An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

On extendibility of unavoidable sets

A subset X of a free monoid A¹ is said to be unavoidable if all but finitely many words in A¹ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ? X and a ? A such that (X ? {x}) ∪ {xa} is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.     

A self-normalized invariance principle for a ϕ-mixing sequence

In this paper, we investigate a self-normalized invariance principle for a ?-mixing stationary sequence {X _j , j ≥ 1} of random variables such that L(x):= E(X ₁ ² I{|X ₁| ≤ x}) is a slowly varying function at ∞.     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

Tournois infinis et critiques

Given a tournament T=(V,A), a subset X of V is an interval of T provided that for every a,b∈X and x∈V?X, (a,x)∈A if and only if (b,x)∈A. For example, ?, {x} (x∈V) and V are intervals of T, called trivial intervals. A tournament all the intervals of which are trivial is called indecomposable; otherwise, it is decomposable. An indecomposable tournament T=(V,A) is then said to be critical if for each x∈V, T(V?{x}) is decomposable and if there are x≠y∈V such that T(V?{x,y}) is indecomposable. We introduce the operation of expansion which allows us to describe a process of construction of critical and infinite tournaments. It follows that, for every critical and infinite tournament T=(V,A), there are x≠y∈V such that T and T(V?{x,y}) are isomorphic. To cite this article: I. Boudabbous, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d _X ) and (Y,d _Y ) be pointed compact metric spaces with distinguished base points e _X and e _Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e _X to 0 is denoted by Lip₀(X). The peripheral range of a function f ∈ Lip₀(X) is the set Ran_µ(f) = {f(x): |f(x)| = ‖f‖_∞} of range values of maximum modulus. We prove that if T ₁, T ₂: Lip₀(X) → Lip₀(Y) and S ₁, S ₂: Lip₀(X) → Lip₀(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip₀(X), then there are mappings φ₁φ₂: Y → $\mathbb{K}$ with φ₁(y)φ₂(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T _j (f)(y) = φ _j (y)S _j (f)(ψ(y)) for all f ∈ Lip₀(X), y ∈ Y, and j = 1, 2. In particular, if S ₁ and S ₂ are identity functions, then T ₁ and T ₂ are weighted composition operators.     

Intersection properties of finite sets

Let X₁, X₂, …, X_m be finite sets. The present paper is concerned with the m² ? m intersection numbers |X_i ∩ X_j| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T₁, T₂, …, T_m be finite sets and let m ? 3. We assume that each of the elements in the set union T₁ ∪ T₂ ∪ … ∪ T_m occurs in at least two of the subsets T₁, T₂, …, T_m. We further assume that every pair of sets T_i and T_j (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set T_k (k ≠ i, k ≠ j) such that T_k intersects both T_i and T_j. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Orthogonalities in linear spaces and difference operators

Summary. Quite recently C. Alsina, P. Cruells and M. S. Tomás [2], motivated by F. Suzuki's property of isosceles trapezoids, have proposed the following orthogonality relation in a real normed linear space (X, ||·||) (X, \Vert \cdot \Vert) : two vectors x,y ? X x,y \in X are T-orthogonal whenever¶||z-x ||² + ||z-y ||² = ||z ||² + ||z-(x+y) ||² \Vert z-x \Vert^2 + \Vert z-y \Vert^2 = \Vert z \Vert^2 + \Vert z-(x+y) \Vert^2 ¶for every z ? X z \in X . A natural question arises whether an analogue of T-orthogonality may be defined in any real linear space (without a norm structure). Our proposal reads as follows. Given a functional j \varphi on a real linear space X we say that two vectors x,y ? X x,y \in X are j \varphi -orthogonal (and write x^_jy x\perp_{\varphi}y ) provided that D_x,yj = 0 \Delta_{x,y}\varphi = 0 (D_h1,h2 \Delta_{h_1,h_2} stands here and in the sequel for the superposition D_h1 °D_h2 \Delta_{h_1} \circ \Delta_{h_2} of the usual difference operators).¶We are looking for necessary and/or sufficient conditions upon the functional j \varphi to generate a j \varphi -orthogonality such that the pair X,^_j X,\perp_{\varphi} forms an orthogonality space in the sense of J. Rätz (cf. [6]). Two new characterizations of inner product spaces as well as a generalization of some results obtained in [2] are presented.     

Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.      14. Linear Maps Preserving Operators of Local Spectral Radius Zero      Constantin Costara 《Integral Equations and Operator Theory》2012,73(1):7-16 Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let rT(x) = limsupn ? ￥ || Tnx|| 1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r T (x) = 0 if and only if rj(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r T (By) = 0 if and only if rj( T) (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.      15. Linear and nonlinear methods of relief approximation      K. I. Oskolkov 《Journal of Mathematical Sciences》2008,155(1):129-152 In this paper, we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ?(x) in the metric {ie129-03}, where {ie129-04} is the unit ball |x| ≤ 1 in the plane ?2. The notation we use is the following: {fx129-01}. Here {ie129-05} is the set of all N-term linear combinations of functions of the plane-wave type {fx129-02} with arbitrary profiles W j (x), x ∈ ?1 and transmission directions {θ j } 1 N ; {ie129-06} is the subset of {ie129-07} associated with N equidistant directions; {fx129-03} denotes the subspace of algebraic polynomials of degree less than or equal to N ? 1 in two real variables. Obviously, the inequalities {ie129-08} hold.We state the following model problem. What are the functions which satisfy the relation {ie129-09}, i.e., where the nonlinear approximation {ie129-10} is more effective than a linear one? This effect has been proved for harmonic functions, namely, for any ε > 0 there exists c ε > 0 such that if Δ?(x) = 0, |x| < 1, and ? ∈ {ie129-11}, then {fx129-04}. On the other hand, {ie129-12}. Thus, {ie129-13} has an “almost squared effectiveness” of {ie129-14} for ? = ?harm. However, this ultra-high order of approximation is obtained via a collapse of wave vectors.On the other hand, the nonlinearity of {ie129-15} which corresponds to the freedom of choice of wave vectors does not much improve the order of approximation, for instance, for all the radial functions. If {ie129-16}, then {ie129-17} and {ie129-18}.The technique we use is the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on {ie129-19}) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov-Nikolskii [14] for trigonometric polynomial classes.      16. Solution of Functional Equations Related to Elliptic Functions      A. A. Illarionov 《Proceedings of the Steklov Institute of Mathematics》2017,299(1):96-108 Functional equations of the form f(x + y)g(x ? y) = Σ j=1 n α j (x)β j (y) as well as of the form f1(x + z)f2(y + z)f3(x + y ? z) = Σ j=1 m φ j (x, y)ψ j (z) are solved for unknown entire functions f, g,α j , β j : ? → ? and f1, f2, f3, ψ j : ? → ?, φ j : ?2 → ? in the cases of n = 3 and m = 4.      17. Two remarks on normality preserving Borel automorphisms of \boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}      K R PARTHASARATHY 《Proceedings Mathematical Sciences》2013,123(1):75-84 Let T be a bijective map on ? n such that both T and T ???1 are Borel measurable. For any θ?∈?? n and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? n with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ j ???θ 0, 1?≤?j?≤?n } is a basis for ? n . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε j ?>???1 for every j and {u j , 1?≤?j?≤?n } is a basis of unit vectors in ? n with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u j . Suppose N(0, I)T ???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 n diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E s } is a collection of 2 n Borel subsets of ? n such that { E s } and {V s U (E s )} are partitions of ? n modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).      18. Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality      《Journal of multivariate analysis》1986,19(1):88-112 A complete comparison is made between the value V(X1,…, Xn) = sup{EXt: t is a stop rule for X1,…,Xn} and E(maxj ≤ nXj) for all uniformly bounded sequences of i.i.d. random variables X1, …, Xn. Specifically, the set of ordered pairs {(x,y): x = V(X1, …, Xn) and y = E(maxj≤nXj) for some i.i.d.r.v.'s X1,…, Xn taking values in [0, 1]} is precisely the set {(x, y): x≤y≤Γn(x); 0 ≤x≤1}, where the upper boundary function Γn is given in terms of recursively defined functions. The result yields families of inequalities for the “prophet” problem, relating the motal's value of a game V(X1, …, Xn) to the prophet's value of the game E(maxj≤nXj). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the “prophet” regions and inequalities is also given.      19. Legendrian Links, Causality, and the Low Conjecture      Vladimir Chernov  Stefan Nemirovski 《Geometric And Functional Analysis》2010,19(5):1320-1333 Let (X m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of \mathbbRm{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres \mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if \mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in \mathbbRm{\mathbb{R}^{m}} .      20. On the P4-structure of perfect graphs III. Partner decompositions      《Journal of Combinatorial Theory, Series B》1987,43(3):349-353 Vertices x and y in the same graph are called partners if there is a set S of three vertices such that both S ∪ {x} and S ∪ {y} induce a chordless path. If the vertices of G can be distributed into two disjoint classes in such a way that every two partners belong to the same class and that each class induces in G a perfect graph then G is perfect. This theorem generalizes two theorems that were proved before, one by Chvátal and Hoàng, and the other by Hoàng alone.

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Linear and nonlinear methods of relief approximation

In this paper, we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ?(x) in the metric {ie129-03}, where {ie129-04} is the unit ball |x| ≤ 1 in the plane ?². The notation we use is the following: {fx129-01}. Here {ie129-05} is the set of all N-term linear combinations of functions of the plane-wave type {fx129-02} with arbitrary profiles W _j (x), x ∈ ?¹ and transmission directions {θ _j } ₁ ^N ; {ie129-06} is the subset of {ie129-07} associated with N equidistant directions; {fx129-03} denotes the subspace of algebraic polynomials of degree less than or equal to N ? 1 in two real variables. Obviously, the inequalities {ie129-08} hold.We state the following model problem. What are the functions which satisfy the relation {ie129-09}, i.e., where the nonlinear approximation {ie129-10} is more effective than a linear one? This effect has been proved for harmonic functions, namely, for any ε > 0 there exists c _ε > 0 such that if Δ?(x) = 0, |x| < 1, and ? ∈ {ie129-11}, then {fx129-04}. On the other hand, {ie129-12}. Thus, {ie129-13} has an “almost squared effectiveness” of {ie129-14} for ? = ?_harm. However, this ultra-high order of approximation is obtained via a collapse of wave vectors.On the other hand, the nonlinearity of {ie129-15} which corresponds to the freedom of choice of wave vectors does not much improve the order of approximation, for instance, for all the radial functions. If {ie129-16}, then {ie129-17} and {ie129-18}.The technique we use is the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on {ie129-19}) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov-Nikolskii [14] for trigonometric polynomial classes.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Two remarks on normality preserving Borel automorphisms of \boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}

Let T be a bijective map on ? ⁿ such that both T and T ^???1 are Borel measurable. For any θ?∈?? ⁿ and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? ⁿ with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ _j ???θ ₀, 1?≤?j?≤?n } is a basis for ? ⁿ . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε _j ?>???1 for every j and {u _j , 1?≤?j?≤?n } is a basis of unit vectors in ? ⁿ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u _j . Suppose N(0, I)T ^???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 ⁿ diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E _s } is a collection of 2 ⁿ Borel subsets of ? ⁿ such that { E _s } and {V s U (E _s )} are partitions of ? ⁿ modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E _s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).     

Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality

A complete comparison is made between the value V(X₁,…, X_n) = sup{EX_t: t is a stop rule for X₁,…,X_n} and E(max_{j ≤ n}X_j) for all uniformly bounded sequences of i.i.d. random variables X₁, …, X_n. Specifically, the set of ordered pairs {(x,y): x = V(X₁, …, X_n) and y = E(max_j≤nX_j) for some i.i.d.r.v.'s X₁,…, X_n taking values in [0, 1]} is precisely the set {(x, y): x≤y≤Γ_n(x); 0 ≤x≤1}, where the upper boundary function Γ_n is given in terms of recursively defined functions. The result yields families of inequalities for the “prophet” problem, relating the motal's value of a game V(X₁, …, X_n) to the prophet's value of the game E(max_j≤nX_j). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the “prophet” regions and inequalities is also given.     

Legendrian Links, Causality, and the Low Conjecture

Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of \mathbbR^m{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X ^m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in \mathbbR^m{\mathbb{R}^{m}} .     

On the P4-structure of perfect graphs III. Partner decompositions

Vertices x and y in the same graph are called partners if there is a set S of three vertices such that both S ∪ {x} and S ∪ {y} induce a chordless path. If the vertices of G can be distributed into two disjoint classes in such a way that every two partners belong to the same class and that each class induces in G a perfect graph then G is perfect. This theorem generalizes two theorems that were proved before, one by Chvátal and Hoàng, and the other by Hoàng alone.