The Invariance Principle for p-θ^→ Chain

There are two parts in this paper. In the first part we construct the Markov chain in random environment（MCRE）, the skew product Markov chain and p-θ^→ chain from a random transition matrix and a two-dimensional probability distribution, and in the second part we prove that the invarianee principle for p-θ^→ chain, a more complex non-homogeneous Markov chain, is true under some reasonable conditions. This result is more powerful.     

L-Factor of Irreducible x1×x2(×)σ

The L-factor of irreducible x1×x2(×)σ defined by Piatetski-Shapiro is computed by using non-split Bessel functional.     

The coarse Lipschitz geometry of \ell_p\oplus\ell_q

We show that for 1 < p < ∞ with p ≠ 2 the space L _p (0,1) is not uniformly homeomorphic to . We also show that if 1 < p < 2 < q < ∞ the space has unique uniform structure, answering a question of Johnson, Lindenstrauss and Schechtman (Geom. Funct. Anal. 6:430–470, 1996). The first author was supported by NSF grant DMS-0555670 and the second author was supported by NSF grant DMS-0701097.     

A note on the gl constant ofE \mathop \otimes \limits^ \vee F F

LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to (2) The local unconditional structure constant of the space of bounded operatorsL (X*_k,Y _k) tends to infinity for every increasing sequence and of finite-dimensional subspaces ofX andY respectively.     

On the Evaluation of the Integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$

The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G.     

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$-manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat \(\partial \overline \partial \)-manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat \(\partial \overline \partial \)-manifolds.     

Single‐cell fourth‐order difference approximations for $${\font\twelvesl=cmsl10\partial {\sl\bf u}\over\partial {\sl\bf x}},{\partial \sl\bf u\over\partial\sl\bf y}$$ , and $${\font\twelvesl=cmsl10\partial {\sl\bf u}\over\partial{\sl\bf z}}$$ of the three‐dimensional quasi‐linear elliptic equation

Finite difference methods of O(h⁴) are proposed for obtaining estimates of first‐order partial derivatives of the solution of three‐dimensional quasi‐linear elliptic equation with mixed derivative terms subject to Dirichlet boundary conditions on a uniform cubic grid. In all the cases, we use a single computational cell and the methods are applicable to the problems both in cartesian and polar coordinates. The utility of the new methods is shown by testing the methods on three‐dimensional poisson solvers in polar coordinates. Some numerical examples are provided to demonstrate the accuracy and efficiency of the methods discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 417–425, 2000     

On New Functional Characteristics of Domains $$\Omega\in\mathbb R^n$$

Mathematical Notes - A system of new differential-geometric notions as the result of an analysis of the Trudinger–Wang inequalities is proposed. Their naturalness in multivariate analysis and...