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In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for dual include polarer Raum [Ha

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We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e., the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends an earlier result of the first and third authors. We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category and to generalize results such as Witt's cancellation theorem, Springer's theorem, the weak Hasse principle, and finiteness of genus to systems of sesquilinear forms over hermitian categories.

Pacific Journal Mathematics2014

MATHICSE Technical Report : A priori error for unilateral contact problems with Lagrange multipliers and IsoGeometric Analyis

Pablo Antolin Sanchez, Annalisa Buffa, Mathieu Jonathan Fabre

In this paper, we consider unilateral contact problem without friction between a rigid body_and deformable one in the framework of isogeometric analysis. We present the theoretical_analysis of the mixed problem using an active-set strategy and for a primal space of NURBS_of degree

p

and

p-2

for a dual space of B-Spline. A inf--sup stability is proved to ensure_a good property of the method. An optimal a priori error estimate is demonstrated without_assumption on the unknown contact set. Several numerical examples in two- and threedimensional_and in small and large deformations demonstrate the accuracy of the proposed_method.

MATHICSE2017

Isogeometric mortar methods

Annalisa Buffa

The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf-sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects. © 2014 Elsevier B.V.

2015

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