Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

Simultaneous weighted sums of elements of finitely generated multiplicative groups

Let {G_j}_jεJ be a finite set of finitely generated subgroups of the multiplicative group of complex numbers C^x. Write H=∩ _jεJ G_j. Let n be a positive integer and a_ij a complex number for i = 1, ..., n and j ε J. Then there exists a set W with the following properties. The cardinality of W depends only on {G_j}_jεJ and n. If, for each jεJ, α has a representation α = Σ _iⁿ = ₁a _ijg_ij in elements g_ij of G_j, then α has a representation a= Σ_k=1ⁿ w_kh_k with w_kεW, h_k εH for k = 1,..., n. The theorem in this note gives information on such representations.     

On a characterization of the exponential distribution by properties of order statistics

A continuous c.d.f. F(x), strictly increasing for x > 0, is exponential if and only if X_{j, n} - X_{i, n} and X_{j−i, n−i} have identical distribution for some i, n, j = j₁, j₂, 1 ⩽ i < j₁ <j₂ ⩽ n, n ⩾ 3. A new proof of this characterization is given, since in Ahsanullah (1975) where it was stated first, an implicit assumption in the proof is that F is NBU or NWU.     

A balancing strategy

Suppose x₁, x₂,…, is a sequence of vectors in R^k, 6X_n6⩽1, where 6(x₁,…,x_k)6 = max_j|x_j|. An algorithm is given for choosing a corresponding sequence ε₁, ε₂,…, of numbers, ε_n = ±1, so that 6ε₁x₁+ … +ε_nx_n6 remains small.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

A relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.     

The distribution of the ‘finitely many times’ in the strong law of large numbers

Consider independent, identically distributed random variables X₁, X₂,… with common mean μ. Let S_n = Σ₁ⁿX₁. and for ε > 0 let N(ε) count the number of values of n for which n⁻¹S_n − μ > ε. This note discusses the distribution of N(ε), thereby elucidating the fluctuations associated with the Strong Law of Large Numbers.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

Ordering Functions of Random Vectors, with Application to Partial Sums

It is known that the sums of the components of two random vectors (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) ordered in the multivariate (s ₁,s ₂,…,s _n )-increasing convex order are ordered in the univariate (s ₁+s ₂+?+s _n )-increasing convex order. More generally, real-valued functions of (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) are ordered in the same sense as long as these functions possess some specified non-negative cross-derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S ₁,S ₂,…,S _n ) and (T ₁,T ₂,…,T _n ) where S _j =X ₁+?+X _j and T _j =Y ₁+?+Y _j and we show that these random vectors are ordered in the multivariate (s ₁,s ₁+s ₂,…,s ₁+?+s _n )-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.     

A note on symmetric doubly-stochastic matrices

Each extreme point in the convex set Δ¹_n of all n×n symmetric doubly-stochastic matrices is shown to have the form $\text{1}\text{2}\text{(P + P}{}^{\mathrm{t}}\text{)}$, for some n×n permutation matrix P. The convex hull Σ_n of the integral points in Δ¹_n (i.e., the symmetric permutation matrices) is shown to consist of those matrices, X = (x_ij) in Δ¹_n satisfying Σ_i∈SΣ_j∈S?{i}x_ij ? 2k, for each subset S of {1, 2, …, n} having cardinality 2k + 1, for some k > 0.     

Continuity of Operators Intertwining with Convolution Operators

Let G be a locally compact abelian group, let μ be a bounded complex-valued Borel measure on G, and let T_μ be the corresponding convolution operator on L¹(G). Let X be a Banach space and let S be a continuous linear operator on X. Then we show that every linear operator Φ: X→L¹(G) such that ΦS=T_μΦ is continuous if and only if the pair (S,T_μ) has no critical eigenvalue.     

On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

Let X,X ₁,X ₂,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S _[nt], 0 ≦ t ≦ 1} where S _n = Σ _j=1 ⁿ X _j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S _[nt]=V _n , 0 ≦ t ≦ 1}, where V _n ² = Σ _j=1 ⁿ X _j ² . L _p approximations of self-normalized partial sum processes are also discussed.     

Linear transformations on matrices: the invariance of sets of ranks

Let R_E denote the set of all m × n matrices over an algebraically closed field F whose ranks lie in the set E, where E is a subset of {1,2,…,m}. Let T be a linear transformation which maps R_E into itself. Under some restrictions on E, or when T is nonsingular, there are nonsingular matrices U and V such that T(A) = UAV for every m × n matrix A.     

Successes,runs and longest runs

The probability distribution of the number of success runs of length k (⩾1) in n (⩾1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let S_n and L_n, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(L_n ⩽ k | S_n = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of X_{n, Ln}^(k) is established, where X_{n, Ln}^(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.     

Classes of matrices associated with the optimal assignment problem

Let E=[e_ij] be a matrix with integral elements, and let x be an indeterminate defined over the rational field Q. We investigate matrices of the form X=[x^e_ij] (i = 1,…, m; j = 1,…, n; m ⩽ n). We may multiply the lines (rows or columns) of the matrix X by suitable integral powers of x in various ways and thereby transform X into a matrix Y=[x^f_ij] such that the f_ij are nonnegative integers and each line of Y contains at least one element x⁰ = 1. We call Y a normalized form of X, and we denote by S(X) the class of all normalized forms associated with a given matrix X. The classes S(X) have a fascinating combinatorial structure, and the present paper is a natural outgrowth and extension of an earlier study. We introduce new concepts such as an elementary transformation called an interchange. We prove, for example, that two matrices in the same class are transformable into one another by interchanges. Our analysis of the class S(X) also yields new insights into the structure of the optimal assignments of the matrix E by way of the diagonal products of the matrix X.     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.     

Successes,runs and longest runs

The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( ⩾ 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let S_n and L_n, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(L_n ⩽ k | S_n = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of X_{n, Ln}^(k) is established, where X_{n, Ln}^(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.     

Ultrametric subsets with large Hausdorff dimension

It is shown that for every ε∈(0,1), every compact metric space (X,d) has a compact subset S?X that embeds into an ultrametric space with distortion O(1/ε), and $$\dim_H(S)\geqslant (1-\varepsilon)\dim_H(X),$$ where dim _H (?) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.     

Reconstituting the distribution of the components of a random Vector from distributions of certain statistics

Let (X₁, ..., X_n) be a random vector with independent components. It is proven in this paper that, under certain restrictions, the distributions of the pairS ₁=sup (a ₁X₁, ..., a_nX_n) andS ₂=sup (b₁X₁,...,b_nX_n) univocally define the distribution function of the components X_j.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 889–892, June, 1973.     

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e ₁, . . . , e _? ] be a partition of n. Denote by X ^e the set of ?-tuples (P ₁, . . . , P _? ) of disjoint nonsingular curvilinear subschemes of X of orders (e ₁, . . . , e _? ). We show that the group Aut(X) acts transitively on X ^e . This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.