Abstract:
We introduce ellipticity criteria for random walks in i.i.d. random environments under which we can extend the ballisticity conditions of Sznitman's and the polynomial effective criteria of Berger, Drewitz and Ramirez originally defined for uniformly elliptic random walks. We prove under them the equivalence of Sznitman's (T') condition with the polynomial effective criterion (P)_M, for M large enough. We furthermore give ellipticity criteria under which a random walk satisfying the polynomial effective criterion, is ballistic, satisfies the annealed central limit theorem or the quenched central limit theorem.

Abstract:
We settle the problem of the uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations $\bar\partial f(z)=H(z, \partial f(z))$. It turns out that the uniqueness holds under definite and explicit bounds on the ellipticity at infinity, but not in general.

Abstract:
We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.

Abstract:
Materials with extreme mechanical anisotropy are designed to work near a material instability threshold where they display stress channelling and strain localization, effects that can be exploited in several technologies. Extreme couple stress solids are introduced and for the first time systematically analyzed in terms of several material instability criteria: positive-definiteness of the strain energy (implying uniqueness of the mixed b.v.p.), strong ellipticity (implying uniqueness of the b.v.p. with prescribed kinematics on the whole boundary), plane wave propagation, ellipticity, and the emergence of discontinuity surfaces. Several new and unexpected features are highlighted: (i.) Ellipticity is mainly dictated by the 'Cosserat part' of the elasticity and (ii.) its failure is shown to be related to the emergence of discontinuity surfaces; (iii.) Ellipticity and wave propagation are not interdependent conditions (so that it is possible for waves not to propagate when the material is still in the elliptic range and, in very special cases, for waves to propagate when ellipticity does not hold). The proof that loss of ellipticity induces stress channelling, folding and faulting of an elastic Cosserat continuum (and the related derivation of the infinite-body Green's function under antiplane strain conditions) is deferred to Part II of this study.

Abstract:
Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, that is, the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable or circular correlation models, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparametric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable.

Abstract:
By combining the Minkowski inequality and the quantum Chernoff bound, we derive easy-to-compute upper bounds for the error probability affecting the optimal discrimination of Gaussian states. In particular, these bounds are useful when the Gaussian states are unitarily inequivalent, i.e., they differ in their symplectic invariants.

Abstract:
These notes are dedicated to whom may be interested in algorithms, Markov chain, coupling, and graph theory etc. I present some preliminaries on coupling and explanations of the important formulas or phrases, which may be helpful for us to understand D. Weitz's paper "Combinatorial Criteria for Uniqueness of Gibbs Measures" with ease.

Abstract:
The capacity region of the two-user Gaussian Interference Channel (IC) is studied. Three classes of channels are considered: weak, one-sided, and mixed Gaussian IC. For the weak Gaussian IC, a new outer bound on the capacity region is obtained that outperforms previously known outer bounds. The sum capacity for a certain range of channel parameters is derived. For this range, it is proved that using Gaussian codebooks and treating interference as noise is optimal. It is shown that when Gaussian codebooks are used, the full Han-Kobayashi achievable rate region can be obtained by using the naive Han-Kobayashi achievable scheme over three frequency bands (equivalently, three subspaces). For the one-sided Gaussian IC, an alternative proof for the Sato's outer bound is presented. We derive the full Han-Kobayashi achievable rate region when Gaussian codebooks are utilized. For the mixed Gaussian IC, a new outer bound is obtained that outperforms previously known outer bounds. For this case, the sum capacity for the entire range of channel parameters is derived. It is proved that the full Han-Kobayashi achievable rate region using Gaussian codebooks is equivalent to that of the one-sided Gaussian IC for a particular range of channel parameters.

Abstract:
We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$

Abstract:
In this paper we derive tight bounds on the expected value of products of {\em low influence} functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multi-dimensional Gaussian distributions. The results derived here have a number of applications to the theory of social choice in economics, to hardness of approximation in computer science and to additive combinatorics problems.