Let V be the representation of the quantized enveloping algebra of gl(n) which is the q-analogue of the vector representation and let V ∗ be the dual representation. We construct a basis for ⊗r (V ⊕ V ∗) with favorable properties similar to those of Lusztig’s dual canonical basis. In particular our basis is invariant under the bar involution and contains a basis for the subspace of invariant te...

Last lecture we studied duality of linear programs (LP), specifically how to construct the dual, the relation between the optimum of an LP and its dual, and some duality applications. In this lecture, we will talk about another application of duality to prove one of the theorems in combinatorics so called Maximum Flow-Minimum Cut Problem. The theorem roughly says that in any graph, the value of...

We describe a method to try to construct non-Desarguesian projective planes of a given finite order using a computer program. If a projective plane of order n exists, then it can be constructed from a dual linear space by a sequence of one-line extensions. We analyze this extension process, and find a criterion to limit the search for possible extensions. Up to isomorphism, there are 105 dual l...

We introduce a model-based excessive gap technique to analyze first-order primaldual methods for constrained convex minimization. As a result, we construct new primal-dual methods with optimal convergence rates on the objective residual and the primal feasibility gap of their iterates separately. Through a dual smoothing and prox-function selection strategy, our framework subsumes the augmented...

The N-extended supersymmetric self-dual Poincaré supergravity equations provide a natural local description of supermanifolds possessing hyperkähler structure. These equations admit an economical formulation in chiral superspace. A reformulation in harmonic superspace encodes self-dual supervielbeins and superconnections in a graded skew-symmetric supermatrix superfield prepotential satisfying ...

Abstract. In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreo...

Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the SU(∞) Nahm equations and the continuous Toda theory, we construct the Moyal deformations of the self dual membrane in terms of the Moyal deformations of the continuous Toda theory. This is performed by using the WeylWigner-Moyal quantization technique of the 3D continuous To...

Assuming the continuum hypothesis, we construct a universally weakly measurable function from [0,1] into a dual of some weakly compactly generated Banach space, which is not Pettis integrable. This (partially) solves a problem posed by Riddle, Saab and Uhl [13]. We prove two results related to Pettis integration in dual Banach spaces. We also contribute to the problem whether it is consistent t...

In this paper, upper bounds on the minimum Euclidean weights of Type I codes over Z6 , and self-dual codes over Z8 and Z10 , are derived for modest lengths. The notion of extremality for Euclidean weights is also introduced. We construct new extremal self-dual codes over these rings. Most of these codes are obtained via the double circulant and quasitwisted constructions. New extremal odd unimo...

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analysing, step by step, the action of homogeneous transformations on the homogeneous part of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and show that two systems are equivale...