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 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.     

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.     

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 minimal number of basic elements in a multiset antichain

Let M be a finite set consisting of k_i elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x₁, x₂,…,x_n) with integral coefficients x_i satisfying 0 ? x_i ? k_i, i = 1, 2,…, n. An antichain

is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities x_i ? y_i, i = 1, 2,…, n all hold). Let $\text{∥Y}\textcolor{red}{}\text{∥}$ denote the number of vectors in S which are contained in at least one vector in

and let $\text{∥B}\textcolor{red}{}\text{∥=∑}{}_{\mathrm{x\in }}\text{(X}{}_1\text{+X}{}_2\text{+?+X}{}_{\mathrm{n}}\text{)}$, the number of basic elements in

. For given m we give procedures for calculating min $\text{∥Y}\textcolor{red}{}\text{∥}$ and min $\text{∥B}\textcolor{red}{}\text{∥}$, where the minima are taken over all m-element antichains

in S.     

Majors of geometric strong maps

Most of the constructions in the theory of combinatorial geometries take place in the category of pregeometries and strong maps. In this present paper, we study these constructions and the structure of pregeometries by factoring strong maps into elementary maps, after the work of Dowling and Kelly. Using the modular cuts of the factorization to determine certain single-element extensions, we associate each factorization Φ to a unique labeled pregeometry, called the major of Φ. Every major of a factorization of a strong map f:H→G has H and G as distinguished minors: H as a subgeometry; G as a contraction.A partial order is defined on the set of all factorizations of f,

, and a compatible partial order is defined on the set of all majors of f,

. The map Φ :

→

is shown to be an increasing surjection, while the map Y :

→

is shown to be an increasing injection. The composition Φ ° Y is shown to be the identity on

, while Y ° Φ is a decreasing map on

, defining a co-closure on majors. The partial order on

, although seemingly more restrictive than necessary, is shown to be necessary to reflect the order on factorizations.Special consideration is given the zero element of

, called the Higgs factorization of f, which is constructed using the lift construction of Higgs. We explore the connection between the majors of this theory and a particular major constructed by Higgs, and show that this Higgs major is the major of the Higgs factorization. As the unique smallest major of any strong map, the Higgs major thus defines a unary operator on pregeometries: Y(G) is the Higgs major of f :

→G.     

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.     

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.)     

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.     

Snakes: Oriented families of periodic orbits,their sources,sinks, and continuation

Poincaré observed that for a differential equation $\text{x′ = ?(x, α)}$ depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in (x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index φ and a “center” index

defined at certain stationary points. We show that genetically there are two types of Hopf bifurcation, those we call “sources” ($\textcolor{red}{}\text{= 1}$) and “sinks” ($\textcolor{red}{}\text{= ?1}$). Generically if the set Q is bounded in (x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter “snake” of orbits. A “snake” is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1.     

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.     

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).     

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.     

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.     

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.     

Some remarks on C1-convexity

Recently the study of completely positive maps has become important to the results of Brown, Douglas, and Fillmore on Ext(

),

a C¹-algebra. Attempts to solve questions related to Ext have often turned into questions about the matrix algebras M_n. In this paper we wish to discuss a notion of C¹-convexity related to completely positive linear maps, to state some facts about C¹-convexity, and to ask some questions about C¹-convexity. To a large degree, the tone of this paper is expository.     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.