed boundary conditions in addition to s⁽ⁿ⁾(a) = 1. Such a spline has at most n + 1 ?r + ∑_{j = 1}^LR_j zeros in (a, b) which fulfill an interlacing condition with the knots if s(x) ? = 0 everywhere. Conversely, given a set of n ?r + ∑_{j = 1}^LR_j zeros then for any choice η₁ < ··· < η_L of the knot locations which fulfills the interlacing condition with the zeros, the unique spline s(x) possessing these knots and zeros and satisfying the boundary conditions is such that s⁽ⁿ⁾(x) vanishes nowhere and changes sign at η_j if and only if R_j is odd. Moreover there exists a choice of the knot locations, not necessarily unique, which makes $\text{¦s}{}^{\mathrm{(n)}}\text{(x)¦ ≡ 1}$. In particular, this establishes the existence of monosplines and perfect splines with knots of given multiplicities, satisfying the mixed boundary conditions and possessing a prescribed maximal zero set. An application is given to double-precision quadrature formulas with mixed boundary terms and a certain polynomial extremal problem connected with it.     

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).     

A note on the similarity depth

In this note we demonstrate the existence of E0L forms F and G which are n-similar, i.e. $\text{L}$ⁿ(F) = $\text{L}$ⁿ(G) but $\text{L}$ⁿ⁺¹(F)≠$\text{L}$ⁿ⁺¹(G) for n ∈ {2, 3}. This partially solves an open problem from [3].     

The maximal sizes of faces and vertices in an arrangement

Let A be an arragement of n lines in the plane. Suppose that F₁,…,F_r are faces of A and that V,…,V_s are vertices of A. Suppose also that each F_i is a (V_j) of the lines of A intersect at V_j. Then we show that

$\text{∑}\text{i=1}\text{r}\text{t(F}{}_{\mathrm{i}}\text{+}\text{∑}\text{j=1}\text{s}\text{t(V}{}_{\mathrm{j}}\text{)?n+4}\text{r}\text{2}\text{+}\text{s}\text{2}\text{+ 2rs}$

.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

On tilings of the binary vector space

Let n be a positive integer, L a subset of {0, 1,…,n}. We discuss the existence of partitions (or tilings) of the n-dimensional binary vector space F_n into L-spheres. By a L-sphere around an x in F_n we mean {y ? F_n, d(x, y) ? L}, d(x, y) being the Hamming distance betwe en x and y. These tilings are generalizations of perfect error correcting codes. We show that very few such tilings exist (Theorem 2) and characterize them all for any L ? {0, 1,…,[$\text{1}\text{2}$n]}.     

A bound for the number of tournaments with specified scores

Let $\text{T}$(R) denote the set of all tournaments with score vector R = (r₁, r₂,…, r_n). R. A. Brualdi and Li Qiao (“Proceedings of the Silver Jubilee Conference in Combinatorics at Waterloo,” in press) conjectured that if R is strong with r₁ ≤ r₂ ≤ … ≤ r_n, then |$\text{T}$(R)| ≥ 2^n?2 with equality if and only if R = (1, 1, 2,…, n ? 3, n ? 2, n ? 2). In this paper their conjecture is proved, and this result is used to establish a lower bound on the cardinality of $\text{T}$(R) for every R.     

A theorem on products of matrices

A new result on products of matrices is proved in the following theorem: let M_i (i=1,2,…) be a bounded sequence of square matrices, and K be the l.u.b. of the spectral radii ρ(M_i). Then for any positive number ε there is a constant A and an ordering p(j) (j = 1,2,…) of the matrices such that

$\text{∏}\text{j=1}\text{n}\text{M}{}_{\mathrm{p(j)}}\text{?A·(K+ε)}{}^{\mathrm{n}}\text{(n = 1,2,…)}$

. The ordering is well defined by p(j), a one-to-one mapping on the set of positive integers. In general the inequality does not hold for any ordering p(j) (a counterexample is provided); however, some sufficient conditions are given for the result to remain true irrespective of the order of the matrices.     

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.     

Best upper bounds for integrals with respect to measures allowed to vary under conical and integral constraints

We consider the general problem

$\text{sup}\text{м?M}\text{∫?dм|∫?dм=z}{}_1\text{(i=1…n)}$

, where the integrals are over an abstract space Ω, the functions $\text{?}{}_{\mathrm{i}}\text{(}\text{i}\text{=0)}\text{…..n}\text{)}$ are defined on that space, and where μm varies in a cone M of measures defined on the space. The integral on the left of the bar has to be maximized. The equalities on the right of the bar are further constraints on μ. The solution of this primal problem goes via the solution of an associated dual problem. The particular cases where M is the cone of positive measures and the cone of positive unimodal measures with fixed mode are investigated in more detail.Only two simple illustrations are given, but several actuarial applications are planned by De Vylder, Goovaerts and Haezendonck.     

Characterization of similarities between two n-frames of projectors

An n-frame on a Banach space $\text{X}$ is E=(E₁,?, E_n) where the E_j's are bounded linear operators on $\text{X}$ such that E_j≠0,

$\text{∑}\text{j=1}\text{n}\text{E}{}_{\mathrm{j}}$

, and E_jE_k=δ_jkE_k (j, k=1,?, n). It is known that if two n-frames E and F are sufficiently close to each other, then they are similar, that is, F_j=TE_jT^-1 with T a bounded linear operator. Among the operators which realize the similarity of the two frames, there is the balanced transformation U(F, E)=(Σⁿ_j=1F_jE_j)(Σⁿ_j=1E_jF_jE_j)^{$\text{-1}\text{2}$}. One of our main results is a local characterization of the balanced transformation. Another operator which implements the similarity between E and F is the direct rotation R(F, E). It comes up in connection with the study of the set of all n-frames as a Banach manifold with an affine connection. Finally, it is shown that for quite a large set of pairs of 2-frames, the direct rotation has a global characterization.     

On hypergraphs without two edges intersecting in a given number of vertices

Let X be a finite set of n-melements and suppose t ? 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family $\text{F}$ = {F₁,…, F_m}, F_i ? X, such that ∥F_i ∩ F_j∥ ≠ t for 1 ? i ≠ j ? m. This problem is solved for n ? n₀(t). Let us mention that the case t = 0 is trivial, the answer being 2^{n ? 1}. For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.     

Partitions of large multipartites with congruence conditions. II

For 1 ≦ l ≦ j, let $\text{a}$_l = ?_h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) ≦ M, and let $\text{a}$_l^′ = $\text{a}$_l ∪ {0}. Let p(n : $\text{B}$) be the number of partitions of n = (n₁,…,n_j) where, for 1 ≦ l ≦ j, the lth component of each part belongs to $\text{B}$_l and let p¹(n : $\text{B}$) be the number of partitions of n into different parts where again the lth component of each part belongs to $\text{B}$_l. Asymptotic formulas are obtained for p(n : $\text{a}$), p¹(n : $\text{a}$) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : $\text{a}$′), p¹(n ; $\text{a}$′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.     

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F1 be its multiplicative group, and $\text{H}$ = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let D_n be the dihedral group of degree n, H be a nontrivial group in $\text{H}$, and τ_n(H) = {α = (α₁, α₂,…, α_n):α_i?H}. For σ?D_n and α?τ_n(H), let P(σ, α) be the matrix whose (i,j) entry is α_iδ_iσ(j) (i.e., a generalized permutation matrix), and

$\text{P(D}{}_{\mathrm{n}}\text{, H) = {P(σ, α):σ?D}{}_{\mathrm{n}}\text{, α?τ}{}_{\mathrm{n}}\text{(H)}}$

. Let M_n(F) be the vector space of all n×n matrices over F and $\text{T}$P(D_n, H) = {T:T is a linear transformation on M_n (F) to itself and T(P(D_n, H)) = P(D_n, H)}. In this paper we classify all T in $\text{T}$P(D_n, H) and determine the structure of the group $\text{T}$P(D_n, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.     

Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.