Narrow Results

We study a class of extensions of the $k$-contracted Poincaré algebra under the hypothesis of generalizing the Bargmann algebra and its central charge. As we will see this type of contractions will lead in a natural way to consider the codajoint Poincaré algebra and some of their contractions. Among them there is one such that considering the quotient of it by a suitable ideal, the (stringy) $p$-brane Galilei algebra is recovered.

In [C. G. Weaver Found. Phys. 51, 1 (2021)], I showed that Boltzmann’s H-theorem does not face a significant threat from the reversibility paradox. I argue that my defense of the H-theorem against that paradox can be used yet again for the purposes of resolving the recurrence paradox without having to endorse heavy-duty statistical assumptions outside of the hypothesis of molecular chaos. As in [C. G. Weaver Found. Phys. 51, 1 (2021)], lessons from the history and foundations of physics reveal precisely how such resolution is achieved.

This work continues the ideas presented in the author's book, The Universality of the Radon Transform (Oxford, 2003), which deals with the group SL(2,R). The complication that arises for G = SL(3,R) comes from the fact that there are now two fundamental representations. This has the consequence that the wave operator, which plays a central role in our work on SL(2,R) is replaced by an overdetermined system of partial differential equations. The analog of the wave operator is defined using an MN invariant orbit of G acting on the direct sum of the symmetric squares of the fundamental representations. The relation of orbits, or, in general, of any algebraic variety, to a system of partial differential equations comes via the Fundamental Principle, which shows how Fourier transforms of functions or measures on an algebraic variety correspond to solutions of the system of partial differential equations defined by the equations of the variety. In particular, we can start with the sum T of the delta functions of the orbit of the group Γ = SL(2,Z) on the light cone. We then take its Fourier transform, using a suitable quadratic form. We then decompose the Fourier transform under the commuting group of G. In this way, we obtain a Γ invariant distribution which has a natural restriction to the orbit G/K, which is the symmetric space of G. This restriction is (essentially) the nonanalytic Eisenstein series. We can compute the periods of the Eisenstein series over various orbits of subgroups of G by means of the Euclidean Planchere formula. A more complicated form of these ideas is needed to define Poincaré series.

Thanks to the introduction of a general Lagrangian, we derive a simple expression for Mercury perihelion advance (MPA) which can be applied to many historical examples, such as Poincaré’s 1906 application of his relativistic action, Einstein’s probable test of his 1912 metric, Einstein-Besso’s 1914 application of the Entwurf … as well as to modern presentations. We argue that Einstein’s failure to derive in his Zurich Notebook the right equations of general relativity has a physical rather than mathematical origin, which amounts to an incomplete application of the 1895 Lorentz Transformations to his 1911 formulation of the Equivalence Principle. MPA could have been already in his hands in 1912.

Biomedical association studies are increasingly done using clinical concepts, and in particular diagnostic codes from clinical data repositories as phenotypes. Clinical concepts can be represented in a meaningful, vector space using word embedding models. These embeddings allow for comparison between clinical concepts or for straightforward input to machine learning models. Using traditional approaches, good representations require high dimensionality, making downstream tasks such as visualization more difficult. We applied Poincaré embeddings in a 2-dimensional hyperbolic space to a large-scale administrative claims database and show performance comparable to 100-dimensional embeddings in a euclidean space. We then examine disease relationships under different disease contexts to better understand potential phenotypes.

From the principle of relativity with two universal invariant parameters c and l, 24 possible kinematical (including geometrical and static) algebras can be obtained. Each algebra is of 10 dimensional, generating the symmetry of a 4 dimensional homogeneous space-time or a pure space. In addition to the ordinary Poincaré algebra, there is another Poincaré algebra among the 24 algebras. New 4d geometries with the new Poincaré symmetry are presented. The motion of free particles on one of the new space-times is discussed.