Affine arithmetic (AA) is widely used in range analysis in word-length optimization of hardware designs. To reduce the uncertainty in the AA and achieve efficient and accurate range analysis of multiplication, this paper presents a novel refined affine approximation method, Approximation Affine based on Space Extreme Estimation (AASEE). The affine form of multiplication is divided into two part...

0. Introduction 5 0.1. Objectives and main results 5 0.1.1. Affine Satake isomorphisms 5 0.1.2. Whittaker functions 6 0.1.3. The setting of the paper 7 0.2. Dunkl operators via DAHA 7 0.2.1. Families of Dunkl operators 8 0.3. The technique of spinors 9 0.3.1. Connections to AKZ 9 0.3.2. Isomorphism theorems 10 0.3.3. The localization functor 10 0.3.4. The Whittaker limit 11 0.4. On Langlands’ p...

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G= SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of sym...

Given continuous functions M and N of two variables, it is shown that if in a continuous iteration semigroup with only (M,N)-convex or (M,N)-concave elements there are two (M,N)-affine elements, then M = N and every element of the semigroup is M -affine. Moreover, all functions in the semigroup either are M -convex or M -concave.

It should be noted that the requirement of f to be convex in the definition of f -divergence is essential. In Euclidean spaces any convex function can be represented as the pointwise supremum of a family of affine functions and vice versa, every supremum of a family of affine functions produces a convex function. Take f(x) = 12 |x− 1| as an example. We see that it can be written as a pointwise ...

In this paper, we show that the property of tight affine frame decomposition of functions in L can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Qj on j-th levels to any function f in a Sobolev space reveals the detailed information Qjf...

It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, Fn, over IF n 2 , is naturally regarded as the 2 n dimensional vector space, IF n 2 . Thus, while analyzing the transformations acting on Fn, S22n , the group of all bijective mappings, defined from IF 2 2 onto itself should be considered. As it is shown i...

This paper presents an efficient approach to the classification of the affine equivalence classes of cosets of the first order ReedMuller code with respect to cryptographic properties such as correlationimmunity, resiliency and propagation characteristics. First, we apply the method to completely classify all the 48 classes into which the general affine group AGL(2, 5) partitions the cosets of ...

This paper considers least squares estimators for regression problems over convex, uniformly bounded, uniformly Lipschitz function classes minimizing the empirical risk over max-affine functions (the maximum of finitely many affine functions). Based on new results on nonlinear nonparametric regression and on the approximation accuracy of maxaffine functions, these estimators are proved to achie...