Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

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.     

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

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

.     

Some limit theorems on the eigenvectors of large dimensional sample covariance matrices

Let {v_ij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let V_n = (v_ij) i = 1, 2,…, n; j = 1, 2,…, s = s(n), where $\text{(}\text{n}\text{s}\text{) → y > 0}$ as n → ∞, and let $\text{M}{}_{\mathrm{n}}\text{= (}\text{1}\text{s}\text{)V}{}_{\mathrm{n}}\text{V}{}_{\mathrm{n}}{}^{\mathrm{T}}$. Previous results [7, 8] have shown the eigenvectors of M_n to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X_n on [0, 1], constructed from the eigenvectors of M_n, is known to converge weakly, as n → ∞, on D[0, 1] to Brownian bridge when v₁₁ is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. X_n(F_n(x)), where F_n is the empirical distribution function of the eigenvalues of M_n. The theorems assume certain conditions on the moments of v₁₁ including E(v₁₁⁴) = 3, the latter being necessary for the theorems to hold.     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

Nonuniform right definiteness

The multiparameter eigenvalue problem W_m(λ) x_m = x_m, $\text{W}{}_{\mathrm{m}}\text{(λ) = T}{}_{\mathrm{m}}\text{+}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}$, m = 1,…, k, where /gl /gE $\text{C}$^k, x_m is a nonzero element of the separable Hilbert space H_m, and T_m and V_mn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λⁱ ? $\text{C}$^k for which the i_mth greatest eigenvalue of W_m(λⁱ) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (y_m, V_mny_m) for all nonzero y_m?H_m, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors x_mⁱ is also given, thereby connecting the abstract oscillation theory with a result of Atkinson.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

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.     

The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

Abstract multiparameter theory,I

In this paper we continue our investigation of multiparameter spectral theory. Let H₁,…, H_k be separable Hilbert spaces and H = ?_{r = 1}^kH_r, be their tensor product. In each space H_r we have densely defined self-adjoint operators T_r and continuous Hermitian operators V_rs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ₁,…, λ_n) ?R^k and eigenvectors $\text{? = ?}{}_1\text{? ··· ? ?}{}_{\mathrm{k}}\text{? H}$ such that $\text{T}{}_{\mathrm{r}}\text{?}{}_{\mathrm{r}}\text{+ ∑}{}_{\mathrm{s\; =\; 1}}{}^{\mathrm{k}}\text{λ}{}_{\mathrm{s}}\text{V}{}_{\mathrm{rs}}\text{?}{}_{\mathrm{r}}\text{= 0}$. We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations.     

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number $\text{D(M) =}\text{inf}\text{{}\text{sup}\text{f(x)}\text{f(y)}\text{: x, y ? M}}$, where the infimum is over all f in $\text{L}{}^1$ which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u₁, …, u_p} and V = {v₁, …, v_q} such that there exist positive coefficients a_i and b_i for which Σ_{i = 1}^pa_iu_i = Σ_{i = 1}^qb_iv_i; (3) let $\text{r =}\text{max}\text{{}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{a}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{b}{}_{\mathrm{i}}\text{,}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{b}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{a}{}_{\mathrm{i}}\text{}}$; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

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.     

Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s₁,…,s_n)]^?k where β is a polynomial of degree n being linear in each s_i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x₁,…,x_n)$\text{exp}\text{(}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{θ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)}\text{f(θ}{}_1\text{,…,θ}{}_{\mathrm{n}}\text{)}$ is negative binomial if and only if its univariate marginals are negative binomial. Let S_t, t = 1,…, m, be subsets of {s₁,…, s_n} with empty ∩_t=1^mS_t. Then an n-variate pgf is of a negative binomial if and only if for all s in S_t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.     

Monotone Splitting of matrices

Given the iterative scheme xⁱ⁺¹ = BTxⁱ + r where B, T are fixed n×:n real matrices, r a fixed real n-vector and xⁱ a real n-vector we investigate the convergence and monotonicity of schemes of the type

$\text{v}{}^{\mathrm{i+1}}\text{w}{}^{\mathrm{i+1}}\text{=}\text{B}\text{O}\text{O}\text{B}\text{S}{}_{11}\text{?S}{}_{12}\text{?S}{}_{21}\text{S}{}_{22}\text{v}{}^{\mathrm{i}}\text{w}{}^{\mathrm{i}}\text{+}\text{r}\text{r}$

, where S_ij are n×:n real matrices related to T. The n-vector iterates vⁱ,wⁱ bracket in a certain sense solutions x of x = BTx + r. We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself xⁱ⁺¹ = BTxⁱ + r. This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.     

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.     

Weak convergence of compound stochastic process,I

Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {ξ^v(t):t?0}, v=1,2,…,M. At each epoch of the renewal process {A(t):t?0} we initiate a random number of each of the M types. Let m_l:l?1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is

, where the ξ^v_lj() are independent copies of ξ^v,m_lv is the vth component of m and {τ_l:l?1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t?0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.