In the first part of this article we review the general theory of Cherednik algebras and non-symmetric Macdonald polynomials, including a formulation and proof of the fundamental duality theorem in its proper general context. In the last section we summarize some of the combinatorial results in this area which we have recently obtained in collaboration with J. Haglund and N. Loehr. Mathematics ...

Uncertainty theory is a branch of axiomatic mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. In this paper, portfolio selection problems in uncertainty environment is solved using uncertainty programming methods. The concept of quadratic entropy is introduced in the model to measure risk of securities. Furthermore, a hybrid intelli...

SL(2,Z) duality transformations in asymptotically AdS4 x S7 act nontrivially on the three-dimensional superconformal field theory of coincident M2-branes on the boundary. We show how S-duality acts away from the IR fixed point.We develop a systematic method to holographically obtain the deformations of the boundary CFT and show how electric-magnetic duality relates different deformations. We an...

Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This comes from consequence of wave-particle duality matter giving statement that position and momentum cannot be measured in same time. article tries to mathematics background behind expression using supported such as Fourier transform, transform integral, probability Gaussian distribution it ends up...

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite di↵erent. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the form...

We establish some sufficient conditions for the profinite and prop completions of an abstract group G of type FPm (resp of finite cohomological dimension, of finite Euler characteristics) to be of type FPm over the field Fp for a fixed natural prime p (resp. of finite cohomological p-dimension, of finite Euler p-characteristics). We apply our methods for orientable Poincaré duality groups G of ...

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that altho...

Recently, developments in the understanding of low-energy N = 1 supersymmetric gauge theory have revealed two important phenomena: the appearance of new four-dimensional superconformal field theories and a non-Abelian generalization of electric-magnetic duality at the IR fixed point. This report is a pedagogical introduction to these phenomena. After presenting some necessary background materia...

In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick’s family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brøndsted-Rockafellar property. In this ...

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with ...