Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic Kähler manifold, we define the symplectic Wirtinger number W(X). We show that and the equality is reached if and only if the subvariety is trianalytic, i.e. compatible with the hyperkähler structure on M. For a sequence of immersions of simple holomorphic symplectic manifolds, we show that      

t-closed rings of formal power series

Let A ì BA\subset B be rings. We say that A is t-closed in B, if for each a ? Aa\in A and b ? Bb\in B such that b²-ab,b³-ab² ? Ab^2-ab,b^3-ab^2\in A, then b ? Ab\in A. We present a sufficient condition for the ring A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] to be t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]]. By an example, we show that our condition is not necessary. Even though the question is still open, some important cases are treated. For example, if A ì BA\subset B is an integral extension, or if A is p-injective, then A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] is t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]] if and only if A is t-closed in B.     

Localization and Locality for Resistance Forms

For a resistance form ${(X, \mathcal{D}(\varepsilon),\varepsilon)}For a resistance form (X, D(e),e){(X, \mathcal{D}(\varepsilon),\varepsilon)} and a point x₀ ? X{x_0 \in X} as boundary, on the space X₀:=X \{x₀}{X_0:=X {\setminus}\{x_0\}} we consider the Dirichlet space D_x0:={f ? D(e) | f(x₀)=0}{\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}} and we develop a good potential theory. For any finely open subset D of X ₀ we consider a localized resistance form (D_{X0 \ D},e_D{\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}) where D_{X0 \ D}:={f ? D_x0 | f=0{\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0} on X₀ \ D}, e_D(f,g):=e(f,g){X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)} for all f,g ? D_{X0 \ D}{f,g\in\mathcal{D}_{X_0 {\setminus} D}}. The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

Lebesgue property for convex risk measures on Orlicz spaces

We present a robust representation theorem for monetary convex risk measures r: X ? \mathbbR{\rho : \mathcal{X} \rightarrow \mathbb{R}} such that

Empirical properties of forecasts with the functional autoregressive model

We study the finite sample performance of predictors in the functional (Hilbertian) autoregressive model X_n+1 = Y(X_n)+e_n{X_{n+1} = \Psi(X_n)+\varepsilon_n}. Our extensive empirical study based on simulated and real data reveals that predictors of the form [^(Y)](X_n){\hat\Psi(X_n)} are practically optimal in a sense that their prediction errors are comparable with those of the infeasible perfect predictor Ψ(X _n ). The predictions [^(Y)](X_n){\hat\Psi(X_n)} cannot be improved by an improved estimation of Ψ, nor by a more refined prediction approach which uses predictive factors rather than the functional principal components. We also discuss the practical limits of predictions that are feasible using the functional autoregressive model. These findings have not been established by theoretical work currently available, and may serve as a practical reference to the properties of predictors of functional data.     

Some convergence theorems on a supercritical Galton-Watson process

Summary LetX=(X _n; n≧0,X ₀=1) be a supercritical Galton-Watson process. The limiting distribution of ) where is the m.l.e. of the offspring mean, is derived. As an application of this result, some limit theorems leading ultimately to a parameter free result of statistical interest, are also established.     

Summation of <Emphasis Type="Italic">p</Emphasis>-Faber series by the Abel–poisson method in the integral metric

We establish conditions on the boundary G \Gamma of a bounded simply connected domain W ì \mathbbC \Omega \subset \mathbb{C} under which the p-Faber series of an arbitrary function from the Smirnov space E_p( W),1 \leqslant p < ￥ {E_p}\left( \Omega \right),1 \leqslant p < \infty , can be summed by the Abel–Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space L_p( G) {L_p}\left( \Gamma \right) .     

Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term

Given a bounded open regular set W ì \mathbbR²{\Omega \subset \mathbb{R}^2} and x₁, x₂, ?, x_m ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem

The Law of Iterated Logarithm of Rescaled Range Statistics for AR（1） Model

Let {Xn,n ≥ 0} be an AR（1） process. Let Q（n） be the rescaled range statistic, or the R/S statistic for {Xn} which is given by （max1≤k≤n（∑j=1^k（Xj - ^-Xn）） - min 1≤k≤n（∑j=1^k（ Xj - ^Xn ））） /（n ^-1∑j=1^n（Xj -^-Xn）^2）^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q（n） for AR（1） model.     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Let W ì \mathbbRⁿ \Omega \subset \mathbb{R}^n be an open set and l(x) | u |_p,l = ( ò_W l^p (x)| u(x) |^p dx )^1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}     

Mass equidistribution of Hilbert modular eigenforms

Let \mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \operatornameGL₂/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight (k₁,?,k_{[\mathbbF:\mathbbQ]})(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k₁,?,k_{[\mathbbF:\mathbbQ]}) ? ￥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty.     

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let W ì \BbbR²\Omega \subset \Bbb{R}^2 denote a bounded domain whose boundary ?W\partial \Omega is Lipschitz and contains a segment G₀\Gamma_0 representing the austenite-twinned martensite interface. We prove inf_{u ? W(W)} ò_W j(?u(x,y))dxdy=0\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Analytic subvarieties with many rational points

We give a generalization of the classical Bombieri–Schneider–Lang criterion in transcendence theory. We give a local notion of LG-germ, which is similar to the notion of E-function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let ${K \subset \mathbb{C}}We give a generalization of the classical Bombieri–Schneider–Lang criterion in transcendence theory. We give a local notion of LG-germ, which is similar to the notion of E-function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let K ì \mathbbC{K \subset \mathbb{C}} be a number field and X a quasi-projective variety defined over K. Let γ : M → X be an holomorphic map of finite order from a parabolic Riemann surface to X such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every p ? X(K)?g(M){p\in X(K)\cap\gamma(M)} the formal germ of M near P is an LG-germ, then we prove that X(K)?g(M){X(K)\cap\gamma(M)} is a finite set. Then we define the notion of conformally parabolic K?hler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact K?hler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that A is conformally parabolic variety of dimension m over \mathbbC{\mathbb{C}} with K?hler form ω and γ : A → X is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then m. Suppose that for every p ? X(K)?g(A){p\in X(K)\cap \gamma (A)} , the image of A is an LG-germ. then we prove that there exists a current T on A of bidegree (1, 1) such that ò_ATùw^m-1{\int_AT\wedge\omega^{m-1}} explicitly bounded and with Lelong number bigger or equal then one on each point in γ ⁻¹(X(K)). In particular if A is affine γ ⁻¹(X(K)) is not Zariski dense.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

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)}.