The quantization dimension of the self-affine measures on general Sierpi��ski carpets

Let??? be a self-affine measure on a general Sierpi??ski carpet E. We give a characterization for the upper and lower quantization dimension of??? in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D _r (??) of??? exists for all r > 0 under an additional condition. We establish an explicit formula for D _r (??) and show that it increases to the box-counting dimension ${dim_B^* \mu}$ of??? as r tends to infinity. For a class of Sierpi??ski carpets E and the uniform measures??? on E, we show that the quantization dimension of??? coincides with its box-counting dimension and that the D _r (??)-dimensional upper and lower quantization coefficient of??? are both positive and finite.     

A deletion-invariance property for random measures satisfying the Ghirlanda–Guerra identities

We show that if a discrete random measure on the unit ball of a separable Hilbert space satisfies the Ghirlanda–Guerra identities then by randomly deleting half of the points and renormalizing the weights of the remaining points we obtain the same random measure in distribution up to rotations.     

Necessary convergence conditions for series ΣαnSn in the case of identically distributed independent random quantities

Necessary (in some cases also sufficient) conditions are obtained for convergence of the series a _n S _n whereS _n = ₁ ⁿ _k k are independent random quantities. The cases in which k are symmetrical or identically distributed quantities are investigated in more detail.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 529–536, October, 1976.     

Rigidity of measures—The high entropy case and non-commuting foliations

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures. To Hillel Furstenberg with friendship and admiration Manfred Einsiedler is partially supported by the NSF Grant DMS 0400587. Anatole Katok is partially supported by the NSF Grant DMS 0071339.     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

On the KŁR conjecture in random graphs

The K?R conjecture of Kohayakawa, ?uczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G _n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Fractional Operators on Morrey—Lorentz Spaces and the Olsen Inequality

Mathematical Notes - Under the Morrey norm, the fractional integral operator and the fractional maximal operator behave similarly as was initially proved by Adams and Xiao. Later on, Tanaka...     

Quantum diffusion of the random Schrödinger evolution in the scaling limit

We consider random Schrödinger equations on R ^d for d???3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and \(\psi_t\) the solution with initial data \(\psi_0\). The space and time variables scale as \(x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)}\). We prove that, in the limit λ?→?0, the expectation of the Wigner distribution of \(\psi_t\) converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data.The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their “naive size” by an extra λ ^c factor per non-(anti)ladder vertex for some c?>?0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.     

Dimension of the Fractal Curve in Plane and Its Derivative of the Fractional Order

Ｄｉｍｅｎｓｉｏｎ　ｏｆ　ｔｈｅ　Ｆｒａｃｔａｌ　Ｃｕｒｖｅ　ｉｎ　Ｐｌａｎｅ　ａｎｄ　Ｉｔｓ　Ｄｅｒｉｖａｔｉｖｅ　ｏｆ　ｔｈｅ　Ｆｒａｃｔｉｏｎａｌ　ＯｒｄｅｒＤｅｎｇＧｕａｎｔｉｅ（邓冠铁）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｈｕ...     

On dilworth’s theorem in the in finite case

The following theorem is proved: Letc be an infinite cardinal. There exists a partially ordered set of cardinalc, which contains no infinite independent subset, and which is not decomposable into less thanc chains.     

The Brezis–Nirenberg Problem for the Fractional p-Laplacian in Unbounded Domains

In this paper we study the existence of nontrivial solutions to the well-known Brezis–Nirenberg problem involving the fractional p-Laplace operator in unbounded cylinder type domains.By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue λp,s(■) with respect to the domain■. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence res...     

A convolution metric in the space of measures and ε-isometries in Lp

For >0, 2,4,6,... on the set of those Borel measures on , such that x^pd(x)<, one introduces a metric, characterizing the nearness of the convolutions x^p* and x^p*. From the convergence of a sequence of probability measures in this metric there follows its convergence in the Kantorovich-Rubinshtein metric. From here one derives theorems on the approximation of -isometries: if H is a finite-dimensional subspace in L_p, then there exists a continuous function _H(), such that for any linear -isometric operator THL_p there exists a linear isometry UH L_p, such that T–U<_H().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 151–156, 1987.