Partitions of finite affine geometries into caps

In a finite geometry of order q² we define a (q^mq^r)-affine cap to be a set of cardinality q^m which is a disjoint union ot q^m affine subgeometrics AG(r,q). such that no three points are coliinear unless contained in the same AG(r,q).

Given a PG(n,q²), where n = 2t + 1 or 2t + 2, and an n + 1 by n + 1 Hermitian matrix H over Gh(q²) with minimal polynomial (x - λ)^{n + 1}. we show that H induces a partition of the AG(n, q²) obtained by deleting a distinguished hyperplane from the PG, into (qⁿ,q^{l + 1})-affine caps; these caps can be viewed as the "large points" of an AG (n,q) with a natural incidence relation. It is also shown that H induces another partition of AG(n,q²), into q^{n - l 1}-caps, constituting the "large points" of an affine geometry AG(n + t + 1,q).

Also, the collineation C of PG(n, q²) given by x^c = H^Tx induces collineations on the AG(n,q) and AG(n + t + 1,q).     

The packing of pairs by quadruples

Let X be a finite set of size ν, further let λ be a positive integer and let σ(4,λν) denote the maximum number of quadruples such that each pair of elements of X is contained in at most λ of them. The value of σ(4, 1;ν) has been determined by Brouwer (1979) for all v4. The value of σ(4,λν) has been determined by Billington, Stanton and Stinson (1984) for all ν≡0 (mod 3) and λ>1. In this paper we complete the determination of σ(4,λν) for all ν4 and λ>1.     

Linear mappings which are rank-k nonincreasing, II

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let S_n(F) denote the space of all n × n symmetric matrices with entries in F, and let T:S_n(F)→S_n(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεF^n,n such that T(A)=λPAP^t for all AεS_n(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.     

On the extreme eigenvalues of hermitian (block) toeplitz matrices

We are concerned with the behavior of the minimum (maximum) eigenvalue λ₀⁽ⁿ⁾ (λ_n⁽ⁿ⁾) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix T_n(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ₀⁽ⁿ⁾ — min ƒ = O(1/n^2k) in the case where ƒ C^2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ₀⁽ⁿ⁾ to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L¹ functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.     

The Hoffman number of a graph

For a connected graph G with n vertices, let {λ₁,λ₂,…,λ_r} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number μ(G) of G is defined by μ(G)=λ₁λ₂…λ_r/n. In this paper, we study some properties and applications of the Hoffman number.     

Two theorems on Euclidean distance matrices and Gale transform

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones

where is the elliptope (set of correlation matrices) and is the (closed convex) cone of EDMs.

The characterization is given using the Gale transform of the points generating D. We also show that given points , for any scalars λ₁,λ₂,…,λ_n such that

∑_j=1ⁿλ_jp^j=0, ∑_j=1ⁿλ_j=0,

we have

∑_j=1ⁿλ_jpⁱ−p^j2= forall i=1,…,n,

for some scalar independent of i.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

The kernels of irreducible representations of the general linear group

A derivation for the kernel of the irreducible representation T_(λ) of the general linear group GL_n(C) is given. This is then applied to the problem of determining necessary and sufficient conditions under which T_(λ)(A) = T_(λ)(B), where A and B are linear transformations, not necessarily invertible. Finally, conditions are obtained under which normality of T_(λ)(A) implies normality of A.     

ContributionsThe existence of perfect Mendelsohn designs with block size 7

Necessary conditions for existence of a (v,k,λ) perfect Mendelsohn design (or PMD) are v k and λv(v − 1) ≡ 0 mod k. When k = 7, this condition is satisfied if v ≡ 0 or 1 mod 7 and v 7 when λ 0 mod 7 and for all v 7 when λ ≡ 0 mod 7. Bennett, Yin and Zhu have investigated the existence problem for k = 7, λ = 1 and λ even; here we provide several improvements on their results and also investigate the situation for λ odd. We reduce the total number of unknown (v,7,λ)-PMDs to 36,31 for λ = 1 and 5 for λ > 1. In particular, v = 294 is the largest unknown case for λ = 1, and the only unknown cases for λ > 1 are for v = 42, λ [2,3,5,9] and v = 18, λ = 7.     

On a conjecture of C. Johnson

Given a pair of n×n matricesA and B, one may form a polynomial P(A,B,λ) which generalizes the characteristic polynomial of BP(B,λ). In particular, when A=I (identity), P(A, B,λ) = P(B,λ), the characteristic polynomial of B. C. Johnson has conjectured [1] (among other things) that when A and B are hermitian and A is positive definite, then P(A,B,λ) has real roots. The case n=2 can be done by hand. In this paper we verify the conjecture for n=3.     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1for some C = C^-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A^-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

Designs with no three mutually disjoint blocks

Let D(v,b,r,k,λ) be any quasi-symmetric block design with block intersection numbers 0 and y. Suppose D has no three mutually disjoint blocks. We show that for a given value of y, there are only finitely many parameter sets of such designs. Moreover, the ‘extremal’ designs D have one of the following parameter sets: (1) v = 4y, k = 2y, λ = 2y − 1 (y 2) (2) v = y(y²+3y+1), k = y(y+1), λ =y²+y−1(y2) (3) v = (y+1)(y²+2y−1), k = y(y+1), λ =y² (y2) A computer search revealed only three parameter sets in the range 1y199, which are not of the above types.     

Poisson evolution in word selection

This paper presents the finding that the invocation of new words in human language samples is governed by a slowly changing Poisson process. The time dependent rate constant for this process has the form

λ(t) = λ₁(1−λ₂t)e^-λ₂t+λ₃(1−λ₄t)e^-λ₄t+λ₅

, where

λ_i > 0, I=1,…,5

.

This form implies that there are opening, middle and final phases to the introduction of new words, each distinguished by a dominant rate constant, or equivalently, rate of decay. With the occasional exception of the phase transition from beginning to middle, the rate λ(t) decays monotonically. Thus, λ(t) quantifies how the penchant of humans to introduce new words declines with the progression of their narratives, written or spoken.     

ContributionsSign-balanced covering matrices

A q × n array with entries from 0, 1,…,q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,… q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1]^t, a λq × n array will be said to form a (t, q, λ, Δ) sign-balanced matrix if for each choice C₁, C₂,…, C_t of t columns and for each choice = (₁,…,_t) Δ of signs, the linear combination ∑_j=1^t _jC_j contains (mod q) each entry of [0, 1,…, q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k(n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice (₁, …, _t) of signs in δ, the linear combination ∑_j=1^t _jC_j contains each entry of [0, 1,…, q t- 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k(n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.     

An Extension of an Inequality Involving Symmetric Products with an Application to Permanents

Let A be an nk × nk positive semi-definite symmetric matrix partitioned into blocks A^ij each of which is an n × n matrix. In [2] Mine states a conjecture of Marcus that per(A) ≥ per(G) where G is the k × k matrix [per(A^ij)]. In this paper we prove a weaker inequality namely that per(A) ≥ (k!)^-1per(G).     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ₁. We consider the power method, i.e. that of choosing a vector v₀ and setting v_k = A^kv₀; then the Rayleigh quotients R_k = (Av_k, v_k)/(v_k, v_k) usually converge to λ₁ as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close R_k is to λ₁. They are both based on a bound on λ₁ − R_k involving the difference of two consecutive Rayleigh quotients and a quantity ω_k. While we do not know how to directly calculate ω_k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ₁ − R_k which is proportional to (λ₂/λ₁)^2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ).     

On the jordan form of the tensor product over fields of prime characteristic

Let A, B denote the companion matrices of the polynomials x^m,xⁿ over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A ^TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.     

Impulsive effects on global existence of solutions for degenerate semilinear parabolic equations

Let q be a nonnegative real number, and λ and σ be positive constants. This article studies the following impulsive problem: for n = 1, 2, 3,…,

. The number λ^* is called the critical value if the problem has a unique global solution u for λ < λ^*, and the solution blows up in a finite time for λ > λ^*. For σ < 1, existence of a unique λ^* is established, and a criterion for the solution to decay to zero is studied. For σ > 1, existence of a unique λ^* and three criteria for the blow-up of the solution in a finite time are given respectively. It is also shown that there exists a unique T^* such that u exists globally for T> T^*, and u blows up in a finite time for T < T^*.