Monitoring process variability using auxiliary information

In this study a Shewhart type control chart namely V _r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V _r chart is developed and its comparison is made with the well-known Shewhart control chart namely S ² chart used for the same purpose. Using power curves as a performance measure it is observed that V _r chart outperforms the S ² chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ _yx . These efficiency conditions on ρ _yx are also obtained for V _r chart in this study.     

Testing the tail–dependence based on the radial component

Throughout this article we assume that the df H of a random vector (X,Y) is in the max-domain of attraction of an extreme value distribution function (df) G with reverse exponential margins. Therefore, the asymptotic dependence structure of H can be represented by a Pickands dependence function D with D = 1 representing the case of asymptotic independence. One of our aims is to test the null hypothesis of tail-dependence against the alternative of tail-independence. Thus we want to prove the validity of the model where D = 1. The test is based on the radial component X + Y. Under a certain spectral expansion it is verified that the df of X + Y, conditioned on X + Y > c, converges to F(t) = t, as c ↑0, if D ≠ 1 and, respectively, to F(t) = t ^1 + ρ , if D = 1, where ρ > 0 determines the rate at which independence is attained. Based on the limiting dfs we find a uniformly most powerful test procedure for testing tail-dependence against rates of tail-independence. In addition, an estimator of the parameter ρ is proposed. The relationship of ρ to another dependence measure, given in the literature, is indicated.      

On upper and lower D*-supercontinuous multifunctions

We define a multifunction F: X ⇝ Y to be upper (lower) D*-supercontinuous if F ⁺(V) (F ⁻(V)) is d*-open in X for every open set V of Y. We obtain some characterizations and several properties concerning upper (lower) D*-supercontinuous multifunctions.      

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R ^k →Y is a C ²-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R ¹ that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ₀)∈X×R ^k and we describe the solution set of the studied equation in a small neighbourhood of this point.     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed&gt;1 and {0} if d&gt;1.Let Bcd be a cd-homogeneous C^*-algebra over S^2&#215;T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2&#215;Z^m-2) in Bcd &#215;C^*(Z^m-3) by a totally skew multiplier ρ on T^2&#215;Z^m-2。It is shown that Sρ^cd&#215;Mρ∞ is isomorphic to C(S^2)&#215;C^*(T^2&#215;Z^m-2,ρ）&#215; Mcd(C)&#215;Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

Let X,X ₁,X ₂,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S _[nt], 0 ≦ t ≦ 1} where S _n = Σ _j=1 ⁿ X _j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S _[nt]=V _n , 0 ≦ t ≦ 1}, where V _n ² = Σ _j=1 ⁿ X _j ² . L _p approximations of self-normalized partial sum processes are also discussed.     

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.     

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

Blow up of balls and coverings in metric spaces

We obtain some refinements and extensions of the Basic Covering Theorem in a metric space (X, ρ). The properties of the metric ρ are used to define an inclusion coefficient k in this theorem, and this is related to the supremum of numbers t such that ρ ^t is a metric in X. The inclusion coefficient k characterizes ultrametric spaces.     

Campedelli surfaces with fundamental group of order 8

Let S be a Campedelli surface (a minimal surface of general type with p _g  = 0, K ² = 2), and an etale cover of degree 8. We prove that the canonical model of Y is a complete intersection of four quadrics . As a consequence, Y is the universal cover of S, the covering group G = Gal(Y/S) is the topological fundamental group π ₁ S and G cannot be the dihedral group D ₄ of order 8. The first author is a member of the Centre for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Lisboa. The second is a member of G.N.S.A.G.A.–I.N.d.A.M.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

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 +∞.     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

Convergence in law of partial sum processes in p-variation norm

Let X ₁, X ₂, … be a sequence of independent identically distributed real-valued random variables, S _n be the nth partial sum process S _n (t) ≔ X ₁ + ⋯ X _⌊tn⌋, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n ^−1/2 S _n converges in law to σW as n → ∞ in p-variation norm if and only if EX ₁ = 0 and σ ² = EX ₁² < ∞. The result is applied to test the stability of a regression model. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-21/07     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .