Two inequalities for the sum of elements of a matrix

The inequalities su A^t su A^m ≥ su A^p su A^q and A^t+ su A^m≥ su A^p+ su A^q are studied and generalized. Here su A denotes the sum of elements of the square matrix A.     

Sums and products of matrices with prescribed similarity classes

Let F be a field and let A and n × n matrices over F. We study some properties of A^' + B^' and A^'B^', when A^' and B^' run over the sets of the matrices similar to A and B, respectively.     

Nonnegative matrices having same nonnegative moore-penrose and group inverses

Nonnegative matrices A whose Moore-Penrose generalized inverse A⁺ is nonnegative and has any one of the three equivalent properties (i) AA⁺ = A⁺A (ii) A⁺ = A, the group inverse, (iii) A⁺ = p(A), some polynomial in A with scalar coefficients, are characterized. This characterization generalizes known results on nonnegative matrices Awhose Moore-Penrose generalized inverse is equal to some power of A.     

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).     

Chung’s law of the iterated logarithm for subfractional Brownian motion

Let X~H= {X~H(t), t ∈ R_+} be a subfractional Brownian motion in R~d. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that X~H has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of X~H, we establish Chung's law of the iterated logarithm for X~H.     

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.     

Regular automorphisms of order p2

Let G be a group, and let α be a regular automorphism of order p² of G, where p is a prime. If G is polycyclic-by-finite and the map ϕ : G →G defined by g^ϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then C_G(α^p) and G/[G,α^p] are both nilpotent-by-finite.     

Hecke特征值相邻极大值的均值估计

取f为权为偶数k的全模群Γ = SL(2,Z)的Hecke特征型.定义Asymmf(n)为与f关联的m阶对称幂次L-函数的Dirichlet展开式的第n个正规化系数.本文中,我们给出了Σn0为一个正数.     

Doubly stochastic matrices whose squares are idempotent

Formulas for the doubly stochastic square roots of the idempotents are presented. Such square roots must necessarily satisfy per A²≤perA.     

Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

An inequality for non-negative matrices

Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there exist integers r, s with 0 r s 2ⁿ such that A^r A^s. We prove that 2ⁿ cannot be replaced with e√n log n. We also give an application to the theory of formal languages.     

On infinite area for complex exponential function

This paper shows via a reduced family of examples, the relaxed Newton's method is applied to complex exponential function F(z)=ze^z and F(z)=ze^z2, the basin of roots has infinite area. In addition, we examined their computer pictures which are fractals for the relaxed Newton's basin. In fact, computer experiments F(z)=P(z)e^z and F(z)=P(z)e^z2, indicate this to hold for arbitrary non-constant polynomial P(z).     

Decomposition of a rational matrix into partial fractions

In this paper a technique is developed for the expansion of a rational matrix F(λ)=G(λ)L^-1(λ)(orF(λ)=L^-1(λ)G(λ)into block partial fractions when the denominator has multiple roots. The method consists in the construction of interpolating matrix polynomials and their properties. Moreover, the approach is extended when L(λ) has nonlinear divisors.     

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.     

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).     

Hilbert's theorem 90 in matrix rings

The fact that every unitary matrix can be expressed in the form A^-1A^* (observed by C. R. Johnson) is interpreted as a "Hilbert Theorem 90" in rings with an involution. Several observations connected with this are included.     

An example of a space L on which the Hardy-Littlewood maximal operator is not bounded

We give an example of a function p such that the Hardy-Littlewood maximal operator is not bounded on the generalized Lebesgue space L^p(x).     

The doubly stochastic matrix equations x:aty= A and X Aty=t

For a doubly stochastic matrix A, each of the equations x:a^ty= A and X A^ty=^t is shown to have doubly stochastic solutions X and Y if and only if A lies in a subgroup of the semigroup of all doubly stochastic matrices of a given order. All elements of this semigroup which are left regular, right regular, or intra-regular are identified.     

Choquet simplexes in finite dimension — A survey

This survey covers various geometric results related to Choquet simplexes in the Euclidean space E^d; it describes the known properties of Choquet simplexes and marks still open problems.     

On the exponential map of borei subgroups in a semi-simple lie group

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element g^m lies on a one-parameter subgroup in G and the element gⁿ lies on a one-parameter subgroup in B. We denote these integers by ind_G(g) and ind_B(g). In this note we prove the conjecture ind_G(g) = ind_B(g), if g is regular.