Abstract: The divergence function in information geometry, and the discrete Lagrangian in discrete 1 geometric mechanics each induce a differential geometric structure on the product manifold QˆQ. 2 We aim to investigate the relationship between these two objects, and the fundamental role that 3 duality, in the form of Legendre transforms, play in both fields. By establishing an analogy between...

Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to SL(n) and tensor products of the standard and dual representations. In this paper, we place their geometric setups in a natural, general framework. For any complex reductive group and any sequence of minu...

We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of generalized Thompson groups Fn. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the so...

A generalization of the exponentiated Weibull geometric model called the transmuted exponentiated Weibull geometric distribution is proposed and studied. It includes as special cases at least ten models. Some of its structural properties including order statistics, explicit expressions for the ordinary and incomplete moments and generating function are derived. The estimation of the model param...

Let A = {x1, . . . , xn} be a subspace arrangement with a geometric lattice such that codim(x) ≥ 2 for every x ∈ A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum x 1 + . . . + x n is a direct sum. The homotopy type of M(A) is also given : it is a product of odd dimensional spheres. Finally, some other equivalent conditions are gi...

We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, un...

Using Euler’s formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discreti...

When grasping an object using three ngered end-e ectors, a certain geometrical entity can be created based on the grasping points and a de nition of an auxiliary point. This geometric con guration is referred to as the invariant con guration of the three ngered grasp. This paper exploits the geometric properties of such con guration using screw theory and inner product spaces. The results are s...

Modelling of deteriorating systems with preventive repair during warranty using geometric processes was carried out in the previous chapter. In this chapter the independent and identically distributed assumption of geometric processes is relaxed to assume only the independent successive inter arrival times. Surveying the literature, one is struck by the fact that there has been so little resear...

We present an ontology of objects, functions, and generic shape representation that supports form-function reasoning. By reasoning from the mechanical and other functions of objects to their geometric shape requirements, we deduce the generic shape representation of objects, which we represent as a partial boundary representation composed of primitive geometric shape elements and their spatial ...