The language of modern mechanics is calculus on manifolds, and exterior calculus is an important part of that. It consists of objects like differential forms, general tensors and vector fields on manifolds, and operators that act on these. While the smooth exterior calculus has a long history going back to Cartan, Lie, Grassmann, Hodge, de Rham and many others, the need for a discrete calculus ...

The dual simplex algorithm is an attractive alternative as a solution method for linear programming problems. Since the addition of new constraints to a problem typically breaks primal feasibility but not dual feasibility, the dual simplex can be deployed for rapid reoptimization, without the need of finding new primal basic feasible solutions. This is especially useful in integer programming, ...

Considering a general vector optimization problem, we attach to it two new vector duals by means of perturbation theory. These vector duals are constructed with the help of the recent Wolfe and Mond-Weir scalar duals for general optimization problems proposed by R.I. Boţ and S.-M. Grad, by exploiting an idea due to W. Breckner and I. Kolumbán. Constrained and unconstrained vector optimization p...

We introduce a notion of Homological Projective Duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are Homologically Projectively Dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial c...

The aim of this paper is devoted to investigate the duality and existence of solutions for a class of bilevel vector pseudomonotone equilibrium problems without involving the information about the solution set of the lower-level equilibrium problem. Firstly, we propose the dual formulations of bilevel vector equilibrium problems (BVEP). Secondly, the primal-dual relationships are derived under ...

Solving analytic systems using inversion can be implemented in a variety of ways. One method is to use Lagrange inversion and variations. Here we present a different approach, based on dual vector fields. For a function analytic in a neighborhood of the origin in the complex plane, we associate a vector field and its dual, an operator version of Fourier transform. The construction extends natur...

In this paper we consider a vector optimization problem; we present some scalarization techniques for finding all the vector optimal points of this problem and we discuss the relationships between these methods. Moreover, in the linear case, the study of dual variables is carried on by means of sensitivity analysis and also by a parametric approach. We also give an interpretation of the dual va...

We argue that the AdS dual of the three dimensional critical O(N) vector model can be evaluated using the Legendre transform that relates the generating functionals of the free UV and the interacting IR fixed points of the boundary theory. As an example, we use our proposal to evaluate the minimal bulk action of the scalar field that it is dual to the spin-zero “current” of the O(N) vector mode...

It is proved that there exist a vector representation of Dirac’s spinor field and in one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can be considered as a generalization of Cartan’s idea of triality to Dirac’s spinors. In the vector representation the first order Dirac Lagrangian is dual equivalent to the two order Lagrangian of topologically mass...