On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

For a sample of iid observations {(X_i, Y_i)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y_[]=(Y_[1], Y_[2], …, Y_[n]) and the stochastic order of subsets of Y_[] are studied. If (X, Y) is totally positive dependent of order 2, Y_[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, h_YX(yx), is decreasing in x for every y, Y_[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order.     

A Kernel Estimator of a Conditional Quantile

Let (X₁, Y₁), (X₂, Y₂), …, be two-dimensional random vectors which are independent and distributed as (X, Y). For 0<p<1, letξ(px) be the conditionalpth quantile ofYgivenX=x; that is,ξ(px)=inf{y : P(YyX=x)p}. We consider the problem of estimatingξ(px) from the data (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n). In this paper, a new kernel estimator ofξ(px) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained.     

On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

On the conditional probability density functions of multivariate uniform random vectors and multivariate normal random vectors

It is shown that the conditional probability density function of Y₁ given (1/n) Σ_i=1ⁿ Y_i=1Y_i^t = Σ, where Y₁, Y₂,…, Y_n are i.i.d, p-variate uniform random vectors with mean 0 equals to that of Y₁ given (1/n) Σ_i=1ⁿ Y_iY_i^t,…, Y_n are i.i.d, p-variate normal random vectors with mean 0 and covariance matrix Σ.     

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T ^*(J ^(r,s,q)(Y, R ^1,1)₀)^*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T ^*(J ^(r,s) (Y, R)₀)^*→R for any Y as above.     

Peirce gradings and Peirce inner ideals in JBW*-triple factors

A Peirce inner ideal J in an anisotropic Jordan*-triple A gives rise to a Peirce grading (J ₀, J ₁, J ₂) of A by defining

Spreads, arcs, and multiple wavelength codes

We present several new families of multiple wavelength (2-dimensional) optical orthogonal codes (2D-OOCs) with ideal auto-correlation λ_a=0 (codes with at most one pulse per wavelength). We also provide a construction which yields multiple weight codes. All of our constructions produce codes that are either optimal with respect to the Johnson bound (J-optimal), or are asymptotically optimal and maximal. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q).     

Explicit constructions of separating hash families from algebraic curves over finite fields

Let X be a set of order n and Y be a set of order m. An (n,m,{w ₁, w ₂})-separating hash family is a set of N functions from X to Y such that for any with , |X ₁| = w ₁ and |X ₂| = w ₂, there exists an element such that . In this paper, we provide explicit constructions of separating hash families using algebraic curves over finite fields. In particular, applying the Garcia–Stichtenoth curves, we obtain an infinite class of explicitly constructed (n,m,{w ₁,w ₂})–separating hash families with for fixed m, w ₁, and w ₂. Similar results for strong separating hash families are also obtained. As consequences of our main results, we present explicit constructions of infinite classes of frameproof codes, secure frameproof codes and identifiable parent property codes with length where n is the size of the codes. In fact, all the above explicit constructions of hash families and codes provide the best asymptotic behavior achieving the bound , which substantially improve the results in [ 8, 15, 17] give an answer to the fifth open problem presented in [11].     

Any two irreducible Markov chains are finitarily orbit equivalent

Two invertible dynamical systems (X, gA, μ, T) and (Y, ℬ, ν, S), where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X ₀ of X of full measure to a subset Y ₀ of Y of full measure such that ϕ|x ₀ is continuous in the relative topology on X ₀, ϕ ⁻¹|Y ₀ is continuous in the relative topology on Y ₀ and ϕ(Orb _T (x)) = Orb _Sϕ (x) for μ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains.     

Generically finite morphisms and formal neighborhoods of arcs

Let f : X → Y be a morphism of pure-dimensional schemes of the same dimension, with X smooth. We prove that if is an arc on X having finite order e along the ramification subscheme R _f of X, and if its image δ = f _∞(γ) on Y does not lie in J _∞(Y _sing), then the induced map T _γ J _∞(X) → T _δ J _∞(Y) is injective, with a cokernel of dimension e. In particular, if Y is smooth too, and if we denote by and the formal neighborhoods of and , then the induced morphism is a closed embedding of codimension e.      

Prophet regions and sharp inequalities for pth absolute moments of martingales

Exact comparisons are made relating E|Y₀|^p, E|Y_n−1|^p, and E(max_j≤n−1 |Y_j|^p), valid for all martingales Y₀,…,Y_n−1, for each p ≥ 1. Specifically, for p > 1, the set of ordered triples {(x, y, z) : X = E|Y₀|^p, Y = E |Y_n−1|^p, and Z = E(max_j≤n−1 |Y_j|^p) for some martingale Y₀,…,Y_n−1} is precisely the set {(x, y, z) : 0≤x≤y≤z≤Ψ_n,p(x, y)}, where Ψ_n,p(x, y) = xψ_n,p(y/x) if x > 0, and = a_n−1,py if x = 0; here ψ_n,p is a specific recursively defined function. The result yields families of sharp inequalities, such as E(max_j≤n−1 |Y_j|^p) + ψ_n,p^*(a) E |Y₀|^p ≤ aE |Y_n−1|^p, valid for all martingales Y₀,…,Y_n−1, where ψ_n,p^* is the concave conjugate function of ψ_n,p. Both the finite sequence and infinite sequence cases are developed. Proofs utilize moment theory, induction, conjugate function theory, and functional equation analysis.     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.     

Chinese remainder codes

Chinese remainder codes are constructed by applying weak block designs and the Chinese remainder theorem of ring theory. The new type of linear codes take the congruence class in the congruence class ring R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n for the information bit, embed R/J _i into R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n, and assign the cosets of R/J _i as the subring of R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n and the cosets of R/J _i in R/I ₁ ∩ I ₂ ∩ ··· ∩ I _n as check lines. Many code classes exist in the Chinese remainder codes that have high code rates. Chinese remainder codes are the essential generalization of Sun Zi codes. Selected from Journal of Mathematical Research and Exposition, 2004, 24(2): 347–352     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Twisting geometric codes

The aim of this paper is to explain how, starting from a Goppa code C(X,G,P₁,…,P_n) and a cyclic covering π:Y→X of degree m, one can twist the initial code to another one C(X,G+D_χ,P₁,…,P_n), where D_χ is a non-principal degree 0 divisor on X associated to a character χ of Gal(Y/X), in the hope that ℓ_X(G+D_χ)>ℓ_X(G). We give, using a MAGMA program, several examples where this occurs, and where both the initial and twisted codes have same minimum distance, so that initial codes have been improved.     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

Special moments

In this article, we show that a linear combination of n independent, unbiased Bernoulli random variables {X_k} can match the first 2n moments of a random variable Y which is uniform on an interval. More generally, for each p2, each X_k can be uniform on an arithmetic progression of length p. All values of lie in the range of Y, and their ordering as real numbers coincides with dictionary order on the vector (X₁,…,X_n). The construction involves the roots of truncated q-exponential series. It applies to a construction in numerical cubature using error-correcting codes [G. Kuperberg, Numerical cubature using error-correcting codes, arXiv:math.NA/0402047]. For example, when n=2 and p=2, the values of are the 4-point Chebyshev quadrature formula.     

The Grothendieck ring of varieties and piecewise isomorphisms

Let K ₀(Var _k ) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X ₁, ..., X _n , Y ₁, ..., Y _n into locally closed subvarieties such that X _i is isomorphic to Y _i for all i ≤ n), then [X] = [Y] in K ₀(Var _k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.