We present a regularity lemma for Boolean functions f : {−1, 1}n → {−1, 1} based on noisy influence, a measure of how locally correlated f is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a “homogenized” version stating that there is a set of input bits so that most restrictions of f on those bi...

A vexing challenge when using the utility-maximization framework to estimate consumers’ decisions on which set of goods to purchase and how much quantity to buy, is obtaining a functional form of the utility that satisfies three criteria: tractability, flexibility and global-regularity. Flexibility refers to the ability of utility function to impose minimal prior restrictions on demand elastici...

We develop high-order numerical methods for solving wave equations with van der Pol type nonlinear boundary conditions. Based on the wave reflection on the boundaries, we first solve the corresponding Riemann invariants by constructing two iterative mappings, and then, regarding the regularity of boundary conditions, propose two different high-order numerical approaches to the system. When the ...

By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to a fair range of different statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression ou...

We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bo...

The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming. Having in mind the topological structure of the positive se...

We solve an important open problem by deriving some readily verifiable necessary and sufficient conditions for a multivariate non-Gaussian linear process to be time-reversible, under two sets of regularity conditions on the contemporaneous dependence structure of the innovations. One set of regularity conditions concerns the case of independent-component innovations, in which case a multivariat...

We revisit regularity of SLE trace, for all κ 6= 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Vik...

Green [Geometric and Functional Analysis 15 (2005), 340–376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

    M. and M.   Abstract: Existence and uniqueness of solution for singular 2-D systems depends on regularity condition. Simple regularity implies regularity and under this assumption, the generalized wave model (GWM) is introduced to cast singular 2-D system of equations as a family of non-singular 1-D models with variable structure.These index dependent models, along with a set of boundary co...