Verdier specialization via weak factorization

Let X?V be a closed embedding, with V?X nonsingular. We define a constructible function ψ _X,V on X, agreeing with Verdier’s specialization of the constant function 1 _V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–W?odarczyk. The main property of ψ _X,V is a compatibility with the specialization of the Chern class of the complement V?X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart Ψ _X,V in a motivic group. The function ψ _X,V and the corresponding Chern class c _SM(ψ _X,V ) and motivic aspect Ψ _X,V all have natural ‘monodromy’ decompositions, for any X?V as above. The definition also yields an expression for Kai Behrend’s constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.     

A comparison theorem for l-adic cohomology

We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F_n)_n on, one has . As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X^rig, the l-adic cohomology of X and X^rig agree.     

Exact distribution of MLE of covariance matrix in a GMANOVA-MANOVA model

For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T B2Z2 T E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.     

Marcinkiewicz–Zygmund Type Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

Etemadi (in Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122, 1981) proved that the Kolmogorov strong law of large numbers holds for pairwise independent identically distributed (pairwise i.i.d.) random variables. However, it is not known yet whether the Marcinkiewicz–Zygmund strong law of large numbers holds for pairwise i.i.d. random variables. In this paper, we obtain the Marcinkiewicz–Zygmund type strong law of large numbers for pairwise i.i.d. random variables {X _n ,n≥1} under the moment condition E|X ₁| ^p (loglog|X ₁|)^2(p?1)<∞, where 1<p<2.     

Perturbation of spectra for a class of 2×2 operator matrices

In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M _X = ( _{0 B} ^{A X} ) and M _Z = ( _{Z B} ^{A C} ) on the Hilbert space H ?? K and the sets $\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} $ and $\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} $ , where R(C) is a closed subspace, are characterized     

Quasikonvexe Multiplikatoren starker Konvergenz

пУсть {f _k; f _k ^* ?X×X^* — пОлНАь БИОРтОгОНАльНАь сИс тЕМА В БАНАхОВОМ пРОстРАН стВЕ X (X^* — сОпРьжЕННОЕ пРОст РАНстВО). пУсть (?→+0) $$\begin{gathered} S_n f = \sum\limits_{k = 0}^n {f_k^* (f)f_k ,} K(f,t) = \mathop {\inf }\limits_{g \in Z} (\left\| {f - g} \right\|_x + t\left| g \right|_z ), \hfill \\ X_0 = \{ f \in X:\mathop {\lim }\limits_{n \to \infty } \left\| {S_n f - f} \right\|_x = 0\} ,X_\omega = \{ f \in X:K(f,t) = 0(\omega (t))\} , \hfill \\ \end{gathered} $$ гДЕZ?X — НЕкОтОРОЕ пОД пРОстРАНстВО с пОлУН ОРМОИ ¦·¦ И Ω — МОДУль НЕпРЕРыВНО стИ УДОВлЕтВОРьУЩИИ Усл ОВИУ sup Ω(t)/t=∞. пОслЕДОВАтЕ льНОстьΤ={Τ _k} кОМплЕксНых ЧИ сЕл НАжыВАЕтсь МНОжИтЕл ЕМ сИльНОИ схОДИМОст И ДльX _Τ, жАпИсьΤ?М[X _Τ,X _Τ], ЕслИ Д ль кАжДОгО ЁлЕМЕНтАf?X _Τ сУЩЕстВ УЕт тАкОИ ЁлЕМЕНтf ^τ?х₀, ЧтОf _k ^* (f ^τ)=Τ_kf _k ^* (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {f_k; f _k ^* } —Н ЕкОтОРыИ (с, 1)-БАжИс тАк ОИ, ЧтО ВыпОлНьУтсь НЕРАВЕН стВА тИпА НЕРАВЕНстВА ДжЕ ксОНА с пОРьДкОМ O(?_n) u тИ пА НЕРАВЕНстВА БЕРНшmЕИ НА с пОРьДкОМ O(1/?_n). ЕслИ пОслЕДОВАтЕл ьНОсть Τ кВАжИВыпУкл А И ОгРАНИЧЕНА, тО $$\tau \in M[X_{\omega ,} X_0 ] \Leftrightarrow \omega (\varphi _n )\tau _n \left\| {S_n } \right\|_{[X,X]} = o(1).$$ ЁтОт ОБЩИИ пОДхОД НЕМ ЕДлЕННО ДАЕт клАссИЧ ЕскИЕ РЕжУльтАты, ОтНОсьЩИ Есь к ОДНОМЕРНыМ тРИгОНОМЕтРИЧЕскИМ РьДАМ. НО тЕпЕРь ВОжМО жНы ДАльНЕИшИЕ пРИлОжЕН Иь, НАпРИМЕР, к РАжлОжЕНИьМ пО пОлИ НОМАМ лЕжАНДРА, лАгЕР РА ИлИ ЁРМИтА.     

Iterated vanishing cycles

If A ^? is a bounded, constructible complex of sheaves on a complex analytic space X, and ${f : X \rightarrow \mathbb{C}}$ and ${g : X \rightarrow \mathbb{C}}$ are complex analytic functions, then the iterated vanishing cycles φ _g [?1](φ _f [?1]A ^?) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ _g [?1]φ _f [?1]A ^?) _x in terms of relative polar curves, algebra, and Morse modules of A ^?.     

VC-sets and generic compact domination

Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra technical hypothesis, this also holds if X is constructible. We deduce from this generic compact domination for definably amenable NIP groups.     

On the exponential growth of graded Capelli polynomials

In a free superalgebra over a field of characteristic zero we consider the graded Capelli polynomials Cap _M+1[Y,X] and Cap _L+1[Z,X] alternating on M+1 even variables and L+1 odd variables, respectively. Here we compute the superexponent of the variety of superalgebras determinated by Cap _M+1[Y,X] and Cap _L+1[Z,X]. An essential tool in our computation is the generalized-six-square theorem proved in [3].     

Hyperspaces of direct limits of locally compact metric spaces

For a tower X₁ ⊂ X₂ ⊂ ⋯ of locally compact metric spaces, let X_∞ = ∪^∞₁ X_n denote the direct limit space. We show that the hyperspace 2^X_∞ of nonempty compact subsets of X_∞, with the Vietoris topology, is homeomorphic to the direct limit of the tower of hyperspaces 2^X₁∪2^X₂∪⋯. Consequently, if each X_n is a generalized Peano continuum, with X_n closed and nowhere dense in X_n+1, then 2^X_∞ is homeomorphic to the direct limit of Hilbert cubes.     

Resonance between automorphic forms and exponential functions

Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum Xn 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Diagonals of separately pointwise Lipschitz mappings

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X ²? →?Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X?×?Y?→?Z, where X, Y, and Z are metric spaces, is investigated.     

Effective non-vanishing of global sections of multiple adjoint bundles for polarized 3-folds

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this paper, we provide a lower bound for h⁰(m(K_X+L)) under the assumption that κ(K_X+L)≥0. In particular, we get the following: (1) if 0≤κ(K_X+L)≤2, then h⁰(K_X+L)>0 holds. (2) If κ(K_X+L)=3, then h⁰(2(K_X+L))≥3 holds. Moreover we get a classification of (X,L) with κ(K_X+L)=3 and h⁰(2(K_X+L))=3 or 4.     

On the Stability of One Characterization of Stable Distributions

In 1923, G. Polya proved that if X ₁ and X ₂ are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X ₁ and (X ₁+X ₂)/ $\sqrt 2$ are coincidental iff X ₁ has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X ₁ and (X ₁+X ₂)/2^1/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X ₁ and (X ₁+X ₂)/2^1/α with a similar condition, namely, requiring, in addition, for the distributions of X ₁ and (X ₁+X ₂+X ₃)/3^1/α to be coincident (here X ₁,X ₂ and X ₃ are i.i.d.r.v.), P. Lévy has proved that X ₁ and X ₂ have a strictly stable distribution. The stability of this characterization in a metric λ₀ (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

Un survol de la theorie de l'integrale stochastique

The objective of this paper is to present the principal results of a large part of stochastic calculus in a manner that should be comprehensible to readers having only the general notions of stochastic processes. Not all the theorems are proved in detail, but all the fundamental theorems are explained with clarity and precision, and with special attention to the motivations behind them.Given two real valued stochastic processes X and Y, the basic problem is to give a meaning to Z = ∫ Y dX in such a way that the integral sign is not misused. If X is a process whose paths are of bounded variation, then Z should coincide with the ordinary Lebesgue-Stieltjes integral taken path by path. If Y is a left continuous step function, then Z should coincide with the obvious choice: if Y is constant on ]t, u], then Z_u-Z_t is that constant times X_u-X_t. And finally, the Lebesgue dominated convergence theorem should hold: if the processes Yⁿ converge to Y and all the Yⁿ are dominated by a process Y' for which ∝ Y' dX is well defined, then Z_n = ∫ Yⁿ dX should converge to Z = ∫ Y dX in some sense.Starting with these requirements, it is shown that, if ∫ Y dX is defined for all predictable Y, then X must be a semimartingale. Conversely, the integral is well defined for all predictable Y and all semimartingales X.With the integrals defined, a number of their important properties are discussed. In particular, the integral Z is a semimartingale, and a change of variable formula (Ito's formula) holds for $\text{?(Z)}$. Finally, stochastic integral equations are introduced, and a general theorem is given on the existence and uniqueness of solutions.A bibliography with commentaries supplements the text for the benefit of those who would like to go deeper into the subject.     

Realizing cohomology classes as Euler classes

For a space X, let E _k (X), E _k ^s (X) and E _k ^?? (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in H ^k (X; ?). We prove some general facts about the sets E _k (X), E _k ^s (X) and E _k ^?? (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P(m, 1) and obtain partial computations in the case that X is a product of spheres.     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.