Conditional expectations and submartingale sequences of random Schwartz distributions

Let (Ω, Σ, P) be a fixed complete probability space, $\text{D}$ the real Schwartz space, and $\text{D′}$ its strong dual. $\text{D}$ and $\text{D′}$ are partially ordered by $\text{C}$ and $\text{C′}$ respectively, where $\text{C}$ is the positive cone of nonnegative functions in $\text{D}$ and $\text{C′}$ its dual in $\text{D′}$. $\text{C}$ is a strict $\text{B}$-cone and $\text{C′}$ is normal, where $\text{B}$ is the family of all bounded subsets of $\text{D}$. If X, Y are two random Schwartz distributions, then X ≤ Y if and only if Y(ω) ? X(ω) ∈ $\text{D′}$ for almost all $\text{ω ∈ Ω(P)}$. Integrability of random Schwartz distributions and properties of such integrals are discussed. The monotone convergence theorem, the dominated convergence theorem, and Fatou's lemma are proved. The existence of conditional expectations of integrable random Schwartz distributions relative to a given sub σ-field of Σ is shown. Properties of conditional expectations are discussed and the conditional form of the monotone convergence theorem is proved. Sub(super)-martingale sequences are defined via the partial order relations introduced above, and a convergence theorem is given. The notion of a potential is introduced and the Riesz decomposition theorem is proved.     

Partition theorems for subspaces of vector spaces

The principal result of this paper provides a nearly complete answer to the following question. For which cardinal numbers $\text{t}$, $\text{m}$, $\text{n}$, $\text{q}$ and $\text{r}$ is it true that whenever the $\text{t}$-dimensional subspaces of an $\text{n}$-dimensional vector space V over a field of $\text{q}$ elements are partitioned into $\text{r}$ classes, there must be some $\text{m}$-dimensional subspace of V, all of whose $\text{t}$-dimensional subspaces lie in the same class? This question is answered completely if $\text{r}$ ≤ $\text{N}$₀. The contributions of this paper are in the form of negative answers, since it turns out that all affirmative answers (which we have) were already known or easily deducible from known results.     

Two relations between oblique and Λ-orthogonal projectors

Let $\text{L}$ and $\text{M}$ be any complementary subspaces. In this article, two relations established by T.N.E. Greville between the projector P_{$\text{L}$|$\text{M}$} on $\text{L}$ along $\text{M}$ and the orthogonal projectors on $\text{L}$ and $\text{M}$ are generalized by admitting any Λ-orthogonal projectors, with Λ being a positive definite matrix. Also, two representations of Λ are found for which, given $\text{L}$ and $\text{M}$, Λ-orthogonal projectors on $\text{L}$ become identical with P_{$\text{L}$|$\text{M}$}.     

Identifying defective sets using queries of small size

We examine the following version of a classic combinatorial search problem introduced by Rényi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y$ of $X$, which is known to have exactly $d$ elements, by means of testing, for as few as possible subsets $A$ of $X$, whether $A$ intersects $Y$ or not. We are primarily concerned with the non-adaptive model, where the family of test sets is specified in advance, in the case where each test set is of size at most some given natural number $k$. Our main results are nearly tight bounds on the minimum number of tests necessary when $d$ and $k$ are fixed and $n$ is large enough.     

Prime ideals and automatic continuity problems for Banach algebras

Conditions are given for Banach algebras $\text{U}$ and commutative Banach algebras $\text{B}$ which insure that every homomorphism v from $\text{U}$ into $\text{B}$ is continuous. Similar results are obtained for derivations which either map the algebra $\text{U}$ into itself or map the algebra into a suitable $\text{U}$-module.     

Relations and homomorphisms of n-hypergroups

In this paper, some types of homomorphisms of $n$-hypergroups is introduced and several properties are found and examples are presented. Homomorphisms of $n$-hypergroups are a generalization of homomorphisms of hypergroups. Homomorphism between $n$-hypergroups and equivalence relations on $n$-hypergroups are closely related. Also, we consider an equivalence relation $\rho$ on an $n$-hypergroup $H$ and define an $n$-hyperoperation on $H/\rho$ and prove some results in this respect.     

Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.     

Unitary block designs

Given a projective plane $\text{E}$ over the field of q² elements and a unitary polarity π of $\text{E}$ it is possible to construct the well-known unitary design $\text{U}$ whose points are the absolute points of π and whose blocks are the non-absolute lines of π. A relation of perpendicularity is defined between blocks and it is shown that this relation can be described in terms of the incidence structure of $\text{U}$. The projective plane $\text{E}$ together with the polarity π can then be reconstructed from the design $\text{U}$ in such a way that any automorphism of $\text{U}$ extends to a collineation of $\text{U}$ which commutes with π.     

Unbounded derivations tangential to compact groups of automorphisms,II

Let G be a compact abelian group, and τ an action of G on a C¹-algebra $\text{U}$, such that $\text{U}$^τ(γ)$\text{U}$^τ(γ)¹ = $\text{U}$^τ(0) $\text{U}$^τ for all $\text{γ ?}\text{G}\text{?}$, where $\text{U}$^τ(γ) is the spectral subspace of $\text{U}$ corresponding to the character γ on G. Derivations δ which are defined on the algebra $\text{U}$_F of G-finite elements are considered. In the special case δ¦_{$\text{U}$^τ} = 0 these derivations are characterized by a cocycle on $\text{G}\text{?}$ with values in the relative commutant of $\text{U}$^τ in the multiplier algebra of $\text{U}$, and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded. Under various restrictions on G and τ properties of the cocycle are deduced which again give characterizations of δ in terms of decompositions into generators of one-parameter subgroups of τ(G) and approximately inner derivations. Finally, a perturbation technique is devised to reduce the case δ($\text{U}$_F) ? $\text{U}$_F to the case δ($\text{U}$_F) ? $\text{U}$_F and δ¦_{$\text{U}$^τ} = 0. This is used to show that any derivation δ with D(δ) = $\text{U}$_F is wellbehaved and, if furthermore G = T¹ and δ($\text{U}$_F) ? $\text{U}$_F the closure of δ generates a one-parameter group of ¹-automorphisms of $\text{U}$. In the case G = T^d, d = 2, 3,… (finite), and δ($\text{U}$_F) ? $\text{U}$_F it is shown that δ extends to a generator of a group of ¹-automorphisms of the σ-weak closure of $\text{U}$ in any G-covariant representation.     

On the ∂-Neumann problem for generalized functions

Using the new theory of generalized functions developed by one of the authors the $\text{?}$ equation in $\text{C}$ⁿ is studied. In particular it is proven that if G is any generalized function on $\text{C}$ (in the above sense) then there is a generalized function S on $\text{C}$ such that $\text{?S}\text{?}\text{z}\text{?}\text{= G}$. Several other results are proven valid in polydiscs of $\text{C}$ⁿ, for which differential forms whose coefficients are generalized functions are introduced.     

Convex polyhedra of doubly stochastic matrices: II. Graph of Ωn

Properties of the graph $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are studied. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is not a k-dimensional rectangular parallelotope for k ≥ 2, then G($\text{F}$) is Hamilton connected. Prime factor decompositions of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ relative to Cartesian product are investigated. In particular, if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then the number of prime graphs in any prime factor decomposition of G($\text{F}$) equals the number of connected components of the neighborhood of any vertex of G($\text{F}$). Distance properties of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ are obtained. Faces $\text{F}$ of $\text{Ω}{}_{\mathrm{n}}$ for which G($\text{F}$) is a clique of $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ are investigated.     

A criterion for the existence of common invariant subspaces of matrices

It is shown that square matrices $\text{A}$ and $\text{B}$ have a common invariant subspace W of dimension $\text{k⩾1}$ if and only if for some scalar s, $\text{A+sI}$ and $\text{B+sI}$ are invertible and their kth compounds have a common eigenvector, which is a Grassmann representative for $\text{W}$. The applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated.     

ALPS: Matrices with nonpositive off-diagonal entries

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices having nonpositive off-diagonal entries. The major classes considered are the sets of diagonally stable, stable, and semipositive matrices, denoted respectively by $\text{A}$, $\text{L}$, and $\text{S}$. The conditions defining these classes are weakened, and the resulting classes are examined. Their relationship to the classes of real matrices $\text{P}$ and $\text{P}$₀, whose off-diagonal entries are nonpositive and whose principal minors are respectively all positive and all nonnegative, is also included.     

Natural quotients on split supercotangent bundles and their canonical supersymplectic structures

The Batchelor model of the supercotangent bundle of a given base supermanifold is studied. Under the assumption that the supercotangent bundle splits, two different fibrations over the given base can be globally defined. The total spaces of these fibrations are in turn quotient supermanifolds of the supercotangent bundle, and each of them is equipped with a supersymplectic structure. Their corresponding supersymplectic $\text{2}$-forms are actually exact, and $\text{Z}{}_2$-homogeneous of different degrees. The $\text{Z}{}_2$-homogeneous $\text{1}$-forms from which they come from are natural with respect to Batchelor trivializations. Each of these $\text{1}$-forms can be pulled back to the supercotangent bundle via the quotient maps, and can be added together in the supercotangent bundle to produce a nonhomogeneous $\text{1}$-form there. Such a $\text{1}$-form in the supercotangent bundle is canonical; it is characterized by the fact that the pullback of it under any $\text{1}$-form on the base supermanifold yields the same $\text{1}$-form on the base. The exterior derivative of this canonical $\text{1}$-form is degenerate. Its radical produces an example of an involutive subsheaf, which is not integrable. This phenomenon is explained at the light of Frobenius Theorem for supermanifolds. The radicals of its homogeneous components, on the other hand, taken separately, do produce two globally defined foliations on the supercotangent bundle, and the corresponding spaces of leaves are precisely the two quotients of the supercotangent bundle we started with     

On the number of unlabeled vertices in edge-friendly labelings of graphs

Let $G$ be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, and $f$ be a $0?1$ labeling of $E\left(G\right)$ so that the absolute difference in the number of edges labeled $1$ and $0$ is no more than one. Call such a labeling $f$ edge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident to more edges labeled $1$ than $0$, are labeled $1$, and vertices which are incident to more edges labeled $0$ than $1$, are labeled $0$. Vertices that are incident to an equal number of edges of both labels we call unlabeled. Call a procedure on a labeled graph a label switching algorithm if it consists of pairwise switches of labels. Given an edge-friendly labeling of $K_n$, we show a label switching algorithm producing an edge-friendly relabeling of $K_n$ such that all the vertices are labeled. We call such a labeling opinionated.     

Self-adjoint extensions of the Laplace operator with respect to electric and magnetic boundary conditions

Systems of the type {$\text{A, KJ, 2iK1, I}$} are studied where $\text{J  J1}$ is unitary and $\text{KJK1 }\text{Im}\text{A}$. A complete realization theory as well as state space isomorphism theorem are given. Coupling problems are considered.     

Minimal codes in abelian group algebras

In this paper we show that two minimal codes $\text{M}$₁ and $\text{M}$₂ in the group algebra $\text{F}$₂[G] have the same (Hamming) weight distribution if and only if there exists an automorphism θ of G whose linear extension to $\text{F}$₂[G] maps $\text{M}$₁ onto $\text{M}$₂. If θ(M₁) = M₂, then $\text{M}$₁ and $\text{M}$₂ are called equivalent. We also show that there are exactly τ(l) inequivalent minimal codes in $\text{F}$₂[G], where ? is the exponent of G, and τ(?) is the number of divisors of ?.     

Kernels by properly colored paths in arc-colored digraphs

A kernel by properly colored paths of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for digraphs with no intersecting cycles, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.     

Isometries of norm ideals of compact operators

Let $\text{J}$ be a symmetric norm ideal of compact operators on Hilbert space $\text{H}$, and assume that the finite rank operators are dense in $\text{J}$ and that $\text{J}$ is not the ideal of Hilbert-Schmidt operators. A linear transformation τ on $\text{J}$ is an isometry of $\text{J}$ onto itself if and only if there are unitary operators U and V on $\text{H}$ such that either τ(X) = UXV or τ(X) = UX^tV, where X^t denotes the transpose of X with respect to a fixed orthonormal basis of $\text{H}$.