Decomposable bilinear numerical radii

Let A be an n-square normal matrix over $\text{C}$, and Q_{m, n} be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Q_{m, n} denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i₁,…,i_k, i_k+1,…,i_m, such that α(i_j)=β(i_j), j=1,…,k, and {α(i_k+1),…,α(i_m)};∩{β(i_k+1),…,β(i_m)}=ø. Let

be the group of n-square unitary matrices. Define the nonnegative number

$\text{?}{}_{\mathrm{k}}\text{(A)=}\text{max}\text{U∈}\textcolor{red}{}\text{|}\text{det}\text{(U}{}^1\text{AU) [α|β]|}$

, where |α∩β|=k. Theorem 1 establishes a bound for ?_k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that

$\text{?}{}_{\mathrm{m}}\text{(A)??}{}_{\mathrm{m?1}}\text{(A)????}{}_0\text{(A)}$

.     

Some explicit formulae in simultaneous padé approximation

Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P₀(z) and P₀(z)f_i(z) ? P_i(z), where P_i(z) are polynomials of degree σ ? ρ_i (i=0,1,…,n), P₀(z)f_i(z) ? P_i(z) are power series in which the terms with z^k, 0?k?σ, vanish (i=1,2,…,n), (ρ₀,ρ₁,…,ρ_n) is an (n+1)-tuple of nonnegative integers, σ=ρ₀+ρ₁+?+ρ_n, and {f_i}ⁿ_i=1 is the set of hypergeometric functions {₁F₁(1;c_i;z)}ⁿ_i=1$\text{(c}{}_{\mathrm{i}}\text{?}\text{Z}\text{z.drule;}\text{N}\text{, c}{}_{\mathrm{i}}\text{? c}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ or {₂F₀(a_i,1;z)}ⁿ_i=1$\text{(a}{}_{\mathrm{i}}\text{?}\text{Z}\text{?}\text{N}\text{, a}{}_{\mathrm{i}}\text{? a}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ under the condition ρ₀?ρ_i ? 1 (i=1,2,…,n).     

The behaviour of (k,…,kn,n−ik)ci/i! is asymptotically normal

Let N(n,i) = (k,…,kⁿ,n?ik)c^i/i, i = O.…,[n/k]. We prove that the random variable X_n such that $\text{P(X}{}_{\mathrm{<mtext>n</mtext>}}\text{= i) =}\text{N(n, i)}\text{Σ}{}_{\mathrm{<mtext>j</mtext>}}\text{N(n, j)}$ has asymptotically (n → ∞) a normal distribution and we give some combinatorial applications of this result.We also improve a result of Godsil [3] dealing with matchings in graph.     

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.     

Estimating functions of canonical correlation coefficients

Let ρ²₁,…,ρ²_p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ₁,…,δ_p, where $\text{δ}{}_{\mathrm{i}}\text{=}\text{ρ}{}^2{}_{\mathrm{i}}\text{(1 ? ρ}{}^2{}_{\mathrm{i}}\text{)}$, i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ₁,…,δ_p, given a random matrix F whose eigenvalues have the same distribution as $\text{r}{}^2{}_{\mathrm{i}}\text{(1 ? r}{}^2{}_{\mathrm{i}}\text{)}$, i = 1,…,p, where r₁,…,r_p are the sample canonical correlation coefficients.     

Matrix extensions of Liouville-Dirichlet-type integrals

The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x₁,…,x_k: x_i?0, i=1,…,k, s?∑^k₁x_i?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes $\text{Ω={V}{}_1\text{,…,V}{}_{\mathrm{k}}\text{: V}{}_{\mathrm{i}}\text{>0, i=1,…,k, 0?∑V}{}_{\mathrm{i}}\text{?t}}$, where now each V_i is a p×p symmetric matrix and A?B means that A?B is positive semidefinite.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

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.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

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.     

On fair coin-tossing games

Let Ω be a finite set with k elements and for each integer $\text{n ≧ 1}$ let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}\text{= {(a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) | (a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) ∈ Ω}{}_{\mathrm{n}}$ and a_j ≠ a_j+1 for some 1 ≦ j ≦ n ? 1}. Let {Y_m} be a sequence of independent and identically distributed random variables such that P(Y₁ = a) = k^?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in $\text{Ω}{}_{\mathrm{n}}$ and in Ω?_n with respect to the stochastic process {Y_m}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.     

On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

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

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

.     

Enumeration of rises and falls by position

Let π=(π₁, π₂,…,π_n) denote a permutation of Z_n = {1, 2,…, n}. The pair (π_i, π_i+1) is a rise if π_i<π_i+1 or a fall if π_i>π_i+1. Also a conventional rise is counted at the beginning of π and a conventional fall at the end. Let k be a fixed integer ≥ 1. The rise π_i,π_i+1 is said to be in a in a j (mod k) position if i ≡ j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let $\text{P}{}_{\mathrm{n}}\text{≡P}{}_{\mathrm{n}}\text{(r}{}_0\text{,…,r}{}_{\mathrm{k}}\text{?1,?}{}_0\text{,…,?;}{}_{\mathrm{k}}\text{?1)}$ denote the number of permutations having r_i rises i (mod k) positions and ?;_i falls in i (mod k) positions. A generating function for P_n is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4.     

An optimum testing algorithm for some symmetric coherent systems

A symmetric coherent system (or k-out-of-n system) is a system composed of n components CO₁, CO₂ …, CO_n, each component existing in either a working or failing state. Such a system is in a working state if and only if k or more of its components are working, where 1 ? k ? n. It is assumed that the components can only be tested individually, and every test gives perfect information as to whether the tested component is working or failing. Let P_i, be the a priori probability that the component CO_i is working and C_i, be the cost of testing component CO_i. An optimal (minimum total expected cost) testing algorithm is an algorithm to determine the condition of a given symmetric coherent system by testing some of its components individually. In general, such an algorithm is a sequential process, that is, the next component to be tested is a function of the outcomes of the tests already applied. Every (optimal) testing algorithm corresponds to a (optimal) feasible testing policy which is basically a binary rooted tree with some component assigned to each node. In this paper an algorithm is presented for constructing an optimal feasible testing policy for symmetric coherent systems, where $\text{C}{}_{\mathrm{i}}\text{P}{}_{\mathrm{i}}\text{≠}\text{C}{}_{\mathrm{j}}\text{P}{}_{\mathrm{j}}$ and $\text{C}{}_{\mathrm{i}}\text{(1 ? P}{}_{\mathrm{i}}\text{)}\text{≠}\text{C}{}_{\mathrm{j}}\text{(1 ? P}{}_{\mathrm{j}}\text{)}$ whenever i ≠ j. This algorithm can be implemented as an optimal testing algorithm with polynomial complexity. Moreover, it is proven that any optimal testing algorithm corresponds to some feasible testing policy which can be generated by this algorithm.     

A partition identity

Let n₁+n₂+?+n_m=n where the n_i's are integers (possibly negative or greater than n). Let p=(k₁,…,k_m), where k₁+k₂+?+k_m=k, be a partition of the nonnegative integer k into m nonnegative integers and let P denote the set of all such partitions. For m?2, we prove the combinatorial identity

$\text{∑}\text{p∈P}\text{∏}\text{i=1}\text{m}\text{n}{}_{\mathrm{i}}\text{+1?k}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{=}\text{∑}\text{i?0}\text{j+m?2}\text{m?2}\text{n+1?k?2}{}_{\mathrm{j}}\text{k?2}{}_{\mathrm{j}}$

which implies the surprising result that the left side of the above equation depends on n but not on the n_i's.     

Sperner systems consisting of pairs of complementary subsets

It was proved by Erdös, Ko, and Radó (Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser.12 (1961), 313–320.) that if $\text{A}$ = {;A₁,…, A_l}; consists of k-subsets of a set with n > 2k elements such that A_i ∩ A_j ≠ ? for all i, j then l ? (_k?1^n?1). Schönheim proved that if A₁, …, A_l are subsets of a set S with n elements such that A_i ? A_j, A_i ∩ A_j ≠ ø and A_i ∪ A_j ≠ S for all i ≠ j then $\text{l ? (}{}_{\mathrm{<mtext>[</mtext><mtext>n</mtext><mtext>2</mtext><mtext>]\; ?\; 1</mtext>}}{}^{\mathrm{n\; ?\; 1}}\text{)}$. In this note we prove a common strengthening of these results.     

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.