Friedrichsand Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H 1(Ω) and Poincaré-type inequalities in some subspaces of W1,p(Ω). The dependencies of the ine...

I consider theories of Dirac fermions with different regularizations, leading to different chiral anomalies. I have investigated the Poincaré invariance properties of these theories. I find that both Lorentz and translational invariance are anomalously violated in general, but that they are respected in a theory that is regularized to give a Wess-Zumino consistent anomaly (the Bardeen anomaly)....

We investigate the relationship between the generalized uncertainty principle in quantum gravity and the quantum deformation of the Poincaré algebra. We find that a deformed Newton-Wigner position operator and the generators of spatial translations and rotations of the deformed Poincaré algebra obey a deformed Heisenberg algebra from which the generalized uncertainty principle follows. The resu...

The star product technique translates the framework of local fields on noncommutative space–time into nonlocal fields on standard space– time. We consider the example of fields on κ– deformed Minkowski space, transforming under κ–deformed Poincaré group with noncommutative parameters. By extending the star product to the tensor product of functions on κ–deformed Minkowski space and κ-deformed P...

This paper deals with the conversion of a digital image of a person into a portrait engraved on the surface of a material (white foam with a dark surface). In this approach, a series of holes of varying diameter are drilled into the surface. A mapping is made from a set of numbers (RGB values) in the input image to a set of diameters for the holes to be drilled. The holes expose a white inner c...

1.1.1. Poincaré group. The Poincaré group is the group of isometries of (the affine space) M . Besides translations, we have the Lorentz transformations O(3, 1), i.e. the set of linear transformations L such that g(LX,LX) = g(X,X), or in other words, LαβL αγ = δ β . The first postulate is that all the laws of special relativity should be invariant (in fact co-variant) under Poincaré transformat...

SOME evidence has come to light which makes it possible to relate three familiar portraits of Dr William Harvey more closely to one another and to suggest a common point of origin from an important branch of the Harvey family. These are the Kent and Canterbury Hospital portrait, the King's Weston version, and the SurrendenDering portrait. Anyone venturing into the field of Harvey portraiture mu...