Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

A topological property enjoyed by near points but not by large points

Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near.     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

A generalization of a result of Sylvester's

Let v₁,…,v_n be vectors in Zⁿ with D = det(v₁,…,v_n) > 0. Let v_{n + 1} be in the cone generated by v₁,…,v_n and such that v₁,…,v_v, v_{n + 1} generate Zⁿ as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v₁,…,v_n, v_{n + 1} and χ = Dv_{n + 1} ? (v₁ + … v_n + v_{n + 1}).     

Simple components of group algebras Q[G] central over real quadratic fields

Let k be a real quadratic field, and $\text{U}$ a central division quaternion algebra over k. In this paper sufficient conditions are given to insure that $\text{U}$ appears in a simple component of the group algebra Q[G] of some finite group G over the rational field Q. In particular, when k is assumed to be Q(√2) or Q(√5), the necessary and sufficient conditions for $\text{U}$ to appear in some Q[G] are given.     

Location change in marginal distributions of linear functions of random vectors

Suppose X and Y are n × 1 random vectors such that l′X + f(l) and l′Y have the same marginal distribution for all n × 1 real vectors l and some real valued function f(l), and the existence of expectations of X and Y is not necessary. Under these conditions it is proven that there exists a vector M such that f(l) = l′M and X + M and Y have the same joint distribution. This result is extended to Banach-space valued random vectors.     

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.     

Properties of Euclidean and non-Euclidean distance matrices

A distance matrix D of order n is symmetric with elements $\text{?}\text{1}\text{2}\text{d}{}_{\mathrm{ij}}{}^2$, where d_ii=0. D is Euclidean when the $\text{1}\text{2}\text{n(n?1)}$ quantities d_ij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether e^TD^?e is not or is zero, the generating points lie in either p=ρ?1 dimensions, in which case they lie on a hypersphere, or in p=ρ?2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

LU-decomposition of a matrix with entries of different kinds

Let F ? K be fields, and consider a matrix A over F whose entries not belonging to K are algebraically independent transcendentals over K. It is shown that if det $\text{A ε}\text{K}{}^1\text{( =}\text{K}\text{? {O})}$, the matrix A, with suitable permutations of its rows and columns, is decomposed into LU-factors with the entries of the U-factor belonging to K.     

Antichains in the set of subsets of a multiset

A set F of distinct subsets x of a finite multiset M (that is, a set with several different kinds of elements) is a c-antichain if for no c + 1 elements x₀, x₁, …, x_c of F does x₀ ? x₁ ? ··· ? x_c hold. The weight of F, wF, is the total number of elements of M in the various elements x of F. For given integers f and c, we find min wF, where the minimum is taken over all f-element c-antichains F. Daykin [9, 10] has solved this problem for ordinary sets and Clements [3] has solved it for multisets, but only for c = 1.     

A unique exchange property for bases

We define the concept of unique exchange on a sequence (X₁,…, X_m) of bases of a matroid M as an exchange of x ? X_i for y ? X_j such that y is the unique element of X_j which may be exchanged for x so that (X_i ? {x}) ∪ {y} and (X_j ? {y}) ∪ {x} are both bases. Two sequences X and Y are compatible if they are on the same multiset. Let UE(1) [UE(2)] denote the class of matroids such that every pair of compatible basis sequences X and Y are related by a sequence of unique exchanges [unique exchanges and permutations in the order of the bases]. We similarly define UE(3) by allowing unique subset exchanges. Then UE(1),UE(2), and UE(3) are hereditary classes (closed under minors) and are self-dual (closed under orthogonality). UE(1) equals the class of series-parallel networks, and UE(2) and UE(3) are contained in the class of binary matroids. We conjecture that UE(2) contains the class of unimodular matroids, and prove a related partial result for graphic matroids. We also study related classes of matroids satisfying transitive exchange, in order to gain information about excluded minors of UE(2) and UE(3). A number of unsolved problems are mentioned.     

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Imprimitive distance-regular graphs and projective planes

We investigate a class of (imprimitive) covering graphs Γ of complete bipartite graphs K_k,k and show that they are in one-to-one correspondence with triples (P, l, P), where P is a projective plane of order k and (l, P) is a distinguished flag of P. If Γ is distance-transitive, then P ? l is a self-dual rank three translation plane and may be coordinatised by a semifield.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$