Elasticity of factorizations in R 0 + X R1 + … + X lRl [X]

For an atomic domain R the elasticity ρ(R) is defined by ρ(R) = sup{m/n ¦ u₁ … u _m = v ₁ … v_n where u_i, v_i ∈ R are irreducible}. Let R ₀ ? … ? R _l be an ascending chain of domains which are finitely generated over ? and assume that R _l is integral over R ₀. Let X be an indeterminate. In this paper we characterize all domains D of the form D = R ₀ + XR₁ + … + X^lR_l[X] whose elasticity ρ(D) is finite.     

Interpolation and Parallel Lines

Here we consider 3 interpolation problems for homogeneous polynomials in n n + 1 variables (i.e. for zero-dimensional subschemes Z of Pⁿ) in which the scheme Z is contained in a “ small number ” of “ parallel lines ”; here a finite union D₁ ∪ … ∪ D _x ? Pⁿ of lines is called a set of parallel lines if there is P ∈ Pⁿ such that P ∈ D _i for all i.     

Arrangements in unitary and orthogonal geometry over finite fields

Let V be an n-dimensional vector space over F_q. Let Φ be a Hermitian form with respect to an automorphism σ with σ² = 1. If σ = 1 assume that q is odd. Let $\text{A}$ be the arrangement of hyperplanes of V which are non-isotropic with respect to Φ, and let L be the intersection lattice of $\text{A}$. We prove that the characteristic polynomial of L has n ? v roots 1, q,…, q^{n ? v? 1} where v is the Witt index of Φ.     

A symmetry relationship for a class of partitions

Let S(n, k, v) denote the number of vectors (a₀,…, a_n?1) with nonnegative integer components that satisfy a₀ + … + a_{n ? 1} = k and Σ_i=0^n?1ia_i ≡ v (mod n). Two proofs are given for the relation S(n, k, v) = S(k, n, v). The first proof is by algebraic enumeration while the second is by combinatorial construction.     

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.     

On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

On certain subgraphs of a complete transitively directed graph

Let G_n be a complete transitively directed graph with n + 1 vertices v₀, v₁, …, v_n. Let ψ(n) be the number of subgraphs H of G_n where each vertex in H lies along a directed path from v₀ to v_n in H. ψ(n) and some related quantities are calculated.     

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.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

A note on the proportion of singular (0, 1)-matrices in a class associated with fractional 2n factorial designs

Let the set T = {(x₁, x₂,…, x_n): x_i = 0, 1}. Since the elements of T can be seen as binary representations of integers, we order them with their corresponding integer values. Let $\text{Γ}{}^1$ be the set of (n + 1) × (n + 1) matrices of the form [1 … D], where the n + 1 rows of D are distinct ordered elements of T. We show that the proportion of singular matrices in $\text{Γ}{}^1$ approaches 0 as n → ∞.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

On an approximation property for nondiscrete inequivalent representations of finitely generated groups

Let R(Γ, G) be the variety of representations of a finitely generated group Γ in a simple complex algebraic group G. We establish some sufficient conditions for the image of the diagonal representation ϱ = (ϱ₁, …, ϱ_t), ϱ_i ε R(Γ, G), to be dense in G^f in the complex topology (“weak approximation”).     

Tail bounds for the height and width of a random tree with a given degree sequence

Fix a sequence c = (c₁,…,c_n) of non‐negative integers with sum n ? 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v₁,…,v_n so that for each 1 ≤ i ≤ n, v_i has exactly c_i children. Let ${\mathcal T}Fix a sequence c = (c₁,…,c_n) of non‐negative integers with sum n ? 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v₁,…,v_n so that for each 1 ≤ i ≤ n, v_i has exactly c_i children. Let ${\mathcal T}$ be a plane tree drawn uniformly at random from among all plane trees with child sequence c . In this note we prove sub‐Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of ${\mathcal T}$. These bounds are optimal up to the constant in the exponent when c satisfies $\sum_{i=1}^n c_i^2=O(n)$; the latter can be viewed as a “finite variance” condition for the child sequence. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

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.     

Alternating eulerian trails with prescribed degrees in two edge-colored complete graphs

Let K_n be the complete graph with vertex set {v₁, v₂, …, v_n} and let g=(g₁, …, g_n) be a sequence of positive integers. Color each edge of this K_n red or blue. In this paper necessary and sufficient conditions are given which guarantee the existence of a connected spanning subgraph F in K_n (as colored) with both red degree and blue degree in F at vertex v₁ equal to g_i. When each g_i = 1 this answers a question of Erdös proved in this special case in [1].