Related papers: The Pseudo-Hyperbolic Functions and the Matrix Rep…
We review the teory of the pseudo-iperbolic functions on the basis of an algebraic point of view which employs the Eisenstein group. We frame the teory within the general context of the number decomposition and discuss the importance of…
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the…
We discuss the most elementary properties of the hyperbolic trigonometry and show how they can be exploited to get a simple, albeit interesting, geometrical interpretation of the special relativity. It yields indeed a straightforword…
We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in $\mathrm{PO}(n,1)$ with $n>3$. We further show that under an additional assumption (satisfied in all known cases), the covolumes of these…
In our joint papers [FL1-FL2] we revive quaternionic analysis and show deep relations between quaternionic analysis, representation theory and four-dimensional physics. As a guiding principle we use representation theory of various real…
A system of commutative hyperbolic complex numbers in 2 dimensions is studied in this paper. Exponential and trigonometric forms are obtained for these hyperbolic twocomplex numbers. Expressions are given for the elementary functions of…
An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In…
Hyperbolic problems can at times be solved employing symbolic arguments. This is especially true for the construction of forward (and backward) fundamental solutions. We formulate a corresponding abstract scheme and illustrate its…
Hyperbolic metamaterials were originally introduced to overcome the diffraction limit of optical imaging. Soon thereafter it was realized that hyperbolic metamaterials demonstrate a number of novel phenomena resulting from the broadband…
Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic…
Polaritons travelling along a hyperbolic medium's surface have recently sparked significant interest in nanophotonics for the unprecedented manipulation ability on light at the nanoscale in a planar way, promising potential nano-optical…
A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis…
In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic…
This is a lightning introduction to some modern techniques used in the study of the statistical properties of hyperbolic dynamical systems. The emphasis is not in presenting a comprehensive theory but rather in fleshing out the main ideas…
Recent papers in the graph machine learning literature have introduced a number of approaches for hyperbolic representation learning. The asserted benefits are improved performance on a variety of graph tasks, node classification and link…
Hyperbolic representations are effective in modeling knowledge graph data which is prevalently used to facilitate multi-hop reasoning. However, a rigorous and detailed comparison of the two spaces for this task is lacking. In this paper,…
We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of $\mathrm{SL}(2,\mathbb{R})$ due to Bernstein and Reznikov to study triple products of Eisenstein…
An exponential representation of the S-matrix provides a natural framework for understanding the semi-classical limit of scattering amplitudes. While sharing some similarities with the eikonal formalism it differs from it in details.…
We present a mathematical model for a physical theory that is compatible with Einstein's Special Relativity Theory. Our model consists of three pseudo-complex dimensions, representing three real dimensions of space, dual to what could be…
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a…