Finite embedding theorems for partial Steiner triple systems

Let P be a finite set and (P,

₁), (P,

₂),…, (P,

_k) any collection of mutually disjoint partial Steiner triple systems. Then these partial triple systems can be embedded in finite mutually disjoint triple systems (S,

₁), (S,

₂),…, (S,

_k). This result is then used to prove the following more general result. If (P,

₁), (P,

₂),…, (P,

_k) are any collection of finite partial Steiner triple systems, then these partial triple systems can be embedded in finite triple systems (S,

₁), (S,

_j =

_i ∩

_j for all i ≠ j = 1, 2,…, k.     

Generalized inverse of linear transformations: a geometric approach

A generalized inverse of a linear transformation A:

→

, where

and

are arbitrary finite dimensional vector spaces, is defined using only geometrical concepts of linear transformations. The inverse is uniquely defined in terms of specified subspaces $\text{L}$ ?

, $\text{M}$ ?

and a linear transformation N satisfying some conditions. Such an inverse is called the $\text{LM}$N-inverse. A Moore-Penrose type inverse is obtained by choosing N = 0. Some optimization problems are considered by choosing

and

as inner product spaces. Our results extend without any major modification of proofs to bounded linear operators with closed range on Hilbert spaces.     

The superposition of the backward and forward processes of a renewal process

A fixed sampling point O is chosen independently of a renewal process

on the whole real line. The distances Y₁, Y₂, … from O to the renewal points of

, when they are measured either forwards or backwards in time, define a point process

. The process

is a folding over of the past of

onto its future. It is the superposition of two equilibrium renewal processes which are known to be independent only when

is a Poisson process. The joint distribution of Y₁, Y₂, …, Y_k is found. The marginal distribution of 2Y_k is shown to be the same as that of the distance from O to the k^th following point of

. The intervals of

are shown to have a stationarity property, and it is proved that if any pair of adjacent intervals of

are independent, then

is a Poisson process.     

Recovering a frame from its sheaves of algebras

Let

be a frame, α an element of

and T a finitary algebraic theory. In this paper we compare the category

of sheaves of T-algebras on

with the category Sh(α↓,T) of sheaves of T-algebras on α↓ (where α↓ is the initial segment {β|β<α}). This comparison suggests the definition of formal initial segments of the category

. For a large class of theories to be called ‘integral’ (examples of which are sets, monoids, groups, rings, modules on a integral domain, boolean algebras,hellip;) the formal initial segments of

coincide with the usual initial segments of

: this means that we are able to recover

axiomatically from

.When

is the initial frame {0, 1}, the frame of formal initial segments of

is the frame of open subsets of a compact space Spec T, called the spectrum of the theory T. When T is the theory of modules on some ring R, we recover in this way various well known notions of spectra and their corresponding sheaf-representation of the ring.     

The completion of a B-convex normed Riesz space is reflexive

If

is a B-convex normed Riesz space, then the topological completion of

is a closed subspace of

⁷, the second Banach dual of

. If $\text{N=}\textcolor{red}{}{}^7$ or $\text{N=}\textcolor{red}{}{}^{\mathrm{7x}}$, then $\text{N}$ is a B-convex σ-Dedekind complete normed Riesz space which is the Banach dual of a normed Riesz space. In such a $\text{N}$, if u₁ ? u₂ ? … ? 0 and inf_n{u_n} = 0, then lim_n∥u_n∥ = 0. This is the key step in verifying that Ogasawara's criteria that a normed Riesz space be reflexive are satisfied by

⁷. Thus the topological completion of

as a closed subspace of

⁷ is also reflexive.     

The Frobenius rank equality for morphisms

Let $\text{U}$(

,ε) be an abelian category with Serre class

and Euler-Poincaré mapping ε. For η a morphism in $\text{U}$(

,ε) with Imη a member of

, let rank _εη =ε(Imη). A proof is given of the Frobenius rank equality: if αβγ is a composition of three morphisms in $\text{U}$(

,ε) and Imβ is a member of

, then rank_εαβ+rank_εβγ+rank_ε(kerβγ)β(cokαβ)=rank_εβ+rank_εαβγ.     

Self-similar singularities of the 3D Euler equations

Self-similar solutions are considered to the incompressible Euler equations in

, where the similarity variable is defined as

, β ≥ 0. It is shown that the scaling exponent is bounded above: β ≤ 1. Requiring |u|

² < ∞ and allowing more than one length scale, it is found β ϵ [2/5, 1]. This new result on the self-similar singularity is consistent with known analytical results for blow-up conditions.     

Stochastic matrices. I. Similarity via stochastic transforms

Suppose two stochastic matrices A and B of order n are similar in the set

of all matrices of order n over a real field R. We obtain sufficient conditions in order that A and B be right similar, left similar, and similar in the set

of all stochastic matrices of order n over R. As a corollary, we obtain the known result that two doubly stochastic matrices of order n which are similar in

are also similar in the set

of all doubly stochastic matrices of order n over R. Examples are given to show that these sufficient conditions are not necessary and are also not vacuous. Finally, we give an application of some of these results to the transition probability matrices of stationary finite Markov processes.     

Scrambled Vandermonde convolutions of Gaussian polynomials

It is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the

statistic on monotone paths in a rectangular grid. We introduce two new statistics,

and

; attach “ornaments” to the grid that scramble the values of

in specific fashion; and re-evaluate these statistics, in order to argue that all scrambled versions of the

statistic are equidistributed with

. Our main result is a representation of the generating function for the bi-statistic

as a new, two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.     

Proof of a conjecture of Burr,Grünbaum,and Sloane

We establish a duality principle for arrangements of pseudolines in the projective plane, and thereby prove the conjecture of Burr, Grünbaum, and Sloane that the solution

of the “orchard problem” for pseudoline arrangements and the solution t?(p) of the dual problem are equal.     

Exact radial solutions for a nonlinear eigenvalue problem

We obtain explicit radial solutions of the nonlinear problem

when Ω ⊂

is the unit ball, N ≥ 2, 2α > −N, λ* ≥ λ 0, λ* = (2α + N)^N/N.     

Some results in combinatorial geometry

Let

denote an n× n square array of lattice points in the plane let f(n) denote the least number of points that can be selected from

such that every point of

is visible from at least one of the points selected. It is proved that, for n sufficiently large,

$\frac{log?n}{2log?log?n}<f(n)<4log?n$

.     

Toeplitz-Hausdorff systems

The numerical range of an n × n matrix T is the image of T under a certain set of linear functionals—a set that comprises the extreme points among the states (i.e., norm-one, positive linear functionals) on the n × n matrices—and is convex, by the Toeplitz-Hausdorff theorem. One can view this convexity as a consequence of T's numerical range being equal to a manifestly convex set, the image of T under all states. Taking this view leads us to ask whether a similar result holds when we replace the n × n matrices by a finite dimensional Banach space

, the states by a closed, convex subset Σ of

1, and the extreme states by the extreme points of Σ. When it does, we call the pair (

, Σ) a Toeplitz-Hausdorff system. In this paper, we show that if

is what we term a nullifying subspace of the n×n matrices, and if Σ is the closed unit ball in

1, then (

, Σ) is a Toeplitz-Hausdorff system. (Both the upper and lower triangular matrices form nullifying subspaces.)     

Note on a characterization of exponential distributions

Let U be uniformly distributed on (0,1) and let Y and Y′

Y be random vectors with nonnegative] components, U, Y and Y independent. It is shown that the relation Y

U(Y+Y′) is satisfied if and only if the components of Y are multiples of a single exponentially distributed random variable.     

Theorems of Stein-Rosenberg type. III. The singular case

In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J_ω and the successive overrelaxation (SOR) matrix

, in the case when the Jacobi matrix J₁ is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J_ω and

have been established which do not depend on the assumption that J₁ is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix A with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that J_ω and

simultaneously semiconverge for all ω in the real interval [0,1).     

Analysis over Discrete spaces

Let

be a σ-finite nonatomic measure space. We think of the customary analysis based upon

as Continuum Analysis. In contrast, we regard Discrete Analysis as being based upon a countable subset of X rather than upon X itself. The particular version of Discrete Analysis introduced here—we shall call it Poisson Analysis—treats simultaneously all countable subsets of X and distinguishes between them probabalisticly via a counting process whose values are random variables with Poisson distributions. The present paper demonstrates the viability of this idea by applying it to the formal theory of a free Boson field. The customary version of this theory may be phrased in terms of

and the related normal Wiener process N. The alternate version introduced here uses, in place of N, what we shall call a Gaussian jump process. We interpret the resulting theory as Discrete Analysis and compare it with the customary theory. In particular we show that the discrete theory already contains the usual one inside it (see Theorem 1), and that, in the discrete case, symmetries are much more readily implementable by unitary transformations.     

Endomorphism semigroups of finite subset algebras

For an arbitrary set X, if $\textcolor{red}{}\text{(X)}$ denotes the collection of finite subsets of X, then $\text{(}\textcolor{red}{}\text{(X),∩,∪,-)}$ is called the finite subset algebra of X (which we also denote by $\textcolor{red}{}\text{(X))}$. Because of the many natural occurrences of finite subset algebras, the intent of this paper is to initiate an investigation of the semigroup, End $\textcolor{red}{}\text{(X)}$, of endomorphisms of $\textcolor{red}{}\text{(X)}$.Several representations of End $\textcolor{red}{}\text{(X)}$ are obtained and these representations are in turn used to obtain information about the structure of End $\textcolor{red}{}\text{(X)}$. In particular, the Green relations are characterized; for finite X, the maximal subgroups of End $\textcolor{red}{}\text{(X)}$ are found and the number of idempotents is given. As a side result, an abstract characterization of a certain semigroup of (0,1)-matrices is obtained.     

Oscillations of a class of difference equations with continuous arguments

This paper is concerned with the difference equations of the form

Sufficient conditions for all solutions of this equations to be oscillatory are obtained.     

Existence results for a model of nonlinear beam on elastic bearings

We study the existence of solutions of the nonlinear fourth-order equation of Kirchhoff type

under nonlinear boundary conditions which models the deformations of beams on elastic bearings.