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.     

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 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.     

Random functions of Poisson type

Let $\text{(X,}\textcolor{red}{}\text{, μ)}$ be a σ-finite nonatomic measure space. We think of the customary analysis based upon $\text{(X,}\textcolor{red}{}\text{, μ)}$ as continuum analysis. By contrast discrete analysis is based upon an arbitrary countable subset of X, rather than upon X itself, and all countable subsets are treated alike with a Poisson process used to distinguish among them probabilisticly. The sort of functions appropriate for discrete analysis are the Campbell functions, or, as they are called in the present paper, the random functions of Poisson type. The paper presents an account of the ideas underlying discrete analysis and treats briefly the specifies of representation, stochastic integrals, and duality theory for random functions of Poisson type. It is chiefly concerned, however, with those random functions which occur in connection with the discrete analysis of Brownian motion, (for example, with Gaussian noise). In particular it shows that there is a completely positive map which carries such discrete processes onto an algebraic version of Wiener's Brownian motion process, and that under this map, random functions of Poisson type go over to the appropriate random functions of Wiener type. It also shows that the map carries random variables into noncommuting operators characteristic of quantum theory.     

Generic periodic solutions of functional differential equations

Consider the class of retarded functional differential equations

$\text{x(t) = f(x}{}_{\mathrm{t}}\text{)}$

, (1) where x_t(θ) = x(t + θ), ?1 ? θ ? 0, so x_t?C = C([?1, 0], Rⁿ), and $\text{f∈}\textcolor{red}{}\text{=C}{}^{\mathrm{r}}\text{(C,R}{}^{\mathrm{n}}\text{)}$. Let 2 ? r ? ∞ and give $\text{X}$ the appropriate (Whitney) topology. Then the set of $\text{f∈}\textcolor{red}{}$ such that all fixed points and all periodic solutions of (¹) are hyperbolic is residual in

.     

Near-boolean algebras I: Combinatorial aspects

An algebra $\textcolor{red}{}\text{=(A; ?, ?, ′, 0, 1)}$ is near-boolean if every two generated subalgebra is a boolean algebra. It is shown that for n ? 4 there is a near-boolean algebra of size n ≡ 4 or 8 (mod 12), that for n ? 16 and n ≡ 4 or 8 (mod 12) there is a near-boolean algebra of size n which is not a boolean algebra, and that the smallest near-boolean algebra which is not a boolean algebra has 16 elements.     

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.     

On Cremona transformations of prime order

We prove that an automorphism of order 5 of the Del Pezzo surface S of degree 5 is conjugate through a birational map $S\text{<mglyph src="https://ars.els-cdn.com/content/image/1-s2.0-S1631073X04003164-fx001.gif"></mglyph>}{\mathbb{P}}^2$ to a linear automorphism of ${\mathbb{P}}^2$. This completes the classification of conjugacy classes of elements of prime order in the Cremona group. To cite this article: A. Beauville, J. Blanc, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

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 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.     

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 $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation

In this paper, we show that the initial boundary value problem for the (singular) nonlinear EPD (Euler-Poisson-Darboux) equation

does not possess global solutions for arbitrary choices of $\text{u(x, 0). (x ? Ω ? R}{}^{\mathrm{n}}\text{, Ω}\text{bounded}\text{, Δ}{}_{\mathrm{n}}\text{= n}\text{dimensional Laplacian}$) when 0 < k ? 1 for a wide class of nonlinearities $\text{T}$, which includes all the even powers of u and the functions u^{2n + 1}, n = 1, 2,…. The solutions are assumed to vanish on the “walls” of the spacetime cylinder and satisfy $\text{?u}\text{?t(x, 0)}\text{= 0, x ? Ω}$. The result is independent of the space dimension.     

Approximation numbers of Sobolev imbeddings over unbounded domains

Let Ω ? $\text{R}$^N be an open set with $\text{dist}\text{(x, ?Ω) = O(¦ x ¦}{}^{\mathrm{?l}}\text{)}\text{for}\text{x ? Ω}$ and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers α_n of Sobolev imbeddings

over these quasibounded domains Ω. Here

denotes the Sobolev space obtained by completing $\text{C}{}_0{}^{\mathrm{staggered\infty }}\text{(Ω)}$ under the usual Sobolev norm. We prove $\text{α}{}_{\mathrm{n}}\text{(I}{}_{\mathrm{p,q}}{}^{\mathrm{m}}\text{)}\text{\$}\text{?}\text{n}{}^{\mathrm{?\gamma }}$, where

. There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of I_p,q^m. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded domain of the above type.     

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.     

Répartition des densités des sous-suites d'une suite d'entiers

Let $\text{A}$ denote a strictly increasing sequence of integers; for any integer n, define A(n) to be the number of positive elements of $\text{A}$ not exceeding n. The upper and lower asymptotic densities of $\text{A}$ are defined by

We describe the set of pairs (d$\text{B}$, $\text{d}$$\text{B}$), where $\text{B}$ runs over all subsequences of $\text{A}$, as being a closed convex region of the plane. The converse statement is also proved.