A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper improves convergence rates of curvature approximations using Regge elements.
The paper analyzes finite element methods on manifolds with approximate metrics.
We give a constructive proof that the Regge symmetry is a scissors congruence in hyperbolic space. The main tool is Leibon's construction for computing the volume of a general hyperbolic tetrahedron. The proof consists of identifying the key elements in Leibon's construction and permuting them.
New finite element method for complex forms in any dimension.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
The paper generalizes Riemann curvature for manifolds with discontinuous metrics.
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
Rust library solves complex equations on abstract simplicial complexes.
We construct a discrete form of Hamilton's Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einstein's theory of general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation is naturally asso…
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
Constructs finite element spaces for -forms, excluding one subspace.
The paper studies curve evolution using the PLR equation and its solutions.
The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geom…
Study curvature of piecewise metrics using moving frames.
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
Study approximates Riemannian manifolds using polyhedra.
The Ponzano-Regge model of three-dimensional quantum gravity is well-defined when the observables satisfy a certain condition involving the twisted cohomology. In this case, the partition function is defined in terms of the Reidemeister torsion. Some consequences for the special cases of planar graphs and knots are giv…
We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action prese…
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…
We propose a definition of center of mass for asymptotically flat manifolds satisfying Regge-Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our definition is consistent both with the one proposed by Corvino and Schoen and another by …
In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the g…
Improved formulation of spinfoam quantum gravity with cosmological constant, ensuring all amplitudes are finite and providing semiclassical asymptotics.
First we express the holonomy along a boundary curve as the integral on the domain, of an expression which is linear in the curvature. Then we provide a rigorous justification of the definition of curvature in Regge calculus.
We define a concept of Lorentzian angle that works even when one or both of the directions involved is null (lightlike). Such angles play a role in Regge-Calculus, in the boundary- and corner- terms for the gravitational action, and in the Lorentzian Gauss-Bonnet theorem (for which we provide a proof).
We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class…
Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version…
In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate t…
We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy-momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig-Ó Murchadha-Regge-T…
We construct knot invariants on the basis of ascribing Euclidean geometric values to a triangulation of sphere S^3 where the knot lies. The main new feature of this construction compared to the author's earlier papers on manifold invariants is that now nonzero "deficit angles" (in the terminology of Regge calculus) can…
In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}.…
Study of a generalized Konno--Oono system with conservation laws and surface immersions.
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and…
We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliat…
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application t…
We study time-like surfaces in the three-dimensional Minkowski space with diagonalizable second fundamental form. On any time-like W-surface we introduce locally natural principal parameters and prove that such a surface is determined uniquely (up to motion) by a special invariant function, which satisfies a natural no…
We derive a system of equations governing the coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime. The equations are derived in the context of global non-linear stability of Reissner-Nordström under axially symmetric polarized perturbations, as a generalization of the recent work on …
We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern-Simons theory on . In particular we analyze its asymptotic behaviour in the double-scaling limit in which both the representation labels and the Chern-Simons coupling are taken to be large, but with fixed ratio. When the Wilson gr…
The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal sur…
On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as …
Study timelike surfaces with parallel mean curvature in Minkowski 4-space.
We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms …
The center of mass in General Relativity is hard to define due to coordinate freedom.
Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance <D_k(N)> from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, …
We prove boundedness and polynomial decay statements for solutions to the spin generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $…
The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper…
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…