Sieve-equivalence and explicit bijections

Suppose A₁,…, A_n are subsets of a finite set A, and B₁,…, B_n are subsets of a finite set B. For each subset S of N = {1, 2,…, n}, let A_s = ∩_i?SA_i and B_S = ∩_i?SB_i. It is shown that if explicit bijections f_S:A_S → B_S for each S ? N are given, an explicit bijection h:A-∪_i=1A_i→B-∪_i=1B_i can be constructed. The map h is independent of any ordering of the elements of A and B, and of the order in which the subsets A_i and B_i are listed.     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

Direct decompositions in free commutative monoids

Let N be the positive integers; let C be a subset closed under taking divisors; and let A and B be subsets of C such that every member of AB (= {ab: a?A, b?B) is uniquely representable in the form ab, and also AB contains C. Given C, all such pairs (A, B) are found. The result is obtained in a slightly more general setting, and the pairs are replaced by arbitrarily large families.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

A minimum principle for superharmonic functions subject to interface conditions

Let D be a bounded domain in R² with smooth boundary. Let B₁, …, B_m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪_{i ? 1}^mB_i respect to D. Suppose that $\text{u ? C}{}^2\text{(D′) ∩ C(}\text{D}\text{?}$) and Δu ? 0 in D′ (where Δ is the Laplacian), while across each “interface” B_i, i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ? 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.     

A necessary and sufficient condition for a product relation to be total

Let A, B, and C be sets, let ? be a relation on A × B, and let σ be a relation on B × C. A necessary and sufficient condition for ? ° σ to be total is provided in terms of a DeMorgan algebra defined on B.     

On a problem of Katona on minimal completely separating systems with restrictions

Let S be a set of n elements, and k a fixed positive integer $\text{<}\text{1}\text{2}\text{n}$. Katona's problem is to determine the smallest integer m for which there exists a family $\text{A}$ = {A₁, …, A_m} of subsets of S with the following property: |_i| ? k (i = 1, …, m), and for any ordered pair x_i, x_i ∈ S (i ≠ j) there is A₁ ∈ $\text{A}$ such that x_i ∈ A₁, x_j ? A₁. It is given in this note that $\text{m = ?}\text{2n}\text{k}\text{?}\text{if}\text{1}\text{2}\text{k}{}^2\text{? 2}$.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Small solutions of linear Diophantine equations

Let Ax = B be a system of m × n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m × m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (x_i) in nonnegative integers wity x_i ⩽ X for n - m variables and x_i ⩽ (m - m + 1)Y for m variables. This improves previous results of the authors and others.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

Separation of fuzzy sets in fuzzy minimal spaces

This paper deals with the concepts of fuzzy minimal separation and fuzzy closed minimal separation. Some criterion for m-separatedness and C_m-separatedness of two fuzzy sets in a fuzzy minimal space are achieved. Further, it is shown that for any fuzzy sets C and D in Y, A × C and B × D are fuzzy m-separated (C_m-separated) in X × Y, if A and B are fuzzy m-separated (C_m-separated) sets in X. Moreover, c.A and c.B are fuzzy m-separated (C_m-separated) if and only if A and B are fuzzy m-separated (C_m-separated).     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Symmetric positive systems in corner domains

Consider the symmetric positive system of n equations in m + 2 variables,

$\text{A}\text{?u}\text{?x}\text{+ B}\text{?u}\text{?y}\text{+}\text{∑}\text{i=1}\text{m}\text{C}{}_{\mathrm{i}}\text{?u}\text{?z}{}_{\mathrm{i}}\text{+ Du = ?}$

in the corner domain x > 0, y > 0, ? ∞ < z_i < ∞, with homogeneous data on x = 0 and y = 0. The n × n matrices A, B, C_i are symmetric and D is sufficiently positive. On the boundary surfaces the matrix coefficients A, B, C_i satisfy certain “torsion” conditions. For ? with square integrable first-order derivatives, the strong solution with first-order strong derivatives is derived for the boundary value problem. For less restricted ?, the partially differentiable strong solution is established, provided more severe torsion conditions are satisfied on the boundaries. Also, the partially differentiable strong solution is obtained for the case that the torsion conditions are satisfied on one side of the boundary only.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

Retractions and contractibility in hyperspaces

Given a metric continuum X, let X² denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of X² having at most k components. Consider mappings (where B∈Cm(X)) and both defined by A?A∪B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φB). The conditions are related to the contractibility of the corresponding hyperspaces.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P_m be a path of m vertices. Let {B_i:1?i?m} be a set of generalized Bethe trees. Let P_m{B_i:1?i?m} be the tree obtained from P_m and the trees B₁,B₂,…,B_m by identifying the root vertex of B_i with the i-th vertex of P_m. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P_m{B_i:1?i?m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B₁=B₂=…=B_m.