Pseudo-Index of Fano Manifolds and Smooth Blow-Ups

Suppose π: X → Y is a smooth blow-up along a submanifold Z of Y between complex Fano manifolds X and Y of pseudo-indices i_X and i_Y respectively (recall that i_X is defined by i_X :=min {−K_X·C | C is a rational curve of X}). We prove that if 2 dim (Z) < dim (Y)+i_Y −1 and show that this result is optimal by classifying the ‘boundary’ cases. As expected, these results are obtained by studying rational curves on X and Y.     

Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A⁻¹X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A⁻¹X| where X is n × k matrix such that X′X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A⁻¹X, X′B²X(X′BCX)⁻¹ X′C²X(X′BCX)⁻¹ where X and Y are n × k matrices such that X′X = Y′Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.     

Complemented copies of ℓ<Subscript>p</Subscript> spaces in tensor products

We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗_π Y, their injective tensor product X ⊗_ɛ Y, or the dual (X ⊗_π Y)^* contain complemented copies of ℓ_p.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

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Also some applications of the above mentioned result will be presented.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

An ergodic Markov chain is not determined by any p-marginals

We prove that if X = (X_n)_nεZ is a finite state space ergodic Markov chain, then for any natural number p, there exist ergodic non-Markovian processes Y = (Y_n)_nεZ with positive entropy, such that for all integers n₁, …, n_p, the joint distribution of Y_n1, …, Y_np is identical to the joint distribution of X_n1, …, X_np.     

Global nonparametric estimation of conditional quantile functions and their derivatives

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (p − m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X₁, Y₁), …, (X_n, Y_n), that achieves the optimal nonparametric rate of convergence n^−r in L_q-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)^−r in L_∞-norm restricted to compacts.     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Consistency of a recursive nearest neighbor regression function estimate

Let (X, Y) be an ^d × -valued random vector and let (X₁, Y₁),…,(X_N, Y_N) be a random sample drawn from its distribution. Divide the data sequence into disjoint blocks of length l₁, …, l_n, find the nearest neighbor to X in each block and call the corresponding couple (X_i^*, Y_i^*). It is shown that the estimate m_n(X) = Σ_{i = 1}ⁿ w_niY_i^*/Σ_{i = 1}ⁿ w_ni of m(X) = E{Y|X} satisfies E{|m_n(X) − m(X)|^p} 0 (p ≥ 1) whenever E{|Y|^p} < ∞, l_n ∞, and the triangular array of positive weights {w_ni} satisfies sup_{i ≤ n}w_ni/Σ_{i = 1}ⁿ w_ni 0. No other restrictions are put on the distribution of (X, Y). Also, some distribution-free results for the strong convergence of E{|m_n(X) − m(X)|^p|X₁, Y₁,…, X_N, Y_N} to zero are included. Finally, an application to the discrimination problem is considered, and a discrimination rule is exhibited and shown to be strongly Bayes risk consistent for all distributions.     

Comparaison des deux composantes d''un subordinateur bivarié, puis étude de l''enveloppe supérieure d''un processus de Lévy

Soit (Y,Z) un subordinateur bivarié. Nous donnons une condition suffisante pour que Y_t/Z_t converge vers zéro quand t tend vers 0 ou +∞. Ceci généralise partiellement des résultats de Bertoin et de Kesten–Erickson.Soit X un processus de Lévy et S_t=sup{X_s: st}. Soit f une fonction sous-additive. En appliquant le résultat précédent au subordinateur bivarié d'échelle, nous donnons des conditions nécéssaires et suffisantes pour que et égalent 0 ou +∞.Let (Y,Z) be a bivariate subordinator. Generalizing theorems of Bertoin and Kesten–Erickson, we give a sufficient condition for Y_t/Z_t to converge to 0 when t tends either to 0 or +∞.Let X be a Lévy process. Denote by S_t=sup{X_s: st} and let f be any sub-additive function. Applying our first result to the bivariate ladder process, we give necessary and sufficient conditions for and to be either 0 or +∞.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Exponential bounds of mean error for the kernel estimates of regression functions

Let (X, Y), (X₁, Y₁), …, (X_n, Y_n) be i.d.d. R^r × R-valued random vectors with E|Y| < ∞, and let Q_n(x) be a kernel estimate of the regression function Q(x) = E(Y|X = x). In this paper, we establish an exponential bound of the mean deviation between Q_n(x) and Q(x) given the training sample Zⁿ = (X₁, Y₁, …, X_n, Y_n), under conditions as weak as possible.     

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.     

Ordered Compactifications of Products of Two Totally Ordered Spaces

We describe the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y where X and Y are certain totally ordered topological spaces, and where _o Z denotes the Stone–ech ordered- or Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y for arbitrary totally ordered topological spaces X and Y. Such products X × Y provide many counterexamples in the theory of ordered compactifications.     

On the fluctuations of independent copies of moving maxima

Let {X_n, n1} be a sequence of independent random variables (r.v.'s) with a common distribution function (d.f.) F. Define the moving maxima Y_k(n)=max(X_n−k(n)+1,X_n−k(n)+2,…,X_n), where {k(n), n1} is a sequence of positive integers. Let Y_k(n)¹ and Y_k(n)² be two independent copies of Y_k(n). Under certain conditions on F and k(n), the set of almost sure limit points of the vector consisting of properly normalised Y_k(n)¹ and Y_k(n)² is obtained.     

A Remark on Self-Normalization for Dependent Random Variables

Let (X _t , t ∈ Z) be a stationary process, and let S _n = ∑_{1⩽ i⩽n} X _i . In this paper, we consider the central limit theorem for the self-normalized sequence S _n /U _n , where U _n ² = ∑_1⩽j⩽N Y _j ² , Y _j = ∑_{(j−1)m<i⩽jm} X _i , n = mN. We show how such a self-normalization works for AR(1) and MA(q) processes.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 173–183, April–June, 2005.     

Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité

We prove Malgrange’s conjecture on the absence of confluence phenomena for integrable meromorphic connections. More precisely, if Y→X is a complex-analytic fibration by smooth curves, Z a hypersurface of Y finite over X, and ∇ an integrable meromorphic connection on Y with poles along Z, then the function which attaches to x∈X the sum of the irregularities of the fiber ∇_(x) at the points of Z _x is lower semicontinuous. The proof relies upon a study of the formal structure of integrable meromorphic connections in several variables.     

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗̌_ɛ Y be the injective tensor product of X and Y.

Weak local homeomorphisms and <Emphasis Type="Italic">B</Emphasis>-favorable spaces

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f ⁻¹ (G) of a set G open in Y is an F _σ-set in X can be represented in the form of the pointwise limit of continuous mappings f _n : X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.